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Cardiac Modeling
Cardiac Modeling A Bueno-Orovio, O Britton, A Muszkiewicz, and B Rodriguez, University of Oxford, Oxford, UK r 2016 Elsevier Inc. All rights reserved.
State-of-the-Art in Cardiac Cell Modeling In 1960, Denis Noble developed the first mathematical model of the electrophysiological activity of a cardiac cell (Noble, 1960). Since then, the field of computational cardiovascular science has developed into a broad and productive area of research with growing demonstration of translational potential into industrial, regulatory, and clinical applications (Sager et al., 2014). Computational heart models are multiscale both spatially and temporally, and integrate information across subcellular, cellular, tissue, and whole organ levels, and temporal scales (ranging from picoseconds to years, from ion channels opening and closing to remodeling in disease). Through continuous use and refinement, computational cardiac cell models have built a strong basis for their usefulness and credibility in a variety of settings, including investigations of genetic mutations, disease, and pharmacology (Clancy et al., 2007; Sarkar and Sobie, 2011; O’Hara and Rudy, 2012; Wilhelms et al., 2012a,b; Britton et al., 2013; Zemzemi et al., 2013; Polak et al., 2014). Cardiac cells (cardiomyocytes) are electrically excitable cells due to the presence of proteins in their membrane which act as mechanisms of ionic transport, such as ion channels, pumps, and exchangers. The transmembrane transport of ions such as potassium, sodium, and calcium results in electrical and concentration gradients between the intracellular and the extracellular media. Under physiological conditions with no external electrical stimulation, cardiomyocytes maintain a negative voltage relative to their environment. However, when stimulated, sodium channels in the cell membrane are activated, altering the balance of currents into the cell and acting to rapidly change the electrical potential across the membrane. This triggers the release of calcium ions from a cellular compartment known as the sarcoplasmic reticulum, which raises the bulk cytoplasmic concentration of calcium and signals the myocyte to contract. The ion channels initially responsible for firing the action potential inactivate, while others (potassium) act to repolarize the myocyte back to its resting potential. This allows the cardiomyocyte to fire another action potential during the next heartbeat. In this manner, the cardiac action potential, first observed by Coraboeuf and Weidmann (1949), allows the heart to control its own pattern of muscular contraction without external stimuli. The action potential generated by each cardiomyocyte can excite nearby myocytes, and thus the electrical signals propagate through the heart. In the last of a landmark series of papers, Hodgkin and Huxley (1952) showed that their mathematical model of the ionic currents of the isolated squid giant axon could recreate many of the axon's key electrical properties. This was the first mathematical model of a neuronal action potential. Their formulation of ion channel kinetics was promptly adapted to create the first action potential model of a cardiac cell (Noble, 1960). The Hodgkin–Huxley model formulates each ionic current as the product of the permeability of the cell
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membrane to the ion type of that current (more commonly called the conductance of that current), multiplied by the difference between the membrane potential and the equilibrium potential for that current (the membrane potential at which the net current flow across the membrane for that ion is zero). The conductances are functions that may depend on membrane potential and time, as well as channel-specific constants. They represent the opening, closing, and inactivation of the ion channels that carry current across the cell membrane. Cardiac cell electrophysiology models, from the very beginning of the field up to the present day, have used the Hodgkin–Huxley formulation to describe the ionic currents in the model. An alternative approach to describe ion channel gating is the use of Markov chain models (Greenstein et al., 2000; Mazhari et al., 2001; Clancy and Rudy, 2002), which model the transition between the open, closed, and inactive states of the channel, based on the probabilistic transition rates between channel states at different transmembrane potentials. From the earliest models that were very closely based on the original three current Hodgkin–Huxley model, new models have increased in complexity, based on our growing understanding of cardiac cellular electrophysiology, for example, the discovery of calcium channels. In general, the additions made in each new model have been in one or more of the following forms: (1) additions of new currents carrying ions such as calcium, (2) addition of new gating mechanisms for existing currents, (3) subdividing a single current into multiple currents either because of different time scales for the parts of the current (e.g., the fast and late sodium currents) and/or because different ion channel proteins carry the different components (e.g., slow and rapid delayed rectifier potassium currents), (4) addition of mechanisms relating to intracellular ionic concentrations (e.g., adding ion transporters such as the sodium–potassium pump or new cellular compartments such as the sarcoplasmic reticulum for calcium handling), and finally (5) adding biochemical mechanisms for ion channel modification, contraction, or signaling, such as, for example, models of beta-adrenergic stimulation (Kuzumoto et al., 2008; Heijman et al., 2011) or ion channel regulation by calcium/calmodulin-dependent protein kinase II (Christensen et al., 2009; O’Hara et al., 2011; Koval et al., 2012). The formulation of the different ion channels and transporters is usually inherited between models, although particularized for the specific cell type and species under study (Niederer et al., 2009; Bueno-Orovio et al., 2014b). This is illustrated in Figure 1 for the particular case of the sodium– potassium pump. Computational cardiac models have played a pivotal historical role in the discovery of new ionic mechanisms through successes and failures in reproducing and challenging experimental evidence (Noble, 2011). As in silico models and simulation techniques developed both conceptually and
Encyclopedia of Cell Biology, Volume 4
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Figure 1 Inheritance in the formulation of the sodium–potassium pump current in a selection of models of cellular cardiac electrophysiology. Models are referred by first author's name and year of publication. The complete list of references can be found in (Noble et al., 2012). Reproduced with permission from Bueno-Orovio, A., Sánchez, C., Pueyo, E., Rodriguez, B., 2014b. Na/K pump regulation of cardiac repolarization: Insights from a systems biology approach. Pflügers Archiv: European Journal of Physiology 466, 183–193.
technically, the questions of verification, validation, and uncertainty quantification have become crucial for their uptake beyond scientific research into industrial, clinical, and regulatory arenas (Carusi et al., 2012; Sager et al., 2014; Wallman et al., 2014; Pathmanathan et al., 2015). The process of validation of in silico models and simulations is complex and still under investigation, but it is essential to increase the credibility of in silico methods as powerful techniques to augment the information extracted from experimental and clinical investigations. The paper by Carusi et al. (2012) specifically explores the issue of validation of computational physiological models in the context of their construction and applications. It highlights the iterative nature of the process of validation and the fact that it needs to consider the ensemble of equations, parameters, simulation techniques, software, and experiments used for a particular study or purpose, namely the Model-Simulation-Experiment system. The aim is to augment the information on the physiological system or process under investigation, such as, for example, the mode of action of a new medicine on specific organs. The dynamics of the iterative process are driven by advances in both experimental and computational techniques, as well as investigations on their combined use. The legacy of such intense work is over 40 available mathematical models of cardiac cellular electrophysiology, for
different species and cell types (see Noble et al. (2012) for a comprehensive historical overview). The iterative process between models, simulations, and experiments leads to the dynamic replacement, refinement, and expansion of these models. In this context, initiatives such as the CellML repository provide the platform that allows the exchange and widespread use of cardiac cell models within the scientific community (Beard et al., 2009).
