Understanding, treating and avoiding hematological disease: better medicine through mathematics?

Understanding Hematopoietic Stem Cell Dynamics—Insights from Mathematical Modelling

Purpose of review

Hematopoietic stem cells (HSCs) drive blood-cell production (hematopoiesis). Out-competition of HSCs by malignant cells occurs in many hematologic malignancies like acute myeloid leukemia (AML). Through mathematical modelling, HSC dynamics and their impact on healthy blood cell formation can be studied, using mathematical analysis and computer simulations. We review important work within this field and discuss mathematical modelling as a tool for attaining biological insight.

Recent findings

Various mechanism-based models of HSC dynamics have been proposed in recent years. Key properties of such models agree with observations and medical knowledge and suggest relations between stem cell properties, e.g., rates of division and the temporal evolution of the HSC population. This has made it possible to study how HSC properties shape clinically relevant processes, including engraftment following an HSC transplantation and the response to different treatment.

Summary

Understanding how properties of HSCs affect hematopoiesis is important for efficient treatment of diseases. Mathematical modelling can contribute significantly to these efforts.

A Blueprint for Identifying Phenotypes and Drug Targets in Complex Disorders with Empirical Dynamics

A central challenge in medicine is translating from observational understanding to mechanistic understanding, where some observations are recognized as causes for the others. This can lead not only to new treatments and understanding, but also to recognition of novel phenotypes. Here, we apply a collection of mathematical techniques (empirical dynamics), which infer mechanistic networks in a model-free manner from longitudinal data, to hematopoiesis. Our study consists of three subjects with markers for cyclic thrombocytopenia, in which multiple cells and proteins undergo abnormal oscillations. One subject has atypical markers and may represent a rare phenotype. Our analyses support this contention, and also lend new evidence to a theory for the cause of this disorder. Simulations of an intervention yield encouraging results, even when applied to patient data outside our three subjects. These successes suggest that this blueprint has broader applicability in understanding and treating complex disorders.

Characterizing Chemotherapy-Induced Neutropenia and Monocytopenia Through Mathematical Modelling

In spite of the recent focus on the development of novel targeted drugs to treat cancer, cytotoxic chemotherapy remains the standard treatment for the vast majority of patients. Unfortunately, chemotherapy is associated with high hematopoietic toxicity that may limit its efficacy. We have previously established potential strategies to mitigate chemotherapy-induced neutropenia (a lack of circulating neutrophils) using a mechanistic model of granulopoiesis to predict the interactions defining the neutrophil response to chemotherapy and to define optimal strategies for concurrent chemotherapy/prophylactic granulocyte colony-stimulating factor (G-CSF). Here, we extend our analyses to include monocyte production by constructing and parameterizing a model of monocytopoiesis. Using data for neutrophil and monocyte concentrations during chemotherapy in a large cohort of childhood acute lymphoblastic leukemia patients, we leveraged our model to determine the relationship between the monocyte and neutrophil nadirs during cyclic chemotherapy. We show that monocytopenia precedes neutropenia by 3 days, and rationalize the use of G-CSF during chemotherapy by establishing that the onset of monocytopenia can be used as a clinical marker for G-CSF dosing post-chemotherapy. This work therefore has important clinical applications as a comprehensive approach to understanding the relationship between monocyte and neutrophils after cyclic chemotherapy with or without G-CSF support.

The timing of cyclic cytotoxic chemotherapy can worsen neutropenia and neutrophilia

Despite recent advances in immunotherapies, cytotoxic chemotherapy continues to be a first-line treatment option for the majority of cancers. Unfortunately, a common side effect in patients undergoing chemotherapy treatment is neutropenia. To mitigate the risk of neutropenia and febrile neutropenia, prophylactic treatment with granulocyte-colony stimulating factor (G-CSF) is administered. Extensive pharmacokinetic/pharmacodynamic modelling of myelosuppression during chemotherapy has suggested avenues for therapy optimization to mitigate this neutropenia. However, the issue of resonance, whereby neutrophil oscillations are induced by the periodic administration of cytotoxic chemotherapy and the coadministration of G-CSF, potentially aggravating a patient's neutropenic/neutrophilic status, is not well-characterized in the clinical literature. Here, through analysis of neutrophil data from young acute lymphoblastic leukaemia patients, we find that resonance is occurring during cyclic chemotherapy treatment in 26% of these patients. Motivated by these data and our previous modelling studies on adult lymphoma patients, we examined resonance during treatment with or without G-CSF. Using our quantitative systems pharmacology model of granulopoiesis, we show that the timing of cyclic chemotherapy can worsen neutropenia or neutrophilia, and suggest clinically-actionable schedules to reduce the resonant effect. We emphasize that delaying supportive G-CSF therapy to 6-7 days after chemotherapy can mitigate myelosuppressive effects. This study therefore highlights the importance of quantitative systems pharmacology for the clinical practice for developing rational therapeutic strategies.

