Abstract
Existence of myocyte–fibroblast coupling in the human heart is still a controversial question. This study aims at investigating in a biophysical model how much coupling would be necessary to perturb significantly the electrical propagation of the cardiac impulse.
A one-dimensional model representing a strand of myocytes covered by a layer of fibroblasts was formulated by reinterpreting the coupled myocyte–fibroblast system as a single unit and connecting these units using a monodomain approach. The myocyte membrane kinetics was described by the Bondarenko mouse cell model and the fibroblast response was based on an experimentally measured current–voltage curve and took into account the delayed activation of that current. Conduction and maximal upstroke velocities were reported for different fibroblast densities and myocyte–fibroblast coupling strengths during paced rhythm. A reduction in conduction and maximum upstroke velocities was observed for increasing coupling and fibroblast density, in agreement with cell culture experiments. This effect was because of an increase in the myocyte resting potential and of the fibroblasts acting as a current sink. At least 10 fibroblasts with capacitance 4.5 pF had to be connected to each myocyte with capacitance 153.4 pF to slow down the conduction by >10%.
Coupling with fibroblasts affects the myocyte resting potential and the impulse propagation, but microstructural changes and myocyte decoupling are needed to explain slow conduction in fibrotic tissue.
Introduction
The increase of atrial fibrillation prevalence with age has prompted the investigation of the effects of structural and functional changes associated with aging, such as fibrosis.1–4 Cardiac fibrosis is marked by the formation of fibrous tissue in the lining and in the heart muscle.5 Although the fibroblasts composing this fibrous tissue are non-excitable, their membrane contains voltage-dependent ion channels.6,7 In addition, fibroblasts can be coupled through gap junctions to other fibroblasts as well as to myocytes.8 Because of this coupling, fibroblasts may act as current sinks or sources for the electrical activity of the myocytes and therefore disturb the propagation of the cardiac impulse. Gaudesius et al.9 showed in a cultured strand of cardiomyocytes that the cardiac impulse can propagate along an insert of fibroblasts over distances up to 300 μm. Using a similar experimental setup, Miragoli et al.10 observed a modulation in conduction velocity (CV) because of the coupling with myofibroblasts of cardiac origin. By combining immunohistochemical labelling and confocal microscopy, Camelliti et al.11 examined a rabbit sinoatrial node (a tissue rich in fibroblasts) and identified gap junctions linking fibroblasts and myocytes. However, it remains unclear whether electrical connections between myocytes and fibroblasts also exist in the human working atrial myocardium in vivo.
In the situations where experimental data are controversial, difficult to interpret, or when they are simply lacking, biophysical models may be used to evaluate what values of the parameter of interest would have a significant impact on the phenomenology. Several attempts have been made to apply this methodology to the myocyte–fibroblast interaction. Kohl et al.12 incorporated the effect of coupling with a fibroblast in a sinoatrial pacemaker cell model in order to estimate its impact on the depolarization rate. In another paper, Kohl et al.8 used a two-dimensional tissue model, including a current sink, to illustrate the occurrence of unidirectional block induced by fibrosis. In a one-dimensional fibre model, Jacquemet13 studied the possible spontaneous activations or pacemaker activity that may result from the interaction with fibroblasts. Recently, MacCannell et al.14 developed a detailed model of fibroblast electrophysiology to analyse how electrotonic coupling with fibroblast could alter the myocyte action potential morphology.
In this paper, a one-dimensional computer model was used to simulate a cardiomyocytes strand covered by a layer of fibroblasts. The model was specifically designed to study the impact of electrotonic coupling with fibroblasts on the propagation of the cardiac impulse in a way similar to the Miragoli et al.10 The Bondarenko et al.'s15 cardiac cell model was coupled through gap junctions to a simple fibroblast model including a delayed activation of the membrane current. The mathematical formulation used here made it possible to reinterpret the coupled system of myocyte–fibroblasts as a new single-cell model in order to facilitate its integration in a tissue model. A myocyte–fibroblast pair was first simulated in order to determine the parameters of the model by comparing the output signals with the available experimental recordings. Then, in a cable model representing a strand of cells covered by a layer of fibroblasts, electrophysiological parameters such as the CV and the maximum upstroke velocity (dV/dt)max were computed as a function of the fibroblast density and the myocyte–fibroblast coupling conductance and compared with the rare available experimental data.
