Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1D vascular network

Abstract

In this work, we present and analyze a mathematical model for tumor growth incorporating ECM erosion, interstitial flow, and the effect of vascular flow and nutrient transport. The model is of phasefield or diffused-interface type in which multiple phases of cell species and other constituents are separated by smooth evolving interfaces. The model involves a mesoscale version of Darcy's law to capture the flow mechanism in the tissue matrix. Modeling flow and transport processes in the vasculature supplying the healthy and cancerous tissue, one-dimensional (1D) equations are considered. Since the models governing the transport and flow processes are defined together with cell species models on a three-dimensional (3D) domain, we obtain a 3D-1D coupled model. We show some mathematical results on the existence of weak solutions. Furthermore, simulation results are presented illustrating the evolution of tumors and the effects of ECM erosion.

Modeling and simulation of vascular tumors embedded in evolving capillary networks

“…Here J pv and J σv are the fluid flux and nutrient flux as described in Subsection 2.3. We consider the following choices of the coupling source functions; see [19],…”

“…We consider a class of 3D-1D vascular tumor models [19] that approximates the flow within the blood vessels by one-dimensional flow based on the Poiseuille law; effectively reducing the flow in the threedimensional vessels to the flow in a network of one-dimensional vessel segments. While coupling the flow in the vessels and tissue, the blood vessels' three-dimensional nature is retained by approximating the vessels as a network of straight-cylinders and applying the fluid exchange at the walls of cylindrical segments.…”

“…From a mathematical and computational point of view, a complicating factor is the use of one-dimensional characterizations of vessel segments in the vascular network embedded in three-dimensional domains of the tissue and the tumor within the tissue and the assignment of mechanical models to this 3D-1D system to depict interstitial flow and pressure fluctuations. Mathematical analysis showing well-posedness and existence of weak solutions for the class of 3D-1D model considered in this work is performed in a recent paper [19].…”

“…Especially, we note the comprehensive developments of diffuse-interface multispecies models presented in [18,62], the ten species models derived in [33], and the multispecies nonlocal models of adhesion and promotes a tumor invasion due to ECM degradation described in [20]. Angiogenesis models embedded in models of hypoxic and cell growth are presented in [19,33,64,65]. Related models of extracellular matrix (ECM) degradation due to matrix-degenerative enzymes (MDEs) released by hypoxic cell concentrations and subsequent tumor invasion and metastasis are discussed in [10,11,16,20].…”