Relationships of linear and nonlinear ultrasound parameters with porosity and trabecular spacing in trabecular-bone-mimicking phantoms Open Access

Osteoporosis is one of the most common skeletal diseases characterized by low bone mass and microarchitectural deterioration of bone tissue, leading to enhanced bone fragility and a consequent increase in fracture risk.¹ Quantitative ultrasound (QUS) is now an accepted tool for the diagnosis of osteoporosis.² Most of the current bone sonometers measure linear ultrasound parameters, such as the speed of sound (SOS) and the normalized broadband ultrasound attenuation (nBUA), at easily accessible peripheral sites consisting of trabecular bone, typically at the calcaneus, by using a transmitter and a receiver placed along the mediolateral axis on opposite sides of the skeletal site to be tested.³ Recently, the nonlinear parameter (B/A) that is defined as the first-order pressure derivative of the elastic modulus was found to have significant correlations with the bone mineral density and the microarchitectural parameters in 28 bovine femoral trabecular bone samples, suggesting that the B/A may be a useful index of trabecular bone properties.⁴ 

Despite the clinical utility of the QUS parameters, the underlying mechanisms responsible for their variations in trabecular bone are still not fully understood because of high porosity and heterogeneity of trabecular bone. Trabecular-bone-mimicking phantoms designed with some desirable characteristic features similar to those of trabecular bone enable us to understand the interaction between ultrasound and trabecular bone.^5–11 For instance, Chiarelli et al. developed hydrogel phantoms by using polyvinylalcohol dissolved into de-ionized water to investigate the influence of the water volume fraction on the wave velocity and the attenuation.⁷ Zhang et al. measured the phase velocity and the attenuation coefficient of the slow wave in trabecular-bone-mimicking phantoms consisting of water-saturated aluminum foams, and compared the measured velocities with the predictions by Biot's theory that predicts the existence of three kind waves in fluid-saturated porous media: Two compressional waves, called fast and slow waves, and one shear wave.^8,12,13

The present study aims to provide insight into the relationships of linear (SOS and nBUA) and nonlinear (B/A) ultrasound parameters with the porosity and the trabecular spacing in trabecular bone. The three ultrasound parameters were measured in 18 trabecular-bone-mimicking phantoms consisting of water-saturated aluminum foams. Their correlations with the porosity and the trabecular spacing were examined.

A total of 18 trabecular-bone-mimicking phantoms consisting of cellular aluminum foams (custom built by Wide, Siheung, Republic of Korea) were interrogated. Figure 1 shows the photograph of a cellular aluminum foam. The aluminum foams in the shape of cylinders had a diameter of 100 mm and a thickness of 30 mm. They had an open-celled structure consisting of an interconnected network of aluminum ligaments (simulating trabeculae) with a large volume fraction of void spaces. The diameter and the separation of the aluminum ligaments correspond to the trabecular thickness (Tb.Th) and the trabecular separation (Tb.Sp), respectively. The structural properties of the phantoms are described by two parameters: The porosity and the trabecular spacing. The aluminum volume fraction (AVF), the ratio of aluminum volume to foam volume, of each phantom was measured to determine its porosity that is given by 1 – AVF. The trabecular spacing between two adjacent aluminum ligaments is given by Tb.Th + Tb.Sp. The mean values for the Tb.Th and the Tb.Sp of the individual phantoms were measured on 30 different locations in their micrographs obtained by using a scanning electron microscope (SEM).

Before ultrasound measurements, the phantoms were vacuum-degassed underwater in a desiccator. The aluminum foams were interrogated in a water tank filled with degassed water at room temperature (20 °C) by using two ultrasonic transducers facing each other, one transmitting and the other receiving. A one-cycle tone burst wave was generated by using a function generator (33250A, Agilent Technologies, Santa Clara, CA) and amplified by using a power amplifier (75A250A, AR, Souderton, PA). Received radio-frequency signals were digitized by using a digital oscilloscope (WS44Xs, LeCroy, Chestnut Ridge, NY) and stored on a personal computer for off-line analysis.