Methods
Modelling a myocyte surrounded by fibroblasts
(A) Schematic representation of the model of myocyte surrounded by several fibroblasts. (B) Current–voltage relationships for fibroblast membrane. The third-order polynomial model is displayed as a thick solid line. The experimental data points were extracted with permission from Figure 1 in Chilton et al.7 (Am J Physiol Heart Circ Physiol 2005;288:H2931–H2939) and Figure 1B in Shibukawa et al.6 (Biophys J 2005;88:3924–35).
The formulation of the fibroblast membrane current will be discussed in the next subsection. The myocyte membrane kinetics is described by the recently developed Bondarenko et al.15 model of mouse cardiac cell. Its membrane capacitance is computed from its membrane area: Cm = 1 μF/cm2 × 1.534 × 10−4 cm2 = 153.4 pF. Because this model reproduces rodent cell electrophysiological properties, it is expected to be well suited to help understand the findings in cell culture experiments. In addition, it includes a detailed formulation of the fast sodium current based on a Markov model fitted to recent patch clamp data.15 Simulating pathological conduction in a fibrotic tissue may require the incorporation of this sophisticated description of the upstroke phase of the action potential.
Membrane currents in fibroblasts
Only relatively sparse and sometimes inconsistent data are currently available about the electrophysiological properties of fibroblasts, possibly because of technical experimental difficulties or simply to a natural variability. The total membrane capacitance of these cells was found to be 4.5 ± 0.4 pF in Shibukawa et al.6 and 6.3 ± 1.7 pF in Chilton et al.7 The large variability in capacitance was assumed to be related to variations in fibroblast size and shape. Their membrane input resistance consistently lies in the gigaOhms range.8 Capacitance values of 0.51–3.817 5.5 ± 0.6,6 and 10.7 ± 2.3 GΩ7 have already been reported. Resting potentials ranging from −70 to 0 mV have been measured (−15 ± 19 mV in Kohl et al.,8 −22 ± 1.9 mV in Kiseleva et al.,17 and −58 ± 3.9 mV in Shibukawa et al.6). However, the existence of a stable resting potential in fibroblasts might be questionable.
If the steady-state membrane current is reached instantaneously, i.e. Iion,f = If,ss, the fibroblast membrane current becomes an increasing function of the membrane potential Vf. This would mean that the fibroblast is a passive cell. In contrast, Kohl et al. measured the membrane potentials in a myocyte–fibroblast pair.5,12 In their recording, the fibroblast membrane potential increased as a result of the activation of the myocyte (elicited by injecting current in the myocyte). However, this increase continued after the fibroblast membrane potential curve crossed that of the myocyte membrane potential, a response that is not possible in a passive cell. If Vm and Vf are equal and larger than the fibroblast resting potential, the coupling current Icoupl vanishes and the membrane current If,ss is positive. Therefore, by Eq. (4), dVf/dt < 0, so that the fibroblast membrane potential cannot continue increasing. Assuming that their records were reliable and that mechanosensitivity was not a key factor in this phenomena, this would mean that fibroblasts are active units. In order to attempt to reproduce this electrophysiological behaviour, a delayed activation of the fibroblast membrane current was introduced.
Inactivation kinetics of fibroblast membrane currents has been investigated by Shibukawa et al.6 The steady-state inactivation curve was shown to follow approximately the function.
The associated time constant was found to be in the range of 1500–3000 ms.6 A gating variable16 could be included in the model to account for this inactivation kinetics. However, its impact on the time course of the membrane potential would be negligible during normal activation because the time the membrane potential spends above −24.3 mV is very small in a mouse cell (<5 ms) compared with the time constant.15
A cardiac cell model incorporating coupling with fibroblasts
Two new state variables are included, namely Vf and Iion,f. The corresponding differential equations (4) and (8) are added to the system.
