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Incompressible Limit of a Mechanical Model for Tissue Growth with Non-Overlapping Constraint.
Sophie Hecht1, Nicolas Vauchelet2
1Francis Crick Institute, 1 Midland Rd, Kings Cross, London NW1 1AT, UK - Imperial College London, South Kensington Campus London SW7 2AZ, UK (sophie.hecht15@imperial.ac.uk).
Communications in Mathematical Sciences
|November 7, 2017
Summary
This study models tissue growth, showing that a singular pressure law ensures cells do not overlap. The research confirms this leads to the same Hele-Shaw free boundary problem in the incompressible limit.
Area of Science:
- Mathematical Biology
- Biophysics
- Free Boundary Problems
Background:
- Tissue growth models often involve cell density dynamics influenced by pressure and proliferation.
- Existing models converge to Hele-Shaw free boundary problems at the incompressible limit.
Purpose of the Study:
- To investigate the impact of a non-overlapping constraint on tissue growth models.
- To prove that a singular pressure law maintains the Hele-Shaw free boundary problem convergence.
Main Methods:
- Development of a mathematical model for cell density dynamics.
- Analysis of pressure forces and proliferation effects.
- Mathematical proof of convergence under a singular pressure law.
Main Results:
- A singular pressure law effectively enforces a non-overlapping constraint for cells.
- The model's incompressible limit converges to the established Hele-Shaw free boundary problem, even with the singularity.
Conclusions:
- The use of a singular pressure law is a viable method to ensure non-overlapping cell constraints in tissue growth models.
- This approach preserves the expected Hele-Shaw free boundary problem dynamics, broadening its applicability.
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