Automatic Optimization of an in Silico Model of Human iPSC Derived Cardiomyocytes Recapitulating Calcium Handling Abnormalities

The growing importance of human induced pluripotent stem cell-derived cardiomyoyctes (hiPSC-CMs), as patient-specific and disease-specific models for studying cellular cardiac electrophysiology or for preliminary cardiotoxicity tests, generated better understanding of hiPSC-CM biophysical mechanisms and great amount of action potential and calcium transient data. In this paper, we propose a new hiPSC-CM in silico model, with particular attention to Ca²⁺ handling. We used (i) the hiPSC-CM Paci2013 model as starting point, (ii) a new dataset of Ca²⁺ transient measurements to tune the parameters of the inward and outward Ca²⁺ fluxes of sarcoplasmic reticulum, and (iii) an automatic parameter optimization to fit action potentials and Ca²⁺ transients. The Paci2018 model simulates, together with the typical hiPSC-CM spontaneous action potentials, more refined Ca²⁺ transients and delayed afterdepolarizations-like abnormalities, which the old Paci2013 was not able to predict due to its mathematical formulation. The Paci2018 model was validated against (i) the same current blocking experiments used to validate the Paci2013 model, and (ii) recently published data about effects of different extracellular ionic concentrations. In conclusion, we present a new and more versatile in silico model, which will provide a platform for modeling the effects of drugs or mutations that affect Ca²⁺ handling in hiPSC-CMs.

Introduction

Human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) are cardiac cells derived from stem cells, which have been produced by donor's differentiated cells by means of reprogramming (Takahashi et al., 2007). The role of hiPSC-CMs, has become more and more pervasive in basic electrophysiological studies as well as in applied research, such as pharmacological tests, since their discover in 2007. As an in vitro human model, hiPSC-CMs strongly impacted the study of biophysical mechanisms underlying cardiac electrophysiology at cellular level, both in control and diseased conditions. Especially hiPSC-CMs' patient- and disease-specificity is fundamental to assess the effects of genetic mutations, such as Long QT (LQT) (Moretti et al., 2010; Lahti et al., 2012; Ma et al., 2013), catecholaminergic polymorphic ventricular tachycardia (CPVT) (Kujala et al., 2012) and hypertrophic cardiomyopathy (HCM) (Ojala et al., 2016), on the functionality of cardiomyocytes. Ever since the beginning of the Comprehensive In vitro Proarrhythmic Assay (CIPA) (http://cipaproject.org/) in 2013, hiPSC-CMs have had a dramatic impact on pharmacology, serving as a powerful in vitro model to test the in silico model predictions regarding cardiac safety or drug toxicity at cellular level.

During the last 10 years, many progresses were done in terms of efficiency of hiPSC-CM production and availability of commercial cell lines. This enabled also the hiPSC-CM electrophysiological and pharmacological evaluation by means of medium-throughput (Rajamohan et al., 2016) or even high-throughput systems (Entcheva and Bub, 2016; Klimas et al., 2016), where the use of voltage- and calcium-sensitive dyes has been combined with hiPSC-CM optogenetic stimulation. Together with the availability of these experimental data, new methods have been developed to process them (Björk et al., 2017; Ahola et al., 2018).

The importance of Ca²⁺ cycling to basic cardiac functionality, together with the growing availability of hiPSC-CM Ca²⁺ cycling data, makes Ca²⁺ transients, and biomarkers computed onto them, as interesting as action potential (AP) measurements. Ca²⁺ is fundamental in the heart excitation-contraction (EC) coupling, i.e., how the electrical and the mechanical properties of the heart are linked together and how the AP leads to the cardiomyocyte contraction. The elements involved in this phenomenon are the L-type Ca²⁺ channels, the sarcoplasmic reticulum (SR) and the sarcomeres, i.e., the contractile unit of the cardiomyocyte. During the AP upstroke, the L-type Ca²⁺ channels open and Ca²⁺ flows into the cardiomyocyte. This Ca²⁺ influx is sufficient to trigger the Ca²⁺ release from SR through the ryanodine-sensitive Ca²⁺ channels, which increases the cytosolic Ca²⁺. Such amount of Ca²⁺ allows starting the crossbridge cycle, which is at the basis of the cardiomyocite contraction and continues until Ca²⁺ is restored to its basal cytosolic concentration. This is mainly done by the SERCA-2 pump, which reabsorbs Ca²⁺ from the cytosol into the SR. Moreover, cytosolic Ca²⁺ is extruded into the extracellular space by the Na⁺/Ca²⁺ exchanger (I_NaCa) and by the sarcolemmal Ca²⁺ pump (I_pCa). All these mechanisms make the intracellular Ca²⁺ concentration change, thus producing Ca²⁺ transients associated to the APs (Walker and Spinale, 1999).

In 2013 we published the first in silico hiPSC-CMs model (Paci et al., 2013), based on our previous model of cardiomyocytes derived from human embryonic stem cells (Paci et al., 2012) and on the experimental data by Ma et al. (2011). This model has been widely used for computational studies, such as (i) the prediction of drug effects on cardiac electrophysiology (Paci et al., 2015; Lei et al., 2017), (ii) the model extension to multielectrode array simulations (Raphel et al., 2017), and (iii) the assessment of hiPSC-CM electrophysiological variability in control and mutant cells, by means of populations of in silico hiPSC-CMs (Paci et al., 2017). However, one of the Paci2013 model limitations resides in its formulations of the Ca²⁺ handling system: especially the Ca²⁺ release from the sarcoplasmic reticulum (SR) is formulated with the functional but quite elementary Ca²⁺ release from the TenTusscher2004 model (ten Tusscher et al., 2004).

