Periodic Hematological Disorders: Quintessential Examples of Dynamical Diseases

Periodic hematological disorders: Quintessential examples of dynamical diseases

Cite as: Chaos 30, 063123 (2020); https://doi.org/10.1063/5.0006517 Submitted: 04 March 2020 . Accepted: 21 May 2020 . Published Online: 08 June 2020

Michael C. Mackey

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

Periodic hematological disorders: Quintessential examples of dynamical diseases

Cite as: Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 Submitted: 4 March 2020 · Accepted: 21 May 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: [email protected]. URL: https://www.mcgill.ca/mathe matical-physiology-lab/

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

The concept of dynamical disease first appeared in 1977, and since the operation of a basically normal control system in a region of that time numerous investigators have searched for examples physiological parameters that produces pathological behavior.” (The that might fulfill the requirements of this hypothesized clini- concept of dynamical disease was preceded by a similar idea related cal entity. Here, I argue that some hematological disorders are to schizophrenia.13) The concept was later elaborated in Ref. 14 with beautiful examples of dynamical diseases and discuss the insights many examples from the biological and medical sciences. that have been obtained into the origin of cyclic neutropenia Here, I summarize the evidence that two periodic hematologi- (and its treatment) and periodic chronic myelogenous leukemia. I cal diseases (cyclic neutropenia and periodic chronic myelogenous also briefly discuss cyclic thrombocytopenia and periodic autoim- leukemia) are perfect examples of dynamical diseases and briefly mune hemolytic anemia. consider two others (periodic autoimmune hemolytic anemia and cyclic thrombocytopenia) for which the evidence is still incomplete.

I. INTRODUCTION II. OUTLINE OF HUMAN HEMATOPOIESIS AND ITS Nearly 2400 years ago, Hippocrates associated disease with a change in the regularity of a physiological process. Present day REGULATION clinical medicine often focuses on diseases in which these changes Blood cells are formed from a hematopoietic stem cell (HSC) in occur on time scales ranging from milliseconds to hours, for exam- a process known as hematopoiesis. In humans, hematopoiesis pro- ple, the generation of cardiac and respiratory arrhythmias, tremors, duces the equivalent of our body weight in red blood cells, white and seizures. More puzzling have been those diseases, collectively blood cells, and platelets every decade of life.15 Throughout, this referred to as “periodic diseases,” in which symptoms recur in an process usually proceeds flawlessly, implying the existence of robust approximately periodic fashion.1 Among the latter are the periodic control mechanisms. This cellular renewal system is thus ideal for hematological diseases, i.e., cyclic neutropenia (CN, also known as the study of the normal regulation of tissue proliferation and differ- periodic hematopoiesis),2–5 cyclic thrombocytopenia (CT),6,7 and the entiation from the single cell to whole organ level, and the study of periodic variants of chronic myelogenous leukemia (PCML)8,9 and derangements of these processes. autoimmune hemolytic anemia.10,11 Though hematopoiesis is incredibly complicated,17,18 the broad In 1977, it was proposed12 that some periodic diseases (not just outlines can be summarized as in Fig. 1, which schematically in hematology) might be “dynamical diseases . . . characterized by shows the major aspects of the mammalian platelet, red blood cell,

