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Finite Element Simulations for the Design of Therapeutic Approaches for Retinal Diseases

The retinal disease age-related macular degeneration is the most common cause of vision loss in industrialized countries. In this thesis, motivated by the drug (antibody) treatment of this disease, we designed long-term three dimensional Finite Element simulations of the drug distribution in the healthy human eye. The underlying model consists of a time-dependent convection-diffusion equation coupled to a stationary Darcy equation describing the flow of the aqueous humor through the vitreous medium. We replaced the in general used Dirichlet boundary condition for the pressure with an inhomogeneous Neumann boundary condition for the velocity to obtain a more realistic description of the flow. The influence of collagen fibers inside the vitreous on drug distribution is accounted for by anisotropic diffusion and the gravity via an additional transport term. The resulting coupled model was solved in a decoupled way: first the Darcy equation by using mixed finite elements, then the convection-diffusion equation by using trilinear Lagrange elements. Krylov subspace methods are used to solve the resulting algebraic system. To cope with large time steps we applied the strong A-stable fractional step theta scheme. With this strategy we achieved second order of convergence in time and space. The developed simulations were used for process optimization for which specific output functionals were evaluated. We found out that an unfavorable injection angle can result in 38% less drug reaching the macula. This thesis provides the first steps toward virtual exploration and improvement of therapy for age-related macular degeneration and for other retinal diseases.

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@phdthesis{doi:10.17170/kobra-202305128014,
  author    ={Dörsam, Simon},
  title    ={Finite Element Simulations for the Design of Therapeutic Approaches for Retinal Diseases},
  keywords ={510 and Finite-Elemente-Methode and Numerische Strömungssimulation and Senile Makuladegeneration and Arzneimittelverteilung and Glaskörper and Konvektions-Diffusionsgleichung and Filtergesetz and Neumann-Problem},
  copyright  ={http://creativecommons.org/licenses/by-nc/4.0/},
  language ={en},
  school={Kassel, Universität Kassel, Fachbereich Mathematik und Naturwissenschaften, Institut für Mathematik},
  year   ={2022}
}