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The authors have declared that no competing interests exist.

Over the past decade, several targeted therapies (e.g. imatinib, dasatinib, nilotinib) have been developed to treat Chronic Myeloid Leukemia (CML). Despite an initial response to therapy, drug resistance remains a problem for some CML patients. Recent studies have shown that resistance mutations that preexist treatment can be detected in a substantial number of patients, and that this may be associated with eventual treatment failure. One proposed method to extend treatment efficacy is to use a combination of multiple targeted therapies. However, the design of such combination therapies (timing, sequence, etc.) remains an open challenge. In this work we mathematically model the dynamics of CML response to combination therapy and analyze the impact of combination treatment schedules on treatment efficacy in patients with preexisting resistance. We then propose an optimization problem to find the best schedule of multiple therapies based on the evolution of CML according to our ordinary differential equation model. This resulting optimization problem is nontrivial due to the presence of ordinary different equation constraints and integer variables. Our model also incorporates drug toxicity constraints by tracking the dynamics of patient neutrophil counts in response to therapy. We determine optimal combination strategies that maximize time until treatment failure on hypothetical patients, using parameters estimated from clinical data in the literature.

Targeted therapy using imatinib, nilotinib or dasatinib has become standard treatment for chronicle myeloid leukemia. A minority of patients, however, fail to respond to treatment or relapse due to drug resistance. One primary driving factor of drug resistance are point mutations within the driving oncogene. Laboratory studies have shown that different leukemic mutants respond differently to different drugs, so a promising way to improve treatment efficacy is to combine multiple targeted therapies. We build a mathematical model to predict the dynamics of different leukemic mutants with imatinib, nilotinib and dasatinib, and employ optimization techniques to find the best treatment schedule of combining the three drugs sequentially. Our study shows that the optimally designed combination therapy is more effective at controlling the leukemic cell burden than any monotherapy under a wide range of scenarios. The structure of the optimal schedule depends heavily on the mutant types present, growth kinetics of leukemic cells and drug toxicity parameters. Our methodology is an important step towards the design of personalized optimal therapeutic schedules for chronicle myeloid leukemia.

Chronic Myeloid Leukemia (CML) is an acquired hematopoietic stem cell disorder leading to the over-proliferation of myeloid cells and an increase in cellular output from the bone marrow that is often associated with splenomegaly. The most common driving mutation in CML is a translocation between chromosomes 9 and 22 that produces a fusion gene known as BCR-ABL. The BCR-ABL protein promotes proliferation and inhibits apoptosis of myeloid progenitor cells and thereby drives expansion of this cell population. By targeting the BCR-ABL oncoprotein, imatinib (brand name Gleevec) is able to induce a complete cytogenetic remission in the majority of chronic phase CML patients. A minority of patients, however, either fail to respond or eventually develop resistance to treatment with imatinib [

The goal of this work is to leverage the differential responses of CML mutant strains to design novel sequential combination treatment schedules using dasatinib, imatinib and nilotinib that optimally control leukemic burden and delay treatment failure due to preexisting resistance. We develop and parametrize a mathematical model for the evolution of both wild-type (WT) CML and mutated (resistant) CML cells in the presence of each therapy. Then we formulate the problem as a discrete optimization problem in which a sequence of monthly treatment decisions is optimized to identify the temporal sequence of imatinib, dasatinib and nilotinib administration that minimizes the total CML cell population over a long time horizon.

There has been a significant amount of work done in the past to mathematically model CML. For example, in [

Another line of research closely related to the current work is the use of optimal control techniques in the design of optimal temporally continuous drug concentration profiles (see, e.g., review articles [

We consider an ODE model of the differentiation hierarchy of hematopoietic cells, adapted from [_{l,i}(_{l,i}(^{j}, where Δ^{m} is the cell abundance at the beginning of month ^{j} under drug

The neutrophils (squares) are part of the terminally differentiated cells.

Here we describe the function of each parameter of this model. For a detailed discussion of how these parameters were estimated from biological data, please see Section A of _{i}(_{1} (resp. _{2}) is computed from the equilibrium abundance of normal (resp. leukemic WT) stem cells assuming only normal (resp. leukemic WT) cells are present, and we set _{i} = _{2} for each _{1} (resp. _{2}) being the equilibrium abundance of normal (resp. leukemic WT) stem cells assuming only normal (resp. leukemic WT) cells are present.

