A convergent explicit finite difference scheme for a mechanical model for tumor growth

Abstract

Abstract. Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on imaging analysis. This work deals with the numerical approximation of a mechanical model for tumor growth and the analysis of its dynamics. The system under investigation is given by a multi-phase flow model: The densities of the different cells are governed by a transport equation for the evolution of tumor cells, whereas the velocity field is given by a Brinkman regularization of the classical Darcy's law. An efficient finite difference scheme is proposed and shown to converge to a weak solution of the system. Our approach relies on convergence and compactness arguments in the spirit of Lions [23].

Analysis and Simulations on a Model for the Evolution of Tumors Under the Influence of Nutrient and Drug Application

“…Proof. The proof of this lemma is similar to the proof of Lemma 3.2 in [16]. All that needs to be done is replacing the function G on the right hand side of the equation by Φ and checking that it satisfies the right growth conditions.…”

“…Related work on the mathematical analysis of mechanical models of Hele-Shaw-type have been presented by Perthame et al [14,15]. In [16], Trivisa and Weber presented a convergent explicit finite difference scheme for the numerical approximation of a Hele-Shawtype system for the evolution of cancerous cells and presents numerical observations in two space dimensions. The work [16] is according to our knowledge the first article that presents rigorous analytical results on the global existence of general weak solutions to Hele-Shaw-type systems.…”

“…In [16], Trivisa and Weber presented a convergent explicit finite difference scheme for the numerical approximation of a Hele-Shawtype system for the evolution of cancerous cells and presents numerical observations in two space dimensions. The work [16] is according to our knowledge the first article that presents rigorous analytical results on the global existence of general weak solutions to Hele-Shaw-type systems. The present article extends the analysis in [16] significantly by investigating the delicate interplay of nutrient and drug application in the treatment of cancer.…”

“…The work [16] is according to our knowledge the first article that presents rigorous analytical results on the global existence of general weak solutions to Hele-Shaw-type systems. The present article extends the analysis in [16] significantly by investigating the delicate interplay of nutrient and drug application in the treatment of cancer. We refer the reader to [7,8,4] where a class of relevant tumor growth models with nutrient and drug application are presented and to the manuscript [1] which provides an overview of mathematical methods and tools for modeling cancer phenomena.…”

“…Next we recall Lemma 3.3 from [16] which gives us higher order regularity estimates for W ε . Lemma 3.7.…”