Optimal administration strategies for EPO based on the model for erythropoiesis involving structured population equations

Abstract

This diploma thesis deals with a hyperbolic partial differential equation (PDE), developed by the Renal Research Institute in New York. It models the population density of CFU-E cells, which depends on time as well as on the maturity attribute of the cells, with respect to external administration of erythropoietin (EPO). CFU-E cells represent one level of development from stem cells to red blood cells and play a role by investigating optimal treatment of dialysis patients, amongst others.First, the PDE is normalized followed by a discretization of the maturity variable - representing the space variable in this context - which is carried out using Legendre polynomials. This leads to an ordinary differential equation (ODE).Utilizing a L^2-objective functional an optimal control problem is expressed. Since injections of EPO are only possible at several fixed days the control obtains a discrete nature.The theoretical background about solvability of the underlying ODE is illuminated and optimality criteria are formulated.Subsequently, different optimization techniques are examined, including the theta-method for computing the state and the gradient and BFGS method combined with a modified Armijo step size strategy for the optimization process. Next, they are integrated in a model predictive control (MPC) framework. At the end of the thesis, numerical tests - some of which are based on actual problem settings occurring at the treatment of dialysis patients - are executed and results are presented.