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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., Competing Interests: The authors have declared that no competing interests exist.

Key Points: Mathematical and computational models of cardiac physiology have been an integral component of cardiac electrophysiology since its inception, and are collectively known as the Cardiac Physiome. We identify and classify the numerous sources of variability and uncertainty in model formulation, parameters and other inputs that arise from both natural variation in experimental data and lack of knowledge. The impact of uncertainty on the outputs of Cardiac Physiome models is not well understood, and this limits their utility as clinical tools. We argue that incorporating variability and uncertainty should be a high priority for the future of the Cardiac Physiome. We suggest investigating the adoption of approaches developed in other areas of science and engineering while recognising unique challenges for the Cardiac Physiome; it is likely that novel methods will be necessary that require engagement with the mathematics and statistics community., Abstract: The Cardiac Physiome effort is one of the most mature and successful applications of mathematical and computational modelling for describing and advancing the understanding of physiology. After five decades of development, physiological cardiac models are poised to realise the promise of translational research via clinical applications such as drug development and patient-specific approaches as well as ablation, cardiac resynchronisation and contractility modulation therapies. For models to be included as a vital component of the decision process in safety-critical applications, rigorous assessment of model credibility will be required. This White Paper describes one aspect of this process by identifying and classifying sources of variability and uncertainty in models as well as their implications for the application and development of cardiac models. We stress the need to understand and quantify the sources of variability and uncertainty in model inputs, and the impact of model structure and complexity and their consequences for predictive model outputs. We propose that the future of the Cardiac Physiome should include a probabilistic approach to quantify the relationship of variability and uncertainty of model inputs and outputs., (© 2016 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society.)

Cardiac electrophysiology models have been developed for over 50years, and now include detailed descriptions of individual ion currents and sub-cellular calcium handling. It is commonly accepted that there are many uncertainties in these systems, with quantities such as ion channel kinetics or expression levels being difficult to measure or variable between samples. Until recently, the original approach of describing model parameters using single values has been retained, and consequently the majority of mathematical models in use today provide point predictions, with no associated uncertainty. In recent years, statistical techniques have been developed and applied in many scientific areas to capture uncertainties in the quantities that determine model behaviour, and to provide a distribution of predictions which accounts for this uncertainty. In this paper we discuss this concept, which is termed uncertainty quantification, and consider how it might be applied to cardiac electrophysiology models. We present two case studies in which probability distributions, instead of individual numbers, are inferred from data to describe quantities such as maximal current densities. Then we show how these probabilistic representations of model parameters enable probabilities to be placed on predicted behaviours. We demonstrate how changes in these probability distributions across data sets offer insight into which currents cause beat-to-beat variability in canine APs. We conclude with a discussion of the challenges that this approach entails, and how it provides opportunities to improve our understanding of electrophysiology., (Copyright © 2015 The Authors. Published by Elsevier Ltd.. All rights reserved.)

Modeling of the heart ventricles is one of the most challenging tasks in soft tissue mechanics because cardiac tissue is a strongly anisotropic incompressible material with an active component of stress. In most current approaches with active force, the number of degrees of freedom (DOF) is limited by the direct method of solution of linear systems of equations. We develop a new approach for high-resolution heart models with large numbers of DOF by: (1) developing a hex-dominant finite element mixed formulation and (2) developing a Krylov subspace iterative method that is able to solve the system of linearized equations for saddle-point problems with active stress. In our approach, passive cardiac tissue is modeled as a hyperelastic, incompressible material with orthotropic properties, and mixed pressure-displacement finite elements are used to enforce incompressibility. Active stress is generated by a model with force dependence on length and velocity of muscle shortening. The ventricles are coupled to a lumped circulatory model. For efficient solution of linear systems, we use Flexible GMRES with a nonlinear preconditioner based on block matrix decomposition involving the Schur complement. Three methods for approximating the inverse of the Schur complement are evaluated: inverse of the pressure mass matrix; least squares commutators; and sparse approximate inverse. The sub-matrix corresponding to the displacement variables is preconditioned by a V-cycle of hybrid geometric-algebraic multigrid followed by correction with several iterations of GMRES preconditioned by sparse approximate inverse. The overall solver is demonstrated on a high-resolution two ventricle mesh based on a human anatomy with roughly 130 K elements and 1.7 M displacement DOF. Effectiveness of the numerical method for active contraction is shown. To the best of our knowledge, this solver is the first to efficiently model ventricular contraction using an iterative linear solver for the mesh size demonstrated and therefore opens the possibility for future very high-resolution models. In addition, several relatively simple benchmark problems are designed for a verification exercise to show that the solver is functioning properly and correctly solves the underlying mathematical model. Here, the output of the newly designed solver is compared to that of the mechanics component of Chaste ('Cancer, Heart and Soft Tissue Environment'). These benchmark tests may be used by other researchers to verify their newly developed methods and codes.

