^{1}

^{2}

^{*}

^{3}

^{*}

Conceived and designed the experiments: PK PL DL. Performed the experiments: PL. Analyzed the data: PK PL DL. Wrote the paper: PK PL DL. Developed the mathematical models, simulated them, and analyzed the results: DL PK.

The authors have declared that no competing interests exist.

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.

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 developed a mathematical model to gain insights into the dynamics and potential impact of the resulting anti-leukemia immune response on CML. 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.

Chronic myelogenous leukemia (CML) results from the uncontrolled growth of white blood cells due to up-regulation of the abl tyrosine kinase

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

In a subsequent paper

Both

As an alternative approach, Komorova and Wodarz develop a model that focuses on the drug resistance of leukemia cells

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

Recent experiments by Chen

The paper is organized as follows. In the

In the

The work of

In

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

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

A state diagram that corresponds to Equations 1 and 2 is shown in _{0}, _{1}, _{2}, and _{3} 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 _{0}, _{1}, _{2}, and _{3} denote the respective concentrations of leukemia cells with resistance mutations. The rate constants _{0}, _{1}, _{2}, and _{3}, respectively. The constant

(A) Cancer cells. The parameters _{y}_{y}_{y}_{z}a_{z}_{z}_{z}

The variable _{i}_{i}_{i}_{i}

The coefficient _{0} is the probability that a T cell engages the cancer cell upon interaction, and _{c}_{n}

It is now well established that cancer suppresses the host immune system in various ways

In Equation 2, _{T}_{n}_{τ} and _{n}_{τ} are the time delayed cancer and T cell concentrations respectively. The coefficient _{T}

The method of modeling T cell proliferation in Equation 2 is similar to what we have previously used in

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

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.

The values of the parameters pertaining to the growth, differentiation, and mutation rates of leukemia cells are taken from _{y}_{y}_{y}_{y}_{z}_{z}_{z}_{z}_{i}

Determining what fraction

For the kinetic coefficient ^{−1} day^{−1} which was originally drawn in _{0} = 0.8, the probability of cancer dying is _{0}_{C}_{0}_{T}_{C}_{T}

In

A summary of the estimated parameters is provided in

Parameter | Description | Estimate | Source |

Fractional adjustment constant | 0.75 | Estimate | |

_{0} | SC death rate | 0.003 | |

_{1} | PC death rate | 0.008 | |

_{2} | DC death rate | 0.05 | |

_{3} | TC death rate | ||

_{y} | Growth rate for nonresistant cells | 0.008/day | |

_{y} | Rates without imatinib treatment | 1.6 | |

_{y} | 10 | ||

_{y} | 100 | ||

Rates during imatinib treatment | _{y} | ||

_{y} | |||

_{y} | |||

_{z} | Growth rate for resistant cells | 0.023/day | |

_{z} | Rates for resistant cells | _{y} | |

_{z} | _{y} | ||

_{z} | _{y} | ||

Mutation rate per division | 4×10^{−8}/division | ||

Kinetic coefficient | 1 (^{−1} day^{−1} | ||

_{0} | Prob. T cell engages cancer cell | 0.8 | |

_{C} | Prob. cancer cell dies from encounter | 0.75 | |

_{T} | Prob. T cell survives encounter | 0.5 | |

Duration of one T cell division | 1 day |

The data from _{T}_{T}_{n}_{0}(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.

Patient | P1 | P4 | P12 |

Pre-treatment leukemia load ( | 73.0 | 23.1 | 116.8 |

P1 | Time (months) | 0 | 5 | 30 | 35 | 46 | |||

SFCs/well | 3 | 29 | 25 | 25 | 9 | ||||

P4 | Time (months) | 0 | 6 | 9 | 18 | 24 | 32 | 34 | 42 |

SFCs/well | 1 | 16.5 | 33 | 30 | 26 | 11 | 15 | 12 | |

P12 | Time (months) | 0 | 2 | 5 | 9 | 13 | 15 | 24 | 30 |

SFCs/well | 11 | 42 | 39 | 71 | 36.5 | 43 | 5 | 6 |

SFCs/well for leukemia bearing+remission PBMCs. The measurement for time 0 corresponds to pre-treatment leukemia bearing PBMCs. (See

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 _{T}^{3} daughter cells

To estimate the range of the T cell death rate, _{T}

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

Parameter | Description | Estimate | Source |

Average number of T cell divisions | 1< | ||

_{T} | Anti-leukemia T cell death rate | <0.02/day | |

_{T} | Anti-leukemia T cell supply rate | ? ( | Based on _{T} |

_{n} | Decay rate of immune responsivity | ? (^{−1} | Based on patient data |

_{0}(0) | Initial concentration of leukemia stem cells | ? ( | Based on patient data |

