Comprehensive Uncertainty Quantification and Sensitivity Analysis for Cardiac Action Potential Models

Computational modeling of cardiac electrophysiology and arrhythmogenesis: toward clinical translation

The complexity of cardiac electrophysiology, involving dynamic changes in numerous components across multiple spatial (from ion channel to organ) and temporal (from milliseconds to days) scales, makes an intuitive or empirical analysis of cardiac arrhythmogenesis challenging. Multiscale mechanistic computational models of cardiac electrophysiology provide precise control over individual parameters, and their reproducibility enables a thorough assessment of arrhythmia mechanisms. This review provides a comprehensive analysis of models of cardiac electrophysiology and arrhythmias, from the single cell to the organ level, and how they can be leveraged to better understand rhythm disorders in cardiac disease and to improve heart patient care. Key issues related to model development based on experimental data are discussed, and major families of human cardiomyocyte models and their applications are highlighted. An overview of organ-level computational modeling of cardiac electrophysiology and its clinical applications in personalized arrhythmia risk assessment and patient-specific therapy of atrial and ventricular arrhythmias is provided. The advancements presented here highlight how patient-specific computational models of the heart reconstructed from patient data have achieved success in predicting risk of sudden cardiac death and guiding optimal treatments of heart rhythm disorders. Finally, an outlook toward potential future advances, including the combination of mechanistic modeling and machine learning/artificial intelligence, is provided. As the field of cardiology is embarking on a journey toward precision medicine, personalized modeling of the heart is expected to become a key technology to guide pharmaceutical therapy, deployment of devices, and surgical interventions.

Global sensitivity analysis with multifidelity Monte Carlo and polynomial chaos expansion for vascular haemodynamics

Computational models of the cardiovascular system are increasingly used for the diagnosis, treatment, and prevention of cardiovascular disease. Before being used for translational applications, the predictive abilities of these models need to be thoroughly demonstrated through verification, validation, and uncertainty quantification. When results depend on multiple uncertain inputs, sensitivity analysis is typically the first step required to separate relevant from unimportant inputs, and is key to determine an initial reduction on the problem dimensionality that will significantly affect the cost of all downstream analysis tasks. For computationally expensive models with numerous uncertain inputs, sample-based sensitivity analysis may become impractical due to the substantial number of model evaluations it typically necessitates. To overcome this limitation, we consider recently proposed Multifidelity Monte Carlo estimators for Sobol' sensitivity indices, and demonstrate their applicability to an idealized model of the common carotid artery. Variance reduction is achieved combining a small number of three-dimensional fluid-structure interaction simulations with affordable one- and zero-dimensional reduced-order models. These multifidelity Monte Carlo estimators are compared with traditional Monte Carlo and polynomial chaos expansion estimates. Specifically, we show consistent sensitivity ranks for both bi- (1D/0D) and tri-fidelity (3D/1D/0D) estimators, and superior variance reduction compared to traditional single-fidelity Monte Carlo estimators for the same computational budget. As the computational burden of Monte Carlo estimators for Sobol' indices is significantly affected by the problem dimensionality, polynomial chaos expansion is found to have lower computational cost for idealized models with smooth stochastic response.

Uncertainty quantification and sensitivity analysis of neuron models with ion concentration dynamics

This paper provides a comprehensive and computationally efficient case study for uncertainty quantification (UQ) and global sensitivity analysis (GSA) in a neuron model incorporating ion concentration dynamics. We address how challenges with UQ and GSA in this context can be approached and solved, including challenges related to computational cost, parameters affecting the system's resting state, and the presence of both fast and slow dynamics. Specifically, we analyze the electrodiffusive neuron-extracellular-glia (edNEG) model, which captures electrical potentials, ion concentrations (Na+, K+, Ca2+, and Cl-), and volume changes across six compartments. Our methodology includes a UQ procedure assessing the model's reliability and susceptibility to input uncertainty and a variance-based GSA identifying the most influential input parameters. To mitigate computational costs, we employ surrogate modeling techniques, optimized using efficient numerical integration methods. We propose a strategy for isolating parameters affecting the resting state and analyze the edNEG model dynamics under both physiological and pathological conditions. The influence of uncertain parameters on model outputs, particularly during spiking dynamics, is systematically explored. Rapid dynamics of membrane potentials necessitate a focus on informative spiking features, while slower variations in ion concentrations allow a meaningful study at each time point. Our study offers valuable guidelines for future UQ and GSA investigations on neuron models with ion concentration dynamics, contributing to the broader application of such models in computational neuroscience.

Virtual clinical QT exposure-response studies - A translational computational approach

BACKGROUND AND PURPOSE

A recent paradigm shift in proarrhythmic risk assessment suggests that the integration of clinical, non-clinical, and computational evidence can be used to reach a comprehensive understanding of the proarrhythmic potential of drug candidates. While current computational methodologies focus on predicting the incidence of proarrhythmic events after drug administration, the objective of this study is to predict concentration-response relationships of QTc as a clinical endpoint.

EXPERIMENTAL APPROACH

Full heart computational models reproducing human cardiac populations were created to predict the concentration-response relationship of changes in the QT interval as recommended for clinical trials. The concentration-response relationship of the QT-interval prolongation obtained from the computational cardiac population was compared against the relationship from clinical trial data for a set of well-characterized compounds: moxifloxacin, dofetilide, verapamil, and ondansetron.

