# Cell-based models

Cell-based models are mathematical models that represent biological cells as a discrete entities. They are used in the field of computational biology for simulating the biomechanics of multicellular structures such as tissues. Their main advantage is the easy integration of cell level processes such as cell division, intracellular processes and single-cell variability within a cell population.[1]

## Model types

Cell-based models can be divided into on- and off-lattice models. On-lattice models such as cellular automata or cellular potts restrict the spatial arrangement of the cells to a fixed grid. The mechanical interactions are then carried out according to literature-based rules (cellular automata)[2] or by minimizing the total energy of the system (cellular potts),[3] resulting in cells being displaced from one grid point to another.

### Off-lattice

Off-lattice models allow for continuous movement of cells in space and evolve the system in time according to force laws governing the mechanical interactions between the individual cells. Examples of off-lattice models are center-based models, vertex-based models, models based on the immersed boundary method[4] and the subcellular element method.[5] They differ mainly in the level of detail with which they represent the cell shape. As a consequence they vary in their ability to capture different biological mechanisms, the effort needed to extend them from two- to three-dimensional models and also in their computational cost.[6]

The simplest off-lattice model, the center-based model, depicts cells as spheres and models their mechanical interactions using pairwise potentials.[7][8] It is easily extended to a large number of cells in both 2D and 3D.[9]

### Vertex

Vertex-based models track the cell membrane as a set of polygonal points and update the position of each vertex according to tensions in the cell membrane resulting from cell-cell adhesion forces and cell elasticity.[10] They are more difficult to implement and also more costly to run. As cells move past one another during a simulation, regular updates of the polygonal edge connections are necessary.[11]

## Applications

Since they account for individual behavior at the cell level such as cell proliferation, cell migration or apoptosis, cell-based models are a useful tool to study the influence of these behaviors on how tissues are organised in time and space.[1] Due in part to the increase in computational power, they have arisen as an alternative to continuum mechanics models[12] which treat tissues as viscoelastic materials by averaging over single cells.

Cell-based mechanics models are often coupled to models describing intracellular dynamics, such as an ODE representation of a relevant gene regulatory network. It is also common to connect them to a PDE describing the diffusion of a chemical signaling molecule through the extracellular matrix, in order to account for cell-cell communication. As such, cell-based models have been used to study processes ranging from embryogenesis[13] over epithelial morphogenesis[14] to tumour growth[15] and intestinal crypt dynamics[16]

## Simulation frameworks

There exist several software packages implementing cell-based models, e.g.

## References

1. ^ a b Van Liedekerke P, Palm MM, Jagiella N, Drasdo D (1 December 2015). "Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results". Computational Particle Mechanics. 2 (4): 401–444. doi:10.1007/s40571-015-0082-3.
2. ^ Peirce SM, Van Gieson EJ, Skalak TC (April 2004). "Multicellular simulation predicts microvascular patterning and in silico tissue assembly". FASEB Journal. 18 (6): 731–3. doi:10.1096/fj.03-0933fje. PMID 14766791.
3. ^ Graner F, Glazier JA (September 1992). "Simulation of biological cell sorting using a two-dimensional extended Potts model". Physical Review Letters. 69 (13): 2013–2016. doi:10.1103/PhysRevLett.69.2013. PMID 10046374.
4. ^ Rejniak KA (July 2007). "An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development". Journal of Theoretical Biology. 247 (1): 186–204. doi:10.1016/j.jtbi.2007.02.019. PMID 17416390.
5. ^ Newman TJ (July 2005). "Modeling multicellular systems using subcellular elements". Mathematical Biosciences and Engineering. 2 (3): 613–24. doi:10.1007/978-3-7643-8123-3_10. PMID 20369943.
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7. ^ Meineke FA, Potten CS, Loeffler M (August 2001). "Cell migration and organization in the intestinal crypt using a lattice-free model". Cell Proliferation. 34 (4): 253–66. doi:10.1046/j.0960-7722.2001.00216.x. PMID 11529883.
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10. ^ Fletcher AG, Osterfield M, Baker RE, Shvartsman SY (June 2014). "Vertex models of epithelial morphogenesis". Biophysical Journal. 106 (11): 2291–304. doi:10.1016/j.bpj.2013.11.4498. PMID 24896108.
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13. ^ Tosenberger A, Gonze D, Bessonnard S, Cohen-Tannoudji M, Chazaud C, Dupont G (9 June 2017). "A multiscale model of early cell lineage specification including cell division". NPJ Systems Biology and Applications. 3 (1): 16. doi:10.1038/s41540-017-0017-0. PMID 28649443.
14. ^ Fletcher AG, Cooper F, Baker RE (May 2017). "Mechanocellular models of epithelial morphogenesis". Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences. 372 (1720): 20150519. doi:10.1098/rstb.2015.0519. PMID 28348253.
15. ^ Drasdo D, Dormann S, Hoehme S, Deutsch A (2004). "Cell-Based Models of Avascular Tumor Growth". Function and Regulation of Cellular Systems. Basel: Birkhäuser: 367–378. doi:10.1007/978-3-0348-7895-1_37.
16. ^ De Matteis G, Graudenzi A, Antoniotti M (June 2013). "A review of spatial computational models for multi-cellular systems, with regard to intestinal crypts and colorectal cancer development". Journal of Mathematical Biology. 66 (7): 1409–62. doi:10.1007/s00285-012-0539-4. PMID 22565629.
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19. ^ Swat MH, Thomas GL, Belmonte JM, Shirinifard A, Hmeljak D, Glazier JA (1 January 2012). "Multi-scale modeling of tissues using CompuCell3D". Methods in Cell Biology. 110: 325–66. doi:10.1016/B978-0-12-388403-9.00013-8. PMC 3612985. PMID 22482955.
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21. ^ Starruß J, de Back W, Brusch L, Deutsch A (May 2014). "Morpheus: a user-friendly modeling environment for multiscale and multicellular systems biology". Bioinformatics. 30 (9): 1331–2. doi:10.1093/bioinformatics/btt772. PMC 3998129. PMID 24443380.
22. ^ Merks RM, Guravage M, Inzé D, Beemster GT (February 2011). "VirtualLeaf: an open-source framework for cell-based modeling of plant tissue growth and development". Plant Physiology. 155 (2): 656–66. doi:10.1104/pp.110.167619. PMID 21148415.
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24. ^ Delile J, Herrmann M, Peyriéras N, Doursat R (January 2017). "A cell-based computational model of early embryogenesis coupling mechanical behaviour and gene regulation". Nature Communications. 8: 13929. doi:10.1038/ncomms13929. PMID 28112150.