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How do bones grow? A mathematical description of the mechanobiological behavior of the epiphyseal plate

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Abstract

Growth modulation is an emerging method for the treatment of skeletal deformities originating in the long bones or the vertebral bodies. It requires the controlled application of mechanical loads to the affected bone, causing an alteration of the growth and ossification process occurring in a cartilaginous region called epiphyseal growth plate or physis. In order to avoid the possibility of under- or over-correction, quantification of the applied forces is necessary. Pursuing this goal, here we propose a phenomenological model of mechanobiological effects on the epiphyseal growth plate, based on the observed similarity between the mechanobiologically induced growth and viscoelastic material behavior. The model incorporates mechanical loading effects on growth direction, growth rate and ossification speed; it also allows to evaluate the occurrence of transient effects. Model consistency was tested against a rather large set of experiments existing in the literature. A generic simplified geometrical model of bones was established for this. Analytical solutions for growth and ossification evolution were obtained for different loading conditions, allowing to test the ability of the model to describe bone growth under various kinds of mechanical loading conditions. Model-predicted changes regarding epiphyseal growth plate thickness as well as longitudinal growth speed are consistent with experiments in which static tension or compression were applied to long bones. Results suggest that when the mechanical load is sinusoidally variable, conflicting data existing in the literature could be explained by a previously unconsidered effect of the the applied load initial phase. The model can accurately fit data regarding torsional loads effects on growth. Mechanobiological data for humans is very scarce. For this reason, when possible, the model parameters values were estimated, for the proposed generic geometry, after growth measurements in animal models available in the literature. Although it is not possible to assert their validity for humans, the proposed model along with the obtained parameters values give a rational foundation to be used in more advanced computational studies.

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References

  • Abad V, Meyers JL, Weise M et al (2002) The role of the resting zone in growth plate chondrogenesis. Endocrinology 143(5):1851–1857. https://doi.org/10.1210/endo.143.5.8776

    Article  Google Scholar 

  • Alberty A, Peltonen J, Ritsilä V (1993) Effects of distraction and compression on proliferation of growth plate chondrocytes: a study in rabbits. Acta Orthop Scand 64(4):449–455. https://doi.org/10.3109/17453679308993665

    Article  Google Scholar 

  • Alonso M, Bertolino G, Yawny A (2020) Mechanobiological based long bone growth model for the design of limb deformities correction devices. J Biomech 109:109–905. https://doi.org/10.1016/j.jbiomech.2020.109905

    Article  Google Scholar 

  • Alonso M, Yawny A, Bertolino G (2021) A tool for solving bone growth related problems using finite elements adaptive meshes. J Mech Behav Biomed Mater. https://doi.org/10.1016/j.jmbbm.2021.104946

    Article  Google Scholar 

  • Anderson M, Green W, Messner M (1963) Growth and predictions of growth in the lower extremities. J Bone Joint Surg 45:1–14

    Article  Google Scholar 

  • Arkin AM, Katz JF (1956) The effects of pressure on epiphyseal growth; the mechanism of plasticity of growing bone. J Bone Joint Surg Am 38–A(5):1056–1076

    Article  Google Scholar 

  • Beaupré GS, Orr TE, Carter DR (1990) An approach for time-dependent bone modeling and remodeling. application: a preliminary remodeling simulation. J Orthop Res 8:662–670

    Article  Google Scholar 

  • Bonnel F, Dimeglio A, Baldet P et al (1984) Biomechanical activity of the growth plate. Clin Anat 6:53–61

    Article  Google Scholar 

  • Burdan F, Szumiło J, Korobowicz A et al (2009) Morphology and physiology of the epiphyseal growth plate. Folia Histochem Cytobiol. https://doi.org/10.2478/v10042-009-0007-1

    Article  Google Scholar 

  • Bursac P, Obitz TW et al (1999) Confined and unconfined stress relaxation of cartilage: appropiateness of a transversely isotropic analysis. J Biomech 32:1125–1130

    Article  Google Scholar 

  • Bush PG, Hall AC, Macnicol MF (2008) New insights into function of the growth plate. J Bone Joint Surg Br 90–B(12):1541–1547. https://doi.org/10.1302/0301-620x.90b12.20805

    Article  Google Scholar 

  • Cancel M, Grimard G, Thuillard-Crisinel D et al (2009) Effects of in vivo static compressive loading on aggrecan and type II and x collagens in the rat growth plate extracellular matrix. Bone 44(2):306–315. https://doi.org/10.1016/j.bone.2008.09.005

