Chaos 30, 063123 (2020); https://doi.org/10.1063/5.0006517 30, 063123

© 2020 Author(s).

Periodic hematological disorders:Quintessential examples of dynamicaldiseasesCite as: Chaos 30, 063123 (2020); https://doi.org/10.1063/5.0006517Submitted: 04 March 2020 . Accepted: 21 May 2020 . Published Online: 08 June 2020

Michael C. Mackey

Chaos ARTICLE scitation.org/journal/cha

Periodic hematological disorders: Quintessentialexamples of dynamical diseases

Cite as: Chaos 30, 063123 (2020); doi: 10.1063/5.0006517

Submitted: 4 March 2020 · Accepted: 21May 2020 ·

Published Online: 8 June 2020 View Online Export Citation CrossMark

Michael C. Mackeya)

AFFILIATIONS

Department of Physiology, Department of Physics, and Department of Mathematics McGill University, Montreal,Quebec H4X 2C1, Canada

Note: This paper is part of the Focus Issue on Dynamical Disease: A Translational Perspective.a)Author to whom correspondence should be addressed: michael.mackey@mcgill.ca. URL: https://www.mcgill.ca/mathematical-physiology-lab/

ABSTRACT

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

Published under license by AIP Publishing. https://doi.org/10.1063/5.0006517

The concept of dynamical disease first appeared in 1977, and sincethat time numerous investigators have searched for examplesthat might fulfill the requirements of this hypothesized clini-cal entity. Here, I argue that some hematological disorders arebeautiful examples of dynamical diseases and discuss the insightsthat have been obtained into the origin of cyclic neutropenia(and its treatment) and periodic chronic myelogenous leukemia. Ialso briefly discuss cyclic thrombocytopenia and periodic autoim-mune hemolytic anemia.

I. INTRODUCTION

Nearly 2400 years ago, Hippocrates associated disease witha change in the regularity of a physiological process. Present dayclinical medicine often focuses on diseases in which these changesoccur on time scales ranging from milliseconds to hours, for exam-ple, the generation of cardiac and respiratory arrhythmias, tremors,and seizures. More puzzling have been those diseases, collectivelyreferred to as “periodic diseases,” in which symptoms recur in anapproximately periodic fashion.1 Among the latter are the periodichematological diseases, i.e., cyclic neutropenia (CN, also known asperiodic hematopoiesis),2–5 cyclic thrombocytopenia (CT),6,7 and theperiodic variants of chronic myelogenous leukemia (PCML)8,9 andautoimmune hemolytic anemia.10,11

In 1977, it was proposed12 that some periodic diseases (not justin hematology) might be “dynamical diseases . . . characterized by

the operation of a basically normal control system in a region ofphysiological parameters that produces pathological behavior.” (Theconcept of dynamical disease was preceded by a similar idea relatedto schizophrenia.13) The concept was later elaborated in Ref. 14 withmany examples from the biological and medical sciences.

Here, I summarize the evidence that two periodic hematologi-cal diseases (cyclic neutropenia and periodic chronic myelogenousleukemia) are perfect examples of dynamical diseases and brieflyconsider two others (periodic autoimmune hemolytic anemia andcyclic thrombocytopenia) for which the evidence is still incomplete.

II. OUTLINE OF HUMAN HEMATOPOIESIS AND ITS

REGULATION

Blood cells are formed from a hematopoietic stem cell (HSC) ina process known as hematopoiesis. In humans, hematopoiesis pro-duces the equivalent of our body weight in red blood cells, whiteblood cells, and platelets every decade of life.15 Throughout, thisprocess usually proceeds flawlessly, implying the existence of robustcontrol mechanisms. This cellular renewal system is thus ideal forthe study of the normal regulation of tissue proliferation and differ-entiation from the single cell to whole organ level, and the study ofderangements of these processes.

Though hematopoiesis is incredibly complicated,17,18 the broadoutlines can be summarized as in Fig. 1, which schematicallyshows the major aspects of the mammalian platelet, red blood cell,

Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-1

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

monocyte, and granulocyte production. Control is mediated by alarge family of growth factors and cytokines. Three of the majorplayers are thrombopoietin (TPO), erythropoietin (EPO), and gran-ulocyte colony stimulating factor (G-CSF), which also have local reg-ulatory (LR) effects within the HSC population. All three cytokineswill play a major role in our discussion of the hematological dis-ease. CFU/BFU refers to the various in vitro analogs of the in vivocommitted stem cells.

From a mathematical standpoint all of these are negative feed-back mechanisms in the sense that a fall in a peripheral circulatingcell numbers leads to a consequent increase in production of theimmature precursor and this response is mediated by a specificcytokine or group of them. The mathematics is further complicatedby the fact that there are significant delays (often state dependent)between when a cytokine acts and the resulting effect is felt in thecirculation.

