The effect of trabecular material properties on the frequency dependence of backscatter from cancellous bone

Abstract

Previous experimental studies indicate that backscatter coefficient for human calcaneal trabecular bone varies approximately as frequency cubed. This frequency dependence has been shown to be consistent with a model in which trabeculae are thought of as long thin cylinders composed of a substance with the same material properties as hydroxyapatite. The true material properties of human trabecular bone are not known however. Based on reported measurements of material properties of many bones and bone-like substances, it is possible that the density and longitudinal sound speed of trabecular bone material are far lower than the hydroxyapatite model would suggest. In this letter, it is shown that the frequency dependence of backscatter is still expected to be approximately cubic for wide ranges for density and longitudinal sound speed (encompassing the conceivable ranges for trabecular bone).

Introduction

The investigation of scattering from trabecular bone is important for two reasons. First, it can help elucidate mechanisms responsible for attenuation (the combined result of absorption and scattering), which has been shown to have great diagnostic utility (Langton et al., 1984; Rossman et al., 1989; Zagzebski et al., 1991; Glüer et al., 1996; Hans et al., 1996; Langton et al., 1996; Bauer et al., 1997; Bouxsein et al., 1997; Chappard et al., 1997; Laugier et al., 1997a). Second, scattering measurements have shown some diagnostic promise in their own right in studies in vitro (Roberjot et al. 1996; Wear and Armstrong, 2000; Hoffmeister et al., 2000) and in vivo (Wear and Garra 1997; Giat et al. 1997; Laugier et al., 1997b; Wear and Garra 1998, Wear and Armstrong, 2001; Roux et al. 2001). Backscatter provides information regarding size, shape, number density, and elastic properties of scatterers (Faran 1951). In cancellous bone, trabeculae are likely candidates for scattering sites due to the high contrast in acoustic properties between mineralized trabeculae and marrow (Wear 1999; Luo et al. 1999). The diminished number and thicknesses of trabeculae that accompany aging and increased fracture risk would be expected to reduce backscatter.

In a previous investigation (Wear, 1999), measurements of average frequency-dependent backscatter from human calcaneal trabecular samples conformed fairly well to a power law dependence on frequency (throughout a typical diagnostic range: 300 – 700 kHz) with an exponent slightly greater than 3. (Due to the high attenuation coefficient of bone, the useful frequency band is much lower than that normally used for soft tissues). A subsequent study reported a similar frequency dependence (Chaffai et al., 2000). These findings have implications regarding the relative roles of scattering and absorption in determining attenuation in trabecular bone. Attenuation for trabecular bone in the 300 – 700 kHz range has been found in numerous studies to be approximately proportional to frequency to the first power. These two different co-existing frequency dependences could be consistent only if absorption is a much larger component of attenuation than scattering.

Empirical estimates of the exponent of the power law of frequency-dependent backscatter have been demonstrated to be consistent with a model in which trabeculae correspond to cylinders that are long (compared to the beam width), thin (relative to the wavelength), and oriented approximately perpendicular to the ultrasound propagation direction (Wear, 1999). Other models have also been shown to be useful for explaining experimental observations. One model assumes an isotropic statistically homogeneous scattering model, in which weak scattering structures are randomly distributed (Chaffai et al., 2000). The other model portrays bone as a random continuum containing identical scatterers and assumes that scattering is proportional to the mean fluctuation in sound speed (Nicholson et al., 2000).

Previously-reported theoretical predictions (Wear, 1999) of frequency-dependent backscatter assumed that the properties of the trabecular bone material are the same as those for hydroxapatite, as measured by Grenoble et al. (1972). In reality, the true parameter values appropriate for trabecular bone are not known with great accuracy. There exist wide ranges of reported material property values for bone and bone-like substances. (See Table I). Density and sound speed for hydroxyapatite are at the high ends of these ranges. This fact opens the previous study to the criticism that its conclusions were based on extreme parameter values which may differ considerably from the true values. The purpose of this letter is to explore the extent to which theoretically predicted approximate cubic frequency dependence of backscatter may be extended to other assumed values for density and sound speed that may be more appropriate for trabecular bone.

Table I.

Material properties of bone and bone-like substances. The parameters listed are longitudinal sound velocity (c_l), shear velocity (c_s), density (ρ) and Poisson’s ratio (ν).

Reference	Substance	cl (m/s)	cs (m/s)	ρ (g/cc)	ν
Hosokawa & Otani, 1997	Bovine cancellous bone	3800	2000	1.95	0.32
Anderson et al., 1998	Hydroxyapatite	6790		3.22	0.28
Luo et al., 1998	Cortical bone	2900	1303	1.85
Gong et al., 1964	Bovine cancellous bone			1.93
Lang et al., 1970	Bovine cortical bone			1.96	0.32
Williams, 1992	Bovine cancellous bone	3800 *
Rho et al., 1993	Bovine cancellous bone	2898±85
Ashman et al., 1993	Human cancellous bone	2639–2754		1.73–1.80
-	Bovine cancellous bone	2501		1.74
Grenoble et al., 1972 †	Powdered human femur	3917	2020		0.33
-	Fresh bovine femur	4890	2495		0.34
-	Hydroxyapatite (mineral)	5565	4765		0.27
-	Hydroxyapatite (synth.)	5628	4818		0.28

^*Extrapolated (to zero porosity) from bulk measurements over a range of porosities (where porosity is the volume fraction of marrow). See (J. L. Williams, 1992), figure 2.

