bidomain model

File:Bidomain domain.png

The bidomain domain is principally represented by two main regions: the cardiac cells, called intracellular domain, and the space surrounding them, called extracellular domain. Moreover, usually another region is considered, called extramyocardial region.

The intracellular and extracellular domains, which are separate by the cellular membrane, are considered to be a unique physical space representing the heart ($\backslash mathbb\; H$), while the extramyocardial domain is a unique physical space adjacent of them ($\backslash mathbb\; T$). The extramyocardial region can be considered as a fluid bath, especially when one wants to simulate experimental conditions, or as a human torso to simulate physiological conditions.

The boundary of the two principal physical domains defined are important to solve the bidomain model. Here the heart boundary is denoted as $\backslash partial\backslash mathbb\; H$ while the torso domain boundary is $\backslash partial\backslash mathbb\; T.$

The unknowns in the bidomain model are three, the intracellular potential $v\_i$, the extracellular potential $v\_e$ and the transmembrane potential $v$, which is defined as the difference of the potential across the cell membrane $v\; =\; v\_i\; -\; v\_e$.

Moreover, some important parameters need to be taken in account, especially the intracellular conductivity tensor matrix {{nowrap|$\backslash mathbf\backslash Sigma\_i$,}} the extracellular conductivity tensor matrix {{nowrap|$\backslash mathbf\backslash Sigma\_e$.}} The transmembrane current flows between the intracellular and extracellular regions and it is in part described by the corresponding ionic current over the membrane per unit area $I\_\backslash text\{ion\}$.

Moreover, the membrane capacitance per unit area $C\_m$ and the surface to volume ratio of the cell membrane $\backslash chi$ need to be considered to derive the bidomain model formulation, which is done in the following section.

The bidomain model is defined through two partial differential equations (PDE) the first of which is a reaction diffusion equation in terms of the transmembrane potential, while the second one computes the extracellular potential starting from a given transmembran potential distribution.

Thus, 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)- I_{s_1} \\[1ex]

&\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}

\end{alignat}

where $I\_\{s\_1\}$ and $I\_\{s\_2\}$ can be defined as applied external stimulus currents.

The ionic current is usually represented by an ionic model through a system of ordinary differential equations (ODEs). Mathematically, one can write $I\_\backslash text\{ion\}\; =\; I\_\backslash text\{ion\}(v,\backslash mathbf\; w)$ where $\backslash mathbf\; w$ is called ionic variable. Then, in general, for all $t\; >\; 0$, the system reads{{cite journal |last1=Boulakia |first1=Muriel |last2=Cazeau |first2=Serge |last3=Fernández |first3=Miguel A. |last4=Gerbeau |first4=Jean-Frédéric |last5=Zemzemi |first5=Nejib |title=Mathematical Modeling of Electrocardiograms: A Numerical Study |journal=Annals of Biomedical Engineering |date=24 December 2009 |volume=38 |issue=3 |pages=1071–1097 |doi=10.1007/s10439-009-9873-0|pmid=20033779 |s2cid=10114284 |url=https://hal.inria.fr/inria-00400490/file/RR-6977.pdf }}

$$\backslash begin\{cases\}$$

\dfrac{\partial \mathbf w}{\partial t} = \mathbf{F}(v,\mathbf w) &\text{in } \mathbb{H} \\

\mathbf w(t = 0) = \mathbf{w}_0 &\text{in } \mathbb{H}

\end{cases}

Different ionic models have been proposed:

• phenomenological models, which are the simplest ones and used to reproduce macroscopic behavior of the cell.
• physiological models, which take into account both macroscopic behaviour and cell physiology with a quite detailed description of the most important ionic current.

In some cases, an extramyocardial region is considered. This implies the addition to the bidomain model of an equation describing the potential propagation inside the extramyocardial domain.

Usually, this equation is a simple generalized Laplace equation of type

$$-\backslash nabla\; \backslash cdot\; (\backslash mathbf\; \backslash Sigma\_0\; \backslash nabla\; v\_0)\; =\; 0\; \backslash quad\; \backslash mathbf\; x\; \backslash in\; \backslash mathbb\{T\}$$

where $v\_0$ is the potential in the extramyocardial region and $\backslash mathbf\; \backslash Sigma\_0$ is the corresponding conductivity tensor.

