Recent mathematical models have been developed to study the dynamics of chronic myelogenous leukemia (CML) under imatinib treatment. None of these models incorporates the anti-leukemia immune response. Recent experimental data show that imatinib treatment may promote the development of anti-leukemia immune responses as patients enter remission. Using these experimental data we develop a mathematical model to gain insights into the dynamics and potential impact of the resulting anti-leukemia immune response on CML. We model the immune response using a system of delay differential equations, where the delay term accounts for the duration of cell division. The mathematical model suggests that anti-leukemia T cell responses may play a critical role in maintaining CML patients in remission under imatinib therapy. Furthermore, it proposes a novel concept of an "optimal load zone" for leukemic cells in which the anti-leukemia immune response is most effective. Imatinib therapy may drive leukemic cell populations to enter and fall below this optimal load zone too rapidly to sustain the anti-leukemia T cell response. As a potential therapeutic strategy, the model shows that vaccination approaches in combination with imatinib therapy may optimally sustain the anti-leukemia T cell response to potentially eradicate residual leukemic cells for a durable cure of CML. The approach presented in this paper accounts for the role of the anti-leukemia specific immune response in the dynamics of CML. By combining experimental data and mathematical models, we demonstrate that persistence of anti-leukemia T cells even at low levels seems to prevent the leukemia from relapsing (for at least 50 months). As a consequence, we hypothesize that anti-leukemia T cell responses may help maintain remission under imatinib therapy. The mathematical model together with the new experimental data imply that there may be a feasible, low-risk, clinical approach to enhancing the effects of imatinib treatment.
Among all mathematical models of CML, our approach is unique in the sense that the experimentally observed anti-leukemia immune response is incorporated into the model. With the addition of the T cell response in our model, persistence of anti-leukemia T cells even at low levels seems to prevent the leukemia from relapsing (for at least 50 months). We therefore hypothesize that anti-leukemia T cells responses may help maintain remission under imatinib therapy. Therapy with imatinib (and other targeted therapies being developed) has the advantage to target leukemic cells more selectively than non-specific therapies such as chemotherapy and radiation. As such, host immune function, including antigen presentation, may be restored more rapidly than after chemotherapy, due to alleviation of leukemia-induced immune suppression. Importantly, normalization of host immune function, while leukemia antigens are still present, may optimally drive anti-leukemia immune responses.
Our model suggests that the balance between immune down-regulation and T cell stimulation by leukemic cells determines the effectiveness of the anti-leukemia T cell response. Studying the optimal level of stimulation led us to define the novel concept of an "optimal load zone" as the range of leukemic cell concentrations where the T cell stimulation rate is optimal. In general, imatinib causes the leukemic cell population to fall into the optimal load zone, stimulating a T cell response most efficiently and to the highest amount before it drops out of this zone. At a certain threshold below the optimal load zone, leukemic cells become essentially invisible to T cells due to low interaction rates, and the immune response contracts. At this point, one would need exogenous stimulation to maintain T cell proliferation.
This led us to hypothesize that cryopreserved autologous leukemic cells, inactivated by irradiation, may be given to patients in remission as vaccines to enhance T cells responses. To study this approach, we added inactivated leukemic cells (unable to proliferate or exert immune suppression) to our model. A strategy of immunotherapy and imatinib treatment for each patient was constructed using an optimization algorithm. Our model predicts that the timing and pacing of the vaccinations are crucial.
Although vaccination optimizations are presented for particular patients, it may be possible to devise a more general strategy that works most of the time. Furthermore, the parameter fitting can be more refined to consider several likely parameter sets, and the optimization problem can be expanded to consider variable vaccination dosages qV,1, qV,2,..., qV,n under the constraint .
Another question is whether the effects of vaccination can be clinically observed. Since most leukemia patients taking imatinib undergo cytogenetic remission, but not molecular remission (P.P.L., unpublished data), it is possible to observe whether vaccinations can further drive the leukemia to molecular remission. The thresholds for cytogenetic and molecular remission are 108 and 106 leukemia cells in the body, respectively. Assuming that an average person has 6 L of blood, these counts correspond to leukemia concentrations around 10?2 and 10?4 k/µL, respectively. Thus, the model predicts that a series of vaccinations will not only drive the leukemia population below the molecular remission level, but will actually drive it to extinction.
To clinically implement these treatments, one would also need to have a criterion for starting the vaccinations. From the model, we observed that vaccinations are best administered just prior to the peak of the T cell response; however, in practice, it may be hard to determine the T cell peak times. We observe that for all patients, the T cell peaks occurred around 10 months after starting the imatinib treatment, while they entered complete and major cytogenetic remissions a few months earlier. Determining whether there is a correlation between remission times and T cell peak times will prove useful in carrying out treatments, and may be the goal of future studies. Such a study would require simultaneously measuring the T cell and the leukemia levels over time, perhaps at the molecular level.
An important issue is whether stem cells are immunologically privileged. In principle, T cells are known to have the capacity of killing leukemia stem cells as evidenced by the success of allogeneic bone marrow transplants. It is unknown whether the autologous immune response can produce similar results. It is also possible that leukemia cells may down-regulate target molecules for the anti-leukemia T cells. However, this rate is probably much slower than the rate of acquiring imatinib resistance. In any case, even if stem cells or mutated leukemia cells were immunologically privileged, what we propose may still substantially delay the leukemia relapse. Indeed,  and  show that an active immune response in conjunction with imatinib plays a significant role in the elimination of leukemia. These papers describe experiments in which imatinib was given to CML patients who relapsed after allogenic bone marrow transplants, resulting in sustained remission.
