Skip to main content

Advertisement

Log in

Growth mixture model of distraction osteogenesis: effect of pre-traction stresses

  • Original Paper
  • Published:
Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

Abstract

In tensional studies of bone fragments during limb lengthening, it is usually assumed that the stress level in the gap tissue before each distraction step (pre-traction stress) is rather modest. However, during the process of distraction osteogenesis, a large interfragmentary gap is generated and these pre-traction stresses may be important. To date, to the authors’ knowledge, no computational study has been developed to assess the effect of stress accumulation during limb lengthening. In this work, we present a macroscopic growth mixture formulation to investigate the influence of pre-traction stresses on the outcome of this clinical procedure. In particular, the model is applied to the simulation of the regeneration of tibial defects by means of distraction osteogenesis. The evolution of pre-traction forces, post-traction forces and peak forces is evaluated and compared with experimental data. The results show that the inclusion of pre-traction stresses in the model affects the evolution of the regeneration process and the corresponding reaction forces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Aarnes GT, Steen H, Kristiansen LP, Ludvigsen P, Reikerås O (2002) Tissue response during monofocal and bifocal leg lengthening in patients. J Orthop Res 20: 137–141

    Article  Google Scholar 

  • Akizuki S, Mow VC, Müller F, Pita JC, Howell DS, Manicourt DH (1986) Tensile properties of human knee joint cartilage: I. influence of ionic conditions, weight bearing, and fibrillation on the tensile modulus. J Orthop Res 4: 379–392

    Article  Google Scholar 

  • Alastrué V, Peña E, Martínez MA, Doblaré M (2007) Assessing the use of the “opening angle method” to enforce residual stresses in patient-specific arteries. Ann Biomed Eng 35: 1821–1837

    Article  Google Scholar 

  • Armstrong CG, Mow VC (1982) Variations in the intrinsic mechanical properties of human articular cartilage with age, degeneration, and water content. J Bone Joint Surg Am 64: 88–94

    Google Scholar 

  • Bailón-Plaza A, van der Meulen MC (2003) Beneficial effects of moderate, early loading and adverse effects of delayed or excessive loading on bone healing. J Biomech 36: 1069–1077

    Article  Google Scholar 

  • Bertram JE, Polevoy Y, Cullinane DM (1998) Mechanics of avian fibrous periosteum: tensile and adhesion properties during growth. Bone 22: 669–675

    Article  Google Scholar 

  • Boccaccio A, Pappalettere C, Kelly DJ (2007) The influence of expansion rates on mandibular distraction osteogenesis: a computational analysis. Ann Biomed Eng 35: 1940–1960

    Article  Google Scholar 

  • Breward CJ, Byrne HM, Lewis CE (2003) A multiphase model describing vascular tumour growth. Bull Math Biol 65: 609–640

    Article  Google Scholar 

  • Brunner UH, Cordey J, Schweiberer L, Perren SM (1994) Force required for bone segment transport in the treatment of large bone defects using medullary nail fixation. Clin Orthop Relat Res 301: 147–155

    Google Scholar 

  • Carter DR, Beaupre GS, Giori NJ, Helms JA (1998) Mechanobiology of skeletal regeneration. Clin Orthop Relat Res 355S: S41–S55

    Article  Google Scholar 

  • Charlebois M, McKee MD, Buschmann MD (2004) Nonlinear tensile properties of bovine articular cartilage and their variation with age and depth. J Biomech Eng 126: 129–137

    Article  Google Scholar 

  • Chen Y, Hoger A (2000) Constitutive functions for elastic materials in finite growth and deformation. J Elast 59: 175–193

    Article  MATH  Google Scholar 

  • Chuong CJ, Fung YC (1986) On residual stresses in arteries. J Biomech Eng 108: 189–192

    Article  Google Scholar 

  • Claes LE, Heigele CA (1999) Magnitudes of local stress and strain along bony surfaces predict the course and type of fracture healing. J Biomech 32: 255–266

    Article  Google Scholar 

  • Davol A, Bingham MS, Sah RL, Klisch SM (2008) A nonlinear finite element model of cartilage growth. Biomech Model Mechanobiol 7: 295–307

