Numerical prediction of osteoporosis evolution is a challenging objective in medicine, particularly when one desires to account for patient dependency. The use of statistical methods to reconstruct bone microstructure distribution could be a helpful tool for this prediction, as they are able to provide different types of microstructures that can be optimized to fit with each patient. An initial bone sample was obtained from high-resolution X-ray computed tomography (HRμCT). Its microstructure evolution in time using a previously developed degradation model was used as the ground truth. Statistical bone microstructures were reconstructed at different stages of this evolution using two-point correlation functions (TPCFs). A blind search approach is used to find the optimized statistical microstructures, and the optimized coefficient showed less than 2% TPCF error between the statistical reconstruction and the degraded model. The statistical models also showed less than 13% error in the corresponding mechanical properties. The results showed a good correlation between the developed approach and the ground truth. The method could be extrapolated to account for the physical characterization of patient dependency to predict bone density loss over time.
HardyRCooperM.Bone loss in inflammatory disorders. J Endocrinol2009; 201(3): 309–320.
2.
SugawaraYKamiokaHHonjoT, et al. Three-dimensional reconstruction of chick calvarial osteocytes and their cell processes using confocal microscopy. Bone2005; 36(5): 877–883.
3.
HadjidakisDJAndroulakis. II. Bone remodeling. Ann N Y Acad Sci2006; 1092(1): 385–396.
4.
Klein-NulendJBakkerADBacabacRG, et al. Mechanosensation and transduction in osteocytes. Bone2013; 54(2): 182–190.
5.
Florencio-SilvaRSassoGRSasso-CerriE, et al. Biology of bone tissue: structure, function, and factors that influence bone cells. BioMed Res Int2015; 2015: 421746.
6.
VeniMACRajathiP.Interaction between bone cells in bone remodeling. J Acad Dent Educ2017; 2: 1–6.
7.
StoltzJ-FMagdalouJGeorgeD, et al. Influence of mechanical forces on bone: introduction to mechanobiology and mechanical adaptation concept. J Cellul Immunother2018; 4: 10–12.
8.
StoltzJ-FChenYMagdalouJ, et al. Response of bone cells to mechanical stimulations: an overview. In: StoltzJ-FChenYMagdalouJ, et al. (eds) Stem cells and regenerative medicine (Biomedical and health research), vol. 79. Amsterdam: IOS Press, 2019, pp. 71–76.
9.
WangQSeemanE.Skeletal growth and peak bone strength. Best Pract Res Clin Endocrinol Metab2008; 22(5): 687–700.
10.
ThudiumC.Development of novel models for studying osteoclasts. Doctoral Dissertation, Faculty of Medicine, Lund University, Lund, 2014, p. 111.
11.
BakalovaLAndreasenCMThomsenJS, et al. Intracortical bone mechanics are related to pore morphology and remodeling in human bone. J Bone Miner Res2018; 33(12): 2177–2185.
LangTFLeblancADEvansHJ, et al. Adaptation of the proximal femur to skeletal reloading after long-duration spaceflight. J Bone Miner Res2006; 21(8): 1224–1230.
14.
ChenHZhouXFujitaH, et al. Age-related changes in trabecular and cortical bone microstructure. Int J Endocrinol2013; 2013: 213234.
15.
CarterDVan der MeulenMCBeaupreGS.Mechanical factors in bone growth and development. Bone1996; 18(1): S5–S10.
16.
CasanovaRMoukokoDPithiouxM, et al. Temporal evolution of skeletal regenerated tissue: what can mechanical investigation add to biological?Med Biol Eng Comput2010; 48(8): 811–819.
17.
HuiskesRRuimermanRVan LentheGH, et al. Effects of mechanical forces on maintenance and adaptation of form in trabecular bone. Nature2000; 405(6787): 704–706.
18.
WolffY.Das Gesetz der Transformation der Knochen. Berlin: Verlag von August Hirschwald, 1892.
19.
CowinSC.Wolff’s law of trabecular architecture at remodeling equilibrium. J Biomed Eng1986; 108(1): 83–88.
20.
FrostHM.Bone “mass” and the “mechanostat”: a proposal. Anat Rec1987; 219(1): 1–9.
21.