The initial concentration of leukemia stem cells, _{0}(0), is the most straightforward parameter to estimate, since its value can be derived directly from the initial leukemia load measured in

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 _{0}(0) and the universal parameters given in

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 ^{−9}

Population | Value ( | Reason |

_{0}(0) | ? | Determined by patient data |

_{1}(0) | _{y}y_{0}/_{1} | Steady state |

_{2}(0) | _{y}y_{1}/_{2} | Steady state |

_{3}(0) | _{y}y_{2}/_{3} | Steady state |

_{0}(0) | 0 or 10^{−9} | Correspond to values in |

_{1}(0) | _{z}z_{0}/_{1} | Steady state |

_{2}(0) | _{z}z_{1}/_{2} | Steady state |

_{3}(0) | _{z}z_{2}/_{3} | Steady state |

_{T}_{T} | Steady state |

To calculate _{0}(0), we set the pre-treatment leukemia loads listed in _{0}(0).

The T cell death rate, _{T}_{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 _{T}_{T}_{T}_{T}

The rate _{n}_{n}

Given the T cell death rate _{T}_{T}_{n}

The remaining parameter

To fit the patient-dependent parameters, we convert the data from ^{5} PBMCs were used in each well. However, only a fraction of the PBMCs are T cells, and measurements of TNF-

In _{T}_{T}_{0}(0), to 0. The remaining parameters for each of the three patients are given in

Pre-treatment leukemia load→_{0}(0) | _{T} | _{T} | _{n} | ||

P1 | 73→7.6×10^{−6} | 1.2 | 0.001 | 1.2×10^{−6} | 1 |

P4 | 23.1→2.4×10^{−6} | 2.2 | 0.0022 | 9×10^{−7} | 7 |

P12 | 116.8→1.2×10^{−5} | 1.17 | 0.007 | 3.08×10^{−5} | 0.8 |

For any given _{T}_{T}_{T}_{T}

Graphs of the solutions of the mathematical model that correspond to patients P1, P4, P12, along with the measured data points are displayed in

(A) P1, (B) P4, and (C) P12. The measurements of SFCs/well from

We estimate the approximate concentration corresponding to complete cytogenetic remission, based on ^{12} 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 10^{10} cells, and cytogenetic remission corresponds to roughly 10^{8} cells. If the average person has 6 l of blood, cytogenetic remission corresponds to a blood concentration of 10^{8}/6 l = 1/60

Regarding the no-immune-response case, Michor _{y}_{0}. 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 ^{−4}, 7.8×10^{−5}, and 2.2×10^{−4}

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

In

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

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.

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 _{n}_{T}

Optimal loads are the cancer concentrations

^{1} (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

The level of immune down-regulation, _{n}_{n}_{n}_{n}

As shown previously at the beginning of treatment, imatinib causes the leukemia population to drop into the optimal load zone, stimulating an immune response. However, under continued treatment, the leukemia population quickly drops below the optimal load zone, and the T cell population contracts due to lack of stimulus. A strategy to maintain the leukemia population within the optimal load zone or to surrogately stimulate anti-leukemia T cells may help in driving the leukemia population to zero.

The experimental results of

To study the feasibility of this approach, we introduce inactivated leukemia cells into our model (Equations 1 and 2). Inactivated leukemia cells (whose number is denoted by _{V}_{V}_{n}_{τ} = _{T}_{n}_{τ} in Equation 4.

The leukemia cells used in vaccinations can be inactivated via irradiation. Since they are in the process of dying, we estimate that they do not survive much longer than 24 to 72 hours, so we set the decay rate _{V}_{V}_{V}_{V}

Parameter | Description | Estimate |

Presumably vaccinations begin after time 0 | ||

_{V} | Decay rate of inactivated leukemia cells | 0.35/day |

_{V} | Vaccination dosage | To be optimized |

_{V} | Duration of delivery | 0.01 day |

_{V} | Vaccination supply rate | _{i} |

Since it is unclear how many vaccinations will be required to eliminate the cancer, we will optimize the treatment strategy according to the following method:

At the end, we select the vaccination strategy that attains the lowest minimum leukemia concentration with the fewest vaccinations. We implement this optimization strategy, because it is more efficient than attempting to globally optimize several vaccinations of varying dosage and irregular time intervals at once. Indeed, it is a one-dimensional search problem, as opposed to a higher dimensional problem. For this assessment, we also assume that there are no mutations, i.e.

