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'.

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'.

Mechanism of doxorubicin cardiotoxicity evaluated by integrating multiple molecular effects into a biophysical model.

BACKGROUND AND PURPOSE: Doxorubicin (DOX) is an effective cancer therapeutic agent but causes therapy-limiting cardiotoxicity. The effects of DOX and its metabolite doxorubicinol (DOXL) on individual channels have been well characterized in isolation. However, it is unknown how the action and interaction of affected channels combine to generate the phenotypic cardiotoxic outcome. We sought to develop an in silico model that links drug effects on channels to action potential duration (APD) and intracellular Ca2+ concentration in order to address this gap in knowledge. EXPERIMENTAL APPROACH: We first propose two methods to obtain, from published values, consensus drug effects on the currents of individual channels, transporters and pumps. Separately, we obtained equivalent values for APD and Ca2+ concentration (the readouts used as surrogates for cardiotoxicity). Once derived, the consensus effects on the currents were incorporated into established biophysical models of the cardiac myocyte and were refined adjusting the sarcoplasmic reticulum Ca2+ leak current (ILeak ) until the consensus effects on APD and Ca2+ dynamics were replicated. Using factorial analysis, we then quantified the relative contribution of each channel to DOX and DOXL cardiotoxicity. KEY RESULTS: The factorial analysis identified the rapid delayed rectifying K+ current, the L-type Ca2+ current and the sarcoplasmic reticulum ILeak as the targets primarily responsible for the cardiotoxic effects on APD and Ca2+ dynamics. CONCLUSIONS AND IMPLICATIONS: This study provides insight into the mechanisms of DOX-induced cardiotoxicity and a framework for the development of future diagnostic and therapeutic strategies.

Image-Based Personalization of Cardiac Anatomy for Coupled Electromechanical Modeling.

Computational models of cardiac electromechanics (EM) are increasingly being applied to clinical problems, with patient-specific models being generated from high fidelity imaging and used to simulate patient physiology, pathophysiology and response to treatment. Current structured meshes are limited in their ability to fully represent the detailed anatomical data available from clinical images and capture complex and varied anatomy with limited geometric accuracy. In this paper, we review the state of the art in image-based personalization of cardiac anatomy for biophysically detailed, strongly coupled EM modeling, and present our own tools for the automatic building of anatomically and structurally accurate patient-specific models. Our method relies on using high resolution unstructured meshes for discretizing both physics, electrophysiology and mechanics, in combination with efficient, strongly scalable solvers necessary to deal with the computational load imposed by the large number of degrees of freedom of these meshes. These tools permit automated anatomical model generation and strongly coupled EM simulations at an unprecedented level of anatomical and biophysical detail.

Interpreting genetic effects through models of cardiac electromechanics.

Multiscale models of cardiac electromechanics are being increasingly focused on understanding how genetic variation and environment underpin multiple disease states. In this paper we review the current state of the art in both the development of specific models and the physiological insights they have produced. This growing research body includes the development of models for capturing the effects of changes in function in both single and multiple proteins in both specific expression systems and in vivo contexts. Finally, the potential for using this approach for ultimately predicting phenotypes from genetic sequence information is discussed.

At the heart of computational modelling.

The link between experimental data and biophysically based mathematical models is key to computational simulation meeting its potential to provide physiological insight. However, despite the importance of this link, scrutiny and analysis of the processes by which models are parameterised from data are currently lacking. While this situation is common to many areas of physiological modelling, to provide a concrete context, we use examples drawn from detailed models of cardiac electro-mechanics. Using this biophysically detailed cohort of models we highlight the specific issues of model parameterization and propose this process can be separated into three stages: observation, fitting and validation. Finally, future research challenges and directions in this area are discussed.

Computational modelling of cardiac electrophysiology: explanation of the variability of results from different numerical solvers.

