Nonparametric estimation of the size-metastasis relationship in solid cancers

This paper is concerned with the relationship between the occurrence of metastases and the size of primary cancers. We consider two probabilistic characterizations of this relationship. First is the distribution function of tumor sizes at the point of metastatic transition; second is the probability that detectable metastases are present when the cancer comes to medical attention. The equation relating these two functions is developed and conditions for their being identical are explored. Since the tumor size at the point of metastasis is not usually observable, estimation of the first distribution requires the use of the EM algorithm. Nonparametric methods of estimating both functions are explored, with attention to the fact that tumors often fail to be measured, particularly those that are known to be metastatic. The methods are applied to the estimation of primary tumor size at the point of distant metastasis in lung cancer (epidermoid and adenocarcinoma) and colorectal cancer and at the point of nodal metastasis in breast cancer. Monte Carlo experiments confirm that the bias inherent in the methodology is acceptably small.