## Bidomain Properties at the Tissue Level

Perhaps the most challenging aspect of setting up the bidomain model is determining the components of the conductivity tensors associated with the intra- and extracellular spaces that are consistent with the underlying structure. Often the cellular structure is either not precisely known or it cannot be determined a priori. In some cases, the properties are determined by making a series of measurements on the preparation and interpreting the data using the bidomain model. More often the properties are simply assumed using literature values from measurements from simpler preparations or from other modeling studies.20

Intuitively, the conductivity tensors, Di and De, are related to the geometry, coupling, orientation, and degree of packing of cells in the tissue. The tensors are symmetric and positive definite matrices where the three eigenvalues of the tensor are associated with the average electrical properties in the directions given by the three orthogonal eigenvectors.

We can obtain a simple estimate for the magnitude of the components of the tensor by assuming cardiac tissue is a lattice of coupled, identical rectangular parallelepiped cells (Fig. 1a). We assume that each unit cell is isotropic with conductivity a\ surrounded by a uniform layer of fluid with conductivity ae, and connected to six neighbors through small tubules on the end and lateral faces. Although clearly an idealization, this cell arrangement gives rise to reasonable estimates for tissue conductivities for reasonable cell dimensions and gap resistances.

As shown in Fig. 2, we can imagine each unit cell has a width, dcell, height dceii, and length Zcen such that the total cell volume is:

Vol cell = dcell X dcell X Zcell.

Figure 2: Dimension of a unit cell: width, dcell; length, lcell, and the thickness of the extracellular fluid,Ae. The unit cell has a total intracellular resistance (cytoplasmic and gap junction) along the cell, R;l = Rcl + Rj, and across the cell, Rit = Rct + Rj

The fluid that surrounds each cell is relatively small and has a thickness of Ae such that the total volume occupied by the cell and extracellular fluid is

where

"tot = "cell + 2Ae; ¿tot = ¿cell + 2 Ae •

The fraction of intracellular space is given by:

Volt

Because Ae, dcell tot ^ ¿celljtot, the intracellular fraction is usually defined as the ratio of the cross-sectional areas, namely

The extracellular fraction is f ^ "cell fi ~ "2 •

The total resistance of the cell is the sum of the resistance of the cytoplasm along the cell, Rcl, or across the cell, Rct, namely

Mo or Rc cell

and the resistance of the gap junction, Rj. If we assume a typical connexon has a conductance gj, then the total resistance of the gap junction is

where N is the number of connexons in the gap junction. For simplicity, we assume that the number of connexons is the same on the end and lateral faces. Thus, the total resistance of the cell and junction along the cell is

+ Rj cell and across the cell is

The total interstitial resistance along the cell is k

cell and across the cell is

Re dt

^e(dtotltot _ dcelllcell)

If we assume the intracellular space and the extracellular space occupy the same total volume, Vtot, then we can compute an effective conductivity (i.e., the bidomain conductivity) that would yield the same total resistance as that computed for the actual intracellular and interstitial volumes. For example, if intrinsic intracellular or interstitial resistance along the cell is Ril_el, then the effective conductivity, snjel, that would yield the same resistance in

Vtot is

1 ltot

Ril,eldtot/ ltot Ril,eldtot

Similarly, the effective conductivity across the cell is

Rit,et dtot ltot/dtot Rit,etltot

Substituting (53) and (55) into (57), we obtain ltot _ (dcel^ dto0 C

0 0