On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport

We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of matrix-degenerative enzyme (MDE) and extracellular matrix (ECM), together with chemotaxis, haptotaxis, apoptosis, nutrient distribution, and cell-to-matrix adhesion. We provide a rigorous proof of the existence of solutions of the coupled system with gradient-based and adhesion-based haptotaxis effects. In addition, we discuss finite element discretizations of the model, and we present the results of numerical experiments designed to show the relative importance and roles of various effects, including cell mobility, proliferation, necrosis, hypoxia, and nutrient concentration on the generation of MDEs and the degradation of the ECM.

Download Full-text

Convergence of solutions of a non-local phase-field system

Discrete & Continuous Dynamical Systems - S ◽

10.3934/dcdss.2011.4.653 ◽

2011 ◽

Vol 4 (3) ◽

pp. 653-670 ◽

Cited By ~ 6

Author(s):

Stig-Olof Londen ◽

◽

Hana Petzeltová ◽

Keyword(s):

Phase Field ◽

Local Phase ◽

Convergence Of Solutions ◽

Field System ◽

Non Local ◽

Phase Field System

Download Full-text

Phase-field modelling and computing for a large number of phases

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2018075 ◽

2019 ◽

Vol 53 (3) ◽

pp. 805-832

Author(s):

Élie Bretin ◽

Roland Denis ◽

Jacques-Olivier Lachaud ◽

Édouard Oudet

Keyword(s):

Phase Field ◽

Field Model ◽

Phase Field Model ◽

Field Representation ◽

High Resolution Data ◽

Phase Field Models ◽

Efficient Storage ◽

Non Local ◽

Discretization Point ◽

Field Approaches

We propose a framework to represent a partition that evolves under mean curvature flows and volume constraints. Its principle follows a phase-field representation for each region of the partition, as well as classical Allen–Cahn equations for its evolution. We focus on the evolution and on the optimization of problems involving high resolution data with many regions in the partition. In this context, standard phase-field approaches require a lot of memory (one image per region) and computation timings increase at least as fast as the number of regions. We propose a more efficient storage strategy with a dedicated multi-image representation that retains only significant phase field values at each discretization point. We show that this strategy alone is unfortunately inefficient with classical phase field models. This is due to non local terms and low convergence rate. We therefore introduce and analyze an improved phase field model that localizes each phase field around its associated region, and which fully benefits of our storage strategy. To demonstrate the efficiency of the new multiphase field framework, we apply it to the famous 3D honeycomb problem and the conjecture of Weaire–Phelan’s tiling.

Download Full-text