Uncertainty Quantification in Computational Models: From Foundations to Biomedical Applications

Abstract

This article provides a comprehensive overview of Uncertainty Quantification (UQ) methodologies and their critical applications in computational science and biomedicine. It explores foundational UQ concepts, including the distinction between aleatory and epistemic uncertainty, and details advanced techniques like polynomial chaos, ensembling, and Bayesian inference. The content covers practical implementation strategies for drug discovery and biomedical models, addresses common troubleshooting scenarios with limited data, and examines Verification, Validation, and Uncertainty Quantification (VVUQ) frameworks for building credibility. Aimed at researchers, scientists, and drug development professionals, this guide synthesizes current UQ practices to enhance model reliability and support risk-informed decision-making in precision medicine and therapeutic development.

Understanding Uncertainty Quantification: Core Concepts and Critical Importance

Defining Aleatory vs. Epistemic Uncertainty in Scientific Models

In the realm of computational modeling for scientific research, particularly in high-stakes fields like drug development, the precise characterization and quantification of uncertainty is not merely an academic exercise—it is a fundamental requirement for model reliability and regulatory acceptance. Uncertainty permeates every stage of model development, from conceptualization through implementation to prediction. The distinction between aleatory and epistemic uncertainty provides a crucial philosophical and practical framework for categorizing and addressing these uncertainties systematically [1]. While both types manifest as unpredictability in model outputs, their origins, reducibility, and implications for decision-making differ profoundly.

Aleatory uncertainty (from Latin "alea" meaning dice) represents the inherent randomness, variability, or stochasticity natural to a system or phenomenon. This type of uncertainty is irreducible in principle, as it stems from the fundamental probabilistic nature of the system being modeled, persisting even under perfect knowledge of the underlying mechanisms [2]. In contrast, epistemic uncertainty (from Greek "epistēmē" meaning knowledge) arises from incomplete information, limited data, or imperfect understanding on the part of the modeler. This form of uncertainty is theoretically reducible through additional data collection, improved measurements, or model refinement [3] [4]. The ability to distinguish between these uncertainty types enables researchers to allocate resources efficiently, focusing reduction efforts where they can be most effective while acknowledging inherent variability that cannot be eliminated.

Conceptual Foundations and Distinctions

Defining Characteristics and Properties

The conceptual distinction between aleatory and epistemic uncertainty extends beyond their basic definitions to encompass fundamentally different properties and implications for scientific modeling. These characteristics determine how each uncertainty type should be represented, quantified, and ultimately addressed within a modeling framework.

Aleatory uncertainty embodies the concept of intrinsic randomness or variability that would persist even with perfect knowledge of system mechanics. This category includes stochastic processes such as thermal fluctuations in chemical reactions, quantum mechanical phenomena, environmental variations affecting biological systems, and the inherent randomness in particle interactions [2]. In pharmaceutical contexts, this might manifest as inter-individual variability in drug metabolism or random fluctuations in protein folding dynamics. The irreducible nature of aleatory uncertainty means it cannot be eliminated by improved measurements or additional data collection, though it can be precisely characterized through probabilistic methods.

Epistemic uncertainty represents limitations in knowledge, modeling approximations, or incomplete information that theoretically could be reduced through better science. This encompasses uncertainty about model parameters, structural inadequacies in mathematical representations, insufficient data for reliable estimation, and limitations in experimental measurements [3] [1]. In drug development, epistemic uncertainty might arise from limited understanding of a biological pathway, incomplete clinical trial data, or simplification of complex physiological processes in pharmacokinetic models. Unlike aleatory uncertainty, epistemic uncertainty can potentially be minimized through targeted research, improved experimental design, or model refinement.

Table 1: Fundamental Characteristics of Aleatory and Epistemic Uncertainty

Characteristic	Aleatory Uncertainty	Epistemic Uncertainty
Origin	Inherent system variability or randomness	Incomplete knowledge or information
Reducibility	Irreducible in principle	Reducible through additional data or improved models
Representation	Probability distributions	Confidence intervals, belief functions, sets of distributions
Data Dependence	Persistent with infinite data	Diminishes with increasing data
Common Descriptors	Random variables, stochastic processes	Model parameters, structural uncertainty

Practical Implications of the Distinction

The classification of uncertainties as either aleatory or epistemic carries significant practical implications for modeling workflows, resource allocation, and decision-making processes. From a pragmatic standpoint, this distinction helps modelers identify which uncertainties have the potential for reduction through targeted investigation [1]. When epistemic uncertainties dominate, resources can be directed toward data collection, model refinement, or experimental validation. Conversely, when aleatory uncertainties prevail, efforts may be better spent on characterizing variability and designing robust systems that perform acceptably across the range of possible outcomes.

The distinction also critically influences how dependence among random events is modeled. Epistemic uncertainties can introduce statistical dependence that might not be properly accounted for if their character is not correctly modeled [1]. For instance, in a system reliability problem, shared epistemic uncertainty about material properties across components creates dependence that significantly affects system failure probability estimates. Similarly, in time-variant reliability problems, proper characterization of both uncertainty types is essential for accurate risk assessment over time.

From a decision-making perspective, the separation of uncertainty types enables more informed risk management strategies. In pharmaceutical development, understanding whether uncertainty about a drug's efficacy stems from inherent patient variability (aleatory) versus limited clinical data (epistemic) directly impacts regulatory strategy and further development investments. This distinction becomes particularly crucial in performance-based engineering and risk-based decision-making frameworks where uncertainty characterization directly influences safety factors and design standards [1].

Quantitative Representation and Mathematical Frameworks

Mathematical Representations and Propagation

The quantitative representation and propagation of aleatory and epistemic uncertainties require distinct mathematical frameworks that respect their fundamental differences. For aleatory uncertainty, conventional probability theory with precisely known parameters typically suffices. However, when epistemic uncertainty is present, more advanced mathematical structures are necessary to properly represent incomplete knowledge.

Dempster-Shafer (DS) structures provide a powerful framework for representing epistemic uncertainty by assigning belief masses to intervals or sets of possible values rather than specific point estimates [2]. In this representation, epistemic uncertainty in a parameter (x) might be expressed as (x \sim {([\underline{x}i, \overline{x}i], pi)}{i=1}^n), where each interval ([\underline{x}i, \overline{x}i]) receives a probability mass (p_i). This structure naturally captures the idea of having limited or imprecise information about parameter values.

For systems involving both uncertainty types, a hierarchical representation emerges where aleatory uncertainty is modeled through conditional probability distributions parameterized by epistemically uncertain variables. The propagation of these combined uncertainties through system models follows a two-stage approach. First, aleatory uncertainty is modeled conditional on epistemic parameters, often through stochastic differential equations or conditional probability densities such as (p(t,x∣θ)≈\mathcal{N}(x; μ(θ), σ^2(θ))), where (θ) represents epistemically uncertain parameters [2]. Second, epistemic uncertainty is propagated through moment evolution equations, which for polynomial systems can be derived using Itô's lemma:

[ \dot{M}k∣{e0} = -k \sumi αi m{i+k-1}∣{e0} + \frac{1}{2} k(k-1) q^2 m{k-2}∣{e_0} ]

where statistical moments (M_k) and parameters become interval-valued due to epistemic uncertainty [2].

Table 2: Mathematical Representations for Different Uncertainty Types

Uncertainty Type	Representation Methods	Key Mathematical Structures
Purely Aleatory	Probability theory	Random variables, stochastic processes, probability density functions
Purely Epistemic	Evidence theory, Interval analysis	Dempster-Shafer structures, credal sets, p-boxes
Mixed Uncertainties	Hierarchical probabilistic models	Second-order probability, Bayesian hierarchical models

Output Representation and Decision Aggregation

After propagating mixed uncertainties through a system model, the resulting uncertainty in system response is typically expressed using probability boxes (p-boxes) within a Dempster-Shafer structure: ({([Fl(x), Fu(x)], pi)}), where each ([Fl(x), F_u(x)]) bounds the cumulative distribution function envelopes induced by the propagated moment intervals for each focal element [2]. This representation preserves the separation between aleatory variability (captured by the CDFs) and epistemic uncertainty (captured by the interval-valued CDFs and their assigned masses).

Prior to decision-making, this second-order uncertainty is often "crunched" into a single actionable distribution through transformations such as the pignistic transformation:

[ P{\text{Bet}}(X ≤ x) = \frac{1}{2} \sumi \left( \underline{N}i(x) + \overline{N}i(x) \right) p_{D,i} ]

which converts set-valued belief structures into a single cumulative distribution function for expected utility calculations and risk analysis [2]. Quantitative indices such as the Normalized Index of Decision Insecurity (NIDI) or the ignorance function ((I_g)) can be computed to assess residual ambiguity and guide confidence-aware decision policies, providing metrics for how much epistemic uncertainty remains in the final analysis.

Experimental Protocols for Uncertainty Quantification

Protocol 1: Bayesian Neural Networks for Epistemic Uncertainty Quantification

Purpose: To quantify epistemic uncertainty in deep learning models used for scientific applications, such as quantitative structure-activity relationship (QSAR) modeling in drug development.

Theoretical Basis: In Bayesian deep learning, epistemic uncertainty is captured through distributions over model parameters rather than point estimates [4]. This approach treats the weights (W) of a neural network as random variables with a prior distribution (p(W)) that is updated through Bayesian inference to obtain a posterior distribution (p(W|X,Y)) given data ((X,Y)).

Materials and Reagents:

• TensorFlow Probability or PyTorch with Bayesian layers: Enables implementation of variational inference for neural networks
• Dataset: Domain-specific dataset (e.g., chemical compounds with associated biological activities)
• High-performance computing resources: GPUs for efficient sampling and training

Procedure:

• Model Specification: Implement a neural network with probabilistic layers. For example, using TensorFlow Probability's DenseVariational layer, which places distributions over weights rather than point estimates [4].
• Prior Definition: Define appropriate prior distributions for network parameters, typically Gaussian priors with specified mean and variance.
• Variational Inference: Approximate the true posterior (p(W|X,Y)) using a variational distribution (q_θ(W)) parameterized by (θ).
• Loss Optimization: Minimize the negative Evidence Lower Bound (ELBO) loss function: [ \mathcal{L}(θ) = \mathbb{E}{qθ(W)}[\log p(Y|X,W)] - \text{KL}(q_θ(W) \| p(W)) ] which balances data fit with regularization toward the prior.
• Uncertainty Estimation: For prediction on a new sample (x^), approximate the predictive distribution: [ p(y^|x^,X,Y) ≈ \int p(y^|x^*,W)q_θ(W)dW ] using Monte Carlo sampling from the variational posterior.
• Epistemic Uncertainty Quantification: Compute the standard deviation of predictions across multiple stochastic forward passes as a measure of epistemic uncertainty.

Interpretation: The epistemic uncertainty, quantified by the variability in predictions under different parameter samples, decreases as more data becomes available and the posterior distribution over weights tightens [4].

Protocol 2: Aleatoric Uncertainty Quantification with Probabilistic Regression

Purpose: To quantify aleatoric uncertainty in regression tasks, capturing inherent noise in the data generation process that persists regardless of model improvements.

Theoretical Basis: Aleatoric uncertainty is modeled by making the model's output parameters of a probability distribution rather than point predictions [4]. For continuous outcomes, this typically involves predicting both the mean and variance of a Gaussian distribution, with the variance representing heteroscedastic aleatoric uncertainty.

Materials and Reagents:

• Deep learning framework with probabilistic capabilities (TensorFlow Probability, PyTorch)
• Dataset with observed input-output pairs, ideally with replication to estimate inherent variability
• Standard computing resources: Aleatoric uncertainty quantification is computationally less demanding than full Bayesian inference

Procedure:

• Model Architecture: Design a neural network with two output units – one predicting the mean (μ(x)) and another predicting the variance (σ^2(x)) of the target distribution.
• Distribution Layer: Implement a DistributionLambda layer that constructs a Gaussian distribution parameterized by the network's outputs: [ p(y|x) = \mathcal{N}(y; μ(x), σ^2(x)) ]
• Loss Function: Use the negative log-likelihood as the loss function: [ \mathcal{L} = -\sum{i=1}^N \log p(yi|x_i) ] which naturally balances mean prediction accuracy with uncertainty calibration.
• Model Training: Optimize all network parameters simultaneously using stochastic gradient descent.
• Aleatoric Uncertainty Extraction: For new predictions, the predicted variance represents the aleatoric uncertainty, which captures how much noise is expected in the outcome for the given input.

Interpretation: Unlike epistemic uncertainty, aleatoric uncertainty does not decrease with additional data from the same data-generating process [4]. The predicted variance reflects inherent noise or variability that cannot be reduced through better modeling or more data collection.

Protocol 3: Distinguishing Epistemic and Aleatoric Uncertainty in Language Models

Purpose: To identify and separate epistemic from aleatoric uncertainty in large language models (LLMs) applied to scientific text generation or analysis.

Theoretical Basis: In language models, token-level uncertainty mixes both epistemic and aleatoric components [5]. Epistemic uncertainty reflects the model's ignorance about factual knowledge, while aleatoric uncertainty stems from inherent unpredictability in language (multiple valid ways to express the same concept).

Materials and Reagents:

• Two language models of different capacities (e.g., LLaMA 7B and LLaMA 65B)
• Text corpora from relevant scientific domains
• Linear probing implementation for model activations

Procedure:

• Contrastive Setup: Use a large, powerful model (e.g., LLaMA 65B) as a reference for "knowable" information, assuming it has less epistemic uncertainty than a smaller model (e.g., LLaMA 7B).
• Token Classification: For each token generated by the small model, classify uncertainty type based on the entropy difference between models:
  • Compute next-token predictive entropy for both small ((HS)) and large ((HL)) models
  • Flag tokens where (HS) is high but (HL) is low as primarily epistemic uncertainty
  • Tokens where both models show high entropy indicate primarily aleatoric uncertainty
• Probe Training: Train linear classifiers on the small model's internal activations to predict the epistemic uncertainty labels derived from the contrastive analysis.
• Unsupervised Alternative: For cases where a large reference model is unavailable, implement unsupervised methods that detect epistemic uncertainty through analysis of activation patterns.

Interpretation: This approach allows for targeted improvement of language model reliability in scientific applications by identifying when model uncertainty stems from lack of knowledge (potentially fixable) versus inherent language ambiguity (unavoidable) [5].

Research Toolkit for Uncertainty Quantification

Table 3: Essential Computational Tools for Uncertainty Quantification in Scientific Models

Tool/Reagent	Type/Category	Function in Uncertainty Quantification
TensorFlow Probability	Software library	Implements probabilistic layers for aleatoric uncertainty and Bayesian neural networks for epistemic uncertainty [4]
Dempster-Shafer Structures	Mathematical framework	Represents epistemic uncertainty through interval-valued probabilities and belief masses [2]
Bayesian Neural Networks	Modeling approach	Quantifies epistemic uncertainty through distributions over model parameters [4]
Probabilistic Programming	Programming paradigm	Enables flexible specification and inference for complex hierarchical models with mixed uncertainties
Linear Probes	Diagnostic tool	Identifies epistemic uncertainty in internal model representations [5]
P-Boxes (Probability Boxes)	Output representation	Visualizes and quantifies mixed uncertainty in prediction outputs [2]

Applications in Scientific Domains

Drug Development and Pharmaceutical Applications

In pharmaceutical research and development, the distinction between aleatory and epistemic uncertainty directly impacts decision-making across the drug discovery pipeline. In early-stage discovery, epistemic uncertainty often dominates due to limited understanding of novel biological targets, incomplete structure-activity relationship data, and simplified representations of complex physiological systems in silico models. Targeted experimental designs can systematically reduce these epistemic uncertainties, focusing resources on the most influential unknown parameters.

As compounds progress through development, aleatory uncertainty becomes increasingly significant, particularly in clinical trials where inter-individual variability in drug response, metabolism, and adverse effects manifests as irreducible randomness. Proper characterization of this variability through mixed-effects models and population pharmacokinetics allows for robust dosing recommendations and safety profiling. The regulatory acceptance of model-based drug development hinges on transparent quantification of both uncertainty types, with epistemic uncertainty determining the "credibility" of model predictions and aleatory uncertainty defining the expected variability in real-world outcomes [1].

Engineering and Risk Assessment

In engineering applications, particularly structural reliability and risk assessment, the proper treatment of aleatory and epistemic uncertainties significantly influences safety factors and design standards [1]. Aleatory uncertainty in material properties, environmental loads, and usage patterns defines the inherent variability that designs must accommodate. Epistemic uncertainty in model form, parameter estimation, and experimental data introduces additional uncertainty that can be reduced through research, testing, and model validation.

The explicit separation of these uncertainty types enables more rational risk-informed decision-making. When epistemic uncertainties dominate, resources can be allocated to research and testing programs that reduce ignorance. When aleatory uncertainties prevail, the focus shifts to robust design strategies that perform acceptably across the range of possible conditions. This approach is particularly valuable in performance-based engineering, where understanding the sources and character of uncertainties allows for more efficient designs without compromising safety [1].

Methodological Workflow and Decision Framework

The systematic quantification and management of aleatory and epistemic uncertainties follows a structured workflow that transforms raw uncertainties into actionable insights for scientific decision-making. The process begins with uncertainty identification and classification, followed by appropriate mathematical representation, propagation through system models, and finally interpretation for specific applications.

Uncertainty Quantification and Decision Workflow

This workflow emphasizes the critical branching point where uncertainties are classified as either aleatory or epistemic, determining their subsequent mathematical treatment. The convergence of both pathways at the propagation stage acknowledges that most practical problems involve mixed uncertainties that must be propagated jointly through system models. The final decision analysis step incorporates measures of residual epistemic uncertainty (ambiguity) to enable confidence-aware decision-making.

The power of this structured approach lies in its ability to provide diagnostic insights throughout the modeling process. By maintaining the separation between uncertainty types, modelers can identify whether limitations in predictive accuracy stem from fundamental variability (suggesting acceptance or robust design) versus reducible ignorance (suggesting targeted data collection or model refinement). This diagnostic capability is particularly valuable in resource-constrained research environments where efficient allocation of investigation efforts can significantly accelerate scientific progress.

Verification, Validation, and Uncertainty Quantification (VVUQ) constitutes a systematic framework essential for establishing credibility in computational modeling and simulation. As manufacturers increasingly shift from physical testing to computational predictive modeling throughout product life cycles, ensuring these computational models are formed using sound procedures becomes paramount [6]. VVUQ addresses this need through three interconnected processes: Verification determines whether the computational model accurately represents the underlying mathematical description; Validation assesses whether the model accurately represents real-world phenomena; and Uncertainty Quantification (UQ) evaluates how variations in numerical and physical parameters affect simulation outcomes [6] [7]. This framework is particularly crucial in fields like drug discovery and precision medicine, where computational decisions guide expensive and time-consuming experimental processes, making trust in model predictions fundamental [8] [9] [10].

The paradigm of scientific computing is undergoing a fundamental shift from deterministic to nondeterministic simulations, explicitly acknowledging and quantifying various uncertainty sources throughout the modeling process [11]. This shift profoundly impacts risk-informed decision-making across engineering and scientific disciplines, enabling researchers to quantify confidence in predictions, optimize solutions stable across input variations, and reduce development costs and unexpected failures [7]. This document outlines structured protocols and application notes for implementing VVUQ within computational models, with particular emphasis on pharmaceutical applications and molecular design.

Theoretical Foundations of VVUQ

Core Definitions and Relationships

The VVUQ framework systematically addresses different aspects of model credibility. Verification is the process of determining that a computational model accurately represents the underlying mathematical model and its solution [6] [11]. Also described as "solving the equations right," verification activities include code review, comparison with analytical solutions, and convergence studies [7]. Validation, by contrast, is the process of determining the degree to which a model accurately represents the real-world system from the perspective of its intended uses [6] [11]. This "solving the right equations" process involves comparing simulation results with experimental data and assessing model performance [7]. Uncertainty Quantification is the science of quantifying, characterizing, tracing, and managing uncertainties in computational and real-world systems [7]. UQ seeks to address problems associated with incorporating real-world variability and probabilistic behavior into engineering and systems analysis, moving beyond single-point predictions to assess likely outcomes across variable inputs [7].

Uncertainty Taxonomy

Uncertainties within VVUQ are broadly classified into two fundamental categories based on their inherent nature:

• Aleatoric Uncertainty: Also known as stochastic uncertainty, this represents inherent variations in physical systems or natural randomness in observed phenomena. Derived from the Latin "alea" (rolling of dice), this uncertainty is irreducible through additional data collection as it represents an intrinsic property of the system [11] [9]. Examples include material property variations, manufacturing tolerances, and stochastic environmental conditions [7].

• Epistemic Uncertainty: Arising from lack of knowledge or incomplete information, this uncertainty is theoretically reducible through additional data collection or improved modeling. Derived from the Greek "episteme" (knowledge), this uncertainty manifests in regions of parameter space where data is sparse or models are inadequately calibrated [11] [9]. Examples include model form assumptions, numerical approximation errors, and unmeasured parameters [7].

Table 1: Uncertainty Classification and Characteristics

Uncertainty Type	Nature	Reducibility	Representation	Examples
Aleatoric	Inherent randomness	Irreducible	Probability distributions	Material property variations, experimental measurement noise [11] [9]
Epistemic	Lack of knowledge	Reducible	Intervals, belief/plausibility	Model form assumptions, sparse data regions, numerical errors [11] [9]

Additional uncertainty sources include approximation uncertainty from model incompetence in fitting complex data, though this is often considered negligible for universal approximators like deep neural networks [9]. Numerical uncertainty arises from discretization, iteration, and computer round-off errors addressed through verification techniques [11].

VVUQ Workflow Diagram

The following diagram illustrates the comprehensive VVUQ workflow, integrating verification, validation, and uncertainty quantification processes into a unified framework for establishing model credibility.

VVUQ Application in Drug Discovery

The Uncertainty Challenge in Pharmaceutical Development

In drug discovery, decisions regarding which experiments to pursue are increasingly influenced by computational models for quantitative structure-activity relationships (QSAR) [8]. These decisions are critically important due to the time-consuming and expensive nature of wet-lab experiments, with typical discovery cycles extending over 3-6 years and costing millions of dollars. Accurate uncertainty quantification becomes essential to use resources optimally and improve trust in computational models [8] [9]. A fundamental challenge arises from the fact that computational methods for QSAR modeling often suffer from limited data and sparse experimental observations, with approximately one-third or more of experimental labels being censored (providing thresholds rather than precise values) in real pharmaceutical settings [8].

The problem of human trust represents one of the most fundamental challenges in applied artificial intelligence for drug discovery [9]. Most in silico models provide reliable predictions only within a limited chemical space covered by the training set, known as the applicability domain (AD). Predictions for compounds outside this domain are unreliable and potentially dangerous for drug-design decision-making [9]. Uncertainty quantification addresses this by enabling autonomous drug designing through confidence level assessment of model predictions, quantitatively representing prediction reliability to assist researchers in molecular reasoning and experimental design [9].

Uncertainty Quantification Methods for Drug Discovery

Multiple UQ approaches have been deployed in drug discovery projects, each with distinct theoretical foundations and implementation considerations:

• Similarity-Based Approaches: These methods operate on the principle that if a test sample is too dissimilar to training samples, the corresponding prediction is likely unreliable [9]. This category includes traditional applicability domain definition methods such as bounding boxes, convex hull approaches, and k-nearest neighbors distance calculations [9]. These methods are more input-oriented, considering the feature space of samples with less emphasis on model structure.

• Bayesian Methods: These approaches treat model parameters and outputs as random variables, employing maximum a posteriori estimation according to Bayes' theorem [9]. Bayesian neural networks provide a principled framework for uncertainty decomposition but often require specialized implementations and can be computationally intensive for large-scale models.

• Ensemble-Based Strategies: These methods leverage the consistency of predictions from various base models as an estimate of confidence [9]. Techniques include bootstrap aggregating (bagging) and deep ensembles, which have demonstrated strong performance in molecular property prediction tasks while maintaining implementation simplicity.

Table 2: Uncertainty Quantification Methods in Drug Discovery

Method Category	Core Principle	Representative Techniques	Advantages	Limitations
Similarity-Based	Predictions for samples dissimilar to training set are unreliable	Bounding Box, Convex Hull, k-NN Distance [9]	Intuitive interpretation, model-agnostic	Limited model-specific insights, dependence on feature representation
Bayesian	Parameters and outputs treated as random variables	Bayesian Neural Networks, Monte Carlo Dropout [9]	Principled uncertainty decomposition, strong theoretical foundation	Computational intensity, implementation complexity
Ensemble-Based	Prediction variance across models indicates uncertainty	Bootstrap Aggregating, Deep Ensembles [8] [9]	Implementation simplicity, strong empirical performance	Computational cost multiple models, potential correlation issues

Advanced UQ Protocol: Handling Censored Regression Labels

Pharmaceutical data often contains censored labels where precise measurement values are unavailable, instead providing thresholds (e.g., "greater than" or "less than" values). Standard UQ approaches cannot fully utilize this partial information, necessitating specialized protocols.

Protocol 3.1: Censored Regression with Uncertainty Quantification

• Objective: Adapt ensemble-based, Bayesian, and Gaussian models to learn from censored regression labels for reliable uncertainty estimation in pharmaceutical settings.

• Materials and Data Requirements:

  • Experimental data with both precise and censored labels (typically ≥30% censored in pharmaceutical applications)
  • Implementation of Tobit model from survival analysis
  • Computational environment: Python 3.11 with PyTorch 2.0.1 or equivalent deep learning framework

• Methodology:

  • Data Preprocessing: Identify and flag censored labels in the dataset, distinguishing left-censored (below detection threshold), right-censored (above detection threshold), and precise measurements.
  • Model Adaptation: Implement Tobit likelihood function for each model type:
    • For ensemble methods: Modify loss function to incorporate censored information across ensemble members
    • For Bayesian networks: Implement censored-aware posterior estimation
    • For Gaussian models: Adapt variance estimation to account for censored regions
  • Temporal Evaluation: Assess model performance on time-split data to simulate real-world deployment conditions and evaluate temporal generalization.
  • Uncertainty Calibration: Validate uncertainty estimates using proper scoring rules and calibration metrics specific to censored data scenarios.

• Validation Metrics:

  • Ranking ability: Correlation between uncertainty estimates and prediction errors (Spearman correlation for regression)
  • Calibration ability: Agreement between predicted confidence intervals and empirical error distributions
  • Temporal performance: Model degradation assessment over time with changing data distributions

• Implementation Notes: This protocol has demonstrated essential improvements in reliably estimating uncertainties in real pharmaceutical settings where substantial portions of experimental labels are censored [8].

VVUQ in Molecular Design and Digital Twins

Uncertainty-Aware Molecular Design Framework

Molecular design presents unique challenges for uncertainty quantification, particularly when optimizing across expansive chemical spaces where models must extrapolate beyond training data distributions. The integration of UQ with graph neural networks (GNNs) enables more reliable exploration of chemical space by quantifying prediction confidence for novel molecular structures [12].

Protocol 4.1: UQ-Enhanced Molecular Optimization with Graph Neural Networks

• Objective: Integrate uncertainty quantification with directed message passing neural networks (D-MPNNs) and genetic algorithms for efficient molecular design across broad chemical spaces.

• Computational Resources:

  • Graph neural network implementation (Chemprop recommended)
  • Tartarus and GuacaMol platforms for benchmarking
  • Genetic algorithm framework for molecular optimization

• Experimental Workflow:

  • Surrogate Model Development: Train D-MPNN models on molecular structure-property data to predict target properties and their associated uncertainties.
  • Uncertainty Integration: Implement probabilistic improvement optimization (PIO) to guide molecular exploration based on the likelihood that candidate molecules exceed predefined property thresholds.
  • Multi-Objective Optimization: Balance competing design objectives using uncertainty-weighted selection criteria, particularly advantageous when objectives are mutually constraining.
  • Validation and Selection: Synthesize and experimentally characterize top candidate molecules identified through the uncertainty-aware optimization process.

• Key Implementation Considerations:

  • The PIO approach is particularly effective for practical applications where molecular properties must meet specific thresholds rather than extreme values
  • Multi-objective tasks benefit substantially from UQ integration, balancing exploration and exploitation in chemically diverse regions
  • Benchmark against uncertainty-agnostic approaches using established molecular design platforms

The following diagram illustrates the integrated workflow for uncertainty-aware molecular design combining GNNs with genetic algorithms:

VVUQ for Digital Twins in Precision Medicine

Digital twins in precision medicine represent virtual representations of individual patients that simulate health trajectories and interventions, creating demanding requirements for VVUQ implementation [10]. The VVUQ framework is essential for ensuring safety and efficacy when integrating digital twins into clinical practice.

• Verification Challenges: Code verification for multi-scale physiological models spanning cellular to organ-level processes, with particular emphasis on numerical accuracy and solution convergence for coupled differential equation systems.

• Validation Methodologies: Development of personalized trial methodologies and patient-specific validation metrics comparing virtual predictions with clinical observations across diverse patient populations.

• Uncertainty Quantification: Characterization of parameter uncertainties, model form uncertainties, and intervention response variabilities across virtual patient populations.

• Standardization Needs: Establishment of standardized VVUQ processes specific to medical digital twins, addressing regulatory requirements and clinical acceptance barriers [10].

Research Reagent Solutions

Table 3: Essential Computational Tools for VVUQ Implementation

Tool/Category	Function	Example Applications	Implementation Notes
ASME VVUQ Standards	Terminology and procedure standardization	Terminology (VVUQ 1-2022), Solid Mechanics (V&V 10-2019), Medical Devices (V&V 40-2018) [6]	Provides standardized frameworks for credibility assessment
UQ Software Platforms	Uncertainty propagation and analysis	SmartUQ for design of experiments, calibration, statistical comparison [7]	Offers specialized tools for uncertainty propagation and sensitivity analysis
Graph Neural Networks	Molecular representation learning	D-MPNN in Chemprop for molecular property prediction [12]	Enables direct operation on molecular graphs with uncertainty quantification
Bayesian Inference Tools	Probabilistic modeling and inference	Bayesian neural networks, Monte Carlo dropout methods [9]	Provides principled uncertainty decomposition
Benchmarking Platforms	Method evaluation and comparison	Tartarus (materials science), GuacaMol (drug discovery) [12]	Enables standardized performance assessment across methods
Censored Data Handlers	Management of threshold-based observations	Tobit model implementations for censored regression [8]	Essential for pharmaceutical data with detection limit censoring

Concluding Remarks

The VVUQ framework represents a fundamental shift from deterministic to probabilistically rigorous computational modeling, enabling credible predictions for high-consequence decisions in drug discovery, molecular design, and precision medicine. Successful implementation requires systematic attention to verification principles, validation against high-quality experimental data, and comprehensive uncertainty quantification addressing both aleatoric and epistemic sources. The protocols and applications outlined herein provide actionable guidance for researchers implementing VVUQ in computational models, with particular relevance to pharmaceutical and biomedical applications. As computational models continue to increase in complexity and scope, further development of standardized VVUQ methodologies remains essential for bridging the gap between simulation and clinical or industrial application.

Uncertainty quantification (UQ) provides a structured framework for understanding how variability and errors in model inputs and assumptions propagate to affect biomedical research outputs and clinical decisions [13]. In healthcare, clinical decision-making is a critical process that directly affects patient outcomes, yet inherent uncertainties in medical data, patient responses, and treatment outcomes pose significant challenges [13]. These uncertainties stem from various sources, including variability in patient characteristics, limitations of diagnostic tests, and the complex nature of diseases [13].

The three pillars of model credibility in computational biomedicine are verification, validation, and uncertainty quantification [13]. While verification ensures the computational implementation correctly solves the model equations and validation confirms the model matches experimental behavior, UQ addresses how uncertainties in inputs affect outputs, making it equally crucial for establishing model trustworthiness [13]. As biomedical research increasingly relies on complex computational models and data-driven approaches, systematically analyzing uncertainties becomes essential for improving the precision and reliability of medical evaluations.

UQ Applications in Biomedical Research: Protocols and Data Analysis

Biomarker Discovery and Validation for Neurological Diseases

Experimental Protocol: Biomarker Identification and Tracking for Motor Neuron Disease

• Objective: To discover and validate biomarkers for improving diagnosis, monitoring progression, and guiding treatment decisions in motor neuron disease (MND) [14].
• Materials and Reagents:
  • Patient blood samples for plasma isolation
  • DNA/RNA extraction kits
  • Next-generation sequencing reagents
  • ELISA kits for target protein quantification
  • Cell culture materials for extracellular vesicle isolation
  • MRI contrast agents (where applicable)
• Methodology:
  • Patient Cohort Selection: Recruit MND patients and age-matched healthy controls following ethical approval and informed consent. Document disease stage, progression history, and genetic background [14].
  • Multimodal Sample Collection: Collect blood samples for molecular analysis (cell-free DNA, proteins, extracellular vesicles) and schedule brain MRI scans using standardized protocols [14].
  • Molecular Profiling:
    • Extract and sequence cell-free DNA to identify genetic signatures and mutations [14].
    • Isulate extracellular vesicles from plasma and analyze cargo (proteins, miRNAs) using targeted proteomics and sequencing [14].
    • Quantify candidate protein biomarkers in serum using validated ELISA assays [14].
  • Neuroimaging:
    • Perform advanced MRI scans (structural, functional, diffusion tensor imaging) to identify brain and spinal cord changes [14].
    • Apply computational methods to extract quantitative features from images (e.g., cortical thickness, white matter integrity) [14].
  • Data Integration and Biomarker Validation:
    • Apply machine learning and bioinformatics approaches to identify biomarker patterns from multimodal datasets [14].
    • Correlate biomarker levels with clinical scores and progression rates.
    • Validate candidate biomarkers in an independent patient cohort to assess reproducibility and clinical utility [14].

Table 1: Quantitative Data Analysis in MND Biomarker Discovery

Biomarker Type	Measurement Technique	Data Variability Source	UQ Method Applied	Key Outcome Metric
Genetic Biomarkers	Next-Generation Sequencing	Sequencing depth, alignment errors	Confidence intervals for mutation frequency	Sensitivity/Specificity for disease subtyping
Protein Biomarkers	ELISA/MS-based Proteomics	Inter-assay precision, biological variation	Error propagation from standard curves	Correlation with disease progression (R² value)
Imaging Biomarkers	Advanced MRI	Scanner variability, patient movement	Test-retest reliability analysis	Effect size in differentiating patient groups
Metabolic Biomarkers	Metabolomics Platform	Instrument drift, peak identification	Principal component analysis with uncertainty	Predictive accuracy for treatment response

Uncertainty-Aware Diagnostic Imaging Analysis

Application Note: Quantifying Uncertainty in Medical Image Processing for Clinical Decision Support

Medical image processing algorithms often serve as either self-contained models or components within larger simulations, making UQ for these tools critical for clinical adoption [13]. For example, an algorithm quantifying extravasated blood volume in cerebral haemorrhage patients directly influences treatment decisions, where understanding measurement uncertainty is essential [13].