GiorgioISpagnuoloMAndreausU, et al. In-depth gaze at the astonishing mechanical behavior of bone: a review for designing bio-inspired hierarchical metamaterials. Math Mech Solids2020; 26(7): 1074–1103.
22.
PivonkaPZimakJSmithDW, et al. Model structure and control of bone remodeling: a theoretical study. Bone2008; 43(2): 249–263.
23.
ScalaISpingarnCRémondY, et al. Mechanically-driven bone remodeling simulation: application to LIPUS treated rat calvarial defects. Math Mech Solids2017; 22(10): 1976–1988.
24.
MadeoAGeorgeDLekszyckiT, et al. A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodeling. CR Mécanique2012; 340(8): 575–589.
25.
GiorgioIAndreausUAlzahraniF, et al. On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon. Biomech Model Mechanobiol2019; 18(3): 1639–1663.
26.
ScerratoDBersaniAGiorgioI.Bio-inspired design of a porous resorbable scaffold for bone reconstruction: a preliminary study. Biomimetics2021; 6(1): 18.
27.
GiorgioIAndreausUDell’IsolaF, et al. Viscous second gradient porous material for bones reconstructed with bio-resorbable grafts. Extreme Mech Lett2017; 13: 141–147.
28.
Dell’IsolaFEremeyevVAPorubovAV.Advances in mechanics of microstructured media and structures (Advanced structured materials), vol. 87. Cham: Springer, 2018.
29.
GiorgioIDe AngeloMTurcoE, et al. A Biot–Cosserat two-dimensional elastic nonlinear model for a micromorphic medium. Contin Mech Thermodyn2020; 32(5): 1357–1369.
30.
La ValleGMassoumiS. A new deformation measure for micropolar plates subjected to in-plane loads. Contin Mech Thermodyn2022; 34(1): 243–257.
AndreausUCollocaMIacovielloD, et al. Optimal-tuning PID control of adaptive materials for structural efficiency. Struct Multidiscipl Optim2011; 43(1): 43–59.
33.
PrendergastPTaylorD.Prediction of bone adaptation using damage accumulation. J Biomech1994; 27(8): 1067–1076.
34.
DoblaréMGarcıaJ.Anisotropic bone remodeling model based on a continuum damage-repair theory. J Biomech2002; 35(1): 1–17.
35.
Dell’IsolaFAndreausUPlacidiL.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 Solids2015; 20(8): 887–928.
36.
MisraAPoorsolhjouyP.Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math Mech Complex Syst2015; 3(3): 285–308.
37.
PlacidiLAndreausUCorteAD, et al. Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Z Angew Math Phys2015; 66(6): 3699–3725.
38.
AbaliBEMüllerWH.Theory and computation of higher gradient elasticity theories based on action principles. Arch Appl Mech2017; 87(9): 1495–1510.
39.
Dell’IsolaFCorteADGiorgioI.Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math Mech Solids2017; 22(4): 852–872.
40.
RémondYAhziSBaniassadiM, et al. Applied RVE reconstruction and homogenization of heterogeneous materials. New York: Wiley Online Library, 2016.
41.
MartinMLemaireTHaiatG, et al. A thermodynamically consistent model of bone rotary remodeling: a 2D study. Comput Methods Biomech Biomed Eng2017; 20(Suppl. 1): 127–128.
42.
GeorgeDSpingarnCDissauxC, et al. Examples of multiscale and multiphysics numerical modeling of biological tissues. Biomed Mater Eng2017; 281: S15–S27.
43.
SpingarnCWagnerDRémondY, et al. Multiphysics of bone remodeling: a 2D mesoscale activation simulation. Biomed Mater Eng2017; 281: S153–S158.
44.
KazempourMBaniassadiMShahsavariH, et al. Homogenization of heterogeneous brain tissue under quasi-static loading: a visco-hyperelastic model of a 3D RVE. Biomech Model Mechanobiol2019; 18(4): 969–981.
45.
HashemiMSBaniassadiMBaghaniM, et al. A novel machine learning based computational framework for homogenization of heterogeneous soft materials: application to liver tissue. Biomech Model Mechanobiol2020; 19: 1131–1142.
46.
HollisterSJBrennanJMKikuchiN.A homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress. J Biomech1994; 27(4): 433–444.
