Monodomain model

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 model, it is still adequate in some cases, and has reduced complexity. ^[1]

Formulation

Being ${\displaystyle \mathbb {T} }$ the domain boundary of the model, the monodomain model can be formulated as follows ^[2]

$${\displaystyle {\frac {\lambda }{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right)\quad \quad {\text{in }}\mathbb {T} ,}$$

where ${\displaystyle \mathbf {\Sigma } _{i}}$ is the intracellular conductivity tensor, ${\displaystyle v}$ is the transmembrane potential, ${\displaystyle I_{\text{ion}}}$ is the transmembrane ionic current per unit area, ${\displaystyle C_{m}}$ is the membrane capacitance per unit area, ${\displaystyle \lambda }$ is the intra- to extracellular conductivity ratio, and ${\displaystyle \chi }$ is the membrane surface area per unit volume (of tissue). ^[1]

Derivation

The monodomain model can be easily derived from the bidomain model. This last one can be written as ^[1]

$${\displaystyle {\begin{aligned}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)&=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right)\\\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\left(\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e}\right)\nabla v_{e}\right)&=0\end{aligned}}}$$

Assuming equal anisotropy ratios, i.e. ${\displaystyle \mathbf {\Sigma } _{e}=\lambda \mathbf {\Sigma } _{i}}$, the second equation can be written as ^[1]

$${\displaystyle \nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=-{\frac {1}{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right).}$$

Then, inserting this into the first bidomain equation gives the unique equation of the monodomain model ^[1]

$${\displaystyle {\frac {\lambda }{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right).}$$

Boundary conditions

Differently from the bidomain model, usually the monodomain model is equipped with an isoltad boundary condition, which means that it is assumed that there is not current that can flow from or to the domain (usually the heart). ^{[3] [4]} Mathematically, this is done imposing a zero transmembrane potential flux, i.e.: ^[4]

${\displaystyle (\mathbf {\Sigma } _{i}\nabla v)\cdot \mathbf {n} =0\quad \quad {\text{on }}\partial \mathbb {T} }$

where ${\displaystyle \mathbf {n} }$ is the unit outward normal of the domain and ${\displaystyle \partial \mathbb {T} }$ is the domain boundary.

See also

• Bidomain model
• Forward problem of electrocardiology

References

1. Pullan, Andrew J.; Buist, Martin L.; Cheng, Leo K. (2005). Mathematically modelling the electrical activity of the heart : from cell to body surface and back again. World Scientific. ISBN   978-9812563736.
2. Keener J, Sneyd J (2009). Mathematical Physiology II: Systems Physiology (2nd ed.). Springer. ISBN   978-0-387-79387-0.
3. Rossi, Simone; Griffith, Boyce E. (1 September 2017). "Incorporating inductances in tissue-scale models of cardiac electrophysiology". Chaos: An Interdisciplinary Journal of Nonlinear Science. 27 (9): 093926. doi:10.1063/1.5000706. ISSN   1054-1500. PMC   5585078 . PMID   28964127.
4. Boulakia, Muriel; Cazeau, Serge; Fernández, Miguel A.; Gerbeau, Jean-Frédéric; Zemzemi, Nejib (24 December 2009). "Mathematical Modeling of Electrocardiograms: A Numerical Study" (PDF). Annals of Biomedical Engineering. 38 (3): 1071–1097. doi:10.1007/s10439-009-9873-0. PMID   20033779. S2CID   10114284.