Protocol: UQ for Tumor Volume Segmentation in MRI

• Objective: To quantify segmentation uncertainty in MRI-based tumor volume measurements and its impact on treatment monitoring.
• Input Data Requirements: Multi-parametric MRI scans (T1, T2, FLAIR, contrast-enhanced T1) with standardized acquisition parameters.
• Processing and Analysis:
  • Multi-observer Annotation: Have multiple expert radiologists manually segment tumor volumes to establish ground truth with inter-observer variability [13].
  • Algorithmic Segmentation: Apply deep learning-based segmentation models (e.g., U-Net variants) to generate primary volume estimates.
  • Uncertainty Quantification:
    • Implement test-time augmentation to assess model robustness to input variations.
    • Use Monte Carlo dropout during inference to estimate model uncertainty.
    • Calculate volume difference metrics between algorithmic and expert segmentations.
  • Uncertainty Propagation: Model how segmentation uncertainty affects subsequent clinical decisions, such as determining treatment response based on volume changes.

Table 2: Uncertainty Sources in Diagnostic Imaging Models

Uncertainty Category	Source Example	Impact on Model Output	Mitigation Strategy
Data-Related (Aleatoric)	MRI image noise, partial volume effects	Irreducible variability in pixel intensity	Characterize noise distribution, use robust loss functions
Model-Related (Epistemic)	Limited training data for rare findings, model architecture choices	Poor generalization to new datasets	Bayesian neural networks, ensemble methods, data augmentation
Coupling-Related	Geometry extraction from segmentation for surgical planning	Errors in 3D reconstruction from 2D slices	Surface smoothing algorithms, manual review checkpoints

Enhancing Clinical Trial Design Through UQ

Protocol: Incorporating Biomarkers and UQ in Clinical Trial Outcomes

Researchers at the UQ Centre for MND Research focus on developing biomarkers that provide clear, data-driven readouts of whether a therapy is working, helping to accelerate and refine MND clinical trials [14]. The integration of UQ in this process allows for better trial design and more nuanced interpretation of results.

Methodology:

• Endpoint Selection: Identify and validate quantitative biomarkers (imaging, blood-based, or physiological) as secondary or primary endpoints alongside clinical scores [14].
• Uncertainty Characterization: For each biomarker endpoint, quantify measurement precision, biological variability, and assay performance metrics.
• Power Analysis: Use uncertainty estimates to perform more accurate sample size calculations, potentially reducing required patient numbers while maintaining statistical power.
• Adaptive Design: Implement futility analyses and dose adjustment rules based on biomarker trajectories and their confidence intervals during the trial.
• Subgroup Identification: Apply machine learning methods to uncertainty-aware biomarker data to identify patient subgroups with distinct treatment responses [14].

Visualization of UQ Workflows in Biomedicine

UQ-Integrated Biomedical Research Workflow

Diagram 1: UQ workflow for biomedical research.

Diagram 2: Uncertainty sources affecting clinical decisions.

The Scientist's Toolkit: Essential Research Reagents and Materials

Table 3: Key Research Reagent Solutions for Biomedical UQ Studies

Reagent/Material	Function in UQ Studies	Application Example
DNA/RNA Extraction Kits	Isolate high-quality nucleic acids for genomic biomarker studies; lot-to-lot variability contributes to measurement uncertainty.	Genetic biomarker discovery in MND using cell-free DNA [14].
ELISA Assay Kits	Quantify protein biomarker concentrations; standard curve precision directly impacts uncertainty in concentration estimates.	Validation of inflammatory protein biomarkers in patient serum [14].
Extracellular Vesicle Isolation Kits	Enrich for vesicles from biofluids; isolation efficiency affects downstream analysis and introduces variability.	Studying vesicle cargo as potential disease biomarkers [14].
MRI Contrast Agents	Enhance tissue contrast in imaging; pharmacokinetic variability between patients affects intensity measurements.	Quantifying blood-brain barrier disruption in neurological diseases.
Cell Culture Reagents	Maintain consistent growth conditions; serum lot variations contribute to experimental uncertainty in cell models.	Developing in vitro models for disease mechanism studies.
Next-Generation Sequencing Reagents	Enable high-throughput sequencing; reagent performance affects base calling quality and variant detection confidence.	Whole genome sequencing for identifying genetic risk factors [14].

Uncertainty quantification provides an essential framework for advancing biomedical research from exploratory science to clinical application. By systematically addressing data-related, model-related, and coupling-related uncertainties, researchers can develop more reliable diagnostic tools, biomarkers, and treatment optimization strategies. The protocols and analyses presented here demonstrate practical approaches for implementing UQ across various biomedical domains, ultimately supporting the development of more robust, clinically relevant research outcomes that can better inform patient care decisions. As biomedical models grow in complexity, integrating UQ from the initial research stages will be crucial for building trustworthiness and accelerating translation to clinical practice.

In computational modeling, particularly within biomedical and drug development research, Uncertainty Quantification (UQ) transforms model predictions from deterministic point estimates into probabilistic statements that characterize reliability. The process involves representing input parameters as random variables with specified probability distributions and propagating these uncertainties through computational models to quantify their impact on outputs. [15] [16] This forward UQ process enables researchers to compute key statistics—including means, variances, sensitivities, and quantiles—that describe the resulting probability distribution of model outputs. These statistics provide critical insights for risk assessment, decision-making, and model validation in preclinical drug development. [15] [17]

Table 1: Definitions of Key UQ Statistics

Statistic	Mathematical Definition	Interpretation in Biomedical Context
Mean	E[u_N(p)]	Expected value of model output (e.g., average drug response)
Variance	E[(u_N(p) - E[u_N(p)])²]	Spread or variability of model output around the mean
Median	Value m where P(u_N ≤ m) ≥ ½ and P(u_N ≥ m) ≥ ½	Central value where half of output distribution lies above/below
Quantiles	Value q where P(u_N ≥ q) ≥ 1-δ and P(u_N ≤ q) ≥ δ for δ ∈ (0,1)	Threshold values defining probability boundaries (e.g., confidence intervals)
Total Sensitivity	S_T,ℐ = V(ℐ)/Var(u_N) for subset ℐ of parameters	Fraction of output variance attributable to a parameter subset
Global Sensitivity	S_G,ℐ = [V(ℐ) - ∑∅≠𝒥⊂ℐV(𝒥)]/Var(u_N)	Main effect contribution of parameters to output variance
Local Sensitivity	∇u_N(p̃) at fixed parameter value p̃	Local rate of change of output with respect to parameter variations

Computational Methodologies for UQ Statistics

Various computational approaches exist for estimating UQ statistics, each with distinct strengths and computational requirements. The choice of methodology depends on model complexity, computational cost per evaluation, and dimensional complexity.

Non-Intrusive Polynomial Chaos (PC) Methods

Polynomial Chaos expansions build functional approximations (emulators) that map parameter values to model outputs using orthogonal polynomials tailored to input distributions. [15] The UncertainSCI software implements modern PC techniques utilizing weighted Fekete points and leverage score sketching for near-optimal sampling. [15] Once constructed, the PC emulator enables rapid computation of output statistics without additional costly model evaluations:

• Means and moments are obtained analytically from PC coefficients
• Sensitivities are computed via variance decomposition
• Quantiles are calculated numerically by sampling the cheap-to-evaluate emulator [15]

Sampling-Based Approaches

Monte Carlo (MC) and Latin Hypercube Sampling (LHS) methods propagate input uncertainties by evaluating the computational model at numerous sample points. [16] While conceptually straightforward, these methods typically require thousands of model evaluations to achieve statistical convergence. Advanced variants include:

• Multifidelity Monte Carlo (MFMC): Uses control variates from low-fidelity models to reduce estimator variance, accelerating mean estimation by almost four orders of magnitude compared to standard MC. [18]
• Importance Sampling: Preferentially places samples in important regions (e.g., near failure boundaries) to efficiently estimate rare event probabilities. [16]
• Sequential Monte Carlo (SMC): Employed for Bayesian data assimilation in dynamical systems like epidemiological ABMs, enabling parameter estimation with streaming data. [19]

Stochastic Expansion Methods

Beyond PC, other expansion techniques include Stochastic Collocation (SC) and Functional Tensor Train (FTT), which form functional approximations between inputs and outputs. [16] These methods provide analytic response moments and variance-based sensitivity metrics, with PDFs/CDFs computed numerically by sampling the expansion.

Diagram 1: UQ Statistical Analysis Workflow

Experimental Protocols for UQ Analysis

Protocol: Polynomial Chaos-Based UQ Analysis

This protocol outlines the procedure for implementing non-intrusive polynomial chaos expansion for uncertainty quantification in computational models, adapted from UncertainSCI methodology. [15]

Research Reagent Solutions:

• UncertainSCI Python Suite: Open-source software for building PC emulators with near-optimal sampling strategies [15]
• Parameter Distributions: Probability distributions characterizing input uncertainties (normal, uniform, beta, etc.) [16]
• Forward Model: Existing computational simulation code (e.g., bioelectric cardiac models, drug response models)
• Sampling Ensemble: Set of parameter values determined via weighted Fekete points or randomized subsampling

Procedure:

• Parameter Distribution Specification
  • Define probabilistic input parameters p = (p₁, p₂, ..., p_d) with joint distribution μ
  • Select appropriate polynomial basis functions orthogonal to input distributions (e.g., Hermite for normal, Legendre for uniform)

• Experimental Design Generation

  • Generate parameter sample ensemble {p^(1), p^(2), ..., p^(N)} using weighted Fekete points
  • Utilize leverage score sketching for near-optimal sampling in high-dimensional spaces

• Forward Model Evaluation

  • Execute computational model at each parameter sample: u(p^(i)) for i = 1, ..., N
  • Collect output responses, potentially including field values in high-dimensional spaces

• Polynomial Chaos Emulator Construction

  • Solve for PC expansion coefficients using regression or projection methods
  • Build surrogate model: u_N(p) = ∑_{α∈Λ} c_α Ψ_α(p) where Ψ_α are multivariate orthogonal polynomials

• Statistical Quantification

  • Compute mean from zeroth-order coefficient: E[u_N] ≈ c_0
  • Calculate variance from higher-order coefficients: Var(u_N) ≈ ∑_{α≠0} c_α²
  • Determine sensitivity indices via Sobol' decomposition of variance
  • Estimate quantiles by sampling the PC surrogate and computing empirical quantiles

• Validation and Error Assessment

  • Compare emulator predictions with additional forward model evaluations
  • Assess convergence of statistical estimates with increasing sample size

Protocol: Multifidelity Global Sensitivity Analysis

This protocol describes the Multifidelity Global Sensitivity Analysis (MFGSA) method for efficiently computing variance-based sensitivity indices, leveraging both high-fidelity and computationally cheaper low-fidelity models. [18]

Research Reagent Solutions:

• High-Fidelity Model: Accurate but computationally expensive computational model
• Low-Fidelity Models: Approximate models with correlated outputs but reduced computational cost
• MFGSA MATLAB Toolkit: Open-source implementation for multifidelity sensitivity analysis [18]
• Correlation Assessment: Methods to quantify output correlation between model fidelities

Procedure:

• Model Fidelity Characterization
  • Identify computational costs for each model fidelity: C_1, C_2, ..., C_K where C_1 is high-fidelity cost
  • Quantify correlation structure between model outputs across fidelities

• Optimal Allocation Design

  • Determine optimal number of evaluations for each model fidelity to minimize estimator variance
  • Allocate computational budget according to relative costs and correlations

• Multifidelity Sampling

  • Generate input samples according to specified parameter distributions
  • Evaluate both high-fidelity and low-fidelity models at allocated sample counts

• Control Variate Estimation

  • Form multifidelity estimators for variance components using low-fidelity models as control variates
  • Compute corrected estimates that leverage correlations between model outputs

• Sensitivity Index Calculation

  • Calculate main effect (first-order) sensitivity indices: S_i = Var[E[Y|X_i]]/Var[Y]
  • Compute total effect indices: S_Ti = E[Var[Y|X_~i]]/Var[Y] where X_~i denotes all parameters except X_i
  • Rank parameters by contribution to output uncertainty

• Variance Reduction Assessment

  • Compare statistical precision of MFGSA with traditional single-fidelity approaches
  • Quantify computational speed-up achieved through multifidelity framework

Table 2: UQ Method Selection Guide

Method	Optimal Use Case	Computational Cost	Key Statistics	Implementation Tools
Polynomial Chaos Expansion	Smooth parameter dependencies, moderate dimensions	50-500 model evaluations [15]	Means, variances, sensitivities, quantiles [15]	UncertainSCI [15], UQLab [20]
Multifidelity Monte Carlo	Models with correlated low-fidelity approximations	10-1000x acceleration over MC [18]	Means, variances, sensitivity indices [18]	MFMC MATLAB Toolbox [18]
Latin Hypercube Sampling	General purpose, non-smooth responses	100s-1000s model evaluations [16]	Full distribution statistics	Dakota [16]
Sequential Monte Carlo	Dynamic systems with streaming data	Varies with state dimension	Time-varying parameter distributions	Custom Jax implementations [19]
Importance Sampling	Rare event probability estimation	More efficient than MC for rare events	Failure probabilities, risk metrics	Dakota [16]

Applications in Drug Development and Biomedical Research

Uncertainty quantification statistics play critical roles in various biomedical applications, from preclinical drug development to clinical treatment planning.

Preclinical Drug Efficacy Assessment

In preclinical drug development, UQ statistics quantify confidence in therapeutic efficacy predictions. For example, in rodent pain models assessing novel analgesics, UQ can determine how parameter uncertainties (e.g., dosage timing, bioavailability) affect predicted pain reduction metrics. [17] Variance-based sensitivity indices identify which pharmacological parameters contribute most to variability in efficacy outcomes, guiding experimental refinement.

Bioelectric Field Modeling

In computational models of bioelectric phenomena (e.g., cardiac potentials or neuromodulation), UQ statistics quantify how tissue property variations affect simulation results. [15] Mean and variance estimates characterize expected ranges of induced electric fields, while quantiles define safety thresholds for medical devices. Sensitivity analysis reveals critical parameters requiring precise measurement.

Diagram 2: UQ in Biomedical Decision Support

Disease Model Calibration

For epidemiological models of disease transmission, UQ statistics facilitate model calibration to observational data. [19] Sequential Monte Carlo methods assimilate streaming infection data to update parameter distributions, with mean estimates providing expected disease trajectories and quantiles defining confidence envelopes for public health planning. Sensitivity analysis identifies dominant factors controlling outbreak dynamics.

The comprehensive quantification of means, variances, sensitivities, and quantiles provides the statistical foundation for credible computational predictions in drug development and biomedical research. These UQ statistics transform deterministic simulations into probabilistic forecasts with characterized reliability, enabling evidence-based decision-making under uncertainty. Modern computational frameworks like UncertainSCI, Dakota, and multifidelity methods make sophisticated UQ analysis accessible to researchers, supporting robust preclinical assessment and therapeutic development. As computational models grow increasingly complex, the rigorous application of these UQ statistical measures will remain essential for translating in silico predictions into real-world biomedical insights.

Parametric Uncertainty Quantification (Parametric UQ) is a fundamental process in computational modeling that involves treating uncertain model inputs as random variables with defined probability distributions and propagating this uncertainty through the model to quantify its impact on outputs [21]. This approach replaces the traditional deterministic modeling paradigm, where inputs and outputs are fixed values, with a probabilistic framework that provides a more comprehensive understanding of system behavior and model predictions. In fields such as drug development and physiological modeling, this is particularly crucial as model parameters often exhibit uncertainty due to measurement limitations and natural physiological variability [21].

The process consists of two primary stages: Uncertainty Characterization (UC), which involves quantifying uncertainty in model inputs by determining appropriate probability distributions, and Uncertainty Propagation (UP), which calculates the resultant uncertainty in model outputs by propagating the input uncertainties through the model [21]. This probabilistic approach enables researchers to assess the robustness of model predictions, identify influential parameters, and make more informed decisions that account for underlying uncertainties.

Key Methodological Approaches

Parametric UQ employs several computational techniques, each with distinct strengths and applications. The table below summarizes the primary methods used in computational modeling research:

Table 1: Key Methodological Approaches for Parametric Uncertainty Quantification

Method	Core Principle	Primary Applications	Key Advantages	Limitations
Monte Carlo Simulation	Uses repeated random sampling from input distributions to compute numerical results [22] [23]	Project forecasting, risk analysis, financial modeling, physiological systems [21] [23]	Handles nonlinear and complex systems; conceptually straightforward; parallelizable [22]	Computationally intensive (convergence rate: N⁻¹/²); requires many model evaluations [22]
Sensitivity Analysis (Sobol Method)	Variance-based global sensitivity analysis that decomposes output variance into contributions from individual inputs and interactions [24]	Factor prioritization, model simplification, identification of key drivers of uncertainty [25]	Quantifies both individual and interactive effects; model-independent; provides global sensitivity measures [24] [25]	Computationally demanding; complexity increases with dimensionality [24]
Bayesian Inference with Surrogate Models	Combines prior knowledge with observed data using Bayes' theorem; often uses surrogate models (Gaussian Processes, PCE) to approximate complex systems [26] [27]	Parameter estimation for complex models with limited data; clinical decision support systems [27]	Incorporates prior knowledge; provides full posterior distributions; quantifies epistemic uncertainty [27]	Computationally challenging for high-dimensional problems; requires careful prior specification [27]
Conformal Prediction	Distribution-free framework that provides finite-sample coverage guarantees without strong distributional assumptions [28]	Uncertainty quantification for generative AI, human-AI collaboration, changepoint detection [28]	Provides distribution-free guarantees; valid under mild exchangeability assumptions; computationally efficient [28]	Requires appropriate score functions; confidence sets may be uninformative with poor scores [28]

Advanced and Hybrid Approaches

Recent methodological advances have focused on increasing computational efficiency and expanding applications to complex systems. Physics-Informed Neural Networks with Uncertainty Quantification (PINN-UU) integrate the space-time domain with uncertain parameter spaces within a unified computational framework, demonstrating particular value for systems with scarce observational data, such as subsurface water bodies [26]. Similarly, conformal prediction methods have been extended to generative AI settings through frameworks like Conformal Prediction with Query Oracle (CPQ), which connects conformal prediction with the classical missing-mass problem to provide coverage guarantees for black-box generative models [28].

Experimental Protocols and Implementation

Protocol: Variance-Based Global Sensitivity Analysis Using Sobol Method

Table 2: Key Parameters for Variance-Based Sensitivity Analysis

Parameter	Description	Typical Settings	Notes
First-Order Sobol Index (Sᵢ)	Measures the contribution of a single input parameter to the output variance [24]	Range: 0 to 1	Values near 1 indicate parameters that dominantly control output uncertainty [24]
Total Sobol Index (Sₜ)	Measures the overall contribution of an input parameter, including both individual effects and interactions with other variables [24]	Range: 0 to 1	Reveals parameters involved in interactions; Sₜ ≫ Sᵢ indicates significant interactive effects [24]
Sample Size (N)	Number of model evaluations required	Typically 1,000-10,000 per parameter	Convergence should be verified by increasing sample size [24]
Sampling Method	Technique for generating input samples	Latin Hypercube Sampling (LHS) [24]	LHS provides more uniform coverage of parameter space than random sampling [24]

Workflow Implementation:

• Define Input Distributions: For each uncertain parameter, specify a probability distribution representing its uncertainty (e.g., normal, uniform, log-normal) based on experimental data or expert opinion [21].

• Generate Sample Matrix: Create two independent sampling matrices (A and B) of size N × k, where N is the sample size and k is the number of parameters, using Latin Hypercube Sampling [24].

• Construct Resampling Matrices: Create a set of matrices where each parameter in A is replaced sequentially with the corresponding column from B, resulting in k additional matrices.

• Model Evaluation: Run the computational model for all sample points in matrices A, B, and the resampling matrices, recording the output quantity of interest for each evaluation.

• Calculate Sobol Indices: Compute first-order and total Sobol indices using variance decomposition formulas:

  • First-order index: ( Si = \frac{V[E(Y|Xi)]}{V(Y)} )
  • Total index: ( S{Ti} = 1 - \frac{V[E(Y|X{-i})]}{V(Y)} ) where ( V[E(Y|X_i)] ) is the variance of the conditional expectation [24].

• Interpret Results: Parameters with high first-order indices (( Si > 0.1 )) are primary drivers of output uncertainty and should be prioritized for further measurement. Parameters with low total indices (( S{Ti} < 0.01 )) can potentially be fixed at nominal values to reduce model complexity [25].

Figure 1: Workflow for Variance-Based Global Sensitivity Analysis

Protocol: Consistent Monte Carlo Uncertainty Propagation

Principle: In distributed or sequential uncertainty analyses, consistent Monte Carlo methods must preserve dependencies of random variables by ensuring the same sequence is used for a particular quantity regardless of how many times or where it appears in the analysis [22].

Implementation Requirements:

• Unique Stream Identification: Assign a unique random number stream to each uncertain input variable in the system, maintained across all computational processes and analysis stages.

• Seed Management: Implement a reproducible seeding strategy that ensures identical sequences are regenerated for the same input variables in subsequent analyses.

• Dependency Tracking: Maintain a mapping between input variables and their corresponding sample sequences, particularly when reusing previously computed quantities in further analyses.

Validation Step: To verify consistency, compute the sample variance of a composite function ( Z = h(X, Y) ) where ( Y = g(X) ), ensuring that the same sequence ( {xn}{n=1}^N ) is used in both evaluations. Inconsistent sampling, where independent sequences are used for the same variable, will produce biased variance estimates [22].

Protocol: Probability of Success Assessment in Drug Development

Table 3: Probability of Success Assessment Framework

Component	Description	Data Sources	Application Context
Design Prior	Probability distribution capturing uncertainty in effect size for phase III [29]	Phase II data, expert elicitation, real-world data, historical clinical trials [29]	Critical for go/no-go decisions at phase II/III transition [29]
Predictive Power	Probability of rejecting null hypothesis given design prior [29]	Phase II endpoint data, association between biomarker and clinical outcomes [29]	Sample size determination for confirmatory trials [29]
Assurance	Bayesian equivalent of power using mixture prior distributions [29]	Combination of prior beliefs and current trial data [29]	Incorporating historical information into trial planning [29]

Implementation Workflow:

• Define Success Criteria: Specify the target product profile, including minimum acceptable and ideal efficacy results required for regulatory approval and reimbursement [29].

• Construct Design Prior: Develop a probability distribution for the treatment effect size in phase III, incorporating phase II data on the primary endpoint. When phase II uses biomarker or surrogate outcomes, leverage external data (e.g., real-world data, historical trials) to establish relationship with clinical endpoints [29].

• Calculate Probability of Success: Compute the probability of demonstrating statistically significant efficacy in phase III, integrating over the design prior to account for uncertainty in the true effect size [29].

• Decision Framework: Use the computed probability of success to inform portfolio management decisions, with typical thresholds ranging from 65-80% for progression to phase III, depending on organizational risk tolerance and development costs [29].

The Scientist's Toolkit: Essential Research Reagents and Computational Solutions

Table 4: Essential Research Reagents and Computational Solutions for Parametric UQ

Category	Item	Function/Application	Implementation Notes
Computational Algorithms	Sobol Method	Variance-based sensitivity analysis quantifying parameter contributions to output uncertainty [24]	Implemented in UQ modules of COMSOL, SAS, R packages (sensitivity) [24]
	Polynomial Chaos Expansion (PCE)	Surrogate modeling for efficient uncertainty propagation and sensitivity analysis [24]	Adaptive PCE automates surrogate model creation; direct Sobol index computation [24]
	Gaussian Process Emulators	Bayesian surrogate models for computationally intensive models [27]	Accelerates model calibration; enables UQ for complex models in clinically feasible timeframes [27]
	Conformal Prediction	Distribution-free uncertainty quantification with finite-sample guarantees [28]	Applied to generative AI, changepoint detection; requires appropriate score functions [28]
Software Tools	COMSOL UQ Module	Integrated platform for screening, sensitivity analysis, and reliability analysis [24]	Provides built-in Sobol method, LHS sampling, and automated surrogate modeling [24]
	Kanban Monte Carlo Tools	Project forecasting incorporating uncertainty and variability [23]	Uses historical throughput data for delivery date and capacity predictions [23]
Data Resources	Real-World Data (RWD)	Informs design priors for probability of success calculations [29]	Patient registries, historical controls; improves precision of phase III effect size estimation [29]
	Historical Clinical Trial Data	External data for biomarker-endpoint relationships [29]	Quantifies association between phase II biomarkers and phase III clinical endpoints [29]

Applications in Pharmaceutical Development and Biomedical Research

Temporal Distribution Shifts in Pharmaceutical Data

Real-world pharmaceutical data often exhibits significant temporal distribution shifts that impact the reliability of UQ methods. A comprehensive evaluation of QSAR models under realistic temporal shifts revealed:

• Magnitude Connection: The extent of distribution shift correlates strongly with the nature of the assay, with some assays showing pronounced shifts in both label and descriptor space over time [30].
• Performance Impairment: Pronounced distribution shifts impair the performance of popular UQ methods used in QSAR models, highlighting the challenge of identifying techniques that remain reliable under real-world data conditions [30].
• Calibration Impact: Temporal shifts significantly impact post hoc calibration of uncertainty estimates, necessitating regular reassessment and adjustment of UQ approaches throughout model deployment [30].

Cardiac Electrophysiology Models

Comprehensive UQ/SA in cardiac electrophysiology models demonstrates the feasibility of robust uncertainty assessment for complex physiological systems:

• Robustness Demonstration: Action potential simulations can be fully robust to low levels of parameter uncertainty, with a range of emergent dynamics (including oscillatory behavior) observed at larger uncertainty levels [21].
• Influential Parameter Identification: Comprehensive analysis revealed that five key parameters were highly influential in producing abnormal dynamics, providing guidance for targeted parameter measurement and model refinement [21].
• Model Failure Analysis: The framework enables systematic analysis of different behaviors that occur under parameter uncertainty, including "model failure" modes, enhancing model reliability in safety-critical applications [21].

Figure 2: Parametric UQ Methodologies and Research Applications

Pulmonary Hemodynamics Modeling

Bayesian parameter inference with Gaussian process emulators enables efficient UQ for complex physiological systems:

• Clinical Timeframes: GP emulators accelerate model calibration, enabling estimation of microvascular parameters and their uncertainties within clinically feasible timeframes [27].
• Disease Correlation: In chronic thromboembolic pulmonary hypertension (CTEPH), changes in inferred parameters strongly correlate with disease severity, particularly in lungs with more advanced disease [27].
• Heterogeneous Adaptation: CTEPH leads to heterogeneous microvascular adaptation reflected in distinct parameter shifts, enabling more targeted treatment strategies [27].

Parametric UQ, through modeling inputs as random variables, provides an essential framework for robust computational modeling in pharmaceutical and biomedical research. The methodologies outlined—from variance-based sensitivity analysis to consistent Monte Carlo approaches and Bayesian inference—offer structured protocols for implementing comprehensive uncertainty assessment. Particularly in drug development, where resources are constrained and decisions carry significant consequences, these approaches enable more informed decision-making by explicitly quantifying and propagating uncertainty through computational models. The integration of real-world data and advanced computational techniques continues to enhance the applicability and reliability of parametric UQ across the biomedical domain, supporting the development of more credible and clinically relevant computational models.

UQ's Role in Model-Informed Drug Development (MIDD)

Uncertainty Quantification (UQ) is a field of study that focuses on understanding, modeling, and reducing uncertainties in computational models and real-world systems [31]. In the context of Model-Informed Drug Development (MIDD), UQ provides a critical framework for quantifying the impact of uncertainties in pharmacological models, thereby making drug development decisions more robust and reliable [31] [32]. The U.S. Food and Drug Administration (FDA) has recognized the value of MIDD approaches, implementing a dedicated MIDD Paired Meeting Program that affords sponsors the opportunity to discuss MIDD approaches in medical product development [32]. This program aims to advance the integration of exposure-based, biological, and statistical models derived from preclinical and clinical data sources in drug development and regulatory review [32].

Uncertainties in drug development models arise from multiple sources, which UQ systematically characterizes and manages [31] [33]. In engineering and scientific modeling, uncertainties are broadly categorized as either epistemic uncertainty (stemming from incomplete knowledge or lack of data) or aleatoric uncertainty (originating from inherent variability in the system or environment) [31]. Both types must be accurately modeled to ensure robust predictions, particularly in high-stakes scenarios like human drug trials where minimizing the probability of incorrect decisions is essential [31] [32].

Table: Fundamental Uncertainty Types in Pharmacological Modeling

Uncertainty Type	Source	UQ Mitigation Approach	MIDD Application Example
Epistemic	Incomplete knowledge or data gaps [31]	Model ensembling, multi-fidelity methods [18] [34]	Extrapolating dose-response beyond tested doses
Aleatoric	Natural variability in biological systems [31]	Quantile regression, probabilistic modeling [34]	Inter-patient variability in drug metabolism
Model Structure	Incorrect model form or assumptions	Bayesian model averaging, discrepancy modeling [31] [33]	Structural uncertainty in PK/PD model selection
Parameter	Uncertainty in model parameter estimates	Bayesian inference, sensitivity analysis [18] [33]	Uncertainty in clearance and volume of distribution

Core UQ Methodologies for MIDD

Multi-Fidelity Uncertainty Propagation

Multi-fidelity UQ methods leverage multiple approximate models of varying computational cost and accuracy to accelerate uncertainty quantification tasks [18]. Rather than just replacing high-fidelity models with low-fidelity surrogates, multi-fidelity UQ methods use strategic recourse to high-fidelity models to establish accuracy guarantees on UQ results [18]. In drug development, this approach enables researchers to combine rapid, approximate screening models with computationally expensive, high-fidelity physiological models.

The Multifidelity Monte Carlo (MFMC) method uses a control variate formulation to accelerate the estimation of statistics of interest using multiple low-fidelity models [18]. This approach optimally allocates evaluations among models with different fidelities and costs, minimizing the variance of the estimator for a given computational budget [18]. For estimating the mean, MFMC can achieve almost four orders of magnitude improvement over standard Monte Carlo simulation using only high-fidelity models [18]. The mathematical formulation of the MFMC estimator for the expected value of a high-fidelity model output 𝔼[Q_{HF}] is:

$\hat{Q}{MFMC} = \frac{1}{N{HF}} \sum{i=1}^{N{HF}} Q{HF}^{(i)} + \alpha \left( \frac{1}{N{LF}} \sum{j=1}^{N{LF}} Q{LF}^{(j)} - \frac{1}{N{HF}} \sum{i=1}^{N{HF}} Q_{LF}^{(i)} \right)$

where $Q{HF}$ and $Q{LF}$ represent high-fidelity and low-fidelity model outputs, $N{HF}$ and $N{LF}$ are sample counts, and $\alpha$ is an optimal control variate coefficient that minimizes estimator variance [18].

Bayesian Calibration and Inference

Bayesian methods provide a natural framework for quantifying parameter uncertainty in pharmacological models and updating beliefs as new data becomes available [33]. Sandia National Laboratories' UQ Toolkit (UQTk) implements Bayesian calibration and parameter estimation methods that have been applied to assess the accuracy of thermodynamic models and propagate associated model errors into derived quantities such as process efficiencies [33]. This assessment enables evaluation of the trade-off between model complexity, computational cost, input data accuracy, and confidence in overall predictions.

For complex models with large numbers of uncertain parameters, multifidelity statistical inference approaches use a two-stage delayed acceptance Markov Chain Monte Carlo (MCMC) formulation [18]. A reduced-order model is used in the first step to increase the acceptance rate of candidates in the second step, with high-fidelity model outputs computed in the second step used to adapt the reduced-order model [18]. This approach is particularly valuable in MIDD for calibrating complex physiologically-based pharmacokinetic (PBPK) models where full model evaluation is computationally expensive.

Diagram: Multi-fidelity MCMC for Model Calibration

Sensitivity Analysis for Factor Prioritization

Variance-based sensitivity analysis quantifies and ranks the relative impact of uncertainty in different inputs on model outputs [18]. Standard Monte Carlo approaches for estimating sensitivity indices for d parameters require N(d+2) samples, which can be prohibitively expensive for complex pharmacological models [18]. The Multifidelity Global Sensitivity Analysis (MFGSA) method expands upon the MFMC control variate approach to accelerate the computation of variance and variance-based sensitivity indices with the same computational budget [18].

In MIDD applications, sensitivity analysis helps identify which parameters contribute most to output uncertainty, guiding resource allocation for additional data collection or experimental refinement. For example, in PBPK model development, sensitivity analysis can determine whether greater precision is needed in measuring tissue partition coefficients, metabolic clearance rates, or binding affinities to reduce uncertainty in predicted human exposure profiles.

Table: Multi-fidelity UQ Methods for MIDD Applications

UQ Method	Key Mechanism	Computational Advantage	MIDD Use Case
Multifidelity Monte Carlo (MFMC) [18]	Control variate using low-fidelity models	10-1000x speedup for mean estimation [18]	Population PK/PD analysis
Multifidelity Importance Sampling (MFIS) [31] [18]	Biasing density from low-fidelity models	Efficient rare event probability estimation [31]	Probability of critical adverse events
Langevin Bi-fidelity IS (L-BF-IS) [31]	Score-function-based sampling	High-dimensional (>100) input spaces [31]	High-dimensional biomarker models
Multifidelity GSA [18]	Control variate for Sobol indices	10x speedup for factor prioritization [18]	PBPK model factor screening

Experimental Protocols for UQ in MIDD

Protocol: Multi-fidelity PBPK Model Calibration

Purpose: To efficiently calibrate a PBPK model using multi-fidelity data sources while quantifying parameter uncertainty.

