en>Luismanu: /* d-Partitions */

2014-01-22T05:13:11Z

d-Partitions

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{{unreferenced|date=October 2013}}
The '''monodomain model''' is a reduction of the [[bidomain model]] of the electrical propagation in myocardial tissue.
The reduction comes from assuming that the intra- and extracellular domains have equal anisotropy ratios.
Although not as physiologically accurate as the [[Bidomain|bidomain model]], it is still adequate in some cases, and has reduced complexity.

==Formulation==
The monodomain model can be formulated as follows
: <math>
\frac{\lambda}{1+\lambda} \nabla \cdot \left(\mathbf\Sigma_i \nabla v \right) = \chi \left( C_m \frac{\partial v}{\partial t} + I_\text{ion} \right)
,</math>
where <math>\mathbf\Sigma_i</math> is the intracellular conductivity tensor, <math>v</math> is the transmembrane potential, <math>I_\text{ion}</math> is the transmembrane ionic current per unit area, <math>C_m</math> is the membrane conductivity per unit area, <math>\lambda</math> is the intra- to extracellular conductivity ratio, and <math>\chi</math> is the membrane surface to volume ratio.

==Derivation==
The [[Bidomain|bidomain model]] model can be written as
: <math>
\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_\text{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}
</math>

Assuming equal anisotropy ratios, i.e. <math>\mathbf\Sigma_e = \lambda\mathbf\Sigma_i</math>, the second equation can be written
: <math>
\nabla \cdot \left(\mathbf\Sigma_i\nabla v_e\right) = -\frac{1}{1+\lambda}\nabla\cdot\left(\mathbf\Sigma_i\nabla v\right)
.</math>

Inserting this into the first bidomain equation gives
: <math>
\frac{\lambda}{1+\lambda} \nabla \cdot \left(\mathbf\Sigma_i \nabla v \right) = \chi \left( C_m \frac{\partial v}{\partial t} + I_\text{ion} \right)
.</math>

==References==
{{reflist}}

[[Category:Cardiac electrophysiology]]
{{Applied-math-stub}}

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en>Luismanu: /* d-Partitions */