A cardiac tissue model incorporating coupling with fibroblasts
In a three-dimensional tissue model, the value of γ is not independent of κ, the parameter expressing the quantity of fibroblasts present in the tissue, because the fibroblasts occupy some space. However, in this paper, the parameter γ was kept constant when κ was increased. This situation corresponds to a monolayer of cultured cells coated with fibroblasts.10
Simulation protocols
A myocyte–fibroblast pair (Nf = 1) was first simulated for different values of the parameters. The capacitance of the myocyte was set to 153.4 pF and that of the fibroblast to 4.5 pF (the value reported by Shibukawa et al.).6 The coupling conductance was varied between 0 and 0.5 nS/pF, and the time constant was varied between 1 μs and 100 ms. The system was allowed to evolve without any stimulation for at least 400 ms in order to ensure that the myocyte–fibroblast pair has reached steady state. Then, a 400 μA/cm2 current was passed in the myocyte for 0.1 ms. The membrane potential of both the myocyte and the fibroblast were simultaneously monitored and presented in a display similar to the experimental recordings shown in Fig. 4 in Kohl et al.12
The propagation of the cardiac impulse in the presence of fibroblasts was studied using a cable comprising 50 myocytes, each of 100 μm length and covered by a layer of fibroblasts, simulated using the approach presented above. The parameter κ representing the relative quantity of fibroblasts in terms of capacitance was varied between 0 and 1.5, and the coupling conductance gc was varied between 0 and 0.9 nS/pF. From the definition of κ, the myocyte–fibroblast pair actually corresponds to the case κ = 0.03 (Nf = 1). Equation (12) was solved using a forward Euler approach with a time step of 5 μs using the Cardiowave software platform.19
Similar to the myocyte–fibroblast pair, a 200 μA/cm2 current was passed in the first myocyte of the cable for 1 ms to elicit an activation wave, after the system has reach steady state. The membrane potential was recorded along the cable during the propagation of the cardiac impulse. The maximum upstroke velocity, (dVm/dt)max, was documented for the cell located in the middle of the fibre. The myocyte action potential morphology was characterized by its resting potential (membrane potential at steady-state just before the application of the stimulus), amplitude, and duration (measured with a threshold at −60 mV, a value approximately corresponding to the reversal potential of the fibroblast ionic current). The average CV was computed based on the activation times extracted from the action potentials measured at seven equidistant cells along the cable. The simulation was repeated for each set of parameters considered.
Results
Myocyte–fibroblast pair
Figure 2 shows the membrane potentials in a myocyte–fibroblast pair for different values of the parameters gc (the coupling between the two cells) and τf (the time constant controlling the delayed activation of the fibroblast membrane current). The time course of the coupling current and the fibroblast ionic current (magnified by a factor 10) is displayed below each action potential. The values chosen for gc, namely 0, 0.02, 0.05, 0.1, 0.2, and 0.5 nS/pF, correspond to, respectively, 0, 3, 7.5, 15, 30, and 75 gap junctions linking the two cells, assuming that a gap junction has a conductance of 30 pS.12
Simulated membrane potentials in a pair myocyte–fibroblast. A 90-ms segment of each signal is represented. The thick lines represent fibroblast membrane potentials. The coupling current per unit fibroblast capacitance Icoupl/Cf and the fibroblast ionic current Iion,f (magnified by a factor of 10) are displayed below each action potential. The dotted lines represent the x-axis (zero current).
In the absence of coupling (first column of Figure 2), the fibroblast remains at its resting potential −58 mV and the myocyte is unaffected by the presence of the fibroblast. As the coupling is increased, the steady-state potential of the fibroblast becomes closer to that of the myocyte and the fibroblast tends to mimic the electrical activity of the myocyte: the amplitude of its activity increases, as well as its maximum slope. This is because of an increase in the coupling current and in the fibroblast ionic current whose amplitude is ∼10 times smaller than the coupling current (Figure 2). For the myocyte, the coupling current is an inward current at steady state (in this case Icoupl/Cf = Iion,f), an outward current during the upstroke, and again an inward current during repolarization. The global shape of the myocyte action potential remains almost identical when this myocyte is coupled to a fibroblast. The action potential amplitude (APA) is reduced by <1 mV and its action potential duration (APD) is prolonged by <1 ms when a coupling conductance up to gc = 0.5 nS/pF is introduced. In addition, the peak sodium current of the myocyte is not significantly affected. Coupling with a larger number or higher capacitance, fibroblasts would be needed to alter significantly the myocyte action potential and its propagation, as it will be discussed in the next subsection.