We also observe that the more demanding vaccination strategies for each patient P1, P4, and P12 require total doses of 2.3 k/µL, 1.0 k/µL, and 2.0 k/µL, respectively. These samples can be obtained from 6 L×2.3/73 = 190 mL, 6 L×1.0/23.1 = 160 mL, 6 L×2.0/116.8 = 100 mL of pre-treatment blood from P1, P4, and P12, respectively. Since we are only interested in collecting leukemia cells prior to imatinib treatment, these samples can be gathered by filtering the white blood cell component of the patient's blood. For reference, we point out that one US pint is about 500 ml, and this quantity of whole blood is routinely collected from healthy individuals.
An issue that was not investigated directly in this study is the functional form of immune downregulation. In our model, we chose to use the form , i.e., an exponential decay. It will be difficult to conduct a sensitivity analysis with respect to the function form. However, as implied by the previous discussion, the functional form does not greatly affect outcome of vaccination strategies as long as there is very low residual downregulation after cancer remission. In other words, when leukemia drops below remission, immune cells are no longer effectively downregulated. Since downregulation is usually hypothesized to be the result of contact-dependent mechanisms or suppression by negative cytokine signaling, it follows that the effects of downregulation will most likely disappear or at least become greatly reduced once the source of downregulation is removed.
As a final point, we note that in the Michor model leukemia relapses at 15 to 24 months despite continued imatinib therapy with the Michor model . This results from imatinib's inability to control leukemic stem cells - a conclusion of this previous work . However, this contradicts clinical observations in imatinib-treated patients , who generally remain in remission well beyond 30-40 months. With the addition of the T cell response in our model, persistence of anti-leukemia T cells even at low levels prevented leukemia from relapsing for up to 50 months. We therefore hypothesize that anti-leukemia T cell responses may help maintain remission under imatinib therapy. Therapy with imatinib (and other targeted therapies being developed) has the advantage to target leukemic cells more selectively than non-specific therapies such as chemotherapy and radiation . As such, host immune function may be restored more rapidly than after chemotherapy, due to alleviation of leukemia-induced immune suppression. Importantly, normalization of host immune function, while leukemia antigens are still present, may optimally drive anti-leukemia immune responses. It should be noted that imatinib was shown to have some immunomodulatory activity in a mouse arthritis model .
An alternative model of CML dynamics was recently published by Roeder et al. . In this model, stem cells exist in non-proliferating or proliferating states. The likelihood for a stem cell to proliferate and differentiate or to return to dormancy is determined by an internal mechanism, called the affinity function. After imatinib treatment, which in this model can target proliferating but not non-proliferating stem cells, most remaining stem cells are dormant, resulting in a much longer remission and a slower relapse than the Michor model . In view of our present work, it is important to note that the model in  does not include the immune response. However, quantitative data for the non-proliferating or proliferating states are not available. In either case, the models do not significantly diverge for the first few years and our analysis focuses within this time period when the anti-leukemia immune response is still active. Hence, the anti-leukemia immune response that we observed experimentally and modeled is consistent with both models.
The approach presented in this paper accounts for the role of the anti-leukemia specific immune response in the dynamics of CML. By combining experimental data and mathematical models we demonstrate that persistence of anti-leukemia T cells even at low levels seems to prevent the leukemia from relapsing (for at least 50 months). Consequently, we hypothesize that anti-leukemia T cells responses may help maintain remission under imatinib therapy.
The mathematical model together with the experimental data of  imply that there may be a feasible, low risk, clinical approach to enhancing the effects of imatinib treatment. These conclusions rest on the hypotheses that imatinib induces an innate immune response and that the patient's immune system functions alongside imatinib to sustain cytogenetic remission for up to several years.
The mathematical modeling of experimental data provides insights, suggesting that these responses may play a critical role in maintaining remission. Our model suggests that properly timed vaccinations with autologous leukemic cells, in combination with imatinib, can sustain the T cell response and potentially eradicate leukemic cells. It also shows that vaccinations must be optimally timed in relation to host anti-leukemia T cell responses. A key assumption in the model is that anti-leukemia T cells can target all leukemic cells (including stem cells and cells that develop resistance to imatinib). Such an assumption is supported by the graft-versus-leukemia response of allogeneic stem cell transplantation , suggesting that leukemic stem cells can be eliminated by the immune response. Resistance to imatinib, such as via abl mutations, could render a leukemic cell even more susceptible to immune targeting. Even if this is not the case, the proposed therapeutic strategy could still potentially result with a substantial increase of the expected time to a relapse. Combining imatinib with optimally timed vaccinations could lead to a potential cure of CML. While cancer vaccines is not a new concept, the importance of optimal timing of vaccinations in relation to the underlying endogenous immune response (which the vaccine attempts to boost) is novel and not previously suggested in the field of cancer immunotherapy. This approach may be transferable to other cancers, as other molecular targeted therapies become available.
While it is still too early to begin human clinical trials with our novel immunotherapy strategies, our immediate goal is to refine and validate our model predictions with additional patient measurements, and only then propose a clinical trial. There is still no good animal model of CML to validate our model predictions or test various vaccination conditions. As such, we are continuing to analyze samples from additional patients - at higher resolution time points guided by our results thus far. We will particularly focus on patients that relapse on imatinib to study their immune responses before, during, and after the relapse period. Such patients are now being put on next generation molecular targeted drugs such as dasatinib, which will bring 80% of patients with imatinib-resistant leukemia back into remission. We will analyze the immune responses in these patients. At all time points, we will obtain accurate measurements of the leukemia load via real-time PCR. This will allow us to validate our predictions for the optimal load zone.