Variations in mineralization affect the stress and strain distributions in cortical and trabecular bone

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1 ARTICLE IN PRESS Journal of Biomechanics 40 (2007) Variations in mineralization affect the stress and strain distributions in cortical and trabecular bone L.J. van Ruijven, L. Mulder, T.M.G.J. van Eijden Department of Functional Anatomy, Academic Centre for Dentistry Amsterdam (ACTA), Universiteit van Amsterdam and Vrije Universiteit, Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands Accepted 1 June 2006 Abstract The mechanical properties of bone depend largely on its degree and distribution of mineralization. The present study analyzes the effect of an inhomogeneous distribution of mineralization on the stress and strain distributions in the human mandibular condyle during static clenching. A condyle was scanned with a micro-ct scanner to create a finite element model. For every voxel the degree of mineralization (DMB) was determined from the micro-ct scan. The Young s moduli of the elements were calculated from the DMB using constant, linear, and cubic relations, respectively. Stresses, strains, and displacements in cortical and trabecular bone, as well as the condylar deformation (extension along the antero-posterion axis) and compliance were compared. Over 90% of the bone mineral was located in the cortical bone. The DMB showed large variations in both cortical bone (mean: 884, SD: 111 mg/cm 3 ) and trabecular bone (mean: 738, SD: 101 mg/cm 3 ). Variations of the stresses and the strains were small in cortical bone, but large in trabecular bone. In the cortical bone an inhomogeneous mineral distribution increased the stresses and the strains. In the trabecular bone, however, it decreased the stresses and increased the strains. Furthermore, the condylar compliance remained relatively constant, but the condylar deformation doubled. It was concluded that neglect of the inhomogeneity of the mineral distribution results in a large underestimation of the stresses and strains of possibly more than 50%. The stiffness of trabecular bone strongly influences the condylar deformation. Vice versa, the condylar deformation largely determines the magnitude of the strains in the trabecular bone. r 2006 Elsevier Ltd. All rights reserved. Keywords: Mineralization; Stress; Strain; Mandibular condyle 1. Introduction Bone tissue is constantly being renewed. In new bone tissue the degree of mineralization (DMB) is lower than in mature bone and this difference can be as large as 30% (Guo, 2001; Grynpas et al., 1993). The renewal process of bone results in an inhomogeneous distribution of mineralization (Paschalis et al., 1997; Camacho et al., 1999; Mulder et al., 2005). Corresponding author. Tel.: ; fax: address: L.J.vanRuijven@AMC.UvA.Nl (L.J. van Ruijven). The Young s modulus of bone tissue depends largely on the DMB. This dependency has been described by both linear and cubic relations (Currey, 1988, 2001). Therefore, the Young s modulus in bone will show similar inhomogeneities as the DMB. In two recent finite element studies it has been shown that arbitrarily applied inhomogeneities affected the apparent stiffness of trabecular bone (van der Linden et al., 2001; Jaasma et al., 2002). Also an increment from 20% to 50% of the variation of the DMB caused a 28-fold increase of the amount of failed tissue (Jaasma et al., 2002). In addition, the apparent stiffness was predicted better when inhomogeneities were included (Bourne and van der Meulen, 2004). These studies only analyzed trabecular bone. When cortical bone is also included, larger /$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi: /j.jbiomech

2 1212 ARTICLE IN PRESS L.J. van Ruijven et al. / Journal of Biomechanics 40 (2007) effects can be expected because in cortical bone the DMB as well as the Young s modulus is on average higher than in trabecular bone (Guo, 2001). The best tool to analyze stresses and strains in bone during in vivo loading is large-scale finite element analysis (van Rietbergen et al., 1995, 1996). To obtain reliable results with this method the resolution of the model should be at most one-fourth the mean thickness of the trabecular structures (van Rietbergen et al., 1995; Guldberg et al., 1998; Niebur et al., 1999). For several joints this method has been used to determine the stress and strain distributions (van Rietbergen et al., 1999; Ulrich et al., 1999; Pistoia et al., 2002; Homminga et al., 2004). In almost all these simulations, however, the Young s modulus was assumed homogeneous. Recently, in a model of the human femoral head different homogeneous moduli were used for cortical and trabecular bone (van Rietbergen et al., 2003). Therefore, the effect of variations in Young s modulus remains unclear. Due to its relatively small size the mandibular condyle is ideal for large-scale finite element analysis. Furthermore, its high bone turnover rate (Sodek, 1977) and its relatively large loads (Koolstra et al., 1988) make it ideal to analyze bone adaptation. Histomorphometric studies have shown that its trabecular structure consists of sagittally oriented plates that are connected by mediolaterally oriented rods (Giesen and van Eijden, 2000; van Eijden et al., 2004, 2006; van Ruijven et al., 2005). The present study will analyze the mandibular condyle during simulated maximal static clenching using a large-scale finite element model, which includes both the cortical and trabecular structure and the DMB as measured with micro-ct. To analyze the influence of the inhomogeneous DMB on the mechanical behavior of the condyle three different models were used. The first model uses a constant Young s modulus. In the other two models the relationship between the Young s modulus and the DMB was assumed linear and cubic, respectively. The analysis will describe the distributions of the equivalent strains, the von Mises stresses, the principal strains, and the displacements for cortical and trabecular bone separately, as well as the condylar compliance and deformation. 