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Biology Articles » Biomathematics » A mathematical model for electrical stimulation of a monolayer of cardiac cells » Methods

Methods
- A mathematical model for electrical stimulation of a monolayer of cardiac cells

The equation governing a two-dimensional sheet of tissue perfused by a three-dimensional bath is similar to the equation derived by Rattay for a one-dimensional nerve axon [13]. The tissue obeys the conservation of current,

where β is the ratio of membrane surface area to tissue volume, Jm is the membrane current density, and Jix and Jiy represent x and y components of the intracellular current density. Ohm's Law gives

Jm = Gm Vm,     (4)

where Vi is the intracellular potential, Vm is the transmembrane potential, Gm is membrane conductivity per unit area, and σix and σiy are the intracellular conductivities parallel to (x) and perpendicular to (y) the fiber axis. Substituting Eqs. (2), (3), and (4) into Eq. (1), and letting Vi = Vm + Ve, we find that

where λx and λy are defined as

In order to solve Eq. (5), we must first determine the extracellular potential, Ve. We assume that Ve is from a point electrode in an infinite, homogeneous bath

where , σe is the conductivity of the bath, I is the stimulus current, and d is the distance from the tissue sheet (z = 0) to the electrode.

We discretize Eq. (5) using a finite difference formula, and solve it using a relaxation method. The number of nodes in each direction is 100, the space step is 0.2 mm (implying a tissue size of 20 × 20 mm), σe is 1 S/m, I is 1 mA, and d is 1 mm.


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