Materials and Computational Tools:

• High-fidelity model: Full PBPK model with detailed physiological representation
• Low-fidelity models: Simplified PBPK, QSP models, or QSAR predictions
• UQ software: UQ Toolkit (UQTk) or custom MFMC implementation [33]
• Data: Preclinical PK data (in vitro, in vivo), physicochemical properties, and early clinical data if available

Procedure:

• Model Preparation:
  • Develop a high-fidelity PBPK model with identified uncertain parameters
  • Create low-fidelity approximations through model simplification or surrogate modeling
  • Define prior distributions for all uncertain parameters based on literature and expert knowledge

• Multi-fidelity Experimental Design:

  • Determine optimal allocation of evaluations between model fidelities using MFMC allocation formulas [18]
  • Generate input samples using Latin Hypercube Sampling or Sobol sequences

• Model Evaluation:

  • Execute high-fidelity and low-fidelity model runs according to the experimental design
  • Record output quantities of interest (e.g., AUC, C~max~, T~max~)

• Uncertainty Propagation:

  • Apply MFMC estimator to compute statistics of interest (means, variances)
  • Calculate variance-based sensitivity indices using MFGSA approach

• Bayesian Calibration:

  • Implement delayed acceptance MCMC with low-fidelity pre-screening
  • Update parameter distributions using observed experimental data
  • Validate calibrated model against withheld data

• Decision Support:

  • Quantify uncertainty in key model predictions (e.g., human dose projection)
  • Perform value of information analysis to guide additional data collection

Expected Outcomes: A calibrated PBPK model with quantified parameter uncertainty, identification of most influential parameters, and projections of human pharmacokinetics with confidence intervals.

Protocol: Quantile Regression for Clinical Trial Simulation

Purpose: To predict confidence intervals for clinical trial outcomes using quantile regression to capture aleatoric uncertainty in patient responses.

Materials and Computational Tools:

• Drug-trial-disease model: Integrated pharmacology and disease progression model
• Patient population simulator: Virtual population generator with covariate distributions
• Quantile regression implementation: Modified neural network architecture with dual readout layers [34]

Procedure:

• Model Configuration:
  • Modify the drug-trial-disease model to include two readout layers with opposite penalization
  • Set asymmetric loss functions for 5th and 95th percentile targets

• Training Phase:

  • Train the model on historical clinical trial data or simulated training data
  • Use quantile loss function: $L_\tau(y, \hat{y}) = \begin{cases} \tau \cdot (y - \hat{y}) & \text{if } y > \hat{y} \ (1 - \tau) \cdot (\hat{y} - y) & \text{otherwise} \end{cases}$
  • Where $\tau$ is the target quantile (0.05 for lower bound, 0.95 for upper bound)

• Trial Simulation:

  • Generate virtual patient populations representing the target clinical population
  • Simulate trial outcomes for each virtual patient
  • Collect predicted quantiles for primary and secondary endpoints

• Uncertainty Quantification:

  • Calculate 90% confidence intervals as the difference between 95th and 5th percentile predictions
  • Aggregate results across virtual trials to estimate probability of trial success

• Scenario Analysis:

  • Evaluate confidence intervals under different trial designs, dosing regimens, and inclusion criteria
  • Optimize trial design to maximize probability of success while accounting for uncertainty

Expected Outcomes: Prediction intervals for clinical trial endpoints, quantitative assessment of trial success probability under different designs, and identification of optimal trial configurations that balance risk and potential benefit.

Regulatory Framework and Implementation

FDA MIDD Paired Meeting Program

The FDA's MIDD Paired Meeting Program provides a formal pathway for sponsors to discuss MIDD approaches, including UQ, for specific drug development programs [32]. The program includes an initial meeting and a follow-up meeting scheduled within approximately 60 days of receiving the meeting package [32]. For fiscal years 2023-2027, FDA grants 1-2 paired-meeting requests quarterly, with the possibility of additional proposals depending on resource availability [32].

Key eligibility criteria include:

• Drug/biologics development company with an active IND or PIND number
• Consortia or software/device developers must partner with a drug development company
• Proposals focused on dose selection, clinical trial simulation, or predictive safety evaluation are prioritized [32]

The FDA specifically recommends that meeting requests include an assessment of model risk, considering both the model influence (weight of model predictions in the totality of data) and the decision consequence (potential risk of making an incorrect decision) [32]. This aligns directly with UQ principles of quantifying how model uncertainties propagate to decision uncertainties.

Diagram: FDA MIDD Paired Meeting Program Workflow

UQ Implementation Strategy for Regulatory Submissions

Successful implementation of UQ in MIDD for regulatory submissions requires careful planning and documentation:

• Context of Use Definition: Clearly specify how the model will be used to inform regulatory decisions, whether for dose selection, trial design optimization, or providing mechanistic insight [32].

• Uncertainty Source Characterization: Systematically identify and document sources of uncertainty, including model structure uncertainty, parameter uncertainty, and experimental variability [31] [33].

• Method Selection and Justification: Choose UQ methods appropriate for the specific application and provide scientific justification for the selection. For example, multi-fidelity methods for computationally expensive models [18] or quantile regression for capturing data distribution uncertainties [34].

• Model Risk Assessment: Evaluate and document model risk based on the decision context, with higher-risk applications requiring more comprehensive UQ [32].

• Visualization and Communication: Develop clear visualizations of uncertainty information that effectively communicate the confidence in model predictions to regulatory reviewers.

Table: Essential UQ Tools and Resources for MIDD Applications

Tool/Resource	Type	Key Features	MIDD Application
UQ Toolkit (UQTk) [33]	Software library	Bayesian calibration, sensitivity analysis, uncertainty propagation	General pharmacological model UQ
Multifidelity Monte Carlo Codes [18]	MATLAB implementation	Optimal model allocation, control variate estimation	PBPK/PD model analysis
LM-Polygraph [35]	Open-source framework	Unified UQ and calibration algorithms, benchmarking	Natural language processing of medical literature
Readout Ensembling [34]	UQ method	Computational efficiency, epistemic uncertainty capture	Foundation model finetuning
Quantile Regression [34]	UQ method	Aleatoric uncertainty quantification, confidence intervals	Clinical trial outcome prediction
FDA MIDD Program [32]	Regulatory pathway	Agency feedback on MIDD approaches, including UQ	Regulatory strategy development

Uncertainty Quantification provides an essential methodological foundation for building confidence in Model-Informed Drug Development approaches. By systematically characterizing, quantifying, and propagating uncertainties through pharmacological models, UQ enables more robust decision-making throughout the drug development process. The integration of multi-fidelity methods, Bayesian inference, and sensitivity analysis creates a powerful framework for addressing the complex uncertainties inherent in predicting drug behavior in humans.

Regulatory agencies increasingly recognize the value of these quantitative approaches, as evidenced by the FDA's MIDD Paired Meeting Program [32]. As MIDD continues to evolve, UQ will play an increasingly critical role in establishing the credibility of model-based predictions and ensuring that drug development decisions are made with a clear understanding of associated uncertainties. The protocols, methods, and resources outlined in this document provide a foundation for researchers to implement rigorous UQ within their MIDD programs, ultimately contributing to more efficient and reliable drug development.

UQ Methods in Action: Techniques and Real-World Biomedical Applications

Non-Intrusive Polynomial Chaos for Efficient Uncertainty Propagation

Uncertainty Quantification (UQ) is indispensable for ensuring the reliability of computational models used to design and analyze complex systems across scientific and engineering disciplines. Traditional UQ methods, particularly Monte Carlo (MC) simulations, often become computationally prohibitive when dealing with expensive, high-fidelity models. Non-Intrusive Polynomial Chaos Expansion (NIPC) has emerged as a powerful surrogate modeling technique that overcomes this limitation by constructing a computationally efficient mathematical metamodel of the original system. Unlike intrusive methods, NIPC treats the deterministic model as a black box, requiring no modifications to the underlying code, thus facilitating its application to complex, legacy, or commercial simulation software [36]. This approach represents the stochastic model output as a series expansion of orthogonal polynomials, the choice of which is determined by the probability distributions of the uncertain inputs [37]. By enabling rapid propagation of input uncertainties, NIPC provides researchers and engineers with a robust framework for obtaining statistical moments and global sensitivity measures, supporting critical decision-making in risk assessment and design optimization.

Fundamental Principles of NIPC

The NIPC method approximates a stochastic model output using a truncated series of orthogonal polynomials. Consider a computational model represented as ( f = \mathcal{F}(u) ), where ( \mathcal{F} ) is the deterministic model, ( u \in \mathbb{R}^d ) is the input vector, and ( f ) is the scalar output. When the inputs are uncertain and represented by a random vector ( U ), the model output ( f(U) ) becomes stochastic. The Polynomial Chaos Expansion (PCE) seeks to represent this output as:

[ f(U) \approx \sum{i=0}^{q} \alphai \Phi_i(U) ]

Here, ( \Phii(U) ) are the multivariate orthogonal polynomial basis functions, and ( \alphai ) are the corresponding PCE coefficients to be determined [37]. The basis functions are selected based on the distributions of the uncertain inputs (e.g., Hermite polynomials for Gaussian inputs, Legendre for uniform) to achieve optimal convergence [37]. The number of terms in the truncated expansion, ( q+1 ), depends on the number of stochastic dimensions ( d ) and the maximum polynomial order ( p ), and is given by ( (d+p)!/(d!p!) ) [37].

The "non-intrusive" nature of the method lies in how the coefficients ( \alpha_i ) are calculated. The deterministic model ( \mathcal{F} ) is executed at a carefully selected set of training points (input samples), and the resulting outputs are used to fit the surrogate model. Two prevalent non-intrusive approaches are:

• Integration (Quadrature) Approach: Leverages the orthogonality of the polynomial basis. The coefficients are computed via numerical integration using quadrature rules (e.g., full-grid Gauss quadrature, sparse grids) [37].
• Regression Approach: Solves a linear least-squares problem to find the coefficients that best fit the model responses at a set of sampling points [38].

Application Notes: NIPC in Practice

The following table summarizes quantitative findings and parameters from recent, successful applications of NIPC across different engineering fields, demonstrating its versatility and effectiveness.

Table 1: Summary of NIPC Applications in Engineering Research

Application Field	Key Uncertain Inputs (Distribution)	Quantities of Interest (QoIs)	NIPC Implementation & Performance
Rotary Blood Pump Performance Analysis [38]	Operating points: Speed [0–5000] rpm, Flow [0–7] l/min	Pressure head, Axial force, 2D velocity field	Polynomial Order: 4Training Points: ≥20Accuracy: Mean Absolute Error = 0.1 m/s for velocity data
Nuclear Fusion Reactor Fault Transients [39]	Varistor parameters: ( K \in [8.134, 13.05] ) (Uniform), ( \beta \in [0.562, 0.595] ) (Uniform)	Coil peak voltage, Deposited FDU energy, Joule power in coil casing	Method: Integration-based using chaospy (v4.3.12)Validation: Benchmarked against Monte Carlo and Unscented Transform
Aircraft Design (Multidisciplinary Systems) [40] [37]	4 to 6 uncertain parameters (aleatoric/epistemic)	System performance metrics (implied)	Method: Graph-accelerated NIPC with partially tensor-structured quadratureResult: >40% reduction in computational cost vs. full-grid Gauss quadrature

Key Insights from Applications

• Parameter Selection is Critical: The blood pump study established that a polynomial order of 4 and a minimum of 20 training points were sufficient for accurate surrogate models [38]. This highlights the need for systematic parameter studies to ensure model fidelity without unnecessary computational expense.
• Handling Discontinuities: A notable challenge identified in the blood pump research is the difficulty in modeling discontinuous data, which is often relevant for clinically realistic operating points. This underscores the importance of assessing data smoothness prior to NIPC application [38].
• Efficiency in High Dimensions: For multidisciplinary systems in aircraft design, the standard full-grid quadrature approach scales poorly with dimensions. The graph-accelerated NIPC method, which uses computational graph transformations (AMTC) and tailored quadrature rules, demonstrated significant cost reductions, making UQ feasible for more complex problems [40] [37].

Detailed Experimental Protocol for NIPC

This protocol outlines the steps for performing uncertainty propagation using the non-intrusive polynomial chaos expansion, based on the methodologies successfully employed in the referenced studies. The workflow is divided into three main phases: Pre-processing, NIPC Construction, and Post-processing.

Phase 1: Pre-processing (Problem Definition)

Step 1: Define Input Uncertainties

• Identify all ( d ) uncertain parameters in the computational model. These can be operating conditions, material properties, or model parameters.
• Assign a probability distribution ( \rho(u) ) to each uncertain parameter. For example, in the fusion reactor study, the varistor parameters ( K ) and ( \beta ) were modeled as uniform distributions [39].

Step 2: Select Polynomial Chaos Basis

• Choose the appropriate family of orthogonal polynomials for the expansion. The selection is dictated by the Askey scheme to match the input distributions (e.g., Legendre polynomials for uniform inputs, Hermite for Gaussian) [37].

Step 3: Generate Training Samples

• Select a sampling strategy to define the points at which the high-fidelity model will be evaluated. Common choices include:
  • Full-Tensor Gauss Quadrature: Efficient for very low-dimensional problems (typically ( d \leq 3 )) but suffers from the curse of dimensionality [37].
  • Sparse Grids (Smolyak): Reduces the number of points compared to full-tensor grids for moderate dimensions.
  • Designed Quadrature/Monte Carlo Sampling: Used for higher-dimensional problems or when the quadrature points do not need to conform to a specific grid structure [37].

Phase 2: NIPC Construction (Surrogate Training)

Step 4: Run Deterministic Model

• Execute the original computational model at each of the training samples ( u^{(k)} ) generated in Step 3 to obtain the corresponding model outputs ( f^{(k)} ). This is typically the most computationally expensive step.

Step 5: Compute PCE Coefficients

• Calculate the coefficients ( \alphai ) of the expansion. Using the integration approach, this is done by exploiting orthogonality: [ \alphai = \frac{1}{\langle \Phii^2 \rangle} \int f(u) \Phii(u) \rho(u) du \approx \frac{1}{\langle \Phii^2 \rangle} \sumk w^{(k)} f^{(k)} \Phi_i(u^{(k)}) ] where ( w^{(k)} ) are the quadrature weights [37]. The regression approach involves solving a linear least-squares problem.

Step 6: Build Surrogate Model

• Construct the final PCE surrogate model by assembling the computed coefficients and the polynomial basis into the expression ( f(U) \approx \sum{i=0}^{q} \alphai \Phi_i(U) ). This surrogate is a closed-form mathematical expression that is cheap to evaluate.

Phase 3: Post-processing (UQ Analysis)

Step 7: Exploit the Surrogate Model

• Use the surrogate for intensive computational tasks. Since evaluating the polynomial surrogate is extremely fast, it becomes feasible to perform massive Monte Carlo sampling (e.g., millions of samples) on the surrogate to estimate the full statistical distribution of the output [39].

Step 8: Extract Statistics and Sensitivities

• Compute Statistical Moments: The mean and variance of the output can be directly derived from the PCE coefficients. The mean is ( \alpha0 ), and the variance is ( \sum{i=1}^{q} \alphai^2 \langle \Phii^2 \rangle ) [37].
• Global Sensitivity Analysis: Calculate Sobol' indices via post-processing of the PCE coefficients to quantify the contribution of each input uncertainty to the total variance of the output. This provides valuable insights for robust design and model simplification.

The Scientist's Toolkit: Essential Research Reagents

For researchers implementing NIPC, the "reagents" are the computational tools and software libraries that facilitate the process. The following table lists key resources.

Table 2: Key Computational Tools for NIPC Implementation

Tool / Resource	Type	Primary Function in NIPC	Application Example
chaospy [39]	Python Library	Provides a comprehensive framework for generating PCE basis, quadrature points, and computing coefficients via integration or regression.	Used for uncertainty propagation in nuclear fusion reactor fault transients [39].
OpenModelica [39]	Modeling & Simulation Environment	Serves as the high-fidelity deterministic model (e.g., for electrical circuit simulation) that is treated as a black box by the NIPC process.	Modeling the power supply circuit of DTT TF coils [39].
3D-FOX [39]	Finite Element Code	Acts as the high-fidelity model for electromagnetic simulations, the evaluations of which are used to build the surrogate.	Calculating eddy currents and Joule power in TF coil casing [39].
Designed Quadrature [37]	Algorithm/Method	Generates optimized quadrature rules that can be more efficient than standard Gauss rules, especially when paired with graph-acceleration.	Achieving >40% cost reduction in 4D and 6D aircraft design UQ problems [37].
AMTC Method [40] [37]	Computational Graph Transformer	Accelerates model evaluations on tensor-grid inputs by eliminating redundant operations, crucial for making quadrature-based NIPC feasible.	Graph-accelerated NIPC for multidisciplinary aircraft systems [40].

Non-Intrusive Polynomial Chaos Expansion stands as a powerful and efficient methodology for uncertainty propagation in complex computational models. Its principal advantage lies in decoupling the uncertainty analysis from the underlying high-fidelity simulation, enabling robust statistical characterization at a fraction of the computational cost of traditional Monte Carlo methods. As demonstrated by its successful application in fields ranging from biomedical device engineering to nuclear fusion energy and aerospace design, NIPC provides researchers and industry professionals with a rigorous mathematical tool for risk assessment and design optimization. The ongoing development of advanced techniques, such as graph-accelerated evaluation and tailored quadrature rules, continues to expand the boundaries of NIPC, making it applicable to increasingly complex and higher-dimensional problems. By adhering to the structured protocols and leveraging the essential tools outlined in this document, scientists can effectively integrate NIPC into their research workflow, enhancing the reliability and predictive power of their computational models.

Uncertainty Quantification (UQ) is a critical component for establishing trust in Neural Network Potentials (NNPs), which are machine learning interatomic potentials trained to approximate the energy landscape of atomic systems. The black-box nature of neural networks and their inherent stochasticity often deter researchers, especially when considering foundation models trained across broad chemical spaces. Uncertainty information provided during prediction helps reduce this aversion and allows for the propagation of uncertainties to extracted properties, which is particularly vital in sensitive applications like drug development [34] [41] [42].

Within this context, readout ensembling has emerged as a computationally efficient UQ method that provides information about model uncertainty (epistemic uncertainty). This approach is distinct from, and complementary to, methods like quantile regression, which primarily captures aleatoric uncertainty inherent in the underlying training data [34]. For researchers and drug development professionals, implementing readout ensembling is essential for identifying poorly learned or out-of-domain structures, thereby ensuring the reliability of NNP-driven simulations in molecular design and material discovery [12].

Theoretical Foundation

Uncertainty Quantification in NNPs

In atomistic simulations, errors on out-of-domain structures can compound, leading to inaccurate probability distributions, incorrect observables, or unphysical results. UQ helps mitigate this risk by providing a confidence measure for model predictions [34]. Two primary types of uncertainty are relevant:

• Epistemic Uncertainty: Uncertainty in the model parameters, arising from a lack of training data or knowledge. This uncertainty can be reduced with more data.
• Aleatoric Uncertainty: Uncertainty inherent in the training data itself, due to noise or stochasticity (e.g., from Density Functional Theory (DFT) calculations). This uncertainty is irreducible [34] [42].

Readout ensembling is primarily designed to quantify epistemic uncertainty, though it can also capture some aleatoric components [34].

Readout Ensembling: Core Concept

Readout ensembling is a technique that adapts the traditional model ensembling approach to reduce its prohibitive computational cost, especially for foundation models. A foundation model is first trained on a large, structurally diverse dataset at significant computational expense. Instead of training multiple full models from scratch, readout ensembling involves creating an ensemble of models where each member shares the same core foundation model parameters but possesses independently fine-tuned readout layers (the final layers responsible for generating the prediction) [34] [43].

Stochasticity is introduced by fine-tuning each model's readout layers on different, randomly selected subsets of the full training set. The ensemble's prediction is the mean of all members' predictions, and the uncertainty is typically quantified as the standard deviation of these predictions. This method approximates the model posterior, providing a measure of how much the model's parameters are uncertain for a given input [34].

Comparative Analysis of UQ Methods

The following table summarizes the key characteristics of readout ensembling against other prominent UQ methods.

Table 1: Comparison of Uncertainty Quantification Methods for Neural Network Potentials

Method	Type	Uncertainty Captured	Key Principle	Computational Cost	Key Advantage
Readout Ensembling	Multi-model	Primarily Epistemic (Model)	Fine-tunes readout layers of a foundation model on different data subsets [34].	Moderate (lower than full ensembling)	High accuracy; better for generalization and model robustness [44].
Quantile Regression	Single-model	Aleatoric (Data)	Uses an asymmetric loss function to predict value ranges (e.g., 5th and 95th percentiles) [34].	Low	Accurately reflects data noise; tends to scale with system size [34].
Full Model Ensembling	Multi-model	Epistemic & Aleatoric	Trains multiple independent models with different initializations [34] [44].	Very High	Considered a robust and high-performing benchmark for UQ [44].
Deep Evidential Regression	Single-model	Epistemic & Aleatoric	Places a prior distribution over model parameters and outputs a higher-order distribution [44].	Low	Does not consistently outperform ensembles in atomistic simulations [44].
Dropout-based UQ	Single-model	Epistemic (Approximate)	Uses dropout at inference time to simulate an ensemble [34].	Low	Less reliable than ensemble-based methods for NNP active learning [34].

Application Protocol: Readout Ensembling for NNP Foundation Models

This protocol details the application of readout ensembling to the MACE-MP-0 NNP foundation model, as demonstrated in recent research [34]. The workflow is designed to be executed on a high-performance computing (HPC) cluster.

Experimental Workflow

The following diagram illustrates the end-to-end process for implementing readout ensembling.

Step-by-Step Methodology

Step 1: Foundation Model Selection and Preparation

• Action: Select a pre-trained NNP foundation model, such as MACE-MP-0, which was trained on the broad Materials Project Trajectory (MPtrj) Dataset [34].
• Rationale: The foundation model provides a robust, general-purpose initialization of the core network parameters, having already learned general relationships across a wide swath of chemical space.

Step 2: Dataset Splitting and Subset Generation

• Action: From the target training dataset (e.g., MPtrj or a task-specific dataset for fine-tuning), randomly generate N unique subsets. In the referenced study, 7 subsets were used, each containing 90,000 structures. Each subset should be further split into training (e.g., 80,000 structures) and validation (e.g., 10,000 structures) partitions [34].
• Rationale: Using different data subsets for each ensemble member introduces stochasticity and ensures diversity in the fine-tuned readout layers, which is crucial for a meaningful uncertainty estimate.

Step 3: Readout Layer Fine-Tuning

• Action: For each of the N data subsets, create a copy of the foundation model. Freeze all parameters of the core network and only fine-tune the weights of the final readout layers using the assigned data subset.
• Training Configuration:
  • Loss Function: Huber loss, which is a piecewise function that switches between Mean Squared Error (MSE) and Mean Absolute Error (MAE) depending on a set threshold [34].
  • Hardware: Due to the reduced computational load, each model can be trained on a single high-performance GPU (e.g., NVIDIA P100 or equivalent) [34].
• Rationale: This step is computationally efficient as it updates only a small subset of the model's total parameters. It adapts the general foundation model to the specific data distribution of each subset, creating a diverse ensemble.

Step 4: Inference and Uncertainty Calculation

• Action: For a new input structure, pass it through each of the N fine-tuned models to obtain a set of predictions {P₁, P₂, ..., Pₙ}.
• Calculation:
  • The final model prediction is the mean of the ensemble's outputs.
  • The uncertainty is quantified as the standard deviation of the ensemble's outputs. Confidence intervals can also be computed using the Student's t-distribution [34].
• Rationale: The standard deviation directly measures the dispersion of the predictions, providing a quantitative estimate of the model's uncertainty for that specific input.

Performance and Validation

Quantitative Performance Metrics

The performance of readout ensembling on the MACE-MP-0 model, tested on a common set of 10,000 MPtrj structures, is summarized below. Errors are reported in meV per electron (meV/e⁻) to remove size-extensive effects [34].

Table 2: Performance Metrics for Readout Ensembling on MACE-MP-0

Metric	Readout Ensemble	Quantile Regression (Single-Model)
Energy MAE (meV/e⁻)	0.721	0.890
Uncertainty-Error Relationship	Tends to increase with error, but magnitude is orders of magnitude lower than the error [34].	More accurately reflects model prediction ability [34].
Scaling Behavior	N/A	Tends to increase with system size [34].
Primary Use Case	Identifying out-of-domain structures (epistemic uncertainty) [34].	Capturing variations in chemical complexity (aleatoric uncertainty) [34].

Interpretation of Results

The data indicates that readout ensembling produces highly accurate energy predictions (lower MAE than quantile regression). However, a critical finding is that the ensemble can be overconfident, meaning the calculated uncertainty, while correlated with error, is often much smaller than the actual error. This underscores the importance of calibrating uncertainty estimates for specific applications. In contrast, quantile regression provides a more reliable measure of prediction reliability, especially for larger systems [34].

The Scientist's Toolkit

Table 3: Essential Research Reagents and Computational Tools

Item	Function in Readout Ensembling	Example/Note
Pre-trained NNP Foundation Model	Provides the core, frozen network parameters that encode general chemical knowledge.	MACE-MP-0 [34], CHGNet [34], ANI-1 [34].
Large-Scale Training Dataset	Source for generating random subsets to fine-tune ensemble members and introduce diversity.	Materials Project Trajectory (MPtrj) [34], Open Catalyst Dataset [34].
High-Performance Computing (HPC) Cluster	Enables parallel fine-tuning of multiple ensemble members, drastically reducing total computation time.	Clusters with multiple GPUs (e.g., NVIDIA P100, A100) [34].
Huber Loss Function	The training objective used during fine-tuning; robust to outliers.	A piecewise function combining MSE and MAE advantages [34].
Uncertainty Metric Calculator	Scripts to compute the standard deviation and confidence intervals from the ensemble's predictions.	Custom Python scripts using libraries like NumPy and SciPy.

Quantile Regression for Capturing Data Distribution Uncertainty

Quantile Regression (QR) is a powerful statistical technique that extends beyond traditional mean-based regression by modeling conditional quantiles of a response variable. This approach provides a comprehensive framework for characterizing the entire conditional distribution, making it particularly valuable for uncertainty quantification in computational models. Unlike ordinary least squares regression that estimates the conditional mean, QR enables direct estimation of the τ-th quantile, defined as qτ(Y|X = x) = inf{y: F(y|X = x) ≥ τ}, where F represents the conditional distribution function [45]. This capability allows researchers to detect distributional features such as asymmetry and heteroscedasticity that are often masked by expectation-based methods [46].

In the context of uncertainty quantification, QR offers distinct advantages for capturing both aleatoric (inherent data noise) and epistemic (model uncertainty) components. While traditional methods often rely on Gaussian assumptions, QR operates without requiring specific distributional assumptions about the target variable or error terms, making it robust for real-world datasets frequently exhibiting non-Gaussian characteristics [45] [47]. This flexibility is especially crucial in drug discovery and development, where decision-making depends on accurate uncertainty estimation for optimal resource allocation and improved trust in predictive models [8].

Fundamental Concepts and Mathematical Framework

Core Principles of Quantile Regression

The mathematical foundation of quantile regression revolves around minimizing a loss function based on the check function, which asymmetrically weights positive and negative residuals. For a given quantile level τ ∈ (0,1), the loss function is defined as:

ρτ(u) = u · (τ - I(u < 0))

where u represents the residual (y - ŷ), and I is the indicator function. This loss function enables QR to estimate any conditional quantile of the response distribution by solving the optimization problem:

minβ ∑ ρτ(yi - xiβ)

This formulation allows QR to capture the conditional quantiles qτ(Y|X = x) without assuming a parametric distribution for the error terms, thus providing greater flexibility in modeling real-world data where normality assumptions often fail [45] [47].

Comparison with Traditional Uncertainty Quantification Methods

Quantile regression addresses several limitations of traditional uncertainty quantification approaches. While methods like Gaussian processes assume homoscedasticity and specific error distributions, QR naturally handles heteroscedasticity and non-Gaussian distributions. Similarly, compared to Bayesian methods that often require complex sampling techniques and substantial computational resources, QR provides a computationally efficient framework for full distributional estimation [48] [49].

Table 1: Comparison of Uncertainty Quantification Methods

Method	Uncertainty Type Captured	Distributional Assumptions	Computational Efficiency
Quantile Regression	Aleatoric (via conditional quantiles)	Non-parametric	High
Gaussian Processes	Both (via predictive variance)	Gaussian	Low to Moderate
Bayesian Neural Networks	Both (via posterior)	Prior specification required	Low
Ensemble Methods	Epistemic (via model variation)	Varies with base models	Moderate
Evidential Learning	Both (via higher-order distributions)	Prior likelyhood required	Moderate

Computational Implementation Approaches

Quantile Regression Neural Networks (QRNN)

The Quantile Regression Neural Network modifies standard neural network architectures by replacing the traditional single-output layer with a multi-output layer that simultaneously predicts multiple quantiles. As demonstrated in spatial analysis of wind speed prediction, a SmaAt-UNet architecture can be adapted where the final convolutional layer is modified from single-channel to a 10-channel output, with each channel corresponding to specific quantile levels τp ∈ {5%, 15%, ..., 95%} for p = 1, 2, ..., 10 [46]. This approach shares feature extraction weights across the encoder-decoder architecture while providing comprehensive distributional coverage. The optimization target for QRNN is given by:

ℒQRNN = 𝔼n,g,p[ρτp(𝐘n,g - 𝐘̂n,gτp)]

where n, g, and p index samples, spatial locations, and quantiles respectively [46].

Quantile Regression Forests (QRF)

Quantile Regression Forests represent a non-parametric approach that extends random forests to estimate full conditional distributions. Unlike standard random forests that predict conditional means, QRF estimates the conditional distribution by weighting observed response values. The algorithm involves generating T unpruned regression trees based on bootstrap samples from the original data, with each node of the trees using a random subset of features [45].

For a given input x, the conditional distribution is estimated as:

F̂(y|X = x) = ∑i=1n ωi(x) I(Yi ≤ y)

where the weights ωi(x) are determined by the frequency with which data points fall into the same leaf node as x across all trees in the forest [45]. The τ-th quantile is then predicted as:

q̂τ(Y|X = x) = inf{y: F̂(y|X = x) ≥ τ}

This method has demonstrated superior performance in drug response prediction applications, achieving higher prediction accuracy compared to traditional elastic net and ridge regression approaches [45].

Advanced Hybrid Approaches

Recent advancements combine QR with other uncertainty quantification frameworks to leverage complementary strengths. Deep evidential learning for Bayesian quantile regression represents a cutting-edge approach that enables estimation of quantiles of a continuous target distribution without Gaussian assumptions while capturing both aleatoric and epistemic uncertainty through a single deterministic forward-pass model [48]. Similarly, Quantile Ensemble methods provide model-agnostic uncertainty quantification by combining predictions from multiple quantile regression models, offering improved calibration and sharpness in clinical applications such as predicting antibiotic concentrations in critically ill patients [49].

Applications in Drug Discovery and Development

Drug Response Prediction

Quantile regression has demonstrated significant utility in predicting drug response for cancer treatment personalization. In applications using the Cancer Cell Line Encyclopedia (CCLE) dataset, Quantile Regression Forests have outperformed traditional point-estimation methods by providing prediction intervals in addition to point estimates [45]. This capability is particularly valuable in precision medicine, as it enables clinicians to assess not only the expected drug response but also the reliability of these predictions through prediction interval length. At identical confidence levels, shorter intervals indicate more reliable predictions, supporting more informed treatment decisions [45].

The three-step QRF approach for drug response prediction involves: (1) preliminary feature screening using Pearson correlation coefficients to filter potentially important genomic features; (2) variable selection using random forests to identify a small subset of variables based on importance scores; and (3) building quantile regression forests using the selected features to generate comprehensive prediction intervals [45]. This methodology has proven particularly effective for modeling drug response metrics such as activity area, which simultaneously captures efficacy and potency of drug sensitivity.

Pharmacokinetic Prediction

In therapeutic drug monitoring, quantile regression enables prediction of antibiotic plasma concentrations with uncertainty quantification in critically ill patients. Research on piperacillin plasma concentration prediction demonstrates that machine learning models (CatBoost) enhanced with Quantile Ensemble methods provide clinically useful individualized uncertainty predictions [49]. This approach outperforms homoscedastic methods like Gaussian processes in clinical applications where uncertainty patterns are often heteroscedastic.

The Quantile Ensemble method proposed for this application can be applied to any model optimizing a quantile function and provides distribution-based uncertainty quantification through two key metrics: Absolute Distribution Coverage Error (ADCE) and Distribution Coverage Error (DCE) [49]. These metrics enable objective evaluation of uncertainty quantification calibration, with lower values indicating better performance. Implementation of this approach has shown that models incorporating quantile-based uncertainty quantification achieve RMSE values of approximately 31.94-33.53 with R² values of 0.60-0.64 in internal evaluations for piperacillin concentration prediction [49].

Censored Data Modeling in Early Drug Discovery

Quantile regression frameworks have been adapted to handle censored regression labels commonly encountered in pharmaceutical assay-based data. In early drug discovery, approximately one-third or more of experimental labels may be censored, providing only thresholds rather than precise values [8]. Traditional uncertainty quantification methods cannot fully utilize this partial information, leading to suboptimal uncertainty estimation.

Adapted ensemble-based, Bayesian, and Gaussian models incorporating tools from survival analysis (Tobit model) enable learning from censored labels, significantly improving reliability of uncertainty estimates in real pharmaceutical settings [8]. This approach is particularly valuable for temporal evaluation under distribution shift, a common challenge in drug discovery pipelines where model performance may degrade over time as compound libraries evolve.

Experimental Protocols and Implementation

Protocol 1: QRF for Drug Response Prediction

Objective: Implement Quantile Regression Forests to predict drug response (activity area) from genomic features with uncertainty quantification.

Materials and Reagents:

• Cancer Cell Line Encyclopedia (CCLE) dataset: Contains expression profiles of 20,089 genes, mutation status of 1,667 genes, copy number variation of 16,045 genes for 947 human cancer cell lines, and 8-point dose-response curves for 24 chemical drugs across 479 cell lines [45].
• Computational environment: R or Python with scikit-learn, scikit-garden, or quantile-forest libraries.

Procedure:

• Data Preprocessing:
  • Download CCLE dataset from http://www.broadinstitute.org/ccle
  • Extract activity area values as drug sensitivity measurement
  • Perform quality control to remove cell lines with excessive missing data
  • Impute missing values using appropriate methods (e.g., k-nearest neighbors)

• Feature Screening:

  • Calculate Pearson correlation coefficients between genomic features and drug responses
  • Perform two-sided t-tests for significance of PCCs
  • Retain features with p-value < 0.05 (approximately 2,000 genes)
  • Repeat for each drug independently

• Variable Selection:

  • Train random forest with 25,000 trees on screened features
  • Calculate variable importance through permutation testing
  • Select variables with importance values > 2 × standard deviation above mean

• Quantile Regression Forest Implementation:

  • Build QRF with 15,000 unpruned regression trees
  • Set m (number of features considered per split) to M/3, where M is total features
  • Set minimum node size to 10 training samples
  • Specify quantile levels τ ∈ {0.05, 0.1, 0.15, ..., 0.95}

• Model Validation:

  • Perform out-of-bag validation to assess predictive accuracy
  • Calculate prediction intervals for each sample at confidence levels (e.g., 90%)
  • Compare point predictions (mean and median) with observed values
  • Evaluate using metrics including RMSE, correlation coefficients, and interval coverage

Troubleshooting Tips:

• For computational efficiency concerns, reduce tree count to 5,000-10,000 while monitoring performance
• If prediction intervals are too wide, increase feature selection stringency
• For memory limitations, implement incremental learning approaches

Protocol 2: Quantile Ensemble for Clinical Concentration Prediction

Objective: Develop quantile ensemble model for predicting piperacillin plasma concentrations with uncertainty quantification in critically ill patients.