Investigations of whole animal dynamic behavior of cellularreplication systems is hampered by the lack of good quality temporaldata on cell numbers and cytokine levels in response to perturbation.Ironically, the best source of data currently available comes fromclinical studies of patients with hematological disease. Of the vastarray of documented hematological pathologies, the periodic hema-tological diseases (periods from weeks to months) have been someof the most instructive in terms of elucidating the control mech-anisms regulating hematopoiesis.19 See Fig. 2 for an illustration offour of the most studied of these disorders which form the focus ofthis paper.

Periodic hematological diseases fall into two broad classes. Thefirst, with oscillations in numbers of a single circulating cell type, isprobably due to a destabilization of a peripheral control mechanism,e.g., cyclic thrombocytopenia with periods of 13–65 days7,23,24 andautoimmune hemolytic anemia.25 The second type has several cir-culating cell types and seemingly involves the stem cells. Examplesare cyclic neutropenia with periods of 14–45 days26,27 and periodicchronic myelogenous leukemia.28,29. Both classes of disorders illu-minate aspects of hematopoietic regulation that would never havebeen discovered in a laboratory setting because of the time scalesinvolved.

III. MATHEMATICAL MODELING IN HEMATOLOGY

Since this is a non-technical survey, detailed consideration ofthe variety of mathematical modeling techniques that have beenemployed to understand normal and pathological hematopoiesis isinappropriate. It suffices to simply note that there is an excellentrecent survey30 of modeling efforts in the area over the past halfcentury. These techniques range from differential equation modelsthrough delay differential equations, partial differential equations,and also agent based models.

In work with my collaborators, we have typically utilized non-linear differential delay equations of variable complexity depend-ing on the question under consideration. Nonlinearities arisebecause of the stoichiometry of cytokine receptor interactions,and the delays typically reflect maturation and cell cycle times.

FIG. 1. The architecture and control ofmammalian hematopoiesis. All blood cellsare formed from hematopoietic stem cells(HSCs), and this figure summarizes mam-malian platelet (P), red blood cell (RBC),monocyte, and granulocyte (G/M includ-ing neutrophil, basophil, and eosinophil)production. Control over these processesis mediated by a variety of cytokines[e.g., thrombopoietin (TPO), erythropoi-etin (EPO), and granulocyte colony stim-ulating factor (G-CSF) are the main onesbut over 50 have so far been identi-fied], and there are also local regula-tory (LR) effects within the HSC pop-ulation. CFU/BFU refers to the variousin vitro analogs of the in vivo commit-ted stem cells. Reprinted with permissionfromC. Haurie et al., Blood 92, 2629–2640(1998). Copyright 1998 American Societyof Hematology.

Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-2

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 2. Examples of data for four periodic hematological diseases. AIHA: reticulocyte numbers (×104 cells/µl) in an AIHA subject.20 CT: cyclic fluctuations in platelet counts(×103 cells/µl).21 CN: circulating neutrophils (×103 cells/µl), platelets (×105 cells/µl), and reticulocytes (×104 cells/µl) in a cyclic neutropenic patient.22 PCML: whiteblood cells (top) (×104 cells/µl), platelets (middle) (×105 cells/µl), and reticulocyte (bottom) (×104 cells/µl) counts in a PCML patient.9 Reprinted with permission fromC. Foley and M. C. Mackey, J. Math. Biol. 58, 285–322 (2009). Copyright 2009 Springer.

Representative examples of these models are easily found27,29,31 as arereviews.5,16,19,32–35

IV. PERIODIC HEMATOLOGICAL DISEASE AS

DYNAMICAL DISEASES

A. Cyclic neutropenia

The number of circulating neutrophils is normally relativelyconstant with an average absolute neutrophil count (ANC) of about5.0 × 109 cells/l. Neutropenia is characterized by a low number ofneutrophils, thus indicating that the individual is less effective atfighting infections. Cyclic neutropenia is defined by oscillations inthe number of neutrophils from normal to very low levels (less than0.5 × 109 cells/l). The period of these oscillations is usually around3 weeks for humans, although periods up to 45 days are found.36

CN is effectively treated with daily administration of G-CSF, whichreduces the period of the oscillations and increases both the oscilla-tion amplitude and the value of the ANC nadir.27 G-CSF decreasesthe duration of severe neutropenia, which is clinically desirable.

Understanding CN has been aided by the existence of a natu-rally occurring animal model in gray collies.37 The canine disordershows the same characteristics as in humans, except that the periodof the oscillations is between 11 and 15 days. The existence of this

animal model has allowed for the collection of a variety of data thatwould have been difficult to obtain in humans.

A major characteristic of CN is that the oscillations are notonly present in neutrophils, but often are observed in platelets,monocytes, and reticulocytes,16 thus, CN is sometimes called peri-odic hematopoiesis.38 This observation suggests that the source ofthe oscillations may lie in, or involve, the stem cell compartment.

