Modeling Cardiac Fibroblasts, their Interactions with Myocytes and their Impact on Impulse Propagation

Abstract

Aims

The 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 significantly perturb the electrical propagation of the cardiac impulse.

Methods

A one-dimensional model representing a strand of myocytes covered by a layer of fibroblasts was formulated by reinterpreting the coupled system myocyte-fibroblast 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 that of the fibroblast was based on an experimentally-measured current-voltage curve and took into account the delayed activation of that current. Conduction velocity and maximal upstroke velocity were reported for different fibroblast densities and myocyte-fibroblast coupling strengths during paced rhythm.

Results

A reduction in conduction velocity and maximum upstroke velocity was observed for increasing coupling and fibroblast density, in agreement with cell culture experiments. This effect was due to an increase of the myocyte resting potential and to 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 conduction by more than 10%.

Conclusion

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 the muscle of the heart [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 a current sink or source for the electrical activity of the myocytes and therefore disturb the propagation of the cardiac impulse. Gaudesius et al. showed in a cultured strand of cardiomyocytes that the cardiac impulse can propagate along an insert of fibroblasts over distances up to 300 μm [9]. Using a similar experimental setup, Miragoli et al. observed a modulation in conduction velocity due to the coupling with myofibroblasts of cardiac origin [10]. By combining immunohistochemical labeling and confocal microscopy, Camelliti et al. examined a rabbit sinoatrial node (a tissue rich in fibroblasts) and identified gap junctions linking fibroblasts and myocytes [11]. 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 value 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. incorporated the effect of coupling with a fibroblast in a sinoatrial pacemaker cell model in order to estimate its impact on the depolarization rate [12]. In another paper, Kohl et al. used a two-dimensional tissue model including a current sink to illustrate the occurrence of unidirectional block induced by fibrosis [8]. In a one-dimensional fiber model, Jacquemet studied the possible spontaneous activations or pacemaker activity that may result from the interaction with fibroblasts [13]. Recently, MacCannell et al. developed a detailed model of fibroblast electrophysiology to analyze how electrotonic coupling with fibroblast could alter the myocyte action potential morphology [14].

In this paper, a one-dimensional computer model was used to simulated a strand of cardiomyocyte 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. experiment [10]. The Bondarenko et al. cardiac cell model [15] 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 myocyte-fibroblasts as a new single cell model in order to facilitate its integration in a tissue model. A pair myocyte-fibroblast 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 conduction velocity and the maximal upstroke velocity (dV/dt)_max were computed as a function of the fibroblast density and the myocyte-fibroblast coupling conductance and compared to the rare available experimental data.

Methods

Modeling a myocyte surrounded by fibroblasts

In order to evaluate the effect of electrotonic coupling with fibroblasts on electrical propagation in a cardiac tissue, a biophysical model is first considered, which consists of a myocyte coupled to N_f identical fibroblasts (see Fig. 1A). The parameter N_f actually represents the ratio of the number of fibroblasts to the number of myocytes, and may, therefore, not necessarily be an integer. The electric potential inside the myocyte is denoted by ϕ_i, and the potential inside the fibroblasts by ϕ_f. The extracellular potential, ϕ_o, is assumed to be spatially uniform in the vicinity of the myocyte. The membrane potentials are defined by:

$$V_m={\varphi }_i-{\varphi }_o$$ (1)

$$V_f={\varphi }_f-{\varphi }_o.$$ (2)

Figure 1.

(A) Schematic representation of the model of myocyte surrounded by several fibroblasts. (B) Current-voltage relationships for fibroblast membrane. The experimental data points were extracted from Fig. 1 in [7] and Fig.1B in [6]. The 3rd order polynomial model is displayed as a thick solid line.

For the sake of simplicity, all the fibroblasts coupled to the myocyte will be assumed to be in the same state, that is, the time course of the potential ϕ_f is supposed to be the same in all these fibroblasts. The membrane capacitance (in pF) of the myocyte is denoted by C_m and that of each fibroblast by C_f. Whenever useful, the fibroblast size will be considered to be in direct relation to its capacitance. The evolution of the membrane potentials V_m and V_f is governed by the equations [16]:

$$C_m\frac{\mathrm{d}{V}_m}{\mathrm{d}t}=-C_m{I}_{\mathrm{ion},m}-N_f{I}_{\mathrm{coupl}}$$ (3)

$$C_f\frac{\mathrm{d}{V}_f}{\mathrm{d}t}=-C_f{I}_{\mathrm{ion},f}+I_{\mathrm{coupl}},$$ (4)

where I_ion,m and I_ion,f are the (outward) ionic current per unit membrane capacitance of the myocyte and the fibroblast respectively. The coupling current I_coupl (flowing from the myocyte toward the fibroblasts) resulting from the presence of gap junctions connecting the fibroblasts to the myocyte is supposed to follow Ohm’s law:

$$I_{\mathrm{coupl}}=G_c({\varphi }_i-{\varphi }_f)=G_c(V_m-V_f)$$ (5)

where G_c is the coupling conductance in nS.

