Physics in Cancer Research Therapies with Diverse Mechanisms of Action Kill Cells by a Similar Exponential Process in Advanced Cancers

Successful cancer treatments are generally defined as those that decrease tumor quantity. In many cases, this decrease occurs exponentially, with deviations from a strict exponential being attributed to a growing fraction of drug-resistant cells. Deviations from an exponential decrease in tumor quantity can also be expected if drugs have a nonuniform spatial distribution inside the tumor, for example, because of interstitial pressure inside the tumor. Here, we examine theoretically different models of cell killing and analyze data from clinical trials based on these models.We show that the best description of clinical outcomes is byfirst-order kineticswith exponential decrease of tumor quantity. We analyzed the total tumor quantity in a diverse group of clinical trials with various cancers during the administration of different classes of anticancer agents and in all cases observed that the models that best fit the data describe the decrease of the sensitive tumor fraction exponentially. The exponential decrease suggests that all drug-sensitive cancer cells have a single rate-limiting step on the path to cell death. If there are intermediate steps in the path to cell death, they are not rate limiting in the observational time scale utilized in clinical trials—tumor restaging at 6to 8-week intervals. On shorter time scales, there might be intermediate steps, but the rate-limiting step is the same. Our analysis, thus, points to a common pathway to cell death for cancer cells in patients. See all articles in this Cancer Research section, "Physics in Cancer Research." Cancer Res; 74(17); 4653–62. 2014 AACR. Introduction In a landmark paper published in Cancer Research in 1964 that described a mouse leukemia model, Skipper formulated two laws of tumor kinetics that have been influential in designing treatment schedules and combination therapies (1, 2). The first law states that dividing cancer cells if left undisturbed grow exponentially. The second law states that when treated with a fixed amount of a therapy, the fraction of cancer cells killed remains the same irrespective of the total number of cancer cells. Although the cellularmechanismof the first law can be traced to symmetric cancer cell division, the molecular roots of the second law are still unknown. For example, as a tumor becomes smaller one could expect a larger fraction might be killed if a fixed amount of a therapy is being used, or if the drug is not uniformly distributed in a tumor, onemay anticipate that some parts of the tumor will be more susceptible than others. The second law might also be in conflict with the fact that tumors are heterogeneous. Recent evidence highlighting the genetic diversity of tumors (3) suggests that the cells comprising a tumor have different susceptibilities to killing by a given therapy. Therefore, one might argue a less susceptible fraction is left behind, and the fraction of cells killed with the same dose should decrease with each consecutive drug administration. In agreement with Skipper (Fig. 1), we have previously shown through analysis of tumor measurements obtained from patients that tumor growth occurs exponentially and that the rate of this exponential growth is constant (4–8). The rates are far slower than those of the leukemia models used by Skipper, but the exponential nature of growthwas confirmed.Wehave further shown that in patients with advanced metastatic cancer, the growth rate constant correlates well with overall survival. Although these data have additionally suggested that the regression of tumors is also exponential, wewished to examine other possibilities for the regression of tumor, given the existence of numerous proposed models of how therapies affect tumors. To do this, we mathematically modeled the occurrence of cancer cell death in several ways and analyzed tumor measurements or biomarker data from clinical trials in which a diverse group of anticancer agents was studied in advanced cancers. In analyzing data from these trials, we will show that although the therapies are mechanistically different, ranging, for example, from microtubule-targeting agents to activated T cells, in all National Science Foundation, Arlington, Virginia. Department of Radiology, Massachusetts General Hospital, Harvard Medical School and the Antinula Martinos Center for Biomedical Imaging, Charlestown, Massachusetts. Medical Oncology, Center for Cancer Research, National Cancer Institute, National Institutes of Health, Bethesda, Maryland. Hebrew University, Jerusalem, Israel. Surgery Branch, Center for Cancer Research, National Cancer Institute, National Institutes of Health, Bethesda, Maryland. Note: Supplementary data for this article are available at Cancer Research Online (http://cancerres.aacrjournals.org/). Corresponding Author: Krastan B. Blagoev, Department of Radiology, Massachusetts General Hospital, HarvardMedical School and the Antinula Martinos Center for Biomedical Imaging, Building 149, 13th Street, Charlestown, MA 02129. E-mail: [email protected] doi: 10.1158/0008-5472.CAN-14-042