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Understanding and Treating Cytopenia Through Mathematical Modeling

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A Systems Biology Approach to Blood

Part of the book series: Advances in Experimental Medicine and Biology ((AEMB,volume 844))

Abstract

Here, we briefly review how the study of dynamic hematological diseases with mathematical modeling tools has led to a better understanding of the origin of some types of neutropenia and thrombocytopenia and to improved treatment strategies. In addition, we have briefly discussed how these models suggest improved ways to minimize and/or treat cytopenia induced by chemotherapy.

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References

  1. Glass L, Mackey MC. From clock to chaos. Princeton: Princeton University Press; 1988.

    Google Scholar 

  2. Foley C, Bernard S, Mackey MC. Cost-effective G-CSF therapy strategies for cyclical neutropenia: mathematical modeling based hypotheses. J Theor Biol. 2006;238:754–63. doi:10.1016/j.jtbi.2005.06.021.

    Article  CAS  PubMed  Google Scholar 

  3. Colijn C, Mackey MC. A mathematical model of hematopoiesis: II. cyclical neutropenia. J Theor Biol. 2005;237:133–46.

    Article  PubMed  Google Scholar 

  4. Haurie C, Dale DC, Mackey MC. Cyclical neutropenia and other periodic hematological diseases: a review of mechanisms and mathematical models. Blood. 1998;92:2629–40.

    CAS  PubMed  Google Scholar 

  5. Haurie C, Dale DC, Mackey MC. Occurrence of periodic oscillations in the differential blood counts of congenital, idiopathic and cyclical neutropenic patients before and during treatment with G-CSF. Exp Hematol. 1999;27:401–9.

    Article  CAS  PubMed  Google Scholar 

  6. Haurie C, Dale DC, Rudnicki R, Mackey MC. Modeling complex neutrophil dynamics in the grey collie. J Theor Biol. 2000;204:505–19.

    Article  CAS  PubMed  Google Scholar 

  7. Fortin P, Mackey MC. Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications. Br J Haematol. 1999;104:336–45.

    Article  CAS  PubMed  Google Scholar 

  8. Mackey MC, Glass L. Oscillation and chaos in physiological control systems. Science. 1977;197:287–9.

    Article  CAS  PubMed  Google Scholar 

  9. Swinburne J, Mackey MC. Cyclical thrombocytopenia: characterization by spectral analysis and a review. J Theor Med. 2000;2:81–91.

    Article  Google Scholar 

  10. Apostu R, Mackey MC. Understanding cyclical thrombocytopenia: a mathematical modeling approach. J Theor Biol. 2008;251:297–316.

    Article  PubMed  Google Scholar 

  11. Hsieh MM, Everhart JE, Byrd-Holt DD, Tisdale JF, Rodgers GP. Prevalence of neutropenia in the U.S. population: age, sex, smoking status, and ethnic differences. Ann Intern Med. 2007;146:486–92.

    Article  PubMed  Google Scholar 

  12. Gill M, Ockelford P, Morris A, Bierre T, Kyle C. Diagnostic handbook-the interpretation of laboratory tests. Auckland: Diagnostic Medlab; 2000.

    Google Scholar 

  13. Haurie C, Person R, Dale DC, Mackey MC. Hematopoietic dynamics in grey collies. Exp Hematol. 1999;27:1139–48.

    Article  CAS  PubMed  Google Scholar 

  14. Reimann HA, de Beradinis CT. Periodic(cyclic) neutropenia. an entity. A collection of sixteen cases. Blood. 1949;4:1109–16.

    CAS  PubMed  Google Scholar 

  15. Morley AA, Carew JP, Baikie AG. Familial cyclical neutropenia. Br J Haematol. 1967;13:719–38.

    Article  CAS  PubMed  Google Scholar 

  16. Palmer SE, Stephens K, Dale DC. Genetics, phenotype, and natural history of autosomal dominant cyclic hematopoiesis. Am J Med Genet. 1996;88:335–40.

    Google Scholar 

  17. Norwitz M, Benson K, Person R, Aprikyan A, Dale D. Mutations in ELA2, encoding neutrophil elastase, define a 21-day biological clock in cyclic haematopoiesis. Nat Genet. 1999;23:433–6.

