Author Affiliations
 Min Tang^{1},^{*},
 Jasmine Foo^{2},^{*},
 Mithat Gönen^{3},
 Joëlle Guilhot^{4},
 FrançoisXavier Mahon^{5} and
 Franziska Michor^{1}
 ^{1}Department of Biostatistics and Computational Biology, DanaFarber Cancer Institute, and Department of Biostatistics, Harvard School of Public Health, Boston, MA, USA
 ^{2}School of Mathematics, University of Minnesota, Minneapolis, MN, USA
 ^{3}Department of Epidemiology and Biostatistics, Memorial SloanKettering Cancer Center, New York, NY, USA
 ^{4}INSERM Centre d'Investigation Clinique CIC 0802, Centre Hospitalier Universitaire de Poitiers, France
 ^{5}Laboratoire d'Hématologie et Service des Maladies du Sang, CHU de Bordeaux, Bordeaux, France, Universite Victor Ségalen Bordeaux 2, Bordeaux, France, and INSERM U876, Bordeaux, France
 Correspondence: Franziska Michor, Department of Biostatistics and Computational Biology, DanaFarber Cancer Institute, Boston, MA 02115, USA. Phone: international +617.6325045. Fax: international +617.6322444. Email: michor{at}jimmy.harvard.edu
 FrançoisXavier Mahon, Laboratoire Hematopoïese Leucemique et cible therapeutique, INSERM U876, Université Victor Segalen Bordeaux 2, 146, rue Leo Saignat, 33076 Bordeaux Cedex, France. Phone: international +33.05.57571524. Fax: international + 33.05.56938884. Email: francoisxavier.mahon{at}ubordeaux2.fr
Abstract
Background Chronic myeloid leukemia is successfully managed by imatinib therapy, but the question remains whether treatment must be administered indefinitely. Imatinib discontinuation trials have led to two distinct outcomes: about 60% of patients experienced disease relapse within 6 months of treatment cessation, while the remaining 40% remained diseasefree throughout the duration of followup. We aimed to investigate the mechanisms underlying these disparate clinical outcomes.
Design and Methods We utilized molecular data from the “Stop Imatinib” trial together with a mathematical framework of chronic myeloid leukemia, based on a fourcompartment model that can explain the kinetics of the molecular response to imatinib. This approach was complemented by statistical analyses to estimate system parameters and investigate whether chronic myeloid leukemia can be cured by imatinib therapy alone.
Results We found that there are insufficient followup data from the “Stop Imatinib” trial in order to conclude whether the absence of a relapse signifies cure of the disease. We determined that selection of less aggressive leukemic phenotypes by imatinib therapy recapitulates the trial outcomes. This postulated mechanism agrees with the observation that most patients who have a complete molecular response after discontinuation of imatinib continue to harbor minimal residual disease, and might work in concert with other factors suppressing leukemic cell expansion when the tumor burden remains low.
Conclusions Our analysis provides evidence for a mechanistic model of chronic myeloid leukemia selection by imatinib treatment and suggests that it may not be safe to discontinue therapy outside a clinical trial.
