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Computer Methods in Biomechanics and Biomedical Engineering, Vol. 11, No. 2, April 2008, 147–157
Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale QUENTIN GRIMAL†‡*, KAY RAUM{k, ALF GERISCH§# and PASCAL LAUGIER†‡** †Laboratoire d’Imagerie Parame´trique, Universite´ Pierre et Marie Curie-Paris6, Paris F-75005, France ‡CNRS, LIP, Paris F-75006, France {Q-BAM Group, Department of Orthopedics, Martin-Luther-Universita¨t Halle-Wittenberg, Magdeburger Straße 22, 06097 Halle (Saale), Germany §Institut fu¨r Mathematik, Martin-Luther-Universita¨t Halle-Wittenberg, Postfach, D-06099 Halle (Saale), Germany (Received 7 December 2006; in final form 23 July 2007)
This paper addresses the relationships between the microscopic properties of bone and its elasticity at the millimetre scale, or mesoscale. A method is proposed to estimate the mesoscale properties of cortical bone based on a spatial distribution of acoustic properties at the microscopic scale obtained with scanning acoustic microscopy. The procedure to compute the mesoscopic stiffness tensor involves (i) the segmentation of the pores to obtain a realistic model of the porosity; (ii) the construction of a field of anisotropic elastic coefficients at the microscopic scale which reflects the heterogeneity of the bone matrix; (iii) finite element computations of mesoscopic homogenized properties. The computed mesoscopic properties compare well with available experimental data. It appears that the tissue anisotropy at the microscopic level has a major effect on the mesoscopic anisotropy and that assuming the pores filled with an incompressible fluid or, alternatively, empty, leads to significantly different mesoscopic properties. Keywords: Cortical bone; Elastic properties; Multi-scale; Homogenization; Finite elements; Acoustic microscopy Msc: 92C10 (Biomechanics); 74Q99 (Homogenization, determination of effective properties); 78M10 (Finite element methods)
1. Introduction The mechanical properties of cortical bone depend on its microstructure, i.e. the shape, number and dimensions of the pores, and on the intrinsic material properties of the bone matrix (Martin et al. 1998, Weiner et al. 1999, Cowin 2001, Currey 2002). The most significant structural pattern at the micrometer scale in human lamellar bone is the osteonal pattern: an arrangement of hollow cylinders (osteons) of diameter about 250 mm predominantly aligned with the longitudinal axis of bone, in the peripheral skeleton. Between the nanometer and the micrometer scales, the bone matrix elementary constituents—mineral, collagen and water—are arranged at several sub-levels with specific patterns (Weiner and Traub 1986, Weiner et al. 1999).
In the framework of continuum mechanics (Eringen 1967), each material point is a volume of a homogeneous material which is mathematically shrunk to zero from the macroscopic point of view but which represents a volume of finite microscopic dimension and possesses a microstructure. The scale of the material point is referred to as the mesoscale. In bone biomechanics, a lot of effort is dedicated to the investigation of the mechanical properties at the mesoscale because the macroscopic behaviour is often critically dependant on that scale (Stoneham and Harding 2003). The mesoscale is not to be confused with the macroscale: A macroscopic homogeneous sample has identical properties at each of its constitutive material points, hence its macroscopic and mesoscopic properties are the same. On the other hand, the material properties
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[email protected] Computer Methods in Biomechanics and Biomedical Engineering ISSN 1025-5842 print/ISSN 1476-8259 online q 2008 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/10255840701688061
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of a macroscopic heterogeneous sample may vary from point to point with a characteristic length of variation larger than the mesoscopic length-scale; in that case, the macroscopic properties are some average of the mesoscopic properties. For bone, being commonly described with five or more levels of organization, the definition of a mesoscopic scale is not obvious and the term does not have a wellestablished usage. For Racila and Crolet (2006), the mesoscale is the scale of the lamellae, i.e. a few micrometers; Hoc et al. (2006) associate the mesoscale in cortical bone to a volume containing a group of 30– 50 osteons, i.e. a few millimetres; Watzky and Naili (2004), following Harrigan et al. (1988), considered in trabecular bone a mesoscopic length scale of a few millimetres. Except for Racila and Crolet, the definition retained for the mesoscale is a scale which includes all levels of organisation up to the osteonal pattern (included). In the present work, we follow most of the authors and use the term “mesoscopic” to refer to a length scale of one millimetre, which corresponds to a volume containing ten to twenty osteons in cortical bone. The characterization of the mesoscopic elastic properties of bone requires the measurement of five or nine constants depending on whether bone is assumed to be transverse isotropic (hexagonal symmetry), or orthotropic, respectively (Yoon and Katz 1976a, Katz and Yoon 1984). Most of the available data was obtained with contact ultrasonic measurements in the 2 – 5 MHz range on samples cut with up to six pairs of parallel faces, with characteristic dimensions of about 5 mm (Yoon and Katz 1976b, van Buskirk et al. 1981). Other groups measured five elastic constants with mechanical methods (Reilly and Burstein 1975, Dong and Guo 2004). Both contact ultrasonic measurements and mechanical methods have several drawbacks: (i) the samples can not be as small as one millimetre so that the mesoscale heterogeneity of the sample material cannot be fully assessed; (ii) the preparation of the samples is very demanding since many parallel faces must be obtained in small volumes of the material; (iii) finally, the mechanical measurements require to prepare several samples, one for each loading mode (longitudinal, shear) and loading direction, so that the properties measured do not strictly correspond to the same volume of bone. This paper introduces a new method to obtain the elasticity tensor of a cortical bone sample at the mesoscale (1 mm-scale) from a mapping of its microscale elasticity. The microscale data are obtained with a calibrated scanning acoustic microscope (SAM) (Katz and Meunier 1997, Raum et al. 2003) with a spatial resolution of 23 mm. The mesoscale properties are then estimated based on a finite element homogenization procedure. Such a method, once validated, should give a straightforward access to mesoscale bone properties, thus avoiding many drawbacks of the standard measurement techniques. In addition, it should prove to be a versatile tool to investigate the micro – macro relationships in cortical bone elasticity.