    Article  Google Scholar 

  18. Cohen T, Cooney DP. Cyclical thrombocytopenia: case report and review of literature. Scand J Haemat. 1974;16:133–8.

    Google Scholar 

  19. Beutler E, Lichtman MA, Coller BS, Kipps T. Williams hematology. New York: McGraw-Hill; 1995.

    Google Scholar 

  20. Balduini C, Stella C, Rosti V, Bertolino G, Noris P, Ascari E. Acquired cyclic thrombocytopenia thrombocytosis with periodic defect of platelet function. Brit J Haematol. 1993;85:718–22.

    Article  CAS  Google Scholar 

  21. Bernard J, Caen J. Purpura thrombopénique et megacaryocytopénie cycliques mensuels. Nouv Rev franc Hémat. 1962;2:378–86.

    CAS  PubMed  Google Scholar 

  22. Dan K, Inokuchi K, An E, Nomura T. Cell mediated cyclic thrombocytopenia treated with azathioprine. Brit J Haematol. 1991;77:365–79.

    Article  CAS  Google Scholar 

  23. Engstrom K, Lundquist A, Soderstrom N. Periodic thrombocytopenia or tidal platelet dysgenesis in a man. Scand J Haemat. 1966;3:290–2.

    Article  CAS  PubMed  Google Scholar 

  24. Hoffman R, Bridell RA, van Besien K, Srour EF, Guscar T, Hudson NW, et al. Acquired cyclic amegakaryocytic thrombocytopenia associated with an immunoglobulin blocking the action of granulocyte-macrophage colony-stimulating factor. N Engl J Med. 1989;321:97–102.

    Article  CAS  PubMed  Google Scholar 

  25. Lewis ML. Cyclical Thrombocytopenia: a thrombopoietin deficiency? J Clin Path. 1974;27:242–6.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  26. Kimura F, Nakamura Y, Sato K, Wakimoto N, Kato T, Tahara T, et al. Cyclic change of cytokines in a patient with cyclic thrombocytopenia. Br J Haemat. 1996;94:171–4.

    Article  CAS  Google Scholar 

  27. Santillán M, Bélair J, Mahaffy JM, Mackey MC. Regulation of platelet production: the normal response to perturbation and cyclical platelet disease. J Theor Biol. 2000;206:585–603. doi:10.1006/jtbi.2000.2149.

    Article  PubMed  Google Scholar 

  28. Ranlov P, Videbaek A. Cyclic haemolytic anaemia synchronous with Pel-Ebstein fever in a case of Hodgkin’s disease. Acta Medica Scandinavica. 1963;100.

    Google Scholar 

  29. Gordon RR, Varadi S. Congenital hypoplastic anemia (pure red-cell anemia) with periodic erythroblastopenia. Lancet. 1962;1:296–9.

    Article  CAS  PubMed  Google Scholar 

  30. Gurney CW, Simmons EL, Gaston EO. Cyclic erythropoiesis in W/Wv mice following a single small dose of 89Sr. Exp Hemat. 1981;9:118–22.

    CAS  PubMed  Google Scholar 

  31. Gibson CM, Gurney CW, Gaston EO, Simmons EL. Cyclic erythropoiesis in the S1/S1d mouse. Exp Hemat. 1984;12:343–8.

    CAS  PubMed  Google Scholar 

  32. Gibson CM, Gurney CW, Simmons EL, Gaston EO. Further studies on cyclic erythropoiesis in mice. Exp Hemat. 1985;13:855–60.

    CAS  PubMed  Google Scholar 

  33. Vácha J, Znojil V. Application of a mathematical model of erythropoiesis to the process of recovery after acute X-irradiation of mice. Biofizika. 1975;20:872–9.

    PubMed  Google Scholar 

  34. Vácha J. Postirradiational oscillations of erythropoiesis in mice. Acta Sc Nat Brno. 1982;16(2):1–52.

    Google Scholar 

  35. Orr JS, Kirk J, Gray KG, Anderson JR. A study of the interdependence of red cell and bone marrow stem cell populations. Br J Haemat. 1968;15:23–4.

