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

Effects of Anacetrapib in Patients with Atherosclerotic Vascular Disease

BACKGROUND: Patients with atherosclerotic vascular disease remain at high risk for cardiovascular events despite effective statin-based treatment of low-density lipoprotein (LDL) cholesterol levels. The inhibition of cholesteryl ester transfer protein (CETP) by anacetrapib reduces LDL cholesterol levels and increases high-density lipoprotein (HDL) cholesterol levels. However, trials of other CETP inhibitors have shown neutral or adverse effects on cardiovascular outcomes. METHODS: We conducted a randomized, double-blind, placebo-controlled trial involving 30,449 adults with atherosclerotic vascular disease who were receiving intensive atorvastatin therapy and who had a mean LDL cholesterol level of 61 mg per deciliter (1.58 mmol per liter), a mean non-HDL cholesterol level of 92 mg per deciliter (2.38 mmol per liter), and a mean HDL cholesterol level of 40 mg per deciliter (1.03 mmol per liter). The patients were assigned to receive either 100 mg of anacetrapib once daily (15,225 patients) or matching placebo (15,224 patients). The primary outcome was the first major coronary event, a composite of coronary death, myocardial infarction, or coronary revascularization. RESULTS: During the median follow-up period of 4.1 years, the primary outcome occurred in significantly fewer patients in the anacetrapib group than in the placebo group (1640 of 15,225 patients [10.8%] vs. 1803 of 15,224 patients [11.8%]; rate ratio, 0.91; 95% confidence interval, 0.85 to 0.97; P=0.004). The relative difference in risk was similar across multiple prespecified subgroups. At the trial midpoint, the mean level of HDL cholesterol was higher by 43 mg per deciliter (1.12 mmol per liter) in the anacetrapib group than in the placebo group (a relative difference of 104%), and the mean level of non-HDL cholesterol was lower by 17 mg per deciliter (0.44 mmol per liter), a relative difference of -18%. There were no significant between-group differences in the risk of death, cancer, or other serious adverse events. CONCLUSIONS: Among patients with atherosclerotic vascular disease who were receiving intensive statin therapy, the use of anacetrapib resulted in a lower incidence of major coronary events than the use of placebo. (Funded by Merck and others; Current Controlled Trials number, ISRCTN48678192 ; ClinicalTrials.gov number, NCT01252953 ; and EudraCT number, 2010-023467-18 .)