    Article  Google Scholar 

  • DiCarlo A, Quiligotti S (2002) Growth and balance. Mech Res Commun 29: 449–456

    Article  MATH  MathSciNet  Google Scholar 

  • Doblaré M, García-Aznar JM (2006) On numerical modelling of growth, differentiation and damage in structural living tissues. Arch Comput Methods Eng 13: 471–513

    Article  MATH  Google Scholar 

  • Epstein M, Maugin GA (2000) Thermomechanics of volumetric growth in uniform bodies. Int J Plast 16: 951–978

    Article  MATH  Google Scholar 

  • Forriol F, Goenaga I, Mora G, Viñolas J, Canadell J (1997) Measurement of bone lengthening forces; an experimental model in the lamb. Clin Biomech 12: 17–21

    Article  Google Scholar 

  • Fung YC (1984) Biodynamics: circulation. Springer, New York

    Google Scholar 

  • Fung YC (1985) What principle governs the stress distribution in living organs? In: Fung YC, Fukada E, Wang JJ (eds) Biomechanics in China, Japan, and USA. Science Press, Beijing, pp 1–13

    Google Scholar 

  • Fung YC (1990a) Biomechanics: motion, flow, stress, and growth. Springer, New York

    MATH  Google Scholar 

  • Fung YC (1990b) Biomechanics: mechanical properties of living tissues. Springer, Berlin

    MATH  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: 1467–1476

    Article  Google Scholar 

  • Gardner TN, Evans M, Simpson H, Kenwright J (1998) Force–displacement behaviour of biological tissue during distraction osteogenesis. Med Eng Phys 20: 708–715

    Article  Google Scholar 

  • Garikipati K, Arruda EM, Grosh K, Narayanan H, Calve S (2004) A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics. J Mech Phys Solids 52: 1595–1625

    Article  MATH  MathSciNet  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: 105–119

    Article  Google Scholar 

  • Gómez-Benito MJ, García-Aznar JM, Kuiper JH, Doblaré M (2006) A 3d computational simulation of fracture callus formation: influence of the stiffness of the external fixator. J Biomech Eng 128: 290–299

    Article  Google Scholar 

  • Gregersen H, Kassab GS, Fung YC (2000) The zero stress state of the gastrointestinal tract: biomechanical and functional implications. Dig Dis Sci 45: 2271–2281

    Article  Google Scholar 

  • Han HC, Fung YC (1996) Direct measurement of transverse residual strains in aorta. Am J Physiol 270: H750–H759

    Google Scholar 

  • Hibbit K, Sorensen I (2008) Theory manual, vol 6.8. HKS inc, Pawtucket

    Google Scholar 

  • Himpel G, Kuhl E, Menzel A, Steinmann P (2005) Computational modelling of isotropic multiplicative growth. Comput Model Eng Sci 8: 119–134

    MATH  Google Scholar 

  • Hoger A (1997) Virtual configurations and constitutive equations for residually stressed bodies with material symmetry. J Elast 48: 125–144

    Article  MATH  MathSciNet  Google Scholar 

  • Hori RY, Lewis JL (1982) Mechanical properties of the fibrous tissue found at the bone-cement interface following total joint replacement. J Biomed Mater Res 16: 911–927

    Article  Google Scholar 

  • Huang CY, Mow VC, Ateshian GA (2001) The role of flow-independent viscoelasticity in the biphasic tensile and compressive responses of articular cartilage. J Biomech Eng 123: 410–417

    Article  Google Scholar 

  • Huang Z (1994) The equilibrium equations and constitutive equations of the growing deformable body in the framework of continuum theory. Int J Non-Linear Mech 39: 951–962

    Article  Google Scholar 

  • Humphrey JD (2001) Stress, strain, and mechanotransduction in cells. J Biomech Eng 123: 638–641

    Article  Google Scholar 

  • Humphrey JD, Rajagopal KR (2003) A constrained mixture model for arterial adaptations to a sustained step change in blood flow. Biomech Model Mechanobiol 2: 109–126

