Abstract
We develop a mathematical model of platelet, megakaryocyte, and thrombopoietin dynamics in humans. We show that there is a single stationary solution that can undergo a Hopf bifurcation, and use this information to investigate both normal and pathological platelet production, specifically cyclic thrombocytopenia. Carefully estimating model parameters from laboratory and clinical data, we then argue that a subset of parameters are involved in the genesis of cyclic thrombocytopenia based on clinical information. We provide model fits to the existing data for both platelet counts and thrombopoietin levels by changing four parameters that have physiological correlates. Our results indicate that the primary change in cyclic thrombocytopenia is an interference with, or destruction of, the thrombopoietin receptor with secondary changes in other processes, including immunemediated destruction of platelets and megakaryocyte deficiency and failure in platelet production. This study contributes to the understanding of the origin of cyclic thrombocytopenia as well as extending the modeling of thrombopoiesis.
This is a preview of subscription content, access via your institution.
References
Apostu R, Mackey MC (2008) Understanding cyclical thrombocytopenia: a mathematical modeling approach. J Theor Biol 251(2):297–316
Aster RH (1966) Pooling of platelets in the spleen: role in the pathogenesis of “hypersplenic” thrombocytopenia. J Clin Invest 45(5):645
Bélair J, Mackey MC (1987) A model for the regulation of mammalian platelet production. Ann N Y Acad Sci 504(1):280–282
Bellen A, Zennaro M (2003) Numerical methods for delay differential equations. Oxford University Press, London
Bellen A, Guglielmi N, Maset S (2006) Numerical methods for delay models in biomathematics. In: Quarteroni A, Formaggia L, Veneziani A (eds) Complex systems in biomedicine. Springer, Milan, pp 147–185
Bellen A, Maset S, Zennaro M, Guglielmi N (2009) Recent trends in the numerical solution of retarded functional differential equations. Acta Numer 18:1–110
Bernard S, Bélair J, Mackey MC (2003a) Oscillations in cyclical neutropenia: new evidence based on mathematical modeling. J Theor Biol 223(3):283–298
Bernard S, PujoMenjouet L, Mackey MC (2003b) Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data. Biophys J 84(5):3414–3424
Bruin M, Tijssen MR, Bierings M, de Haas M (2005) Juvenile cyclic amegakaryocytic thrombocytopenia: a novel entity. J Pediatr Hematol Oncol 27(3):148–152
Cohen T, Cooney DP (1974) Cyclic thrombocytopenia. Case report and review of literature. Scand J Haematol 12:9–17
Colijn C, Mackey MC (2005a) A mathematical model of hematopoiesisI. Periodic chronic myelogenous leukemia. J Theor Biol 237(2):117–132
Colijn C, Mackey MC (2005b) A mathematical model of hematopoiesisII. Cyclical neutropenia. J Theor Biol 237(2):133–146
Colijn C, Mackey MC (2007) Bifurcation and bistability in a model of hematopoietic regulation. SIAM J Appl Dyn Syst 6(2):378–394
Colijn C, Dale DC, Foley C, Mackey MC (2006) Observations on the pathophysiology and mechanisms for cyclic neutropenia. Math Model Nat Phenom 1:45–68
Connor DE, Joseph JE (2011) Cyclic thrombocytopenia associated with marked rebound thrombocytosis and fluctuating levels of endogenous thrombopoietin and reticulated platelets: a case report. Am J Hematol [Letter] 87:120–122
Craig M, Humphries AR, Mackey MC (2016) A mathematical model of granulopoiesis incorporating the negative feedback dynamics and kinetics of GCSF/neutrophil binding and internalisation. Bull Math Biol 78(12):2304–2357
Cryer CW, Tavernini L (1972) The numerical solution of Volterra functional differential equations by Euler’s method. SIAM J Numer Anal 9(1):105–129
de Sauvage FJ, CarverMoore K, Luoh SM, Ryan A, Dowd M, Eaton DL, Moore MW (1996) Physiological regulation of early and late stages of megakaryocytopoiesis by thrombopoietin. J Exp Med 183(2):651–656
Debili N, Wendling F, Cosman D, Titeux M, Florindo C, DusanterFourt I, Schooley K, Methia N, Charon M, Nador R (1995) The Mpl receptor is expressed in the megakaryocytic lineage from late progenitors to platelets. Blood 85(2):391–401
Engström K, Lundquist A, Söderström N (1966) Periodic thrombocytopenia or tidal platelet dysgenesis in a man. Scand J Haematol 3(4):290–294
Finch CA, Harker LA, Cook JD (1977) Kinetics of the formed elements of human blood. Blood 50(4):699–707
Foley C, Mackey MC (2009) Dynamic hematological disease: a review. J Math Biol 58:285–322
Giles C (1981) The platelet count and mean platelet volume. Br J Haematol 48(1):31–37
Glass L, Mackey MC (1988) From clocks to chaos: the rhythms of life. Princeton University Press, Princeton
Go RS (2005) Idiopathic cyclic thrombocytopenia. Blood Rev 19(1):53–59
Grozovsky R, Hoffmeister KM, Falet H (2010) Novel clearance mechanisms of platelets. Curr Opin Hematol 17(6):585–589
Hairer E, Norsett SP, Wanner G (1993) Solving ordinary differential equations I nonstiff problems. Springer Series in Computational Mathematics, vol 8, 2nd edn. Springer, Berlin
Harker LA, Finch CA (1969) Thrombokinetics in man. J Clin Invest 48(6):963
Haurie C, Dale DC, Mackey MC (1998) Cyclical neutropenia and other periodic hematological diseases: a review of mechanisms and mathematical models. Blood 92:2629–2640
Helleberg C, Taaning E, Hansen PB (1995) Cyclic thrombocytopenia successfully treated with low dose hormonal contraception. Am J Hematol 48(1):62–63
Hitchcock IS, Kaushansky K (2014) Thrombopoietin from beginning to end. Br J Haematol 165(2):259–268
