William E. Annan, PhD

Healthy eye

Rod and cone cell

Detahed retina

PhD Thesis: Modeling the spread of retina detachment and its effect on the dynamics of rod outer segment (ROS) renewal

Introduction

Rod and cone cells in the retina play a crucial role in vision by converting light energy into electrical signals. This process relies on the daily renewal of their outer segments, involving the addition of new discs at the base and removal at the top. However, retinal detachment disrupts this renewal process, potentially leading to blindness if left untreated. Despite its significance, the precise effects of retinal detachment on this renewal process, as well as the survival time of photoreceptor cells in a detached retina and the rate of progression of detachment, remain unclear.

Objectives

To address these gaps, we have developed a mathematical model consisting of ordinary differential equations (ODEs) to describe the renewal of the rod outer segment during retinal detachment. Our aim is to elucidate the specific impact of detachment on this renewal process and determine the survival time of a rod cell in a detached retina. Additionally, using a fluid-structure interaction model, we have created a partial differential equation (PDE) model to track the progression of retinal detachment. This model aims to estimate the time frame for effective treatment of retinal detachment.

Results

Simulation and analysis of the ODE model revealed that retinal detachment inhibits the addition of new discs at the base while allowing for some removal mechanism, a finding that challenges existing hypotheses in the literature. Furthermore, the survival time of a rod cell in a detached retina was found to be dependent on the rate of disc removal. Currently, we are conducting simulations with the PDE model to determine the rate of progression of retinal detachment.

By investigating these fundamental questions, we aim to advance our understanding of retinal detachment and contribute to the development of more effective treatment strategies.

Example of pattern in nature

Simulated pattern for sensory precussor cells

Research Assistantship project: Understanding the dynamics of filopodia interaction in pattern formation

Introduction

The intricate patterns observed in biological species, such as the stripes on zebrafish, the radial color patterns in flowers, and the organization of sensory organ precursor cells in fruit flies, all arise during development, where individual cells assume distinct fates and functions. Central to this process is the interaction between neighboring cells mediated by the Notch-Delta signaling pathway. Recent research suggests that filopodia, protrusions extending from cell surfaces, play a crucial role in facilitating long-range interactions that contribute to the formation of well-defined and organized patterns. However, the precise mechanism by which filopodia mediate this interaction remains unclear.

Objectives

In this study, we aim to employ mathematical modeling, complemented by experimental data, to precisely elucidate how filopodia interactions lead to Notch activation and subsequent pattern formation of sensory precursor cells in the thorax of fruit flies.

Results

We are currently conducting simulations to investigate the intricate dynamics of filopodia-mediated Notch activation and its role in pattern formation. By addressing this fundamental question, we seek to deepen our understanding of the mechanisms underlying pattern formation in biological systems .

Application of FEM to one-dimensional advection diffusion equation with nonlinear source term.

Performance Numerical Schemes with Finite Element Method

Introduction

Partial differential equations (PDEs) manifest across diverse mathematical domains. However, the scope of PDEs amenable to closed-form or analytic solutions is constrained. This limitation exacerbates in higher dimensions, particularly with intricate geometries. The finite element method (FEM) has demonstrated adeptness in handling complex geometries, outperforming finite difference schemes in this regard. FEM primarily addresses spatial discretization, necessitating the selection of an appropriate numerical scheme for temporal derivatives.

Objectives

This project aims to scrutinize the performance of various numerical schemes when applied to temporal derivatives subsequent to employing FEM for spatial discretization, yielding a system of ordinary differential equations (ODEs). My motivation for undertaking this project stems primarily from a desire to refine my computational and analytical proficiencies. As an applied mathematician, adeptness in model-solving and comprehensive analysis is paramount.

Results

To date, I have effectively applied FEM in solving numerous PDEs both in one and two dimensions, with the intention to extend its application to three dimensions. Preliminary analysis suggests that the Exponential Time Differencing (ETD) scheme outperforms alternatives such as Imex-BDF2 and Backwards Euler.

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