Assessment of the pore geometry of stereolithographic models by high resolution MRI.

It is becoming increasingly evident that material strength and other mechanical properties depend not only on the density but also on the internal architecture of the structure in question. The internal structures of five different stereolithographic samples were examined noninvasively using high resolution magnetic resonance imaging (MRI). The image analysis techniques of segmentation applied to the acquired images allowed the quantification of the structural parameters mean pore size and pore separation. These parameters were quantified along many angular orientations with a spatial resolution of 0.12 mm, and their distributions clearly revealed the structural anisotropy of the samples.

Fractal dimension predicts broadband ultrasound attenuation in stereolithography models of cancellous bone.

There has been considerable debate on the relative dependence of broadband ultrasound attenuation (nBUA, dB MHz(-1) cm(-1)) upon the density and structure of cancellous bone. A nonlinear relationship between nBUA and porosity has recently been demonstrated using stereolithography models, indicating a high structural dependence for nBUA. We report here on the measurement of trabecular perimeter and fractal dimension on the two-dimensional images used to create the stereolithography models. Adjusted coefficients of determination (R2) with nBUA were 94.4% (p < 0.0001) and 98.4% (p < 0.0001) for trabecular perimeter and fractal dimension respectively. The feature of fractal dimension representing both the porosity and connectivity of a given structure is most exciting. Further work is required to determine the relationship between broadband ultrasound attenuation and fractal dimension in complex three-dimensional cancellous bone structures.

A comparison of porosity, fabric and fractal dimension as predictors of the Young's modulus of equine cancellous bone.

The purpose of this study was to compare the structural parameters of fabric and fractal dimension as predictors of the Young's modulus of equine cancellous bone. Eight 15 mm cubes of cancellous bone were obtained from three equine third metacarpal bones. Young's modulus was determined for the three orthogonal directions. The fabric and fractal dimension were calculated for each of the six exposed faces of each cube. Fractal dimension plus porosity provided a higher explanatory power for Young's modulus (R2 = 78.7%. P < 0.0001) than fabric plus porosity (R2 = 69.2%, P < 0.0001). Fractal dimension was also significantly correlated with fabric (R2 = 53.8%, P < 0.0001). Although this novel method for combining fractal dimension data into a pseudo-directionally dependent predictor of Young's modulus requires further validation over a greater range of porosities and differing cancellous bone tissues, its potential has been demonstrated.

The ability of ultrasound velocity to predict the stiffness of cancellous bone in vitro.

The mechanical status of bones is an important consideration in skeletal pathological conditions such as osteoporosis, which result in fracture at predominantly cancellous bone sites. Density is a good predictor of the stiffness and strength of cancellous bone. However, these mechanical properties are also dependent on the cancellous bone's architecture. The objective of this work was to investigate the ability of ultrasound velocity to predict the Young's modulus of elasticity of cancellous bone. The cancellous bone specimens were 20 mm cubes from bovine femur and 21 mm diameter mediolateral cylinders cored from human calcaneus. Ultrasound velocity (V) and Young's modulus (E) were determined in three orthogonal directions for the bovine cubes [anteroposterior (AP), mediolateral (ML), and proximodistal (PD)], and mediolaterally in the calcaneus. Apparent density (p) was determined after the other tests. Density alone explains 87.6% of the variance of Young's modulus in human calcaneal and bovine femoral bone tested in the PD direction only. Velocity, however, explains 95% and a combination of density and velocity 97%. Velocity and stiffness are not random with respect to the three directions in the bovine specimens. Further, for each cube we obtained the mean of the three values of E and of V, and characterized each value of E and V by their deviation from their mean. There is an extremely strong positive correlation (r = 0.80) showing that the degree of deviation is consistent for E and V, and of the same sign. These results demonstrate that the velocity of ultrasound in cubes of cancellous bone can give structure-specific information. In particular, knowledge of both density and velocity allows better predictions of stiffness than do density or ultrasound velocity on their own. Because there are noninvasive methods of measuring density that do not depend on ultrasonic measurement the combination of these two measurements promises, eventually, to give improved assessment of a bone's weakness and liability to fracture.

The in vitro measurement of ultrasound in cancellous bone.