Jackson CW, Brown LK, Somerville BC, Lyles SA, Look AT (1984) Twocolor flow cytometric measurement of dna distributions of rat megakaryocytes in unfixed, unfractionated marrow cell suspensions. Blood 63(4):768–778
Kaufman RM, Airo R, Pollack S, Crosby WH (1965) Circulating megakaryocytes and platelet release in the lung. Blood 26(6):720–731
Kaushansky K (1995) Thrombopoietin: the primary regulator of platelet production. Blood 86(2):419–431
Kaushansky K, Lichtman MA, Kipps TJ, Seligsohn U, Prchal JT, Beutler E (2012) Williams hematology, 8th edn. McGrawHill Higher Education, New York
Keeling MJ, Rohani P (2008) Modeling infectious diseases in humans and animals. Princeton University Press, Princeton
Kimura F, Nakamura Y, Sato K, Wakimoto N, Kato T, Tahara T, Yamada M, Nagata N, Motoyoshi K (1996) Cyclic change of cytokines in a patient with cyclic thrombocytopenia. Br J Haematol 94:171–174
Kosugi S, Tomiyama Y, Shiraga M, Kashiwagi H, Nakao H, Kanayama Y, Kurata Y, Matsuzawa Y (1994) Cyclic thrombocytopenia associated with IgM antiGPIIbIIIa autoantibodies. Br J Haematol 88(4):809–815
Kuter DJ (2009) Thrombopoietin and thrombopoietin mimetics in the treatment of thrombocytopenia. Annu Rev Med 60:193–206
Kuter DJ (2013) The biology of thrombopoietin and thrombopoietin receptor agonists. Prog Hematol 98(1):10–23
Kuter DJ, Greenberg SM, Rosenberg RD (1989) Analysis of megakaryocyte ploidy in rat bone marrow cultures. Blood 74(6):1952–1962
Li J, Xia Y, Kuter DJ (1999) Interaction of thrombopoietin with the platelet cMpl receptor in plasma: binding, internalization, stability and pharmacokinetics. Br J Haematol 106(2):345–356
Mackey MC (2001) Cell kinetic status of haematopoietic stem cells. Cell Prolif 34(2):71–83
Mahaffy JM (1982) A test for stability of linear differential delay equations. Q Appl Math 40(2):193–202
Mahaffy JM, Bélair MC, Mackey J (1998) Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis. J Theor Biol 190(2):135–146
Majka M, JanowskaWieczorek A, Ratajczak J, Kowalska M, Vilaire G, Pan Z, Honczarenko M, Marquez L, Poncz M, Ratajczak M (2000) Stromalderived factor 1 and thrombopoietin regulate distinct aspects of human megakaryopoiesis. Blood 96(13):4142–4151
Marjoram P, Molitor J, Plagnol V, Tavaré S (2003) Markov chain Monte Carlo without likelihoods. Proc Natl Acad Sci 100(26):15,324–15,328
Maset S, Torelli L, Vermiglio R (2005) Runge Kutta methods for retarded functional differential equations. Math Models Methods Appl Sci 15(08):1203–1251
Mason KD, Carpinelli MR, Fletcher JT, Collinge JE, Hilton AA, Ellis S, Kelly PN, Ekert PG, Metcalf D, Roberts AW, Huang DCS, Kile BT (2007) Programmed anuclear cell death delimits platelet life span. Cell 128(6):1173–1186
Mathworks (2015) MATLAB 2015a. Mathworks, Natick, Massachusetts
McClatchey KD (2002) Clinical laboratory medicine. Lippincott Williams & Wilkins, Baltimore
Morley A (1969) A platelet cycle in normal individuals. Australas Ann Med 18(2):127
Nakeff A (1977) Colonyforming unit, megakaryote (CFUM): its use in elucidating the kinetics and humoral control of the megakaryocytic committed progenitor cell compartment. In: Experimental hematology today. Springer, New York, pp 111–123
Nakeff A, Ingram M (1970) Platelet count: volume relationships in four mammalian species. J Appl Physiol 28(4):530–533
Nomura S, Ogami K, Kawamura K, Tsukamoto I, Kudo Y, Kanakura Y, Kitamura Y, Miyazaki H, Kato T (1997) Cellular localization of thrombopoietin mRNA in the liver by in situ hybridization. Exp Hematol 25(7):565–572
Pedersen N (1978) Occurrence of megakaryocytes in various vessels and their retention in the pulmonary capillaries in man. Scand J Haematol 21(5):369–375
Qian S, Fu F, Li W, Chen Q, de Sauvage FJ (1998) Primary role of the liver in thrombopoietin production shown by tissuespecific knockout. Blood 92(6):2189–2191
Rocha R, Horstman L, Ahn YS, Mylvaganam R, Harrington WJ (1991) Danazol therapy for cyclic thrombocytopenia. Am J Hematol 36(2):140–143
Santillan M, Mahaffy JM, Bélair J, Mackey MC (2000) Regulation of platelet production: the normal response to perturbation and cyclical platelet disease. J Theor Biol 206(4):585–603
Scholz M, Gross A, Loeffler M (2010) A biomathematical model of human thrombopoiesis under chemotherapy. J Theor Biol 264(2):287–300
Sender R, Fuchs S, Milo R (2016) Revised estimates for the number of human and bacteria cells in the body. PLoS Biol 14(8):e1002533
Skoog WA, Lawrence JS, Adams WS (1957) A metabolic study of a patient with idiopathic cyclical thrombocytopenic purpura. Blood 12(9):844–856
Swinburne J, Mackey MC (2000) Cyclical thrombocytopenia: characterisation by spectral analysis and a review. J Theor Med 2:81–91
Tamada Feese T M D, Kato Y, Maeda Y, Hirose M, Matsukura Y, Shigematsu H, Muto T, Matsumoto A, Watarai H et al (2004) Structure of the receptorbinding domain of human thrombopoietin determined by complexation with a neutralizing antibody fragment. PNAS 101(7):1816–1821
Tavernini L (1971) Onestep methods for the numerical solution of Volterra functional differential equations. SIAM J Numer Anal 8(4):786–795
Tomer A, Harker L (1996) Measurements of in vivo megakaryocytopoiesis: studies in nonhuman primates and patients. Stem Cells 14(S1):18–30
Trowbridge EA, Martin JF, Slater DN (1982) Evidence for a theory of physical fragmentation of megakaryocytes, implying that all platelets are produced in the pulmonary circulation. Thromb Res 28(4):461–475
Tsan MF (1984) Kinetics and distribution of platelets in man. Am J Hematol 17(1):97–104