In this work, we propose an updated version of the Paci2013 hiPSC-CM ventricular-like model with a more flexible Ca²⁺ handling formulation. The Ca²⁺ transients produced by the model were calibrated on experimental Ca²⁺ imaging data recorded in our laboratory on hiPSC-CMs. Moreover, the fine tuning of the model parameters was performed by means of an automatic optimization technique (Fabbri et al., 2017), in order to reproduce realistic AP and Ca²⁺ transient shapes and to speed up the parameter tuning phase. Parameter optimization affected only the parameters representing the Ca²⁺ SR fluxes and a very limited set of parameters of membrane currents in order to be consistent with the voltage clamp experiment fitting done by Paci et al. (Paci et al., 2013). Finally, the resulting updated model was validated against ion current blocking data.

Materials and Methods

New Ca²⁺ Handling System Formulation

The main limitation of the original Paci2013 model (Paci et al., 2013) and its following minor updates (Paci et al., 2015, 2017) is the simplified description of the Ca²⁺ release from SR, i.e., the release current I_rel. I_rel was formulated as in the TenTusscher2004 model (ten Tusscher et al., 2004) in the following way:

$\begin{array}{l}{I}_{rel}=K_{rel}·\left(c_{rel}+\frac{a_{rel}·C{a}_{SR}^2}{b_{rel}^2+C{a}_{SR}^2}\right)·d·g\end{array}$

where K_rel, a_rel, b_rel, and c_rel are constants, Ca_SR is the Ca²⁺ concentration in the SR reticulum, d is the L-type Ca²⁺ current (I_CaL) voltage-dependent activation gate, and g is the I_rel specific inactivation gate. Therefore, I_rel is not activated by the cytosolic Ca²⁺ concentration Ca_i sensed by the Ryanodine-sensitive receptor located on the SR membrane, but it is triggered the same way as I_CaL. Due to this mechanism, the model was not able to produce proarrhythmic triggers, such as delayed afterdepolarization (DADs) (Fink et al., 2011).

We reformulated I_rel according to the formulation used by Koivumäki et al. (2011) for the human atrial myocyte:

$\begin{array}{l}{I}_{rel}=I_{rel,max}·Ry{R}_{CaSR}·Ry{R}_o·Ry{R}_c·(C{a}_{SR}-C{a}_i)\end{array}$

where I_rel,max represents the maximum Ca²⁺ release from SR, RyR_CaSR is the dependence on Ca_SR, RyR_o is the open (activation) gating variable, RyR_c the closed (inactivation) gating variable. A third gating variable, RyR_a, was used to modulate the working point (adaptation) of the RyR_o and RyR_c gates according to the cytosolic Ca²⁺ concentration as in Koivumäki et al. (2011). The formulation of the single gating variables is the following

$\begin{array}{l}Ry{R}_{CaSR}=1-\frac{1}{1+e^{\frac{C{a}_{SR}-0.3}{0.1}}}\\ Ry{R}_{a,ss}=Ry{R}_{a1}-\frac{Ry{R}_{a2}}{1+e^{\frac{C{a}_i-Ry{R}_{a,half}}{Ry{R}_{a,k}}}}\\ Ry{R}_{o,ss}=1-\frac{1}{1+e^{\frac{C{a}_i-(Ry{R}_a+Ry{R}_{o,half})}{Ry{R}_{o,k}}}}\\ Ry{R}_{c,ss}=\frac{1}{1+e^{\frac{C{a}_i-(Ry{R}_a+Ry{R}_{c,half})}{Ry{R}_{c,k}}}}\end{array}$

where the subscript ss indicates the steady state value of the gating variable and the subscripts half and k indicate the half Ca²⁺ concentration and the slope, respectively, of the gating variable steady state. The values of the constants RyR_a1, RyR_a2 and the gating variable half activations RyR_a,half, RyR_o,half and RyR_c,half were then optimized (together with other parameters) as described in section “Parameter Optimization.” We report the full set of equations in section 1 of the Supplementary Material.

The formulation of the other SR fluxes, namely the SERCA pump (I_up) and the leakage current (I_leak) were not changed with respect to the Paci2013 model (Paci et al., 2013). However, their parameters went through the optimization process, as we describe in section Parameter Optimization. In the following we will refer as Paci2013 to the model version presented in Paci et al. (2017).

Parameter Optimization

The parameter optimization process was adapted from Fabbri et al. (2017).

Shortly, parameter optimization was done using the Matlab© function fminsearch, which implements the Nelder-Mead Simplex Method, as reported in Fabbri et al. (2017). Such function minimizes a cost function built on the experimental biomarkers we want the model to simulate. The following equations show the cost function structure (Fabbri et al., 2017):

$\begin{array}{l}Cost={\displaystyle \sum _{i=1}^{N_{biomarkers}}}Cos{t}_i\\ Cos{t}_i=\{\begin{array}{ll}\frac{\left|b_{i,Exp}-b_{i,Sim}\right|-S{D}_i}{S{D}_i}·w_i\hfill & if\left|b_{i,Exp}-b_{i,Sim}\right|>SD\hfill \\ 0\hfill & otherwise\hfill \end{array}\end{array}$

where Cost_i and SD_i represent the cost and the standard deviation for a single biomarker b_i.

