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A Mathematical Model Relating Response Durations to Amount of Subclinical Resistant Disease¹

Imperial Cancer Research Fund Clinical Oncology Unit, Guy's Hospital, London SE1 9RT [W. M. G., M. A. R.]; Imperial Cancer Research Fund Department of Medical Oncology, St. Bartholomew's and Homerton Hospitals, London EC1A 7BE [M. L. S.]; and Department of Oncology, University College and Middlesex School of Medicine, London WC1E 6AU [R. L. S.], England

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 responseduration 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 10⁴ 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.

¹ Supported by the Imperial Cancer Research Fund and the Cancer Research Campaign.

² To whom requests for reprints should be addressed.

Received 2/22/90. Accepted 12/ 5/90.

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