von Schulthess GK, Gessner U (1986) Oscillating platelet counts in healthy individuals: experimental investigation and quantitative evaluation of thrombocytopoietic feedback control. Scand J Haematol 36(5):473–479
Wang B, Nichol JL, Sullivan JT (2004) Pharmacodynamics and pharmacokinetics of AMG 531, a novel thrombopoietin receptor ligand. Clin Pharmacol Ther 76(6):628–638
Wang YMC, Krzyzanski W, Doshi S, Xiao JJ, PerezRuixo JJ, Chow AT (2010) Pharmacodynamicsmediated drug disposition (PDMDD) and precursor pool lifespan model for single dose of romiplostim in healthy subjects. AAPS J 12(4):729–740
Wichmann HE, Gerhardts MD, Spechtmeyer H, Gross R (1979) A mathematical model of thrombopoiesis in the rat. Cell Tissue Kinet 12:551–567
Wichmann HE, Loeffler M (1985) Mathematical modeling of cell proliferation: stem cell regulation in hemopoiesis. CRC Press, Boca Raton
Wilkinson T, Firkin B (1966) Idiopathic cyclical acute thrombocytopenic purpura. Med J Aust 1(6):217
Yanabu M, Nomura S, Fukuroi T, Kawakatsu T, Kido H, Yamaguchi K, Suzuki M, Kokawa T, Yasunaga K (1993) Periodic production of antiplatelet autoantibody directed against GPIIIa in cyclic thrombocytopenia. Acta Haematol 89(3):155–159
Zauli G, Vitale M, Falcieri E, Gibellini D, Bassini A, Celeghini C, Columbaro M, Capitani S (1997) In vitro senescence and apoptotic cell death of human megakaryocytes. Blood 90(6):2234–2243
Zent CS, Ratajczak J, Ratajczak MZ, Anastasi J, Hoffman PC, Gewirtz AM (1999) Relationship between megakaryocyte mass and serum thrombopoietin levels as revealed by a case of cyclic amegakaryocytic thrombocytopenic purpura. Br J Haematol 105(2):452–458
Acknowledgements
This research was supported by the NSERC (National Sciences and Engineering Research Council) of Canada through Discovery grants to JB, ARH, and MCM, and PGSD program to MC. SRS thanks McGill University for a Science Undegraduate Research Award. GPL and MCM are especially grateful to Dr. Jayson Potts (UBC) for his initial contact that prompted the initiation of this research. We thank Prof. Jiguo Cao (SFU) for introducing us to LW.
Author information
Affiliations
Corresponding author
Appendices
Appendix 1: Parameter estimation and constraints
This extensive appendix contains the details of the parameter estimation procedure for this model largely based on experimental data. First, in “Homeostasis relationships” section we consider the model at homeostasis. We then use TPOknockout experimental observations in “TPO knockout relationships” section to derive further parameter constraints. In “Parameters estimated from experimental data” section we provide estimates for other parameters directly from experimental data. Finally, in “Parameters calculated from experimental data and the model” section we calculate remaining parameters using experimental data and the relationships derived in “Homeostasis relationships” and “TPO knockout relationships” sections.
Homeostasis relationships
Let \(Q^{*}\) denote the stem cell concentration, \(M_{e}^{*}\) the total megakaryocyte volume, \(P^{*}\) the platelet concentration, \(T^{*}\) the thrombopoietin concentration, \(\eta _{m}^{*}\) and \(\eta _{e}^{*}\), the rate of mitosis and endomitosis, respectively, and \(\tau _{m}\) and \(\tau _{e}\), the average time megakaryoblasts and megakaryocytes spend in the mitotic and endomitotic stages, respectively, at homeostasis. At this steady state, the equations for megakaryocyte production rate (3), platelet production rate (13), megakaryocyte volume (8), platelet balance (9), and thrombopoietin balance (10) become
TPO knockout relationships
The elimination of TPO gene or its receptor in mice reduces megakaryocyte and platelet levels to approximately 10% of normal (Sauvage et al. 1996), a finding also observed in humans (Kaushansky, private communication). Therefore, the model must have a steady state solution at 10% normal platelet and megakaryocyte levels when the thrombopoietin production rate and level are both zero, giving
and
Using Eqs. (26) and (27), we rewrite these two relationships to obtain
Parameters estimated from experimental data
Megakaryocyte compartment
Mackey (2001) estimated the homeostatic concentration of HSCs using data from cats and mice, giving an estimate of \(Q^{*}=1.1\times 10^{6}\hbox { cells/kg of body weight}\). We assume that humans have roughly this same number of stem cells per kg of body weight. We estimate the parameter \(\kappa _{P}\), the rate at which stem cells commit to the megakaryocyte lineage, from the model of stem cells dynamics Mackey (2001) and Bernard et al. (2003a) developed, and the assumption that stem cells differentiate at an equal rate into all blood lineages. This gives, to four significant digits, an estimate of \(\kappa _{P}\approx 0.0072419\hbox { cells/kg of body weight per day}\). See Craig et al. (2016) for more details.
Tomer and Harker (1996) measured the diameters of megakaryocytes in the bone marrow of 10 healthy individuals. They found that megakaryocytes of ploidy 2N (those megakaryocytes which have not yet undergone endomitosis) had a mean diameter of \(21 \pm 4\,\upmu \hbox {m}\), while the average megakaryocyte had a mean diameter of \(37 \pm 4\,\upmu \hbox {m}\). We set the average volume of a megakaryocyte of ploidy 2N to be approximately that of a sphere with mean diameter \(21\, \upmu \hbox {m}\), and hence, set \(V_{m}=4\pi (21)^{3}/24\,\hbox {fL}\). The parameter \(\tau _{e}\), the time a megakaryocyte spends in endomitosis, is estimated by various sources to range from 5 to 7 days (Finch et al. 1977; Kuter 2013). We take \(\tau _{e}=5\,\hbox {days}\).
