1.

Introduction

Osteoporosis and fractures stemming from it have emerged as chronic diseases adversely affecting the health of the elderly.¹ As populations age, the annual global increase in the number of fracture patients is $\sim 8.9\text{million}$, leading to a considerable rise in public health costs.^2–4 Therefore, early diagnosis of osteoporosis not only can prevent fractures but also substantially reduce healthcare expenditures and resource usage.

Compared to dense bone, the microstructure and chemical composition of cancellous bone are more sensitive to osteoporosis, as evidenced by reduced trabecular thickness, connectivity, and number, along with increased lipid content,^5–8 suggesting that cancellous bone is ideally suited as a diagnostic site for early osteoporosis. However, the unique structural characteristics of cancellous bone make its detection and quantitative evaluation challenging, thereby complicating the early diagnosis of osteoporosis. The primary detection methods employed in clinical and research settings for osteoporosis include dual-energy X-ray absorptiometry (DEXA), quantitative computed tomography (QCT), magnetic resonance imaging (MRI), and quantitative ultrasound (QUS).^9–12 DEXA serves as the “gold standard,” primarily because it provides the best predictor of osteoporotic fractures through bone mineral density (BMD) information.¹³ However, DEXA accounts for only 60% to 70% of changes in bone strength and lacks information on microstructure and elasticity.¹⁴ QCT extends bone analysis from two to three dimensions, providing both volumetric BMD and microstructural characteristics.^15,16 However, its utility is hampered by radiation hazards and a lack of chemical information. MRI can identify alterations in bone marrow fat content and microstructure, facilitating early diagnosis.^17–19 Nonetheless, its high cost and operational complexity limit its widespread use.^20,21 QUS, being radiation-free, quick, affordable, and user-friendly, has gained traction as a powerful tool for screening bone quality.^5,22,23 Initially centered on cortical bone,^24–26 QUS is increasingly being used to explore cancellous bone characteristics due to its heightened sensitivity to osteoporotic changes. It provides physical information on BMD, bone microstructure, and mechanical properties, mainly by detecting the speed of sound (SoS) and broadband ultrasound (US) attenuation.^12,27 However, it falls short of detecting changes in the organic chemical composition of bone tissue.

Photoacoustic (PA) techniques offer both chemical and physical insights into biological tissues and have been utilized for tissue identification and disease detection, including osteoporosis.^28–32 During the past decade, considerable advancements have been made in PA-based bone evaluation. Mandelis and Lashkari³³ established a set of photoacoustic-ultrasound (PA-US) backscattering detection systems, successfully detecting minute changes in the BMDs of both cancellous and cortical bones. Their study indicated that the apparent integral backscattering coefficient decreases as the collagen content decreases.^34–36 Wang^37,38 employed three-dimensional PA imaging (PAI) and power spectral analysis to achieve both qualitative and quantitative evaluations of bone microstructure. He also introduced multispectral PA decoupling and thermal PA methods for quantitative evaluation of the organic and inorganic chemical components in cancellous bone.^37–39 Additionally, a combined PA-US system was developed, verifying the in vivo feasibility of assessing human calcaneus microstructure and chemical composition.⁴⁰ Steinberg developed a multispectral PA-US dual-mode system capable of in vivo detection of fat and blood ratios in tibial bone marrow.⁴¹ The SoS for the first arriving wave in the tibia showed a strong correlation with the SoS value based on QUS.⁴² Feng et al.⁴³ simulated various aspects of PA in skeletal tissues, including light attenuation and distribution, along with PA signal generation and propagation, based on a three-dimensional model. They proposed a PA physicochemical spectral method for assessing changes in the chemical composition and microstructure of cancellous bone.^44,45 Our group has leveraged the MWPA time-frequency spectral analysis method to evaluate both the chemical content (minerals and lipids) and microstructure of bone tissues, based on the distinct optical absorption characteristics and sizes of various chromophores present.^46,47 The preceding research in this field demonstrates the successful application of various PAI and spectral analysis methods for osteoporosis diagnosis, making it possible to assess BMD, bone microstructure, and chemical composition in a comprehensive manner.⁴⁸

