A. Odgaard

A NEW METHOD TO DETERMINE TRABECULAR BONE ELASTIC PROPERTIES AND LOADING USING MICROMECHANICAL FINITE-ELEMENT MODELS

The apparent mechanical behavior of trabecular bone depends on properties at the tissue or trabecular level. Many investigators have attempted to determine trabecular tissue properties and loading. However, accuracy and applicability of all methods reported are limited. The small size of the trabeculae and a possible size effect are complicating factors when using traditional testing methods on single trabeculae. Other methods reported, using models that describe the trabecular structure, are of limited value because they consider bone as a repetitive structure in order to describe a reasonably large region of bone. The present study introduces a new finite-element method strategy that enables analysis of reasonably large regions of trabecular bone in full detail. The method uses three-dimensional serial reconstruction techniques to construct a large-scale FE model, by directly converting voxels to elements. A 5 mm cube of trabecular bone was modeled in this way, resulting in a FE model that consists of 296,679 elements. Special strategies were developed to solve the set of equations that results from the FE approach. Using this model in combination with experimental apparent data taken from the literature, the upper and lower boundaries for the tissue modulus were calculated to be 10.1 and 2.23 GPa, respectively. From the local stress and strain distributions it was concluded that the deformation mode of the trabeculae in the present cube was predominantly in bending. It was concluded that the method developed offers new perspectives for the study of trabecular bone.

Mechanical and textural properties of pelvic trabecular bone

So far, virtually nothing is known about the mechanical properties of pelvic trabecular bone. In this study, several techniques have been used to establish some insight in these properties. Dual-energy quantitative computer tomography (DEQCT) was used to look at the distribution of bone densities throughout the pelvic bone and nondestructive mechanical testing was used to obtain Youngs moduli and Poissons ratios in three orthogonal directions for cubic specimens of pelvic trabecular bone. The same specimens were then used for stereological measurements to obtain volume fractions and the spatial orientations of the mean intercept lengths. The combined data on the mechanical tests and the stereological measurements made it possible to calculate Youngs moduli and Poissons ratios for the specimens principal material axes. DEQCT showed that bone densities within a pelvic bone are significantly higher in the superior part of the acetabulum, extending to the sacroiliac joint area and, secondly, in the area of the pubic symphysis. Volume fractions found for the specimens did not exceed 20%. This may be considered rather low when compared to values reported in the literature for trabecular bone of femoral or tibial origin, but the values do lie in the same range as vertebral trabecular bone. With the volume fraction as its primary predictor, values of Youngs moduli were also low. For most specimens these values were not higher than 100 MPa, with an occasional peak of 250 MPa. Looking at the ratio of the highest and lowest Youngs modulus or at the components of the fabric tensor, it can be concluded that pelvic trabecular bone is not highly anisotropic. On an average, Poissons ratio was found to be closer to 0.2 rather than 0.3, which is in accordance with other studies on Poissons ratio of trabecular bone.

The underestimation of Young's modulus in compressive testing of cancellous bone specimens

In order to determine the accuracy of measurements of Youngs modulus of cancellous bone by conventional compression testing, two independent strain measurements were made simultaneously during non-destructive uniaxial compression to 0.8% strain of rectangular specimens (n = 18). Strain was measured by an extensometer attached to the compression anvils close to the specimen and by an optical system covering the central half of the specimens. Mean Youngs modulus determined by the extensometer technique was 689 MPa, but was 871 MPa when determined by the optical technique (mean difference = 182 MPa, SED = 50 MPa, p less than 0.002). Uneven strain distribution due to lack of support of cut vertical trabeculae at the anvil-specimen interface is believed to be causing the underestimation of Youngs modulus measured by the extensometer technique. The influence of friction at the specimen-anvil interface was studied by performing a finite element analysis. It is concluded that Youngs modulus of specimens of the chosen geometry on average is underestimated by about 20% by conventional compressing testing. The underestimation seems not to be dependent upon specimen density.

Tensile and compressive properties of cancellous bone

The relationship between the mechanical properties of trabecular bone in tension and compression was investigated by non-destructive testing of the same specimens in tension and compression, followed by random allocation to a destructive test in either tension or compression. There was no difference between Youngs modulus in tension and compression, and there was a strong positive correlation between the values (R = 0.97). Strength, ultimate strain and work to failure was significantly higher in tensile testing than in compressive testing.

A direct method for fast three-dimensional serial reconstruction

A method for accurate three‐dimensional reconstruction of openly connected porous structures is described. The method is based on embedding of a specimen in a contrast coloured epoxy resin and serial sectioning in a standard hard tissue microtome. A PC‐based image processing system is used for direct digitization of the cut surface, and by thresholding two‐phase images are obtained. The process is fully automated, and about 170 sections can be produced, digitized, dichotomized, and stored per hour. As an example of its applications, the method is used on trabecular bone, which is an anisotropic porous structure.

