Modeling imatinib-treated chronic myelogenous leukemia: reducing the complexity of agent-based models

Dynamical models of mutated chronic myelogenous leukemia cells for a post-imatinib treatment scenario: Response to dasatinib or nilotinib therapy

Targeted inhibition of the oncogenic BCR-ABL1 fusion protein using the ABL1 tyrosine kinase inhibitor imatinib has become standard therapy for chronic myelogenous leukemia (CML), with most patients reaching total and durable remission. However, a significant fraction of patients develop resistance, commonly due to mutated ABL1 kinase domains. This motivated development of second-generation drugs with broadened or altered protein kinase selectivity profiles, including dasatinib and nilotinib. Imatinib-resistant patients undergoing treatment with second-line drugs typically develop resistance to them, but dynamic and clonal properties of this response differ. Shared, however, is the observation of clonal competition, reflected in patterns of successive dominance of individual clones. We present three deterministic mathematical models to study the origins of clinically observed dynamics. Each model is a system of coupled first-order differential equations, considering populations of three mutated active stem cell strains and three associated pools of differentiated cells; two models allow for activation of quiescent stem cells. Each approach is distinguished by the way proliferation rates of the primary stem cell reservoir are modulated. Previous studies have concentrated on simulating the response of wild-type leukemic cells to imatinib administration; our focus is on modelling the time dependence of imatinib-resistant clones upon subsequent exposure to dasatinib or nilotinib. Performance of the three computational schemes to reproduce selected CML patient profiles is assessed. While some simple cases can be approximated by a basic design that does not invoke quiescence, others are more complex and require involvement of non-cycling stem cells for reproduction. We implement a new feedback mechanism for regulation of coupling between cycling and non-cycling stem cell reservoirs that depends on total cell populations. A bifurcation landscape analysis is also performed for solutions to the basic ansatz. Computational models reproducing patient data illustrate potential dynamic mechanisms that may guide optimization of therapy of drug resistant CML.

Long-term treatment effects in chronic myeloid leukemia

We propose and analyze a simplified version of a partial differential equation (PDE) model for chronic myeloid leukemia (CML) derived from an agent-based model proposed by Roeder et al. This model describes the proliferation and differentiation of leukemic stem cells in the bone marrow and the effect of the drug Imatinib on these cells. We first simplify the PDE model by noting that most of the dynamics occurs in a subspace of the original 2D state space. Then we determine the dominant eigenvalue of the corresponding linearized system that controls the long-term behavior of solutions. We mathematically show a non-monotonous dependence of the dominant eigenvalue with respect to treatment dose, with the existence of a unique minimal negative eigenvalue. In terms of CML treatment, this shows that there is a unique dose that maximizes the decay rate of the CML tumor load over long time scales. Moreover this unique dose is lower than the dose that maximizes the initial tumor load decay. Numerical simulations of the full model confirm that this phenomenon is not an artifact of the simplification. Therefore, while optimal asymptotic dosage might not be the best one at short time scales, our results raise interesting perspectives in terms of strategies for achieving and improving long-term deep response.

Optimization and Control of Agent-Based Models in Biology: A Perspective

Agent-based models (ABMs) have become an increasingly important mode of inquiry for the life sciences. They are particularly valuable for systems that are not understood well enough to build an equation-based model. These advantages, however, are counterbalanced by the difficulty of analyzing and using ABMs, due to the lack of the type of mathematical tools available for more traditional models, which leaves simulation as the primary approach. As models become large, simulation becomes challenging. This paper proposes a novel approach to two mathematical aspects of ABMs, optimization and control, and it presents a few first steps outlining how one might carry out this approach. Rather than viewing the ABM as a model, it is to be viewed as a surrogate for the actual system. For a given optimization or control problem (which may change over time), the surrogate system is modeled instead, using data from the ABM and a modeling framework for which ready-made mathematical tools exist, such as differential equations, or for which control strategies can explored more easily. Once the optimization problem is solved for the model of the surrogate, it is then lifted to the surrogate and tested. The final step is to lift the optimization solution from the surrogate system to the actual system. This program is illustrated with published work, using two relatively simple ABMs as a demonstration, Sugarscape and a consumer-resource ABM. Specific techniques discussed include dimension reduction and approximation of an ABM by difference equations as well systems of PDEs, related to certain specific control objectives. This demonstration illustrates the very challenging mathematical problems that need to be solved before this approach can be realistically applied to complex and large ABMs, current and future. The paper outlines a research program to address them.