    Article  CAS  Google Scholar 

  36. Mackey MC. Periodic auto-immune hemolytic anemia: an induced dynamical disease. Bull Math Biol. 1979;41:829–34.

    Article  CAS  PubMed  Google Scholar 

  37. O’Dwyer M, Druker BJ, Mauro M, Talpaz M, Resta D, Peng B, et al. STI571: a tyrosine kinase inhibitor for the treatment of CML. Ann Oncol. 2000;11:155.

    Article  Google Scholar 

  38. Melo J. The diversity of BCR-ABL fusion proteins and their relationship to leukemia phenotype. Blood. 1996;88:2375.

    CAS  PubMed  Google Scholar 

  39. Grignani F. Chronic myelogenous leukemia. Crit Rev Oncol Hematol. 1985;4:31–66.

    Article  CAS  PubMed  Google Scholar 

  40. Henderson ES, Lister TA, Greaves MF, editors. Leukemia. Philadelphia: Saunders; 1996.

    Google Scholar 

  41. Foley C, Mackey MC. Dynamic hematological disease: a review. J Math Biol. 2009;58:285–322.

    Article  PubMed  Google Scholar 

  42. Wichard ZL, Sarkar CA, Kimmel M, Corey SJ. Hematopoiesis and its disorders: a systems biology approach. Blood. 2011;115:2339–47.

    Article  Google Scholar 

  43. Mackey MC, Haurie C, Bélair J. Cell replication and control. In: Beuter A, Glass L, Mackey MC, Titcombe MS, editors. Nonlinear dynamics in physiology and medicine. New York: Springer; 2003, pp. 233–69.

    Chapter  Google Scholar 

  44. Lei J, Mackey MC. Multistability in an age-structured model of hematopoiesis: cyclical neutropenia. J Theor Biol. 2011;270:143–53.

    Article  PubMed  Google Scholar 

  45. Mackey MC. Unified hypothesis for the origin of aplastic anemia and periodic haematopiesis. Blood. 1978;51:941–56.

    CAS  PubMed  Google Scholar 

  46. Mackey MC. Mathematical models of hematopoietic cell replication and control. In: Othmer H, Adler F, Lewis M, Dallon J, editors. Case studies in mathematical modeling-ecology, physiology, and cell biology. New Jersey: Prentice-Hall; 1996. pp. 149–78.

    Google Scholar 

  47. Hoffbrand AV, Pettit JE, Moss PAH. Essential haematology. 4th ed. Milan: Blackwell Science; 2011.

    Google Scholar 

  48. Mahaffy JM, Bélair J, Mackey MC. Hematopoietic model with moving boundary condition and state dependent delay: application in erythropoiesis. J Theor Biol. 1998;190:135–46.

    Article  CAS  PubMed  Google Scholar 

  49. Bélair J. Mackey MC. A model for the regulation of mammalian platelet. Ann N Y Acad Sci. 1987;504:280–2.

    Article  Google Scholar 

  50. Adamson JW. The relationship of erythropoietin and iron metabolism to red blood cell production in humans. Semin Oncol. 1994;2:9–15.

    Google Scholar 

  51. Price TH, Chatta GS, Dale DC. Effect of recombinant granulocyte colony-stimulating factor on neutrophil kinetics in normal young and elderly humans. Blood. 1996;88:335–40.

    CAS  PubMed  Google Scholar 

  52. Ratajczak MZ, Ratajczak J, Marlicz W, Jr Pletcher WC, Machalinshi B, Moore J, et al. Recombinant human thrombopoietin(TPO) stimulates erythropoiesis by inhibiting erythroid progenitor cell apoptosis. Br J Haematol. 1997;98:8–17.

    Article  CAS  PubMed  Google Scholar 

  53. Tanimukai S, Kimura T, Stakabe H, Ohmizono Y, Kato T, Miyazaki H, et al. Recomninant human c-Mpl ligand (thrombopoietin) not only acts on megakaryocyte progenitors, but also on erythroid and multipotential progenitors in vitro. Exp Hematol. 1997;25:1025–33.