The technical aspects of measurements of the linear ultrasound parameters (SOS and nBUA) in transmission have been extensively developed in previous studies and will not be presented in detail here.³ Two opposing coaxially aligned transducers with a diameter of 25.4 mm and a center frequency of 0.5 MHz (V301, Panametrics, Waltham, MA) were separated by twice the near-field distance (108 mm), and the phantoms were positioned at the near-field distance of the transducer (54 mm). In order to determine the SOS, we used the arrival times of the envelope maxima of the signals transmitted with and without the phantom in the acoustic path. The nBUA (in units of dB/cm/MHz) was determined from a slope of the linear regression fit on the frequency-dependent attenuation coefficient over the frequency bandwidth from 0.2 to 0.6 MHz, where the attenuation coefficient increased approximately linearly with increasing frequency.

A finite-amplitude through-transmission method was used to measure the nonlinear ultrasound parameter (B/A) by using a transmitting transducer with a diameter of 25.4 mm and a center frequency of 0.5 MHz (V301, Panametrics, Waltham, MA) and a receiving transducer with a diameter of 12.7 mm and a center frequency of 1.0 MHz (V303, Panametrics, Waltham, MA).⁴ Two opposing coaxially aligned transducers were separated by twice the near-field distance of the 0.5-MHz transmitting transducer (108 mm), and the phantoms were positioned very close to the 1.0-MHz receiving transducer. Transmitted signals were recorded both with and without the phantom in the acoustic path. The B/A was determined by using¹⁴ 

where the subscript “p” refers to the phantom and the subscript “w” refers to the degassed water. $A2p$ and $A2w$ are the amplitudes of the second harmonic of the transmitted signals with and without the phantom in the acoustic path, respectively. $L$ is the distance between the two opposing transducers, and $d$ is the thickness of the phantom. $α1$ and $α2$ are the attenuation coefficients of the phantom at the first (0.5 MHz) and the second (1.0 MHz) harmonic frequencies, respectively. $ρ$ is the density, and $c$ is the speed of sound. $(B/A)w$ is the nonlinear parameter of distilled water (approximately 5.2 °C at 20 °C). $Twp=2ρpcp/(ρwcw+ρpcp)$ and $Tpw=2ρwcw/(ρwcw+ρpcp)$ are the transmission coefficients at the water/phantom and the phantom/water interfaces, respectively.

Linear regression fits were performed between the ultrasound parameters (SOS, nBUA, and B/A) and the structural properties (porosity and trabecular spacing) to obtain Pearson's correlation coefficients.

Figure 2 shows the time-domain signals transmitted with and without a phantom in the acoustic path. The phantom signal exhibited a later arrival time than the reference signal through water only. This phantom signal corresponds to the slow wave (the second kind of compressional wave of Biot's theory) generated from the out-of-phase motion between the fluid and the solid.^12,13 The very weak fast wave (the first kind of compressional wave of Biot's theory) arriving before the slow wave could only be revealed after significant amplification. The strong slow wave and the very weak fast wave were consistently observed in the signals transmitted through all of the phantoms. This indicates the fact that the ultrasound parameters of the phantoms used in the present study are governed by the slow wave, consistent with the behavior in cellular aluminum foams used by Zhang et al.⁸ 

Table 1 shows the descriptive statistics of the porosity, the trabecular spacing, and the ultrasound parameters (SOS, nBUA, and B/A) for the 18 trabecular-bone-mimicking phantoms. As seen in Table 1, the mean porosity of the 18 phantoms was 88.6% ± 3.7% (mean ± standard deviation), which corresponded to the porosity range reported in the human calcaneus, 86%–98%.¹⁵ The mean trabecular spacing of the aluminum ligaments measured on 30 different locations in the SEM micrographs of each phantom was 1084 ± 145 μm, which was reasonably close to the mean trabecular spacing reported in the human calcaneus, 811 μm.¹⁵ One should note that the mean SOS of the 18 phantoms was 1473 ± 4 m/s ranging from 1467 to 1480 m/s and lower than the speed of sound in distilled water (1482 m/s at 20 °C).

Figure 3 shows the measurements of the ultrasound parameters (SOS, nBUA, and B/A) plotted versus the porosity (open circles) and the trabecular spacing (asterisks) for the 18 trabecular-bone-mimicking phantoms. The black and the gray solid lines represent the linear regression fits to the data. As observed in Fig. 3(a), the SOS increased as the porosity and the trabecular spacing increased, showing positive correlation coefficients of r = 0.82 and 0.85 (see Table 2). This finding is contrary to the behavior in human trabecular bone.^15,16 For instance, Wear et al. reported that the phase velocity decreased monotonically with increasing porosity for porosities from 86% to 98% in 53 human calcaneal trabecular bone samples.¹⁵ Wear et al. also observed that the SOS was negatively correlated with the porosity (r = −0.77) and the trabecular spacing (r = −0.25) in 29 human calcaneal trabecular bone samples.¹⁶ As predicted by using Biot's theory for elastic wave propagation in fluid-saturated porous media, the slow wave velocity in trabecular bone increases with increasing porosity, and the fast wave exhibits an opposite dependence on the porosity.¹⁷ Hence, the positive correlations of the SOS with the porosity and the trabecular spacing obtained in the present study underpins the fact that the signals transmitted through the phantoms consisting of water-saturated aluminum foams correspond to the slow wave.