Human Models of Cardiac Cell Electrophysiology Computer modeling of the human heart has been a key priority for the scientific community as it provides a vital tool for a better understanding of human health and disease in clinical and industrial applications. Over the last decades a number of models of human adult electrophysiology have been proposed for both ventricular and atrial cells. Biophysically detailed models represent biophysical processes such as ion transport across the membrane and intracellular calcium handling, which are described by equations that represent individual ionic currents and fluxes. A different approach is based on phenomenological models, which reproduce the electrophysiological behavior of cardiac cells without detailed modeling of the underlying biophysical processes. These models
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Figure 2 Schematic diagram representations of models of human cellular cardiac electrophysiology. (a) O’Hara–Rudy model of the human ventricular cardiomyocyte. (b) Comparison of different models of human atrial cellular electrophysiology. Reproduced with permission from O’Hara, T., Virág, L., Varró, A., Rudy, Y., 2011. Simulation of the undiseased human cardiac ventricular action potential: Model formulation and experimental validation. PLoS Computational Biology 7, e1002061; Wilhelms, M., Hettmann, H., Maleckar, M.M., et al., 2012a. Benchmarking electrophysiological models of human atrial myocytes. Frontiers in Physiology 3, 487.
require fewer equations and parameters, and are ideal for applications that do not necessarily require the representation of ionic processes, such as the impact of cellular level properties on the cardiac electrical propagation at the tissue level. Several human ventricular and atrial cell models have been proposed and compared over the last two decades. Their structure is illustrated in Figure 2, through a schematic representation of the O’Hara–Rudy human ventricular model (O’Hara et al., 2011), and a graphical comparison of the ionic currents included in different human atrial models (Wilhelms et al., 2012a,b). Human ventricular cell models include the biophysically detailed Priebe and Beuckelmann (Priebe and Beuckelmann, 1998), Iyer (Iyer et al., 2004), ten Tusscher and Panfilov (ten Tusscher et al., 2004; ten Tusscher and Panfilov, 2006), Grandi (Grandi et al., 2010), Carro (Carro et al., 2011), O'Hara–Rudy (O’Hara et al., 2011), and Asakura (Asakura et al., 2014) models, as well as phenomenological models for human ventricular myocardium (Bueno-Orovio et al., 2008; Bueno-Orovio et al., 2012). Equivalently, several models of human atrial electrophysiology have been published such as the Nygren (Nygren et al., 1998), Courtemanche (Courtemanche et al., 1998), Maleckar (Maleckar et al., 2009), Grandi (Grandi et al., 2011), and Koivumäki (Koivumäki et al., 2011) models. The phenomenological Bueno-Orovio model has also been adapted to reproduce characteristic action potentials of human atrial tissue (Weber et al., 2008). Comprehensive comparisons between these ventricular and atrial human models can be found, for example, in Bueno-Orovio et al. (2008), Wilhelms et al. (2012a,b), and Elshrif and Cherry (2014). Modern human cardiac cell models consist of a system of ordinary differential equations, generally based on the Hodgkin and Huxley or Markov chain formulations for ionic gating variables. As shown in Figure 2, a number of biophysical components are represented, including ionic currents and cellular compartments. Ionic currents that are usually included in human ventricular cell models are the sodium current (INa), sometimes divided on its fast (INaf) and late (INaL) components; the L-type calcium current (ICaL); the transient outward potassium current (Ito); the rapid (IKr) and slow (IKs)
delayed rectifier potassium currents; the inward rectifier potassium current (IK1); the sodium–calcium exchanger current (INCX); and the sodium–potassium pump current (INaK). Human atrial cell models incorporate additional ionic currents specific to atrial electrophysiology, such as the ultra-rapid delayed rectifier (IKur) or the acetylcholine activated (IACh) potassium currents, as well as differences in calcium handling between ventricular and atrial cells (Figure 2). Usually these models contain at least three compartments, the bulk cytoplasm, encompassing the bulk of the cell and the majority of the cell membrane, where most ionic currents enter or exit the cell and where intracellular sodium, calcium, and potassium ionic concentrations are modeled; a cytoplasmic subspace representing the space near the T-tubules, through which the L-type calcium current flows; and one or more compartments comprising the sarcoplasmic reticulum, with modeled uptake of calcium ions from the bulk cytoplasm by sarco/endoplasmic reticulum Ca2 þ -ATPase (SERCA) and release into the subspace compartment by calcium-induced calcium-release through ryanodine receptors. Other currents, biochemical and signaling processes, and cellular subdivisions exist, but these differ from model to model.
Modeling Variability in Cardiac Cell Electrophysiology Variability is essential to biology. No two measurements are the same, no two individuals are the same, and even the same individual will change over time due to aging, disease, or the environment. This variability is exhibited at all biological levels, from cells to organs, with broad implications in our individual response to disease or drug action. However, intersubject variability is usually combined with measurement errors in experiments, and expressed as part of the error in mean values of measured quantities. This permeates into computational models of cardiac cellular electrophysiology, which are commonly fitted to the mean data, therefore representing an ‘average’ individual. As discussed in the previous section, cellular cardiac models consist of a system of ordinary differential equations, with
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a set of parameters controlling the magnitude and kinetics of the ionic currents included in the model. A common assumption is that variability enters the model at the level of parameters and not at the level of equations. That is, for a specific cell type under investigation, the biophysical processes are the same between individuals, but the quantitative properties of their ionic currents may differ. Two main approaches probing variability in this manner have been described in the literature: these are sensitivity analysis and population of models-based methods. Sensitivity analysis methods are those where one or more parameters are varied around an existing parameter set (most often the original published parameter set of the model), to quantify the effect this has on model outputs. This approach has been used in cardiac electrophysiological studies. Romero et al. (2009) performed a single parameter sensitivity analysis and compared the results to experimental measurements, in order to determine the impact of variability in human ventricular ionic properties on important preclinical biomarkers of arrhythmic risk. In contrast, Sobie (2009) varied multiple parameters simultaneously in a variety of ventricular cell models by sampling their values from a statistical log-normal distribution, centered at the original parameter values (Sarkar and Sobie, 2011; Sarkar et al., 2012). Following this, they used a multivariate regression analysis to relate changes in parameters to model outputs. Using similar methods, Walmsley et al. (2013) investigated the role of mRNA expression levels (genetic messengers of ion channel expression in the cell membrane) on electrophysiological remodeling in failing human hearts. More recently, these techniques were used by Cummins et al. (2014), who compared 13 ventricular cell models in the context of drug action at different stimulation frequencies. The second methodology used for modeling intersubject variability is to create and analyze experimentally calibrated populations of models (Britton et al., 2013). To generate a population of models, a subset of the model parameters are varied within the population, based on the parameters known or hypothesized to vary most between individuals, and sampled using random methods such as Monte Carlo or Latin hypercube sampling (McKay et al., 1979). These parameter sets are subsequently inserted into the model equations and simulated. Experimental calibration is then conducted by imposing the requirement that all the considered action potential properties must lay within the experimentally observed bounds, for all the simulated protocols. Therefore, models that are not fully consistent with the experimental data are discarded from the population. This is referred to as the calibration step, and is the major difference between the population of models and sensitivity analysis-based methods. The ensemble of models in full agreement with the experimental data is termed the experimentally calibrated population of models, and used for further investigations. Figure 3 illustrates simulated action potential traces using an experimentally calibrated population of human atrial models (Sánchez et al., 2014). The population of models approach was originally developed in the field of neuroscience (Prinz et al., 2003; Taylor et al., 2009; Marder and Taylor, 2011). Its use in cardiac electrophysiology was pioneered by the work of Britton et al.