    CAS  PubMed  Google Scholar 

  54. Silva M, Grillot D, Benito A, Richard C, Nunez G, Fernandez-Luna J. Erythropoietin can promote erythroid progenitor survival by repressing apoptosis through bcl-1 and bcl-2. Blood. 1996;88:1576–82.

    CAS  PubMed  Google Scholar 

  55. Ritchie A, Gotoh A, Gaddy J, Braun S, Broxmeyer H. Thrombopoietin upregulates the promoter conformation of p53 in a proliferation-independent manner coincident with a decreased expression of bax: potential mechanisms for survival enhancing effects. Blood. 1997;90:4394–402.

    CAS  PubMed  Google Scholar 

  56. Kaushansky K, Lin N, Grossmann A, Humes J, Sprugel K, Broudy V. Thrombopoietin expands erythoid, granulocyte-macrophage, and megakaryocyte progenitor cells in normal and myelosuppressed mice. Exp Hematol. 1996;24:256–69.

    Google Scholar 

  57. Kearns CM, Wang WC, Stute N, Ihle J, Evans W. Disposition of recombinant human granulocyte colony-stimulating factor in children with severe chronic neutropenia. J Pediatr. 1993;123(3):471–9.

    Article  CAS  PubMed  Google Scholar 

  58. Mempel K, Pietsch T, Menzel T, Zeidler C, Welte K. Increased serum levels of granulocyte colony stimulating factor in patients with severe congenital neutropenia. Blood. 1991;77:1919–22.

    CAS  PubMed  Google Scholar 

  59. Takatani H, Soda H, Fukuda M, Watanabe M, Kinoshita A, Nakamura T, et al. Levels of recombinant human granulocyte colony stimulating factor in serum are inversely correlated with circulating neutrophil counts. Antimicrob Agents Chemother. 1996;40:988–91.

    CAS  PubMed Central  PubMed  Google Scholar 

  60. Watari K, Asano S, Shirafuji N, Kodo H, Ozawa K, Takaku F, et al. Serum granulocyte colony stimulating factor levels in healthy volunteers and patients with various disorders as estimated by enzyme immunoassay. Blood. 1989;73:117–22.

    CAS  PubMed  Google Scholar 

  61. Roeder I. Quantitative stem cell biology: computational studies in the hematopoietic system. Curr Opin Hematol. 2006;13:222–8.

    Article  PubMed  Google Scholar 

  62. Viswanathan S, Zandstra PW. Towards predicitive models of stem cell fate. CytoTechnol. 2003;41:75–92.

    Article  CAS  Google Scholar 

  63. Bélair J, Mackey MC, Mahaffy JM. Age-structuredand two-delay models for erythropoiesis. Math Biosci. 1995;128:317–46.

    Article  PubMed  Google Scholar 

  64. Colijn C, Mackey MC. A mathematical model of hematopoiesis: I. periodic chronic myelogenous leukemia. J Theor Biol. 2005;237:117–32.

    Article  PubMed  Google Scholar 

  65. Foley C, Mackey MC. Mathematical model for G-CSF administration after chemotherapy. J Theor Biol. 2009;257:27–44.

    Article  CAS  PubMed  Google Scholar 

  66. Brooks G, Provencher-Langlois G, Lei J, Mackey MC. Neutrophil dynamics after chemotherapy and G-CSF: the role of pharmacokinetics in shaping the response. J Theor Biol. 2012;315:97–109.

    Article  CAS  PubMed  Google Scholar 

  67. Colijn C, Dale DC, Foley C, Mackey MC. Observations on the pathophysiology and mechanisms for cyclic neutropenia. Math Model Nat Phenomena. 2006;1:45–69.

    Article  Google Scholar 

  68. Dunn CDR. Cyclical hematopoiesis: the biomathematics. Exp Hematol. 1983;11:779–91.

    CAS  PubMed  Google Scholar 

  69. Fisher G. An introduction to chaos theory and some haematological applications. Comp Haematol Int. 1993;3:43–51.

    Article  Google Scholar 

  70. Kazarinoff ND, van den Driessche P. Control of oscillations in hematopoiesis. Science. 1979;203:1348–50.

    Article  CAS  PubMed  Google Scholar 

  71. King-Smith EA, Morley A. Computer simulation of granulopoiesis: normal and impaired granulopoiesis. Blood. 1970;36:254–62.

