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Introduction

Chronic myelogenous leukemia (CML) results from the uncontrolled growth of white blood cells due to up-regulation of the abl tyrosine kinase [1]. The standard first-line therapy against CML is imatinib, a molecular-targeted drug that inhibits the abl tyrosine kinase [2]. Under imatinib, nearly all patients attain hematologic remission (HR) [3] and 75% achieve cytogenetic remission (CR) [4]. However, imatinib does not completely eliminate residual leukemia cells and patients inevitably relapse after stopping treatment [4]. We note that for a hematologic remission (also known as complete hematologic response) the following must be present: Platelet count 450,000/µL, WBC count <10,000/µL, WBC differential: no immature granulocytes and <5% basophils, Spleen nonpalpable. Cytogenetic remission (or response) is defined with the following sub-categories. None: Ph+ cells >95%; Minimal: Ph+ cells 66–95%; Minor: Ph+ cells 36–65%; Partial: Ph+ cells 1–35%; Complete: Ph+ cells 0%.

In this paper, we model the dynamics of T cell responses to CML. Insights gained from this model were used to develop a possible combination between imatinib and immunotherapy, in the form of cancer vaccines, to enhance the efficacy of imatinib treatment and potentially eliminate leukemia for a durable cure.

Various papers have proposed hypotheses concerning the effects of imatinib treatment on leukemia cells from a dynamical systems perspective. In a recent work, Michor et al. develop a model for the interaction between leukemia and imatinib [5]. In their model, they assume that leukemia cells differentiate through four stages of their life cycle, beginning with leukemia stem cells. Imatinib functions by reducing the rate at which leukemia cells pass from one stage to the next, causing a rapid drop in the leukemia population. Based on their assumptions and analysis, they propose that leukemia inevitably persists, because imatinib hinders the differentiation of differentiated leukemia cells, but does not affect the leukemia stem cells. In particular, Michor et al. hypothesize that there is always a steadily growing population of leukemia stem cells despite imatinib treatment. As a result, based on their model, the leukemia population under imatinib eventually relapses, regardless of whether the model considers imatinib resistance mutations.

In a subsequent paper [6], Roeder et al. develop a similar model of CML and imatinib. However, they subdivide the leukemia stem cells into two compartments: proliferating and quiescent cells. Proliferating leukemia stem cells are affected by imatinib, while quiescent leukemia stem cells are not affected. Due to this additional assumption, the leukemia population under imatinib does not relapse without the effects of imatinib resistance mutations. Instead, under imatinib treatment, the leukemia stem cell population restabilizes at lower equilibrium level and does not continue growing as in the Michor model.

Both [5] and [6] propose that imatinib does not eliminate the leukemia stem cell population. Consequently, the papers conclude that imatinib therapy should be combined with an additional treatment that either directly impacts leukemia stem cells or causes leukemia stem cells to become vulnerable to imatinib.

As an alternative approach, Komorova and Wodarz develop a model that focuses on the drug resistance of leukemia cells [7]. In their model, they implicitly assume that imatinib affects all leukemia cells including stem cells and that inevitable relapse is a result of acquired imatinib resistance mutations. Komorova and Wodarz consider the possibility of treating patients with multiple drugs to reduce the probability of any leukemia cell eventually acquiring resistance-mutations to all drugs. They determine that a treatment strategy consisting of three leukemia-targeted drugs of different specificity might have a strong chance of eliminating the disease.

The three approaches discussed above present a variety of hypotheses for the dynamics of imatinib treatment on leukemia cells. These papers also propose potential treatment strategies to enhance the effectiveness of imatinib. However, the difficulty with these treatments is that it is unclear what kind of drug could be used to target leukemia stem cells or what alternative drugs could be used in addition to imatinib for a multiple-drug strategy.

