Representation of Multiple Cellular Phenotypes Within TissueLevel Simulations of Cardiac Electrophysiology
Abstract
Distinct electrophysiological phenotypes are exhibited by biological cells that have differentiated into particular cell types. The usual approach when simulating the cardiac electrophysiology of tissue that includes different cell types is to model the different cell types as occupying spatially distinct yet coupled regions. Instead, we model the electrophysiology of wellmixed cells by using homogenisation to derive an extension to the commonly used monodomain or bidomain equations. These new equations permit spatial variations in the distribution of the different subtypes of cells and will reduce the computational demands of solving the governing equations. We validate the homogenisation computationally, and then use the new model to explain some experimental observations from stem cellderived cardiomyocyte monolayers.
Keywords
Homogenisation Monodomain Bidomain Cardiac electrophysiology Stem cellderived cardiomyocytes1 Introduction
Since its inception in the 1960s, the field of computational cardiac electrophysiology has contributed to many advances in understanding the links between the flow of ions, transmembrane potential and electromechanical activity of the heart under control, pathological and druginfluenced conditions. In particular, much attention has been devoted to modelling the action potential within cardiac tissue—that is, the transmembrane potential at a given location, as a function of time, during a given cardiac cycle—and the extracellular potential, allowing simulation of electrocardiograms. Mathematical models are now available of the action potentials observed in many different species and cardiac cell types (Noble and Rudy 2001; Fink et al. 2011).
In this paper, we develop methods for simulating a system that is of particular interest for safety pharmacology—monolayers of human stem cellderived cardiomyocytes (hSCCMs). The Comprehensive in vitro Proarrhythmia Assay (CiPA) initiative has proposed a series of complementary cardiac safety assays to improve upon the current methods of assessing the arrhythmic risk associated with novel pharmaceutical compounds (Sager et al. 2014; Gintant et al. 2016). The use of hSCCMs will form a key component of the CiPA paradigm through multicellular assays such as the microelectrode array (Harris et al. 2013; Clements and Thomas 2014).
1.1 Characteristics of Human Stem CellDerived Cardiomyocytes
Human stem cellderived cardiomyocytes are electrophysiologically and structurally immature, with some of their properties resembling neonatal cells rather than their adult counterparts. They are small and rounded, with diameters of approximately 10–50 µm [see for example Snir et al. (2003), Gherghiceanu et al. (2011) or Fig. 1 in Ma et al. (2011)]. hSCCMs typically beat spontaneously, with multicellular cultures exhibiting a focus of activation, or pacemaker region, which triggers excitation in the remainder of the culture. The activation wavefronts travel at slower speeds than observed in adult cardiac tissue, usually in the region of 2–20 cm/s (Burridge et al. 2011; Mehta et al. 2011; Lee et al. 2012).
As this system is made up of coupled oscillators, from a mathematical perspective we might expect to observe synchronous activation throughout the monolayer (Mirollo and Strogatz 1990). However, finite conduction velocities are observed.
Analysis of hSCCM action potentials from single cells has indicated that three subpopulations, or phenotypes, may be present within a given sample of cells: atriallike, ventricularlike and nodallike (He et al. 2003; Zhang et al. 2009; Ma et al. 2011). The phenotypes are named to reflect the similarity with the action potentials found in the respective regions of the adult heart, and are usually defined in terms of metrics based on the duration of the action potential, although alternatives have also been proposed (LopezRedondo et al. 2016). Precise statistics on the relative abundance of each phenotype are difficult to obtain due to different methods of classification and inherent variability within each of the phenotypes (PekkanenMattila et al. 2010). At the present time, there are differing views on the spatial organisation of these phenotypes within tissue. Zhu et al. (2016) and Vestergaard et al. (2017) reported regions of different action potential morphology within some, but not all, clusters of human embryonic stem cellderived cardiomyocytes. However, Du et al. (2015) did not detect such spatial organisation in their studies of monolayers of humaninduced pluripotent stem cellderived cardiomyocytes, instead reporting a spectrum of action potential morphologies throughout the tissue.