    CAS  PubMed  Google Scholar 

  72. MacDonald N. Cyclical neutropenia: models with two cell types and two time lags. In: Valleron AJ, Macdonald PDM, editors. Biomathematics and cell kinetics. Amsterdam: Elsevier; 1978. pp. 287–95.

    Google Scholar 

  73. Morley A. A platelet cycle in normal individuals. Aust Ann Med. 1969;18:127–9.

    CAS  PubMed  Google Scholar 

  74. Morley A. Blood-cell cycles in polycythaemia vera. Aust Ann Med. 1969;18:124.

    CAS  PubMed  Google Scholar 

  75. Morley A, Stohlman F. Cyclophosphamide induced cyclical neutropenia. N Engl J Med. 1970;282:643–6.

    Article  CAS  PubMed  Google Scholar 

  76. Morley A. Cyclic hemopoiesis and feedback control. Blood Cells.1979;5:283–96.

    CAS  PubMed  Google Scholar 

  77. Reeve J. An analogue model of granulopoiesis for the analysis of isotopic and other data obtained in the non-steady state. Br J Haematol. 1973;25:15–32.

    Article  CAS  PubMed  Google Scholar 

  78. Schmitz S. Ein mathematisches Modell der zyklischen Haemopoese. Ph.D thesis, Universitat Koln; 1988.

    Google Scholar 

  79. Schmitz S, Franke H, Brusis J, Wichmann HE. Quantification of the cell kinetic effects of G-CSF using a model of human granulopoiesis. Exp Hematol. 1993;21:755–60.

    CAS  PubMed  Google Scholar 

  80. Schmitz S, Franke H, Loeffler M, Wichmann HE, Diehl V. Reduced variance of bone-marrow transit time of granulopoiesis: a possible pathomechanism of human cyclic neutropenia. Cell Prolif. 1994;27:655–67.

    Article  Google Scholar 

  81. Schmitz S, Loeffler M, Jones JB, Lange RD, Wichmann HE. Synchrony of bone marrow proliferation and maturation as the origin of cyclic haemopoiesis. Cell Tissue Kinet. 1990;23:425–41.

    CAS  PubMed  Google Scholar 

  82. Schmitz S, Franke H, Wichmann HE, Diehl V. The effect of continuous G-CSF application in human cyclic neutropenia: a model analysis. Br J Haematol. 1995;90:41–7.

    Article  CAS  PubMed  Google Scholar 

  83. Shvitra D, Laugalys R, Kolesov YS. Mathematical modeling of the production of white blood cells. In: Marchuk G, Belykh LN, editors. Mathematical modeling in immunology and medicine. Amsterdam: North-Holland; 1983. pp. 211–23.

    Google Scholar 

  84. von Schulthess GK, Mazer NA. Cyclic neutropenia(CN): a clue to the control of granulopoiesis. Blood. 1982;59:27–37.

    CAS  PubMed  Google Scholar 

  85. Wichmann HE, Loeffler M, Schmitz S. A concept of hemopoietic regulation and its biomathematical realization. Blood Cells.1988; 14:411–29.

    CAS  PubMed  Google Scholar 

  86. Hearn T, Haurie C, Mackey MC. Cyclical neutropenia and the peripherial control of white blood cell production. J Theor Biol. 1998;192:167–81.

    Article  CAS  PubMed  Google Scholar 

  87. Bernard S, Bélair J, Mackey MC. Oscillations in cyclical neutropenia: new evidence based on mathematical modeling. J Theor Biol. 2003;223:283–98. doi:10.1016/S0022-5193(03)00090-0.

    Article  PubMed  Google Scholar 

  88. Colijn C, Foley C, Mackey MC. G-CSF treatment of canine cyclical neutropeina: a comprehensive mathematical model. Exp Hematol. 2007;37:898–907.