Nevertheless, the intricacy of PA signals, which arise from the two-phase, solid–liquid porous structure of cancellous bone, presents a significant challenge in using PA techniques for imaging of bone marrow distribution in cancellous bone.⁴⁹ First, PA signals generated from various chemical clusters within cancellous bone with different microscales have complex optical and ultrasonic spectral distributions. In addition, these chemical clusters have uneven spatial distributions, adding another layer of complexity to the spatial distribution of PA sources. Thus the PA sources of cancellous bone possess a threefold complexity. Second, as these PA signals navigate through the cancellous bone, they undergo multiple scattering and attenuation events. Therefore, the final PA signal received by the transducers is a composite of broadband signals from distributed PA sources and the acoustic propagation characteristics specific to the porous cancellous bone. Decoupling these signals could yield valuable multidimensional insights into cancellous bone.

In this paper, we do a focused study—both theoretical and experimental—on the high-frequency broadband PA signal propagation generated by bone marrow in cancellous bone. Theoretically, we apply high-frequency and viscous corrections to Biot’s theory, tailoring it for bone marrow PA signal propagation in two-phase solid–liquid porous media. The numerical simulations of PA wave modes and velocity and attenuation coefficients are also employed for validation. In the method, we propose a PA differential attenuation spectrum (PA-DAS) method to remove the contribution of PA sources on PA signals in the frequency domain. Subsequently, ex vivo experiments are performed to measure PA-DAS, and a quantitative PA-DAS parameter is extracted to evaluate BMD and diagnose osteoporosis. Our results highlight the potential utility of this method for comprehensive bone quality assessment and lay a foundation for the further PAI study of bone marrow distribution in cancellous bone.

2.

Modified Biot’s Theory

Cancellous bone is a complex porous medium composed of solid trabecular bone interspersed with fluid-filled bone marrow clusters, making it a typical example of a porous, elastic, viscous medium. Biot and Willis^50–55 established the elastic theory for understanding acoustic propagation in such saturated solid–liquid two-phase porous media, thereby enabling further theoretical research in the field. Biot’s theory has subsequently found applications in the nondestructive evaluation of bone using biomedical US.^56–62 However, Biot’s theory is limited to cases where the acoustic frequency is below the medium’s critical frequency $f_{\mathrm{c}}$, and the flow of liquid through the pores is well-described by Poiseuille flow. The critical frequency $f_{\mathrm{c}}$ is defined as⁶³

In addition, the viscous nature of the fluid bone marrow leads to energy dissipation due to the relative motion between the fluid and the solid trabecular framework, further indicating the need for viscosity corrections. To account for the dissipation of acoustic wave propagation in a solid–liquid two-phase porous medium, the governing equations of motion can be expressed as follows:^54,55

The Fourier transform solution of velocities for the shear wave ($c_{\mathrm{T}}^{*}$) and longitude wave ($c_{\mathrm{L}1}^{*}$ and $c_{\mathrm{L}2}^{*}$) in solid–liquid two-phase porous media with viscous losses can be expressed as follows:⁷⁰

The velocity of shear wave ($c_{\text{T}}$) and two types of longitude wave ($c_{\text{L}1}$ and $c_{\text{L}2}$) in solid–liquid two-phase porous media without viscosity is given by^54,55

By comparing the velocities in solid–liquid two-phase porous media with and without viscosity, we observe that the viscosity loss alters Biot’s mass coefficients ${\rho }_{mn}$ into complex quantities. This results in the shear wave velocity and yields the fast and slow longitudinal wave velocities that are complex numbers. As sound waves propagate through a dissipative solid–liquid two-phase porous medium, their amplitudes decay progressively with increased propagation distance.

