monodomain model

Being $\backslash mathbb\; T$ the spatial domain, and $T$ the final time, the monodomain model can be formulated as follows{{cite book

|vauthors=Keener J, Sneyd J

| year = 2009

| edition = 2nd

| title = Mathematical Physiology II: Systems Physiology

| publisher = Springer

| isbn = 978-0-387-79387-0

}}

\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) \quad \quad \text{in }\mathbb T \times (0, T)

,

where $\backslash mathbf\backslash Sigma\_i$ is the intracellular conductivity tensor, $v$ is the transmembrane potential, $I\_\backslash text\{ion\}$ is the transmembrane ionic current per unit area, $C\_m$ is the membrane capacitance per unit area, $\backslash lambda$ is the intra- to extracellular conductivity ratio, and $\backslash chi$ is the membrane surface area per unit volume (of tissue).

The monodomain model can be easily derived from the bidomain model. This last one can be written as

\begin{align}

\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{align}

Assuming equal anisotropy ratios, i.e. $\backslash mathbf\backslash Sigma\_e\; =\; \backslash lambda\backslash mathbf\backslash Sigma\_i$, the second equation can be written as

\nabla \cdot \left(\mathbf\Sigma_i\nabla v_e\right) = -\frac{\lambda}{1+\lambda}\nabla\cdot\left(\mathbf\Sigma_i\nabla v\right)

.

Then, inserting this into the first bidomain equation gives the unique equation of the monodomain model

\frac{1}{1+\lambda} \nabla \cdot \left(\mathbf\Sigma_i \nabla v \right) = \chi \left( C_m \frac{\partial v}{\partial t} + I_\text{ion} \right)

.

Differently from the bidomain model, the monodomain model is usually equipped with an isolated boundary condition, which means that it is assumed that there is not current that can flow from or to the domain (usually the heart).{{cite journal |last1=Rossi |first1=Simone |last2=Griffith |first2=Boyce E. |title=Incorporating inductances in tissue-scale models of cardiac electrophysiology |journal=Chaos: An Interdisciplinary Journal of Nonlinear Science |date=1 September 2017 |volume=27 |issue=9 |pages=093926 |doi=10.1063/1.5000706 |pmid=28964127 |issn=1054-1500|pmc=5585078 }}{{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 }} Mathematically, this is done imposing a zero transmembrane potential flux (homogeneous Neumann boundary condition), i.e.:

:$(\backslash mathbf\; \backslash Sigma\_i\; \backslash nabla\; v)\backslash cdot\; \backslash mathbf\; n\; =\; 0\; \backslash quad\; \backslash quad\; \backslash text\{on\; \}\backslash partial\backslash mathbb\; T\; \backslash times\; (0,\; T)$

where $\backslash mathbf\; n$ is the unit outward normal of the domain and $\backslash partial\; \backslash mathbb\; T$ is the domain boundary.

• Bidomain model
• Forward problem of electrocardiology

{{reflist}}

Category:Cardiac electrophysiology

Category:Electrophysiology

Category:Partial differential equations

Category:Biological theorems

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