The formulation of the fibroblast membrane current will be discussed in the next subsection. The myocyte membrane kinetics will be described by the recently developed Bondarenko et al. model of mouse cardiac cell [15]. Its membrane capacitance is computed from its membrane area [15]: C_m = 1 μF/cm² × 1.534 · 10⁻⁴ cm² = 153.4 pF. Since 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 due to 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 GΩ range [8]. Values of 0.51–3.8 GΩ [17], 5.5±0.6 GΩ [6] and 10.7±2.3 GΩ [7] have 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 is questionable.

Despite these limitations, a linear model of the electrical response of a fibroblast can be formulated based on these data. Kohl et al. [12] showed using such a model that the firing rate of a pacemaker cell is altered when it is coupled to a fibroblast. When the fibroblast membrane potential varies significantly, a non-linear current response is expected. A few studies document the membrane currents in fibroblasts and myofibroblasts [6, 7, 18]. Shibukawa et al. [6] and Chilton et al. [7] both observed a non-linear, rectifying relationship between the steady-state fibroblast membrane current per unit fibroblast capacitance and the fibroblast membrane potential. Chilton et al. [7] also considered myofibroblasts. All these experimental steady-state current-voltage (I-V) relationship are displayed in Fig. 1B. A 3rd degree polynomial model fitted to these data points was used to incorporate a non-linear I-V relationship in the biophysical model. The steady-state membrane current (in pA/pF) is formulated as

$$I_{f,\mathrm{ss}}\left(V_f\right)=c_3{V}_f^3+c_2{V}_f^2+c_1{V}_f+c_0,$$ (6)

where the coefficients are given by

$$c_3=1.182\cdot {10}^{-5},c_2=2.87\cdot {10}^{-3},c_1=0.259,c_0=7.675$$ (7)

and V_f is in mV. This curve is plotted in Fig. 1B as a continuous line. The coefficients of the polynomial were selected in such a way that they enables to reproduce the resting state and nearresting state electrophysiological properties reported in Shibukawa et al. [6]. The current I_f,ss is zero at V_f = −58 mV, which corresponds to the value −58.0 ± 3.9 mV for the resting potential [6]. In addition, the membrane conductance (g_m = dI_f,ss/dV_f) at V_f = −58 mV is g_m = 0.045 nS/pF. The time constant is therefore τ_m = g_m⁻¹ = 22 ms. This value is in agreement with Shibukawa et al. [6], in which the membrane capacitance was C_f = 4.5 ± 0.4 pF and the membrane input resistance was R_m = 5.5 ± 0.6 GΩ, leading to a time constant of τ_m = R_mC_f = 24.8 ± 4.9 ms.

If the steady-state membrane current is reached instantaneously, i.e. I_ion,f = I_f,ss. The fibroblast membrane current becomes an increasing function of the membrane potential V_f. This would mean that the fibroblast is a passive cell. In contrast, Kohl et al.measured the membrane potentials in a pair myocyte-fibroblast [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). But this increase continued after the fibroblast membrane potential curve crossed that of the myocyte membrane potential. This is not possible in a passive cell. If V_m and V_f are equal and larger than the fibroblast resting potential, the coupling current I_coupl vanishes and the membrane current I_f,ss is positive. Therefore, by Eq. (4), dV_f/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 behavior, a delayed activation of the fibroblast membrane current was introduced.

Little is known about the activation kinetics of fibroblast membrane currents. Voltage-dependent time constants in the range 20–60 ms have been reported [6], but only the situation V_f > 0 mV was investigated. In order to take into account the slow activation process, the current I_ion,f was computed by solving the equation:

$$\frac{\mathrm{d}{I}_{\mathrm{ion},f}}{\mathrm{d}t}=\frac{I_{f,ss}\left(V_f\right)-I_{\mathrm{ion},f}}{{\tau }_f},$$ (8)

involving only one parameter, namely the time constant τ_f.

