  We propose a curve-based Riemannian-geometric approach for general
shape-based statistical analyses of tumors obtained from radiologic images. A
key component of the framework is a suitable metric that (1) enables
comparisons of tumor shapes, (2) provides tools for computing descriptive
statistics and implementing principal component analysis on the space of tumor
shapes, and (3) allows for a rich class of continuous deformations of a tumor
shape. The utility of the framework is illustrated through specific statistical
tasks on a dataset of radiologic images of patients diagnosed with glioblastoma
multiforme, a malignant brain tumor with poor prognosis. In particular, our
analysis discovers two patient clusters with very different survival, subtype
and genomic characteristics. Furthermore, it is demonstrated that adding tumor
shape information into survival models containing clinical and genomic
variables results in a significant increase in predictive power.