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(2013), who also demonstrated its quantitative predictive power under pharmacological action. The approach has later been applied to investigating sources of variability in cellular repolarization in rabbit ventricular electrophysiology (Gemmell et al., 2014), mechanisms of human intersubject variability in sinus rhythm versus chronic atrial fibrillation as illustrated in Figure 3 (Sánchez et al., 2014), and the investigation of pro-arrhythmic mechanisms in human hypertrophic cardiomyopathy (Passini et al., 2014), among others. A study by Davies et al. (2012) also used a population of models approach to create representative parameter sets for each of the individuals in their dataset. These studies illustrate some of the different ways in which populations of models can be used to address diverse research questions, depending on the question being asked and the experimental data that is available. Another source of variability in cardiac electrophysiology is the temporal or beat-to-beat variability of the action potential, which is modulated by disease and pharmacological drug action (Hondeghem et al., 2001; Thomsen et al., 2004; Myles et al., 2008; Hinterseer et al., 2010; Johnson et al., 2010). Recent simulation studies using stochastic cardiac models provide supporting evidence to the contribution of the intrinsically stochastic dynamics of cardiac ion channels to beat-to-beat variability and repolisation abnormalities under pharmacological action (Lemay et al., 2011; Pueyo et al., 2011; Heijman et al., 2013). Interestingly, these instabilities are modulated and suppressed by intercellular tissue coupling, highlighting the multiscale nature of sources and modulators of cardiac variability (Pueyo et al., 2011). These results also highlight the importance of multiscale simulations in elucidating the complex interplay of mechanisms determining the spatio-temporally dynamic and variable response of the human heart.
Whole Organ Cardiac Modeling Cardiac cell models are often embedded in tissue and whole organ preparations in order to investigate the impact of ionic alterations on the propagation of electrical excitation in health and disease. Tissue and whole organ models have been extensively used in computational cardiovascular science during the last 15 years, as recently reviewed, for example, in Trayanova (2011) and Smaill et al. (2013). Simulations of cardiac electrophysiology at the tissue and whole organ levels require mathematical modeling of the processes underlying intercellular electrical coupling and propagation of electrical excitation from cell to cell through cardiac tissue. The cardiac bidomain model is currently the gold standard, and arises by representing the cardiac tissue as a continuum with the intracellular and the extracellular spaces electrically connected through the membrane of cardiomyocytes (Tung, 1978). The bidomain model consists of two partial differential equations representing the tissue electrical properties in the intracellular and extracellular domains, coupled through the system of ordinary differential equations representing the membrane kinetics (as described in the previous sections). The partial differential equations assume that cardiac tissue is a continuum and are derived using two fundamental laws: Ohm’s law (relating electrical potential to the
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Figure 3 Experimentally calibrated population of human atrial action potential models for sinus rhythm (SR) and chronic atrial fibrillation (cAF). (a): Human atrial action potentials simulated using an initially unrestricted set of model parameters, constructed by sampling ion channel conductances from 7100% of their original values. (b): Human atrial action potential populations after calibration against experimental bounds for SR and cAF samples. (c), (d): Histograms corresponding to action potential duration at 90, 50, and 20% of repolarization (APD90, APD50, and APD20), in both the populations of models (c) and experimental measurements (d). Reproduced with permission from Sánchez, C., Bueno-Orovio, A., Wettwer, E., et al., 2014. Inter-subject variability in human atrial action potential in sinus rhythm versus chronic atrial fibrillation. PLoS ONE 9, e105897.
net flow of transmembrane, intracellular, and extracellular currents) and Kirchhoff’s law (for conservation of charge). Generating solutions of the bidomain equations is computationally expensive, requiring application of sophisticated numerical techniques, such as the finite element method. Simplified formulations of the bidomain equations have been derived to decrease the high computational expense of numerical simulations, and alternative formulations include the monodomain and the Eikonal equations, as well as graphbased methods (Potse et al., 2006; Wallman et al., 2012). Novel modeling approaches considering fractional diffusion models have been recently applied to cardiac modeling (Bueno-Orovio et al., 2014a). Pushing the boundaries beyond the perfect continuum assumption behind the bidomain equations, fractional diffusion cardiac models aim to capture
the effects of the heterogeneous microstructure of cardiac tissue, and therefore allow to investigate implications of heterogeneity in cardiac electrophysiological behavior. Multiscale whole organ simulations of cardiac electrophysiological behavior also require the definition of spatial tissue characteristics such as its geometry, which are defined by a mesh. Anatomical meshes of the ventricles and the atria are obtained from cardiac magnetic resonance, computer tomography, or histology datasets, using image processing and mesh generation techniques (Bishop et al., 2010; Vadakkumpadan et al., 2010; Bordas et al., 2011; Zhao et al., 2012). The processed images are then used to generate a volumetric mesh by applying a discretization process in space, determined by the numerical method used to solve the propagation model equations (e.g., the finite element method for the bidomain equations).
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Figure 4 Multiscale simulations of drug-induced effects on the human heart: from drug–ionic current interactions to body surface potentials. (a) Cross-section of the human ventricular model showing the distribution of simulated activation times across the ventricular wall and on the endocardium. (b) Distribution of the body surface potential in the human torso model. (c)–(h) Human ventricular action potentials (c) and electrocardiograms ((d)–(h)), simulated for different degrees of block of L-type calcium and hERG potassium currents (drug dose indicated as a function of the half maximal inhibitory concentration, IC50). Reproduced with permission from Zemzemi, N., Rodriguez, B., 2015. Effects of L-type calcium channel and human ether-a-go-go related gene blockers on the electrical activity of the human heart: A simulation study. Europace 17, 326–333.
