×

Normal and pathological dynamics of platelets in humans. (English) Zbl 1380.37146

Recent laboratory and clinical data suggest to develop a more physiologically realistic model for the regulation of mammalian platelet production concentrating on humans, which takes into account both megararyocytes and platelets and the effects on trombopoietin (TPO). The presented mathematical model describes the dynamics of the megararyocytes as an age-structured model, which is divided in two stages: mitosis, and endomitosis. Platelet and trombopoietin dynamics deal with endomitosis. The model of mitosis simply gives an effective proliferation rate that includes both cellular birth and death. The choice of the Hill function in the model reflects the fact that TPO has a stimulatory, yet saturating, effect on the process.
The platelet population dynamics are governed by the balance between platelet production and destruction.
The authors model the TPO dynamics as the balance between production and destruction of platelets. Here the same Hill function is used, because it is assumed that the endogenous removal rate is proportional to the saturable Hill function.
Thus, the model of thrombopoiesis consists of two integro-differential equations with constant delays and an integral equation. The two differential equations model the dynamics of platelets and TPO, while the integral equation models the volume of megakaryocytes in the bone marrow.
In contrast to previous models the given model incorporates the regulation mechanisms and the dynamics of megararyocytes and trombopoietin.
The authors extend linear techniques to this model and develop numerical methods to perform a stability analysis.
The mathematical analysis of the nonlinear model indicates that there remain details to be understood, which could be explored further and possibly give insight into the transitions from the stable normal state to the diseased state, specifically in patients with cyclic thrombocytopenia.

MSC:

37N25 Dynamical systems in biology
92C30 Physiology (general)
92C35 Physiological flow
92C50 Medical applications (general)