KEY RESULTS

Computationally derived concentration-response relationships of QT interval changes for three of the four drugs had slopes within the confidence interval of clinical trials (dofetilide, moxifloxacin and verapamil) when compared to placebo-corrected concentration-ΔQT and concentration-ΔQT regressions. Moxifloxacin showed a higher intercept, outside the confidence interval of the clinical data, demonstrating that in this example, the standard linear regression does not appropriately capture the concentration-response results at very low concentrations. The concentrations corresponding to a mean QTc prolongation of 10 ms were consistently lower in the computational model than in clinical data. The critical concentration varied within an approximate ratio of 0.5 (moxifloxacin and ondansetron) and 1 times (dofetilide, verapamil) the critical concentration observed in human clinical trials. Notably, no other in silico methodology can approximate the human critical concentration values for a QT interval prolongation of 10 ms.

CONCLUSION AND IMPLICATIONS

Computational concentration-response modelling of a virtual population of high-resolution, 3-dimensional cardiac models can provide comparable information to clinical data and could be used to complement pre-clinical and clinical safety packages. It provides access to an unlimited exposure range to support trial design and can improve the understanding of pre-clinical-clinical translation.

Computational Modeling of Cardiac Electrophysiology

Mathematical modeling and simulation are well-established and powerful tools to integrate experimental data of individual components of cardiac electrophysiology, excitation-contraction coupling, and regulatory signaling pathways, to gain quantitative and mechanistic insight into pathophysiological processes and guide therapeutic strategies. Here, we briefly describe the processes governing cardiac myocyte electrophysiology and Ca2+ handling and their regulation, as well as action potential propagation in tissue. We discuss the models and methods used to describe these phenomena, including procedures for model parameterization and validation, in addition to protocols for model interrogation and analysis and techniques that account for phenotypic variability and parameter uncertainty. Our objective is to provide a summary of basic concepts and approaches as a resource for scientists training in this discipline and for all researchers aiming to gain an understanding of cardiac modeling studies.

Uncertainty assessment of proarrhythmia predictions derived from multi-level in silico models

In silico methods can be used for an early assessment of arrhythmogenic properties of drug candidates. However, their use for decision-making is conditioned by the possibility to estimate the predictions' uncertainty. This work describes our efforts to develop uncertainty quantification methods for the predictions produced by multi-level proarrhythmia models. In silico models used in this field usually start with experimental or predicted IC50 values that describe drug-induced ion channel blockade. Using such inputs, an electrophysiological model computes how the ion channel inhibition, exerted by a drug in a certain concentration, translates to an altered shape and duration of the action potential in cardiac cells, which can be represented as arrhythmogenic risk biomarkers such as the APD90. Using this framework, we identify the main sources of aleatory and epistemic uncertainties and propose a method based on probabilistic simulations that replaces single-point estimates predicted using multiple input values, including the IC50s and the electrophysiological parameters, by distributions of values. Two selected variability types associated with these inputs are then propagated through the multi-level model to estimate their impact on the uncertainty levels in the output, expressed by means of intervals. The proposed approach yields single predictions of arrhythmogenic risk biomarkers together with value intervals, providing a more comprehensive and realistic description of drug effects on a human population. The methodology was tested by predicting arrhythmogenic biomarkers on a series of twelve well-characterised marketed drugs, belonging to different arrhythmogenic risk classes.

Myocardial infarction detection using ITD, DWT and deterministic learning based on ECG signals

Nowadays, cardiovascular diseases (CVD) is one of the prime causes of human mortality, which has received tremendous and elaborative research interests regarding the prevention issue. Myocardial ischemia is a kind of CVD which will lead to myocardial infarction (MI). The diagnostic criterion of MI is supplemented with clinical judgement and several electrocardiographic (ECG) or vectorcardiographic (VCG) programs. However the visual inspection of ECG or VCG signals by cardiologists is tedious, laborious and subjective. To overcome such disadvantages, numerous MI detection techniques including signal processing and artificial intelligence tools have been developed. In this study, we propose a novel technique for automatic detection of MI based on disparity of cardiac system dynamics and synthesis of the standard 12-lead and Frank XYZ leads. First, 12-lead ECG signals are synthesized with Frank XYZ leads to build a hybrid 4-dimensional cardiac vector, which is decomposed into a series of proper rotation components (PRCs) by using the intrinsic time-scale decomposition (ITD) method. The novel cardiac vector may fully reflect the pathological alterations provoked by MI and may be correlated to the disparity of cardiac system dynamics between healthy and MI subjects. ITD is employed to measure the variability of cardiac vector and the first PRCs are extracted as predominant PRCs which contain most of the cardiac vector's energy. Second, four levels discrete wavelet transform with third-order Daubechies (db3) wavelet function is employed to decompose the predominant PRCs into different frequency bands, which combines with three-dimensional phase space reconstruction to derive features. The properties associated with the cardiac system dynamics are preserved. Since the frequency components above 40 Hz are lack of use in ECG analysis, in order to reduce the feature dimension, the advisable sub-band (D4) is selected for feature acquisition. Third, neural networks are then used to model, identify and classify cardiac system dynamics between normal (healthy) and MI cardiac vector signals. The difference of cardiac system dynamics between healthy control and MI cardiac vector is computed and used for the detection of MI based on a bank of estimators. Finally, experiments are carried out on the PhysioNet PTB database to assess the effectiveness of the proposed method, in which conventional 12-lead and Frank XYZ leads ECG signal fragments from 148 patients with MI and 52 healthy controls were extracted. By using the tenfold cross-validation style, the achieved average classification accuracy is reported to be 98.20%. Results verify the effectiveness of the proposed method which can serve as a potential candidate for the automatic detection of MI in the clinical application.

