The respective and dependent effects of scattering and bone matrix absorption on ultrasound attenuation in cortical bone

Cortical bone is characterized by a dense solid matrix permeated by fluid-filled pores. Ultrasound scattering has potential for the non-invasive evaluation of changes in bone porosity. However, there is an incomplete understanding of the impact of ultrasonic absorption in the solid matrix on ultrasound scattering. In this study, maps were derived from scanning acoustic microscopy images of human femur cross-sections. Finite-difference time domain ultrasound scatter simulations were conducted on these maps. Pore density, diameter distribution of the pores, and nominal absorption values in the solid and fluid matrices were controlled. Ultrasound pulses with a central frequency of 8.2 MHz were propagated, both in through-transmission and backscattering configurations. From these data, the scattering, bone matrix absorption, and attenuation extinction lengths were calculated. The results demonstrated that as absorption in the solid matrix was varied, the scattering, absorption, and attenuation extinction lengths were significantly impacted. It was shown that for lower values of absorption in the solid matrix (less than 2 dB mm−1), attenuation due to scattering dominates, whereas at higher values of absorption (more than 2 dB mm−1), attenuation due to absorption dominates. This will impact how ultrasound attenuation and scattering parameters can be used to extract quantitative information on bone microstructure.


Introduction
Long bones of humans and large mammals consist of both trabecular and cortical bone.Human cortical bone is characterized by highly dense material, permeated by pores that range in diameter from the sub-micron level to more than 100 μm (Wang et al 2003, Milovanovic et al 2017, Armbrecht et al 2021).Osteoporosis is a bone disease that weakens bone and which is characterized by the loss of bone mass and bone mineral density (BMD) (Feng et al 2011, Richards et al 2008).At the tissue level, these losses are predominantly caused by (i) a rarefication of the trabecular network, (ii) an increase of the intracortical pore size (iii) a decrease of the cortical bone thickness (Dvorak et al 2011, Bala et al 2014, Harrison et al 2020).
In screening for osteoporosis, discerning the bone mineral density (BMD) is useful, as it has been shown that fracture risk is inversely related to BMD (Ross et al 1990, Marshall et al 1996, Alarkawi et al 2016).The primary method for evaluating BMD is dual energy x-ray absorptiometry (DXA).However, DXA utilizes ionizing rays, and when capturing changes in BMD, cortical bone loss due to structural deterioration is poorly captured (Schuit et al 2004, Siris et al 2004, Wainwright et al 2005).However, around 70% of all bone loss after the age of 65 years occurs at peripheral sites and occurs in cortical bone, which makes up around 80% of the human skeleton and is responsible for the majority of bone strength (Hernlund et al 2013, Zebaze et al 2015).The introduction of high-resolution peripheral computed tomography into clinical research has drawn attention to the crucial role that cortical bone plays in bone strength and underscored the significance of evaluating cortical bone for enhanced clinical management of individuals with osteoporosis (Zebaze et al 2010).This resurgence in research has reinvigorated efforts related to quantitative ultrasound (QUS) of cortical bone, leading to a substantial increase in the evolution and clinical implementation of technologies aimed at assessing cortical bone.These developments encompass improvements in axial transmission, scattering, pulse-echo techniques, and imaging methodologies during the last decade (Karbalaeisadegh et al 2022, Raum et al 2022).QUS is a potential alternative, for ultrasound utilizes nonionizing mechanical waves, is relatively inexpensive, and is readily available.Iori et al (2021) has proposed a theoretical cortical bone backscatter model (CortBS) that in combination with multi-directional 3D acquisition and data processing scheme allows the assess microstructural properties in cortical bone.They demonstrated that despite around 70% of ultrasound energy being reflected at the bone interface with