Improved Modeling of Glioblastoma Proliferation and Necrosis using Growth Parameters Derived from MRI
Vishal Patel1 and Leith Hathout2

1University of California, Los Angeles, Los Angeles, CA, United States, 2Harvard University, Boston, MA, United States

Synopsis

We outline a new model for the proliferation and necrosis of glioblastoma multiforme. Our approach uniquely accounts both for the anisotropic migration of tumor cells through brain parenchyma and for central tumor necrosis. Model parameters relating to cell division, cell migration, and tumor necrosis are estimated directly from serial MR imaging to generate a customized growth profile for each tumor. The proposed model is shown to replicate observed tumor growth more closely than existing techniques. We anticipate that improved modeling of tumor growth profiles will enable more effective tailoring of treatment regimens.

Purpose

Glioblastoma multiforme (GBM) represents the most aggressive and most common primary brain neoplasm in adults. Though the prognosis is generally poor, there is considerable variability in response to treatment, suggesting that the optimal therapeutic approach likely incorporates an individualized assessment of expected tumor proliferation. Here, we develop a novel model for GBM evolution which more accurately replicates the observed progression patterns of known tumors. In particular, our model accounts for tumor cell proliferation, migration of tumor cells along white matter fiber bundles, and, uniquely, tumoral necrosis. The model relies on parameters derived from each patient's MR imaging to produce a growth profile that is unique to the anatomy and tumor characteristics of each case.

Methods

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 tensor1 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 indices2 as in prior works3. Mean cell diffusivity ($D$) and $\rho$ are estimated directly from MR imaging of each tumor via the Fisher approximation4, 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 T2 and post-contrast T1 tumor radii correspond to approximately 16% and 80% of the maximum tumor cell density, respectively5 (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/s2.

Results and Discussion

Figure 2 depicts T2-weighted and post-contrast T1-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 mm2/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 model6 that treats tumor cell diffusion as isotropic and does not account for tumor necrosis (panels C and D), results from a more recently-described model3 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.

Conclusion

We have developed a new computational model for GBM proliferation. The proposed model uniquely accounts for both anisotropic migration of tumor cells as well as central tumoral necrosis. We have demonstrated that growth profiles generated using this model match those of observed tumors more closely than those produced using existing methods. Given the variability of GBM response to treatment, the availability of accurate models of tumor proliferation is likely to be of utility in guiding individualized treatment protocols. Further work will be required to determine whether particular cytogenetic properties of GBM correspond to the observed rates of diffusion, proliferation, and necrosis.

Acknowledgements

No acknowledgement found.

References

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.

Figures

Figure 1: Parameter estimation. Previously reported empiric data suggests that the T2 and post-contrast T1 tumor detection thresholds correspond to 16% and 80% of maximum tumor cell density, respectively. In combination with the Fisher approximation (see Methods), this relationship is used to estimate the values of $D$ and $\rho$ for an individual tumor.

Figure 2: Glioblastoma multiforme proliferation in one patient. Post-contrast T1-weighted (left) and unenhanced T2-weighted (right) images obtained at initial presentation (above) and at 30-day follow-up (below). Tumor radii were measured on these volumes and used to estimate the model parameters for proliferation, migration, and necrosis.

Figure 3: Model evaluation. For comparison, we reproduce the post-contrast T1-weighted images of GBM in a single patient acquired across a 30-day interval (A and B). The remaining columns depict tumor growth simulations in a healthy subject overlaid on the fractional anisotropy map. Three models were evaluated: isotropic cell migration, no necrosis (C and D); anisotropic cell migration, no necrosis (E and F); anisotropic cell migration, with necrosis (the proposed model, G and H).

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