Two fundamental issues regarding the modelling of cortical bone at the mesoscopic scale are addressed: (i) authors, in the past, used contradictory assumptions, considering filled or, alternatively, empty pores in their models. In the present work, the impact of the assumption used for the pores on the mesoscopic material properties is investigated thoroughly; (ii) the impact of the elastic symmetry assumed for the bone matrix on the mesoscopic behaviour has been debated recently (Sevostianov and Kachanov 2000, Currey and Zioupos 2001, Kachanov and Sevostianov 2001). In the present work, the influence of the assumed symmetry of the bone matrix is quantified. With this introduction as background, section 2 of the paper introduces the notations and the formalism of the mechanical problem. Section 3 is a detailed description of the method which includes the processing of the experimental data, the construction of the representative mesoscopic volumes for four cases of material definition, and the homogenization technique. The results in terms of average mesoscale effective properties are given in section 4, and a discussion is provided in section 5.
2. Definitions and notations 2.1 Geometry Most of cortical bone is found in the diaphysis of long bones which geometry can be modelled as a hollow cylinder. Let O denote a point of the cylinder axis. We will use the local reference Cartesian frame RðO; x 1 ; x 2 ; x 3 Þ, where O is the origin of the Cartesian coordinate system and ðx 1 ; x 2 ; x 3 Þ is an orthonormal basis where x 3 is oriented in the direction of the cylinder axis and x1 and x2 are chosen arbitrarily in the plane normal to x 3. 2.2 Elasticity tensor Bone properties are investigated in the framework of linear elasticity (Eringen 1967). The components of the Cauchy stress tensor and of the infinitesimal strain tensor are denoted sij and 1ij, respectively. The generalized Hooke’s law is sij ¼ Cijkl1kl where Cijkl denotes the components of the stiffness tensor of rank four. The notations associated with the matrix representation used for the stiffness tensor throughout the paper are given in Appendix. 2.3 Mesoscopic effective properties The homogenization procedure consists of finding an effective stiffness tensor, which can be used for the description of the elastic properties at the mesoscale. An estimation of this tensor is obtained via a certain average of the microscopic material information. The method is based on the consideration of two length scales: the mesoscopic length scale L and the characteristic length scale of the microscopic data l, where L is supposed to be “much larger” than l. For the computations of the present
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work we hypothesized that L is large enough compared to l (this hypothesis is discussed a posteriori in section 5). The effective elastic response of the heterogeneous material is k½slV ¼ ½C * k½1lV ;
ð1Þ
Ð where k·lV ¼ ð1=jVjÞ ·dV is the average of the quantity V
between brackets over a statistically representative volume element (RVE) of volume jVj with characteristic dimension L, and [C *] is by definition the effective stiffness tensor (Zohdi and Wriggers 2005).
3. Method 3.1 Microscopic data An acoustic impedance image of a fresh human radius section cut in the (x1,x2)-plane was obtained with a 50-MHz scanning acoustic microscope. The sample preparation and experimental procedure have been reported elsewhere (Raum et al. 2005). The spatial resolution of the image was 23 mm and the spatial increment between two adjacent image pixels was set to 20 mm. Hence the image consists of a number of pixels with coordinates (x1,x2), each representing a 20 £ 20 mm2 surface. A calibration procedure was applied for defocus correction and to convert the reflectivity signal into acoustical impedance values (Raum et al. 2004). The pixel value is then directly related to the acoustic impedance of the material within the pixel area. 3.2 Representative volume elements (RVE) In a first step, 58 RVE realizations u1 ; u2 ; . . . ; ui ; . . . ; un of the microscopic data were generated using the impedance spatial distribution of the entire radius section. The impedance data for one realization ui is an ensemble of values, denoted Z(ui;x1,x2), given at points (x1,x2) which lie within a square of side length L ¼ 1 mm, where L is the characteristic dimension introduced in section 2.3. The square domains were selected manually on the radius cross section, preferentially close to the periosteal region and avoiding the largest resorption cavities close to the endosteal boundary as shown in figure 1. In a previous project, a threshold impedance value was determined which allows the segmentation of the pores from the matrix (Raum et al. 2005). After thresholding, the RVE can be seen as a composite material made of a heterogeneous bone matrix and pores. Considering the spatial resolution and pixel size, only the Haversian pores and resorption cavities can be separated from the remaining tissue. Smaller pores—osteocytes lacunae, Volkman canals and canaliculi—cannot be resolved with the resolution of the device used, and thus the measured impedance values may be slightly affected by the presence of small unresolved pores. Accordingly, the smallest pores
Figure 1. An example of the selection of eight mesoscopic RVE realizations in a cortical bone image obtained with a 50 MHz scanning acoustic microscope. Each cross represents the diagonals of a square RVE of side length 1 mm.
are considered to contribute to the mechanical properties of the bone matrix. Figure 2 shows a typical RVE realization (u31) before and after thresholding. The mean porosity (area of pores divided by total RVE area) of the population of 58 RVE realizations was 9.6% with a SD of 2.8%. In order to compute the 21 coefficients of the effective stiffness matrix [C *(ui)] for one realization of the microscopic data, a 3D RVE is constructed upon extrusion in the x3-direction of the 2D spatial distribution of impedance along the distance d3, which was chosen arbitrarily for the computations (d3 ¼ 20 mm in the present project). This construction is consistent with the actual organisation of bone, i.e. osteons have their axis aligned preferentially with the x 3-direction. Finally, the impedance data for one RVE realization ui is an ensemble of values, denoted Zðui ; x1 ; x2 ; x3 Þ, given at points ðx1 ; x2 ; x3 Þ, which lie within a parallelepiped of volume 1 £ 1 £ d3 mm3. 3.3 Case studies Impedance values Zðx1 ; x2 ; x3 Þ of the bone matrix were exp converted to stiffness values C33 ðx1 ; x2 ; x3 Þ using an
Figure 2. A typical RVE realization (u31) before (a) and after (b) thresholding. Pores appear in black and the bone matrix in grey.
