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Evaluation of damage to trabecular bone of the osteoporotic human acetabulum at small strains using nonlinear micro-finite element analyses

DING, Hai; ZHU, Zhen-an; DAI, Ke-rong

doi: 10.3760/cma.j.issn.0366-6999.2009.17.015
Original article
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Background With advance of age, alterations in bone quality, quantity and microarchitecture render osteoporotic trabecular bone become more sensitive to local failure. The aims of the present study were to clarify the extent to which the distribution of tissue-level stresses and strains was affected by structural changes and the extent to which osteoporotic acetabular trabecular bone was damaged at small strains.

Methods Using a DAWING 4000A supercomputer, nonlinear micro-finite element (μFE) analyses were performed to calculate the tissue-level strains and stresses for each element in the trabecular bone of one osteoporotic acetabulum at small strains to quantify the tissue-level damage accumulation and mechanical properties.

Results In contour plots of the tissue, maximum principal logarithmic strains, high tissue-level strains, both compressive and tensile, were observed in the osteoporotic trabecular bone at small apparent strains from 0.2% to 0.5% strain. The compressive apparent stress-strain curve showed typical nonlinear behavior and tangent modulus reduction with increasing strains. The microdamage curve suggested that microdamage began at 0.2% apparent strain in the osteoporotic trabecular bone and increased sharply, although very few microfractures occurred. The quartiles of the maximum principal logarithmic strains, minimum principal logarithmic strains and Von Mises stresses increased nonlinearly. For the inter-quartile range of the Von Mises stresses, a leap occurred at small strains ranging from 0.2% to 0.3% while microdamage commenced.

Conclusions Extensive microdamage was primarily responsible for the large loss in apparent mechanical properties that occurred in the trabecular bone of the osteoporotic acetabulum at small strains. With increasing apparent strains, continuous nonlinear increments of tissue-level strains and stresses resulted in microdamage that propagated throughout the specimen with very few microfractures.

Chin Med J 2009;122(17):2041–2047

Department of Orthopaedics, Ninth People's Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200011, China (Ding H, Zhu ZA and Dai KR)

Correspondence to: Dr. ZHU Zhen-an, Ninth People's Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200011, China (Tel: 86-21-63138341. Fax: 86-21-63137020. Email: zhuzhenan2006@126.com)

This research was supported by grants from the National High Technology Research and Development Program of China (No. 2006AA02A137) and the Postgraduate Creativity Foundation of Shanghai Jiao Tong University (No. BXJ0730).

(Received December 10, 2008)

Edited by HAO Xiu-yuan and PAN Cheng

Faced with the clinical need of better understanding the effects of daily activities on the elasticity and strength properties of trabecular bone, researchers have begun to investigate trabecular bone damage at small strains. Damage including degradation in mechanical properties and microdamage in trabecular bone following loading is a physical disruption to the trabecular microstructure.1–3 Since damage can serve as a biological response4,5 and decrease bone quality,6 such evidence motivates further studies on the loading magnitudes that cause trabecular bone damage. Investigating whether damage occurs at small strains will address the possible existence of a damage threshold for osteoporotic acetabular trabecular bone and aid in defining the damage behavior at strain magnitudes representative of those experienced with aging in vivo. Although apparent strains have not been measured in trabecular bone in vivo until now, the peak surface strains recorded in human cortical bone during vigorous activity have not exceeded 0.3%7 and the peak strains in femoral head trabecular bone during walking have been quantified at less than 0.5%.8 Therefore, characterizing the role of small strains in damage induction will help to further elucidate the relationships between microdamage and biological repair responses in osteoporotic trabecular bone.

A few previous studies have addressed the mechanical properties of pelvic trabecular bone at the continuum level. Vasu et al9 and Rapperport et al10 used density observations from roentgenograms to estimate the Young's moduli. In 1993, Dalstra et al11 used three different techniques to obtain better insights into the material properties of pelvic trabecular bone and concluded that pelvic trabecular bone was not highly anisotropic. However, these studies were performed to examine the trabecular bone of the whole pelvis without focusing on the apparent mechanical properties of trabecular bone of the acetabulum where the trabecular bone density is highest. It is important to note that pelvic trabecular bone was studied as a homogenized continuum, which means that these methods can provides stresses and strains at the homogenized level but not in individual trabeculae. Owing to the lack of stress and strain information at this detailed level, the extent to which the distribution of tissue-level stresses and strains is affected by structural changes caused by osteoporosis and the extent to which osteoporotic acetabular trabecular bone is damaged at small strains remain unknown.