To account for high-frequency dissipation, Biot’s mass coefficients can be transformed as follows:

The attenuation coefficients are determined by the imaginary parts of the complex wave numbers:

The high-frequency dissipation coefficient ${\alpha }^{*}$ is influenced by several factors, including the porosity $\varphi$, the average pore size $a$ of the solid–liquid two-phase porous medium, the frequency $f$ of the acoustic wave, and the fluid viscosity $\eta$. Therefore, the porosity $\varphi$ of cancellous bone can be inferred based on the frequency-dependent attenuation coefficient.

3.

Numerical Simulations

3.2.

Numerical Simulation Results

Based on the modified Biot’s theory outlined above, we establish a computational model of semi-infinite cancellous bone and numerically simulate the propagation of PA waves in cancellous bone generated by bone marrow on the bone surface irradiated by a pulsed spotlight with a diameter of 2 mm. Figures 1(a) and 1(b) demonstrate the PA field at $17.92\mu \mathrm{s}$ in the cancellous bone with 0.72 and 0.83 porosity, respectively, including fast longitude wave (fast $P$ wave), slow longitude wave (slow $P$ wave), shear wave, and Rayleigh waves (for the complete 0 to $17.92\mu \mathrm{s}$ sound field simulation, see Video 1 for a porosity of 0.72 and Video 2 for a porosity of 0.83). The numerical simulation shows that the speed and amplitude of fast $P$ waves are much larger than those of other wave modes and other wave modes, which means that we can extract very clean fast $P$ waves from the PA signal in the time domain.

Fig. 1

Numerical simulation results of the sound field of PA wave generated by photoexcited bone marrow clusters on the boundary of semi-infinite cancellous bone propagating in cancellous bone with porosities of (a) 0.72 and (b) 0.83 at $17.92\mu \mathrm{s}$ (Video 1, mp4, 3.165 MB [URL: https://doi.org/10.1117/1.JBO.29.S1.S11526.s1]; Video 2, mp4, 3.094 MB [URL: https://doi.org/10.1117/1.JBO.29.S1.S11526.s2]).

Figures 2(a) and 2(b) illustrate the calculated trend of the velocity of fast and slow $P$ waves propagating in solid–liquid two-phase porous media with porosity and frequency, respectively. Similarly, in the calculated porosity range of 0.72 to 0.90, the velocity of fast $P$ wave ($c_{\text{f}}$) is much higher than that of slow $P$ wave ($c_{\mathrm{s}}$) at the same porosity. Notably, both fast and slow $P$ wave velocities are insensitive to frequency change, that is, there is almost no dispersion, which is very advantageous for PA detection because the bandwidth of the PA signal is usually broad. However, both velocities decrease significantly with increasing porosity, which is also consistent with Fig. 1. This is mainly because the fact that as the porosity increases, the proportion of the solid phase decreases and the solid–liquid coupling becomes weaker, leading to a decrease in the velocity of the fast and slow waves, respectively.

Fig. 2

Numerical simulation results. SoS of (a) fast and (b) slow longitude waves as a function of frequency at different porosities.

Figures 3(a) and 3(b) showcase the viscous absorption attenuation coefficients for fast and slow $P$ waves as they travel through solid–liquid two-phase porous media, respectively. In the calculated porosity range of 0.72 to 0.90, the absorption attenuation coefficient for the fast $P$ wave is much smaller than that of the slow $P$ wave with the same porosity. In addition, these attenuation coefficients are influenced by both the porosity of the cancellous bone and the frequency of the acoustic waves. When the porosity is constant, the absorption attenuation coefficients for both fast and slow waves increase with increasing frequency. The attenuation coefficients exhibit a shift from fast to slow changes, eventually tending toward a linear pattern within the frequency range of 1 to 6 MHz as the frequency increases. Conversely, when the frequency is held constant, the absorption attenuation coefficient decreases as the porosity increases. This is mainly because as the porosity of cancellous bone increases, the solid–liquid interface area decreases, which in turn leads to a reduction in energy dissipation caused by the relative motion between the solid bone trabecular frame and the liquid bone marrow.