The structure of the cost function Cost is consistent with the cost function used in Fabbri et al. (2017). The contribution of each biomarker to the overall cost is zero if the biomarker is within its ±SD range, otherwise it grows linearly. Moreover, each contribution Cost_i was weighted according to the respective weight w_i. This results in a non-linear cost function, which is zero if all the biomarkers are within the experimental ranges and greater than zero if at least one biomarker is out of range. The AP experimental biomarkers considered for the cost function were: AP amplitude (APA), maximum diastolic potential (MDP), cycle length (CL), maximum upstroke velocity (V_max), AP duration at 10, 30, and 90% of repolarization (APD₁₀, APD₃₀, APD₉₀) and AP shape factor (Triangulation). Triangulation is a shape factor used by Ma et al. (2011) to discriminate between atrial-like (Triangulation < 1.5) and ventricular-like (Triangulation>1.5) APs, and it is computed as:

$\begin{array}{l}Triangulation=\frac{AP{D}_{30}-AP{D}_{40}}{AP{D}_{70}-AP{D}_{80}}\end{array}$

The Ca²⁺ transient experimental biomarkers were: duration of the Ca²⁺ transient (DURATION), time to peak (TPEAK), rise time from 10 to 50% (RT₁₀₅₀), rise time from 10 to 90% (RT₁₀₉₀), decay time from 90 to 10% (DT₉₀₁₀) and the Ca²⁺ transient rate (FREQ). The full list of the parameters optimized by this method and their original values are reported in Table 1. For optimization, we only chose parameters related to Ca²⁺ handling and additional parameters of I_NaCa and the Na⁺/K⁺ pump (I_NaK), i.e., those currents for which the Ma et al. dataset (Ma et al., 2011) did not provide experimental data. The parameter values were constrained in a range [−20%, +20%] with respect to their nominal value in the Paci2013 model, in order to avoid non-physiological values, such as negative conductances. The initial parameter values for RyR_a1, RyR_a2, RyR_a,half, RyR_o,half, RyR_c,half were rescaled to 1/10 of their values in Koivumäki et al. (2011) before the optimization procedure, in order to adapt them to the Ca²⁺ concentrations in the Paci2013 model, and then optimized as the other parameters. All the other ionic current parameters were kept as in Paci et al. (2013) and Paci et al. (2017) for the late Na⁺ current (I_NaL) only.

TABLE 1

Table 1. Parameters chosen for optimization.

To constrain the model parameters, two different experimental datasets were used. AP data were taken from Ma et al. (2011). This is the same dataset used to calibrate the Paci2013 model and it includes biomarkers computed on spontaneous ventricular-like hiPSC-CM APs recorded at 35–37°C. A second, new, dataset was obtained through Ca²⁺ transient recordings performed at the BioMediTech Institute (Tampere, Finland).

Calcium Recordings in hiPSC-CMs

This study was carried out in accordance with the recommendations of Guidelines of the Ethics Committee of Pirkanmaa Hospital District (Tampere, Finland). The protocol was approved by the Ethics Committee of Pirkanmaa Hospital District (Aalto-Setälä R08070). All subjects gave written informed consent in accordance with the Declaration of Helsinki. New Ca²⁺ transient dataset were recorded at the BioMediTech Institute from healthy control hiPSC-CMs at 35–37°C. The generation and characterization of the control hiPSC line and cardiac differentiation were done as described earlier (Ojala et al., 2016). Cardiomyocytes plated on a coverslip were loaded with 4 μM Fluo-4 AM (Thermo Fisher Scientific) for 30 min and de-esterified for 10 min in perfusate medium: (in mM) 137 NaCl, 5 KCl, 0.44 KH₂PO₄, 20 HEPES, 4.2 NaHCO₃, 5 D-glucose, 2 CaCl₂, 1.2 MgCl₂, and 1 Na-pyruvate dissolved in H₂O (all from Sigma Aldrich). pH of the perfusate medium was adjusted to 7.4 with NaOH (Sigma Aldrich). Perfusate was heated with an inline heater SH-27B controlled with a TC-324B controller unit and input into a RC-25 imaging chamber (all Warner instruments Inc., CT, USA). Calcium kinetics of spontaneously beating cardiomyocytes were imaged with an inverted Olympus IX70 microscope using UApo/340 0,75NA 20x air objective (Olympus, Tokyo, Japan) and recorded with ANDOR iXon 885 EM-CCD camera (Andor Technology, Belfast, Northern Ireland) using 2 × 2 binning and synchronized with a Polychrome V light source by a real time DPS control unit. LiveAcquisition software (TILL Photonics, Munich, Germany) was used to control light source and camera during recording. Fluo-4 was excited at 490 nm wavelength and the emission was recorded through Olympus U-MF2 Alexa 488 band-pass filter cube (ex.470–495, em.525/50 nm).

Data were recorded from 15 cells for a total of 218 transients. For each biomarker mean value and standard deviation (SD) were computed for model calibration. The experimental values for the biomarkers and their weights are reported in Table 2.

TABLE 2

Table 2. Experimental and simulated values of the biomarkers considered for the parameter optimization.

Model Validation With Current Blocker Simulations

In order to validate the new hiPSC-CM model after the introduction of the new I_rel formulation and the parameters optimization by means of our Ca²⁺ transient data, we chose to replicate the current blocker simulations performed in Paci et al. (2013). The model was paced at 1 Hz for 800 s to reach its steady state, then the current blocker was simulated by reducing the maximum conductance of the affected current and finally the AP biomarkers were computed after 400 s. The current blockers considered in the simulations were the same used by Ma et al. (2011): tetrodotoxine (TTX, I_Na blocker), nifedipine (NIFED, I_CaL blocker), E4031 (I_Kr blocker) and 3R4S-Chromanol 293B (CHR, I_Ks blocker). For TTX, NIFED and E4031 we considered the following IC₅₀ values: 0.64 μM (Ma et al., 2011), 0.038 μM (Ma et al., 2011) and 100 nM (Sanguinetti and Jurkiewicz, 1990; Gerlach et al., 2010) respectively. In Ma et al. (2011) CHR had small effects on the AP biomarkers, therefore we tested a range of block levels: 30, 50, 70, and 90%. The stimulus current was 550 pA for NIFED, E4031 and CHR, while we chose 750 pA for TTX in order to trigger APs also at the highest blocker concentrations.

