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Integrating molecular and cellular kinetics into a coupled continuum mechanobiological stimulus for bone reconstruction

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Abstract

The development of multiphysics numerical models to predict bone reconstruction is a very challenging task as it is a complex phenomenon where many biological, chemical and mechanical processes occur at different lengths and timescales. We present here a mechanobiological theoretical numerical model accounting for both the mechanical and biological environments to predict the bone reconstruction process through the use of a global stimulus integrating the contributions of applied external mechanical loads, cellular activities and cellular nutriments such as oxygen and glucose supply. The bone density evolution will hence depend on the overall stimulus and evolve accordingly to the intensities of each of its individual constituents. We show their specific influences and couplings on a simple two-dimensional geometry and confirm that, although the mechanics plays a crucial role in the bone reconstruction process, it is still highly dependent on the occurring biological events and will evolve accordingly.

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References

  1. Frost, H.: Bone “mass” and the “mechanostat”: a proposal. J. Anat. Rec. 219, 1–9 (1987)

    Article  Google Scholar 

  2. Cowin, S.C.: Wolff’s law of trabecular architecture at remodeling equilibrium. J. Biomed. Eng. 108(1), 83–88 (1986)

    Google Scholar 

  3. Beaupré, G.S., Orr, T.E., Carter, D.R.: An approach for time-dependent bone modeling and remodeling–application: a preliminary remodeling simulation. J. Orth. Res. 8(5), 662–670 (1990)

    Article  Google Scholar 

  4. Turner, C.H.: Three rules for bone adaptation to mechanical stimuli. Bone 23(5), 399–407 (1998)

    Article  Google Scholar 

  5. Pivonka, P., Zimak, J., Smith, D.W., Gardiner, B.S., Dunstan, C.R., Sims, N.A.: Model structure and control of bone remodeling: a theoretical study. Bone 43(2), 249–263 (2008)

    Article  Google Scholar 

  6. Pivonka, P., Komarova, S.V.: Mathematical modeling in bone biology: from intracellular signaling to tissue mechanics. Bone 47(2), 181–189 (2010)

    Article  Google Scholar 

  7. Lekszycki, T.: Modeling of bone adaptation based on an optimal response hypothesis. Meccanica 37, 343–354 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Madeo, A., Lekszycki, T., Dell’Isola, F.: A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. C.R. Mécanique 339, 625–640 (2011)

    Article  ADS  Google Scholar 

  9. Madeo, A., George, D., Lekszycki, T., Nierenberger, M., Rémond, Y.: A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodeling. C.R. Mécanique 340, 575–589 (2012)

    Article  ADS  Google Scholar 

  10. Lekszycki, T., Dell’Isola, F.: A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM 92, 426–444 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Madeo, A., George, D., Rémond, Y.: Second-gradient models accounting for some effects of microstructure on remodelling of bones reconstructed with bioresorbable materials. Comp. Meth. Biomech. Biomed. Eng. 16(sup1), 260–261 (2013)

    Article  Google Scholar 

  12. Andreaus, U., Giorgio, I., Lekszycki, T.: A 2-D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time. ZAMM 94, 978–1000 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola. Math. Mech. Sol 20(8), 887–928 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Comp. Syst. 3(3), 285–308 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ganghoffer, J.F.: Spatial and material stress tensors in continuum mechanics of growing solid bodies. Math. Mech. Comp. Syst. 3(4), 341–363 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Placidi, L., Andreaus, U., Della Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeit. Ang. Math. Phys. 66(6), 3699–3725 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. George, D., Spingarn, C., Madeo, A., Rémond, Y.: Effects of mechanical loading conditions on 3D bone reconstruction: a theoretical numerical study for application to Maxillo-facial surgery. In: Proceedings of the 9th European Solid Mechanics Conference, Madrid, Spain (2015)

  18. Giorgio, I., Andreaus, U., Scerrato, D., Dell’Isola, F., et al.: A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials. Biomech. Model Mechanobiol. 15, 1325–1343 (2016)

    Article  Google Scholar 

  19. Scala, I., Spingarn, C., Rémond, Y., Madeo, A., George, D.: Mechanically-driven bone remodeling simulation: application to LIPUS treated rat calvarial defects. Math. Mech. Sol. 22(10), 1976–1988 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Spingarn, C., Rémond, Y., George, D.: Techniques de changement d’échelles pour la modélisation de tissus biologiques, Conf. MECAMAT, Mécanique pour le vivant, Aussois, France (2016)