    Article  Google Scholar 

  • Carrier J, Aubin C, Villemure I et al (2004) Biomechanical modelling of growth modulation following rib shortening or lenghtening in adolescent idiopathic scoliosis. Med Bio Eng Comput 42:541–548

    Article  Google Scholar 

  • Carriero A, Jonkers I, Shefelbine S (2011) Mechanobiological prediction of proximal femoral deformities in children with cerebral palsy. Comput Methods Biomech Biomed Eng 14(3):253–262

    Article  Google Scholar 

  • Carter D, Beaupré G (2001) Skeletal function and form. Mechanobiology of skeletal development, aging and regeneration. Cambridge University Press, Cambridge

    Google Scholar 

  • Carter DR, Wong M (1988) The role of mechanical loading histories in the development of diarthrodial joints. J Orthop Res 6(6):804–816. https://doi.org/10.1002/jor.1100060604

    Article  Google Scholar 

  • Choi K, Lea Kuhn J (1990) The elastic moduli of human subchondral trabecular, and cortical bone tissue and the size-dependency of cortial bone modulus. J Biomech 23:1103–1113

    Article  Google Scholar 

  • Cobanoglu M, Cullu E, Kilimci FS et al (2016) Rotational deformities of the long bones can be corrected with rotationally guided growth during the growth phase. Acta Orthop 87(3):301–305. https://doi.org/10.3109/17453674.2016.1152450

    Article  Google Scholar 

  • Cohen B, Chorney GS, Phillips DP et al (1994) Compressive stress relaxation behavior of bobine growth plate may be described by the nonlinear biphasic theory. J Orthop Res 12:804–813

    Article  Google Scholar 

  • D’Andrea CR, Alfraihat A, Singh A et al (2021) Part 1. Review and meta-analysis of studies on modulation of longitudinal bone growth and growth plate activity: a macro-scale perspective. J Orthop Res 39(5):907–918. https://doi.org/10.1002/jor.24976

    Article  Google Scholar 

  • Fung YC (1990) Biomechanics. Springer, New York. https://doi.org/10.1007/978-1-4419-6856-2

    Book  MATH  Google Scholar 

  • Fung YC (1993) Biomechanics. Springer, New York. https://doi.org/10.1007/978-1-4757-2257-4

    Book  MATH  Google Scholar 

  • Gao J, Williams JL, Roan E (2016) Multiscale modeling of growth plate cartilage mechanobiology. Biomech Model Mechanobiol 16(2):667–679. https://doi.org/10.1007/s10237-016-0844-8

    Article  Google Scholar 

  • Giorgi M, Carriero A, Shefelbine S et al (2014) Mechanobiological simulations of prenatal joint morphogenesis. J Biomech 47:989–995

    Article  Google Scholar 

  • Giorgi M, Carriero A, Shefelbine S et al (2015) Effects of normal and abnormal loading conditions on morphogenesis of the prenatal hip joint: application to hip dysplasia. J Biomech 48:3390–3397

    Article  Google Scholar 

  • Heegaard JH, Beaupré GS, Carter DR (1999) Mechanically modulated cartilage growth may regulate joint surface morphogenesis. J Orthop Res 17(4):509–517. https://doi.org/10.1002/jor.1100170408

    Article  Google Scholar 

  • Hunziker E (1994) Mechanism of longitudinal bone growth and its regulation by growth plate chondrocites. Microsc Res Tech 28:505–519

    Article  Google Scholar 

  • Kaviani R, Londono I, Parent S, et al (2015) Growth plate cartilage shows different strain patterns in response to static versus dynamic mechanical modulation. Biomech Model Mechanobiol

  • Khatib N, Parisi C et al (2021) Differential effect of frequency and duration of mechanical loading on fetal chick cartilage and bone development. Eur Cell Mater 41:531–545. https://doi.org/10.22203/ecm.v041a34

    Article  Google Scholar 

  • Lee D, Erickson A, Dudley AT et al (2018) Mechanical stimulation of growth plate chondrocytes: previous approaches and future directions. Exp Mech 59(9):1261–1274. https://doi.org/10.1007/s11340-018-0424-1

    Article  Google Scholar 

  • Lerner AL, Kuhn JL, Hollister SJ (1998) Are regional variations in bone growth related to mechanical stress and strain parameters? J Biomech 31:327–335

    Article  Google Scholar 

  • Lin H, Aubin C, Parent S et al (2009) Mechanobiological bone growth: comparative analysis of two biomechanical modelling approaches. Med Biol Eng Compu 47:357–366