In this paper, we model the anti-leukemia immune response in CML patients on imatinib therapy. Biological insights from the model lead us to propose a novel approach that incorporates the leukemia specific immune response into the mathematical models. We show that the model of Michor et al. [5], when extended in time, predicts a relapse approximately three years after the start of treatment. However, a three-year relapse conflicts with clinical observations as patients under imatinib often remain in cytogenetic remission for several years. The models of Roeder et al. and Komorova and Wodarz present alternative models that may explain the long-term remission typically observed in patients; however, none of these approaches consider the dynamics and impact of the immune response to CML.

Recent experiments by Chen et al., observe that some CML patients under imatinib-induced remission develop a robust but transient anti-leukemia immune response involving both CD4+ and CD8+ T cells [8]. The results of Chen et al. extend the findings of Wang et al. pertaining to antigen-presenting cells and CD4+ T cells in CML [9]. By developing a model that combines imatinib and immune dynamics, we formulate an alternative hypothesis about how remission is sustained and propose a novel treatment strategy to enhance the effectiveness of imatinib.

The paper is organized as follows. In the Materials & Methods section we develop a mathematical model for the dynamics of CML, imatinib, and the imatinib-induced immune response to CML. This model is written as a system of delay differential equations (DDEs) where the delay accounts for T cell division. As part of the model presentation, we pay considerable attention to discussing the parameter estimates. This discussion is divided into two parts. First we deal with the estimation of the universal parameters, i.e., the parameters for which we assume that their range is identical for all patients. We then proceed to discuss the estimation of the three patient specific parameters. This estimation is done by fitting the simulations of the model to the experimental data from [8].

In the Results section we use simulations of our model to discuss the brief anti-leukemia immune response that occurs during imatinib-induced remission. We hypothesize that the immune response serves to sustain leukemia remission longer than it would last otherwise. At the same time, we do point out that this immune response dies off too quickly to be effective at completely eliminating CML.

The work of [8] has also indicated that when an anti-leukemia immune response is not detectable, it can be re-stimulated by in vitro incubation with irradiated autologous leukemia cells or lysates (available from cryopreserved blood from the patient before imatinib therapy). We hypothesize that a similar stimulation of the anti-leukemia immune response can be also obtained in-vivo. We refer to such a procedure as a “cancer vaccine”. We modify our mathematical model to include terms that account for the cancer vaccines. Through mathematical simulations of this new model we show that if indeed a similar response to what was seen in vitro can be also obtained in patients, one can possibly use properly timed vaccines to develop an anti-leukemia response that will be of sufficient magnitude and duration to eradicate all residual leukemia cells. The timing of the vaccine and the doses are tailored to the specific measurable parameters of the immune response of each patient. We study the number of vaccines, their doses, and their timing. We also study the sensitivity of the model to the patient specific parameters. Comments on various aspects of the proposed treatment strategy are provided in the concluding Discussion section.

Materials and Methods

A Mathematical Model of the Immune Response to CML

In [8] Chen et al. conducted an experimental study involving fourteen patients under imatinib treatment. During the course of treatment, they conducted IFN-γ ELISPOT analysis at multiple time points to measure the evolution of the anti-leukemia T cell responses of each patient. All patients achieved HR within 1–3 months. Ten patients achieved complete CR, and 4 achieved major CR. All patients also achieved at least major molecular responses, and sustained molecular as well as cytogenetic responses over time (up to 60 months), except patient 9 (P9), who relapsed after 3 years, and P13, who relapsed after 4 years (after stopping treatment due to imatinib intolerance). We note that a complete molecular response is when the BCR-ABL transcript is non-detectable and non-quantifiable. A major molecular response is defined as BCR-ABL/control gene ratio 0.001.

To study the dynamics of the imatinib-induced immune response, we formulate a mathematical model for leukemia cells and anti-leukemia T cells. The leukemia growth and the response to imatinib follows [5] to which we add interactions with anti-leukemia T cells. Leukemia cells may be killed by interactions with T cells. Also, T cells interacting with leukemia cells may be stimulated to proliferate or to become anergic and die. The T cell interactions are modeled in the same way as in our previous paper [10].

