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KAUST-IAMCS Workshop on Modeling and Simulation of Wave Propagation and Applications

Fatma Zohra Nouri, Badji Mokhtar University (Algeria)
Mathematical Modeling and Brain Tumor Growth


Mathematical tumor growth models have started to attract attention from the medical image analysis community in last years. These models could help better understanding of the mechanical influence and the diffusion process of gliomas. For the clinical applications, they would provide tools to identify the invaded areas that are not visible in the MR images in order to better adapt the resection in surgery or the irradiation margins in radiotherapy. As one of the most important goals, they would give the opportunity to identify from patent images some model parameters that could help characterizing the tumor and perhaps predict its future evolution.

Research conducted on brain tumor growth modeling can be coarsely classified into two large groups:

Microscopic models
Observations in the microscopic scale as a result formulate the growth phenomena at the cellular level
Macroscopic models
Observations at the macroscopic scale like the ones provided by medical images, formulation of the average behavior of tumor cells and their interactions with underlying tissue structures, which are visible at this scale of observation (e.g., MRI, MR-DTI), detecting real boundaries (grey matter, white matter, bones...)

Almost all diffusive macroscopic models use the reaction-diffusion formalism. This formalism models the invasive tumor by adding a diffusion team to simple solid tumor growth models, which formulates proliferation of cells. However, because of the different nature of the brain tissues (see Figure 1), the change of tumor cell density at a point \(u\left(x\right)\) should be described by an anisotropic diffusion and a nonlinear reaction process. We propose: \[\begin{cases} \frac{\partial u}{\partial t} = \nabla \cdot \left( D \left( x \right)\nabla u \right) + \rho \ldotp R \left( u \right) \\ D \left( x \right) \nabla u \cdot \vec{\eta} = 0 \end{cases}\]

The infiltration of tumor cells is explained by the diffusion process \(\nabla \left( D \left( x \right) \nabla u \right)\), which is characterized by the diffusion tensor \(D\). The proliferation of tumor cells are embedded in the reaction part \(\rho \ldotp R \left( u \right)\) with the motosis right \(\rho\) and \(R \left( u \right)\) a nonlinear function. The Neumann boundary conditions dictate that tumor cells will not pass through the skull nor the ventricles. (See numerical results in Figure 2).

An image of a cross-sectional human brain scan.
Figure 1: Diffusion properties of diffent tissues in the brain.
An image displaying numerical results.
Figure 2: From (a) to (h), the wavefront splits to pass around and meets again after the obstacles, developing a shock at the intersections. Here, the reaction term is taken to be \(R \left( u \right) = u \left( 1 - u \right)\).


  1. A. Giese, L. Kluwe, B. Laube, H. Meissner, M. Berens, and M. Westphal, . Migration of human glioma cells on myelin. Neurosurgery, 38.
  2. A. Giese, O. Clatz, M. Sermesant, P. Bondiau, H. Delingette, S. Warfield, G. Malandain, and N. Ayache, . Realistic simulation of the 3d growth of brain tumors in mr images coupling diffusion with biomechanical deformation. IEEE TMI, 24.
  3. E. Konukoglu, O. Clatz, P. Bondiau, M. Sermesant, H. Delingette, and N. Ayache, . Towards an identification of tumor growth parameters from time series of images. Miccai .
  4. F.Z. Nouri in collaboration with C. Bell, E. Chang, A. Foss, L. Hazelwood, J. O'Flaherty, C. Please, G. Richardson, B. Gorilla, A. Setchi, R. Shipley, J. Siggers, M. Tindall, and J. Ward, Mechanisms and localised treatment for complex heart rhythm disturbances. UK MMSG Cardiac Arrhythmias .