Modeling Cell Movement and Chemotaxis Using Pseudopod-Based Feedback

Abstract

A computational framework is presented for the simulation of eukaryotic cell migration and chemotaxis. An empirical pattern formation model, based on a system of non-linear reaction-diffusion equations, is approximated on an evolving cell boundary using an Arbitrary Lagrangian Eulerian surface finite element method (ALE-SFEM). The solution state is used to drive a mechanical model of the protrusive and retractive forces exerted on the cell boundary. Movement of the cell is achieved using a level set method. Results are presented for cell migration with and without chemotaxis. The simulated behaviour is compared with experimental results of migrating Dictyostelium discoideum cells.

A Fully Discrete Evolving Surface Finite Element Method

“…Partial differential equations on moving surfaces appear in a wide range of applications such as material science [3,7], fluid flow [10,12], and biology and biophysics [2,8,14,16]. Numerical approaches include level set methods, surface finite elements, finite volume methods, and diffuse interface methods; see [1,4,5,9,13,15,17] and the references cited therein.…”

Particle methods to solve modelling problems in wound healing and tumor growth

“…Similar modelling principles can be found in, among others, [4,7,20,37,41,49]. Cellular automata models, which form an important and interesting class of modelling approaches, see for instance [19,34], will not be dealt with in the present paper.…”

“…The modelling of the cell through a domain with obstacles is very interesting and in our own modelling approaches, we also have some preliminary results where cells even have to adjust their geometry in order to go through an obstacle. Another cell-shape evolutionary model is described in [36,37], where a set of reaction-transport equations is solved over the cell boundary using an ALE formulation within a finite-element method with linear elements. The actual movement of the cell boundary is determined using a level-set method.…”

“…Further, our method does not require the discretisation of the partial differential equations for the level-set method over the entire domain of computation. Further, no partial differential equations have to be solved over the cell boundary in our methods opposed to the studies in [36,37]. The model in [47] treats migration of cells in larger quantities with applications to fibroblasts.…”

“…An example of a system of ordinary differential equations is given in [55] where angiogenesis in the context of solid cancer growth is modelled; -Compartmental level, where one models the dynamics in several components of one cell, this approach could be partial differential equations based where one models diffusion over or through cell membranes, such as in [32]; -Cellular-Based Models -Cell-shape evolutionary models, where one models cell deformation and migration or other processes of one single cell or at most a low number of cells, here one uses either partial differential equations, large coupled systems of ordinary differential equations, or cellular automata models (e.g. Cellular Potts Modelling), see for instance the work in [4,33,34,37,48]; -Constant cell-shape models, where one models a large number of cells for migration and sometimes including differentiation, division and death, using (stochastic) ordinary differential equations sometimes combined with partial differential equations for the distribution of chemicals interacting with the cells. Here also cellular automata models are used.…”