We numerically solve the system given by Equations 2–4 using the DDE solver ‘dde23’ from Matlab 7.0. For each run, we evaluate the solution up to day 400. We use parameter sets from

For each fixed dosage, we find optimal vaccination delivery times (up to a day), and our goal is to drive the cancer below 1 cell/6 ^{−10}

We assume that the criterion (Equation 5) represents cancer elimination. Since this model is a continuous deterministic system, in reality, Equation 2 never allows the cancer population to actually reach 0.

Using the aforementioned method of optimization, we optimize the timing of a series of vaccinations of varying dosages. We measure dosages in units of concentration (^{9} inactivated leukemia cells. For the parameters from

Dose ( | Timing | Pacing | Number | log_{10} [Min cancer load] |

0.1, 6.0×10^{8} | 233 | 10 | 5 | −10.5 |

1.0, 6.0×10^{9} | 240 | - | 1 | −10.4 |

For each dosage, the timing indicates the day on which the first vaccination is given, and the Pacing indicates the number of days between subsequent vaccinations. The Number indicates the number of vaccinations administered, and the final column indicates the base 10 logarithm of the minimum cancer concentration attained after the final vaccination. Values of less than 10^{−10} correspond to fewer than one cell in the body and denote cancer elimination.

On one hand, five vaccinations of dosage 0.1

The time evolution of the cancer and T cell populations for the treatment strategy in row 1 of

(A) Time evolution of cancer and T cell populations. Vaccinations are delivered on days 233, 243, 253, 263, and 273. (B) Time evolution of the four types of leukemia cells: stem cells (SC), progenitors (PC), differentiated cells (DC), and terminally differentiated cells (TC). Concentrations are shown on a logarithmic scale.

In addition, analogous tables of vaccination strategies for P1 and P12 are shown in

Dose ( | Timing | Pacing | Number | log_{10} [Min cancer load] |

0.1, 6.0×10^{8} | 202 | 5 | 12 | −10.7 |

2.3, 6.0×10^{10} | 209 | - | 1 | −10.2 |

Dose ( | Timing | Pacing | Number | log_{10} [Min cancer load] |

0.1, 6.0×10^{8} | 195 | 4 | 11 | −10.1 |

2.0, 1.2×10^{10} | 199 | - | 1 | −10.4 |

In all cases, the first vaccinations are given before the peak of the T cell responses to boost the response. The peaks of the T cell populations fall between months 9 and 10 (see

Thereafter, the following vaccinations serve to sustain the immune response over an extended time, so the gaps between these vaccinations depend on how long it takes for the previous vaccination to clear out of the system.

In the data of

The timing and pacing of the vaccination strategies are critical to the success of the outcome. For example, consider the alternative vaccination strategies in

Dose ( | Timing | Pacing | Number | log_{10} [Min cancer load] |

0.1, 6.0×10^{8} | 1–30 | 5 | 12 | −3.3 |

300 | 5 | 12 | −6.2 | |

233 | 1 | 12 | −8.0 | |

233 | 20 | 12 | −5.4 | |

2.3, 1.4×10^{10} | 1–30 | - | 1 | −3.3 |

300 | - | 1 | −7.3 |

Dose ( | Timing | Pacing | Number | log_{10} [Min cancer load] |

0.1, 6.0×10^{8} | 1–30 | 10 | 5 | −3.2 |

300 | 10 | 5 | −7.6 | |

233 | 1 | 5 | −6.7 | |

233 | 20 | 5 | −9.4 | |

1.0, 1.2×10^{10} | 1–30 | - | 1 | −3.2 |

300 | - | 1 | −8.6 |

Dose ( | Timing | Pacing | Number | log_{10} [Min cancer load] |

0.1, 6.0×10^{8} | 1–30 | 4 | 11 | −3.1 |

300 | 4 | 11 | −5.9 | |

195 | 1 | 11 | −8.3 | |

195 | 20 | 11 | −5.9 | |

2.0, 1.2×10^{10} | 1–30 | - | 1 | −3.1 |

300 | - | 1 | −6.7 |

Note that if vaccinations are initiated within 30 days of the start of imatinib treatment, the effect of the vaccinations is insignificant. There is hardly any anti-leukemia immune response, and the decline in the leukemia population is mainly due to natural death under imatinib. This happens since within the first 30 days, the leukemia population is still well above the optimal load zone, and in general, vaccinations are ineffective when the leukemic load is above the optimal load zone (where the immune suppression is too strong). However, once the leukemic population is sufficiently low, the optimal load is no longer an issue, since inactivated leukemic cells that have no immunosuppressive effects are used for vaccinations.