Platelet compartment
Giles (1981) measured the mean platelet count and volume in 1011 healthy human adult blood specimens. He found the mean platelet count to be \(290\times 10^{9}\) platelets/L of blood and the mean platelet volume to be 8.6 fL, so we set \(\beta _{P}\) to be 8.6 fL. Since, on average, one third of the total platelet mass in the body is sequestered by the spleen (Aster 1966), we approximate the mean platelet count in the body is 1.5 times this amount. The Hill coefficient for the plateletdependent removal of platelets is assumed to take the value \(n_P = 2\). Supposing that a healthy adult has roughly 5 L of blood per 70 kg of body weight gives \(P^{*}=1.5\times (5/70)\times (290\times 10^{9})\approx 3.1071\times 10^{10}\hbox { platelets/kg of body weight}\).
Thrombopoietin compartment
Normal TPO concentrations in humans range from 50 to 150 pg/ml of blood (Kuter 2013), and so we select the middle of the range as the homeostatic concentration, \(T^{*}=100\) pg/ml. Since only platelets in circulation, and not in the spleen, contribute to the binding of TPO, we set \(k_{S}=2/3\), which is the fraction of the platelet mass in circulation (Aster 1966). As there are two thrombopoietin binding sites on a TPO receptor (Hitchcock and Kaushansky 2014), we set \(n_{T}=2\) for the binding coefficient of thrombopoietin.
Parameters calculated from experimental data and the model
Megakaryocyte and platelet compartments
At equilibrium, the total megakaryocyte volume is
wherein \(\eta _{e}^{*} = \eta _e(T^*)\). Tomer and Harker (1996) found that the average megakaryocyte in humans has a mean diameter of \(37 \pm 4 \,\upmu \hbox {m}\), giving an approximate mean volume of \(4\pi (37)^3/24\,\hbox {fL}\). Assuming the average megakaryocyte volume predicted by our model (total volume of megakaryocytes divided by the number of megakaryocytes) equals this value, we have from (33) that
which can be rearranged as
Using the MATLAB (Mathworks 2015) function fsolve, which solves the equation \(F(x) = 0\) for x for some function F, we solved Eq. (35) for \(\tau _{e}\eta _{e}^{*}\), yielding \(\tau _{e}\eta _{e}^{*} \approx 2.788\). Since \(\tau _{e}\) is known, we have \(\eta _{e}^{*} \approx 0.5576\).
Using \(^{111}\) InOxine and \(^{111}\) Intropolone (more reliable markers than the previously used \(^{51}\)Cr label), Tsan (1984) measured the mean platelet survival time \(\tau _P\) to be \(8.4 \pm 0.25\) days. We assume that the platelet production rate replenishes the full platelet population (those circulating and in the spleen) in about \(\tau _{P}\) days. Therefore,
and, in particular,
Solving for \(\alpha _{P}\), we have
All parameters in (38) except for \(b_P\) and \(\gamma _P\) are known. The rates of removal of the platelets from the blood should be positive, implying that \(\gamma _P\) and \(\alpha _P > 0\). The latter requires that
One megakaryocyte sheds 1000–3000 platelets (Harker and Finch 1969). Assuming, on average, that one megakaryocyte sheds 2000 platelets, then the rate of production of megakaryocytes times the number of platelets shed per megakaryocyte equals roughly the rate of production of platelets:
giving a rate of production of megakaryocytes of \(1.85\times 10^{6}\,\hbox {megakaryocytes/kg/day}\), which is close to the value of \(2 \times 10^{6}\,\hbox {megakaryocytes/kg/day}\) estimated to be the normal production rate of megakaryocytes (Finch et al. 1977).
The parameter \(D_0\), the fraction of megakaryocytes shedding platelets, can be solved for in (36) by equating (40) with (36) and using (27). This gives
Rearranging (40) and solving for \(\eta _{m}^{*}\tau _{m}\), we get
Substituting in Eq. (31) and solving for \(\eta ^{min}_{*}\), we have
Using Eq. (1) at homeostasis to solve for \(\eta ^{max}_m\) in Eq. (31) gives
We can now use the steady state equation for the platelet numbers in absence of thrombopoietin, Eq. (32), in combination with the expressions (40) and (41) to get
Solving for \(\eta _{e}^{min}\eta _{e}^{*}\), we get
We can solve for \(\eta ^{max}_{e}\) via Eq. (5) at steady state:
Thrombopoietin compartment
Using Eq. (30), the homeostasis relationship for the thrombopoietin concentration, we can solve for \(\alpha _T\), yielding
Parameters fit from experimental data
From the above calculations, it remains to estimate eight more parameters: \(\tau _m\), \(b_m\), \(b_{e}\), \(\gamma _P\), \(b_P\), \(T_{prod}\), \(\gamma _T\), and \(k_T\). The first five parameters pertain to megakaryocyte and platelet dynamics, while the last three pertain to thrombopoietin dynamics.
We digitized data from Wang et al. (2010) for the circulating platelet and TPO levels in healthy patients following a \(1\,\upmu \text{ g/kg } \text{ of } \text{ bodyweight }\) intravenous infusion of Romiplostim, a TPO mimetic with similar physiological activity to TPO. We then fitted the parameters by simulating the response of our model (11)–(14) to an infusion of \(1\, \upmu \hbox {g/kg}\) of TPO and minimizing the squared error between data and simulation.
Specifically, the platelet and TPO data points (and error bars, when available) were interpolated and evaluated at 1000 points in each time interval (ranging from 0 to 42 days for the platelet data and 0 to 1 day for the TPO data), yielding the vectors of points \({\mathbf {P}}_{data}\) and \({\mathbf {T}}_{data}\). To simulate the response of our model to a \(1\,\upmu \text{ g/kg }\) intravenous infusion of thrombopoietin, we ran the numerical algorithm described in “Appendix 7” with initial history functions \((P_h, T_h) = (P^*, T^*)\) and initial conditions \(P_0 = P^*\) and \(T_0 = T_{data}(t=0)\). We then evaluated the solution of our model at the interpolated points \({\mathbf {P}}_{data}\) and \({\mathbf {T}}_{data}\) to obtain the model points \({\mathbf {P}}_{model}\) and \({\mathbf {T}}_{model}\), respectively.