    Article  Google Scholar 

  89. Santillan M, Mahaffy JM, Bélair J, Mackey MC. Regulation of platelet productin: the normal response to perturbation and cyclical platelet disease. J Theor Biol. 2000;206:585–603.

    Article  CAS  PubMed  Google Scholar 

  90. Zhuge C, Lei J, Mackey MC. Neutrophil dynamics in response to chemotherapy and G-CSF. J Theor Biol. 2012;293:111–20.

    Article  CAS  PubMed  Google Scholar 

  91. Hannun Y. Apoptosis and the dilemma of cancer chemotherapy. Blood. 1997;89:1845–53.

    CAS  PubMed  Google Scholar 

  92. Hammond WP, Price TH, Souza LM, Dale DC. Treatment of cyclic neutropenia with granulocyte colony stimulating factor. N Engl J Med. 1989;320:1306–11.

    Article  PubMed  Google Scholar 

  93. Koury MJ. Programmed cell death(apoptosis) in hematopoiesis. Exp Hematol. 1992;20:391–4.

    CAS  PubMed  Google Scholar 

  94. Park JR. Cytokine regulation of apoptosis in hematopoietic precursor cells. Curr Opin Hematol. 1996;3:191–6.

    Article  CAS  PubMed  Google Scholar 

  95. Migliaccio AR, Migliaccio G, Dale DC, Hammond WP. Hematopoietic progenitors in cyclic neutropenia: effect of granulocyte colony stimulating factor in vivo. Blood. 1990;75:1951–9.

    CAS  PubMed  Google Scholar 

  96. Williams G, Smith C. Molecular regulation of apoptosis: genetic controls on cell death. Cell. 1993;74:777–9.

    Article  CAS  PubMed  Google Scholar 

  97. Crawford J, Dale DC, Lyman GH. Chemotherapy-induced neutropenia: risks, consequences, and new directions for its management. Cancer. 2003;100:228–37.

    Article  Google Scholar 

  98. Ozer H, Armitage JO, Bennett CL, et al. Update of recommendations for the use of hematopoietic colony-stimulating factors: evidence-based, clinical practice guidelines. J Clin Oncol. 2000;18:3558–85.

    CAS  PubMed  Google Scholar 

  99. Clark OA, Lyman GH, Castro AA, Clark LG, Djulbegovic B. Colony-stimulating factors for chemotherapy-induced febrile neutropenia: a meta-analysis of randomized controlled trials. J Clin Oncol. 2005;23:4198–214.

    Article  CAS  PubMed  Google Scholar 

  100. Bennett CL, Weeks JA, et al. Use of hematopoietic colony-stimulating factors: comparison of the 1994 and 1997 American Society of Clinical Oncology surveys regarding ASCO clinical practice guidelines. Health Services Research Committee of the American Society of Clinical Oncology. J Clin Oncol. 1999;17:3676–81.

    CAS  PubMed  Google Scholar 

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Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council (NSERC, Canada) and the Mathematics of Information Technology and Complex Systems (MITACS, Canada), and carried out in Beijing and Montreal.

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Authors and Affiliations

  1. Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084, China

    Jinzhi Lei

  2. Department of Physiology and CAMBAM, Room 1124, 3655 Promenade Sir William Osler, Monteral, QC H3G 1Y6, Canada

    Michael C. Mackey

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  1. Jinzhi Lei
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  2. Michael C. Mackey
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Correspondence to Michael C. Mackey .

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Editors and Affiliations

  1. Departments of Pediatrics and Cell & Molecular Biology, Northwestern University Feinberg School of Medicine and Lurie Children’s Hospital of Chicago, Chicago, Illinois, USA

    Seth Joel Corey

  2. Department of Statistics, Rice University, Houston, Texas, USA

    Marek Kimmel

  3. Northwestern University Comprehensive Cancer Center, Evanston, Illinois, USA

    Joshua N. Leonard

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Lei, J., Mackey, M. (2014). Understanding and Treating Cytopenia Through Mathematical Modeling. In: Corey, S., Kimmel, M., Leonard, J. (eds) A Systems Biology Approach to Blood. Advances in Experimental Medicine and Biology, vol 844. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2095-2_14

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