Periodic hematological disorders: Quintessential examples of dynamical diseases

This paper summarizes the evidence supporting the classification of cyclic neutropenia as a dynamical disease and periodic chronic myelogenous leukemia is also considered. The unsatisfactory state of knowledge concerning the genesis of cyclic thrombocytopenia and periodic autoimmune hemolytic anemia is detailed.

A general mathematical framework for understanding the behavior of heterogeneous stem cell regeneration

Stem cell heterogeneity is essential for homeostasis in tissue development. This paper establishes a general mathematical framework to model the dynamics of stem cell regeneration with cell heterogeneity and random transitions of epigenetic states. The framework generalizes the classical G0 cell cycle model and incorporates the epigenetic states of individual cells represented by a continuous multidimensional variable. In the model, the kinetic rates of cell behaviors, including proliferation, differentiation, and apoptosis, are dependent on their epigenetic states, and the random transitions of epigenetic states between cell cycles are represented by an inheritance probability function that describes the conditional probability of cell state changes. Moreover, the model can be extended to include genotypic changes and describe the process of gene mutation-induced tumor development. The proposed mathematical framework provides a generalized formula that helps us to understand various dynamic processes of stem cell regeneration, including tissue development, degeneration, and abnormal growth.

A mathematical model of stem cell regeneration with epigenetic state transitions

In this paper, we study a mathematical model of stem cell regeneration with epigenetic state transitions. In the model, the heterogeneity of stem cells is considered through the epigenetic state of each cell, and each epigenetic state defines a subpopulation of stem cells. The dynamics of the subpopulations are modeled by a set of ordinary differential equations in which epigenetic state transition in cell division is given by the transition probability. We present analysis for the existence and linear stability of the equilibrium state. As an example, we apply the model to study the dynamics of state transition in breast cancer stem cells.

Pharmacokinetic-pharmacodynamic modelling of neutrophil response to G-CSF in healthy subjects and patients with chemotherapy-induced neutropenia

AIM

The objective of the present study was to use pharmacokinetic-pharmacodynamic modelling to characterize the effects of chemotherapy on the granulopoietic system and to predict the absolute neutrophil counts (ANCs) for patients with chemotherapy-induced neutropenia treated with filgrastim and pegfilgrastim.

METHODS

Data were extracted from 10 phase I-III studies conducted in 110 healthy adults, and 618 adult and 52 paediatric patients on chemotherapy following administration of filgrastim or pegfilgrastim. The structural model accounted for ANC dynamics and the effects of filgrastim and pegfilgrastim, chemotherapy and corticosteroids. The impact of neutrophils on drug disposition was based on a drug receptor-binding model that assumed quasi-equilibrium and stimulation of the production and maturation of neutrophils upon treatment. The chemotherapy and corticosteroid effects were represented by kinetic-pharmacodynamic-type models, where chemotherapy stimulated elimination of neutrophil precursors at the mitotic stage, and corticosteroids stimulated neutrophil production.

RESULTS

The systemic half-lives of filgrastim (2.6 h) and pegfilgrastim (10.1 h) were as expected. The effective half-life of chemotherapy was 9.6 h, with a 2-day killing effect. The rate of receptor elimination from mitotic compartments exhibited extreme interindividual variability (% coefficient of variation >200), suggesting marked differences in sensitivity to chemotherapy effects on ANCs. The stimulatory effects of pegfilgrastim were significantly greater than those of filgrastim. Model qualification confirmed the predictive capability of this model.

CONCLUSION

This qualified model simulates the time course of ANC in the absence or presence of chemotherapy and predicts nadir, time to nadir and time of recovery from different grades of neutropenia upon treatment with filgrastim and pegfilgrastim.