a central frequency of 6.6 MHz, it was possible to analyze waves transmitted into the cortical shell and the subsequent backscatter from intracortical pores.The procedure uses the power spectrum of the signal reflected from the bone surface to compensate for soft-tissue attenuation and quantifies frequency-dependent attenuation and backscatter coefficients, denoted as α(f) and BSC(f), respectively, at the tibia.It then obtains the cortical pore-size distribution (Ct.Po.Dm.D) by fitting a theoretical backscatter coefficient to the experimentally determined BSC(f).The method has been validated ex vivo and in vivo on postmenopausal women (Armbrecht et al 2021).CortBS represents the first quantitative bone imaging method capable of measuring microstructural tissue deteriorations in cortical bone, occurring through normal aging and osteoporosis development, and the in vivo investigation indicated a superior discriminatory efficacy of CortBS measurements in comparison to BMD.Nevertheless, additional research is required to disentangle the influences of matrix tissue absorption and scattering on α(f).In porous media, ultrasound propagation is characterized by attenuation, due to both absorption and scattering (Kaufman et al 2003, Pinton et al 2012).While absorption can be neglected in trabecular bone, it cannot be neglected in cortical bone; most attenuation in trabecular bone occurs due to scattering (Bossy et al 2007, Mézière et al 2014, Yousefian et al 2021).Despite the contribution that absorption makes to the attenuation of the system, absorption in the solid matrix is not often accounted for when modeling ultrasound propagation and scattering in porous media (Zhou et al 2021, Luppé et al 2022, White et al 2022).
Another significant contributor to attenuation in porous media is scattering by material discontinuities (Kaufman et al 2003, Pinton et al 2012).As ultrasound propagates through bone, it encounters pores of fluid that permeate the bone.The acoustic impedance of the fluid filled pores differs greatly from that of bone, and significant scattering occurs at these sites (Karbalaeisadegh et al 2019).Multiple scattering, i.e. the occurrence of scattering of a scattered wave by other neighboring scatterers, has been shown to occur in cortical bone (Mézière et al 2014, Karbalaeisadegh et al 2019, Yousefian et al 2021).Numerical studies have previously shown that the diffusion constant (D) and scattering extinction length (l scatt ) can be extracted from analyzing the backscattered signals in different media (Tourin et al 2000, Aubry et al 2007).White et al showed that the Independent Scattering Approximation (ISA) and the Waterman and Truell models both can be fit to experimental ultrasound data acquired in human femoral samples (Waterman et al 1961, Yousefian et al 2019, White et al 2021, White et al 2022).The majority of prior work in bone pertaining to ultrasound backscatter has been done in trabecular bone; however, the microstructure of cortical bone also warrants due investigation, for the majority of bone loss occurs in cortical bone (Bala et al 2014).The impact of absorption by the solid bone matrix on the diffusion constant and scattering extinction length has yet to be fully unraveled, although it has been shown that absorption plays a significant role in attenuation (Pinton et al 2012, Yousefian et al 2021).The combined effect of absorption and scattering is nonlinear and highly complex (Yousefian et al 2021).The objective of this study is to investigate the impact of absorption by the solid matrix on multiple scattering ultrasound parameters such as the diffusion constant and the scattering, absorption, and attenuation extinction lengths.
In this numerical study, the diffusion constant is calculated via backscattering analysis for multiple maps of cortical bone derived from Scanning Acoustic Microscopy (SAM) images of human femur cross sections.Independently, the attenuation extinction length (l att ) is calculated for these maps.Once the effects of map- specific and other considerations are accounted for (l other ), the absorption extinction length (l abs ) is obtained, and the impact of absorption in the solid matrix on the diffusion constant and extinction lengths is isolated and discussed.