Fig. 3

Numerical simulation results. Viscous absorption attenuation coefficients of (a) fast and (b) slow longitude waves as a function of frequency at different porosities.

These simulation results indicate that the fast $P$ wave with high speed and low attenuation is more suitable for the PA detection of cancellous bone, and the porosity $\varphi$ of porous media can be evaluated based on the attenuation of acoustic waves across various frequencies.

4.

PA-DAS Method

As discussed earlier, the PA signal arriving at the transducers is a complex mixture of broadband signals originating from distributed PA sources and the inherent acoustic propagation characteristics of porous cancellous bone. The relationship of PA signals received by the transducers, $p_1(t_1)$ and $p_2(t_2)$, in the time domain, can be represented as

Figure 3 shows that the attenuation coefficient is approximately linear within the 1 to 6 MHz range. Thus the acoustic propagation characteristics within this frequency band can be linearly fitted to quantify attenuation across different frequencies.

5.

Ex vivo Experiments on Rabbit Bone Specimens

Based on the results of theoretical and numerical simulations, the PA experiment was conducted on a rabbit osteoporosis model to verify the feasibility of bone evaluation based on PA-DAS.

5.1.

Animal Model and Bone Tissue Sample

In our study, we used a sample of eleven 5-month-old female New Zealand white rabbits. Six were randomly selected to undergo ovariectomy, forming the experimental group, while the remaining five received a sham operation to serve as the control group. After 5 months of identical living conditions, all rabbits were euthanized. The left metaphyseal region was then carefully dissected and sectioned into slices with a uniform thickness of 1.5 mm. Dense peripheral bone tissue was removed, and the slices were further trimmed to a standard width $D_b$ equal to 10 mm, as shown in Fig. 4(a), making them suitable for the PA experiments.

Fig. 4

Micro-CT results: (a) 3D images of normal bone and osteoporotic bone. (b) Statistical analysis results of BMD of bone specimens from the osteoporotic group ($n=6$) and control group ($n=5$) (**$p<0.01$). (c) ROI used to calculate porosity. (d) Statistical analysis results of porosity of bone specimens from the osteoporotic groups ($n=6$) and control group ($n=5$) (*$p<0.05$).

5.3.

PA Experimental Setup

To measure the frequency-dependent attenuation of PA signals in cancellous bone, we propose an eccentric excitation differential detection system specifically for PA experiments. Figure 5 illustrates the schematic layout of the experimental setup. A tunable optical parametric oscillator laser (Phocus Mobile, OPOTEK, Carlsbad, California, United States) produces laser pulses with durations ranging from 2 to 5 ns. We selected a laser wavelength of 1730 nm, which corresponds to the significant specific absorption wavelength of lipids^78,79—a major component of bone marrow clusters⁸⁰—to irradiate the bone samples and thereby excite PA signals.^81,82 To compensate for variations in laser energy over time, a spectrophotometer with a 9:1 transmittance-to-reflectance ratio was used to split the laser beam into two paths. One path was focused using a convex lens to irradiate a blackbody, and the other was weakly focused on one side of the bone tissue sample surface, tangent to the side of the sample, forming a light spot with a diameter $D_l$ of $\sim 2\mathrm{mm}$. The bone sample was placed on a 5 cm thick phantom to mitigate any direct light or sound reflections from the platform. As shown in Fig. 5, a needle hydrophone T1 (HNC1500, ONDA Corporation, Sunnyvale, California, United States) with a bandwidth of 0 to 20 MHz was placed on the side of the sample near the light spot to receive the unattenuated PA signals. These signals were then amplified by 25 dB using an amplifier (5072PR, Olympus Corporation, Tokyo, Japan) and subjected to 1 MHz high-pass filtering to remove low-frequency noise. On the opposite side of the light source, a planar transducer with a center frequency of 2.25 MHz (Olympus Corporation, Tokyo, Japan) was positioned to receive the PA signals transmitted through the cancellous bone. These signals were amplified by another 25 dB using an amplifier (5073PR, Olympus Corporation, Tokyo, Japan). Both transducers were acoustically coupled to the bone sample using a transparent ultrasonic coupling agent. Concurrently, a 1 MHz focusing transducer (Olympus Corporation, Tokyo, Japan) was employed to receive the PA signal generated by the blackbody. Data acquisition was performed using a digital oscilloscope (HDO6000, Teledyne Lecroy, United States) set at a sampling rate of 250 MHz, which was deemed sufficient for our experimental requirements. Due to the anisotropic properties of cancellous bone, we employed a translation stage to move the bone sample longitudinally in 2 mm increments, for a total of three. This enabled us to capture PA signals at four distinct positions, thus providing a comprehensive evaluation of signal attenuation throughout the cancellous bone. To improve the signal-to-noise ratio, PA signals were recorded 50 times at each position and subsequently averaged.