Delayed Afterdepolarizations

In order to trigger DADs in spontaneous APs, Ca²⁺ overload had to be simulated: we chose to increase the superfusate Ca²⁺ concentration (Volders et al., 2000). To ensure that Ca²⁺ overload was simulated only once the model was in steady state, we ran a 800 s simulation without external pacing. From these steady state conditions, we increased the extracellular Ca²⁺ concentration by setting it to 3.945 mM and again we simulated the spontaneous APs. We ran this protocol with both the Paci2013 and the Paci2018 models.

Results

Spontaneous Action Potentials and Ca²⁺ Transients

The automatic optimization process provided a new set of parameters, which is reported in Table 1, fourth column. The simulated spontaneous APs and Ca²⁺ transients were in agreement with all the biomarker variability ranges (mean ± SD), as reported in the last column of Table 2. Figure 1 shows a comparison between the Paci2018 model (in solid blue) and the Paci2013 model (in dashed red). Figure 2 illustrates the simulated Ca²⁺ transients and traces from four illustrative cells: the comparison highlights that the simulated Ca²⁺ transient contour is fully in agreement with our experiments. In Figure 3 simulated spontaneous AP and Ca²⁺ traces in steady state are reported, together with the main ionic currents and concentrations. I_rel gating variables, together with the Ca²⁺ transients, are detailed in Figure 4. It shows that the Ca²⁺ release from SR is not directly dependent on the I_CaL activation, but on the cytosolic Ca²⁺ concentration Ca_i, which rules the behavior of the RyR_o, RyR_c, and RyR_a gating variables.

FIGURE 1

Figure 1. Comparison between the APs and Ca²⁺ transients simulated by the new Paci2018 model (solid blue) and the Paci2013 model (dashed red). (A) membrane potential. (B) Ca²⁺ transient.

FIGURE 2

Figure 2. Illustrative experimental Ca²⁺ transients from four cells (blue, cyan, green, magenta) and the Ca²⁺ transient simulated by the Paci2018 model (black). The y-axis of the experimental traces (originally ΔF/F) was normalized between the minimum and maximum values of the simulated trace for comparison. The inset contains only the decay part (from 90 to 10% of the transient amplitude) of the experimental and simulated transients reported in the main Figure.

FIGURE 3

Figure 3. Simulated spontaneous action potentials and ionic currents. (A,G) membrane potential. (B) Fast Na⁺ current (I_Na). (C) L-type Ca²⁺ current (I_CaL). (D) Funny current (I_f) and Late Na⁺ current (I_NaL). (E) Transient outward K⁺ current (I_to) and Na⁺/K⁺ pump (I_NaK). (F) Rapid delayed (I_Kr), slow delayed (I_Ks) and inward (I_K1) rectifier K⁺ currents. (H) Na⁺/Ca²⁺ exchanger (I_NaCa). (I) Release current from sarcoplasmic reticulum (I_rel). (J) Na⁺ cytosolic concentration (Na_i). (K) Cytosolic Ca²⁺ concentration (Ca_i). (L) Sarcoplasmic Ca²⁺ concentration (Ca_SR).

FIGURE 4

Figure 4. Details on the new formulation of the Ca²⁺ release from sarcoplasmic reticulum. (A) Time-course of the three sarcoplasmic release current gates during spontaneous action potentials. (B) Ca²⁺ transients during spontaneous action potentials.

Current Blocker Simulations

We compared in Table 3 the current blocker effects on the AP biomarkers (i) experimentally recorded by Ma et al. (2011), (ii) simulated by means of the Paci2013 model (Paci et al., 2013), and (iii) simulated by the Paci2018 model. Figure 5 shows that, despite the changes in the I_rel formulation and parameter optimization, the behavior of the Paci2018 model is still consistent with the Paci2013 simulations (Paci et al., 2013) and the experimental data by Ma et al. (2011). TTX affected the upstroke phase, reducing the AP V_max and delaying the AP Peak. NIFED, by acting on I_CaL, reduced the Ca²⁺ influx through the cell membrane thus reducing the AP duration. Conversely, E4031 reduced the K⁺ efflux through the cell membrane, thus prolonging the AP. Finally, CHR showed little effect on the AP biomarkers, in agreement with the experiments.

TABLE 3

Table 3. Quantitative effects of current blockers on stimulated APs.

FIGURE 5

Figure 5. Simulation of current block effects on hiPSC-CMs paced at 1 Hz. (A) Tetrodotoxine blocks I_Na, slowing down the upstroke phase. (B) Nifedipine blocks I_CaL, shortening APD and triangulating AP profile. (C) E4031 blocks selectively I_Kr, increasing APD. (D) I_Ks block by 3R4S-Chromanol 293B does not affect significantly the AP shape.