Although rare, cyclic neutropenia is probably the most exten-sively studied periodic hematological disorder. One of the firstmodels to consider cyclic neutropenia is found in a computer study39

which inspired a mathematical analysis.40 Rubinow and Lebowitz41

offered a comprehensive formulation of the regulation of neutrophilproduction and these studies have been nicely complemented byother contributions.42–48

Guided by the observation of oscillations in all of the circulatingblood cell types, I had initially thought that the origin of the oscil-lations must originate from a defect in the stem cell compartment49

and, therefore, used the Burns and Tannock50 G0 model for the cellcycle (see Fig. 3) to investigate this. Based on a rudimentary bifurca-tion analysis (see Fig. 4), I concluded that the only way in whichthe characteristics of cyclic neutropenia could occur was throughan abnormally high rate γ of apoptosis in the proliferating phase ofthe stem cells. Referring to Fig. 4, the hypothesis went roughly likethis. Neutropenia was due to a very large value of γ > γ2 and there

Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-3

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 3. The Burns and Tannock50 model for the cell cycle consisting of a restingphase (G0) and the proliferating phase P with the sub-phases G1, S (DNA syn-thesis), G2, and M (mitosis and cytokinesis). It is assumed that cells can die fromthe proliferating phase at a rate γ and exit into the differentiation pathway fromG0 at a rate δ.

were no oscillations, but for somewhat lower values of γ ∈ (γ1, γ2],oscillations of variable amplitude and period could exist. This waspartially consistent with the findings of the effects of exogenousG-CSF administered to gray collies37 as well as humans,26 sinceG-CSF decreases apoptosis.

However, there is a problem because the model predicts thatever higher doses of G-CSF (decreased apoptosis rate γ ) should leadto a decreased amplitude of oscillation but ever increasing periodand eventually no oscillation. Although it was observed that oscilla-tions could be obliterated in gray collies with administration of largedoses of G-CSF, it was never observed that the amplitude increasedto a maximum and then started to decrease as G-CSF levels wereincreased! Thus, as always, the devil is in the details.

This conundrum was eventually solved52 by considering themore complete model of Fig. 5. The results of a bifurcation anal-ysis indicated that cyclic neutropenia was indeed probably due toelevated levels of apoptosis, but not in the stem cells. Thus, theresolution of the issue suggests that cyclic neutropenia is causedby abnormally high levels of apoptosis53 in the proliferating neu-trophil precursors. This leads to a fall in the level of circulatingmature neutrophils, and this in turn triggers a surge of G-CSF,which increases the rate of differentiation out of the G0 phase ofthe stem cell compartment into the neutrophil line. This elevationof the differentiation rate destabilizes the stem cell dynamics leadingto oscillations through a Hopf bifurcation.

Building on these ideas, Colijn and Mackey27 developed a morecomplete model for human hematopoiesis that included all threemajor cell lines (Fig. 1). Using data from nine gray collies37 aswell as from 27 neutropenia patients in a clinical trial of G-CSF26

and assuming a set of normal parameters (separate for dogs andhumans), it was shown that the most significant parameter changesfrom normal to mimic the cyclic neutropenia results were a largeincrease in the level of apoptosis in the neutrophil precursors anda less significant decrease in the maximal rate of re-entry of G0

phase stem cells into the proliferative phase. These findings wereconsistent between both gray collie and patient data.

FIG. 4. The results of a rudimentary bifurcation analysis of the Burns andTannock50model for the cell cycle showing the behavior of the stem cell efflux (toppanel) from G0 as a function of the apoptosis rate γ in the proliferating phase. Atγ1 and γ2, there are supercritical Hopf bifurcations, and the amplitude and periodof the oscillations are shown in the middle and bottom panels. S (stable), U (unsta-ble), Normal (0 ≤ γ < γ1), Cycling NP (cyclic neutropenia, γ1 ≤ γ < γ2), andNP (neutropenia, γ ≥ γ2). A recent

51 and very complete bifurcation analysis hasrevealed the rich dynamical complexity of the Burns and Tannock model for thecell cycle.

Data were also available for both dogs and patients while underG-CSF therapy. Not unexpectedly, the modeling indicated that themost significant change in both was a significant decrease in the rateof neutrophil precursor apoptosis due to G-CSF.

Thus, the results of Colijn and Mackey27 are consistent with theconclusion52 that cyclic neutropenia is primarily a disorder causedby an abnormally high level of apoptosis in the proliferating neu-trophil precursors, and cyclic neutropenia seems to be an example of

Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-4

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 5. A diagrammatic representation of the relationships between stem cellsand neutrophils. Resting G0 phase stem cells (S) may either stay in that phaseindefinitely, re-enter the proliferative phase at a rate K(S) dependent on S ordifferentiate into the neutrophil line at a rate F(N), where N is the circulatingnumber of mature neutrophils. (Differentiation into the erythrocyte and plateletlines is neglected in this treatment, but it is unimportant for the considerationshere.) Following differentiation into the neutrophil line, a stem cell undergoes anamplification (box with a peak) A during a proliferation and maturation periodlasting τN days before being released into the circulation (bottom circle) as amature neutrophil and dying at a random rate α. Reprinted with permission fromBernard et al., J. Theor. Biol. 223, 283–298 (2003). Copyright 2003 Elsevier.

a dynamical disease. The control system is apparently working nor-mally but in a parameter range (increased rate of apoptosis) givingpathological behavior. Much is known about the molecular basis forthis increased apoptosis,5 but will not be discussed here.

