Theoretical and experimental study of attenuation in cancellous bone

Abstract. Significance Photoacoustic (PA) technology shows great potential for bone assessment. However, the PA signals in cancellous bone are complex due to its complex composition and porous structure, making such signals challenging to apply directly in bone analysis. Aim We introduce a photoacoustic differential attenuation spectrum (PA-DAS) method to separate the contribution of the acoustic propagation path to the PA signal from that of the source, and theoretically and experimentally investigate the propagation attenuation characteristics of cancellous bone. Approach We modified Biot’s theory by accounting for the high frequency and viscosity. In parallel with the rabbit osteoporosis model, we build an experimental PA-DAS system featuring an eccentric excitation differential detection mechanism. Moreover, we extract a PA-DAS quantization parameter—slope—to quantify the attenuation of high- and low-frequency components. Results The results show that the porosity of cancellous bone can be evaluated by fast longitude wave attenuation at different frequencies and the PA-DAS slope of the osteoporotic group is significantly lower compared with the normal group (**p<0.01). Conclusions Findings demonstrate that PA-DAS effectively differentiates osteoporotic bone from healthy bone, facilitating quantitative assessment of bone mineral density, and osteoporosis diagnosis.

1 Introduction 1 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 approximately 8.9 million, leading to a considerable rise in public health costs [2][3][4] .Therefore, early diagnosis of osteoporosis can not only prevent fractures but also substantially reduce healthcare expenditures and resource usage.
Compared to dense bone, the microstructure and chemical composition of cancellous bone is more sensitive to osteoporosis, as evidenced by reduced trabecular thickness, connectivity, and number, along with increased lipid content [5][6][7][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][10][11][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%-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][18][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][25][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 attenuation 12,27 .Unfortunately, it falls short in 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 .Over the past decade, significant advancements have been made in PA-based bone evaluation.Mandelis and Lashkari established a set of photoacoustic-ultrasound (PA-US) backscattering detection systems, successfully detecting minute changes in the BMD of both cancellous and cortical bones 33 .Their work indicated that the apparent integral backscattering coefficient decreases with a decrease in collagen content [34][35][36] .Wang employed threedimensional PA imaging (PAI) and power spectrum analysis to achieve both qualitative and quantitative evaluations of bone microstructure 37,38 .He further introduced multispectral PA decoupling and thermal PA methods for quantitative evaluation of the organic and inorganic chemical components in cancellous bone [37][38][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 and  In this paper, we propose a PA differential attenuation spectrum (PA-DAS) method designed to remove the contribution of PA sources on PA signals in the frequency domain.This allows for a focused study-both theoretical and experimental-on the acoustic propagation characteristics of cancellous bone for bone quality assessment.Theoretically, we apply highfrequency and viscous corrections to Biot's theory, tailoring it for two-phase, solid-liquid porous media, and employ numerical simulations to validate.Subsequently, ex vivo experiments are performed to measure PA-DAS, and a quantitative PA-DAS parameter is extracted for evaluating BMD and diagnosing osteoporosis.Our results highlight the potential utility of this method for comprehensive bone quality assessment.

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 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 [50][51][52][53][54][55] .Biot's theory has subsequently found applications in the nondestructive evaluation of bone via biomedical ultrasound (US) [56][57][58][59][60][61] .However, Biot's theory is limited to cases where the acoustic frequency is below the medium's critical frequency  c , and the flow of liquid through the pores is well-described by Poiseuille flow.The critical frequency   is defined as 62 : where  denotes the porosity,  denotes the fluid viscosity,  f represents the fluid density, and  denotes the permeability.For cancellous bone, filled with viscous bone marrow,  c typically ranges between 1-10 kHz 63,64 .The sizes of the trabecular bone vary from 50 to 200 μm, while the trabecular spaces (bone marrow clusters) range from 0.2 to 3 mm 65 .The SoS in trabecular bone and bone marrow are 3200 m/s and 1500 m/s, respectively 66 .Notably, the frequencies of PA signals generated in these structures exceed 220 kHz 67 , far surpassing  c .
This means that the laminar flow condition described by Poiseuille's law no longer holds, necessitating modifications to Biot's theory for high-frequency applications.
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 : where  ⃑ denotes the solid skeleton displacement vector,  ⃑ ⃑ represents the liquid displacement vector.The elastic constants   and   characterize the frame's material elasticity and also depend on structural parameters such as porosity . and  are additional elastic constants, with  being Biot's constant that describes the coupling between the liquid and solid phases.The Fourier transform solution of velocities for the shear wave ( T * ) and longitude wave ( L1 * and  L2 * ) in solid-liquid two-phase porous media with viscous losses can be expressed as follows 68 : 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   into complex quantities.
This results in the shear wave velocity, as well as the fast and slow longitudinal wave velocities, becoming complex numbers.As sound waves propagate through a dissipative solid-liquid twophase porous medium, their amplitudes decay progressively with increased propagation distance.
To account for high-frequency dissipation, the Biot's mass coefficients can be transformed as follows: Where, () = is a complex number that represents the deviation from Poiseuille's law as frequency increases, reflecting the phase difference between velocity and friction forces 54 .The frequency-dependent dissipation coefficient  = ( , where  is the average pore size in the porous medium.
The attenuation coefficients are determined by the imaginary parts of the complex wave numbers: The high-frequency dissipation coefficient  * is influenced by several factors, including the porosity  , the average pore size  of the solid-liquid two-phase porous medium, the frequency  of the acoustic wave, and the fluid viscosity .Therefore, the porosity  of cancellous bone can be inferred based on the frequency-dependent attenuation coefficient.