Fig. 5

Schematic of experimental setup for PA measurements of bone samples.

5.4.

PA Signal Processing

As shown in Fig. 5, the PA signal received by the hydrophone T1 closest to the light source serves as the unattenuated signal generated by the cancellous bone. Conversely, the PA signal received by transducer T2 located farther from the light source is a composite, reflecting both the inherent properties of the PA source and the acoustic propagation medium. By calculating the PA-DAS for both transducers, we can isolate and understand the frequency-related propagation characteristics specific to cancellous bone.

In the experiment, only longitudinal wave signals could be picked up because the gel was used as coupling agent. Moreover, the numerical results in Sec. 3.2 show that the attenuation of the slow $P$ wave is much greater than that of the fast $P$ wave, and the speed is much smaller than that of the fast $P$ wave. Therefore, it is more meaningful to analyze the fast $P$ wave signals with low attenuation than the complete signal.

It is necessary to select a reasonable time window for more accurate spectral analysis. Thus we introduce the SoS for bone marrow $c_{\text{m}}=1500\mathrm{m}/\mathrm{s}$, which is larger than slow $P$ wave and smaller than fast $P$ wave, and bone trabecular $c_{\text{t}}=3200\mathrm{m}/\mathrm{s}$ (Ref. 66), which is larger than fast $P$ wave, to calculate the start and end time of the picking window for fast-wave signals. As delineated by the red dotted box in Fig. 6(a), the direct PA signal duration for T1 is given by $t1=\frac{D_{\text{l}}}{c_{\text{m}}}=1.33\mu \mathrm{s}$. The earliest and latest times for the PA signal of T2 were selected by $\frac{D_{\text{b}}-D_{\text{l}}}{c_{\text{t}}}$, and $\frac{D_{\text{b}}}{c_{\text{m}}}$, respectively. The length of the PA signal as captured by T2 is $t2=\frac{D_{\text{b}}}{c_{\text{m}}}-\frac{D_{\text{b}}-D_{\text{l}}}{c_{\text{t}}}=4.17\mu \mathrm{s}$.

Fig. 6

PA signals generated by bones received by (a) the transducer near the light source T1 and (b) the transducer away from the light source T2.

The frequency–response curve of the transducer T2, centered at 2.25 MHz, is presented in Fig. 7(a). As evident in this curve, the transducer produces its maximum output at the central frequency of 2.25 MHz and attenuates on either side, leading to distortion in the frequency domain of the PA signal as displayed by the oscilloscope. To more accurately investigate the spectral characteristics of the PA signal, it is necessary to correct the observed output signals in line with the frequency–response curve of the transducer. Corresponding to a 6 dB decline from the maximum frequency response to prevent excessive correction of high-frequency noise, a fitting frequency band ranging from 1.46 to 2.96 MHz was selected. After applying a 1 MHz high-pass filter to eliminate the influence of low-frequency noise, the spectral characteristics of the PA signal received by the distant transducer—both before and after minus the frequency response of transducer T2 for correction based on the frequency response provided by the Olympus Corporation (Tokyo, Japan)—are illustrated by the blue and red curves in Fig. 7(b). Although the frequency–response curve of the broadband hydrophone T1 is uniform, the signals received by T1 do not need correction. Given the data in Fig. 3, the linear fitting for the frequency curve was confined to the 1.46 to 2.96 MHz range.

