Mathematical Modelling of Spatio-temporal Cell Dynamics Observed During Bone Remodelling

  • Madalena M. A. Peyroteo
  • Jorge BelinhaEmail author
  • Susana Vinga
  • R. Natal Jorge
  • Lúcia Dinis
Part of the Lecture Notes in Computational Vision and Biomechanics book series (LNCVB, volume 35)


Bone is an active tissue capable to adapt its activity according to external stimuli. The consequent changes in mass, composition and shape are a result of this remodelling process, in which old bone is continuously replaced by new tissue. In order to mathematically describe this remodelling process, several mathematical models have been proposed. In this chapter, a novel model simulating the biological events that occur during bone remodelling is presented, based on Komarova’s (Bone 33:206–215, 2003 [1]) and Ayati’s (Biol Direct 5:28, 2010 [2]) models. Also, a thorough temporal and spatial analysis is performed using numerical methods, namely the finite element method (FEM), the radial point interpolation method (RPIM) and the natural neighbour radial point interpolation method (NNRPIM), being the last two meshless approaches. Results show that the combination of this model with distinct numerical approaches allows an accurate reproduction of the biological event, as described in the literature. Besides this, although all numerical techniques have been successfully applied, better quality results have been obtained with the NNRPIM.


  1. 1.
    Komarova SV, Smith RJ, Dixon SJ et al (2003) Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling. Bone 33:206–215. Scholar
  2. 2.
    Ayati BP, Edwards CM, Webb GF, Wikswo JP (2010) A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease. Biol Direct 5:28. Scholar
  3. 3.
    Hadjidakis DJ, Androulakis II (2006) Bone remodeling. Ann NY Acad Sci 1092:385–396. Scholar
  4. 4.
    Lemaire T, Naili S, Lemaire T (2013) Multiscale approach to understand the multiphysics phenomena in bone adaptation. Stud Mechanobiol Tissue Eng Biomater 14:31–72. Scholar
  5. 5.
    Post TM, Cremers SCLM, Kerbusch T, Danhof M (2010) Bone physiology, disease and treatment. Clin Pharmacokinet 49:89–118. Scholar
  6. 6.
    Lemaire V, Tobin FL, Greller LD et al (2004) Modeling the interactions between osteoblast and osteoclast activities in bone remodeling. J Theor Biol 229:293–309. Scholar
  7. 7.
    Tsukii K, Shima N, Mochizuki S et al (1998) Osteoclast differentiation factor mediates an essential signal for bone resorption induced by 1α,25-Dihydroxyvitamin D3, Prostaglandin E2, or parathyroid hormone in the microenvironment of bone. Biochem Biophys Res Commun 246:337–341. Scholar
  8. 8.
    Simmons DJ, Grynpas M (1990) Mechanisms of bone formation in vivo. In: Hall BK (ed) Bone, vol I, A treatrise. CRC Press, Boca RatonGoogle Scholar
  9. 9.
    Eriksen EF (2010) Cellular mechanisms of bone remodeling. Rev Endocr Metab Disord 11:219–227. Scholar
  10. 10.
    Greenfield EM, Bi Y, Miyauchi A (1999) Regulation of osteoclast activity. Life Sci 65:1087–1102. Scholar
  11. 11.
    Pettermann HE, Reiter TJ, Rammerstorfer FG (1997) Computational simulation of internal bone remodeling. Arch Comput Methods Eng 4:295–323. Scholar
  12. 12.
    García-Aznar JM, Rueberg T, Doblare M (2005) A bone remodelling model coupling microdamage growth and repair by 3D BMU-activity. Biomech Model Mechanobiol 4:147–167. Scholar
  13. 13.
    Pivonka P, Zimak J, Smith DW et al (2008) Model structure and control of bone remodeling: a theoretical study. Bone 43:249–263. Scholar
  14. 14.
    Pivonka P, Zimak J, Smith DW et al (2010) Theoretical investigation of the role of the RANK-RANKL-OPG system in bone remodeling. J Theor Biol 262:306–316. Scholar
  15. 15.
    Savageau MA (1976) Biochemical systems analysis—a study of function and design in molecular biology. Addison-Wesley Pub. Co., MassachusettsGoogle Scholar
  16. 16.
    Noor M, Shoback D (2000) Paget’s disease of bone: diagnosis and treatment update. Curr Rheumatol Rep 2:67–73. Scholar
  17. 17.
    Ryser MD, Nigam N, Komarova SV (2009) Mathematical modeling of spatio-temporal dynamics of a single bone multicellular unit. J Bone Miner Res 24:860–870. Scholar
  18. 18.
    Pivonka P, Komarova SV (2010) Mathematical modeling in bone biology: from intracellular signaling to tissue mechanics. Bone 47:181–189. Scholar
  19. 19.
    Pepper D, Kassab A, Divo E (2014) Introduction to finite element, boundary element, and meshless methods: with applications to heat transfer and fluid flow. ASME Press, USACrossRefGoogle Scholar
  20. 20.