Numerical simulation parameters
Based on the above modified Biot's theory, we numerically simulated the propagation of acoustic waves in porous cancellous bone using MATLAB software.The parameters used for these simulations are detailed in Tables 1 and 2. It should be noted that, in general, the porosity of normal cancellous bone is about 0.73 69,70 , and that of osteoporotic cancellous is larger than this value, so we mainly carried out research on the porosity range of 0.72-0.90.We specifically examined the influences of porosity and sound frequency on sound velocity and attenuation.

Numerical simulation results
Based on the modified Biot's theory 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 spot light with a diameter of 2 mm. Figure 1   Similarly, in the calculated porosity range of 0.72 to 0.90, the velocity of fast P-wave (c f ) is much high than that of slow P-wave ( s ) at 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 wide.
However, both velocities decrease significantly with increasing porosity, which is also consistent with Fig. 1.This is mainly due to 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.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 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 frequency is held constant, the absorption attenuation coefficient decreases as porosity increases.This is mainly due to the fact that the higher the porosity of cancellous bone, the smaller the solid-liquid interface area, 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.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  of porous media can be evaluated based on the attenuation of acoustic waves across various frequencies.

Photoacoustic differential attenuation spectrum method
As discussed earlier, the PA signal arriving at the transducers is a complex mixture of where (),  0 (), and () denote the Fourier transforms of (), ( 0 ), and ℎ(), respectively.By setting  0 to 0, we can derive the frequency-related acoustic propagation characteristics of cancellous bone using the following Eq.( 13) to calculate the differential attenuation spectrum of the PA signals received by transducers at different times: According to Fig. 3, it is evident that the attenuation coefficient is approximately linear within the 1-6 MHz range.Thus, the acoustic propagation characteristics within this frequency band can be linearly fitted to quantify attenuation across different frequencies.

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.

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 five 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   of 10 mm, as shown in Fig. 4(a), making them suitable for the PA experiments.

Gold standard -BMD
Following the PA experiments, all eleven bone samples from both the experimental and control groups were subjected to micro-computed tomography scanning (Micro-CT, venus001, Avatar3, Life Medical Technology).Three-dimensional images of both osteoporotic and normal bone are presented in Fig. 4(a).As evidenced by Fig. 4(b), the BMD of the normal bones in the control group was significantly higher than that of the osteoporotic bones in the experimental group (<0.01).In Fig. 4(d), the statistical analysis of the region of interest (ROI) from Fig. 4(c) reveals a significantly higher porosity in the osteoporotic group compared to the control group (  <0.05).Literature suggests that higher BMD is inversely related to porosity 69,70,73,74 .Therefore, it can be concluded that osteoporotic bone exhibits higher porosity compared to normal bone.

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, CA) produces laser pulses with durations ranging from 2 to 5 ns.We selected a laser wavelength of 1730 nm, which corresponds to the specific absorption wavelength of lipids-a major component of bone marrow clusters-to irradiate the bone samples and thereby excite PA signals 75,76 .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, while 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   of approximately 2 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 Corp., Sunnyvale, CA) with a bandwidth of 0-20 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 Corp, Tokyo, Japan) and subjected to 1 MHz highpass 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 Corp, 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 Corp, Tokyo, Japan).Both transducers were acoustically coupled to the bone sample using a transparent ultrasonic coupling agent.
Concurrently, a 1 MHz focusing transducer (Olympus Corp, Tokyo, Japan) was employed to receive the PA signal generated by the blackbody.Data acquisition was performed using a digital oscilloscope (HDO6000, Teledyne Lecroy, USA) 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.

