A Review of Mathematical Models for Leukemia and Lymphoma

Dynamically adjusted cell fate decisions and resilience to mutant invasion during steady-state hematopoiesis revealed by an experimentally parameterized mathematical model

A major next step in hematopoietic stem cell (HSC) biology is to enhance our quantitative understanding of cellular and evolutionary dynamics involved in undisturbed hematopoiesis. Mathematical models have been and continue to be key in this respect, and are most powerful when parameterized experimentally and containing sufficient biological complexity. In this paper, we use data from label propagation experiments in mice to parameterize a mathematical model of hematopoiesis that includes homeostatic control mechanisms as well as clonal evolution. We find that nonlinear feedback control can drastically change the interpretation of kinetic estimates at homeostasis. This suggests that short-term HSC and multipotent progenitors can dynamically adjust to sustain themselves temporarily in the absence of long-term HSCs, even if they differentiate more often than they self-renew in undisturbed homeostasis. Additionally, the presence of feedback control in the model renders the system resilient against mutant invasion. Invasion barriers, however, can be overcome by a combination of age-related changes in stem cell differentiation and evolutionary niche construction dynamics based on a mutant-associated inflammatory environment. This helps us understand the evolution of e.g., TET2 or DNMT3A mutants, and how to potentially reduce mutant burden.

Impact of Resistance on Therapeutic Design: A Moran Model of Cancer Growth

Resistance of cancers to treatments, such as chemotherapy, largely arise due to cell mutations. These mutations allow cells to resist apoptosis and inevitably lead to recurrence and often progression to more aggressive cancer forms. Sustained-low dose therapies are being considered as an alternative over maximum tolerated dose treatments, whereby a smaller drug dosage is given over a longer period of time. However, understanding the impact that the presence of treatment-resistant clones may have on these new treatment modalities is crucial to validating them as a therapeutic avenue. In this study, a Moran process is used to capture stochastic mutations arising in cancer cells, inferring treatment resistance. The model is used to predict the probability of cancer recurrence given varying treatment modalities. The simulations predict that sustained-low dose therapies would be virtually ineffective for a cancer with a non-negligible probability of developing a sub-clone with resistance tendencies. Furthermore, calibrating the model to in vivo measurements for breast cancer treatment with Herceptin, the model suggests that standard treatment regimens are ineffective in this mouse model. Using a simple Moran model, it is possible to explore the likelihood of treatment success given a non-negligible probability of treatment resistant mutations and suggest more robust therapeutic schedules.

Early dynamics of chronic myeloid leukemia on nilotinib predicts deep molecular response

Chronic myeloid leukemia (CML) is a myeloproliferative disorder caused by the BCR-ABL1 tyrosine kinase. Although ABL1-specific tyrosine kinase inhibitors (TKIs) including nilotinib have dramatically improved the prognosis of patients with CML, the TKI efficacy depends on the individual patient. In this work, we found that the patients with different nilotinib responses can be classified by using the estimated parameters of our simple dynamical model with two common laboratory findings. Furthermore, our proposed method identified patients who failed to achieve a treatment goal with high fidelity according to the data collected only at three initial time points during nilotinib therapy. Since our model relies on the general properties of TKI response, our framework would be applicable to CML patients who receive frontline nilotinib or other TKIs.

Differential Response to Cytotoxic Drugs Explains the Dynamics of Leukemic Cell Death: Insights from Experiments and Mathematical Modeling

This study presents a framework whereby cancer chemotherapy could be improved through collaboration between mathematicians and experimentalists. Following on from our recently published model, we use A20 murine leukemic cells transfected with monomeric red fluorescent proteins cells (mCherry) to compare the simulated and experimental cytotoxicity of two Federal Drug Administration (FDA)-approved anticancer drugs, Cytarabine (Cyt) and Ibrutinib (Ibr) in an in vitro model system of Chronic Lymphocytic Leukemia (CLL). Maximum growth inhibition with Cyt (95%) was reached at an 8-fold lower drug concentration (6.25 μM) than for Ibr (97%, 50 μM). For the proposed ordinary differential equations (ODE) model, a multistep strategy was used to estimate the parameters relevant to the analysis of in vitro experiments testing the effects of different drug concentrations. The simulation results demonstrate that our model correctly predicts the effects of drugs on leukemic cells. To assess the closeness of the fit between the simulations and experimental data, RMSEs for both drugs were calculated (both RMSEs < 0.1). The numerical solutions of the model show a symmetrical dynamical evolution for two drugs with different modes of action. Simulations of the combinatorial effect of Cyt and Ibr showed that their synergism enhanced the cytotoxic effect by 40%. We suggest that this model could predict a more personalized drug dose based on the growth rate of an individual’s cancer cells.

Spatial structure governs the mode of tumour evolution

Characterizing the mode-the way, manner or pattern-of evolution in tumours is important for clinical forecasting and optimizing cancer treatment. Sequencing studies have inferred various modes, including branching, punctuated and neutral evolution, but it is unclear why a particular pattern predominates in any given tumour. Here we propose that tumour architecture is key to explaining the variety of observed genetic patterns. We examine this hypothesis using spatially explicit population genetics models and demonstrate that, within biologically relevant parameter ranges, different spatial structures can generate four tumour evolutionary modes: rapid clonal expansion, progressive diversification, branching evolution and effectively almost neutral evolution. Quantitative indices for describing and classifying these evolutionary modes are presented. Using these indices, we show that our model predictions are consistent with empirical observations for cancer types with corresponding spatial structures. The manner of cell dispersal and the range of cell-cell interactions are found to be essential factors in accurately characterizing, forecasting and controlling tumour evolution.