Numerical simulations
The finite-difference, time domain (FDTD) SimSonic research freeware (www.simsonic.fr)was used to complete all simulations (Bossy et al 2004).SimSonic is computationally very fast and has been used extensively for the simulation of sound propagation through cortical bone (Karbalaeisadegh et al 2019, Nguyen et al 2020, Iori et al 2021, Yousefian et al 2021, Minh et al 2022).It allows for user control of the simulation medium and its properties (elastic constants, density and absorption), the distribution of emitters and receivers, and the transmitted pulses.The location of any emitters or receivers used to transmit or receive signals does not affect the ultrasound propagation, which allows for a variety of techniques to quantify ultrasound propagation, such as the time-distance-matrix-approach (TDMA) method (Yousefian et al 2019).SimSonic was used in this study to simulate 2D ultrasonic wave propagation in numerical models of cortical bone.

Cortical bone maps and simulations
In this study, bone geometries were generated as described in Karbalaeisadegh et al (Iori et al 2018, Karbalaeisadegh et al 2019).A total of 24 structures were created, using polydisperse pore distributions with varying average pore diameters ranging from 30 to 100 microns, in increments of 10 microns.For each average pore diameter, three structures were created, which shared porosity properties, but had different pore distributions.All structures had a pore density of 10 pores mm −2 .The properties of these pore distributions are shown in table 1.
For each of these 24 structures, 8 different sets of material absorption were used for bone and water, resulting in a total of 192 different sets of simulations.One group of 24 simulations did not incorporate any material absorption in the bone or pores.All other simulations incorporated a fixed, frequency-independent absorption for the pores (0.01 dB mm −1 ), and varying degrees of frequency-independent absorption for the bone matrix, ranging between 0.5 and 10 dB mm −1 .Table 2 shows the densities and elastic constants used for bone and pores in all simulations (Bossy et al 2004).Table 3 shows all nominal bone absorptions used in all simulations.Figure 1 shows three example simulation maps with different average pore diameters.
Perfectly matched layers of 7.5 mm were added to all boundaries of the maps.This was done to ensure minimal reflections of the signal against the boundaries.A grid step of 10 microns was used, which meets the 20 points per wavelength requirement (Bossy 2012).A convergence study confirmed that such a grid step was enough to model scattering by small pores (Karbalaeisadegh et al 2019).The time step used for all simulations was chosen by: where x D is the spatial grid step, c max is the speed of sound in the bone matrix, and d is the dimension.The sampling frequency is the inverse of the time step.For the grid step and material properties used in these simulations, the sampling frequency was 571.4 MHz.
Table 1.Properties of pore distributions in maps.

Map name
For all simulation maps, an ultrasonic transducer with 64 elements was placed along the left side of the simulation map.The widths and pitches of all transducer elements were 0.3 mm.Each array element was successively excited with a Gaussian pulse whose central frequency was 8.2 MHz.The −3 dB transmit bandwidth of this pulse was 3.5 MHz, and the −6 dB bandwidth was 4.9 MHz.Note this central frequency is far below the sampling frequency of 571.4 MHz.One individual element of the transducer array was used to emit a pulse, and all 64 receive elements recorded the reflected signals from each transmit, according to a Full Synthetic Aperture sequence (Jensen et al 2006).This process was repeated for all transducer elements, yielding an inter-  element response matrix (IRM) H containing 4096 elements (64 × 64) for every point in time.This matrix is necessarily symmetric such that H ij = H ji , where the indices i and j correspond to the transmit and receive elements, respectively.Eight IRMs were generated from each of the 24 maps-one for each combination of nominal absorption values shown in table 3. From these different IRMs, different values for l scatt were obtained for identical structures, but for whose absorptive properties differed.