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As a representative example, Figure 4 illustrates simulations of the effects on the body surface electrocardiogram of drug action in calcium and potassium channels, using a multiscale human whole ventricular model with biophysically detailed representation of membrane kinetics, and embedded within a human torso (Zemzemi and Rodriguez, 2015). This figure exemplifies the potential that these novel technologies have in shaping the future of cardiovascular science. In silico modeling in biomedical applications is a research priority that can contribute to improvements in achieving a faster and cheaper drug development process, the reduction of medication errors and adverse side events, and the design of a more personalized healthcare.
Conclusions and Outlook In this article we have described how cardiac modeling has developed from single cell models of a few cell types into a field with a wide variety of biophysically detailed single cell, tissue, and whole organ models, with a particular focus on models of human physiology. We have also described the importance of phenomena such as intersubject variability in cellular cardiac electrophysiology and heterogeneity in the structure of cardiac tissue, which contribute to variability in the responses of different hearts and patients to drugs and disease, and we have provided an overview of recently developed methods designed to probe both the sources and the consequences of such intersubject differences. These recent advances in both models and modeling frameworks allow us to investigate how different phenomena interact and affect cardiac behavior at multiple spatio-temporal scales, ranging from single cells, through tissue, to the whole heart, as well as from picoseconds to years, thus opening new avenues to identify key determinants of the electrophysiological activity of the heart in health and disease. A major focus in cardiac modeling is its integration with experimental and clinical research in order to better explain and predict the effects of complex drugs, diseases, and genetic mutations on the human heart. In line with this, computational cardiovascular science is evolving beyond the traditional single physiological human paradigm to construct patient-specific or subpopulation-specific models of whole organ and single cells, respectively. In doing so, we hope to better understand the currently unpredictable risks of lethal pathologies in patients, and enhance prevention and treatment strategies.
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Vertical Integration: Modeling: Cardiac Modeling Hinterseer, M., Beckmann, B.-M., Thomsen, M.B., et al., 2010. Usefulness of shortterm variability of QT intervals as a predictor for electrical remodeling and proarrhythmia in patients with nonischemic heart failure. American Journal of Cardiology 106, 216–220. Hodgkin, A.L., Huxley, A.F., 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology 117, 500–544. Hondeghem, L.M., Carlsson, L., Duker, G., 2001. Instability and triangulation of the action potential predict serious proarrhythmia, but action potential duration prolongation is antiarrhythmic. Circulation 103, 2004–2013. Iyer, V., Mazhari, R., Winslow, R.L., 2004. A computational model of the human leftventricular epicardial myocyte. Biophysical Journal 87, 1507–1525. Johnson, D.M., Heijman, J., Pollard, C.E., et al., 2010. IKs restricts excessive beatto-beat variability of repolarization during beta-adrenergic receptor stimulation. Journal of Molecular and Cellular Cardiology 48, 122–130. Koivumäki, J.T., Korhonen, T., Tavi, P., 2011. Impact of sarcoplasmic reticulum calcium release on calcium dynamics and action potential morphology in human atrial myocytes: A computational study. PLoS Computational Biology 7, e1001067. Koval, O.M., Snyder, J.S., Wolf, R.M., et al., 2012. Ca2 þ /calmodulin-dependent protein kinase II-based regulation of voltage-gated Na þ channel in cardiac disease. Circulation 126, 2084–2094. Kuzumoto, M., Takeuchi, A., Nakai, H., et al., 2008. Simulation analysis of intracellular Na þ and Cl homeostasis during beta 1-adrenergic stimulation of cardiac myocyte. Progress in Biophysics and Molecular Biology 96, 171–186. Lemay, M., de Lange, E., Kucera, J.P., 2011. Effects of stochastic channel gating and distribution on the cardiac action potential. Journal of Theoretical Biology 281, 84–96. Maleckar, M.M., Greenstein, J.L., Giles, W.R., Trayanova, N.A., 2009. K þ current changes account for the rate dependence of the action potential in the human atrial myocyte. American Journal of Physiology. Heart and Circulatory Physiology 297, H1398–H1410. Marder, E., Taylor, A.L., 2011. Multiple models to capture the variability in biological neurons and networks. Nature Neuroscience 14, 133–138. Mazhari, R., Greenstein, J.L., Winslow, R.L., Marbán, E., Nuss, H.B., 2001. Molecular interactions between two long-QT syndrome gene products, HERG and KCNE2, rationalized by in vitro and in silico analysis. Circulation Research 89, 33–38. McKay, M.D., Beckman, R.J., Conover, W.J., 1979. Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21 (2), 239–245. Myles, R.C., Burton, F.L., Cobbe, S.M., Smith, G.L., 2008. The link between repolarisation alternans and ventricular arrhythmia: Does the cellular phenomenon extend to the clinical problem? Journal of Molecular and Cellular Cardiology 45, 1–10. Niederer, S.A., Fink, M., Noble, D., Smith, N.P., 2009. A meta-analysis of cardiac electrophysiology computational models. Experimental Physiology 94, 486–495. Noble, D., 1960. Cardiac action and pacemaker potentials based on the Hodgkin− Huxley equations. Nature 188, 495–497. Noble, D., 2011. Successes and failures in modeling heart cell electrophysiology. Heart Rhythm 8, 1798–1803. Noble, D., Garny, A., Noble, P.J., 2012. How the Hodgkin−Huxley equations inspired the Cardiac Physiome Project. Journal of Physiology 590, 2613–2628. Nygren, A., Fiset, C., Firek, L., et al., 1998. Mathematical model of an adult human atrial cell: The role of K þ currents in repolarization. Circulation Research 82, 63–81. O’Hara, T., Rudy, Y., 2012. Quantitative comparison of cardiac ventricular myocyte electrophysiology and response to drugs in human and nonhuman species. American Journal of Physiology. Heart and Circulatory Physiology 302, H1023–H1030. O’Hara, T., Virág, L., Varró, A., Rudy, Y., 2011. Simulation of the undiseased human cardiac ventricular action potential: Model formulation and experimental validation. PLoS Computational Biology 7, e1002061. Passini, E., Rodriguez, B., Mincholé, A., et al., 2014. Late sodium current inhibition counteracts pro-arrhythmic mechanisms in human hypertrophic cardiomyopathy. Computing in Cardiology 41, 861–864. Pathmanathan, P., Shotwell, M.S., Gavaghan, D.J., Cordeiro, J.M., Gray, R.A., 2015. Uncertainty quantification of fast sodium current steady-state inactivation for multi-scale models of cardiac electrophysiology. Progress in Biophysics and Molecular Biology. doi:10.1016/j.pbiomolbio.2015.01.008. Polak, S., Wiśniowska, B., Fijorek, K., Glinka, A., Mendyk, A., 2014. In vitro–in vivo extrapolation of drug-induced proarrhythmia predictions at the population level. Drug Discovery Today 19, 275–281.