In the present study, we aimed to find answers to these issues by quantifying the tissue-level stresses and strains in trabecular bone from the superior area of one Chinese osteoporotic human acetabulum using a micro-finite element (μFE) analysis that can represent individual trabeculae. In this technique, high-resolution sequential images obtained by microcomputed tomography (μCT) scanners are used as a basis for the geometry of a three-dimensional (3D) μFE model that can represent the detailed microstructure of the trabecular bone. As models generated in this way will generally contain more than half a million degrees of freedom, many researchers have been forced to use custom codes that utilize element-by-element iterative solvers. Owing to the complexity of nonlinear μFE analyses, these special iterative FE solvers have been limited to linear elastic analyses. However, many questions regarding the nonlinear mechanical behavior of trabecular bone remain unanswered. With the steady development in computational power, the feasibility of using this nonlinear μFE analysis to calculate tissue-level damage as well as tissue-level stresses and strains in trabecular bone by ABAQUS is explored here.

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METHODS

Materials

Bone mineral density values of the right hip joint were measured in situ in ten cadavers from an anatomic dissection course using a DXA scanner (Discovery A; Hologic, UK). None of the donors had a known history of bone and joint disease. Based on these measurements, the osteoporotic acetabulum of one Chinese male donor (78 years; T-score: -2.8) with a body weight of 61 kg, height of 165 cm and acetabular diameter of 50 mm was selected.

From this acetabulum, one cylindrical specimen (10 mm in diameter and 10 mm in length) was cut from the anterior-superior region of the acetabular trabecular bone using a standard biopsy drill without damage to the trabeculae (Figure 1). The axis was aligned with the principal axis based on the results of X-ray projections. The cortical shell was not included in the bone column.

Figure 1.

Figure 1.

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μFE Modeling

A μCT system (μCT-80; Scanco Medical AG, Bassersdorf, Switzerland) was used to scan the microarchitecture of the trabecular column. The spatial resolution for the specimen scanning was set to 37 μm. During each scan, the column was scanned continuously with a thickness and increment of 37 μm for 217 slices. The voxel size was 37 μm × 37 μm × 37 μm. After scanning, μCT images of the trabecular body were obtained. A region of interest was selected as a 5 mm × 5 mm × 5 mm cube in the center of the trabecular column to exclude boundary artifacts, and resulted in one trabecular cube. A μCT reconstruction model was used to construct one μFE analysis model by directly converting image voxels representing hard tissue (37 μm in size) to eight-node brick finite elements (37 μm in size) (Figure 2). The trabecular microarchitectural parameters of trabecular bone volume fraction (BV/TV), trabecular thickness (Tb. Th), trabecular spacing (Tb. Sp) and structural model index (SMI) were calculated using the 37 μm-micro model (Table 1).

Figure 2. A:

Figure 2. A:

Table 1

Table 1

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Nonlinear μFE analyses

For the osteoporotic trabecular cube, the mesh quantity of the model was 417 654 elements and 567 646 nodes. Trabecular tissue was modeled as an elastic-plastic material (Young's modulus = 18 GPa; Poisson's ratio = 0.3) with a composite yield surface possessing different tensile and compressive yield strains (0.412% and 0.825%, respectively).13 The post-yield modulus was 0.9 GPa.

A fully nonlinear (geometric and material) μFE analysis (ABAQUS 6.6; Abaqus Inc., Pawtucket, RI, USA) was conducted in which the specimen was simulated to be loaded in uniaxial compression to 0.5% apparent strain on the longitudinal direction.13 A fixed displacement boundary condition was chosen for the μFE analysis as follows: all nodes at the bone-platen interface were constrained in the plane of the platen, with all other surfaces unconstrained, to simulate the boundary conditions that have been used in other mechanical testing procedures.

The μFE analysis was performed with ABAQUS/Standard, which is optimized for easy parallel processing and solving of both geometrically and materially nonlinear models. Sixteen processors of a DAWING 4000A supercomputer (Shanghai Supercomputer Center, China), with a total of 2128 processors and 4256 GB of memory, were used for the calculations. The memory requirement for the osteoporotic acetabular trabecular cube was 17 GB. The total central process unit (CPU) time for solving the osteoporotic acetabular trabecular bone model was about 8.5 hours, and the total wall-clock time was approximately 11 hours.