Scattering extinction length
With these IRMs, the diffusion constant for each structure and set of parameters may be obtained.The method used for obtaining the diffusion constant was initially proposed by Aubry and Derode and exploited for bone by Karbalaeisadegh et al (Aubry et al 2007, Karbalaeisadegh et al 2019).All elements of the IRM were shifted in time such that the analysis of all recorded response signals began at the same time relative to a signal arriving at any particular receiver element.This time shifting step is necessary to compensate for differences in the signal arrival times at each element, caused by the bone curvature and the variety of distances between any two elements of the transducer.These time-shifted signals are comprised of two primary parts: the signal that traveled directly from any emitter to a particular receiver (denoted as 'Direct Wave' in figure 2) and the signal backscattered by the medium (denoted as 'Backscattered Wave' in figure 2).A threshold of 0.2 times the maximum of the signal amplitude was used to denote the start of each signal, and that was shifted to time T = 0.After this, any part of the signal occurring before T = 0 was cut out.The same thresholding process was used to time-shift the remaining signals from the IRM.An example of this time shift is shown in figure 2. The red, bolded part of the line is the part of the signal cut before analysis.
After the time-shifting procedure was completed, the signal was partitioned into 0.5 microsecond intervals T. These intervals half-overlap with each other; for instance, one interval encompasses the signal recorded between 0 and 0.5 microseconds, while the next interval encompasses the signal recorded between 0.25 and 0.75 microseconds.The time-dependent, backscattered intensity was obtained for each time interval by integrating the squared value of signals over each time window, using the following expression: where T is the time at the center of the time window and D is the width of each time window.Using this, the backscattered intensity I X T , ( )was obtained by averaging I ij over the emitter-receiver pairs separated by the same distance, where the distance X X X , where X emitter and X receiver respectively denote the location of the center of the emitter and receiver elements.The initial part of the reflected intensity corresponds to the reflected intensity at the surface of the water-bone interface; as such, the data in the first time window is removed from the intensity matrix.As is described in Tourin et al, within each time window, there is both a coherent and incoherent contribution to the backscattered intensity (Tourin et al 2000).Over time, the incoherent scattering contribution increases, and the incoherent intensity becomes larger for larger X distances.This rate of spatial growth is directly related to the diffusion constant via the relationship expressed in equation (7) (Tourin et al 2000).To separate the coherent and incoherent contributions of the backscattered intensity, an antisymmetrization method was used (Aubry et al 2007).The IRM H A was defined to be antisymmetric using the following criteria: It has been shown that the incoherent contribution of the backscattered intensity is where I A is the backscattered intensity as described in equation (2), but derived from H A (Aubry et al 2007).The incoherent intensity may be approximated using the following expression: where D is the diffusion constant.To obtain D, a Gaussian function was fit to the function I X T , inc ( )at each time window.This function was then normalized with respect to its maximum value, and its variance was obtained.The variance of the Gaussian curve linearly increases in time at a rate of 2D.Identifying this rate of increase allowed for the diffusion constant to be calculated for all different combinations of pore diameters and absorptions.Figure 3 shows a graphic depiction of how the diffusion constant was obtained.For 2D diffusion, the diffusion constant can be related to the scattering extinction length by the following expression: where v is the velocity speed of sound in the medium and l* is the transport mean free path (Busch et al 1994, Van Rossum et al 1999, Tourin et al 2000, Mohanty et al 2017).The transport mean free path is related to the scattering extinction length by the following equation: where l scatt is the scattering extinction length and cos ( ) q is the average scattering angle by one scatterer given by Tourin et al (2000): To determine the average scattering angle cos ( ) q by one scatterer, simulations were set up as described in Yousefian et al (2019).Plane waves were emitted incident upon single pores.Simulations were run for pores ranging between 30 and 120 microns, in steps of 10 microns, at a central frequency of 8 MHz.The scattering cross sections were obtained for all angles, and the cosine of the angle of the scattering cross sections was averaged for all pore diameters in the range.All values for cos ( ) q were close to zero (maximum value of 0.17, average value of −0.02, standard deviation of 0.06), so from equation (9), it may be inferred that l l scatt * » for the frequencies, diameters and material properties used in these simulations.
From this, it may be inferred that Total extinction length-attenuation Scattering and absorption in the solid matrix both contribute to the overall attenuation of the wave.In order to isolate the effects of scattering and absorption in the solid matrix on the overall attenuation, the following expression is used (Tourin et al 2000): where l , att l , abs and l other are the attenuation, absorption, and 'other' extinction lengths, respectively.The value of l other is the extinction length due to factors that are not absorption by the solid matrix or scattering, such as the geometry of the bone and diffraction.To find l , att simulations were completed with the same maps, signals, and nominal absorption values as were used in the simulations to calculate the diffusion constants.Instead of recording an IRM using pulses transmitted by individual elements, a plane emitter was placed along the entire left side of the map, and a plane receiver was placed along part of the right side of the map.The receiver was placed between the 1500th and 2500th gridded rows of the simulation medium, for the bone thickness between these rows is relatively uniform horizontally.Perfectly matched layers of 7.5 mm were again placed along all boundaries of the system.Each simulation was run for 22 microseconds to ensure that the signal had sufficiently decayed before the simulation was done.A reference simulation was also run with these same conditions, with a reference map only containing only water and no bone.Figure 4 below shows snapshots of the plane wave as it propagates through the system.
The following expression was used to isolate l : where j is the signal of the attenuated system, ref j is the signal of the reference system, and L is the length through which the wave travels through the attenuating medium.The magnitude of the received signal at each point in time was squared and integrated over the length of the entire simulation.The average length L of the bone was measured along all rows containing the receiver, and the average length L was measured to be 3.3 mm, or about 330 grid steps.From this, l att was able to be obtained for each map.
Once l scatt and l att are estimated, l abs and l other are the two remaining unknowns.The value of l other may be calculated by analyzing simulations that do not have any nominal absorption in the bone.Because bone absorption does not contribute to the attenuation for simulations containing no bone absorption, it may be assumed that for simulations containing no nominal absorption in the bone, the value of l abs is infinite.If this is the case, then equation (12) simplifies to: where l att and l scatt are known.This allows the value of l other to be isolated for each map, independent of any bone absorption in the map.Once l other is known for each map, then l abs can be calculated using equation (12).All data processing was performed using Matlab (R2020b, The MathWorks, Inc., Natick, MA).

