Mechanistic modeling of vascular tumor growth: an extension of Biot’s theory to hierarchical bi-compartment porous medium systems

Abstract

Existing continuum multiphase tumor growth models typically do not include microvasculature, or if present, this is modeled as a non-deformable network of vessels. Vasculature behavior and blood flow are usually non-coupled with the underlying tumor phenomenology from the mechanical viewpoint; hence, phenomena like vessel compression/occlusion modifying microcirculation and oxygen supply cannot be taken into account. Here, the tumor tissue is modeled as a reactive bi-compartment porous medium: the extracellular matrix constitutes the solid scaffold; blood flows in the vascular porosity, whereas the extravascular porous compartment is saturated by two cell phases and interstitial fluid (mixture of water and nutrient species). The pressure difference between blood and the extravascular overall pressure is sustained by vessel walls and drives shrinkage or dilatation of the vascular porosity. Model closure is achieved thanks to a consistent non-conventional definition of the Biot’s effective stress tensor. Angiogenesis is modeled by introducing a vascularization state variable and accounting for tumor angiogenic factors and endothelial cells. Closure relationships and mass exchange terms related to vessel formation are detailed in a numerical example reproducing the principal features of angiogenesis. This example is preceded by a first pedagogical numerical study on one-dimensional bio-consolidation. Results demonstrate that the bi-compartment poromechanical model is fully coupled (the external loads impact fluid flow in both porous compartments) and that it can serve as a basis for further applications like modeling of drug delivery and tissue ulceration.