2. Materials and methods 2.1. Specimen The model was constructed from a dried human mandible of an adult female (age: 25 years) with full dentition and no signs of malocclusion. The use of the mandible conforms to a written protocol that was reviewed and approved by the Department of Anatomy and Embryology of the Academic Medical Center of the University of Amsterdam. The right condyle was separated from the mandible with an electric saw (Fig. 1) Micro-CT After the condyle was submerged in 70% ethanol and subjected to a vacuum to remove all the air from the cancellous bone, it was scanned in a micro-ct scanner (mct 40, SCANCO Medical AG, Bassersdorf, Switzerland). The size of the scan was voxels, the resolution 36 mm, the voltage 45 kv, and the integration time 1500 ms. The DMB of every voxel was calculated using a new beam-hardening correction that was designed specially for bone scans (SCANCO Medical AG, Bassersdorf, Switzerland). Using the same configuration the DMB was measured in a phantom containing known hydroxyapatite densities of 100, 200, 400, and 800 mg/cm 3 ; the noise (relative standard deviation of the gray values) was less than 6%, and the error (relative difference between the measured and real mineral density) was less than 3%. A histogram was made from the gray values and its minimum was used as a threshold to separate bone from marrow. Cortical bone was identified with a separate segmentation step using a custom Gaussian filter. The trabecular bone was defined as all the remaining bone tissue. Architectural parameters for the trabecular bone were calculated using morphometric software (Software version 3.2, SCANCO Medical AG, Bassersdorf, Switzerland). The trabecular bone of the condyle had a bone volume fraction of 0.24, a mean trabecular thickness of mm, and a mean trabecular separation of mm Finite element analysis With a voxel conversion technique (van Rietbergen et al., 1995) a finite element mesh was created identical to the segmented scan. Its resolution of 36 mm was well below the requirement of one-fourth of the mean trabecular thickness (Niebur et al., 1999). The mesh consisted of 22 million elements. The positions of all nodes in the horizontal saw plane were fixed. To simulate joint loading during maximal static clenching, the exterior surface of the upper half of the condyle was identified, and for all its nodes a force was applied perpendicular to the surface. The surface normal was calculated using singular value decomposition of the coordinates of all exterior nodes within a small neighborhood. For the magnitude of the force the positions of 6 control points (Fig. 1) were used. Let d i ¼ jjx p i jj be the distance between a node at position x and control point i, then the force F on the node was F ¼ C expð P 6 i¼0 d2 i Þ, where C is a scaling constant. The control points were positioned such that the load

3 ARTICLE IN PRESS L.J. van Ruijven et al. / Journal of Biomechanics 40 (2007) distribution resembled physiological loading (Koolstra and van Eijden, 2005). To keep the number of boundary conditions limited, amplitudes below 0.1 N were neglected. Finally, all forces were scaled to obtain a total force of 300 N (Koolstra et al., 1988) Young s modulus Three models were analyzed with different relations for the tissue moduli (Fig. 2, lower panel). In the first model a constant tissue stiffness of E ¼ 10:0 GPa (van Ruijven et al., 2003) was assigned to all elements of the model. In the second model a linear dependence on the DMB of the voxels was assumed (Currey 1988; van der Linden et al., 2001): E i ¼ E ðdmb i =DMB mean Þ. In the third simulation a cubic relation was assumed (Currey 1988; van der Linden et al., 2001): E i ¼ E ðdmb i =DMB mean Þ 3. In the second and third model the stiffnesses were also scaled to ensure that the mean stiffness in all simulations was 10 GPa. In all three simulations the Poisson ratio of the bone tissue was 0.3. Using specific software (van Rietbergen et al., 1995) and 16 parallel processors on a SGI Origin 3800, each model was solved in approximately 5 days (1800 h CPU time). Fig. 1. Antero-medial view of the finite element model of the condyle (individual elements are not visible, because they are too small). The horizontal saw-plane was fixed (see arrow), and the articular surface was loaded. The colors depict how the force was distributed over the surface. In total the force on the surface was 300 N. The gray region at the bottom was not considered for the analysis. Above the condyle the six control points (and their projections on the surface) are shown. These were used to define the articular load distribution. Fig. 2. The distributions of the mineral in the cortical and trabecular bone of the condyle (upper panel) and the dependencies of the Young s modulus on the degree of mineralization for the three models (lower panel).

