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Modelling cortical bone using the method of asymptotic homogenization
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Journal o f Biomechanics 2006, Vol. 39 (Suppl 1)
and physical chemical properties (carbonate and protein content, apatite crystal size, crystallized apatite/total phosphate and non-apatite phosphate/total phosphate) were assessed on femur bone of young to old Wistar rat (1, 4, 9, 12, 18 and 24 months old). Mechanical properties were found to increase during growth (p<0.05), to decrease (p < 0.05) and to stabilize (p > 0.05) during senescence. Microporosity was found to decrease during growth (p < 0.05) and then to stabilize (p > 0.05). Composition analysis show no evolution of apatite and non apatite content, significant increase of carbonate content, a decrease of protein content (p<0.05), and an increase of the apatite crystal size. During ageing, no signicant variations of the bone matrix composition were detected although crystal size increases slowly. The correlation between parameters show the impact of the maturation of bone matrix (decrease of collagen ratio and increase of the carbonate ratio) and the decrease of microporosity on the increase of bone mechanical properties. However, the significant decrease of mechanical properties during senescence do not seem related neither to matrix nor to microporosity which do not exhibit significant evolution. In comparison with human bone, adult Wistar rat cortical bone exhibits similar values of mechanical and physical-chemical properties, although microporosity and tissue structure is different (no haversian system). The bone mechanical evolution during growth seems similar, but not at senescence. These results suggested that rat bone may be useful as model of human bone maturation but for senescence, the ageing mechanism is different. 4709 Th, 16:15-16:30 (P45) Modelling cortical bone using the method of asymptotic homogenization W.J. Parnell 1, Q. Grimal 2, I.D. Abrahams 1, P. Laugier 2. 1School of Mathematics, University of Manchester, Manchester, UK, 2Laboratoire d'lmagerie Param6trique, Universit6 Pierre et Marie Curie Paris 6; UMR CNRS 7623, Paris, France The hierarchical structure of cortical bone gives rise to an effective stiffness tensor that depends on the properties of constituent phases and their organization. Mathematical models are useful as a means of testing how changes in the microstructure and phase characteristics affect macroscopic behaviour. Previous models based on the asymptotic homogenization technique [1] lead to complicated formulations of the effective properties which are hard to reproduce in practice. We propose a model to estimate the stiffness tensor of cortical bone based on the asymptotic homogenization method, stemming from the work of Parnell and Abrahams [2]. We treat cortical bone as a two phase fibrous composite and model the propagation of elastic waves through the structure. The small parameter in the scheme is the ratio of the microstructural lengthscale to a characteristic propagating wavelength. We derive novel effective wave equations and thus easily computable closed form expressions of the material properties as functions of the isotropic phase characteristics and geometry are deduced. Furthermore the anisotropy of the homogenized composite naturally emerges from the effective equations; a slight alteration in microstructure geometry provides a natural change in anisotropy in these equations. We compare results with our own experiments and other models. In particular, the consequences of disregarding matrix anisotropy in the model are investigated. The asymptotic approach means that lower level microstructures can be built in by using three scale asymptotic expansions in two small parameters for example. Furthermore it is simple to model the perturbation of the structure from a periodic lattice and hence build in disorder. References [1] J.M Crolet, B. Aoubiza, A. Meunier. Compact bone: numerical simulation of mechanical characteristics. J. Biomech. 1993; 26(6): 677~87. [2] W.J. Parnell, I.D. Abrahams. Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves. Wave Motion, submitted July 2005. 5311 Th, 16:30-16:45 (P45) Ageing of compact bone tissue I. Knets, V. Vitins, M. Dobelis. Institute of Biomaterials and Biomechanics of the Riga Technical University, Riga, Latvia Different changes in the structure and biochemical composition of compact bone tissue have been found due to ageing. As a result, the changes are taking place in the bone tissue mechanical properties and mechanical behaviour also. Investigation of the changes of mechanical properties over six zones of cross section of human tibia in different age groups revealed the specific peculiarities. During the period up to 18-25 years there is gradual increase of stiffness (characterised by moduli of elasticity and shear moduli) and strength in tension, compression and torsion has been observed. After the age of 25 these parameters of the mechanical behaviour of compact bone tissue start to decrease gradually. For example, the modulus of elasticity of bone tissue