    Article  Google Scholar 

  • Ménard AL, Grimard G, Valteau B et al (2014) In vivo dynamic loading reduces bone growth without histomorphometric changes of the growth plate. J Orthop Res 32(9):1129–1136. https://doi.org/10.1002/jor.22664

    Article  Google Scholar 

  • Meng Y, Untaroiu CD (2018) A review of pediatric lower extremity data for pedestrian numerical modeling: injury epidemiology, anatomy, anthropometry, structural, and mechanical properties. Appl Bionics Biomech. Vol. 2018

  • Mikic B (1996) Epigenetic influences on long bone growth, development, and evolution. UMI

  • Mikic B, Clark RT, Battaglia TC et al (2004) Altered hypertrophic chondrocyte kinetics in gdf-5 deficient murine tibial growth plate. J Orthop Res 22:552–556

    Article  Google Scholar 

  • Moreland M (1980) Morphological effects of torsion applied to growing bone. An in vivo study in rabbits. J Bone Joint Surg Br 62(2):230–237. https://doi.org/10.1302/0301-620x.62b2.6988435

    Article  Google Scholar 

  • Nilsson O, Baron J (2004) Fundamental limits on longitudinal bone growth: growth plate senescence and epiphyseal fusion. Trends Endocrionol Metab 15:370–374

    Article  Google Scholar 

  • Peterson HA (2012) Physeal injury other than fracture. Springer, London

    Book  Google Scholar 

  • von Pfeil DJ, Decamp CE (2009) The epiphyseal plate: physiology, anatomy, and trauma. Compend Contin Educ Vet 31(8):E1–E12

    Google Scholar 

  • Piszczatowski S (2011) Material aspects of growth plate modelling using carter’s and stokes’s approaches. Acta Bioeng Biomech 13:3–13

    Google Scholar 

  • Piszczatowski S (2012) Geometrical aspects of growth plate modelling using carter’s and stokes’s approaches. Acta Bioeng Biomech 1:93–106

    Google Scholar 

  • Pous JG, Dimeglio A, Baldet P, et al (1980) Cartilage de conjugaison et croissance. Notions fondamentales en orthopédie, vol 28. Doin éditeurs

  • Radhakrishnan P, Lewis N, Mao J (2004) Zone specific micromechanical properties of the extracellular matrices of growth plate cartilage. Anal Biomed Eng 32(2):284–291

    Article  Google Scholar 

  • Robling A, Duijvelaar K, Geevers J et al (2001) Modulation of appositional and longitudinal bone growth in the rat ulna by applied static and dynamic force. Bone 29(2):105–113. https://doi.org/10.1016/s8756-3282(01)00488-4

    Article  Google Scholar 

  • Rolian C (2008) Developmental basis of limb length in rodents: evidence for multiple divisions of labor in mechanisms of endochondral bone growth. Evol Dev 10(1):15–28. https://doi.org/10.1111/j.1525-142x.2008.00211.x

    Article  Google Scholar 

  • Rozbruch SR, Ilizarov S (2007) Limb lenghthening and reconstruction surgery. Informa healthcare, New York

    Google Scholar 

  • Sadeghian SM, Lewis CL, Shefelbine SJ (2019) Predicting growth plate orientation with altered hip loading: potential cause of cam morphology. Biomech Model Mechanobiol 19(2):701–712. https://doi.org/10.1007/s10237-019-01241-2

    Article  Google Scholar 

  • Sergerie K, Parent S, Beauchemin PF et al (2010) Growth plate explants respond differently to in vitro static and dynamic loadings. J Orthop Res 29(4):473–480. https://doi.org/10.1002/jor.21282

    Article  Google Scholar 

  • Shefelbine SJ, Carter DR (2004) Mechanobiological predictions of growth front morphology in developmental hip dysplasia. J Orthop Res 22(2):346–352. https://doi.org/10.1016/j.orthres.2003.08.004

    Article  Google Scholar 

  • Shim KS (2015) Pubertal growth and epiphyseal fusion. Ann Pediatr Endocrinol Metab 20(1):8. https://doi.org/10.6065/apem.2015.20.1.8

    Article  Google Scholar 

  • Stevens SS, Beaupré GS, Carter D (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 IA, Laible JP (1989) Three-dimensional osseo-ligamentous model of the thorax representing initiation of scoliosis by assymetric growth. J Biomech 23:589–595

    Article  Google Scholar 

  • Stokes IA, Mente PL, Iatridis JC et al (2002) Enlargement of growth plate chondrocytes modulated by sustained mechanical loading. J Bone Joint Surg Am 84(10):1842–1848

