A New Material Mapping Procedure for Quantitative Computed Tomography-Based, Continuum Finite Element Analyses of the Vertebra

Ginu U Unnikrishnan; Elise F Morgan

doi:10.1115/1.4004190

. 2011 Jul 13;133(7):0710011–0710018. doi: 10.1115/1.4004190

A New Material Mapping Procedure for Quantitative Computed Tomography-Based, Continuum Finite Element Analyses of the Vertebra

Ginu U Unnikrishnan, Elise F Morgan ¹

PMCID: PMC3379558 NIHMSID: NIHMS382652 PMID: 21823740

Abstract

Inaccuracies in the estimation of material properties and errors in the assignment of these properties into ﬁnite element models limit the reliability, accuracy, and precision of quantitative computed tomography (QCT)-based ﬁnite element analyses of the vertebra. In this work, a new mesh-independent, material mapping procedure was developed to improve the quality of predictions of vertebral mechanical behavior from QCT-based ﬁnite element models. In this procedure, an intermediate step, called the material block model, was introduced to determine the distribution of material properties based on bone mineral density, and these properties were then mapped onto the ﬁnite element mesh. A sensitivity study was ﬁrst conducted on a calibration phantom to understand the inﬂuence of the size of the material blocks on the computed bone mineral density. It was observed that varying the material block size produced only marginal changes in the predictions of mineral density. Finite element (FE) analyses were then conducted on a square column-shaped region of the vertebra and also on the entire vertebra in order to study the effect of material block size on the FE-derived outcomes. The predicted values of stiffness for the column and the vertebra decreased with decreasing block size. When these results were compared to those of a mesh convergence analysis, it was found that the inﬂuence of element size on vertebral stiffness was less than that of the material block size. This mapping procedure allows the material properties in a ﬁnite element study to be determined based on the block size required for an accurate representation of the material ﬁeld, while the size of the ﬁnite elements can be selected independently and based on the required numerical accuracy of the ﬁnite element solution. The mesh-independent, material mapping procedure developed in this study could be particularly helpful in improving the accuracy of ﬁnite element analyses of vertebroplasty and spine metastases, as these analyses typically require mesh reﬁnement at the interfaces between distinct materials. Moreover, the mapping procedure is not speciﬁc to the vertebra and could thus be applied to many other anatomic sites.

Keywords: bone mineral density, ﬁnite element analysis, geometry, material assignment, mesh size

1. Introduction

Computationally based, noninvasive assessment of vertebral strength and stiffness holds great promise for the study of age-related bone fragility, assessment of therapeutic interventions for osteoporosis, and characterization of vertebral failure mechanisms [1–4]. Computational tools such as quantitative computed tomography (QCT)-based ﬁnite element analysis (e.g., Ref. [5]) can integrate patient-speciﬁc vertebral geometry, inhomogeneous distributions of material properties, and simulation of clinically relevant load conditions in order to provide better predictions of vertebral mechanical behavior than those obtained from more idealized models [5–11]. The reliability, accuracy, and precision of this type of analysis are limited primarily by the approximations of the vertebral geometry, by inaccuracies in the estimation of material properties from QCT images, and by errors in the assignment of these properties into the ﬁnite element model. Even though specimen-speciﬁc vertebral models for ﬁnite element simulations have been developed that accurately represent the bone geometry [12,13], implementation of image-based estimates of material properties into the ﬁnite element model is still an unresolved issue [14]. In this work, an improved method for the assignment of material properties in a QCT-based ﬁnite element model of the vertebra is developed.

QCT-based ﬁnite element models of vertebral specimens have been created either by voxel-based [3,7,11] or by geometry-based [1,2,15] methods. In the voxel-based method, the geometry is obtained directly from the images without using any surfaces or solid bodies, and the ﬁnite element mesh is developed by assigning hexahedral elements that each enclose a predeﬁned cubic volume of image voxels. The material properties of the elements are calculated from the average intensity of the image voxels within each hexahedral element. Element sizes on the order of 1–5 mm on a side [4,7,11,16] have typically been used for the voxel-based method. In the geometry-based method, a geometric model of vertebra is created from a set of image-based surface points and a mesh is generated with hexahedral or tetrahedral elements. The material properties of the elements are obtained by assigning either a weighted average of the voxel information from the nodal values of an element [17] or by averaging information from a ﬁxed number of voxels around the elemental centroid or around the Gauss integration points [1,2,18,19]. Mapping algorithms that use information from all the voxels within the element boundaries, similar to the voxel-based methods, have also been implemented [20,21] for the geometry-based methods.