Assume the initial population of each cell type is known. Our goal is to select a treatment plan to minimize the tumor size at the end of the planning horizon. We call this the Optimal Treatment Plan problem (OTP). Each treatment plan is completely characterized by a temporal sequence of monthly treatment decisions over a long time horizon. Between each monthly treatment decision, the dosing regimen is identical from day to day. The standard regimens for each drug, which we will utilize throughout the work, are 300mg twice daily for nilotinib, 100mg once daily for dasatinib, and 400mg once daily for imatinib [

We introduce the binary decision variables ^{m,j} to indicate whether drug ^{m,j} variables that satisfy all constraints in the optimization model gives a feasible treatment plan.

Note the total leukemic cell abundance at day _{l≥1}∑_{i≥2} _{l,i}(

To summarize the previous display, in

The OTP problem is a mixed-integer nonlinear optimization problem, in which some constraints are specified by the solution to a nonlinear system of ODEs

It should be noted that our model does not explicitly consider the phenomena of TKI resistance acquired during therapy. Our model suggests a method for designing optimized treatment plans, beyond monotherapies, at the beginning of treatment. In addition, this optimization procedure can be re-run and modified during the course of treatment, with updated inputs from each patient’s response to the treatment—including the presence of acquired mutations.

Below we summarize our notation for the ease of the reader.

Δ

_{1}: the equilibrium abundance of normal stem cells when only normal cells are present.

_{2}: the equilibrium abundance of leukemic WT stem cells when only leukemic cells are present.

In this work we consider the dynamics of CML response to single-agent and combination schedules utilizing the standard therapies imatinib, dasatinib and nilotinib.

We first utilize the model to demonstrate the dynamics of CML populations with preexisting BCR-ABL mutations under monotherapy with the standard therapies imatinib, dasatinib and nilotinib. Recall that the standard dosing regimens are 300mg twice daily for nilotinib, 100mg once daily for dasatinib, and 400mg once daily for imatinib [^{12} cells [

In the first example we consider a patient harboring a low level of the BCR-ABL mutant F317L before the initiation of TKI therapy. According to the

normal cell | Wild-type | F317L | |
---|---|---|---|

SC | 7.34 × 10^{4} |
2.80 × 10^{5} |
1.48 × 10^{4} |

PC | 1.61 × 10^{7} |
3.87 × 10^{7} |
2.04 × 10^{6} |

DC | 3.24 × 10^{9} |
1.03 × 10^{10} |
5.40 × 10^{8} |

TC | 3.24 × 10^{11} |
1.03 × 10^{12} |
5.40 × 10^{10} |

We plot in

The unit of vertical axis is number of cells. The dynamics of healthy normal PC, DC and TC with imatinib monotherapy (green) and no drug (orange) coincide, since the birth and death rates of normal PC, DC and TC under imatinib monotherapy happen to be the same as the corresponding birth and death rates of normal PC, DC and TC with no drug. Initial conditions are provided in

In the next example we consider a patient with BCR-ABL mutant type M351T preexisting therapy. In contrast to the previous example, this commonly occurring mutant has been found to be partially sensitive in varying degrees to all three therapies. The initial conditions are given in

Normal cell | Wild-type | M351T | |
---|---|---|---|

SC | 7.34 × 10^{4} |
2.80 × 10^{5} |
1.48 × 10^{4} |

PC | 1.61 × 10^{7} |
3.87 × 10^{7} |
2.04 × 10^{6} |

DC | 3.24 × 10^{9} |
1.03 × 10^{10} |
5.40 × 10^{8} |

TC | 3.24 × 10^{11} |
1.03 × 10^{12} |
5.40 × 10^{10} |

In

The unit of vertical axis is number of cells. The dynamics of healthy normal PC, DC and TC with imatinib monotherapy (green) and no drug (orange) coincide, since the birth and death rates of normal PC, DC and TC under imatinib monotherapy happen to be the same as the corresponding birth and death rates of normal PC, DC and TC with no drug. Initial conditions are provided in

We next solve the discrete optimization problem to identify sequential combination therapies utilizing imatinib, dasatinib and nilotinib to optimally treat CML patients with preexisting BCR-ABL mutations. We consider schedules in which a monthly treatment decision is made between one of four choices: imatinib, dasatinib, nilotinib, and drug holiday. During months in which one of the three drugs is administered, the dosing regimen is fixed at 300mg twice daily for nilotinib, 100mg once daily for dasatinib, and 400mg once daily for imatinib. In the following we optimize over feasible treatment decision sequences that result in a minimal leukemic cell burden after 3 years. Each treatment plan is completely characterized by a temporal sequence of drugs over a long time horizon.

