Abstract
The mechanically driven biological stimulus in bone tissues regulates and controls the action of special cells called osteoblasts and osteoclasts. Different models have been proposed to describe the important and not yet completely understood phenomena related to this ‘feedback’ process. In Lekszycki and dell’Isola (2012) an integro-differential system of equations has been studied to describe the remodeling process in reconstructed bones where the biological stimulus in a given instant t depends on the deformation state of the tissue at the same instant. Instead biological knowledge suggests that the biological stimulus, once produced, is ‘diffused’ in bone tissue to reach the target cells. In this paper, we propose a model for de-scribing biological stimulus diffusion in remodeling tissues in which diffusive time dependent phenomena are taken into account. Some preliminary numerical simulations are presented which suggest that this model is promising and deserves further investigations.
Keywords
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References
Abali BE, Völlmecke C, Woodward B, Kashtalyan M, Guz I, Müller WH (2012) Numerical modeling of functionally graded materials using a variational formulation. Continuum Mechanics and Thermodynamics 24(4-6):377–390
Abeyaratne R, Knowles JK (2006) Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge
Agerbaek MO, Eriksen EF, Kragstrup J, Mosekilde L, Melsen F (1991) A reconstruction of the remodelling cycle in normal human cortical iliac bone. Bone and mineral 12(2):101–112
Allena R, Cluzel C (2018) Heterogeneous directions of orthotropy in three-dimensional structures: finite element description based on diffusion equations. Mathematics and Mechanics of Complex Systems 6(4):339–351
Altenbach H, Eremeyev V (2009) Eigen-vibrations of plates made of functionally graded material. Computers, Materials, & Continua 9(2):153–178
Altenbach H, Eremeyev V (2015) On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Mathematics and Mechanics of Complex Systems 3(3):273–283
Altenbach H, Eremeyev VA (2014) Vibration analysis of non-linear 6-parameter prestressed shells. Meccanica 49(8):1751–1761
Altenbach H, Eremeyev VA, Naumenko K (2015) On the use of the first order shear deformation plate theory for the analysis of three-layer plates with thin soft core layer. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 95(10):1004–1011
Amar MB, Goriely A (2005) Growth and instability in elastic tissues. Journal of the Mechanics and Physics of Solids 53(10):2284–2319
Ambrosi D, Guillou A (2007) Growth and dissipation in biological tissues. Continuum Mechanics and Thermodynamics 19(5):245–251
Ambrosi D, Preziosi L, Vitale G (2010) The insight of mixtures theory for growth and remodeling. Zeitschrift für angewandte Mathematik und Physik 61(1):177–191
Andreaus U, Giorgio I, Lekszycki T (2014) 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-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 94(12):978–1000
Ashby MF, Evans AG, Fleck NA, Gibson LJ, Hutchinson JW, Wadley HNG (2000) Metal Foams: a Design Guid. Butterworth-Heinemann, Boston
Ateshian GA (2007) On the theory of reactive mixtures for modeling biological growth. Biomechanics and Modeling in Mechanobiology 6(6):423–445
Balobanov V, Khakalo S, Niiranen J (2016) Isogeometric analysis of gradient-elastic 1D and 2D problems. In: Altenbach H, Forest S (eds) Generalized Continua as Models for Classical and Advanced Materials, Advanced Structured Materials, vol 42, Springer, Cham
Beaupre GS, Orr TE, Carter DR (1990a) An approach for time-dependent bone modeling and remodeling—application: A preliminary remodeling simulation. Journal of Orthopaedic Research 8(5):662–670
Beaupre GS, Orr TE, Carter DR (1990b) An approach for time-dependent bone modeling and remodeling—theoretical development. Journal of Orthopaedic Research 8(5):651–661
Bednarczyk E, Lekszycki T (2016) A novel mathematical model for growth of capillaries and nutrient supply with application to prediction of osteophyte onset. Zeitschrift für angewandte Mathematik und Physik 67(4):94
Berezovski A, Engelbrecht J, Maugin GA (2008) Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, New Jersey et al.