Fig. 7

(a) Frequency–response curve of T2 provided by the Olympus Corporation (Tokyo, Japan). (a) Frequency–response curve of T2. Red intersection point denotes the frequency corresponding to the maximum value decreasing by 6 dB. (b) Spectrum of the PA signal of bone received by T2 (solid blue line) and spectrum after frequency–response correction (solid red line).

To account for the spectral differences of the PA source caused by the anisotropy of cancellous bone, we calculated the PA-DAS $\mathrm{As}(f)$ based on Eq. (14) to evaluate the acoustic propagation characteristics of the bone:

Eq. (14)

$$\mathrm{As}(f)=10\lg \left(\frac{{\mathrm{PSD}}_2(f)}{{\mathrm{PSD}}_1(f)}\right).$$

Here ${\mathrm{PSD}}_1$ and ${\mathrm{PSD}}_2$ represent the power spectra of PA signals received by transducers 1 and 2, respectively, as depicted in Fig. 8(a). Notably, the PA signal from the transducer closer to the light source includes a higher concentration of high-frequency components due to the absence of propagation attenuation. In contrast, the PA signal from the transducer situated farther from the light source was mainly composed of low-frequency components owing to significant propagation attenuation in cancellous bone.

Fig. 8

(a) Spectrum of PA signals received by the two transducers. (b) Differential spectrum of PA signal received by the two transducers.

Linear fitting was performed for the PA-DAS in the frequency range of 1.46 to 2.96 MHz. From this, we extracted the absolute value of slope—slope—as a quantization parameter, enabling us to obtain the frequency-related attenuation of the PA signal as it travels through the bone.

5.5.

Results

Figure 9 presents the slopes for both normal and osteoporotic bones, revealing a significantly lower slope for osteoporotic bones when compared with normal ones (**$p<0.01$). The average slope of the normal group is $8.77\mathrm{dB}/\mathrm{cm}/\mathrm{MHz}$ and the osteoporosis is $2.55\mathrm{dB}/\mathrm{cm}/\mathrm{MHz}$. The attenuation coefficient of trabecular bone given in literature is $9.94\mathrm{dB}/\mathrm{cm}/\mathrm{MHz}$.⁸³ Ignoring the attenuation of bone marrow, the attenuation coefficients of cancellous bone with porosity $\varphi$ can be calculated by $9.94\times (1-\varphi )\mathrm{dB}/\mathrm{cm}/\mathrm{MHz}$. It is the same order of magnitude as our experimental results but lower due to the ignorance of bone marrow attenuation and scattering attenuation, proving the rationality of our experiment design and implementation. Given that normal bone has higher BMD and lower porosity, the attenuation of high-frequency components is stronger during PA signal propagation through the cancellous bone structure. This results in a decreased high-to-low frequency ratio in the spectral distribution, leading to a larger slope for the control group. Conversely, osteoporotic bones have a slower slope, due to the weakening of the solid–liquid interface coupling and the decrease of viscosity attenuation caused by lower BMD and higher porosity. The experimental results qualitatively agree with the numerical simulation outcomes, quantitative comparisons are discussed in Sec. 6.

Fig. 9

Statistical results of bone slope in the two studied groups (**$p<0.01$).

6.