We first assume that the mutant M351T preexists therapy. The initial cell populations are given in

Optimal combination | 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 |

Initial conditions are provided in

In ^{7} with the proposed combination therapy and 5.92 × 10^{7} with dasatinib (the best monotherapy). We can see that the proposed optimal treatment schedule leads to more than 50% reduction on final leukemic cell abundances over the best monotherapy.

Next we consider a patient with preexisting mutant F317L instead of M351T. The initial cell abundances are given in

M351T preexisting | 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 |

F317L preexisting | 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |

In ^{7} and 9.48 × 10^{7} under the propose schedule and monotherapy with nilotinib, respectively. The combination therapy performs better in reducing final leukemic cell population than three monotherapies, but the improvement is marginal in this case. Again the optimal schedule uses dasatinib to reduce the wild-type progenitor cell population, but switches to nilotinib much earlier to reduce the wild-type differentiated cell and F317L cell populations.

Lastly we investigate how much gain can be expected from combination therapy if more than one mutant type preexists before initiation of therapy. We again consider an optimization model over a 36-month horizon. We assume mutants M351T and F317L preexist therapy at a low level (each consists of 5% of the total leukemic cell population); the initial conditions are given in

Normal cell | Wild-type | M351T | F317L | |
---|---|---|---|---|

SC | 7.34 × 10^{4} |
2.66 × 10^{5} |
1.48 × 10^{4} |
1.48 × 10^{4} |

PC | 1.61 × 10^{7} |
3.66 × 10^{7} |
2.04 × 10^{6} |
2.04 × 10^{6} |

DC | 3.24 × 10^{9} |
9.72 × 10^{9} |
5.40 × 10^{8} |
5.40 × 10^{8} |

TC | 3.24 × 10^{11} |
9.72 × 10^{11} |
5.40 × 10^{10} |
5.40 × 10^{10} |

We now assume that the two mutants present are E255K and F317L. According to the

M351T | 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 |

F317L | 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |

M351T & F317L | 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |

F317L & E255K | 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 |

One of the most common side effects of TKIs in CML is neutropenia, or the condition of abnormally low neutrophils in the blood. Neutropenia is defined in terms of the absolute neutrophil count (ANC). To incorporate toxicity constraints we develop a model of the dynamics of the patient’s ANC in response to each therapy schedule. We then constrain our optimization problem by considering only schedules during which the patient’s ANC stays above an acceptable threshold level _{anc}. Typically, ANC at diagnosis is within normal limits (between 1500–8000/mm^{3}); thus we set each patient’s initial ANC to be 3000/mm^{3}. Treatment with imatinib, dasatinib and nilotinib all result in reduction of the ANC at varying rates. Neutropenia is defined as an ANC level below _{anc} = 1000/mm^{3}. If a patient’s ANC falls below the threshold, a drug holiday is required at the next monthly treatment decision stage. During a drug holiday, ANC level will recover back to safe levels.

To model this process, we assume the patient’s ANC decreases at rate _{anc,j} per month taking drug _{anc} per month but never exceeds the normal level of _{anc} = 3000/mm^{3}. More specifically, let ^{m} denote the ANC level at the beginning of month ^{m,j} indicate whether drug ^{m,j} = 1 (resp. ^{m,j} = 0) indicates drug ^{m} takes a drug holiday in month ^{m} + _{anc} if ^{m} + _{anc} is not higher than the normal level _{anc}, or _{anc} if ^{m} + _{anc} exceeds _{anc}. If the patient instead takes nilotinib in month ^{m} − _{anc,1}. The parameters governing ANC rate of change are provided in Section A of

There is a substantial literature on mathematical modeling of ANC levels, with many sophisticated nonlinear mathematical models, see e.g. [

We next study how drug toxicity affects the optimal therapy, in particular with the drug toxicity constraint introduced in the _{anc}. The ANC decreases at a constant rate each month under each drug, and increases at a constant rate without drug. The ANC never exceeds the normal level of _{anc}. We first assume that nilotinib has a higher toxicity than dasatinib, and dasatinib has a higher toxicity than imatinib. In particular, the monthly decrease rates of ANC for nilotinib, dasatinib, and imatinib are 350/mm^{3}, 300/mm^{3}, and 250/mm^{3}, respectively, and ANC increases by 2000/mm^{3} with one month drug holiday. Please refer to Section A of

We incorporated the toxicity constraints into the preexisting M351T mutant scenario described previously, i.e. initial cell populations are given in _{anc}.