Optimal harvesting for a predator-prey agent-based model using difference equations

In this paper, a method known as Pareto optimization is applied in the solution of a multi-objective optimization problem. The system in question is an agent-based model (ABM) wherein global dynamics emerge from local interactions. A system of discrete mathematical equations is formulated in order to capture the dynamics of the ABM; while the original model is built up analytically from the rules of the model, the paper shows how minor changes to the ABM rule set can have a substantial effect on model dynamics. To address this issue, we introduce parameters into the equation model that track such changes. The equation model is amenable to mathematical theory—we show how stability analysis can be performed and validated using ABM data. We then reduce the equation model to a simpler version and implement changes to allow controls from the ABM to be tested using the equations. Cohen's weighted κ is proposed as a measure of similarity between the equation model and the ABM, particularly with respect to the optimization problem. The reduced equation model is used to solve a multi-objective optimization problem via a technique known as Pareto optimization, a heuristic evolutionary algorithm. Results show that the equation model is a good fit for ABM data; Pareto optimization provides a suite of solutions to the multi-objective optimization problem that can be implemented directly in the ABM.

Etiology and treatment of hematological neoplasms: stochastic mathematical models

Leukemias are driven by stemlike cancer cells (SLCC), whose initiation, growth, response to treatment, and posttreatment behavior are often "stochastic", i.e., differ substantially even among very similar patients for reasons not observable with present techniques. We review the probabilistic mathematical methods used to analyze stochastics and give two specific examples. The first example concerns a treatment protocol, e.g., for acute myeloid leukemia (AML), where intermittent cytotoxic drug dosing (e.g., once each weekday) is used with intent to cure. We argue mathematically that, if independent SLCC are growing stochastically during prolonged treatment, then, other things being equal, front-loading doses are more effective for tumor eradication than back loading. We also argue that the interacting SLCC dynamics during treatment is often best modeled by considering SLCC in microenvironmental niches, with SLCC-SLCC interactions occurring only among SLCC within the same niche, and we present a stochastic dynamics formalism, involving "Poissonization," applicable in such situations. Interactions at a distance due to partial control of total cell numbers are also considered. The second half of this chapter concerns chromosomal aberrations, lesions known to cause some leukemias. A specific example is the induction of a Philadelphia chromosome by ionizing radiation, subsequent development of chronic myeloid leukemia (CML), CML treatment, and treatment outcome. This time evolution involves a coordinated sequence of > 10 steps, each stochastic in its own way, at the subatomic, molecular, macromolecular, cellular, tissue, and population scales, with corresponding time scales ranging from picoseconds to decades. We discuss models of these steps and progress in integrating models across scales.

A Review of Mathematical Models for Leukemia and Lymphoma

Recently, there has been significant activity in the mathematical community, aimed at developing quantitative tools for studying leukemia and lymphoma. Mathematical models have been applied to evaluate existing therapies and to suggest novel therapies. This article reviews the recent contributions of mathematical modeling to leukemia and lymphoma research. These developments suggest that mathematical modeling has great potential in this field. Collaboration between mathematicians, clinicians, and experimentalists can significantly improve leukemia and lymphoma therapy.

Complex system modelling for veterinary epidemiology

The use of mathematical models has a long tradition in infectious disease epidemiology. The nonlinear dynamics and complexity of pathogen transmission pose challenges in understanding its key determinants, in identifying critical points, and designing effective mitigation strategies. Mathematical modelling provides tools to explicitly represent the variability, interconnectedness, and complexity of systems, and has contributed to numerous insights and theoretical advances in disease transmission, as well as to changes in public policy, health practice, and management. In recent years, our modelling toolbox has considerably expanded due to the advancements in computing power and the need to model novel data generated by technologies such as proximity loggers and global positioning systems. In this review, we discuss the principles, advantages, and challenges associated with the most recent modelling approaches used in systems science, the interdisciplinary study of complex systems, including agent-based, network and compartmental modelling. Agent-based modelling is a powerful simulation technique that considers the individual behaviours of system components by defining a set of rules that govern how individuals ("agents") within given populations interact with one another and the environment. Agent-based models have become a recent popular choice in epidemiology to model hierarchical systems and address complex spatio-temporal dynamics because of their ability to integrate multiple scales and datasets.