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empirical law derived in a previous study (Raum et al. 2006) exp C 33 ðx1 ; x2 ; x3 Þ ¼ 1:31 þ 0:075Zðx1 ; x2 ; x3 Þ
þ 0:5Z 2 ðx1 ; x2 ; x3 Þ; where the superscript “exp” refers to the experimentally derived stiffness values. The stiffness values exp ðx1 ; x2 ; x3 Þ were then used to derive an isotropic matrix C 33 material and an anisotropic matrix material, as described in sections 3.5 and 3.6, respectively. The pores were considered, alternatively, as empty or filled with an incompressible fluid, as described in section 3.4. In total, four composite material model cases were defined: – PEMI: pores empty and matrix isotropic; – PFMI: pores filled with liquid and matrix isotropic; – PEMA: pores empty and matrix anisotropic; – PFMA: pores filled with liquid and matrix anisotropic. 3.4 Pores The elastic properties of pores are best defined using the bulk modulus k and the second Lame´ constant m. When there is no matter in pores, k and m should be zero. When the pores are filled with a water-like non-viscous fluid which cannot flow, the pore properties should be k ¼ 2.2 GPa and m ¼ 0, where 2.2 GPa is the bulk modulus of water (Lide 1993). The implementation in the finite element program used to compute the effective properties requires that the elastic constants be set in terms of Young modulus E and Poisson ratio n and not in terms of k and m. Accordingly, pore properties were set to n ¼ 0.01 and E ¼ 1026 GPa in cases PEMI and PEMA, and n ¼ 0.49 and E ¼ 0.13 GPa in cases PFMI and PFMA. 3.5 Isotropic matrix For the isotropic models (PFMI and PEMI), the matrix is characterized by its Young modulus E and its Poisson ratio n. Following several experimental and numerical works (Kabel et al. 1999, Zysset et al. 1999, Rho et al. 2002), the Poisson ratio was assumed to have a constant value of n ¼ 0.3, while E was calculated from exp Eðx1 ; x2 ; x3 Þ ¼ C 33 ðx1 ; x2 ; x3 Þ
ð1 þ nÞð1 2 2nÞ : ð1 2 nÞ
ð2Þ
of these five coefficients at point ðx1 ; x2 ; x3 Þ is derived through a procedure described below, which combines the exp experimental value C 33 ðx1 ; x2 ; x3 Þ with a micromechanical model of the bone matrix proposed by Hellmich et al. (2004) (referred to as “Concept I” in the cited paper). The micromechanical model yields the stiffness tensor CIJ ðx1 ; x2 ; x3 ; f col ; f w ; f ha Þ of the bone matrix as a function of the volume fractions of the three elementary constituents, water, collagen and hydroxyapatite, denoted by fw, fcol and fha, respectively. The model assumes fixed values for the elastic properties of water, collagen and mineral. The specific values of fw, fcol and fha at a given point ðx1 ; x2 ; x3 Þ are unknown, but we hypothesise that they are somehow related to the measured stiffness exp C 33 ðx1 ; x2 ; x3 Þ. Since the volume fractions add to unity ðf w þ f col þ f ha ¼ 1Þ, CIJ ðx1 ; x2 ; x3 ; f col ; f w ; f ha Þ is a function of only two independent input variables. In order to reduce the number of independent input variables to one, a relationship established by Raum et al. (2006), using experimental data of Broz et al. (1995), was used f col ¼ 0:36 þ 0:084 exp ð6:7f ha Þ: fw
ð3Þ
Finally, the stiffness tensor of the bone matrix given by the micromechanical model can be rewritten as a function of only one of the volume fractions chosen to be fcol: C IJ ðx1 ; x2 ; x3 ; f col Þ. For each point ðx1 ; x2 ; x3 Þ, the equation exp C 33 ðx1 ; x2 ; x3 ; f col Þ ¼ C 33 ðx1 ; x2 ; x3 Þ was solved for fcol and the solution value was used to derive the rest of the stiffness constants C IJ ðx1 ; x2 ; x3 ; f col Þ. The anisotropic stiffness values derived for the bone matrix following the above procedure were compared with experimental data obtained by Hofmann et al. (2006) upon combining acoustic microscopy and nanoindentation. Hofmann et al. obtained mean values of Young and shear moduli in osteon lamellae of E1 ¼ 23.4 GPa, E3 ¼ 26.5 GPa and G12 ¼ 10.4 GPa, respectively. Using the assumption of a constant Poisson ratio n ¼ 0.3 for all material directions, the corresponding stiffness coefficients are derived: C11 ¼ 30.62 GPa, C33 ¼ 34.28 GPa, C12 ¼ 12.62 GPa, C13 ¼ 12.97 GPa, C55 ¼ 10.4 GPa and C66 ¼ 9 GPa. On the other hand, for a typical value of the bone matrix impedance Z ¼ 8.5 Mrayl, the procedure described above yields C11 ¼ 29.5 GPa, C33 ¼ 38.1 GPa, C12 ¼ 11.0 GPa, C13 ¼ 11.9 GPa, C55 ¼ 10.1 GPa and C66 ¼ 9.3 GPa. This comparison with experimental values suggests that the procedure proposed to derive the local anisotropic stiffness tensor C IJ ðx1 ; x2 ; x3 Þ of the bone matrix yields realistic values.