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

$$a_{\infty }\left(V_f\right)=\frac{1}{1+\exp \left(\right(V_f+24.3)∕7.0)}$$ (9)

The associated time constant was found to be in the range 1500–3000 ms [6]. A gating variable [16] 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) as compared to the time constant [15].

A cardiac cell model incorporating coupling with fibroblasts

The Bondarenko mouse cell model was modified in order to naturally incorporate the effect of electrotonic coupling with surrounding fibroblasts. For that purpose, Eq. (3) and (4) are interpreted as the equations governing the evolution of a single cardiac cell model. Equation (3) is reformulated as

$$C_m\frac{\mathrm{d}{V}_m}{\mathrm{d}t}=-C_m{I}_{\mathrm{ion}}\text{where}{I}_{\mathrm{ion}}=I_{\mathrm{ion},m}+\frac{N_f}{C_m}{I}_{\mathrm{coupl}}.$$ (10)

Two new state variables are included, namely V_f and I_ion,f. Their corresponding differential equations (4) and (8) are added to the system.

In this framework, the electrical coupling between the myocyte and the surrounding fibroblasts is fully determined by two parameters: the coupling conductance per unit fibroblast membrane capacitance g_c = G_c/C_f (in ms⁻¹) and the total capacitance of the surrounding fibroblast expressed as a fraction of the myocyte capacitance,

$$\kappa =\frac{N_f{C}_f}{C_m}.$$ (11)

The parameter g_c and the time constant τ_f will be selected by comparing the membrane potential time course obtained with this model to in vitro experiments. The parameter κ will be considered as a control parameter to study the impact of fibroblasts on the propagation of the cardiac impulse.

A cardiac tissue model incorporating coupling with fibroblasts

Cells described by the model presented in the previous subsection can be coupled together in order to form a tissue model. This approach ignores the coupling between fibroblasts. The advantage, however, is that coupling with fibroblasts within a cardiac tissue can be introduced as a straightforward extension of the bidomain model [16]. In this paper, electrical propagation will be simulated in a cable with uniform conduction properties. Under these assumptions, the bidomain model reduces to a monodomain model [16] governed by the equation

$$C_m\frac{\partial V_m}{\partial t}={\gamma }^{-1}\nabla \cdot \sigma \nabla V_m-C_m{I}_{\mathrm{ion},m}-C_m\kappa g_c(V_m-V_f)+I_{\mathrm{stim}},$$ (12)

where γ = 1.304 · 10⁷ cm⁻³ is the number of myocyte per unit volume (C_m is in pF), σ = 0.2 S/m is the tissue conductivity, and I_stim is an externally driven stimulation current. The value of γ chosen corresponds to a surface-to-volume ratio of 2000 cm⁻¹. Extending the formulation of the last subsection, Eq. (12) can be interpreted as the monodomain propagation equation for a membrane model whose ionic current is given by I_ion = I_ion,m + κ g_c(V_m – V_f).

In a three-dimensional tissue, 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 κ was kept constant when κ was increased. This situation corresponds to a monolayer of cultured cells coated with fibroblasts [10].

Simulation protocols

A pair myocyte-fibroblast (N_f = 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 g_c was varied between 0 and 0.5 nS/pF, and the time constant τ_f 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 pair myocyte-fibroblast has reached steady-state. Then a current of strength 400 μA/cm² and duration 0.1 ms was injected in the myocyte. The membrane potential of both the myocyte and the fibroblast were simultaneously monitored and presented in a display similar to the Kohl et al. [12] experimental recordings.

The propagation of the cardiac impulse in the presence of fibroblasts was studied in a cable comprising 50 myocytes, each of length 100 μm 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 g_c was varied between 0 and 0.9 nS/pF. From the definition of κ, the pair myocyte-fibroblast actually corresponds to the case κ = 0.03 (N_f = 1). Equation (12) was solved using a forward Euler approach with a time step of 5 μs. Similarly to the pair myocyte-fibroblast, a current of strength 200 μA/cm² and duration 1 ms was injected in the first myocyte of the cable 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, (dV_m/dt)max, was documented for the cell located in the middle of the fiber. The myocyte action potential morphology was characterized by its resting potential (membrane potential at steady-state just before the application of the stimulus), its amplitude and its duration (measured with a threshold at −60 mV, a value corresponding approximately to the reversal potential of the fibroblast ionic current). The average conduction velocity was computed based on the activation times extracted from the action potentials measured at 7 equidistant cells along the cable. The simulation was repeated for each set of parameters considered.