The mathematical model is formulated as a system of DDEs as follows:
(1)

(2)
where

A state diagram that corresponds to Equations 1 and 2 is shown in Figure 1. The system of Equation 1 is a modification of the model of [5] for which in each question we added a term that accounts for the death of leukemic cells as a result of an interaction with T cells. The variables y₀, y₁, y₂, and y₃ denote the concentrations of leukemia hematopoietic stem cells (SC), progenitors (PC), differentiated cells (DC), and terminally differentiated cells (TC) without resistance mutations to imatinib. The variables z₀, z₁, z₂, and z₃ denote the respective concentrations of leukemia cells with resistance mutations. The rate constants a, b, and c are given with indices corresponding to non-resistant and resistant leukemia populations. The death rates of the four cell categories are given by d₀, d₁, d₂, and d₃, respectively. The constant u is the rate of resistance mutation per cell division.

Figure 1. A State diagram for the model in Equations 2 and 3.

(A) Cancer cells. The parameters a_y, b_y, c_y correspond to the rates of differentiation of leukemia cells without imatinib treatment, whereas the parameters , , correspond to the rates of differentiation under imatinib treatment. In the case of imatinib-resistant cancer cells, the growth and differentiation rates in the diagram are replaced by r_za_z, b_z, c_z. (B) T cells.

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The variable C denotes the total concentration of all leukemia cells with and without resistance mutations. The variable T denotes the concentration of anti-leukemia T cells. The final terms in each of the equations in Equation 1 are of the form (or ). We assume the law of mass action, stating that two cell populations interact at a rate proportional to the product of their concentrations. Hence, the component kTy_i (or kTz_i) is the rate of interaction between T cells and the leukemia subpopulation y_i (or z_i) where k is the kinetic coefficient.

The coefficient p₀ is the probability that a T cell engages the cancer cell upon interaction, and q_c is the probability that the cancer cell dies from the T cell response. Furthermore, leukemia cells suppress anti-leukemia immune responses, and while the precise mechanism is unknown, we assume that the level of down-regulation depends on the current cancer population. In particular, we model the probability that a T cell engages a cancer cell decays exponentially as a function of the cancer concentration, i.e., the probability of a productive T cell interaction with a cancer cell is where c_n is the rate of exponential decay due to negative pressure.

It is now well established that cancer suppresses the host immune system in various ways [11]. Leukemia is particularly immunosuppressive as leukemic cells grow within the bone marrow, and can directly suppress both growth and function of normal blood cells. As such, leukemia patients are known to be at higher risk for infections and other cancers [12],[13]. While the mechanisms are varied, we recently showed that cancer patients may have a defect in the interferon signaling pathway [14]. Interferon is an important cytokine in driving immune responses.

In Equation 2, s_T denotes the constant supply rate of T cells into the system from stem cells. The second term is the natural death rate of T cells. The third term is the rate at which T cells engage leukemia cells and commit to n rounds of division. The final term represents the population increase due to n divisions of stimulated T cells where τ is the average duration of one division, and C_nτ and T_nτ are the time delayed cancer and T cell concentrations respectively. The coefficient q_T is the probability that a T cell survives the encounter with an activated leukemia cell.

The method of modeling T cell proliferation in Equation 2 is similar to what we have previously used in [10]. Once a T cell is stimulated, it exits the collection of interacting T cells and reenters the system nτ time units later after n divisions. This approach ensures that the T cell population does not double faster than once every nτ days. It is an alternative to using the Michaelis-Menten expression or other saturating terms.

Parameter Estimates

A considerable amount of effort is devoted to estimating the parameters that appear in our mathematical model (Equations 1 and 2). The discussion is divided into two parts. First, we present the methods for estimating the universal parameters, i.e., the parameters we assume have ranges of values that are similar for all patients. Following the work of [5] we assume that the time-dynamics of cancer is universal, i.e., we describe the evolution of the cancer cells in their various stages of development using parameters that are assumed to be identical for all patients. Clearly, there is no reason to believe that the dynamics of cancer is identical for all patients (as commonly done in mathematical models). Nevertheless, it does serve, in our case, as a way of simplifying the computations in addition to a way to connect between our work and previous works.

