This study investigated the relationship between morphology and finite-element derived elastic moduli in μMR images of TB subvolumes from the distal tibia both in specimen images acquired under in vivo
scanning conditions and in vivo
in a small cohort of human subjects. Elastic constants derived from μMRI are comparable in magnitude to values reported by other investigators, including data obtained by direct mechanical testing of samples from the proximal tibia (elastic moduli range: 10–1500 MPa) (35
) and elastic constants determined from μFE analysis of HR-pQCT images of the distal tibia (mean elastic modulus: 720±297 MPa) (1
). Our results show that variations in μFE-derived elastic constants associated with anatomical location and test direction were not adequately explained by bone volume fraction in either the specimen or in vivo
data, exhibiting a strong regional and test-direction dependence. These findings agree with previous studies conducted on the basis of high-resolution μCT imaging evaluating the relationship between morphological measures and elasticity (2
When including the MIL-derived fabric tensor information via Cowin’s model of orthotropic materials, the nine orthotropic μFE-calculated elastic constants could be predicted with high coefficients of determination (mean R2adj: 0.92 in specimen, 0.82 in vivo) independent of sub-volume location and test direction. Additionally, the model-predicted elastic properties were found to distinguish post-menopausal women with reduced bone mass from younger healthy subjects with significance levels similar to those achieved by μFE modeling.
The dependence of the orthotropic elastic constants on volume fraction and fabric has previously been shown in high-resolution reconstructions of TB specimen image data (R2adj
ranging from 0.88 to 0.99) (13
). However, until now, the validity of Cowin’s model had not been demonstrated on the basis of in vivo
The 18 empirically determined model parameters did not appear to change substantially with input images from the same anatomical location, as evidenced by their ability to predict elastic constants derived from in vivo images using the specimen-derived model parameters. This result suggests the possibility of developing a model on the basis of a large set of specimen datasets, potentially in concert with mechanical experiments, and then applying the model to structural measures assessed from images acquired in vivo to predict bone mechanical properties in patients. This finding is of particular relevance considering the spread in BV/TV values and moduli between specimen and in vivo images.
The difference in mean bone volume fraction (and thereby elastic moduli) found between specimen and in vivo
data is somewhat surprising and demands some scrutiny. One possible reason is that spin dephasing at trabecular bone boundaries caused by the susceptibility difference between the slightly paramagnetic fixing solution and diamagnetic bone is likely to inflate BV/TV values in the specimens relative to values obtained in vivo
) (since the bone-fixing solution susceptibility difference is greater than that between bone and marrow).
The validity of Cowin’s model for the current data is particularly noteworthy. The degree to which TB complies with the notion of orthotropic symmetry was evaluated in terms of the orthotropic error ε (see Eq. 
). Forcing orthotropic symmetry was associated with a ~17% error in the calculated elastic constants. This error is considerably larger than the 6% error found in μCT images of (4mm)3
TB specimens from seven distinct human metaphyseal locations by Zysset et al (35
). One explanation for the relatively large mean error in the present work is the proximity of the TB subvolumes to the cortical boundary where the structural, and thus the mechanical environment, is expectedly more heterogeneous. Trabecular bone subvolumes used in Zysset et al’s work were likely extracted from more centrally located regions of the skeleton where the architecture is more homogeneous. Deviations from orthotropic symmetry also impact the degree of alignment between fabric and orthotropic symmetry axes. The average offset angles Ωavg
were 11.2° (specimen) and 9.2° (in vivo
), data that are in excellent agreement with those found by Zysset et al (~11°) (35
). Increasing angular offsets Ωavg
were associated with increasing orthotropic error for specimen and in vivo
data (see ). Therefore, the error in the orthotropic elastic constants and the model prediction were augmented by the degree to which the TB sub-volume adhered to orthotropic symmetry. This source of error was minimized by excluding TB sub-volumes with orthotropic errors greater than 4% (12
). As mean errors in the measured Young’s (Shear) moduli of 9.5% (1.1%) were found for offset angles of 10° (38
), it would have been necessary to exclude a substantial number of datasets from this study, as it was in Turner’s study, if utilizing such an extreme cut-off value. Instead, all datasets were retained at the expense of lower accuracy in the model prediction.
Figure 8 Mean offset angle Ωavg and standard deviations relative to range in orthotropic error ε (%) (see Eq. ) for specimen (a) and in vivo (b) datasets. Numbers indicate the number of datasets falling into the associated range of ε. (more ...)
The spatial variations in architecture, notably in terms of structural orientation and trabecular density between the three subvolumes examined (), is substantial. Thus, inclusion of a larger volume (> 1–2 cm in width) would likely violate the condition of orthotropy and the underlying theory would be inapplicable. For this reason, prediction of mechanical parameters in terms of structural orientation are limited to relatively small volumes (subvolumes with sides ranging from 0.4 to 1cm), which may not be representative of the bone’s overall mechanical properties. Moreover, larger volumes are now readily amenable to image-based FE analysis. However, the method provides insight into the regional structural and mechanical make-up of the trabecular network. By decomposing the mechanical behavior of TB into contributions from volume fraction and fabric, its underlying causes can be inferred. For example, it is understood that structural anisotropy increases with age and osteoporosis, thereby rendering the bone more likely to fail due to buckling (see, for example (15
)). Further, it has previously been shown that Cowin’s model parameters do not substantially differ for normal and osteoporotic bone on the basis of micro-computed tomography (39
). It would therefore be of interest to examine these structural alterations during disease progression and treatment.
This work is, to the authors’ knowledge, the first application of Cowin’s model to images of the distal tibia acquired either under conditions matching those in vivo or actual images obtained in human subjects by μMRI. The data emphasize the role of fabric in bone mechanical properties, and that the relationship between fabric and mechanical constants can be assessed in the regime of limited resolution and SNR of high-resolution magnetic resonance. The findings should be equally relevant to other imaging modalities, notably HR-pQCT, that have similar point-spread function limited resolution.