Introduction
Imatinib mesylate (Gleevec^{®}, Novartis Pharmaceuticals, formerly STI571) is the current standard of care for patients with chronic myeloid leukemia (CML), inducing clinical, cytogenetic and molecular remissions and prolonging progressionfree survival.1,2 The phase III multicenter IRIS trial studied 1,106 previously untreated patients with chronic phase disease who were randomized to receive imatinib or interferonα plus cytarabine (AraC). The superiority of imatinib over interferonα plus cytarabine was proven after a median of 19 months' followup. Five years after the initiation of imatinib therapy, 40% of the patients in chronic phase had achieved a complete molecular response (CMR). CMR is defined as the state when residual disease cannot be detected by quantitative reverse transcriptase polymerase chain reaction analysis.3 The estimated overall survival rate at 5 years is 89%, while it is 85% at 8 years.4
The achievement of CMR is not, however, a guarantee of disease eradication since undetectable minimal residual disease may persist, and there is little evidence on whether treatment must be administered indefinitely to prevent recurrence of disease. To determine whether imatinib can be discontinued safely without eliciting a loss of CMR, a pilot study and two clinical trials of imatinib cessation have been conducted.57 About 60% of patients who discontinue imatinib after a period of stable CMR relapse within 6 months of treatment cessation, while approximately 40% of patients continue to present with undetectable disease with a followup beyond 18 months5 (Figure 1A). This dichotomy between early relapse and stable CMR may support the hypothesis that in a minority of patients, prolonged imatinib therapy can lead to eradication of the disease. Alternatively, leukemic cells may persist below the detection limit of quantitative reverse transcriptase polymerase chain reaction assays, but may be prevented from expanding to cause molecular relapse by immunological or other mechanisms.5,8
In this study, we performed statistical analyses to investigate whether the data from the Stop Imatinib (STIM) trial8 support the conclusion that a subset of CML patients can be cured by administration of imatinib alone. We then performed a comprehensive investigation of a mathematical framework of CML treatment response. These analyses led us to propose the hypothesis that imatinib therapy exerts a selection pressure, which leads to a prevalence of leukemic clones that have different growth characteristics from those of the predominant clone at the start of treatment. This work is part of a growing body of literature on theoretical investigations of CML treatment response.918
Design and Methods
Testing for a cure
We first sought to determine whether the STIM trial data, which represent an interim report of this clinical trial, are consistent with the hypothesis that a subset of patients can be cured by imatinib therapy alone. Based on the KaplanMeier curve of the relapse data, in which 59 out of 100 patients lost CMR during the followup period of the trial (Figure 1A), we hypothesized that a statistical cure model may fit the data; such a model has the underlying assumption that a fraction of patients succeeded in eradicating their disease. However, since the distributions of relapse and censoring times partially overlap, it remains possible that the followup of the clinical trial is not long enouchg to provide information about the possibility of a cure. We thus probed whether the data on relapsefree survival were consistent with a model in which a fraction of patients can be cured. Using the nonparametric method based on the KaplanMeier curve, we found that there is no significant evidence that the followup is sufficiently long enough in the STIM trial (see Online Supplementary Information for details). This analysis led us to conclude that it may be possible that imatinib can eradicate CML in a subset of patients; however, the followup in the STIM trial is not long enough to provide statistical support to the existence of a cure. We, therefore, investigated two scenarios: (i) the situation in which a subset of patients (the “cure fraction”) can be cured by imatinib (Figure 1B), and (ii) the situation in which all patients will eventually relapse, i.e. in which the cure fraction is zero. We conducted separate analyses for these scenarios.
The mathematical framework
We utilized a mathematical model of the treatment response of CML cells to imatinib therapy,9,19 which describes four layers of the differentiation hierarchy of the hematopoietic system. Stem cells give rise to progenitors, which produce differentiated cells, which in turn produce terminally differentiated cells. This hierarchy applies to both normal and leukemic cells. Only stem cells have the potential for indefinite selfrenewal, but progenitor and differentiated cells possess the capability to undergo limited reproduction, which, together with differentiation, leads to an expansion of the cell number at each level of the differentiation hierarchy.
