Modelling the Effect of Gap Junctions on Tissue-Level Cardiac Electrophysiology

Abstract

When modelling tissue-level cardiac electrophysiology, a continuum approximation to the discrete cell-level equations, known as the bidomain equations, is often used to maintain computational tractability. Analysing the derivation of the bidomain equations allows us to investigate how microstructure, in particular gap junctions that electrically connect cells, affect tissue-level conductivity properties. Using a one-dimensional cable model, we derive a modified form of the bidomain equations that take gap junctions into account, and compare results of simulations using both the discrete and continuum models, finding that the underlying conduction velocity of the action potential ceases to match up between models when gap junctions are introduced at physiologically realistic coupling levels. We show that this effect is magnified by: (i) modelling gap junctions with reduced conductivity; (ii) increasing the conductance of the fast sodium channel; and (iii) an increase in myocyte length. From this, we conclude that the conduction velocity arising from the bidomain equations may not be an accurate representation of the underlying discrete system. In particular, the bidomain equations are less likely to be valid when modelling certain diseased states whose symptoms include a reduction in gap junction coupling or an increase in myocyte length.

Appendix

1.1 Proof of the Symmetry of the Homogenised Conductivity Tensors

We will show that the conductivity tensor given in Eq. (8) is symmetric by demonstrating that $\mathbf{S_{i}}^{(1,2)}$ = $\mathbf{S_{i}}^{(2,1)}$, with the symmetry of the other components following in a similar way. The extracellular conductivity tensor given in Eq. (12) can then be shown to be symmetric using the same logic. Throughout, we will write W for W ⁽ⁱ⁾. For reference, the intracellular tensor is given by

$$\mathbf{S_{i}} = \frac{1}{V_{\text{unit}}}\int_{\varOmega_i} \sigma_i \biggl(I + \frac{\partial W}{\partial\mathbf{z}} \biggr) \, \mathrm{d} V_{\mathbf{z}}. $$

As the identity matrix is symmetric, it remains to show that $\int \sigma_{i} \frac{\partial W}{\partial\mathbf{z}} \, \mathrm{d} V_{\mathbf {z}}$ is symmetric. Using integration by parts, we have

$$\begin{aligned} \iiint\sigma \biggl( \frac{ \partial W_1}{\partial z_2} - \frac {\partial W_2}{\partial z_1} \biggr) \, \mathrm{d} V_{\mathbf{z}} =& \iiint\frac{\partial(\sigma W_1)}{\partial z_2} - \frac{\partial (\sigma W_2)}{\partial z_1} \\ &{}- \frac{\partial\sigma}{\partial z_2} W_1 + \frac{\partial\sigma}{\partial z_2} W_2 \, \mathrm{d} V_{\mathbf{z}}, \end{aligned}$$

(24)

where σ _i =σ(z) and, for discontinuous σ, the derivative is taken to be the weak derivative. Using Eq. (9), this becomes

$$\iiint\nabla_{\mathbf{z}} \cdot(-\sigma W_2,\sigma W_1,0) + \nabla _{\mathbf{z}} \cdot(\sigma\nabla_{\mathbf{z}} W_2) W_1 - \nabla _{\mathbf{z}} \cdot(\sigma \nabla_{\mathbf{z}} W_1) W_2 \, \mathrm{d} V_{\mathbf{z}}, $$

which upon rearrangement can be expressed as

$$ \iiint\nabla_{\mathbf{z}} \cdot \bigl[ (-\sigma W_2,\sigma W_1,0) + \sigma W_1 \nabla_{\mathbf{z}} W_2 - \sigma W_2 \nabla_{\mathbf{z}} W_1 \bigr] \, \mathrm{d} V_{\mathbf{z}}. $$

(25)

Applying the divergence theorem then gives

$$\iint\sigma(-W_2 n_1 + W_1 n_2) + \sigma W_1 (\nabla_{\mathbf{z}} W_2) \cdot\mathbf{n} - \sigma W_2 (\nabla_{\mathbf{z}} W_1) \cdot\mathbf {n} \, \mathrm{d} S_{\mathbf{z}}, $$

and by grouping terms, we see that this may be written

$$\iint\sigma W_1 \bigl(n_2 + (\nabla_{\mathbf{z}} W_2) \cdot\mathbf{n}\bigr) - \sigma W_2 \bigl(n_1 + (\nabla_{\mathbf{z}} W_1) \cdot\mathbf{n} \bigr) \, \mathrm {d} S_{\mathbf{z}}. $$

We see from Eq. (10) that this quantity is zero, so that $\iiint\sigma \frac{ \partial W_{1}}{\partial z_{2}} \, \mathrm{d} V_{\mathbf{z}} = \iiint\sigma\frac{\partial W_{2}}{\partial z_{1}} \, \mathrm{d} V_{\mathbf{z,}}$ and hence $\mathbf{S_{i}}^{(1,2)} = \mathbf{S_{i}}^{(2,1)}$ as required.

1.2 Derivation of the Intracellular Conductivity Tensor in the Case Where Gap Junctions are Present

We now give a derivation of Eq. (23). We consider the governing equations for the weight functions in the cell and the gap junction separately, and again write W for W ⁽ⁱ⁾. As each compartment has homogeneous conductivity, W ₁ will satisfy Laplace’s equation in each, and so a solution that satisfies the governing PDE and boundary conditions has a derivative

$$ \frac{\partial W_1}{\partial x} = \begin{cases} B_1, & 0 < x < \delta, \\ B_2, & \delta< x < 1, \end{cases} $$

(26)

where B ₁ and B ₂ are constants. To find B ₁ and B ₂, we can integrate the governing equation for $W^{i}_{1}$, given in (9), across the boundary between the cell and the gap junction at x=δ to give

$$\biggl[ \sigma\frac{\partial W_1}{\partial x} \biggr]^{\delta^+}_{\delta ^-} = [ - \sigma ]^{\delta^+}_{\delta^-}, $$

and, therefore, obtain the relation between B ₁ and B ₂:

$$\sigma_i B_2 - \sigma_g B_1 = \sigma_g - \sigma_i. $$

We then use the fact that W ₁ has zero mean in x to give us

$$\int_0^1 \frac{\partial W_1}{\partial x} \, \mathrm{d} x = \delta B_1 + (1-\delta) B_2 = 0. $$

Solving for B ₁ and B ₂ gives

$$B_1 = \frac{\delta(\sigma_g-\sigma_i)}{\delta\sigma_i + (1-\delta)\sigma_g} , \qquad B_2 = \frac{(\delta-1)(\sigma_g-\sigma_i)}{\delta\sigma_i + (1-\delta) \sigma_g} , $$

and substituting the above into (8) gives

$$\mathbf{S}_i^{(1,1)} = \frac{V_{\text{intra}}}{V_{\text{cell}}} \biggl( \int _0^{\delta}\sigma_g(1+B_1)\, \mathrm{d}x + \int_{\delta}^1 \sigma_i(1+B_2) \, \mathrm{d}x \biggr)= \frac{f_i \sigma_i \sigma_g }{\delta\sigma_i + (1-\delta) \sigma_g}. $$