The influence of the time constant τf on the fibroblast membrane potential is less marked, revealing that the electrical activity of the fibroblast is dominated by the coupling current. The parameter τf determines the amplitude of the fibroblast ionic current (see Figure 2). As a result, it has an impact on the rise time of the electrical activity of the fibroblast (i.e. the time interval between the stimulus and position of the peak fibroblast potential) (Table 1). A larger τf delays the maximum fibroblast activity. It also enables the fibroblast potential to increase a little further when the myocyte membrane potential becomes more negative than that of the fibroblast, as observed by Kohl et al.12 The set of parameter (gc = 0.2 nS/pF, τf = 20 ms) was found to compare well in terms of rise time and amplitude with the experimental data shown in Kohl et al.'s study.12 The value τf = 20 ms, which lies within the experimental range of 20–60 ms reported for positive membrane potentials,6 was the one selected for further investigations in a cable.
Rise time of the electrical activity of the fibroblast as a function of the parameters gc and τf corresponds to that of Figure 2
| τf | gc (nS/pF) | ||||
|---|---|---|---|---|---|
| 0.02 (ms) | 0.05 (ms) | 0.1 (ms) | 0.2 (ms) | 0.5 (ms) | |
| 1 μs | 12.5 | 10.3 | 7.7 | 5.0 | 3.3 |
| 1 ms | 12.3 | 10.1 | 7.5 | 4.9 | 3.3 |
| 20 ms | 15.0 | 11.9 | 8.9 | 5.8 | 3.6 |
| 100 ms | 17.5 | 13.1 | 9.6 | 6.0 | 3.6 |
| τf | gc (nS/pF) | ||||
|---|---|---|---|---|---|
| 0.02 (ms) | 0.05 (ms) | 0.1 (ms) | 0.2 (ms) | 0.5 (ms) | |
| 1 μs | 12.5 | 10.3 | 7.7 | 5.0 | 3.3 |
| 1 ms | 12.3 | 10.1 | 7.5 | 4.9 | 3.3 |
| 20 ms | 15.0 | 11.9 | 8.9 | 5.8 | 3.6 |
| 100 ms | 17.5 | 13.1 | 9.6 | 6.0 | 3.6 |
Rise time of the electrical activity of the fibroblast as a function of the parameters gc and τf corresponds to that of Figure 2
| τf | gc (nS/pF) | ||||
|---|---|---|---|---|---|
| 0.02 (ms) | 0.05 (ms) | 0.1 (ms) | 0.2 (ms) | 0.5 (ms) | |
| 1 μs | 12.5 | 10.3 | 7.7 | 5.0 | 3.3 |
| 1 ms | 12.3 | 10.1 | 7.5 | 4.9 | 3.3 |
| 20 ms | 15.0 | 11.9 | 8.9 | 5.8 | 3.6 |
| 100 ms | 17.5 | 13.1 | 9.6 | 6.0 | 3.6 |
| τf | gc (nS/pF) | ||||
|---|---|---|---|---|---|
| 0.02 (ms) | 0.05 (ms) | 0.1 (ms) | 0.2 (ms) | 0.5 (ms) | |
| 1 μs | 12.5 | 10.3 | 7.7 | 5.0 | 3.3 |
| 1 ms | 12.3 | 10.1 | 7.5 | 4.9 | 3.3 |
| 20 ms | 15.0 | 11.9 | 8.9 | 5.8 | 3.6 |
| 100 ms | 17.5 | 13.1 | 9.6 | 6.0 | 3.6 |
Strand of myocytes coated with fibroblasts
Figure 3 displays the CV in a cable as a function of the fibroblast density κ for different values of the coupling conductance gc, namely 0, 0.02, 0.1, 0.2, 0.4, and 0.9 nS/pF. When the fibroblast density tends to 0, the CV tends to the baseline value 39.6 cm/s. When the fibroblast density increases, CV reduces, in agreement with Miragoli et al.10 This effect is more pronounced in the case of a larger coupling conductance gc. In the absence of fibroblast–myocyte coupling, even a large number of fibroblasts have no effect on the propagation of the cardiac impulse. Note that this is only true if the myocyte membrane surface-to-volume ratio as well as the intra- and extracellular tissue conductance are assumed to be unaffected by those fibroblasts (as it might be the case when the fibroblasts create a coat over a monolayer of myocytes), and if there is no fibroblast–fibroblast coupling.
