Mathematical Modeling of Glioblastoma Growth and Response to Treatment

Abstract

PURPOSE: Glioblastoma is a devastating disease with generally poor prognosis despite multimodal therapy. However, there is significant variability in individual patient outcomes. Thus, the goal of the current work is to use mathematical modeling in combination with serial MR imaging to personalize prognostication on a patient-by-patient level and to help develop individualized, optimized therapy. METHODS: Mathematical modeling was done using a reaction-diffusion partial differential equation to account for glioblastoma cell diffusion as well as proliferation. To individualize the model for each patient, tumor-specific diffusion and proliferation parameters can be derived for each patient using contrast-enhanced T1 and T2 MR imaging from as few as two days. Subsequent projects, as detailed in the attached publications, focused on either (1) improving the accuracy of the model or (2) exploring applications of the model. With regards to improving the model’s accuracy, 3D-DTI data was incorporated to model anisotropic tumor growth in 3 dimensions. Additionally, the model was extended to incorporate the effect of tumor cell necrosis, which is an important component of glioblastoma growth and progression. With regards to applications, the model was used to describe the tumor cell concentration gradient beyond imaging boundaries, optimization of radiation therapy using the technique of genetic algorithms, and evaluation of the effect of the extent of surgical resection on patient survival. RESULTS: There were several important results from this work, as detailed in the attached publications. Incorporation of 3D-DTI data was qualitatively demonstrated to more accurately reproduce tumor growth and response to radiation therapy than the more traditional, one-dimensional tumor model. Additionally, the current work demonstrated the ability to utilize initial tumor location as a personalized parameter in addition to the tumor-specific diffusion and proliferation parameters used in the model. The incorporation of necrosis into the 3D-DTI model was qualitatively and quantitatively demonstrated to reproduce tumor morphology more accurately than one-dimensional, isotropic tumor growth model as well as the 3D model without necrosis. With regards to model applications, it was found that tumors of the same size on MR imaging may have significantly different tumor cell concentration gradients below the threshold of imaging detection. Specifically, tumors with higher diffusion:proliferation ratios were found to have a greater amount of subthreshold disease burden and a greater spatial extent of tumor cells throughout the brain. The current work also demonstrated that the extent of resection required to improve patient outcomes depends on the tumor-specific proliferation coefficient. CONCLUSIONS: The combination of mathematical modeling and serial MR imaging contributes to a more complete understanding of glioblastoma growth and response to treatments including surgical resection and radiation therapy. The ability to personalize the model to individual patients allows for more tailored therapy in order to optimize patient outcomes.