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Potse, M., Dubé, B., Richer, J., Vinet, A., Gulrajani, R.M., 2006. A comparison of monodomain and bidomain reaction−diffusion models for action potential propagation in the human heart. IEEE Transactions on Bio-Medical Engineering 53, 2425–2435. Priebe, L., Beuckelmann, D.J., 1998. Simulation study of cellular electric properties in heart failure. Circulation Research 82, 1206–1223. Prinz, A.A., Billimoria, C.P., Marder, E., 2003. Alternative to hand-tuning conductance-based models: Construction and analysis of databases of model neurons. Journal of Neurophysiology 90, 3998–4015. Pueyo, E., Corrias, A., Virág, L., et al., 2011. A multiscale investigation of repolarization variability and its role in cardiac arrhythmogenesis. Biophysical Journal 101, 2892–2902. Romero, L., Pueyo, E., Fink, M., Rodriguez, B., 2009. Impact of ionic current variability on human ventricular cellular electrophysiology. American Journal of Physiology. Heart and Circulatory Physiology 297, H1436–H1445. Sager, P.T., Gintant, G., Turner, J.R., Pettit, S., Stockbridge, N., 2014. Rechanneling the cardiac proarrhythmia safety paradigm: A meeting report from the Cardiac Safety Research Consortium. American Heart Journal 167, 292–300. Sánchez, C., Bueno-Orovio, A., Wettwer, E., et al., 2014. Inter-subject variability in human atrial action potential in sinus rhythm versus chronic atrial fibrillation. PLoS ONE 9, e105897. Sarkar, A.X., Christini, D.J., Sobie, E.A., 2012. Exploiting mathematical models to illuminate electrophysiological variability between individuals. Journal of Physiology 590, 2555–2567. Sarkar, A.X., Sobie, E.A., 2011. Quantification of repolarization reserve to understand interpatient variability in the response to proarrhythmic drugs: A computational analysis. Heart Rhythm 8, 1749–1755. Smaill, B.H., Zhao, J., Trew, M.L., 2013. Three-dimensional impulse propagation in myocardium: Arrhythmogenic mechanisms at the tissue level. Circulation Research 112, 834–848. Sobie, E.A., 2009. Parameter sensitivity analysis in electrophysiological models using multivariable regression. Biophysical Journal 96, 1264–1274. Taylor, A.L., Goaillard, J.-M., Marder, E., 2009. How multiple conductances determine electrophysiological properties in a multicompartment model. Journal of Neuroscience 29, 5573–5586. Thomsen, M.B., Verduyn, S.C., Stengl, M., et al., 2004. Increased short-term variability of repolarization predicts d-sotalol-induced torsades de pointes in dogs. Circulation 110, 2453–2459. Trayanova, N.A., 2011. Whole-heart modeling: Applications to cardiac electrophysiology and electromechanics. Circulation Research 108, 113–128. Tung, L., 1978. A bi-domain model for describing ischemic myocardial d-c potentials (Ph.D. thesis). Massachusetts: Massachusetts Institute of Technology Massachusetts ten Tusscher, K.H.W.J., Noble, D., Noble, P.J., Panfilov, A.V., 2004. A model for human ventricular tissue. American Journal of Physiology. Heart and Circulatory Physiology 286, H1573–H1589. ten Tusscher, K.H.W.J., Panfilov, A.V., 2006. Alternans and spiral breakup in a human ventricular tissue model. American Journal of Physiology. Heart and Circulatory Physiology 291, H1088–H1100. Vadakkumpadan, F., Arevalo, H., Prassl, A.J., et al., 2010. Image-based models of cardiac structure in health and disease. Wiley Interdisciplinary Reviews. Systems Biology and Medicine 2, 489–506. Wallman, M., Smith, N.P., Rodriguez, B., 2012. A comparative study of graph-based, eikonal, and monodomain simulations for the estimation of cardiac activation times. IEEE Transactions on Bio-Medical Engineering 59, 1739–1748. Wallman, M., Smith, N.P., Rodriguez, B., 2014. Computational methods to reduce uncertainty in the estimation of cardiac conduction properties from electroanatomical recordings. Medical Image Analysis 18, 228–240. Walmsley, J., Rodriguez, J.F., Mirams, G.R., et al., 2013. mRNA expression levels in failing human hearts predict cellular electrophysiological remodeling: A population-based simulation study. PLoS ONE 8, e56359. Weber, F.M., Lurz, S., Keller, D.U.J., et al., 2008. Adaptation of a minimal four-state cell model for reproducing atrial excitation properties. Computers in Cardiology 35, 61–64. Wilhelms, M., Hettmann, H., Maleckar, M.M., et al., 2012a. Benchmarking electrophysiological models of human atrial myocytes. Frontiers in Physiology 3, 487. Wilhelms, M., Rombach, C., Scholz, E.P., Dössel, O., Seemann, G., 2012b. Impact of amiodarone and cisapride on simulated human ventricular electrophysiology and electrocardiograms. Europace 14, v90–v96. Zemzemi, N., Bernabeu, M.O., Saiz, J., et al., 2013. Computational assessment of drug-induced effects on the electrocardiogram: From ion channel to body surface potentials. British Journal of Pharmacology 168, 718–733.
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Zemzemi, N., Rodriguez, B., 2015. Effects of L-type calcium channel and human ether-a-go-go related gene blockers on the electrical activity of the human heart: A simulation study. Europace 17, 326–333. Zhao, J., Butters, T.D., Zhang, H., et al., 2012. An image-based model of atrial muscular architecture: Effects of structural anisotropy on electrical activation. Circulation. Arrhythmia and Electrophysiology 5, 361–370.
Relevant Website http://www.cellml.org/ CellML.