Conduction velocity in a cable as a function of the fibroblast density κ. If the fibroblast capacitance is assumed to be 4.5 pF, a fibroblast density of κ = 0.5, 1, and 1.5 corresponds to Nf = 17, 34, and 51 fibroblasts per myocyte. Solid lines: gc = 0, 0.02, 0.1, 0.2, 0.4, and 0.9 nS/pF (top to bottom); dashed line: cable without a coat of fibroblast (gc = 0 nS/pF), but with a capacitance Cm multiplied by 1 + κ.
The effect of adding fibroblasts was compared with the effect of increasing the myocyte capacitance. As far as capacity is concerned, incorporating fibroblasts with density κ is equivalent to multiplying the myocyte capacitance by (1 + κ). Note that the myocyte membrane currents will be assumed to be increased by the same factor since the Bondarenko model specifies them per unit membrane capacitance. The dashed line of Figure 3 represents the CV in a tissue without fibroblasts, but with increased capacitance. This curve is approximately of the form CV ∞ (1+κ)−1/2. The cross indicates that the stimulus (strength 200 μA/cm2, duration 1 ms) did not initiate a propagation for κ > 1, i.e. if Cm > 306.8 pF. This occurred only for very large values of fibroblast density (κ > 2) in a tissue with normal myocyte capacitance.
Figure 4 shows the maximum upstroke velocity (dV/dt)max measured in the middle of the cable, similar to Figure 3. When the coupling conductance gc of the fibroblast density is higher, the upstroke is slower, again in agreement with Miragoli et al.10 In contrast, when the myocyte capacitance was increased without inclusion of fibroblast, (dV/dt)max increased slightly, as described by Spach et al.20 If the cumulative capacitance of all fibroblasts coupled to the myocyte become of the order of the myocyte capacitance, the myocyte action potential is affected significantly. Along with a decrease in (dV/dt)max, a reduction in the peak sodium current INa,peak was consistently observed. The correlation coefficient between the two variables (dV/dt)max and INa,peak was found to be 0.94. The decrease in CV was also associated with an increase in the myocyte resting potential (Figure 5). Although the correlation coefficient was high (−0.89), the variation in myocyte resting potential explains only part of the reduction in CV. The source-sink role of the fibroblast in the upstroke phase is also a factor that modulates CV. This effect clearly depends on the coupling conductance.
Maximal upstroke velocity (dV/dt)max in a cable as a function of the fibroblast density κ. If the fibroblast capacitance is assumed to be 4.5 pF, a fibroblast density of κ = 0.5, 1, and 1.5 corresponds to Nf = 17, 34, and 51 fibroblasts per myocyte. Solid lines: gc = 0, 0.02, 0.1, 0.2, 0.4, and 0.9 nS/pF (top to bottom); Dashed line: cable without a coat of fibroblast (gc = 0 nS/pF), but with a capacitance Cm multiplied by 1 + κ.
Conduction velocity as a function of the myocyte resting potential. The dotted lines show data points associated with the same coupling conductance: gc = 0.02, 0.1, 0.2, 0.4, and 0.9 nS/pF from top to bottom. When gc = 0 nS/pF, the myocyte resting potential is −82.3 mV.