3.6 Anisotropic matrix For the anisotropic models (PFMA and PEMA) the matrix was considered to be transverse isotropic with (x1,x 2) being the plane of isotropy. At a given point ðx1 ; x2 ; x3 Þ, the matrix is characterized by five stiffness coefficients C 11 ðx1 ; x2 ; x3 Þ, C33 ðx1 ; x2 ; x3 Þ, C 12 ðx1 ; x2 ; x3 Þ, C 13 ðx1 ; x2 ; x3 Þ, and C 44 ðx1 ; x2 ; x3 Þ. The value of each
3.7 Formulation of the homogenization problem for one RVE realization Solving the homogenization problem for one RVE, i.e. for one realization Zðui ; x1 ; x2 ; x3 Þ of the microstructure, consists of calculating the effective stiffness tensor [C *(ui)] as defined in equation (1), where V ¼ V(ui)
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Derivation of the mesoscopic elasticity tensor of cortical bone
is the parallelepipedic domain corresponding to the RVE realization ui. The procedure consists of applying six linearly independent loadings with specified displacement conditions on the RVE boundaries ›V (Hollister and Kikuchi 1992, Zohdi and Wriggers 2005). The boundary value problem to solve in the parallelepiped V is defined by the equilibrium equation and the generalized Hooke law 3 X
sij; j ¼ 0;
ð4Þ
j¼1
½sðx1 ; x2 ; x3 Þ ¼ ½Cðui ; x1 ; x2 ; x3 Þ½1ðx1 ; x2 ; x3 Þ;
ð5Þ
where ½Cðui ; x1 ; x2 ; x3 Þ is the stiffness matrix of the heterogeneous material, which components are specified according to the above subsections for each of the four cases PEMI, PFMI, PEMA, and PFMA. The six loadings split in three longitudinal loadings and three 2 shear loadings. Let ›Vþ n and ›Vn be one pair of opposite faces with normal vector n. In this paragraph, we give the mathematical expressions assuming that the RVE is aligned with ðx 1 ; x 2 ; x 3 Þ. For a longitudinal loading in direction x1, the displacements are prescribed according to u1 c›Vþ1 ¼ d, u1 c›V21 ¼ 0 and u2 c›Vþ2 ¼ u2 c›V22 ¼ u3 c›Vþ3 ¼ u3 c›V23 ¼ 0, where the choice of d is arbitrary because the problem is linear. For the shear loading in the (x1,x2)-plane, the boundary conditions are u2 c›Vþ1 ¼ d, u2 c›V21 ¼ 0, and u1 ›cVþ2 ¼ d, u1 ›cV22 ¼ 0. Conditions for longitudinal loadings in directions x2 and x3 and for the shear loadings in planes (x2, x3) and (x1, x3) are obtained by circular permutation of the indices. For any of the longitudinal (resp. shear) loading case, the mean strain state k½1lV in volume jVj is very close to a pure longitudinal (resp. a pure shear) strain state, referred to with letter K: k½1KI lV < 1dIK , where 1 is the applied strain amplitude, dIK are the components of the Kronecker tensor, and K stands for the Kth strain state. In the following, we will assume strict equality, i.e. k½1KI lV ¼ 1dIK . The validity of this assumption was checked a posteriori with the results of the computations (strains that are assumed to be zero were at least two orders of magnitude less than the other strains). (Note that pure longitudinal or pure shear strain states are obtained without assumption if the applied displacement boundary conditions are u›V ¼ k½1lV x (Zohdi and Wriggers 2005); in the present work we have preferred a slightly different set of boundary conditions in order to allow the formulation of equivalent 2D-boundary value problems as explained in the next section.) Equation (1) yields k½sKI lV ¼ ½Cðui Þ*raw IJ k½1KJ lV ¼ ½Cðui Þ*raw IK 1:
151
column ½Cðui Þraw *IK , I ¼ 1, . . . ,6, of the effective stiffness matrix can be derived, once the stress field [s ]K in volume V is known. 3.8 Finite element computations and post-processing The boundary value problems for the longitudinal loadings in directions x1 and x2 and the shear loading in the plane (x1,x2) are, formally, 2D plane strain problems. Hence the finite element computations consist of solving three 2D problems and three 3D problems, thus reducing the computational cost by almost a factor two compared to a full 3D formulation. A typical finite element mesh used for a 2D computation is shown in figure 3. The largest element dimension for both 2D and 3D computations was 22 mm, which is slightly smaller than the image resolution (23 mm). Lagrange quadratic elements and a direct finite element solver were used. Computations were performed with a commercially available finite element program (Comsol Multiphysicsq v3.2, Comsol France, Grenoble, France). Computations for one of the four cases over 58 RVE realizations took approximately 8 h on a desktop PC computer. Due to the construction of the 3D-RVE with an extrusion in the x3-direction, the “raw” effective stiffness tensor ½Cðui Þ*raw corresponds to a monoclinic material with one plane of symmetry of normal x3. This tensor with a priori 13 independent elastic constants (Royer and Dieulesaint 1999) was approximated by a tensor [C(ui)*] with hexagonal symmetry using a method proposed by Franc¸ois et al. (1998) and the associated software SYMETRIC (Marc Franc¸ois, LMT-Cachan, France). In addition to the “closest” stiffness tensor with a given symmetry (here, hexagonal), the method of Franc¸ois et al. provides an estimation of the “distance” d between the raw
ð6Þ
Here we use the subscript “raw” because we choose to reserve the notation [C(ui)*] for an effective stiffness tensor possessing hexagonal symmetry that is derived from ½Cðui Þ*raw , as described in the next subsection. Equation (6) states that from the Kth loading case, the Kth
Figure 3. Map of Young modulus (model PFMI) and finite element mesh for the RVE realisation (u31).