B. Periodic chronic myelogenous leukemia

Leukemia is a malignancy of the blood characterized byan abnormal proliferation of blood cells, usually leucocytes.Chronic myelogenous leukemia (CML) is distinguished from otherleukemias by the presence of a genetic abnormality in blood cells,called the Philadelphia chromosome.54

A variant of particular interest is periodic chronic myeloge-nous leukemia (PCML), characterized by oscillations in circulatingcell numbers, primarily in leucocytes, but often also in the plateletsand reticulocytes.28 The leucocyte count varies periodically betweenvalues of 30 and 200 × 109 cells/l with a period ranging from 40to 80 days. Oscillations in platelet and reticulocyte numbers, whenpresent, occur with the same period as the leucocytes, around nor-mal or elevated numbers.28,55 The hypothesis that the disease origi-nates from the stem cell compartment is supported by the presenceof the Philadelphia chromosome in all of the mature cells as well asoscillations in more than one cell lineage.

My first efforts to understand periodic leukemia came in apaper by Mackey and Glass12 when we put forward the dynamicaldisease concept. The model in that paper is embarrassingly naïvefrom a biological point of view, but rather remarkably has generateda huge amount of interest and is now known as the “Mackey–Glassequation.”56

The reason for this is connected with the fact that for ratherwide ranges of parameter values, the solutions appear to be “chaotic”in that numerically they have no well defined period. Thus, thesolutions to this model have served as paradigmatic examples fornumerical analysts developing techniques to detect chaotic behav-ior. Understandably, this has generated considerable interest amongmathematicians, but unfortunately concrete analytic results havebeen few and far between.57

Going forward, as my knowledge of the clinical aspects ofchronic myelogenous leukemia improved, more realistic modelingefforts appeared,58–60 culminating in Ref. 29. In that study, the Colijnand Mackey27 model was used to analyze the data from 11 periodicchronic myelogenous leukemia patients. Assuming initially normalparameter values, it was determined for each of the patients whatparameters had to be changed to match the clinical hematologicaldata. It was a consistent finding that there needed to be a decrease inthe level of neutrophil precursor apoptosis, an increase in the max-imal differentiation rate from the G0 stem cells into the neutrophilline, and to a lesser extent a decrease in the rate of stem cell apop-tosis. More recent studies61 have extended this work, but basicallyconfirmed the original findings.

It is noteworthy that the available data on chronic myelogenousleukemia patients is not being augmented, thanks to the appearanceof a highly effective pharmacological treatment62 and the success ofstem cell transplant when feasible. Therefore, the historical data thatwe have is all that we are likely to have for this fascinating disorder.

Is periodic chronic myelogenous leukemia a dynamical disease?The evidence is inconclusive, but my guess is that it probably is.However, it is also likely that we will never know for sure.

C. Cyclic thrombocytopenia

Platelets take part in the clotting process, and thrombocytope-nia denotes a reduced platelet (thrombocyte) count. In cyclic throm-bocytopenia (CT), platelet counts oscillate generally from very lowvalues (1 × 109 cells/l) to normal (150–450 × 109 platelets/L blood)or above normal levels (2000 × 109 cells/l).7 These oscillations havebeen observed with periods varying between 20 and 40 days.6

Our own efforts to understand platelet dynamics have evolvedover the years,23,24,63,64 but I think it is fair to say that we do not yet

Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-5

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

really understand the genesis of cyclic thrombocytopenia. To com-plicate matters, though it was initially thought that cyclic thrombo-cytopenia only involved oscillations in platelet numbers, a patientwas recently discovered,65 who had statistically significant oscilla-tions in both platelets and neutrophils at the same period. Thisraises the possibility that we have been incorrect in thinking thatcyclic thrombocytopenia only involves the platelet line. With thisinformation, it has recently been shown,66 using a model similar toRef. 29, that one can find parameter values consistent with oscil-lations in only platelets, as well as other parameters for which bothplatelets and neutrophils oscillate. Thus, there is much scope for fur-ther research here, and it will be especially valuable because it hasproved almost impossible to utilize TPO as a therapeutic tool in thetreatment of thrombocytopenia (cyclic or not).