1.2 Existing Methods for Tissue Simulations Containing Multiple Electrophysiology Phenotypes
Tissuelevel cardiac electrophysiology is usually modelled using the monodomain or bidomain equations (Keener and Sneyd 2009). When modelling multiple phenotypes, the tissue is usually partitioned into regions containing only one phenotype. However, this method becomes computationally infeasible if the phenotypes are wellmixed within the tissue, as the tissue must be partitioned into many very small regions where just a single phenotype is present. Under these conditions, we may utilise the extended bidomain (or tridomain) model. The extended bidomain model adds a second intracellular domain for a second phenotype and has been used to simulate mixtures of cardiomyocytes and fibroblasts (Sachse et al. 2009) and gastrointestinal electrophysiology (Buist and Poh 2010). The two intracellular domains represent continuously linked regions of each of the two types of cell; a third intracellular domain would be required if it were to be used for simulating the three cell types reported in hSCCM cultures. The extended bidomain model is wellsuited for simulating thoroughly mixed cell types (Corrias et al. 2012), with the two interconnected intracellular domains providing a natural method by which two cell types can be considered to occupy a small unit of space. To model spatial variation in phenotype proportions in the extended bidomain model, we could adjust the surface area of each domain per unit volume of tissue (\(\chi \)). But, as Sachse et al. (2009) observed, “it is unclear [how we should adjust the intra and interdomain gap junction conductivities to model] the density and arrangement of myocytes and fibroblasts”. These authors linearly scaled the conductivities in each domain from values in tissues with 100% myocytes or 100% fibroblasts according to volume fraction of each phenotype. Based on the assumption that within each domain there are always connections between cells of a given phenotype, the number of connections is proportional to the volume fraction. These assumptions may not hold in regions with low proportions of a phenotype: with only say 10% of a given cell type present, a typical cell of that type does not ‘touch’ and share gap junctions with any cells of the same type, there may be no continuous domain of this cell type through which currents can flow.
We therefore take an alternative and perhaps simpler approach and develop a modified derivation of the bidomain equations that assumes a mixture of cell types within the repeating homogenisation unit that is used in their derivation.
1.3 Outline of Study
The overall goal of this study is to model the electrophysiological properties of cardiac tissue containing multiple cellular subpopulations by extending the derivation of the standard bidomain equations to permit the modelling of more than one cell type. The equations governing this model are derived in “Appendix A” and summarised in Sect. 2. In Sect. 3, we propose a suite of simulations, designed with two aims in mind: to verify the derivation of the model, and to illustrate some key properties of systems that contain more than one cellular population. We present the results of the simulations in Sect. 4. Finally, in Sect. 5, we conclude by discussing how these simulations can inform investigation of hSCCM monolayers in a twodimensional domain.
2 The Mathematical Model
3 Description of Simulations
3.1 Simulation Sets
Set 1 This set of simulations is designed to test whether the action potentials of the PP model tend towards those of the HP model as the size of the unit that we homogenise over is decreased; that is, in the limit \(\delta \rightarrow 0\), where \(\delta \) is defined in “Appendix A”. This is achieved by varying n, the number of repeating units that the domain is divided into. The layout of phenotypes is shown in Fig. 3. Four different combinations of parameterisations of the FitzHugh–Nagumo model are investigated, with each pair having different combinations of positive or negative \(\alpha \) values. The simulations are run until the action potential on each cardiac cycle is identical to that on the previous cycle.
Set 2 In a further stage of verification of the homogenisation, we use a small size of partitioned unit and alter the relative proportions of the two model phenotypes, \(\rho _1\) and \(\rho _2\), throughout the series of simulations, and compare the beat rates from both models. We see in Eq. (9) that varying \(\rho _1\) and \(\rho _2\) alters the excitability properties of the HP model, and so this set of simulations allows us to verify that the excitability properties of the HP model are correctly predicted. The layout of phenotypes is shown in Fig. 3.
Set 6 Our final set of simulations closely follows the design of those in Set 5, but a physiological cell model is used rather than a phenomenological model. The first cell model is the ventricularlike model of Paci et al. (2013), while the second model is the atriallike model from the same paper. In addition to altering the parameter A, we also alter the number of units, n, that the fibre is partitioned into. Example phenotype distributions can be found in Fig. 4.