The abundances of normal hematopoietic stem cells, progenitors, differentiated cells, and terminally differentiated cells can be denoted by x_{0}, x_{1}, x_{2}, and x_{3}. Their respective leukemic abundances are given by y_{0}, y_{1}, y_{2}, and y_{3}. We assume that homeostatic mechanisms maintain the hematopoietic stem cell population at a constant level and, therefore, introduce a density dependence term, 0, in the stem cell production rate. The BCRABL oncogene is present in all leukemic cells, leading to slow clonal growth of leukemic stem cells and accelerating the rate at which leukemic progenitors and differentiated cells are generated. The system containing stem cells (SC), progenitor cells (PC), differentiated cells (DC) and terminally differentiated cells (TC) is, therefore, described by:
Here density dependence in the stem cell population is given by ϕ = 1/[1+p_{x} (x_{0} + y_{0})] and φ = 1/[1+p_{y} (x_{0} + y_{0})]. The potentially different carrying capacities of normal and leukemic stem cells are represented by the parameters p_{x} and p_{y}. Imatinib therapy reduces the production rates of leukemic progenitors and differentiated cells, and potentially also inhibits the expansion of leukemic stem cells. This change in rates leads to a biphasic decline of the leukemic cell burden. The parameters during imatinib therapy are denoted by r_{y}', a_{y}', b_{y}' etc, and the parameters after imatinib cessation are denoted by r_{y}”, a_{y}”, b_{y}” etc. This mathematical framework can be used to study the dynamics of CML from its inception to diagnosis and its response to treatment as well as posttreatment behavior.9,19
The BCRABL/ABL ratio, for comparison with the clinical data, is calculated by taking into account the abundances of normal and leukemic terminally differentiated cells in the mathematical framework; since these cells are several orders of magnitude more frequent in peripheral blood than other, less differentiated cell types, the latter can be ignored when calculating the BCRABL/ABL ratio.
In the basic version of the framework, we model the leukemic stem cell pool as a homogeneous population whose growth and differentiation kinetics follow particular distributions; these distributions account for the heterogeneity within the population. However, the framework can be extended to explicitly describe separate stem cell types to account for heterogeneity. This more detailed framework provides more specifics than the general framework but does not significantly change the results. See the Online Supplementary Information for details of the mathematical framework and its extensions.
Results
We hypothesized that selective pressure exerted by imatinib therapy, which acts differently on different leukemic phenotypes and may differ in effect between patients, generates variability among the population of patients with regards to the time of loss of CMR after discontinuation of imatinib treatment. We characterized the extent of selection on the leukemic cell population by performing a comprehensive investigation of the parameter space in our mathematical framework to determine the joint density and range of parameters that can explain the observed variation in relapse times, both for the cure and the noncure cases. We focused on the five parameters that influence the time of loss of CMR most significantly: the production rates of leukemic progenitors, differentiated and terminally differentiated cells after imatinib cessation as well as the growth rate of leukemic stem cells both during and after imatinib therapy (see Online Supplementary Information for details).
Our mathematical model represents betweenpatient heterogeneity via variability in patientspecific cell growth and differentiation kinetics. Two patients with identical parameter values will have identical modelpredicted cell growth profiles over time. Given a particular set of parameter values, the mathematical model can be computationally solved to evaluate the resulting relapse time; however, from a given relapse time it is not possible to determine a unique set of corresponding parameter values. In addition, the characteristics of the underlying parameter distributions for the growth and differentiation kinetics are unknown. For these reasons, we utilized a retrospective approach to determine the parameter distributions given the observed relapse time data. The first step in this process is to identify the distribution that best fits the data; this distribution depends on whether it is possible to cure CML under the assumptions of the model, as outlined in the following two sections.
The cure model
To investigate the scenario in which a cure may be possible, we first fitted the relapse data of the STIM trial using parametric cure models for censored data. Of the parametric distributions tested, the loglogistic distribution exhibited the best fit (Figure 1B). We then numerically solved the differential equation system to obtain the relapse time, defined as the time when the percentage of leukemic cells in peripheral blood exceeds 10^{5}, for different parameter sets (see Online Supplementary Information for details). A fine grid (over 260 million samples) was used to sample the fivedimensional parameter space, and the corresponding times of loss of CMR or followup were determined using the mathematical model. Here CMR is defined as the time when the BCRABL/ABL transcript level decreases to 10^{5}; this cutoff can be modulated to investigate alternative definitions. Such variations may lead to small changes in the estimated parameter values but do not alter our main conclusions (data not shown). We then selected the subset of outcomes from these times that recapitulated the fitted loglogistic cure curve up until the maximum followup and then retrospectively found S1, the set of parameters in the mathematical model that resulted in this subset of times. The set S1 then represents the sets of samples from the joint density of the five parameters in the mathematical framework for the scenario in which a cure may be possible.