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P58 * * 2 ½s ¼ ð1=57Þ i¼1 ð½Cðui Þ 2 ½C Þ , where the square operation applies to each component of the tensors.
tensor and the closest hexagonal material tensor as dðui Þ ¼
kCðui Þ*raw 2 Cðui Þ* k : kCðui Þ*raw k
ð7Þ 4. Results
where kk is the Euclidian norm. The value of d was typically a few percent for the RVE realizations considered in the present study. For illustration purposes, the post-processing is detailed below for the realization u31 (case PFMI). The “raw” stiffness matrix was 0
28:004 12:27 B B 12:270 27:941 B B 12:314 12:296 B * ½Cðu31 Þraw ¼ B B 2D 2D B B B 2D 2D @ 0:040
0:026
In this section, the values of the stiffness coefficients, denoted as C 11 ; C22 ; . . . ; C 66 , and obtained from the effective matrix [C *] using the notations defined in equation (A.3), are presented for the four different model cases. Note that we have dropped the star “*” in the labels 1 12:309 e e 0:061 C 12:292 e e 0:192 C C 29:789 e e 0:051 C C C; ð8Þ e 14:545 e 2D C C C e e 14:623 2D C A 0:015 e e 15:328
where “2D” indicates the components which are not computed in the 2D calculations (columns 1, 2 and 6) and “e” indicates values less than 1023. Columns 3, 4 and 5 are obtained with 3D computations. Note that this matrix form is consistent with that of a monoclinic material and is close to the form of a stiffness matrix for a hexagonal material: C11 < C22, C13 < C23, C44 < C55 and (C11 2 C22)/2 ¼ 7.86 < C66. The “closest” material stiffness tensor corresponding to hexagonal symmetry was found to be 0
27:879
B B 12:364 B B 12:303 B * ½Cðu31 Þ ¼ B B 0 B B B 0 @ 0
of the effective stiffness components CIJ, since in the following only effective properties are discussed. Results are compared with three reference measurement data sets obtained with contact ultrasonic methods in the MHzrange on macroscopic human cortical bone samples (a few millimetres). Set 1 (Yoon and Katz 1976) provides the five constants of a hexagonal material. Sets 2 (Rho 1996) and 3 (Ashman et al. 1984) provide the nine constants of an orthotropic material. For comparison, the orthotropic
12:364
12:303
0
0
0
27:879
12:303
0
0
0
12:303
29:789
0
0
0
0
0
14:584
0
0
0
0
0
14:584
0
0
0
0
0
15:515
with an error of d(u31) ¼ 0.5%. 3.9 Mean effective stiffness matrix of cortical bone In a macroscopic bone volume, the mesoscale properties vary from point to point over distances of a few millimetres. This is in particular the case through the cortical thickness due to the porosity gradient from the periosteal to the endosteal surfaces (Bousson et al. 2001). In order to compare the mesoscopic properties computed in the present work to available experimental data, the effective mesoscopic stiffness coefficients obtained for the different realizations were averaged. For each case (PEMI, PFMI, PEMA, PFMA), the mean [C *] of the stiffness matrix over Pthe 58 realizations were * calculated as ½C * ¼ ð1=58Þ 58 i¼1 ½Cðui Þ , and the SD as
1 C C C C C C C C C C A
ð9Þ
values were “converted” to hexagonal values upon looking for the closest stiffness tensor corresponding to an hexagonal material using the method of Franc¸ois et al. (1998), as described in section 3.8. The bone samples in Set 1 were dried before measurements while bones in Sets 2 and 3 were kept moist during measurements. Table 1 gives the values of the effective stiffness coefficients obtained for the four cases and SDs, together with the macroscopic values of experimental Sets 1– 3. The anisotropy ratios (ARs) defined as C33/C11 are also given in table 1. Table 2 shows a quantification of the differences between the stiffness values obtained with the filled pores and empty pores assumptions. Table 3 shows a quantification of the differences between the stiffness values obtained with the isotropic and anisotropic matrix assumptions.
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Table 1. Average (SD) mesoscopic stiffness coefficients for the four cases PFMI, PEMI, PFMA and PEMA defined in section 3.3. Isotropic matrix
C11 C33 C12 C13 C44 C66 Anistropy ratio (AR) C33/C11
Anisotropic matrix
Macroscopic experimental data
PFMI (SD)
PEMI (SD)
PFMA (SD)
PEMA (SD)
Set 1
Set 2
Set 3
23.31 (2.52) 26.91 (1.85) 10.68 (0.95) 10.55 (0.98) 5.34 (1.01) 6.31 (0.79) 1.15
19.83 (3.21) 25.16 (2.05) 7.71 (1.43) 7.71 (1.43) 5.28 (1.08) 6.06 (0.89) 1.27
19.7 (1.98) 27.32 (3.13) 8.76 (0.73) 8.88 (1.87) 4.91 (0.94) 5.47 (0.67) 1.39
16.23 (2.51) 25.78 (2.02) 5.8 (0.97) 6.62 (1.04) 4.73 (0.99) 5.21 (0.77) 1.59
23.4 32.5 9.06 9.11 8.71 7.17 1.39
19.8 31 11.5 12.7 5.4 4.1 1.56
19.08 27.6 10 10.4 5.92 4.54 1.45
Stiffness coefficients as measured by different groups are given; Set 1 (Yoon and Katz 1976), Set 2 (Ashman et al. 1984) and Set 3 (Taylor et al. 2002). The anisotropy ratio (AR) C33/C11 is given for each case.
