Is There a Causal Relationship Between Quantitative Ultrasound and the Elastic Constants of Cancellous Bone?


To the Editor:

Two statements in a recent paper in the Journal(1) about the use of quantitative ultrasound (QUS) to determine mechanically induced damage to cancellous bone in the human calcaneus are at odds with many previous publications. The authors of the paper concluded that:

  • (1) “… there is no causal relationship between elastic modulus and quantitative ultrasound measurements in human cancellous bone from the calcaneus;” and(1)
  • (2) “ … our findings question the validity of Biot's theory in cancellous bone, because this theoretical approach predicts the elastic modulus of the cancellous framework to be a major determinant of velocity.”(1)

These statements are also not consistent with the finding that ultrasound transmission velocity reflects bone density independently of the bone elastic constants, and that the slope of the bone ultrasound attenuation (BUA) versus frequency reflects bone elastic constants independently of bone density in the heel.(2)

The authors arrived at their conclusions because their QUS values remained unchanged while their mechanically determined elastic moduli decreased with the extent of postyield damage. I would respectfully suggest that drawing the above two conclusions from this result is a bit like throwing the baby out with the bath water. There are several aspects to the ultrasonically determined elastic constants, which have not received much attention in the biomechanics literature, but which are relevant to these results(1):

  • (1) The average strain amplitude incurred during the propagation of an ultrasound stress wave is in the low 10−6 (i.e., 10−4%) range for a solid like silver.(3) To my knowledge it has not been determined for bone but is probably not very different. This must be compared with the average strain used to determine the elastic constants by mechanical testing or 0.4% to 0.6% in the authors' work.(1)
  • (1) The strain rate incurred during the propagation of a stress wave at ultrasound speed is probably much higher than the strain rate used in the determination of Young's modulus in a “quasistatic” mechanical test. The evidence for this is admittedly indirect, and I do not know of any study where this has been measured for ultrasound testing of bone. It has been shown(4) that Young's modulus for Perspex (polymethyl methacrylate [PMMA]) determined ultrasonically is twice the quasistatic value determined by mechanical testing. The same relationship between ultrasonic and quasistatic Young's modulus was found for bone cement PMMA and for composites made of intact cancellous bone and PMMA.(5) This discrepancy has been attributed to differences between the strain rates in an ultrasound test and a quasistatic mechanical test.(4, 5) By comparison, Young's modulus of cortical bone that is tested at high strain rates (1500 1/s) is approximately twice the value at quasistatic strain rates (0.001 1/s).(6)

In the case of cancellous bone from the calcaneus, the authors have shown that QUS measurements are unaffected by in vitro postyield damage caused by a compression test. One explanation for these results is that ultrasonic testing provides deformations of insufficient magnitude and duration to engage the damaged regions and provides elastic constants under conditions that are not relevant to physiologic strain levels and strain rates. Similar issues may also hold for undamaged cancellous bone, because ultrasonically derived elastic constants may differ from those measured at physiologic strain magnitudes and rates. However, as the authors point out, there is also the question of the nature of the damage caused by the fixed-platen compression test. There is the possibility that damage occurred only at the compressed cut ends of the trabeculae. If so, there could be a large decrease in Young's modulus, because Young's modulus was determined using the platen-to-platen displacements; but one would expect only a small change in ultrasound velocity. The velocity would not change much because the sound waves spend proportionately more time in transit in the undamaged part of the bone, which has a longer path length. On the other hand, damage that is highly localized internal to the structure may not significantly change the arrival time of the ultrasound wave either.

The authors(1) have shown that their QUS is not useful as a measure of unspecified damage in cancellous bone. Whether the damage is localized at the cut ends or internal to the sample, or diffuse, is critical for clinical and theoretical interpretation of these results, but this paper shows that QUS is not useful in detecting such damage. It is quite likely that ultrasonically derived constants are not equivalent to those obtained at substantially different strain magnitudes and rates.(5) It is also possible that the strain-magnitude and strain-rate dependence of Young's modulus may become more pronounced after yielding. However, I do not think the results provide any evidence to support the broad conclusions that there is no causal relationship between ultrasound and the elastic constants of cancellous bone in the calcaneus and that simple mixture law theories and Schoenberg's theory are therefore more consistent with these results than is Biot's theory.