Nilotinib | 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 |

Dasatinib | 2 2 2 2 2 2 0 2 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 2 0 2 2 2 2 2 2 |

Imatinib | 3 3 3 3 3 3 3 0 3 3 3 3 3 3 3 3 0 3 3 3 3 3 3 3 3 0 3 3 3 3 3 3 3 3 0 3 |

Combination | 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 1 1 1 1 |

The cell dynamics of three monotherapies and the proposed combination therapy are given in

Since it is not clear whether nilotinib or dasatinib result in higher toxicity effects, we also switched the monthly ANC depletion rates to nilotinib—300/mm^{3}, dasatinib—350/mm^{3}, and imatinib—250/mm^{3}, so that dasatinib has the highest toxicity. Other conditions are kept the same. The recommended combination therapy is shown in

Nilotinib>Dasatinib | 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 1 1 1 1 |

Dasatinib>Nilotinib | 2 0 2 2 2 2 2 0 3 2 2 2 2 2 0 3 2 2 2 2 2 0 3 2 2 2 2 2 0 3 3 1 1 1 1 1 |

Here it is assumed that the ANC reduction rate during dasatinib treatment is higher than during nilotinib treatment.

The extent to which TKIs affect leukemic stem cells is currently unknown. Several studies have demonstrated that these cells may in fact be resistant to TKIs, see e.g. [

We assume that the TKIs reduce the production rate of each leukemic stem cell by

0 | 5 | 10 | 20 | |
---|---|---|---|---|

Combination therapy | 2.75 × 10^{7} |
2.65 × 10^{7} |
2.56 × 10^{7} |
2.34 × 10^{7} |

Best monotherapy (dasatinib) | 5.92 × 10^{7} |
5.70 × 10^{7} |
5.47 × 10^{7} |
5.03 × 10^{7} |

We assume that the TKIs reduce the production rate of each leukemic stem cell by

0 | 5 | 10 | 20 | |
---|---|---|---|---|

Combination therapy | 7.46 × 10^{7} |
7.33 × 10^{7} |
7.20 × 10^{7} |
6.94 × 10^{7} |

Best monotherapy (nilotinib) | 9.48 × 10^{7} |
9.38 × 10^{7} |
9.28 × 10^{7} |
9.09 × 10^{7} |

In this work we have considered the problem of finding optimal treatment schedules for the administration of a variety of TKIs for treating chronic phase CML. We modeled the evolution of wild-type and mutant leukemic cell populations with a system of ordinary differential equations. We then formulated an optimization problem to find the sequence of TKIs that lead to a minimal cancerous cell population at the end of a fixed time horizon of 36 months. The 36-month therapeutic horizon is clinically meaningful since it appears that the risk of therapeutic failure and disease progression to blast crisis is highest within the first two years from diagnosis [

At first glance the optimization problem studied in this work (OTP) is quite challenging. It is a mixed-integer nonlinear optimization problem, where the nonlinear constraints are specified by the solution to a nonlinear system of differential equations. However, one factor mitigating the complexity of the problem is the assumption that the TKIs do not effect the stem cell compartment. This has the effect of making the evolution of the stem cell compartment independent of the TKI schedule chosen. In addition, the remaining layers in the cellular hierarchy are modeled by linear differential equations. We can thus numerically solve the differential equation governing the stem cell layer, and treat this function as an inhomogeneous forcing term in the linear differential equation governing the progenitor cells. This allows us to approximate the nonlinear constraints specified by the differential equations by linear constraints with high accuracy. Then the OTP problem can be approximated by a mixed-integer linear optimization problem, which we are able to solve with state-of-the-art optimization software CPLEX [

Throughout this work we have observed that the structure of the optimal therapy depends heavily on model parameters, e.g., cellular growth rates and ANC decay rates. It is likely that each individual patients will have unique model parameters, and therefore a unique best schedule. An exciting application of this work would be the development of personalized optimal therapeutic schedules. Determination of (i) the mutant types (if any) present in a patient’s leukemic cell population, (ii) growth kinetics of their leukemic cell populations, and (iii) patient ANC level responses under various TKIs, would enable our optimization framework to build treatment schedules in a patient-specific setting. At the start of treatment for acute lymphoblastic leukemia (ALL), Quantitative RT-PCR or similar techniques are sometimes used to perform mutational analysis to identify the preexisting mutant types. Indeed, studies have demonstrated that BCR-ABL mutants are present at the time of diagnosis in many ALL patients, and as sequencing technologies improve, smaller and smaller subclones with resistant phenotypes will likely be discovered [

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