PA signal processing
As shown in Fig. 5, the PA signal received by 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 can be picked up because the gel is used as coupling agent.Moreover, the numerical results in Section 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, so we introduce the SoS for bone marrow   = 1500 m/ which is larger than slow P-wave and smaller than fast P-wave, and bone trabecular   = 3200 /s 66 which is larger than fast Pwave, as the rising and falling edges, respectively, of the window for picking up fast-wave   To account for the spectral differences of the PA source caused by the anisotropy of cancellous bone, we calculated the PA-DAS () was calculated, based on Eq. ( 14) to evaluate the acoustic propagation characteristics of the bone.
Here,  1 and  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 is mainly composed of low-frequency components, owing to significant propagation attenuation in cancellous bone.
Linear fitting was performed for the PA-DAS in the frequency range of 1 to 3.6 MHz.From this, we extracted the slope as a quantization parameter, enabling us to obtain the frequencyrelated attenuation of the PA signal as it travels through the bone.

Results
Control group Osteoporotic group Figure 9 presents the slopes for both normal and osteoporotic bones, revealing a significantly larger slope for osteoporotic bones when compared to normal ones (<0.05).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, which in turn leads to a smaller slope for the control group.Conversely, osteoporotic bones have a larger slope, which is due to the weakening of the solid-liquid interface coupling and the decrease of viscosity attenuation caused by lower BMD and higher porosity.

Conclusion and discussion
The propagation of acoustic waves through the cancellous bone, a solid-liquid two-phase porous medium was studied theoretically, numerically, and experimentally in this paper.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.Also, 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 employed 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.
Cancellous bone's acoustic propagation properties are predominantly governed by absorption and scattering of viscous solid-liquid two-phase porous structure 77 .Absorption attenuation is an energy dissipation process caused by viscous friction at the fluid-solid interface, while scattering attenuation 78 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 79 .This work focuses 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.
When considering the clinical application of PA-DAS, it is crucial to account for the layers of cortical bone and soft tissue that envelop cancellous bone.Since 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 .By using the speed of sound 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 [80][81][82] .

2 𝒦
=  11 +  22 +  33 =  •  ⃑ denotes the normal strains for the solid,  =  11 +  22 +  33 =  •  ⃑ ⃑ denotes the strain for fluid, while   are mass coefficients related to the porosity  and the mass densities  s and  f of the solid and fluid  12 is the mass coupling.The parameter  =  serves as a dissipation factor and is a function of the porosity .
(a) and 1(b) demonstrate the PA field with the frequency of 2.3 MHz in the cancellous bone with 0.72 porosity and 0.83 porosity, respectively, at 17.92 µs, including fast longitude wave (fast P-wave), slow longitude wave (slow P-wave), shear wave, and Rayleigh wave.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
Fig. 1 Numerical simulation results of the sound field of 2.3 MHz acoustic wave propagating in cancellous bone with (a) 0.72 porosity and (b) 0.83 porosity at 17.92 µs.

Figure 2 (
Figure 2(a) and (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.

Fig. 2
Fig. 2 Numerical simulation results.Speed of sound of (a) fast and (b) slow longitude waves with different porosity and frequency.

Figure 3 (
Figure 3(a) and (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 ~ 0.90, the absorption attenuation coefficient for the fast P-

Fig. 3
Fig. 3 Numerical simulation results.Viscous absorption attenuation coefficients of (a) fast and (b) slow longitude waves with different porosity and frequency.
broadband signals originating from distributed PA sources and the inherent acoustic propagation characteristics of porous cancellous bone.The PA signal received by the transducers at time  can be represented as: () = ( 0 ) * ℎ( −  0 ) (11) where * denotes convolution operator, ( 0 ) represents the PA signal generated by bone tissue received by the transducers at time  0 , ℎ() denotes the systematic response of bone tissue.Applying the Fourier transform to Eq. (11) yields the spectrum of PA signals arriving at the transducers at time : () =  0 ()()  0 (12)

Fig. 4
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) Region of interest to calculate porosity.(d) Statistical analysis results of porosity of bone specimens from the osteoporotic group (n=6) and control group (n=5) (* p<0.05).

Fig. 5
Fig. 5 Schematic of experimental setup for PA measurement of bone samples.
signals.As delineated in the red dotted box in Fig. 6(a), the direct PA signal duration for T1 is given by 1 =     = 1.33 μs.The earliest and latest times for the PA signal of T2 were selected by The length of the PA signal as captured by T2 is 2 =

Fig. 6
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 20 dB decline

Fig. 7
Fig. 7 (a) Frequency-response curve of transducer.Red intersection point is frequency corresponding to maximum value decreasing by 20 dB.(b) Spectrum of bone photoacoustic signal received by transducer (solid blue line) and spectrum after frequency-response correction (solid red line).

Fig. 8
Fig. 8 (a) Spectrum of PA signals received by the two transducers.(b) Differential spectrum of PA signal received by the two transducers.

Table 1
65ructural and acoustic parameters of cancellous bone65.