Conclusion and Discussion

PA technology holds immense promise for bone assessment and imaging. Researchers have proposed various spectral analysis methods in PA bone assessment, such as physicochemical⁴⁴ and time-frequency spectrum,^84,85 to discern characteristics of the bone microstructure and chemical composition of bone. However, achieving high-precision PAI to visualize the distribution of bone marrow clusters in the bone remains challenging due to the ultrawideband spectrum (from 220 kHz to 13.5 MHz in cancellous bone, see Appendix A2 in the Supplementary Material) and high frequency of PA signals generated by cancellous bone. Comprehending the attenuation of PA signals in cancellous bone is a pivotal cornerstone for high-resolution PAI of cancellous bone. The propagation of PA waves generated by bone marrow (mainly lipids) through the cancellous bone, a solid–liquid two-phase porous medium was studied theoretically, numerically, and experimentally in this study. Biot’s theory was modified to account for high frequencies and viscosity, providing an analytical solution for the broadband PA signal’s behavior in cancellous bone. Numerical simulation results show that when porosity is greater than 0.72, the viscous absorption attenuation of PA signals increases with frequency but decreases with greater porosity. Additionally, the fast $P$ wave with high speed and low attenuation is more suitable for the detection of cancellous bone. In PA detection, to isolate the acoustic propagation properties of porous cancellous bone from the aliasing with the PA source, a PA-DAS method was proposed and a parameter slope was extracted to quantify frequency-specific acoustic attenuation. Experimental data on a rabbit osteoporotic model demonstrate that PA-DAS can effectively distinguish normal from osteoporotic bone, validating its utility for bone evaluation.

As shown in Fig. 9, the average slope of the normal group measured $8.77\mathrm{dB}/\mathrm{cm}/\mathrm{MHz}$, and the osteoporosis group exhibited a slope of $2.55\mathrm{dB}/\mathrm{cm}/\mathrm{MHz}$. Nevertheless, the slopes obtained from numerical simulations were $0.61\mathrm{dB}/\mathrm{cm}/\mathrm{MHz}$ for the normal bone and $0.19\mathrm{dB}/\mathrm{cm}/\mathrm{MHz}$ for the osteoporosis bone, as detailed in Appendix A1 in the Supplementary Material. Although the qualitative agreement between the experimental and simulated results is evident, quantitative disparities still exist. This is because cancellous bone’s acoustic propagation properties are predominantly governed by absorption and scattering of viscous solid–liquid two-phase porous structure.⁸⁶ Absorption attenuation is an energy dissipation process caused by viscous friction at the fluid–solid interface, whereas scattering attenuation⁸⁷ arises from reflections and scattering due to acoustic disparities between solid trabecular bone and liquid bone marrow, leading to the reduction of acoustic energy along the original propagation direction.⁸⁸ This study focused on high-frequency viscous absorption attenuation in porous media, refining Biot’s theory to address higher frequency bands in PA signals and the viscosity of bone marrow. However, for more accurate quantification of scattering characteristics of cancellous bone, future studies should consider the anisotropy and heterogeneity intrinsic to its structure. Thankfully, the scattering attenuation in cancellous bone also adheres to a linear law with frequency.^87,89 The proposed method PA-DAS in this paper can be readily extended due to this approximation.

When considering the clinical applications of PA-DAS, it is crucial to account for the layers of cortical bone and soft tissue that envelop cancellous bone. As the transducers are located on opposite sides of the bone, the thickness of the dense bone and soft tissue on both sides of the bone should be considered. Our previous studies successfully isolated PA signals originating from soft tissue, cortical bone, and cancellous bone in the time domain.⁴⁵ Using the SoS to calculate the thickness of these three types of media and employing the exponential attenuation law to consider the optical and ultrasonic propagation attenuation of cortical bone and soft tissue, we can compensate the PA signals in further studies to isolate and analyze the pure PA signal corresponding to cancellous bone.^90–92 In addition, the diameter of calcaneus bone in the human body is in the range of 2 to 3 cm.⁴³ The attenuation of the cancellous bone to the high-frequency PA signal is stronger, and the received signal is mainly concentrated in the lower frequency band between 220 kHz and 1 MHz. According to the numerical simulation results shown in Fig. 3, the relationship between the attenuation coefficient and frequency is nonlinear at frequencies lower than 1 MHz. Therefore, it is necessary to fit the variation of low-frequency attenuation coefficient with frequency using a nonlinear function before applying this method to clinical practice.