Modeling biological tissue growth: Discrete to continuum representations

Abstract

There is much interest in building deterministic continuum models from discrete agent-based models governed by local stochastic rules where an agent represents a biological cell. In developmental biology, cells are able to move and undergo cell division on and within growing tissues. A growing tissue is itself made up of cells which undergo cell division, thereby providing a significant transport mechanism for other cells within it. We develop a discrete agent-based model where domain agents represent tissue cells. Each agent has the ability to undergo a proliferation event whereby an additional domain agent is incorporated into the lattice. If a probability distribution describes the waiting times between proliferation events for an individual agent, then the total length of the domain is a random variable. The average behavior of these stochastically proliferating agents defining the growing lattice is determined in terms of a Fokker-Planck equation, with an advection and diffusion term. The diffusion term differs from the one obtained Landman and Binder [J. Theor. Biol. 259, 541 (2009)] when the rate of growth of the domain is specified, but the choice of agents is random. This discrepancy is reconciled by determining a discrete-time master equation for this process and an associated asymmetric nonexclusion random walk, together with consideration of synchronous and asynchronous updating schemes. All theoretical results are confirmed with numerical simulations. This study furthers our understanding of the relationship between agent-based rules, their implementation, and their associated partial differential equations. Since tissue growth is a significant cellular transport mechanism during embryonic growth, it is important to use the correct partial differential equation description when combining with other cellular functions.

Spatial and temporal dynamics of cell generations within an invasion wave: A link to cell lineage tracing

“…Only the non-growing case has been discussed here. However, both CA and PDE models have been used to simulate ENC migration and proliferation on a growing domain (Binder et al, 2008;Cheeseman et al, 2014;Hywood et al, 2013). In addition, we have verified that the GDGW process and estimation technique for q i (t) remain valid for the growing domain case.…”

Celebrating Soft Matter's 10th Anniversary: Cell division: a source of active stress in cellular monolayers

“…There are many models for collective cell migration in the literature [21][22][23][24][25] . The cultures of cells are often regarded as a continuous medium due to strong contacts between the cells 15,16 .…”

Inference of cell–cell interactions from population density characteristics and cell trajectories on static and growing domains

“…This growth mechanism, which we shall refer to as a 'pushing' mechanism, has been successfully used in modelling avian gut growth [19], and to compare the accuracy of different stochastic update schemes for discrete models [29,32]. Our pushing growth mechanism is as follows: at each time step one site is randomly selected from each row of the domain to undergo growth.…”

“…We choose linear domain growth as it has been shown to be present in many biological systems [2,19,33,34]. However, this growth mechanism is equally applicable to other types of isotropic domain growth, such as exponential growth [29]. If site (i, j) is selected for growth, the new site is inserted at (i, j), and the 'selected' site is moved to (i, j + 1), taking its contents (i.e.…”

“…Importantly, we derive continuum equivalents of our discrete models for the spatial and temporal evolution of the cell population density, and the expected displacement of tagged cells on both growing and static domains [21][22][23]29]. These continuum approximations in many cases are extensions of previously derived continuum models [21][22][23].…”