Fiber Organization has Little Effect on Electrical Activation Patterns during Focal Arrhythmias in the Left Atrium

Over the past two decades there has been a steady trend towards the development of realistic models of cardiac conduction with increasing levels of detail. However, making models more realistic complicates their personalization and use in clinical practice due to limited availability of tissue and cellular scale data. One such limitation is obtaining information about myocardial fiber organization in the clinical setting. In this study, we investigated a chimeric model of the left atrium utilizing clinically derived patient-specific atrial geometry and a realistic, yet foreign for a given patient fiber organization. We discovered that even significant variability of fiber organization had a relatively small effect on the spatio-temporal activation pattern during regular pacing. For a given pacing site, the activation maps were very similar across all fiber organizations tested.

Dynamical behavior analysis of the heart system by the bifurcation structures

The functioning of the heart rhythm can exhibit a wide variety of dynamic behaviours under certain conditions. In the case of rhythm disorders or cardiac arrhythmias, the natural rhythm of the heart is usually involved in the sinoatrial node, the atrioventricular node, the atria of the carotid sinus, etc. The study of heart related disorders requires an important analysis of its rhythm because the regularity of cardiac activity is conditioned by a large number of factors. The cardiac system is made up of a combination of nodes ranging from the sinus node, the atrioventricular node to its Purkinje bundles, which interact with each other via communicative aspects. Due to the nature of their respective dynamics, the above are treated as self-oscillating elements and modelled by nonlinear oscillators. By modelling the cardiac conduction system as a model of three nonlinear oscillators coupled by delayed connections and subjected to external stimuli depicting the behavior of a pacemaker, its dynamic behavior is studied in this paper by nonlinear analysis tools. From an electrocardiogram (ECG) assessment, the heart rhythm reveals normal and pathological rhythms. Three forms of ventricular fibrillation, ventricular flutter, ventricular tachycardia and atrial fibrillation are observed. The results are confirmed by the respective maximum Lyapunov exponents. Considering the cardiac nodes as microchips, using microcontroller simulation technology, the cardiac conduction system was modelled as a network of four ATmega 328P microcontrollers. A similarity with the results obtained numerically can be observed.

Credibility assessment of patient-specific computational modeling using patient-specific cardiac modeling as an exemplar

Reliable and robust simulation of individual patients using patient-specific models (PSMs) is one of the next frontiers for modeling and simulation (M&S) in healthcare. PSMs, which form the basis of digital twins, can be employed as clinical tools to, for example, assess disease state, predict response to therapy, or optimize therapy. They may also be used to construct virtual cohorts of patients, for in silico evaluation of medical product safety and/or performance. Methods and frameworks have recently been proposed for evaluating the credibility of M&S in healthcare applications. However, such efforts have generally been motivated by models of medical devices or generic patient models; how best to evaluate the credibility of PSMs has largely been unexplored. The aim of this paper is to understand and demonstrate the credibility assessment process for PSMs using patient-specific cardiac electrophysiological (EP) modeling as an exemplar. We first review approaches used to generate cardiac PSMs and consider how verification, validation, and uncertainty quantification (VVUQ) apply to cardiac PSMs. Next, we execute two simulation studies using a publicly available virtual cohort of 24 patient-specific ventricular models, the first a multi-patient verification study, the second investigating the impact of uncertainty in personalized and non-personalized inputs in a virtual cohort. We then use the findings from our analyses to identify how important characteristics of PSMs can be considered when assessing credibility with the approach of the ASME V&V40 Standard, accounting for PSM concepts such as inter- and intra-user variability, multi-patient and “every-patient” error estimation, uncertainty quantification in personalized vs non-personalized inputs, clinical validation, and others. The results of this paper will be useful to developers of cardiac and other medical image based PSMs, when assessing PSM credibility.

Patient-specific models are computational models that have been personalized using data from a patient. After decades of research, recent computational, data science and healthcare advances have opened the door to the fulfilment of the enormous potential of such models, from truly personalized medicine to efficient and cost-effective testing of new medical products. However, reliability (credibility) of patient-specific models is key to their success, and there are currently no general guidelines for evaluating credibility of patient-specific models. Here, we consider how frameworks and model evaluation activities that have been developed for generic (not patient-specific) computational models, can be extended to patient specific models. We achieve this through a detailed analysis of the activities required to evaluate cardiac electrophysiological models, chosen as an exemplar field due to its maturity and the complexity of such models. This is the first paper on the topic of reliability of patient-specific models and will help pave the way to reliable and trusted patient-specific modeling across healthcare applications.

