Bevacizumab reduces the growth rate constants of renal carcinomas: a novel algorithm suggests early discontinuation of bevacizumab resulted in a lack of survival advantage

Abstract

Background: To hasten cancer drug development, new paradigms are needed to assess therapeutic efficacy. In a randomized phase II study in patients with renal cell carcinoma, 10 microg/kg bevacizumab (Avastin; Genentech, Inc., South San Francisco, CA) administered every 2 weeks resulted in a longer time to progression but a statistically significant difference in overall survival could not be demonstrated.

Methods: We developed a novel two-phase equation to estimate concomitant rates of tumor regression (regression rate constant) and tumor growth (growth rate constant). This method allows us to assess therapeutic efficacy using tumor measurements gathered while a patient receives therapy in a clinical trial.

Results: The growth rate constants of renal cell carcinomas were significantly lower during therapy with 10 microg/kg bevacizumab than those of tumors in patients receiving placebo. In all cohorts the tumor growth rate constants were correlated with survival. That a survival advantage was not demonstrated with bevacizumab appears to have been a result of early discontinuation of bevacizumab.

Conclusions: Single-agent bevacizumab significantly affects the growth rate constants of renal cell carcinoma. Extrapolating from the growth rate constants, we conclude that the failure to demonstrate a survival advantage in the original study was a result of premature discontinuation of bevacizumab. The mathematical model described herein has applications to many tumor types and should aid in evaluating the relative efficacies of different therapies. Quantitating tumor growth rate constants using data gathered while patients are enrolled in a clinical trial, as in the present study, may streamline and assist in drug development.

Figures

Figure 1

Theoretical plots for the regression/growth model. (A): The curve labeled “Regression” describes that fraction of the tumor that is regressing (decaying) during treatment. The curve depicted is the prediction of this equation with parameter g set to zero (i.e., regression only). The curve labeled “Growth” describes that fraction that grows continuously. The curve depicted is the prediction of this equation with parameter d set to zero (i.e., growth only). The curve labeled “Sum of Growth and Regression” gives the (net) sum of these two processes. The curve depicted is the prediction of the full regression/growth model of equation (1) in the text, with rate constant g set at 100 per day and d set at 10 per day. (B): The sums of concomitant regression and growth for several model tumors with varying regression rate (d) and growth rate (g) constants are depicted, showing that tumor measurements will vary depending on the extent of regression and growth that is occurring concurrently. Several curves are depicted for the “Sum of Growth and Regression.” These curves were generated using the theoretical half-times for growth and regression indicated in the box and demonstrate what the outcome will be for a tumor in which either growth or regression predominates even as the opposite effect is also occurring.

Figure 2

The sum of the perpendicular diameters (as a fraction of the value at the start of treatment assigned a value of one) against time in days for four patients of the 102 for whom sufficient data were available to attempt a full analysis (the full set can be found in online supplementary Figs. S1 and S2). The median number of data points was three per patient and the median time over which data were collected was 147 days. Initially the dataset from each patient was subjected to curve-fitting analysis. For 98 datasets, either g or d (or both) had an associated p < .05. (A) shows the pattern found in all but four of the patients randomized to the placebo arm that fit equation (3)—a pattern of growth either initially as shown in (A) or after a delay. For some of the patients randomized to receive bevacizumab, especially those randomized to the low-dose bevacizumab arm, a similar pattern was obtained. (B) and (C) depict the pattern found in the majority of patients randomized to the high-dose bevacizumab arm that fit equation (1)—regression followed by subsequent regrowth. (D) shows an example where the data showed much scatter and the model did not fit the observed data well—this was only observed in four of the 102 patients, two each from the low-dose and high-dose arms. The lines drawn are the best-fit theoretical predictions of the appropriate equations.

