Resistance models to EGFR inhibition and chemotherapy in non-small cell lung cancer via analysis of tumour size dynamics

Target lesion measurements of patients from 10 arms of 9 clinical studies were collected. References of the studies used can be found in the first column of Table 1. All data except for the studies using gefitinib were available in ProjectDataSphere [5, 6]. Gefitinib data were obtained from AstraZeneca. Only patients who had a response to treatment, i.e., those who were classified as CR, PR, or SD at the first visit, were taken forward for further analysis. This allowed us to look at response followed by resistance (hence, patient relapse), the objective of this study. On average, patients had 4–5 CT scans measured every 6–8 weeks and were followed on treatments. Gefitinib was dosed orally 250 mg daily, whereas the chemotherapy was dosed every 3 weeks; for further details of the studies, we refer the reader to the original studies (note that there was no information within the data sets on whether the tumour size measurements were below the limit of quantification).

The time-series models used to analyse individual lesion dynamics are graphically depicted in Fig. 1. The radiologically measured longest diameter is used as a surrogate for tumour size and so tumour cell population. For the de-novo resistance model, four parameters needed to be estimated: (i) initial size of the drug sensitive part of a lesion, Y₁(0); (ii) initial size of the drug resistant part, Y₂(0); (iii)–(iv) the corresponding rates of decay, d, and of growth, g. For the acquired resistance model, again, four parameters needed to be estimated: (i) the initial longest diameter, Y(0); the adaptation (mutation) rate, c; (iii) the decay rate, d; (iv) the growth rate, g. Therefore, the two mathematical models of shared common parameters and structure, with one difference; however: one model allowed for the selection of a pre-existing resistant clone during treatment; the other allowed for conversion to resistant disease during treatment (without further granularity in the data, we were not able to model a combination of both scenarios). Full derivation of model equations for both de-novo and acquired resistance models can be found in the supplemental methods, analytical solutions are provided in the following:

The two models are essentially sum of exponentials but with different parameterisations.

The resultant models were fit to the longest diameter data by placing them within a non-linear mixed-effect framework. All parameters were assumed to follow a log-normal distribution. Residual errors were assumed to be additive. Diagnostic plots (standardised residuals versus individual fitted; individual fitted versus observed; and standardised residuals over time) and parameter values for the final models are reported in the supplemental results.

A model competition-based approach, as used in survival prediction competitions [7], was implemented to assess which model best-described the time-series of individual lesion longest diameters in the following way. 1000 bootstrap samples [8] of the exact sample size of the original study arms were generated from the original data by sampling at patient level. Both models were fitted to each sample with knowledge of which lesion belonged to which patient. Bayesian Information Criterion (BIC), which provides information on the degree to which the model describes the data, was recorded for each model. Using these values across all samples, we calculated Bayes Factors to assess how likely one model was, over another, at describing the data [7] for that particular study arm. Bayes Factors are calculated in the following way which is based on the method used by Guinney et al. [7] Two models M1 and M2 are fit to a bootstrap sample Bi (with i varying from 1 to 1000). Therefore, for each sample, i, we generate a Bayesian Information Criteria value for both models, BIC-M1i and BIC-M2i, for, respectively, models M1 and M2. To generate a Bayes’ factor, we calculate the number of samples for which BIC-M1i > BIC-M2i and divide this quantity by the number of samples for which BIC-M1i < BIC-M2i. This then provides us with a score of how often model M1 provides a better fit over model M2 across the bootstrap samples. Thus, if the value of Bayes’ factor is 1, then both models are equally as good as each other in describing the data. If the value is different to 1, then one model is preferred over the other; the larger the deviation from 1, the stronger the evidence for one model over another. From the resultant resistance model, we assessed the effect of treating lesions independently from each other versus accounting for which lesion belonged to which patient, nesting between- and within-patient random effects, using the same mixed-effects approach.

In addition to performing a model-based analysis, we also assessed whether differences in dynamics between study arms could be visualised from the raw data. Our approach involved the use of a first-order autoregressive (AR) model [9], Y_n+1 = αY_n, where Y_n is the size of an individual lesion at visit n, and hence, α represents the relative change in individual lesion size between current and previous visits. For each lesion in each patient in each study arm, we generated a series of α values over time that described the relative change in individual lesion size from one visit to the next. To visualise how these α values changed over time, we employed the following approach. For the study arms which we wanted to compare, we visualised the frequency of α values over time, to ensure that data collection times were consistent between the study arms being compared. If frequency distributions overlapped, it implied data collection between the two study arms was independent of time. We then proceeded to calculate an ROC AUC value (area under the receiver-operating characteristic curve) at each post-baseline visit, which calculates how well the α values over two consecutive visits could discriminate between those visits. The series of ROC AUC values together with their 95% confidence intervals were plotted over visit number from the study arms of interest and compared visually. The resultant graph highlighted how tumour size changed from one visit to the next and how this differed between two treatments.

All analyses were performed in R v3.1.1. [10] The mixed-effects modelling analysis was conducted using the nlme package [11].