D. Periodic autoimmune hemolytic anemia

Autoimmune hemolytic anemia (AIHA) results from an abnor-mality of the immune system that produces autoantibodies thatattack red blood cells as if they were foreign to the body, leading toan abnormally high destruction rate of the red blood cells. PeriodicAIHA is a rare form of hemolytic anemia in humans10 characterizedby oscillatory reticulocyte and/or (more rarely) erythrocyte numbersabout a depressed level. The origin of the disease is unclear. PeriodicAIHA, with a period of 16 to 17 days in hemoglobin and reticulo-cyte counts, has been experimentally induced in rabbits by using redblood cell auto-antibodies.20

Considering the fact that erythropoietin was the first of themajor cytokines that was identified and sequenced, it is astonishingthat so little has been published related to the regulation of ery-thropoiesis. The exception has been in iron metabolism and storagemodeling which received extensive consideration by modelers in themid-20th century.67,68

The earliest work that I am aware of is in a computer study,69

and this was followed by my attempt25 to understand the periodicautoimmune hemolytic anemia data in Ref. 20. It is not an under-statement to say that the modeling that has been done with thesedata in mind has been primitive to say the least and that there istremendous scope for further work in this area. It is unclear if thenaturally occurring form of periodic autoimmune hemolytic anemiais a dynamical disease, but the proliferation of investigations lookingat chemotherapy effects on erythropoiesis offer significant modelingplatforms with which to investigate this question.

V. SUMMARY

I have offered a personal view of dynamical diseases, concen-trating on potential examples from periodic hematological disor-ders. I argue that cyclic neutropenia has a very strong likelihood ofbeing a dynamical disease due to a fundamental defect in the mech-anism governing neutrophil precursor apoptosis. Periodic chronicmyelogenous leukemia is, in my opinion, another example but wemay never know for sure.

For other examples like cyclic thrombocytopenia and peri-odic autoimmune hemolytic anemia, the jury is still out, and it isincumbent on future generations to figure this out with models forthe regulation of platelet and erythrocyte production that are more

sophisticated and biologically realistic. It is possible that our cur-rent models for thrombopoiesis and erythropoiesis are inaccuratebecause of undiscovered, but biologically important, factors.

Implicit in the concept of dynamical disease is the corollary thatone might be able to manipulate physiological parameters to oblit-erate the symptoms of a dynamical disease. In cyclic neutropenia,the clinical experience with G-CSF bears this out in the sense thatG-CSF treatment increases the nadir of neutrophil counts and thusattenuates or eliminates the most annoying symptoms of cyclic neu-tropenia. However, it does not eliminate the cycling in totality, andfrom the analysis of our models for cyclic neutropenia, it is likelythat this would be difficult to do because of the multistability dis-played by the system. These points have been extensively discussedelsewhere52,70 and likely pertain to periodic leukemia as well as cyclicthrombocytopenia.

One clearly wonders if these efforts have had any significantimpact on clinical practice. I can do no better than to cite therole that modeling has played in the dosing of G-CSF for cyclicneutropenia71 as well as the insight gleaned from the modeling intothe deleterious effects of periodically administered chemotherapyand the consequent neutropenia.72,73

As a final note, it is worth pointing out that models for cyclicneutropenia27,52 predict the co-existence of a locally stable limit cycle(the cyclic neutropenia state) and a locally stable steady state, andthere is experimental evidence for this in grey collies.71 This raisesthe possibility of a “single-shot” therapy for cyclic neutropenia inwhich G-CSF is used to move the dynamics to the locally stablesteady state. This is precisely the one-shot birth control proposal ofWinfree74 but has never been tested clinically on cyclic neutropeniapatients. If it worked, it would involve substantial financial savingsin their treatment.

ACKNOWLEDGMENTS

I would like to acknowledge the NSERC (Natural Sciences andEngineering Research Council of Canada), MITACS (Mathematicsof Information Technology and Complex Systems), and the Alexan-der von Humboldt Stiftung for their generous research support overthe past decades as well as Professor Dr. Klaus Pawelzik, UniversitätBremen, Germany for his hospitality during the time this was writ-ten. Leon Glass and I started this journey together in 1976, and Ithink that we have both benefited immeasurably from our mutualinteractions and collaborations. Thank you Leon! My colleaguesTyler Cassidy, Morgan Craig, and Jinzhi Lei provided invaluablecomments on this manuscript, and Professor Hans–Otto Walthervery kindly made Ref. 57 available. Additionally, I would like tothank my many research collaborators for all of the fun and excite-ment we have shared in discovering the secrets of Nature and herderangements. It has been a wonderful and joyful voyage and wouldnot have been possible without the generous way in which Pro-fessor David C. Dale, University of Washington, shared data fromboth cyclic neutropenia patients and his grey collies. Last, but notleast, McGill University and my home Department of Physiologyhave been remarkably tolerant of an unconventional physiologist foralmost half a century for which I am profoundly grateful.

Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-6

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data werecreated or analyzed in this study.