Accuracy of the geometry- and voxel-based methods is heavily dependent on the size of the ﬁnite elements. In general, decreasing the element size does not always lead to a better solution due to the dependence of the computed material properties on element size. For example, in mapping algorithms based on average voxel information from the nodes or from a ﬁxed number of voxels around a point, signiﬁcant errors are introduced when the element size is larger than the volume that is occupied by the chosen voxels. Mapping algorithms that consider all voxels within an element (e.g., Ref. [15]) give accurate results when the elements are larger than the voxels; however, the accuracy is severely compromised when the element size is comparable to or smaller than the voxel size [22]. To overcome these limitations, it is therefore necessary to have a material mapping procedure that is independent of the ﬁnite element mesh.

The main objective of this work is to develop a robust, mesh-independent, material mapping procedure for QCT-based ﬁnite element analysis of the vertebra. The new mapping procedure is based on the introduction of an intermediate step, called the material block model, for calculating the material properties. The material block model is created following a semi-automated, image segmentation process that deﬁnes the vertebral geometry from QCT data. Bone mineral density and the corresponding material constants are determined from the material block model and are then mapped onto the ﬁnite element mesh. The material mapping procedure is described in more detail in Sec. 2. In Sec. 3, the effect of the size of the material blocks on the calculation of material properties and the results from ﬁnite element simulations of the vertebra under clinically relevant conditions are presented. This paper concludes with a discussion of the results in Sec. 4.

2. Materials and Methods

An L1 spine segment with adjacent intervertebral discs was harvested from fresh-frozen spines by making a transverse cut just above the inferior endplate of T12 and below the superior end-plate of L2. The spine segment was scanned using a 64-row detector system (GE Lightspeed VCT, GE Healthcare, Waukesha, WI) with the following acquisition parameters: 120 kV, 240 mA, pixel size of 0.31 mm × 0.31 mm, and slice thickness of 0.625 mm. The specimen was held in an acrylic ﬁxture to ensure the slices were in the transverse anatomic plane of the specimen with the ﬁxture submerged in a container ﬁlled with degassed water. A calcium hydroxyapatite calibration phantom (Image Analysis, Columbia, KY) was included in each scan to correct for scanner drift and for the estimation of bone mineral density.

2.1. Model Generation.

The QCT images were imported into AMIRA (AMIRA 5.2, Visage Imaging, Inc., San Diego, CA), and the external surface of the vertebral body was deﬁned and converted to a set of contours (DXF ﬁle) by using a semi-automated segmentation technique. The plane of the images corresponded to the geometric x–y plane, the z-axis corresponded to the superior-inferior (SI) direction, and the origin was set to the ﬁrst slice of the stack. The cortical shell and the endplates of the vertebra were included in the geometric model. The superior and inferior surfaces were assumed to be planar surfaces in order to avoid any artifacts that would arise when applying nonuniform external boundary conditions on an uneven surface. A solid geometric representation of the vertebra was created in RHINOCEROS (RHINOCEROS 4.0, Robert McNeel & Associates, Seattle, WA) by ﬁrst lofting a surface through the set of contours and then converting the closed surface to a solid. The solid geometry was exported from RHINOCEROS as a SAT ﬁle (Fig. 1) and was directly imported into the ABAQUS preprocessor (ABAQUS V6.7, SIMULIA, Providence, RI) to create the ﬁnite element model.

Fig. 1 — Development of solid geometry from QCT images

2.2. Material Property Determination.

The computed tomography (CT) attenuation data in Hounsﬁeld units (HU) were converted to equivalent values of bone mineral density (in mg/cubic centimeter (cc)) using the calibration phantom. A linear relationship between HU and mineral density was derived by calculating the mean HU values at regions corresponding to 0 mg/cc, 75 mg/cc, and 150 mg/cc in the calibration phantom. The following experimentally determined relationship [23] was used to convert the bone mineral density to the elastic modulus along the SI direction:

E_{z z} = - 34.7 + 3.230 ρ_{QCT}

(1)

where E_zz is the elastic modulus in the SI direction (MPa) and ρ_QCT is the bone mineral density in mg/cc. Any negative modulus obtained from the above-presented relationship was converted to a preset value of 0.1 kPa. The remaining orthotropic elastic properties were determined by assuming the following ﬁxed ratios and values [24,25]:

\begin{matrix} \frac{E_{x x}}{E_{z z}} = 0.333 \frac{E_{y y}}{E_{z z}} = 0.333 \\ \frac{G_{x y}}{E_{z z}} = 0.121 \frac{G_{x z}}{E_{z z}} = 0.157 \frac{G_{y z}}{E_{z z}} = 0.157 \\ v_{x y} = 0.381 v_{x y} = 0.104 v_{x y} = 0.104 \end{matrix}

(2)

2.3. Material Property Assignment.

In order to remove the dependence of the assigned material properties on the ﬁnite element (FE) mesh, and also to allow for mesh reﬁnement without affecting the assessment of the material properties, an intermediate step called the material block model was introduced. The material properties for the ﬁnite element analysis were derived from the material block model using a centroid-based mapping algorithm. This process works as follows.

2.3.1. Material Block Model.

In the ﬁrst stage of the process, the solid geometry was meshed using linear hexahedral elements, called material blocks, in ABAQUS. It is important to emphasize that the material block model is distinct from the mesh used for the ﬁnite element analysis (FEA mesh). The material block model creates a discrete ﬁeld for each material property, and these properties are then mapped onto the FEA mesh. The material blocks can be nonuniform in size and arbitrarily oriented. Collectively, the material blocks map the entire vertebral geometry, including the cortical region and the endplates. In this study, the block dimensions were signiﬁcantly greater than the typical thickness of the cortical shell (∼0.35 mm) so that the material properties of the peripheral blocks contained contributions from both the outer cortical shell and inner trabecular region. Such an approach avoids the need to explicitly model the cortical shell, thereby eliminating any shell-speciﬁc image processing. An in-house MATLAB code was used to read the ABAQUS input ﬁle, and the mineral density for each material block was calculated from the QCT images. The mineral density of a given block was obtained by averaging the HU units for all the voxels within the block and then converting this average into mineral density using the previously derived HU-density relationship.

2.3.2. Centroid-Based Mapping Algorithm.

An FEA mesh with ﬁnite elements smaller than the material blocks was created in ABAQUS. The location of the centroid of each element of the FEA mesh was calculated and then mapped onto the material block model, and the material properties at that point were assigned to the entire element. This approach allows the investigator to vary the element size at a speciﬁc region of the FEA mesh without drastically affecting the assigned material properties at that point. A limitation of this approach arises when a ﬁne FEA mesh is mapped onto a coarse material block model. This leads to a condition wherein the centroids of some of the FEA mesh elements on the edges do not have corresponding points in the material block model. In such cases, those elements were identiﬁed and material properties of the nearest centroid were assigned.

2.4. Verification and Sensitivity Analysis.

The aim of this analysis was to verify the accuracy of the material mapping procedure and to understand the inﬂuence of the size of the material blocks on the predicted bone mineral density. A geometric model of the calibration phantom was created from the QCT images, and the bone mineral density was calculated for material block sizes ranging from 1.25 mm to 5 mm in length. The minimum material block size was restricted to 1.25 mm in order to have at least two image slices (0.625 mm thickness) for the calculation of material properties.

2.5. Test Case—Finite Element Analysis of a Vertebral Column Model.

The size of the material blocks can affect the results of the ﬁnite element analysis of the vertebra in two ways. First, the block size controls the resolution with which the spatial variation of material properties is represented. Second, the block size inﬂuences how well the material block model can map the material properties in the periphery of the irregularly shaped vertebra, with smaller block sizes allowing for a better match of the material block model to the vertebral geometry. In order to isolate and examine the ﬁrst of these effects, a columnar region of square cross section (15 mm × 15 mm) and oriented along the SI direction was isolated from the QCT images of the vertebra (Fig. 2(a)). The effect of material block size on the FE-derived stiffness of the column was studied by conducting ﬁnite element analyses with block sizes ranging from approximately 1.25 mm to 5 mm. For these analyses, the FEA mesh was the same as the material block model. Axial compression was simulated with a uniform displacement on the superior surface and a fully constrained inferior surface. The axial stiffness of the column was computed for each of the block sizes by dividing the total reaction force on the inferior surface by the displacement applied to the superior surface. The vertebral column model was also used to examine the inﬂuence of the FEA mesh size on the predicted patterns of deformation. FE analyses were carried out for multiple FEA mesh sizes but a single material block size of 4.0 mm. The resulting distributions of minimum principal strain were plotted, as this distribution is often used for the prediction of the onset of vertebral fracture [6].