4 1214 ARTICLE IN PRESS L.J. van Ruijven et al. / Journal of Biomechanics 40 (2007) Statistical analysis Frequency distributions of the equivalent strain, the von Mises stress, and the principal strains were calculated for the trabecular bone, the cortical bone, and the total bone structure. For the equivalent strains and the von Mises stress the means, standard deviations, and modes (i.e. the value which occurs most frequently) were calculated. Also the means and standard deviations of the displacements of the elements were calculated, where applicable values are calculated as mean SD. In addition, for all simulations the compliance (amount to which the condyle is distorted by the load), the vertical height, and the antero-posterior length were calculated for the mid-sagittal cross-section (see Fig. 5). To avoid artifacts from the boundary conditions, the lower 200 voxels ð¼ 7:2mmÞ of the scan were excluded from all results derived from the finite element analysis. 3. Results Fig. 2 (upper panel) shows a histogram of the mineral distribution of the complete condyle (including the surface voxels). Excluding the surface voxels, the DMB was still 20% higher in cortical bone ð mg=cm 3 Þ than in trabecular bone ð mg=cm 3 Þ. The volume of cortical bone (913 mm 3 ) was eight times higher than that of trabecular bone (117 mm 3 ). The cutoff at a DMB just below 500 mg/cm 3 was due to surface voxels, which are not always completely filled with bone. Without the surface voxels, both distributions resembled symmetric Gaussian distributions and voxels with a DMB of 500 mg/ cm 3 were not present anymore. Fig. 3 shows a sagittal section of the condyle to illustrate the spatial distribution of the largest principal strains obtained using a cubic relation between the Young s modulus and the DMB. The frequency distributions of the equivalent strain, von Mises stress, and principal strains are shown in Fig. 4; they were constructed with 11.3 million cortical and 2.2 million trabecular voxels. Table 1 summarizes the statistics of the equivalent strains, von Mises stresses, and displacements. None of the distributions was symmetric and they rather resembled Gamma distributions than symmetric Gaussian distributions. The distributions for the total condyle (cortical plus trabecular bone) strongly resembled the distributions for the cortical bone. In the trabecular bone the distributions had relatively long right tails. For instance, the modal equivalent strain was highest in cortical bone, but equivalent strains above 8000 mstrain still occurred in trabecular bone but not in cortical bone. It should be noted that both the means and the standard deviations of these distributions are increased significantly by these tails. Fig. 3. A sagittal section of the condyle, showing the spatial distribution of the largest principal strains occurring with the cubic relation between the Young s modulus and the degree of mineralization. The largest compressive strains occurred in supero-inferiorly oriented trabeculae, and largest tensile strains in antero-posteriorly oriented trabeculae. Long and straight trabecular structures were most heavily strained. Table 1 shows that for the constant model, compared with trabecular bone, cortical bone had a lower mean strain (cortical: 2257 mstrain, trabecular: 2770 mstrain), lower mean stress (cortical: 11 MPa, trabecular: 13 MPa), higher modal strain (cortical: 2401 mstrain, trabecular: 1918 mstrain), and higher modal stress (cortical: 13 MPa, trabecular: 9 MPa) (Table 1). Compared to the constant model, in the linear model the mean Young s modulus was higher in cortical bone (10.3 GPa) and lower in trabecular bone (7.7 GPa). Furthermore, in cortical bone increased mean stresses and strains, and decreased modal stresses and strains were observed in the linear model. In trabecular bone the mean and modal stresses were decreased, while the mean and modal strains increased. Compared to the constant model, the cubic model showed similar differences as the linear model, but much larger. In the trabecular bone the mean strain increased by 70%. The mean displacements within the condyle were largest along the supero-inferior and the antero-posterior direction (Table 1). Compared to these displacements the displacements in the medio-lateral direction

5 ARTICLE IN PRESS L.J. van Ruijven et al. / Journal of Biomechanics 40 (2007) Fig. 4. Distributions of the equivalent strain (left column), von Mises stress (middle column), and principal strains (right column) in the trabecular bone (bottom row), cortical bone (middle row), and all the bone (top row). Note that the distributions for total bone strongly resemble the distributions for cortical bone, but not the distributions for trabecular bone. were negligibly small. The linear and cubic relationships for the Young s modulus sharply increased the mediolateral displacements, especially in the trabecular bone, where the cubic relation resulted in a three-fold increase of the medio-lateral displacements. The effect on the other displacements was negligibly small. Fig. 5 shows the condylar compliances and deformations. The compliances in the three models were almost identical. The deformation consisted of compression along the vertical axis, ranging from 0.20% for the constant relationship to 0.26% for the cubic relationship, and extension along the antero-posterior direction, ranging from 0.36% for the constant relationship to 0.65% for the cubic relationship. This extension was mainly caused by bulging of the posterior cortex. 