Cancer recurrence times from a branching process model

Author summary The majority of cancer related deaths are due to the development of secondary tumors called metastases. However, the dynamics of metastases establishment and growth and their relation with the primary tumor evolution are still not clear. A standard treatment starts with the resection of the primary tumor. At this time metastases may have already formed and still be too small to be detected. The presence of only undetectable metastases poses a challenge for deciding on the follow-up therapy. These small metastases could grow to a detectable size-thus leading to a recurrence of the disease-some time after surgery. We are interested in this time until cancer relapse. We present a mathematical model of metastases formation using tools from probability theory and estimate the model parameters for five different cancer types. Our predictions for the probability of visible metastases present at surgery and the mean time to relapse when no visible metastases are found at surgery are both in agreement with clinical data. As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases. This is the main cause of cancer related mortality. Here we investigate a conceptually simple model of metastasis formation where metastatic lesions are initiated at a rate which depends on the size of the primary tumor. The evolution of each metastasis is described as an independent branching process. We assume that the primary tumor is resected at a given size and study the earliest time at which any metastasis reaches a minimal detectable size. The parameters of our model are estimated independently for breast, colorectal, headneck, lung and prostate cancers. We use these estimates to compare predictions from our model with values reported in clinical literature. For some cancer types, we find a remarkably wide range of resection sizes such that metastases are very likely to be present, but none of them are detectable. Our model predicts that only very early resections can prevent recurrence, and that small delays in the time of surgery can significantly increase the recurrence probability.