Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 224

Data Assimilation in Brain Tumor Models

Joshua McDaniel, Eric Kostelich, Yang Kuang, John Nagy, Mark C. Preul,
Nina Z. Moore, and Nikolay L. Matirosyan

1 Introduction

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 ]orgenetic
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

J. McDaniel ( ) ￿ E. Kostelich ￿ Y. Kuang
School of Mathematical and Statistical Sciences, Arizona State University,
Tempe, AZ 85287, USA
e-mail: [e-mail] ; [e-mail] ; [e-mail]
J. Nagy
Department of Biology, Scottsdale Community College, Scottsdale, AZ 85256-2626, USA
e-mail: [e-mail]
M.C. Preul ￿ N.Z. Moore ￿ N.L. Matirosyan
Department of Neurosurgery Research, Barrow Neurological Institute, St. Josephs Hospital
and Medical Center, Phoenix, AZ 85013, USA
e-mail: [e-mail] ; [e-mail] ; [e-mail]