3.2 Parameters Used in the Simulations
Values of the FitzHugh–Nagumo model parameters \(\alpha , \beta \) and \(\epsilon \)
Name  \(\alpha \)  \(\beta \)  \(\epsilon \)  Rate  \(\hbox {APD}_{90}\)  MDP 

Model S1  \(\) 0.12  \(2\times 10^{7}\)  0.002  0.0019  122  \(\) 0.433 
Model S2  \(\) 0.08  \(3\times 10^{7}\)  0.003  0.0027  81.3  \(\) 0.410 
Model S3  \(\) 0.06  \(4\times 10^{7}\)  0.004  0.0034  61.6  \(\) 0.400 
Model E1  0.12  \(2\times 10^{7}\)  0.002  N/A  84.6  \(\) 0.281 
Model E2  0.08  \(3\times 10^{7}\)  0.003  N/A  67.1  \(\) 0.310 
Model E3  0.06  \(4\times 10^{7}\)  0.004  N/A  54.3  \(\) 0.326 
Tissuelevel parameters for monodomain simulations
Parameter  Value (FHN simulations)  Value (Paci et al. (2013) simulations) 

\(C_m\)  1  \(1\,\upmu \hbox {F}\,\hbox {cm}^{2}\) 
\(\chi \)  1  \(1400\,\hbox {cm}^{1}\) 
\(\varSigma \)  1  \(0.3\, \hbox {mS}\,\hbox {cm}^{1}\) 
xdomain  \(0{}100\)  \(0{}1\,\hbox {cm}\) 
xstep  0.013  0.00052 cm 
Simulation duration  8000  20 s 
Time step (PDE)  \(2^{10}\)  \(2.5\times 10^{4}\,\hbox {s}\) 
Time step (ODE)  \(2^{10}\)  \(5\times 10^{6}\,\hbox {s}\) 
Initial conditions  \(v=1\times 10^{3}\)  As listed in the supplement of 
\(w=0\)  Paci et al. (2013)  
Stimulus period  500  N/A 
Stimulus duration  2  N/A 
Stimulus magnitude  \(\,0.4\) at \(0<x<x_\text {end}/30\)  N/A 
The monodomain problem was solved numerically using a custom implementation of the piecewise linear finite element method in Matlab. The systems of ordinary differential equations from either the FitzHugh–Nagumo or Paci et al. (2013) models were solved using the Forward Euler method. Accuracy of the solver was checked by comparing output against the analytical solution of an example onedimensional monodomain problem from Pathmanathan and Gray (2014). Convergence studies were performed on systems based on the first set of simulations. The selected values of the space and timesteps are listed in Table 2, along with other relevant simulation parameters.
4 Results of Simulations
We now perform the simulations described in Sect. 3.
4.1 Set 1: Variation in the Size of the Partitioned Unit

Models S1 and S3 Both selfexciting, with \(\alpha _H=0.09\);

Models S1 and E2 Selfexciting and excitable, respectively, with \(\alpha _H=0.02\);

Models S3 and E2 Also selfexciting and excitable, but with \(\alpha _H=0.02\); and

Models E1 and E3 Both excitable, with \(\alpha _H=0.09\).
4.1.1 \(\hbox {APD}_{90}\) and Maximum Diastolic Potential
The results shown in Figs. 6 and 7 generally follow a smooth trend in that, as the size of the partitioned unit decreases, the \(\hbox {APD}_{90}\) and MDP of the PP model approach those of the HP model. This confirms that the homogenisation process has worked as expected.
In Figs. 6 and 7, we note that the \(\hbox {APD}_{90}\) and MDP vary across the fibre. This variation becomes more marked near to the boundaries. This is because a travelling wave action potential, i.e. \(V=f(xct)\) (where c may depend on all variables and parameters in the model) is unable to satisfy the boundary condition given by Eq. (6), as has previously been noted by Cherry and Fenton (2011) in a singlephenotype study. We investigate this phenomenon in more detail in Sect. 4.3, but first make some comments that may be explained using these initial simulations.