    Article  Google Scholar 

  • Isaksson H, Comas O, van Donkelaar CC, Mediavilla J, Wilson W, Huiskes R, Ito K (2007) Bone regeneration during distraction osteogenesis: mechano-regulation by shear strain and fluid velocity. J Biomech 40: 2002–2011

    Article  Google Scholar 

  • Isaksson H, Wilson W, van Donkelaar CC, Huiskes R, Ito K (2006) Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing. J Biomech 39: 1507–1516

    Article  Google Scholar 

  • Klisch SM, Chen SS, Sah RL, Hoger A (2003) A growth mixture theory for cartilage with application to growth-related experiments on cartilage explants. J Biomech Eng 125: 169–179

    Article  Google Scholar 

  • Klisch SM, Sah RL, Hoger A (2005) A cartilage growth mixture model for infinitesimal strains: solutions of boundary-value problems related to in vitro growth experiments. Biomech Model Mechanobiol 3: 209–223

    Article  Google Scholar 

  • Klisch SM, Van Dyke TJ, Hoger A (2001) A theory of volumetric growth for compressible elastic biological materials. Math Mech Solids 6: 551–575

    Article  MATH  Google Scholar 

  • Kuhl E, Menzel A, Steinmann P (2003) Computational modeling of growth. Comput Mech 32: 71–88

    Article  MATH  Google Scholar 

  • Kuhl E, Steinmann P (2003) Mass- and volumetric-specific views on thermodynamics for open systems. Proc R Soc London 459A: 2547–2568

    MathSciNet  Google Scholar 

  • Lacroix D, Prendergast PJ (2002) A mechano-regulation model for tissue differentiation during fracture healing: analysis of gap size and loading. J Biomech 35: 1163–1171

    Article  Google Scholar 

  • Lappa M (2005) A CFD level-set method for soft tissue growth: theory and fundamental equations. J Biomech 38: 185–190

    Google Scholar 

  • Leong JC, Ma RY, Clark JA, Cornish LS, Yau AC (1979) Viscoelastic behavior of tissue in leg lengthening by distraction. Clin Orthop Relat Res 139: 102–109

    Google Scholar 

  • Leong PL, Morgan EF (2008) Measurement of fracture callus material properties via nanoindentation. Acta Biomater 4: 1569–1575

    Article  Google Scholar 

  • Levick JR (1987) Flow through interstitium and other fibrous matrices. Q J Exp Physiol 72: 409–438

    Google Scholar 

  • Li G, Bronk JT, An KN, Kelly PJ (1987) Permeability of cortical bone of canine tibiae. Microvasc Res 34: 302–310

    Article  Google Scholar 

  • Li G, Simpson HRW, Kenwright J, Triffitt JT (1997) Assessment of cell proliferation in regenerating bone during distraction osteogenesis at different distraction rates. J Orthop Res 15: 765–772

    Article  Google Scholar 

  • Liu SQ, Fung YC (1988) Zero-stress states of arteries. J Biomech Eng 110: 82–84

    Article  Google Scholar 

  • Lubarda VA, Hoger A (2002) On the mechanics of solids with a growing mass. Int J Solids Struct 39: 4627–4664

    Article  MATH  Google Scholar 

  • Matsushita T, Nakamura K, Kurokawa T (1999) Tensile force in limb lengthening: histogenesis or only mechanical elongation. Orthopedics 22: 61–63

    Google Scholar 

  • Menzel A (2005) Modelling of anisotropic growth in biological tissues. A new approach and computational aspects. Biomech Model Mechanobiol 3: 147–171

    Article  Google Scholar 

  • Menzel A (2006) Anisotropic remodeling of biological tissue. In: Holzapfel, Ogden (eds) Mechanics of biological tissue. Springer, Berlin, pp 91–104

    Chapter  Google Scholar 

  • Nafei A, Danielsen C, Linde F, Hvid I (2000) Properties of growing trabecular ovine bone. Part I: mechanical and physical properties. J Bone Joint Surg Br 82: 910–920

    Article  Google Scholar 

  • Nakamura K, Matsushita T, Okazaki H, Kurokawa T (1997) Soft tissue responses to limb lengthening. J Orthop Sci 2: 191–197

