Inferring pore radius and density from ultrasonic attenuation using physics-based modeling

Abstract

This work proposes the use of two physics-based models for wave attenuation to infer the microstructure of cortical bone-like structures. One model for ultrasound attenuation in porous media is based on the independent scattering approximation (ISA) and the other model is based on the Waterman Truell (WT) approximation. The microstructural parameters of interest are pore radius and pore density. Attenuation data are simulated for three-dimensional structures mimicking cortical bone using the finite-difference time domain package SimSonic. These simulated structures have fixed sized pores (monodisperse), allowing fine-tuned control of the microstructural parameters. Structures with pore radii ranging from 50 to $100\hspace{0.17em}\hspace{0.17em}\mu \mathrm{m}$ and densities ranging from 20 to 50 pores/mm³ are generated in which only the attenuation due to scattering is considered. From here, an inverse problem is formulated and solved, calibrating the models to the simulated data and producing estimates of pore radius and density. The estimated microstructural parameters closely match the values used to simulate the data, validating the use of both the ISA and WT approximations to model ultrasonic wave attenuation in heterogeneous structures mimicking cortical bone. Furthermore, this illustrates the effectiveness of both models in inferring pore radius and density solely from ultrasonic attenuation data.

I. INTRODUCTION

The present work aims to characterize the microstructure of complex heterogeneous media mimicking cortical bone using ultrasonic interrogation. Doing so provides a step toward developing a noninvasive technique for quantifying levels of osteoporosis present in cortical bone. Osteoporosis is one of the most common bone diseases and leads to the degradation of both trabecular and cortical bone, resulting in an, on average, increase in both pore size (radius) and the number of pores present (density; Chen et al., 2013; Yerramshetty and Akkus, 2012), although for later stages, merging of very large pores can lead to a decrease in pore density (Iori et al., 2019). This degradation results from aging, menopause, as well as certain medications, and leads to an increase in susceptibility to fracture (Augat and Schorlemmer, 2006; Chen et al., 2013; Hoc et al., 2006). On average, 20% of men and 33% of women over the age 50 will experience osteoporotic fractures (Melton et al., 1998; Melton et al., 1992) with more than 8.9 × 10⁶ fractures worldwide annually (Johnell and Kanis, 2006). Furthermore, studies show that in women, 80% of appendicular bone loss during menopause is attributed to cortical bone degradation (Bjørnerem et al., 2018). Early diagnosis and treatment can reduce the risk of osteoporosis related injuries (Delmas, 2005).

The classical approach to quantifying markers of osteoporosis, such as bone mineral density (BMD) evaluation, has been shown to be inadequate in predicting fractures due to a significant BMD overlap between subjects who experienced a fracture and subjects who did not (Marshall et al., 1996). Furthermore, magnetic resonance imaging (MRI) can be used to characterize cortical bone porosity (Rajapakse et al., 2015), but it is costly, has a poor resolution, and is infeasible in practice at large scales for screening purposes. Thus, a noninvasive, nonionizing method for quantifying osteoporosis is needed. Quantitative ultrasound (QUS) can potentially provide just that. Furthermore, it has been shown in Chaffaî et al. (2002) and Du et al. (2017) that ultrasound parameters, such as speed of sound, and the backscattering coefficient relate to the microstructural parameters of bone, including trabecular thickness and pore volume fraction, albeit for trabecular bone (Bossy et al., 2005; Wear, 2008; Nguyen Minh et al., 2020). Among all the QUS parameters, it has been shown that changes in pore size and density in cortical bone have a significant impact on attenuation as it affects the scattering of waves. This was done using phenomenological modeling of the ultrasonic attenuation and provides information regarding porosity at the resolution of low, medium, or high porosity but was unable to capture more detailed microstructural properties (Yousefian et al., 2018). In comparison, physics-based models, such as the independent scattering approximation (ISA), have been shown to predict attenuation in trabecular bone samples (Méziére et al., 2014) as well as in simulated two-dimensional (2D) cortical bone samples (Yousefian et al., 2019). A higher order model used to calculate wave attenuation, the Waterman Truell (WT) model, has been used for heterogeneous structures such as three-dimensional (3D) acoustic metamaterials (Brunet et al., 2015). Furthermore, quantifying the size and density of air voids entrapped in cement using ultrasonic attenuation has been researched. Specifically, simplex methods were used to calibrate the ISA to the experimental data and produce microstructural parameter estimates (Punurai, 2006).

In this work we aim to quantify pore size and density for structures mimicking cortical bone using ultrasonic attenuation. To do so, we generate frequency dependent attenuation data using the SimSonic finite-difference time domain (FDTD) MATLAB package (Bossy, 2012) to simulate wave propagation in monodisperse structures with pores randomly distributed. We consider pore radii ranging 50 to $100\hspace{0.17em}\hspace{0.17em}\mu \mathrm{m}$ and densities ranging from 20 to 50 pores/mm³. These values correspond to porosity values reported in the literature for cortical bone (Evans and Bang, 1967; Thomas et al., 2006). To retrieve the micro-architectural properties of porosity from attenuation data, we then formulate and solve an inverse problem, fitting the WT and ISA models to the simulated data sets. This produces estimates for pore radius and density, which are compared to the nominal values used to generate the data. Estimates closely matching the nominal values indicate that both models are capable of inferring the microstructural parameters associated with random heterogeneous media similar to that of cortical bone.

