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A reaction–diffusion model for long bones growth

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Abstract

Bone development is characterized by differentiation and growth of chondrocytes from the proliferation zone to the hypertrophying one. These two cellular processes are controlled by a complex signalling regulatory loop between different biochemical signals, whose production depends on the current cell density, constituting a coupled cell-chemical system. In this work, a mathematical model of the process of early bone growth is presented, extending and generalizing other earlier approaches on the same topic. A reaction–diffusion regulatory loop between two chemical factors: parathyroid hormone-related peptide (PTHrP) and Indian hedgehog (Ihh) is hypothesized, where PTHrP is activated by Ihh and inhibits Ihh production. Chondrocytes proliferation and hypertrophy are described by means of population equations being both regulated by the PTHrP and Ihh concentrations. In the initial stage of bone growth, these two cellular proceses are considered to be directionally dependent, modelling the well known column cell formation, characteristic of endochondral ossification. This coupled set of equations is solved within a finite element framework, getting an estimation of the chondrocytes spatial distribution, growth of the diaphysis and formation of the epiphysis of a long bone. The results obtained are qualitatively similar to the actual physiological ones and quantitatively close to some available experimental data. Finally, this extended approach allows finding important relations between the model parameters to get stability of the physiological process and getting additional insight on the spatial and directional distribution of cells and paracrine factors.

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References

  • Baker RE, Maini PK (2007) A mechanism for morphogen-controlled domain growth. J Math Biol 54(5): 597–622

    Article  MATH  MathSciNet  Google Scholar 

  • Ballock RT, O’Keefe RJ (2003) The biology of the growth plate. J Bone Joint Surg Am 85-A(4): 715–726

    Google Scholar 

  • Brouwers JEM, van Donkelaar CC, Sengers BG, Huiskes R (2006) Can the growth factors PTHrP, Ihh and VEGF, together regulate the development of a long bone?. J Biomech 39(15): 2774– 2782

    Article  Google Scholar 

  • Bailón-Plaza A, van der Meulen MC (2001) A mathematical framework to study the effects of growth factor influences on fracture healing. J Theor Biol 212(2): 191–209

    Article  Google Scholar 

  • Carter DR, Van der Meulen MCH, Beaupré GS (1996) Mechanical factors in bone growth and development. Bone 18(1): 5–10

    Article  Google Scholar 

  • Carter DR, Mikic B, Padian K (1998) Epigenetic mechanical factors in the evolution of long bone epiphyses. Zool J Linn Soc 123(2): 163–178

    Article  Google Scholar 

  • Chitty LS, Altman DG (2002) Charts of fetal size: limb bones. BJOG Int J Obstet Gynecol 109: 919–929

    Article  Google Scholar 

  • Chung U, Schipani E, McMahon AP, Kronenberg HM (2001) Indian Hedgehog couples chondrogenesis to osteogenesis in endochondral bone development. J Clin Invest 107(3): 295–304

    Article  Google Scholar 

  • Crampin EJ, Maini PK (2001) Reaction–diffusion models for biological pattern formation. Methods Appl Anal 8(3): 415–428

    MATH  MathSciNet  Google Scholar 

  • Crampin EJ, Hackborn WW, Maini PK (2002) Pattern formation in reaction–diffusion models with nonuniform domain growth. Bull Math Biol 64: 747–769

    Article  Google Scholar 

  • Darling EM, Zauscher S, Guilak F (2006) Viscoelastic properties of zonal articular chondrocytes measured by atomic force microscopy. Osteoarthr Cartil 14: 571–579

    Article  Google Scholar 

  • de Crombrugghe B, Lefebvre V, Nakashima K (2001) Regulatory mechanisms in the pathways of cartilage and bone formation. Curr Opin Cell Biol 13(6): 721–728

    Article  Google Scholar 

  • Dillon R, Othmer HG (1999) A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud. J Theor Biol 197(3): 295–330

