Mathematical Analysis of Glioma Growth in a Murine Model

Abstract

Five immunocompetent C57BL/6-cBrd/cBrd/Cr (albino C57BL/6) mice were injected with GL261-luc2 cells, a cell line sharing characteristics of human glioblastoma multiforme (GBM). The mice were imaged using magnetic resonance (MR) at five separate time points to characterize growth and development of the tumor. After 25 days, the final tumor volumes of the mice varied from 12 mm3 to 62 mm3, even though mice were inoculated from the same tumor cell line under carefully controlled conditions. We generated hypotheses to explore large variances in final tumor size and tested them with our simple reaction-diffusion model in both a 3-dimensional (3D) finite difference method and a 2-dimensional (2D) level set method. The parameters obtained from a best-fit procedure, designed to yield simulated tumors as close as possible to the observed ones, vary by an order of magnitude between the three mice analyzed in detail. These differences may reflect morphological and biological variability in tumor growth, as well as errors in the mathematical model, perhaps from an oversimplification of the tumor dynamics or nonidentifiability of parameters. Our results generate parameters that match other experimental in vitro and in vivo measurements. Additionally, we calculate wave speed, which matches with other rat and human measurements.

A Spatio-Temporal Model of Macrophage-Mediated Drug Resistance in Glioma Immunotherapy

“…The model could be extended to include cancer stem cells to investigate the heterogeneous response of cancer cells to immunotherapies. Moreover, the in vivo data used to estimate the model parameters are indispensable to a more realistic model because the previous relevant works have showed that the slight changes in the model parameter might significantly influence the prediction results (47)(48)(49). In future work, we will develop data-driven A B D C Figure 6.…”

Traveling Waves of a Go-or-Grow Model of Glioma Growth

“…In our previous work, we have investigated the applicability of a single-equation reaction-diffusion equation model (Rutter et al [24]) and a single-equation densitydependent diffusion model (Stepien et al [31]) to experimental data as well as analyzed the existence of traveling wave solutions in the latter paper. Stepien et al [31] and others such as Scribner and Fathallah-Shaykh [26] have shown that incorporating more complex dynamics into a single equation model can result in behavior characteristic of GBM invasion.…”

Estimating intratumoral heterogeneity from spatiotemporal data

“…Mathematically, we should quantify how much information is needed in order to recover the parameter distributions. Since we are, in effect, estimating many parameters to describe a joint distribution (instead of just 2 parameters for the constant-coefficient reaction-diffusion equation), it is imperative that we understand how much data is necessary to avoid the issue of parameter identifiability (which is a known issue in estimating parameters for the reaction-diffusion equation using only one type of imaging data [49]). Quantifying the uncertainty for the distributions estimated using the Prohorov metric framework remains to be investigated.…”