Mathematical modeling of mass transfer in microvascular wall and interstitial space

A one-dimensional, unsteady-state mathematical model was developed to describe the transfer of macromolecules across a microvascular wall and into the interstitial space. The proposed theoretical model accounts for both molecular diffusion and convective transfer through the microvascular wall as well as in the interstitial space. The resulting partial differential equations were simultaneously solved using the Laplace transform method. The inversion of the Laplace transformed equations was obtained by using contour integration in the complex region. The final solution is represented by two equations expressing the macromolecule concentration in the microvascular wall region and in the interstitial space, respectively, as functions of time, spatial coordinate, macromolecule concentration in the microvascular wall at the plasma-wall interface, wall thickness, wall-interstitial space equilibrium constant for the macromolecules, ratio of the cross-sectional area of the two regions, sieving coefficients, diffusivity coefficients, and average fluid velocity terms in the two regions. Plots of the macromolecule concentration in both regions as a function of time are presented and discussed for selected values of the parameters. An analytical expression for the total amount of mass which has accumulated in a portion of the interstitial space at any given time was also derived and used to determine the average fluid velocity term and the diffusivity coefficient for each of the two regions from published experimental data (A. Y. Bekker, A. B. Ritter, and W. N. Durán, 1989, Microvasc. Res. 34, 200-216). A numerical nonlinear regression method was used for this purpose. The values for the diffusivity coefficients found in this work for this particular data set compare favorably with the results previously obtained by other workers in similar systems. It is expected that our model will be used in the future to describe the dynamics of mass transfer across a microvascular wall and into the interstitial space, on the basis of the molecular diffusion and/or convective transport mechanisms, thus contributing to the solution of the controversy regarding the nature of the transfer mechanism controlling macromolecule transport in living systems.