Delayed Afterdepolarizations

In Figure 6 we compared the behavior of the Paci2013 and the Paci2018 models in conditions of high extracellular Ca²⁺ concentration (Ca_o = 3.945 mM instead of 1.8 mM) and no external stimulation, i.e., both models produced spontaneous APs. As shown in Figure 6B, for t = [14, 30] s I_rel underwent a small reactivation, slowing down the decay of the Ca²⁺ transients (Figure 6E), affecting the inward component of I_NaCa (Figure 6D) and triggering DADs in the membrane potential (Figures 6A,F). Around t = 30 s, I_rel underwent a full reactivation, triggering an anticipated Ca²⁺ transient, which resulted in a full anticipated beat (Volders et al., 2000). We reported the extended AP and Ca²⁺ transient traces in the Supplementary Figure 1. Conversely, the old Paci2013 model did not show anomalies in the membrane potential (Figure 6G) or Ca²⁺ transients (Figure 6J) as consequence of the high extracellular Ca²⁺ concentration, while the Ca²⁺ in SR grew dramatically. The improvements introduced in the Paci2018 model allowed to simulate DADs not only in case of Ca²⁺ overload, but offered flexibility to simulate behaviors such as the spontaneous premature Ca²⁺ releases reported by Kim et al. (2015) in their Figures 2C, 4B and their Supplemental Figure S1B. We tested also the effect of a rate increment on the generation of DADs in case of hypercalcemia, by providing external pacing. In Supplementary Figure 2, we paced the Paci2018 model at its spontaneous basal rate (38 bpm, in solid blue) and at a basal rate increased by 50% (57 bpm, in dashed red). Despite the shorter diastolic depolarization phase, also at 57 bpm we obtained a fully developed DAD.

FIGURE 6

Figure 6. Comparison of the behavior of the Paci2018 (in blue, A–F) and the Paci2013 model (in red, G–K) in case of spontaneous APs and Ca²⁺ overload induced by hypercalcemia (Ca_o = 3.945 mM). In this condition both models can produce spontaneous APs, reported in (A,G). The Paci2018 model shows a saturation of the sarcoplasmic Ca²⁺ concentration (C), after which reactivation of I_rel (B) induces a distortion in the shape of the Ca²⁺ transients (E) and in the inward component of I_NaCa (D), which corresponds to DADs in the membrane potential (A,F) in between t = [14, 30] s. At t = 30 s a strong spontaneous release of Ca²⁺ from SR is shown, which induces a spurious AP. On the contrary, the Paci2013 is not affected in these conditions and does not produce repolarization abnormalities (G).

Figure 7 shows the spontaneous APs, Ca²⁺ transients and I_NaCa, together with the release Ca²⁺ flux (I_rel), characterized with premature Ca²⁺ releases. These traces were obtained with a normal extracellular Ca²⁺ concentration (Ca_o = 1.8 mM) but simulating more “immature” RyR machinery (namely, by shifting RyR_o,half and RyR_c,half by −0.002 and 0.002 mM respectively, doubling RyR_o time constant and reducing to half of its nominal value RyR_c time constant). In these conditions of more “immature” RyR machinery, we tested also the effects of I_NaCa block on DADs generation. In Supplementary Figure 3, we tested three I_NaCa block levels (50, 70, and 90%) while pacing the Paci2018 model at 1 Hz. With no I_NaCa block (solid blue trace), APs show small DADs comparable to those reported in Figure 7 in t = [10, 25] s. By blocking 50% of I_NaCa (dashed orange trace), we observed that the amplitude of DADs increases. Finally, for the higher block levels, we did not observe any DAD.

FIGURE 7

Figure 7. Repolarization abnormalities with standard extracellular Ca²⁺ concentrations. Such behavior was obtained with a control extracellular Ca²⁺ concentration Ca_o = 1.8 mM, shifting RyR_o,half and RyR_c,half by −0.002 and 0.002 mM respectively, doubling RyR_o time constant and reducing to half of its nominal value RyR_c time constant. These traces show the ability of the Paci2018 to simulate pathological conditions affecting the Ca²⁺ from SR. (A) Membrane potential. (B) I_NaCa. (C) Cytosolic Ca²⁺ concentration. (D) I_rel. A similar morphology of the cytosolic Ca²⁺ time-course is reported in Kim et al. (2015) in their Figure 4B.

Comparison With Other Experimental Data

In order to compare the Paci2018 model with the experimental data, we challenged our model in the following conditions: (i) I_f block by ivabradine, (ii) hyperkalemia, (iii) hypocalcemia.

We simulated I_f block by 3 μM ivabradine as a 41% block of I_f maximum conductance (Yaniv et al., 2012; Koivumäki et al., 2018). As shown in Figure 8, I_f block induced only a slight reduction of the frequency of spontaneous APs (−2.3%), in agreement with (Kim et al., 2015), where 3 μM had virtually no effects on the spontaneous activity.

FIGURE 8

Figure 8. Effect of 3 μM ivabradine on the spontaneous APs. Administration of ivabradine (in dashed red) slightly slows down the rate of the spontaneous APs in comparison to control (in solid blue).

Kim et al. (2015) reported a slowdown of the spontaneous activity in conditions of hyperkalemia (from 4 to 8 mM the median frequency dropped from about 1.2 to 0.2 Hz) and a further increase in extracelluar K⁺ to 12 mM stopped the spontaneous activity. Our single cell model showed a similar trend (although with less sensitivity on K⁺): for K_o = 8 mM and K_o = 16 mM the Ca²⁺ transient spontaneous rate dropped by 11% and 15% respectively. An extracellular K⁺ concentration equal to 20 mM stopped the hiPSC-CM spontaneous activity as in Kim et al. (2015).