A recent verification study compared 11 large-scale cardiac electrophysiology solvers on an unambiguously defined common problem. An unexpected amount of variation was observed between the codes, including significant error in conduction velocity in the majority of the codes at certain spatial resolutions. In particular, the results of the six finite element codes varied considerably despite each using the same order of interpolation. In this present study, we compare various algorithms for cardiac electrophysiological simulation, which allows us to fully explain the differences between the solvers. We identify the use of mass lumping as the fundamental cause of the largest variations, specifically the combination of the commonly used techniques of mass lumping and operator splitting, which results in a slightly different form of mass lumping to that supported by theory and leads to increased numerical error. Other variations are explained through the manner in which the ionic current is interpolated. We also investigate the effect of different forms of mass lumping in various types of simulation.

Calcium dynamics in the ventricular myocytes of SERCA2 knockout mice: A modeling study.

We describe a simulation study of Ca²(+) dynamics in mice with cardiomyocyte-specific conditional excision of the sarco(endo)plasmic reticulum calcium ATPase (SERCA) gene, using an experimental data-driven biophysically-based modeling framework. Previously, we reported a moderately impaired heart function measured in mice at 4 weeks after SERCA2 gene deletion (knockout (KO)), along with a >95% reduction in the level of SERCA2 protein. We also reported enhanced Ca²(+) flux through the L-type Ca²(+) channels and the Na(+)/Ca²(+) exchanger in ventricular myocytes isolated from these mice, compared to the control Serca2(flox/flox) mice (flox-flox (FF)). In the current study, a mathematical model-based analysis was applied to enable further quantitative investigation into changes in the Ca²(+) handling mechanisms in these KO cardiomyocytes. Model parameterization based on a wide range of experimental measurements showed a 67% reduction in SERCA activity and an over threefold increase in the activity of the Na(+)/Ca²(+) exchanger. The FF and KO models were then validated against experimentally measured [Ca²(+)](i) transients and experimentally estimated sarco(endo)plasmic reticulum (SR) function. Simulation results were in quantitative agreement with experimental measurements, confirming that sustained [Ca²(+)](i) transients could be maintained in the KO cardiomyocytes despite severely impaired SERCA function. In silico analysis shows that diastolic [Ca²(+)](i) rises sharply with progressive reductions in SERCA activity at physiologically relevant pacing frequencies. Furthermore, an analysis of the roles of the compensatory mechanisms revealed that the major combined effect of the compensatory mechanisms is to lower diastolic [Ca²(+)](i). Finally, by using a comprehensive sensitivity analysis of the role of all cellular calcium handling mechanisms, we show that the combination of upregulation of the Na(+)/Ca²(+) exchanger and increased L-type Ca²(+) current is the most effective means to maintain diastolic and systolic calcium levels after loss of SERCA function.

Coupling multi-physics models to cardiac mechanics.

We outline and review the mathematical framework for representing mechanical deformation and contraction of the cardiac ventricles, and how this behaviour integrates with other processes crucial for understanding and modelling heart function. Building on general conservation principles of space, mass and momentum, we introduce an arbitrary Eulerian-Lagrangian framework governing the behaviour of both fluid and solid components. Exploiting the natural alignment of cardiac mechanical properties with the tissue microstructure, finite deformation measures and myocardial constitutive relations are referred to embedded structural axes. Coupling approaches for solving this large deformation mechanics framework with three dimensional fluid flow, coronary hemodynamics and electrical activation are described. We also discuss the potential of cardiac mechanics modelling for clinical applications.

A mathematical model of the murine ventricular myocyte: a data-driven biophysically based approach applied to mice overexpressing the canine NCX isoform.