Figure 6 illustrates the effect of a strong coupling with fibroblasts on the action potential morphology. In addition to a decrease in CV, maximum upstroke velocity, and peak sodium current, a stronger coupling or a coupling with more or higher capacitance fibroblasts resulted in a decrease in APA and in a prolongation of the APD. The reduction in peak sodium current was associated with an increase in the myocyte resting potential (resulting from the coupling with fibroblasts). For the sake of comparison, the resting potential of the Bondarenko's model is −82.3 mV and in a myocyte–fibroblast pair the myocyte resting potential remains below −82 mV even for strong coupling (gc = 0.9 nS/pF). The corresponding APA is 115.1 mV in the absence of coupling and 114.6 mV for strong coupling. Its APD is 14.4 ms without coupling and 14.8 ms for strong coupling.
Simulated membrane potentials in a strand of myocytes coated by fibroblasts for different values of the coupling conductance gc and the fibroblast density κ. A 100-ms segment of each signal is represented. If the fibroblast capacitance is assumed to be 4.5 pF, a fibroblast density of κ = 0.3, 0.6, and 0.9 corresponds to Nf = 10, 20, and 31 fibroblasts per myocyte. Plain line represents myocyte membrane potentials and thick lines fibroblast membrane potentials. The myocyte resting potential (Vrest), the action potential amplitude (APA) and the action potential duration (APD) are documented in each case. The corresponding coupling current per unit myocyte capacitance κIcoupl/Cf is shown below each action potential.
Table 2 reports the excitability threshold computed as the minimum current (applied to the first cell of the cable) that initiates propagation along the cable. When coupling is weak (gc < 0.3 nS/pF), the excitability threshold is decreased by the fibroblasts acting as a current source (their membrane potential is higher than that of the myocyte). In contrast, at high coupling (gc > 0.3 nS/pF), the myocyte resting potential is significantly affected by the fibroblasts. The sodium channels at this higher resting potential (see Figure 6) are in a less excitable state, resulting in an increase in the excitation threshold.
Excitation threshold (in μA/cm2) when the first cell of the cable is excited by a 1-ms square pulse. A myocyte membrane area of 1.534 × 10−4 cm2 is assumed15
| κ-value | gc (nS/pF) | |||||
|---|---|---|---|---|---|---|
| 0 | 0.01 | 0.1 | 0.2 | 0.4 | 0.9 | |
| 0 | 137 | 137 | 137 | 137 | 137 | 137 |
| 0.3 | 137 | 135 | 133 | 135 | 138 | 143 |
| 0.6 | 137 | 134 | 130 | 132 | 138 | 149 |
| 0.9 | 137 | 132 | 126 | 130 | 139 | 154 |
| 1.2 | 137 | 130 | 122 | 127 | 139 | 158 |
| 1.5 | 137 | 129 | 118 | 126 | 141 | 164 |
| κ-value | gc (nS/pF) | |||||
|---|---|---|---|---|---|---|
| 0 | 0.01 | 0.1 | 0.2 | 0.4 | 0.9 | |
| 0 | 137 | 137 | 137 | 137 | 137 | 137 |
| 0.3 | 137 | 135 | 133 | 135 | 138 | 143 |
| 0.6 | 137 | 134 | 130 | 132 | 138 | 149 |
| 0.9 | 137 | 132 | 126 | 130 | 139 | 154 |
| 1.2 | 137 | 130 | 122 | 127 | 139 | 158 |
| 1.5 | 137 | 129 | 118 | 126 | 141 | 164 |
Excitation threshold (in μA/cm2) when the first cell of the cable is excited by a 1-ms square pulse. A myocyte membrane area of 1.534 × 10−4 cm2 is assumed15
| κ-value | gc (nS/pF) | |||||
|---|---|---|---|---|---|---|
| 0 | 0.01 | 0.1 | 0.2 | 0.4 | 0.9 | |
| 0 | 137 | 137 | 137 | 137 | 137 | 137 |
| 0.3 | 137 | 135 | 133 | 135 | 138 | 143 |
| 0.6 | 137 | 134 | 130 | 132 | 138 | 149 |
| 0.9 | 137 | 132 | 126 | 130 | 139 | 154 |
| 1.2 | 137 | 130 | 122 | 127 | 139 | 158 |
| 1.5 | 137 | 129 | 118 | 126 | 141 | 164 |
| κ-value | gc (nS/pF) | |||||
|---|---|---|---|---|---|---|
| 0 | 0.01 | 0.1 | 0.2 | 0.4 | 0.9 | |
| 0 | 137 | 137 | 137 | 137 | 137 | 137 |
| 0.3 | 137 | 135 | 133 | 135 | 138 | 143 |
| 0.6 | 137 | 134 | 130 | 132 | 138 | 149 |
| 0.9 | 137 | 132 | 126 | 130 | 139 | 154 |
