Finite element simulation of the pharmacodynamic model for aflibercept and ranibizumab for the treatment of age related macular degeneration

Age related macular degeneration is known to be one of the major causes of irreversible blindness among the older generation. We present a mathematical model of partial differential equations for the therapy of this disease, which is based on the intravitreal injection of a drug into the vitreous body. For the treatment to work, the drug has to travel past the inner-limiting membrane into the retina and reduce the free vascular endothelial growth factor (VEGF) concentration by binding to at least one of the two binding sites of the VEGF molecule. Therefore, our model consists of two compartments, the vitreous and the retina. In the vitreous we employ four coupled convection-diffusion-reaction equations with an additional coupling to the underlying aqueous humor flow and four coupled diffusion-reaction equations in the retina. The resulting PDE system is solved numerically in a realistic 3D eye geometry. Temporal discretization is based on one-step theta schemes and spatial discretization is done using the Finite Element method. The numerical results are used to demonstrate the therapy concept and to analyze the drug efficacy of aflibercept and ranibizumab. The results show, among other things, that only about 20 % of the drug reaches the retina through the inner-limiting membrane and that 50 % of the VEGF concentration has been rebuilt in the retina after 38.19 days for a single ranibizumab injection.