Bidomain model

The bidomain model is a mathematical model for the electrical properties of cardiac muscle that takes into account the anisotropy of both the intracellular and extracellular spaces. It is formed of the bidomain equations.

The bidomain model was developed in the late 1970s. ^{[1] [2] [3] [4] [5] [6] [7] [8]} It is a generalization of one-dimensional cable theory. The bidomain model is a continuum model, meaning that it represents the average properties of many cells, rather than describing each cell individually. ^[9]

Many of the interesting properties of the bidomain model arise from the condition of unequal anisotropy ratios. The electrical conductivity in anisotropic tissue is different parallel and perpendicular to the fiber direction. In a tissue with unequal anisotropy ratios, the ratio of conductivities parallel and perpendicular to the fibers is different in the intracellular and extracellular spaces. For instance, in cardiac tissue, the anisotropy ratio in the intracellular space is about 10:1, while in the extracellular space it is about 5:2. ^[10] Mathematically, unequal anisotropy ratios means that the effect of anisotropy cannot be removed by a change in the distance scale in one direction. ^[11] Instead, the anisotropy has a more profound influence on the electrical behavior. ^[12]

Three examples of the impact of unequal anisotropy ratios are

• the distribution of transmembrane potential during unipolar stimulation of a sheet of cardiac tissue,^[13]
• the magnetic field produced by an action potential wave front propagating through cardiac tissue,^[14]
• the effect of fiber curvature on the transmembrane potential distribution during an electric shock.^[15]

The bidomain model is now widely used to model defibrillation of the heart.

Formulation

Standard formulation

The bidomain model can be formulated as follows:

$\begin{alignat}{2} \nabla \cdot \left(\mathbf\Sigma_i \nabla v \right) + \nabla \cdot \left(\mathbf\Sigma_i \nabla v_e \right) & = \chi \left( C_m \frac{\partial v}{\partial t} + I_\mathrm{ion} \right) \\ \nabla \cdot \left( \mathbf\Sigma_i \nabla v \right) + \nabla \cdot \left( \left( \mathbf\Sigma_i + \mathbf\Sigma_e \right) \nabla v_e \right) & = 0 \end{alignat}$

where $\chi$ is the membrane surface area per unit volume (of tissue), $C_m$ is the electrical capacitance of the membrane per unit area, $v=v_i-v_e$ where $v_i$ is the interstitial voltage and $v_e$ is the extracellular voltage, and $I_\mathrm{ion}$ is the ionic current over the membrane per unit area.

Formulation with boundary conditions and surrounding tissue

The surrounding tissue $\mathbb T$ can be included to give reasonable boundary conditions to make the system solvable:

$\begin{alignat}{4} \nabla \cdot \left(\mathbf\Sigma_i \nabla v \right) + \nabla \cdot \left(\mathbf\Sigma_i \nabla v_e \right) & = \chi \left( C_m \frac{\partial v}{\partial t} + I_\mathrm{ion} \right) & \,\,\,\,\,\,\, & \mathbf x \in \mathbb H \\ \nabla \cdot \left( \mathbf\Sigma_i \nabla v \right) + \nabla \cdot \left( \left( \mathbf\Sigma_i + \mathbf\Sigma_e \right) \nabla v_e \right) & = 0 && \mathbf x \in \mathbb H \\ \nabla \cdot \left( \mathbf\Sigma_0 \nabla v_0 \right) & = 0 && \mathbf x \in \mathbb T \\ \vec n \cdot \left( \mathbf\Sigma_0 \nabla v_0 \right) & = 0 && \mathbf x \in \partial \mathbb T \\ \vec n \cdot \left( \mathbf\Sigma_0 \nabla v_0 \right) - \vec n \cdot \left( \mathbf\Sigma_e \nabla v_e \right) & = 0 && \mathbf x \in \partial \mathbb H \\ \vec n \cdot \left( \mathbf\Sigma_i \nabla v \right) + \vec n \cdot \left( \mathbf\Sigma_i \nabla v_e \right) & = 0 && \mathbf x \in \partial \mathbb H \end{alignat}$

Derivation

Let $\mathbb H$ with boundary $\partial \mathbb H$ be the set of all points $\mathbf x$ in the heart. In each point in $\mathbb H$ there is an intra- and extracellular voltage and current, denoted by $v_i$, $v_e$, $J_i$ and $J_e$ respectively. Let $\mathbf\Sigma_i$ and $\mathbf\Sigma_e$ be the intra- end extracellular conductivity tensor matrices respectively.

