Improved Modeling of Glioblastoma Proliferation and Necrosis using Growth Parameters Derived from MRI

Vishal Patel^{1} and Leith Hathout^{2}

We conceptualize the tumor cell concentration within a particular voxel as influenced by: 1) the intrinsic multiplication rate ($$$\rho$$$) of tumor cells within that voxel, 2) the influx of tumor cells migrating from adjacent voxels as driven by the cell diffusion tensor ($$$\mathbf{D}$$$), and 3) tumor necrosis once cell density outstrips the supporting capacity of the underlying substrate. The change in glioma cell concentration $$$c$$$ in the voxel at position $$$\mathbf{x}$$$ over a time interval $$$t$$$ is then given by:$$\frac{\partial c}{\partial t}=\begin{cases}{\rho}c\left(1-\frac{c}{K}\right)+\mathbf{D}\frac{\partial^{2}c}{\partial\mathbf{x}^2}&c\lt0.8K\\nc&c\ge0.8K\end{cases}$$where $$$K$$$ represents the maximum cell density, and $$$n\lt{1}$$$ denotes the cell survival rate once the necrosis threshold is achieved.

Given that cell migration is more strongly influenced by local tissue architecture than is the diffusion of water, we first estimate the water diffusion tensor^{1} and then transform it into the desired cell diffusion tensor via differential scaling of the eigenvalues based on Westin's linear ($$$c_l$$$), planar ($$$c_p$$$), and spherical ($$$c_s$$$) anisotropy indices^{2} as in prior works^{3}. Mean cell diffusivity ($$$D$$$) and $$$\rho$$$ are estimated directly from MR imaging of each tumor via the Fisher approximation^{4}, which states that the tumor margin progresses with constant velocity ($$$v=2\sqrt{D\rho}$$$), solved using two time points and established empiric evidence that the T_{2} and post-contrast T_{1} tumor radii correspond to approximately 16% and 80% of the maximum tumor cell density, respectively^{5} (see Figure 1). The necrosis parameter $$$n$$$ is also tuned to match the observed velocity.

To validate this model, we simulate tumor growth in the brain of a healthy adult volunteer using kinetic parameters derived from the imaging of a GBM patient and compare the observed tumor profile to those produced by existing models. All images were acquired on a 3 T scanner with diffusion-weighted images acquired over 64 distinct gradient directions at a *b*-value of 1000 mm/s^{2}.

Figure 2 depicts T_{2}-weighted and post-contrast T_{1}-weighted images, acquired 30 days apart, from a patient who later underwent resection with final histology confirming GBM. Using the tumor radii measured on these images, the cell diffusion coefficient ($$$\mathbf{D}$$$ = 0.0825 mm^{2}/day), proliferation rate ($$$\rho$$$ = 0.33/day) and necrosis parameter ($$$n$$$ = 0.91) were determined.

In Figure 3, we illustrate attempts to reproduce this observed tumor profile *de novo* using anatomical information from MR imaging of a healthy, age and gender matched volunteer. We compare, over a simulated 30 day interval, results from an early model^{6} that treats tumor cell
diffusion as isotropic and does not account for tumor necrosis (panels C and D), results from a more recently-described model^{3} that incorporates anisotropy into cell migration profiles but does not consider necrosis (panels E and F), and finally, results from our proposed model which accounts for both anisotropic tumor cell diffusion and central tumoral necrosis (panels G and H). We note a marked improvement in similarity between the current model's results and the actual observed tumor growth (panels A and B).

The results demonstrate that the consideration of anisotropic cell migration and central necrosis are both essential in generating realistic GBM growth profiles. We note in particular that the omission of a necrosis term implies the persistence of a high density nidus of cells at the tumor core which may have unintended implications for treatment planning.

1. Basser PJ, Mattiello J, LeBihan D. MR diffusion tensor spectroscopy and imaging. Biophysical Journal. 1994;66(1):259–67.

2. Westin CF, Peled S, Gudbjartsson H, Kikinis R, Jolesz FA. Geometrical diffusion measures for MRI from tensor basis analysis. International Society for Magnetic Resonance in Medicine. 1997:1742.

3. Jbabdi S, Mandonnet E, Duffau H, Capelle L, Swanson KR, Pélégrini-Issac M, Guillevin R, Benali H. Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging. Magnetic Resonance in Medicine. 2005;54(3):616–24.

4. Wang CH, Rockhill JK, Mrugala M, Peacock DL, Lai A, Jusenius K, Wardlaw JM, Cloughesy T, Spence AM, Rockne R, Alvord EC Jr, Swanson KR. Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model. Cancer Research. 2009;69(23):9133–40.

5. Swanson KR, Rostomily RC, Alvord EC Jr. A mathematical modelling tool for predicting survival of individual patients following resection of glioblastoma: a proof of principle. British Journal of Cancer. 2008;98(1):113–9.

6. Tracqui P, Cruywagen GC, Woodward DE, Bartoo GT, Murray JD, Alvord EC Jr. A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Proliferation. 1995;28(1):17–31.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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