There are two reasons for the boundary effects that can be observed in plots of \(\hbox {APD}_{90}\) across the domain in Fig. 6 where we have spatially alternating phenotype partitions. Initially we consider regions distant from any boundaries. If a phenotype A has a longer singlecell \(\hbox {APD}_{90}\) than a phenotype B then, upon repolarisation in the PP tissue simulation, current flows from more depolarised to less depolarised regions, which means that phenotype B’s repolarisation is delayed by both its neighbouring A phenotypes. The same currents cause phenotype A’s repolarisation to be encouraged by both its neighbouring B phenotypes, and the overall effect is to smooth the \(\hbox {APD}_{90}\) along the fibre. However, a phenotype on the boundary has just one neighbouring phenotype partition, with a noflux boundary condition at the other side, which means that these smoothing effects are reduced and its \(\hbox {APD}_{90}\) phenotype can become more dominant. The difference between the singlecell \(\hbox {APD}_{90}\) values of the phenotypes themselves (\(\hbox {APD}_{90}\) shown in Table 1) then dictates the magnitude of this effect (S1–E2 have a large \(\hbox {APD}_{90}\) difference of approximately 55 ms, and large edge effects; whereas S3–E2 have a difference of only 5.5 ms and much smaller edge effects).
We also see edge effects due to wave propagation: an action potential reaching a boundary exhibits a shortened \(\hbox {APD}_{90}\) due to the noflux condition instead of the presence of a more depolarised wave ahead; and conversely prolonged \(\hbox {APD}_{90}\) when an action potential originates on a boundary (as studied in detail by Cherry and Fenton (2011)). This second effect occurs in a homogenous phenotype situation as well, and so we deduce it is the dominant cause of the boundary effects in the lower two cases of Fig. 6 as both the HP and PP models exhibit similar edge effects.
As the MDP is a property of the action potential during the hyperpolarised or resting phase, the influence of the pacemaker location on its value is smaller than for the \(\hbox {APD}_{90}\). As we observed for the \(\hbox {APD}_{90}\), the nature of the boundary conditions pulls the MDP higher or lower than would otherwise be expected at the boundary. The one exception is again related to the pacemaking site, with the minimum value of MDP being slightly higher than expected at the righthand side of the S1–S3 fibre.
There are, however, two exceptions to the smooth trends that we now explain. The first exception is in the two lower plots on the righthand side of Fig. 6. We note that the \(\hbox {APD}_{90}\) seen across the region \(35< x < 65\) does still exhibit variations, albeit small variations, as the size of the partitioned unit decreases. This is because both of these simulations are excitable, rather than selfexciting, and therefore require a stimulus (artefact of stimulus edge can be seen on far left). Since the subsequent behaviour is asymmetric the influence of the stimulus prolongation can still be seen in the \(35< x < 65\) domain.
The second exception is in the third panel down on the right of Fig. 6, where we see an outlying result. For a large partitioned unit, where we may not expect the homogenisation to be valid, this PP model was selfexciting despite \(\alpha _H\) being positive. The region of the selfexciting Model S3 closest to the boundary was able to spontaneously depolarise as it was separated from the influence of nonselfexciting Model E2 by the entire large length of the partitioned unit. As a stimulus was also applied, the action potentials switched between spontaneous and stimulated. The change in effective beat rate has an impact on the \(\hbox {APD}_{90}\), which can be seen in Fig. 6: the final beat in the simulation with the secondlargest partitioned unit was spontaneous rather than triggered by the stimulus.
4.1.2 Conduction Velocity
In the nonselfexciting stimulated fibres (the lower panels of Fig. 8), we observe good agreement between the conduction velocity for the HP and PP models for all lengths of partitioned unit.
4.2 Set 2: The Beat Rate and Excitability Condition When Phenotype Proportions Vary
In our second stage of the verification of the homogenisation, we investigate whether the beating condition of the FitzHugh–Nagumo model (\(\alpha <0\) for spontaneous beating) holds, and compare the beat rates of fibres simulated using the PP and HP models.