Merging concepts - coupling an agent-based model of hematopoietic stem cells with an ODE model of granulopoiesis

Background

Hematopoiesis is a complex process involving different cell types and feedback mechanisms mediated by cytokines. This complexity stimulated various models with different scopes and applications. A combination of complementary models promises to provide their mutual confirmation and to explain a broader range of scenarios. Here we propose a combination of an ordinary differential equation (ODE) model of human granulopoiesis and an agent-based model (ABM) of hematopoietic stem cell (HSC) organization. The first describes the dynamics of bone marrow cell stages and circulating cells under various perturbations such as G-CSF treatment or chemotherapy. In contrast to the ODE model describing cell numbers, our ABM focuses on the organization of individual cells in the stem population.

Results

We combined the two models by replacing the HSC compartment of the ODE model by a difference equation formulation of the ABM. In this hybrid model, regulatory mechanisms and parameters of the original models were kept unchanged except for a few specific improvements: (i) Effect of chemotherapy was restricted to proliferating HSC and (ii) HSC regulation in the ODE model was replaced by the intrinsic regulation of the ABM. Model simulations of bleeding, chronic irradiation and stem cell transplantation revealed that the dynamics of hybrid and ODE model differ markedly in scenarios with stem cell damage. Despite these differences in response to stem cell damage, both models explain clinical data of leukocyte dynamics under four chemotherapy regimens.

Conclusions

ABM and ODE model proved to be compatible and were combined without altering the structure of both models. The new hybrid model introduces model improvements by considering the proliferative state of stem cells and enabling a cell cycle-dependent effect of chemotherapy. We demonstrated that it is able to explain and predict granulopoietic dynamics for a large variety of scenarios such as irradiation, bone marrow transplantation, chemotherapy and growth factor applications. Therefore, it promises to serve as a valuable tool for studies in a broader range of clinical applications, in particular where stem cell activation and proliferation are involved.

Quantitative modeling of chronic myeloid leukemia: insights from radiobiology

Mathematical models of chronic myeloid leukemia (CML) cell population dynamics are being developed to improve CML understanding and treatment. We review such models in light of relevant findings from radiobiology, emphasizing 3 points. First, the CML models almost all assert that the latency time, from CML initiation to diagnosis, is at most ∼10 years. Meanwhile, current radiobiologic estimates, based on Japanese atomic bomb survivor data, indicate a substantially higher maximum, suggesting longer-term relapses and extra resistance mutations. Second, different CML models assume different numbers, between 400 and 10(6), of normal HSCs. Radiobiologic estimates favor values>10(6) for the number of normal cells (often assumed to be the HSCs) that are at risk for a CML-initiating BCR-ABL translocation. Moreover, there is some evidence for an HSC dead-band hypothesis, consistent with HSC numbers being very different across different healthy adults. Third, radiobiologists have found that sporadic (background, age-driven) chromosome translocation incidence increases with age during adulthood. BCR-ABL translocation incidence increasing with age would provide a hitherto underanalyzed contribution to observed background adult-onset CML incidence acceleration with age, and would cast some doubt on stage-number inferences from multistage carcinogenesis models in general.

Fitness conferred by BCR-ABL kinase domain mutations determines the risk of pre-existing resistance in chronic myeloid leukemia

Chronic myeloid leukemia (CML) is the first human malignancy to be successfully treated with a small molecule inhibitor, imatinib, targeting a mutant oncoprotein (BCR-ABL). Despite its successes, acquired resistance to imatinib leads to reduced drug efficacy and frequent progression of disease. Understanding the characteristics of pre-existing resistant cells is important for evaluating the benefits of first-line combination therapy with second generation inhibitors. However, due to limitations of assay sensitivity, determining the existence and characteristics of resistant cell clones at the start of therapy is difficult. Here we combined a mathematical modeling approach using branching processes with experimental data on the fitness changes (i.e., changes in net reproductive rate) conferred by BCR-ABL kinase domain mutations to investigate the likelihood, composition, and diversity of pre-existing resistance. Furthermore, we studied the impact of these factors on the response to tyrosine kinase inhibitors. Our approach predicts that in most patients, there is at most one resistant clone present at the time of diagnosis of their disease. Interestingly, patients are no more likely to harbor the most aggressive, pan-resistant T315I mutation than any other resistance mutation; however, T315I cells on average establish larger-sized clones at the time of diagnosis. We established that for patients diagnosed late, the relative benefit of combination therapy over monotherapy with imatinib is significant, while this benefit is modest for patients with a typically early diagnosis time. These findings, after pre-clinical validation, will have implications for the clinical management of CML: we recommend that patients with advanced-phase disease be treated with combination therapy with at least two tyrosine kinase inhibitors.