We then proceed to describe the methods we used for estimating the remaining three model parameters. These parameters characterize the individual immune response. Consequently they are allowed to vary from patient to patient.

Universal parameters.

The values of the parameters pertaining to the growth, differentiation, and mutation rates of leukemia cells are taken from [5] without modification. These parameters are r_y, a_y, b_y, c_y, r_z, , , , a_z, b_z, c_z, and u. The death rates from [5] correspond to the natural death rates of the leukemia populations under imatinib. However, in our model, we distinguish between the natural death rate of leukemia and the death rate due to the cytotoxic T cell response. Hence, our natural death rates, d_i, should be a fraction, λ, of the combined death rates estimated in [5].

Determining what fraction λ of the leukemia death rates from [5] result from non-immune versus immune causes is difficult and requires some assumptions. First, we assume that λ is greater than 0.5, so that the anti-leukemia immune response contributes to less than half of the decline in leukemia under imatinib treatment. Due to the lack of data on λ we set it as λ = 0.75. A discussion on the sensitivity of the results to the choice of λ will follow.

For the kinetic coefficient k, we use the same value of 1 (k/µL)⁻¹ day⁻¹ which was originally drawn in [10] from the rate constant of virus elimination in [15]. For T cell-cancer interactions, we apply the following assumptions from [10]: 20% of the time nothing happens, and both cells survive and depart; 20% of the time cancer lives, and the T cell becomes anergic or dysfunctional; 40% of the time cancer dies, and the T cell survives and moves on; 20% of the time both cancer and the T cell die. From these assumptions, we deduce that the probability of any sort of interaction is p₀ = 0.8, the probability of cancer dying is p₀q_C = 0.6, and the probability of a T cell surviving is p₀q_T = 0.4. Hence, q_C = 0.6/0.8 = 0.75 and q_T = 0.4/0.8 = 0.5.

In [15], Luzyanina et al. estimate that T cell divisions take between 0.4 to 2 days, and their best fit estimate is 0.6 days. Also, Janeway estimates that primed T cells divide 2 to 4 times per day [16], which corresponds to a duration of 0.25 to 0.5 days. Combining these sources, we conclude that T cell divisions take between 0.25 and 2 days. Since the anti-leukemia T cells are emerging from an environment of immune down-regulation, we assume they divide at the more conservative rate of one division per day.

A summary of the estimated parameters is provided in Table 1.

Table 1. Estimates of universal parameters.

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Patient-Dependent Parameters.

The data from [8] for three patients, P1, P4, and P12, each consists of at least five time points per patient. Hence, we focus on these patients when fitting the model to patient data. Tables 2–3 summarize the data from [8] for P1, P4, and P12. Since the duration and the magnitude of the immune responses vary greatly across the three patients, we fit the parameters s_T, d_T, c_n, n, y₀(0) to each patient independently and do not attempt to come up with universal estimates of these values. These five parameters denote the supply rate of anti-leukemia T cells, the death rate of anti-leukemia T cells, the level of immune down-regulation by leukemia cells, the average number of T cell divisions upon stimulation, and the initial concentration of leukemia stem cells, respectively.

Table 2. Pre-treatment leukemia load.

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Table 3. Patient data from ELISPOT assay from [8] for P1, P4, and P12.

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Since even for these three patients only few data points are available, we do not apply a formal method to fit the five patient-dependent parameters to the data. Rather, we use certain features of the data sets, such as the peak height of the T cell response, to estimate the patient-dependent parameters.

We use known information from the literature to determine reasonable ranges for n and d_T. To determine an upper bound for the average number of T cell divisions, n, we consider that when naïve CD8+ T cells are primed for the first time, they go through several cycles of division. An analysis of experimental data by Antia et al. showed that stimulation of naïve CD8+ cells result with up to 8 divisions in vitro [17]. In addition, Janeway estimates that the proliferation of primed CD8+ cells leads to about 10³ daughter cells [16], which implies about 10 divisions. Primed CD8+ T cells continue to divide as long as they receive stimulus, but not as many times as during the initial stimulation. Hence, we conclude that primed CD8+ T cells divide fewer than 10 times and most likely fewer than 8 times per stimulation.