Oral Presentations along the longitudinal axis of tibia at the age of 95 has decreased by 21% to compare with that at the age of 25. Specific reaction to ageing has ultimate strain and ultimate strain energy in tension and torsion. For example, at the moment of fracture ultimate strain decrease by 82% comparing bone tissue at the age of one year and 80 years. Very specific is the change of the shear modulus G12 over these six zones of the tibia cross section due to ageing. The largest value of G12 starting from the age of 19 is always in one of 3 corner zones, but the lowest - always in one of intermediate zones laying just next to the zone with the largest value of G12. During increase of age the maximum value of G12 is gradually transferring to the next corner zone, but the minimum - to the next intermediate zone. During creep of bone tissue under constant longitudinal tension the non-uniformity of creep strain over zones of tibia cross section is also decreasing with age. There have been found certain correlation between parameters of biomechanical properties and values of the compounds of biochemical composition. 6089 Th, 16:45-17:00 (P45) Viscoelastic/plastic finite element simulation of nanoindentation mechanical properties o f bone J. Zhang, G. Niebur, T. Ovaert. Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA It has been shown from nanoindentation and other mechanical test experiments that a specific constitutive material model that includes both viscoelastic and plastic behavior is necessary to characterize the mechanical properties of bone. Viscoelastic materials, in particular, are sensitive to loading and unloading rates, thus, typical elastic models are of limited accuracy. In addition, permanent deformation in bone can occur during a nanoindentation test, thus the need to account for plasticity effects. To accomplish this, a four-parameter constitutive equation has been developed and implemented in a finite element simulation of nanoindentation. The constitutive equation is based on a rheological model that includes spring (modulus), dashpot (viscosity), and Ramberg-Osgood elasto-plasticity (which adds two additional non-dimensional plasticity parameters). A nanoindentation creep test was used in the experiments, having a maximum load of 100 ~tN, constant loading and unloading rates (10 ~tN/s), and a constant holding time (10 s). The four material parameters for the bone sample were then determined by an iterative process matching the experimental load vs. deformation test data with the finite element simulation results. The data yielded average modulus and viscosity values of approximately 11 GPa and 1.5 GPa s, respectively, which are reasonable values for bone and which validate the use of the proposed constitutive model. 7531 Th, 17:00-17:15 (P45) Minimal size of structural ~tFE models o f trabecular bone to predict the apparent stiffness U. Simon, J. Abel, U. Wolfram, L. Claes. Institute of Orthopaedic Research and Biomechanics, University of UIm, Germany Introduction: The mechanical properties of trabecular bone are strongly determined by its complex structure. Physiological and pathological remodeling processes (osteoporosis) can change this structure with serious consequences to its strength. High-resolution computer tomography (~tCT) provides data to generate Finite Element models (~tFE models) [1,2] to study these effects. Regarding the quality of such micro-structural models it is still unclear, which minimal size they should have to be able to predict apparent mechanical properties. It was the aim of this work to answer this question. Method: A biopsy from the proximal ovine tibia was scanned (30~tm-CT). Using this data, we generated 1000 cubic ~tFE models of 20 different sizes (edge length from 200 to 2400~tm) at 50 different positions within the biopsy. The linear-elastic models consisted of up to 600,000 elements. The apparent stiffness of the models was determined in a compression analysis. We determined the median and variation of the predicted stiffnesses depending on the model size. Results: The variation of the predicted apparent stiffnesses decreases with an increasing edge length (from 2000% with 200~tm to 8.3% with 2400~tm). The median of the stiffness converges to a value of E = 379 MPa. Only above a critical edge size of 2100 ~tm (4.7 times pore size), 90% of the models predicted the stiffness with an error smaller than 15%. Discussion: Using the presented method the size effect of structural ~tFE models for predicting the apparent stiffness was studied the first time. For trabecular bone (biopsy from the proximal ovine tibia) we determined a critical edge length of about five times the pore size. Harrigan et al. [3] studied the continuum assumption in cancellous bone with a one-dimensional linescanning method and found a similar critical length of three to five times the pore size. The presented method serves essential fundamentals, which should be used to optimize or evaluate the quality of micro-structural models. References [1] Homminga et al. (2001). J Biomech: 513-517. [2] Kabel et al. (1999). Bone: 115-120.