Table 2. Percentage difference between elastic properties computed for pores filled (PFMI and PFMA) and pores empty (PEMI and PEMA). Isotropic matrix 100[Cij(PFMI) 2 Cij(PEMI)]/Cij(PFMI)
Anisotropic matrix 100[Cij(PFMA) 2 Cij(PEMA)]/Cij(PFMA)
14.91 6.47 27.88 22.67 1.16 3.94
17.61 5.64 33.79 25.45 3.67 4.75
DC11 DC33 DC12 DC13 DC44 DC66
Table 3. Percentage difference between elastic properties computed for the anisotropic matrix model (PFMA and PEMA) and the isotropic matrix model (PFMI and PEMI). Pores filled 100[Cij(PFMI) 2 Cij(PEMA)]/Cij(PFMI)
Pores empty 100[Cij(PFMI) 2 Cij(PEMA)]/Cij(PEMI)
15.47 21.54 18.01 15.81 8.00 13.33
18.15 22.44 24.73 18.84 10.34 14.06
DC11 DC33 DC12 DC13 DC44 DC66
5. Discussion The computed effective stiffness values for the four cases PEMI, PFMI, PEMA, and PFMA (table 1) are all in the range of the experimental values of the reference Sets 1– 3, as well as values reported by others, e.g. for human bone (Reilly and Burstein 1975, Taylor et al. 2002) and bovine bone (Ambardar and Ferris 1978, van Buskirk et al. 1981, Pithioux et al. 2002). Anisotropy ratios (ARs) have often been used to compare models and experiments (Crolet et al. 1993) and to test hypotheses on the organization of bone tissue at various scales (Takano et al. 1996, Weiner et al. 1999). In the present study, the ARs obtained for the isotropic matrix (PEMI and PFMI) are quite low compared to the experimental values, while ARs obtained for the anisotropic matrix (PEMA and PFMA) are in the range of those found for experimental values (table 1). Compared to the isotropic assumption, the anisotropic assumption increases ARs
by 20 and 25% for the filled and empty pore cases, respectively. Note that the ARs predicted for empty pores are larger than for filled pores, both for the isotropic and anisotropic matrix cases. This is due to the relatively small value of C11 in the empty pores assumption. Both for empty and for filled pores the assumption of matrix anisotropy has a significant impact on the overall properties (table 3). Only the impact on C33 is negligible due to the construction of the anisotropic model which requires that the values of C 33 ðx1 ; x2 ; x3 Þ at the matrix level be identical in the anisotropic and isotropic cases (see section 3.6). These results indicate that the anisotropy of the bone matrix should be taken into account to derive a realistic stiffness tensor for cortical bone. These results are consistent with the discussions of, e.g.Currey and Zioupos (2001) and Weiner et al. (1999) who suspected a significant impact of the matrix anisotropy on the whole bone anisotropy. The results of the present study quantify this impact: the increase of AR due to matrix anisotropy is found to be about 20%. For both isotropic and anisotropic matrix models, the assumption made for the pores (filled or empty) has a significant impact on the overall properties (table 2) and this effect is different for the different components of the stiffness tensor. The impact is more important for the nondiagonal terms C12 and C13, which is probably due to the influence of the Poisson ratio assumed for the filled pores (incompressibility): the effective Poisson ratio increases when the pores are filled with liquid. On the other hand, the impact of the pore assumption is very small on the shear stiffness coefficients C44 and C66, which is related to the fact that the fluid in the pores is assumed to have zero shear modulus. Sevostianov and Kachanov (2000) proposed a micromechanics model to study the influence of porosity on the anisotropy of cortical bone. They concluded that the differences between the cases of empty pores and pores filled with soft material were insignificant. This is in contrast to our results presented in table 1. Sevostianov and Kachanov (2000) stated that the material filling the pores has a very low Young modulus compared to that of the bone matrix. However, the increase in stiffness due to filled pores which is observed in comparing cases PEMI and PFMI or PEMA and PFMA, is not due to the assumed value of the fluid Young modulus—which was also very low in our case—but
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rather to the high value of the Poisson ratio. Indeed, the soft material is assumed not to flow and is almost incompressible, so that compression in one direction also stresses the material in the directions transversal to the load. Hence the observed increase in stiffness is due to the Poisson effect in the pores. Crolet et al. (1993) also used empty pores in the homogenization model, but considered this hypothesis as a limitation of the model (Aoubiza et al. 1996), which is equivalent to the assumption that the liquid in the pores is free to flow. The assumption of pores filled with a rather incompressible soft material is in line with the assumption used in the micromechanics models proposed by Hellmich and Ulm (2004). The best assumption—filled or empty pores—should depend on the in vivo biomechanical application considered at the macro scale: if the macroscopic rate of loading is rather low, then the fluid may be assumed to flow freely in the pores and the empty assumption is probably the best. On the contrary, if the rate of loading is rather high, the filled pore assumption is presumably more reliable. The thresholds on the rate of loading for which one or the other pore model should be used can be discussed in the framework of poroelasticity (Cowin 1999). Note that the elasticity of reference Set 1 obtained on dried samples is not less than the elasticity measured for samples in Sets 2 and 3 on wet samples. This apparent inconsistency with the results given by our model is presumably due to differences in experimental setups and physiological variation in the bone properties among the donors. Furthermore, this discussion should be balanced by the fact that the bone used for the present study was a human radius while the bone samples in the reference sets came from human tibia (Set 2) and human femur (Sets 1 and 3). For the three experimental data Sets 1–3 and others (Reilly and Burstein 1975, Ambardar and Ferris 1978, van Buskirk et al. 1981, Pithioux et al. 2002, Taylor et al. 2002) the relationship C66 , C55 ¼ C44 always holds. However, this is not the case for the computed effective stiffness coefficients C55 and C66 presented in table 1, although other stiffness components are in accordance with experimental data. The reason for this discrepancy is not fully understood. An error in the implementation of the method has been discarded since we found good agreement of the program results in special cases with several analytic solutions, e.g. the academic Eshelby inclusion problems (Zaoui 2002). It seems that either C66 is intrinsically overestimated or C44 ¼ C55 is intrinsically underestimated due to the homogenization method used. Furthermore, the values of ½Cðui Þ*raw 44 and ½Cðui Þ*raw 55 on the one hand and the values ½Cðui Þ*raw 66 on the other hand are very close in values and it was observed numerically that the approximations performed to obtain the closest hexagonal stiffness tensor tend to decrease ½Cðui Þ*raw 44 and ½Cðui Þ*raw 55 , and to increase ½Cðui Þ*raw 66 , which may explain a part of the discrepancies with experimental values. It is important to note that the effective properties of one RVE as defined through equation (1), are not unique. In particular, the effective properties computed using