    Hrennikoff A (1941) Solution of problems in elasticity by the frame work method. J Appl Mech 8:169–175Google Scholar
  21. 21.
    McHenry D (1943) A lattice analogy for the solution of plane stress problems. J Inst Civ Eng 21:59–82. Scholar
  22. 22.
    Courant R (1943) Variational methods for the solution of problems of equilibrium and vibrations. Bull Am Math Soc 49:1–24. Scholar
  23. 23.
    Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method: its basis and fundamentals. Elsevier, AmsterdamCrossRefGoogle Scholar
  24. 24.
    Nguyen VP, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79:763–813. Scholar
  25. 25.
    Belinha J (2014) Meshless methods in biomechanics—bone tissue remodelling analysis, 1st edn. Springer International Publishing Switzerland, ChamCrossRefGoogle Scholar
  26. 26.
    Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318. Scholar
  27. 27.
    Belytschko T, Gu L, Lu YY (1994) Fracture and crack growth by element free Galerkin methods. Model Simul Mater Sci Eng 2:519–534. Scholar
  28. 28.
    Liu GR, Gu YT (2001) A point interpolation method for two-dimensional solids. Int J Numer Methods Eng 50:937–951.;2-XCrossRefGoogle Scholar
  29. 29.
    Liu GR, Gu YT (2001) A local point interpolation method for stress analysis of two-dimensional solids. Struct Eng Mech 11:221–236. Scholar
  30. 30.
    Liu GR, Gui-R (2010) Meshfree methods : moving beyond the finite element method. CRC Press, Boca RatonCrossRefGoogle Scholar
  31. 31.
    Liu GR, Zhang GY, Gu YT, Wang YY (2005) A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Comput Mech 36:421–430. Scholar
  32. 32.
    Wang JG, Liu GR (2002) On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Comput Methods Appl Mech Eng 191:2611–2630. Scholar
  33. 33.
    Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Methods Eng 54:1623–1648. Scholar
  34. 34.
    Liu GR, Gu YT (2001) A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. J Sound Vib 246:29–46. Scholar
  35. 35.
    Gu YT, Liu GR (2001) A local point interpolation method for static and dynamic analysis of thin beams. Comput Methods Appl Mech Eng 190:5515–5528. Scholar
  36. 36.
    Liu GR, Yan L, Wang JG, Gu YT (2002) Point interpolation method based on local residual formulation using radial basis functions. Struct Eng Mech 14:713–732. Scholar
  37. 37.
    Belinha J, Dinis LMJS, Jorge RMN (2013) Composite laminated plate analysis using the natural radial element method. Compos Struct 103:50–67. Scholar
  38. 38.
    Belinha J, Dinis LMJS, Jorge RMN (2013) Analysis of thick plates by the natural radial element method. Int J Mech Sci 76:33–48. Scholar
  39. 39.
    Belinha J, Dinis LMJS, Jorge RMN (2013) The natural radial element method. Int J Numer Methods Eng 93:1286–1313. Scholar
  40. 40.
    Dinis LMJS, Jorge RMN, Belinha J (2008) Analysis of plates and laminates using the natural neighbour radial point interpolation method. Eng Anal Bound Elem 32:267–279. Scholar
  41. 41.
    Dinis LMJS, Jorge RMN, Belinha J (2010) An unconstrained third-order plate theory applied to functionally graded plates using a meshless method. Mech Adv Mater Struct 17:108–133. Scholar
  42. 42.
    Moreira S, Belinha J, Dinis LMJS, Jorge RMN (2014) Análise de vigas laminadas utilizando o natural neighbour radial point interpolation method. Rev Int Métodos Numéricos para Cálculo y Diseño en Ing 30:108–120. Scholar
  43. 43.
    Belinha J, Araújo AL, Ferreira AJM et al (2016) The analysis of laminated plates using distinct advanced discretization meshless techniques. Compos Struct 143:165–179. Scholar
  44. 44.
    Dinis LMJS, Jorge RMN, Belinha J (2011) A natural neighbour meshless method with a 3D shell-like approach in the dynamic analysis of thin 3D structures. Thin-Walled Struct 49:185–196. Scholar
  45. 45.
    Dinis LMJS, Jorge RMN, Belinha J (2011) Static and dynamic analysis of laminated plates based on an unconstrained third order theory and using a radial point interpolator meshless method. Comput Struct 89:1771–1784. Scholar
  46. 46.
    Dinis LMJS, Jorge RMN, Belinha J (2010) Composite laminated plates: a 3D natural neighbor radial point interpolation method approach. J Sandw Struct Mater 12:119–138. Scholar
  47. 47.
    Dinis LMJS, Jorge RMN, Belinha J (2010) A 3D shell-like approach using a natural neighbour meshless method: isotropic and orthotropic thin structures. Compos Struct 92:1132–1142. Scholar
  48. 48.
    Dinis LMJS, Jorge RMN, Belinha J (2009) The natural neighbour radial point interpolation method: dynamic applications. Eng Comput 26:911–949. Scholar
  49. 49.
