ABSTRACT
Glioblastoma multiforme (GBM) is a malignant brain cancer with a tendency to both migrate and proliferate. We propose modeling GBM with heterogeneity in cell phenotypes using a random differential equation version of the reaction-diffusion equation, where the parameters describing diffusion (D) and proliferation ([Formula: see text]) are random variables. We investigate the ability to perform the inverse problem to recover the probability distributions of D and [Formula: see text] using the Prohorov metric, for a variety of probability distribution functions. We test the ability to perform the inverse problem for noisy synthetic data. We then examine the predicted effect of treatment, specifically, chemotherapy, when assuming such a heterogeneous population and compare with predictions from a homogeneous cell population model.
Subject(s)
Brain Neoplasms/pathology , Glioblastoma/pathology , Models, Biological , Antineoplastic Agents/therapeutic use , Brain Neoplasms/drug therapy , Cell Movement , Cell Proliferation , Computer Simulation , Glioblastoma/drug therapy , Humans , Mathematical Concepts , Neoplasm Invasiveness/pathology , Phenotype , Probability , Spatio-Temporal Analysis , Tumor Burden/drug effectsABSTRACT
We consider nonparametric estimation of probability measures for parameters in problems where only aggregate (population level) data are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades [24, 5, 12, 28, 16]. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other) least squares estimates and estimators for the measure estimation problem with specific application to random PDEs.