Berezovski A, Yildizdag ME, Scerrato D (2018) On the wave dispersion in microstructured solids. Continuum Mechanics and Thermodynamics pp 1–20, https://doi.org/10.1007/s00161-018-0683-1
Bonewald LF, Johnson ML (2008) Osteocytes, mechanosensing and Wnt signaling. Bone 42(4):606–615
Bonucci E (2009) The osteocyte: the underestimated conductor of the bone orchestra. Rendiconti Lincei 20(3):237–254
Camar-Eddine M, Seppecher P (2001) Non-local interactions resulting from the homogenization of a linear diffusive medium. Comptes Rendus de l’Academie des Sciences Series I Mathematics 332(5):485–490
Carlen EA, Carvalho MC, Esposito R, Lebowitz JL, Marra R (2009) Droplet minimizers for the Gates–Lebowitz–Penrose free energy functional. Nonlinearity 22(12):2919
Carpentier VT, Wong J, Yeap Y, Gan C, Sutton-Smith P, Badiei A, Fazzalari NL, Kuliwaba JS (2012) Increased proportion of hypermineralized osteocyte lacunae in osteoporotic and osteoarthritic human trabecular bone: Implications for bone remodeling. Bone 50(3):688–694
Cazzani A, Malagù M, Turco E (2016a) Isogeometric analysis of plane-curved beams. Mathematics and Mechanics of Solids 21(5):562–577
Cazzani A, Malagù M, Turco E, Stochino F (2016b) Constitutive models for strongly curved beams in the frame of isogeometric analysis. Mathematics and Mechanics of Solids 21(2):182–209
Chatzigeorgiou G, Javili A, Steinmann P (2014) Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Mathematics and Mechanics of Solids 19(2):193–211
Chen AE, Ginty DD, Fan CM (2005) Protein kinase A signalling via CREB controls myogenesis induced by Wnt proteins. Nature 433(7023):317
Cluzel C, Allena R (2018) A general method for the determination of the local orthotropic directions of heterogeneous materials: application to bone structures using μCT images. Mathematics and Mechanics of Complex Systems 6(4):353–367
Colangeli M, De Masi A, Presutti E (2016) Latent heat and the Fourier law. Physics Letters A 380(20):1710–1713
Colangeli M, De Masi A, Presutti E (2017) Microscopic models for uphill diffusion. Journal of Physics A: Mathematical and Theoretical 50(43):435,002
Contrafatto L, Cuomo M (2006) A framework of elastic–plastic damaging model for concrete under multiaxial stress states. International Journal of Plasticity 22(12):2272–2300
Cowin SC (1999) Bone poroelasticity. Journal of Biomechanics 32(3):217–238
Cowin SC (ed) (2001) Bone Mechanics Handbook, 2nd edn. CRC Press, Boca Raton
Cuomo M, Contrafatto L, Greco L (2014) A variational model based on isogeometric interpolation for the analysis of cracked bodies. International Journal of Engineering Science 80:173–188
Dallas SL, Bonewald LF (2010) Dynamics of the transition from osteoblast to osteocyte. Annals of the New York Academy of Sciences 1192(1):437–443
De Masi A, Ferrari PA, Lebowitz JL (1986) Reaction-diffusion equations for interacting particle systems. Journal of Statistical Physics 44(3-4):589–644
De Masi A, Gobron T, Presutti E (1995) Travelling fronts in non-local evolution equations. Archive for Rational Mechanics and Analysis 132(2):143–205
dell’Isola F, Seppecher P, Madeo A (2012) How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert”. Zeitschrift für angewandte Mathematik und Physik 63(6):1119–1141
Di Carlo A, Quiligotti S (2002) Growth and balance. Mechanics Research Communications 29(6):449–456
Diebels S, Steeb H (2003) Stress and couple stress in foams. Computational Materials Science 28(3–4):714–722
Engelbrecht J, Berezovski A (2015) Reflections on mathematical models of deformation waves in elastic microstructured solids. Mathematics and Mechanics of Complex Systems 3(1):43–82
Epstein M, Maugin GA (2000) Thermomechanics of volumetric growth in uniform bodies. International Journal of Plasticity 16(7):951–978
Eremeyev VA, PietraszkiewiczW(2009) Phase transitions in thermoelastic and thermoviscoelastic shells. Archives of Mechanics 61(1):41–67
Eremeyev VA, Pietraszkiewicz W (2011) Thermomechanics of shells undergoing phase transition. Journal of Mechanics and Physics of Solids 59(7):1395–1412
