The relation between Krogh and compartmental transport models

The meaning of compartmental concentration is intuitively simple in well-stirred compartment models with uniform concentration but not so in the general case. This meaning may be better understood by relating the general compartmental model to a spatially explicit one such as the Krogh cylinder model. We show that compartmental equations result directly by spatial averaging of the Krogh partial differential equations. Intercompartmental transport is usually stated as flux is proportional to the capillary-tissue concentration difference or $\text{J = ? (}\text{C}{}_{\mathrm{c}}\text{-}\text{C}{}_{\mathrm{T}}\text{)}$. We verify this simple equation by showing that flux in the Krogh model is also approximately first order and then derive the transport coefficient, ?, in terms of the geometry and diffusivity in the Krogh model. A relation between capillary and venous concentration or, for a gas, partial pressure is needed. For the highly diffusable, metabolic gas C0₂ this relation is $\text{P}{}_{\mathrm{c}}\text{= P}{}_{\mathrm{v}}\text{?}\text{M}\text{20K}{}_{\mathrm{c}}$ that is, capillary and venous partial pressures differ by one half the metabolic generation.