Results

Pair myocyte-fibroblast

Figure 2 shows the membrane potentials in a pair myocyte-fibroblast for different values of the parameters g_c (the coupling between the two cells) and τ_f (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) are displayed below each action potential. The values chosen for g_c, 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].

Figure 2.

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 I_coupl/C_f and the fibroblast ionic current I_ion,f (magnified by a factor 10) are displayed below each action potential. The dotted lines represent the x-axes (zero current).

In the absence of coupling (first column of Fig. 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 due to an increase of both the coupling current and the fibroblast ionic current, whose amplitude is about 10 times smaller than the coupling current, as shown on Fig. 2. For the myocyte, the coupling current is an inward current at steady-state (in this case I_coupl/C_f = I_ion,f), an outward 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 is reduced by less than 1 mV and its action potential duration is prolonged by less than 1 ms when a coupling conductance up to g_c = 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 τ_f on the fibroblast membrane potential is less marked, revealing that the electrical activity of the fibroblast is dominated by the coupling current. The parameter τ_f determines the amplitude of the fibroblast ionic current (see Fig. 2). As a result, it has an impact on the rise time of the electrical activity of the fibroblast (that is, the time interval between the stimulus and position of the peak fibroblast potential), as shown in Table 1. A larger τ_f 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 in Kohl et al. [12]. The set of parameter (g_c = 0.2 nS/pF, τ_f = 20 ms) was found to compare well in terms of rise time and amplitude with the experimental data shown in [12]. The value τ_f = 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.

Table 1.

Rise time of the electrical activity of the fibroblast as a function of the parameters g_c and τ_f. The display correspond to that of Fig. 2.

	gc [nS/pF]
	0.02	0.05	0.1	0.2	0.5
τf = 1 μs	12.5 ms	10.3 ms	7.7 ms	5.0 ms	3.3 ms
τf = 1 ms	12.3 ms	10.1 ms	7.5 ms	4.9 ms	3.3 ms
τf = 20 ms	15.0 ms	11.9 ms	8.9 ms	5.8 ms	3.6 ms
τf = 100 ms	17.5 ms	13.1 ms	9.6 ms	6.0 ms	3.6 ms

Strand of myocytes coated with fibroblasts

Figure 3 displays the conduction velocity (CV) in a cable as a function of the fibroblast density κ for different values of the coupling conductance g_c, namely 0, 0.02, 0.1, 0.2, 0.4, 0.9 nS/pF. When the fibroblast density tends to zero, the CV tends to the baseline value 39.6 cm/s. When the fibroblast density increases, CV is reduced, in agreement with Miragoli et al. [10]. This effect is more pronounced in the case of a larger coupling conductance g_c. 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.

Figure 3.

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, 1.5 corresponds to N_f = 17, 34, 51 fibroblasts per myocyte. Solid lines: g_c = 0, 0.02, 0.1, 0.2, 0.4, 0.9 nS/pF (top to bottom); Dashed line: cable without a coat of fibroblast (g_c = 0 nS/pF), but with a capacitance C_m multiplied by (1 + κ).

The effect of adding fibroblasts was compared to 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 assume to be increased by the same factor since the Bondarenko model specifies them per unit membrane capacitance. The dashed line of Fig. 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, that is, if C_m > 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, with the same display as Fig. 3. When the coupling conductance g_c 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, a phenomenon described by Spach et al. [19]. If the cumulative capacitance of all fibroblasts coupled to the myocyte become of the order of the myocyte capacitance, the myocyte action potential is significantly affected. Along with a decrease in (dV/dt)_max, a reduction in the peak sodium current I_Na,peak was consistently observed. The correlation coefficient between the two variables (dV/dt)_max and I_Na,peak was found to be 0.94. The decrease in CV was also associated with an increase of the myocyte resting potential, as shown in Fig. 5. Although the correlation coefficient was high (−0.89), the variation in myocyte resting potential only explains 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.

Figure 4.

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, 1.5 corresponds to N_f = 17, 34, 51 fibroblasts per myocyte. Solid lines: g_c = 0, 0.02, 0.1, 0.2, 0.4, 0.9 nS/pF (top to bottom); Dashed line: cable without a coat of fibroblast (g_c = 0 nS/pF), but with a capacitance C_m multiplied by (1 + κ).

Figure 5.