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Apostu R, Mackey MC (2008) Understanding cyclical thrombocytopenia: a mathematical modeling approach. J Theor Biol 251(2):297-316 · Zbl 1398.92107 · doi:10.1016/j.jtbi.2007.11.029
[2] Aster RH (1966) Pooling of platelets in the spleen: role in the pathogenesis of “hypersplenic” thrombocytopenia. J Clin Invest 45(5):645 · doi:10.1172/JCI105380
[3] Bélair J, Mackey MC (1987) A model for the regulation of mammalian platelet production. Ann N Y Acad Sci 504(1):280-282 · doi:10.1111/j.1749-6632.1987.tb48740.x
[4] Bellen A, Zennaro M (2003) Numerical methods for delay differential equations. Oxford University Press, London · Zbl 1038.65058 · doi:10.1093/acprof:oso/9780198506546.001.0001
[5] Bellen, A.; Guglielmi, N.; Maset, S.; Quarteroni, A. (ed.); Formaggia, L. (ed.); Veneziani, A. (ed.), Numerical methods for delay models in biomathematics, 147-185 (2006), Milan · Zbl 1387.92007 · doi:10.1007/88-470-0396-2_5
[6] 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 · Zbl 1178.65078 · doi:10.1017/S0962492906390010
[7] Bernard S, Bélair J, Mackey MC (2003a) Oscillations in cyclical neutropenia: new evidence based on mathematical modeling. J Theor Biol 223(3):283-298 · Zbl 1464.92062 · doi:10.1016/S0022-5193(03)00090-0
[8] Bernard S, Pujo-Menjouet L, Mackey MC (2003b) Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data. Biophys J 84(5):3414-3424 · doi:10.1016/S0006-3495(03)70063-0
[9] 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 · doi:10.1097/01.mph.0000157299.89406.ce
[10] Cohen T, Cooney DP (1974) Cyclic thrombocytopenia. Case report and review of literature. Scand J Haematol 12:9-17 · doi:10.1111/j.1600-0609.1974.tb00174.x
[11] Colijn C, Mackey MC (2005a) A mathematical model of hematopoiesis-I. Periodic chronic myelogenous leukemia. J Theor Biol 237(2):117-132 · Zbl 1440.92024 · doi:10.1016/j.jtbi.2005.03.033
[12] Colijn C, Mackey MC (2005b) A mathematical model of hematopoiesis-II. Cyclical neutropenia. J Theor Biol 237(2):133-146 · Zbl 1440.92025 · doi:10.1016/j.jtbi.2005.03.034
[13] Colijn C, Mackey MC (2007) Bifurcation and bistability in a model of hematopoietic regulation. SIAM J Appl Dyn Syst 6(2):378-394 · Zbl 1210.37033 · doi:10.1137/050640072
[14] 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 · Zbl 1337.92104 · doi:10.1051/mmnp:2008004
[15] 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 · doi:10.1002/ajh.22186
[16] Craig M, Humphries AR, Mackey MC (2016) A mathematical model of granulopoiesis incorporating the negative feedback dynamics and kinetics of G-CSF/neutrophil binding and internalisation. Bull Math Biol 78(12):2304-2357 · Zbl 1361.92022 · doi:10.1007/s11538-016-0179-8
[17] 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 · Zbl 0244.65085 · doi:10.1137/0709012
[18] de Sauvage FJ, Carver-Moore 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 · doi:10.1084/jem.183.2.651
[19] Debili N, Wendling F, Cosman D, Titeux M, Florindo C, Dusanter-Fourt 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
[20] Engström K, Lundquist A, Söderström N (1966) Periodic thrombocytopenia or tidal platelet dysgenesis in a man. Scand J Haematol 3(4):290-294 · doi:10.1111/j.1600-0609.1966.tb02373.x
[21] Finch CA, Harker LA, Cook JD (1977) Kinetics of the formed elements of human blood. Blood 50(4):699-707
[22] Foley C, Mackey MC (2009) Dynamic hematological disease: a review. J Math Biol 58:285-322 · Zbl 1161.92338 · doi:10.1007/s00285-008-0165-3
[23] Giles C (1981) The platelet count and mean platelet volume. Br J Haematol 48(1):31-37 · doi:10.1111/j.1365-2141.1981.00031.x
[24] Glass L, Mackey MC (1988) From clocks to chaos: the rhythms of life. Princeton University Press, Princeton · Zbl 0705.92004
[25] Go RS (2005) Idiopathic cyclic thrombocytopenia. Blood Rev 19(1):53-59 · doi:10.1016/j.blre.2004.05.001
[26] Grozovsky R, Hoffmeister KM, Falet H (2010) Novel clearance mechanisms of platelets. Curr Opin Hematol 17(6):585-589 · doi:10.1097/MOH.0b013e32833e7561
[27] 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 · Zbl 0789.65048
[28] Harker LA, Finch CA (1969) Thrombokinetics in man. J Clin Invest 48(6):963 · doi:10.1172/JCI106077
[29] 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
[30] Helleberg C, Taaning E, Hansen PB (1995) Cyclic thrombocytopenia successfully treated with low dose hormonal contraception. Am J Hematol 48(1):62-63 · doi:10.1002/ajh.2830480117
[31] Hitchcock IS, Kaushansky K (2014) Thrombopoietin from beginning to end. Br J Haematol 165(2):259-268 · doi:10.1111/bjh.12772
[32] Jackson CW, Brown LK, Somerville BC, Lyles SA, Look AT (1984) Two-color flow cytometric measurement of dna distributions of rat megakaryocytes in unfixed, unfractionated marrow cell suspensions. Blood 63(4):768-778
[33] Kaufman RM, Airo R, Pollack S, Crosby WH (1965) Circulating megakaryocytes and platelet release in the lung. Blood 26(6):720-731
[34] Kaushansky K (1995) Thrombopoietin: the primary regulator of platelet production. Blood 86(2):419-431
[35] Kaushansky K, Lichtman MA, Kipps TJ, Seligsohn U, Prchal JT, Beutler E (2012) Williams hematology, 8th edn. McGraw-Hill Higher Education, New York
[36] Keeling MJ, Rohani P (2008) Modeling infectious diseases in humans and animals. Princeton University Press, Princeton · Zbl 1279.92038
[37] 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 · doi:10.1046/j.1365-2141.1996.d01-1783.x