Building mean field ODE models using the generalized linear chain trick & Markov chain theory

The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on [0,∞) that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

Optimal CAR T-cell Immunotherapy Strategies for a Leukemia Treatment Model

CAR T-cell immunotherapy is a new development in the treatment of leukemia, promising a new era in oncology. Although so far, this procedure only helps 50–90% of patients and, like other cancer treatments, has serious side effects. In this work, we have proposed a controlled model for leukemia treatment to explore possible ways to improve immunotherapy methodology. Our model is described by four nonlinear differential equations with two bounded controls, which are responsible for the rate of injection of chimeric cells, as well as for the dosage of the drug that suppresses the so-called “cytokine storm”. The optimal control problem of minimizing the cancer cells and the activity of the cytokine is stated and solved using the Pontryagin maximum principle. The five possible optimal control scenarios are predicted analytically using investigation of the behavior of the switching functions. The optimal solutions, obtained numerically using BOCOP-2.2.0, confirmed our analytical findings. Interesting results, explaining, why therapies with rest intervals (for example, stopping injections in the middle of the treatment interval) are more effective (within the model), rather than with continuous injections, are presented. Possible improvements to the mathematical model and method of immunotherapy are discussed.

A Mathematical Model of the Transition from Normal Hematopoiesis to the Chronic and Accelerated-Acute Stages in Myeloid Leukemia

A mathematical model given by a two-dimensional differential system is introduced in order to understand the transition process from the normal hematopoiesis to the chronic and accelerated-acute stages in chronic myeloid leukemia. A previous model of Dingli and Michor is refined by introducing a new parameter in order to differentiate the bone marrow microenvironment sensitivities of normal and mutant stem cells. In the light of the new parameter, the system now has three distinct equilibria corresponding to the normal hematopoietic state, to the chronic state, and to the accelerated-acute phase of the disease. A characterization of the three hematopoietic states is obtained based on the stability analysis. Numerical simulations are included to illustrate the theoretical results.

Generalizations of the 'Linear Chain Trick': incorporating more flexible dwell time distributions into mean field ODE models

In this paper we generalize the Linear Chain Trick (LCT; aka the Gamma Chain Trick) to help provide modelers more flexibility to incorporate appropriate dwell time assumptions into mean field ODEs, and help clarify connections between individual-level stochastic model assumptions and the structure of corresponding mean field ODEs. The LCT is a technique used to construct mean field ODE models from continuous-time stochastic state transition models where the time an individual spends in a given state (i.e., the dwell time) is Erlang distributed (i.e., gamma distributed with integer shape parameter). Despite the LCT’s widespread use, we lack general theory to facilitate the easy application of this technique, especially for complex models. Modelers must therefore choose between constructing ODE models using heuristics with oversimplified dwell time assumptions, using time consuming derivations from first principles, or to instead use non-ODE models (like integro-differential or delay differential equations) which can be cumbersome to derive and analyze. Here, we provide analytical results that enable modelers to more efficiently construct ODE models using the LCT or related extensions. Specifically, we provide (1) novel LCT extensions for various scenarios found in applications, including conditional dwell time distributions; (2) formulations of these LCT extensions that bypass the need to derive ODEs from integral equations; and (3) a novel Generalized Linear Chain Trick (GLCT) framework that extends the LCT to a much broader set of possible dwell time distribution assumptions, including the flexible phase-type distributions which can approximate distributions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^+$$\end{document}R+ and can be fit to data.

Study of cohabitation and interconnection effects on normal and leukaemic stem cells dynamics in acute myeloid leukaemia

On the basis of recent studies, understanding the intimate relationship between normal and leukaemic stem cells is very important in leukaemia treatment. The authors' aim in this work is to clarify and assess the effect of coexistence and interconnection phenomenon on the healthy and cancerous stem cell dynamics. To this end, they perform the analysis of two time-delayed stem cell models in acute myeloid leukaemia. The first model is based on decoupled healthy and cancerous stem cell populations (i.e. there is no interaction between cell dynamics) and the second model includes interconnection between both population's dynamics. By using the positivity of both systems, they build new linear functions that permit to derive global stability conditions for each model. Moreover, knowing that most common types of haematological diseases are characterised by the existence of oscillations, they give conditions for the existence of a limit cycle (oscillations) in a particularly interesting healthy situation based on Poincare-Bendixson theorem. The obtained results are simulated and interpreted to be significant in understanding the effect of interconnection and would lead to an improvement in leukaemia treatment.

A study on the predictability of acute lymphoblastic leukaemia response to treatment using a hybrid oncosimulator

Efficient use of Virtual Physiological Human (VPH)-type models for personalized treatment response prediction purposes requires a precise model parameterization. In the case where the available personalized data are not sufficient to fully determine the parameter values, an appropriate prediction task may be followed. This study, a hybrid combination of computational optimization and machine learning methods with an already developed mechanistic model called the acute lymphoblastic leukaemia (ALL) Oncosimulator which simulates ALL progression and treatment response is presented. These methods are used in order for the parameters of the model to be estimated for retrospective cases and to be predicted for prospective ones. The parameter value prediction is based on a regression model trained on retrospective cases. The proposed Hybrid ALL Oncosimulator system has been evaluated when predicting the pre-phase treatment outcome in ALL. This has been correctly achieved for a significant percentage of patient cases tested (approx. 70% of patients). Moreover, the system is capable of denying the classification of cases for which the results are not trustworthy enough. In that case, potentially misleading predictions for a number of patients are avoided, while the classification accuracy for the remaining patient cases further increases. The results obtained are particularly encouraging regarding the soundness of the proposed methodologies and their relevance to the process of achieving clinical applicability of the proposed Hybrid ALL Oncosimulator system and VPH models in general.