Results
Figure 5 shows the evolution of the diffusion constant as the normalized frequency kf varies, where k is the wavenumber of the emitted pulse at its central frequency and f is the average pore diameter in the bone used in any given map.It is seen that for lower bone absorption values (below 2 dB mm −1 ), the correlation between the diffusion constant and normalized frequency changes from mostly negative to mostly positive as the absorption by the bone matrix value increases, whereas at higher absorption values (above 2 dB mm −1 ), there is a monotonic increase in the value of the diffusion constant for any particular normalized frequency.
Figures 6(a)-(c) shows l , scatt l , att and l abs as a function of bone absorption.Each plot contains trend lines, with each trend line corresponding to a different average pore diameter.Figure 6(d) shows the trends for the scattering, absorption, and attenuation extinction lengths as absorption in the bone matrix increases.For each value of bone absorption, the extinction lengths obtained from all 24 maps are averaged and plotted.Absorption values in the bone matrix between 2 and 10 dB mm −1 are shown on the plot below, for at lower bone absorption values, absorption contributes little to the overall attenuation of the wave, and the values for l abs are rather large and vary a lot, rendering the graph difficult to read and glean trends from.

Absorption versus scattering
Although the values for the aforementioned extinction lengths significantly change as absorption in the bone matrix changes, no mention has yet been made of how these values relate to each other.It is relevant to note the   values themselves, as shown in figure 6(d).As is the result of equation ( 12), the value of l att is lower than the values of l , scatt l , abs or l .other One item of note is that the value of l att is relatively constant across the entire domain of possible bone absorption values.Although absorption in the bone matrix necessarily impacts l , att the numerical value of l att does not vary as much as might be expected.It is possible that other factors would likely influence the value of l att to a larger extent, such as the exterior profile of the bone.This is demonstrated by the values of l .
other As seen in figure 1, the bone itself is not a uniform slab, but is curved.The nonuniform nature of the exterior profile impacts the reflection of the wave, and the lack of a uniform bone thickness affects the distance through which a part of the wave might travel through the bone, thus affecting the degree to which scattering impacts attenuation.This likely contributes to the values of l .other Of note is the relative behavior of l scatt and l .
abs At the lower values of bone absorption (i.e.0-2 dB mm −1 ), l abs is larger than l .
scatt This indicates that for lower bone absorption values, scattering has a larger impact on the overall attenuation of the system than does absorption.Intuitively, this is not surprising, for if the bone absorption is quite low, then its effect will be muted.However, as the absorption value in the bone matrix increases to larger values (i.e.>2 dB mm −1 ), then l abs decreases to values lower than l , scatt indicating that at higher bone absorption values, absorption has a greater impact than scattering on the overall attenuation of the system.
It is, of course, no surprise that l abs tends to have a negative correlation with absorption in the bone matrix.It appears that l scatt also tends to have a positive correlation with absorption.It is known that for a fixed value of l , att l abs and l scatt must move inversely to each other to satisfy equation (12).As l scatt and l abs would move inversely to each other, it is perhaps less surprising that l att remains relatively unchanged as the bone absorption is varied.
Physically, this may be explained by a competing effect of absorption and scattering.At lower values of absorption in the solid matrix, scattering attenuates most of the signal that is propagating through the medium.
As the bone absorption increases, absorption contributes increasingly to the overall attenuation of the signal.
The signal thus becomes increasingly diminished as it remains in the medium for more time, meaning that scattering does not have the opportunity to attenuate as much of the signal, for the signal is weaker as it continually reaches future scatterers.At increasingly higher bone absorption values, the signal sees a decreasing number of scatterers, so the impact of scattering correspondingly decreases.
In figure 5, the values of D for different absorption values in the solid matrix are plotted as a function of the normalized frequency k .