As seen in Fig. 3(b), the nBUA decreased as the porosity and the trabecular spacing increased, showing negative correlation coefficients of r = −0.92 and −0.83 (see Table 2). These results may help explain findings by other researchers investigating the relationships of the nBUA with the microarchitectural parameters in human trabecular bone.^16,18 For instance, Padilla et al. observed that the nBUA was highly correlated with the porosity (r = −0.83) and the trabecular spacing (r = −0.79) in 37 human femoral trabecular bone samples.¹⁸ Wear et al. also found that the nBUA was highly correlated to the porosity (r = −0.85) but not significantly to the trabecular spacing (r = −0.22) in 29 human calcaneal trabecular bone samples.¹⁶ As found in Figs. 3(a) and 3(b), the dependence of the SOS on the porosity and the trabecular spacing exhibited a behavior opposite to that of the nBUA. This can be explained as follows: As the porosity and the trabecular spacing increases, both the cell and the pore sizes of the cellular aluminum foams increase. The widening of the pores might enhance the fluid motion and the decrease the friction of ultrasound propagation, thereby, increasing the SOS and reducing the attenuation.

As shown in Fig. 3(c), the B/A decreased as the porosity and the trabecular spacing increased, showing negative correlation coefficients of r = −0.70 and −0.75 (see Table 2), consistent with the behavior in bovine trabecular bone.⁴ One should note that the mean B/A of the 18 phantoms was 43.1 ± 5.8 ranging from 30.5 to 50.0 (see Table 1). The measurements of the B/A in the phantoms may be compared with previously reported measurements of the B/A in human trabecular bone.^19,20 For instance, Renaud et al. measured a value of 22 for the B/A in a low-density trabecular bone region and 142 in a high-density region of a human calcaneal trabecular bone sample by using a time-of-flight modulation method.²⁰ Hence, the phantom data obtained in the present study fall within the range of the human calcaneus data. In contrast, Lee measured the values of the B/A ranging from 61.4 to 125.1 in 28 bovine femoral trabecular bone samples with relatively higher density and thicker trabeculae compared to the osteoporotic human calcaneus.⁴ The nonlinear acoustic response might be due to the presence of microcracks with a typical size of tens to hundreds micrometers or fractured trabeculae in the bone tissue.²⁰ Hence, assuming that the crack density is homogeneous in the solid phase, the B/A increases with increasing bone density.

Table 2 shows the Pearson's correlation coefficients (r) between the ultrasound parameters (SOS, nBUA, and B/A) and the structural properties (porosity and trabecular spacing) for the 18 trabecular-bone-mimicking phantoms (p < 0.0001 for all). As found in Table 2, all three ultrasound parameters exhibited high correlation coefficients with the porosity and the trabecular spacing. The highest correlation was observed between the nBUA and the porosity (r = −0.92).

In conclusion, we have investigated the relationships of the linear (SOS and nBUA) and the nonlinear (B/A) ultrasound parameters with the porosity and the trabecular spacing in 18 trabecular-bone-mimicking phantoms consisting of water-saturated aluminum foams. The SOS increased as the porosity and the trabecular spacing increased. In contrast, both the nBUA and the B/A showed opposite dependences on the porosity and the trabecular spacing. All three ultrasound parameters exhibited high correlation coefficients with the porosity and the trabecular spacing. These results suggest that the B/A, in addition to the existing diagnostic parameters (SOS and nBUA) could be useful for the assessment of bone status and osteoporosis. In clinical applications in vivo, the cortical shell enclosing the trabecular bone would be expected to reduce the accuracy of the B/A measurements due to the creation of a significant second harmonic energy in the cortical shell. Hence, this source of artifact requires future in vivo studies on human subjects.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (No. 2014R1A1A1A05002187).