# Parametric FEM for Shape Optimization applied to Golgi Stack

## TL;DR

This paper develops a parametric finite element method for shape optimization of Golgi stacks, modeling cisternae as elastic vesicles to understand their morphological evolution.

## Contribution

It introduces a novel FEM-based shape optimization model for Golgi stacks, incorporating biological constraints and handling multiple vesicles simultaneously.

## Key findings

- The model successfully simulates Golgi stack shapes.
- Numerical results align with observed Golgi morphologies.
- The approach offers insights into Golgi cisternae dynamics.

## Abstract

The thesis is about an application of the shape optimization to the morphological evolution of Golgi stack. Golgi stack consists of multiple layers of cisternae. It is an organelle in the biological cells. Inspired by the Helfrich Model \cite{Helfrich}, which is a model for vesicles typically applied to biological cells, a new model specially designed for Golgi stack is developed and then implemented using FEM in this thesis. In the Golgi model, each cisternae of the Golgi stack is viewed as a closed vesicle without topological changes, and our model is adaptable to both single-vesicle case and multiple-vesicle case. The main idea of the math model is to minimize the elastic energy(bending energy) of the vesicles, with some constraints designed regarding the biological properties of Golgi stack. With these constraints attached to the math model, we could extend this model to an obstacle-type problem. Hence, in the thesis, not only the simulations of Golgi stack are shown, but some interesting examples without biological meanings are also demonstrated. Also, as multiple cisternaes are considered as a whole, this is also a model handling multiple objects. A set of numerical examples is shown to compare with the observed shape of Golgi stack, so we can lay down some possible explanations to the morphological performance of trans-Golgi cisternae.