Discussion
This new biologically based model demonstrates that it is possible to determine, in a logical and structured fashion, interrelationships between treatment and biological factors of the primary tumor, via their effects on the crucial parameters of resistant sub-clinical tumor burden and growth rates.
The model we developed resulted in additional results and hypotheses over and above those obtained from conventional trial analyses. Women with ER-negative breast cancer are estimated to have more rapid tumor growth rates; they may, therefore, receive benefit from more frequent chemotherapy, as was demonstrated with accelerated chemotherapy in the CALGB 9741 study [
10]. Further evidence to support this comes from an analysis of the CALGB 8541 trial, which tested lower doses/shorter durations of adjuvant CAF chemotherapy [
10]. In that study, for women with ER-negative disease, who would be expected to have more rapidly proliferating tumors, 4 cycles of standard CAF chemotherapy was superior to 6 cycles of lower dose CAF; by contrast, for those with more slowly proliferating ER-positive disease lower dose CAF given for a longer duration was equally effective [
10]. Differential effects of treatment by ER status are particularly pronounced in both breast cancer datasets, though similar personalizing of therapy could be considered within different histological grade, or genomic subgroups. This model will outperform standard models such as the Cox model where factors relate to growth rate differences, which are unlikely to fit the proportional hazards assumption.
What distinguishes this new model from the classical statistical approach of the Cox model is the incorporation of an a priori understanding of biological mechanisms; it does not require the often flawed assumption of proportional hazards, yet produces substantially better model fits in these breast cancer DFS/IDFS datasets than does the equivalent Cox model (see Fig.
3 and supplementary Fig. 1). Interactions with time, and stratification by non-proportional hazards variables such as ER status, can, of course, be incorporated into Cox models to avoid the assumption that hazard ratios are constant across time, but at the cost of added complexity often making the results difficult to understand. This model, in contrast, produces results that relate to the very biological parameters that the researchers are usually trying to affect in the trial design, and are therefore meaningful and directly interpretable. The distinction between these two approaches is significant since our model offers the prospect of developing a clinically useful framework for cancer treatment that incorporates the increasingly sophisticated understanding of tumor biology gained from experimental studies. Such a framework could include features such as measures of tumor genome heterogeneity [
19] and consequent evolutionary dynamics [
20]. The former might be reflected, for example, in the parameters describing resistant volume and growth rate. Such models could incorporate factors such as pharmacokinetic parameters and have the potential to inform personalized treatment regimens [
21,
22]. Because this approach is mathematical, it offers a rigorous and quantitative approach to integrating various aspects of tumor biology and determining the consequences for therapy [
23].
Applying the insights gained from the CALGB study, we can consider the AZURE results in the light of factors relating to tumor growth rate. Firstly, the correlation between ER status and tumor grade again supports the conclusion derived from the model analyses that ER-negative women have faster growing tumors. Then considering the treatment effect by ER status, a fall-off in the IDFS curves at 5 years is only seen in the women with ER-positive disease not in those with ER-negative disease (Fig.
5a, b). We have not found departures such as this to be common, and we have now had considerable experience of using this model across a range of cancers. This analysis therefore suggests that 5 years of zoledronic acid treatment is adequate in women with the faster growing ER-negative tumors, but perhaps insufficient for those with more slowly proliferating ER-positive tumors who continue to relapse at a significant rate well beyond the 5 year time point. This conclusion would not have been forthcoming from the Cox model, which does not deal well with growth rate effects. Although zoledronic acid has a long half-life in bone, there may be a threshold of residual activity and inhibition of bone turnover required for benefit to occur. Tumor dormancy is also a much more prominent feature in ER-positive disease and may well be of relevance to these findings, with >50 % of relapses beyond 5 years reflecting the late, unexplained, emergence of disseminated tumor cells from the dormant state [
24]. Continuous long-term application of treatment appears necessary to prevent this re-awakening of quiescent ER-positive cells unlike the highly proliferative ER-negative disseminated cells which are typically either eradicated by adjuvant treatments or initiate clinically detectable metastases over a much shorter timeframe (typically <5 years). It is intriguing that recent data suggest longer duration of treatment may also benefit patients receiving adjuvant tamoxifen therapy [
25‐
27]. We do not, therefore, assume this effect to be specific to bisphosphonates. The findings are consistent with the long duration of therapy that is now the standard of care in hormone-receptor positive breast cancer, with 5 years as a minimum standard, and with several studies supporting extended therapy to 10 years [
28,
29]. Thus, the model may lead to the generation of novel treatment strategies, as did the original univariate model [
1] in acute lymphoblastic leukemia, Hodgkin’s disease, multiple myeloma, and breast cancer [
6].
It is well recognized that cancers are not homogenous with regard to biology and natural history. This was first documented in breast cancer with the recognition that women with ER-positive disease were more at risk of late relapse than those with ER-negative disease. As molecular and genetic understanding grows in this and other cancers, identification of patients with different outcomes will be increasingly possible. The model demonstrates that it is possible to determine these differences, and their potential interactions with treatment, by modeling the “shape” of the DFS/IDFS curves. Furthermore, with that insight, the mathematical model allows predictions to be made of the long-term benefit beyond the period of follow-up of the study and generate hypotheses that can be tested in prospective randomized trials. This model, as with all such models, represents a considerable simplification of tumor dynamics, since it does not, at present, explicitly incorporate current understanding of immune signaling, angiogenesis, stem-cell subpopulations, and other important host-tumor interactions, but unless they are thought to act other than on the net tumor doubling time and resistant disease volume, their effects could still be estimated. As more data are generated on the biological and genetic characteristics of cancers, this approach may be further developed and additional complexity included as required.
Concepts derived from mathematical modeling have been productive in generating new trials and treatment approaches, for instance of alternating non-cross resistant drug combinations as proposed by Goldie et al. [
30], or hypotheses regarding proportional cell-kill and its exploitation as suggested by Skipper et al. [
31,
32]. More recently, and since the publication of the univariate version of this model, Norton [
14] and Day et al. [
33] used the Gompertzian model of tumor growth in breast cancer and derived fits to observed response duration curves using simulation techniques [
14], and numerical integration [
33]. The Gompertzian model provided the background for the dose-dense chemotherapy schedules [
9,
10] which have since been tested in clinical trials [
11,
12] and implemented in clinical practice. As with all pre-clinical hypotheses in cancer, not all of these insights have been confirmed in prospective clinical studies, but the mathematical model described herein is a continuation of these lines of thought, designed to quantify and add scientific rigor to these concepts by analytically fitting such a model. This brings with it related statistical techniques such as significance testing and confidence intervals, as well as enabling a multivariate form to become a practical tool for trial data interpretation. This approach may be especially relevant in the era of rationally designed targeted therapies, defining subgroups where therapeutic interactions occur. The application of the model warrants confirmation in future prospectively designed clinical trials.