For parameter estimation, we minimized the fitting error
by using the fmincon function in MATLAB (Mathworks 2015) to find the set of parameters that minimizes (47). (The factor of 2 / 3 accounts for the fraction of platelets that circulate in blood in our model.) The fit obtained from this procedure is shown in Fig. 9.
Appendix 2: Proof of existence and uniqueness of a positive steady state solution
Here, we prove that the model has a unique, positive steady state solution by showing that Eqs. (9) and (10) have a unique positive fixed point.
As discussed in “Appendix 1”, Eqs. (9) and (10) at steady state (homeostasis) are given by
and
where
and
Unlike most population and blood cell regulation models, notice that \((P^*= 0, T^*=0)\) is not an equilibrium of Eqs. (48) and (49). This is because of the nonzero constant production rate of thrombopoietin reflected in the first term of (49), and is analogous to infectious disease models for which there is a constant influx of susceptibles (see, for example, Sect. 2.1.2 of Keeling and Rohani 2008).
Although finding \((P^*,T^*)\) involves solving two highly nonlinear equations in two unknowns, we can use Eq. (49) to solve for \(P^*\) explicitly in terms of \(T^*\), yielding
Denote the righthandside of Eq. (50) by \(\mathscr {L}(T^*)\). Writing
and
we can rewrite Eq. (50) as
We now show that \(\mathscr {L}\) is a monotone decreasing function of \(T^*\), and thus defines an injective function of \(P^*\). First, notice that \(\mathscr {L}_1\) is a monotone decreasing function of \(T^*\), approaching \(+\infty \) as \(T^* \rightarrow 0\), becoming negativevalued when \(T^*~>~ T_{prod}/\gamma _T\), and approaching \(\infty \) as \(T^* \rightarrow \infty \). Second, as both \(\eta _m(T^*)\) and \(\eta _e(T^*)\) are monotone increasing in \(T^*\) (taking values in the intervals \((\eta _m^{min},\eta _m^{max})\) and \((\eta _e^{min},\eta _e^{max})\), respectively), the terms
in \(\mathscr {L}_2(T^*)\) are both monotone increasing whenever \(\tau _e\eta ^{*}_{e}(T^*) ~>~ 0\), which holds by definition in our model. As \(\mathscr {L}_2\) is the product of two monotone increasing functions, it is also a monotone increasing function. Taken together, these results show that \(\mathscr {L}\) defines an injective function of \(P^*\).
Now consider Eq. (48) and denote its lefthandside and righthandside by \(g(T^*)\) and \(h(P^*)\), respectively. The function h is clearly monotone increasing in \(P^*\), starting from 0 and approaching \(+\infty \) as \(P^* \rightarrow \infty \). Using the previous argument on \(\eta ^{*}_m(T^*)\) and \(\eta ^{*}_e(T^*)\), we have that g is a monotone increasing function of \(T^*\) and uniformly bounded above and below by
Finding a positive stationary solution in \((P^*, T^*)\) therefore amounts to finding a value of \(T^*\) satisfying
and then defining the corresponding value of \(P^*\) being given by Eq. (50). As \(h(\mathscr {L}(T^*))\) is monotone decreasing from \(+\infty \) to negative values and \(g(T^*)\) is monotone increasing between two positive values, there is a unique positive solution to Eq. (51). That is, Eqs. (17) and (18) have a unique positive steady state solution \((P^*,T^*)\).
Appendix 3: Linearization of the thrombopoiesis equations and bifurcation analysis
We take the model in Sect. 4, which has a differential equation for the platelets, (17), and one for the thrombopoietin, (18), depending only on P and T. From “Appendix 2”, the model has a unique positive equilibrium, \((P^*, T^*)\). Linearizing about the equilibrium, we let \(x(t) = P(t)P^*\) and \(y(t) = T(t)T^*\). We use Taylor expansions for both the exponential function and \(\eta _m(T(s))\) to obtain the following linear approximation:
Similarly,
and
Linearizing the integral product in Eq. (18), we obtain the following approximation:
These results can be used to find the linearization of the platelet and thrombopoietin equations. The platelet equation with only the constant and linear terms (higher order terms dropped) is given by
where
The thrombopoietin equation can be written as
where
The thrombopoietin equation with only the constant and linear terms (higher order terms dropped) is given by:
By the definition of an equilibrium, the constant terms in Eqs. (52) and (53) sum to zero, yielding the linear equations for platelets, (19), and thrombopoietin, (21), given in Sect. 4.1.
Details for the characteristic equation
We examine the integral terms in Eqs. (19) and (21), using the exponential form for \(y(t) = \mathrm {e}^{\lambda t}\). There are four integrals, which we evaluate below:
and
These expressions are used in the terms \(L_2(\lambda )\) and \(L_4(\lambda )\) in the characteristic equation for the linear functional Eq. (23).
If Eq. (24) is multiplied by \(\lambda (\lambda  \eta _e)\), then the terms in the denominator can be eliminated (at the expense of introducing the roots \(\lambda = 0\) and \(\eta _e\)). The first polynomial piece becomes
Next we consider the portion \(L_4(\lambda )  C_1\) in (24)
We take the previous expression and multiply by \(\lambda + L_1\) and define
The results are
and
Multiplying by \(\lambda (\lambda \eta _e)\) produces a quartic exponential polynomial in the eigenvalues, which can be analyzed using techniques we have developed earlier (Mahaffy 1982). The characteristic equation can be written as
We examine the terms above and obtain the following coefficients:
Appendix 4: Parameter sensitivity of the model for healthy subjects
The characteristic equation (24) from the linear analysis is used to study the sensitivity of each parameter near its normal value. We developed MATLAB programs from the linear analysis to compute the leading eigenvalues of the model. With the normal parameters of Table 1 we obtained the equilibrium \((P^*, T^*) = (31.071, 100)\), which has the leading pair of eigenvalues \(\lambda _1 = 0.058953 \pm 0.053015i\). It follows that the model with these eigenvalues is asymptotically stable. It should be noted that the frequency is equivalent to a period of 118.5 days, which is far from the observed oscillation periods in either the healthy subjects or patients with cyclic thrombocytopenia.