Conduction velocity as a function of the myocyte resting potential. The dotted lines connect data points associated with the same coupling conductance: g_c = 0.02, 0.1, 0.2, 0.4, 0.9 nS/pF from top to bottom. When g_c = 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 conduction velocity, maximum upstroke velocity and peak sodium current, a stronger coupling or a coupling with more or higher-capacitance fibroblasts resulted in a decrease in action potential amplitude (APA) and in a prolongation of the action potential duration (APD). The reduction of peak sodium current was associated with an increase of the myocyte resting potential (resulting from the coupling with fibroblasts). For the sake of comparison, the resting potential of the Bondarenko model is −82.3 mV and in a pair myocyte-fibroblast the myocyte resting potential remains below −82 mV even for strong coupling (g_c = 0.9 nS/pF). The corresponding action potential amplitude is 115.1 mV in the absence of coupling and 114.6 mV for strong coupling. Its action potential duration is 14.4 ms without coupling and 14.8 ms for strong coupling.

Figure 6.

Simulated membrane potentials in a strand of myocytes coated by fibroblasts for different values of the coupling conductance g_c 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, 0.9 corresponds to N_f = 10, 20, 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 I_coupl/C_f 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 a propagation along the cable. When the coupling is weak (g_c < 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 (g_c > 0.3 nS/pF), the myocyte resting potential is significantly affected by the fibroblasts. The sodium channels at this higher resting potential (see Fig. 6) are in a less excitable state resulting in an increase of the excitation threshold.

Table 2.

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 assumed [15].

	gc [nS/pF]
	0	0.01	0.1	0.2	0.4	0.9
κ = 0	137	137	137	137	137	137
κ = 0.3	137	135	133	135	138	143
κ = 0.6	137	134	130	132	138	149
κ = 0.9	137	132	126	130	139	154
κ = 1.2	137	130	122	127	139	158
κ = 1.5	137	129	118	126	141	164

Discussion

A one-dimensional model was formulated by reinterpreting the coupled system myocyte-fibroblast as a single unit and connecting these units using a standard monodomain formulation. Since the presence of this myocyte-fibroblast coupling in the human heart is still a controversial question, our model was aimed at determining how much coupling would be necessary to significantly perturbs 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 g_c and the fibroblast density κ, that is, 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 g_c and fibroblast density κ. In the Miragoli et al. experiment [10], the CV also slowed down from a baseline value of 40 cm/s down to CVs slower than 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 up κ to 1.5 when g_c = 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·C_m (230 pF for the Bondarenko model). If each fibroblast has a capacitance of 4.5 pF (resp. 6.3 pF to compare with [14]), N_f = κC_m/C_f = 51 fibroblasts (resp. 36) per myocyte would be needed to induce the same reduction in CV as the extreme case of the experiment. Using the Pandit model of rat ventricular cell [20] which has a myocyte capacitance C_m of 100 pF would further reduce to 24 the required number of fibroblasts per myocyte. In any case, the cumulative capacitance of the fibroblasts needs to be non-negligible as compared to 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 due to the coupling with fibroblasts. First, because the myocyte resting potential (when coupled to fibroblasts) is higher than that of an isolated myocyte (Fig. 6), the sodium channels are in a less excitable state, resulting in a decrease of the peak sodium current and a slower CV. Second, 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, which can be an arrhythmogenic factor.

Effect of fibroblasts on the myocyte action potential

Coupling with fibroblasts resulted in a prolongation of the myocyte action potential duration, a decrease of its amplitude and its upstroke velocity. The variations were significant only for strong coupling with a sufficiently high number of fibroblasts (about 10, κ = 0.3 in Fig. 6). In contrast, in another recent simulation study, MacCannell et al. observed a dramatic shortening of the action potential duration (from 259 ms down to 155 ms) when a myocyte was connected to only 4 active fibroblasts through a coupling conductance similar to those used in this paper [14]. 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 about 20 ms (near −80 to −70 mV) up to 160 ms near V_f = −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 (τ_f = 20 ms) the we used was the same order of magnitude as the myocyte action potential duration. This brief comparison suggests that the effect of fibroblast may be action potential dependent, at least through the ratio of activation time constant to action potential duration. This may confuse the clinical significance of cell culture 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 below that of the fibroblast (resp. above), the fibroblast acts as a current source (resp. 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 are reported in the literature [14]. We chose the value −58 mV from [6] 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-myocyte are introduced.

Our fibroblast model is merely a nonlinear 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 modeling provides a way to analyze these factors independently in order to help make the link between cell culture experiments and in vivo human heart electrophysiology.