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Med Phys. 2012 Jul;39(7):4444-59. doi: 10.1118/1.4722749.

Biophysical modeling of brain tumor progression: from unconditionally stable explicit time integration to an inverse problem with parabolic PDE constraints for model calibration.

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Institute of Medical Engineering, University of Lübeck, Ratzeburger Allee 160, Lübeck,



A novel unconditionally stable, explicit numerical method is introduced to the field of modeling brain cancer progression on a tissue level together with an inverse problem (IP) based on optimal control theory that allows for automated model calibration with respect to observations in clinical imaging data.


Biophysical models of cancer progression on a tissue level are in general based on the assumption that the spatiotemporal spread of cancerous cells is determined by cell division and net migration. These processes are typically described in terms of a parabolic partial differential equation (PDE). In the present work a parallelized implementation of an unconditionally stable, explicit Euler (EE(⋆)) time integration method for the solution of this PDE is detailed. The key idea of the discussed EE(⋆) method is to relax the strong stability requirement on the spectral radius of the coefficient matrix by introducing a subdivision regime for a given outer time step. The performance is related to common implicit numerical methods. To quantify the numerical error, a simplified model that has a closed form solution is considered. To allow for a systematic, phenomenological validation a novel approach for automated model calibration on the basis of observations in medical imaging data is developed. The resulting IP is based on optimal control theory and manifests as a large scale, PDE constrained optimization problem.


The numerical error of the EE(⋆) method is at the order of standard implicit numerical methods. The computing times are well below those obtained for implicit methods and by that demonstrate efficiency. Qualitative and quantitative analysis in 12 patients demonstrates that the obtained results are in strong agreement with observations in medical imaging data. Rating simulation success in terms of the mean overlap between model predictions and manual expert segmentations yields a success rate of 75% (9 out of 12 patients).


The discussed EE(⋆) method provides desirable features for image-based model calibration or hybrid image registration algorithms in which the model serves as a biophysical prior. This is due to (i) ease of implementation, (ii) low memory requirements, (iii) efficiency, (iv) a straightforward interface for parameter updates, and (v) the fact that the method is inherently matrix-free. The explicit time integration method is confirmed via experiments for automated model calibration. Qualitative and quantitative analysis demonstrates that the proposed framework allows for recovering observations in medical imaging data and by that phenomenological model validity.

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