Materials:

• Patient data: Demographics, biochemistry, SOFA scores, APACHE II scores, creatinine clearance, plasma concentrations [49]
• Software: Python with CatBoost, NumPy, pandas, and scikit-learn

Procedure:

• Data Collection and Curation:
  • Prospectively collect blood samples for piperacillin analysis from critically ill patients
  • Record patient covariates: serum creatinine, albumin, platelets, lactate, white blood cells, bilirubin
  • Calculate creatinine clearance from 8-hour urinary collection or estimate via CKD-EPI equation
  • Document TZP dosing regimen: loading dose (4/0.5 g/30 min) followed by continuous infusion

• Model Architecture Design:

  • Implement CatBoost as base model with quantile loss function
  • Configure Quantile Ensemble to output multiple quantiles (τ = 0.05, 0.25, 0.5, 0.75, 0.95)
  • Set hyperparameters via grid search: learning rate (0.01-0.1), depth (6-10), iterations (1000-5000)

• Uncertainty Quantification Implementation:

  • Calculate prediction intervals from quantile estimates: PIα(x) = [q̂α/2(x), q̂1-α/2(x)]
  • Implement distribution coverage error metrics:
    • DCE = |Coverage - Expected Coverage|
    • ADCE = ∑|I(yi ∈ PI(xi)) - (1-α)| / n
  • Optimize model calibration using DCE as objective function

• Model Evaluation:

  • Perform internal validation using bootstrap sampling or cross-validation
  • Compare with population pharmacokinetic model performance
  • Assess external validation on dataset from different medical center
  • Evaluate both point prediction (RMSE, R²) and uncertainty quantification (DCE, sharpness)

Interpretation Guidelines:

• Clinically acceptable prediction intervals should cover 90-95% of observed concentrations
• Sharp intervals (narrow width) with proper calibration indicate high-quality uncertainty quantification
• Models with ADCE < 0.05 demonstrate excellent calibration performance
• Deterioration in external validation suggests limited generalizability to different dosing regimens

Visualization and Workflow Diagrams

Quantile Regression Forest Workflow

Figure 1: QRF Implementation Workflow

Quantile Ensemble Method for Clinical Prediction

Figure 2: Quantile Ensemble Clinical Implementation

Research Reagent Solutions

Table 2: Essential Research Materials for Quantile Regression Implementation

Resource	Specifications	Application Context	Access Information
CCLE Dataset	Gene expression (20,089 genes), mutation status (1,667 genes), copy number variation (16,045 genes), drug response (24 compounds)	Drug response prediction, biomarker identification	http://www.broadinstitute.org/ccle [45]
Clinical Pharmacokinetic Data	Patient demographics, biochemistry, SOFA/APACHE-II scores, antibiotic concentrations	Therapeutic drug monitoring, concentration prediction	Institutional collection protocols required [49]
Quantile Regression Software	Python (scikit-learn, CatBoost, PyTorch) or R (quantreg, grf) packages	Method implementation, model development	Open-source repositories (GitHub, PyPI, CRAN) [50]
Uncertainty Quantification Toolkits	UQ360, Chaospy, Pyro, Uncertainty Toolbox	Advanced uncertainty quantification, model comparison	Open-source repositories [50]

Performance Metrics and Evaluation Framework

Quantitative Assessment of Uncertainty Quantification

Rigorous evaluation of quantile regression models requires specialized metrics beyond traditional point prediction assessment. The following metrics provide comprehensive evaluation of both predictive accuracy and uncertainty quantification quality:

Point Prediction Metrics:

• Root Mean Squared Error (RMSE): √(1/n ∑(yi - ŷi)²)
• Mean Absolute Error (MAE): 1/n ∑|yi - ŷi|
• R² Coefficient of Determination: 1 - (∑(yi - ŷi)²)/(∑(yi - ȳ)²)

Uncertainty Quantification Metrics:

• Continuous Ranked Probability Score (CRPS): Measures squared difference between forecast CDF and empirical CDF of observation [47]
• Prediction Interval Coverage Probability (PICP): Proportion of observations falling within prediction intervals
• Mean Prediction Interval Width (MPIW): Average width of prediction intervals, assessing sharpness
• Distribution Coverage Error (DCE): |PICP - Expected Coverage|, evaluating calibration [49]

Table 3: Performance Benchmark of Quantile Regression Methods

Method	Application Domain	Point Prediction (RMSE)	Uncertainty Quantification (CRPS)	Computational Efficiency
Quantile Regression Forests	Drug response prediction	Superior to elastic net/ridge regression	Excellent through prediction intervals	Moderate (15,000 trees) [45]
Quantile Gradient Boosting	NO2 pollution forecasting	Best performance among 10 models	Best distributional calibration	High [47]
Quantile Neural Networks	Wind speed prediction	Comparable to deterministic models	Realistic spatial uncertainty	Moderate [46]
Quantile Ensemble (CatBoost)	Clinical concentration prediction	RMSE: 31.94-33.53, R²: 0.60-0.64	Clinically useful individualized uncertainty	High [49]

Quantile regression represents a versatile and powerful framework for uncertainty quantification in computational models, particularly in drug discovery and development applications. Its non-parametric nature, ability to capture heteroscedasticity, and computational efficiency make it well-suited for real-world challenges where distributional assumptions are frequently violated. The methodologies and protocols outlined provide researchers with practical implementation guidelines across various application scenarios.

Future research directions include integration of quantile regression with deep learning architectures for unstructured data, development of causal quantile methods for intervention analysis, and adaptation to federated learning environments for privacy-preserving model development. As uncertainty quantification continues to gain importance in regulatory decision-making and clinical applications, quantile regression methodologies are poised to play an increasingly critical role in advancing pharmaceutical research and personalized medicine.

Bayesian Inference for Parameter Calibration and Estimation

Bayesian inference provides a powerful probabilistic framework for calibrating parameters and quantifying uncertainty in computational models. This approach is fundamentally rooted in Bayes' theorem, which updates prior beliefs about model parameters with new observational data to obtain a posterior distribution [51] [52]. The theorem is formally expressed as:

[ P(\theta \mid D) = \frac{P(D \mid \theta) \cdot P(\theta)}{P(D)} ]

Where ( P(\theta \mid D) ) is the posterior distribution of parameters ( \theta ) given data ( D ), ( P(D \mid \theta) ) is the likelihood function, ( P(\theta) ) is the prior distribution, and ( P(D) ) is the marginal likelihood [52] [53]. In computational model calibration, this framework enables researchers to systematically quantify uncertainty from multiple sources, including measurement error, model structure discrepancy, and parameter identifiability issues [54] [55].

The strength of Bayesian methods lies in their explicit treatment of uncertainty, making them particularly valuable for complex computational models where parameters cannot be directly observed and must be inferred from indirect measurements [54]. This approach has demonstrated significant utility across diverse fields, from pulmonary hemodynamics modeling in cardiovascular research to drug development and rare disease studies [54] [56] [57].

Foundational Concepts and Theoretical Framework

Core Components of Bayesian Analysis

Bayesian parameter calibration relies on three fundamental components that together form the analytical backbone of the inference process:

• Prior Distribution (( P(\theta) )): Encapsulates existing knowledge about parameters before observing new data. Priors can be informative (based on historical data or expert knowledge) or weakly informative (diffuse distributions that regularize inference without strong directional influence) [51] [55]. In regulatory settings like drug development, prior specification requires careful justification to avoid introducing undue subjectivity [56] [57].

• Likelihood Function (( P(D \mid \theta) )): Quantifies how probable the observed data is under different parameter values. The likelihood connects the computational model to empirical observations, serving as the mechanism for data-driven updating of parameter estimates [52] [53]. For complex models, evaluating the likelihood often requires specialized techniques such as approximate Bayesian computation when closed-form expressions are unavailable.

• Posterior Distribution (( P(\theta \mid D) )): Represents the updated belief about parameters after incorporating evidence from the observed data. The posterior fully characterizes parameter uncertainty, enabling probability statements about parameter values and their correlations [51] [52]. In practice, the posterior is often summarized through credible intervals, posterior means, or highest posterior density regions [55].

Bayesian Uncertainty Quantification

A critical advantage of Bayesian methods is their natural capacity for comprehensive uncertainty quantification [54] [55]. The posterior distribution inherently captures both parameter uncertainty (epistemic uncertainty about model parameters) and natural variability (aleatory uncertainty inherent in the system) [55]. This dual capability makes Bayesian approaches particularly valuable for safety-critical applications where understanding the full range of possible outcomes is essential [56] [57].

For computational models, Bayesian inference also facilitates propagation of uncertainty through model simulations. By drawing samples from the posterior parameter distribution and running the model forward, researchers can generate predictive distributions that account for both parameter uncertainty and model structure [54]. This approach provides more realistic uncertainty bounds compared to deterministic calibration methods that yield single-point estimates [55].

Computational Implementation Protocols

Workflow for Bayesian Parameter Estimation

Implementing Bayesian inference for parameter calibration follows a systematic workflow that integrates computational modeling with statistical inference:

Markov Chain Monte Carlo (MCMC) Methods

For most practical applications, the posterior distribution cannot be derived analytically and must be approximated numerically. Markov Chain Monte Carlo methods represent the gold standard for this purpose [51] [53]. MCMC algorithms generate correlated samples from the posterior distribution through a random walk process that eventually converges to the target distribution [52] [55].

Table: Common MCMC Algorithms for Bayesian Parameter Estimation

Algorithm	Key Mechanism	Optimal Use Cases	Convergence Considerations
Metropolis-Hastings	Proposal-accept/reject cycle	Models with moderate parameter dimensions	Sensitive to proposal distribution tuning
Gibbs Sampling	Iterative conditional sampling	Hierarchical models with conditional conjugacy	Efficient when full conditionals are available
Hamiltonian Monte Carlo	Hamiltonian dynamics with gradient information	High-dimensional, complex posterior geometries	Requires gradient computations; less sensitive to correlations
No-U-Turn Sampler (NUTS)	Adaptive path length HMC variant	General-purpose application; default in Stan	Automated tuning reduces user intervention

Implementation of MCMC requires careful convergence diagnostics to ensure the algorithm has adequately explored the posterior distribution. Common diagnostic measures include the Gelman-Rubin statistic (comparing within-chain and between-chain variance), effective sample size (measuring independent samples equivalent), and visual inspection of trace plots [55]. For complex models, convergence may require millions of iterations, making computational efficiency a practical concern [54] [55].

Gaussian Process Emulation for Computational Efficiency

When dealing with computationally intensive models where a single evaluation takes minutes to hours, direct MCMC sampling becomes infeasible. In such cases, Gaussian process (GP) emulation provides a powerful alternative [54]. GP emulators act as surrogate models that approximate the computational model's input-output relationship using a limited number of model evaluations.

The protocol for GP emulation involves:

• Designing an experimental strategy over the parameter space (e.g., Latin Hypercube sampling)
• Running the computational model at selected design points
• Fitting a GP model to the input-output data
• Using the emulator in place of the full model during MCMC sampling

This approach can reduce computational requirements by several orders of magnitude while maintaining accurate uncertainty quantification [54]. In pulmonary hemodynamics modeling, for example, GP emulation enabled parameter estimation for a one-dimensional fluid dynamics model within a clinically feasible timeframe [54].

Experimental Design and Data Requirements

Calibration Experiment Design

Effective Bayesian parameter estimation requires carefully designed experiments or observational protocols that provide sufficient information to identify parameters. Optimal experimental design principles can be applied to maximize the information content of data used for calibration:

• Identifiability Analysis: Before data collection, perform theoretical (structural) and practical identifiability analysis to determine which parameters can be uniquely estimated from available measurements [55]. Non-identifiable parameters may require stronger priors or modified experimental designs.

• Sequential Design: For iterative calibration, employ sequential experimental design where preliminary parameter estimates inform subsequent data collection to maximize information gain [56]. This approach is particularly valuable in adaptive clinical trial designs where accumulating data guides treatment allocation [56] [57].

• Multi-fidelity Data Integration: Combine high-precision, low-throughput measurements with lower-precision, high-throughput data to constrain parameter space efficiently [54] [57]. Bayesian methods naturally accommodate data with heterogeneous quality through appropriate likelihood specification.

Specifying appropriate prior distributions requires systematic approaches, especially in regulatory environments where subjectivity must be minimized [56] [57]:

Table: Prior Elicitation Methods for Parameter Calibration

Method	Procedure	Application Context	Regulatory Considerations
Historical Data Meta-analysis	Analyze previous studies using meta-analytic predictive priors	Drug development, engineering systems	FDA encourages when historical data are relevant [56]
Expert Elicitation	Structured interviews with domain experts using encoding techniques	Novel systems with limited data	Requires documentation of expert selection and justification [57]
Weakly Informative Priors	Use conservative distributions that regularize without strongly influencing	Exploratory research, preliminary studies	Default choice when substantial prior knowledge is lacking [55]
Commensurate Priors	Dynamically adjust borrowing from historical data based on similarity	Incorporating external controls in clinical trials	FDA draft guidance addresses appropriateness determination [58]

Application Protocols Across Domains

Biomedical Model Calibration

In cardiovascular modeling, Bayesian methods have been successfully applied to estimate microvascular parameters in pulmonary hemodynamics using clinical measurements from a dog model of chronic thromboembolic pulmonary hypertension (CTEPH) [54]. The protocol involves:

• Model Specification: A one-dimensional fluid dynamics model representing pulmonary blood flow
• Data Collection: Pressure and flow measurements under baseline and CTEPH conditions
• Prior Definition: Based on physiological constraints and previous experimental results
• Posterior Computation: Using MCMC with GP emulation to accelerate inference
• Validation: Comparing parameter estimates with independent markers of disease severity

This approach identified distinct parameter shifts associated with CTEPH development and demonstrated strong correlation with clinical disease markers [54].

Drug Development Applications

Bayesian methods are increasingly employed throughout the drug development pipeline, with specific protocols tailored to different phases [56]:

For rare disease applications where traditional randomized trials are infeasible, Bayesian approaches enable more efficient designs through historical borrowing and extrapolation [57]. The protocol for a hypothetical Phase III trial in Progressive Supranuclear Palsy (PSP) demonstrates how to reduce placebo group size using data from three previous randomized studies [57]:

• Prior Derivation: Apply meta-analytic-predictive approach to placebo data from historical trials
• Trial Design: Use 2:1 randomization (treatment:placebo) instead of conventional 1:1
• Analysis: Compute posterior probability of treatment effect exceeding clinically meaningful threshold
• Sample Size: Reduce from 170 total patients (85 per arm) to 128 (85 treatment, 43 placebo)

This design maintains statistical power while reducing placebo group exposure, addressing ethical concerns in rare disease research [57].

Validation and Diagnostic Framework

Calibration Assessment Protocols

Ensuring Bayesian inference is properly calibrated requires rigorous validation against empirical data [55]. The following protocol assesses calibration reliability:

• Posterior Predictive Checks: Generate replicated datasets from the posterior predictive distribution and compare with observed data using discrepancy measures [55] [53]. Systematic differences indicate model misfit.

• Coverage Analysis: Compute the proportion of instances where credible intervals contain true parameter values in simulation studies. Well-calibrated 95% credible intervals should contain the true parameter approximately 95% of the time [55].

• Cross-Validation: Employ leave-one-out or k-fold cross-validation to assess predictive performance on held-out data, using proper scoring rules that account for uncertainty [55].

• Sensitivity Analysis: Evaluate how posterior conclusions change with different prior specifications, likelihood assumptions, or model structures [56] [55].

Computational Diagnostics

For MCMC-based inference, comprehensive diagnostic checking is essential [55]:

Table: Essential MCMC Diagnostics for Bayesian Parameter Estimation

Diagnostic	Computation Method	Interpretation Guidelines	Remedial Actions
Effective Sample Size (ESS)	Spectral analysis of chains	ESS > 200 per chain recommended	Increase iterations; improve sampler
Gelman-Rubin Statistic (R̂)	Between/within chain variance ratio	R̂ < 1.05 indicates convergence	Run longer chains; multiple dispersed starting points
Trace Plot Inspection	Visual assessment of chain mixing	Stationary, well-mixed fluctuations indicates convergence	Adjust sampler parameters; reparameterize model
Monte Carlo Standard Error	ESS-based estimate of simulation error	MCSE < 5% of posterior standard deviation	Increase iterations for desired precision
Divergent Transitions	Hamiltonian dynamics discontinuities	No divergences in well-specified models	Reduce step size; reparameterize; simplify model

Research Reagent Solutions

Implementing Bayesian parameter calibration requires both computational tools and methodological components. The following table details essential "research reagents" for effective implementation:

Table: Essential Research Reagents for Bayesian Parameter Estimation

Reagent Category	Specific Tools/Functions	Implementation Purpose	Usage Considerations
Probabilistic Programming Languages	Stan, PyMC3, JAGS	Specify models and perform efficient posterior sampling	Stan excels for complex models; PyMC3 offers Python integration [51] [55]
Diagnostic Packages	Arviz, shinystan, coda	Assess MCMC convergence and model fit	Arviz provides unified interface for multiple programming languages [55]
Prior Distribution Families	Normal/gamma (conjugate), half-t (weakly informative), power priors (historical borrowing)	Encode pre-existing knowledge while maintaining computational tractability	Power priors require careful weighting of historical data [56] [57]
Emulation Methods	Gaussian processes, Bayesian neural networks	Approximate computationally intensive models for feasible inference	GP emulators effective for smooth responses; require careful kernel selection [54]
Divergence Metrics	Kullback-Leibler divergence, Wasserstein distance	Quantify differences between prior and posterior distributions	Large changes may indicate strong data influence or prior-posterior conflict
Sensitivity Measures	Prior sensitivity index, likelihood influence measures	Quantify robustness of conclusions to model assumptions	High sensitivity warrants more conservative interpretation of results [55]

Bayesian inference provides a coherent framework for parameter calibration and estimation that naturally accommodates uncertainty quantification, prior knowledge integration, and sequential learning. The protocols outlined in this document offer researchers structured approaches for implementing these methods across diverse application domains, from biomedical modeling to drug development. Proper application requires attention to computational diagnostics, model validation, and careful prior specification to ensure results are both statistically sound and scientifically meaningful. As computational models grow in complexity and impact, Bayesian methods offer a principled approach to parameter estimation that fully acknowledges the inherent uncertainties in both models and data.

In the early stages of drug discovery, decisions regarding which experiments to pursue are critically influenced by computational models due to the time-consuming and expensive nature of the experiments [59]. Accurate Uncertainty Quantification (UQ) in machine learning predictions is therefore becoming essential for optimal resource allocation and improved trust in models [59]. Computational methods in drug discovery often face challenges of limited data and sparse experimental observations. However, additional information frequently exists in the form of censored labels, which provide thresholds rather than precise values of observations [59]. For instance, when a fixed range of compound concentrations is used in an assay and no response is observed within this range, the experiment may only indicate that the response lies above or below the tested concentrations, resulting in a censored label [59].

While standard UQ approaches cannot fully utilize these censored labels, recent research has adapted ensemble-based, Bayesian, and Gaussian models with tools from survival analysis, specifically the Tobit model, to learn from this partial information [60] [59]. This advancement demonstrates that despite the reduced information in censored labels, they are essential to accurately and reliably model the real pharmaceutical setting [60] [59].

Theoretical Foundations: Uncertainty in Drug Discovery

In machine learning for drug discovery, uncertainty is typically categorized into two primary types:

• Aleatoric uncertainty: Refers to the inherent stochastic variability within experiments, often considered irreducible because it cannot be mitigated through additional data or model improvements [59]. In drug discovery, this can reflect the inherent unpredictability of interactions between certain molecular compounds due to biological stochasticity or human intervention [59].

• Epistemic uncertainty: Encompasses uncertainties related to the model's lack of knowledge, which can stem from insufficient training data or model limitations [59]. Unlike aleatoric uncertainty, epistemic uncertainty can be reduced by acquiring additional data or through model improvements [59].

The Challenge of Censored Data

Censored labels arise naturally in pharmaceutical experiments where measurement ranges are exceeded, preventing recording of exact values [59]. While these labels can be easily included in classification tasks by categorizing observations as active or inactive, integrating them into regression models that predict continuous values is far less trivial [59]. Prior to recent advancements, this type of data had not been properly utilized in regression tasks within drug discovery, despite its potential to enhance model accuracy and uncertainty quantification [59].

Protocol: Implementing Censored Regression for Molecular Property Prediction

Research Reagent Solutions

Table 1: Essential Materials and Computational Tools for Censored Regression Implementation

Item	Function/Description	Application Context
Internal Pharmaceutical Assay Data	Provides realistic temporal evaluation data; preferable to public datasets which may lack relevant experimental timestamps [59].	Model training and evaluation using project-specific target-based assays and cross-project ADME-T assays [59].
Censored Regression Labels	Partial information in the form of thresholds rather than precise values; provides crucial information about measurement boundaries [59].	Incorporated into loss functions (MSE, NLL) to enhance model accuracy and uncertainty estimation [59].
Tobit Model Framework	Statistical approach from survival analysis adapted to handle censored regression labels in machine learning models [59].	Implementation of censored-aware learning in ensemble, Bayesian, and Gaussian models [59].
Ensemble Methods	Multiple model instances are combined to improve predictive performance and uncertainty estimation [59].	Generation of robust predictive models with improved uncertainty quantification capabilities [59].
Graph Neural Networks (GNNs)	Neural network architecture specifically designed to operate on graph-structured data, such as molecular structures [61].	Molecular property prediction with automated architecture search (AutoGNNUQ) for enhanced UQ [61].

Adapted Modeling Approaches

The methodology adapts several modeling frameworks to incorporate censored labels:

• Ensemble-based models: Modified to learn from additional partial information available in censored regression labels [59].
• Bayesian models: Adapted through the Tobit framework to properly handle censored data in probabilistic predictions [59].
• Gaussian mean-variance estimators: Extended to incorporate censored labels for improved uncertainty estimation [59].

The core adaptation involves deriving extended versions of the mean squared error (MSE) and Gaussian negative log-likelihood (NLL) to account for censored labels, potentially using a one-sided squared loss approach [59].

Experimental Workflow

The following diagram illustrates the complete experimental workflow for implementing censored regression in molecular property prediction:

Data Considerations and Temporal Evaluation

The analysis should be performed on data from internal biological assays, categorized into two distinct groups:

• Project-specific target-based assays: Modeling either IC₅₀ or EC₅₀ values [59].
• Cross-project ADME-T assays: Important for testing pharmacokinetic profile and safety of drug candidates [59].

A comprehensive temporal evaluation using internal pharmaceutical assay-based data is crucial, as it better approximates real-world predictive performance compared to random or scaffold-based splits [59]. Public benchmarks often lack relevant temporal information, as timestamps in public data (e.g., ChEMBL) relate to when compounds were added to the public domain rather than when experiments were performed [59].

Application Notes: Implementation Framework

Model Architecture and Uncertainty Decomposition

The following diagram illustrates the conceptual architecture for uncertainty-aware molecular property prediction with censored data handling:

Quantitative Performance Comparison

Table 2: Comparison of UQ Methods in Molecular Property Prediction

Method	Censored Data Handling	Aleatoric Uncertainty	Epistemic Uncertainty	Key Advantages
Censored Ensemble Models [59]	Direct integration via Tobit loss	Estimated	Estimated via model variance	Utilizes partial information from censored labels
Censored Bayesian Models [59]	Probabilistic treatment	Quantified	Naturally captured in posterior	Coherent probabilistic framework
Censored Gaussian MVE [59]	Adapted likelihood	Explicitly modeled	Limited	Efficient single-model approach
AutoGNNUQ [61]	Not specified in results	Separated via variance decomposition	Separated via variance decomposition	Automated architecture search
Standard Ensemble Methods [59]	Cannot utilize	Estimated	Estimated via model variance	Established baseline
Direct Prompting (LLMs) [62]	Not applicable	Not quantified	Not quantified	Simple implementation

Practical Implementation Considerations

When implementing censored regression for molecular property prediction:

• Data Preparation: Identify and properly label censored observations in assay data. Common scenarios include concentration values reported as "greater than" or "less than" detectable limits [59].
• Model Selection: Choose appropriate base models (ensemble, Bayesian, or Gaussian) based on available computational resources and required uncertainty decomposition level [59].
• Loss Function Adaptation: Implement censored-aware versions of MSE or NLL using the Tobit model framework to properly handle the partial information [59].
• Evaluation Metrics: Adapt available evaluation methods to compare models trained with and without additional censored labels, using temporal splits for realistic assessment [59].

Incorporating censored regression labels through the Tobit model framework significantly enhances uncertainty quantification in drug discovery applications [60] [59]. Despite the partial information available in censored labels, they are essential to accurately and reliably model the real pharmaceutical setting [59]. The adapted ensemble-based, Bayesian, and Gaussian models demonstrate improved predictive performance and uncertainty estimation when leveraging this previously underutilized data source [59]. This approach enables more informed decision-making in resource-constrained drug discovery pipelines by providing better quantification of predictive uncertainty, ultimately contributing to more efficient and reliable molecular property prediction.

Model-Informed Drug Development (MIDD) represents an essential framework for advancing pharmaceutical development and supporting regulatory decision-making through quantitative approaches [63]. The fit-for-purpose (FFP) concept provides a strategic methodology for closely aligning modeling and uncertainty quantification (UQ) tools with specific scientific questions and contexts of use throughout the drug development lifecycle [63]. This methodology ensures that modeling resources are deployed efficiently to address the most critical development challenges while maintaining scientific rigor.

A model or method is considered not FFP when it fails to adequately define its context of use, lacks proper verification and validation, or suffers from unjustified oversimplification or complexity [63]. The FFP approach requires careful consideration of multiple factors, including the key questions of interest, intended context of use, model evaluation criteria, and the potential influence and risk associated with model predictions [63]. This strategic alignment promises to empower development teams to shorten development timelines, reduce costs, and ultimately benefit patients by delivering innovative therapies more efficiently.

Table 1: Key Components of Fit-for-Purpose Model Implementation

Component	Description	Implementation Considerations
Question of Interest	Specific scientific or clinical problem to be addressed	Determines appropriate modeling methodology and level of complexity required
Context of Use	Specific application and decision-making context	Defines regulatory requirements and validation stringency
Model Evaluation	Assessment of model performance and predictive capability	Varies based on development stage and risk associated with decision
Influence and Risk	Impact of model results on development pathway	Determines appropriate level of model verification and validation

Uncertainty Quantification Methodologies and Tools

Core UQ Tools in Drug Development

Uncertainty quantification provides the mathematical foundation for evaluating model reliability and predictive performance in MIDD. The Verification, Validation, and Uncertainty Quantification (VVUQ) framework has emerged as a critical discipline for assessing uncertainties in mathematical models, computational solutions, and experimental data [64]. Recent advances in VVUQ have become particularly important in the context of artificial intelligence and machine learning applications in drug development [64].

For computational models, verification ensures that the mathematical model is solved correctly, while validation determines whether the model accurately represents reality [64]. Uncertainty quantification characterizes the limitations of model predictions by identifying various sources of uncertainty, including parameter uncertainty, structural uncertainty, and data uncertainty [64]. In large language models and other AI approaches, recent research has demonstrated that entropy- and consistency-based methods effectively estimate model uncertainty, even in the presence of data uncertainty [65].

UQ Tools Alignment with Development Stages

The appropriate application of UQ tools varies significantly across drug development stages, requiring careful alignment with the specific context of use [63]. Early discovery phases may employ simpler UQ approaches with broader uncertainty bounds, while later stages demand more rigorous quantification to support regulatory decisions [63]. This progressive refinement of UQ strategies ensures efficient resource allocation while maintaining appropriate scientific standards.

Table 2: UQ Tools and Their Applications Across Drug Development Stages

Development Stage	Primary UQ Tools	Key Applications	Uncertainty Focus
Discovery	QSAR, AI/ML approaches	Target identification, lead compound optimization	Structural uncertainty, model selection uncertainty
Preclinical Research	PBPK, QSP/T, FIH Dose Algorithms	Preclinical prediction accuracy, first-in-human dose selection	Inter-species extrapolation uncertainty, parameter uncertainty
Clinical Research	PPK/ER, Semi-Mechanistic PK/PD, Bayesian Inference	Clinical trial design optimization, dosage optimization, exposure-response characterization	Population variability, covariate uncertainty, data uncertainty
Regulatory Review	Model-Integrated Evidence, Virtual Population Simulation	Bioequivalence demonstration, subgroup analysis	Model form uncertainty, extrapolation uncertainty
Post-Market Monitoring	Model-Based Meta-Analysis, Adaptive Trial Design	Label updates, safety monitoring, comparative effectiveness	Real-world evidence reliability, long-term uncertainty

Experimental Protocols for UQ in Model Development

Protocol for Definitive Quantitative Model Validation

Definitive quantitative models require rigorous validation to establish their fitness for purpose in regulatory decision-making. The following protocol outlines the key steps for establishing model credibility:

Step 1: Define Context of Use and Acceptance Criteria

• Clearly document the specific regulatory or development decision the model will inform
• Establish predefined acceptance limits for model performance based on the context of use
• Define quantitative metrics for evaluating model accuracy and precision

Step 2: Characterize Model Performance

• Conduct sensitivity analysis to identify influential parameters
• Perform uncertainty analysis to quantify parameter uncertainty and variability
• Evaluate model robustness through stress-testing under extreme conditions

Step 3: Assess Predictive Performance

• Utilize accuracy profile methodology incorporating total error (bias + intermediate precision)
• Establish β-expectation tolerance intervals (typically 95%) for future measurements
• Verify that a specified percentage of future measurements will fall within predefined acceptance limits [66]

Step 4: Document and Report Validation Results

• Compile comprehensive validation report including all experimental data
• Document model limitations and boundaries of applicability
• Provide evidence linking model performance to predefined acceptance criteria

For definitive quantitative methods, recommended performance standards include evaluation of both precision (% coefficient of variation) and accuracy (mean % deviation from nominal concentration) [66]. Repeat analyses of pre-study validation samples should typically vary by <15-25%, depending on the specific application and biomarker characteristics [66].

Protocol for Qualitative and Semi-Quantitative Model Validation

Qualitative and categorical models require different validation approaches focused on classification accuracy rather than numerical precision:

Step 1: Establish Classification Performance Metrics

• Define sensitivity, specificity, and accuracy targets based on context of use
• Establish positive and negative predictive value requirements
• Determine acceptable rates of false positives and false negatives

Step 2: Validate Using Appropriate Reference Standards

• Utilize well-characterized samples with known classification
• Ensure representative coverage of all expected categories or classes
• Include borderline cases to assess model performance boundaries

Step 3: Assess Robustness and Reproducibility

• Evaluate inter-operator and inter-instrument variability
• Test model stability over time and across conditions
• Assess performance against relevant biological and technical variations

Visualization of FFP Modeling Framework

FFP Modeling and UQ Implementation Workflow

Research Reagent Solutions for UQ Implementation

Table 3: Essential Research Reagents and Computational Tools for UQ Studies

Tool Category	Specific Solutions	Function in UQ Implementation
Modeling Platforms	PBPK Software (GastroPlus, Simcyp), QSP Platforms	Provide mechanistic frameworks for quantifying interspecies and inter-individual uncertainty
Statistical Analysis Tools	R, SAS, NONMEM, MONOLIX	Enable population parameter estimation, variability quantification, and covariance analysis
UQ Specialized Software	DAKOTA, UNICOS, UQLab	Implement advanced uncertainty propagation methods including polynomial chaos and Monte Carlo
Data Management Systems	Electronic Lab Notebooks, Clinical Data Repositories	Ensure data integrity and traceability for regulatory submissions
Visualization Tools	MATLAB, Python Matplotlib, Spotfire	Create informative visualizations of uncertainty distributions and sensitivity analysis results
Benchmark Datasets	Public Clinical Trial Data, Biomarker Reference Sets	Provide reference data for model validation and comparison

UQ Application Across Drug Development Stages

Early Development UQ Strategies

During early discovery and preclinical development, UQ focuses primarily on parameter uncertainty and model selection uncertainty. Quantitative Structure-Activity Relationship (QSAR) models employ UQ to assess prediction confidence for lead compound optimization [63]. Physiologically Based Pharmacokinetic (PBPK) models utilize UQ to quantify uncertainty in interspecies extrapolation and first-in-human dose prediction [63].

The fit-for-purpose approach in early development emphasizes iterative model refinement rather than comprehensive validation. As development progresses, models undergo continuous improvement through the incorporation of additional experimental data [63]. This iterative process allows for efficient resource allocation while building model credibility progressively.

Late-Stage Clinical Development UQ

In later development stages, UQ requirements become more stringent to support regulatory decision-making. Population PK/PD models employ UQ to characterize between-subject variability and covariate uncertainty [63]. Exposure-response models utilize UQ to quantify confidence in dose selection and benefit-risk assessment [63].

Clinical trial simulations incorporate UQ to assess the probability of trial success under various scenarios and design parameters [63]. This approach enables more robust trial designs and helps quantify the risk associated with different development strategies. Adaptive trial designs leverage UQ to make informed modifications based on accumulated data while controlling type I error [63].

UQ Tool Progression Through Development Stages

Regulatory Considerations and Compliance

The regulatory landscape for model-informed drug development has evolved significantly with recent guidelines such as the ICH M15 guidance, which aims to standardize MIDD practices across different regions [63]. Regulatory agencies recognize that the level of model validation should be commensurate with the model's context of use and potential impact on regulatory decisions [63].

For 505(b)(2) applications and generic drug development, model-integrated evidence generated through PBPK and other computational approaches plays an increasingly important role in demonstrating bioequivalence and supporting waiver requests [63]. The fit-for-purpose approach ensures that the level of evidence generated matches the regulatory requirements for each specific application.

Successful regulatory interactions require clear documentation of the model context of use, validation evidence, and uncertainty quantification [63]. Regulatory agencies expect transparent reporting of model limitations and the potential impact of uncertainties on model conclusions [63]. This transparency enables informed regulatory decision-making based on a comprehensive understanding of model capabilities and limitations.

