On the unsteady Darcy-Forchheimer-Brinkman equation in local and nonlocal tumor growth models

TL;DR

This paper presents a comprehensive mathematical analysis of local and nonlocal tumor growth models incorporating Darcy-Forchheimer-Brinkman equations, including existence proofs, sensitivity analysis, and numerical simulations to understand tumor dynamics.

Contribution

It introduces a complete existence analysis for coupled tumor growth models with convective velocity and long-range interactions, along with novel sensitivity analysis methods.

Findings

Sensitivity analyses yield similar conclusions on key quantities.

Numerical simulations demonstrate model behavior for tumor growth.

Parameter variations significantly affect model outputs.

Abstract

A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy-Forchheimer-Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observational data. Remarkably, the two approaches yield very similar conclusions on sensitivity for certain quantities of interest. The work concludes with the presentation of numerical approximations of solutions of the governing equations and results of numerical experiments on tumor growth produced using finite element discretizations of the full tumor model for representative cases.