Eremeyev VA, Pietraszkiewicz W (2016) Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Mathematics and Mechanics of Solids 21(2):210–221
Eremeyev VA, Lebedev LP, Altenbach H (2013) Foundations of Micropolar Mechanics. Springer, Berlin
Eremeyev VA, Lebedev LP, Cloud MJ (2015) The Rayleigh and Courant variational principles in the six-parameter shell theory. Mathematics and Mechanics of Solids 20(7):806–822
Eriksen EF (2010) Cellular mechanisms of bone remodeling. Reviews in Endocrine and Metabolic Disorders 11(4):219–227
Franciosi P, Spagnuolo M, Salman OU (2019) Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Continuum Mechanics and Thermodynamics 31(1):101–132
Frost HM (1987) Bone “mass” and the “mechanostat”: a proposal. The Anatomical Record 219(1):1–9
Fung YC (2006) Biomechanics. Mechanical Properties of Living Tissues, 2nd edn. Springer, New York
Ganghoffer JF (2012) A contribution to the mechanics and thermodynamics of surface growth. application to bone external remodeling. International Journal of Engineering Science 50(1):166– 91
Garikipati K, Olberding JE, Narayanan H, Arruda EM, Grosh K, Calve S (2006) Biological remodelling: stationary energy, configurational change, internal variables and dissipation. Journal of the Mechanics and Physics of Solids 54(7):1493–1515
George D, Allena R, Remond Y (2017) Mechanobiological stimuli for bone remodeling: mechanical energy, cell nutriments and mobility. Computer Methods in Biomechanics and Biomedical Engineering 20(S1):91–92
George D, Allena R, Remond Y (2018) A multiphysics stimulus for continuum mechanics bone remodeling. Mathematics and Mechanics of Complex Systems 6(4):307–319
Gibson LJ, Ashby MF (1997) Cellular Solids: Structure and Properties, 2nd edn. Cambridge Solid State Science Series, Cambridge University Press, Cambridge
Giorgio I, Andreaus U, Scerrato D, dell’Isola F (2016) A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials. Biomechanics and Modeling in Mechanobiology 15(5):1325–1343
Giorgio I, Andreaus U, dell’Isola F, Lekszycki T (2017a) Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts. Extreme Mechanics Letters 13:141–147
Giorgio I, Andreaus U, Lekszycki T, Della Corte A (2017b) The influence of different geometries of matrix/scaffold on the remodeling process of a bone and bioresorbable material mixture with voids. Mathematics and Mechanics of Solids 22(5):969–987
Giorgio I, Andreaus U, Scerrato D, Braidotti P (2017c) Modeling of a non-local stimulus for bone remodeling process under cyclic load: Application to a dental implant using a bioresorbable porous material. Mathematics and Mechanics of Solids 22(9):1790–1805
Goda I, Assidi M, Ganghoffer JF (2014) A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomechanics and Modeling in Mechanobiology 13(1):53–83
Gong Y, Slee RB, Fukai N, et al (2001) LDL receptor-related protein 5 (LRP5) affects bone accrual and eye development. Cell 107(4):513–523
Goriely A, Robertson-Tessi M, Tabor M, Vandiver R (2008) Elastic growth models. In: Mondaini RP, Pardalos PM (eds) Mathematical Modelling of Biosystems, Applied Optimization, vol 102, Springer, pp 1–44
Hambli R (2014) Connecting mechanics and bone cell activities in the bone remodeling process: an integrated finite element modeling. Frontiers in Bioengineering and Biotechnology 2(6):1–12
Himeno-Ando A, Izumi Y, Yamaguchi A, Iimura T (2012) Structural differences in the osteocyte network between the calvaria and long bone revealed by three-dimensional fluorescence morphometry, possibly reflecting distinct mechano-adaptations and sensitivities. Biochemical and Biophysical Research Communications 417(2):765–770
Holzapfel GA, Ogden RW (eds) (2006) Mechanics of Biological Tissue. Springer, Berlin
van Hove RP, Nolte PA, Vatsa A, Semeins CM, Salmon PL, Smit TH, Klein-Nulend J (2009) Osteocyte morphology in human tibiae of different bone pathologies with different bone mineral density — Is there a role for mechanosensing? Bone 45(2):321–329
Imatani S, Maugin GA (2002) A constitutive model for material growth and its application to threedimensional finite element analysis. Mechanics Research Communications 29(6):477–483
