In vivo measurement of local tissue characteristics by modern bioimaging techniques such as positron emission tomography (PET) provides the opportunity to analyze quantitatively the role that tissue heterogeneity may play in understanding biological function. This paper develops a statistical measure of the heterogeneity of a tissue characteristic that is based on the deviation of the distribution of the tissue characteristic from a unimodal elliptically contoured spatial pattern. An efficient algorithm is developed for computation of the measure based on volumetric region of interest data. The technique is illustrated by application to data from PET imaging studies of fluorodeoxyglucose utilization in human sarcomas. A set of 74 sarcoma patients (with five-year follow-up survival information) were evaluated for heterogeneity as well as a number of other potential prognostic indicators of survival. A Cox proportional hazards analysis of these data shows that the degree of heterogeneity of the sarcoma is the major risk factor associated with patient death. Some theory is developed to analyze the asymptotic statistical behavior of the heterogeneity estimator. In the context of data arising from Poisson deconvolution (PET being the prime example), the heterogeneity estimator, which is a non-linear functional of the PET image data, is consistent and converges at a rate that is parametric in the injected dose.