A model of fluid flow in solid tumors.

Solid tumors consist of a porous interstitium and a neoplastic vasculature composed of a network of capillaries with highly permeable walls. Blood flows across the vasculature from the arterial entrance point to the venous exit point, and enters the tumor by convective and diffusive extravasation through the permeable capillary walls. In this paper, an integrated theoretical model of the flow through the tumor is developed. The flow through the interstitium is described by Darcy's law for an isotropic porous medium, the flow along the capillaries is described by Poiseuille's law, and the extravasation flux is described by Starling's law involving the pressure on either side of the capillaries. Given the arterial, the venous, and the ambient pressure, the problem is formulated in terms of a coupled system of integral and differential equations for the vascular and interstitial pressures. The overall hydrodynamics is described in terms of hydraulic conductivity coefficients for the arterial and venous flow rates whose functional form provides an explanation for the singular behavior of the vascular resistance observed in experiments. Numerical solutions are computed for an idealized case where the vasculature is modeled as a single tube, and charts of the hydraulic conductivities are presented for a broad range of tissue and capillary wall conductivities. The results in the physiological range of conditions are found to be in good agreement with laboratory observations. It is shown that the assumption of uniform interstitial pressure is not generally appropriate, and predictions of the extravasation rate based on it may carry a significant amount of error.

Kinematics of surface growth.

In this paper a general mathematical framework is developed to describe cases of fixed and moving growth surfaces. This formulation has the mathematical structure suggested by Skalak (1981), but is extended herein to include discussion of possible singularities, incompatibilities, residual stresses and moving growth surfaces. Further, the general theoretical equations necessary for the computation of the final form of a structure from the distribution of growth velocities on a growth surface are presented and applied in a number of examples. It is shown that although assuming growth is always in a direction normal to the current growth surface is generally sufficient, growth at an angle to the growth surface may represent the biological reality more fully in some respects. From a theoretical viewpoint, growth at an angle to a growth surface is necessary in some situations to avoid postulating singularities in the growth velocity field. Examples of growth on fixed and moving surfaces are developed to simulate the generation of horns, seashells, antlers, teeth and similar biological structures.