We first examine the discrepancy in beat rate between the HP and PP models in both panels of Fig. 9. The beat rates of the HP and PP models differ more when the partitioned units are large (top panel) than when the partitioned units are small (bottom panel). In general the condition that the PP model is selfexciting only when \(\alpha _H>0\) is adhered to. The only places where this condition is not met is around \(\alpha _H=0\), for the case where \(n=60\). This problem disappears when \(n=240\), i.e. when the size of the partitioned unit decreases and \(\delta \) approaches zero, and we are closer to the limit in which our homogenisation is valid.
4.3 Set 3: Boundary Effects
In Sect. 4.1.1, we noted the presence of boundary effects in both the PP and HP models. To examine this effect, we alter the model phenotype that is located at the boundary of three otherwise similar phenotype layouts for the PP model. We use Models S1 and S3 for this investigation as their spontaneous activity is representative of the beating of hSCCMs. In the first case, where we have an even number of partitioned units, the outer model phenotype on the lefthand side of the fibre is Model S1 (with a slower natural frequency), while that at the right is Model S3 (with a faster natural frequency). In the other two cases, we have an odd number of units (one more than in the evenn cases). In one of these, Model S1 is present at both boundary units, while in the other Model S3 is present at both boundaries. We test these three patterns of model phenotypes with partitioned units in a range of sizes.
Throughout the simulations shown in Fig. 10, the origin of activation is consistent as the number of partitioned units is increased. The activation wave always originates from a region of the fasterbeating model phenotype; if Model S3 is present at one of the boundaries, the activation wave originates there. In the central case, activation begins at the central instance of Model S3, as the slowerbeating Model S1 takes longer to reach the activation threshold at the boundaries than it does elsewhere. The increase in conduction velocity as the size of the partitioned unit decreases shows that the action potentials tend towards synchronisation in the homogenised limit, as we saw in Fig. 8.
4.4 Set 4: Regular Spatial Variation in Phenotype Proportion
4.5 Set 5: Random Spatial Variation in Phenotype Proportion
In contrast to the simulations for a regular variation in \(\rho _1\) and \(\rho _2\) shown in Fig. 11, we see that introducing randomness into \(\rho _1\) and \(\rho _2\) can induce differences between the PP and HP models. Specifically, the location of the pacemaker region may differ between these models, particularly in the simulations where the smooth variation is small (i.e. A is small), as the variations due to random error are then relatively large when compared to A. We note, however, that the gradients of all activation plots are very similar. Hence, although the pacemaker region may not be accurately located using the HP model, the conduction velocity is consistent. We return to this point when considering a physiological cell model in Sect. 4.6.
4.6 Set 6: Simulations with Physiological Action Potential Models
In this set of simulations, we extend the simulations in Sect. 4.5 to use the Paci et al. (2013) models of atriallike and ventricularlike hSCCM electrophysiology. The atriallike model has a faster beat rate and shorter \(\hbox {APD}_{90}\) than the ventricularlike model, as is shown previously in Fig. 1. We arrange these two phenotypes in a similar manner to that used in the previous series of simulations, which can be seen in the second panel of Fig. 4. We investigate the impact of varying the parameter A, which sets the amount of variation in phenotype across the fibre.
In the previous section, we noted that the HP model is able to capture the overall behaviour of the PP model very well when there is substantial variation in phenotype, i.e. a high value of A. We make similar observations to those made in Fig. 13—as A is progressively increased, keeping n fixed, we see that the pacemaker region is accurately located by the HP model. Even if the pacemaker region is not accurately located, it is seen that the conduction velocity is accurately predicted, as can be seen by the gradient of the plot of activation times. We also observe that the pacemaker region is more accurately located as n increases, as expected. Finally, we note from the gradient of the activation plots that the conduction velocity is around \(19\,\text {cm/s}\) for all values of A, which is similar to that discussed in Sect. 1.1.
5 Conclusions
We have investigated two models for including multiple cellular phenotypes within simulations of cardiac tissue. In the partitioned phenotypes (PP) model, the simulated domain contains distinct regions where a singlemodel phenotype is present. The homogenised phenotypes (HP) model assumes a wellmixed sample of cells, which we represent as a homogenised system. We have verified that the electrical activity generated by the PP model tends towards that of the HP model as the size of the partitions decreases. The HP model is therefore a good approximation to the PP model when the length scale of regions containing a mixture of cell types is small.
