Exponential decay nonlinear regression analysis of patient survival curves: Preliminary assessment in non-small cell lung cancer

Abstract

Background For processes that follow first order kinetics, exponential decay nonlinear regression analysis (EDNRA) may delineate curve characteristics and suggest processes affecting curve shape. We conducted a preliminary feasibility assessment of EDNRA of patient survival curves. Methods EDNRA was performed on Kaplan-Meier overall survival (OS) and time-to-relapse (TTR) curves for 323 patients with resected NSCLC and on OS and progression-free survival (PFS) curves from selected publications. Results and Conclusions In our resected patients, TTR curves were triphasic with a “cured” fraction of 60.7% (half-life [t1/2] >100,000 months), a rapidly-relapsing group (7.4%, t1/2=5.9 months) and a slowly-relapsing group (31.9%, t1/2=23.6 months). OS was uniphasic (t1/2=74.3 months), suggesting an impact of co-morbidities; hence, tumor molecular characteristics would more likely predict TTR than OS. Of 172 published curves analyzed, 72 (42%) were uniphasic, 92 (53%) were biphasic, 8 (5%) were triphasic. With first-line chemotherapy in advanced NSCLC, 87.5% of curves from 2-3 drug regimens were uniphasic vs only 20% of those with best supportive care or 1 drug (p<0.001). 54% of curves from 2-3 drug regimens had convex rapid-decay phases vs 0% with fewer agents (p<0.001). Curve convexities suggest that discontinuing chemotherapy after 3-6 cycles “synchronizes” patient progression and death. With postoperative adjuvant chemotherapy, the PFS rapid-decay phase accounted for a smaller proportion of the population than in controls (p=0.02) with no significant difference in rapid-decay t1/2, suggesting adjuvant chemotherapy may move a subpopulation of patients with sensitive tumors from the relapsing group to the cured group, with minimal impact on time to relapse for a larger group of patients with resistant tumors. In untreated patients, the proportion of patients in the rapid-decay phase increased (p=0.04) while rapid-decay t1/2 decreased (p=0.0004) with increasing stage, suggesting that higher stage may be associated with tumor cells that both grow more rapidly and have a higher probability of surviving metastatic processes than in early stage tumors. This preliminary assessment of EDNRA suggests that it may be worth exploring this approach further using more sophisticated, statistically rigorous nonlinear modelling approaches. Using such approaches to supplement standard survival analyses could suggest or support specific testable hypotheses.

The Effects of Tissue Regenerative Status on Hormesis in Dogs Irradiated during Their Lifespan

“…where S was survival in relative units, t was the elapsed time in years, and k was the hazard rate (instantaneous death rate per year) [32].…”

Membrane Carbonic Anhydrase IX Expression and Relapse Risk in Resected Stage I–II Non–Small-Cell Lung Cancer

“…The 2- and 5-year probabilities of maintaining freedom from recurrence were 80.7% (95% confidence interval [CI]:(75.3%, 86.4%)) and 74.6% (95% CI:(68.6%, 81.2%)), respectively. When TTR was replotted as an exponential decay (log-linear) curve and subjected to nonlinear regression analysis as previously described, 6 those on the terminal flat portion of the curve (the “cured” fraction) constituted 70% of the overall population, and the estimated half-life to relapse for the 30% relapsing was 20 months (Supplementary Figure 3). …”

The need for speed in advanced non‐small cell lung cancer: A population kinetics assessment

“…8,11 Using loglinear relationships is already used in standard statistical calculations (e.g., in determination of hazard ratios), and population kinetic approaches to OS and PFS data analysis can also be useful in a variety of other ways. [7][8][9][10][11] While concern might be raised about using digitized data from published Kaplan-Meier curves, our experience suggests that this approach is highly reliable. For example, PFS and OS half-lives derived from digitized curve data correlate very highly with their respective median values, with R 2 ≥ 0.96, and gain in PFS half-life is a better predictor of OS gain than is gain in median PFS or PFS hazard ratio.…”

“…For most cancers, OS and progression-free survival (PFS) approximate first order kinetics and can be fit by exponential decay nonlinear regression models. 7,8 This means that one can use a population kinetics approach to calculate OS and PFS half-lives (t 1/2 : the time to death or progression of half of all remaining patients). OS and PFS halflives correlate very strongly with OS and PFS medians, 8 but conceptualizing them as half-lives has advantages.…”

“…In previous publications, we have illustrated examples of curves that fit 2 phase EDNLRA models. 7,9,11 We have also previously published examples of how some individual EDNLRA curves compare to the corresponding Kaplan-Meier curves (Supplementary online figure 1 from Ref. [8]).…”