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Data Assimilation in Brain Tumor Models

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Mathematical Methods and Models in Biomedicine

Abstract

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.

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Notes

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Acknowledgments

Portions of this work were funded by the Barrow Neurological Institute Women’s Foundation and by funds from the Newsome Family Endowed Chair of Neurosurgery Research held by Dr. Preul. J.M. was supported in part by an Achievement Reward for College Scientists Scholarship. Y.K. was supported by NSF grants DMS-0436341 and DMS-0920744.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Joshua McDaniel, Eric Kostelich & Yang Kuang

  2. Department of Biology, Scottsdale Community College, Scottsdale, AZ, 85256-2626, USA

    John Nagy

  3. Department of Neurosurgery Research, Barrow Neurological Institute, St. Josephs Hospital and Medical Center, Phoenix, AZ, 85013, USA

    Mark C. Preul, Nina Z. Moore & Nikolay L. Matirosyan

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  1. Joshua McDaniel
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Correspondence to Joshua McDaniel .

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Editors and Affiliations

  1. , Mathematics and Statistics, Southern Illinois University Edwardsvill, Edwardsville, 62026, Illinois, USA

    Urszula Ledzewicz

  2. , Electrical and Systems Engineering, Washington University, Brookings Drive 1, Saint Louis, 63130, Missouri, USA

    Heinz Schättler

  3. Mathematical Biosciences Institute, Ohio State University, W. 18th Avenue 231, Columbus, 43210-1292, Ohio, USA

    Avner Friedman

  4. Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Eugene Kashdan

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McDaniel, J. et al. (2013). Data Assimilation in Brain Tumor Models. In: Ledzewicz, U., Schättler, H., Friedman, A., Kashdan, E. (eds) Mathematical Methods and Models in Biomedicine. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4178-6_9

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