When comparing the KaplanMeier curve obtained from the set S1 to the KaplanMeier survival curve of the clinical data,5 we found that there was no significant difference between the simulated curve and the clinical data (P value = 0.98 based on the logrank test, Figure 1B). Thus, under the assumption of a cure, the joint density implied by the sample set S1 results in a KaplanMeier survival curve matching the clinical survival curve for relapse times. The characteristics of the five parameters are summarized in Table 1, and their marginal densities are shown in Figure 2AE. The densities of the production rates of progenitors and differentiated cells after imatinib cessation are correlated (Figure 2F), but no other parameters show any significant associations.
Using this approach, we determined that the selection pressure of imatinib on leukemic stem cells leads to moderate changes in the growth rate of this population after discontinuation of the treatment; the growth rate of leukemic stem cells after cessation of therapy (mean of 0.0062, median of 0.0040 per day) is smaller than that before initiation of treatment (considered to be 0.008 per day for all patients9,19). The growth kinetics of progenitor, differentiated cell and terminally differentiated cell populations are sculpted to a larger extent by imatinib treatment. Progenitors, for instance, were found to have a mean posttreatment production rate of 0.2102 (median 0.1100) while the rate before initiation of therapy was 0.8.9,19 Table 1 shows the full set of estimated parameter values. Note that differential growth rates and differentiation kinetics of leukemic cell types between patients also lead to disparate abundances of these types, which contributes to the variability in the relapse kinetics among the patients.
The noncure model
We then sought to investigate the scenario in which eventual relapse is assumed for all patients, since their disease cannot be cured by the administration of imatinib alone. We first fitted the relapse data of the STIM trial using parametric survival models for censored data. Of the parametric distributions tested, the lognormal distribution exhibited the best fit (Figure 1C). We then again numerically solved the differential equation system to obtain the relapse time for different parameter sets, and selected the subset of outcomes from these times that recapitulated the fitted lognormal survival curve up until the maximum followup. The set S2 therefore represents the sets of samples from the joint density of the five parameters for the case in which a cure cannot be achieved.
When comparing the KaplanMeier curve obtained from the set S2 to the KaplanMeier survival curve of the clinical data,5 we found that there was no significant difference between the simulated curve and the clinical data (P value = 0.72 based on the logrank test, Figure 1C). Thus, under the assumption of no cure, the joint density implied by the sample set S2 results in a KaplanMeier survival curve matching the clinical survival curve for relapse times. The characteristics of the five parameters are summarized in Table 2, and their marginal densities are shown in Figure 3AE. Again, the densities of the production rates of progenitors and differentiated cells after cessation of imatinib treatment are correlated (Figure 3F), but no other parameters showed any significant associations.
Using this approach, we determined that the selection pressure of imatinib on leukemic stem cells leads to very minor changes in the growth rate of this population after discontinuation; the growth rate of leukemic stem cells after cessation of therapy (mean of 0.008, median of 0.007 per day) is similar to that before initiation of treatment (0.008 per day for all patients). The growth kinetics of progenitor, differentiated cell and terminally differentiated cell populations, however, are sculpted to a larger extent by imatinib treatment. For instance, the posttreatment production rate of progenitors was 0.177 (mean; median 0.070) while the rate before initiation of therapy was 0.8. Table 2 shows the full set of estimated parameter values.