State-of-the-Art in Cardiac Cell Modeling In 1960, Denis Noble developed the first mathematical model of the electrophysiological activity of a cardiac cell (Noble, 1960). Since then, the field of computational cardiovascular science has developed into a broad and productive area of research with growing demonstration of translational potential into industrial, regulatory, and clinical applications (Sager et al., 2014). Computational heart models are multiscale both spatially and temporally, and integrate information across subcellular, cellular, tissue, and whole organ levels, and temporal scales (ranging from picoseconds to years, from ion channels opening and closing to remodeling in disease). Through continuous use and refinement, computational cardiac cell models have built a strong basis for their usefulness and credibility in a variety of settings, including investigations of genetic mutations, disease, and pharmacology (Clancy et al., 2007; Sarkar and Sobie, 2011; O’Hara and Rudy, 2012; Wilhelms et al., 2012a,b; Britton et al., 2013; Zemzemi et al., 2013; Polak et al., 2014). Cardiac cells (cardiomyocytes) are electrically excitable cells due to the presence of proteins in their membrane which act as mechanisms of ionic transport, such as ion channels, pumps, and exchangers. The transmembrane transport of ions such as potassium, sodium, and calcium results in electrical and concentration gradients between the intracellular and the extracellular media. Under physiological conditions with no external electrical stimulation, cardiomyocytes maintain a negative voltage relative to their environment. However, when stimulated, sodium channels in the cell membrane are activated, altering the balance of currents into the cell and acting to rapidly change the electrical potential across the membrane. This triggers the release of calcium ions from a cellular compartment known as the sarcoplasmic reticulum, which raises the bulk cytoplasmic concentration of calcium and signals the myocyte to contract. The ion channels initially responsible for firing the action potential inactivate, while others (potassium) act to repolarize the myocyte back to its resting potential. This allows the cardiomyocyte to fire another action potential during the next heartbeat. In this manner, the cardiac action potential, first observed by Coraboeuf and Weidmann (1949), allows the heart to control its own pattern of muscular contraction without external stimuli. The action potential generated by each cardiomyocyte can excite nearby myocytes, and thus the electrical signals propagate through the heart. In the last of a landmark series of papers, Hodgkin and Huxley (1952) showed that their mathematical model of the ionic currents of the isolated squid giant axon could recreate many of the axon's key electrical properties. This was the first mathematical model of a neuronal action potential. Their formulation of ion channel kinetics was promptly adapted to create the first action potential model of a cardiac cell (Noble, 1960). The Hodgkin–Huxley model formulates each ionic current as the product of the permeability of the cell
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membrane to the ion type of that current (more commonly called the conductance of that current), multiplied by the difference between the membrane potential and the equilibrium potential for that current (the membrane potential at which the net current flow across the membrane for that ion is zero). The conductances are functions that may depend on membrane potential and time, as well as channel-specific constants. They represent the opening, closing, and inactivation of the ion channels that carry current across the cell membrane. Cardiac cell electrophysiology models, from the very beginning of the field up to the present day, have used the Hodgkin–Huxley formulation to describe the ionic currents in the model. An alternative approach to describe ion channel gating is the use of Markov chain models (Greenstein et al., 2000; Mazhari et al., 2001; Clancy and Rudy, 2002), which model the transition between the open, closed, and inactive states of the channel, based on the probabilistic transition rates between channel states at different transmembrane potentials. From the earliest models that were very closely based on the original three current Hodgkin–Huxley model, new models have increased in complexity, based on our growing understanding of cardiac cellular electrophysiology, for example, the discovery of calcium channels. In general, the additions made in each new model have been in one or more of the following forms: (1) additions of new currents carrying ions such as calcium, (2) addition of new gating mechanisms for existing currents, (3) subdividing a single current into multiple currents either because of different time scales for the parts of the current (e.g., the fast and late sodium currents) and/or because different ion channel proteins carry the different components (e.g., slow and rapid delayed rectifier potassium currents), (4) addition of mechanisms relating to intracellular ionic concentrations (e.g., adding ion transporters such as the sodium–potassium pump or new cellular compartments such as the sarcoplasmic reticulum for calcium handling), and finally (5) adding biochemical mechanisms for ion channel modification, contraction, or signaling, such as, for example, models of beta-adrenergic stimulation (Kuzumoto et al., 2008; Heijman et al., 2011) or ion channel regulation by calcium/calmodulin-dependent protein kinase II (Christensen et al., 2009; O’Hara et al., 2011; Koval et al., 2012). The formulation of the different ion channels and transporters is usually inherited between models, although particularized for the specific cell type and species under study (Niederer et al., 2009; Bueno-Orovio et al., 2014b). This is illustrated in Figure 1 for the particular case of the sodium– potassium pump. Computational cardiac models have played a pivotal historical role in the discovery of new ionic mechanisms through successes and failures in reproducing and challenging experimental evidence (Noble, 2011). As in silico models and simulation techniques developed both conceptually and
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Figure 1 Inheritance in the formulation of the sodium–potassium pump current in a selection of models of cellular cardiac electrophysiology. Models are referred by first author's name and year of publication. The complete list of references can be found in (Noble et al., 2012). Reproduced with permission from Bueno-Orovio, A., Sánchez, C., Pueyo, E., Rodriguez, B., 2014b. Na/K pump regulation of cardiac repolarization: Insights from a systems biology approach. Pflügers Archiv: European Journal of Physiology 466, 183–193.
technically, the questions of verification, validation, and uncertainty quantification have become crucial for their uptake beyond scientific research into industrial, clinical, and regulatory arenas (Carusi et al., 2012; Sager et al., 2014; Wallman et al., 2014; Pathmanathan et al., 2015). The process of validation of in silico models and simulations is complex and still under investigation, but it is essential to increase the credibility of in silico methods as powerful techniques to augment the information extracted from experimental and clinical investigations. The paper by Carusi et al. (2012) specifically explores the issue of validation of computational physiological models in the context of their construction and applications. It highlights the iterative nature of the process of validation and the fact that it needs to consider the ensemble of equations, parameters, simulation techniques, software, and experiments used for a particular study or purpose, namely the Model-Simulation-Experiment system. The aim is to augment the information on the physiological system or process under investigation, such as, for example, the mode of action of a new medicine on specific organs. The dynamics of the iterative process are driven by advances in both experimental and computational techniques, as well as investigations on their combined use. The legacy of such intense work is over 40 available mathematical models of cardiac cellular electrophysiology, for
different species and cell types (see Noble et al. (2012) for a comprehensive historical overview). The iterative process between models, simulations, and experiments leads to the dynamic replacement, refinement, and expansion of these models. In this context, initiatives such as the CellML repository provide the platform that allows the exchange and widespread use of cardiac cell models within the scientific community (Beard et al., 2009).
Human Models of Cardiac Cell Electrophysiology Computer modeling of the human heart has been a key priority for the scientific community as it provides a vital tool for a better understanding of human health and disease in clinical and industrial applications. Over the last decades a number of models of human adult electrophysiology have been proposed for both ventricular and atrial cells. Biophysically detailed models represent biophysical processes such as ion transport across the membrane and intracellular calcium handling, which are described by equations that represent individual ionic currents and fluxes. A different approach is based on phenomenological models, which reproduce the electrophysiological behavior of cardiac cells without detailed modeling of the underlying biophysical processes. These models
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Figure 2 Schematic diagram representations of models of human cellular cardiac electrophysiology. (a) O’Hara–Rudy model of the human ventricular cardiomyocyte. (b) Comparison of different models of human atrial cellular electrophysiology. Reproduced with permission from O’Hara, T., Virág, L., Varró, A., Rudy, Y., 2011. Simulation of the undiseased human cardiac ventricular action potential: Model formulation and experimental validation. PLoS Computational Biology 7, e1002061; Wilhelms, M., Hettmann, H., Maleckar, M.M., et al., 2012a. Benchmarking electrophysiological models of human atrial myocytes. Frontiers in Physiology 3, 487.