  21. Abali, B.E., Müller, W.H., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. App. Mech. 87(9), 1495–1510 (2017)

    Article  Google Scholar 

  22. Turco, E.: Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. Eur. J. Env. Civ. Eng. 21(5), 509–554 (2017)

    Article  MathSciNet  Google Scholar 

  23. dell’Isola, F., Della Corte, A., Giorgio, I.: Higher gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Sol. 22(4), 852–872 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Templet, G.J., Steigmann, D.J.: On the theory of diffusion and swelling in finitely deforming elastomers. Math. Mech. Comp. Syst. 1(1), 105–128 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. George, D., Rémond, Y.: Multiscale mechanobiology of human tissue: experiment and modelling. In: 6th International Symposium Europe China, Molecular, cellular and tissue engineering, and clinical applications, Nancy, France (2016)

  26. George, D., Spingarn, C., Dissaux, C., Rémond, Y.: Understanding bone mechanobiology to predict bone reconstruction kinetics: application to maxillo-facial surgery, XII Rencontres du Vietnam, Mechanobiology, from molecules to tissue, Quy Nhon, Vietnam (2016)

  27. Spingarn, C., Wagner, D., Rémond, Y., George, D.: Multiphysics of bone remodeling: a 2D mesoscale activation simulation. Bio-Med. Mat. Eng. 28(s1), S153–S158 (2017)

    Google Scholar 

  28. Cuomo, M.: Forms of the dissipation function for a class of viscoplastic models. Math. Mech. Comp. Syst. 5(3–4), 217–237 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Giorgio, I., Andreaus, U., Dell’Isola, F., Lekszycki, T.: Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts. Ext. Mech. Let. 13, 141–147 (2017)

    Article  Google Scholar 

  30. George, D., Spingarn, C., Dissaux, C., Nierenberger, M., Rahman, R.A., Rémond, Y.: Examples of multiscale and multiphysics numerical modeling of biological tissues. Bio-Med. Mat. Eng. 28(S1), S15–S27 (2017)

    Google Scholar 

  31. Lemaire, T., Capiez-Lernout, E., Kaiser, J., Naili, S., Sansalone, V.: What is the importance of multiphysical phenomena in bone remodelling signals expression? A multiscale perspective. J. Mech. Behav. Biom. Mat. 4(6), 909–920 (2011)

    Article  Google Scholar 

  32. Sansalone, V., Gagliardi, D., Descelier, C., Haiat, G., Naili, S.: On the uncertainty propagation in multiscale modeling of cortical bone elasticity. Comp. Meth. Biom. Biomed. Eng. 18, 2054–2055 (2015)

    Article  Google Scholar 

  33. Bednarczyk, E., Lekszycki, E.: A novel mathematical model for growth of capillaries and nutrient supply with application to prediction of osteophyte onset. ZAMP 67, 94 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  34. Lu, Y., Lekszycki, T.: A novel coupled system of non-local integro-differential equations modelling Young’s modulus evolution, nutrients’ supply and consumption during bone fracture healing. ZAMP 67, 111 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  35. Moya, A., Paquet, J., Deschepper, M., et al.: Human mesenchymal stem cell failure to adapt to glucose shortage and rapidly use intracellular energy reserves through glycolysis explains poor cell survival after implantation. Stem Cells 36, 363–376 (2018). https://doi.org/10.1002/stem.2763

    Article  Google Scholar 

  36. Paquet, J., Deschepper, M., Moya, A., et al.: Oxygen tension regulates human mesenchymal stem cell paracrine functions. Stem Cells Trans. Med. 4(7), 809–821 (2015)

    Article  Google Scholar 

  37. Deschepper, M., Oudina, K., David, B., Myrtil, V., Collet, C., Bensidhoum, M., Logeart-Avramoglou, D., Petite, H.: Survival and function of mesenchymal stem cells (MSCs) depend on glucose to overcome exposure to long-term, severe and continuous hypoxia. J. Cell. Mol. Med. 15(7), 1505–14 (2011)

    Article  Google Scholar 

  38. Farrel, M.J., Shin, J.I., Smith, L.J., Mauck, R.L.: Functional consequences of glucose and oxygen deprivation on engineered mesenchymal stem cell-based cartilage constructs. Osteoarthr. Cartil. 23(1), 134–42 (2015)

    Article  Google Scholar 

  39. Cisewski, S.E., Zhang, L., Kuo, J., Wright, G.J., Wu, Y., Kem, M.J., Yao, H.: The effects of oxygen level and glucose concentration on the metabolism of porcine TMJ disc cells. Osteoarthr. Cartil. 23(10), 1790–6 (2015)

    Article  Google Scholar 

  40. Wagner, D., Bolender, Y., Rémond, Y., George, D.: Mechanical equilibrium of forces and moments applied on orthodontic brackets of a dental arch : correlation with litterature data on two and three adjacent teeth. Bio-Med. Mat. Eng. 28, S169–S177 (2017)

    Google Scholar 

  41. Martin, M., Lemaire, T., Haiat, G., Pivonka, P., Sansalone, V.: A thermodynamically consistent model of bone rotary remodeling: a 2D study. Comp. Meth. Biomech. Biomed. Eng. 20(S1), 127–128 (2017)