f In agreement with previous work, l scatt decreases with an increased normalized frequency when the bone absorption value in the simulations is equal to zero (Karbalaeisadegh et al 2019).This had led to the conclusion that the diffusion constant could be used to evaluate bone porosity (Karbalaeisadegh et al 2019).However, as the absorption in the bone matrix increases from 0 to about 2 dB mm −1 , the trend lines change their shape to yield lines with generally positive slopes, rather than lines with a negative slope as described in previous work (Karbalaeisadegh et al 2019).This rotation stops at about 2 dB mm −1 , at which point, each line maintains a roughly similar trend and translates upward on the figure.This behavior further indicates a competing effect between absorption and scattering, as there is a change in trend.At lower bone absorption values, before this monotonic increase occurs, the normalized frequency at which the highest value of the diffusion constant occurs for a given bone absorption level continually increases.However, as absorption increases in the bone matrix to the point it begins to dominate the overall attenuation, the values of D shift monotonically upward for any normalized frequency, and the peak values of D occur at similar normalized frequencies.Figure 6(d) compares the scattering, attenuation, and absorption extinction lengths.It is shown that at lower values for bone absorption, scattering dominates absorption, whereas at higher values of bone absorption, absorption dominates scattering.This change in the cause of attenuation could account for the change in trend of the scattering extinction length curves versus the pore normalized frequency.This must be taken into account when trying to infer bone porosity from diffusion constant or l scatt values.
Relationship between scattering and absorption in the clinically relevant range Absorption alone has never been measured in cortical bone, precisely because it is very difficult to disentangle it from scattering attenuation.It is expected that the scattering attenuation coefficient would be in the range between 2 and 5 dB mm −1 for bone at 8.2 MHz (Yousefian et al 2021).The literature cites a total attenuation in the range of 1 dB mm −1 MHz −1 , which includes both absorption and scattering (Sasso et al 2007).Therefore, absorption attenuation coefficients in bone in the range of 3-6 dB mm −1 at 8.2 MHz, the central frequency used for these simulations, were used.It stands to reason that practically, the values obtained relating bone absorptions between 4 and 6 dB mm −1 would be of the greatest interest.In figure 6(d), the values of l abs at both 4 and 6 dB mm −1 do not exceed those of l , scatt indicating that absorption in the solid matrix has a greater impact on the overall attenuation of the system than does scattering at these values for nominal bone absorption.However, within this range of bone absorption, scattering and absorption both contribute meaningfully to the overall attenuation, and ultrasound scattering remains of interest for the noninvasive evaluation of bone porosity.We also note that, at these clinically relevant absorption values in bone, the trend of the diffusion constant with porosity is positive, contrary to what was observed in previous work at zero absorption (Karbalaeisadegh et al 2019).In this range, the expected value for l scatt would range between 2 and 2.5 mm.Based on the relatively constant values of l , att it can be inferred that for through transmission, the results should be relatively independent of the absorption value in bone.However, when measuring attenuation through backscattering, it is shown that the scattering extinction length varies significantly, and this information could be useful in screening for osteopenia and osteoporosis.
Limitations of the study Equation (8) assumes linearity between the effects of attenuation due to scattering, absorption, and other considerations.However, it has been demonstrated for the attenuation that scattering and absorption are nonlinearly related to each other, so this assumption is questionable (Yousefian et al 2021).Furthermore, equation (8) assumes that all attenuation that does not come from scattering or absorption gets lumped together into 'other' considerations, such as the geometry of the bone shape and simulation artifacts.Effectively, all attenuation causes get combined into the l other term, and l other does not represent a single, physical phenomenon.
SimSonic is a limiting tool insomuch as it is computationally expensive to execute.Thousands of individual simulations were run to acquire the IRMs needed to complete this study; as such, the 2D version of SimSonic was used, rather than the 3D version.The 3D version is inherently a more accurate program for modeling scattering, as scattering is a three-dimensional phenomenon, and out-of-plane scattering occurs in any real system, but the computational resources that the 3D version of SimSonic would require, are massive.However, the 2D version of SimSonic still serves as a useful tool for evaluating the relationship between scattering and absorption.Additionally, SimSonic is limiting in its capacity to model ultrasonic absorption, as absorption may only be modeled as a frequency independent phenomenon, whereas absorption tends to increase as a function of frequency.In the future, spectroscopy will be used to account for this, where multiple simulations will be run at varying absorption levels to account for this.