Fig. 2 — Finite element geometry of (a) column model derived from the vertebra and (b) nonuniform axial compressive loading of vertebral superior surface with an elliptical rigid body

2.6. Test Case—Finite Element Analysis of a Vertebra.

In this second test case, ﬁnite element analyses of the entire vertebra were carried out using (a) displacement and (b) force boundary conditions on the superior surface. The displacement boundary conditions consisted of a uniform, axial, compressive displacement applied to the superior surface of the vertebra with the displacements on the inferior surface fully constrained in all directions. The force boundary conditions consisted of nonuniform, axial, compressive loading as described in more detail in the following.

Analyses involving displacement boundary conditions were ﬁrst performed to isolate the effect of the size of the material blocks on the material mapping of the irregularly shaped vertebra. Spatially varying material properties (as determined by the spatial variations in HU) were assigned with block sizes ranging from 2 mm to 5 mm. The material block and the FEA mesh were one and the same in each of these simulations. Hence, block sizes of less than 2.0 mm were not used, because below this minimum size some elements of the FEA mesh were of poor quality due to the complex vertebral geometry. A second set of analyses was then performed with spatially homogeneous material properties. Block size was again varied from 2 mm to 5 mm, and in each case the FEA mesh was identical to the material block model. For comparison between the two analyses, the homogeneous SI modulus was chosen as 358.5 MPa (average SI elastic modulus of the vertebra calculated with a material block size of 5.00 mm) and the remaining material constants were given by Eq. (2). Finally, the inﬂuence of the density of the FEA mesh on the FE-derived vertebral stiffness and principal strain distribution was studied, using a material block size of 4.0 mm and both hexahedral and tetrahedral FEA meshes.

In the analysis involving force boundary conditions, nonuniform, axial, compressive loading was applied on the superior surface with the inferior surface fully constrained in all directions. Nonuniform loading was chosen for this analysis because nonuniform vertebral loading can occur in vivo due to disc degeneration and due to the presence of prosthetic disc implants [26,27], and also because the nonuniformity creates regions of large stress and strain gradients on the vertebral surface. The ability of the material mapping procedure to enable the FE analyses to capture these gradients was thus studied. The ﬁnite element model consisted of the vertebral body and an elliptical, rigid surface plate smaller than the superior surface (Fig. 2(b)). The elliptically shaped loading surface was selected solely to simplify the computational procedure by deﬁning regions of large gradients prior to the simulation. At the reference point of the rigid plate an axial, compressive load of 750 N (less than the in vivo standing load of 1000 N, Ref. [28]) was applied, and the other degrees of freedom were restrained. The material properties were derived from the material block model with block size of 4.0 mm. Both a coarse and a ﬁne FEA mesh were considered in order to understand the inﬂuence of the mesh size on the predicted results.

3. Results

3.1. Verification and Sensitivity Analysis.

The material mapping procedure produced accurate reproductions of the density variations in the hydroxyapatite phantom (Fig. 3(a)). Decreasing the size of the material blocks resulted in variations of the bone mineral density predictions by ±5 mg/cc for the 150 mg/cc and 0 mg/cc regions and ±10 mg/cc for the 75 mg/cc region (Fig. 3(b)). The effect of mesh size was most pronounced at the interfaces between the distinct density regions.

3.2. Test Case—Finite Element Analysis of a Vertebral Column Model.

When the size of the material blocks in the column model was increased, the axial column stiffness increased (Fig. 4(a) ), and signiﬁcant changes were observed in the minimum principal strain distribution (Fig. 4(b)). In contrast, ﬁxing the size of the material blocks at 4.00 mm while varying the density of the ﬁnite element mesh resulted only in minor changes to the column stiffness (Fig. 5(a) ). As the ﬁnite element mesh density was increased, smoother distributions of minimum principal strain were observed.

Fig. 5 — (a) Vertebral column stiffness and (b) distribution of minimum principal strain in the column model for different sizes of the ﬁnite element mesh. All models used a material block size of 4.00 mm.