Moreover, an isolated domain assumption is considered, which means that the following boundary conditions are added

$$(\backslash mathbf\; \backslash Sigma\_0\; \backslash nabla\; v\_0)\; \backslash cdot\; \backslash mathbf\{n\}\_0\; =\; 0\; \backslash quad\; \backslash mathbf\; x\; \backslash in\; \backslash partial\; \backslash mathbb\{T\},$$

$\backslash mathbf\{n\}\_0$ being the unit normal directed outside of the extramyocardial domain.

If the extramyocardial region is the human torso, this model gives rise to the forward problem of electrocardiology.

The bidomain equations are derived from the Maxwell's equations of the electromagnetism, considering some simplifications.

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.

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

$$\backslash mathbf\{E\}\; =\; -\; \backslash nabla\; \backslash varphi.$$

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

\begin{alignat}{2}

J_i & = -\mathbf\Sigma_i \nabla v_i \\

J_e & = -\mathbf\Sigma_e \nabla v_e.

\end{alignat}

where the subscript $i$ and $e$ represent the intracellular and extracellular quantities respectively.

The second assumption is that the heart is isolated so that the current that leaves one region 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 sign, and can be defined as the product of the surface to volume ratio of the cell membrane and the transmembrane ionic current density $I\_m$ per unit area, which means that

$$-\backslash nabla\; \backslash cdot\; J\_i\; =\; \backslash nabla\; \backslash cdot\; J\_e\; =\; \backslash chi\; I\_m.$$

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

{{NumBlk|:|$$

\begin{alignat}{2}

\nabla \cdot (\mathbf\Sigma_i \nabla v_i) &= \chi I_m \\

\nabla \cdot (\mathbf\Sigma_e \nabla v_e) &= - \chi I_m

\end{alignat}

|{{EquationRef|1}}}}

from which, summing the two equations

:$$\backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\_i)\; =\; -\; \backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_e\; \backslash nabla\; v\_e).$$

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

From here, it is easy to find the second equation of the bidomain model subtracting $\backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\_e)$ from both sides. In fact,

$$\backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\_i)\; -\; \backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\_e)\; =\; -\; \backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_e\; \backslash nabla\; v\_e)\; -\backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\_e)$$

and knowing that the transmembral potential is defined as $v\; =\; v\_i\; -\; v\_e$

$$\backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v)\; =\; -\; \backslash nabla\; \backslash cdot\; ((\backslash mathbf\backslash Sigma\_i\; +\; \backslash mathbf\backslash Sigma\_e)\; \backslash nabla\; v\_e).$$

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,

{{NumBlk|:| $$

I_m = \chi \left( C_m \frac{\partial v}{\partial t} + I_\text{ion} \right)

,|{{EquationRef|2}}}}

Combining equations ({{EquationNote|1}}) and ({{EquationNote|2}}) gives

\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 $\backslash nabla\; \backslash cdot\; (\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\_e)$ on the left and rearranging $v\; =\; v\_i\; -\; v\_e$, one can get the first equation of the bidomain model

\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 transmembrane potential in time.

The final formulation described in the standard formulation section is obtained through a generalization, considering possible external stimulus which can be given through the external applied currents $I\_\{s\_1\}$ and $I\_\{s\_2\}$.

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

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

$$(\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\_i\; )\; \backslash cdot\; \backslash mathbf\; n\; =\; 0\; \backslash quad\; \backslash mathbf\; x\; \backslash in\; \backslash partial\; \backslash mathbb\; H$$

where $\backslash mathbf\; n$ 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 $v\; =\; v\_i\; -\; v\_e$, namely

$$(\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v)\; \backslash cdot\; \backslash mathbf\; n\; =\; -\; (\backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\_e)\; \backslash cdot\; \backslash mathbf\; n\; \backslash quad\; \backslash mathbf\; x\; \backslash in\; \backslash partial\; \backslash mathbb\; H.$$

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

$$\backslash left(\; \backslash mathbf\backslash Sigma\_e\; \backslash nabla\; v\_e\; \backslash right)\; \backslash cdot\; \backslash mathbf\; n\_e\; =\; -\backslash left(\; \backslash mathbf\backslash Sigma\_0\; \backslash nabla\; v\_0\; \backslash right)\; \backslash cdot\; \backslash mathbf\; n\_0\; \backslash quad\; \backslash mathbf\; x\; \backslash in\; \backslash partial\; \backslash mathbb\; H\; .$$