^†Assuming ρ = 1.96 gm/cc.

Methods

Faran developed a model for elastic scattering from a cylinder in a fluid (Faran, 1951). This model may be used to predict the intensity of backscatter from a cylindrical scatterer exposed to an incident plane wave. The axis of the cylinder is assumed to be oriented perpendicular to the acoustic propagation direction. Experimental measurements of scattering from a wire target suggest that Faran’s expression is approximately proportional to the measured differential scattering cross section (from which the backscatter coefficient is derived) (Wear, 1999). Faran’s model requires assumed values for the densities and longitudinal velocities for the scattering and embedding media. In addition, Poisson’s ratio for the scattering medium must be specified. Finally, the diameter of the cylinder, the frequency of sound, and the angle between incident and observation directions are required.

In the present analysis, Faran’s model was used to predict frequency-dependent backscatter. A wide range of assumed material properties for bone was employed. See Table I. Poisson’s ratio seems to be fairly consistent for a variety of bone and bone-like substances and was taken to be 0.3. Trabecular diameter was assumed to be 120 microns (Hausler et al, 1999). The embedding medium was assumed to be water (as is commonly used in in vitro experiments), with a longitudinal velocity of 1480 m/s and a density of 1 g/cc. For more details regarding computations, see (Wear, 1999).

Backscattered intensity (e.g. the response from a unit incident plane wave) I(f) was assumed to obey a power law dependence on frequency, f, so that I(f) = Afⁿ over a range of frequencies corresponding to the usable bandwidth of the transducer (300 – 700 kHz). Linear fits of the form, log[I(f)] = log A + nlogf were performed. The frequency dependence of backscatter was characterized by the exponent n.

Results

Frequency-dependent backscatter, numerically obtained using the theory of Faran, for four combinations of assumed density and longitudinal sound speed (near the extremes from Table I) are shown in Figure 1. These parameter values encompass the conceivable range for trabecular bone. It may be seen that over the frequency band of interest for experimental measurements (300 – 700 kHz), the frequency dependence of backscatter was similar for all parameter choices, despite the wide range of densities and longitudinal sound speeds employed.

Figure 1.

Backscattered intensity from a cylindrical scatterer (diameter = 120 microns) immersed in water for various choices of density and longitudinal sound velocity. The frequency dependence of backscatter is approximately the same for all pairs of parameter choices. The vertical dashed lines correspond to the frequency band of analysis for experimental measurements (300 – 700 kHz).

In Figure 2, exponents of power law fits to backscattered intensity versus frequency (over the range from 300 – 700 kHz) are shown as functions of the assumed value for longitudinal velocity of trabecular material. Over the plausible ranges for densities (1.7 g/cc < ρ < 3.2 g/cc) and longitudinal velocities (2500 m/s < c < 7000 m/s) for trabecular bone material (see Table I), the exponent remained very close to 3 (ranging from 2.93 – 2.97). The exponent only began to deviate substantially from 3 when material values departed substantially from the expected range for bone and approached levels for water (ρ < 1.5 g/cc and c < 1800 m/s).

Figure 2.

The exponent of a power law fit (n) to backscattered intensity (I) versus frequency (f) from a cylindrical scatterer immersed in water. It has been assumed that I(f) = Afⁿ. The exponent of the power law fit is equal to the slope of the linear fit of log-transformed data (log[I(f)] = log A + nlogf). Fits were performed over the range from 300 – 700 kHz.

Conclusion

Over a wide range of assumed values for density and sound speed for trabecular bone material, Faran’s cylinder model predicts that backscatter varies approximately as frequency cubed, consistent with experimental measurements. So the conclusions of the previous study (Wear, 1999) are not restricted to a narrow choice of parameter values.

Measurements of power law exponent (n) for human calcaneus in the low frequency range have been reported as 3.26 (Wear, 1999) and 3.38 (Chaffai et al., 2000), which are a little higher than would be predicted by the cylinder model. The difference of 0.3 – 0.4 between theory and experiment may be due in part to the fact that trabecular bone has a more complex geometry than the cylinder model would suggest. Trabecular bone contains plate-like structures in addition to rod-like trabeculae. In addition, trabeculae are not perfectly straight (as assumed in the model) but are somewhat curved and jagged. Nevertheless, cylinder-like objects (trabeculae) contribute a substantial portion of the scattering. Moreover, only those plate-like structures oriented approximately perpendicular to the ultrasound propagation direction can measurably affect the frequency dependence of scattering. Finally, plates are comparatively rare in bones from older subjects, upon which the data from the studies mentioned above are based.

The potential effects of coherent scattering, multiple scattering, small measurement volumes (often only a few wavelengths deep), ultrasonic beam distortion (due to propagation through an inhomogeneous medium, bone), and phase cancellation compromise accuracy of measurements of frequency-dependent backscatter. Given these numerous complicating factors, it may not be possible to ascertain at present the extent to which any discrepancy between measurements and theoretical predictions is attributable to imperfect modeling as opposed to experimental uncertainty.