    Article  Google Scholar 

  42. Rémond, Y., Ahzi, S., Baniassadi, M., Garmestani, M.: Applied RVE reconstruction and homogenization of heterogeneous materials, Ed. Wiley-ISTE ISBN: 978-1-84821-901-4 (2016)

  43. Burr, D.B., Allen, M.R.: Basic and Applied Bone Biology, pp. 85–86. Academic Press, Cambridge (2013)

    Google Scholar 

  44. Lemaire, T., Naïli, S., Rémond, A.: Multiscale analysis of the coupled effects governing the movement of interstitial fluid in cortical bone. Biomech. Mod. Mech. 5(1), 39–52 (2006)

    Article  Google Scholar 

  45. Lemaire, T., Naili, S., Sansalone, V.: Multiphysical modelling of fluid transport through osto-articular media. An. Ac. Bras. Ciências 82(1), 127–144 (2010)

    Article  MATH  Google Scholar 

  46. Lemaire, T., Kaiser, J., Naili, S., Sansalone, V.: Three-scale multiphysics modeling of transport phenomena within cortical bone. Math. Prob. Eng. 398970 (2015)

  47. Allena, R., Maini, P.K.: Reaction–diffusion finite element model of lateral line primordium migration to explore cell leadership. Bull. Math. Biol. 76(12), 3028–3050 (2014)

    Article  MATH  Google Scholar 

  48. Schmitt, M., Allena, R., Schouman, T., Frasca, S., Collombet, J.M., Holy, X., Rouch, P.: Diffusion model to describe osteogenesis within a porous titanium scaffold. Comp. Meth. Biomech. Biomed. Eng. 19(2), 171–179 (2015)

    Article  Google Scholar 

  49. Frame, J., Rohan, P.Y., Corté, L., Allena, R.: A mechano-biological model of mulit-tissue evolution in bone, Cont. Mech. Thermo. 1–13 (2017) (in press). https://doi.org/10.1007/s00161-017-0611-9

  50. George, D., Allena, R., Rémond, Y.: Mechanobiological stimuli for bone remodeling: mechanical energy, cell nutriments and mobility. Comp. Meth. Biomech. Biomed. Eng. 20, S91–S92 (2017)

    Article  Google Scholar 

  51. Hambli, R., Rieger, R.: Physiologically based mathematical model of transduction of mechanobiological signals by osteocytes. Biomech. Model Mechanobiol. 11(1–2), 83–93 (2012)

    Article  Google Scholar 

  52. Hambli, R., Kourta, A.: A theory for internal bone remodeling based on interstitial fluid velocity stimulus function. App. Math. Mod. 39(12), 3525–3534 (2015)

    Article  MathSciNet  Google Scholar 

  53. Currey, J.D.: The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. J. Biomech. 21(2), 131–139 (1988)

    Article  Google Scholar 

  54. Rho, J.Y., Ho Ba Tho, M.C., Ashman, R.B.: Relations of mechanical properties to density and CT numbers in human bone. Med. Eng. Phys. 17(5), 347–355 (1995)

    Article  Google Scholar 

  55. Gibson, L.J.: The mechanical behaviour of cancellous bone. J. Biomech. 18(5), 317–328 (1985)

    Article  Google Scholar 

  56. Lai, Y.-S., Chen, W.-C., Huang, C.-H., Cheng, C.-K., Chan, K.-K., Chang, T.-K.: The effect of graft strength on knee laxity and graft in-situ forces after posterior cruciate ligament reconstruction. PLoS One 10(5), e0127293 (2015)

    Article  Google Scholar 

  57. Renders, G.A.P., Mulder, L., Van Ruijven, L.J., Van Eijden, T.M.G.J.: Porosity of human mandibular condylar bone. J. Anat. 210(3), 239–248 (2007)

    Article  Google Scholar 

  58. Burr, D.B., Allen, M.R.: Basic and Applied Bone Biology. Elsevier Inc, Amsterdam (2013)

    Google Scholar 

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Authors and Affiliations

  1. ICUBE, CNRS, University of Strasbourg, 2 Rue Boussingault, 67000, Strasbourg, France

    D. George & Y. Rémond

  2. Arts et Métiers ParisTech, LBM/Institut de Biomécanique Humaine Georges Charpak, 151 bd de l’Hôpital, 75013, Paris, France

    R. Allena

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  1. D. George
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  2. R. Allena
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  3. Y. Rémond
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Correspondence to D. George.

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Communicated by Andreas Öchsner.

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George, D., Allena, R. & Rémond, Y. Integrating molecular and cellular kinetics into a coupled continuum mechanobiological stimulus for bone reconstruction. Continuum Mech. Thermodyn. 31, 725–740 (2019). https://doi.org/10.1007/s00161-018-0726-7

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