Bidomain model

The bidomain equations are derived from the Maxwell's equations of the electromagnetism, considering some semplifications.[12]

The first assumption is that the intracellular current can flow only between the intracellular and extracellular regions, while the intracellular and extramyocardial regions can comunicate between them, so that the current can flow into and from the extramyocardial regions but only in the extracellular space.[12]

Using Ohm's law and a quasi-static assumpion, the gradient of a scalar potential field ${\displaystyle \varphi }$ can describe an electrical field ${\displaystyle \mathbf {E} }$, which means that[12]

${\displaystyle \mathbf {E} =-\nabla \varphi .}$

Then, if ${\displaystyle J}$ represent the current density of the electric field ${\displaystyle \mathbf {E} }$, two equations can be obtained[12]

${\displaystyle {\begin{alignedat}{2}J_{i}&=-\mathbf {\Sigma } _{i}\nabla v_{i}\\J_{e}&=-\mathbf {\Sigma } _{e}\nabla v_{e}.\end{alignedat}}}$

where the subscript ${\displaystyle i}$ and ${\displaystyle e}$ represent the intracellular and extracellular quantities respectively.[12]

The second assumption is that the heart is isolated, so that the current that leaves one regione need to flow into the other. Then, the current density in each of the intracellular and extracellular domain must be equal in magnitude but opposite in signe, and can be defined as the product of the surface to volume ratio of the cell membrane and the transmembrane ionic current density ${\displaystyle I_{m}}$ per unit area, which means that[12]

${\displaystyle -\nabla \cdot J_{i}=\nabla \cdot J_{e}=\chi I_{m}.}$

By combining the previous assumptions, the conservation of current densities is obtained, namely[12]

${\displaystyle {\begin{alignedat}{2}\nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{i})&=\chi I_{m}\\\nabla \cdot (\mathbf {\Sigma } _{e}\nabla v_{e})&=-\chi I_{m}\end{alignedat}}}$

(1)

from which, summing the two equations[12]

${\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{i})=-\nabla \cdot (\mathbf {\Sigma } _{e}\nabla v_{e}).}$

This equation states exaclty that all currents exiting one domain must enter the other.[12]

From here, it is easy to find the second equation of the bidomain model subtracting ${\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}$ from both sides. Infact,[12]

${\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{i})-\nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})=-\nabla \cdot (\mathbf {\Sigma } _{e}\nabla v_{e})-\nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}$

and knowing that the transmembral potential is defined as ${\displaystyle v=vi-v_{e}}$[12]

${\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v)=-\nabla \cdot ((\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e})\nabla v_{e}).}$

Then, knowing the transmembral potential, one can recover the extracellular potential.

Then, the current that flows across the cell membrane can be modelled with the cable equation,[12]

${\displaystyle I_{m}=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{ion}\right),}$

(2)

Combining equations (1) and (2) gives[12]

${\displaystyle \nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{i}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{ion}\right).}$

Finally, adding and subtracting ${\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}$ on the left and rearranging ${\displaystyle v=vi-v_{e}}$, one can get the first equation of the bidomain model[12]

${\displaystyle \nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right),}$

which describes the evolution of the transmembral potential in time.

The final formulation described in the standard formulation section is obtained throught a generalization, considering possible external stimulus which can be given through the external applied currents ${\displaystyle I_{s_{1}}}$ and ${\displaystyle I_{s_{2}}}$.[12]

In order to solve the model, boundary conditions are needed. The more classical boundary conditions are the following ones, formulated by Tung.[6]

First of all, as state before in the derive section, there ca not been any flow of current between the intracellular and extramyocardial domains. This can be mathematically described as[12]

${\displaystyle (\mathbf {\Sigma } _{i}\nabla v_{i})\cdot \mathbf {n} =0\quad \mathbf {x} \in \partial \mathbb {H} }$

where ${\displaystyle \mathbf {n} }$ is the vector that represents the outwardly unit normal to the myocardial surface of the heart. Since the intracellular potential is not explicitily presented in the bidomain formulation, this condition is usually described in terms of the transmembrane and extracellular potential, knowing that ${\displaystyle v=v_{i}-v_{e}}$, namely[12]

${\displaystyle (\mathbf {\Sigma } _{i}\nabla v)\cdot \mathbf {n} =-(\mathbf {\Sigma } _{i}\nabla v_{e})\cdot \mathbf {n} \quad \mathbf {x} \in \partial \mathbb {H} .}$

For the extracellular potential, if the myocardial region is presented, a balance in the flow between the extracellular and the extramyocardial regions is considered[12]

${\displaystyle \left(\mathbf {\Sigma } _{e}\nabla v_{e}\right)\cdot \mathbf {n} _{e}=-\left(\mathbf {\Sigma } _{0}\nabla v_{0}\right)\cdot \mathbf {n} _{0}\quad \mathbf {x} \in \partial \mathbb {H} .}$

Here the normal vectors from the perspective of both domains are considered, thus the negative sign are necessary. Moreover, a perfect trasmission of the potential on the cardiac boundary is necessary, which gives[12]

${\displaystyle v_{e}=v_{0}\quad \mathbf {x} \in \partial \mathbb {H} }$.

Instead, if the heart is considered as isoleted, which means that no myocardial region is presented, a possible boundary condition for the extracellular problem is

${\displaystyle \left(\mathbf {\Sigma } _{i}\nabla v\right)\cdot \mathbf {n} =-\left((\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e})\nabla v_{e}\right)\cdot \mathbf {n} \quad \mathbf {x} \in \partial \mathbb {H} .}$[12]

By assuming equal anisotropy ratios for the intra- and extracellular domains, i.e. ${\displaystyle \mathbf {\Sigma } _{i}=\lambda \mathbf {\Sigma } _{e}}$ for some scalar ${\displaystyle \lambda }$, the model can be reduced to one single equation, called monodomain equation

${\displaystyle \nabla \cdot (\mathbf {\Sigma } \nabla v)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{i}on\right)-I_{s}}$

where the only variable is now the transmembrane potential, and the conductivity tensor ${\displaystyle \mathbf {\Sigma } }$ is a combination of ${\displaystyle \mathbf {\Sigma } _{i}}$ and ${\displaystyle \mathbf {\Sigma } _{e}.}$[12]

If the heart is considered as an isolated tissue, which means that no current can flow outside of it, the final formulation with boundary conditions reads[12]

${\displaystyle {\begin{cases}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right)-I_{s_{1}}&\mathbf {x} \in \mathbb {H} \\\nabla \cdot \left(\left(\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e}\right)\nabla v_{e}\right)=-\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+I_{s_{2}}&\mathbf {x} \in \mathbb {H} \\\mathbf {\Sigma } _{i}(\nabla v+\nabla v_{e})\cdot \mathbf {n} =0&\mathbf {x} \in \partial \mathbb {H} \\\left[\mathbf {\Sigma } _{i}(\nabla v+\nabla v_{e})+\mathbf {\Sigma } _{e}\nabla v_{e}\right]\cdot \mathbf {n} =0&\mathbf {x} \in \partial \mathbb {H} \end{cases}}}$

There are lots of different possible techniques to solve the bidomain equations. Between them, one can find finite difference schemes, finite element schemes and also finite volume schemes. Special considerations can be made for the numerical solution of these equations, due to the high time and space resolution needed for numerical convergence.[20][21]

• Monodomain model
• Forward problem of electrocardiology

• Scholarpedia article about the bidomain model