REFERENCES1H. Reimann, Periodic Diseases (F.A. Davis, Philadelphia, PA, 1963).2D. G. Wright, D. C. Dale, A. S. Fauci, and S. M. Wolff, “Human cyclic neu-tropenia: Clinical review and long term follow up of patients,” Medicine 60, 1–13(1981).3R. Lange, “Cyclic hematopoiesis: Human cyclic neutropenia,” Exp. Hematol. 11,435–451 (1983).4D. C. Dale and W. P. Hammond, “Cyclic neutropenia: A clinical review,” BloodRev. 2, 178–185 (1988).5D. C. Dale and M. C. Mackey, “Understanding, treating and avoiding hematolog-ical disease: Better medicine through mathematics?,” Bull. Math. Biol. 77, 739–757(2015).6T. Cohen and D. P. Cooney, “Cyclical thrombocytopenia: Case report and reviewof literature,” Scand. J. Haemat. 16, 133–138 (1974).7J. Swinburne and M. Mackey, “Cyclical thrombocytopenia: Characterization byspectral analysis and a review,” J. Theor. Med. 2, 81–91 (2000).8R. A. Gatti, W. A. Robinson, A. S. Deinard, M. Nesbit, J. J. McCullough, M.Ballow, and R. A. Good, “Cyclic leukocytosis in chronic myelogenous leukemia,”Blood 41, 771–782 (1973).9G. Chikkappa, G. Borner, H. Burlington, A. D. Chanana, E. P. Cronkite, S. Ohl,M. Pavelec, and J. S. Robertson, “Periodic oscillation of blood leukocytes, platelets,and reticulocytes in a patient with chronic myelocytic leukemia,” Blood 47,1023–1030 (1976).10P. Ranlov and A. Videbaek, “Cyclic haemolytic anaemia synchronous withPel-Ebstein fever in a case of Hodgkin’s disease,” Acta Medica Scand. 174(5),583–588 (1963).11R. R. Gordon and S. Varadi, “Congenital hypoplastic anemia (pure red cellanemia) with periodic erythroblastopenia,” Lancet 296–299 (1962).12M. C. Mackey and L. Glass, “Oscillations and chaos in physiological controlsystems,” Science 197, 287–289 (1977).13J. Cronin-Scanlon, “A mathematical model for catatonic schizophrenia,” Ann.N.Y. Acad. Sci. 231, 112–122 (1974).14L. Glass and M. Mackey, From Clocks to Chaos: The Rhythms of Life (PrincetonUniversity Press, Princeton, NJ, 1988).15M. C. Mackey, “Cell kinetic status of haematopoietic stem cells,” Cell Prolif. 34,71–83 (2000).16C. Haurie, D. Dale, and M. C. Mackey, “Cyclical neutropenia and other peri-odic hematological diseases: A review of mechanisms and mathematical models,”Blood 92, 2629–2640 (1998).17E. Laurenti and B. Göttgens, “From haematopoietic stem cells to complexdifferentiation landscapes,” Nature 553, 418–426 (2018).18E. B. Rankin and K. M. Sakamoto, “The cellular and molecular mechanisms ofhematopoiesis,” in Bone Marrow Failure (Springer, 2018), pp. 1–23.19C. Foley and M. C. Mackey, “Dynamic hematological disease: A review,”J. Math. Biol. 58, 285–322 (2009).20J. S. Orr, J. Kirk, K. Gray, and J. Anderson, “A study of the interdependenceof red cell and bone marrow stem cell populations,” Br. J. Haematol. 15, 23–24(1968).21M. Yanabu, S. Nomura, T. Fukuroi, T. Kawakatsu, H. Kido, K. Yamaguchi, M.Suzuki, T. Kokawa, and K. Yasunaga, “Periodic production of antiplatelet autoan-tibody directed against GP IIIa in cyclic thrombocytopenia,” Acta Haematol. 89,155–159 (1993).22D. Guerry, D. Dale, M. Omine, S. Perry, and S. Wolff, “Periodic hematopoiesisin human cyclic neutropenia,” J. Clin. Invest. 52, 3220–3230 (1973).23R. Apostu and M. C. Mackey, “Understanding cyclical thrombocytopenia: Amathematical modeling approach,” J. Theor. Biol. 251, 297–316 (2008).24G. P. Langlois, M. Craig, A. R. Humphries, M. C. Mackey, J. M. Mahaffy,J. Bélair, T. Moulin, S. R. Sinclair, and L. Wang, “Normal and pathologicaldynamics of platelets in humans,” J. Math. Biol. 75, 1411–1462 (2017).