In the previous section, we tested the model in hypercalcemia conditions to simulate Ca²⁺ overload and the generation of DADs. We also challenged the model with an extracellular Ca²⁺ concentration of 0.1 mM (hypocalcemia) as in Kim et al. (2015): again we observed a trend qualitatively in agreement with the experimental data, since the rate of the spontaneous Ca²⁺ transients dropped by 17%.

Discussion

In this paper we present an updated version of the Paci2013 model of ventricular-like hiPSC-CM (Paci et al., 2013), here named Paci2018, developed by exploiting new Ca²⁺ transients data and an automatic algorithm for the parameter optimization.

The Paci2018 model aims to overcome an important limitation of the Paci2013 model, i.e., the simple formulation of the Ca²⁺ release from SR, I_rel, initially presented by the TenTusscher2004 model of human adult ventricular cardiomyocyte (ten Tusscher et al., 2004). The peculiarity of such I_rel is the direct link between the activation of I_CaL and the activation of I_rel (through the voltage-dependent activation gating variable d). This I_rel formulation was then chosen for its simplicity instead of more complex formulations, however it presumes I_rel to be directly dependent on the intracellular Ca²⁺ concentration and it prevents the model to be able to reproduce Ca²⁺-related anomalies such as DADs. As reported in Fink et al. (2011), models where the open probability of the RyR-sensitive channels is directly dependent on the opening of I_CaL are unable to produce DADs. Indeed, DADs start from a repolarized membrane, at potentials where the I_CaL gates are closed. Therefore, RyR-sensitive channels have to be able to open even when I_CaL is zero, which is not possible in the old I_rel formulation. Moreover, the presence of the c_rel constant in the TenTussher2004 I_rel formulation could allow a Ca²⁺ flux also from an empty SR, which is clearly not possible.

We chose a quite simple I_rel formulation, inspired by the Koivumäki2011 model of human adult atrial cardiomyocyte (Koivumäki et al., 2011), characterized by three gating variables. We did not choose more complex formulations, such as the Markov formulation in TenTusscher2006 model (ten Tusscher and Panfilov, 2006), since the latter was implemented mainly to correctly take into account the presence of T-tubules and the associated microdomains in the human adult ventricular cell. However, hiPSC-CMs produced so far with current differentiation protocols have not shown functional T-tubules (Li et al., 2013).

In order to keep the Paci2018 model the most consistent with its predecessor and with the Ma et al. dataset (Ma et al., 2011) of ionic currents and AP biomarkers, we did not change the formulation of I_Na, I_CaL, I_f, I_to, I_Kr, I_Ks, and I_K1, which were carefully fitted for the development of the Paci2013 model. In order to tune the Paci2018 model on the new data of Ca²⁺ transients, we decided to act only on those parameters defining the formulation of currents for which the Ma et al. (2011) dataset did not provide direct experimental data.

The result is a new model where spontaneous APs match the Ma et al. dataset and where Ca²⁺ transients are in agreement with the new Ca²⁺ dataset. In this paper, we did not aim to propose only modifications to a previous model in order to simulate a specific phenomenon of interest, but we wanted to develop a new cardiac cell model to substantially increase modeling accuracy. Therefore, it was fundamental to check the Paci2018 model capability to reproduce all the experimental data used for the Paci2013 model validation. In detail, we tested the Paci2018 model's responses to the same prototypical current blockers used to validate the Paci2013 model, showing that the behavior of the two models is consistent and in agreement with the experimental data. We further validated the model to simulate the experimentally observed phenomenon of DADs and other specific results obtained in different experimental conditions, being able to replicate all these experiments. In particular, the Paci2018 model can reproduce Ca²⁺-related abnormalities such as DAD-like anomalies that were not available in old formalism. We observed that the small or full I_rel reactivations increase the cytosolic Ca²⁺ concentration and this produces an increased activity of the inward component of I_NaCa, which then translates in DAD-like abnormalities in the membrane potential. This is particularly clear in Supplementary Figure 3, where for high levels of I_NaCa block (70 and 90%) we do not observe DADs. It is interesting to note that the 50% block is not enough to cancel DADs: in fact, the inward I_NaCa component is still strong enough and further enhanced by the cytosolic Ca²⁺ accumulation, due to the very same I_NaCa block (minimum diastolic Ca²⁺: 0.028 vs. 0.017 μM; average diastolic Ca²⁺: 0.060 vs. 0.027 μM, in case of 50% I_NaCa block or no I_NaCa block, respectively). This results in even larger DADs than in the no block case. Similar DAD-like abnormalities in the electrical properties of hiPSC-CMs were also observed in mutant hiPSC lines, such as those derived from patients with HCM (Ojala et al., 2016) or CPVT (Kujala et al., 2012). Of note, in this paper we do not aim to simulate these mutations, which would deserve a specific analysis by their own, but to provide a model which can enable there in silico modeling. Notably, the membrane potential and cytosolic Ca²⁺ time-courses obtained by our model are very similar to the experimental ones, as seen by comparison of our Figures 6, 7 with Figures 2C, 4B and S1 in Kim et al. (2015). In particular, Figure 4B in Kim et al. (2015) shows the effect of isoproterenol, a non-selective β adrenoreceptor agonist, which first induces small delayed spontaneous Ca²⁺ releases and then triggers repetitive high rate firing. In our Figures 6A,E we show a very similar pattern in conditions of hypercalcemia (see Supplementary Figure 1 for an expanded version of Ca²⁺ and AP time-courses), where the spontaneous Ca²⁺ releases, which trigger DADs, finally cause a transition to a higher rate of Ca²⁺ transients and APs.