Uncertainty quantification (UQ) has become an essential component of computational modeling, enabling researchers to quantify the effect of variability and uncertainty in model parameters on simulation outputs. In biomedical research, where model parameters often represent physical features, material coefficients, and physiological effects that lack well-established fixed values, UQ is particularly valuable for increasing model reliability and predictive power [15] [67]. The development of open-source UQ tools has made these sophisticated analyses accessible to a broader range of scientists, facilitating extension and modification to meet specific research needs.

This application note focuses on two prominent open-source UQ toolkits—UncertainSCI and the Uncertainty Quantification Toolkit (UQTk)—with particular emphasis on their applications in biomedical research. We provide a comparative analysis of their capabilities, detailed protocols for implementation, and specific examples of their use in cardiac and neural bioelectric simulations.

UncertainSCI

UncertainSCI is a Python-based software suite specifically designed with an emphasis on needs for biomedical simulations and applications [68]. It implements non-intrusive forward UQ methods by building polynomial chaos expansion (PCE) emulators through modern, near-optimal techniques for parameter sampling and PCE construction [15] [67]. The unique technology employed by UncertainSCI involves recent advances in high-dimensional approximation that ensures the construction of near-optimal emulators for general polynomial spaces in evaluating uncertainty [15]. Its non-intrusive pipeline allows users to leverage existing software libraries and suites to accurately ascertain parametric uncertainty without modifying their core simulation code.

UQTk

The UQ Toolkit (UQTk) is a collection of libraries and tools for the quantification of uncertainty in numerical model predictions, implemented primarily in C++ with Python interfaces [69]. It offers capabilities for representing random variables using Polynomial Chaos Expansions, intrusive and non-intrusive methods for propagating uncertainties through computational models, tools for sensitivity analysis, methods for sparse surrogate construction, and Bayesian inference tools for inferring parameters and model uncertainties from experimental data [69] [70]. UQTk has been applied to diverse fields, including fusion science, fluid dynamics, and Earth system land models [71].

Table 1: Core Capability Comparison Between UncertainSCI and UQTk

Feature	UncertainSCI	UQTk
Primary Language	Python	C++ with Python interfaces (PyUQTk)
Distribution Support	Various types of distributions [15]	Various types of distributions [69]
UQ Methods	Non-intrusive PCE with weighted approximate Fekete points [15]	Intrusive and non-intrusive PCE; Bayesian inference [69]
Sensitivity Analysis	Global and local sensitivity indices [15]	Global sensitivity analysis [71]
Inverse Problems	Not currently addressed [15]	Bayesian inference tools available [69]
Model Error Handling	Not specified	Framework for representing model structural errors [71]
Key Innovation	Weighted max-volume sampling with mean best-approximation guarantees [15]	Sparse surrogate construction; embedded model error correction [71]

Table 2: Statistics and Sensitivities Computable from UQ Emulators

Computable Quantity	Mathematical Definition	Application Context
Mean	𝔼[uN(p)]	Expected value of model output
Variance	𝔼[(uN(p) - 𝔼[uN(p)])²]	Spread of output values around mean
Quantiles	Value q such that ℙ(uN ≥ q) ≥ 1-δ and ℙ(uN ≤ q) ≥ δ	Confidence intervals for output predictions
Total Sensitivity	ST,ℐ = V(ℐ)/Var(uN)	Fraction of variance explained by parameter set ℐ
Global Sensitivity	SG,ℐ = [V(ℐ) - ∑∅≠𝒥⊂ℐV(𝒥)]/Var(uN)	Main effect contribution of parameter set ℐ

The Scientist's Toolkit: Essential Research Reagents

Table 3: Core Software and Computational Tools for UQ in Biomedical Research

Tool/Component	Function	Implementation in UQ
Polynomial Chaos Expansions	Functional representations of the relationship between parameters and outputs	Surrogate modeling to replace computationally expensive simulations [15] [71]
Weighted Approximate Fekete Points	Near-optimal parameter sampling strategy	Efficiently selects parameter combinations for forward model evaluations [15]
Global Sensitivity Analysis	Identifies dominant uncertain model inputs across parameter space	Determines which parameters most influence output variability [71]
Bayesian Inference	Statistical method for parameter estimation from data	Infers parameters and model uncertainties from experimental data [69]
Model Error Correction	Embedded stochastic terms to represent structural errors	Accounts for discrepancies between model and physical system [71]

Protocols for UQ in Biomedical Applications

Protocol 1: Uncertainty Quantification in Cardiac Simulations Using UncertainSCI

Background: Electrocardiographic imaging (ECGI) involves estimating cardiac potentials from measured body surface potentials, where cardiac geometry parameters significantly influence simulation outcomes [15]. Shape variability due to imaging and segmentation pipelines introduces uncertainty that can be quantified using UncertainSCI.

Materials:

• UncertainSCI Python package (installed via pip or from source)
• Cardiac simulation software (e.g., existing simulation pipeline for bioelectric potentials)
• Parameter distributions representing cardiac geometry variability

Procedure:

• Installation and Setup

  • Install UncertainSCI: pip install UncertainSCI
  • Verify installation by running basic examples from documentation
  • Establish interface between UncertainSCI and existing cardiac simulation code

• Parameter Distribution Definition

  • Identify uncertain parameters in cardiac model (e.g., tissue conductivity, geometry dimensions)
  • Define probability distributions for each parameter based on experimental data or literature values
  • Initialize distributions in UncertainSCI:

• Polynomial Chaos Setup

  • Specify polynomial space (typically total order polynomial basis)
  • Define the number of samples (typically scales with number of parameters and polynomial order)
  • Initialize PCE model:

• Sampling and Model Evaluation

  • Generate parameter samples using weighted approximate Fekete points:

  • For each sample, run cardiac simulation to compute output of interest
  • Collect simulation outputs corresponding to each parameter sample

• Emulator Construction and Analysis

  • Build PCE emulator using simulation data:

  • Compute output statistics (mean, variance, quantiles)
  • Perform global sensitivity analysis to identify dominant parameters

Troubleshooting Tips:

• If emulator accuracy is poor, increase polynomial order or number of samples
• For high-dimensional parameter spaces, use sensitivity analysis to focus on most influential parameters
• Verify interface with cardiac simulation code by testing with simplified model first

Protocol 2: Brain Stimulation Analysis with UQTk

Background: In transcranial electric stimulation simulations, the width and conductivity of the cerebrospinal fluid layer surrounding the brain significantly impact predicted electric fields [15] [67]. UQTk can quantify how uncertainty in these parameters affects stimulation predictions.

Materials:

• UQTk installed from GitHub repository
• Brain stimulation simulation software (e.g., SimNIBS or custom FEM solver)
• Parameter distributions for tissue properties

Procedure:

• UQTk Installation

  • Clone UQTk repository: git clone https://github.com/sandialabs/UQTk
  • Follow build instructions in docs/UQTk_manual.pdf
  • Verify installation by running tests: ctest

• Parameter Distribution Specification

  • Define probability distributions for tissue conductivity parameters
  • Use UQTk's distribution objects:

• Sparse Grid Sampling

  • Generate parameter samples using sparse grid techniques
  • Use UQTk's sparse grid functionality for high-dimensional efficiency
  • Export parameter sets for external simulations

• Forward Model Evaluation

  • Run brain stimulation simulation for each parameter sample
  • Record electric field values at locations of interest
  • Format simulation outputs for UQTk processing

• Surrogate Construction and Bayesian Analysis

  • Build sparse PCE surrogate using simulation data
  • Perform forward UQ to compute statistics of electric field predictions
  • Optionally, use Bayesian inference to calibrate parameters against experimental measurements

Validation Steps:

• Compare surrogate model predictions with full simulations at test points
• Check convergence of statistics with increasing number of samples
• Verify physical plausibility of sensitivity analysis results

Case Studies and Applications

Cardiac Uncertainty Quantification

In a study quantifying uncertainty in cardiac simulations, UncertainSCI was used to analyze the role of myocardial fiber direction in epicardial activation patterns [68]. The research demonstrated that UncertainSCI could efficiently identify which fiber architecture parameters had the greatest influence on activation patterns, providing insights important for understanding cardiac arrhythmia mechanisms. Similarly, another study used UncertainSCI to quantify uncertainty in simulations of myocardial ischemia, helping to establish confidence intervals for model predictions used in clinical decision support [68].

Neural Stimulation Uncertainty

In brain stimulation applications, UncertainSCI has been employed to quantify uncertainty in transcranial electric stimulation simulations [68]. The study focused on how variability in tissue conductivity parameters affects the predicted electric fields in the brain, with implications for treatment planning and dosing. The analysis provided sensitivity indices that ranked the influence of different tissue types on stimulation variability, helping researchers prioritize parameter measurement efforts.

UQTk for Biological and Healthcare Applications

While UQTk has traditionally been applied to physical and engineering systems, its methodologies are highly relevant to biological and healthcare applications. The toolkit's capabilities for Bayesian inference and model error quantification are particularly valuable for biological systems where model misspecification is common [72]. As biological digital twins become more prevalent, UQTk's comprehensive UQ framework can help establish model credibility by quantifying various sources of uncertainty.

UncertainSCI and UQTk provide complementary capabilities for uncertainty quantification in biomedical research. UncertainSCI offers a lightweight, Python-based solution with state-of-the-art sampling techniques specifically tailored for biomedical applications, while UQTk provides a comprehensive C++ framework with additional capabilities for inverse problems and Bayesian inference. By implementing the protocols outlined in this application note, biomedical researchers can systematically quantify how parameter variability affects their simulation outcomes, leading to more robust and reliable models for drug development and clinical decision support. As the field moves toward increased use of digital twins in healthcare, these UQ tools will play an essential role in establishing model credibility and translating computational predictions into clinical applications.

Overcoming UQ Challenges: Optimization Strategies for Complex Models

In computational science and engineering, the pursuit of accurate predictions is often hampered by the formidable computational expense of high-fidelity models. This challenge is particularly acute in the field of uncertainty quantification (UQ), where thousands of model evaluations may be required to propagate input uncertainties to output quantities of interest [73]. Model reduction and surrogate modeling have emerged as two pivotal strategies for mitigating these costs. While model reduction techniques, such as reduced-order modeling (ROM), aim to capture the essential physics of a system in a low-dimensional subspace, surrogate models provide computationally inexpensive approximations of the input-output relationship of complex models [74] [75]. These approaches are not mutually exclusive and are often integrated to achieve even greater efficiencies [76]. Framed within a broader thesis on UQ, this article details the application of these methods, providing structured protocols and resources to aid researchers, especially those in drug development and computational engineering.

Background and Core Concepts

The Computational Bottleneck in Uncertainty Quantification

UQ tasks—such as forward propagation, inverse problems, and reliability analysis—fundamentally require numerous model evaluations. When a single evaluation of a high-fidelity, physics-based model can take hours or even days, conducting a comprehensive UQ study becomes computationally prohibitive [74] [75]. This "curse of dimensionality" is exacerbated as the number of stochastic input parameters grows, leading to an exponential expansion of the input space that must be explored [74] [73].

The Role of Model Reduction and Surrogate Modeling

Model Reduction, including techniques like Proper Orthogonal Decomposition (POD) and reduced-basis methods, addresses cost by projecting the high-dimensional governing equations of a system onto a low-dimensional subspace. This results in a Reduced-Order Model (ROM) that is faster to evaluate while preserving the physics-based structure of the original model [77] [74]. For example, in cloud microphysics simulations, ROMs can efficiently simulate the evolution of high-dimensional systems like droplet-size distributions [78].

Surrogate Modeling takes a different approach. A surrogate model (or metamodel) is a data-driven approximation of the original computational model's input-output map. It is constructed from a limited set of input-output data and serves as a fast-to-evaluate replacement for the expensive model during UQ analyses [74] [75]. Popular surrogate models include Kriging (Gaussian Process Regression), Polynomial Chaos Expansion, and neural networks.

The synergy between them is powerful: model reduction can first simplify the system, and a surrogate can then be built for the reduced model, further accelerating computations [76].

Application Notes and Protocols

This section provides detailed, actionable protocols for implementing these strategies.

Protocol 1: Constructing a Projection-Based Reduced-Order Model

This protocol outlines the process for creating a ROM using the Proper Orthogonal Decomposition (POD) method, a common projection-based technique.

• Objective: To derive a low-dimensional, physics-based model that approximates the behavior of a high-fidelity computational model for rapid UQ.
• Principle: A low-dimensional basis is extracted from snapshots of the high-fidelity model. The full-order system is then projected onto this basis to create a reduced system of equations [74].

Step-by-Step Workflow:

• Generate Snapshot Matrix: Execute the high-fidelity model for a representative set of input parameters ( {\mathbf{x}1, \mathbf{x}2, ..., \mathbf{x}N} ). Collect the resulting system states (e.g., field solutions) as column vectors ( \mathbf{u}(\mathbf{x}i) ) to form the snapshot matrix ( \mathbf{S} = [\mathbf{u}(\mathbf{x}1), \mathbf{u}(\mathbf{x}2), ..., \mathbf{u}(\mathbf{x}_N)] ).
• Perform Dimensionality Reduction: Apply Singular Value Decomposition (SVD) to the snapshot matrix: ( \mathbf{S} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^T ). The columns of ( \mathbf{U} ) are the POD basis vectors. Select the first ( r ) basis vectors ( \mathbf{U}_r ) to form the reduced basis, where ( r ) is chosen based on the decay of singular values in ( \boldsymbol{\Sigma} ) to capture the dominant system dynamics.
• Project Governing Equations: Express the state variable as ( \mathbf{u} \approx \mathbf{U}r \mathbf{u}r ), where ( \mathbf{u}_r ) is the vector of reduced coordinates. Substitute this approximation into the full-order model's governing equations (e.g., a partial differential equation) and project onto the reduced subspace via Galerkin or Petrov-Galerkin projection to obtain the ROM.
• Validate the ROM: Test the ROM on a new set of input parameters not used in the training snapshots. Compare its outputs against the full-order model to quantify accuracy and speedup.

The following diagram illustrates the core workflow and logical relationships of this protocol:

Protocol 2: Building a Kriging Surrogate Model with Functional Dimension Reduction

This protocol describes constructing a Kriging surrogate model for dynamical systems, enhanced by dimension reduction to handle high-dimensional output spaces, such as time-series data [79].

• Objective: To create a fast and accurate surrogate model for a computationally expensive dynamical system, enabling efficient forward and inverse UQ.
• Principle: The system's time-varying responses are treated as functions. Functional data analysis is used to reduce the output dimension before building the Kriging model in the latent functional space [79].

Step-by-Step Workflow:

• Training Data Generation: Sample the input space ( \mathcal{X} ) using a design of experiments (e.g., Latin Hypercube Sampling). For each sample ( \mathbf{x}i ), run the high-fidelity model to obtain the time-series output ( \mathbf{y}i(t) ).
• Functional Dimension Reduction:
  • Represent the output functions ( \mathbf{y}(t) ) using a set of basis functions (e.g., polynomials, splines). A roughness regularization term can be added to handle noisy data [79].
  • Apply functional Principal Component Analysis (fPCA) to identify a few key latent functions ( \boldsymbol{\psi}(t) ) that capture the majority of the variance in the output data.
  • Map each high-dimensional output ( \mathbf{y}i ) to its low-dimensional latent representation ( \boldsymbol{\psi}i ).
• Construct Kriging Surrogates: Build independent Kriging (Gaussian Process) surrogate models ( \mathcal{M}_{KR}^j ) for each component ( \psi^j ) of the latent vector ( \boldsymbol{\psi} ), mapping the input parameters ( \mathbf{x} ) to the latent space.
• Predict and Reconstruct: For a new input ( \mathbf{x}^* ), the Kriging models predict the latent vector ( \boldsymbol{\psi}^* ). The full output ( \mathbf{y}^(t) ) is then reconstructed from ( \boldsymbol{\psi}^ ) using the inverse of the functional dimension reduction mapping.
• UQ Analysis: Use the trained surrogate model for efficient UQ tasks, such as Monte Carlo simulation, by repeatedly evaluating the fast surrogate instead of the original model.

The logical flow of this advanced surrogate modeling technique is shown below:

Quantitative Performance Data

The efficacy of model reduction and surrogate modeling is demonstrated by significant reductions in computational cost and resource requirements across various fields. The following tables summarize quantitative findings from the literature.

Table 1: Performance Gains in Drug Discovery Applications

Application / Method	Key Performance Metric	Reported Outcome	Source Context
AI-Driven Drug Discovery (Uncertainty-Guided)	Cost Reduction	75% reduction in discovery costs	[80]
AI-Driven Drug Discovery (Uncertainty-Guided)	Speed Acceleration	10x faster discovery process	[80]
AI-Driven Drug Discovery (Uncertainty-Guided)	Data Efficiency	60% less training data required	[80]
Molecular Property Prediction (Censored Data)	Data Utilization	Reliable UQ with ~33% censored labels	[8]

Table 2: Performance of General Surrogate and Reduced-Order Modeling Techniques

Method / Technique	Application Domain	Key Advantage / Performance	Source Context
Dimensionality Reduction as Surrogate (DR-SM)	High-Dimensional UQ	Serves as a baseline; avoids reconstruction mapping; handles high-dimensional input.	[76]
Post-hoc UQ for ROMs (Conformal Prediction)	Cloud Microphysics	Model-agnostic UQ; provides prediction intervals for latent dynamics & reconstruction.	[78]
Kriging with Functional Dimension Reduction (KFDR)	Dynamical Systems	Accurate UQ for systems with limited training samples; handles noisy data.	[79]

The Scientist's Toolkit: Research Reagents and Computational Solutions

This section lists key computational tools and methodologies that form the essential "reagents" for implementing the protocols discussed in this article.

Table 3: Essential Computational Tools for Model Reduction and Surrogate Modeling

Tool / Method	Category	Primary Function	Relevant Protocol
Proper Orthogonal Decomposition (POD)	Model Reduction	Extracts an optimal low-dimensional basis from system snapshot data to create a ROM.	Protocol 1 (ROM Construction)
Singular Value Decomposition (SVD)	Linear Algebra	The core numerical algorithm used to compute the basis in POD and other dimension reduction techniques.	Protocol 1 (ROM Construction)
Kriging / Gaussian Process Regression	Surrogate Modeling	Constructs a probabilistic surrogate model that provides a prediction and an uncertainty estimate.	Protocol 2 (Kriging Surrogate)
Functional Principal Component Analysis (fPCA)	Dimension Reduction	Reduces the dimensionality of functional data (e.g., time series) by identifying dominant modes of variation.	Protocol 2 (Kriging Surrogate)
Polynomial Chaos Expansion (PCE)	Surrogate Modeling	Represents the model output as a series of orthogonal polynomials, useful for moment-based UQ.	General UQ Surrogates [74]
Conformal Prediction	Uncertainty Quantification	Provides model-agnostic, distribution-free prediction intervals for any black-box model or ROM.	UQ for ROMs [78]
Latin Hypercube Sampling (LHS)	Experimental Design	Generates a space-filling sample of input parameters for efficient training data collection.	Protocol 2, General Use

Model reduction and surrogate modeling are indispensable strategies for managing the prohibitive computational costs associated with rigorous uncertainty quantification in complex systems. The protocols and data presented herein provide a concrete foundation for researchers in drug development and computational engineering to implement these techniques. By adopting projection-based model reduction to create fast, physics-preserving ROMs, or leveraging advanced surrogate models like functional Kriging, scientists can achieve order-of-magnitude improvements in efficiency. This enables previously infeasible UQ studies, leading to more reliable predictions, robust designs, and accelerated discovery cycles, as evidenced by the dramatic cost and time reductions reported in the pharmaceutical industry. The integration of robust UQ methods, such as conformal prediction, further ensures that the uncertainties in these accelerated computations are properly quantified, fostering greater trust in computational predictions for high-consequence decision-making.

Strategies for UQ with Limited or Sparse Data

Uncertainty Quantification (UQ) is a critical component of predictive computational modeling, providing a framework for assessing the reliability of model-based predictions in the presence of various sources of uncertainty. While UQ methodologies have advanced significantly across scientific and engineering disciplines, conducting robust UQ remains particularly challenging when dealing with limited or sparse data—a common scenario in many real-world applications. Data sparsity can arise from high costs of data collection, physical inaccessibility of sampling locations, or inherent limitations in measurement technologies. This application note synthesizes current strategies and protocols for performing credible UQ under data constraints, drawing from recent advances across multiple domains including environmental science, nuclear engineering, and materials design.

The fundamental challenge in sparse-data UQ lies in the tension between model complexity and informational constraints. Without sufficient data coverage, traditional UQ methods may produce unreliable uncertainty estimates, potentially leading to overconfident predictions in unsampled regions. This note presents a structured approach to addressing these challenges through methodological adaptations, surrogate modeling, and specialized sampling strategies that maximize information extraction from limited data.

Theoretical Foundations of Sparse-Data UQ

In sparse data environments, epistemic uncertainty (resulting from limited knowledge about the system) typically dominates aleatoric uncertainty (inherent system variability) [81] [82]. Epistemic uncertainty manifests prominently in regions of the input space with few or no observations, where models must extrapolate rather than interpolate. This type of uncertainty is reducible in principle through additional data collection, though practical constraints often prevent this.

The characterization of uncertainty sources is particularly important when working with sparse datasets [82]:

• Data uncertainty: Arises from measurement noise, observational errors, and sampling biases that become more pronounced with limited data.
• Model structure uncertainty: Results from potential misspecification of the underlying physical relationships, which becomes harder to detect and correct with sparse data.
• Parametric uncertainty: Stems from imperfect knowledge of model parameters that cannot be well-constrained with limited observations.
• Extrapolation uncertainty: Emerges when predictions are required outside the convex hull of the training data, a common scenario with sparse spatial or temporal coverage.

Mathematical Frameworks for Sparse-Data UQ

Bayesian methods provide a natural mathematical foundation for UQ with sparse data by explicitly representing uncertainty through probability distributions over model parameters and outputs [83] [82]. The Bayesian formulation allows for the incorporation of prior knowledge, which can partially compensate for data scarcity. For a model with parameters θ and data D, the posterior distribution is given by:

[ P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)} ]

where the prior ( P(\theta) ) encodes existing knowledge before observing data D. With sparse data, the choice of prior becomes increasingly influential on the posterior estimates, requiring careful consideration of prior selection.

Frequentist approaches, particularly conformal prediction (CP) methods, offer an alternative framework that provides distribution-free confidence intervals without requiring strong distributional assumptions [82]. These methods can be particularly valuable when prior knowledge is limited or unreliable.

Computational Strategies and Protocols

Surrogate Modeling for Computational Efficiency

Surrogate modeling replaces computationally expensive high-fidelity models with cheap-to-evaluate approximations, enabling UQ tasks that would otherwise be prohibitively expensive [84] [85]. This approach is especially valuable in data-sparse environments where many model evaluations may be needed for uncertainty propagation.

Protocol 3.1.1: Sparse Polynomial Chaos Expansion Surrogate Modeling

Objective: Construct an accurate surrogate model using limited training data.

Materials: High-fidelity model, parameter distributions, computing resources.

Procedure:

• Experimental Design: Generate training samples using Latin Hypercube Sampling (LHS) or sparse grid designs to maximize information from limited runs [84].
• Basis Selection: Employ a basis-adaptive least-angle-regression strategy to identify important polynomial terms while inducing sparsity [84].
• Coefficient Estimation: Compute polynomial coefficients using regression or projection methods.
• Validation: Assess surrogate accuracy using cross-validation or holdout samples, with emphasis on extrapolation performance.

Applications: This method has demonstrated 30,000-fold computational savings for parameter estimation in complex systems with 20 uncertain parameters [84].

Protocol 3.1.2: Sensitivity-Driven Dimension-Adaptive Sparse Grids

Objective: Enable UQ in high-dimensional problems with limited computational budget.

Materials: Computational model, sensitivity analysis tools, adaptive grid software.

Procedure:

• Initialization: Begin with a coarse sparse grid covering the parameter space.
• Sensitivity Analysis: Compute global sensitivity indices (Sobol indices) for each parameter.
• Adaptive Refinement: Prioritize grid refinement along dimensions with highest sensitivity [85].
• Iteration: Repeat steps 2-3 until computational budget is exhausted or convergence criteria are met.

Applications: This approach reduced the required simulations by two orders of magnitude in fusion plasma turbulence modeling with eight uncertain parameters [85].

Bayesian Deep Learning for Spatial UQ

Bayesian deep learning provides a framework for quantifying uncertainty in data-driven models, particularly valuable when transferring models to un-sampled regions [83].

Protocol 3.2.1: Last-Layer Laplace Approximation (LLLA) for Neural Networks

Objective: Efficiently quantify predictive uncertainty in deep learning models with limited data.

Materials: Pre-trained neural network, transfer region data, Laplace approximation software.

Procedure:

• Model Training: Train a conventional neural network on the available data.
• Last-Layer Bayesian Treatment: Freeze all network weights except the final layer.
• Laplace Approximation: Approximate the posterior distribution of the last-layer weights using a multivariate Gaussian distribution.
• Predictive Distribution: Propagate uncertainty through the network to obtain predictive intervals.

Applications: Successfully applied to soil property prediction where models trained in one region were transferred to geographically separate regions with similar characteristics [83].

Uncertainty-Driven Data Imputation and Balancing

Data sparsity often coincides with imbalanced datasets, where certain output values are significantly underrepresented [81] [86].

Protocol 3.3.1: Uncertainty-Quantification-Driven Imbalanced Regression (UQDIR)

Objective: Improve model accuracy for imbalanced regression problems common with sparse data.

Materials: Imbalanced dataset, machine learning model with UQ capability.

Procedure:

• Uncertainty Estimation: Train initial model and compute epistemic uncertainty for each prediction.
• Rare Sample Identification: Identify samples corresponding to rare output values based on high epistemic uncertainty [81].
• Weight Assignment: Assign resampling weights using a UQ-based weight function.
• Model Retraining: Retrain model using the restructured dataset.

Applications: Effective for metamaterial design and other engineering applications where the output distribution is naturally imbalanced [81].

Protocol 3.3.2: Temporal Interpolation for Sparse Time Series Data

Objective: Construct complete input datasets from sparse temporal observations.

Materials: Sparse time series data, interpolation software.

Procedure:

• Method Selection: Compare interpolation methods (linear, spline, moving average) using available data.
• Gap Filling: Apply selected method to construct continuous time series.
• Uncertainty Propagation: Quantify how interpolation uncertainty affects model outputs.
• Validation: Compare model outputs using different interpolation strategies.

Applications: Hydrodynamic-water quality modeling where monthly measurements were interpolated to daily inputs, with linear interpolation showing superior performance for gap filling [86].

Practical Implementation Tools

UQ Method Selection Guide

Table 1: Comparative Analysis of UQ Methods for Sparse Data

Method	Data Requirements	Computational Cost	Uncertainty Types Captured	Best-Suited Applications
Laplace Approximation	Moderate	Low	Epistemic, Model	Model transfer to new domains [83]
Sparse Polynomial Chaos	Low-Moderate	Medium	Parametric, Approximation	Forward UQ, Sensitivity analysis [84]
Gaussian Processes	Low	Low-Medium	Data, Approximation	Spatial interpolation, Small datasets [82]
Monte Carlo Dropout	Moderate	Low	Model, Approximation	Deep learning applications [82]
Deep Ensembles	Moderate-High	High	Model, Data, Approximation	Complex patterns, Multiple data sources [82]
Conformal Prediction	Low-Moderate	Low	Data, Model misspecification	Distribution-free confidence intervals [82]

Research Reagent Solutions

Table 2: Essential Computational Tools for Sparse-Data UQ

Tool Category	Specific Examples	Function	Implementation Considerations
Surrogate Models	Sparse PCE, Kriging	Replace expensive models	Balance accuracy vs. computational cost [84]
Bayesian Inference Libraries	Pyro, Stan, TensorFlow Probability	Posterior estimation	Choose based on model complexity and data size [83]
Sensitivity Analysis	Sobol indices, Morris method	Identify important parameters	Global vs. local methods depending on linearity [85]
Adaptive Sampling	Sensitivity-driven sparse grids	Maximize information gain	Prioritize uncertain regions [85]
UQ in Deep Learning	MC Dropout, Deep Ensembles	Quantify DL uncertainty	Architecture-dependent implementation [82]

Case Studies and Applications

Spatial UQ for Soil Property Prediction

In digital soil mapping, researchers faced the challenge of predicting soil properties in under-sampled regions [83]. Using a Bayesian deep learning approach with Laplace approximations, they quantified spatial uncertainty when transferring models from well-sampled to data-sparse regions. The methodology successfully identified areas where model predictions were reliable versus areas requiring additional data collection, demonstrating the value of spatial UQ for prioritizing sampling efforts.

Turbulent Transport Modeling in Fusion Research

In computationally expensive turbulence simulations for fusion plasma confinement, researchers employed sensitivity-driven dimension-adaptive sparse grid interpolation to conduct UQ with only 57 high-fidelity simulations despite eight uncertain parameters [85]. This approach exploited the anisotropic coupling of uncertain inputs to reduce computational effort by two orders of magnitude while providing accurate uncertainty estimates and an efficient surrogate model.

Hydrodynamic-Water Quality Modeling with Sparse Inputs

For water quality modeling in the Mississippi Sound and Mobile Bay, researchers compared interpolation methods for constructing daily inputs from sparse monthly measurements [86]. Through systematic evaluation of linear interpolation, spline methods, and moving averages, they quantified how input uncertainty propagated to model outputs, enabling more informed decisions about data collection and model calibration.

Visual Guides and Workflows

Comprehensive UQ Workflow for Sparse Data

Diagram Title: Comprehensive UQ Workflow for Sparse Data

Uncertainty-Driven Data Balancing Process

Diagram Title: Uncertainty-Driven Data Balancing Process

This application note has outlined principal strategies and detailed protocols for conducting uncertainty quantification with limited or sparse data. The presented methodologies—including surrogate modeling, Bayesian deep learning, uncertainty-driven data balancing, and adaptive sampling—provide a toolkit for researchers facing data scarcity across various domains. As computational models continue to grow in complexity and application scope, the ability to rigorously quantify uncertainty despite data limitations becomes increasingly critical for credible predictive science.

Future directions in sparse-data UQ include the development of hybrid methods that combine physical knowledge with data-driven approaches, more efficient transfer learning frameworks for leveraging related datasets, and automated UQ pipelines that can adaptively select appropriate methods based on data characteristics and modeling goals. By adopting the strategies outlined in this document, researchers can enhance the reliability of their computational predictions even under significant data constraints, leading to more informed decision-making across scientific and engineering disciplines.

Addressing Overconfidence in Ensemble Predictions

Ensemble models significantly enhance predictive performance by combining multiple machine learning models. However, they are not immune to overconfidence, where models produce incorrect but highly confident predictions, a critical issue in high-stakes fields like drug development [87] [88]. Within uncertainty quantification (UQ) computational research, overconfidence represents a failure to properly quantify predictive uncertainty, potentially leading to misguided decisions based on unreliable model outputs [89] [90].

This document details protocols for diagnosing and mitigating overconfidence in ensembles, providing researchers with practical tools to enhance model reliability. We focus on methodologies that distinguish between data (aleatoric) and model (epistemic) uncertainty, crucial for developing robust predictive systems in scientific domains [91] [90].

Understanding Overconfidence in Ensembles

Key Concepts and Mechanisms

Overconfidence in ensemble models arises when the combined prediction exhibits high confidence that is not aligned with actual accuracy. The variance in the predictions across ensemble members is a common heuristic for quantifying this uncertainty; low variance suggests high confidence, while high variance indicates low confidence [90]. However, research on neural network interatomic potentials shows that in Out-of-Distribution (OOD) settings, uncertainty estimates can behave counterintuitively, often plateauing or even decreasing as predictive errors grow, highlighting a fundamental limitation of current UQ approaches [89].

Primary Causes of Overconfidence

Several factors contribute to overconfident ensemble predictions [87] [88]:

• Excessive Model Complexity: Overly complex ensembles with numerous parameters can overfit training data noise.
• Data Bias and Insufficient Training Data: Biased or limited training datasets prevent models from learning generalizable patterns, leading to overgeneralization.
• Imbalanced Class Distributions: Ensembles can become biased toward the majority class, resulting in overconfident predictions for that class.
• Over-Optimization on Training Data: Hyperparameter tuning focused solely on training performance can inadvertently encourage overfitting.
• Lack of Model Diversity: If ensemble members are highly correlated, the ensemble cannot properly capture the uncertainty, leading to overconfident collective predictions [92].

Quantitative Framework and Data Presentation

Comparison of Ensemble UQ Methods

The table below summarizes quantitative characteristics of different ensemble-based UQ methods, aiding in the selection of appropriate techniques for mitigating overconfidence.

Table 1: Uncertainty Quantification Methods for Ensemble Models

Method Category	Specific Technique	Key Mechanism	Strengths	Limitations / Computational Cost	Best-Suited Uncertainty Type
Sampling-Based	Monte Carlo Dropout [90]	Applies dropout during inference for multiple stochastic forward passes.	Computationally efficient; no re-training required.	Approximate inference; may yield over-confident estimates on OOD data [93].	Model (Epistemic)
Bayesian Methods	Bayesian Neural Networks [93] [90]	Treats model weights as probability distributions.	Principled UQ; rigorous uncertainty decomposition.	High computational cost; complex approximate inference [93].	Model (Epistemic) & Data (Aleatoric)
Ensemble Methods	Deep Ensembles [90]	Trains multiple models with different initializations.	High-quality uncertainty estimates; easy to implement.	High computational cost (requires multiple models).	Model (Epistemic) & Data (Aleatoric)
	Bootstrap Aggregating [94] [92]	Trains models on different data subsets (bootstrapping).	Reduces variance; robust to overfitting.	Requires multiple models; can be memory-intensive.	Model (Epistemic)
Frequentist Methods	Discriminative Jackknife [93]	Uses influence functions to estimate a jackknife sampling distribution.	Provides theoretical coverage guarantees; applied post-hoc.	Computationally intensive for large datasets.	Model (Epistemic)
Conformal Prediction	Conformal Forecasting [93]	Uses a calibration set to provide distribution-free prediction intervals.	Model-agnostic; provides finite-sample coverage guarantees.	Requires a held-out calibration dataset; intervals can be conservative.	Model (Epistemic) & Data (Aleatoric)

Impact of Overconfidence Across Industries

Understanding the real-world impact of overconfidence underscores the importance of robust UQ.