Estimation of structural anisotropy based on volume orientation: a new concept

The quantification of anisotropy—its main direction and the degree of dispersion around it—is desirable in numerous research fields dealing with physical structures. Conventional methods are based on the orientation of interface elements. The results of these methods do not always agree with perceived anisotropy, and anisotropic structures do not necessarily turn out to be ‘anisotropic’ using these methods. In the present paper, we propose an alternative to curve and surface orientation, namely volume orientation. Using trabecular bone as an example of a two‐phase anisotropic structure, the new concept is studied in some detail. In particular, a parametric method of estimating volume orientation from sections is presented and discussed.

Compressive axial strain distributions in cancellous bone specimens

The compressive axial strain distribution in cylindrical trabecular bone specimens was studied using digitized images of the specimen surface. Specimens were tested with strain rate 0.00015 s-1. Images were taken at 0, 1, 2, 3, 4, 6, 8 and 10% strain. Using an optical illusion of movement by rapidly changing succeeding images, failures were classified as transverse (33%) or oblique collapses (67%). The location of failure was not determined by the specimen density gradient. Local axial strain in the distal, intermediate and proximal third was measured throughout the compression in the transversely failing specimens, whereas local strain in the obliquely failing specimens was measured in the pre-failure phase only. Axial strain inhomogeneity was observed in the pre-failure as well as in the post-failure phase. In the pre-failure phase the intermediate third was strained significantly less than the thirds near the ends. In the post-failure phase specimen strain occurred solely in the collapsed part. Ultimate strain of the transversely failing specimens was 2.5% and ultimate strain of the failing third was 3.7%. At failure less than 1% strain was observed in the intermediate third and at 10% specimen strain 1.5% local strain was found in the intermediate third. The results indicate unreliability of conventional stiffness and strain measurements in trabecular bone specimens probably due to lack of trabecular constraint at the end surfaces. Conventional measurements tend to underestimate stiffness and, by giving an average value of strain in spite of considerable strain inhomogeneity, to underestimate failure strain.

Star length distribution: a volume‐based concept for the characterization of structural anisotropy

Determination and quantification of anisotropy is of great interest in research fields dealing with physical structures or surface textures. In this paper, a volume‐based method is presented, which essentially determines the mean object length in a certain direction for a typical point within a structure or texture. The mean object lengths for all orientations together form the so‐called star length distribution (SLD). The validity and the accuracy of the SLD method are investigated, and illustrated by applying it to trabecular bone. By using a line sampling algorithm, the relation with other anisotropy measures could be studied analytically. Preliminary tests suggest that with SLD a more exact description of the mechanical properties of porous structures may be obtained than with other anisotropy measures. However, due to possible secondary orientations that become apparent with SLD, a fabric tensor must be of rank higher than two in order to properly describe an orthogonal structure mathematically.

Physical bone changes in carragheenin-induced arthritis evaluated by quantitative computed tomography.

Repeated non-invasive measurements were performed in dogs of trabecular bone density (TBD), low density bone area (LDBA), and high density bone area (HDBA) in chronic arthritis using quantitative computed tomography (QCT). Unilateral chronic arthritis of the knee had been induced by weekly instillation of 2 ml carragheenin into the right knee joint for 12 weeks with the left knee serving as a control. CT scanning of the distal femoral condyles was performed in 12 mature dogs with chronic arthritis. Another 6 dogs underwent a longitudinal CT study starting immediately prior to induction of arthritis. During induction of arthritis TBD decreased (P<0.01), LDBA increased (P<0.05) and HDBA decreased (P<0.01) in the arthritic bone. Opposite changes were found on the control side, i.e. TBD increased (P<0.01), LDBA decreased (P<0.01) and HDBA increased (P<0.01). The chronic arthropathic bone showed 20% lower TBD (P< 0.0001), greater LDBA (P<0.0001) and lower HDBA (P<0.0001) as compared with the control bone. Reproducibility tests of TBD showed a coefficient of variation of 0.8%. Indentation tests and histomorphometric analyses confirmed the bone density changes as measured by CT.

Complete assessment of elastic properties of trabecular bone architecture from 3D reconstruction images

A method is presented that allows for a complete mechanical evaluation of trabecular bone architecture directly from three-dimensional computer reconstruction images. With this method, the reconstruction images are used as a basis for microstructural FE-analyses. From the results of these analyses the full stiffness matrix of bone specimens is obtained, using a standard mechanics approach. An optimization procedure is then used to find the best orthotropic representation and principal directions of this matrix. The method is demonstrated here relative to two trabecular bone specimens. With the development of in vivo reconstructions and the methods demonstrated here, even in vivo measurements will be possible.