Gliomas are well known for their potential for aggressive proliferation as well as their diffuse invasion of the normal-appearing parenchyma peripheral to the bulk lesion. This review presents a history of the use of mathematical modeling in the study of the proliferative-invasive growth of gliomas, illustrating the progress made in understanding the in vivo dynamics of invasion and proliferation of tumor cells. Mathematical modeling is based on a sequence of observation, speculation, development of hypotheses to be tested, and comparisons between theory and reality. These mathematical investigations, iteratively compared with experimental and clinical work, demonstrate the essential relationship between experimental and theoretical approaches. Together, these efforts have extended our knowledge and insight into in vivo brain tumor growth dynamics that should enhance current diagnoses and treatments.

Glioblastoma multiforme is a primary brain tumor known for its rapid proliferation, diffuse invasion, and prominent neovasculature and necrosis. This study explores the in vivo link between these characteristics and hypoxia by comparing the relative spatial geometry of developing vasculature inferred from gadolinium-enhanced T1-weighted MRI (T1Gd), edematous tumor extent revealed on T2-weighted MRI (T2), and hypoxia assessed by 18F-fluoromisonidazole PET (18F-FMISO). Given the role of hypoxia in upregulating angiogenic factors, we hypothesized that the distribution of hypoxia seen on 18F-FMISO is correlated spatially and quantitatively with the amount of leaky neovasculature seen on T1Gd. Methods A total of 24 patients with glioblastoma underwent T1Gd, T2, and 18F-FMISO—11 studies preceded surgical resection or biopsy, 7 followed surgery and preceded radiation therapy, and 11 followed radiation therapy. Abnormal regions seen on the MRI scan were segmented, including the necrotic center (T0), the region of abnormal blood–brain barrier associated with disrupted vasculature (T1Gd), and infiltrating tumor cells and edema (T2). The 18F-FMISO images were scaled to the blood 18F-FMISO activity to create tumor-to-blood ratio (T/B) images. The hypoxic volume (HV) was defined as the region with T/Bs greater than 1.2, and the maximum T/B (T/Bmax) was determined by the voxel with the greatest T/B value. Results The HV generally occupied a region straddling the outer edge of the T1Gd abnormality and into the T2. A significant correlation between HV and the volume of the T1Gd abnormality that relied on the existence of a large outlier was observed. However, there was consistent correlation between surface areas of all MRI-defined regions and the surface area of the HV. The T/Bmax, typically located within the T1Gd region, was independent of the MRI-defined tumor size. Univariate survival analysis found the most significant predictors of survival to be HV, surface area of HV, surface area of T1Gd, and T/Bmax. Conclusion Hypoxia may drive the peripheral growth of glioblastomas. This conclusion supports the spatial link between the volumes and surface areas of the hypoxic and MRI regions; the magnitude of hypoxia, T/Bmax, remains independent of size.

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