For every increment of 0.05% apparent strain, the equivalent plastic strains, maximum principal logarithmic strains, minimum principal logarithmic strains and Von Mises stresses were calculated for each element. The maximum and minimum principal logarithmic strains, both compressive and tensile, were observed at small apparent strains to clarify the extent to which each element yielded in either compression or tension at the tissue level. On the basis of the strain-based microdamage criterion, if the value of the equivalent plastic strain was not 0, particular trabecula was defined as damaged at the tissue level. This microdamage analysis was repeated for each of the ten selected applied apparent strains. Microdamage quantity was defined as the percentage of trabeculae for which the equivalent plastic strains were not 0.14

To identify the bone tissue at risk of microfracture, the tissue-level maximum and minimum principal logarithmic strains were calculated at every increment of 0.05% apparent strain. Any bone element with either its maximum or minimum principal logarithmic strain beyond 8.8% strain was identified as being microfractured.15

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RESULTS

In contour plots of the tissue-level maximum principal logarithmic strains, high tissue-level strains, both compressive and tensile, were observed in the osteoporotic trabecular bone at small apparent strains from 0.2% to 0.5% strain (Figure 3). The focal regions of tissue yield were found to be interspersed among regions experiencing mild or very low tissue-level strains.

Figure 3.

Figure 3.

For the osteoporotic trabecular bone, the compressive apparent stress-strain curve showed typical nonlinear behavior and tangent modulus reduction with increasing strains. This nonlinearity resulted in a significant non-zero reduction in the tangent modulus from 0.2% to 0.5% strain. The tangent modulus was greater at 0.2% strain than at 0.4% strain, and the reduction was 3.399% (Figure 4).

Figure 4.

Figure 4.

Extrapolation of the microdamage curve suggested that microdamage began at 0.2% apparent strain in the osteoporotic trabecular bone. The microdamage of the osteoporotic trabecular bone accumulated rapidly at low strains (Figure 5). Even at an apparent strain as low as 0.250%, 0.069% of the specimen yielded in either compression or tension. Thereafter, the percentage increased to 0.858% at an apparent strain of 0.5%. The curve of the incidence of microdamage and the applied apparent strains also indicated typical nonlinear behavior.

Figure 5.

Figure 5.

While the median values of the maximum and minimum principal logarithmic strains and the Von Mises stresses increased, the quartiles of these values increased nonlinearly with increasing applied strains (Figure 6). For the inter-quartile range of the Von Mises stresses, in particular, a leap occurred at small strains ranging from 0.2% to 0.3% strain (Figure 7) while microdamage commenced.

Figure 6.

Figure 6.

Figure 7.

Figure 7.

Microfracture did not occur until 0.35% apparent strain using the fracture criterion and the number of microfractured elements was only one. At 0.5% apparent strain, there were four microfractured elements (Table 2).

Table 2

Table 2

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DISCUSSION

As efforts to quantify the occurrences and consequences of in vivo damage in trabecular bone increase, there is a need to investigate the damage behavior at the smaller strains encountered in daily life, as opposed to traumatic strain or overloading. The overall goals of the present study were to characterize the biomechanical roles of microdamage and the mechanical property changes in the trabecular bone of the osteoporotic human acetabulum at small strains. Our data demonstrated that microdamage occurred in trabecular bone at 0.2% strain while a leap in the inter-quartile range of the Von Mises stresses took place. Even at these small apparent strains, tissue-level strains and stresses were sufficiently high in regions of the specimen to cause local tissue yielding and trabecular microdamage, although very few microfractures were observed. Therefore, the negative consequences on the apparent mechanical properties of the trabecular cube at small strains are best explained by widespread accumulation of microdamage within the trabecular bone.

The results of this nonlinear μFE analysis are plausible for a number of reasons. First, this μFE model was based on a high-resolution 3D model of acetabular trabecular bone scanned by μCT, indicating that this μFE model can be considered to represent an effective surrogate for real specimens.16,17 This kind of μFE model can successfully describe in detail the thickness, length, orientation of individual trabeculae and 3D microarchitecture of the trabecular bone. Second, the trabecular cube was scanned at 37 μm, and 37 μm is sufficiently high to obtain precise 3D images of the trabecular microstructure. Therefore, based on this high-resolution 3D model, the μFE analysis can accurately describe the tissue-level biomechanical properties and changes of the trabecular bone. A few previous studies that investigated the effects of voxel size on bone microstructural parameters calculated from μCT images and on the computed values of bone properties in μFE simulations proved that high-resolution scanning and reconstruction were important for achieving precise results using μFE analyses.18,19 Third, since this finite element formulation included geometric and material nonlinearity, the nonlinear μFE analyses could accurately describe the nonlinear behavior of the trabeculae and the strength asymmetry at the level of individual trabeculae.20 For highly porous bone, such as osteoporotic trabecular bone, the effects of geometric nonlinearity should be great. Owing to the material nonlinearity, our tissue-level constitutive model allowed strain softening (a negative modulus). Therefore, this approach included the physics essential for predicting apparent level strain softening. Finally, the results of our study agreed well with previous experimental studies on trabecular damage to human vertebral bone, which indicated that microdamage was the prevalent damage for modulus reduction versus small apparent strains.21,22