There are two notable observations surrounding the clinical study characteristics (Table 1). First, the numbers of deaths due to progression are low; less than 10% in each study. This suggests that patients who have a CR, PR, or SD response at their first visit are unlikely to die before their disease radiologically progresses. Given that imaging time-series ceased to be collected once a patient’s disease had progressed, these data show that the time-series drop-out mechanism is not informative of survival. Rather, it is informative of when a patient stops taking one treatment and moves on to the next.

Second, the distribution of the longest diameter sizes across all lesions within a study is consistent across all studies. This shows that there is a degree of consistency, based on the initial size, in the choice of lesions by radiologists across all these studies.

Results of the resistance competition between the de-novo and acquired models show a degree of consistency for the same drug within the same line of treatment (Table 2). In the first-line (treatment naive) setting, we found that, for paclitaxel/carboplatin, the acquired model was preferred, whereas, for gefitinib, the de-novo model was preferred. Simulated mean profiles, from the preferred models, with 95% CI overlaid on top of the raw data can be seen in the top row of Fig. 2. A comparison of those simulated profiles between gefitinib and paclitaxel/carboplatin can be seen in the middle row of Fig. 2. The comparison confirms the resistance competition findings: tumour size dynamics differ between gefitinib and paclitaxel/carboplatin for treatment naive patients. We see that gefitinib shrinks tumours quicker than paclitaxel/carboplatin, log(d) is − 4.47 (95% CI − 4.58, − 4.35) for gefitinib versus − 5.47 (95% CI − 5.56, − 5.38) for paclitaxel/carboplatin. In addition, the rate of re-growth appears to be faster under paclitaxel/carboplatin than gefitinib (log(g) is − 5.42 (95% CI − 5.56, − 5.28) for gefitinib versus − 4 (95% CI − 4.11, − 3.88)for paclitaxel/carboplatin), which is further confirmed by analysing the decay and growth rate parameters (Supplemental Tables S4–S7). Furthermore, these results appear to be consistent across two independent studies for both treatments, middle row of Fig. 2. Thus, the results suggest that resistance under gefitinib treatment leads to a less aggressive resistant clone than resistance under paclitaxel/carboplatin in the treatment naive setting.

Finally, we assessed the size of the resistant fraction under gefitinib treatment in IFUM and IPASS, bottom row of Fig. 2: it is approximately 0.3 in IFUM (EGFR mutant selected Caucasian patients) and 0.2 in IPASS (unselected Asian patients). Furthermore, the within-patient variability in the resistant fraction is similar to the between-patient variability across both studies.

In the second-line setting for docetaxel, we found the acquired model was generally preferred; most convincingly in VITAL and less convincingly in INTEREST. The mean model simulation from the preferred model together with the 95% CI overlaid on top of the raw data can be seen in the top row of Fig. 3. On comparing the simulations, see bottom row of Fig. 3, we find that the dynamics are consistent and that the uncertainty in the re-growth phase varies between studies; VITAL showing the lowest degree of uncertainty, followed by ZODIAC and then INTEREST.

Three study arms (from IDEAL1, ZEST, and SUNITINIB) with EGFR inhibitors in the second-line setting could also be examined. Although the data could be described well by the model, as shown in Figs. S8–S13, we found that one model was not more likely than the other, in describing the time-series dynamics. Since no winner could be found, further analyses of these treatments were not conducted.

In studies where a preferred model was found, that model was taken forward into the next set of analyses; these explored whether dynamics across lesions, within a patient, were correlated. We found that knowledge of “which lesion belongs to which patient” is important. Thus, there is a degree of correlation in time-series dynamics across lesions, within a patient.

In addition to the model-based analyses described to this point, we also assessed whether these differences in dynamics could have been detected visually. The study chosen to assess this was IPASS, since (i) we found that the dynamics were different for the two treatment arms of that study, and (ii) the data collection was similar across the two arms (Supplemental Fig. S15). The process undertaken to analysing the time-series using an autoregressive technique is shown pictorially in Fig. 4.

The raw data with mean model simulations overlaid can be seen in the top row of Fig. 4. The distribution of alpha values, defined as relative change in tumour size from one visit to the next, across visits can be seen in the middle row of Fig. 4. The overall trend in the distribution of alpha values from visit to visit is subtly different. This difference becomes more apparent in the plot at the bottom row of Fig. 4. It shows how well alpha values can discriminate between two consecutive on-treatment visits. For example, at the 3 v 2 point, the ROC AUC values correspond to how well alpha values can discriminate between whether the values are from Visit 2 versus Visit 3. Overall, the dynamics under paclitaxel/carboplatin treatment change continuously until we reach Visit 6. In contrast, for gefitinib, the largest changes in dynamics occur early on; there is then a period of no-change, followed by a subtle change, as we move from Visit 5 to Visit 6. These results show that the dynamics in tumour size, assessed via an autoregressive visualisation approach, are markedly different for gefitinib versus paclitaxel/carboplatin, in further support of the findings using the resistance model competition.