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Modeling biological tissue growth: discrete to continuum representations.
Jack D Hywood1, Emily J Hackett-Jones, Kerry A Landman
1Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 16, 2013
Summary
This study models tissue growth using agent-based simulations, revealing how cell proliferation influences tissue expansion. It reconciles discrepancies in continuum models by developing a master equation for stochastic processes in growing tissues.
Area of Science:
- Mathematical Biology
- Developmental Biology
- Computational Biology
Background:
- Deterministic continuum models are often derived from discrete agent-based models.
- Biological cells exhibit stochastic behaviors like movement and division within developing tissues.
- Tissue growth itself acts as a cellular transport mechanism.
Purpose of the Study:
- To develop a discrete agent-based model for simulating tissue growth.
- To derive and analyze the average behavior of stochastically proliferating agents using a Fokker-Planck equation.
- To reconcile discrepancies between existing models and provide a more accurate description of tissue growth.
Main Methods:
- Developed a discrete agent-based model with proliferating domain agents.
- Derived a Fokker-Planck equation with advection and diffusion terms to describe average behavior.
- Determined a discrete-time master equation and an asymmetric nonexclusion random walk to reconcile model discrepancies.
- Validated theoretical findings with numerical simulations.
Main Results:
- The agent-based model captures tissue growth through cell proliferation.
- A Fokker-Planck equation describes the average behavior of proliferating agents.
- A novel master equation and random walk model resolve discrepancies in diffusion terms for growing domains.
- Numerical simulations confirm the theoretical results.
Conclusions:
- This work enhances understanding of the link between agent-based rules and continuum partial differential equations.
- Accurate partial differential equation descriptions are crucial for modeling cellular transport during embryonic development.
- The developed model provides a more robust framework for studying tissue growth dynamics.
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