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A Continuum Mathematical Model of Substrate-Mediated Tissue Growth.
Maud El-Hachem1, Scott W McCue1, Matthew J Simpson2
1School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
Bulletin of Mathematical Biology
|March 3, 2022
Summary
This study introduces a mathematical model for tissue formation in bioscaffolds, revealing distinct sharp and smooth-fronted cell growth patterns. The model accurately simulates experimental observations, offering insights into tissue development dynamics.
Area of Science:
- Mathematical Biology
- Tissue Engineering
- Biophysics
Background:
- Biological tissue formation is crucial for regenerative medicine and understanding development.
- Recent experiments show thin tissue growth in 3D-printed bioscaffolds.
- Existing models may not fully capture the dynamics of cell migration and proliferation influenced by extracellular substrates.
Purpose of the Study:
- To develop and analyze a continuum mathematical model for biological tissue formation.
- To investigate the mechanisms behind sharp and smooth-fronted tissue growth observed in experiments.
- To provide a deeper understanding of cell migration, proliferation, and substrate dynamics.
Main Methods:
- Developed a substrate model involving partial differential equations for tissue density and extracellular substrate concentration.
- Incorporated nonlinear diffusion for cell migration and logistic growth for proliferation.
- Analyzed travelling wave solutions, including sharp-fronted and smooth-fronted waves, using mathematical techniques and geometric interpretation via slow manifold reduction.
- Performed preliminary numerical simulations.
Main Results:
- The substrate model successfully recapitulates key features of experimental tissue growth, including sharp front formation.
- The model supports both sharp-fronted travelling wave solutions with a minimum speed and faster smooth-fronted travelling wave solutions.
- A geometric interpretation explains the distinction between sharp and smooth-fronted solutions.
- Approximations for travelling wave shapes in various limits were developed and tested.
Conclusions:
- The developed mathematical model provides a robust framework for studying biological tissue formation.
- The model elucidates the emergence of different tissue front morphologies based on underlying biophysical processes.
- The findings offer valuable insights for designing and optimizing 3D-printed bioscaffolds for tissue engineering applications.
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