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Coarse graining in simulated growing cell populations
Interdisciplinary Centre for Bioinformatics
University of Leipzig
Michael Block, Eckehard Schöll
Institute for Theoretical Physics
Technical University of Berlin
Dept. of Mathematics
University of Nottingham
Hospital Paul Brousse
University Pierre and Marie Curie
Background: The phenotype of cell populations growing in-vitro can be very different and range from sparsely distributed to dense, and from monolayer to multi-layers. Typical cell population sizes of a single cell patch are ~100000 cells. Some cell lines show extensive migration activities. Multi-cellular spheroids can have up to ~500000 cells, in-vivo Xenografts and solid tumors about 109-1011 cells. Large cell population sizes of ~106 cells or more, cell populations that show extensive migration activity, the systematic sensitivity analyses of the effect of migration, cell-cell, cell-substrate adhesion, or other parameters, or the consideration of cell-internal degrees of freedom (such as molecular regulation and differentiation mechanisms) require models that can deal with a high complexity at small run-times.
Models: We developed a class of cellular automaton models on regular lattices and (random) Dirichlet tessellations. The rules of the cellular automaton have been chosen according to simulated observations in a single-cell-based off-lattice model in which a cell is parameterized according to measurable cell-biological and cell-biophysical quantities (Drasdo, 2005). The parameters of the cellular automaton in which each cell occupy a lattice site can be related to the parameters of the off-lattice models. The run-time in the lattice models is about 1000-times smaller than in the off-lattice models and permits a systematic sensitivity analysis. The knowledge of the system behaviour of the individual-cell-based models permits to derive continuum models with the same system behavior.
In continuum models local densities of cells rather than individual cells are considered. As the final step of the model hierarchy phenomenological growth laws are considered that capture characteristic features of growing cell populations and can easily be applied by biologists and physicians. Our procedures permit to relate the model parameters on larger scales to those on smaller scales.
Results: The cellular automaton simulations show growth scenarios that suggest that many phenotypes observed experimentally may represent transient states of a small numbers of growth scenarios (Drasdo, 2005). If the cell-cell adhesion is large both the cell population size and the spatial spread (diameter) of both, monolayers and multi-cellular spheroids grow exponentially fast. If the cell-cell adhesion is very small such that cells can easily detach, and if the migration activity of cells is large then the initial spatial spread is diffusive (the diameter grows as the square root of the time) while the cell population size still grows exponentially fast. Eventually in both cases the spatial spread of growing cell population becomes linear while the number of cells grows as a power law in time where the exponent is the spatial dimension of the cell population (two for monolayers, three for multicellular spheroids).
In case the cell-cell adhesion is small a systematic coarse graining method can be used to derive a stochastic partial differential equation for the local cell density which turns out to be a stochastic form of the Fisher-KPP equation. The expansion velocity is in two dimensions and complies with the relation for the minimum wave velocity of the deterministic Fisher-KPP equation. ΔR is the effective (true) width of the proliferative rim, I the cell diameter,Τ the cycle time, Φ the hopping rate and α a lattice-dependent correction factor. The functional dependence of the expansion velocity from the thickness of the proliferation rim and the diffusion constant complies well with the findings of the cellular automaton models.
Benefit: The projects aim at a hierarchy of models to investigate the growth behaviour of expanding cell populations such as tumors which permits qualitative and quantitative predictions on spatial scales from the single cell up to populations of in-vivo tumours and eventually permits to be used in the optimization of therapy protocols in cancer treatment.
Outlook: (1) The Fisher-KPP equation does not capture the initial growth kinetics of the cell population diameter if the cell-cell adhesion is large. Currently this case is studied (co-operation with Helen Byrne). (2) So far most of the models assume that all cells have intrinsically the same properties. By mutations, differentiation, or regulation the properties at later stages may markedly differ between cells within large cell populations. This has to be included properly in the model hierarchy. (3) So far the main focus was on growing monolayers. In a next step we consider multi-cellular spheroids and in-vivo tumors (co-operation with Jan G. Hengstler).