    Dinis LMJS, Natal Jorge R, Belinha J (2009) Large deformation applications with the radial natural neighbours interpolators. Comput Model Eng Sci 44:1–34Google Scholar
  50. 50.
    Azevedo JMC, Belinha J, Dinis LMJS, Natal Jorge RM (2015) Crack path prediction using the natural neighbour radial point interpolation method. Eng Anal Bound Elem 59:144–158. Scholar
  51. 51.
    Belinha J, Azevedo JMC, Dinis LMJS, Jorge RMN (2016) The natural neighbor radial point interpolation method extended to the crack growth simulation. Int J Appl Mech 08:1650006. Scholar
  52. 52.
    Rossi J-M, Wendling-Mansuy S (2007) A topology optimization based model of bone adaptation. Comput Methods Biomech Biomed Engin 10:419–427. Scholar
  53. 53.
    Jacobs CR, Levenston ME, Beaupré GS et al (1995) Numerical instabilities in bone remodeling simulations: the advantages of a node-based finite element approach. J Biomech 28:449–459. Scholar
  54. 54.
    Weinans H, Huiskes R, Grootenboer HJ (1992) The behavior of adaptive bone-remodeling simulation models. J Biomech 25:1425–1441. Scholar
  55. 55.
    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. Scholar
  56. 56.
    Huiskes R, Weinans H, Grootenboer HJ et al (1987) Adaptive bone-remodeling theory applied to prosthetic-design analysis. J Biomech 20:1135–1150CrossRefGoogle Scholar
  57. 57.
    Carter DR, Fyhrie DP, Whalen RT (1987) Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy. J Biomech 20:785–794. Scholar
  58. 58.
    Belinha J, Jorge RMN, Dinis LMJS (2012) Bone tissue remodelling analysis considering a radial point interpolator meshless method. Eng Anal Bound Elem 36:1660–1670. Scholar
  59. 59.
    Belinha J, Dinis LMJS, Natal Jorge RM (2016) The analysis of the bone remodelling around femoral stems: A meshless approach. Math Comput Simul 121:64–94. Scholar
  60. 60.
    Peyroteo M, Belinha J, Dinis L, Natal Jorge R (2018) The mechanologic bone tissue remodeling analysis: a comparison between mesh-depending and meshless methods. In: Numerical methods and advanced simulation in biomechanics and biological processes. Academic Press, Cambridge, pp 303–323CrossRefGoogle Scholar
  61. 61.
    Logan DL (2011) A first course in the finite element method, 5th edn. Cengage Learning, BostonGoogle Scholar
  62. 62.
    Chao TY, Chow WK, Kong H (2002) A review on the applications of finite element method to heat transfer and fluid flow. Int J Archit Sci 3:1–19Google Scholar
  63. 63.
    Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106. Scholar
  64. 64.
    Atluri SN, Zhu T (1998) A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117–127. Scholar
  65. 65.
    Sibson R (1981) A brief description of natural neighbour interpolation. In: Barnett V (ed) Interpreting multivariate data. Wiley, Chichester, pp 21–36Google Scholar
  66. 66.
    Dinis LMJS, Jorge RMN, Belinha J (2007) Analysis of 3D solids using the natural neighbour radial point interpolation method. Comput Methods Appl Mech Eng 196:2009–2028. Scholar
  67. 67.
    Hardy RL (1990) Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988. Comput Math with Appl 19:163–208. Scholar
  68. 68.
    Seibel MJ, Robins SP BJ (2006) Dynamics of bone and cartilage metabolism: principles and clinical applications, 2nd edn. Academic Press, CambridgeGoogle Scholar
  69. 69.
    Eilertsen K, Vestad A, Geier O, Skretting A (2008) A simulation of MRI based dose calculations on the basis of radiotherapy planning CT images. Acta Oncol (Madr) 47:1294–1302. Scholar
  70. 70.
    Parfitt AM (1994) Osteonal and hemi-osteonal remodeling: the spatial and temporal framework for signal traffic in adult human bone. J Cell Biochem 55:273–286. Scholar
  71. 71.
    Akchurin T, Aissiou T, Kemeny N et al (2008) Complex dynamics of osteoclast formation and death in long-term cultures. PLoS ONE 3:e2104. Scholar
  72. 72.
    Komarova SV (2005) Mathematical model of paracrine interactions between osteoclasts and osteoblasts predicts anabolic action of parathyroid hormone on bone. Endocrinology 146:3589–3595. Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Madalena M. A. Peyroteo
    • 1
  • Jorge Belinha
    • 2
    Email author
  • Susana Vinga
    • 3
  • R. Natal Jorge
    • 4
  • Lúcia Dinis
    • 4
  1. 1.Institute of Mechanical Engineering and Industrial Management (INEGI)PortoPortugal
  2. 2.Department of Mechanical Engineering, School of EngineeringPolytechnic of Porto (ISEP)PortoPortugal
  3. 3.Institute of Mechanical Engineering (IDMEC), ISTLisbonPortugal
  4. 4.Faculty of Engineering, Mechanical Engineering DepartmentUniversity of Porto (FEUP)PortoPortugal

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