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Elastic Staining on Paraffin-embedded Slides of pT3N0M0 Gastric Cancer Tissue

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Growth of form in thin elastic structures.

Salem Al Mosleh¹, Ajay Gopinathan², Christian Santangelo¹

¹Department of Physics, University of Massachusetts Amherst, Amherst, MA 01003, USA. csantang@physics.umass.edu.

|October 12, 2018

Summary

Heterogeneous growth in elastic structures is stabilized by coupling growth laws to local stress and curvature. Mean curvature flow is essential for stabilizing long-wavelength fluctuations in cylindrical shells.

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Area of Science:

Physics
Materials Science
Biophysics

Background:

Heterogeneous growth is crucial for pattern formation in elastic structures like cell walls and flower petals.
Understanding the stability and regulation of this growth is key to predicting and controlling shape evolution.

Purpose of the Study:

To model and analyze the stability of heterogeneous growth in thin elastic structures.
To investigate regulation mechanisms by coupling growth laws to local shape properties.
To identify conditions for stable growth in cylindrical shells, relevant to bacteria.

Main Methods:

Quasi-static time evolution model of a metric with rapid elastic shape relaxation.
Coupling growth laws to local shape properties: curvature and stress.
Analysis of stability for cylindrical shell models.

Main Results:

Coupling growth to curvature alone leads to linear instability in cylindrical shells.
Adding coupling to local stress stabilizes short-wavelength fluctuations.
Mean curvature flow is required to stabilize long-wavelength fluctuations.

Conclusions:

Local stress and curvature coupling offer pathways to stabilize heterogeneous growth.
Mean curvature flow is a critical factor for overall stability in certain geometries.
The framework provides insights into designing stable growth laws for diverse target shapes.