25M. C. Mackey, “Periodic auto-immune hemolytic anemia: An induced dynam-ical disease,” Bull. Math. Biol. 41, 829–834 (1979).26C. Haurie, D. C. Dale, and M. C. Mackey, “Occurrence of periodic oscillationsin the differential blood counts of congenital, idiopathic, and cyclical neutropenicpatients before and during treatment with G-CSF,” Exp. Hematol. 27, 401–409(1999).27C. Colijn and M. Mackey, “A mathematical model of hematopoiesis: I. Cyclicalneutropenia,” J. Theor. Biol. 237, 133–146 (2005).28P. Fortin and M. Mackey, “Periodic chronic myelogenous leukemia: Spectralanalysis of blood cell counts and etiological implications,” Br. J. Haematol. 104,336–245 (1999).29C. Colijn and M. Mackey, “A mathematical model of hematopoiesis: I. Periodicchronic myelogenous leukemia,” J. Theor. Biol. 237, 117–132 (2005).30L. Pujo-Menjouet, “Blood cell dynamics: Half of a century of modelling,” Math.Model. Nat. Phenom. 11, 92–115 (2016).31M. Craig, A. R. Humphries, and M. C. Mackey, “A mathematical model ofgranulopoiesis incorporating the negative feedback dynamics and kinetics of G-CSF/neutrophil binding and internalization,” Bull. Math. Biol. 78, 2304–2357(2016).32J. G. Milton and M. C. Mackey, “Periodic haematological diseases: Mysti-cal entities or dynamical disorders?,” J. Roy. Coll. Phys. (Lond.) 23, 236–241(1989).33M. C. Mackey and J. G. Milton, “Feedback, delays, and the origin of blood celldynamics,” Commun. Theor. Biol. 1, 299–327 (1990).34C. Haurie, D. C. Dale, and M. C. Mackey, “Cyclical neutropenia and otherperiodic hematological disorders: A review of mechanisms and mathematicalmodels,” Blood 92, 2629–2640 (1998).35C. Colijn, D. C. Dale, C. Foley, and M. Mackey, “Observations on the patho-physiology and mechanisms for cyclic neutropenia,” Math. Model. Nat. Phenom.1(2), 45–69 (2006).36C. Haurie, D. C. Dale, R. Rudnicki, and M. C. Mackey, “Modeling complexneutrophil dynamics in the grey collie,” J. Theor. Biol. 204, 504–519 (2000).37C. Haurie, R. Person, D. C. Dale, and M. Mackey, “Haematopoietic dynamics ingrey collies,” Exp. Hematol. 27, 1139–1148 (1999).38S. E. Palmer, K. Stephens, and D. C. Dale, “Genetics, phenotype, and natu-ral history of autosomal dominant cyclichematopoiesis,” Am. J. Med. Genet. 66,413–422 (1996).39E. A. King-Smith and A. Morley, “Computer simulation of granulopoiesis:Normal and impaired granulopoiesis,” Blood 36, 254–262 (1970).40N. D. Kazarinoff and P. van den Driessche, “Control of oscillations inhematopoiesis,” Science 203, 1348–1350 (1979).41S. Rubinow and J. Lebowitz, “A mathematical model of neutrophil productionand control in normal man,” J. Math. Biol. 1, 187–225 (1975).42S. Schmitz, “Ein mathematisches Modell der zyklischen Haemopoese,” Ph.D.thesis (Universität Köln, 1988).43S. Schmitz, M. Loeffler, J. B. Jones, R. D. Lange, and H. E. Wichmann, “Syn-chrony of bone marrow proliferation and maturation as the origin of cyclichaemopoiesis,” Cell Tissue Kinet. 23, 425–441 (1990).44S. Schmitz, H. Franke, J. Brusis, and H. E. Wichmann, “Quantification of the cellkinetic effects of G-CSF using a model of human granulopoiesis,” Exp. Hematol.21, 755–760 (1993).45S. Schmitz, H. Franke, M. Loeffler, H. E. Wichmann, and V. Diehl, “Reducedvariance of bone-marrow transit time of granulopoiesis: A possible pathomecha-nism of human cyclic neutropenia,” Cell Prolif. 27, 655–667 (1994).46S. Schmitz, H. Franke, H. E. Wichmann, and V. Diehl, “The effect of contin-uous G-CSF application in human cyclic neutropenia: A model analysis,” Br. J.Haematol. 90, 41–47 (1995).47M. Scholz, C. Engel, and M. Loeffler, “Modelling human granulopoiesis underpolychemotherapy with G-CSF support,” J. Math. Biol. 50, 397–439 (2005).48M. Scholz, C. Engel, and M. Loeffler, “Model based design of chemotherapeu-tic regimens accounting for heterogeneity in leucopenia,” Br. J. Haematol. 132,723–735 (2006).49M. C. Mackey, “A unified hypothesis for the origin of aplastic anemia andperiodic haematopoiesis,” Blood 51, 941–956 (1978).50F. Burns and I. Tannock, “On the existence of a G0 phase in the cell cycle,” CellTissue Kinet. 3, 321–334 (1970).

Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-7

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

51D. C. De Souza and A. R. Humphries, “Dynamics of a mathematical hematopoi-etic ‘ population model,” SIAM J. Appl. Dyn. Syst. 18, 808–852 (2019).52S. Bernard, J. Belair, and M. Mackey, “Oscillations in cyclical neutropenia: Newevidence based on mathematical modeling,” J. Theor. Biol. 223, 283–298 (2003).53A. A. Aprikyan, W. C. Liles, E. Rodger, M. Jonas, E. Y. Chi, and D. C. Dale,“Impaired survival of bone marrow hematopoietic progenitor cells in cyclicneutropenia,” Blood 97, 147–153 (2001).54M. O’Dwyer, B. J. Druker, M. Mauro, M. Talpaz, D. Resta, B. Peng, E. Buch-dunger, J. Ford, S. F. Reese, R. Capdeville, and C. L. Sawyers, “STI571: A tyrosinekinase inhibitor for the treatment of CML,” Ann. Oncol. 11, 155 (2000).55Leukemia, edited by E. S. Henderson, T. A. Lister, and M. F. Greaves (Saunders,1996).56L. Glass and M. Mackey, “Mackey-Glass equation,” Scholarpedia 5, 6908(2010).57H.-O. Walther, “The impact on mathematics of the paper ‘Oscillation and chaosin physiological control systems’ by Mackey and Glass in Science, 1977,” e-printarXiv.2001.09010v1 (2009).58L. Pujo-Menjouet and M. Mackey, “Contribution to the study of periodicchronic myelogenous leukemia,” C. R. Biol. 327, 235–244 (2004).59L. Pujo-Menjouet, S. Bernard, and M. Mackey, “Long period oscillations in a G0

model of hematopoietic stem cells,” SIAM J. Appl. Dyn. Syst. 4, 312–332 (2005).60M. C. Mackey, C. Ou, L. Pujo-Menjouet, and J. Wu, “Periodic oscillations ofblood cell populations in chronic myelogenous leukemia,” SIAM J. Math. Anal.38, 166–187 (2006).61A. Safarishahrbijari and A. Gaffari, “Parameter identification of hematopoiesismathematical model—Periodic chronic myelogenous leukemia,” Contemp.Oncol. 17, 73 (2013).62M. H. Cohen, G. Williams, J. R. Johnson, J. Duan, J. Gobburu, A. Rahman,K. Benson, J. Leighton, S. K. Kim, R. Wood et al., “Approval summary for ima-tinib mesylate capsules in the treatment of chronic myelogenous leukemia,” Clin.Cancer Res. 8, 935–942 (2002).

63J. Bélair and M. Mackey, “A model for the regulation of mammalian plateletproduction,” Ann. N.Y. Acad. Sci. 504, 280–282 (1987).64M. Santillan, J. Mahaffy, J. Belair, and M. Mackey, “Regulation of plateletproduction: The normal response to perturbation and cyclical platelet disease,”J. Theor. Biol. 206, 585–603 (2000).65G. P. Langlois, D. M. Arnold, J. Potts, B. Leber, D. C. Dale, and M. C. Mackey,“Cyclic thrombocytopenia with statistically significant neutrophil oscillations,”Clin. Case Rep. 6, 1347 (2018).66C. Zhuge, M. C. Mackey, and J. Lei, “Origins of oscillation patterns in cyclicalthrombocytopenia,” J. Theor. Biol. 462, 432–445 (2019).67G. C. Nooney, “Iron kinetics and erythron development,” Biophys. J. 5, 755–765(1965).68G. C. Nooney, “An erythron-dependent model of iron kinetics,” Biophys. J. 6,601–609 (1966).69J. Kirk, J. S. Orr, and C. S. Hop, “A mathematical analysis of red blood celland bone marrow stem cell control mechanism,” Br. J. Haematol. 15, 35–46(1968).70J. Lei and M. C. Mackey, “Multistability in an age-structured model ofhematopoiesis: Cyclical neutropenia,” J. Theor. Biol. 270, 143–153 (2011).71C. Foley, S. Bernard, and M. C. Mackey, “Cost-effectve G-CSF therapy strate-gies for cyclical neutropenia: Mathematical modelling based hypotheses,” J. Theor.Biol. 238, 754–763 (2006).72M. Craig, A. R. Humphries, F. Nekka, J. Bélair, J. Li, and M. C. Mackey,“Neutrophil dynamics during concurrent chemotherapy and G-CSF administra-tion: Mathematical modelling guides dose optimisation to minimise neutropenia,”J. Theor. Biol. 385, 77–89 (2015).73T. Cassidy, A. R. Humphries, M. Craig, and M. C. Mackey, “Characterizingchemotherapy-induced neutropenia and monocytopenia through mathematicalmodelling,” bioRxiv.74A. T. Winfree, “Time and timelessness in biological clocks,” in Temporal Aspectsof Therapeutics (Springer, 1973), pp. 35–57.

Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-8

Published under license by AIP Publishing.