displacement boundary conditions, as done in the present work, theoretically are higher in values than effective properties computed using stress boundary conditions (Hollister and Kikuchi 1992, Torquato 2002, Zohdi and Wriggers 2005). It is also well established that the differences in computed effective stiffness values for different types of boundary conditions are all the more important that (i) the volume considered for the homogenization (RVE) is small compared to heterogeneities and (ii) the heterogeneity of the RVE is important. The applied displacement approach used here is similar to the one used by van Rietbergen et al. (1996) for trabecular bone. Another option to pure stresses or displacements boundary conditions is to use a formulation in the framework of the asymptotic theory of homogenization developed for periodic media (Sanchez-Hubert and Sanchez-Palencia 1992) which yields effective properties lying between the values obtained with stress boundary conditions and with displacement boundary conditions. Such an approach has been used to derive effective properties of trabecular bone by Hollister et al. (1994). The effective properties of a homogeneous isotropic matrix in which cylindrical inclusions (pores) are embedded can be estimated with the solution of an Eshelby-type problem using the Mori-Tanaka scheme (Zaoui 2002). Such estimations usually fall between the values obtained for applied displacements and applied stresses. Figure 4 shows the evolution with porosity of the stiffness coefficients provided by the Mori-Tanaka scheme and those computed for the 58 realizations of the microstructure considering filled pores and an isotropic homogeneous matrix (same properties at each point of the bone matrix). The fact that, in figure 4, the solutions computed for the realizations with different porosity values are close to the analytical solutions suggests that the method proposed here, based on applied displacements, gives a good estimation of the effective properties and that an estimation based on applied stresses would not yield much different values. Accordingly, the hypothesis stated in section 2.3 that the size l of the
Figure 4. Comparison of an analytical solution (Eshelby problem, Mori-Tanaka scheme, Torquato 2002) with the computed stiffness values for pores filled with liquid, and an isotropic, homogeneous, matrix. The lines corresponds to the analytical solution and the dots, crosses and circles to the corresponding values computed for each realization of the mesoscopic RVE.
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Derivation of the mesoscopic elasticity tensor of cortical bone
heterogeneities (essentially the pores) is small enough compared to the size L of the RVE seems to be validated with the data in figure 4: if l were too large compared to L, we would expect a stronger mismatch between the MoriTanaka solutions and the computed values. Still, the quality of the estimation using applied displacement boundary conditions, and the optimal size of the RVE, should be assessed in future studies. In order to construct the homogenization model from the microscopic scale to the mesoscale, we followed a consistent approach, using experimental data when available and models when experimental data were lacking. This was especially important for the model of the anisotropic matrix (PEMA and PFMA). The data derived from microscopic experimental measurements exp C 33 ðui ; x1 ; x2 ; x3 Þ was used to derive a stiffness tensor ½Cðui ; x1 ; x2 ; x3 Þ with five elastic constants for a transversely isotropic material. For this purpose, experimental data derived at the microscopic scale (Broz et al. 1995, Raum et al. 2006) and a micromechanics model based on an idealized representation of the intimate organization of the bone matrix were used (Hellmich et al. 2004). The micromechanics model used is a model of the bone “ultrastructure” which, as defined by Hellmich and Ulm (2002), is a certain arrangement of collagen, mineral and water bonded to mineral crystals and collagen molecules, and has a characteristic length scale of a few micrometers. It should be noted that the ultrastructure does not include the small pores of bone (canaliculi, Volkman canals and lacunae) but is the tissue in which these pores are embedded. In the present work we have used the model of ultrastructure to model the bone matrix itself (as defined in section 3.2) because the contribution to the overall anisotropy of the small pores (lacunae and canaliculi) is likely to be a minor effect. Nevertheless, the model used to derive the local anisotropic matrix should be improved and validated with more experimental data. In contrast to many studies of the elasticity of cortical bone based on idealized microstructures (Crolet et al. 1993, Sevostianov and Kachanov 2000, Lucchinetti 2001, Hellmich and Ulm 2004), the originality of the present work is to propose a method to assess mesoscopic cortical bone properties based on actual microscopic data that includes realistic pore shapes and the elastic heterogeneity of the bone matrix. The proposed method is in line with the approaches developed by Hollister et al. (1991, 1994) and van Rietbergen et al. (1995, 1996) for the study of trabecular bone based on real microstructures obtained with high resolution X-ray techniques. The main limitation of the method is that is it based on a 2D elasticity distribution, while bone is a 3D material. Following several theoretical works (Crolet et al. 1993, Sevostianov and Kachanov 1998, Hellmich 2005) we have hypothesized that the bone material is invariant along the longitudinal axis. This limitation may be partially overcome upon combining acoustic microscopy and high-resolution 3D micro-computed X-ray tomography, which technique can be used to obtain the 3D pore shapes.
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6. Conclusion In this work, a method to assess the mesoscale (1 mm) anisotropic stiffness tensor of cortical bone from a high resolution impedance map has been developed. The developed procedure essentially involves (i) the segmentation of the pores to obtain a realistic model of the cortical bone porosity; (ii) the construction of a local heterogeneous anisotropic stiffness tensor of the bone matrix which reflects its spatial heterogeneity; and (iii) solving a homogenization problem on RVEs with characteristic length scale of 1 mm, with the finite element method. The computed stiffness coefficients compare well with experimental data obtained by others. Furthermore, the simulation results indicate that the microscopic anisotropy should be accounted for in models, and that the properties assumed for the pores have a significant impact on the elasticity predictions. The computation procedure should be validated by comparing measured macroscopic elasticity and computed effective properties derived from the same group of samples.
