Journal of Mathematical Biology

, Volume 72, Issue 1–2, pp 409–433 | Cite as

An inverse problem formulation for parameter estimation of a reaction–diffusion model of low grade gliomas

  • Amir Gholami
  • Andreas Mang
  • George Biros


We present a numerical scheme for solving a parameter estimation problem for a model of low-grade glioma growth. Our goal is to estimate the spatial distribution of tumor concentration, as well as the magnitude of anisotropic tumor diffusion. We use a constrained optimization formulation with a reaction–diffusion model that results in a system of nonlinear partial differential equations. In our formulation, we estimate the parameters using partially observed, noisy tumor concentration data at two different time instances, along with white matter fiber directions derived from diffusion tensor imaging. The optimization problem is solved with a Gauss–Newton reduced space algorithm. We present the formulation and outline the numerical algorithms for solving the resulting equations. We test the method using a synthetic dataset and compute the reconstruction error for different noise levels and detection thresholds for monofocal and multifocal test cases.


Inverse problems Parameter estimation Glioma  Glioblastoma multiforme Tumor growth 

Mathematics Subject Classification

65L09 65F08 



We would like to thank Thomas Hillen for the helpful discussion on the anisotropic diffusion of gliomas. We would like to also thank Florian Tramnitzke for contributing to this work during his internship in our group.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA

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