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

$$v\_e\; =\; v\_0\; \backslash quad\; \backslash mathbf\; x\; \backslash in\; \backslash partial\; \backslash mathbb\; H.$$

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

$$\backslash left(\; \backslash mathbf\backslash Sigma\_i\; \backslash nabla\; v\; \backslash right)\; \backslash cdot\; \backslash mathbf\; n\; =\; -\backslash left(\; (\backslash mathbf\backslash Sigma\_i\; +\; \backslash mathbf\backslash Sigma\_e)\; \backslash nabla\; v\_e\; \backslash right)\; \backslash cdot\; \backslash mathbf\; n\; \backslash quad\; \backslash mathbf\; x\; \backslash in\; \backslash partial\; \backslash mathbb\; H\; .$$

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

$$\backslash nabla\; \backslash cdot\; (\backslash mathbf\; \backslash Sigma\; \backslash nabla\; v\; )\; =\; \backslash chi\; \backslash left(\; C\_m\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; t\}\; +\; I\_\backslash mathrm\{ion\}\backslash right)\; -\; I\_s$$

where the only variable is now the transmembrane potential, and the conductivity tensor $\backslash mathbf\backslash Sigma$ is a combination of $\backslash mathbf\backslash Sigma\_i$ and $\backslash mathbf\; \backslash Sigma\_e.$

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

$$\backslash begin\{cases\}$$

\nabla \cdot \left(\mathbf\Sigma_i \nabla v \right) + \nabla \cdot \left(\mathbf\Sigma_i \nabla v_e \right) = \chi \left( C_m \dfrac{\partial v}{\partial t} + I_\mathrm{ion} \right) - I_{s_1} & \mathbf x \in \mathbb H \\[1ex]

\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 \\[1ex]

\mathbf \Sigma_i(\nabla v + \nabla v_e) \cdot \mathbf n = 0 &\mathbf x \in \partial\mathbb H\\[1ex]

\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 various 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.{{cite journal| last1=Niederer|first1=S. A.| last2=Kerfoot|first2=E.| last3=Benson|first3=A. P.| last4=Bernabeu|first4=M. O.| last5=Bernus|first5=O.| last6=Bradley|first6=C.| last7=Cherry|first7=E. M.| last8=Clayton|first8=R.| last9=Fenton|first9=F. H.| last10=Garny|first10=A.| last11=Heidenreich|first11=E.| last12=Land|first12=S.| last13=Maleckar|first13=M.| last14=Pathmanathan|first14=P.|last15=Plank|first15=G.| last16=Rodriguez|first16=J. F.| last17=Roy|first17=I.| last18=Sachse|first18=F. B.| last19=Seemann|first19=G.| last20=Skavhaug|first20=O.| last21=Smith|first21=N. P.| title=Verification of cardiac tissue electrophysiology simulators using an N-version benchmark| journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences| date=3 October 2011| volume=369| issue=1954| pages=4331–4351| doi=10.1098/rsta.2011.0139| pmid=21969679| pmc=3263775| bibcode = 2011RSPTA.369.4331N }}{{cite journal| last1=Pathmanathan|first1=Pras| last2=Bernabeu|first2=Miguel O.| last3=Bordas|first3=Rafel| last4=Cooper|first4=Jonathan| last5=Garny|first5=Alan| last6=Pitt-Francis|first6=Joe M.| last7=Whiteley|first7=Jonathan P.| last8=Gavaghan|first8=David J.| title=A numerical guide to the solution of the bidomain equations of cardiac electrophysiology| journal=Progress in Biophysics and Molecular Biology| volume=102| issue=2–3| pages=136–155|doi=10.1016/j.pbiomolbio.2010.05.006| pmid=20553747|year=2010| url=https://ora.ox.ac.uk/objects/uuid:8a46ee8e-3c5f-491b-b212-c44fa0305d5e}}

• Monodomain model
• Forward problem of electrocardiology

{{reflist}}

• [http://www.scholarpedia.org/article/The_bidomain_model Scholarpedia article about the bidomain model]

Category:Cardiac electrophysiology

Category:Electrophysiology

Category:Partial differential equations

Category:Mathematical modeling

Category:Numerical analysis