To analyze the sensitivity of the model to the various parameters, we used our MATLAB code to find the new equilibrium and leading eigenvalues as we varied each parameter by ±10%. Table 6 gives a complete listing of how the equilibrium changes and leading eigenvalues shifts with all of the individual parameter changes. We are most interested in stability changes, so the lowest ratio of the real part of the leading eigenvalues gives the largest change in the direction of a Hopf bifurcation.
For the sensitivity analysis, if we only consider the movement of the leading pair of eigenvalues toward the imaginary axis, then the smallest values in the 7th column (ratio Re) give the greatest shift toward instability. The parameter changes that destabilize the model most are (in descending order) increasing \(b_P\), decreasing \(\alpha _P\), decreasing \(k_T\), increasing \(\beta _P\), increasing \(k_S\), increasing \(b_m\), and decreasing \(T_{prod}\).
As an experiment to extend this analysis, we chose to increase or decrease all seven of these parameters by 20% to see what happened to the equilibrium and the leading eigenvalues. The result of all seven changes resulted in the equilibrium \((P^*, T^*) = (20.772, 85.117)\) and leading eigenvalues \(\lambda _1 = 0.02555 \pm 0.06563i\). We note that this more than halves the distance of the real part toward the imaginary axis, and also the frequency shifts the period to 95.74 days.
The analysis shows that near normal, there are several parameters to which the model is very insensitive. Surprisingly, this includes all of the delay parameters, \(\tau _e\) and \(\tau _m\). It is also quite insensitive to changes in \(\gamma _P\), \(\gamma _T\), and \(b_e\). However, our numerical study shows that \(\tau _e\) and \(\tau _m\) can have effects on the imaginary part.
This analysis shows that the leading eigenvalues have the wrong frequency for the observed oscillations in cyclic thrombocytopenia patients. This suggests the need to examine the next, second leading pair of eigenvalues for this model. Its frequency is closer to the range of interest and provides a starting point for a Hopf bifurcation study of our cyclic thrombocytopenia patients. Again, with the normal parameters the second eigenvalues are \(\lambda _2 = 0.11375 \pm 0.35888i\), which gives a quasiperiod near 17.5 days. Table 7 shows the effects on this pair of eigenvalues as the parameters are changed by ±10%.
For the sensitivity analysis, we examine the movement of the second leading pair of eigenvalues, \(\lambda _2\), toward the imaginary axis. Again, the smallest values in the 7th column (ratio Re) give the greatest shift toward instability. The most destabilizing changes for this pair of eigenvalues occur by (in descending order) increasing \(\tau _e\), decreasing \(b_m\), increasing \(k_S\), increasing \(\beta _P\), increasing \(\tau _m\), decreasing \(b_P\), and decreasing \(\gamma _P\). We note that the delays \(\tau _e\) and \(\tau _m\) affect the movement of the real part of these eigenvalues.
To extend this analysis, we chose to increase or decrease all seven of these parameters by 20% to see what happened to the equilibrium and eigenvalues. The result of all seven changes resulted in the equilibrium \((P^*, T^*) = (26.505, 87.236)\) and eigenvalues \(\lambda _2 = 0.09281 \pm 0.3091i\). Note that the distance of the real part toward the imaginary axis is slightly more than for the leading pair with its most significant parameters, and also the frequency shifts the period to 20.33 days.
Appendix 5: Hopf bifurcation for CT patients
This appendix continues the studies of the CT patients from Sect. 5.4. The four parameters are varied linearly between the normal state and the best fitting parameters for several CT patients. Again, our numerical methods tracked the changes in the equilibria and the pairs of eigenvalues, which result in Hopf bifurcations leading to the cyclic behavior observed in the CT patients.
Table 2 in Sect. 5.2 shows the best parameter fit to \(\tau _{e}\), \(\alpha _{P}\), \(\alpha _{T}\), and \(k_{T}\) for the CT patients of Connor and Joseph (2011), Kimura et al. (1996), and Zent et al. (1999). As was done with the CT patient of Bruin et al. (2005) in Sect. 5.4, a hyperline in the 4Dparameter space from the normal parameter values to each of the parameter sets for these three CT patients was followed, and the numerical values of the equilibrium \((P^{*}\), \(T^{*})\) and eigenvalues \(\lambda \) were tracked at each set of parameter values. As before, the eigenvalues tracked were the ones from the second leading pair of the healthy subject, which is the pair that undergoes a Hopf bifurcation as the hyperline extends to any of the CT patients. The results are displayed in Fig. 10.
The equilibrium for the normal parameter set is \((P^{*}, T^{*}) = (31.071, 100.00)\). As the parameter sets move along each of the hyperlines, the equilibria first shift slowly in an arc toward the origin. For all of our examples, this first arc of equilibria takes over 98% of the hyperline, that is, in Eq. (25) if the arc is created by \(t \in [0, t_1)\), then \(t_1 > 0.98\). For an unknown reason (likely a transition in one of the Hill functions), the equilibrium rapidly shifts away from this slow path toward the origin along a different trajectory. This new direction roughly doubles back, but heads to the different states of equilibria for each of the different patients. It is along this rapidly evolving path that the Hopf bifurcation occurs. Figure 10 (and Fig. 8) shows that the cusplike behavior is similar in all cases, but the evolving paths are distinct for each patient. This complicates the interpretation of how the cyclic thrombocytopenia is explained through the parameters. Table 8 gives the values for the equilibria of the Hopf bifurcation along with the equilibria for the best fitting parameters of the different CT patients.