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monocyte, and granulocyte production. Control is mediated by a Periodic hematological diseases fall into two broad classes. The large family of growth factors and cytokines. Three of the major first, with oscillations in numbers of a single circulating cell type, is players are thrombopoietin (TPO), erythropoietin (EPO), and gran- probably due to a destabilization of a peripheral control mechanism, ulocyte colony stimulating factor (G-CSF), which also have local reg- e.g., cyclic thrombocytopenia with periods of 13–65 days7,23,24 and ulatory (LR) effects within the HSC population. All three cytokines autoimmune hemolytic anemia.25 The second type has several cir- will play a major role in our discussion of the hematological dis- culating cell types and seemingly involves the stem cells. Examples ease. CFU/BFU refers to the various in vitro analogs of the in vivo are cyclic neutropenia with periods of 14–45 days26,27 and periodic committed stem cells. chronic myelogenous leukemia.28,29. Both classes of disorders illu- From a mathematical standpoint all of these are negative feed- minate aspects of hematopoietic regulation that would never have back mechanisms in the sense that a fall in a peripheral circulating been discovered in a laboratory setting because of the time scales cell numbers leads to a consequent increase in production of the involved. immature precursor and this response is mediated by a specific cytokine or group of them. The mathematics is further complicated by the fact that there are significant delays (often state dependent) III. MATHEMATICAL MODELING IN HEMATOLOGY between when a cytokine acts and the resulting effect is felt in the Since this is a non-technical survey, detailed consideration of circulation. the variety of mathematical modeling techniques that have been Investigations of whole animal dynamic behavior of cellular employed to understand normal and pathological hematopoiesis is replication systems is hampered by the lack of good quality temporal inappropriate. It suffices to simply note that there is an excellent data on cell numbers and cytokine levels in response to perturbation. recent survey30 of modeling efforts in the area over the past half Ironically, the best source of data currently available comes from century. These techniques range from differential equation models clinical studies of patients with hematological disease. Of the vast through delay differential equations, partial differential equations, array of documented hematological pathologies, the periodic hema- and also agent based models. tological diseases (periods from weeks to months) have been some In work with my collaborators, we have typically utilized non- of the most instructive in terms of elucidating the control mech- linear differential delay equations of variable complexity depend- anisms regulating hematopoiesis.19 See Fig. 2 for an illustration of ing on the question under consideration. Nonlinearities arise four of the most studied of these disorders which form the focus of because of the stoichiometry of cytokine receptor interactions, this paper. and the delays typically reflect maturation and cell cycle times.

FIG. 1. The architecture and control of mammalian hematopoiesis. All blood cells are formed from hematopoietic stem cells (HSCs), and this ﬁgure summarizes mammalian platelet (P), red blood cell (RBC), monocyte, and granulocyte (G/M includ- ing neutrophil, basophil, and eosinophil) production. Control over these processes is mediated by a variety of cytokines [e.g., thrombopoietin (TPO), erythropoietin (EPO), and granulocyte colony stimulating factor (G-CSF) are the main ones but over 50 have so far been identi- ﬁed], and there are also local regula- tory (LR) effects within the HSC population. CFU/BFU refers to the various in vitro analogs of the in vivo committed stem cells. Reprinted with permission fromC.Haurie et al., Blood 92, 2629–2640 (1998). Copyright 1998 American Society of Hematology.

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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: white blood 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 from C. 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 are animal model has allowed for the collection of a variety of data that reviews.5,16,19,32–35 would have been difficult to obtain in humans. A major characteristic of CN is that the oscillations are not only present in neutrophils, but often are observed in platelets, IV. PERIODIC HEMATOLOGICAL DISEASE AS monocytes, and reticulocytes,16 thus, CN is sometimes called peri- DYNAMICAL DISEASES odic hematopoiesis.38 This observation suggests that the source of the oscillations may lie in, or involve, the stem cell compartment. A. Cyclic neutropenia Although rare, cyclic neutropenia is probably the most exten- The number of circulating neutrophils is normally relatively sively studied periodic hematological disorder. One of the first constant with an average absolute neutrophil count (ANC) of about models to consider cyclic neutropenia is found in a computer study39 5.0 × 109 cells/l. Neutropenia is characterized by a low number of which inspired a mathematical analysis.40 Rubinow and Lebowitz41 neutrophils, thus indicating that the individual is less effective at offered a comprehensive formulation of the regulation of neutrophil fighting infections. Cyclic neutropenia is defined by oscillations in production and these studies have been nicely complemented by the number of neutrophils from normal to very low levels (less than other contributions.42–48 0.5 × 109 cells/l). The period of these oscillations is usually around Guided by the observation of oscillations in all of the circulating 3 weeks for humans, although periods up to 45 days are found.36 blood cell types, I had initially thought that the origin of the oscil- CN is effectively treated with daily administration of G-CSF, which lations must originate from a defect in the stem cell compartment49 50 reduces the period of the oscillations and increases both the oscilla- and, therefore, used the Burns and Tannock G0 model for the cell tion amplitude and the value of the ANC nadir.27 G-CSF decreases cycle (see Fig. 3) to investigate this. Based on a rudimentary bifurca- the duration of severe neutropenia, which is clinically desirable. tion analysis (see Fig. 4), I concluded that the only way in which Understanding CN has been aided by the existence of a natu- the characteristics of cyclic neutropenia could occur was through rally occurring animal model in gray collies.37 The canine disorder an abnormally high rate γ of apoptosis in the proliferating phase of shows the same characteristics as in humans, except that the period the stem cells. Referring to Fig. 4, the hypothesis went roughly like of the oscillations is between 11 and 15 days. The existence of this this. Neutropenia was due to a very large value of γ > γ2 and there