4. Discussion In the present study stress and strain distributions in cortical and trabecular bone were analyzed for three relationships between the DMB and the tissue stiffness. The shapes of the distributions are similar to previously published distributions (van Rietbergen et al., 2003; Homminga et al., 2004; Ryan and van Rietbergen, 2005). In cortical bone the mean stress and mean strain were lower, and the stress and strain distributions were narrower. In trabecular bone, however, the modal stress and modal strain were lower. This seems contradictory, but is simply due to the longer tail found in trabecular bone. It is not clear what caused this tail, but since trabecular bone has a larger surface than cortical bone, an overestimation of the stresses and strains in surface voxels cannot be excluded. With the cubic relationship the mean Young s modulus of trabecular bone tissue decreased to 4.6 GPa. Therefore, the low modulus was unavoidable after the decision to use the same mean Young s modulus in the three models in order to facilitate comparison of the results. However, a value twice as large is more likely for trabecular bone (van Ruijven et al., 2003) and scaling of the stresses and strains correspondingly might be considered. Together with the weakening caused by the inhomogeneities (van der Linden et al., 2001; Jaasma et al., 2002; van Rietbergen et al., 2003) we estimate that the cubic relationship

6 1216 ARTICLE IN PRESS L.J. van Ruijven et al. / Journal of Biomechanics 40 (2007) Table 1 Statistics of the equivalent strains, von Mises stresses, and displacements ðmean SDÞ Cortex Trabeculae Total ðn ¼ Þ ðn ¼ Þ Constant model Mean E (GPa) Mean equivalent strain (m) Modal equivalent strain (m) Mean von Mises stress (MPa) Modal von Mises stress (MPa) Displ, x a (mm) Displ, y a (mm) Displ, z a (mm) Linear model Mean E (GPa) Mean equivalent strain (m) Modal equivalent strain (m) Mean von Mises stress (MPa) Modal von Mises stress (MPa) Displ, x a (mm) Displ, y a (mm) Displ, z a (mm) Cubic model Mean E (GPa) Mean equivalent strain (m) Modal equivalent strain (m) Mean von Mises stress (MPa) Modal von Mises stress (MPa) Displ, x a (mm) Displ, y a (mm) Displ, z a (mm) a The orientations of the axes x, y, and z are supero-inferiorly, medio-laterally, and antero-posteriorly, respectively. reduced the apparent stiffness of trabecular bone almost three-fold relative to the constant model. This is in agreement with earlier studies, which have shown that variations in the Young s modulus reduce the apparent stiffness of trabecular bone especially when a cubic relation is used (van der Linden et al., 2001). In this study the cubic relation resulted in a doubling of the bulging, and a 70% increase of the strains in the trabecular bone. The reduction of the stresses in the trabecular bone is most certainly a consequence of this change, just as the higher stiffness of the cortical bone increased the cortical stresses. The deformation of the condyle (antero-posterior extension and vertical compression) ranged from 0.2% to 0.6%. That is in agreement with an earlier study of the mandibular condyle, where the trabecular structure was modeled as a homogeneous material (van Ruijven et al., 2002). Furthermore, an increase of the condylar deformation was found for the linear and cubic relationships. This must be due to the reduction of the mean Young s modulus of the trabecular structure. This demonstrates that the apparent stiffness of trabecular bone largely influences the amount of condylar deformation. This is also illustrated in Fig. 3 where it can be seen that very high tensile strains occur in trabeculae with an antero-posterior orientation. These tensile strains are directly related to the posterior bulging of the cortex. At the same time the results revealed that the linear and cubic models had a small effect on the compliance of the condyle. Apparently, the stiffness of the trabecular structure has a large influence on the amount of bulging and the failure load of the condyle, but a relatively small influence on its compliance. The linear and cubic relationships increased the equivalent strains in cortical and trabecular bone. Especially the number of highly strained elements showed a disproportionately large increase, which suggests that weakening of the trabecular structure strongly increases the risk of failure. Also the mediolateral displacements showed a disproportionately large increase. Apparently, the sagittally oriented plates in the trabecular structure are very sensitive to deformations in the sagittal plane, just as a sheet of paper will bend strongly, when two of its opposing edges are moved slightly in the plane of the sheet. The trabecular structure seems to combine optimal mechanical resistance against cortical deformations with optimal sensitivity for the same cortical deformations.