We numerically solve Eq. (24), starting at the eigenvalues for the normal case with \(\lambda = 0.11375 \pm 0.3588i\). In all cases, the eigenvalues create an arc with the imaginary part decreasing, while the real part first increases then decreases. This arc is created quite slowly and follows the slowly evolving equilibria above. When the equilibria start evolving rapidly, the values of the eigenvalues rapidly shift with increasing real part. Specifically, the real part turns around and increases very rapidly to the Hopf bifurcation. The different CT patients have slightly different changes in their eigenvalues, particularly in what happens to the imaginary part. All cases of the CT patients have their eigenvalues with positive real part, which is to be expected. The frequency of the eigenvalues varies from 0.1641 to 0.2698, yielding periods in the range of 23.3 to 38.3 days, which are consistent with the simulations in Sect. 5.2.
Appendix 6: Fitting of cyclic thrombocytopenia patient data
In this appendix, we describe the statistical procedure used in Sect. 5 to fit the parameters \(\tau _{e}\), \(\alpha _P\), \(\alpha _T\), and \(k_T\) of our model to 15 published platelet and TPO data sets of patients with CT (Bruin et al. 2005; Cohen and Cooney 1974; Connor and Joseph 2011; Engström et al. 1966; Helleberg et al. 1995; Kimura et al. 1996; Kosugi et al. 1994; Morley 1969; Rocha et al. 1991; von Schulthess and Gessner 1986; Skoog et al. 1957; Wilkinson and Firkin 1966; Yanabu et al. 1993; Zent et al. 1999). Of these 15 data sets, only four contained both platelet and TPO data (Bruin et al. 2005; Connor and Joseph 2011; Kimura et al. 1996; Zent et al. 1999). We chose to analyze data sets of untreated CT patients only, as treatments may have altered platelet or TPO dynamics or both and is thus outside the scope of this model.
We first introduce some notation before describing the ABCMCMC algorithm and the fitting procedure (Marjoram et al. 2003). Let \({\theta }\) be the vector of all the parameters, that is, \({\theta }=(\tau _{e}, \alpha _P, \alpha _T, k_T)^\top .\) Let \(\mathscr {D}\) denote the observed data, including \({\mathbf {P}}_{data}\) and \({\mathbf {T}}_{data}\). Let \(\mathscr {D}^{'}\) denote the model response, including \({\mathbf {P}}_{model}\) and \({\mathbf {T}}_{model}\). Let \(\rho (\mathscr {D}^{'}, \mathscr {D})\) denote the distance between \(\mathscr {D}^{'}\) and \(\mathscr {D}\). Let \(\epsilon \) be a prefixed threshold. The gist of the fitting procedure for a data set is as follows.

1.
Choose an initial set of values of parameters for \({\theta }\).

2.
Propose a move from the current value of \(\theta \) to \({\theta }^{'}\) according to a transition kernel \(q(\cdot \theta )\).

3.
Simulate \(\mathscr {D}^{'}\) using the model with parameters \({\theta }^{'}\).

4.
If \(\rho (\mathscr {D}^{'}, \mathscr {D})\le \epsilon \), go to step 5, and otherwise stay at \({\theta }\) and return to step 2.

5.
Calculate
$$\begin{aligned} \alpha ({\theta }, \theta ^{'}) = \min \left( 1, \frac{\pi ({\theta }^{'}) q(\theta {\theta }^{'})}{\pi (\theta ) q({\theta }^{'}\theta )}\right) , \end{aligned}$$update \({\theta }\) to \(\theta ^{'}\) with this probability, and store the value of \({\theta }^{'}\).

6.
Repeat steps 2–5 using \({\theta }'\) as the new initial set of values of parameters for a sufficient number of times, and finally pick \({\theta }'\) that minimizes \(\rho (\mathscr {D}^{'}, \mathscr {D})\) among all stored values of \({\theta }'\).
To measure the distance between the simulated data \(\mathscr {D}^{'}\) and the observed data \(\mathscr {D}\), we use the following sum of squared errors (SSE):
where the factor of 2/3 accounts for the fraction of platelets that circulate in blood in our model.
For the fits in this paper, in step 1 we chose the initial parameters so that the model generated oscillations, as discussed in Sect. 5.1, and gave a rough approximation to the data set. These parameters defined the initial data \(\mathscr {D}\), and the initial SSE was computed using Eq. (54). We fixed the threshold \(\epsilon \) to 1.15 times the initial SSE and computed steps 2–5 for 250 successful iterations before choosing the vector of parameters \(\theta \) minimizing \(\rho (\mathscr {D}^{'}, \mathscr {D})\).
In step 5 of the ABCMCMC implementation, we use a uniform distribution as the prior distribution for the parameters, which leads to \(\pi ({\theta }^{'})/\pi (\theta )=1\). In addition, we choose a Gaussian distribution to be the transition kernel, which implies \(q(\theta {\theta }^{'})/q(\theta ^{'}{\theta })=1\). As a result, \(\alpha (\theta , {\theta }^{'})\) is simplified to 1.
The stationary distribution of the MCMC chain is the posterior distribution of the parameters given the simulated data are close enough to the observed data (\(\rho (\mathscr {D}^{'}, \mathscr {D})\le \epsilon \)). In other words, with a good initial choice of parameter values, this ABCMCMC algorithm guarantees convergence to a good fit.
Appendix 7: Numerical analysis
Numerical simulations of the system of equations (12) and (13) are necessary both to illustrate results for any parameter set, and also as part of the parameter fitting described in “Appendix 6”. Accurate numerical solution of these equations is complicated by their structure as distributed delay differential equations (DDEs) with the terms \(m_{e}(t,\tau _{e})\) and \(M_{e}(t)\) both defined by integrals of the solution functions from time \(t\tau _{e}\tau _{m}\) to t, with \(M_{e}(t)\) requiring the computation of the integral of the product of the exponential of two integrals.