Conclusion
It is useful to provide one set of parameters related to scattering, absorption, and attenuation.With the future goal of quantifying bone porosity, the extinction lengths are proposed.The current numerical study investigates the impact of absorption by the solid matrix on multiple scattering ultrasound parameters.Simulations were conducted with various sets of parameters on bone samples with controlled pore densities, pore diameters, and nominal absorption levels.The diffusion constants and attenuation extinction lengths for attenuation were found for all samples, and subsequently, the scattering and absorption extinction lengths were found in all samples.The extinction lengths were shown to vary significantly with absorption.It has been demonstrated that there is a nonlinear, intricate relationship between absorption and scattering.The implications from these results could lead to better methods for diagnosing osteopenia and osteoporosis.

Figure 3 .
Figure 3. Graphic depiction of obtaining the diffusion constant.

Figure 5 .
Figure 5. Average value of diffusion constant versus normalized frequency.

Figure 6 .
Figure 6.Average extinction lengths versus bone absorption.Figures (a), (b), and (c) respectively compare l , scatt l , att and l abs to bone absorption.Figure (d) shows the trends for the scattering, absorption, and attenuation extinction lengths versus bone absorption.For each value of bone absorption plotted in (d), the extinction lengths obtained from all 24 maps are averaged and plotted.

Table 2 .
Material properties for bone and pores used in simulations (not Including absorption).

Table 3 .
Nominal absorption values used.