A multicellular basis for the origination of blast crisis in chronic myeloid leukemia

Chronic myeloid leukemia (CML) is characterized by a specific chromosome translocation, and its pathobiology is considered comparatively well understood. Thus, quantitative analysis of CML and its progression to blast crisis may help elucidate general mechanisms of carcinogenesis and cancer progression. Hitherto, it has been widely postulated that CML blast crisis originates mainly via cell-autonomous mechanisms such as secondary mutations or genomic instability. However, recent results suggest that carcinogenic transformation may be an inherently multicellular event, in departure from the classic unicellular paradigm. We investigate this possibility in the case of blast crisis origination in CML. A quantitative, mechanistic cell population dynamics model was employed. This model used recent data on imatinib-treated CML; it also used earlier clinical data, not previously incorporated into current mathematical CML/imatinib models. With the pre-imatinib data, which include results on many more blast crises, we obtained evidence that the driving mechanism for blast crisis origination is a cooperation between specific cell types. Assuming leukemic-normal interactions resulted in a statistically significant improvement over assuming either cell-autonomous mechanisms or interactions between leukemic cells. This conclusion was robust with regard to changes in the model's adjustable parameters. Application of the results to patients treated with imatinib suggests that imatinib may act not only on malignant blast precursors, but also, to a limited degree, on the malignant blasts themselves.

Mathematical modeling as a tool for planning anticancer therapy

We review a large volume of literature concerning mathematical models of cancer therapy, oriented towards optimization of treatment protocols. The review, although partly idiosyncratic, covers such major areas of therapy optimization as phase-specific chemotherapy, antiangiogenic therapy and therapy under drug resistance. We start from early cell cycle progression models, very simple but admitting explicit mathematical solutions, based on methods of control theory. We continue with more complex models involving evolution of drug resistance and pharmacokinetic and pharmacodynamic effects. Then, we consider two more recent areas: angiogenesis of tumors and molecular signaling within and among cells. We discuss biological background and mathematical techniques of this field, which has a large although only partly realized potential for contributing to cancer treatment.

Computer modeling describes gravity-related adaptation in cell cultures

Questions about the changes of biological systems in response to hostile environmental factors are important but not easy to answer. Often, the traditional description with differential equations is difficult due to the overwhelming complexity of the living systems. Another way to describe complex systems is by simulating them with phenomenological models such as the well-known evolutionary agent-based model (EABM). Here we developed an EABM to simulate cell colonies as a multi-agent system that adapts to hyper-gravity in starvation conditions. In the model, the cell's heritable characteristics are generated and transferred randomly to offspring cells. After a qualitative validation of the model at normal gravity, we simulate cellular growth in hyper-gravity conditions. The obtained data are consistent with previously confirmed theoretical and experimental findings for bacterial behavior in environmental changes, including the experimental data from the microgravity Atlantis and the Hypergravity 3000 experiments. Our results demonstrate that it is possible to utilize an EABM with realistic qualitative description to examine the effects of hypergravity and starvation on complex cellular entities.

A PDE model for imatinib-treated chronic myelogenous leukemia

We derive a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia (CML). This model is a continuous extension of the agent-based CML model of Roeder et al. (Nat. Med. 12(10), 1181-1184, 2006) and of its recent formulation as a system of difference equations (Kim et al. in Bull. Math. Biol. 70(3), 728-744, 2008). The new model is formulated as a system of partial differential equations that describe various stages of differentiation and maturation of normal hematopoietic cells and of leukemic cells. An imatinib treatment is also incorporated into the model. The simulations of the new PDE model are shown to qualitatively agree with the results that were obtained with the discrete-time (difference equation and agent-based) models. At the same time, for a quantitative agreement, it is necessary to adjust the values of certain parameters, such as the rates of imatinib-induced inhibition and degradation.