Journal o f Biomechanics 2006, Vol. 39 (Suppl 1)
and physical chemical properties (carbonate and protein content, apatite crystal size, crystallized apatite/total phosphate and non-apatite phosphate/total phosphate) were assessed on femur bone of young to old Wistar rat (1, 4, 9, 12, 18 and 24 months old). Mechanical properties were found to increase during growth (p<0.05), to decrease (p < 0.05) and to stabilize (p > 0.05) during senescence. Microporosity was found to decrease during growth (p < 0.05) and then to stabilize (p > 0.05). Composition analysis show no evolution of apatite and non apatite content, significant increase of carbonate content, a decrease of protein content (p<0.05), and an increase of the apatite crystal size. During ageing, no signicant variations of the bone matrix composition were detected although crystal size increases slowly. The correlation between parameters show the impact of the maturation of bone matrix (decrease of collagen ratio and increase of the carbonate ratio) and the decrease of microporosity on the increase of bone mechanical properties. However, the significant decrease of mechanical properties during senescence do not seem related neither to matrix nor to microporosity which do not exhibit significant evolution. In comparison with human bone, adult Wistar rat cortical bone exhibits similar values of mechanical and physical-chemical properties, although microporosity and tissue structure is different (no haversian system). The bone mechanical evolution during growth seems similar, but not at senescence. These results suggested that rat bone may be useful as model of human bone maturation but for senescence, the ageing mechanism is different. 4709 Th, 16:15-16:30 (P45) Modelling cortical bone using the method of asymptotic homogenization W.J. Parnell 1, Q. Grimal 2, I.D. Abrahams 1, P. Laugier 2. 1School of Mathematics, University of Manchester, Manchester, UK, 2Laboratoire d'lmagerie Param6trique, Universit6 Pierre et Marie Curie Paris 6; UMR CNRS 7623, Paris, France The hierarchical structure of cortical bone gives rise to an effective stiffness tensor that depends on the properties of constituent phases and their organization. Mathematical models are useful as a means of testing how changes in the microstructure and phase characteristics affect macroscopic behaviour. Previous models based on the asymptotic homogenization technique [1] lead to complicated formulations of the effective properties which are hard to reproduce in practice. We propose a model to estimate the stiffness tensor of cortical bone based on the asymptotic homogenization method, stemming from the work of Parnell and Abrahams [2]. We treat cortical bone as a two phase fibrous composite and model the propagation of elastic waves through the structure. The small parameter in the scheme is the ratio of the microstructural lengthscale to a characteristic propagating wavelength. We derive novel effective wave equations and thus easily computable closed form expressions of the material properties as functions of the isotropic phase characteristics and geometry are deduced. Furthermore the anisotropy of the homogenized composite naturally emerges from the effective equations; a slight alteration in microstructure geometry provides a natural change in anisotropy in these equations. We compare results with our own experiments and other models. In particular, the consequences of disregarding matrix anisotropy in the model are investigated. The asymptotic approach means that lower level microstructures can be built in by using three scale asymptotic expansions in two small parameters for example. Furthermore it is simple to model the perturbation of the structure from a periodic lattice and hence build in disorder. References [1] J.M Crolet, B. Aoubiza, A. Meunier. Compact bone: numerical simulation of mechanical characteristics. J. Biomech. 1993; 26(6): 677~87. [2] W.J. Parnell, I.D. Abrahams. Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves. Wave Motion, submitted July 2005. 5311 Th, 16:30-16:45 (P45) Ageing of compact bone tissue I. Knets, V. Vitins, M. Dobelis. Institute of Biomaterials and Biomechanics of the Riga Technical University, Riga, Latvia Different changes in the structure and biochemical composition of compact bone tissue have been found due to ageing. As a result, the changes are taking place in the bone tissue mechanical properties and mechanical behaviour also. Investigation of the changes of mechanical properties over six zones of cross section of human tibia in different age groups revealed the specific peculiarities. During the period up to 18-25 years there is gradual increase of stiffness (characterised by moduli of elasticity and shear moduli) and strength in tension, compression and torsion has been observed. After the age of 25 these parameters of the mechanical behaviour of compact bone tissue start to decrease gradually. For example, the modulus of elasticity of bone tissue
