A mathematical model relating response durations to amount of subclinical resistant disease

A mathematical model is presented which seeks to determine, from examination of the response durations of a group of patients with malignant disease, the mean and distribution of the resistant tumor volume. The mean tumor-doubling time and distribution of doubling times are also estimated. The model assumes that in a group of patients there is a log-normal distribution both of resistant disease and of tumor-doubling times and implies that the shapes of certain parts of an actuarial response-duration curve are related to these two factors. The model has been applied to data from two reported acute leukemia trials: (a) a recent acute myelogenous leukemia trial was examined. Close fits were obtained for both the first and second remission-duration curves. The model results suggested that patients with long first remissions had less resistant disease and had tumors with slower growth rates following second line treatment; (b) an historical study of maintenance therapy for acute lymphoblastic leukemia was used to estimate the mean cell-kill (approximately 10(4) cells) achieved with single agent, 6-mercaptopurine. Application of the model may have clinical relevance, for example, in identifying groups of patients likely to benefit from further intensification of treatment.