Traditional Runge–Kutta methods for ordinary differential equations (ODEs) only define a numerical approximation to the solution on a discrete set of time points. This is problematical when the solution is required at off mesh time values, which arises for example for the accurate evaluation of integrals, as is the case in our problem. Continuous Runge–Kutta (CRK) methods were developed (Bellen and Zennaro 2003; Hairer et al. 1993) to produce continuous output suitable for the numerical solution of both ODEs and DDEs. These are the methods currently most often used to solve discrete DDEs, with the Matlab (Mathworks 2015) software package containing built in functions (dde23, ddesd) for the solution of discrete constant delay and statedependent variable delay DDEs. However, these methods are not appropriate for problems with vanishing or distributed delays, because they become fully implicit. To see how this arises suppose the system (12)–(13) is already solved up to time \(t_n\) and consider the computation of the next step. To compute the jth stage of the CRK method for the next step requires the computation of the righthand side of the system of equations at time \(t_n+c_jh\), where \(h=t_{n+1}t_n\) is the stepsize of the method and the \(c_j\) are the abscissa of the CRK method. But this requires us to evaluate \(m_{e}(t_n+c_jh,\tau _{e})\) and \(M_{e}(t_n+c_jh)\), for which we need integrals of T(t) up to time \(t_n+c_jh\), but until all the stages of the current step are computed we only have the solution of T(t) available up to time \(t_n\). This problem does not arise in the first order forward Euler method (which has just one stage with \(c_1=0\)), but all higher order CRK methods become fully implicit. Adhoc methods for approximating the missing integral result in a reduction in the order of the method, and so the only CRK methods that are appropriate for our problem are the first order forward Euler method or higher order implicit methods. To obtain accurate solutions efficiently we do not use such methods.
Methods that remain explicit for distributed DDEs and DDEs with vanishing delays were first proposed by Tavernini (1971), and have more recently been developed into a class of methods called Functional Continuous Runge–Kutta (FCRK) methods (Bellen et al. 2009; Maset et al. 2005). In Bellen et al. (2006) it was proposed to apply FCRK methods to distributed DDEs in biomathematics, but we are not aware of any implementation of FCRK methods for distributed DDEs before the current work.
To solve the system (12)–(13) we implemented the explicit twostage second order Heun FCRK method proposed in Cryer and Tavernini (1972). This has Butcher tableau
which defines the parameters \(c_{i}\), \(b_{i}(\alpha )\) and \(a_{i j}(\alpha )\) in the explicit FCRK method
where the \(K_{i}\) are said to be the stage variables. In the method, \(u_{n}\) is the numerical solution with stepsize h generated by the nth step of the method defined up to time \(t_{n}\) for the delayed functional differential equation
where \(\phi \) is said to be the starting data, \(t_{0}\) is the initial time, and \(u_{t}\) is the continuous function \(u_{t}(\sigma ) = u(t + \sigma )\) for \(\sigma \in [\tau , 0]\), where \(\tau \) is the largest delay in the system (which is \(\tau =\tau _e+\tau _m\) for our problem).
The fundamental difference between CRKs and FCRKs which enables the FCRK methods to remain explicit is that the FCRKs are endowed with a continuous approximation \(Y^i(t)\) associated with each stage, whereas the CRK methods only have the single continuous approximation \(u_{n+1}(t)\) defined once the step is computed.
To implement this FCRK method for the system (12)–(13), the integrals need to be evaluated numerically to sufficient accuracy to maintain the convergence order of the method. Although the composite trapezoidal rule would be sufficient for second order accuracy we used fourth order composite methods, to allow for the possible later implementation of a fourth order FCRK method. It is necessary to evaluate the integrals on the same computational mesh as the underlying FCRK method. Since the numerical approximation \(T_{n+1}(t)\) to T(t) is smooth on each interval \([t_n,t_{n+1}]\), but not differentiable at the mesh points \(t_n\), the functions \(\eta _m(T_{n+1}(t))\) and \(\eta _m(T_{n+1}(t))\) will also not be differentiable at the mesh points, and the convergence theory of the composite quadrature methods will break down unless the mesh points \(t_n\) of the FCRK method are included as quadrature points for the integration. The exact solutions of delayed functional differential equations also have discontinuous derivatives at breaking points as outlined in Bellen and Zennaro (2003), which requires certain time points to be included in the computational mesh. But because of the additional smoothing afforded by the integrals in (12)–(13) the only breaking point that needs to be included in the mesh to obtain second order convergence is the initial point \(t_0\).
Taking account of these considerations we used Simpson’s method to evaluate \(M_e(t_n+c_jh)\) using values of \(m_e(t_n+c_jh,kh)\) and \(m_e(t_n+c_jh,(k+1/2)h)\), where \(c_j=0\) or 1 for Heun’s method, and \(k\in \{0,1,...,N\}\) is an integer with the stepsize h of the FCRK method chosen so that \(\tau _e=Nh\). The necessary values of \(m_e(t,a)\) were obtained from (13), evaluating the integrals in this formula using Milne’s method. To evaluate the nested integrals efficiently we store values of \(\eta _m(T(t))\) and \(\eta _e(T(t))\) at relevant points (\(t_n+kh/4\) for \(k=0,1,2,3\)) and also store integrals which would otherwise be recomputed at multiple steps, including
which appears in \(m_e(t_j+kh,kh)\) and hence is required to evaluate \(M_e(t_{j+k})\) for each of \(k=0,1,\ldots ,N\).
The method is written to return a function handle that is created from the continuous approximation to the solution generated by the method. This is returned to the user, which allows for the evaluation of P(t) and T(t) at all points in the computational interval (not just at mesh points).
The explicit fourth order FCRK method proposed in Cryer and Tavernini (1972) could be similarly implemented, but requires 6 stages to evaluate one step to fourth order, and so is not as advantageous as the 4stage fourth order RK method for ODEs, and would require the computation of some complicated integrals over 6 stages.
Rights and permissions
About this article
Cite this article
Langlois, G.P., Craig, M., Humphries, A.R. et al. Normal and pathological dynamics of platelets in humans. J. Math. Biol. 75, 1411–1462 (2017). https://doi.org/10.1007/s0028501711256
Received:
Revised:
Published:
Issue Date:
Keywords
 Platelet regulation dynamics
 Thrombopoiesis
 Megakaryopoiesis
 Cyclic thrombocytopenia
 Dynamic diseases
 Delay differential equations
Mathematics Subject Classification
 37N25
 92B99
 92C30
 37G15