Mathematical modeling of Ca(2+) dynamics in the heart has the potential to provide an integrated understanding of Ca(2+)-handling mechanisms. However, many previous published models used heterogeneous experimental data sources from a variety of animals and temperatures to characterize model parameters and motivate model equations. This methodology limits the direct comparison of these models with any particular experimental data set. To directly address this issue, in this study, we present a biophysically based model of Ca(2+) dynamics directly fitted to experimental data collected in left ventricular myocytes isolated from the C57BL/6 mouse, the most commonly used genetic background for genetically modified mice in studies of heart diseases. This Ca(2+) dynamics model was then integrated into an existing mouse cardiac electrophysiology model, which was reparameterized using experimental data recorded at consistent and physiological temperatures. The model was validated against the experimentally observed frequency response of Ca(2+) dynamics, action potential shape, dependence of action potential duration on cycle length, and electrical restitution. Using this framework, the implications of cardiac Na(+)/Ca(2+) exchanger (NCX) overexpression in transgenic mice were investigated. These simulations showed that heterozygous overexpression of the canine cardiac NCX increases intracellular Ca(2+) concentration transient magnitude and sarcoplasmic reticulum Ca(2+) loading, in agreement with experimental observations, whereas acute overexpression of the murine cardiac NCX results in a significant loss of Ca(2+) from the cell and, hence, depressed sarcoplasmic reticulum Ca(2+) load and intracellular Ca(2+) concentration transient magnitude. From this analysis, we conclude that these differences are primarily due to the presence of allosteric regulation in the canine cardiac NCX, which has not been observed experimentally in the wild-type mouse heart.

Modelling and measuring electromechanical coupling in the rat heart.

Tension-dependent binding of Ca(2+) to troponin C in the cardiac myocyte has been shown to play an important role in the regulation of Ca(2+) and the activation of tension development. The significance of this regulatory mechanism is quantified experimentally by the quantity of Ca(2+) released following a rapid change in the muscle length. Using a computational, coupled, electromechanics cell model, we have confirmed that the tension dependence of Ca(2+) binding to troponin C, rather than cross-bridge kinetics or the rate of Ca(2+) uptake by the sarcoplasmic reticulum, determines the quantity of Ca(2+) released following a length step. This cell model has been successfully applied in a continuum model of the papillary muscle to analyse experimental data, suggesting the tension-dependent binding of Ca(2+) to troponin C as the likely pathway through which the effects of localized impaired tension generation alter the Ca(2+) transient. These experimental results are qualitatively reproduced using a three-dimensional coupled electromechanics model. Furthermore, the model predicts that changes in the Ca(2+) transient in the viable myocardium surrounding the impaired region are amplified in the absence of tension-dependent binding of Ca(2+) to troponin C.

A meta-analysis of cardiac electrophysiology computational models.

Computational models of cardiac electrophysiology are exemplar demonstrations of the integration of multiple data sets into a consistent biophysical framework. These models encapsulate physiological understanding to provide quantitative predictions of function. The combination or extension of existing models within a common framework allows integrative phenomena in larger systems to be investigated. This methodology is now routinely applied, as demonstrated by the increasing number of studies which use or extend previously developed models. In this study, we present a meta-analysis of this model re-use for two leading models of cardiac electrophysiology in the form of parameter inheritance trees, a sensitivity analysis and a comparison of the functional significance of the sodium potassium pump for defining restitution curves. These results indicate that even though the models aim to represent the same physiological system, both the sources of parameter values and the function of equivalent components are significantly different.

Energetic consequences of mechanical loads.

In this brief review, we have focussed largely on the well-established, but essentially phenomenological, linear relationship between the energy expenditure of the heart (commonly assessed as the oxygen consumed per beat, oxygen consumption (VO2)) and the pressure-volume-area (PVA, the sum of pressure-volume work and a specified 'potential energy' term). We raise concerns regarding the propriety of ignoring work done during 'passive' ventricular enlargement during diastole as well as the work done against series elasticity during systole. We question the common assumption that the rate of basal metabolism is independent of ventricular volume, given the equally well-established Feng- or stretch-effect. Admittedly, each of these issues is more of conceptual than of quantitative import. We point out that the linearity of the enthalpy-PVA relation is now so well established that observed deviations from linearity are often ignored. Given that a one-dimensional equivalent of the linear VO2-PVA relation exists in papillary muscles, it seems clear that the phenomenon arises at the cellular level, rather than being a property of the intact heart. This leads us to discussion of the classes of crossbridge models that can be applied to the study of cardiac energetics. An admittedly superficial examination of the historical role played by Hooke's Law in theories of muscle contraction foreshadows deeper consideration of the thermodynamic constraints that must, in our opinion, guide the development of any mathematical model. We conclude that a satisfying understanding of the origin of the enthalpy-PVA relation awaits the development of such a model.