Computational cardiac physiology for new modelers: Origins, foundations, and future

Mathematical models of the cardiovascular system have come a long way since they were first introduced in the early 19th century. Driven by a rapid development of experimental techniques, numerical methods, and computer hardware, detailed models that describe physical scales from the molecular level up to organs and organ systems have been derived and used for physiological research. Mathematical and computational models can be seen as condensed and quantitative formulations of extensive physiological knowledge and are used for formulating and testing hypotheses, interpreting and directing experimental research, and have contributed substantially to our understanding of cardiovascular physiology. However, in spite of the strengths of mathematics to precisely describe complex relationships and the obvious need for the mathematical and computational models to be informed by experimental data, there still exist considerable barriers between experimental and computational physiological research. In this review, we present a historical overview of the development of mathematical and computational models in cardiovascular physiology, including the current state of the art. We further argue why a tighter integration is needed between experimental and computational scientists in physiology, and point out important obstacles and challenges that must be overcome in order to fully realize the synergy of experimental and computational physiological research.

Design and execution of a verification, validation, and uncertainty quantification plan for a numerical model of left ventricular flow after LVAD implantation

Background

Left ventricular assist devices (LVADs) are implantable pumps that act as a life support therapy for patients with severe heart failure. Despite improving the survival rate, LVAD therapy can carry major complications. Particularly, the flow distortion introduced by the LVAD in the left ventricle (LV) may induce thrombus formation. While previous works have used numerical models to study the impact of multiple variables in the intra-LV stagnation regions, a comprehensive validation analysis has never been executed. The main goal of this work is to present a model of the LV-LVAD system and to design and follow a verification, validation and uncertainty quantification (VVUQ) plan based on the ASME V&V40 and V&V20 standards to ensure credible predictions.

Methods

The experiment used to validate the simulation is the SDSU cardiac simulator, a bench mock-up of the cardiovascular system that allows mimicking multiple operation conditions for the heart-LVAD system. The numerical model is based on Alya, the BSC’s in-house platform for numerical modelling. Alya solves the Navier-Stokes equation with an Arbitrary Lagrangian-Eulerian (ALE) formulation in a deformable ventricle and includes pressure-driven valves, a 0D Windkessel model for the arterial output and a LVAD boundary condition modeled through a dynamic pressure-flow performance curve. The designed VVUQ plan involves: (a) a risk analysis and the associated credibility goals; (b) a verification stage to ensure correctness in the numerical solution procedure; (c) a sensitivity analysis to quantify the impact of the inputs on the four quantities of interest (QoIs) (average aortic root flow Q A o a v g, maximum aortic root flow Q A o m a x, average LVAD flow Q V A D a v g, and maximum LVAD flow Q V A D m a x); (d) an uncertainty quantification using six validation experiments that include extreme operating conditions.

Results

Numerical code verification tests ensured correctness of the solution procedure and numerical calculation verification showed a grid convergence index (GCI)95% <3.3%. The total Sobol indices obtained during the sensitivity analysis demonstrated that the ejection fraction, the heart rate, and the pump performance curve coefficients are the most impactful inputs for the analysed QoIs. The Minkowski norm is used as validation metric for the uncertainty quantification. It shows that the midpoint cases have more accurate results when compared to the extreme cases. The total computational cost of the simulations was above 100 [core-years] executed in around three weeks time span in Marenostrum IV supercomputer.

Conclusions

This work details a novel numerical model for the LV-LVAD system, that is supported by the design and execution of a VVUQ plan created following recognised international standards. We present a methodology demonstrating that stringent VVUQ according to ASME standards is feasible but computationally expensive.

Exploring the Coordination of Cardiac Ion Channels With Action Potential Clamp Technique

The patch clamp technique underwent continual advancement and developed numerous variants in cardiac electrophysiology since its introduction in the late 1970s. In the beginning, the capability of the technique was limited to recording one single current from one cell stimulated with a rectangular command pulse. Since that time, the technique has been extended to record multiple currents under various command pulses including action potential. The current review summarizes the development of the patch clamp technique in cardiac electrophysiology with special focus on the potential applications in integrative physiology.

Space-time shape uncertainties in the forward and inverse problem of electrocardiography

In electrocardiography, the "classic" inverse problem is the reconstruction of electric potentials at a surface enclosing the heart from remote recordings at the body surface and an accurate description of the anatomy. The latter being affected by noise and obtained with limited resolution due to clinical constraints, a possibly large uncertainty may be perpetuated in the inverse reconstruction. The purpose of this work is to study the effect of shape uncertainty on the forward and the inverse problem of electrocardiography. To this aim, the problem is first recast into a boundary integral formulation and then discretised with a collocation method to achieve high convergence rates and a fast time to solution. The shape uncertainty of the domain is represented by a random deformation field defined on a reference configuration. We propose a periodic-in-time covariance kernel for the random field and approximate the Karhunen-Loève expansion using low-rank techniques for fast sampling. The space-time uncertainty in the expected potential and its variance is evaluated with an anisotropic sparse quadrature approach and validated by a quasi-Monte Carlo method. We present several numerical experiments on a simplified but physiologically grounded two-dimensional geometry to illustrate the validity of the approach. The tested parametric dimension ranged from 100 up to 600. For the forward problem, the sparse quadrature is very effective. In the inverse problem, the sparse quadrature and the quasi-Monte Carlo method perform as expected, except for the total variation regularisation, where convergence is limited by lack of regularity. We finally investigate an H1/2 regularisation, which naturally stems from the boundary integral formulation, and compare it to more classical approaches.