We assume Ohmic current-voltage relationship and get

$\begin{alignat}{2} J_i & = -\mathbf\Sigma_i \nabla v_i \\ J_e & = -\mathbf\Sigma_e \nabla v_e. \end{alignat}$

We require that there is no accumulation of charge anywhere in $\mathbb H$, and therefore that

$\begin{alignat}{2} \nabla \cdot \left( J_i + J_e \right) & = 0 \\ \nabla \cdot \left( -\mathbf\Sigma_i \nabla v_i - \mathbf\Sigma_e \nabla v_e \right) & = 0 \end{alignat}$

giving one of the model equations:

$\nabla \cdot \left( \mathbf\Sigma_i \nabla v_i \right) + \nabla \cdot \left( \mathbf\Sigma_e \nabla v_e \right) = 0 .$

(1)

This equation states that all current exiting one domain must enter the other.

The transmembrane current is given by

$J_t = \nabla \cdot \left( \mathbf\Sigma_i \nabla v_i \right) = -\nabla \cdot \left( \mathbf \Sigma_e \nabla v_e \right) .$

(2)

We model the membrane similarly to that of the cable equation,

$J_t = \chi \left( C_m \frac{\partial v}{\partial t} + I_\mathrm{ion} \right) ,$

(3)

where $\chi$ is the membrane surface area per unit volume (of tissue), $C_m$ is the electrical capacitance of the membrane per unit area, $v=v_i-v_e$ and $I_\mathrm{ion}$ is the ionic current over the membrane per unit area.

Combining equations (2) and (3) gives

$\nabla \cdot \left( \mathbf\Sigma_i \nabla v_i \right) = \chi \left( C_m \frac{\partial v}{\partial t} + I_\mathrm{ion} \right) ,$

which can be rearranged using $v=v_i-v_e$ to get another model equation:

$\nabla \cdot \left( \mathbf\Sigma_i \nabla v \right) + \nabla \cdot \left( \mathbf\Sigma_i \nabla v_e \right) = \chi \left( C_m \frac{\partial v}{\partial t} + I_\mathrm{ion} \right) .$

(4)

Boundary conditions

In order to solve the model, boundary conditions are needed. One way to define the boundary condition is to extend the model with a volume $\mathbb T$ with perimeter $\partial \mathbb T$ that surrounds the heart and represent the body tissue.

As was the case for $\mathbb H$, we assume no accumulation of charge in $\mathbb T$, i.e.

$\nabla \cdot \left( \mathbf\Sigma_0 \nabla v_0 \right) = 0 \,\,\,\,\,\,\, \mathbf x \in \mathbb T ,$

(5)

where $\mathbf\Sigma_0$ is the conductance tensor of the body tissue and $v_0$ is the voltage in $\mathbb T$.

Assuming that the body is electrically surrounded from the environment, there can be no current component on the surface $\partial \mathbb T$ in the normal direction, hence:

$\vec n \cdot \left( \mathbf\Sigma_0 \nabla v_0 \right) = 0 \,\,\,\,\,\,\, \mathbf x \in \partial \mathbb T .$

(6)

On the surface of the heart, a common assumption is that there is a direct connection between the surrounding tissue and the extracellular domain. This means that the potentials $v_e$ and $v_0$ must be equal on the heart surface, i.e.

