A mathematical theory of capillary exchange as a function of tissue structure

Capillary net and injection modeling

A former work proved that the physiological Krogh model is valid and can be used as a tool by physicians and biologists in order to simulate drug injections. Here, after a short critical review, a more complete model describing the capillary net is established. A numerical method which can be used to solve the associate mathematical model is explained. Then, the results are discussed and used to explain why the Krogh model is valid despite its simplifying hypotheses. Diffusion and permeability coefficients are also estimated.

Evolution of the concentration field of an antibiotic during an injection

A complete mathematical model is presented which describes a substrate injection into a capillary and its diffusion in tissues. Numerical methods are implemented and compared, simplification is discussed and the results' behaviour is studied. A new experimental technique yields curves which are compared to the numerical solutions of the model and used for estimating diffusion and permeability coefficients. Furthermore it is proved that the physiological Krogh model is valid and can be used as a tool by physicians and biologists.

Transcapillary exchange in the working left ventricle of the dog

Sets of multiple indicator dilution curves were obtained from working hearts of dogs with closed chests. A multiple capillary adaptation of a permeability-limited capillary model which assumes that exchanging material returns to the capillary at its site of escape was tested on these curves. The model has only two lumped parameters: a permeability surface product per accessible extravascular volume, and flow per extravascular volume. Labeled red cells and albumin were used as vascular indicators. The model provided close fits for the curves of diffusion-limited indicators (sucrose, inulin, sulfate, sodium, chloride, and urea), and unique values were obtained for both parameters. The accessible extravascular volume obtained by this method was independent of flow whereas the permeability surface product per accessible extravascular volume (a relatively low value for these indicators) increased with flow. In this first group the outflow patterns varied with the size of the test molecule. For a second group of substances (water, ethanol and antipyrine), the outflow patterns were virtually identical and independent of molecular size (i.e., flow-limited). The modeling did not provide an appropriate description of these curves, and analysis indicated that the exchanging indicator may intercommunicate between capillaries in a random fashion, i.e., that the indicator may not return to each capillary at the same site at which it escaped.