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FIG. 3. The Burns and Tannock50 model for the cell cycle consisting of a resting phase (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 from the proliferating phase at a rate γ and exit into the differentiation pathway from G0 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 was partially consistent with the findings of the effects of exogenous G-CSF administered to gray collies37 as well as humans,26 since G-CSF decreases apoptosis. However, there is a problem because the model predicts that ever higher doses of G-CSF (decreased apoptosis rate γ ) should lead to a decreased amplitude of oscillation but ever increasing period and eventually no oscillation. Although it was observed that oscillations could be obliterated in gray collies with administration of large doses of G-CSF, it was never observed that the amplitude increased to a maximum and then started to decrease as G-CSF levels were increased! Thus, as always, the devil is in the details. This conundrum was eventually solved52 by considering the more complete model of Fig. 5. The results of a bifurcation analysis indicated that cyclic neutropenia was indeed probably due to elevated levels of apoptosis, but not in the stem cells. Thus, the resolution of the issue suggests that cyclic neutropenia is caused FIG. 4. The results of a rudimentary bifurcation analysis of the Burns and 50 by abnormally high levels of apoptosis53 in the proliferating neu- Tannock model for the cell cycle showing the behavior of the stem cell efﬂux (top G trophil precursors. This leads to a fall in the level of circulating panel) from 0 as a function of the apoptosis rate γ in the proliferating phase. At γ1 and γ2, there are supercritical Hopf bifurcations, and the amplitude and period mature neutrophils, and this in turn triggers a surge of G-CSF, of the oscillations are shown in the middle and bottom panels.S(stable),U(unsta- which increases the rate of differentiation out of the G0 phase of ble), Normal (0 ≤ γ < γ1), Cycling NP (cyclic neutropenia, γ1 ≤ γ < γ2), and 51 the stem cell compartment into the neutrophil line. This elevation NP (neutropenia, γ ≥ γ2). A recent and very complete bifurcation analysis has of the differentiation rate destabilizes the stem cell dynamics leading revealed the rich dynamical complexity of the Burns and Tannock model for the to oscillations through a Hopf bifurcation. cell cycle. Building on these ideas, Colijn and Mackey27 developed a more complete model for human hematopoiesis that included all three major cell lines (Fig. 1). Using data from nine gray collies37 as well as from 27 neutropenia patients in a clinical trial of G-CSF26 Data were also available for both dogs and patients while under and assuming a set of normal parameters (separate for dogs and G-CSF therapy. Not unexpectedly, the modeling indicated that the humans), it was shown that the most significant parameter changes most significant change in both was a significant decrease in the rate from normal to mimic the cyclic neutropenia results were a large of neutrophil precursor apoptosis due to G-CSF. increase in the level of apoptosis in the neutrophil precursors and Thus, the results of Colijn and Mackey27 are consistent with the 52 a less significant decrease in the maximal rate of re-entry of G0 conclusion that cyclic neutropenia is primarily a disorder caused phase stem cells into the proliferative phase. These findings were by an abnormally high level of apoptosis in the proliferating neu- consistent between both gray collie and patient data. trophil precursors, and cyclic neutropenia seems to be an example of