    Article  Google Scholar 

  • Stokes IA, Gwadera J, Dimock A et al (2005) Modulation of vertebral and tibial growth by compression loading: Diurnal versus full-time loading. J Orthop Res 23(1):188–195. https://doi.org/10.1016/j.orthres.2004.06.012

    Article  Google Scholar 

  • Stokes IA, Clark KC, Farnum CE et al (2007) Alterations in the growth plate associated with growth modulation by sustained compression or distraction. Bone 41(2):197–205. https://doi.org/10.1016/j.bone.2007.04.180

    Article  Google Scholar 

  • Stokes IAF, Aronsson DD, Dimock AN et al (2006) Endochondral growth in growth plates of three species at two anatomical locations modulated by mechanical compression and tension. J Orthop Res 24:1327–1334

    Article  Google Scholar 

  • Sylvestre P, Villemure I, Aubin CE (2007) Finite element modeling of the growth plate in a detailed spine model. Med Biol Eng Comput 45:977–988

    Article  Google Scholar 

  • Tjørve KMC, Tjørve E (2017) The use of gompertz models in growth analyses, and new gompertz-model approach: an addition to the unified-richards family. PLoS ONE 12(6):e0178691. https://doi.org/10.1371/journal.pone.0178691

    Article  Google Scholar 

  • Ueki M, Tanaka N, Tanimoto K et al (2008) The effect of mechanical loading on the metabolism of growth plate chondrocytes. Ann Biomed Eng 36(5):793–800. https://doi.org/10.1007/s10439-008-9462-7

    Article  Google Scholar 

  • Valteau B, Grimard G, Londono I et al (2011) In vivo dynamic bone growth modulation is less detrimental but as effective as static growth modulation. Bone 49(5):996–1004. https://doi.org/10.1016/j.bone.2011.07.008

    Article  Google Scholar 

  • Villemure I, Stokes IA (2009) Growth plate mechanics and mechanobiology. A survey of present understanding. J Biomech 42(12):1793–1803

    Article  Google Scholar 

  • Wang DC, Deeney V, Roach JW et al (2015) Imaging of physeal bars in children. Pediatr Radiol 45(9):1403–1412. https://doi.org/10.1007/s00247-015-3280-5

    Article  Google Scholar 

  • Wang X, Mao JJ (2002) Accelerated chondrogenesis of the rabbit cranial base growth plate by oscillatory mechanical stimuli. J Bone Miner Res 17(10):1843–1850. https://doi.org/10.1359/jbmr.2002.17.10.1843

    Article  Google Scholar 

  • Wirtz DC, Nea Schiffers (2000) Critical evaluation of known bone material properties to realize anisotropic fe-simulation of the proximal femur. J Biomech 33:1325–1330

    Article  Google Scholar 

  • Yadav P, Shefelbine S, Gutierrez-Farewick EM (2016) Effect of growth plate geometry and growth direction on prediction of proximal femoral morphology. J Biomech

  • Zimmermann EA, Bouguerra S, Londoño I et al (2017) In situ deformation of growth plate chondrocytes in stress-controlled static vs dynamic compression. J Biomech 56:76–82. https://doi.org/10.1016/j.jbiomech.2017.03.008

    Article  Google Scholar 

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Acknowledgements

This research was supported by Comisión Nacional de Energía Atómica, Gobierno de Argentina (RESOL-2021-306-APN-CNEA#MEC), and projects PICT 2018-03300 (FONCyT, Argentina) and SIIP 06/045 335 (2019-2021) (Universidad Nacional de Cuyo, Argentina).

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  1. Alejandro Yawny and Graciela Bertolino have equally contributed to this work.

Authors and Affiliations

  1. División Física de Metales, CNEA, Centro Atómico Bariloche, Bariloche, 8400, Río Negro, Argentina

    Gastón Alonso, Alejandro Yawny & Graciela Bertolino

  2. CONICET, Buenos Aires, Argentina

    Alejandro Yawny & Graciela Bertolino

  3. Instituto Balseiro, Universidad Nacional de Cuyo, Mendoza, Argentina

    Gastón Alonso, Alejandro Yawny & Graciela Bertolino

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  1. Gastón Alonso
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Correspondence to Gastón Alonso.

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Alonso, G., Yawny, A. & Bertolino, G. How do bones grow? A mathematical description of the mechanobiological behavior of the epiphyseal plate. Biomech Model Mechanobiol 21, 1585–1601 (2022). https://doi.org/10.1007/s10237-022-01608-y

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