3.3. Test Case—Finite Element Analysis of Vertebra.

The stiffness for the vertebral model with homogeneous material properties (SI modulus of 358.5 MPa) showed a near uniform stiffness value for different block sizes, while for the HU-based model a decreasing trend with decreasing material block size was observed (Fig. 6). The vertebral stiffness predictions show similar trend to the stiffness values obtained from the column model and highlight the importance of the volume used for calculation of material properties. Smaller block sizes were found to capture the vertebral geometry more accurately than larger block sizes (Fig. 7). Smaller material blocks also led to larger variations in the predicted elastic modulus within the vertebra. As the material block size decreased, both the maximum SI elastic modulus and the number of blocks with unique values of the material properties increased (Table 1). This increase in modulus with a decrease in block size occurred primarily because the smaller block sizes allowed for inclusion of more regions at the endplate and the cortical shell.

Fig. 6 — Inﬂuence of material block size on vertebral stiffness for the models with HU-derived material properties and with homogeneous material properties

Fig. 7 — Inﬂuence of material block size on the distribution of SI elastic modulus in the vertebra

Table 1.

Inﬂuence of material block size on the computed mineral densities and SI moduli for the test case that modeled the entire vertebra

Material block size (mm)	Number of material blocks	Avg. BMD (mg/cc)	Maximum elastic modulus (MPa)
2.00	2360	134.77	1900
3.00	833	136.23	1529
4.00	380	133.15	1447
5.00	188	136.79	1266

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Similar distributions of material properties were observed for the hexahedral and tetrahedral FEA meshes that were obtained from the same material block model (Fig. 8). Good convergence of vertebral stiffness was observed with both element types, with the stiffness nearly constant for the range of mesh densities for the hexahedral elements (Fig. 9). The models with tetrahedral elements predicted higher values for vertebral stiffness than did the corresponding hexahedral element models. A ﬁne FEA mesh produced a smoother distribution of the minimum principal strain when compared to the coarse FEA mesh (Fig. 10). When examining this distribution, a critical value of strain of −0.77% was deﬁned, corresponding to the compressive yield strain for human vertebral trabecular bone [29]. Nonuniform axial compression led to a nonuniform distribution of minimum principal strain on the superior surface of the vertebra (Fig. 11). The large strain gradients in the regions of the superior endplate that were in contact with the rigid plate more clearly followed the elliptical contour of the plate for the ﬁne as compared to coarse FEA mesh. The minimum principal strain distribution inside the vertebra was also modiﬁed by the density of the FEA mesh.

Fig. 8 — Effect of hexahedral and tetrahedral mesh density on the distribution of SI elastic modulus for the material block size of 4.0 mm

Fig. 9 — Vertebral stiffness for different mesh densities for both hexahedral and tetrahedral elements. All models used a material block size of 4.0 mm.

Fig. 10 — Distribution of minimum principal strain for different FEA mesh sizes with a material block size of 4.0 mm. The elements having minimum principal strain less than −0.77% are shaded in black.

Fig. 11 — Distribution of minimum principal strain on the vertebral half-section for nonuniform compression applied to the vertebral superior surface

4. Discussion

The numerical accuracy of image-based computational models of vertebra is severely restricted by approximations of the bone geometry and by simpliﬁcations made when estimating the material properties and assigning these properties into the ﬁnite element model. The accuracy of the ﬁnite element results for image-based models is further restricted if the method of computing the material property depends on the ﬁnite element mesh. This dependence leads to constraints on the size of the ﬁnite element mesh for material assignment methods that consider a ﬁxed number of sample voxels around the elemental centroid [1,2] or the Gauss integration points [19], and also for methods that consider all the voxels within an element [15,30]. For a method that uses a ﬁxed number of voxels, errors are induced if the element size is greater than the voxel volume sampled; while for a method that uses all voxels in an element, small element sizes induce signiﬁcant error in the simulation [22]. To overcome these constraints on mesh size, a new, mesh-independent material mapping procedure for QCT-based ﬁnite element analysis of the vertebra was developed in this work. The use of a material block model disengaged the material property estimation from the FEA mesh, and a number of analyses were conducted in this work to highlight this capability.