    Article  Google Scholar 

  • Farnum CE, Lee R, Ohara K, Urban JPG (2002) Volume increase in growth plate chondrocytes during hypertrophy: the contribution of organic osmolytes. Bone 30(4): 574–581

    Article  Google Scholar 

  • Forriol F, Shapiro F (2005) Bone development. Clin Orthop Relat Res 432: 14–33

    Article  Google Scholar 

  • Gao B, Guo J, She C, Shu A, Yang M, Tan Z, Yang X, Guo S, Feng G, He L (2001) Mutations in IHH, encoding Indian hedgehog, cause brachydactyly type A-1. Nat Genet 28(4): 386–388

    Article  Google Scholar 

  • Ganong WF (2003) Review of medical physiology (LANGE Basic Science), 21st edn. McGraw-Hill Medical, New York

    Google Scholar 

  • García-Aznar JM, Kuiper JH, Gómez-Benito MJ, Doblaré M, Richardson JB (2007) Computational simulation of fracture healing: influence of interfragmentary movement on the callus growth. J Biomech 40(7): 972–980

    Article  Google Scholar 

  • Griffiths SF (2000) Developmental biology, 6th edn. Sinauer Associates Inc, Sunderland

    Google Scholar 

  • Grimsrud CD, Romano PR, D’Souza M, Edward J, Schwarz EM, Reynolds PR, Roiser RN, O’Keefe RJ (2001) BMP signaling stimulates chondrocyte maturation and the expression of Indian hedgehog. J Orthop Res 19(1): 18–25

    Article  Google Scholar 

  • Gómez-Benito MJ, García-Aznar JM, Kuiper JH, Doblaré M (2005) Influence of fracture gap size on the pattern of long bone healing: a computational study. J Theor Biol 235(1): 105–119

    Article  Google Scholar 

  • Henderson JH, Carter DR (2002) Mechanical induction in limb morphogenesis: the role of growth-generated strains and pressures. Bone 31(6): 645–653

    Article  Google Scholar 

  • Herrero MA, Lopez JM (2005) Bone formation: biological aspects and modelling problems. J Theor Med 6(1): 41–55

    MATH  MathSciNet  Google Scholar 

  • Hibbit, Karlsson, Sorensen, (2004) Inc. Abaqus user’s Manual, v. 6.5. HKS inc. Pawtucket, RI, USA

  • Horton WA (1990) The biology of bone growth. Growth Genet Horm 6(2): 1–5

    MathSciNet  Google Scholar 

  • Holzapfel GA (2000) Nonlinear solid mechanics. A continuum approach for engineering. Wiley, New York

    MATH  Google Scholar 

  • Hughes TJR (2000) The finite element method—linear static and dynamic finite element analysis. Dover Publishers, New York

    Google Scholar 

  • Hunzinker EB, Schenk RK, Cruz-Orive LM (2006) Quantification of chondrocyte perfomance in growth-plate cartilage during longitudinal bone growth. Osteoarthr Cartil 14: 571–579

    Article  Google Scholar 

  • Jeanty P, Cousaert E, Cantraine E, Hobbins JC, Tack B, Struyven JA (1984) A longitudinal study of fetal limb growth. Am J Perinatol 1(2): 136–144

    Article  Google Scholar 

  • Johnsen SL, Wilsgaard T, Rasmussen S, Sollien R, Kiserud T (2005) Longitudinal reference charts for growth of the fetal head, abdomen and femur. Eur J Obstet Gynecol Reprod Biol 127(2): 286–297

    Google Scholar 

  • Karp S, Schipani E, St-Jacques B, Hunzelman J, Kronenberg H, McMahon AP (2000) Indian Hedgehog coordinates endochondral bone growth and morphogenesis via Parathyroid Hormone related-Protein-dependent and -independent pathways. Development 127: 543–548