Severe congenital neutropenias

Severe congenital neutropenias are a heterogeneous group of rare haematological diseases characterized by impaired maturation of neutrophil granulocytes. Patients with severe congenital neutropenia are prone to recurrent, often life-threatening infections beginning in their first months of life. The most frequent pathogenic defects are autosomal dominant mutations in ELANE, which encodes neutrophil elastase, and autosomal recessive mutations in HAX1, whose product contributes to the activation of the granulocyte colony-stimulating factor (G-CSF) signalling pathway. The pathophysiological mechanisms of these conditions are the object of extensive research and are not fully understood. Furthermore, severe congenital neutropenias may predispose to myelodysplastic syndromes or acute myeloid leukaemia. Molecular events in the malignant progression include acquired mutations in CSF3R (encoding G-CSF receptor) and subsequently in other leukaemia-associated genes (such as RUNX1) in a majority of patients. Diagnosis is based on clinical manifestations, blood neutrophil count, bone marrow examination and genetic and immunological analyses. Daily subcutaneous G-CSF administration is the treatment of choice and leads to a substantial increase in blood neutrophil count, reduction of infections and drastic improvement of quality of life. Haematopoietic stem cell transplantation is the alternative treatment. Regular clinical assessments (including yearly bone marrow examinations) to monitor treatment course and detect chromosomal abnormalities (for example, monosomy 7 and trisomy 21) as well as somatic pre-leukaemic mutations are recommended.

Towards Quantitative Systems Pharmacology Models of Chemotherapy-Induced Neutropenia

Neutropenia is a serious toxic complication of chemotherapeutic treatment. For years, mathematical models have been developed to better predict hematological outcomes during chemotherapy in both the traditional pharmaceutical sciences and mathematical biology disciplines. An increasing number of quantitative systems pharmacology (QSP) models that combine systems approaches, physiology, and pharmacokinetics/pharmacodynamics have been successfully developed. Here, I detail the shift towards QSP efforts, emphasizing the importance of incorporating systems-level physiological considerations in pharmacometrics.

Response of an oscillatory differential delay equation to a single stimulus

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

A Mathematical Model of Granulopoiesis Incorporating the Negative Feedback Dynamics and Kinetics of G-CSF/Neutrophil Binding and Internalization

We develop a physiological model of granulopoiesis which includes explicit modelling of the kinetics of the cytokine granulocyte colony-stimulating factor (G-CSF) incorporating both the freely circulating concentration and the concentration of the cytokine bound to mature neutrophils. G-CSF concentrations are used to directly regulate neutrophil production, with the rate of differentiation of stem cells to neutrophil precursors, the effective proliferation rate in mitosis, the maturation time, and the release rate from the mature marrow reservoir into circulation all dependent on the level of G-CSF in the system. The dependence of the maturation time on the cytokine concentration introduces a state-dependent delay into our differential equation model, and we show how this is derived from an age-structured partial differential equation model of the mitosis and maturation and also detail the derivation of the rest of our model. The model and its estimated parameters are shown to successfully predict the neutrophil and G-CSF responses to a variety of treatment scenarios, including the combined administration of chemotherapy and exogenous G-CSF. This concomitant treatment was reproduced without any additional fitting to characterize drug-drug interactions.

Stem-cell dynamics and lineage topology from in vivo fate mapping in the hematopoietic system

In recent years, sophisticated fate-mapping tools have been developed to study the behavior of stem cells in the intact organism. These experimental approaches are beginning to yield a quantitative picture of how cell numbers are regulated during steady state and in response to challenges. Focusing on hematopoiesis and immune responses, we discuss how novel mathematical approaches driven by these fate-mapping data have provided insights into the dynamics and topology of cellular differentiation pathways in vivo. The combination of experiment and theory has allowed to quantify the degree of self-renewal in stem and progenitor cells, shown how native hematopoiesis differs fundamentally from post-transplantation hematopoiesis, and uncovered that the diversification of T lymphocytes during immune responses resembles tissue renewal driven by stem cells.

Dynamical disease: Challenges for nonlinear dynamics and medicine

Dynamical disease refers to illnesses that are associated with striking changes in the dynamics of some bodily function. There is a large literature in mathematics and physics which proposes mathematical models for the physiological systems and carries out analyses of the properties of these models using nonlinear dynamics concepts involving analyses of the stability and bifurcations of attractors. This paper discusses how these concepts can be applied to medicine.