The primary advantage of using the material block model is that the size of the material blocks is independent of the size of the ﬁnite elements. The block size can thus be chosen solely through consideration of the size required for accurate representation of the material ﬁeld. The material block size should be larger than the voxel size and should have sufﬁcient volume to reduce the noise from the discrete QCT data. At the same time, the block size should be small enough to capture the heterogeneity in density throughout the vertebral body. The size of the ﬁnite elements can be selected separately, with the main consideration being the numerical accuracy of the FE solution. A comparison between the mesh-dependent methods and the mesh-independent material block model developed in this study is presented using the ﬁnite element analyses on the vertebral column model (Figs. 4 and 5). The material block model was able to disengage the size of the ﬁnite element mesh from the column stiffness and also from the distribution of minimum principal strain in the column. The analyses on an entire vertebral body showed that decreasing the size of the ﬁnite elements led to a better approximation of the minimum principal strain distribution all the while keeping the material properties constant (Figs. 10 and 11). Also, the method is capable of handling higher-order elements that could map the geometry more accurately than linear elements, thereby reducing errors induced due to geometric approximations of complex structures like bones. In this method, the mesh quality could be improved in order to increase the accuracy of the ﬁnite element solutions, but without changing the material properties. Developing computationally intensive, mesh-independent spatial distribution functions of material properties based on HU value [31] can also be avoided by this method. In this method the density information from all of the voxels in a material block was considered for the determination of the material properties. Such an approach is more representative of the actual vertebral body compared to methods that consider only a ﬁxed number of voxels around the centroid or Gauss integration points. The restrictions on element size for ﬁnite element analysis were removed by this method so that element sizes comparable or smaller than the voxels could be used.

Although the new mapping procedure provides a fast and accurate mapping of material properties, the computational model used for veriﬁcation in this study suffers from several drawbacks. The superior and inferior surfaces were considered as uniform plane surfaces for ease of ﬁnite element modeling and also to remove the subjectivity in selecting the displacement and load boundary conditions. In contrast, the entire geometry with uneven superior and inferior surfaces should be considered for patient-speciﬁc stiffness and strength predictions. The geometry and material properties of the intervertebral disc, or at least the force distribution that the disc creates on the vertebral endplate, should also be represented accurately. As implemented in this study, the procedure assigned a single set of material properties to each element, which then requires that the element size in the FEA mesh size be smaller than the size of the material blocks. The material property assignment could be enhanced by replacing the centroid-based method with one that is capable of allowing material property variation within an element, such as variations either at the Gauss points or at the FE nodes. Finally, the procedure developed here requires additional user interaction because of the additional step of creating the material block model apart from the FEA mesh. However, the additional step does not increase the computational costs as the calculation of density-based elastic modulus from the QCT images for the material blocks was the major time consuming step. For a 4.0 mm material block with 380 distinct material properties, the total time taken for the determination of the elastic modulus was less than 5 min on a Core 2Duo Pentium processor of 3.166 GHz speed.

Even with these limitations, this computationally efﬁcient procedure is capable of reducing the inaccuracies in the determination of material properties in geometry-based ﬁnite element analyses of the vertebra. Although the present study was conducted using scans performed ex vivo, the material block model can readily be applied to clinical QCT data. The material block model can be implemented in computational analysis of vertebroplasty [13,32] and spine metastasis [33,34], where signiﬁcant variations in the material properties exist and where mesh reﬁnement is required at the material interfaces for accurate ﬁnite element solutions. Moreover, the material block model could be incorporated in ﬁnite element models of spine segments, because use of the material block model for the vertebra does not place any restrictions on how the intravertebral discs, ligaments, and other soft tissues are modeled. Finally, use of the material block model is not speciﬁc to the vertebra, and thus the mapping procedure could be applied to many other anatomic sites.

5. Conclusions

In this work, a new mesh-independent, material mapping procedure for the QCT-based ﬁnite element analysis of the vertebra was developed. The accuracy of this procedure was veriﬁed, and the implementation of this procedure in computational models of vertebral stiffness and deformation behavior was illustrated through several test cases. This method should help improve the reliability and accuracy of the ﬁnite element predictions and lead to advancements in the noninvasive assessment of whole bone mechanical behavior.

Acknowledgment

Funding for this study was provided by NIH R01 AR054620. The authors thank Dr. Glenn Barest and Amira Hussein for technical assistance.

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PERMALINK

A New Material Mapping Procedure for Quantitative Computed Tomography-Based, Continuum Finite Element Analyses of the Vertebra

Ginu U Unnikrishnan

Elise F Morgan

Abstract

1. Introduction