Table 2: Consequences of Overconfidence in Different Sectors

Industry	Potential Impact of Overconfident Ensemble Models
Healthcare & Drug Development	Misdiagnosis or incorrect prognosis due to models relying on spurious correlations in medical data; failure in predicting drug efficacy or toxicity [87].
Finance	Poor investment decisions or incorrect risk assessments from models that overfit historical market data and fail to predict novel market conditions [87].
Autonomous Systems	Safety-critical failures in self-driving cars due to misclassification of objects or scenarios not well-represented in training data [87].

Experimental Protocols for UQ Assessment

This section provides detailed methodologies for evaluating and mitigating overconfidence in ensemble models.

Protocol 1: Benchmarking Ensemble UQ Methods

Objective: Systematically compare the calibration and accuracy of different ensemble UQ methods on in-distribution (ID) and out-of-distribution (OOD) data.

Materials:

• Research Reagent Solutions:
  • Datasets: Curated datasets with known distribution shifts (e.g., CIFAR-10 (ID) vs. CIFAR-10-C (OOD) [91].
  • Software Frameworks: TensorFlow Probability (for BNNs), PyTorch (for MC Dropout), Scikit-learn (for Random Forests), UQ toolboxes (e.g., Uncertainty Quantification 360).
  • Evaluation Metrics: Expected Calibration Error (ECE), Negative Log-Likelihood (NLL), sharpness of prediction intervals.

Procedure:

• Data Preparation: Split the dataset into training (60%), validation (20%), ID test (10%), and OOD test (10%) sets. The OOD set should represent a realistic covariate or concept shift relevant to the application (e.g., different experimental conditions in drug discovery).
• Model Training:
  • Train multiple ensemble types (e.g., Deep Ensemble, MC Dropout Ensemble, Bootstrap Ensemble) on the training set.
  • Use the validation set for hyperparameter tuning, ensuring to optimize for calibration metrics like ECE, not just accuracy.
• Inference & Prediction: For each ensemble method and each sample in the ID and OOD test sets, collect the predictive mean and a uncertainty statistic (e.g., predictive entropy or variance).
• Metrics Calculation:
  • Expected Calibration Error (ECE): Partition predictions into bins based on confidence. Compute the absolute difference between average accuracy and average confidence across bins. Weight by bin size and sum.
  • Negative Log-Likelihood (NLL): Compute the average negative log of the predicted probability assigned to the true label. A lower NLL indicates better probabilistic calibration.
  • Sharpness: Calculate the average width of the prediction intervals (for regression) or the concentration of the predictive distribution (for classification). Trades off with calibration.
• Analysis: Compare the ranking of UQ methods based on their OOD performance, as this is often where overconfidence is most pronounced [89].

Protocol 2: Mitigating Overconfidence via Conformal Prediction

Objective: Apply conformal prediction to ensemble models to obtain prediction sets with guaranteed coverage, thereby controlling overconfidence.

Materials:

• Research Reagent Solutions:
  • Pre-trained Ensemble Model: An already trained ensemble model.
  • Calibration Dataset: A held-out dataset not used for training, exchangeable with the test data.
  • Conformal Prediction Library: Such as crepes or MAPIE (for Python).

Procedure:

• Data Splitting: Split available data into a proper training set (for initial model training), a calibration set, and a test set.
• Nonconformity Score Calculation: For each sample in the calibration set, pass it through the ensemble to get a set of predictions. Define a nonconformity score, s_i. For classification, a common score is s_i = 1 - f(x_i)[y_i], where f(x_i)[y_i] is the predicted probability for the true class y_i [90].
• Threshold Calculation: Sort the nonconformity scores from the calibration set. For a target coverage rate of 1-α (e.g., 95%, where α=0.05), compute the threshold q as the ⌈(n+1)(1-α)⌉ / n-th quantile of the sorted scores, where n is the size of the calibration set.
• Prediction Set Formation: For a new test sample x_{test}:
  • Compute the nonconformity score s_test(l) for every possible label l.
  • Include label l in the prediction set if s_test(l) <= q.
• Validation: Verify on the test set that the empirical coverage (the proportion of times the true label is contained in the prediction set) is at least 1-α. This provides a frequentist guarantee against overconfidence.

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Tools and Materials for UQ Experiments

Item	Function in UQ Research	Example Tools / Libraries
UQ Software Libraries	Provide implemented algorithms for Bayesian inference, ensemble methods, and conformal prediction.	TensorFlow Probability, PyTorch, PyMC, Scikit-learn, UQ360 (IBM)
Calibration Metrics	Quantitatively measure the alignment between predicted confidence and empirical accuracy.	Expected Calibration Error (ECE), Negative Log-Likelihood (NLL)
Benchmark Datasets	Standardized datasets with defined training and OOD test sets for reproducible evaluation of UQ methods.	CIFAR-10/100-C, ImageNet-A/O, MoleculeNet (for cheminformatics)
Conformal Prediction Packages	Automate the calculation of nonconformity scores and prediction sets for any pre-trained model.	crepes, MAPIE, nonconformist
Visualization Tools	Create reliability diagrams and other plots to diagnose miscalibration visually.	Matplotlib, Seaborn (in Python); custom plotting scripts

Addressing overconfidence is not a single-step process but an integral part of developing trustworthy AI systems for scientific discovery. By integrating the ensemble methods, benchmarking protocols, and calibration techniques outlined in these application notes, researchers can significantly improve the reliability of their predictive models. The future of UQ research lies in developing more computationally efficient and accurate methods, particularly those robust to real-world distribution shifts, ultimately enabling more confident and credible decision-making in drug development and beyond.

Global Sensitivity Analysis for Identifying Key Uncertainty Contributors

Global Sensitivity Analysis (GSA) represents a critical methodology within uncertainty quantification for computational models, particularly in pharmaceutical research and drug development. Unlike local approaches that vary one parameter at a time while holding others constant, GSA examines how uncertainty in model outputs can be apportioned to different sources of uncertainty in the model inputs across their entire multidimensional space [95]. This systematic approach allows researchers to identify which parameters contribute most significantly to outcome variability, thereby guiding resource allocation for parameter estimation and experimental design.

The fundamental principle of GSA involves exploring the entire parameter space simultaneously, enabling the detection of interaction effects between parameters that local methods would miss [95] [96]. This capability is particularly valuable in complex biological systems and pharmacological models where nonlinear relationships and parameter interactions are common. For computational models in drug development, GSA provides a mathematically rigorous framework to quantify how uncertainties in physiological parameters, kinetic constants, and experimental conditions propagate through systems biology models, pharmacokinetic/pharmacodynamic (PK/PD) models, and disease progression models [96].

Theoretical Foundations and Methodological Approaches

Key Properties of Global Sensitivity Analysis

An ideal GSA method should possess several critical properties that distinguish it from local approaches. According to the Joint Research Centre's guidelines, these properties include: (1) coping with the influence of scale and shape, meaning the method should incorporate the effect of the range of input variation and its probability distribution; (2) including multidimensional averaging to evaluate the effect of each factor while all others are varying; (3) maintaining model independence to work regardless of the additivity or linearity of the model; and (4) being able to treat grouped factors as if they were single factors for more agile interpretation of results [95].

These properties ensure that GSA methods can effectively handle the complex, nonlinear models frequently encountered in pharmaceutical research, where interaction effects between biological parameters can significantly impact model predictions. The ability to account for the full distribution of parameter values, rather than just point estimates, makes GSA particularly suitable for quantifying uncertainty in drug development, where many physiological and biochemical parameters exhibit natural variability or measurement uncertainty [96].

Classification of GSA Methods

GSA methods can be broadly categorized into four groups based on their mathematical foundations: variance-based methods, derivative-based methods, density-based methods, and screening designs [97]. Each category offers distinct advantages and is suitable for different stages of the model analysis pipeline in pharmaceutical research.

Table 1: Classification of Global Sensitivity Analysis Methods

Method Category	Key Principles	Representative Techniques	Pharmaceutical Applications
Variance-Based	Decomposition of output variance into contributions from individual parameters and interactions	Sobol' indices, Extended Fourier Amplitude Sensitivity Test (eFAST)	PK/PD modeling, systems pharmacology, clinical trial simulations
Screening Designs	Preliminary factor ranking with minimal computational cost	Morris method, Cotter design, Iterated Fractional Factorial Designs	High-dimensional parameter screening, early-stage model development
Sampling-Based	Statistical analysis of input-output relationships using designed sampling	Partial Rank Correlation Coefficient (PRCC), Standardized Regression Coefficients (SRC)	Disease modeling, biomarker identification, dose-response relationships
Response Surface	Approximation of complex models with surrogate functions for analysis	Gaussian process emulation, polynomial chaos expansion	Complex computational models with long runtimes, optimization problems

Variance-based methods, particularly Sobol' indices, are widely regarded as among the most robust and informative approaches [97]. These methods decompose the variance of model output into contributions attributable to individual parameters and their interactions. The first-order Sobol' index (Si) measures the direct contribution of each input parameter to the output variance, while the total-order index (STi) captures both main effects and all interaction effects involving that parameter [97]. This decomposition is particularly valuable in biological systems where parameter interactions are common and often biologically significant.

Experimental Protocols and Implementation Frameworks

Integrated GSA Workflow for Computational Models

The following diagram illustrates the comprehensive workflow for implementing global sensitivity analysis in computational models for drug development:

Two-Step GSA Framework for High-Dimensional Models

For complex models with numerous parameters, a two-step GSA framework efficiently identifies key uncertainty contributors while managing computational costs [98]. This approach is particularly valuable in pharmaceutical research where computational models may contain dozens or hundreds of parameters with uncertain values.

Step 1: Factor Screening Using Morris Method The first step employs the Morris method, an efficient screening design that provides qualitative sensitivity measures while requiring relatively few model evaluations [95] [98]. The Morris method computes elementary effects (EE_i) for each parameter by measuring the change in model output when parameters are perturbated one at a time from baseline values:

• Parameter Space Discretization: Define a p-dimensional grid for k parameters, where each parameter can take values from {0, 1/(d-1), 2/(d-1), ..., 1} for a given number of levels d
• Trajectory Generation: Generate r independent random trajectories through the parameter space, each consisting of (k+1) points
• Elementary Effects Calculation: For each parameter i in each trajectory, compute the elementary effect: EEi = [y(x1, ..., xi+Δ, ..., xk) - y(x)]/Δ where Δ is a predetermined step size
• Sensitivity Metrics: Calculate μi (mean of absolute elementary effects) and σi (standard deviation of elementary effects) across all trajectories for each parameter
• Factor Ranking: Rank parameters based on μi values, with higher values indicating more influential parameters; σi indicates nonlinear effects or interactions

Step 2: Variance-Based Quantitative Analysis The second step applies variance-based methods (e.g., Sobol' indices) to the subset of influential parameters identified in Step 1, providing quantitative sensitivity measures:

• Sample Generation: Generate input samples using Sobol' sequences or Latin Hypercube Sampling (LHS) for the reduced parameter set [96] [98]
• Model Evaluation: Compute model outputs for all sample points
• Index Calculation: Estimate first-order (Si) and total-order (STi) Sobol' indices using variance decomposition formulas [97]
• Interaction Quantification: Calculate interaction effects by comparing Si and STi values (STi - Si represents total interaction effects for parameter i)

This two-step approach balances computational efficiency with comprehensive sensitivity assessment, making it particularly suitable for complex biological models with potentially influential parameters.

Latin Hypercube Sampling with Partial Rank Correlation Coefficient Analysis

For models with moderate parameter counts (10-50 parameters), the combined LHS-PRCC approach provides a robust screening methodology [96]. The protocol implementation includes:

Sampling Phase

• Define probability distributions for each input parameter based on experimental data or literature values
• Generate LHS matrix with N samples (where N should be at least k+1, with k being the number of parameters, though typically much larger for accuracy) [96]
• For each parameter, divide the cumulative distribution function into N equiprobable intervals
• Randomly select one value from each interval without replacement, ensuring full stratification

Analysis Phase

• Execute model simulations for each of the N parameter combinations from the LHS matrix
• Compute PRCC between each input parameter and model output while controlling for all other parameters
• Assess statistical significance of PRCC values using appropriate hypothesis testing
• Rank parameters based on the magnitude and significance of PRCC values

This method is particularly effective for monotonic but nonlinear relationships common in biological systems, such as dose-response curves and saturation kinetics [96].

Comparative Analysis of GSA Methods

Quantitative Performance Metrics

Table 2: Comparative Analysis of Global Sensitivity Analysis Methods

Method	Computational Cost	Handling of Interactions	Output Information	Implementation Complexity	Optimal Use Cases
Sobol' Indices	High (N×(k+2) to N×(2k+2) model runs)	Explicit quantification of all interactions	First-order, higher-order, and total-effect indices	High	Final analysis of refined parameter sets, interaction quantification
Morris Method	Moderate (r×(k+1) model runs)	Detection but not quantification of interactions	Qualitative ranking with elementary effects statistics	Medium	Initial screening of high-dimensional parameter spaces
PRCC with LHS	Moderate to High (N model runs, N>k)	Implicit through correlation conditioning	Correlation coefficients with significance testing	Medium	Monotonic relationships, nonlinear but monotonic models
eFAST	Moderate (N_s×k model runs)	Quantitative assessment of interactions	First-order and total-effect indices	Medium to High	Oscillatory models, alternative to Sobol' with different sampling
Monte Carlo Filtering	Variable based on filtering criteria	Detection through statistical tests	Identification of important parameter regions	Medium	Factor mapping, identifying critical parameter ranges

The computational requirements represent approximate model evaluations needed, where k is the number of parameters, r is the number of trajectories in the Morris method (typically 10-50), and N is sample size for sampling-based methods (typically hundreds to thousands) [95] [96].

Research Reagent Solutions for GSA Implementation

Table 3: Essential Computational Tools for Global Sensitivity Analysis

Tool/Category	Specific Examples	Function in GSA Implementation	Application Context
Sampling Algorithms	Latin Hypercube Sampling, Sobol Sequences, Morris Trajectory Design	Generate efficient space-filling experimental designs for parameter space exploration	Creating input matrices that efficiently cover parameter spaces with minimal samples
Statistical Software	R (sensitivity package), Python (SALib, PyDREAM), MATLAB (Global Sensitivity Analysis Toolbox)	Compute sensitivity indices from input-output data using various GSA methods	Implementing GSA methodologies without developing algorithms from scratch
Variance Decomposition	Sobol' Indices Calculator, eFAST Algorithm	Decompose output variance into contributions from individual parameters and interactions	Quantifying parameter importance and interaction effects in nonlinear models
Correlation Analysis	Partial Rank Correlation Coefficient, Standardized Regression Coefficients	Measure strength of relationships while controlling for other parameters	Screening analyses and monotonic relationship quantification
Visualization Tools	Sensitivity Heatmaps, Scatterplot Matrices, Interaction Networks	Communicate GSA results effectively to diverse audiences	Result interpretation and presentation to interdisciplinary teams

These computational tools form the essential "wet lab" equivalent for in silico sensitivity analysis, enabling researchers to implement robust GSA workflows without developing fundamental algorithms from scratch [96] [97].

Advanced GSA Frameworks and Emerging Approaches

Optimal Transport Theory for GSA

Recent advances in GSA methodology include the application of optimal transport theory to sensitivity analysis, particularly for energy systems models with potential applications in pharmaceutical manufacturing and bioprocess optimization [99]. This approach quantifies the influence of input parameters by measuring how perturbations in input distributions "transport" the output distribution, providing a comprehensive metric that captures both moment-based and shape-based changes in output distributions.

The optimal transport approach offers advantages in capturing complex changes in output distributions beyond variance alone, making it suitable for cases where output distributions may undergo significant shape changes rather than simple variance increases [99]. While this methodology has been primarily applied in energy systems, its mathematical foundation shows promise for pharmaceutical applications where output distribution shapes carry critical information about biological variability and risk assessment.

Regional Sensitivity Analysis for Local Effect Characterization

Regional Sensitivity Analysis (RSA) complements global approaches by examining parameter sensitivities within specific regions of the output space [100]. This technique is particularly valuable for identifying parameters that drive specific model behaviors of interest, such as:

• Bifurcation Analysis: Identifying parameters that push the system toward critical thresholds or phase changes
• Failure Region Identification: Determining which parameters drive the system toward failure states or undesirable outcomes
• Regime-Specific Sensitivities: Recognizing that parameter importance may vary across different operating regimes or biological conditions

The RSA workflow involves: (1) defining regions of interest in the output space, (2) applying statistical tests (e.g., Kolmogorov-Smirnov) to compare input distributions that lead to different output regions, and (3) quantifying the separation between conditional and unconditional input distributions [100].

Practical Implementation Considerations

Sampling Strategy Selection and Optimization

The sampling strategy forms the foundation of reliable GSA, with significant implications for both computational efficiency and result accuracy. The following diagram illustrates the relationship between different sampling methods and their positioning in the GSA workflow:

Sampling Size Determination Appropriate sample size depends on multiple factors including model complexity, parameter dimensionality, and the specific GSA method employed. As a general guideline:

• Screening Methods: r×(k+1) model runs, where r ranges from 10 to 50 and k is parameter count [95]
• Variance-Based Methods: N×(2k+2) model runs for accurate Sobol index estimation, where N typically ranges from 500 to several thousand [97]
• Sampling-Based Methods: N model runs, where N should be at least 10×k for reasonable correlation estimates [96]

Handling Stochastic Models and Aleatory Uncertainty

Pharmaceutical models often incorporate stochastic elements to represent biological variability, measurement error, or stochastic processes. Traditional GSA methods require adaptation for such models, where output uncertainty arises from both parametric uncertainty (epistemic) and inherent randomness (aleatory) [96].

Two-Stage Sampling Approach

• Inner Loop: For each parameter combination, perform multiple replications with different random seeds to characterize output distribution
• Outer Loop: Sample parameter values using standard GSA sampling techniques
• Output Aggregation: Compute summary statistics (mean, variance, quantiles) across replications for each parameter combination
• Sensitivity Assessment: Apply GSA methods to the relationship between parameters and output statistics

This approach effectively separates the contributions of parametric uncertainty and inherent variability, providing a more nuanced understanding of uncertainty sources in stochastic models [96].

Global Sensitivity Analysis represents an indispensable methodology within uncertainty quantification for computational models in pharmaceutical research and drug development. The structured frameworks presented in this protocol provide researchers with systematic approaches for identifying key uncertainty contributors across various model types and complexity levels. By implementing these GSA methodologies, researchers can prioritize parameter estimation efforts, guide experimental design, and enhance the reliability of model predictions in drug development pipelines.

The choice of specific GSA method should be guided by model characteristics, computational constraints, and the specific research questions being addressed. For high-dimensional models, the two-step approach combining Morris screening with variance-based methods provides an optimal balance between comprehensiveness and computational efficiency. As computational models continue to increase in complexity and impact within pharmaceutical development, robust sensitivity analysis will remain critical for model credibility and informed decision-making.

Physics-Enhanced Machine Learning (PEML), also referred to as scientific machine learning or grey-box modeling, represents a fundamental shift in computational science by integrating physical knowledge with data-driven approaches. This paradigm addresses critical limitations of purely data-driven models, including poor generalization performance, physically inconsistent predictions, and inability to quantify uncertainties effectively [101] [102]. PEML strategically incorporates physical information through various forms of biases—observational biases (e.g., data augmentation), inductive biases (e.g., physical constraints), learning biases (e.g., inference algorithm setup), and model form biases (e.g., terms describing partially known physics) [103].

Within computational drug discovery, PEML provides a robust framework for uncertainty quantification (UQ) by constraining the space of admissible solutions to those that are physically plausible, even with limited data [101]. This capability is particularly valuable in pharmaceutical research where experimental data is often scarce, expensive to obtain, and subject to multiple sources of uncertainty. By embedding physical principles into machine learning architectures, PEML enables more reliable predictions of molecular properties, enhances trust in model outputs, and guides experimental design through improved uncertainty estimates [104] [9].

Quantitative Comparison of Uncertainty Quantification Methods

Performance Metrics for UQ Methods

Evaluating uncertainty quantification methods requires specialized metrics that assess both ranking ability (correlation between uncertainty and error) and calibration ability (accurate estimation of error distribution) [9]. The pharmaceutical and computational chemistry communities have adopted several standardized metrics:

• Spearman Rank Correlation: Measures how well the estimated uncertainty ranks predictions by their error [104].
• ROC AUC: Evaluates how well uncertainty scores separate correct from incorrect predictions in classification tasks [9].
• σ-difference: Quantifies the separation in uncertainty values between correct and incorrect predictions [104].
• Expected Normalized Calibration Error (ENCE): Assesses how well the predicted confidence intervals match the actual observed error distribution [104].

Comparative Performance of UQ Methods

Table 1: Performance comparison of UQ methods across different data splitting strategies in molecular property prediction (adapted from [104])

UQ Method	Friendly Split (Spearman)	Scaffold Split (Spearman)	Random Split (Spearman)	Strengths	Limitations
GP-DNR	0.72	0.68	0.75	Robust across splits; handles high local roughness	Requires DNR calculation
Gaussian Process (GP)	0.61	0.55	0.64	Native uncertainty; theoretical foundations	Struggles with complex SAR
Model Ensemble	0.58	0.52	0.60	Simple implementation; parallel training	Computationally expensive
MC Dropout	0.54	0.49	0.56	Minimal implementation changes	Can underestimate uncertainty
Evidence Regression	0.50	0.45	0.53	Direct uncertainty estimation	Can be over-conservative

Table 2: Taxonomy of UQ methods used in drug discovery applications (based on [9])

UQ Category	Core Principle	Representative Methods	Uncertainty Type Captured	Application Examples
Similarity-based	Reliability depends on similarity to training data	Box Bounding, Convex Hull, k-NN Distance	Primarily Epistemic	Virtual screening, toxicity prediction
Bayesian	Treats parameters and outputs as random variables	Bayes by Backprop, Stochastic Gradient Langevin Dynamics	Epistemic & Aleatoric	Protein-ligand interaction prediction
Ensemble-based	Consistency across multiple models indicates confidence	Bootstrap Ensembles, Random Forests	Primarily Epistemic	Molecular property prediction, ADMET
Hybrid PEML	Integrates physical constraints with data-driven UQ	GP-DNR, Physics-Informed NN with UQ	Epistemic & Aleatoric	Lead optimization, active learning

The performance comparison reveals that the GP-DNR method, which explicitly incorporates local roughness information (a form of physical bias), consistently outperforms other approaches across different data splitting scenarios [104]. This demonstrates the value of integrating domain-specific physical knowledge into uncertainty quantification frameworks. On average, GP-DNR achieved approximately 17% improvement in rank correlation, 10% improvement in ROC AUC, 50% improvement in σ-difference, and 65% improvement in calibration error compared to the next best method [104].

Protocols for PEML Implementation in Drug Discovery

Protocol 1: GP-DNR for Molecular Property Prediction

Background: The GP-DNR (Gaussian Process with Different Neighbor Ratio) method addresses the challenge of quantifying uncertainty in regions of high local roughness within the chemical space, where the structure-activity relationship (SAR) changes rapidly [104].

Materials:

• Molecular structures (SMILES strings or 2D/3D representations)
• Experimental activity/property data
• Computational environment: Python with RDKit, GPyTorch or scikit-learn
• Morgan fingerprints (radius 2, 2048 bits) for molecular representation

Procedure:

• Data Preprocessing:
  • Encode molecules as Morgan fingerprints.
  • Calculate Tanimoto similarity matrix between all compounds.
  • Split dataset using appropriate strategy (friendly, scaffold, or random split).

• DNR Calculation:

  • For each molecule i, identify neighbors within Tanimoto similarity threshold (typically 0.4).
  • Calculate DNR as the proportion of neighbors with significant activity differences (e.g., >2 pIC50 units): DNR_i = count(|y_i - y_j| > threshold) / total_neighbors [104].

• Model Training:

  • Train a Gaussian Process regression model using the molecular fingerprints as inputs and experimental activities as targets.
  • Incorporate the DNR as an additional input feature or use it to modulate the noise parameter in the GP kernel.

• Uncertainty Quantification:

  • The predictive variance from the GP represents the base uncertainty.
  • Combine with DNR-based uncertainty: Total_uncertainty = GP_variance + λ * DNR where λ is a scaling parameter [104].

• Model Validation:

  • Evaluate using Spearman correlation between prediction errors and uncertainty estimates.
  • Assess calibration using expected normalized calibration error (ENCE).

Troubleshooting:

• If DNR values are uniformly low, adjust the similarity threshold or activity difference criterion.
• If GP training is computationally expensive, use sparse Gaussian Process approximations.
• For small datasets, consider Bayesian neural networks as an alternative to GP.

Protocol 2: PEML with Censored Regression Labels

Background: In pharmaceutical experimentation, precise measurements are often unavailable for compounds with very high or low activity, resulting in censored data (e.g., ">10μM" or "<1nM") [8] [105]. Standard UQ methods cannot utilize this partial information.

Materials:

• Experimental data with precise measurements and censored labels
• Python with PyTorch and TensorFlow Probability
• Access to Bayesian inference tools (Pyro, Stan, or equivalent)

Procedure:

• Data Preparation:
  • Identify censored data points and specify censoring type (left, right, or interval censoring).
  • For right-censored data (>value), the true value is known to be greater than the reported value.
  • For left-censored data (<value), the true value is known to be less than the reported value.

• Model Adaptation:

  • Implement the Tobit model for handling censored data within the machine learning framework: y_observed = { y_latent if y_latent ∈ [c_l, c_u], c_l if y_latent < c_l, c_u if y_latent > c_u } where y_latent is the true unobserved activity [8].
  • Modify the loss function to account for censoring using the negative log-likelihood for censored data.

• Uncertainty-Aware Training:

  • For ensemble methods: Train multiple models on bootstrapped samples, each implementing the censored data likelihood.
  • For Bayesian methods: Use variational inference or Markov Chain Monte Carlo (MCMC) to approximate the posterior distribution of parameters given both precise and censored observations.

• Inference and UQ:

  • For prediction, the model provides full posterior distributions accounting for censoring.
  • Epistemic uncertainty is captured through ensemble disagreement or posterior variance.
  • Aleatoric uncertainty is captured through the noise estimate in the Tobit model.

• Validation:

  • Use temporal validation where models trained on earlier data predict later compounds [8].
  • Compare with models ignoring censoring using negative log-likelihood on holdout test data.

Troubleshooting:

• If model fails to converge, check censoring specification and consider increasing model capacity.
• For highly censored datasets (>30% censored points), prioritize methods specifically designed for censored data [8].
• Validate censoring assumptions with domain experts to ensure appropriate model specification.

Workflow Visualization

PEML-UQ Integrated Workflow: The diagram illustrates the integration of physical biases (DNR, censored data handling) with machine learning for enhanced uncertainty quantification in drug discovery.

UQ Method Taxonomy: Classification of uncertainty quantification methods highlighting the position of PEML-enhanced approaches as integrating multiple uncertainty types.

Table 3: Essential research reagents and computational resources for PEML in drug discovery

Category	Item	Specifications	Application/Function
Data Resources	Morgan Fingerprints	Radius 2, 2048 bits	Molecular representation capturing substructure features [104]
	Censored Activity Data	>10μM, <1nM thresholds	Partial information from solubility/toxicity assays [8]
	Temporal Dataset Split	Time-based validation	Real-world model performance assessment [8]
Computational Tools	Gaussian Process Libraries	GPyTorch, scikit-learn	Probabilistic modeling with native uncertainty [104]
	Deep Learning Frameworks	PyTorch, TensorFlow	Flexible model implementation [8]
	Bayesian Inference Tools	Pyro, Stan, TensorFlow Probability	Posterior estimation for UQ [9]
UQ Methodologies	DNR Metric	Tanimoto similarity >0.4, activity difference >2 pIC50	Quantifies local roughness in chemical space [104]
	Tobit Model	Censored regression likelihood	Incorporates partial information from censored data [8]
	Ensemble Methods	5-10 models, diverse architectures	Captures model uncertainty through prediction variance [9]
Validation Metrics	Spearman Correlation	Rank correlation error vs. uncertainty	Assesses UQ ranking capability [104]
	Expected Normalized Calibration Error (ENCE)	Calibration between predicted and observed errors	Evaluates uncertainty reliability [104]
	ROC AUC	Separation of correct/incorrect predictions	Measures classification uncertainty quality [9]

Applications in Drug Discovery

Active Learning for Lead Optimization

PEML-enhanced UQ enables efficient active learning cycles in lead optimization. By identifying compounds with high epistemic uncertainty (representing novelty in chemical space), models can prioritize which compounds to synthesize and test experimentally [104] [9]. Research demonstrates that GP-DNR-guided selection significantly outperforms both random selection and standard GP uncertainty, achieving substantial reduction in prediction error with the same experimental budget [104]. In one implementation, adding only 10% of candidate compounds selected by GP-DNR produced significant MSE reduction, whereas standard GP uncertainty performed similarly to random selection [104].

Handling Censored Data in Early Discovery

During early-stage screening, approximately one-third or more of experimental labels may be censored [8]. Traditional machine learning models discard this valuable information, while PEML approaches specifically adapted for censored regression (e.g., using the Tobit model) can leverage these partial observations. Studies show that models incorporating censored labels provide more reliable uncertainty estimates, particularly for compounds with extreme property values that often represent the most promising or problematic candidates [8] [105].

Uncertainty-Decomposed Decision Support

Different types of uncertainty inform different decisions in drug discovery. High epistemic uncertainty suggests collecting more data in underrepresented chemical regions, while high aleatoric uncertainty indicates inherent measurement noise or complex SAR that may require alternative molecular designs [9]. PEML facilitates this decomposition, enabling nuanced decision support. For instance, in ADMET prediction, well-calibrated uncertainty estimates help researchers balance potency with desirable pharmacokinetic properties while understanding prediction reliability [104] [9].

Physics-Enhanced Machine Learning represents a paradigm shift in uncertainty quantification for computational drug discovery. By integrating physical biases—whether through local roughness measures like DNR, censored data handling, or other domain knowledge—PEML addresses fundamental limitations of purely data-driven approaches. The protocols and methodologies outlined provide researchers with practical frameworks for implementing PEML-UQ strategies that enhance model reliability, guide experimental design, and ultimately accelerate the drug discovery process. As these methods continue to evolve, they promise to further bridge the gap between computational predictions and experimental reality, enabling more efficient and informed decision-making in pharmaceutical research and development.

Handling High-Dimensional Parameter Spaces in Biomedical Models

Mathematical models in immunology and systems biology, such as those describing T cell receptor (TCR) or B cell antigen receptor (BCR) signaling networks, provide powerful frameworks for understanding complex biological processes [106]. These models typically encompass numerous protein-protein interactions, each characterized by one or more unknown kinetic parameters. A model covering even a subset of known interactions may contain tens to hundreds of unknown parameters that must be estimated from experimental data [106]. This high-dimensional parameter space presents significant challenges for parameter estimation and uncertainty quantification (UQ), which are essential for producing reliable, predictive models. The computational burden is further compounded by the potentially large state space (number of chemical species) in models derived from rule-based frameworks, making simulations computationally demanding [106]. This application note addresses these challenges by providing detailed protocols for parameter estimation and UQ, specifically tailored for high-dimensional biomedical models within the broader context of uncertainty quantification for computational models research.

Foundational Concepts and Formulations

Model Specification Standards

Proper model specification is a critical first step in ensuring compatibility with parameter estimation tools. We recommend using standardized formats to enable interoperability with general-purpose software tools:

• BioNetGen Language (BNGL): Particularly useful for rule-based modeling of biomolecular site dynamics, which are common in immunoreceptor signaling systems [106].
• Systems Biology Markup Language (SBML): Widely supported software ecosystem; the Level 3 core specification is recommended, with extension packages used judiciously due to limited support [106].

Conversion tools are available to translate BNGL models to SBML, allowing BNGL models to benefit from SBML-compatible parameterization tools [106].

Objective Function Formulation

The parameter estimation problem is fundamentally an optimization problem that minimizes a chosen objective function measuring the discrepancy between experimental data and model simulations. A common and statistically rigorous choice is the chi-squared objective function:

[ \chi^2(\theta) = \sum{i} \omegai (yi - \hat{y}i(\theta))^2 ]

where (yi) are experimental measurements, (\hat{y}i(\theta)) are corresponding model predictions parameterized by (\theta), and weights (\omegai) are typically chosen as (1/\sigmai^2) with (\sigmai^2) representing the sample variance associated with (yi) [106]. This formulation appropriately weights residuals based on measurement precision.

Parameter Estimation Methodologies

Optimization Algorithms

Table 1: Comparison of Parameter Estimation Methods for High-Dimensional Problems

Method Class	Specific Algorithms	Key Features	Computational Considerations	Ideal Use Cases
Gradient-Based	L-BFGS-B [106], Levenberg-Marquardt [106]	Uses gradient information; fast local convergence; second-order methods avoid saddle points	Requires efficient gradient computation; multiple starts needed for global optimization	Models with computable gradients; medium-scale parameter spaces
Gradient-Free Metaheuristics	Genetic algorithms, particle swarm optimization [106]	No gradient required; global search capability; handles non-smooth objectives	Computationally expensive; requires many function evaluations; convergence not guaranteed	Complex, multi-modal objectives; initial global exploration
Hybrid Approaches	Multi-start with gradient refinement	Combines global search with local refinement	Balanced computational cost	Production-level parameter estimation

Gradient Computation Techniques

For gradient-based optimization, efficient computation of the objective function gradient with respect to parameters is essential. The following table compares approaches:

Table 2: Gradient Computation Methods for ODE-Based Biological Models

Method	Implementation Complexity	Computational Cost	Accuracy	Software Support
Finite Difference	Low	High for many parameters (O(p) simulations)	Approximate, sensitive to step size	Universal
Forward Sensitivity	Medium	High for many parameters/ODEs (solves p×n ODEs)	Exact for ODE models	AMICI [106], COPASI [106]
Adjoint Sensitivity	High	Efficient for many parameters (solves ~n ODEs)	Exact for ODE models	Limited [106]
Automatic Differentiation	Low (user perspective)	Varies; can be inefficient for large, stiff ODEs [106]	Exact	Stan [106]

Protocol 3.2.1: Adjoint Sensitivity Analysis for Large ODE Models

• Model Preparation: Formulate your biological model as a system of ODEs ( \frac{dx}{dt} = f(x,t,\theta) ) with initial conditions ( x(0) = x_0 ), where ( x ) is the state vector and ( \theta ) is the parameter vector.
• Objective Function Definition: Define the objective function as a sum of squared residuals between model predictions and experimental data: ( J(\theta) = \sum{k=1}^{N} (y(tk) - \hat{y}_k)^2 ), where ( y(t) = g(x(t,\theta)) ) is the observation function.
• Adjoint System Definition: Define the adjoint system ( \frac{d\lambda}{dt} = -\left(\frac{\partial f}{\partial x}\right)^T \lambda - \left(\frac{\partial g}{\partial x}\right)^T \rho ), where ( \rhok = 2(y(tk) - \hat{y}_k) ) at observation times.
• Backward Integration: Integrate the adjoint system backward in time from ( tN ) to ( t0 ).
• Gradient Computation: Compute the gradient as ( \frac{dJ}{d\theta} = -\int{t0}^{t_N} \lambda^T \frac{\partial f}{\partial \theta} dt ).
• Validation: Verify gradient accuracy against finite differences for a subset of parameters.