Oral Presentations along the longitudinal axis of tibia at the age of 95 has decreased by 21% to compare with that at the age of 25. Specific reaction to ageing has ultimate strain and ultimate strain energy in tension and torsion. For example, at the moment of fracture ultimate strain decrease by 82% comparing bone tissue at the age of one year and 80 years. Very specific is the change of the shear modulus G12 over these six zones of the tibia cross section due to ageing. The largest value of G12 starting from the age of 19 is always in one of 3 corner zones, but the lowest - always in one of intermediate zones laying just next to the zone with the largest value of G12. During increase of age the maximum value of G12 is gradually transferring to the next corner zone, but the minimum - to the next intermediate zone. During creep of bone tissue under constant longitudinal tension the non-uniformity of creep strain over zones of tibia cross section is also decreasing with age. There have been found certain correlation between parameters of biomechanical properties and values of the compounds of biochemical composition. 6089 Th, 16:45-17:00 (P45) Viscoelastic/plastic finite element simulation of nanoindentation mechanical properties o f bone J. Zhang, G. Niebur, T. Ovaert. Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA It has been shown from nanoindentation and other mechanical test experiments that a specific constitutive material model that includes both viscoelastic and plastic behavior is necessary to characterize the mechanical properties of bone. Viscoelastic materials, in particular, are sensitive to loading and unloading rates, thus, typical elastic models are of limited accuracy. In addition, permanent deformation in bone can occur during a nanoindentation test, thus the need to account for plasticity effects. To accomplish this, a four-parameter constitutive equation has been developed and implemented in a finite element simulation of nanoindentation. The constitutive equation is based on a rheological model that includes spring (modulus), dashpot (viscosity), and Ramberg-Osgood elasto-plasticity (which adds two additional non-dimensional plasticity parameters). A nanoindentation creep test was used in the experiments, having a maximum load of 100 ~tN, constant loading and unloading rates (10 ~tN/s), and a constant holding time (10 s). The four material parameters for the bone sample were then determined by an iterative process matching the experimental load vs. deformation test data with the finite element simulation results. The data yielded average modulus and viscosity values of approximately 11 GPa and 1.5 GPa s, respectively, which are reasonable values for bone and which validate the use of the proposed constitutive model. 7531 Th, 17:00-17:15 (P45) Minimal size of structural ~tFE models o f trabecular bone to predict the apparent stiffness U. Simon, J. Abel, U. Wolfram, L. Claes. Institute of Orthopaedic Research and Biomechanics, University of UIm, Germany Introduction: The mechanical properties of trabecular bone are strongly determined by its complex structure. Physiological and pathological remodeling processes (osteoporosis) can change this structure with serious consequences to its strength. High-resolution computer tomography (~tCT) provides data to generate Finite Element models (~tFE models) [1,2] to study these effects. Regarding the quality of such micro-structural models it is still unclear, which minimal size they should have to be able to predict apparent mechanical properties. It was the aim of this work to answer this question. Method: A biopsy from the proximal ovine tibia was scanned (30~tm-CT). Using this data, we generated 1000 cubic ~tFE models of 20 different sizes (edge length from 200 to 2400~tm) at 50 different positions within the biopsy. The linear-elastic models consisted of up to 600,000 elements. The apparent stiffness of the models was determined in a compression analysis. We determined the median and variation of the predicted stiffnesses depending on the model size. Results: The variation of the predicted apparent stiffnesses decreases with an increasing edge length (from 2000% with 200~tm to 8.3% with 2400~tm). The median of the stiffness converges to a value of E = 379 MPa. Only above a critical edge size of 2100 ~tm (4.7 times pore size), 90% of the models predicted the stiffness with an error smaller than 15%. Discussion: Using the presented method the size effect of structural ~tFE models for predicting the apparent stiffness was studied the first time. For trabecular bone (biopsy from the proximal ovine tibia) we determined a critical edge length of about five times the pore size. Harrigan et al. [3] studied the continuum assumption in cancellous bone with a one-dimensional linescanning method and found a similar critical length of three to five times the pore size. The presented method serves essential fundamentals, which should be used to optimize or evaluate the quality of micro-structural models. References [1] Homminga et al. (2001). J Biomech: 513-517. [2] Kabel et al. (1999). Bone: 115-120.