$v_e = v_0 \,\,\,\,\,\,\, \mathbf x \in \partial \mathbb H .$

(7)

This direct connection also require that all ionic current exiting $\mathbb T$ on the heart surface, must enter the extracellular domain, and vica versa. This gives another boundary condition:

$\vec n \cdot \left( \mathbf\Sigma_0 \nabla v_0 \right) = \vec n \cdot \left( \mathbf\Sigma_e \nabla v_e \right) \,\,\,\,\,\,\, \mathbf x \in \partial \mathbb H .$

(8)

Finally, we assume that there is a complete isolation of the intracellular domain and the surrounding tissue. Similarly to equation (2), we get

$\vec n \cdot \left( \mathbf\Sigma_i \nabla v_i \right) = 0 \,\,\,\,\,\,\, \mathbf x \in \partial \mathbb H$

which can be rewritten using $v=v_i-v_e$ to

$\vec n \cdot \left( \mathbf\Sigma_i \nabla v \right) + \vec n \cdot \left( \mathbf\Sigma_i \nabla v_e \right) = 0 \,\,\,\,\,\,\, \mathbf x \in \partial \mathbb H .$

(9)

Extending the model to include equations (5)-(9) gives a solvable system of equations.

Reduction to monodomain model

By assuming equal anisotropy ratios for the intra- and extracellular domains, i.e. $\mathbf\Sigma_i = \alpha\mathbf\Sigma_e$ for some scalar $\alpha$, the model can be reduced to the monodomain model.

Numerical solution

There are some special considerations for numerical solution of these equations, due to high time and space resolution needed for numerical convergence.^{[16] [17]}

References

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3. Muler AL, Markin VS (1977). "Electrical properties of anisotropic nerve-muscle syncytia-III. Steady form of the excitation front.". Biofizika 22 (4): 671–675. PMID 901827.
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9. Neu JC, Krassowska W (1993). "Homogenization of syncytial tissues.". Critical Reviews of Biomedical Engineering 21: 137–199.
10. Roth BJ (1997). "Electrical conductivity values used with the bidomain model of cardiac tissue.". IEEE Transactions on Biomedical Engineering 44 (4): 326–328. doi:10.1109/10.563303. PMID 9125816.
11. Roth BJ (1992). "How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle.". Journal of Mathematical Biology 30 (6): 633–646. doi:10.1007/BF00948895. PMID 1640183.
12. Henriquez CS (1993). "Simulating the electrical behavior of cardiac tissue using the bidomain model.". Critical Reviews of Biomedical Engineering 21: 1–77.
13. Sepulveda NG, Roth BJ, Wikswo JP Jr (1989). "Current injection into a two-dimensional bidomain.". Biophysical Journal 55 (5): 987–999. Bibcode:1989BpJ....55..987S. doi:10.1016/S0006-3495(89)82897-8. PMC 1330535. PMID 2720084.
14. Sepulveda NG, Wikswo JP Jr (1987). "Electric and magnetic fields from two-dimensional bisyncytia.". Biophysical Journal 51 (4): 557–568. Bibcode:1987BpJ....51..557S. doi:10.1016/S0006-3495(87)83381-7. PMC 1329928. PMID 3580484.
15. Trayanova N, Roth BJ, Malden LJ (1993). "The response of a spherical heart to a uniform electric field: A bidomain analysis of cardiac stimulation.". IEEE Transactions on Biomedical Engineering 40 (9): 899–908. doi:10.1109/10.245611. PMID 8288281.
16. Niederer, S. A.; Kerfoot, E.; Benson, A. P.; Bernabeu, M. O.; Bernus, O.; Bradley, C.; Cherry, E. M.; Clayton, R.; Fenton, F. H.; Garny, A.; Heidenreich, E.; Land, S.; Maleckar, M.; Pathmanathan, P.; Plank, G.; Rodriguez, J. F.; Roy, I.; Sachse, F. B.; Seemann, G.; Skavhaug, O.; Smith, N. P. (3 October 2011). "Verification of cardiac tissue electrophysiology simulators using an N-version benchmark". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369 (1954): 4331–4351. Bibcode:2011RSPTA.369.4331N. doi:10.1098/rsta.2011.0139.
17. Pathmanathan, Pras; Bernabeu, Miguel O.; Bordas, Rafel; Cooper, Jonathan; Garny, Alan; Pitt-Francis, Joe M.; Whiteley, Jonathan P.; Gavaghan, David J. "A numerical guide to the solution of the bidomain equations of cardiac electrophysiology". Progress in Biophysics and Molecular Biology 102 (2-3): 136–155. doi:10.1016/j.pbiomolbio.2010.05.006.

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