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Analysis, Imaging and Visualization in Physiological Systems

James B. Bassingthwaighte
Center for Bioengmeering, WD-12, Univ. of Washington, Seattle, WA 98195 USA

Richard B. King
Center for Bioengineering, WD-12, Univ. of Washington, Seattle, WA 98195, USA

Keith Kroll
Center for Bioengineering, WD-12, Univ. of Washington, Seattle, WA 98195 USA


Beginning with Christian Bohr's [9] demonstration that the concentration of a solute escaping from a long tube should diminish exponentially as a function of distance from the entry to the tube, there has been a continuing development of mathematical models of the exchange processes between the blood in a capillary and the fluid in the surrounding tissue. Early models used compartmental analysis with one compartment representing the blood (or plasma) space and a second representing all the extravascular tissue space, e.g., Sapirstein [22]. Through the years, the modeling of blood-tissue exchange has evolved in several aspects: inclusion of additional anatomical regions (e.g., endothelial and parenchymal cells), distributed models that can exhibit axial (i.e., arterial to venous) concentration gradients, and improved analytical and numerical techniques that increase both accuracy and computational speed.
Sangren and Sheppard [21] first gave the analytic solution for a blood-tissue exchange model containing two regions, capillary and extravascular tissue, separated by a permeable membrane. The solution assumed that axial diffusion in both regions was zero and that radial diffusion was infinitely fast. The return flux from the extravascular region to the capillary was taken into account; thus mass was conserved. Rose, Goresky and Bach [20] developed a solution for the analogous model that included the parenchymal cell and its membrane, a two barrier three region model. This model was extended by Kuikka et al. [15], adding axial diffusion, and by Lumsden and Silverman [17], adding terms for the consumption of the solute in all three regions.

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