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A variant of particular interest is periodic chronic myelogenous leukemia (PCML), characterized by oscillations in circulating cell numbers, primarily in leucocytes, but often also in the platelets and reticulocytes.28 The leucocyte count varies periodically between values of 30 and 200 × 109 cells/l with a period ranging from 40 to 80 days. Oscillations in platelet and reticulocyte numbers, when present, occur with the same period as the leucocytes, around normal or elevated numbers.28,55 The hypothesis that the disease origi- nates from the stem cell compartment is supported by the presence of the Philadelphia chromosome in all of the mature cells as well as oscillations in more than one cell lineage. My first efforts to understand periodic leukemia came in a paper by Mackey and Glass12 when we put forward the dynamical disease concept. The model in that paper is embarrassingly naïve from a biological point of view, but rather remarkably has generated a huge amount of interest and is now known as the “Mackey–Glass equation.”56 The reason for this is connected with the fact that for rather wide ranges of parameter values, the solutions appear to be “chaotic” in that numerically they have no well defined period. Thus, the solutions to this model have served as paradigmatic examples for numerical analysts developing techniques to detect chaotic behavior. Understandably, this has generated considerable interest among mathematicians, but unfortunately concrete analytic results have been few and far between.57 Going forward, as my knowledge of the clinical aspects of chronic myelogenous leukemia improved, more realistic modeling efforts appeared,58–60 culminating in Ref. 29. In that study, the Colijn and Mackey27 model was used to analyze the data from 11 periodic chronic myelogenous leukemia patients. Assuming initially normal parameter values, it was determined for each of the patients what parameters had to be changed to match the clinical hematological data. It was a consistent finding that there needed to be a decrease in FIG. 5. A diagrammatic representation of the relationships between stem cells the level of neutrophil precursor apoptosis, an increase in the max- S and neutrophils. Resting G0 phase stem cells ( ) may either stay in that phase imal differentiation rate from the G0 stem cells into the neutrophil indeﬁnitely, re-enter the proliferative phase at a rate K(S) dependent on S or line, and to a lesser extent a decrease in the rate of stem cell apop- F N N differentiate into the neutrophil line at a rate ( ), where is the circulating tosis. More recent studies61 have extended this work, but basically number of mature neutrophils. (Differentiation into the erythrocyte and platelet lines is neglected in this treatment, but it is unimportant for the considerations confirmed the original findings. here.) Following differentiation into the neutrophil line, a stem cell undergoes an It is noteworthy that the available data on chronic myelogenous ampliﬁcation (box with a peak) A during a proliferation and maturation period leukemia patients is not being augmented, thanks to the appearance lasting τN days before being released into the circulation (bottom circle) as a of a highly effective pharmacological treatment62 and the success of mature neutrophil and dying at a random rate α. Reprinted with permission from stem cell transplant when feasible. Therefore, the historical data that et al. Bernard , J. Theor. Biol. 223, 283–298 (2003). Copyright 2003 Elsevier. we 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. a dynamical disease. The control system is apparently working nor- However, it is also likely that we will never know for sure. mally but in a parameter range (increased rate of apoptosis) giving pathological behavior. Much is known about the molecular basis for C. Cyclic thrombocytopenia this increased apoptosis,5 but will not be discussed here. Platelets take part in the clotting process, and thrombocytopenia denotes a reduced platelet (thrombocyte) count. In cyclic throm- B. Periodic chronic myelogenous leukemia bocytopenia (CT), platelet counts oscillate generally from very low Leukemia is a malignancy of the blood characterized by values (1 × 109 cells/l) to normal (150–450 × 109 platelets/L blood) an abnormal proliferation of blood cells, usually leucocytes. or above normal levels (2000 × 109 cells/l).7 These oscillations have Chronic myelogenous leukemia (CML) is distinguished from other been observed with periods varying between 20 and 40 days.6 leukemias by the presence of a genetic abnormality in blood cells, Our own efforts to understand platelet dynamics have evolved called the Philadelphia chromosome.54 over 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- sophisticated and biologically realistic. It is possible that our cur- plicate matters, though it was initially thought that cyclic thrombo- rent models for thrombopoiesis and erythropoiesis are inaccurate cytopenia only involved oscillations in platelet numbers, a patient because of undiscovered, but biologically important, factors. was recently discovered,65 who had statistically significant oscilla- Implicit in the concept of dynamical disease is the corollary that tions in both platelets and neutrophils at the same period. This one might be able to manipulate physiological parameters to oblit- raises the possibility that we have been incorrect in thinking that erate the symptoms of a dynamical disease. In cyclic neutropenia, cyclic thrombocytopenia only involves the platelet line. With this the clinical experience with G-CSF bears this out in the sense that information, it has recently been