Abstract
In certain discrete models of populations of biological cells, the mechanical forces between the cells are center based or vertex based on the microscopic level where each cell is individually represented. The cells are circular or spherical in a center based model and polygonal or polyhedral in a vertex based model. On a higher, macroscopic level, the time evolution of the density of the cells is described by partial differential equations (PDEs). We derive relations between the modelling on the micro and macro levels in one, two, and three dimensions by regarding the micro model as a discretization of a PDE for conservation of mass on the macro level. The forces in the micro model correspond on the macro level to a gradient of the pressure scaled by quantities depending on the cell geometry. The two levels of modelling are compared in numerical experiments in one and two dimensions.
Highlights
Mathematical modelling for simulation of cell populations is a tool complementing experimental studies to understand the complex biochemical and mechanical interactions between the cells in aggregations of unicellular organisms in bacterial cell colonies (Hellweger et al 2016) and in multicellular organisms forming growing and developing tissues (Liedekerke et al 2015; Lowengrub et al 2010). 75 Page 2 of 31 P
The evolution of the cell population can be modelled by the solution of time dependent, nonlinear partial differential equations (PDEs) for the cellular densities and concentrations of chemical compounds (Brodland et al 2006; Frieboes et al 2010; Humphrey 2003)
By comparing the expressions for the macroscale velocity of the discretization of the PDE in (2) with the microscale velocity in (1) for the CBM, we find that the appropriately scaled pressure agrees with the CBM forces
Summary
Mathematical modelling for simulation of cell populations is a tool complementing experimental studies to understand the complex biochemical and mechanical interactions between the cells in aggregations of unicellular organisms in bacterial cell colonies (Hellweger et al 2016) and in multicellular organisms forming growing and developing tissues (Liedekerke et al 2015; Lowengrub et al 2010)
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