This paper describes our recent findings on the relationships between ultrasonic measurements (velocity and broadband ultrasonic attenuation) and some physical properties of human and bovine cancellous bone (density, trabecular orientation and Young's modulus of elasticity). We have found velocity to be an extremely effective measure of Young's modulus (R2 approximately 95%). When velocity is combined with a measure of apparent density R2 improves to approximately 97%. We demonstrate that this is due to the ability of ultrasound velocity to measure structural anisotropy in the tissue. The findings for broadband ultrasonic attenuation (BUA) are more complex. In the same specimens BUA is not as good as velocity at predicting Young's modulus (R2 approximately 62%). We demonstrate that this is due to a non-linear relationship between BUA and tissue density (porosity). However there is a strong indication that BUA is also affected by variation in cancellous structure.

The non-linear relationship between BUA and porosity in cancellous bone.

There is growing interest in assessing the clinical value of ultrasound in the prediction and management of osteoporosis. However, the mechanism of ultrasound propagation in cancellous bone is not well understood. The Biot theory is one approach to modelling the interaction of sound waves with cancellous structure, and porosity is one of its input parameters. In this paper we report the relationship between broadband ultrasonic attenuation (BUA) corrected for specimen thickness (nBUA) and porosity in a porous Perspex cancellous bone mimic, a stereolithography cancellous bone mimic and in natural human and bovine tissue. nBUA and porosity have a non-linear parabolic relationship. The maximum nBUA value (nBUAmax) occurs at approximately 30% porosity in the Perspex mimic, approximately 70% in the stereolithography mimic and approximately 75% in natural cancellous bone. We discuss the effect of structure on the form of the nBUA-porosity relationship.

Orthogonal relationships between ultrasonic velocity and material properties of bovine cancellous bone.

Osteoporotic fractures follow a period of asymptomatic bone loss and hence bone strength, predominantly in cancellous bone. An effective management of osteoporosis requires an understanding of the mechanical behaviour of cancellous bone including the anisotropic dependence. Ultrasound velocity (V) and elasticity (Young's modulus, E) were measured in the three orthogonal directions in 20 mm cubes of bovine cancellous bone. Student paired t-test analysis showed significant variations in velocity and elasticity for the three orthogonal directions, the highest significance being between proximal-distal (PD) and antero-posterior (AP) directions with t = 5.63 and 4.09 for velocity and elasticity respectively, the lowest significance between medio-lateral (ML) and antero-posterior directions. Elasticity followed a power law relationship with apparent density (p) as reported in the literature, the exponent (b) being direction dependent (b = 1.98 +/- 0.21 for PD, 2.42 +/- 0.24 for AP and 2.03 +/- 0.17 for ML). The adjusted R2 values between elasticity and apparent density were highly significant (79.9% for PD, 81.9% for AP and 85.7% for ML). The relationship between velocity and apparent density is less significant in terms of the amount of variance explained (48.5% for PD, 63.3% for AP and 64.4% for ML). R2 values relating elasticity and velocity were again highly significant (79.4% for PD, 82.9% for AP and 80.5% for ML) and the coefficients, determined by regression analysis, independent of direction. Analysis of velocity, elasticity and density data for a range of reference materials demonstrated that experimentally measured longitudinal wave velocity could be reliably substituted into the bar wave equation (v = square root E/p). This implies that a combination of velocity and apparent density may be an improved indicator of bone fragility than density alone.

Prediction of mechanical properties of the human calcaneus by broadband ultrasonic attenuation.

Broadband ultrasonic attenuation (dB MHz cm-1, nBUA) was determined for specimens from 20 human calcanei, along with apparent density, elasticity (Young's modulus), and compressive strength. The calcanei were modified to provide "whole" (only soft tissue removed), "core" (mediolateral cores corresponding to in vivo measurement region), "can" (cortical end plates removed from core), and "def" (core defatted) samples. The nBUA values for the various modifications were highly correlated. The presence of the cortical endplates creates a significant nBUA, probably due to complex phase interactions. nBUAcan was a good predictor of elasticity (R2 = 75.7%) and strength (R2 = 73.6%). Apparent density was a better predictor of the mechanical variables than nBUA, with R2 values of 88.5% for elasticity and 87.6% for strength. The morphological anisotropy defined by "fabric" for the specimens was extremely uniform. The coefficient of variation in nBUA (40.5%) and compressive strength (64.4%) was significantly greater than for apparent density (23.5%) and fabric (6.7%). It is well known that a power law relationship exists between apparent density and elasticity or strength in cancellous bone. An interesting finding in this work is that there also appears to be a power law relationship between nBUA and apparent density, with an exponent of approximately 2, which, in the light of clinical implications, warrants further investigation.