Measuring and modeling chloride-hydroxyl exchange in the Guinea-pig ventricular myocyte.

Protons are powerful modulators of cardiac function. Their intracellular concentration is regulated by sarcolemmal ion transporters that export or import H+-ions (or their ionic equivalent: HCO3-, OH-). One such transporter, which imports H+-equivalents, is a putative Cl-/OH- exchanger (CHE). A strong candidate for CHE is SLC26A6 protein, a product of the SLC26A gene family of anion transporters, which has been detected in murine heart. SLC26A6 protein is suggested to be an electrogenic 1Cl-/2OH-(2HCO3-) exchanger. Unfortunately, there is insufficient characterization of cardiac CHE against which the properties of heterologously expressed SLC26A6 can be matched. We therefore investigated the proton, Cl-, and voltage dependence of CHE activity in guinea-pig ventricular myocytes, using voltage-clamp, intracellular pH fluorescence, and mathematical modeling techniques. We find that CHE activity is tightly regulated by intracellular and extracellular pH, is voltage-insensitive over a wide range (+/-80 mV), and displays substrate dependence suggestive of electroneutral 1Cl-/1OH- exchange. These properties exclude electrogenic SLC26A6 as sole contributor to CHE. Either the SLC26A6 product in heart is electroneutral, or CHE comprises at least two transporters with oppositely balanced voltage sensitivity. Alternatively, CHE may comprise an H+-Cl- coinflux system, which cannot be distinguished kinetically from an exchanger. Irrespective of ionic mechanism, CHE's pH sensitivity helps to define resting intracellular pH, and hence basal function in the heart.

A quantitative analysis of cardiac myocyte relaxation: a simulation study.

The determinants of relaxation in cardiac muscle are poorly understood, yet compromised relaxation accompanies various pathologies and impaired pump function. In this study, we develop a model of active contraction to elucidate the relative importance of the [Ca2+]i transient magnitude, the unbinding of Ca2+ from troponin C (TnC), and the length-dependence of tension and Ca2+ sensitivity on relaxation. Using the framework proposed by one of our researchers, we extensively reviewed experimental literature, to quantitatively characterize the binding of Ca2+ to TnC, the kinetics of tropomyosin, the availability of binding sites, and the kinetics of crossbridge binding after perturbations in sarcomere length. Model parameters were determined from multiple experimental results and modalities (skinned and intact preparations) and model results were validated against data from length step, caged Ca2+, isometric twitches, and the half-time to relaxation with increasing sarcomere length experiments. A factorial analysis found that the [Ca2+]i transient and the unbinding of Ca2+ from TnC were the primary determinants of relaxation, with a fivefold greater effect than that of length-dependent maximum tension and twice the effect of tension-dependent binding of Ca2+ to TnC and length-dependent Ca2+ sensitivity. The affects of the [Ca2+]i transient and the unbinding rate of Ca2+ from TnC were tightly coupled with the effect of increasing either factor, depending on the reference [Ca2+]i transient and unbinding rate.

Computing work in the ischemic heart.

This study analyzes the effect of a poorly perfused (ischemic) region on active tension and strain at a point in a model of the cardiac ventricles. The cellular active mechanics and electrophysiology were calculated using a coupled electromechanical cell model. The deformation of the heart was calculated combining an anisotropic constitutive law with the active tension generated by the cell models. An ischemic cell model was developed by adapting the Noble 98-HMT cell model parameters. The ischemic model exhibits a raised resting potential, shortened action potential duration and reduced amplitude. Ischemic cells were introduced into the heart model in the anterior wall of the left ventricle. The model was solved with and in the absence of the ischemic region. The stress and strains at a point located in the ischemic region were calculated. The ischemic cell model produced negligible amounts of active tension and this reduced the work load of ischemic cells by several orders of magnitude. The ischemic region had a significant effect on pump function in both the left and right ventricles.