Theoretical and Practical Considerations

We would like to thank Dr. Williams for his interest in our work and for generating the opportunity for additional discussion of these key issues regarding ultrasonic assessment of cancellous bone. In his letter, Dr. Williams states that the conclusions of our recent study1 contradict earlier work. We argue that this is not the case because, to our knowledge, no previous study has specifically tested for causal relationships between acoustic and elastic properties in cancellous bone. In contrast to our study, the majority of previous work has investigated the relationships between ultrasonic and mechanical properties using correlation and regression analysis alone. In this regard, it is important to note that regression analysis is subject to several important limitations. First, in the presence of random errors, adding a second independent variable in a regression model can increase predictive power, even when both parameters reflect the same physical property. Second, if causally irrelevant variables are included in the model or if relevant variables are included in the analysis, multicollinearity (i.e., intercorrelation among the dependent variables) may mean that no reliable inferences about causation can be made, as we have noted elsewhere.2 We feel that our approach, to induce a change in the variable of interest (i.e., the elastic modulus), while keeping other relevant variables constant (i.e., the density and overall architecture), affords a better assessment of causality than do traditional correlation and regression analyses. Dr. Williams makes several interesting points concerning ultrasonically determined elastic constants and the very different strain rate and amplitude occurring during ultrasonic wave propagation compared with quasistatic mechanical testing. Whereas the issues he raised are genuine, we respectfully disagree that these arguments apply to the case of cancellous bone. We believe it is a misconception to think of ultrasonically derived elastic constants at all in the context of fluid-saturated cancellous bone. Ultrasound can only be used to derive elastic constants in materials where the wave propagation is well characterized theoretically. This is the case in cortical bone, for instance, where the porosity is low (typically 10% or less), and the pores themselves are much smaller than the typical wavelengths used. Cortical bone can therefore be treated acoustically as behaving approximately like a homogeneous solid, with velocity determined by bone elastic properties and density alone. In marked contrast to cortical bone, cancellous bone is mostly fluid (typically 85% by volume in the heel), and the solid bone present has a complex architecture with typical dimensions comparable with the ultrasonic wavelength. The theoretical description of wave propagation in cancellous bone remains an area of ongoing work, and a validated comprehensive model does not yet exist. The only cases where a theoretically convincing relationship between acoustic velocity and cancellous bone elastic properties have been reported have been in small bar-shaped specimens, drained of fluid and measured at very low frequencies, conditions under which so-called “bar wave” propagation occurs.3 Dr. Williams is correct to note the possibility that the damage induced in our study may have occurred preferentially in the trabeculae at the ends of the specimens. We agree that this localization of damage may have reduced the sensitivity of ultrasound to any changes. However, we do not believe that this invalidates our findings for the following reasons. First, we observed no evidence at all for changes in the quantitative ultrasound (QUS) parameters, even in the highest damage group, whereas the effect would only tend to reduce the magnitude of any QUS change, not remove it completely. Second, we preconditioned the specimens by applying eight cycles of low-strain compressive load. This preconditioning regimen would be expected to consolidate the trabecular bone in contact with the platens before applying the damage cycle. Third, given that the specimens were relatively small (∼18 mm in length), even if the damage were localized toward the ends of the core, the damaged volume would likely still constitute a high fraction of the total volume. Finally, in light of the above comments, we maintain that our results have a direct bearing on the validity of certain theoretical models for wave propagation in cancellous bone. Using a formulation of Biot's theory proposed by Dr. Williams for bovine bone,4 a 70% reduction in the modulus of the trabecular framework leads to a predicted velocity reduction of approximately 40% for cancellous bone at porosities comparable with our specimens. Ongoing work using a Biot model with input parameters optimized for human calcaneal cancellous bone predicts much smaller velocity reductions of typically 1% (P. H. F. Nicholson, unpublished data, September, 2000). Nevertheless, in our study, the mean velocity changes seen experimentally after the induction of damage were less than 0.2% and were not statistically significant. This strongly suggests that at least for the case of highly porous, water-saturated cancellous bone specimens measured along the mediolateral axis, Biot's theory does not accurately model the relevant acoustic interactions.