Predicting cortical bone adaptation to axial loading in the mouse tibia

The development of predictive mathematical models can contribute to a deeper understanding of the specific stages of bone mechanobiology and the process by which bone adapts to mechanical forces. The objective of this work was to predict, with spatial accuracy, cortical bone adaptation to mechanical load, in order to better understand the mechanical cues that might be driving adaptation. The axial tibial loading model was used to trigger cortical bone adaptation in C57BL/6 mice and provide relevant biological and biomechanical information. A method for mapping cortical thickness in the mouse tibia diaphysis was developed, allowing for a thorough spatial description of where bone adaptation occurs. Poroelastic finite-element (FE) models were used to determine the structural response of the tibia upon axial loading and interstitial fluid velocity as the mechanical stimulus. FE models were coupled with mechanobiological governing equations, which accounted for non-static loads and assumed that bone responds instantly to local mechanical cues in an on–off manner. The presented formulation was able to simulate the areas of adaptation and accurately reproduce the distributions of cortical thickening observed in the experimental data with a statistically significant positive correlation (Kendall's τ rank coefficient τ = 0.51, p < 0.001). This work demonstrates that computational models can spatially predict cortical bone mechanoadaptation to a time variant stimulus. Such models could be used in the design of more efficient loading protocols and drug therapies that target the relevant physiological mechanisms

Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models

Mineralized collagen fibrils have been usually analyzed like a two phase composite material where crystals are considered as platelets that constitute the reinforcement phase. Different models have been used to describe the elastic behavior of the material. In this work, it is shown that, when Halpin-Tsai equations are applied to estimate elastic constants from typical constituent properties, not all crystal dimensions yield a model that satisfy thermodynamic restrictions. We provide the ranges of platelet dimensions that lead to positive definite stiffness matrices. On the other hand, a finite element model of a mineralized collagen fibril unit cell under periodic boundary conditions is analyzed. By applying six canonical load cases, homogenized stiffness matrices are numerically calculated. Results show a monoclinic behavior of the mineralized collagen fibril. In addition, a 5-layer lamellar structure is also considered where crystals rotate in adjacent layers of a lamella. The stiffness matrix of each layer is calculated applying Lekhnitskii transformations and a new finite lement model under periodic boundary conditions is analyzed to calculate the homogenized 3D anisotropic stiffness matrix of a unit cell of lamellar bone. Results are compared with the rule-of-mixtures showing in general good agreement.The authors acknowledge the Ministerio de Economia y Competitividad the financial support given through the project DPI2010-20990 and the Generalitat Valenciana through the Programme Prometeo 2012/023. The authors thank Ms. Carla Gonzalez Carrillo by her help in the development of some of the numerical models.Vercher Martínez, A.; Giner Maravilla, E.; Arango Villegas, C.; Tarancón Caro, JE.; Fuenmayor Fernández, FJ. (2014). Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models. Biomechanics and Modeling in Mechanobiology. 13(2):1-21. https://doi.org/10.1007/s10237-013-0507-yS121132Akiva U, Wagner HD, Weiner S (1998) Modelling the three-dimensional elastic constants of parallel-fibred and lamellar bone. J Mater Sci 33:1497–1509Ascenzi A, Bonucci E (1967) The tensile properties of single osteons. Anat Rec 158:375–386Ascenzi A, Bonucci E (1968) The compressive properties of single osteons. Anat Rec 161:377–392Ashman RB, Cowin SC, van Buskirk WC, Rice JC (1984) A continuous wave technique for the measurement of the elastic properties of cortical bone. J Biomech 17:349–361Bar-On B, Wagner HD (2012) Elastic modulus of hard tissues. J Biomech 45:672–678Bondfield W, Li CH (1967) Anisotropy of nonelastic flow in bone. J Appl Phys 38:2450–2455Cowin SC (2001) Bone mechanics handbook, 2nd edn. CRC Press Boca Raton, FloridaCowin SC, van Buskirk WC (1986) Thermodynamic restrictions on the elastic constant of bone. J Biomech 19:85–86Currey JD (1962) Strength of bone. Nature 195:513Cusack S, Miller A (1979) Determination of the elastic constants of collagen by brillouin light scattering. J Mol Biol 135:39–51Doty S, Robinson RA, Schofield B (1976) Morphology of bone and histochemical staining characteristics of bone cells. In: Aurbach GD (ed) Handbook of physiology. American Physiology Soc, Washington, pp 3–23Erts D, Gathercole LJ, Atkins EDT (1994) Scanning probe microscopy of crystallites in calcified collagen. J Mater Sci Mater Med 5:200–206Faingold A, Sidney RC, Wagner HD (2012) Nanoindentation of osteonal bone lamellae. J Mech Biomech Materials 9:198–206Franzoso G, Zysset PK (2009) Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation. J Biomech Eng 131:021001Gebhardt W (1906) Über funktionell wichtige Anordnungsweisen der eineren und grösseren Bauelemente des Wirbeltierknochens. II. Spezieller Teil. Der Bau der Haversschen Lamellensysteme und seine funktionelle Bedeutung. Arch Entwickl Mech Org 20:187–322Gibson RF (1994) Principles of composite material mechanics. McGraw-Hill, New YorkGiraud-Guille M (1988) Twisted plywood architecture of collagen fibrils in human compact bone osteons. Calcif Tissue Int 42:167–180Gurtin ME (1972) The linear theory of elasticity. Handbuch der Physik VIa/ 2:1–296Halpin JC (1992) Primer on composite materials: analysis, 2nd edn. CRC Press, Taylor & Francis, Boca Raton, FloridaHassenkam T, Fantner GE, Cutroni JA, Weaver JC, Morse DE, Hanma PK (2004) High-resolution AFM imaging of intact and fractured trabecular bone. Bone 35:4–10Hohe J (2003) A direct homogenization approach for determination of the stiffness matrix for microheterogeneous plates with application to sandwich panels. Composites Part B 34:615–626Hulmes DJS, Wess TJ, Prockop DJ, Fratzl P (1995) Radial packing, order, and disorder in collagen fibrils. Biophys J 68:1661–1670Jäger I, Fratzl P (2000) Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles. Biophys J 79:1737–1746Ji B, Gao H (2004) Mechanical properties of nanostructure of biological materials. J Mech Phy Sol 52:1963–1990Landis WJ, Hodgens KJ, Aerna J, Song MJ, McEwen BF (1996) Structural relations between collagen and mineral in bone as determined by high voltage electron microscopic tomography. Microsc Res Tech 33:192–202Lekhnitskii SG (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day, San FranciscoLempriere BM (1968) Poisson’s ratio in orthotropic materials. Am Inst Aeronaut Astronaut J J6:2226–2227Lowenstam HA, Weiner S (1989) On biomineralization. Oxford University, New YorkLusis J, Woodhams RT, Xhantos M (1973) The effect of flake aspect ratio on flexural properties of mica reinforced plastics. Polym Eng Sci 13:139–145Martínez-Reina J, Domínguez J, García-Aznar JM (2011) Effect of porosity and mineral content on the elastic constants of cortical bone: a multiscale approach. Biomech Model Mechanobiol 10:309–322Orgel JPRO, Miller A, Irving TC, Fischetti RF, Hammersley AP, Wess TJ (2001) The in situ supermolecular structure of type I collagen. Structure 9:1061–1069Padawer GE, Beecher N (1970) On the strength and stiffness of planar reinforced plastic resins. Polym Eng Sci 10:185–192Pahr DH, Rammerstofer FG (2006) Buckling of honeycomb sandwiches: periodic finite element considerations. Comput Model Eng Sci 12:229–242Reisinger AG, Pahr DH, Zysset PK (2010) Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods. Biomech Model Mechanobiol 9:499–510Reisinger AG, Pahr DH, Zysset PK (2011) Elastic anisotropy of bone lamellae as a function of fibril orientation pattern. Biomech Model Mechanobiol 10:67–77Rezkinov N, Almany-Magal R, Shahar R, Weiner S (2013) Three-dimensional imaging of collagen fibril organization in rat circumferential lamellar bone using a dual beam electron microscope reveals ordered and disordered sub-lamellar structures. Bone 52(2):676–683Rho JY, Kuhn-Spearing L, Zioupos P (1998) Mechanical properties and the hierarchical structure of bone. Med Eng Phys 20:92–102Rubin MA, Jasiuk I, Taylor J, Rubin J, Ganey T, Apkarian RP (2003) TEM analysis of the nanostructure of normal and osteoporotic human trabecular bone. Bone 33:270–282Suquet P (1987) Lecture notes in physics-homogenization techniques for composite media. Chapter IV. Springer, BerlinWagermaier W, Gupta HS, Gourrier A, Burghammer M, Roschger P, Fratzl P (2006) Spiral twisting of fiber orientation inside bone lamellae. Biointerphases 1:1–5Wagner HD, Weiner S (1992) On the relationship between the microstructure of bone and its mechanical stiffness. J Biomech 25:1311–1320Weiner S, Wagner HD (1998) The material bone: structure-mechanical function relations. Annu Rev Mater Sci 28:271–298Weiner S, Traub W, Wagner H (1999) Lamellar bone: structure-function relations. J Struct Biol 126:241–255Yao H, Ouyang L, Ching W (2007) Ab initio calculation of elastic constants of ceramic crystals. J Am Ceram 90:3194–3204Yoon YJ, Cowin SC (2008b) The estimated elastic constants for a single bone osteonal lamella. Biomech Model Mechanobiol 7:1–11Yuan F, Stock SR, Haeffner DR, Almer JD, Dunand DC, Brinson LC (2011) A new model to simulate the elastic properties of mineralized collagen fibril. Biomech Model Mechanobiol 10:147–160Zhang Z, Zhang YWF, Gao H (2010) On optimal hierarchy of load-bearing biological materials. Proc R Soc B 278:519–525Zuo S, Wei Y (2007) Effective elastic modulus of bone-like hierarchical materials. Acta Mechanica Solida Sinica 20:198–20