Use of the PP model requires that the mesh is sufficiently fine in order to capture the geometry of the partitioned regions as closely as possible. For realistic two and threedimensional simulations with small regions of distinct cell types, this will result in a very large number of nodes, and simulations using this mesh may not be computationally feasible. An advantage of the HP model is that it does not require the mesh to explicitly model the geometry of the partitioned regions, thus significantly reducing the number of nodes in the mesh and eliminating the need for customised versions of the mesh when simulating the same domain with different arrangements of cell types.
Our simulations have demonstrated some experimentally observed properties of hSCCM monolayers. The first two sets of simulations involved fibres with regularly repeating units of alternating phenotype, tending towards a fully mixed system. We observed that changes in values of \(\hbox {APD}_{90}\) and MDP were apparent across the fibre, with the changes being gradual despite clear division between cell types in the PP model. The conduction velocity of the activation wave increased rapidly when a selfactivating cell model was present in fibres simulated with the smallunit PP model or the HP model. More realistic conduction velocities were seen in simulations where there was spatial variation in the distribution of phenotypes. The lack of a dramatic variation in conduction velocity in experimental hSCCM systems, such as those described in Lee et al. (2012), suggests that spatially homogeneous cellular phenotypes are unlikely to occur in cultures of hSCCMs; and that there must be variation in the intrinsic beat rate of hSCCMs in these multicellular cultures. This prediction is consistent with two other recent modelling studies: Abbate et al. (2018) and Tixier et al. (2018) propose that there must be spatial variation in phenotype in hSCCMs to provoke signals of the magnitude observed in microelectrode array experiments. In these papers, different phenotypes were introduced with a partitioned phenotype, and with a smoothly varying parameter set within one model, respectively.
The final three sets of simulations demonstrated how local spatial variability in the relative proportions of the two phenotypes introduced a stable pacemaker region in the HP and smallunit PP models. This observation provides a mechanism by which stable propagation of the activation wave can occur in hSCCMs, even in cultures that only exhibit small amounts of variation in phenotype. The sixth set of simulations utilised physiologically based models of atriallike and ventricularlike hSCCM electrophysiology. We demonstrated similar conduction velocities in cases where both large and small amounts of phenotypic variation were simulated, in the region of values observed experimentally, which vary from approximately 1 cm/s to 20 cm/s depending on maturity (Mehta et al. 2011; Lee et al. 2012; Zhu et al. 2017). We can therefore propose that even a small amount of phenotypic variation removes the system from the fully synchronous regime observed when the HP model was used in the first set of simulations. However, synchronisation may still play a small role in the value of conduction velocity: our observations lead us to the prediction that paced hSCCM monolayers may show slower conduction velocities than they do when left to selfexcite.
In future work, we will compare twodimensional simulation results using this model with experimental measurements from approximately twodimensional monolayers of stem cellderived cardiomyocytes. Such experiments typically use microelectrode arrays to record extracellular potential at a number of sites in the centre of a monolayer in a circular well, and so provide some information on the direction and speed of propagating waves.
In addition to our main focus of human stem cellderived cardiomyocytes, the homogenised phenotypes model may also be useful in other types of cardiac simulation where two or more cell types are present, such as in sinoatrial node where cellular properties are reported to vary based on their position within the pacemaking region. The current interest in uncertainty quantification and variability in biological systems is driven by the need to understand how these factors can affect model output, thus influencing the utility of these models to complement experiments (Elkins et al. 2013; Mirams et al. 2016). Our proposals for simulation of multiple cell types will enable detailed investigation of the impact of variable spatial distributions of cell type on the signals recorded from monolayer cultures of human stem cellderived cardiomyocytes that are part of the proposed Comprehensive in vitro Proarrhythmia Assay initiative (Sager et al. 2014).
Notes
Acknowledgements
This work was supported by the UK Engineering and Physical Sciences Research Council [grant number EP/F500394/1]; and the Wellcome Trust [grant number 101222/Z/13/Z]. The EPSRC supported LAB through the Life Science Interface Doctoral Training Centre; and the Wellcome Trust & Royal Society supported GRM with a Sir Henry Dale Fellowship.
Supplementary material
References
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