47.
TsubotaK-ISuzukiYYamadaT, et al. Computer simulation of trabecular remodeling in human proximal femur using large-scale voxel FE models: approach to understanding Wolff’s law. J Biomech2009; 42(8): 1088–1094.
48.
JangIGKimIY.Computational simulation of simultaneous cortical and trabecular bone change in human proximal femur during bone remodeling. J Biomech2010; 43(2): 294–301.
49.
MarzbanANayeb-HashemiHVaziriA.Numerical simulation of load-induced bone structural remodeling using stress-limit criterion. Comput Methods Biomech Biomed Eng2015; 18(3): 259–268.
50.
KershMEZyssetPKPahrDH, et al. Measurement of structural anisotropy in femoral trabecular bone using clinical-resolution CT images. J Biomech2013; 46(15): 2659–2666.
51.
LianW-DLegrainGCartraudP.Image-based computational homogenization and localization: comparison between X-FEM/levelset and voxel-based approaches. Comput Mech2013; 51(3): 279–293.
52.
KazempourMBagherianASheidaeiA, et al. Numerical simulation of osteoporosis degradation at local scale: a preliminary study on the kinematic loss of mechanical bone stiffness. In: StoltzJ-FChenYMagdalouJ, et al. (eds) Stem cells and regenerative medicine. Amsterdam: IOS Press, 2019, pp. 87–94.
53.
GeorgeDAllenaRRemondY.A multiphysics stimulus for continuum mechanics bone remodeling. Math Mech Complex Syst2018; 6(4): 307–319.
54.
GeorgeDAllenaRRemondY.Integrating molecular and cellular kinetics into a coupled continuum mechanobiological stimulus for bone reconstruction. Contin Mech Thermodyn2019; 31(3): 725–740.
55.
GeorgeDAllenaRBourzacC, et al. A new comprehensive approach for bone remodeling under medium and high mechanical load based on cellular activity. Math Mech Complex Syst2020; 8(4): 287–306.
56.
GodaIAssidiMBelouettarS, et al. A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. J Mech Behav Biomed Mater2012; 16: 87–108.
57.
RäthCBaumTMonettiR, et al. Scaling relations between trabecular bone volume fraction and microstructure at different skeletal sites. Bone2013; 57(2): 377–383.
58.
GodaIAssidiMGanghofferJF.A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech Model Mechanobiol2014; 13(1): 53–83.
59.
WierszyckiMSzajekKŁodygowskiT, et al. A two-scale approach for trabecular bone microstructure modeling based on computational homogenization procedure. Comput Mech2014; 54(2): 287–298.
60.
GodaIGanghofferJFCzarneckiS, et al. Optimal internal architectures of femoral bone based on relaxation by homogenization and isotropic material design. Mech Res Commun2016; 76: 64–71.
61.
BagherianABaghaniMGeorgeD, et al. A novel numerical model for the prediction of patient-dependent bone density loss in microgravity based on micro-CT images. Contin Mech Thermodyn2019; 32: 927–943.
62.
SheidaeiAKazempourMHasanabadiA, et al. Influence of bone microstructure distribution on developed mechanical energy for bone remodeling using a statistical reconstruction method. Math Mech Solids2019; 24(10): 3027–3041.
63.
FamouriSBagherianAShahmohammadiA, et al. Refining anticipation of degraded bone microstructures during osteoporosis based on statistical homogenized reconstruction method via quality of connection function. Int J Comput Mater Sci Eng2020; 9(4): 2050023.
64.
HashinZShtrikmanS.A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids1963; 11(2): 127–140.
65.
GokhaleATewariAGarmestaniH.Constraints on microstructural two-point correlation functions. Scr Mater2005; 53(8): 989–993.
66.
AdamsBLKalidindiSRFullwoodD.Microstructure sensitive design for performance optimization. Oxford: Butterworth-Heinemann, 2012.
67.
FullwoodDKalidindiSRNiezgodaSR, et al. Gradient-based microstructure reconstructions from distributions using fast Fourier transforms. Mater Sci Eng A2008; 494(1–2): 68–72.
68.
GerchbergRW.A practical algorithm for the determination of phase from image and diffraction plane pictures. Optik1972; 35: 237–246.