    Google Scholar 

  • Kindblom JM, Nilsson O, Hurme T, Ohlsson C, Savendahl L (2002) Expression and localization of Indian Hedgehog (Ihh) and parathyroid hormone related protein (PTHrP) in the human growth plate during pubertal development. J Endocrinol 174: R1–R6

    Article  Google Scholar 

  • Kobayashi T, Soegiarto DW, Yang Y, Lanske B, Schipani E, McMahon AP, Kronenberg HM (2005) Indian Hedgehog stimulates periarticular chondrocyte differentiation to regulate growth plate length independently of PTHrP. J Clin Invest 115: 1734–1742

    Article  Google Scholar 

  • Kronenberg HM (2003) Development regulation of the growth plate. Nature 423: 332–336

    Article  Google Scholar 

  • Li T, Dong Y, Ionescu A, Rosier R, Zuscik M, Scwarz E, O’keefe R, Drissi H (2004) Parathyroid hormone-related peptide (PTHrP) inhibits RUNX2 expression through the PKA signaling pathway. Exp Cell Res 299(1): 128–136

    Article  Google Scholar 

  • Liang S, Xu J, Weng L, Dai H, Zhang X, Zhang L (2006) Protein diffusion in agorase hydrogel in situ measured by improved refractiveindex method. J Control Release 115(2): 189–196

    Article  Google Scholar 

  • Long F, Joenga K, Xuand S, Efstratiadisd A, McMahone AP (2006) Independent regulation of skeletal growth by Ihhnext term and IGF signaling. Dev Biol 298(1): 327–333

    Article  Google Scholar 

  • Madzvamuse A (2000) A numerical approach to the study of spatial pattern formation. PhD thesis

  • Madzvamuse A (2006) Time-stepping schemes for moving grid finite elements applied to reaction–diffusion systems on fixed and growing domains. J Comput Phys 214: 239–263

    Article  MATH  MathSciNet  Google Scholar 

  • Madzvamuse A, Maini PK, Wathen AJ (2005) A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J Sci Comput 24: 247–262

    Article  MATH  MathSciNet  Google Scholar 

  • Maini PK (2004) Using mathematical models to help understand biological pattern formation. C R Biol 327: 225–234

    Article  Google Scholar 

  • Marsden JE, Hughes TJ (1983) Mathematical foundations of elasticity. Courier Dover Publications, USA

    MATH  Google Scholar 

  • Menzel A (2006) A fibre reorientation model for orthotropic multiplicative growth: Configurational driving stresses, kinematics-based reorientation, and algorithmic aspects. Biomech Model Mechanobiol 6(5): 303–320

    Article  Google Scholar 

  • Murray JD (2001) Mathematical biology I. Springer, New York

    Google Scholar 

  • Mosley JR, Lanyon LE (2002) Growth rate rather than gender determines the size of the adaptive response of the growing skeleton to mechanical strain. Bone 30(1): 314–319

    Article  Google Scholar 

  • Nowlan NC, Murphy P, Prendergast PJ (2007) Mechanobiology of embryonic limb development. Ann N Y Acad Sci 1101: 389–411

    Article  Google Scholar 

  • Nowlan NC, Murphy P, Prendergast PJ (2008) A dynamic pattern of mechanical stimulation promotes ossification in avian embryonic long bones. J Biomech 41(2): 249–258

    Article  Google Scholar 

  • Ohashi N, Robling AG, Burr DB, Turner CH (2002) The effects of dynamic axial loading on the rat growth plate. J Bone Miner Res 17(2): 284–292

    Article  Google Scholar 

  • Pathia S, Rutenberga JB, Johnson RL, Vortkamp A (1999) Interaction of Ihh term and BMP/Noggin signaling during cartilage differentiation. Dev Biol 209(2): 239–253

    Article  Google Scholar 

  • Philpott C, Rahman N, Kenny N, Undo T, Young R, Barisas G, Roess D (1995) Rotational dynamics of leutinizing hormone receptors and MHC class I antigens on murine Leydig Cells. Biochim Biophys Acta 1235(1): 62–68