Interactive 3D Human Heart Simulations on Segmented Human MRI Hearts

Understanding cardiac arrhythmic mechanisms and developing new strategies to control and terminate them using computer simulations requires realistic physiological cell models with anatomically accurate heart structures. Furthermore, numerical simulations must be fast enough to study and validate model and structure parameters. Here, we present an interactive parallel approach for solving detailed cell dynamics in high-resolution human heart structures with a local PC's GPU. In vitro human heart MRI scans were manually segmented to produce 3D structures with anatomically realistic electrophysiology. The Abubu.js library was used to create an interactive code to solve the OVVR human ventricular cell model and the FDA extension of the model in the human MRI heart structures, allowing the simulation of reentrant waves and investigation of their dynamics in real time. Interactive simulations of a physiological cell model in a detailed anatomical human heart reveals propagation of waves through the fine structures of the trabeculae and pectinate muscle that can perpetuate arrhythmias, thereby giving new insights into effects that may need to be considered when planning ablation and other defibrillation methods.

Bayesian Calibration of Electrophysiology Models Using Restitution Curve Emulators

Calibration of cardiac electrophysiology models is a fundamental aspect of model personalization for predicting the outcomes of cardiac therapies, simulation testing of device performance for a range of phenotypes, and for fundamental research into cardiac function. Restitution curves provide information on tissue function and can be measured using clinically feasible measurement protocols. We introduce novel "restitution curve emulators" as probabilistic models for performing model exploration, sensitivity analysis, and Bayesian calibration to noisy data. These emulators are built by decomposing restitution curves using principal component analysis and modeling the resulting coordinates with respect to model parameters using Gaussian processes. Restitution curve emulators can be used to study parameter identifiability via sensitivity analysis of restitution curve components and rapid inference of the posterior distribution of model parameters given noisy measurements. Posterior uncertainty about parameters is critical for making predictions from calibrated models, since many parameter settings can be consistent with measured data and yet produce very different model behaviors under conditions not effectively probed by the measurement protocols. Restitution curve emulators are therefore promising probabilistic tools for calibrating electrophysiology models.

Scientific and regulatory evaluation of mechanistic in silico drug and disease models in drug development: Building model credibility

The value of in silico methods in drug development and evaluation has been demonstrated repeatedly and convincingly. While their benefits are now unanimously recognized, international standards for their evaluation, accepted by all stakeholders involved, are still to be established. In this white paper, we propose a risk‐informed evaluation framework for mechanistic model credibility evaluation. To properly frame the proposed verification and validation activities, concepts such as context of use, regulatory impact and risk‐based analysis are discussed. To ensure common understanding between all stakeholders, an overview is provided of relevant in silico terminology used throughout this paper. To illustrate the feasibility of the proposed approach, we have applied it to three real case examples in the context of drug development, using a credibility matrix currently being tested as a quick‐start tool by regulators. Altogether, this white paper provides a practical approach to model evaluation, applicable in both scientific and regulatory evaluation contexts.

Enabling forward uncertainty quantification and sensitivity analysis in cardiac electrophysiology by reduced order modeling and machine learning

We present a new, computationally efficient framework to perform forward uncertainty quantification (UQ) in cardiac electrophysiology. We consider the monodomain model to describe the electrical activity in the cardiac tissue, coupled with the Aliev-Panfilov model to characterize the ionic activity through the cell membrane. We address a complete forward UQ pipeline, including both: (i) a variance-based global sensitivity analysis for the selection of the most relevant input parameters, and (ii) a way to perform uncertainty propagation to investigate the impact of intra-subject variability on outputs of interest depending on the cardiac potential. Both tasks exploit stochastic sampling techniques, thus implying overwhelming computational costs because of the huge amount of queries to the high-fidelity, full-order computational model obtained by approximating the coupled monodomain/Aliev-Panfilov system through the finite element method. To mitigate this computational burden, we replace the full-order model with computationally inexpensive projection-based reduced-order models (ROMs) aimed at reducing the state-space dimensionality. Resulting approximation errors on the outputs of interest are finally taken into account through artificial neural network (ANN)-based models, enhancing the accuracy of the whole UQ pipeline. Numerical results show that the proposed physics-based ROMs outperform regression-based emulators relying on ANNs built with the same amount of training data, in terms of both numerical accuracy and overall computational efficiency.

The 'Digital Twin' to enable the vision of precision cardiology

Providing therapies tailored to each patient is the vision of precision medicine, enabled by the increasing ability to capture extensive data about individual patients. In this position paper, we argue that the second enabling pillar towards this vision is the increasing power of computers and algorithms to learn, reason, and build the 'digital twin' of a patient. Computational models are boosting the capacity to draw diagnosis and prognosis, and future treatments will be tailored not only to current health status and data, but also to an accurate projection of the pathways to restore health by model predictions. The early steps of the digital twin in the area of cardiovascular medicine are reviewed in this article, together with a discussion of the challenges and opportunities ahead. We emphasize the synergies between mechanistic and statistical models in accelerating cardiovascular research and enabling the vision of precision medicine.