Figure 3

Dotplots of the distribution of the best-fit regression rate constants (d, left side of each panel, filled circles) or growth rate constants (g, right side of each panel, open circles). The horizontal lines in each set are the median values and the 95% confidence intervals. The ordinate is the logarithm of the derived rate constant. Regression rate constants could be measured in a larger number of patients in the bevacizumab arms and consequently the number of filled circles increases as one moves from placebo on the left to high-dose bevacizumab on the right. The values for both d and g varied over a nearly 50-fold range. The regression rate constants, taking all three arms of the study together, were significantly larger (p = .008) than the corresponding set of growth rate constants, with mean values of 10−2.222 day−1 (standard deviation [SD], 100.345) versus 10−2.372 day−1 (SD, 100.360), respectively. For the patients in the placebo arm, the mean g value was 10−2.231 day−1 (SD = 100.342) compared with mean values of 10−2.330 day−1 (SD, 100.289) and 10−2.561 day−1 (SD, 100.364) for patients in the low-dose and high-dose bevacizumab arms, respectively. For patients in the placebo arm, the mean d value was 10−2.332 day−1 (SD, 100.364) compared with mean values of 10−2.265 day−1 (SD, 100.327) and 10−2.138 day−1 (SD, 100.338) for patients in the low-dose and high-dose bevacizumab arms, respectively.

Figure 4

Dependence of patient survival (y-axis in days) on the log of the growth and regression rate constants for patients randomized to each of the three study arms (x-axis). All x-axes are logarithmic scales. Growth rate constants (g, per day) were derived using equation (1) or equation (3) and regression rate constants (d, per day) were derived using equation (1) or equation (2). Survival was strongly correlated (negatively, a higher growth rate being associated with a poorer survival) with the logarithm of the growth rate constant—(A): Pearson’s r = −0.648; p < .001; d.f. = 32; (C): Pearson’s r = −0.643; p = .002; d.f. = 27; (E): Pearson’s r = −0.657; p < .001; d.f. = 30—but not with the logarithm of the regression (decay) rate constant—(B): Pearson’s r = −0.236; p = 0.438; d.f. = 11; (D): Pearson’s r = −0.446; p = .055; d.f. = 17; (F): Pearson’s r = −0.321; p < .096; d.f. = 26. Note that for the correlation with the regression rate constant the curves, while not statistically significant, appear if anything negatively correlated, which is not what would be expected: a higher rate of tumor regression being associated here, if at all, with a poorer survival.

Figure 5

Theoretical predictions based on the median data for patients in the placebo and high-dose bevacizumab arms of the study. This figure is similar to the examples shown in Figure 1B, with the median results here for the sum of the growth and regression curves shown. The x-axis records the number of days after treatment commenced while the y-axis depicts the tumor quantity. The starting quantity of tumor has been arbitrarily assigned a value of 1. The circles are data for two patients—one in each arm—who happened to have actual measured values that approximate the median values for the entire group. Note that, in an individual patient, this would be the curve drawn as a result of the curve fit analysis—in effect a graphic refinement of the raw data. The growth of tumor beginning from an arbitrary starting point of 1 at the time of enrollment in the study to a maximum value of 11.42 at the time of death is shown. The value of 11.42 was estimated by plotting the expected amount of tumor patients in the placebo arm would have had at the time of their death if their tumor continued to grow “off study” at the same rate that it grew while on study and receiving placebo. The bold curve furthest to the left depicting the exponential curve for patients randomized to the placebo arm reaches a value of 11.42 times the on-study value of 1 at 453 days, the median survival time for this group of patients. (The vertical line is set at 453 days and intersects the growth curve at a relative tumor value of 11.42, depicted by the solid horizontal line.) The thin solid curve furthest to the right depicts the median results of the patients randomized to the high-dose bevacizumab arm, again with the starting tumor quantity arbitrarily set as 1. As this was a randomized study, the actual amount would have been expected to be similar to that of the placebo arm and indeed this was the case (33 cm2 for the placebo arm and 37 cm2 for the high-dose bevacizumab arm; difference not significant at p = .643). The thin solid line is what would be predicted for patients randomized to the high-dose bevacizumab arm had their tumors continued to grow at a rate comparable with that measured while they were enrolled in the study and receiving bevacizumab. This is the median overall survival time that one would have predicted for patients in the high-dose bevacizumab arm if one assumed that death occurred in patients when a given amount of tumor was present and if the on-study growth rate constant remained unchanged. The dashed curve shows what the curve would have looked like for patients randomized to the high-dose bevacizumab arm of the study if, after discontinuing bevacizumab therapy, the growth rate of their tumors increased to a value comparable with that of the patients who were randomized to the placebo arm of the study. The arrow at 490 days is the median survival time in days for the patients randomized to the high-dose bevacizumab arm of the study.