On the other hand, administering vaccinations too late (e.g. at 300 days) is not entirely ineffective, since leukemia has already passed into remission and no longer exerts a great immuno-suppressive effect. However, 300 days after the start of treatment, the initial anti-leukemia T cell response has started to decline, so the response to vaccination is not as strong. By considering early and late vaccinations, we see that optimizing vaccination delivery times depends on a balance between minimizing the immuno-suppressive effect of leukemia and maximizing the available anti-leukemia T cells to respond to the stimulus.

In the same way, in multiple vaccination strategies, there is an optimal pacing between vaccinations that will optimally maintain the immune stimulation over time. As we can see from

In general, the vaccination strategies for each patient will still work if T cells die at slower rates, if the immune-suppression is lower, or if T cells divide more after stimulation. These scenarios correspond to decreasing _{T}_{n}

However, our optimization strategy seeks to find the lowest dosage or smallest number of vaccinations necessary to eliminate cancer. Hence, these strategies are sensitive to underestimates of _{T}_{n}

A full treatment on the optimal way to overload a vaccination strategy leads to more complex optimization problems, which we leave for future work. However, in _{n}_{n}_{T}

Dose ( | Timing | Pacing | Number | Allowed variability for _{n} |

2×0.1 | 202 | 5 | 12 | ±10% |

0.1 | 202 | 5 | 2×12 | ±10% |

2×2.3 | 209 | - | 1 | ±4% |

Dose ( | Timing | Pacing | Number | Allowed variability for _{n} |

2×0.1 | 233 | 10 | 5 | ±10% |

0.1 | 233 | 10 | 2×5 | ±12% |

2×1.0 | 240 | - | 1 | ±5% |

Dose ( | Timing | Pacing | Number | Allowed variability for _{n} |

2×0.1 | 195 | 4 | 11 | ±7% |

0.1 | 195 | 4 | 2×11 | ±8% |

2×2.0 | 199 | - | 1 | ±4% |

From this preliminary analysis, it appears that doubling is more effective for strategies consisting of vaccine low dosages administered multiple times. It is unclear whether it is more effective to double the dosages or to double the number of times vaccinations are administered, since the relative efficacies of each approach vary from patient to patient. In any case, overloading vaccinations seems to be an effective method for increasing the robustness of vaccination strategies against uncertainties in parameter values.

The analysis in the previous sections focused on three particular patients and proposed three different treatment strategies for each case. However, to extend out findings to a general approach, we would like to examine which scenarios favor one vaccination regime over another. Indeed, from

To study the correlation between parameters and the effectiveness of proposed vaccination strategies, we apply the Latin Hypercube sampling (LHS) method

For each LHS simulation, we test one vaccination strategy over a range of 500 randomly sampled parameter sets. The parameters are sampled uniformly over the ranges indicated in

Description | Estimate | Range | PPMC | SROC | |

fractional adjustment constant | 0.75 | 0.5 to 1 | −0.2152 | −0.1395 | |

_{0} | SC death rate | 0.003 | ±25% | −0.0354 | −0.0123 |

_{1} | PC death rate | 0.008 | ±25% | −0.0643 | −0.0066 |

_{2} | DC death rate | 0.05 | ±25% | −0.1497 | −0.0130 |

_{3} | TC death rate | ±25% | 0.0206 | 0.0080 | |

_{y} | Growth rate for nonresistant cells without imatinib treatment | 0.008/day | ±25% | 0.0242 | 0.0174 |