Uncertainty Quantification Frameworks

UQ Methodologies

Table 3: Uncertainty Quantification Methods for Parameter Estimates

Method	Theoretical Basis	Computational Demand	Information Gained	Implementation
Profile Likelihood	Likelihood theory [106]	Medium (1D re-optimization)	Parameter identifiability, confidence intervals	PESTO [106], Data2Dynamics [106]
Bootstrapping	Resampling statistics [106]	High (hundreds of resamples)	Empirical confidence intervals	PyBioNetFit [106], custom code
Bayesian Inference	Bayes' theorem [106]	Very high (MCMC sampling)	Full posterior distribution, model evidence	Stan [106], PyBioNetFit

Practical UQ Protocol

Protocol 4.2.1: Comprehensive Uncertainty Quantification

• Structural Identifiability Analysis:

  • Check if parameters are theoretically identifiable from the perfect, noise-free data.
  • Use symbolic computation (e.g., differential algebra) or profile likelihood approach.
  • For non-identifiable parameters, consider model reduction or reparameterization.

• Practical Identifiability Assessment:

  • Compute profile likelihoods for each parameter by constrained optimization.
  • Determine likelihood-based confidence intervals from profiles.
  • Identify practically non-identifiable parameters (flat profiles).

• Parameter Confidence Estimation:

  • Option A (Profile Likelihood): Use the threshold ( \chi^2(\thetai) - \chi^2(\hat{\theta}) < \Delta{\alpha} ) where ( \Delta_{\alpha} ) is the ( \alpha )-quantile of the χ²-distribution.
  • Option B (Bootstrapping): Generate parametric bootstrap samples by adding noise to best-fit simulations; re-estimate parameters for each sample.
  • Option C (Bayesian): Implement Markov Chain Monte Carlo (MCMC) sampling with appropriate priors.

• Prediction Uncertainty Quantification:

  • Propagate parameter uncertainties to model predictions using sampled parameter sets (from bootstrapping or Bayesian posterior).
  • Report prediction intervals alongside point predictions in results.

The Scientist's Toolkit

Table 4: Essential Research Reagent Solutions for Biomedical Model Parameterization

Tool/Category	Specific Examples	Primary Function	Key Applications
Integrated Software Suites	COPASI [106], Data2Dynamics [106]	All-in-one modeling, simulation, and parameter estimation	General systems biology models; ODE-based signaling pathways
Specialized Parameter Estimation Tools	PESTO [106] (with AMICI), PyBioNetFit [106]	Advanced parameter estimation and UQ	High-dimensional models; rule-based models; profile likelihood analysis
Model Specification Tools	BioNetGen [106], SBML-supported tools	Rule-based model specification; standardized model exchange	Large-scale signaling networks; immunoreceptor models
High-Performance Simulation	AMICI [106], NFsim [106]	Fast simulation of ODE/stochastic models	Large ODE systems; network-free simulation of rule-based models
Statistical Inference Environments	Stan [106]	Bayesian inference with MCMC sampling	Bayesian parameter estimation; hierarchical models

Integrated Workflow and Visualization

The following workflow diagram illustrates the complete parameter estimation and UQ process for high-dimensional biomedical models:

Integrated Workflow for Parameter Estimation and UQ

Special Considerations for High-Dimensional Data

High-dimensional data (HDD) settings, where the number of variables (p) associated with each observation is very large, present unique statistical challenges that extend to parameter estimation in mechanistic models [107]. In biomedical contexts, prominent examples include various omics data (genomics, transcriptomics, proteomics, metabolomics) and electronic health records data [107]. Key considerations include:

• Multiple Testing: When performing statistical tests across many parameters, stringent multiplicity adjustments are required to control false positive findings [107].
• Sample Size Limitations: Standard sample size calculations generally do not apply in HDD settings, and studies are often conducted with inadequate sample size, leading to non-reproducible results [107].
• Dependence Structures: Parameters in biological models are often not independent, requiring methods that account for complex dependence structures [108].
• Computational Complexity: The "large p" scenario presents difficulties for computation and requires specialized algorithms [107] [108].

Protocol 7.1: Handling High-Dimensional Parameter Spaces

• Regularization:

  • Incorporate regularization terms (L₁/L₂ penalty) into the objective function to handle parameter collinearity.
  • Use L₁ regularization (lasso) for parameter selection in overly complex models.

• Parameter Subspace Identification:

  • Perform sensitivity analysis to identify insensitive parameters.
  • Fix insensitive parameters at nominal values to reduce effective parameter space dimension.

• Sequential Estimation:

  • Estimate subsets of parameters from appropriate, targeted datasets.
  • Integrate subset estimates as initial guesses for full model estimation.

• Dimension Reduction:

  • Apply principal component analysis or related techniques to experimental data.
  • Estimate parameters against reduced-dimensionality data representations.

Applications to Immunoreceptor Signaling Models

The methodologies described herein have been successfully applied to systems-level modeling of immune-related phenomena, particularly immunoreceptor signaling networks [106]. These applications include:

• T Cell Receptor (TCR) Signaling: Parameterization of early TCR signaling events, including phosphorylation kinetics and adapter protein recruitment [106].
• B Cell Antigen Receptor (BCR) Signaling: Quantification of signaling thresholds and feedback mechanisms in BCR activation [106].
• FcϵRI Receptor Signaling: Characterization of allergen-induced mast cell activation through FcϵRI signaling cascades [106].

These applications demonstrate the feasibility of the presented protocols for parameterizing biologically realistic models of immunoreceptor signaling, despite the challenges posed by high-dimensional parameter spaces.

Ensuring Model Credibility: VVUQ Frameworks and Comparative Analysis

The adoption of digital twins in precision medicine represents a paradigm shift towards highly personalized healthcare. Defined as a set of virtual information constructs that mimic the structure, context, and behavior of a natural system, dynamically updated with data from its physical counterpart, digital twins offer predictive capabilities that inform decision-making to realize value [109]. In clinical contexts, this involves creating computational models tailored to individuals' unique physiological characteristics and lifestyle behaviors, enabling precise health assessments, accurate diagnoses, and personalized treatment strategies through simulation of various health scenarios [109].

The critical framework ensuring safety and efficacy of these systems is Verification, Validation, and Uncertainty Quantification (VVUQ). When dealing with patient health, trust in the underlying processes is paramount and influences acceptance by regulatory bodies like the FDA and healthcare professionals [109]. VVUQ provides the methodological foundation for building this essential trust. Verification ensures that computational models are correctly implemented, validation tests whether models accurately represent real-world phenomena, and uncertainty quantification characterizes the limitations and confidence in model predictions [109] [10].

Foundational Concepts and Uncertainty Taxonomy

Uncertainty Classification in Digital Twins

Uncertainty in digital twins is categorized into two fundamental types, each requiring distinct quantification approaches [110]:

• Aleatoric uncertainty arises from inherent, irreducible variability in natural systems. Examples include intrinsic sensor noise, physiological fluctuations, and environmental variations in manufacturing. This variability persists regardless of data collection efforts.
• Epistemic uncertainty stems from limited knowledge, data, or understanding of the system being modeled. This includes discrepancies between simulation and experimental data, numerical errors in machine learning models, and gaps in scientific knowledge. Unlike aleatoric uncertainty, epistemic uncertainty can be reduced through additional data collection and model refinement.

Table 1: Uncertainty Types and Their Characteristics in Medical Digital Twins

Uncertainty Type	Origin	Reducibility	Examples in Medical Digital Twins
Aleatoric	Inherent system variability	Irreducible	Physiological fluctuations, sensor noise, genetic expression variability [110]
Epistemic	Limited knowledge/data	Reducible	Model-form error, parametric uncertainty, limited patient data [110] [109]

VVUQ Components and Definitions

The VVUQ framework comprises three interconnected processes essential for establishing digital twin credibility [109]:

• Verification: The process of ensuring that software or system components perform as intended through code solution verification. This involves checking that the computational model correctly solves the intended mathematical equations.
• Validation: Testing models for applicability and accuracy in representing real-world scenarios. This determines under what conditions—such as specific disease types or treatment regimens—model predictions can be trusted.
• Uncertainty Quantification: The formal process of tracking uncertainties throughout model calibration, simulation, and prediction. UQ provides confidence bounds that demonstrate the degree of confidence one should have in predictions, encompassing both aleatoric and epistemic uncertainties [109].

VVUQ Methodologies and Computational Approaches

Uncertainty Quantification Techniques

Multiple computational methods exist for propagating and analyzing uncertainties in complex models. The choice of method depends on the uncertainty type (aleatoric or epistemic) and the model's computational demands [16].

Table 2: Uncertainty Quantification Methods for Digital Twins

Method Category	Specific Methods	Applicable Uncertainty Type	Key Features
Sampling Methods	Monte Carlo, Latin Hypercube Sampling (LHS)	Aleatory	Simple implementation, handles complex models, computationally intensive [16]
Reliability Methods	FORM, SORM, AMV	Aleatory	Efficient for estimating low probabilities, local approximations [16]
Stochastic Expansions	Polynomial Chaos, Stochastic Collocation	Aleatory	Functional representation of uncertainty, efficient with smooth responses [16]
Interval Methods	Interval Analysis, Global/Local Optimization	Epistemic	No distributional assumptions, produces bounds on outputs [16]
Evidence Theory	Dempster-Shafer Theory	Epistemic (Mixed)	Handles incomplete information, produces belief/plausibility measures [16]
Bayesian Methods	Bayesian Calibration, Inference	Both	Updates prior knowledge with data, produces posterior distributions [16]

Machine Learning for UQ in Complex Systems

Machine learning approaches are increasingly important for UQ in digital twins, particularly when dealing with complex, high-dimensional data [110]. Different ML architectures are suited to different data types:

• Gaussian process models are effective for sparse data scenarios commonly encountered in rare diseases or novel treatments.
• Recursive neural networks handle time series data such as continuous physiological monitoring streams.
• Graph neural networks apply to multidimensional applications including molecular interactions and cellular networks.
• Convolutional neural networks process image data such as medical imaging and histopathology slides [110].

For quantifying uncertainty in ML models, Bayesian approaches including Monte Carlo Dropout and Laplace Approximation are particularly amenable to digital twin applications [110]. Recent research has also focused on developing specialized uncertainty metrics for specific data types, such as ordinal classification in medical assessments, where traditional measures like Shannon entropy may be inappropriate [111].

Experimental Protocols for VVUQ in Medical Digital Twins

Protocol 1: VVUQ Framework Implementation for Cardiac Digital Twins

This protocol outlines the procedure for establishing a VVUQ pipeline for cardiac electrophysiological digital twins, used for diagnosing arrhythmias like atrial fibrillation [109].

Workflow Diagram: VVUQ for Cardiac Digital Twins

Materials and Reagents:

• Clinical Imaging Data: Cardiac CT or MRI DICOM images for anatomical reconstruction
• Electrophysiological Recording System: Surface ECG and intracardiac electrogram data
• Computational Platform: High-performance computing resources with cardiac simulation software (e.g., OpenCARP, Simula)
• UQ Software Tools: Dakota UQ toolkit or custom Bayesian inference algorithms [16]

Procedure:

• Anatomical Model Construction
  • Segment cardiac structures from CT/MRI DICOM images to create 3D geometric mesh
  • Incorporate patient-specific tissue properties based on imaging characteristics
  • Document mesh resolution and quality metrics for verification

• Model Personalization and Calibration

  • Adjust electrophysiological parameters to match patient-specific ECG measurements
  • Implement Bayesian calibration to infer parameter distributions consistent with data
  • Record calibrated parameter values and their associated uncertainties

• Verification Procedures

  • Perform code verification to ensure numerical solvers correctly implement mathematical models
  • Conduct solution verification to quantify numerical errors from discretization
  • Document convergence tests for spatial and temporal discretization

• Validation Against Clinical Data

  • Compare model predictions of cardiac electrical activity with measured electrograms
  • Validate against clinical endpoints (e.g., arrhythmia inducibility)
  • Quantitative metrics: Root-mean-square error < 5% of signal amplitude

• Uncertainty Quantification and Propagation

  • Propagate parameter uncertainties through model to predict confidence bounds on outputs
  • Perform sensitivity analysis to identify dominant uncertainty sources
  • Calculate uncertainty intervals for clinical predictions (e.g., tachycardia risk)

• Clinical Implementation and Updating

  • Implement model in clinical workflow for treatment planning
  • Establish schedule for model re-validation with new patient data
  • Update model parameters as new electrophysiological data becomes available

Protocol 2: Multi-scale Cancer Digital Twin Verification

This protocol details VVUQ procedures for multi-scale cancer digital twins integrating cellular systems biology with tissue-level agent-based models for predicting tumor response to therapies [112].

Workflow Diagram: Multi-scale Cancer Digital Twin Framework

Materials and Reagents:

• Multi-omics Data: Genomic, transcriptomic, and proteomic profiles from tumor biopsies
• Medical Imaging: Histopathology slides, MRI, or PET-CT scans for spatial validation
• Computational Resources: High-performance computing cluster for agent-based simulation
• ML Frameworks: TensorFlow or PyTorch for surrogate model development
• UQ Tools: Dakota or custom implementations for sensitivity analysis [16]

Procedure:

• Cellular Systems Biology Modeling
  • Implement mechanistic models of key cancer pathways (e.g., ErbB signaling, p53-mediated DNA damage response)
  • Calibrate pathway models using patient-specific multi-omics data
  • Document parameter ranges and their biological plausibility

• Tissue-Level Agent-Based Model Development

  • Develop rules for cell-agent behaviors (proliferation, death, migration) based on cellular models
  • Incorporate patient-specific tumor microenvironment from imaging data
  • Implement spatial constraints and nutrient gradients

• Multi-scale Model Integration

  • Connect cellular pathway models to agent-based rules
  • Establish communication protocols between modeling scales
  • Verify consistent variable transfer between scales

• Machine Learning Surrogate Development

  • Train neural network surrogates to emulate expensive ABM simulations
  • Validate surrogate accuracy against full model outputs (target R² > 0.9)
  • Use surrogates for global sensitivity analysis and rapid UQ

• Verification and Validation

  • Code Verification: Unit testing for individual model components
  • Solution Verification: Mesh convergence for spatial discretization
  • Validation: Compare spatiotemporal tumor growth predictions against clinical imaging and histology
  • Temporal Validation: Establish schedule for re-validation as tumor evolves

• Uncertainty Quantification

  • Perform global sensitivity analysis to identify dominant uncertainty sources
  • Quantify uncertainty in treatment response predictions
  • Propagate parameter uncertainties through multi-scale framework
  • Generate confidence intervals for clinical predictions (e.g., progression-free survival)

Research Reagents and Computational Tools

Table 3: Essential Research Reagents and Computational Tools for Medical Digital Twin VVUQ

Category	Item	Specifications	Application in VVUQ
Clinical Data	Multi-omics Profiles	Genomics, transcriptomics, proteomics from biopsies	Model personalization and validation [112]
Medical Imaging	Cardiac CT/MRI	DICOM format, 1mm resolution or better	Anatomical model construction [109]
Biosensors	Wearable Monitors	Clinical-grade, real-time data streaming	Dynamic model updating [109]
UQ Software	Dakota Toolkit	SNL-developed, v6.19.0 or newer	Uncertainty propagation and sensitivity analysis [16]
ML Libraries	TensorFlow/PyTorch	With probabilistic layers	Bayesian neural networks for UQ [110]
Modeling Frameworks	Agent-Based Platforms	NetLogo, CompuCell3D	Tissue-level cancer modeling [112]
Cardiac Simulators	OpenCARP	Open-source platform	Cardiac electrophysiology simulation [109]

Signaling Pathways for Medical Digital Twins

Key Signaling Networks in Cancer Digital Twins

Several well-characterized signaling pathways form the foundation for mechanistic models in cancer digital twins:

• ErbB Receptor-Mediated Signaling: Ras-MAPK and PI3K-AKT pathways regulating cell growth and proliferation, frequently dysregulated in multiple cancer types [112]
• p53-Mediated DNA Damage Response: Pathway controlling cell cycle arrest and apoptosis in response to cellular stress, crucial for predicting treatment response [112]
• Cross-talk Networks: Integrated pathway models, such as PI3K-androgen receptor interactions in prostate cancer, essential for understanding therapeutic resistance [112]

GPCR Signaling Networks

G protein-coupled receptors (GPCRs) represent key therapeutic targets in cardiovascular, neurological, and metabolic disorders. Digital twins for precision GPCR medicine integrate genomic, proteomic, and real-time physiological data to create patient-specific virtual models for optimizing receptor-targeted therapies [113].

Signaling Pathway Diagram: GPCR Digital Twin Framework

Implementation Challenges and Future Directions

While VVUQ provides a rigorous framework for digital twin credibility, significant challenges remain in clinical implementation. A major research gap identified in the National Academies report is the need for standardized procedures to build trustworthiness in medical digital twins [109]. Key challenges include:

• Temporal Validation: Determining appropriate schedules for re-validating digital twins as patient conditions evolve [109]
• Regulatory Acceptance: Establishing VVUQ standards that meet FDA requirements for clinical decision support [109]
• Computational Complexity: Managing the significant computational demands of UQ for multi-scale models in clinical timeframes [112]
• Ethical Considerations: Addressing data privacy, equitable access, and transparency in model limitations [113]

Future directions focus on developing personalized trial methodologies, standardized validation metrics, and automated VVUQ processes that can keep pace with real-time data streams from biosensors [109]. The integration of AI explainability with mechanistic models and VVUQ is likely to create new opportunities for risk assessment that are not readily available today [109]. As these frameworks mature, VVUQ will enable digital twins to become reliable tools for simulating interventions and personalizing therapeutic strategies at an unprecedented level of precision.

Code and Solution Verification for Computational Models

Verification, Validation, and Uncertainty Quantification (VVUQ) forms a critical framework for establishing the credibility of computational models. Within this framework, code and solution verification are foundational processes that ensure mathematical models are solved correctly and accurately. This document details application notes and protocols for verification, framed within broader Uncertainty Quantification (UQ) research for computational models. It provides researchers, scientists, and drug development professionals with standardized methodologies to assess and improve the reliability of their simulations, a necessity in fields where predictive accuracy impacts critical decisions from material design to therapeutic development [6].

The discipline of VVUQ is supported by standards from organizations like ASME, which define verification as the process of determining that a computational model correctly implements the intended mathematical model and its solution. Solution verification specifically assesses the numerical accuracy of the obtained solution [6]. This is distinct from validation, which concerns the model's accuracy in representing real-world phenomena.

Standard Definitions and Terminology

Adherence to standardized terminology is essential for clear communication and reproducibility in computational science. The following table defines key terms as established by leading standards bodies like ASME.

Table 1: Standard VVUQ Terminology

Term	Formal Definition	Context of Use
Verification	Process of determining that a computational model accurately represents the underlying mathematical model and its solution [6].	Assessing code correctness and numerical solution accuracy.
Solution Verification	The process of assessing the numerical accuracy of a computational solution [6].	Estimating numerical errors like discretization error.
Validation	Process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model [6].	Comparing computational results with experimental data.
Uncertainty Quantification (UQ)	The process of quantifying uncertainties in computational model outputs, typically stemming from uncertainties in inputs [33].	Propagating input parameter variances to output confidence intervals.
Experimental Standard Deviation of the Mean	An estimate of the standard deviation of the distribution of the arithmetic mean, given by ( s(\bar{x}) = s(x)/\sqrt{n} ) [114].	Reporting the statistical uncertainty of a simulated observable.

Quantitative Metrics and Acceptance Criteria

Quantifying numerical error is the cornerstone of solution verification. The following metrics are widely used to evaluate the convergence and accuracy of computational solutions.

Table 2: Key Metrics for Solution Verification

Metric	Formula/Description	Application Context	Acceptance Criterion
Grid Convergence Index (GCI)	Extrapolates error from multiple mesh resolutions to provide an error band. Based on Richardson Extrapolation [6].	Finite Element, Finite Volume, and Finite Difference methods.	GCI value below an application-dependent threshold (e.g., 5%).
Order of Accuracy (p)	Observed rate at which numerical error decreases with mesh refinement: ( \epsilon \propto h^p ), where ( h ) is a measure of grid size.	Verifying that the theoretical order of convergence of a numerical scheme is achieved.	Observed ( p ) matches the theoretical order of the discretization scheme.
Standard Uncertainty	Uncertainty in a result expressed as a standard deviation. For a mean, this is the experimental standard deviation of the mean [114].	Reporting the confidence interval for any simulated scalar observable.	Uncertainty is small relative to the magnitude of the quantity and its required predictive tolerance.

Experimental Protocols for Verification

Protocol 1: Code Verification via the Method of Manufactured Solutions (MMS)

Objective: To verify that the computational model solves the underlying mathematical equations correctly, free of coding errors.

Workflow:

• Manufacture a Solution: Choose a smooth, non-trivial analytical function ( u_{manufactured}(\vec{x}) ) for the solution field.
• Apply the Operator: Substitute ( u_{manufactured}(\vec{x}) ) into the governing partial differential equation (PDE) ( L(u) = 0 ). This will not equal zero, generating a residual source term ( R(\vec{x}) ).
• Modify the PDE: The new modified PDE is ( L(u) = R(\vec{x}) ).
• Run Simulation: Solve the modified PDE ( L(u) = R(\vec{x}) ) computationally, using ( u_{manufactured} ) as the boundary condition.
• Compute Error: Calculate the numerical error: ( \epsilon = u{computed} - u{manufactured} ).
• Assess Convergence: Refine the mesh/grid and observe the error ( \epsilon ). Correct code will show the error converging to zero at the theoretical order of the numerical scheme.

Protocol 2: Solution Verification via Grid Convergence Study

Objective: To quantify the numerical discretization error in a specific simulation result.

Workflow:

• Generate Solution Series: Run the simulation on three or more systematically refined grids (e.g., coarse, medium, fine). Record a key output observable ( f ) from each.
• Calculate Observed Order: Using the solutions ( f1, f2, f_3 ) from the fine, medium, and coarse grids, compute the observed order of accuracy ( p ).
• Perform Richardson Extrapolation: Use the fine and medium grid solutions and the observed order ( p ) to estimate the exact solution ( f_{exact} ).
• Compute Error Estimates: Calculate the relative error for each grid: ( \epsiloni = |(fi - f{exact}) / f{exact}| ).
• Report GCI: Calculate the Grid Convergence Index for the fine and medium solutions to report a formal error band. The solution is verified when the GCI is sufficiently small for the intended application.

Figure 1: Solution verification workflow for grid convergence.

Protocol 3: Uncertainty Quantification for Sampled Observables

Objective: To properly estimate and report the statistical uncertainty in observables derived from stochastic or correlated data (e.g., from Molecular Dynamics or Monte Carlo simulations).

Workflow:

• Run Simulation & Collect Data: Generate a single, long trajectory or multiple independent trajectories, recording the raw data for the observable of interest.
• Check for Equilibration/Discard Burn-in: Visually inspect the time series to identify and discard the initial non-equilibrated (burn-in) portion of the data.
• Assess Statistical Independence: Calculate the autocorrelation function or the integrated correlation time ( \tau ) for the observable. This quantifies how many steps apart data points must be to be considered independent.
• Compute Effective Sample Size (ESS): Estimate the number of independent samples as ( N_{eff} \approx N / (2\tau) ), where ( N ) is the total number of data points.
• Estimate Statistics: Calculate the arithmetic mean ( \bar{x} ) and the experimental standard deviation ( s(x) ) of the data.
• Report Standard Uncertainty: The final uncertainty for the reported mean is the experimental standard deviation of the mean, computed using the effective sample size: ( s(\bar{x}) = s(x) / \sqrt{N_{eff}} ) [114].

The Scientist's Toolkit: Research Reagent Solutions

This section details essential software tools and libraries that implement advanced UQ and verification methods.

Table 3: Key Software Tools for UQ and Verification

Tool Name	Primary Function	Application in Research
UncertainSCI	Open-source Python suite for non-intrusive forward UQ. Uses polynomial chaos (PC) emulators built via near-optimal sampling to propagate parametric uncertainty [67].	Efficiently computes output statistics (mean, variance, sensitivities) for biomedical models (e.g., cardiac bioelectric potentials) with limited forward model evaluations.
UQ Toolkit (UQTk)	A lightweight, open-source C++/Python library for uncertainty quantification developed at Sandia National Laboratories. Focuses on parameter sampling, sensitivity analysis, and Bayesian inference [33].	Provides modular tools for UQ workflows, including propagating input uncertainties and calibrating models against experimental data in fields like electrochemistry and materials science.
ASME V&V Standards	A series of published standards (e.g., V&V 10 for Solid Mechanics, V&V 20 for CFD) providing terminology and procedures for Verification and Validation [6].	Offers authoritative, domain-specific guidelines and benchmarks for performing and reporting code and solution verification studies.
Polynomial Chaos Emulators	Surrogate models that represent the input-output relationship of a complex model using orthogonal polynomials. Drastically reduce the cost of UQ studies [67].	Replaces computationally expensive simulation models to enable rapid uncertainty propagation, sensitivity analysis, and design optimization.

Integrated VVUQ Workflow for Computational Models

A robust VVUQ process integrates both verification and uncertainty quantification to fully establish model credibility. The following diagram illustrates the logical relationships and workflow between these components, from defining the mathematical model to making informed predictions.

Figure 2: Integrated VVUQ workflow for credible predictions.

Validation Metrics for Model Accuracy Assessment

In computational modeling, particularly for applications in drug development and engineering, validation metrics provide quantitative measures to assess the accuracy of model predictions against experimental reality. Unlike qualitative graphical comparisons, these computable measures sharpen the assessment of computational accuracy by statistically comparing computational results and experimental data over a range of input variables [115]. This protocol outlines the application of confidence interval-based validation metrics and classification accuracy assessments, providing researchers with standardized methodologies for uncertainty quantification in computational models.

Theoretical Foundation of Validation Metrics

Core Definitions and Concepts

Verification and Validation: Code verification ensures the mathematical model is solved correctly, while solution verification quantifies numerical accuracy. Validation assesses modeling accuracy by comparing computational results with experimental data [115].

Validation Metric: A computable measure that quantitatively compares computational results and experimental measurements, incorporating estimates of numerical error, experimental uncertainty, and input parameter uncertainties [115].

Confidence Intervals: Statistical ranges that likely contain the true value of a parameter, forming the basis for rigorous validation metrics [115].

Recommended Features of Validation Metrics

An effective validation metric should:

• Explicitly include or exclude numerical error in the system response quantity (SRQ)
• Incorporate experimental uncertainty in the SRQ
• Include input parameter uncertainties affecting the SRQ
• Provide an objective measure of agreement throughout the validation domain
• Be composable from multiple sources of uncertainty and applicable to multiple SRQs [115]

Confidence Interval-Based Validation Metrics

Pointwise Validation Metric

For a SRQ at a single operating condition, the validation metric estimates an interval containing the modeling error centered at the comparison error with width determined by validation uncertainty [116].

Let:

• (E) = experimental measurement
• (S) = simulation result
• (u_{input}) = uncertainty from input parameters
• (u_{num}) = numerical solution error
• (u_{exp}) = experimental measurement uncertainty
• (u{val}) = validation uncertainty = (\sqrt{u{input}^2 + u{num}^2 + u{exp}^2})

The validation metric interval is: [ \text{Modeling Error} = (S - E) \pm u_{val} ]

Table 1: Validation Metric Components for Pointwise Comparison

Component	Symbol	Description	Estimation Method
Comparison Error	(S - E)	Difference between simulation and experiment	Direct calculation
Input Uncertainty	(u_{input})	Uncertainty from model input parameters	Uncertainty propagation
Numerical Error	(u_{num})	Discretization and solution approximation	Grid convergence studies
Experimental Uncertainty	(u_{exp})	Random and systematic measurement error	Statistical analysis of replicates
Validation Uncertainty	(u_{val})	Combined uncertainty	Root sum square combination

Interpolation-Based Metric for Dense Experimental Data

When experimental data is sufficiently dense over the input parameter range, construct an interpolation function through experimental data points. The validation metric becomes:

[ \text{Modeling Error}(x) = (S(x) - IE(x)) \pm u{val}(x) ]

Where (I_E(x)) represents the interpolated experimental mean at point (x) [115].

Protocol Steps:

• Collect experimental data at numerous setpoints across the parameter space
• Construct interpolation function through experimental means
• Compute comparison error throughout the domain
• Determine validation uncertainty at each point
• Calculate the validation metric interval across the domain

Regression-Based Metric for Sparse Experimental Data

For sparse experimental data, employ regression (curve fitting) to represent the estimated mean:

[ \text{Modeling Error}(x) = (S(x) - RE(x)) \pm u{val}(x) ]

Where (R_E(x)) represents the regression function through experimental data [115].

Protocol Steps:

• Collect available experimental data across the parameter space
• Determine appropriate regression function form
• Fit regression function to experimental data
• Compute comparison error using regression function
• Determine validation uncertainty incorporating regression error
• Calculate validation metric interval

Table 2: Validation Metric Types and Applications

Metric Type	Experimental Data Requirement	Application Context	Key Advantages
Pointwise	Single operating condition	Model assessment at specific points	Simple computation and interpretation
Interpolation-Based	Dense data throughout parameter space	Comprehensive validation across domain	Utilizes full experimental information
Regression-Based	Sparse data throughout parameter space	Practical engineering applications	Works with limited experimental resources

Classification Accuracy Assessment

Confusion Matrix Analysis

For classification models, accuracy assessment quantifies agreement between predicted classes and ground-truth data [117]. The confusion matrix forms the foundation for calculating key accuracy metrics.

Experimental Protocol:

• Partition labeled data into training (≈80%) and testing (≈20%) sets
• Train classifier using training set
• Generate predictions for testing set
• Construct confusion matrix comparing predictions to actual values

Table 3: Binary Classification Confusion Matrix

	Actual Positive	Actual Negative
Predicted Positive	True Positive (TP)	False Positive (FP)
Predicted Negative	False Negative (FN)	True Negative (TN)

Accuracy Metrics from Confusion Matrix

Overall Accuracy: Proportion of correctly classified instances [ \text{Overall Accuracy} = \frac{TP + TN}{\text{Sample Size}} ]

Producer's Accuracy (Recall): Proportion of actual class members correctly classified [ \text{Producer's Accuracy} = \frac{TP}{TP + FN} ]

User's Accuracy (Precision): Proportion of predicted class members correctly classified [ \text{User's Accuracy} = \frac{TP}{TP + FP} ]

Kappa Coefficient: Measures how much better the classification is versus random assignment [ \text{Kappa} = \frac{\text{observed accuracy} - \text{chance agreement}}{1 - \text{chance agreement}} ]

Error Types:

• Omission Error = 100% - Producer's Accuracy
• Commission Error = 100% - User's Accuracy [117]

Experimental Protocols

Protocol 1: Confidence Interval Validation

Application: Computational fluid dynamics, structural mechanics, pharmacokinetic modeling

Materials:

• Computational model of the system
• Experimental apparatus for system response measurement
• Uncertainty quantification framework

Methodology:

• Define system response quantities (SRQs) for validation
• Identify relevant input parameters and operating conditions
• Quantify numerical solution error through grid convergence studies
• Estimate input parameter uncertainties through sensitivity analysis
• Conduct replicated experiments to quantify experimental uncertainty
• Compute validation uncertainty using root sum square combination
• Calculate validation metric intervals at each operating condition
• Assess whether modeling error intervals contain zero within acceptable bounds

Protocol 2: Classification Accuracy Assessment

Application: Image classification, molecular pattern recognition, diagnostic models

Materials:

• Labeled reference dataset (ground truth)
• Classification algorithm
• Computing environment for accuracy assessment

Methodology:

• Implement stratified sampling to partition data into training and testing sets
• Train classification model using training dataset
• Generate predictions for testing dataset
• Construct confusion matrix comparing predictions to reference data
• Calculate overall accuracy, producer's accuracy, and user's accuracy
• Compute kappa coefficient to assess improvement over random classification
• Analyze omission and commission errors by class
• Implement cross-validation to assess metric stability

Visualization of Methodologies

Validation Methodology Workflow

Confusion Matrix and Derived Metrics

Research Reagent Solutions

Table 4: Essential Research Materials for Validation Studies

Item	Function	Application Context
Computational Model	Mathematical representation of physical system	Prediction of system response quantities
Experimental Apparatus	Physical system for empirical measurements	Generation of validation data
Uncertainty Quantification Framework	Statistical analysis of error sources	Quantification of numerical, input, and experimental uncertainties
Reference Datasets	Ground truth measurements with known accuracy	Classification model training and testing
Statistical Software	Implementation of validation metrics	Computation of confidence intervals and accuracy metrics
Grid Convergence Tools	Numerical error estimation	Solution verification and discretization error quantification
Sensitivity Analysis Methods	Input parameter importance ranking	Prioritization of uncertainty sources

Uncertainty Quantification (UQ) has emerged as a critical component in computational models, particularly for high-stakes fields like drug discovery and materials science. UQ methods provide a measure of confidence for model predictions, enabling researchers to distinguish between reliable and unreliable outputs [9]. This is especially vital when models encounter data outside their training distribution, a common scenario in real-world research applications.

In computational drug discovery, for instance, models often make predictions for compounds that reside outside the chemical space covered by the training set (the Applicability Domain, or AD). Predictions for these compounds are unreliable and can lead to costly erroneous decisions in the drug-design process [9]. UQ methods help to flag such unreliable predictions, thereby fostering trust and facilitating more informed decision-making.

UQ techniques are broadly categorized by their architecture into two competing paradigms: ensemble-based methods and single-model methods. Ensemble methods combine predictions from multiple models to yield a collective prediction with an associated uncertainty measure [118]. In contrast, single-model methods, such as Mean-Variance Estimation (MVE) and Deep Evidential Regression, aim to provide uncertainty estimates from a single, deterministic neural network, often at a lower computational cost [119]. This application note provides a comparative analysis of these approaches, offering structured data, detailed protocols, and practical toolkits to guide researchers in selecting and implementing appropriate UQ strategies.