Data-Driven Uncertainty Quantification for Cardiac Electrophysiological Models: Impact of Physiological Variability on Action Potential and Spiral Wave Dynamics

Computational modeling of cardiac electrophysiology (EP) has recently transitioned from a scientific research tool to clinical applications. To ensure reliability of clinical or regulatory decisions made using cardiac EP models, it is vital to evaluate the uncertainty in model predictions. Model predictions are uncertain because there is typically substantial uncertainty in model input parameters, due to measurement error or natural variability. While there has been much recent uncertainty quantification (UQ) research for cardiac EP models, all previous work has been limited by either: (i) considering uncertainty in only a subset of the full set of parameters; and/or (ii) assigning arbitrary variation to parameters (e.g., ±10 or 50% around mean value) rather than basing the parameter uncertainty on experimental data. In our recent work we overcame the first limitation by performing UQ and sensitivity analysis using a novel canine action potential model, allowing all parameters to be uncertain, but with arbitrary variation. Here, we address the second limitation by extending our previous work to use data-driven estimates of parameter uncertainty. Overall, we estimated uncertainty due to population variability in all parameters in five currents active during repolarization: inward potassium rectifier, transient outward potassium, L-type calcium, rapidly and slowly activating delayed potassium rectifier; 25 parameters in total (all model parameters except fast sodium current parameters). A variety of methods was used to estimate the variability in these parameters. We then propagated the uncertainties through the model to determine their impact on predictions of action potential shape, action potential duration (APD) prolongation due to drug block, and spiral wave dynamics. Parameter uncertainty had a significant effect on model predictions, especially L-type calcium current parameters. Correlation between physiological parameters was determined to play a role in physiological realism of action potentials. Surprisingly, even model outputs that were relative differences, specifically drug-induced APD prolongation, were heavily impacted by the underlying uncertainty. This is the first data-driven end-to-end UQ analysis in cardiac EP accounting for uncertainty in the vast majority of parameters, including first in tissue, and demonstrates how future UQ could be used to ensure model-based decisions are robust to all underlying parameter uncertainties.

Sensitivity analysis of an electrophysiology model for the left ventricle

Modelling the cardiac electrophysiology entails dealing with the uncertainties related to the input parameters such as the heart geometry and the electrical conductivities of the tissues, thus calling for an uncertainty quantification (UQ) of the results. Since the chambers of the heart have different shapes and tissues, in order to make the problem affordable, here we focus on the left ventricle with the aim of identifying which of the uncertain inputs mostly affect its electrophysiology. In a first phase, the uncertainty of the input parameters is evaluated using data available from the literature and the output quantities of interest (QoIs) of the problem are defined. According to the polynomial chaos expansion, a training dataset is then created by sampling the parameter space using a quasi-Monte Carlo method whereas a smaller independent dataset is used for the validation of the resulting metamodel. The latter is exploited to run a global sensitivity analysis with nonlinear variance-based indices and thus reduce the input parameter space accordingly. Thereafter, the uncertainty probability distribution of the QoIs are evaluated using a direct UQ strategy on a larger dataset and the results discussed in the light of the medical knowledge.

Deep learning-based reduced order models in cardiac electrophysiology

Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration. More importantly, these models feature coherent structures that propagate over time, such as wavefronts. These systems can hardly be reduced to lower dimensional problems by conventional reduced order models (ROMs) such as, e.g., the reduced basis method. This is primarily due to the low regularity of the solution manifold (with respect to the problem parameters), as well as to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To overcome this difficulty, in this paper we propose a new, nonlinear approach relying on deep learning (DL) algorithms—such as deep feedforward neural networks and convolutional autoencoders—to obtain accurate and efficient ROMs, whose dimensionality matches the number of system parameters. We show that the proposed DL-ROM framework can efficiently provide solutions to parametrized electrophysiology problems, thus enabling multi-scenario analysis in pathological cases. We investigate four challenging test cases in cardiac electrophysiology, thus demonstrating that DL-ROM outperforms classical projection-based ROMs.

Credible practice of modeling and simulation in healthcare: ten rules from a multidisciplinary perspective

The complexities of modern biomedicine are rapidly increasing. Thus, modeling and simulation have become increasingly important as a strategy to understand and predict the trajectory of pathophysiology, disease genesis, and disease spread in support of clinical and policy decisions. In such cases, inappropriate or ill-placed trust in the model and simulation outcomes may result in negative outcomes, and hence illustrate the need to formalize the execution and communication of modeling and simulation practices. Although verification and validation have been generally accepted as significant components of a model’s credibility, they cannot be assumed to equate to a holistic credible practice, which includes activities that can impact comprehension and in-depth examination inherent in the development and reuse of the models. For the past several years, the Committee on Credible Practice of Modeling and Simulation in Healthcare, an interdisciplinary group seeded from a U.S. interagency initiative, has worked to codify best practices. Here, we provide Ten Rules for credible practice of modeling and simulation in healthcare developed from a comparative analysis by the Committee’s multidisciplinary membership, followed by a large stakeholder community survey. These rules establish a unified conceptual framework for modeling and simulation design, implementation, evaluation, dissemination and usage across the modeling and simulation life-cycle. While biomedical science and clinical care domains have somewhat different requirements and expectations for credible practice, our study converged on rules that would be useful across a broad swath of model types. In brief, the rules are: (1) Define context clearly. (2) Use contextually appropriate data. (3) Evaluate within context. (4) List limitations explicitly. (5) Use version control. (6) Document appropriately. (7) Disseminate broadly. (8) Get independent reviews. (9) Test competing implementations. (10) Conform to standards. Although some of these are common sense guidelines, we have found that many are often missed or misconstrued, even by seasoned practitioners. Computational models are already widely used in basic science to generate new biomedical knowledge. As they penetrate clinical care and healthcare policy, contributing to personalized and precision medicine, clinical safety will require established guidelines for the credible practice of modeling and simulation in healthcare.