To estimate the range of the T cell death rate, d_T, we consider the observations and calculations from [18] that primed CD4+ T cells peak nine days after stimulation, initially die with a half-life of 3 days, and slow down to a half-life of 35 days, eight days after the peak of the response. These numbers yield an initial death rate of 0.23/day and an eventual death rate of 0.02/day. In addition, in [18] it is estimated that primed CD8+ T cells die with a half-life of 1.7 days, yielding a death rate of about 0.4/day. The half-lives of memory CD4+ and CD8+ T cells are much higher, i.e. 500 days to lifelong respectively. Since we are looking at data points that were measured over several years, most of the lingering T cells in the anti-leukemia response are probably CD4+ effector cells or memory CD4+ and CD8+ cells. Since we are examining time-scales of several months to a few years, for convenience, we assume that the T cell death rate is constant at 0.02/day or lower and do not take into account the biphasic switch that probably occurs around seventeen days after the beginning of the immune response.

The characteristics of the five patient-dependent parameters are summarized in Table 4.

Table 4. Estimated ranges of patient-dependent parameters.

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The initial concentration of leukemia stem cells, y₀(0), is the most straightforward parameter to estimate, since its value can be derived directly from the initial leukemia load measured in [8].

If we assume that all populations start in their steady states, we can calculate the initial concentrations of all leukemia cell compartments in terms of y₀(0) and the universal parameters given in Table 1. (Likewise, we can calculate the initial concentration of T cells in terms of the T cell supply and death rates.)

If we assume that there are no resistant cells at the start of treatment, the initial concentration of imatinib resistant stem cells is 0. Note that Michor et al. also consider a scenario, in which the initial resistant stem cell count is 10 cells [5]. Assuming that an average person has 6 L of blood, this initial count corresponds to an initial concentration of 10/6 L~10⁻⁹ k/µL. Clearly, this is a very crude estimate as the leukemic cells are distributed within the bone marrow, spleen, and blood. However, as will be shown in the sensitivity study below, the initial concentration plays a rather limited role in the emerging dynamics, and thus even such a crude estimate will suffice. See Table 5 for a list of initial concentrations.

Table 5. Initial concentrations.

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To calculate y₀(0), we set the pre-treatment leukemia loads listed in Table 2 equal to the expression for the total initial leukemia concentration, , and solve for y₀(0).

The T cell death rate, d_T, is estimated from the rate of decline of the anti-leukemia T cell populations after their peak. Hence, the last three data points for P1, the last five data points for P4, and the last five data points for P12 are used to estimate the rate of T cell death d_T.

If we assume that the T cell population is at steady state before treatment, the concentration of anti-leukemia T cells at time 0 is s_T/d_T. By setting this ratio equal to the initial T cell concentrations obtained from the data in Table 3, we can determine s_T in terms of d_T.

The rate c_n of the decay of the immune response due to negative pressure is difficult to estimate. However, the value of c_n affects the number of T cells that are stimulated during the course of imatinib treatment and how soon T cell expansion initiates. Specifically, we can use the data points before the T cell peak to estimate the time of initiation of the anti-leukemia T cell response for each patient. From the data, it is apparent that the T cell response does not initiate immediately, indicating a lingering immunosuppressive effect from the leukemia cells. We assume that the T cell responses start approximately 2.5, 3, and 2 months after the start of imatinib treatment for patients P1, P4, and P12, respectively.

Given the T cell death rate d_T, we can determine the range of cancer concentrations where the T cell growth rate, , exceeds the T cell death rate, d_T. Before the T cell response starts, the leukemia concentration falls solely based on its natural death rate, since there is no active T cell response. Thus, we can further determine the time that the cancer concentration first reaches the point where the T cell growth rate exceeds the T cell death rate. Hence, we can approximate the value of c_n that causes the T cell responses of P1, P4, and P12 to begin expanding around months 2.5, 3, and 2, respectively. We examine this idea more thoroughly when we introduce the “optimal load zone” for T cell stimulation.

