# A Model for Elastic Evolution on Foliated Shapes

## TL;DR

This paper introduces a shape evolution model based on anisotropic elasticity, useful for modeling slow biological shape changes like atrophy, with a focus on layered shapes inspired by cortical structures.

## Contribution

It presents a novel elastic evolution framework using optimal control for shape deformation, including a layered shape specialization inspired by cortical organization.

## Key findings

- Preliminary experiments on synthetic layered shapes demonstrate elasticity effects.
- The model effectively captures slow biological shape changes like atrophy.
- Layered shape decomposition offers new insights into cortical volume deformation.

## Abstract

We study a shape evolution framework in which the deformation of shapes from time t to t + dt is governed by a regularized anisotropic elasticity model. More precisely, we assume that at each time shapes are infinitesimally deformed from a stress-free state to an elastic equilibrium as a result of the application of a small force. The configuration of equilibrium then becomes the new resting state for subsequent evolution. The primary motivation of this work is the modeling of slow changes in biological shapes like atrophy, where a body force applied to the volume represents the location and impact of the disease. Our model uses an optimal control viewpoint with the time derivative of force interpreted as a control, deforming a shape gradually from its observed initial state to an observed final state. Furthermore, inspired by the layered organization of cortical volumes, we consider a special case of our model in which shapes can be decomposed into a family of layers (forming a "foliation"). Preliminary experiments on synthetic layered shapes in two and three dimensions are presented to demonstrate the effect of elasticity.