# Mathematical model to reconstruct the mechanical properties of an elastic medium

Post by Rafael Henriques

I am currently finishing my journey as Master’s student at the Centre for Mathematics of the University of Coimbra (CMUC). As a student at the University of Coimbra, I am very satisfied with the knowledge I have acquired and the quality of teaching during these years. I am proud to be a student of this University that, at the level of teaching and research in Mathematics, is internationally recognized. I can also say that this university provides a strong link between teaching and research with real and industrial applications. In addiction, Coimbra is a city with history and rich in culture that provides a memorable experience.

This year, I had the opportunity to participate in the ElastoOCT project, “Optical Coherence Elastography for imaging of the mechanical properties of the retina” http://miguelmorgado.net/research/projects/on-going/elastooct.html, which gathers a multidisciplinary team, whose objective is to develop an Optical coherence elastography (OCE) technique for measuring in vivo the mechanical properties of the retina on animal models, with the purpose of detecting early signs of neurodegeneration. The work I developed for this project was the theme for my master’s thesis.

OCE is an emerging biomedical imaging technique based in the optical coherence tomography (OCT) imaging modality to form pictures of biological tissue and map its biomechanical properties. OCE combines mechanical excitation with OCT for measuring the corresponding elastic displacement. Applications of this technique vary from skin to the retina, being the latter the one of most interest to the work developed.

In my thesis I started by developing a numerical model for the mechanical deformation of the retina induced by the propagation of acoustic waves, given the parameters that define the mechanical properties of the medium. This mathematical model was used to numerically simulate the displacement field within the retina, given a known excitation, and considering different sets of values for the parameters of the model. This problem is known as the direct problem and the proposed model is based on time-harmonic equations of linear elasticity. To compute the numerical solution of this model I used a finite element method in a three-dimensional domain.

Another task of the project is to investigate a way of obtaining the mechanical properties of the retina given the displacement field, that is, to solve the inverse problem of elastography. In my thesis, this inverse problem is formulated as an optimization program having the mathematical model for solving the direct problem as the computational basis. In practice, the objective is to infer the parameters that characterize the mechanical properties of the medium so that the difference between simulated displacements obtained with the mathematical model for the direct problem with those parameters and the data are minimized.

For the numerical experiments in the inverse problem we used fabricated noise free and noisy data obtained using the numerical solution of the direct problem. In future work, the goal is to use real data from the retina.

I am very grateful to participate in the project since I believe that the work developed is important and will contribute in the area of health more precisely in prevention.