Khalili N, Selvadurai APS (2003) A fully coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity. Geophysical Research Letters 30(24)
Komori T (2013) Functions of the osteocyte network in the regulation of bone mass. Cell and Tissue Research 352(2):191–198
Kühl M, Sheldahl LC, Park M, Miller JR, Moon RT (2000) The Wnt/Ca2+ pathway: a new vertebrate Wnt signaling pathway takes shape. Trends in Genetics 16(7):279–283
Lekszycki T, dell’Isola F (2012) A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 92(6):426–444
Lekszycki T, Bucci S, Del Vescovo D, Turco E, Rizzi NL (2017) A comparison between different approaches for modelling media with viscoelastic properties via optimization analyses. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 97(5):515–531
Lemaire T, Kaiser J, Naili S, Sansalone V (2010) Modelling of the transport in electrically charged porous media including ionic exchanges. Mechanics Research Communications 37(5):495–499
Li J, Slesarenko V, Rudykh S (2019) Microscopic instabilities and elastic wave propagation in finitely deformed laminates with compressible hyperelastic phases. European Journal of Mechanics-A/Solids 73:126–136
Lu Y, Lekszycki T (2017) Modelling of bone fracture healing: influence of gap size and angiogenesis into bioresorbable bone substitute. Mathematics and Mechanics of Solids 22(10):1997–2010
Lu Y, Lekszycki T (2018) New description of gradual substitution of graft by bone tissue including biomechanical and structural effects, nutrients supply and consumption. Continuum Mechanics and Thermodynamics 30(5):995–1009
Lurie S, Solyaev Y, Volkov A, Volkov-Bogorodskiy D (2018a) Bending problems in the theory of elastic materials with voids and surface effects. Mathematics and Mechanics of Solids 23(5):787–804
Lurie SA, Kalamkarov YO A L and Solyaev, Ustenko AD, Volkov AV (2018b) Continuum microdilatation modeling of auxetic metamaterials. International Journal of Solids and Structures 132:188–200
Madeo A, George D, Lekszycki T, Nierenberger M, Remond Y (2012) A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling. Comptes Rendus Mécanique 340(8):575–589
Martin RB (1984) Porosity and specific surface of bone. Critical Reviews™ in Biomedical Engineering 10(3):179–222
Menzel A (2005) Modelling of anisotropic growth in biological tissues. Biomechanics and Modeling in Mechanobiology 3(3):147–171
Misra A, Poorsolhjouy P (2015) Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Mathematics and Mechanics of Complex Systems 3(3):285–308
Misra A, Marangos O, Parthasarathy R, Spencer P (2013) Micro-scale analysis of compositional and mechanical properties of dentin using homotopic measurements. In: Andreaus U, Iacoviello D (eds) Biomedical Imaging and Computational Modeling in Biomechanics. Lecture Notes in Computational Vision and Biomechanics, vol 4, Springer, Dordrecht, pp 131–141
Misra A, Parthasarathy R, Singh V, Spencer P (2015) Micro-poromechanics model of fluidsaturated chemically active fibrous media. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 95(2):215–234
Mlodzik M (2002) Planar cell polarization: do the same mechanisms regulate Drosophila tissue polarity and vertebrate gastrulation? Trends in Genetics 18(11):564–571
Mullender MG, Huiskes R, Weinans H (1994) A physiological approach to the simulation of bone remodeling as a self-organizational control process. Journal of Biomechanics 27(11):1389– 1394
Niiranen J, Niemi AH (2017) Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates. European Journal of Mechanics-A/Solids 61:164–179
Niiranen J, Balobanov V, Kiendl J, Hosseini SB (2019) Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro-and nano-beam models. Mathematics and Mechanics of Solids 24(1):312–335
Park HC, Lakes RS (1986) Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent. Journal of Biomechanics 19(5):385–397
Pinson KI, Brennan J, Monkley S, Avery BJ, Skarnes WC (2000) An LDL-receptor-related protein mediates Wnt signalling in mice. Nature 407(6803):535
Placidi L, Barchiesi E (2018) Energy approach to brittle fracture in strain-gradient modelling. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474(2210):20170,878
