| Various lumped and distributed parameter models have been proposed for the theoretical and experimental analysis of the ion transport and electrophysiological properties of epithelia. The most widely used model to date involves the application, to these systems, of two-dimensional or Bessel cable theory. Since cable theory treats spatially extended electrical systems by a continuous variable formulation, it is, in a strict sense, applicable only to systems that can be considered, physically and mathematically, to be continuous, homogeneous structures.;The second serious short coming of cable theory is that in demanding that all properties be evenly distributed or smeared throughout the model, no detailed fine structure or localized variation is permitted. This does not apply in the case of epithelia. The hallmark of an epithelium is its discontinuous, discrete, periodic structure.;Since epithelia show discrete periodic structural and functional variations, it seems preferable to develop models based on an analytical technique that is itself discrete. The calculus of finite differences was used to develop such a model, based on a finite difference analysis of the ladder network shown.;r(,c), r(,s) and r(,o) are the total transcellular, shunt and bathing fluid resistances, i(,s) and (tau)(,s) are mesh currents. V(,c) is the transepithelial P.D. Using Kirchoff's voltage law to write mesh equations for the S'th loop set, and applying forward and inverse difference operators (E, E('-1), e.g. Ei(,s) = i(,s+1), Ei(,s)('-1) = i(,s-1)) to this network one obtains the equations:;Several features of cable theory should be underscored to emphasize its limited applicability to the analysis of biological systems, particularly epithelia. First, there is no provision in the cable model for the existence of an EMF. All potentials involved in the model must be of passive origin: they are generated as potential differences across dissipative elements due to current flows induced by exteral sources of energy. Biological structures have intrinsic sources of electrical energy. Their inclusion in the cable model as discrete entities is not possible. This limitation is particularly severe for the case of the epithelium for it is necessary not only to include EMF's but to position them discretely and periodically throughout the model structure.;-(r(,c) + 2r(,o) + r) i(,s) + (r + r(,c)E('-1))(tau)(,s) = -V(,c).;(r + r(,c)E) i(,s) - (r + 2r(,o) + r(,c))(tau)(,s) = V(,c).;In applying the model to tissue studies, experimentally imposed electrical constraints provide boundary conditions. From these and the above equations numerical solutions for r(,c), r(,s) and V(,c) in terms of the experimentally established parameters I(,1), V(,1), I(,2), V(,2), and r(,o), can be obtained. The application of this model to the experimental determination of r(,c) and r(,s) in isolated sheets of Necturus gallbladder and bullfrog small intestine is discussed. |
