a b s t r a c tCell invasion involves a population of cells which are motile and proliferative. Traditional discrete models of proliferation involve agents depositing daughter agents on nearestneighbor lattice sites. Motivated by time-lapse images of cell invasion, we propose and analyze two new discrete proliferation models in the context of an exclusion process with an undirected motility mechanism. These discrete models are related to a family of reactiondiffusion equations and can be used to make predictions over a range of scales appropriate for interpreting experimental data. The new proliferation mechanisms are biologically relevant and mathematically convenient as the continuum-discrete relationship is more robust for the new proliferation mechanisms relative to traditional approaches.
A general mathematical model of cell invasion is developed and validated with an experimental system. The model incorporates two basic cell functions: non-directed (diffusive) motility and proliferation to a carrying capacity limit. The model is used here to investigate cell proliferation and motility differences along the axis of an invasion wave. Mathematical simulations yield surprising and counterintuitive predictions. In this general scenario, cells at the invasive front are proliferative and migrate into previously unoccupied tissues while those behind the front are essentially nonproliferative and do not directly migrate into unoccupied tissues. These differences are not innate to the cells, but are a function of proximity to uninvaded tissue. Therefore, proliferation at the invading front is the critical mechanism driving apparently directed invasion. An appropriate system to experimentally validate these predictions is the directional invasion and colonization of the gut by vagal neural crest cells that establish the enteric nervous system. An assay using gut organ culture with chick-quail grafting is used for this purpose. The experimental results are entirely concordant with the mathematical predictions. We conclude that proliferation at the wavefront is a key mechanism driving the invasive process. This has important implications not just for the neural crest, but for other invasion systems such as epidermal wound healing, carcinoma invasion and other developmental cell migrations.
Interpretive and predictive tools are needed to assist in the understanding of cell invasion processes. Cell invasion involves cell motility and proliferation, and is central to many biological processes including developmental morphogenesis and tumor invasion. Experimental data can be collected across a wide range of scales, from the population scale to the individual cell scale. Standard continuum or discrete models used in isolation are insufficient to capture this wide range of data. We develop a discrete cellular automata model of invasion with experimentally motivated rules. The cellular automata algorithm is applied to a narrow two-dimensional lattice and simulations reveal the formation of invasion waves moving with constant speed. The simulation results are averaged in one dimension-these data are used to identify the time history of the leading edge to characterize the population-scale wave speed. This allows the relationship between the population-scale wave speed and the cell-scale parameters to be determined. This relationship is analogous to well-known continuum results for Fisher's equation. The cellular automata algorithm also produces individual cell trajectories within the invasion wave that are analogous to cell trajectories obtained with new experimental techniques. Our approach allows both the cell-scale and population-scale properties of invasion to be predicted in a way that is consistent with multiscale experimental data. Furthermore we suggest that the cellular automata algorithm can be used in conjunction with individual data to overcome limitations associated with identifying cell motility mechanisms using continuum models alone.
Pressure filtration is an important method for removing liquids from a suspension. Previous work used linear models or applied to stable suspensions. Nonlinear models for flocculated suspensions are studied here. The equations governing the consolidation of flocculated suspensions under the influence of an applied pressure are based on the assumption that when the volume fraction is high enough, the network formed from the aggregation of floes possesses a compressive yield stress P,, (4) that is a function of local volume fraction 4 only. There are two modes of operation of the pressure filter-the fluid flux or the applied pressure is specified-and both of these are studied. The resulting nonlinear partial differential equations involve the time-dependent piston position, and in the case of the suspension being initially unnetworked, another internal moving boundary below which the suspension is networked. The small time behavior of these systems is~obtained with an asymptotic method. In general, at later times, the solution can only be found numerically and an algorithm for doing this is discussed. The important parameters and properties of the filter cake are described. The results suggest various ways of controlling the filtration process, which may be useful in the manufacture of ceramics.
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