Data Assimilation in Brain Tumor Models

A typical problem in applied mathematics and science is to estimate the future state of a dynamical system given its current state. One approach aimed at understanding one or more aspects determining the behavior of the system is mathematical modeling. This method frequently entails formulation of a set of equations, usually a system of partial or ordinary differential equations. Model parameters are then measured from experimental data or estimated from computer simulation or other methods, for example chi-squared parameter optimization as done in[26] or genetic algorithms which are frequently used in neuroscience [33]. Solutions to the model are then studied through mathematical analysis and numerical simulation usually for qualitative fit to the dynamical system of interest and any relative time-series data that is available. While mathematical modeling can provide meaningful insight, it may have limited predictive value due to idealized assumptions underlying the model, measurement error in experimental data and parameters, and chaotic behavior in the system. In this chapter we explore a different approach focused on optimal state estimation given a model and observational data of a biological process, while accounting for the relative uncertainty in both. The case explored here is the growth and spread of glioblastoma multiforme (GBM), a very aggressive form of glioma brain tumor which remains extremely difficult to manage clinically. The method employed is different from other approaches used in biology in that it is independent of the mathematical model and seeks an optimal initial condition. This is in contrast to other techniques such as those discussed in [21], which are model dependent and seek to find an optimal model parameterization given the observations and uncertainties in the system of interest.