Perhaps the most mature area of multi-scale systems biology is the modelling of the heart. Current models are grounded in over fifty years of research in the development of biophysically detailed models of the electrophysiology (EP) of cardiac cells, but one aspect which is inadequately addressed is the incorporation of uncertainty and physiological variability. Uncertainty quantification (UQ) is the identification and characterisation of the uncertainty in model parameters derived from experimental data, and the computation of the resultant uncertainty in model outputs. It is a necessary tool for establishing the credibility of computational models, and will likely be expected of EP models for future safety-critical clinical applications. The focus of this paper is formal UQ of one major sub-component of cardiac EP models, the steady-state inactivation of the fast sodium current, INa. To better capture average behaviour and quantify variability across cells, we have applied for the first time an 'individual-based' statistical methodology to assess voltage clamp data. Advantages of this approach over a more traditional 'population-averaged' approach are highlighted. The method was used to characterise variability amongst cells isolated from canine epi and endocardium, and this variability was then 'propagated forward' through a canine model to determine the resultant uncertainty in model predictions at different scales, such as of upstroke velocity and spiral wave dynamics. Statistically significant differences between epi and endocardial cells (greater half-inactivation and less steep slope of steady state inactivation curve for endo) was observed, and the forward propagation revealed a lack of robustness of the model to underlying variability, but also surprising robustness to variability at the tissue scale. Overall, the methodology can be used to: (i) better analyse voltage clamp data; (ii) characterise underlying population variability; (iii) investigate consequences of variability; and (iv) improve the ability to validate a model. To our knowledge this article is the first to quantify population variability in membrane dynamics in this manner, and the first to perform formal UQ for a component of a cardiac model. The approach is likely to find much wider applicability across systems biology as current application domains reach greater levels of maturity., (Published by Elsevier Ltd.)

When modelling tissue-level cardiac electrophysiology, a continuum approximation to the discrete cell-level equations, known as the bidomain equations, is often used to maintain computational tractability. Analysing the derivation of the bidomain equations allows us to investigate how microstructure, in particular gap junctions that electrically connect cells, affect tissue-level conductivity properties. Using a one-dimensional cable model, we derive a modified form of the bidomain equations that take gap junctions into account, and compare results of simulations using both the discrete and continuum models, finding that the underlying conduction velocity of the action potential ceases to match up between models when gap junctions are introduced at physiologically realistic coupling levels. We show that this effect is magnified by: (i) modelling gap junctions with reduced conductivity; (ii) increasing the conductance of the fast sodium channel; and (iii) an increase in myocyte length. From this, we conclude that the conduction velocity arising from the bidomain equations may not be an accurate representation of the underlying discrete system. In particular, the bidomain equations are less likely to be valid when modelling certain diseased states whose symptoms include a reduction in gap junction coupling or an increase in myocyte length.

