Modeling breast cancer survival and metastasis rates from moderate-sized clinical data

Predicting time-dependent survival probability of a breast cancer patient using information such as primary tumor size, grade, node spread status, and patient age at the time of surgery can be of immense help in managing life expectations and strategizing postoperative treatment. However, for moderate-sized clinical datasets the application of standard Kaplan-Meier theory to determine survival probability as a function of multiple cofactors can become challenging when continuous variables like tumor diameter and survival time are segmented into a large number of narrow intervals, a problem commonly termed the curse of dimensionality. We circumvent this problem by modeling the patient-to-patient distribution of primary tumor diameter with a realistic, right-skewed function, and then matching the diameter-marginalized survival with the mean Kaplan-Meier survival for the data. We apply this procedure on a recent clinical data from 1875 breast cancer patients and develop parameters that can be readily used to estimate post-surgery survival for an arbitrary time length. Finally, we show that the observed fraction of node-positive patients can be quantitatively explained within a simple tumor growth and metastasis framework. Employing two different tumor growth models from the literature (i.e., Gompertz and logistic growth models), we utilize the observed fraction-node-positive data to determine metastasis rates from the surface of a primary tumor and its patient-to-patient distribution.