These results suggest that imatinib may select leukemic phenotypes associated with the production of fewer nonselfrenewing cells, thereby leading to a slower expansion of cell numbers throughout the differentiation hierarchy (Figure 4A). Notably, the results of the scenario in which a cure is achievable and the scenario in which patients cannot be cured by imatinib alone lead to very similar results. In both cases, the selection pressure exerted by imatinib enhances the frequency of leukemic stem cell clones with growth properties that are less aggressive than those of the predominant clone at the start of treatment (Figure 4A). This selection effect on leukemic stem cells is more pronounced if we assume that patients who had not yet relapsed at the time of censoring were cured as compared to the assumption in which patients may relapse at a later time. In contrast, the estimated parameters governing the more differentiated cell types are very similar between the two scenarios (Tables 1 and 2). Thus, if a cure is achievable, the selection effect of imatinib on leukemic stem cells is expected to be stronger (mean of 0.0062 after treatment, 0.0080 before treatment) as compared to the case in which imatinib cannot eradicate the disease (mean of 0.0079 after treatment, 0.0080 before treatment). An alternative mechanism that may act in concert with selection of less aggressive phenotypes is the suppression of leukemic cells by immunological, microenvironmental, or densitysensing processes (Figure 4C and D). These processes were not explicitly considered in our mathematical modeling approach since at this time, insufficient quantitative data are available to estimate parameters associated with these mechanisms. These processes are the subject of ongoing investigation.
Discussion
In this paper, we investigated the disparate outcomes in the STIM trial and discussed two scenarios: the possibility of CML patients being cured by imatinib therapy alone (Figure 1B) and the absence of a cure (Figure 1C). Even though the cure model (Figure 1B) visually gives the impression of a better fit than the noncure model (Figure 1C), the followup time in the STIM trial was not long enough to provide the data to distinguish between these two models statistically. We, therefore, discussed both cases. Notably, under the noncure assumption, the survival function eventually decreases to zero. Thus no single parametric density will be able to approximate the tail plateau present in the KaplanMeier curve of the clinical data. We chose the lognormal distribution, which presented the best fit among a set of distributions and our analysis showed that there was no significant difference between the simulated curve and the clinical data (P value = 0.72 based on the logrank test, Figure 1C). Note that all analyses were performed using data from all patients enrolled in the STIM trial and were not based solely on those who did or did not experience a relapse of their disease during the followup period.
Our approach suggests a mechanistic hypothesis explaining the disparate outcomes of imatinib discontinuation trials57 (Figure 4A and B): we propose that the selection pressure exerted by imatinib leads to an increase in the frequency of leukemic clones that have slower growth and differentiation properties as compared to those of the predominant clone at the start of treatment (Figure 4A). This less aggressive phenotype may stem from a lessened capability of leukemic stem cells to produce more differentiated populations and/or a decreased ability of progenitors and differentiated leukemic cells to undergo limited cell division. The selection of clones with altered growth kinetics then leads to variability among patients with regard to the aggressiveness of their disease, thereby generating a distribution of times at which patients experience loss of CMR after imatinib cessation (Figure 4B). This distribution, together with censoring due to limited followup in the STIM trial and a potential cure of a fraction of patients, causes a dichotomous outcome in that some patients relapse within the trial period while others remain in CMR during the period of observation. This postulation of selective pressures exerted by imatinib on different clones within patients represents a novel concept that differs from previously discussed effects of imatinib in that intra as well as interpatient valiability in the growth kinetics of leukemic cells is taken into account.
In many situations, there is marked heterogeneity in phenotype even if cells are genetically identical.2023 Similarly, the leukemic stem cell population may represent a continuum of phenotypes with disparate growth and differentiation kinetics; indeed, experimental evidence suggests that both the amount of BCRABL mRNA and second site mutations alter the fitness of leukemic cells.24,25 Furthermore, it was recently demonstrated that leukemic stem cells in acute lymphoblastic leukemia are highly heterogeneous, harboring clones with varied growth kinetics.26,27 If this is also the case for CML, which remains to be proven experimentally, we might then speculate that imatinib exerts a selection pressure leading to an adaptation of the leukemic stem cell population; this may explain the different kinetics of recurrence after discontinuation of treatment. Based on our results, we propose that imatinib therapy diminishes the clones with the most aggressive growth potential. However, unless treatment is administered in perpetuity, populations with less malignant properties may persist and lead to disease relapse after a variable duration of CMR. As a longer followup of the trial becomes available, we may be able to provide statistical proof that a cure is possible in a subset of these patients; however, at this time such a conclusion cannot be made.