7 ARTICLE IN PRESS L.J. van Ruijven et al. / Journal of Biomechanics 40 (2007) Fig. 5. The deformation and compliance of the loaded condyle magnified 10-fold. The axes have arbitrarily values. The black line depicts the unloaded shape. The red, green, and blue lines correspond to the constant, linear, and cubic relationships between the Young s modulus and the DMB, respectively. Although the force was more vertically oriented, the compliance (BB 0 ) was oriented almost horizontal and was relatively constant with a magnitude of 0.2 mm (2 mm after the magnification). The antero-posterior length (AC) changed from 7.70 mm in the unloaded case to 7.99, 8.05, and 8.21 mm for the constant, linear, and cubic relationships, respectively. The vertical height (distance from B to the saw-plane) went from to 20.98, 20.98, and mm for the three relationships. It may be questioned whether the threshold or the partial volume effect (i.e. the effect that voxels located at the surface are in general not completely filled with bone) have affected the current results. For instance, Fig. 2 shows that a small decrease of the threshold results in a relatively large increase of the number of trabecular bone voxels. It should be noted, however, that below the threshold (not shown in Fig. 2) the frequency distributions of both trabecular and cortical bone start to rise again. This is due to noise in the gray values. Consequently, the increase is rather the result of marrow falsely identified as bone. Also, because these voxels have a low BMD, their Young s modulus will be very low, and their effect on the strength of trabecular structure will also be minimal. Furthermore, in the cubic model the Young s modulus of trabecular bone depends on the threshold. A lower threshold results in a higher number of trabecular voxels, lower mean DMB, and a lower Young s modulus. But Fig. 2 also shows that the lower DMB is also present in the inner voxels of trabecular and cortical bone. They have a DMB of 738 and 884 gr/cm 3, respectively, and this results in Young s moduli of 6.7 and 11.6 GPa, respectively. This corresponds largely with the difference found when all voxels are included. Using 24 human mandibular condyles (age: years) Giesen et al. (2003) found a trabecular thickness of 0.13 mm, a trabecular spacing of 0.64 mm, and a bone volume fraction of In our specimen (age: 33) these values were 0.19, 0.60, and 0.24 mm, respectively. Since the trabecular thickness and the bone volume fraction decrease with age, these differences could have been expected. The mineral concentrations found in this study as well as its variations are below values found in other studies. For instance, quantitative backscattered electron imaging resulted in a calcium density for trabecular bone from the human transiliac of 23 3wt%(Roschger et al., 1998). Using a specific weight of 2.0 g/cm 3 for bone (van Eijden et al., 2004), and assuming that pure hydroxyapatite contains 40 wt% Ca (Roschger et al., 1998), this equals to a mineral density of mg=cm 3. With contact microradiography a mean DMB of mg=cm 3 was found in the trabecular bone of 20 human calcanei (Follet et al., 2004). Using micro-ct, a mean DMB of mg=cm 3 was found in the cortical bone of 10 human mandibular condyles (Renders et al., 2006). It should be noted, however, that the equations used to calculate the Young s moduli only depend on variations in DMB, and not on its mean value. In conclusion, the trabecular bone of the condyle is subjected to a larger range of stresses and strains than the cortical bone. An inhomogeneous mineral distribution reduces the trabecular stiffness substantially, which in turn leads to an increase of the condylar deformation, cortical strains, trabecular strains, and medio-lateral trabecular displacements. Apparently, there is a strong coupling in the trabecular bone between its stiffness and the strains occurring in it, which is likely caused by the sagittal orientation of its plate-like structure. On the other hand, the compliance of the condyle is not affected by an inhomogeneous mineral distribution and seems only dependent on the cortical properties. Acknowledgments This work was sponsored by the National Computing Facilities Foundation (NCF) for the use of supercomputing facilities. This research was institutionally supported by the Inter-University Research School of Dentistry, through the Academic Centre for Dentistry Amsterdam. We would like to thank SARA Computing and Networking Services for their technical support. We are grateful to Jan Harm Koolstra for his comments on the manuscript. References Bourne, B.C., van der Meulen, M.C.H., Finite element models predict cancellous apparent modulus when tissue modulus is scaled from specimen CT-attenuation. Journal of Biomechanics 37,

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