To further validate our model, we challenged it to simulate specific conditions such as hyperkalemia and hypocalcemia, to compare its behavior to published experimental data. The model showed qualitative agreement with such experiments, in spite of the fact that the spontaneous rate reduction is stronger in the experiments. However, such differences can emerge from (i) the multicellular embryoid body culturing and measurements, i.e., on multicellular ensembles, and especially (ii) as consequence of the high inter-cell line and inter-laboratory variability showed by hiPSC-CMs (Knollmann, 2013), especially in beating rate and MDP. In fact, the embryoid bodies examined by Kim et al. (2015) exhibited negligible I_K1 and a much depolarized MDP (−59.1 ± 3.3 mV), which facilitate the spontaneous electrical activity, as shown also by the high frequency of the spontaneous Ca²⁺ transients (1 Hz or higher) in control conditions. On the contrary, the experimental datasets we used in this paper showed slower spontaneous activity and more hyperpolarized MDP, as simulated by our model and reported in Table 2 (MDP = −75.8 mV, FREQ = 0.65 Hz).

We tested also the capability of the Paci2018 model to simulate early afterdepolarizations (EADs) in conditions of I_Kr block: the protocol and the results of the EAD simulations were reported in section 3 of the Supplementary Material. We observed that the Paci2018, as the old Paci2013 model, does not produce EADs. This is not surprising, since we designed the Paci2018 model carefully maintaining the same formulation for the ion currents that we fit on the experimental data by Ma et al. (2011), and that we used for the Paci2013 model (Paci et al., 2013). However, in our previous study (Paci et al., 2016), we demonstrated that I_Kr block triggered EADs in a population of in silico hiPSC-CM based on the Paci2013 model. Replicating here that study is out of the scope of this paper, however we decided to test if two parameter sets, which in Paci et al. (2016) triggered EADs, and one parameter set, which triggered repolarization failure, could induce repolarization abnormalities also in the Paci2018 model. Supplementary Figure 4 shows that, as expected, two derived models produced EADs and the third one repolarization failure. In particular, as reported in the Supplementary Table 1, these parameter sets showed (compared to baseline): (i) greater I_CaL (which corresponds to a greater Ca²⁺ influx during the repolarization phase in case of I_CaL reactivation); (ii) greater I_NaCa (which is inward during repolarization, thus promoting abnormalities in conditions of compromised repolarization as in case of I_Kr block), and (iii) smaller I_K1 (corresponding to a smaller contribution to the membrane potential stabilization). It is also interesting to notice that the three parameter sets showed greater I_Kr compared to baseline: this suggests that in models where I_Kr is highly expressed, a 90% block can have more dramatic effects than in cells expressing smaller I_Kr. This small test suggests, as we reported among the limitations, that the use of more refined modeling approaches, e.g., populations of in silico models, is advisable to simulate specific phenomena (in this case EADs as consequence of I_Kr block) in cell types characterized by high electrophysiological variability. In the perspective of offering researchers a tool to investigate the mechanisms of ventricular arrhythmia development in hiPSC-CMs, we ran also two preliminary tests about the occurrence of alternans or alternans-like patterns in the Paci2018 model (Methods and Results are detailed in section 4 of the Supplementary Material). When pacing the Paci2018 model at 120 bpm (2 Hz), the membrane potential exhibited an alternans-like pattern, characterized by a full AP followed by a smaller one (Supplementary Figure 5A). We tested also the effect of very high pacing rate (200 bpm, 3.33 Hz) in ischemia-like conditions (Supplementary Figure 5B): in this case, the membrane potential was strongly depolarized (−60 mV) and exhibited 2:1 alternans. We consider these results valuable also for more advanced studies regarding arrhythmias propagation in case of extension of the simulations to a monodimensional strand or a bidimensional patch.

