Sensitivity analysis for a parametrized model for erythropoiesis involving structured population equations with one structural variable

Abstract

The base of this master thesis is a hyperbolic partial differential equation (PDE), developed by the Renal Research Institute in New York, USA 2012. The hyperbolic PDE describes the cell population of red blood cells in a specific cell stage (stage of CFU-E cells) depending on the concentration of the hormone erythropoietin (EPO). The model is understood as a parametrized partial differential equation (PPDE). As parameters serve constants, which describe the natural cell death rate and additionally control parameters, which indicate the amount of administered EPO in form of injections. The aim of this work is the sensitivity analysis relating to these parameters. For this purpose, a further parametrized partial differential equation is derived. Its solution describes the sensitivity of the cell population relating to a parameter. Two methods - the finite difference (FD) method and the reduced basis (RB) method - are used to compute a numerical solution of this PPDE. The RBmethod reduces the dimension of the discretization space (of the finite difference method) with a Galerkin method. The reduced basis is generated by a greedy algorithm. Further, an error estimation is used which measures the error between the sensitivities computed by the finite difference method and the sensitivity calculated by the reduced basis method. Next the sensitivities of a linear functional in form of the total cell population and a quadratic cost functional are considered. Additionally, a subset selection method is applied to get a ranking for the parameters according to the degree of their sensitivity. At the end of the thesis the results are presented. Significant differences with regard to the sensitivities of the parameters will be shown. One parameter from the parameter set with the constants is a lot more sensitive than the others and two parameters from the control parameter set with nine parameters are more sensitive. The similar result will be shown for the parameters of the total cell population and of the quadratic cost functional.