    Article  Google Scholar 

  • Provot S, Schipani E (2005) Molecular mechanisms of endochondral bone development. Biochem Biophys Res Commun 328(3): 658–665

    Article  Google Scholar 

  • Robling AG, Duijvelaar KM, Geevers JV, Ohashi N, Turner CH (2001) Modulation of appositional and longitudinal bone growth in the rat ulna by applied static and dynamic force. Bone 29(2): 105–113

    Article  Google Scholar 

  • Sadler T (2004) Langman’s medical embryology, 9 edn. Lippincott Williams & Wilkins, Baltimore

    Google Scholar 

  • Shefelbine SJ (2007) Effect of mechanical loading on bone morphology during growth. J Morphol 268: 1133

    Google Scholar 

  • Skalak R, Dasgupta G, Moss M, Otten E, Dullemeijer P, Vilmann H (1982) Analytical description of growth. J Theor Biol 94: 555–577

    Article  MathSciNet  Google Scholar 

  • Stevens SS, Beaupré GS, Carter DR (1999) Computer model of endochondral growth and ossification in long bones: biological and mechanobiological influences. J Orthop Res 17: 646–653

    Article  Google Scholar 

  • Stokes IAF, Clark K, Farnum CE, Aronsson DD (2007) Alterations in the growth plate associated with growth modulation by sustained compression or distraction. Bone 41: 197–205

    Article  Google Scholar 

  • Tiet TD, Hopyan S, Nadesan P, Gokgoz N, Poon R, Lin AC, Yan T, Andrulis IL, Alman BA, Wunder JS (2006) Constitutive Hedgehog signaling in chondrosarcomera up-regulates tumor cell proliferation. Am J Phisiol 168(1): 321–330

    Google Scholar 

  • van Donkelaar CC, Huiskes R (2007) The PTHrP–Ihh feedback loop in the embryonic growth plate allows pthrp to control hypertrophy and ihh to regulate proliferation. Biomech Model Mechanobiol 6(1–2): 55–62

    Article  Google Scholar 

  • Wilsman NJ, Farnum CE, Leiferman EM, Fry M, Barreto C (1996) Differential growth by growth plates as a function of multiple parameters of chondrocytic kinetics. J Orthop Res 14(6): 927–936

    Article  Google Scholar 

  • Zelzer E, Olsen B (2003) The genetic basis for skeletal diseases. Nature 423: 343–348

    Article  Google Scholar 

  • Zylan T, Murshid KW (2003) An assesment of femur growth parameters in human fetuses and their relationship to gestational age. Turk J Med Sci 33: 27–32

    Google Scholar 

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Authors and Affiliations

  1. Group of Structural Mechanics and Materials Modelling, Aragón Institute of Engineering Research (I3A), University of Zaragoza, Zaragoza, Spain

    D. A. Garzón-Alvarado, J. M. García-Aznar & M. Doblaré

  2. CIBER-BBN Centro de Investigación Biomédica en Red, Bioingeniería, Biomateriales y Nanomedicina, Zaragoza, Spain

    D. A. Garzón-Alvarado, J. M. García-Aznar & M. Doblaré

  3. Department of Mechanical and Mechatronic Engineering, National University of Colombia, Bogotá, Colombia

    D. A. Garzón-Alvarado

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  1. D. A. Garzón-Alvarado
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  2. J. M. García-Aznar
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  3. M. Doblaré
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Correspondence to J. M. García-Aznar.

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Garzón-Alvarado, D.A., García-Aznar, J.M. & Doblaré, M. A reaction–diffusion model for long bones growth. Biomech Model Mechanobiol 8, 381–395 (2009). https://doi.org/10.1007/s10237-008-0144-z

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  • DOI: https://doi.org/10.1007/s10237-008-0144-z

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