| 1.2 | 137 | 130 | 122 | 127 | 139 | 158 |
| 1.5 | 137 | 129 | 118 | 126 | 141 | 164 |
Discussion
A one-dimensional model was formulated by reinterpreting the myocyte–fibroblast coupled system as a single unit and connecting these units using a standard monodomain formulation. Because the presence of this myocyte–fibroblast coupling in the human heart is still controversial, our model was aimed at determining how much coupling would be necessary to significantly perturb the electrical propagation of the cardiac impulse. Although the approach developed did not enable us to study fibroblast–fibroblast interactions, it provides a simple tool to investigate relevant questions about the propagation of the cardiac impulse in the presence of fibroblasts. Moreover, its extension to a three-dimensional heart model would be straightforward. In contrast, integrating in a tissue model, two different types of cells (with significantly different cell size) that are mixed at the micro-scale level may require subcellular discretization and therefore the development of much more complex models.
Effect of fibroblasts on conduction velocity
This simulation study demonstrates in a simple model the potential impact of coupling with fibroblast on the propagation of the cardiac impulse as well as the effects of changing the parameters related to this coupling. The parameters of interest controlling this electrical interaction were the coupling conductance and the fibroblast density κ, i.e. the ratio of the total capacitance of the fibroblasts connected to a myocyte to the capacitance of that myocyte. A reduction in CV was consistently found for increasing coupling and fibroblast density κ. In the Miragoli et al.'s experiment,10 the CV also slowed down from a baseline value of 40 to 15 cm/s when the fibroblast density was increased. In order to obtain a CV inferior to 20 cm/s in the model, it was necessary to increase κ up to 1.5 when gc = 0.9 nS/pF (i.e. 135 gap junctions of 30 pS each). This parameter set corresponds to a situation in which each myocyte is electrically coupled to a number of fibroblasts whose cumulative capacitance is 1.5 × Cm (230 pF for the Bondarenko's model). If each fibroblast has a capacitance of 4.5 pF (resp. 6.3 pF to compare with MacCannell et al.14), Nf = κCm/Cf = 51 fibroblasts (resp. 36) per myocyte would be needed to induce the same reduction in CV as the extreme case of the experiment. The use of the Pandit's model of rat ventricular cell21 which has a myocyte capacitance Cm of 100 pF would further reduce the required number of fibroblasts per myocyte to 24. In any case, the cumulative capacitance of the fibroblasts needs to be non-negligible compared with that of the myocytes in order to significantly alter the CV.
The simulated data suggest that two mechanisms are needed to explain the reduction in CV as a result of the coupling with fibroblasts. First, because the myocyte resting potential (when coupled to fibroblasts) is higher than that of an isolated myocyte (Figure 6), the sodium channels are in a less excitable state, resulting in a decrease in the peak sodium current and a slower CV. Secondly, in the upstroke phase, as soon as the myocyte membrane potential is higher than the fibroblast membrane potential, the fibroblast acts as a current sink to slow down the activation. It is worth noting that it was found crucial to wait until the steady state has been reached before applying a stimulus and measuring CV. When the stimulus was applied while the state of the myocyte was the resting state for an isolated myocyte although coupling with a fibroblast was effective (a condition impossible to reproduce in an experiment), the CV increased slightly as a function of the fibroblast density, because the fibroblasts acted as a pure current source without altering the initial state of the myocyte.