Compared with previous studies on the trabecular bone of the acetabulum, our study is unique because it used nonlinear μFE analyses with ABAQUS to study tissue-level microdamage in context with microfracture in the trabecular bone of one Chinese osteoporotic acetabulum. Our model showed that the tangent modulus reduction of 6.748% was the result of 0.858% of the total trabeculae being microdamaged and only four elements being microfractured. The curve of the percentage of microdamage and applied strains indicated that microdamage commenced at 0.2% strain and increased nonlinearly and sharply with small applied strains. Other studies reported that residual strains and changes in modulus were observed at apparent strains of less than half the apparent compressive yield strain,23 which supports the idea that damage contributes to the nonlinearity of the stress strain curve at small strains.24 The dominating effects of microdamage over the range of strains applied in the present study as well as in other studies provide a mechanical explanation of the modulus reductions that have been shown to occur over such small strains.25 In previous experiments regarding damage in trabecular bone, frequent occurrence of microdamage was observed with very few microfractures, even at an apparent strain of 15%.3,21,22 Our analysis is also consistent with other FE studies on trabecular damage, which proved that microdamage is primarily responsible for the apparent level modulus reduction.

The micromechanics of our model dictate that loads at small apparent strains that are sufficiently large to cause microfractures would similarly induce severe amounts of microdamage in the trabecular bone of the osteoporotic acetabulum. Therefore, the biomechanical effects of microdamage dominated over those of microfracture, suggesting that microfracture should be viewed as an endpoint of microdamage accumulation with increasing apparent strains and providing an explanation for why microfractures were seldom observed in comparison to microdamage.21,22

Our data indicated that microdamage commenced at 0.2% apparent strain. When the relationship between modulus reduction and peak applied strain for human trabecular bone obtained in a previous compressive test is extrapolated,26 modulus reductions are predicted to commence at about 0.2% strain, which coincides with our result. Taken together, these findings may imply that the threshold of microdamage in trabecular bone should be 0.2% apparent strain regardless of differences in bone density, microstructure, anatomical site and age. The quartiles of the maximum principal logarithmic strains, minimum principal logarithmic strains and Von Mises stresses increased nonlinearly with the applied strains. In other words, these similarities imply that the continuous nonlinear increments in tissue-level strains and stresses result in nonlinear accelerated accumulation of microdamage in trabecular bone, whereby modulus reductions are initiated at an early point and then increase with apparent strain as microdamage propagates throughout the specimen.

Some limitations of this study should be mentioned. The first limitation is that the study does not have complementary experimental data regarding microdamage at small strains. With experimental measurements of microdamage, we would be able to gain insights into the physical deformation and bulking phenomena associated with the changes. However, it is very difficult, and perhaps impossible, to noninvasively quantify trabecular damage through biomechanical techniques at the tissue level.25 Some previous studies have indicated that nonlinear μFE analyses can precisely predict the tissue-level mechanical behavior of trabecular bone and that the results agree well with experimental data. Second, the computational requirements for these nonlinear μFE analyses are still excessive. For the analysis, the equivalent of 134 hours of CPU time was used. As a result, a parallel supercomputer is required to perform these analyses within a reasonable amount of wall-clock time. Finally, inhomogeneous FE models are more accurate representations of the heterogeneous distribution of tissue in trabecular bone. Hence, inhomogeneous FE analyses are expected to more precisely estimate the local stresses and strains in trabeculae than homogeneous FE analyses. However, a previous study found that there was less than 9% difference in the estimated stresses and strains between these two kinds of FE analyses.27

Biologically and clinically, the present results provide extra information about the tissue-level biomechanical nature of osteoporotic acetabular trabecular bone. From a biological perspective, our data indicate that microdamage and microfracture can take place in osteoporotic trabecular bone at relatively low apparent strains. As such small strains can be expected to happen during strenuous activity28 or daily life,8 it is possible that some form of subtle microdamage and microfracture can occur habitually, which subsequently induces biological responses that serve as a remodeling stimulus.29 On the contrary, such microdamage can decrease bone quality30 and increase fracture susceptibility31,32 if the microdamage accumulation is beyond the repair ability of the bone with aging. From a clinical perspective, since the trabeculae of cancellous bone have similar mechanical properties to those of cortical bone33,34 and cortical bone is often described as condensed trabeculae,35,36 such microdamage and microfracture may be doomed to occur in the cortical bone of the osteoporotic acetabulum at small apparent strains, which may play important roles in the aseptic loosening of acetabular prostheses. Therefore, further μFE analyses of the whole acetabular bone, including the cortical bone and trabecular bone, may help to provide insights into the biomechanical mechanism underlying the aseptic loosening of acetabular prostheses.

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Keywords:

mechanical property; trabecular bone; nonlinear micro-finite element analysis; acetabulum; osteoporosis

© 2009 Chinese Medical Association