The remaining parameter n, which represents the average number of T cell divisions per stimulation, is estimated by matching the results of the simulation to the data points. In particular, the peak height of the T cell response is a strong indicator of the value of n, since higher n lead to higher T cell peaks.

To fit the patient-dependent parameters, we convert the data from [8] into units of concentration, namely thousands of cells per microliter (k/µL). The data in [8] is originally given in SFCs/well and 10⁵ PBMCs were used in each well. However, only a fraction of the PBMCs are T cells, and measurements of TNF-α and IFN-γ in [8] imply that the standard procedure of measuring the IFN-γ response using the ELISPOT assay may underestimate the strength of a T cell response. Due to these uncertainties, we assume the measurements from the ELISPOT assay indicate relative magnitudes among T cell responses at various time points, but we do not convert the SFCs/well measurements directly into units of concentration (k/µL).

In [8], Chen et al. conducted TNF-α and IFN-γ ELISPOT analyses to measure T cell activity. From this data (in particular the data of patient P4), it is seen that roughly 4% of CD4+ T cells and 1% of CD8+ respond to leukemia at the peak of the T cell response. Hence, we scale the ELISPOT data down by 2500 to obtain T cell concentrations. This corresponds to about 1% of T cells from P4 responding to leukemia at the peak of the response. Furthermore, we use the scaled values for the initial ELISPOT measurements at time 0 to set the steady state T cell concentration, s_T/d_T. The cancer-related parameters are given in Table 1. For our first study, we assume that there are no resistance mutations, so we set the mutation rate, u, and the initial concentration of imatinib-resistant stem cells, z₀(0), to 0. The remaining parameters for each of the three patients are given in Table 6.

Table 6. The parameters for Figure 2.

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Results

Imatinib-Induced Immune Dynamics

Graphs of the solutions of the mathematical model that correspond to patients P1, P4, P12, along with the measured data points are displayed in Figure 2. The cases labeled “no immune response” in Figure 2 are taken from [5] and correspond to setting the T cell concentrations in Equation 1 to 0, i.e., without the immune response. In comparison to the no-immune-response cases, the T cell response contributes to driving the leukemia population lower than with imatinib alone. Furthermore, the persistence of anti-leukemia T cells at low levels keeps the leukemia population from relapsing for up to several years, whereas in the no-immune-response cases, cancer rebounds are noticeable after 15 to 24 months (see Figure 2).

Figure 2. Model solutions fit to data measurements for 3 patients.

(A) P1, (B) P4, and (C) P12. The measurements of SFCs/well from [8] are scaled down by 2500 to show relative magnitudes and are shown as black squares. The dashed lines show the approximate level of complete cytogenetic remission. “No immune response” correspond to the predictions of [5]. “Leukemia” correspond to the results of our model (Equations 1 and 2). The “T cells” curve is obtained with our model after fitting the parameters to the experimental data (shown in blank squares).

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We estimate the approximate concentration corresponding to complete cytogenetic remission, based on [19]. According to [19], there are 10¹² leukemia cells prior to imatinib treatment. As a general medical assumption, there are three layers of remission, hematological, cytogenetic, and molecular, and each layer corresponds to a 2 log, or 100-fold, difference from the previous one. Hence, hematological remission corresponds to roughly 10¹⁰ cells, and cytogenetic remission corresponds to roughly 10⁸ cells. If the average person has 6 l of blood, cytogenetic remission corresponds to a blood concentration of 10⁸/6 l = 1/60 k/µL. The cytogenetic remission level is shown as dashed lines in Figure 2.

Regarding the no-immune-response case, Michor et al. demonstrate that imatinib significantly reduces the populations of differentiated leukemia cells, but does not eliminate leukemia stem cells [5]. As a result, the leukemia population decreases rapidly at the beginning of treatment, while the stem cell population continues to rise exponentially at a much slower rate of r_y – d₀. This phenomenon occurs even in the absence of resistance mutations, making an eventual relapse unavoidable.

