Abstract
This paper deals with the derivation of a collective model of cell populations out of an individual-based description of the underlying physical particle system. By looking at the spatial distribution of cells in terms of time-evolving measures, rather than at individual cell paths, we obtain an ensemble representation stemming from the phenomenological behavior of the single component cells. In particular, as a key advantage of our approach, the scale of representation of the system, i.e., microscopic/discrete vs. macroscopic/continuous, can be chosen a posteriori according only to the spatial structure given to the aforesaid measures. The paper focuses in particular on the use of different scales based on the specific functions performed by cells. A two-population hybrid system is considered, where cells with a specialized/differentiated phenotype are treated as a discrete population of point masses while unspecialized/undifferentiated cell aggregates are represented by a continuous approximation. Numerical simulations and analytical investigations emphasize the role of some biologically relevant parameters in determining the specific evolution of such a hybrid cell system.
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Notes
Throughout the paper we will use the subindex \(t\) to denote dependence on time of measures or measure-related quantities. In no case will this notation stand for the time derivative, which will be indicated by an explicit symbol such as e.g., \(\partial _t\).
Recall that, by definition, population \(2\) has the same initial condition in both cases.
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Acknowledgments
The authors extend warm thanks to Luigi Preziosi for many stimulating and fruitful discussions.
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M. Scianna has been funded by a post-doctoral research scholarship awarded by the National Institute for Advanced Mathematics “F. Severi” (INdAM, Italy).
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Colombi, A., Scianna, M. & Tosin, A. Differentiated cell behavior: a multiscale approach using measure theory. J. Math. Biol. 71, 1049–1079 (2015). https://doi.org/10.1007/s00285-014-0846-z
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DOI: https://doi.org/10.1007/s00285-014-0846-z