The same reduction in CV could be obtained without fibroblast by increasing the membrane capacitance. However, the conduction was less safe in the sense that a conduction block already occurred at a faster CV. Another difference was that the (dV/dt)max increased slightly instead of significantly decreasing. The peak sodium current remained essentially unchanged after modifying the myocyte membrane capacitance. Therefore, by inducing a decrease in the peak Na+ current, coupling with fibroblasts enables very slow but safe conduction, leading to significant delays in activation that can be an arrhythmogenic factor.
Effect of fibroblasts on the myocyte action potential
Coupling with fibroblasts resulted in a prolongation of the myocyte APD, a decrease of its amplitude, and its upstroke velocity. The variations were significant only for strong coupling with a sufficiently high number of fibroblasts (∼10, κ = 0.3 as shown in Figure 6). In contrast, in another recent simulation study, MacCannell et al.14 observed a dramatic shortening of the APD (from 259 ms down to 155 ms) when a myocyte was connected to only four active fibroblasts through a coupling conductance similar to those used in this paper. They used a human ventricular cell model that spends up to 250 ms above the fibroblast resting potential during the plateau phase. The activation time constant was voltage dependent and ranged from ∼20 ms (near −80 to −70 mV) up to 160 ms near Vf = 20 mV. Our mouse cell action potential had a triangular shape and a very short duration. The activation time constant of the fibroblast ionic current (τf = 20 ms) was in the same order of magnitude as the myocyte APD. This brief comparison suggests that the effect of fibroblast may be action potential dependent, at least through the ratio of activation time constant to APD. This may complicate the clinical significance of cell cultured models.
Possible effects of the fibroblast resting potential
The fibroblast resting potential is an important parameter influencing the effect of fibroblasts on the electrophysiological properties of the tissue. It determines the myocyte resting potential at steady state, which, in turn, has an impact on the peak sodium current and therefore on the CV. When the myocyte membrane potential is more negative than that of the fibroblast, the fibroblast acts as a current source. When it is less negative, the fibroblast acts as a current sink. This parameter (along with the coupling strength) was also shown to control the occurrence of pacemaker activity in the myocyte through the same mechanism.13 Unfortunately, a wide range of values is reported in the literature.14 We chose the value −58 mV from6 because this paper provided the most comprehensive set of consistent data available at that time. A fibroblast with a higher resting potential would have a larger impact as a current source and a smaller impact as a current sink. However, as suggested by Table 2, if the myocyte resting potential is significantly different from that of an isolated myocyte, the ionic channels at resting state may be sufficiently altered to affect the impulse propagation. This again is modulated by the coupling conductance.
Limitations
The simulations carried out in this paper assume that the introduction of a layer of fibroblast on top of the cardiac tissue does not alter the myocyte-to-myocyte coupling. This is difficult to ensure in a cell culture experiment. The presence of a fibroblast intercalated between two myocytes may reduce the coupling or even suppress it. In addition, fibroblasts (so with stronger reason a layer of fibroblasts) may have an impact on the extracellular medium, and in particular on the extracellular conductivity. Decreasing the extracellular conductivity would further reduce the CV. Another confusing element is the fact that, in the framework of the bidomain theory, a reduction of the myocyte density or membrane surface-to-volume ratio leads to a faster CV.16 The reason is that this theory assumes that the myocytes occupy all the space within the myocardium, which is not anymore the case when non-myocytes are introduced.
Our fibroblast model is merely a non-linear extension of a linear I–V curve model. The cell is electrophysiologically active only through the introduction of a delayed activation of the membrane current. As a more comprehensive set of data will be made available for fibroblasts, new mathematical models will be developed to better represent their electrical activity.14 However, basic electrical circuit principles such as the effect of the ratio of fibroblasts to myocyte capacitance will still apply.
Thus, in a real heart, a number of factors not included in this study play a role in determining the CV in a cardiac tissue with fibroblasts. However, despite its inherent limitations, computer modelling provides a way to analyse these factors independently in order to help make the link between cell culture experiments and in vivo human heart electrophysiology.
Conflict of interest: none declared.
Funding
This work supported by the Swiss National Science Foundation fellowship PA002-113171 (V.J.) and NIH grant R01HL76 767 (C.S.H.).