[38] Kosugi S, Tomiyama Y, Shiraga M, Kashiwagi H, Nakao H, Kanayama Y, Kurata Y, Matsuzawa Y (1994) Cyclic thrombocytopenia associated with IgM anti-GPIIb-IIIa autoantibodies. Br J Haematol 88(4):809-815 · doi:10.1111/j.1365-2141.1994.tb05121.x
[39] Kuter DJ (2009) Thrombopoietin and thrombopoietin mimetics in the treatment of thrombocytopenia. Annu Rev Med 60:193-206 · doi:10.1146/annurev.med.60.042307.181154
[40] Kuter DJ (2013) The biology of thrombopoietin and thrombopoietin receptor agonists. Prog Hematol 98(1):10-23 · doi:10.3324/haematol.2012.069385
[41] Kuter DJ, Greenberg SM, Rosenberg RD (1989) Analysis of megakaryocyte ploidy in rat bone marrow cultures. Blood 74(6):1952-1962
[42] Li J, Xia Y, Kuter DJ (1999) Interaction of thrombopoietin with the platelet c-Mpl receptor in plasma: binding, internalization, stability and pharmacokinetics. Br J Haematol 106(2):345-356 · doi:10.1046/j.1365-2141.1999.01571.x
[43] Mackey MC (2001) Cell kinetic status of haematopoietic stem cells. Cell Prolif 34(2):71-83 · doi:10.1046/j.1365-2184.2001.00195.x
[44] Mahaffy JM (1982) A test for stability of linear differential delay equations. Q Appl Math 40(2):193-202 · Zbl 0499.34049 · doi:10.1090/qam/666674
[45] 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 · doi:10.1006/jtbi.1997.0537
[46] Majka M, Janowska-Wieczorek A, Ratajczak J, Kowalska M, Vilaire G, Pan Z, Honczarenko M, Marquez L, Poncz M, Ratajczak M (2000) Stromal-derived factor 1 and thrombopoietin regulate distinct aspects of human megakaryopoiesis. Blood 96(13):4142-4151
[47] 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 · doi:10.1073/pnas.0306899100
[48] Maset S, Torelli L, Vermiglio R (2005) Runge Kutta methods for retarded functional differential equations. Math Models Methods Appl Sci 15(08):1203-1251 · Zbl 1079.65079 · doi:10.1142/S0218202505000716
[49] 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 · doi:10.1016/j.cell.2007.01.037
[50] Mathworks (2015) MATLAB 2015a. Mathworks, Natick, Massachusetts
[51] McClatchey KD (2002) Clinical laboratory medicine. Lippincott Williams & Wilkins, Baltimore
[52] Morley A (1969) A platelet cycle in normal individuals. Australas Ann Med 18(2):127 · doi:10.1111/imj.1969.18.2.127
[53] Nakeff A (1977) Colony-forming unit, megakaryote (CFU-M): 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
[54] Nakeff A, Ingram M (1970) Platelet count: volume relationships in four mammalian species. J Appl Physiol 28(4):530-533
[55] 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
[56] 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 · doi:10.1111/j.1600-0609.1978.tb00381.x
[57] Qian S, Fu F, Li W, Chen Q, de Sauvage FJ (1998) Primary role of the liver in thrombopoietin production shown by tissue-specific knockout. Blood 92(6):2189-2191
[58] Rocha R, Horstman L, Ahn YS, Mylvaganam R, Harrington WJ (1991) Danazol therapy for cyclic thrombocytopenia. Am J Hematol 36(2):140-143 · doi:10.1002/ajh.2830360213
[59] 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 · doi:10.1006/jtbi.2000.2149
[60] Scholz M, Gross A, Loeffler M (2010) A biomathematical model of human thrombopoiesis under chemotherapy. J Theor Biol 264(2):287-300 · Zbl 1406.92277 · doi:10.1016/j.jtbi.2009.12.032
[61] 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 · doi:10.1371/journal.pbio.1002533
[62] Skoog WA, Lawrence JS, Adams WS (1957) A metabolic study of a patient with idiopathic cyclical thrombocytopenic purpura. Blood 12(9):844-856
[63] Swinburne J, Mackey MC (2000) Cyclical thrombocytopenia: characterisation by spectral analysis and a review. J Theor Med 2:81-91 · Zbl 0943.92023 · doi:10.1080/10273660008833039
[64] 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 receptor-binding domain of human thrombopoietin determined by complexation with a neutralizing antibody fragment. PNAS 101(7):1816-1821 · doi:10.1073/pnas.0308530100
[65] Tavernini L (1971) One-step methods for the numerical solution of Volterra functional differential equations. SIAM J Numer Anal 8(4):786-795 · Zbl 0231.65070 · doi:10.1137/0708072
[66] Tomer A, Harker L (1996) Measurements of in vivo megakaryocytopoiesis: studies in nonhuman primates and patients. Stem Cells 14(S1):18-30 · doi:10.1002/stem.5530140703
[67] 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 · doi:10.1016/0049-3848(82)90163-3
[68] Tsan MF (1984) Kinetics and distribution of platelets in man. Am J Hematol 17(1):97-104 · doi:10.1002/ajh.2830170114
[69] 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 · doi:10.1111/j.1600-0609.1986.tb02283.x
[70] 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 · doi:10.1016/j.clpt.2004.08.010
[71] Wang YMC, Krzyzanski W, Doshi S, Xiao JJ, Perez-Ruixo JJ, Chow AT (2010) Pharmacodynamics-mediated drug disposition (PDMDD) and precursor pool lifespan model for single dose of romiplostim in healthy subjects. AAPS J 12(4):729-740 · doi:10.1208/s12248-010-9234-9
[72] Wichmann HE, Gerhardts MD, Spechtmeyer H, Gross R (1979) A mathematical model of thrombopoiesis in the rat. Cell Tissue Kinet 12:551-567
[73] Wichmann HE, Loeffler M (1985) Mathematical modeling of cell proliferation: stem cell regulation in hemopoiesis. CRC Press, Boca Raton
[74] Wilkinson T, Firkin B (1966) Idiopathic cyclical acute thrombocytopenic purpura. Med J Aust 1(6):217
[75] 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 · doi:10.1159/000204510
[76] 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
[77] 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 · doi:10.1111/j.1365-2141.1999.01351.x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.