Abstract 5152: Estimating the fraction of a tumor that is killed by a drug.
Authors: Stein Wilfred D. , Wilkerson Julia , Manasanch Elisabet , Zhuang Sen H. , Bates Susan E. , Fojo Tito 
Published: August 2014

Abstract
We analysed mathematically the progress of thousands of patients enrolled in clinical trials with diagnoses of prostate, renal cell, breast, and thyroid carcinomas and multiple myeloma, treated with cytotoxic and targeted drugs, tumor quantity being measured radiologically or by serum markers (see, e.g., Stein et al Clin Cancer Res. 2011,17, 907-17, for prostate cancer). Treatment often results in reduction of tumor quantity, with subsequent regrowth. In some regression does not occur; in others regrowth is not recorded. All these outcomes are described by a simple mathematical formula (see Eq. (1) below) that describes progress of treatment, yielding the rate of tumor decay, d, and rate of regrowth, g. When regrowth is not seen, g is indistinguishable from zero, as is d if tumor quantity fails to decrease. Parameter g is a surrogate measure of overall survival. Median g values can be used to compare the arms of a clinical trial.
In a minority of cases, increase in tumor quantity is delayed for an extended period of time before regrowth occurs, and the simple formula must be modified to include an additional parameter, Ø, the fraction of tumor killed by the drug, see Eq. (2). In a trial of 750 multiple myeloma (MM) patients treated with liposomal doxorubicin plus bortezomib (PLD + B) or bortezomib (B) alone, 114 cases were better fitted by Eq. (2) than by Eq. (1). For these, the median value of Ø was 0.99, meaning that 99% of the tumor was destroyed by drug, only 1% being left to regrow. Were this 1% of tumor to regrow at rate 0.0131 per day (the median for these 114 patients) it would return to the initial tumor quantity, expanding 100-fold, in ln(100)/0.0131 days or 11.5 months. The OS for these patients who died before the study closed was 12.7 (8.9 to 16) months, close to the predicted. If Ø is less than 0.85, and g is larger than 0.02 per day, the predictions of Eqs. 1 and 2 are similar. Slight scatter in the data will not allow a distinction to be made between the models, nor a definitive value of Ø estimated. Nevertheless, aggregating data from numerous patients enables Ø to be estimated for such a data set. Aggregating the data for the 388 MM patients for whom g and/or d but not Ø could be established, Ø was 0.84 for the PLD + B arm and 0.79 for B. To regrow to the initial tumor quantity requires 2.7 doublings for the PLD + B arm and 2.2 for B, a difference that would result in only a trivial increase in OS. However, the difference between the computed tumor growth rates, 0.0019 and 0.0037 for the PLD + B arm and B respectively, would be expected to lead to a noticeable increase in OS were treatment to continue beyond conventional endpoints.
Measuring the fraction of tumor killed by a drug, and the rate of growth recovery, can give insight into drug/tumor interactions and guide development of new therapies.
Equation 1: f(t) = exp (-d • t) + exp (g • t) -1, exp is the base of the natural logarithms, and f(t) the tumor measurement at time t, normalized to the tumor measurement at day 0;
Equation 2: f(t) = Ø • exp (-d • t) + (1-Ø) • exp (g • t)]
Citation Format: Wilfred D. Stein, Julia Wilkerson, Elisabet Manasanch, Sen H. Zhuang, Susan E. Bates, Tito Fojo. Estimating the fraction of a tumor that is killed by a drug. [abstract]. In: Proceedings of the 104th Annual Meeting of the American Association for Cancer Research; 2013 Apr 6-10; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2013;73(8 Suppl):Abstract nr 5152. doi:10.1158/1538-7445.AM2013-5152