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. See all articles in this Cancer Research section, "Physics in Cancer Research."

Figure 1

From Skipper HE. Cancer Res 1964; 24:1295-1302 [1]. “Chart 4” – Hypothetical illustration of the possible importance of drug level and schedule in attempts to achieve “cell cure” in early experimental leukemia (L1210) animals. A. Untreated controls; number of leukemic cells quadruples daily until the number reaches about 10⁹ at which time the animals die. B. Daily drug treatment; low level, long term (until death); plotted to represent a daily 50 per cent “drug kill” of the animals’ leukemic cell population and a daily quadrupling of the surviving leukemic cells. Increase in host life span was achieved, but “cell cure” was not approached. C. Daily drug treatment; moderate level, long term; plotted to represent a daily 75 per cent “drug kill” of the animals’ leukemic cell population; infinite host survival if cumulative drug toxicity, development of drug resistance within the leukemic cell population or the blood brain barrier problem did not intervene. D. Daily drug treatment; high level, short term; plotted to represent a daily 99 per cent “drug kill” of the animals’ leukemic cell population, no other complications and “cure” of a 10⁵ cell inoculum. In our actual experience a single maximum dose has been most effective with this very rapidly proliferating experimental disease.

Figure 2

Tumor surface model of tumor regression where only (principally) cells on the surface of the tumor are killed with each treatment.

Figure 3A

Graphs depicting the solutions of formulas (for constant fraction or exponential model, constant number model and tumor surface model) where tumor volume (or number of cells) is assessed.

Figure 3B

Graphs depicting the solutions of the same formulas as in 3A where the longest linear distance or longest diameter of the tumor is assessed.

Figure 4

Examples of individual data from nine different patients’ “best fit” to any of the seven models discussed in the text and summarized in Table 2

Figure 5

Three scenarios of treatment action: In the scenario shown in the upper panels a treatment affects a constant fraction of 30% of cells (red), and these die. In the scenario depicted in the middle panels some of the treatment-affected cells (green) repair their damage, rejoin the viable tumor cell pool and continue contributing to the growth of the tumor, reducing the fraction of cells killed to 20%. Here, drug resistant cells that increase cell number irrespective of treatment are not shown. Also the sensitive cells (blue) are assumed not to be dividing between treatments. Data obtained at the next assessment cannot distinguish between these two possibilities and only captures the net regression rate. In the lower panels a scenario in which a constant (18) number of cells are killed each time the treatment is administered. At the first treatment the same number of cells is killed in panel one and three, but subsequently the cells are killed at increasingly larger proportions in the third scenario, because the number of cells that are killed stays the same while in the first and second scenario the number of cells killed decreases in time.

Figure 5

Three scenarios of treatment action: In the scenario shown in the upper panels a treatment affects a constant fraction of 30% of cells (red), and these die. In the scenario depicted in the middle panels some of the treatment-affected cells (green) repair their damage, rejoin the viable tumor cell pool and continue contributing to the growth of the tumor, reducing the fraction of cells killed to 20%. Here, drug resistant cells that increase cell number irrespective of treatment are not shown. Also the sensitive cells (blue) are assumed not to be dividing between treatments. Data obtained at the next assessment cannot distinguish between these two possibilities and only captures the net regression rate. In the lower panels a scenario in which a constant (18) number of cells are killed each time the treatment is administered. At the first treatment the same number of cells is killed in panel one and three, but subsequently the cells are killed at increasingly larger proportions in the third scenario, because the number of cells that are killed stays the same while in the first and second scenario the number of cells killed decreases in time.