Abstract A42: A mechanically coupled reaction-diffusion model for predicting in vivo C6 glioma growth in rats
Authors: Hormuth David A. , Weis Jared A. , Miga Michael I. , Yankeelov Thomas E. 
Published: May 2016

Abstract
Introduction: Accurate biophysical models of tumor growth that can be used to provide predictions of future tumor growth and therapy response on an individual basis would be a powerful tool for clinicians to select and guide patient therapy. The reaction-diffusion equation is one such model that has been extensively used to model in vivo glioma growth. This model describes the change in tumor cell number over time due to the net proliferation of tumor cells (reaction) and the random movement of tumor cells (diffusion). One shortcoming of this model is that tumor growth is essentially spatially unrestricted resulting in inaccurate tumor growth predictions. It is well-known, however, that tumor expansion is restricted in response to increased mechanical stresses, and we have incorporated these effects into a more realistic mechanically coupled reaction-diffusion model. Using in vivo MRI measurements of glioma growth, model parameters for both the mechanically and non-mechanically coupled models can be estimated and used to generate individualized tumor growth predictions. Thus, the goal of this study is to evaluate the accuracy of tumor growth predictions made with the mechanically and non-mechanically coupled reaction-diffusion models to in vivo measurements of glioma growth.
Methods: The mechanical reaction-diffusion model alters the standard model by changing the tumor cell diffusion coefficient spatially and temporally as a function of the local von Mises stress. The von Mises stress is determined by calculating the tissue displacement caused by the growing tumor. As the tumor grows and causes increased tissue displacement, the von Mises stress also increases, resulting in a decrease in the tumor cell diffusion coefficient. To evaluate this model, rats (N = 2) with C6 gliomas were imaged over ten days with diffusion-weighted MRI (DW-MRI) and contrast enhanced MRI (CE-MRI). Signal enhancement observed in the CE-MRI datasets were used to identify tumor regions. Tumor cellularity was then estimated within these enhancing regions using DW-MRI data. Model parameters (i.e., tumor cell proliferation rate and tumor cell diffusion coefficient) from the standard and mechanical models were then estimated from day 10, day 12, and day 14's cellularity measurements. For the mechanical model, several weights of mechanical coupling were investigated to determine the optimal value for a given tumor. These estimated parameters were then used to evaluate the model from days 14 through 20 to predict future tumor growth. Predicted tumor growth obtained from the standard and mechanical models were compared quantitatively to the in vivo measurement of tumor growth at the local (concordance correlation coefficient (CCC)) and at the global (percent error in total volume and Dice value) levels.
Results: The standard model predictions ranged from 23.3% to 91.6% error in tumor volume estimates on days 15, 16, 18, and 20. Mechanical model predictions ranged from in 3.6% to 15.5% error on days 15, 16, 18, and 20. The standard model resulted in Dice values ranging from 0.60 to 0.75, while the mechanical model had values ranging from 0.77 to 0.83. However, a poor level of agreement was observed for voxel predictions of tumor cellularity resulting in low CCCs (less than 0.03) for both the standard and mechanical models.
Conclusions: Incorporating the inhibiting effects of mechanical stress on tumor expansion resulted in improved global level predictions (i.e., decreased percent error in tumor volume, and increased Dice values) compared to the standard model. Both models, however, poorly predicted the observed intratumor heterogeneity in cell number. Therefore, the mechanical model is a more accurate description of in vivo C6 glioma growth at the global level, but further improvements are needed to provide a more complete description intratumor cellularity changes.
Citation Format: David A. Hormuth, II, Jared A. Weis, Michael I. Miga, Thomas E. Yankeelov. A mechanically coupled reaction-diffusion model for predicting in vivo C6 glioma growth in rats. [abstract]. In: Proceedings of the AACR Special Conference: Advances in Brain Cancer Research; May 27-30, 2015; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2015;75(23 Suppl):Abstract nr A42.