Our results may also provide insights into other clinical characteristics such as nonresponse and early relapse. Such clinical scenarios, in the context of our framework, could be explained by heterogeneity in the leukemic stem cell population as well as differentiation kinetics both within and between patients, such that patients with an intrinsically more aggressive disease  due to either additional genomic alterations or epigenetic variability  are less likely to show an initial response. The patient and clonespecific response to imatinib as well as immune system interactions could then explain the rates of relapse after treatment cessation.
An alternative mechanism that may act in concert with selection of less aggressive phenotypes is the suppression of leukemic cells by immunological, microenvironmental, or densitysensing processes (Figure 4C and D); in a small proportion of patients, these factors may even lead to disease eradication. There is a clinical distinction, however, between residual disease that can be maintained in perpetuity without treatment, thereby leading to a “cure”, and the maintenance of an unstable equilibrium of the leukemic cell population which may break down to lead to relapses. Identification of the factors that cause either scenario is an important goal.
For simplicity, we assumed that the leukemic cell characteristics are the same in all patients before the initiation of therapy. This modeling choice was made to prevent the need to estimate the distributions of leukemic cell division and differentiation parameters since no data are yet available for this purpose. However, a parsimonious explanation of the trial outcomes is that patients with intrinsically more aggressive disease, as indicated by a high Sokal score (a prognostic test performed at diagnosis to characterize a patient as having a low risk, intermediate risk or high risk based on diagnostic markers including spleen size, platelet count, patient's age, and blast count), have a higher risk of relapsing early. In addition, administration of imatinib therapy can decrease the severity of the disease by selecting for less aggressive clones, which were present already at the beginning of therapy. We did not explicitly incorporate the Sokal score into our mathematical framework since it includes covariates such as age and spleen size, whose relationship to leukemic cell numbers and growth kinetics remain poorly understood. However, it would be of significant interest to incorporate information about the phenotypic characteristics of the leukemic cell burden of a patient into his/her Sokal score to aid the prediction of treatment response and relapse time after treatment discontinuation.
It has been suggested that it is impossible to cure CML using targeted therapy because leukemic stem cells cannot be eradicated.28 The sole treatment likely to succeed is allogeneic stem cell transplantation; however, late molecular relapses have been reported even after this treatment option.29 In the STIM trial, we observed that most patients relapsed after a few months, but several instances of late relapse and fluctuations of BCRABL levels after imatinib discontinuation also occurred. Additionally, 40% of patients did not relapse within our limited time of followup. Our mechanistic model invoking selection of less aggressive phenotypes by imatinib therapy may contribute to an explanation for these disparate outcomes, along with the effects of the immune system and microenvironment. Our work suggests the need to perform experimental investigations into the phenotypes of leukemic cells as well as statistical analyses of the patterns of relapse after nilotinib or dasatinib discontinuation, since these secondgeneration drugs are able to induce CMR in a greater fraction of patients than imatinib.30,31
Acknowledgments
the authors would like to thank Subhajyoti De, Yi Li, Giovanni Parmigiani, and Eric V. Slud for discussion and comments.
Funding: this work was supported by NCI grants R01CA138234 (MT, MG and FM) and U54CA143798 (JF, MG and FM).
Footnotes

↵* These authors contributed equally to this work.

The online version of this article has a Supplementary Appendix.

Authorship and Disclosures: The information provided by the authors about contributions from persons listed as authors and in acknowledgments is available with the full text of this paper at www.haematologica.org.

Financial and other disclosures provided by the authors using the ICMJE (www.icmje.org) Uniform Format for Disclosure of Competing Interests are also available at www.haematologica.org.
 Received January 23, 2012.
 Revision received February 22, 2012.
 Accepted March 8, 2012.
 © 2012 Ferrata Storti Foundation