On the other hand, our model including the anti-leukemia T cell response predicts a substantially slower relapse and provides a fit to the immunological data. Hence, it is possible that a combination of imatinib and an immune response keeps the leukemia population under control and allows patients to remain in cytogenetic remission for several years. Indeed, the model predicts that the patients remain in cytogenetic remission beyond month 50.

In all three patients, the leukemia cells are not eliminated completely by imatinib treatment. In fact, the lowest concentrations obtained by the cancer populations in Figure 2 for P1, P4, and P12, are 1.3×10⁻⁴, 7.8×10⁻⁵, and 2.2×10⁻⁴ k/µL,respectively, which correspond to half a million to a million cells remaining in the body, assuming that an average person has 6 L of blood. As can be observed in Figure 2C, it seems that leukemia starts increasing again about 24 months after the start of treatment. This observation stresses an important point, namely that our model does not predict that CML is eliminated by imatinib treatment alone. It does, however, predict that it takes significantly more time for the disease to relapse (when compared with the Michor model).

Nonetheless, leukemia drops to such a low level that the T cells are no longer stimulated and begin to contract. As a result, the immune response does not expand sufficiently to eliminate the leukemia cells. Unfortunately, although imatinib drives the cancer population to low levels, it does not eliminate the leukemia stem cells [5]. Hence, the low population of leukemia stem cells remain below immune surveillance and out of reach of imatinib, escaping complete elimination.

In Figure 3 we show simulations for the three patients that demonstrate what happens when the imatinib treatment is stopped at month 12. Similar results are observed for all three patients. The removal of imatinib leads to a resurgence of the leukemia population which causes an initial increase in the T cell response; however, the T cell response is never strong enough to overcome the rapidly growing leukemia population. This result is consistent with clinical observations that patients taken off imatinib invariably relapse [5]. For the purposes of this paper, we assume that the patients are always treated by imatinib. The strong immune responses in Figure 3 are induced by imatinib. Indeed, in the absence of any imatinib treatment, no immune response initiates, a scenario that is shown in Figure 4.

Figure 3. A predicted relapse when imatinib is removed at month 12.

The T cell response is never sufficient without imatinib and the removal of imatinib leads to full relapse. (A) P1. (B) P4. (C) P12.

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Figure 4. Solving the model equations without an imatinib treatment.

The T cell responses are fully suppressed and stay flat at their steady state concentrations while cancer grows rapidly. (A) P1. (B) P4. (C) P12.

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We would now like to further elaborate on the various aspects regarding the stimulation of the immune response as reflected in our model (Equations 1 and 2). From Equation 2, the balance between immune down-regulation and T cell stimulation by leukemia cells is given by the term . Hence, the optimal level of T cell stimulation occurs at C = 1/c_n. We define the optimal load zone to be the range of leukemic concentrations where the T cell stimulation rate is faster than the T cell death rate, i.e., , where k is the mass-action coefficient and d_T is the T cell death rate. Figure 5 shows the optimal load zones and stimulus levels of T cells as functions of the leukemia concentrations for the three patients. The anti-leukemia T cell populations begin expanding when the leukemia concentration drops into the optimal load zone and begin contracting when the leukemia concentration drops below the optimal load zone.

Figure 5. Stimulation levels of anti-leukemia T cells versus logs of the cancer concentrations.

Optimal loads are the cancer concentrations C for which the perceived stimulus, , is maximized. Optimal load zones are the range of leukemic concentrations where the T cell stimulation rate is faster than the T cell death rate, i.e. . (A) P1. (B) P4. (C) P12.

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Figure 5 shows that if the cancer concentrations grow beyond approximately 10¹ (for the three patients) the perceived stimulus is so low that the anti-leukemia T cell response begins to contract, allowing the cancer population to expand more rapidly. The expanding cancer population then further suppresses the T cell response, leading to an uncontrolled relapse. Hence, we can say that the relapses in Figure 3 are complete, and the immune responses do not recover.

The level of immune down-regulation, c_n, by leukemia cells is a key parameter that governs how well the immune response can function against the relap