Topology optimization for human proximal femur considering bi-modulus behavior of cortical bones

© Springer International Publishing Switzerland 2015. The material in the human proximal femur is considered as bi-modulus material and the density distribution is predicted by topology optimization method. To reduce the computational cost, the bi-modulus material is replaced with two isotropic materials in simulation. The selection of local material modulus is determined by the previous local stress state. Compared with density prediction results by traditional isotropic material in proximal femur, the bi-modulus material layouts are different obviously. The results also demonstrate that the bi-modulus material model is better than the isotropic material model in simulation of density prediction in femur bone

Development of an in vitro three dimensional loading-measurement system for long bone fixation under multiple loading conditions: a technical description

The purpose of this investigation was to design and verify the capabilities of an in vitro loading-measurement system that mimics in vivo unconstrained three dimensional (3D) relative motion between long bone ends, applies uniform load components over the entire length of a test specimen, and measures 3D relative motion between test segment ends to directly determine test segment construct stiffness free of errors due to potting-fixture-test machine finite stiffness

New Suggestions for the Mechanical Control of Bone Remodeling

Bone is constantly renewed over our lifetime through the process of bone (re)modeling. This process is important for bone to allow it to adapt to its mechanical environment and to repair damage from everyday life. Adaptation is thought to occur through the mechanosensitive response controlling the bone-forming and -resorbing cells. This report shows a way to extract quantitative information about the way remodeling is controlled using computer simulations. Bone resorption and deposition are described as two separate stochastic processes, during which a discrete bone packet is removed or deposited from the bone surface. The responses of the bone-forming and -resorbing cells to local mechanical stimuli are described by phenomenological remodeling rules. Our strategy was to test different remodeling rules and to evaluate the time evolution of the trabecular architecture in comparison to what is known from μ-CT measurements of real bone. In particular, we tested the reaction of virtual bone to standard therapeutic strategies for the prevention of bone deterioration, i.e., physical activity and medications to reduce bone resorption. Insensitivity of the bone volume fraction to reductions in bone resorption was observed in the simulations only for a remodeling rule including an activation barrier for the mechanical stimulus above which bone deposition is switched on. This is in disagreement with the commonly used rules having a so-called lazy zone

Mechanical Strain Stabilizes Reconstituted Collagen Fibrils against Enzymatic Degradation by Mammalian Collagenase Matrix Metalloproteinase 8 (MMP-8)

