A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues. I. Theory.

In order to better understand the mechanisms governing transport of drugs, nanoparticle-based treatments, and therapeutic biomolecules, and the role of the various physiological parameters, a number of mathematical models have previously been proposed. The limitations of the existing transport models indicate the need for a comprehensive model that includes transport in the vessel lumen, the vessel wall, and the interstitial space and considers the effects of the solute concentration on fluid flow. In this study, a general model to describe the transient distribution of fluid and multiple solutes at the microvascular level was developed using mixture theory. The model captures the experimentally observed dependence of the hydraulic permeability coefficient of the capillary wall on the concentration of solutes present in the capillary wall and the surrounding tissue. Additionally, the model demonstrates that transport phenomena across the capillary wall and in the interstitium are related to the solute concentration as well as the hydrostatic pressure. The model is used in a companion paper to examine fluid and solute transport for the simplified case of an axisymmetric geometry with no solid deformation or interconversion of mass.

A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues. II: Factor sensitivity analysis, calibration, and validation.

The treatment of cancerous tumors is dependent upon the delivery of therapeutics through the blood by means of the microcirculation. Differences in the vasculature of normal and malignant tissues have been recognized, but it is not fully understood how these differences affect transport and the applicability of existing mathematical models has been questioned at the microscale due to the complex rheology of blood and fluid exchange with the tissue. In addition to determining an appropriate set of governing equations it is necessary to specify appropriate model parameters based on physiological data. To this end, a two stage sensitivity analysis is described which makes it possible to determine the set of parameters most important to the model's calibration. In the first stage, the fluid flow equations are examined and a sensitivity analysis is used to evaluate the importance of 11 different model parameters. Of these, only four substantially influence the intravascular axial flow providing a tractable set that could be calibrated using red blood cell velocity data from the literature. The second stage also utilizes a sensitivity analysis to evaluate the importance of 14 model parameters on extravascular flux. Of these, six exhibit high sensitivity and are integrated into the model calibration using a response surface methodology and experimental intra- and extravascular accumulation data from the literature (Dreher et al. in J Natl Cancer Inst 98(5):335-344, 2006). The model exhibits good agreement with the experimental results for both the mean extravascular concentration and the penetration depth as a function of time for inert dextran over a wide range of molecular weights.