Predicting left ventricular contractile function via Gaussian process emulation in aortic-banded rats

Cardiac contraction is the result of integrated cellular, tissue and organ function. Biophysical in silico cardiac models offer a systematic approach for studying these multi-scale interactions. The computational cost of such models is high, due to their multi-parametric and nonlinear nature. This has so far made it difficult to perform model fitting and prevented global sensitivity analysis (GSA) studies. We propose a machine learning approach based on Gaussian process emulation of model simulations using probabilistic surrogate models, which enables model parameter inference via a Bayesian history matching (HM) technique and GSA on whole-organ mechanics. This framework is applied to model healthy and aortic-banded hypertensive rats, a commonly used animal model of heart failure disease. The obtained probabilistic surrogate models accurately predicted the left ventricular pump function (R² = 0.92 for ejection fraction). The HM technique allowed us to fit both the control and diseased virtual bi-ventricular rat heart models to magnetic resonance imaging and literature data, with model outputs from the constrained parameter space falling within 2 SD of the respective experimental values. The GSA identified Troponin C and cross-bridge kinetics as key parameters in determining both systolic and diastolic ventricular function. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Considering discrepancy when calibrating a mechanistic electrophysiology model

Uncertainty quantification (UQ) is a vital step in using mathematical models and simulations to take decisions. The field of cardiac simulation has begun to explore and adopt UQ methods to characterize uncertainty in model inputs and how that propagates through to outputs or predictions; examples of this can be seen in the papers of this issue. In this review and perspective piece, we draw attention to an important and under-addressed source of uncertainty in our predictions-that of uncertainty in the model structure or the equations themselves. The difference between imperfect models and reality is termed model discrepancy, and we are often uncertain as to the size and consequences of this discrepancy. Here, we provide two examples of the consequences of discrepancy when calibrating models at the ion channel and action potential scales. Furthermore, we attempt to account for this discrepancy when calibrating and validating an ion channel model using different methods, based on modelling the discrepancy using Gaussian processes and autoregressive-moving-average models, then highlight the advantages and shortcomings of each approach. Finally, suggestions and lines of enquiry for future work are provided. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

An audit of uncertainty in multi-scale cardiac electrophysiology models

Models of electrical activation and recovery in cardiac cells and tissue have become valuable research tools, and are beginning to be used in safety-critical applications including guidance for clinical procedures and for drug safety assessment. As a consequence, there is an urgent need for a more detailed and quantitative understanding of the ways that uncertainty and variability influence model predictions. In this paper, we review the sources of uncertainty in these models at different spatial scales, discuss how uncertainties are communicated across scales, and begin to assess their relative importance. We conclude by highlighting important challenges that continue to face the cardiac modelling community, identifying open questions, and making recommendations for future studies. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Quantification of model and data uncertainty in a network analysis of cardiac myocyte mechanosignalling

Cardiac myocytes transduce changes in mechanical loading into cellular responses via interacting cell signalling pathways. We previously reported a logic-based ordinary differential equation model of the myocyte mechanosignalling network that correctly predicts 78% of independent experimental results not used to formulate the original model. Here, we use Monte Carlo and polynomial chaos expansion simulations to examine the effects of uncertainty in parameter values, model logic and experimental validation data on the assessed accuracy of that model. The prediction accuracy of the model was robust to parameter changes over a wide range being least sensitive to uncertainty in time constants and most affected by uncertainty in reaction weights. Quantifying epistemic uncertainty in the reaction logic of the model showed that while replacing 'OR' with 'AND' reactions greatly reduced model accuracy, replacing 'AND' with 'OR' reactions was more likely to maintain or even improve accuracy. Finally, data uncertainty had a modest effect on assessment of model accuracy. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Creation and application of virtual patient cohorts of heart models

Patient-specific cardiac models are now being used to guide therapies. The increased use of patient-specific cardiac simulations in clinical care will give rise to the development of virtual cohorts of cardiac models. These cohorts will allow cardiac simulations to capture and quantify inter-patient variability. However, the development of virtual cohorts of cardiac models will require the transformation of cardiac modelling from small numbers of bespoke models to robust and rapid workflows that can create large numbers of models. In this review, we describe the state of the art in virtual cohorts of cardiac models, the process of creating virtual cohorts of cardiac models, and how to generate the individual cohort member models, followed by a discussion of the potential and future applications of virtual cohorts of cardiac models. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Practical Use of Regularization in Individualizing a Mathematical Model of Cardiovascular Hemodynamics Using Scarce Data

Individualizing physiological models to a patient can enable patient-specific monitoring and treatment in critical care environments. However, this task often presents a unique "practical identifiability" challenge due to the conflict between model complexity and data scarcity. Regularization provides an established framework to cope with this conflict by compensating for data scarcity with prior knowledge. However, regularization has not been widely pursued in individualizing physiological models to facilitate patient-specific critical care. Thus, the goal of this work is to garner potentially generalizable insight into the practical use of regularization in individualizing a complex physiological model using scarce data by investigating its effect in a clinically significant critical care case study of blood volume kinetics and cardiovascular hemodynamics in hemorrhage and circulatory resuscitation. We construct a population-average model as prior knowledge and individualize the physiological model via regularization to illustrate that regularization can be effective in individualizing a physiological model to learn salient individual-specific characteristics (resulting in the goodness of fit to individual-specific data) while restricting unnecessary deviations from the population-average model (achieving practical identifiability). We also illustrate that regularization yields parsimonious individualization of only sensitive parameters as well as adequate physiological plausibility and relevance in predicting internal physiological states.