_{y} | 1.6 | ±25% | −0.0366 | −0.0259 | |

_{y} | 10 | ±25% | 0.0372 | 0.0087 | |

_{y} | 100 | ±25% | 0.0419 | 0.0411 | |

Rates during imatinib treatment | _{y} | Same as _{y} | - | - | |

_{y} | Same as _{y} | - | - | ||

_{y} | Same as _{y} | - | - | ||

_{z} | Growth rate for resistant cells | 0.023/day | ±25% | 0.0036 | 0.0158 |

_{z} | _{y} | Same as _{y} | - | - | |

_{z} | _{y} | Same as _{y} | - | - | |

_{z} | _{y} | Same as _{y} | - | - | |

Mutation rate per division | 4×10^{−8}/division | ±100% | −0.0156 | 0.0252 | |

Kinetic coefficient | 1(^{−1} day^{−1} | ±25% | −0.1241 | −0.1287 | |

_{0} | Prob. T cell engages cancer cell | 0.8 | ±25% | −0.1328 | −0.1606 |

_{C} | Prob. cancer cell dies from encounter | 0.75 | ±25% | 0.0084 | 0.0105 |

_{T} | Prob. T cell survives encounter | 0.5 | ±25% | −0.0947 | −0.1419 |

Duration of one T cell division | 1 day | 12–24 hrs | 0.0676 | 0.0301 | |

Avg no. of cell divisions | 1.17 to 2.2 | 1 to 3 | −0.4681 | −0.6889 | |

_{T} | T cell death rate | 1–7×10^{−3}/day | 1E-3 to 1E-2 | 0.1786 | 0.2523 |

_{V} | Inactivated leukemia cell decay rate | 0.35/day | Not varied | - | - |

_{T} | T cell supply rate | 1E-5 to 1E-6 | 1E-5 to 1E-6 | −0.0412 | −0.0557 |

_{n} | Decay rate of immune responsivity | 0.8 to 7/day | 0 to 10 | 0.1785 | 0.2623 |

Pre-treatment cancer load | 23.1–116.8 | 20 to 200 | 0.1819 | 0.0717 |

Also, shown are correlations between parameters and minimum cancer concentrations. Correlation coefficients are as follows: Pearson product-moment correlation (PPMC), Spearman rank-order correlation (SROC).

From ^{6} ^{9} leukemia cells.

Total dosage ( | Dosage ( | Schedule | (Number of successes)/500 |

0.5 | 0.1 | 200, 205, 210, 215, 220 | 0.474 |

0.1 | 200, 210, 220, 230, 240 | 0.490 | |

0.1 | 200, 220, 240, 260, 280 | 0.502 | |

0.1 | 200, 240, 280, 320, 360 | 0.488 | |

0.1 | 240, 245, 250, 255, 260 | 0.510 | |

0.1 | 240, 250, 260, 270, 280 | 0.514 | |

0.1 | 240, 260, 280, 300, 320 | 0.520 | |

0.1 | 240, 280, 320, 360, 400 | 0.472 | |

1.0 | 0.1 | 200, 205, 210, 215, 220, 225, 230, 235, 240, 245 | 0.578 |

0.1 | 200, 210, 220, 230, 240, 250, 260, 270, 280, 290 | 0.622 | |

0.1 | 200, 220, 240, 260, 280, 300, 320, 340, 360, 380 | 0.610 | |

0.1 | 240, 245, 250, 255, 260, 265, 270, 275, 280, 285 | 0.630 | |

0.1 | 240, 250, 260, 270, 280, 290, 300, 310, 320, 330 | 0.642 | |

0.1 | 240, 260, 280, 300, 320, 340, 360, 380, 400, 420 | 0.616 | |

0.1 | 200, 210, 220, 230, 240 | 0.570 | |

0.1 | 240, 250, 260, 270, 280 | 0.624 | |

0.1 | 200 | 0.482 | |

0.1 | 240 | 0.532 |

(Assuming the average person has 6 L of blood, the total number of leukemia cells needed for each vaccination strategy is (Total dosage)×6×10^{6}

Among these first eight vaccination strategies with a total dosage of 0.5

Among the ten strategies with total dosage 1.0

In this work we only examined a limited set of possible vaccination strategies. Indeed, there is no reason to require that all vaccinations will be of the same size or that the pacing will remain uniform. However, generalizing our survey of possible strategies poses a challenging optimization problem that is beyond the scope of the current paper.

On the other hand, LHS sampling allows us to determine the statistical correlations between the treatment outcomes and a wide range of model parameters. For each LHS simulation, we measure the correlations between the varied parameters and two indices: the minimum cancer concentration attained during the course of simulation (600 days) and the success of the treatment strategy. For our correlations, we use the Pearson and Spearman rank-order coefficients, and consider a treatment to be successful if it causes the cancer concentration to drop below 10^{−10}

In _{T}_{n}

(A) Scatter plot of LHS simulation results with respect to the T cell death rate, _{T}_{n}

We see from the correlation table that the most sensitive parameter is the average number of T cell divisions per stimulation,

There are three additional parameters that have some influence on the outcome. These are the T cell death rate, _{T}_{n}_{n}_{n}_{n}

All the parameters other than the four discussed above have little influence on the outcome of the treatment. We especially point out that the mutation rate per cell division,

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 _{V}_{,1}, _{V}_{,2},…, _{V,n}

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 10^{8} and 10^{6} 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}

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,

We also observe that the more demanding vaccination strategies for each patient P1, P4, and P12 require total doses of 2.3

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

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

An alternative model of CML dynamics was recently published by Roeder

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

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

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.