require fewer equations and parameters, and are ideal for applications that do not necessarily require the representation of ionic processes, such as the impact of cellular level properties on the cardiac electrical propagation at the tissue level. Several human ventricular and atrial cell models have been proposed and compared over the last two decades. Their structure is illustrated in Figure 2, through a schematic representation of the O’Hara–Rudy human ventricular model (O’Hara et al., 2011), and a graphical comparison of the ionic currents included in different human atrial models (Wilhelms et al., 2012a,b). Human ventricular cell models include the biophysically detailed Priebe and Beuckelmann (Priebe and Beuckelmann, 1998), Iyer (Iyer et al., 2004), ten Tusscher and Panfilov (ten Tusscher et al., 2004; ten Tusscher and Panfilov, 2006), Grandi (Grandi et al., 2010), Carro (Carro et al., 2011), O'Hara–Rudy (O’Hara et al., 2011), and Asakura (Asakura et al., 2014) models, as well as phenomenological models for human ventricular myocardium (Bueno-Orovio et al., 2008; Bueno-Orovio et al., 2012). Equivalently, several models of human atrial electrophysiology have been published such as the Nygren (Nygren et al., 1998), Courtemanche (Courtemanche et al., 1998), Maleckar (Maleckar et al., 2009), Grandi (Grandi et al., 2011), and Koivumäki (Koivumäki et al., 2011) models. The phenomenological Bueno-Orovio model has also been adapted to reproduce characteristic action potentials of human atrial tissue (Weber et al., 2008). Comprehensive comparisons between these ventricular and atrial human models can be found, for example, in Bueno-Orovio et al. (2008), Wilhelms et al. (2012a,b), and Elshrif and Cherry (2014). Modern human cardiac cell models consist of a system of ordinary differential equations, generally based on the Hodgkin and Huxley or Markov chain formulations for ionic gating variables. As shown in Figure 2, a number of biophysical components are represented, including ionic currents and cellular compartments. Ionic currents that are usually included in human ventricular cell models are the sodium current (INa), sometimes divided on its fast (INaf) and late (INaL) components; the L-type calcium current (ICaL); the transient outward potassium current (Ito); the rapid (IKr) and slow (IKs)
delayed rectifier potassium currents; the inward rectifier potassium current (IK1); the sodium–calcium exchanger current (INCX); and the sodium–potassium pump current (INaK). Human atrial cell models incorporate additional ionic currents specific to atrial electrophysiology, such as the ultra-rapid delayed rectifier (IKur) or the acetylcholine activated (IACh) potassium currents, as well as differences in calcium handling between ventricular and atrial cells (Figure 2). Usually these models contain at least three compartments, the bulk cytoplasm, encompassing the bulk of the cell and the majority of the cell membrane, where most ionic currents enter or exit the cell and where intracellular sodium, calcium, and potassium ionic concentrations are modeled; a cytoplasmic subspace representing the space near the T-tubules, through which the L-type calcium current flows; and one or more compartments comprising the sarcoplasmic reticulum, with modeled uptake of calcium ions from the bulk cytoplasm by sarco/endoplasmic reticulum Ca2 þ -ATPase (SERCA) and release into the subspace compartment by calcium-induced calcium-release through ryanodine receptors. Other currents, biochemical and signaling processes, and cellular subdivisions exist, but these differ from model to model.
Modeling Variability in Cardiac Cell Electrophysiology Variability is essential to biology. No two measurements are the same, no two individuals are the same, and even the same individual will change over time due to aging, disease, or the environment. This variability is exhibited at all biological levels, from cells to organs, with broad implications in our individual response to disease or drug action. However, intersubject variability is usually combined with measurement errors in experiments, and expressed as part of the error in mean values of measured quantities. This permeates into computational models of cardiac cellular electrophysiology, which are commonly fitted to the mean data, therefore representing an ‘average’ individual. As discussed in the previous section, cellular cardiac models consist of a system of ordinary differential equations, with
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a set of parameters controlling the magnitude and kinetics of the ionic currents included in the model. A common assumption is that variability enters the model at the level of parameters and not at the level of equations. That is, for a specific cell type under investigation, the biophysical processes are the same between individuals, but the quantitative properties of their ionic currents may differ. Two main approaches probing variability in this manner have been described in the literature: these are sensitivity analysis and population of models-based methods. Sensitivity analysis methods are those where one or more parameters are varied around an existing parameter set (most often the original published parameter set of the model), to quantify the effect this has on model outputs. This approach has been used in cardiac electrophysiological studies. Romero et al. (2009) performed a single parameter sensitivity analysis and compared the results to experimental measurements, in order to determine the impact of variability in human ventricular ionic properties on important preclinical biomarkers of arrhythmic risk. In contrast, Sobie (2009) varied multiple parameters simultaneously in a variety of ventricular cell models by sampling their values from a statistical log-normal distribution, centered at the original parameter values (Sarkar and Sobie, 2011; Sarkar et al., 2012). Following this, they used a multivariate regression analysis to relate changes in parameters to model outputs. Using similar methods, Walmsley et al. (2013) investigated the role of mRNA expression levels (genetic messengers of ion channel expression in the cell membrane) on electrophysiological remodeling in failing human hearts. More recently, these techniques were used by Cummins et al. (2014), who compared 13 ventricular cell models in the context of drug action at different stimulation frequencies. The second methodology used for modeling intersubject variability is to create and analyze experimentally calibrated populations of models (Britton et al., 2013). To generate a population of models, a subset of the model parameters are varied within the population, based on the parameters known or hypothesized to vary most between individuals, and sampled using random methods such as Monte Carlo or Latin hypercube sampling (McKay et al., 1979). These parameter sets are subsequently inserted into the model equations and simulated. Experimental calibration is then conducted by imposing the requirement that all the considered action potential properties must lay within the experimentally observed bounds, for all the simulated protocols. Therefore, models that are not fully consistent with the experimental data are discarded from the population. This is referred to as the calibration step, and is the major difference between the population of models and sensitivity analysis-based methods. The ensemble of models in full agreement with the experimental data is termed the experimentally calibrated population of models, and used for further investigations. Figure 3 illustrates simulated action potential traces using an experimentally calibrated population of human atrial models (Sánchez et al., 2014). The population of models approach was originally developed in the field of neuroscience (Prinz et al., 2003; Taylor et al., 2009; Marder and Taylor, 2011). Its use in cardiac electrophysiology was pioneered by the work of Britton et al.
