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Modeling and Simulation of Coupled Cell Proliferation and Regulation in Heterogeneous Tissue

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Abstract

The primary objective of this work is to develop a computational framework that efficiently simulates the time-transient proliferation of cellular tissue, with heterogeneous microstructure, utilizing two strongly-coupled conservation laws:

  • Conservation Law 1: comprises (a) rate of change of cells, (b) cellular migration, (c) cellular proliferation controlled by a cell mitosis regulating chemical, (d) cell apoptosis and

  • Conservation Law 2: comprises (a) rate of change of the cell mitosis chemical regulator, (b) regulator diffusion, (c) regulator production by cells and (d) regulator decay.

Specifically, a straightforward approach is developed that researchers in the field can easily implement and use as a computationally-efficient tool to study such biological systems. Because multifield coupling is present, a recursive, staggered, temporally-adaptive, Finite Difference Time Domain scheme is developed to resolve the interacting fields. The time-step adaptation allows the numerical scheme to iteratively resolve the changing physical fields by reducing the time-steps during phases of the process when the system is undergoing changes on relatively small time-scales or enlarging the time-steps when the processes are relatively slow. The spatial discretization grids are uniform and dense, and the heterogeneous microstructure, is embedded into the spatial discretization. The regular grid allows one to generate a matrix-free iterative formulation which is amenable to rapid computation and minimal memory requirements, making it ideal for laptop computation. Numerical examples are provided to illustrate the approach.

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Notes

  1. For the class of problems under consideration, due to the quadratic dependency on \(\Delta t\), \(p \approx 1\).

  2. Typically, \(K_d\) is chosen to be between five to ten iterations, although this is problem and analyst dependent.

  3. At the implementation level, since the exact solution is unknown, the following relative error term is used, \(\varpi ^{L+1,K} \mathop {=}\limits ^\mathrm{def}{\varvec{U}}^{L+1,K}-{\varvec{U}}^{L+1,K-1}\).

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  1. Department of Mechanical Engineering, University of California, 6195 Etcheverry Hall, Berkeley, CA, 94720-1740, USA

    T. I. Zohdi

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Correspondence to T. I. Zohdi.

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Associate Editor Estefanía Peña oversaw the review of this article.

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Zohdi, T.I. Modeling and Simulation of Coupled Cell Proliferation and Regulation in Heterogeneous Tissue. Ann Biomed Eng 43, 1666–1679 (2015). https://doi.org/10.1007/s10439-014-1194-2

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