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

Abstract

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 by first-order kinetics with 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 6-to 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.

Drug resistant cells with very large proliferative potential grow exponentially in metastatic prostate cancer

“…In the majority of metastatic solid tumors including prostate cancer, an effective treatment leads initially to a fall or decay in tumor mass followed by growth of the tumor mass after a nadir quantity is reached. We previously showed [12] that the decay part is well characterized by an exponential decay, which confirms that observation. This observation is mathematically represented as M s e -dt = M s /e dt , where M s is the mass of the sensitive cells that are exponentially decaying at rate d over time t. We hypothesize that past the nadir, the majority of drug-sensitive cells have been killed and the tumor now increases in size as the still viable, drug-resistant cancer cells continue to grow.…”

“…In the majority of metastatic solid tumors including prostate cancer, an effective treatment leads initially to a fall or decay in tumor mass that is sensitive to the treatment (M s ) followed by growth of the tumor mass that is resistant to the therapy and is designated M r after a nadir quantity is reached. We previously showed [12] that the decay part is well characterized by an exponential decay, which confirms that observation. This observation is mathematically represented as M s e -dt = M s / e dt , where M s is the mass of the sensitive cells that are exponentially decaying at rate d over time t. After initial decay, the PSA levels start to rise in most patients and the decay term becomes small compared to the growth (g) term and we focus on that term in the scaling analysis.…”

Resisting Resistance

“…This functional form was shown to be a good description of the change in tumor size during therapy in in vitro and in vivo models of targeted therapy in colorectal cancer (Misale et al 2015) and of measures that correlate with tumor volume in prostate cancer (serum prostate-specific antigen) and multiple myeloma (M protein levels) (Blagoev et al 2014). Equation 5 can be used to quantify the concept of primary resistance.…”

The Complexity of Translating Anti-angiogenesis Therapy from Basic Science to the Clinic

“…Although single-agent bevacizumab is active in reducing tumor growth rate constants, as determined by mathematical modeling (Stein et al, 2008), it alone does not induce tumor regression. On the other hand, most chemotherapy agents are cytotoxic and can significantly increase tumor shrinkage rates (Blagoev et al, 2014;Malhotra and Perry, 2003). Therefore, mathematical principle dictates that the combination of the reduced tumor growth rate constants associated with bevacizumab and the increased tumor shrinkage rates associated with chemotherapy will lead to pro-longation of PFS over chemotherapy alone.…”