Separate effects of osteoporosis and density on the strength and stiffness of human cancellous bone.

The question this paper attempts to answer is whether osteoporosis has any effect on the mechanical properties of cancellous bone, except that produced by osteoporotic bone being generally less dense. The general density of the heads of the tibia, which we took as a surrogate measure of osteoporosis, was measured on the tibias of nine women using dual energy X-ray absorptiometry, and cubes cut from these tibias were tested mechanically. Equations were derived relating Young's modulus of elasticity and strength to the densities of the cubes, to their 'fabric' (a measure of anisotropy) and to 'mineral volume density' (a measure of the general density of the bone from the head of each tibia). Density and fabric between them accounted for about 80% of the variance in strength and in stiffness of the cubes, and adding mineral volume density as an explanatory variable had no effect on the explanatory power of the equations. Fabric was completely uncorrelated with density or mineral volume density. We conclude that although, obviously, osteoporosis weakens bones by making them less dense, there is no evidence that osteoporosis weakens bones in any other way. If there is an effect, it must be subtle.

Mechanical properties of microcallus in human cancellous bone.

Until now, the mechanical properties of the microcalluses that form in human cancellous bone have been unexplained. We measured the microhardnesses of microcalluses in cancellous bone, of the trabeculae within the microcalluses, of the trabeculae adjacent to microcalluses, and of trabeculae lacking microcalluses in a human tibia and femur. We observed no important differences between materials at the four different sites. Because the microhardness of bone is very closely related to its stiffness, this finding indicates that microcalluses are likely to stiffen the trabeculae in which they are formed, even though they may surround unhealed fractures of the cancellous trabeculae.

Effects of structural variation on Young's modulus of non-human cancellous bone.

The modulus of elasticity of cubes of cancellous bone, tested in three orthogonal directions, was measured along with apparent density, fabric (measured using image analysis techniques), connectivity and mineral volume fraction. Multiple regression was used to relate Young's modulus and the explanatory variables. The most important explanatory variable was apparent density; fabric was also important. Connectivity was highly correlated with apparent density, and so it is difficult to assign relative weights to these two variables. However, both have a significant, and separate, effect on Young's modulus. The total variance in Young's modulus explained by these three variables was about 85 per cent. This implies that other explanatory variables are either unimportant or highly correlated with the variables listed above. Mineral volume fraction is shown to be an unimportant variable, and arguments are produced why this is to be expected, even though it is highly important in explaining the variation in Young's modulus of compact bone.

The effect of variation in structure on the Young's modulus of cancellous bone: a comparison of human and non-human material.

The Young's modulus of cubes of human cancellous bone was measured in three orthogonal directions. Apparent density and mineral volume fraction were also measured, as were two architectural variables, fabric and connectivity, which were determined using image analysis techniques. Multiple regression was used to relate the Young's modulus to the four explanatory variables. The results from this study are compared with those obtained from a previous investigation using non-human cancellous bone. The relationships revealed by the two studies are very similar. It was possible to explain approximately 93 per cent of the variance in Young's modulus using the four variables in this present study. Apparent density is the major explanatory variable in both studies and shows a strong correlation with connectivity. In common with the non-human study the measure of fabric is a worthwhile explanatory variable; however, connectivity and mineral volume fraction are relatively unimportant. The four explanatory variables contribute to a successful model for the prediction of Young's modulus. Any other candidate variables are likely to be unimportant or be highly correlated with those already investigated.

Hardness, an indicator of the mechanical competence of cancellous bone.

Hardness and calcium content in compact bone are strongly related. Variation in Young's modulus is produced mainly by variations in mineralisation. Therefore, there should be a relationship between hardness and Young's modulus. We demonstrate this. The calcium content of cancellous bone and adjacent compact bone in several species shows little difference, the cancellous bone having approximately 10% less calcium. The hardness of cancellous bone in Bos is approximately 12% less than that of adjacent compact bone, and the calcium is approximately 2% less. These lines of evidence make it unlikely that the Young modulus of cancellous bone material is much different from that of compact bone. Similar evidence suggests that the yield stress of cancellous bone is similar to that of adjacent compact bone.