Higher Dimensional Coupling and Propagation

Cardiac Tissue

The Bidomain Model:

At each point of the cardiac domain there are two comingled regions, the extracellular and the intracellular domains with potentials $\phi_e$ and $\phi_i$, and transmembrane potential

$$\phi = \phi_i - \phi_e.$$

These potentials drive currents,

$$i_e = - \sigma_e\nabla\phi_e, \qquad i_i = - \sigma_i\nabla\phi_i\nonumber$$

where $\sigma_i$ and $\sigma_e$ are conductivity tensors.

Total current is

$$i_T = i_e + i_i = - \sigma_e\nabla\phi_e - \sigma_i\nabla\phi_i.$$

Kirchhoff's laws:

• Total current is conserved
  
  $$\nabla\cdoti_T=\nabla\cdot (\sigma_i\nabla\phi_i + \sigma_e \nabla\phi_e) = 0$$
  
  

• transmembrane current is balanced
  
  
  $$\chi (C_m{\partial\phi \over \partial \tau} + I_m ) = \nabla\cdot (\sigma_i\nabla\phi_i)$$
  
  

Individual terms:

• $C_m{\partial\phi \over \partial \tau}$ capacitive current
• $I_m$ transmembrane ionic current
• $\chi$ surface to volume ratio
• $\nabla\cdot( \sigma_i\nabla\phi_i)$ outward intracellular current

Boundary conditions:

$${\rm n}\cdot\sigma_i \nabla\phi_i = 0,\hspace{0.5in} {\rm n}\cdot\sigma_e\nabla\phi_e = I(t,x)$$

for $x \in \partial\Omega$, the boundary of the cardiac domain, and n is the outward unit normal to the boundary, and zero net current

$$\int_{\partial\Omega} I(t,x) dS = 0.$$