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Patients undergoing radiation therapy (and their physicians alike) are concerned with the probability of cure (long-term recurrence-free survival, meaning the absence of a detectable or symptomatic tumor). This is not what current practice categorizes as "tumor control (TC);" instead, TC is taken to mean the extinction of clonogenic tumor cells at the end of treatment, a sufficient but not necessary condition for cure. In this review, we argue that TC thus defined has significant deficiencies. Most importantly, (1) it is an unobservable event and (2) elimination of all malignant clonogenic cells is, in some cases, unnecessary. In effect, within the existing biomedical paradigm, centered on the evolution of clonogenic malignant cells, full information about the long-term treatment outcome is contained in the distribution Pm(T) of the number of malignant cells m that remain clonogenic at the end of treatment and the birth and death rates of surviving tumor cells after treatment. Accordingly, plausible definitions of tumor control are invariably traceable to Pm(T). Many primary cancers, such as breast and prostate cancer, are not lethal per se; they kill through metastases. Therefore, an object of tumor control in such cases should be the prevention of metastatic spread of the disease. Our claim, accordingly, is that improvements in radiation therapy outcomes require a twofold approach: (a) Establish a link between survival time, where the events of interest are local recurrence or distant (metastatic) failure (cancer-free survival) or death (cancer-specific survival), and the distribution Pm(T) and (b) link Pm(T) to treatment planning (modality, total dose, and schedule of radiation) and tumor-specific parameters (initial number of clonogens, birth and spontaneous death rates during the treatment period, and parameters of the dose-response function). The biomedical, mathematical, and practical aspects of implementing this program are discussed.

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Abstract.We obtain necessary and sufficient conditions on a compact metric space (K, p) that provide a natural isometric isomorphism between completion of the space of Borel measures on K with the Kantorovich-Rubinstein norm and the space [\ix)(K, />))* or equivalently between the spaces Lip(K, p) and (lip(ZY, /»))** . Such metric spaces are studied and related properties of Lipschitz spaces are established.