Background and Purpose: Understanding drug effects on the heart is key to safety pharmacology assessment and anti-arrhythmic therapy development. Here our goal is to demonstrate the ability of computational models to simulate the effect of drug action on the electrical activity of the heart, at the level of the ion-channel, cell, heart and ECG body surface potential., Experimental Approach: We use the state-of-the-art mathematical models governing the electrical activity of the heart. A drug model is introduced using an ion channel conductance block for the hERG and fast sodium channels, depending on the IC(50) value and the drug dose. We simulate the ECG measurements at the body surface and compare biomarkers under different drug actions., Key Results: Introducing a 50% hERG-channel current block results in 8% prolongation of the APD(90) and 6% QT interval prolongation, hERG block does not affect the QRS interval. Introducing 50% fast sodium current block prolongs the QRS and the QT intervals by 12% and 5% respectively, and delays activation times, whereas APD(90) is not affected., Conclusions and Implications: Both potassium and sodium blocks prolong the QT interval, but the underlying mechanism is different: for potassium it is due to APD prolongation; while for sodium it is due to a reduction of electrical wave velocity. This study shows the applicability of in silico models for the investigation of drug effects on the heart, from the ion channel to the ECG-based biomarkers., (© 2012 The Authors. British Journal of Pharmacology © 2012 The British Pharmacological Society.)

The bidomain framework has been extensively used to model tissue electrophysiology in a variety of applications. One limitation of the bidomain model is that it describes the activity of only one cell type interacting with the extracellular space. If more than one cell type contributes to the tissue electrophysiology, then the bidomain model is not sufficient. Recently, evidence has suggested that this is the case for at least two important applications: cardiac and gastrointestinal tissue electrophysiology. In the heart, fibroblasts ubiquitously interact with myocytes and are believed to play an important role in the organ electrophysiology. Along the GI tract, interstitial cells of Cajal (ICC) generate electrical waves that are passed on to surrounding smooth muscle cells (SMC), which are interconnected with the ICC and with each other. Because of the contribution of more than one cell type to the overall organ electrophysiology, investigators in different fields have independently proposed similar extensions of the bidomain model to incorporate multiple cell types and tested it on simplified geometries. In this paper, we provide a general derivation of such an extended bidomain framework applicable to any tissue and provide a generic and efficient implementation applicable to any geometry. Proof-of-concept results of tissue electrophysiology on realistic 3D organ geometries using the extended bidomain framework are presented for the heart and the stomach., (This journal is © The Royal Society of Chemistry 2012)

Ongoing developments in cardiac modelling have resulted, in particular, in the development of advanced and increasingly complex computational frameworks for simulating cardiac tissue electrophysiology. The goal of these simulations is often to represent the detailed physiology and pathologies of the heart using codes that exploit the computational potential of high-performance computing architectures. These developments have rapidly progressed the simulation capacity of cardiac virtual physiological human style models; however, they have also made it increasingly challenging to verify that a given code provides a faithful representation of the purported governing equations and corresponding solution techniques. This study provides the first cardiac tissue electrophysiology simulation benchmark to allow these codes to be verified. The benchmark was successfully evaluated on 11 simulation platforms to generate a consensus gold-standard converged solution. The benchmark definition in combination with the gold-standard solution can now be used to verify new simulation codes and numerical methods in the future.

Simulation of cardiac electrical activity using the bi-domain equations can be a massively computationally demanding problem. This study provides a comprehensive guide to numerical bi-domain modelling. Each component of bi-domain simulations--discretization, ODE-solution, linear system solution, and parallelization--is discussed, and previously-used methods are reviewed, new methods are proposed, and issues which cause particular difficulty are highlighted. Particular attention is paid to the choice of stimulus currents, compatibility conditions for the equations, the solution of singular linear systems, and convergence of the numerical scheme., (2010 Elsevier Ltd. All rights reserved.)

Soft tissue deformation is often modelled using incompressible non-linear elasticity, with solutions computed using the finite element method. There are a range of options available when using the finite element method, in particular the polynomial degree of the basis functions used for interpolating position and pressure, and the type of element making up the mesh. The effect of these choices on the accuracy of the computed solution is investigated, using a selection of model problems motivated by typical deformations seen in soft tissue modelling. Model problems are set up with discontinuous material properties (as is the case for the breast), steeply changing gradients in the body force (as found in contracting cardiac tissue), and discontinuous first derivatives in the solution at the boundary, caused by a discontinuous applied force (as in the breast during mammography). It was found that the choice of pressure basis functions is vital in the presence of a material interface, higher-order schemes do not perform as well as may be expected when there are sharp gradients, and in general it is important to take the expected regularity of the solution into account when choosing a numerical scheme. [ABSTRACT FROM AUTHOR]