Placidi L, Barchiesi E, Misra A (2018a) A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Mathematics and Mechanics of Complex Systems 6(2):77–100
Placidi L, Misra A, Barchiesi E (2018b) Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik 69(3):56
Prakash C, Singh S, Farina I, Fraternali F, Feo L (2018) Physical-mechanical characterization of biodegradable Mg-3Si-HA composites. PSU Research Review 2(2):152–174
Prendergast PJ, Taylor D (1994) Prediction of bone adaptation using damage accumulation. Journal of Biomechanics 27(8):1067–1076
Roux W (1895) Der Kampf der Teile im Organismus. 1881. Leipzig: Engelmann Ruimerman R, Hilbers P, van Rietbergen B, Huiskes R (2005) A theoretical framework for strainrelated trabecular bone maintenance and adaptation. Journal of Biomechanics 38(4):931–41
Sansalone V, Kaiser J, Naili S, Lemaire T (2013) Interstitial fluid flow within bone canaliculi and electro-chemo-mechanical features of the canalicular milieu. Biomechanics and Modeling in Mechanobiology 12(3):533–553
Scala I, Rosi G, Nguyen VH, Vayron R, Haiat G, Seuret S, Jaffard S, Naili S (2018) Ultrasonic characterization and multiscale analysis for the evaluation of dental implant stability: A sensitivity study. Biomedical Signal Processing and Control 42:37–44
Seppecher P (1993) Equilibrium of a Cahn-Hilliard fluid on a wall: influence of the wetting properties of the fluid upon the stability of a thin liquid film. European Journal of Mechanics Series B Fluids 12:69–69
Seppecher P (2000) Second-gradient theory: application to Cahn-Hilliard fluids. In: Maugin GA, Drouot R, Sidoroff F (eds) Continuum Thermomechanics. Solid Mechanics and Its Applications, vol 76, Springer, Dordrecht, pp 379–388
Spagnuolo M, Barcz K, Pfaff A, dell’Isola F, Franciosi P (2017) Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mechanics Research Communications 83:47–52
Spingarn C, Wagner D, Remond Y, George D (2017) Multiphysics of bone remodeling: a 2D mesoscale activation simulation. Bio-medical Materials and Engineering 28(s1):S153–S158
Stern AR, Nicolella DP (2013) Measurement and estimation of osteocyte mechanical strain. Bone 54(2):191–195
Taber LA (1995) Biomechanics of growth, remodeling, and morphogenesis. Applied Mechanics Reviews 48:487–545
Taber LA (2009) Towards a unified theory for morphomechanics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367(1902):3555–3583
Turner CH (1991) Homeostatic control of bone structure: An application of feedback theory. Bone 12(3):203–217
Turner CH (1998) Three rules for bone adaptation to mechanical stimuli. Bone 23(5):399–407
Vatsa A, Breuls RG, Semeins CM, Salmon PL, Smit TH, Klein-Nulend J (2008) Osteocyte morphology in fibula and calvaria—Is there a role for mechanosensing? Bone 43(3):452–458
Yang JFC, Lakes RS (1982) Experimental study of micropolar and couple stress elasticity in compact bone in bending. Journal of Biomechanics 15(2):91–98
Yeremeyev VA, Zubov LM (1999) The theory of elastic and viscoelastic micropolar liquids. Journal of Applied Mathematics and Mechanics 63(5):755–767
Yildizdag ME, Demirtas M, Ergin A (2018) Multipatch discontinuous Galerkin isogeometric analysis of composite laminates. Continuum Mechanics and Thermodynamics pp 1–14, https://doi.org/10.1007/s00161-018-0696-9
Yildizdag ME, Ardic IT, Demirtas M, Ergin A (2019) Hydroelastic vibration analysis of plates partially submerged in fluid with an isogeometric FE-BE approach. Ocean Engineering 172:316– 329
Yoo A, Jasiuk I (2006) Couple-stress moduli of a trabecular bone idealized as a 3D periodic cellular network. Journal of Biomechanics 39(12):2241–2252
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Giorgio, I., Andreaus, U., Alzahrani, F., Hayat, T., Lekszycki, T. (2019). A Diffusion Model for Stimulus Propagation in Remodeling Bone Tissues. In: Altenbach, H., Müller, W., Abali, B. (eds) Higher Gradient Materials and Related Generalized Continua. Advanced Structured Materials, vol 120. Springer, Cham. https://doi.org/10.1007/978-3-030-30406-5_5
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