shown,66 using a model similar to G-CSF treatment increases the nadir of neutrophil counts and thus Ref. 29, that one can find parameter values consistent with oscil- attenuates or eliminates the most annoying symptoms of cyclic neu- lations in only platelets, as well as other parameters for which both tropenia. However, it does not eliminate the cycling in totality, and platelets and neutrophils oscillate. Thus, there is much scope for fur- from the analysis of our models for cyclic neutropenia, it is likely ther research here, and it will be especially valuable because it has that this would be difficult to do because of the multistability dis- proved almost impossible to utilize TPO as a therapeutic tool in the played by the system. These points have been extensively discussed treatment of thrombocytopenia (cyclic or not). elsewhere52,70 and likely pertain to periodic leukemia as well as cyclic thrombocytopenia. One clearly wonders if these efforts have had any significant D. Periodic autoimmune hemolytic anemia impact on clinical practice. I can do no better than to cite the Autoimmune hemolytic anemia (AIHA) results from an abnor- role that modeling has played in the dosing of G-CSF for cyclic mality of the immune system that produces autoantibodies that neutropenia71 as well as the insight gleaned from the modeling into attack red blood cells as if they were foreign to the body, leading to the deleterious effects of periodically administered chemotherapy an abnormally high destruction rate of the red blood cells. Periodic and the consequent neutropenia.72,73 AIHA is a rare form of hemolytic anemia in humans10 characterized As a final note, it is worth pointing out that models for cyclic by oscillatory reticulocyte and/or (more rarely) erythrocyte numbers neutropenia27,52 predict the co-existence of a locally stable limit cycle about a depressed level. The origin of the disease is unclear. Periodic (the cyclic neutropenia state) and a locally stable steady state, and AIHA, with a period of 16 to 17 days in hemoglobin and reticulo- there is experimental evidence for this in grey collies.71 This raises cyte counts, has been experimentally induced in rabbits by using red the possibility of a “single-shot” therapy for cyclic neutropenia in blood cell auto-antibodies.20 which G-CSF is used to move the dynamics to the locally stable Considering the fact that erythropoietin was the first of the steady state. This is precisely the one-shot birth control proposal of major cytokines that was identified and sequenced, it is astonishing Winfree74 but has never been tested clinically on cyclic neutropenia that so little has been published related to the regulation of ery- patients. If it worked, it would involve substantial financial savings thropoiesis. The exception has been in iron metabolism and storage in their treatment. modeling which received extensive consideration by modelers in the mid-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 periodic autoimmune hemolytic anemia data in Ref. 20. It is not an under- ACKNOWLEDGMENTS statement to say that the modeling that has been done with these I would like to acknowledge the NSERC (Natural Sciences and data in mind has been primitive to say the least and that there is Engineering Research Council of Canada), MITACS (Mathematics tremendous scope for further work in this area. It is unclear if the of Information Technology and Complex Systems), and the Alexan- naturally occurring form of periodic autoimmune hemolytic anemia der von Humboldt Stiftung for their generous research support over is a dynamical disease, but the proliferation of investigations looking the past decades as well as Professor Dr. Klaus Pawelzik, Universität at chemotherapy effects on erythropoiesis offer significant modeling Bremen, Germany for his hospitality during the time this was writ- platforms with which to investigate this question. ten. Leon Glass and I started this journey together in 1976, and I think that we have both benefited immeasurably from our mutual interactions and collaborations. Thank you Leon! My colleagues V. SUMMARY Tyler Cassidy, Morgan Craig, and Jinzhi Lei provided invaluable I have offered a personal view of dynamical diseases, concen- comments on this manuscript, and Professor Hans–Otto Walther trating on potential examples from periodic hematological disor- very kindly made Ref. 57 available. Additionally, I would like to ders. I argue that cyclic neutropenia has a very strong likelihood of thank my many research collaborators for all of the fun and excite- being a dynamical disease due to a fundamental defect in the mech- ment we have shared in discovering the secrets of Nature and her anism governing neutrophil precursor apoptosis. Periodic chronic derangements. It has been a wonderful and joyful voyage and would myelogenous leukemia is, in my opinion, another example but we not have been possible without the generous way in which Pro- may never know for sure. fessor David C. Dale, University of Washington, shared data from For other examples like cyclic thrombocytopenia and peri- both cyclic neutropenia patients and his grey collies. Last, but not odic autoimmune hemolytic anemia, the jury is still out, and it is least, McGill University and my home Department of Physiology incumbent on future generations to figure this out with models for have been remarkably tolerant of an unconventional physiologist for the regulation of platelet and erythrocyte production that are more almost half a century for which I am profoundly grateful.

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Chaos 30, 063123 (2020); doi: 10.1063/5.0006517 30, 063123-8 Published under license by AIP Publishing.