Numerical Methods for Two-Dimensional Stem Cell Tissue Growth

Abstract

Growth of developing and regenerative biological tissues of different cell types is usually driven by stem cells and their local environment. Here, we present a computational framework for continuum tissue growth models consisting of stem cells, cell lineages, and diffusive molecules that regulate proliferation and differentiation through feedback. To deal with the moving boundaries of the models in both open geometries and closed geometries (through polar coordinates) in two dimensions, we transform the dynamic domains and governing equations to fixed domains, followed by solving for the transformation functions to track the interface explicitly. Clustering grid points in local regions for better efficiency and accuracy can be achieved by appropriate choices of the transformation. The equations resulting from the incompressibility of the tissue is approximated by high-order finite difference schemes and is solved using the multigrid algorithms. The numerical tests demonstrate an overall spatiotemporal second-order accuracy of the methods and their capability in capturing large deformations of the tissue boundaries. The methods are applied to two biological systems: stratified epithelia for studying the effects of two different types of stem cell niches and the scaling of a morphogen gradient with the size of the Drosophila imaginal wing disc during growth. Direct simulations of both systems suggest that that the computational framework is robust and accurate, and it can incorporate various biological processes critical to stem cell dynamics and tissue growth.

Stochastic dynamics of cell lineage in tissue homeostasis

“…To solve the cell lineage equations Eq. (2, 3, 4, 7), we first transform the spatial domain [0, z max (t)] to a unit domain [0, 1] by a transformation [34]:…”

“…A one-dimensional continuum model [7] shows that the spatial morphogen gradient contributes to the tissue stratification. Studies of two-dimensional models suggest sources for distorted tissue morphologies [34,35]. Through studying a discrete cell model, the selective cell adhesion is found to be a key factor for layer formation and tissue stratification [9].…”

Effects of Matrix Metalloproteinase on Tumour Growth and Morphology via Haptotaxis

“…In a continuum model, the cell species velocity is obtained from the inertia less momentum conservation equation based on Darcy's law, representing the instantaneous equilibrium among forces associated with pressure, cell adhesion, elastic forces, forces exchanged with the ECM leading to haptotaxis and chemo taxis due to gradients of nutrient and other growth factors, and other mechanical effects. The continuum models incorporate the adhesion among cancer cells by a surface tension at the tumour surface [20][21][22]24,37,41,[43][44][45][46][47][48][49][50]. Surface tension also models cell-ECM adhesion and the presence of a membrane, or capsule, formed of ECM macromolecules that may encapsulate tumours in vitro and in vivo.…”

Spheroid Formation of Hepatocarcinoma Cells in Microwells: Experiments and Monte Carlo Simulations

“…The available generic 2D and 3D MC simulations have been focused on the growth and differentiation of stem cells [ 20 ], cell seeding [ 21 ], and formation of cell sheets [ 4 ]. Related theoretical studies concern stem-cell niches [ 22 – 25 ] and scaffold-less biofabrication [ 26 ].…”