A stochastic model for the sizes of detectable metastases

A stochastic entirely mechanistic model of metastatic progression of cancer is developed. Based on this model the joint conditional distribution of the ordered sizes of detectable metastases given their number, n, is computed. It is shown that this distribution coincides with the joint distribution of order statistics for a random sample of size n derived from some probability distribution, and a formula for the latter is obtained. This formula is specialized for the case of exponentially growing primary and secondary tumors and exponentially distributed metastasis promotion times, and identifiability of model parameters is ascertained. These results allow for estimation of the natural history of cancer. As an example, it is estimated for a breast cancer patient with 31 bone metastases of known sizes. The proposed model for the sizes of detectable metastases provided an excellent fit to these data.