Key aspects for effective mathematical modelling of fractional-diffusion in cardiac electrophysiology: a quantitative study

Microscopic structural features of cardiac tissue play a fundamental role in determining complex spatio-temporal excitation dynamics at the macroscopic level. Recent efforts have been devoted to the development of mathematical models accounting for non-local spatio-temporal coupling able to capture these complex dynamics without the need of resolving tissue heterogeneities down to the micro-scale. In this work, we analyse in detail several important aspects affecting the overall predictive power of these modelling tools and provide some guidelines for an effective use of space-fractional models of cardiac electrophysiology in practical applications. Through an extensive computational study in simplified computational domains, we highlight the robustness of models belonging to different categories, i.e., physiological and phenomenological descriptions, against the introduction of non-locality, and lay down the foundations for future research and model validation against experimental data. A modern genetic algorithm framework is used to investigate proper parameterisations of the considered models, and the crucial role played by the boundary assumptions in the considered settings is discussed. Several numerical results are provided to support our claims.

Sensitivity and Uncertainty Analysis of Two Human Atrial Cardiac Cell Models Using Gaussian Process Emulators

Biophysically detailed cardiac cell models reconstruct the action potential and calcium dynamics of cardiac myocytes. They aim to capture the biophysics of current flow through ion channels, pumps, and exchangers in the cell membrane, and are highly detailed. However, the relationship between model parameters and model outputs is difficult to establish because the models are both complex and non-linear. The consequences of uncertainty and variability in model parameters are therefore difficult to determine without undertaking large numbers of model evaluations. The aim of the present study was to demonstrate how sensitivity and uncertainty analysis using Gaussian process emulators can be used for a systematic and quantitive analysis of biophysically detailed cardiac cell models. We selected the Courtemanche and Maleckar models of the human atrial action potential for analysis because these models describe a similar set of currents, with different formulations. In our approach Gaussian processes emulate the main features of the action potential and calcium transient. The emulators were trained with a set of design data comprising samples from parameter space and corresponding model outputs, initially obtained from 300 model evaluations. Variance based sensitivity indices were calculated using the emulators, and first order and total effect indices were calculated for each combination of parameter and output. The differences between the first order and total effect indices indicated that the effect of interactions between parameters was small. A second set of emulators were then trained using a new set of design data with a subset of the model parameters with a sensitivity index of more than 0.1 (10%). This second stage analysis enabled comparison of mechanisms in the two models. The second stage sensitivity indices enabled the relationship between the L-type Ca 2+ current and the action potential plateau to be quantified in each model. Our quantitative analysis predicted that changes in maximum conductance of the ultra-rapid K + channel I Kur would have opposite effects on action potential duration in the two models, and this prediction was confirmed by additional simulations. This study has demonstrated that Gaussian process emulators are an effective tool for sensitivity and uncertainty analysis of biophysically detailed cardiac cell models.

Quantitative analysis of variability in an integrated model of human ventricular electrophysiology and β-adrenergic signaling

In ventricular myocytes, stimulation of β-adrenergic receptors activates critical cardiac signaling pathways, leading to shorter action potentials and increased contraction strength during the "fight-or-flight" response. These changes primarily result, at the cellular level, from the coordinated phosphorylation of multiple targets by protein kinase A. Although mathematical models of the intracellular signaling downstream of β-adrenergic receptor activation have previously been described, only a limited number of studies have explored quantitative interactions between intracellular signaling and electrophysiology in human ventricular myocytes. Accordingly, our objective was to develop an integrative mathematical model of β-adrenergic receptor signaling, electrophysiology, and intracellular calcium (Ca²⁺) handling in the healthy human ventricular myocyte. We combined published mathematical models of intracellular signaling and electrophysiology, then calibrated the model results against voltage clamp data and physiological changes occurring after stimulation of β-adrenergic receptors with isoproterenol. We subsequently: (1) explored how molecular variability in different categories of model parameters translated into phenotypic variability; (2) identified the most important parameters determining physiological cellular outputs in the model before and after β-adrenergic receptor stimulation; and (3) investigated which molecular level alterations can produce a phenotype indicative of heart failure with preserved ejection fraction (HFpEF). Major results included: (1) variability in parameters that controlled intracellular signaling caused qualitatively different behavior than variability in parameters controlling ion transport pathways; (2) the most important model parameters determining action potential duration and intracellular Ca²⁺ transient amplitude were generally consistent before and after β-adrenergic receptor stimulation, except for a shift in the importance of K⁺ currents in determining action potential duration; and (3) decreased Ca²⁺ uptake into the sarcoplasmic reticulum, increased Ca²⁺ extrusion through Na⁺/Ca²⁺ exchanger and decreased Ca²⁺ leak from the sarcoplasmic reticulum may contribute to HFpEF. Overall, this study provided novel insight into the phenotypic consequences of molecular variability, and our integrated model may be useful in the design and interpretation of future experimental studies of interactions between β-adrenergic signaling and cellular physiology in human ventricular myocytes.