Theoretical Foundations of Uncertainty

In the context of machine learning, uncertainty is typically decomposed into two fundamental types, each with a distinct origin and implication for model development.

• Aleatoric Uncertainty: This derives from the inherent noise or randomness in the data itself. It is an irreducible property of the data generation process, such as variability in experimental measurements. Aleatoric uncertainty can be represented as the variance in the observed data around a mean prediction [9].
• Epistemic Uncertainty: This stems from a lack of knowledge or insufficient data in certain regions of the sample space experienced by the model. It is also known as model uncertainty. For example, predictions for a chemical compound that is structurally very different from any molecule in the training set will have high epistemic uncertainty. Unlike aleatoric uncertainty, epistemic uncertainty can be reduced by collecting more relevant data in the under-represented regions of the feature space [9].

A robust UQ method should ideally account for both types of uncertainty to provide a comprehensive confidence estimate for its predictions.

Ensemble-Based Methods

Ensemble learning is a machine learning technique that combines multiple individual models (sometimes called "weak learners") to produce a prediction that is often more accurate and robust than any single constituent model [118]. The core principle is that a group of models working together can correct for each other's errors, leading to improved overall performance.

The primary strength of ensembles lies in their ability to mitigate the bias-variance trade-off, a fundamental challenge in machine learning. By aggregating predictions, ensembles can reduce variance (overfitting) and often achieve a more favorable balance than a single model [118]. For UQ, the variation in predictions across the individual models in an ensemble provides a direct and effective measure of epistemic uncertainty.

Common ensemble techniques include:

• Bagging (Bootstrap Aggregating): Trains multiple versions of the same model on different random subsets of the training data. The final prediction is an average (regression) or majority vote (classification) of all individual predictions. Example: Random Forest [118].
• Boosting: A sequential method that trains models one after another, with each new model focusing on the examples that previous models misclassified. Examples: AdaBoost, Gradient Boosting, XGBoost [118].
• Stacking: Combines multiple models using a meta-learner that is trained to best aggregate the base models' predictions [118].

Despite their effectiveness, a common perceived drawback of ensembles is their computational cost, as they require training and maintaining multiple models. However, research has shown that ensembles of smaller models can match or exceed the accuracy of a single large state-of-the-art model while being more efficient to train and run [120].

Single-Model Methods

Single-model UQ techniques seek to provide uncertainty estimates from a single neural network, thereby avoiding the computational expense of ensembles. These methods can be broadly grouped into antecedent and succedent schemes [119].

• Antecedent Methods: These place priors on the input data or incorporate UQ directly into the training objective.
  • Mean-Variance Estimation (MVE): This method places a Gaussian prior on the input data and trains the network to predict both the mean and the variance for a given input. The predicted variance is then interpreted as the uncertainty [119].
  • Deep Evidential Regression: This approach places a higher-order prior distribution (e.g., a Normal-Inverse-Gamma distribution) over the likelihood function of the data. The network is trained to output the parameters of this evidential distribution, from which the uncertainty can be derived [119].
• Succedent Methods: These estimate uncertainty after the network has been trained, typically by analyzing the network's internal state or feature representations.
  • Gaussian Mixture Models (GMM): This succedent method involves fitting a Gaussian Mixture Model on the latent space representations (the activations of a hidden layer) of a pre-trained neural network. The distance of a new data point to the learned mixture components can then be used as an uncertainty score [119].

Comparative Analysis

Performance and Robustness

A systematic comparison of UQ methods is essential for informed selection. Recent research evaluated ensemble, MVE, evidential regression, and GMM methods across various datasets, including the rMD17 dataset for molecular energies and forces [119]. Performance was measured using metrics that assess how well the predicted uncertainty ranks the true prediction error (e.g., Spearman correlation) and the calibration of the uncertainty estimates.

Table 1: Comparative Performance of UQ Methods on the rMD17 Dataset

UQ Method	Architecture	Prediction Error (Test MAE)	Ranking Performance	Computational Cost (Relative Training Time)	Key Strengths and Weaknesses
Ensemble	Multiple independent models	Lowest [119]	Good across all metrics [119]	High (~5x single model) [119]	Strengths: Superior generalization, robust NNIPs, removes parametric uncertainty. Weaknesses: Higher computational cost. [119]
MVE	Single deterministic NN	Highest [119]	Good for in-domain interpolation [119]	~1x [119]	Strengths: Effective for in-domain points. Weaknesses: Poorer out-of-domain generalization, harder-to-optimize loss. [119]
Evidential Regression	Single deterministic NN	Moderate [119]	Inconsistent (bimodal distribution) [119]	~1x [119]	Strengths: -- Weaknesses: Poor epistemic uncertainty prediction, atom type-dependent parameters. [119]
GMM	Single deterministic NN	Moderate [119]	Better for out-of-domain data [119]	~1x (plus post-training fitting) [119]	Strengths: More accurate and lightweight than MVE/Evidential. Weaknesses: Worst performance in all metrics (though within error bars). [119]

The key finding from this comparative study is that no single UQ method consistently outperformed all others across every metric and dataset [119]. However, ensemble-based methods demonstrated consistently strong performance, particularly for robust generalization and in applications like active learning for molecular dynamics simulations. While single-model methods like MVE and GMM showed promise in specific scenarios (in-domain and out-of-domain, respectively), they could not reliably match the all-around robustness of ensembles [119].

Computational Efficiency

The perception that ensembles are prohibitively expensive is being re-evaluated. Google Research has demonstrated that an ensemble of two smaller models (e.g., EfficientNet-B5) can match the accuracy of a single, much larger model (e.g., EfficientNet-B7) while using approximately 50% fewer FLOPS and significantly less training time (96 TPU days vs. 160 TPU days) [120]. Furthermore, cascades, a subset of ensembles that execute models sequentially and exit early when a prediction is confident, can reduce the average computational cost even further while maintaining high accuracy [120].

Application Protocols

Protocol 1: Comparative Evaluation of UQ Methods

This protocol outlines the steps for a standardized evaluation of different UQ methods on a given dataset, following the methodology used in recent literature [119].

Objective: To empirically compare the performance of ensemble and single-model UQ methods based on prediction accuracy, uncertainty quality, and computational efficiency.

Materials:

• A labeled dataset (e.g., rMD17, QSAR data).
• Computational resources (GPU recommended).
• Deep learning framework (e.g., TensorFlow, PyTorch).

Procedure:

• Data Preparation: Split the dataset into training, validation, and test sets. Ensure the test set includes both in-domain and out-of-domain samples to test extrapolation capability.
• Model Implementation: a. Ensemble: Train 5-10 independent models with different random weight initializations. b. MVE: Implement a network with two output neurons (mean and variance) and train using a negative log-likelihood (NLL) loss function. c. Evidential Regression: Implement a network that outputs four parameters (γ, ν, α, β) of the evidential distribution and train using the maximum a posteriori (MAP) loss. d. GMM: Train a standard network, then fit a GMM on the latent space representations of the training data from a chosen hidden layer.
• Evaluation: a. Prediction Accuracy: Calculate Mean Absolute Error (MAE) on the test set. b. Uncertainty Quality: i. Ranking: Calculate the Spearman correlation between the predicted uncertainties and the true absolute errors for all test points. ii. Calibration: Compute the miscalibration area (the difference between the empirical and predicted confidence levels). c. Computational Cost: Record the total training time and the inference time per sample for each method.

The following workflow diagram illustrates this experimental procedure:

UQ Method Evaluation Workflow

Protocol 2: Active Learning for Robust Interatomic Potentials

This protocol details the use of UQ in an active learning loop to build robust Neural Network Interatomic Potentials (NNIPs), a method that can be adapted for computational chemistry and drug discovery tasks like molecular property prediction [119] [121].

Objective: To iteratively improve the robustness and accuracy of a model by using its uncertainty estimates to selectively acquire new training data from underrepresented regions of the input space.

Materials:

• An initial, small set of high-quality labeled data (e.g., from ab initio calculations).
• A pool of unlabeled data points (e.g., molecular configurations from simulations).
• A UQ-enabled model (e.g., an ensemble).

Procedure:

• Initial Training: Train the initial model on the small labeled dataset.
• Uncertainty Sampling: Use the trained model to make predictions on the large pool of unlabeled data. Calculate the uncertainty for each prediction.
• Query Selection: Select the data points with the highest uncertainty (i.e., those where the model is least confident) for labeling. This can be done by a human expert or through an oracle (e.g., further ab initio calculations).
• Data Augmentation: Add the newly labeled, high-uncertainty points to the training set.
• Model Retraining: Retrain the model on the augmented training set.
• Iteration: Repeat steps 2-5 until a performance plateau is reached or a computational budget is exhausted.

The following diagram visualizes this iterative cycle:

Active Learning Loop with UQ

The Scientist's Toolkit: Research Reagent Solutions

This section outlines key computational "reagents" essential for implementing UQ methods in computational research.

Table 2: Essential Research Reagents for UQ Experiments

Item	Function in UQ Research	Example Usage/Note
Benchmark Datasets	Provides a standardized foundation for training and comparing UQ methods.	rMD17 (molecular dynamics), QSAR datasets (drug discovery). Should include in-domain and out-of-domain splits. [119]
Deep Learning Framework	Provides the programming environment for building and training UQ-enabled models.	TensorFlow, PyTorch, or JAX. Essential for implementing custom loss functions (e.g., for MVE and Evidential Regression). [119]
UQ-Specific Software Libraries	Offers pre-built implementations of advanced UQ techniques, reducing development time.	Libraries such as Uncertainty Baselines or Pyro can provide implementations of ensembles, Bayesian NNs, and evidential methods.
High-Performance Computing (HPC) Resources	Accelerates the training of multiple models (ensembles) and large-scale data generation.	GPU/TPU clusters are crucial for practical training of ensembles and for running active learning loops in a reasonable time. [120]
Latent Space Analysis Tools	Enables the implementation of succedent UQ methods like GMM.	Scikit-learn for fitting GMMs; dimensionality reduction tools (UMAP, t-SNE) for visualizing latent spaces to diagnose model behavior. [119]

Visualization of Uncertainty

Effectively communicating uncertainty is as important as calculating it. In the context of computational models and drug discovery, visualizing uncertainty helps stakeholders interpret model predictions accurately and make risk-aware decisions [122].

• Error Bars and Confidence Intervals: These are the most common techniques, used in bar charts, scatterplots, and line graphs to show variability or a confidence range around a predicted value [122].
• Confidence Bands: Extend confidence intervals across a continuous range, ideal for showing uncertainty in model predictions over an entire input space, such as a dose-response curve [122].
• Probability Distributions: Visualizing the full distribution of possible outcomes (e.g., using histograms, violin plots, or density plots) provides a comprehensive view of uncertainty, showing not just the range but the likelihood of different outcomes [122].
• Visual Property Encoding: Techniques like adjusting the blurriness, transparency, or saturation of data points can intuitively encode uncertainty. For example, a blurry point on a scatter plot of compound efficacy vs. toxicity could indicate a high-uncertainty prediction [122].

Best Practice: Always match the visualization technique to the audience. Use error bars and statistical plots for expert audiences, and more intuitive visual properties like blur or multiple scenario plots for lay audiences [122].

Uncertainty Quantification (UQ) has emerged as a critical discipline within computational biomedical research, particularly for informing regulatory decisions on drugs and biologics. Regulatory bodies globally are increasingly recognizing the value of UQ in assessing the reliability, robustness, and predictive capability of computational models used throughout the medical product lifecycle. The forward UQ paradigm focuses on characterizing how variability and uncertainty in model input parameters affect model outputs and predictions. This approach is especially valuable in regulatory contexts where decisions must be made despite incomplete information about physiological parameters, material properties, and inter-subject variability. By quantifying these uncertainties, researchers can provide regulatory agencies with clearer assessments of risk and confidence in model-based conclusions, ultimately supporting more informed and transparent decision-making processes for therapeutic products [15].

The regulatory landscape for using computational evidence continues to evolve rapidly. Major regulatory agencies including the U.S. Food and Drug Administration (FDA), European Medicines Agency (EMA), and Pharmaceuticals and Medical Devices Agency (PMDA) in Japan have developed frameworks that acknowledge the importance of understanding uncertainty in evidence generation [123]. These developments align with the broader adoption of Real-World Evidence (RWE) in regulatory decision-making, where quantifying uncertainty becomes paramount when analyzing non-randomized data sources. The 21st Century Cures Act in the United States and the European Pharmaceutical Strategy have further emphasized the need for robust methodological standards in evidence generation, including formal approaches to characterize uncertainty in computational models and data sources used for regulatory submissions [123].

Regulatory Landscape and Standards

Global Regulatory Frameworks for Evidence Generation

The global regulatory environment for computational modeling and real-world evidence has matured significantly, with multiple jurisdictions developing specific frameworks and guidance documents. These frameworks establish foundational principles for assessing the reliability and relevance of computational evidence, including requirements for comprehensive uncertainty quantification. The development of these frameworks typically follows a stepwise approach, beginning with general position papers and evolving into detailed practical guidance on data quality, study methodology, and procedural aspects [123].

Table 1: Global Regulatory Frameworks Relevant to UQ in Decision-Making

Regulatory Body	Region	Key Frameworks/Guidance	UQ-Relevant Components
U.S. Food and Drug Administration (FDA)	North America	21st Century Cures Act (2016), PDUFA VII (2022), RWE Framework (2018)	Defines evidentiary standards for model-based submissions; outlines expectations for characterization of uncertainty in computational assessments [123].
European Medicines Agency (EMA)	Europe	Regulatory Science to 2025, HMA/EMA Big Data Taskforce	Emphasizes understanding uncertainty in complex evidence packages; promotes qualification of novel methodologies with defined uncertainty bounds [123].
Health Canada (HC)	North America	Optimizing Use of RWE (2019)	Provides guidance on assessing data reliability and analytical robustness, including uncertainty in real-world data sources [123].
Medicines and Healthcare products Regulatory Agency (MHRA)	United Kingdom	Guidance on RWD in Clinical Studies (2021), RCTs using RWD (2021)	Details methodological expectations for dealing with uncertainty in real-world data and hybrid study designs [123].
National Medical Products Administration (NMPA)	China	RWE Guidelines for Drug Development (2020), Guiding Principles of RWD (2021)	Includes technical requirements for assessing and reporting sources of uncertainty in real-world evidence [123].

Key Regulatory Elements for UQ Implementation

Successful implementation of UQ in regulatory submissions requires attention to three key elements that regulatory agencies have identified as critical. First, data quality guidance establishes standards for characterizing uncertainty in input data, including real-world data sources, and provides frameworks for assessing fitness-for-use. Second, study methods guidance addresses methodological approaches for designing studies that properly account for uncertainty, including specifications for model validation and sensitivity analysis. Third, procedural guidance outlines processes for engaging with regulatory agencies regarding UQ approaches, including submission requirements and opportunities for early feedback on UQ plans [123].

Alignment between regulators and Health Technology Assessment (HTA) bodies on the acceptance of UQ methodologies continues to evolve. Recent initiatives have focused on developing evidentiary standards that satisfy both regulatory and reimbursement requirements, emphasizing the importance of transparently characterizing uncertainty in cost-effectiveness and comparative effectiveness models [123]. This alignment is particularly important for developers seeking simultaneous regulatory approval and reimbursement recommendations based on computationally-derived evidence.

UQ Methodologies and Protocols

Foundational UQ Methods

Uncertainty Quantification employs a diverse set of mathematical and statistical techniques to characterize, propagate, and reduce uncertainty in computational models. The appropriate methodology depends on the model complexity, computational expense, and the nature of the uncertainty sources. For regulatory applications, methods must provide interpretable and auditable results that support decision-making under uncertainty [124].

Table 2: Core UQ Methods for Regulatory Science Applications

Method	Key Principle	Regulatory Application Examples	Implementation Considerations
Monte Carlo Simulation	Uses random sampling to generate probability distributions of model outputs.	Risk assessment for medical devices; pharmacokinetic variability analysis.	Computationally intensive; requires many model evaluations; implementation is straightforward but convergence can be slow [124].
Polynomial Chaos Expansion	Represents model outputs as polynomial functions of input parameters.	Cardiac electrophysiology models; neuromodulation simulations.	More efficient than Monte Carlo for smooth systems; creates computationally inexpensive emulators for sensitivity analysis [15].
Bayesian Inference	Updates prior parameter estimates using new data through Bayes' theorem.	Model calibration using clinical data; adaptive trial designs; meta-analysis.	Incorporates prior knowledge; provides natural uncertainty quantification; computational implementation can be challenging [124].
Sensitivity Analysis	Measures how output uncertainty apportions to different input sources.	Identification of critical quality attributes; parameter prioritization.	Complements other UQ methods; helps focus resources on most influential parameters [15].

Experimental Protocol: UQ for Computational Model Evaluation

The following protocol provides a standardized approach for implementing UQ in computational models intended to support regulatory submissions. This protocol adapts established UQ methodologies specifically for the regulatory context, emphasizing transparency, reproducibility, and decision relevance [15].

Protocol Title: Non-Intrusive Uncertainty Quantification for Computational Models in Regulatory Submissions

Objective: To characterize how parametric uncertainty and variability propagate through computational models to affect key outputs relevant to regulatory decisions.

Materials and Software Requirements:

• Computational model of the physiological system or intervention
• Parameter distributions based on experimental data or literature
• UQ software platform (e.g., UncertainSCI, UQTk, or custom implementation)
• Computing resources adequate for multiple model evaluations

Procedure:

Step 1: Problem Formulation

• Define the model outputs relevant to regulatory decisions (efficacy, safety, performance)
• Identify all uncertain input parameters and classify uncertainty type (aleatory/epistemic)
• Establish parameter probability distributions based on available data or expert knowledge
• Document all assumptions in the uncertainty model

Step 2: Parameter Sampling

• Select appropriate sampling strategy based on model characteristics and computational cost
• For polynomial chaos methods: Generate parameter ensembles using near-optimal sampling techniques such as weighted Fekete points
• For Monte Carlo methods: Ensure sufficient sample size for convergence of output statistics
• Record all parameter values in the ensemble for reproducibility

Step 3: Model Evaluation

• Execute the computational model for each parameter set in the ensemble
• Extract and store all relevant output quantities of interest
• Monitor for numerical errors or non-physical results that may require ensemble adjustment

Step 4: Emulator Construction (if using surrogate modeling)

• Build polynomial chaos emulator using the input-output pairs
• Validate emulator accuracy against additional model evaluations
• Quantify emulator error and incorporate into overall uncertainty assessment

Step 5: Uncertainty Analysis

• Compute output statistics (mean, variance, quantiles) from the ensemble or emulator
• Perform global sensitivity analysis to identify influential parameters
• Calculate Sobol indices or other sensitivity metrics to apportion output variance to inputs
• Visualize uncertainty propagation through probability distributions and statistical summaries

Step 6: Documentation and Reporting

• Prepare comprehensive report of UQ methodology, results, and interpretation
• Document all software tools, version numbers, and computational environment details
• Include visualizations of key results that clearly communicate uncertainty to decision-makers
• Relate uncertainty findings specifically to regulatory questions or criteria

Protocol: Sensitivity Analysis for Regulatory Submissions

Protocol Title: Global Sensitivity Analysis for Model-Informed Drug Development

Objective: To identify and rank model parameters that contribute most significantly to output variability, guiding resource allocation for parameter refinement and model reduction.

Materials:

• Validated computational model with defined parameter distributions
• Sensitivity analysis software (UncertainSCI, SALib, or equivalent)
• Computing resources for multiple model evaluations

Procedure:

• Define output metrics of regulatory interest (e.g., clinical endpoints, surrogate markers)
• Establish plausible ranges for all model parameters through literature review or experimental data
• Select appropriate sensitivity analysis method (variance-based, Morris method, etc.)
• Generate parameter samples using structured design (Sobol sequences, Latin Hypercube)
• Run model simulations for all sample points
• Calculate sensitivity indices (first-order, total-order, or other relevant metrics)
• Interpret results to identify parameters requiring precise quantification
• Document findings for regulatory submission, including methodological justification

Successful implementation of UQ for regulatory decision-making requires both computational tools and conceptual frameworks. The following table summarizes essential resources for researchers developing UQ approaches for regulatory submissions [15].

Table 3: Essential UQ Tools and Resources for Regulatory Science

Tool/Resource	Type	Function	Regulatory Application
UncertainSCI	Open-source software	Implements polynomial chaos expansion for forward UQ tasks; provides near-optimal sampling.	Biomedical simulation uncertainty; cardiac and neural applications; parametric variability assessment [15].
UQTk	Software library	Provides tools for parameter propagation, sensitivity analysis, and Bayesian inference.	Hydrogen conversion processes; electrochemical systems; materials modeling [33].
SPIRIT 2025	Reporting guideline	Standardized protocol items for clinical trials, including UQ-related methodology.	Improving planning and reporting of trial protocols; enhancing reproducibility [125].
Polynomial Chaos Expansion	Mathematical framework	Represents model outputs as orthogonal polynomial expansions of uncertain inputs.	Building efficient emulators for complex models; reducing computational cost for UQ [15].
Sobol Indices	Sensitivity metric	Quantifies contribution of input parameters to output variance through variance decomposition.	Identifying critical parameters; prioritizing experimental refinement; model reduction [15].
Bayesian Calibration	Statistical method	Updates parameter estimates and uncertainties by combining prior knowledge with new data.	Incorporating heterogeneous data sources; sequential updating during product development [124].

Application in Regulatory Decision-Making

UQ for Model-Informed Drug Development

Uncertainty Quantification plays increasingly important roles across the drug development lifecycle through Model-Informed Drug Development (MIDD) approaches. In early development, UQ helps prioritize compound selection by quantifying confidence in preclinical predictions of human efficacy and safety. During clinical development, UQ supports dose selection and trial design by characterizing uncertainty in exposure-response relationships. For regulatory submissions, UQ provides transparent assessment of confidence in model-based inferences, particularly when supporting label expansions or approvals in special populations [123].

Regulatory agencies have specifically highlighted the value of UQ in assessing real-world evidence for regulatory decisions. The FDA's RWE Framework and subsequent guidance documents emphasize the need to understand and quantify uncertainties when using real-world data to support effectiveness claims [123]. This includes characterizing uncertainty in patient identification, exposure classification, endpoint ascertainment, and confounding control. Sophisticated UQ methods such as Bayesian approaches and quantitative bias analysis provide structured frameworks for assessing how these uncertainties might affect study conclusions and their relevance to regulatory decisions.

Decision-Focused UQ Implementation

Effective UQ for regulatory submissions must be decision-focused rather than merely technical. This requires early engagement with regulatory agencies to identify the specific uncertainties most relevant to the decision context and to establish acceptable levels of uncertainty for favorable decisions. The Procedural Guidance issued by various regulatory agencies provides frameworks for these discussions, including opportunities for parallel advice with regulatory and HTA bodies [123].

Visualization of uncertainty is particularly important for regulatory communication. Diagrams and interactive tools that clearly show how uncertainty propagates through models to decision-relevant endpoints facilitate more transparent regulatory assessments. The development of standardized UQ report templates that align with Common Technical Document (CTD) requirements helps ensure consistent presentation of uncertainty information across submissions [123]. These templates should include quantitative summaries of key uncertainties, their potential impact on decision-relevant outcomes, and approaches taken to mitigate or characterize these uncertainties.

Uncertainty Quantification represents an essential capability for modern regulatory science, providing structured approaches to characterize, communicate, and manage uncertainty in computational evidence supporting drug and device evaluations. The evolving regulatory landscape increasingly formalizes expectations for UQ implementation, with major agencies developing specific frameworks and guidance documents. Successful adoption of UQ methodologies requires both technical sophistication in implementation and strategic alignment with regulatory decision processes. The protocols and frameworks presented here provide researchers with practical approaches for implementing UQ in regulatory contexts, ultimately supporting more transparent and robust decision-making for innovative medical products. As regulatory agencies continue to advance their capabilities in evaluating complex computational evidence, researchers who master UQ methodologies will be better positioned to efficiently translate innovations into approved products that benefit patients.

Application Note: Cardiovascular Digital Twin for Pulmonary Hemodynamics

Application Context and Uncertainty Quantification (UQ) Framework

This application note details the development of a patient-specific cardiovascular digital twin for predicting pulmonary artery pressure (PAP), a critical hemodynamic metric in heart failure (HF) management. The model addresses inherent uncertainties from sparse clinical measurements and complex anatomy by implementing a UQ framework to determine the minimal geometric model complexity required for accurate, non-invasive prediction of left pulmonary artery (LPA) pressure [126].

The UQ strategy systematically evaluates uncertainty introduced by the segmentation of patient anatomy from medical images. The core of the UQ involves constructing and comparing three distinct geometric models of the pulmonary arterial tree for each patient, with varying levels of anatomical detail [126]. This approach quantifies how geometric simplification propagates to uncertainty in the final hemodynamic predictions, ensuring model fidelity while maintaining computational efficiency.

Table 1: Uncertainty Quantification in Pulmonary Artery Geometric Modeling

Complexity Level	Anatomical Structures Included	Segmentation Time & Computational Cost	Impact on LPA Pressure Prediction Accuracy
Level 1 (Simplest)	Main Pulmonary Artery (MPA), Left PA (LPA), Right PA (RPA)	Lowest	Determined to be sufficient for accurate prediction [126]
Level 2	Level 1 + First-order vessel branches	Medium	Negligible improvement over Level 1 [126]
Level 3 (Most Detailed)	Level 2 + Second-order vessel branches	Highest (Significant bottleneck)	No significant improvement over Level 1 [126]

Experimental Protocol: Cardiovascular Digital Twin Construction and Validation

Objective: To create and validate a patient-specific digital twin for non-invasive prediction of pulmonary artery pressure, quantifying uncertainty from geometric modeling and boundary conditions [126].

Materials and Software:

• Medical Imaging: High-resolution (0.6 mm) CT pulmonary angiograms [126].
• Hemodynamic Data: Invasively measured data from Right Heart Catheterization (RHC) and Implantable Hemodynamic Monitors (IHM) [126].
• Segmentation Software: For constructing 3D geometric models from CT images [126].
• CFD Solver: HARVEY computational fluid dynamics software [126].

Methodology:

• Patient Cohort & Data Acquisition: A cohort of HF patients with IHM devices is identified. CT pulmonary angiograms and concurrent RHC measurements are collected [126].
• Geometric Model Construction (Multi-Level UQ):
  • For each of the first five patients, segment three distinct 3D models of the pulmonary arteries from CT images [126].
  • Level 3: Segment the entire pulmonary arterial tree down to second-order branches [126].
  • Level 2: Trim the Level 3 model to first-order vessel branches only [126].
  • Level 1: Further simplify to include only the MPA, LPA, and RPA [126].
• CFD Simulation Setup:
  • Import the geometric models into the HARVEY CFD solver [126].
  • Implement patient-specific boundary conditions using RHC-measured data (e.g., mean volumetric flow rate) [126].
  • Initial simulations use a steady-state flow model with no-slip vessel walls and zero outlet pressure for baseline analysis [126].
• Boundary Condition Sensitivity Analysis (UQ): Conduct a systematic study by varying the boundary condition settings at the model outlets. This quantifies the uncertainty in LPA pressure predictions resulting from incomplete knowledge of downstream vascular resistance [126].
• Model Validation: For each geometric complexity level, compare the CFD-predicted LPA pressure against the gold-standard LPA pressure measured directly by the IHM device [126].
• Determination of Minimal Complexity: Analyze the results from steps 4 and 5 to identify the simplest geometric model (Level 1) that maintains predictive accuracy within clinically acceptable limits [126].
• Scaled Application: Apply the validated Level 1 geometric modeling approach to the remaining patients in the cohort to demonstrate scalability [126].

Workflow Visualization: Cardiovascular Digital Twin Pipeline

The Scientist's Toolkit: Cardiovascular Digital Twin Research Reagents

Table 2: Essential Research Reagents and Resources for Cardiovascular Digital Twin Implementation

Item / Resource	Function / Application in the Protocol
CT Pulmonary Angiogram	Provides high-resolution 3D anatomical data for patient-specific geometric model construction [126].
Right Heart Catheterization (RHC)	Provides gold-standard, invasive hemodynamic measurements (e.g., flow rates) used to calibrate model boundary conditions [126].
Implantable Hemodynamic Monitor (IHM)	Provides continuous, direct measurements of LPA pressure for rigorous model validation [126].
HARVEY CFD Solver	Open-source computational fluid dynamics software used to simulate blood flow and pressure in the 3D models [126].
Image Segmentation Software	Software tool (e.g., 3D Slicer, ITK-Snap) used to extract 3D geometric models of the pulmonary arteries from CT images [126].

Application Note: Oncology Digital Twin for Patient-Specific Treatment Modeling

Application Context and Uncertainty Quantification (UQ) Framework

This note explores a predictive digital twin framework for oncology, designed to inform patient-specific clinical decision-making for tumors, such as glioblastoma [127]. The core challenge is the significant uncertainty arising from sparse, noisy, and longitudinal patient data (e.g., non-invasive imaging). The UQ framework is built to formally quantify this uncertainty and propagate it through the model to produce risk-informed predictions [127].

The methodology employs a Bayesian inverse problem approach. A mechanistic model of spatiotemporal tumor progression (a reaction-diffusion PDE) is defined. The statistical inverse problem then infers the spatially varying parameters of this model from the available patient data [127]. The output is not a single prediction but a scalable approximation of the Bayesian posterior distribution, which rigorously quantifies the uncertainty in model parameters and subsequent forecasts due to data limitations [127]. This allows clinicians to evaluate "what-if" scenarios with an understood level of confidence.

Table 3: Uncertainty Quantification in an Oncology Digital Twin

UQ Component	Description	Role in Addressing Uncertainty
Mechanistic Model	Reaction-diffusion model of tumor progression, constrained by patient-specific anatomy [127].	Provides a physics/biology-based structure, reducing reliance purely on noisy data.
Bayesian Inverse Problem	Statistical framework to infer model parameters from sparse, noisy imaging data [127].	Quantifies the probability of different parameter sets being true, given the data.
Posterior Distribution	The output of the inverse problem; a probability distribution over model parameters and predictions [127].	Encapsulates total uncertainty, enabling risk-informed decision making (e.g., via credible intervals).
Virtual Patient Verification	Testing the pipeline on a "virtual patient" with known ground truth and synthetic data [127].	Validates the UQ methodology by confirming it can recover known truths under controlled conditions.

Experimental Protocol: Oncology Digital Twin with Quantified Uncertainty

Objective: To develop a predictive digital twin for a cancer patient that estimates spatiotemporal tumor dynamics and rigorously quantifies the uncertainty in these predictions to support clinical decision-making [127].

Materials and Software:

• Longitudinal Imaging Data: A time-series of non-invasive (e.g., MRI) scans of the patient's tumor [127].
• Mechanistic Model: A pre-defined reaction-diffusion Partial Differential Equation (PDE) model representing tumor growth [127].
• Computational Anatomy: A 3D mesh of the patient's specific anatomy derived from baseline imaging [127].
• High-Performance Computing (HPC) Cluster: For solving the computationally intensive statistical inverse problem [127].

Methodology:

• Data Preprocessing: Segment the tumor volume from each scan in the longitudinal imaging series. Register all images to a common coordinate system based on the patient's anatomical mesh [127].
• Forward Model Definition: Implement the reaction-diffusion tumor growth PDE. The model should be capable of simulating tumor dynamics over time on the patient's specific anatomical domain [127].
• Bayesian Inverse Problem Formulation (Core UQ):
  • Define Priors: Establish prior probability distributions for the unknown, spatially varying model parameters (e.g., tumor cell proliferation rate, diffusion coefficient). These represent belief before seeing the patient's data [127].
  • Define Likelihood: Construct a function that quantifies how well the model output, given a set of parameters, matches the observed longitudinal imaging data. This function accounts for measurement noise [127].
  • Solve for Posterior: Use scalable statistical sampling or variational inference algorithms to compute the posterior distribution. This distribution represents the updated belief about the model parameters after incorporating the patient data [127].
• Virtual Patient Verification (UQ Calibration):
  • Run the entire pipeline on a synthetic "virtual patient" where the true parameters are known.
  • Confirm that the true parameters lie within the high-probability region of the estimated posterior distribution. This step is critical for validating the UQ framework [127].
• Prediction with Uncertainty Quantification:
  • Using the calibrated posterior distribution, propagate the parameter uncertainty forward to generate probabilistic predictions of future tumor progression under different therapeutic interventions [127].
  • Outputs should include metrics like credible intervals for tumor volume over time.
• Optimal Experimental Design (Advanced UQ): Utilize the calibrated twin to answer questions about data value. For example, simulate how reducing uncertainty in predictions depends on the frequency of future imaging scans [127].

Workflow Visualization: Oncology Digital Twin with UQ

The Scientist's Toolkit: Oncology Digital Twin Research Reagents

Table 4: Essential Research Reagents and Resources for Oncology Digital Twin Implementation

Item / Resource	Function / Application in the Protocol
Reaction-Diffusion PDE Model	The core mechanistic model describing the spatiotemporal dynamics of tumor growth and invasion [127].
Longitudinal Medical Imaging	Provides the time-series data (e.g., MRI, CT) essential for informing and calibrating the model to an individual patient [127].
Multi-omics Data (e.g., from TCGA, iAtlas)	Provides population-level genomic, transcriptomic, and immunoprofile data used to define physiologically plausible parameter ranges and validate virtual patient cohorts [128].
High-Performance Computing (HPC) Resources	Necessary for solving the computationally demanding Bayesian inverse problem and performing massive in silico simulations [127].
Bayesian Inference Software	Libraries (e.g., PyMC, Stan, TensorFlow Probability) or custom code for solving the statistical inverse problem and sampling from posterior distributions [127].

Conclusion

Uncertainty Quantification has emerged as an indispensable component of credible computational modeling, particularly in high-stakes fields like biomedicine and drug development. By integrating foundational UQ principles with advanced methodological approaches—from polynomial chaos and ensemble methods to sophisticated Bayesian inference—researchers can transform models from black-box predictors into trusted, transparent tools for decision-making. The rigorous application of VVUQ frameworks provides the necessary foundation for building trust in emerging technologies like digital twins for precision medicine. Future directions will likely focus on scaling UQ methods for increasingly complex multi-scale models, developing standardized VVUQ protocols for regulatory acceptance, and further integrating AI and machine learning with physical principles to enhance predictive reliability. As computational models take on greater significance in therapeutic development and personalized treatment strategies, robust UQ practices will be fundamental to ensuring their safe and effective translation into clinical practice.

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