    Article  Google Scholar 

  • Ochoa JA, Hillberry BM (1992) Permeability of bovine cancellous bone. In: Transactions of the 38th ORS, Washinton, DC

  • Pauwels F (1960) Eine neue Theorie über den Einfluβ mechanischer Reize auf die Differenzierung der Stützgewebe. Z Anat Entwicklungsgesch 121: 478–515

    Article  Google Scholar 

  • Perren SM, Cordey J (1980) The concept of interfragmentary strain. Currect concepts of internal fixation of fractures. Springer, Berlin, pp 63–77

  • Popowics T, Zhu Z, Herring S (2002) Mechanical properties of the periosteum in the pig, sus scrofa. Arch Oral Biol 47: 733–741

    Article  Google Scholar 

  • Preziosi L, Farina A (2002) On Darcy’s law for growing porous media. Int J Non-Linear Mech 37: 485–491

    Article  MATH  Google Scholar 

  • Quiligotti S (2002) On bulk growth mechanics of solid-fluid mixtures: kinematics and invariance requirements. Theor Appl Mech 28(29): 277–288

    Article  MathSciNet  Google Scholar 

  • Ramasubramanian A, Taber LA (2008) Computational modeling of morphogenesis regulated by mechanical feedback. Biomech Model Mechanobiol 7: 77–91

    Article  Google Scholar 

  • Reina-Romo E, Gómez-Benito MJ, García-Aznar JM, Domínguez J, Doblaré M (2009) Modeling distraction osteogenesis: analysis of the distraction rate. Biomech Model Mechanobiol. doi:10.1007/s10237-008-0138-x

  • Richards M, Wineman AS, Alsberg E, Goulet JA, Goldstein SA (1999) Viscoelastic characterization of mesenchymal gap tissue and consequences for tension accumulation during distraction. J Biomech Eng 121: 116–123

    Article  Google Scholar 

  • Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27: 455–467

    Article  Google Scholar 

  • Simpson AH, Cunningham JL, Kenwright J (1996) The forces which develop in the tissues during leg lengthening. A clinical study. J Bone Joint Surg Br 78: 979–983

    Article  Google Scholar 

  • Skalak R (1981) Growth as a finite displacement field. In: Carlson DE, Shield RT (eds) Proceedings of IUTAM symposium on finite elasticity. Martinus Nijhoff, The Hague, pp 347–355

    Google Scholar 

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

    Article  Google Scholar 

  • Taber LA (1998) A model for aortic growth based on fluid shear and fiber stresses. J Biomech Eng 120: 348–354

    Article  Google Scholar 

  • Taber LA, Eggers DW (1996) Theoretical study of stress-modulated growth in the aorta. J Theor Biol 180: 343–357

    Article  Google Scholar 

  • Taber LA, Humphrey JD (2001) Stress-modulated growth, residual stress, and vascular heterogeneity. J Biomech Eng 123: 528–535

    Article  Google Scholar 

  • Taber LA, Perucchio R (2000) Modeling heart development. J Elast 61: 165–197

    Article  MATH  MathSciNet  Google Scholar 

  • Vaishnav RN, Vossoughi J (1983) Estimation of residual strain in aortic segments. In: Hall CW (eds) Biomedical engineering II: recent development. Pergamon Press, Oxford, , pp 330–333

    Google Scholar 

  • Vaishnav RN, Vossoughi J (1987) Residual stress and strain in aortic segments. J Biomech 20: 235–239

    Article  Google Scholar 

  • Volokh KY (2004) A simple phenomenological theory of tissue growth. Mech Chem Biosyst 1: 147–160

    MATH  Google Scholar 

  • Waanders NA, Richards M, Steen H, Kuhn JL, Goldstein SA, Goulet JA (1998) Evaluation of the mechanical environment during distraction osteogenesis. Clin Orthop Relat Res 349: 225–234

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Esther Reina-Romo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reina-Romo, E., Gómez-Benito, M.J., García-Aznar, J.M. et al. Growth mixture model of distraction osteogenesis: effect of pre-traction stresses. Biomech Model Mechanobiol 9, 103–115 (2010). https://doi.org/10.1007/s10237-009-0162-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-009-0162-5

Keywords

Navigation