Acknowledgements The authors are indebted to Marc Franc¸ois (Universite´ Pierre et Marie Curie-Paris6, LMT, Cachan, France) for helpful discussions on anisotropy and for providing the software SYMETRIC. Vincent Liabeuf implemented the micromechanics model used to derive the anisotropic bone properties at the microscopic level. The authors are grateful for the financial supports to: Universite´ Pierre et Marie Curie-Paris6 (BQR funding of the project “Mode´lisation me´canique du tissu osseux”); and the French–German research network “Ultrasound assessment of bone strength from the tissue level to the organ level” (DFG: Grant Ra 1380/1-1; CNRS). References A. Ambardar and C.D. Ferris, “Compact anisotropic bone: elastic constants, in vitro aging effects and numerical results of a mathematical model”, Acta Biologica Academiae Scientiarum Hungaricae, 29(1), pp. 81– 94, 1978. B. Aoubiza, J.M. Crolet and A. Meunier, “On the mechanical characterization of compact bone structure using the homogenization theory”, J. Biomech., 29(12), pp. 1539– 1547, 1996. R.B. Ashman, S.C. Cowin, W.C. van Buskirk and J.C. Rice, “A continuous wave technique for the measurement of the elastic properties of cortical bone”, J. Biomech., 17(5), pp. 349–361, 1984. V. Bousson, A. Meunier, C. Bergot, E. Vicaut, M.A. Rocha, M.H. Morais, A.M. Laval-Jeantet and J.D. Laredo, “Distribution of intracortical porosity in human midfemoral cortex by age and gender”, J. Bone Miner. Res., 16, pp. 1308–1317, 2001. J.J. Broz, S.J. Simske and A.R. Greenberg, “Material and compositional properties of selectively demineralized cortical bone”, J. Biomech., 28(11), pp. 1357–1368, 1995. S.C. Cowin, “Bone poroelasticity”, J. Biomech., 32(3), pp. 217 –238, 1999. S.C. Cowin, Bone Mechanics Handbook, 2nd ed., S.C. Cowin, Ed., Boca Raton, FL: CRC Press, 2001, Vol. 1. J.M. Crolet, B. Aoubiza and A. Meunier, “Compact bone: numerical simulation of mechanical characteristics”, J. Biomech., 26(6), pp. 677–687, 1993.
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Derivation of the mesoscopic elasticity tensor of cortical bone
Appendix With the tensorial basis matrix representation (Helnwein 2001) Hooke’s law for an arbitrary anisotropic material is written 1 0 0 s11 C1111 C 1122 C B B B s22 C B C C 2222 C B 1122 B C B B C B B B s33 C B C1133 C 1133 C B B ½s ¼ ½C½1 ¼ B pffiffiffi C ¼ B pffiffiffi B s23 C B 2C 1123 2C 2223 C B B C B pffiffiffi B p ffiffiffi B s C B 2C 2C 2213 B 13 C B 1113 A @ pffiffiffi @ pffiffiffi s12 2C 1112 2C 2212
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For an isotropic material, C 11 ¼ C33 ¼ l þ 2m, where l and m are the first and second Lame´ constants, respectively, C 44 ¼ C 66 ¼ m, and C12 ¼ C13 ¼ l. The
C 1133 C 2233 C 3333 pffiffiffi 2C3323 pffiffiffi 2C3313 pffiffiffi 2C3312
pffiffiffi 2C 1123 pffiffiffi 2C 2223 pffiffiffi 2C 3323
pffiffiffi 2C 1113 pffiffiffi 2C 2213 pffiffiffi 2C 3313
2C2323
2C 2313
2C2313
2C 1313
2C2312
2C 1312
pffiffiffi 10 1 111 2C1112 CB C pffiffiffi C B 2C2212 C CB 122 C C C B pffiffiffi CB C 2C3312 CB 133 C CB C CB C C B 2C 2312 CB 123 C C CB C C B 2C 1312 CB 113 C C A@ A 112 2C 1212 ðA:1Þ
For a transverse isotropic material with its natural axes aligned with ðx 1 ; x 2 ; x 3 Þ, ðx 1 ; x 2 Þ being the plane of isotropy, Hooke’s law simplifies to 1 0 0 C1111 C 1122 C 1133 s11 0 C B B B s22 C B C1122 C 1111 C 1133 0 C B B C B B C B B 0 B s33 C B C1133 C 1133 C 3333 C B B ¼ ½s ¼ ½C½1 ¼ B C B B s23 C B 0 0 0 2C2323 C B B C B B Bs C B 0 0 0 0 B 13 C B A @ @ s12 0 0 0 0 The stiffness matrix in equation (A.2) can be rewritten so as to introduce the standard notations CIJ for the stiffness coefficients: 1 0 C 11 C12 C 13 0 0 0 BC 0 0 0 C C B 12 C11 C 13 C B C BC C C 0 0 0 13 33 C B 13 C ðA:3Þ ½C ¼ B B 0 0 0 C 0 0 2C44 C B C B B 0 0 C 0 0 0 2C 44 A @ 0 0 0 0 0 2C 66 with C 66 ¼ ðC11 2 C 12 Þ=2.
0 0 0 0 2C 2323 0
0
10
111
1
CB C CB 122 C CB C CB C CB C 0 CB 133 C CB C CB C B 123 C 0 C CB C CB C B1 C 0 C CB 13 C A@ A 112 2C 1212 0
ðA:2Þ
bulk modulus is defined as k ¼ l þ ð2=3Þm. Poisson ratio n and Young modulus E can be written in terms of the Lame´ constants as (Royer and Dieulesaint 1999).
n¼
l 2ðl þ mÞ
E¼
mð3l þ 2mÞ lþm
ðA:4Þ