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(2013), who also demonstrated its quantitative predictive power under pharmacological action. The approach has later been applied to investigating sources of variability in cellular repolarization in rabbit ventricular electrophysiology (Gemmell et al., 2014), mechanisms of human intersubject variability in sinus rhythm versus chronic atrial fibrillation as illustrated in Figure 3 (Sánchez et al., 2014), and the investigation of pro-arrhythmic mechanisms in human hypertrophic cardiomyopathy (Passini et al., 2014), among others. A study by Davies et al. (2012) also used a population of models approach to create representative parameter sets for each of the individuals in their dataset. These studies illustrate some of the different ways in which populations of models can be used to address diverse research questions, depending on the question being asked and the experimental data that is available. Another source of variability in cardiac electrophysiology is the temporal or beat-to-beat variability of the action potential, which is modulated by disease and pharmacological drug action (Hondeghem et al., 2001; Thomsen et al., 2004; Myles et al., 2008; Hinterseer et al., 2010; Johnson et al., 2010). Recent simulation studies using stochastic cardiac models provide supporting evidence to the contribution of the intrinsically stochastic dynamics of cardiac ion channels to beat-to-beat variability and repolisation abnormalities under pharmacological action (Lemay et al., 2011; Pueyo et al., 2011; Heijman et al., 2013). Interestingly, these instabilities are modulated and suppressed by intercellular tissue coupling, highlighting the multiscale nature of sources and modulators of cardiac variability (Pueyo et al., 2011). These results also highlight the importance of multiscale simulations in elucidating the complex interplay of mechanisms determining the spatio-temporally dynamic and variable response of the human heart.
Whole Organ Cardiac Modeling Cardiac cell models are often embedded in tissue and whole organ preparations in order to investigate the impact of ionic alterations on the propagation of electrical excitation in health and disease. Tissue and whole organ models have been extensively used in computational cardiovascular science during the last 15 years, as recently reviewed, for example, in Trayanova (2011) and Smaill et al. (2013). Simulations of cardiac electrophysiology at the tissue and whole organ levels require mathematical modeling of the processes underlying intercellular electrical coupling and propagation of electrical excitation from cell to cell through cardiac tissue. The cardiac bidomain model is currently the gold standard, and arises by representing the cardiac tissue as a continuum with the intracellular and the extracellular spaces electrically connected through the membrane of cardiomyocytes (Tung, 1978). The bidomain model consists of two partial differential equations representing the tissue electrical properties in the intracellular and extracellular domains, coupled through the system of ordinary differential equations representing the membrane kinetics (as described in the previous sections). The partial differential equations assume that cardiac tissue is a continuum and are derived using two fundamental laws: Ohm’s law (relating electrical potential to the
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Figure 3 Experimentally calibrated population of human atrial action potential models for sinus rhythm (SR) and chronic atrial fibrillation (cAF). (a): Human atrial action potentials simulated using an initially unrestricted set of model parameters, constructed by sampling ion channel conductances from 7100% of their original values. (b): Human atrial action potential populations after calibration against experimental bounds for SR and cAF samples. (c), (d): Histograms corresponding to action potential duration at 90, 50, and 20% of repolarization (APD90, APD50, and APD20), in both the populations of models (c) and experimental measurements (d). Reproduced with permission from Sánchez, C., Bueno-Orovio, A., Wettwer, E., et al., 2014. Inter-subject variability in human atrial action potential in sinus rhythm versus chronic atrial fibrillation. PLoS ONE 9, e105897.
net flow of transmembrane, intracellular, and extracellular currents) and Kirchhoff’s law (for conservation of charge). Generating solutions of the bidomain equations is computationally expensive, requiring application of sophisticated numerical techniques, such as the finite element method. Simplified formulations of the bidomain equations have been derived to decrease the high computational expense of numerical simulations, and alternative formulations include the monodomain and the Eikonal equations, as well as graphbased methods (Potse et al., 2006; Wallman et al., 2012). Novel modeling approaches considering fractional diffusion models have been recently applied to cardiac modeling (Bueno-Orovio et al., 2014a). Pushing the boundaries beyond the perfect continuum assumption behind the bidomain equations, fractional diffusion cardiac models aim to capture
the effects of the heterogeneous microstructure of cardiac tissue, and therefore allow to investigate implications of heterogeneity in cardiac electrophysiological behavior. Multiscale whole organ simulations of cardiac electrophysiological behavior also require the definition of spatial tissue characteristics such as its geometry, which are defined by a mesh. Anatomical meshes of the ventricles and the atria are obtained from cardiac magnetic resonance, computer tomography, or histology datasets, using image processing and mesh generation techniques (Bishop et al., 2010; Vadakkumpadan et al., 2010; Bordas et al., 2011; Zhao et al., 2012). The processed images are then used to generate a volumetric mesh by applying a discretization process in space, determined by the numerical method used to solve the propagation model equations (e.g., the finite element method for the bidomain equations).
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Figure 4 Multiscale simulations of drug-induced effects on the human heart: from drug–ionic current interactions to body surface potentials. (a) Cross-section of the human ventricular model showing the distribution of simulated activation times across the ventricular wall and on the endocardium. (b) Distribution of the body surface potential in the human torso model. (c)–(h) Human ventricular action potentials (c) and electrocardiograms ((d)–(h)), simulated for different degrees of block of L-type calcium and hERG potassium currents (drug dose indicated as a function of the half maximal inhibitory concentration, IC50). Reproduced with permission from Zemzemi, N., Rodriguez, B., 2015. Effects of L-type calcium channel and human ether-a-go-go related gene blockers on the electrical activity of the human heart: A simulation study. Europace 17, 326–333.
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As a representative example, Figure 4 illustrates simulations of the effects on the body surface electrocardiogram of drug action in calcium and potassium channels, using a multiscale human whole ventricular model with biophysically detailed representation of membrane kinetics, and embedded within a human torso (Zemzemi and Rodriguez, 2015). This figure exemplifies the potential that these novel technologies have in shaping the future of cardiovascular science. In silico modeling in biomedical applications is a research priority that can contribute to improvements in achieving a faster and cheaper drug development process, the reduction of medication errors and adverse side events, and the design of a more personalized healthcare.
Conclusions and Outlook In this article we have described how cardiac modeling has developed from single cell models of a few cell types into a field with a wide variety of biophysically detailed single cell, tissue, and whole organ models, with a particular focus on models of human physiology. We have also described the importance of phenomena such as intersubject variability in cellular cardiac electrophysiology and heterogeneity in the structure of cardiac tissue, which contribute to variability in the responses of different hearts and patients to drugs and disease, and we have provided an overview of recently developed methods designed to probe both the sources and the consequences of such intersubject differences. These recent advances in both models and modeling frameworks allow us to investigate how different phenomena interact and affect cardiac behavior at multiple spatio-temporal scales, ranging from single cells, through tissue, to the whole heart, as well as from picoseconds to years, thus opening new avenues to identify key determinants of the electrophysiological activity of the heart in health and disease. A major focus in cardiac modeling is its integration with experimental and clinical research in order to better explain and predict the effects of complex drugs, diseases, and genetic mutations on the human heart. In line with this, computational cardiovascular science is evolving beyond the traditional single physiological human paradigm to construct patient-specific or subpopulation-specific models of whole organ and single cells, respectively. In doing so, we hope to better understand the currently unpredictable risks of lethal pathologies in patients, and enhance prevention and treatment strategies.
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