Overall, we believe that the here presented Paci2018 model substantially increases hiPSC-CM modeling accuracy. Of course, as every in silico model, also the Paci2018 is an approximated description of a real system: consequently, new mechanisms can be explored and improved. This is especially true for such a complex cellular system as hiPSC-CM, which is not fully characterized yet. The main limitation that affects the Paci2018 model, as well the other single cell in silico models available in literature, is the fact that the simulated behavior belongs to a theoretical cell, built on averaged experimental data from many cells. In spite of our efforts to build the Paci2018 in a rigorous way, we do not have the presumption to consider it representative of all the hiPSC-CMs available nowadays, both from commercial lines and laboratory-specific lines. This is particularly true in case of an in vitro model like hiPSC-CMs, since the same experiments that we need to build an in silico model are extremely variable, as reported by multiple studies (Knollmann, 2013; Lu et al., 2015; Paci et al., 2017). As an additional example, in Lu et al. (2015) the authors succeeded in culturing iCell hiPSC-CMs that showed limited biomarker variability. Nevertheless, their cells showed way slower spontaneous beating rate than the cells we used to build our model (~15 bpm vs. ~38 bpm), despite both datasets were recorded at the same 37°C temperature. In order to have a better translation from in silico to in vitro models and finally to (pre)clinical applications, more refined modeling techniques exist, e.g., the population of in silico models (Britton et al., 2013; Passini et al., 2017). In addition to this, in terms of translation of drug test results from hiPSC-CMs to clinical applications, more challenges emerged recently. For example, in Abi-Gerges et al. (2017), testing a panel of 30 drugs on non-paced hiPSC-CMs provided on one hand good predictivity of pro-arrhythmic events, but on the other hand limited predictivity as an early QT screening. Therefore, in spite the high potential of hiPSC-CMs as in vitro models for diseases and drug tests, direct translation of results is still a work in progress, as indicated by the fact that advances in phenotype selection, cell maturation and combined recording platforms are currently being made. One inherent limitation of the Paci2018 model consists in the use of two different datasets of experimental data, one for the AP and ionic current data and the other for Ca²⁺ data. This is clear, e.g., by comparing the mean values of APD₉₀ and Ca²⁺ transient DURATION biomarkers: 414.7 and 804.5 ms, respectively. It is known that high affinity Ca²⁺ indicators, such as Fluo-4, can artificially prolong the Ca²⁺ transients and that Ca²⁺ transients last more than APs (Lee et al., 2012); however, the aforementioned Ca²⁺ transient DURATION value is substantially longer than APD₉₀. Therefore, we tried to find an acceptable tradeoff by means of the parameter optimization algorithm, in order to fulfill all the experimental ranges of both the AP and Ca²⁺ transient biomarkers. In particular, in the cost function we chose heavier weights (w_i = 2) only for two critical biomarkers which characterize hiPSC-CMs, i.e., MDP (depolarized compared to adult cardiomyocytes) and the rate of the spontaneous APs. For all the other biomarkers, the weights were set to one. The tradeoff obtained by the parameter optimization produced a model that simulates Ca²⁺ transients whose DURATION is 617.8 ms. Despite this value lies within the experimental range 804.5 ± 188.0 ms, it is very close to its lower bound. Conversely, the DT₉₀₁₀ biomarker properly lies within its experimental range, although slightly smaller than the experimental mean DT₉₀₁₀ (409.8 ± 100.1 vs. 367.6 ms). The inset included in Figure 2 shows only the decay part (from 90 to 10% of the transient amplitude) of the experimental and simulated transients reported in the main Figure 2: our simulated decay lies within the illustrative experimental traces, as well as DT₉₀₁₀ of the simulated transient is included in the experimental interval. Of note, the experimental traces reported in Figure 2 are purely illustrative of our experiments: the global quantitative comparison between experiments and simulations is done in Table 2 in terms of biomarkers. It is also known that temperature affects the spontaneous electrical activity of hiPSC-CMs, e.g., in Laurila et al. (2016) the authors showed how the temperature increments and reductions accelerate or slow down hiPSC-CM spontaneous beating, respectively. The fact that the AP and Ca²⁺ transient experimental datasets, chosen to develop the Paci2018 model, were recorded at the same temperature (35–37°C) further mitigates the differences between the two datasets. Another potential limitation of our model is related to its simple structure, consisting of two compartments only (cytosol and SR). Such compartmentalization does not allow to simulate a spatial inhomogeneous Ca²⁺ distribution, as conversely recently done by others (Koivumäki et al., 2018). We preferred to keep the model formulation simple, without entering in such detailed description, since it would have required a more complex mathematical formulation, a more detailed knowledge of the subcellular organization of hiPSC-CMs and, consequently, a more challenging parameter identification and higher simulation time. However, our formulation is successful in simulating the most important electrophysiological mechanisms at the whole cell level, including DADs elicited from different kind of stresses, without resorting to random RyR openings, as done in Koivumäki et al. (2018). A final limitation of the Paci2018 model is that it describes only the electrophysiology and the Ca²⁺ handling, but it does not take into consideration contractility. In literature, a few models of myofilament are available (Negroni and Lascano, 2008; Rice et al., 2008; Negroni et al., 2015). For instance, in Negroni et al. (2015) the authors presented an integrated rabbit ventricular cardiomyocyte model, which included also the mathematical description of force generation and the myofilament Ca²⁺ kinetics. However, the sarcomeric structure of hiPSC-CMs is immature and the myofibrils in hiPSC-CMs are oriented in multiple directions within the cell: in Bedada et al. (2016) a higher level of sarcomeric organization was reached only by repopulating hiPSC-CMs into a biological cardiac matrix. We acknowledge the potential of an integration of the myofilament description in the Paci2018, nevertheless the translation from a different species model of, e.g., rabbit cardiomyoycte to a hiPSC-CM model is not elementary and out of the scope of this paper. Future developments of this work will include using the Paci2018 model to ran in silico drug tests and to ran monodimensional and bidimensional simulations, as done, e.g., by Raphel et al. with the Paci2013 model (Raphel et al., 2017).

In conclusion, in this work we present an updated and more versatile version of our hiPSC-CM in silico model, based on a new dataset of electrophysiological data. Our model can represents the basis for new in silico studies on the effects of drugs or mutations (e.g., CPVT), which affect the Ca²⁺ handling in hiPSC-CMs. Due to its relatively light formulation (23 ordinary differential equations), our model is suitable also for very large studies on in silico populations, e.g., to support screening of different drug/compounds at various concentrations.

Author Contributions

All the authors conceived and designed the study. R-PP and KP performed the in vitro measurements and analyzed the in vitro data. MP and DC performed the in silico model development and validation. MP and SS analyzed the in silico data, prepared the figures and drafted the manuscript. All the authors interpreted the results and revised the manuscript.

Funding

MP was financially supported by the Academy of Finland (project CardSiPop, decision number 307967). R-PP was financially supported by Paavo Nurmi Foundation. The Heart Group has been supported by the Finnish Cardiovascular Foundation, the Academy of Finland, TEKES—the Finnish Funding Agency for Innovation and Technology, and Pirkanmaa Hospital District.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The reviewer JB and handling Editor declared their shared affiliation.

Acknowledgments

The authors acknowledge CSC—IT Center for Science, Finland, for generous computational resources, Tampere Facility of Electrophysiological Measurements for their service and Dr. Jussi Koivumäki for sharing his code.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphys.2018.00709/full#supplementary-material

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