Collagen, a triple-helical, self-organizing protein, is the predominant structural protein in mammals. It is found in bone, ligament, tendon, cartilage, intervertebral disc, skin, blood vessel, and cornea. We have recently postulated that fibrillar collagens (and their complementary enzymes) comprise the basis of a smart structural system which appears to support the retention of molecules in fibrils which are under tensile mechanical strain. The theory suggests that the mechanisms which drive the preferential accumulation of collagen in loaded tissue operate at the molecular level and are not solely cell-driven. The concept reduces control of matrix morphology to an interaction between molecules and the most relevant, physical, and persistent signal: mechanical strain.The investigation was carried out in an environmentally-controlled microbioreactor in which reconstituted type I collagen micronetworks were gently strained between micropipettes. The strained micronetworks were exposed to active matrix metalloproteinase 8 (MMP-8) and relative degradation rates for loaded and unloaded fibrils were tracked simultaneously using label-free differential interference contrast (DIC) imaging. It was found that applied tensile mechanical strain significantly increased degradation time of loaded fibrils compared to unloaded, paired controls. In many cases, strained fibrils were detectable long after unstrained fibrils were degraded.In this investigation we demonstrate for the first time that applied mechanical strain preferentially preserves collagen fibrils in the presence of a physiologically-important mammalian enzyme: MMP-8. These results have the potential to contribute to our understanding of many collagen matrix phenomena including development, adaptation, remodeling and disease. Additionally, tissue engineering could benefit from the ability to sculpt desired structures from physiologically compatible and mutable collagen

Finite difference calculations of permeability in large domains in a wide porosity range.

Determining effective hydraulic, thermal, mechanical and electrical properties of porous materials by means of classical physical experiments is often time-consuming and expensive. Thus, accurate numerical calculations of material properties are of increasing interest in geophysical, manufacturing, bio-mechanical and environmental applications, among other fields. Characteristic material properties (e.g. intrinsic permeability, thermal conductivity and elastic moduli) depend on morphological details on the porescale such as shape and size of pores and pore throats or cracks. To obtain reliable predictions of these properties it is necessary to perform numerical analyses of sufficiently large unit cells. Such representative volume elements require optimized numerical simulation techniques. Current state-of-the-art simulation tools to calculate effective permeabilities of porous materials are based on various methods, e.g. lattice Boltzmann, finite volumes or explicit jump Stokes methods. All approaches still have limitations in the maximum size of the simulation domain. In response to these deficits of the well-established methods we propose an efficient and reliable numerical method which allows to calculate intrinsic permeabilities directly from voxel-based data obtained from 3D imaging techniques like X-ray microtomography. We present a modelling framework based on a parallel finite differences solver, allowing the calculation of large domains with relative low computing requirements (i.e. desktop computers). The presented method is validated in a diverse selection of materials, obtaining accurate results for a large range of porosities, wider than the ranges previously reported. Ongoing work includes the estimation of other effective properties of porous media

MSH3 polymorphisms and protein levels affect CAG repeat instability in huntington's disease mice

Expansions of trinucleotide CAG/CTG repeats in somatic tissues are thought to contribute to ongoing disease progression through an affected individual's life with Huntington's disease or myotonic dystrophy. Broad ranges of repeat instability arise between individuals with expanded repeats, suggesting the existence of modifiers of repeat instability. Mice with expanded CAG/CTG repeats show variable levels of instability depending upon mouse strain. However, to date the genetic modifiers underlying these differences have not been identified. We show that in liver and striatum the R6/1 Huntington's disease (HD) (CAG)~100 transgene, when present in a congenic C57BL/6J (B6) background, incurred expansion-biased repeat mutations, whereas the repeat was stable in a congenic BALB/cByJ (CBy) background. Reciprocal congenic mice revealed the Msh3 gene as the determinant for the differences in repeat instability. Expansion bias was observed in congenic mice homozygous for the B6 Msh3 gene on a CBy background, while the CAG tract was stabilized in congenics homozygous for the CBy Msh3 gene on a B6 background. The CAG stabilization was as dramatic as genetic deficiency of Msh2. The B6 and CBy Msh3 genes had identical promoters but differed in coding regions and showed strikingly different protein levels. B6 MSH3 variant protein is highly expressed and associated with CAG expansions, while the CBy MSH3 variant protein is expressed at barely detectable levels, associating with CAG stability. The DHFR protein, which is divergently transcribed from a promoter shared by the Msh3 gene, did not show varied levels between mouse strains. Thus, naturally occurring MSH3 protein polymorphisms are modifiers of CAG repeat instability, likely through variable MSH3 protein stability. Since evidence supports that somatic CAG instability is a modifier and predictor of disease, our data are consistent with the hypothesis that variable levels of CAG instability associated with polymorphisms of DNA repair genes may have prognostic implications for various repeat-associated diseases