The ultimate goal is to infer the microstructure of cortical bones in humans in vivo. This present work provides a crucial step toward being able to do so. We validate the use of the ISA along with the higher order WT model to estimate the microstructural parameters, pore radius, and pore density, based solely on attenuation data, which had not been done for cortical bone-like samples. This is important as both models are explicitly dependent on pore radius and density, allowing them to be reformulated to consider polydisperse (pores of varying size) structures, which would be expected from in vitro experiments. Thus, this work provides the foundation for approaching such real-world problems. Furthermore, validating the use of ultrasonic attenuation to perform this analysis is promising as ultrasound is noninvasive, nonionizing, and relatively cheap compared to typical bone imaging techniques such as MRI.

We begin by introducing both the ISA and WT models in Sec. II, explaining how they model attenuation that is dependent on frequency as well as on microstructure. From here, we discuss in detail the numerical simulation that allows us to simulate attenuation data in heterogenous structures in Sec. III. This is followed in Sec. IV by the inverse problem formulation in which our specific objective is to calibrate both models and estimate the “true” or nominal microstructure. Next, in Sec. V, we provide the results of solving these inverse problems, followed by a discussion. Final conclusions are then presented in Sec. VI.

II. MODELING OF ULTRASOUND ATTENUATION

We seek to infer the microstructure of cortical bone from ultrasonic attenuation data. This requires a mathematical model that relates the microstructural parameters of interest to attenuation measurements. Here, we consider two such physics-based models that predict attenuation as a function of pore size and density. The first model considered is based on the ISA, which has been used to characterize trabecular bone (Méziére et al., 2014), cortical bone (Yousefian et al., 2019), as well as air voids in cement (Punurai et al., 2007). The second, higher order model is based on the WT approximation and has been used to describe soft 3D acoustic metamaterials (Brunet et al., 2015). We consider formulations of these models to describe 3D monodisperse structures (fixed pore radius). Here, we give a brief overview of wave propagation in heterogeneous media, describing how attenuation is calculated and its dependence on the approximation (ISA or WT) scheme used.

Consider a time-harmonic wave propagating in the direction x. The energy density, $⟨e⟩$, decays exponentially according to

$$⟨e⟩(x)\propto \hspace{0.17em}\text{exp}\hspace{0.17em}(-\alpha x),$$ (1)

where α is the attenuation coefficient, which we aim to model mathematically. The attenuation due to scattering can be approximated as

$$\alpha =-\text{Im}\left[k_{\text{eff}}\right],$$ (2)

where $k_{\text{eff}}$ represents the complex, effective wavenumber. The ISA and WT models provide varying levels of approximations of this effective wavenumber. Following Méziére et al. (2014), consider that in a random scattering medium, the coherent field, which is characterized by $k_{\text{eff}}$, is the solution of Dyson's equation. Specifically, the “self-energy” in Dyson's equation incorporates all multiple scattering terms. This is referred to a perturbative method, resulting in a Taylor series (Chekroun et al., 2012) solution given by

$$k_{\text{eff}}^2\approx k_0^2+4\pi n_s{f}_0(\omega ;r)+\frac{4{\pi }^2{n}_s^2}{k_{l_0}^2}[f_0^2(\omega ;r)-f_{\pi }^2(\omega ;r)],$$ (3)

where k₀ is the wavenumber and n_s is the pore density. The scattering function, $f_{\theta }$, is dependent directly on the pore radius, r, and the angle of incidence, θ [see Eq. (9)]. The simplest approximation, the ISA, includes only the linear terms. The WT model is higher order, including the quadratic terms.

A. The ISA model

As mentioned, the ISA is considered a first-order approximation in that it considers only the first-order terms in the Taylor series expansion. Thus, it estimates the effective wavenumber as

$$k_{\text{eff}}^2\approx k_0^2+4\pi n_s{f}_0(\omega ;r).$$ (4)

Notice that this representation describes the scattering of the wave as it hits a pore as being independent, i.e., once the wave hits a scatterer (pore), it never returns to the same scatterer. Furthermore, by the optical theorem (Jackson, 1999), we have that the forward scattering function, f₀, can be related to the scattering cross section, ${\gamma }^{\text{scatt}}$, as

$${\gamma }^{\text{scatt}}=\frac{-4\pi }{k_0}\text{Im}(f_0).$$ (5)

It can be shown that for the ISA model we have

$$\frac{1}{2}{n}_s{\gamma }^{\text{scatt}}(\omega ;r)\approx -\text{Im}\left[{\left(k_0^2+4\pi n_s{f}_0(\omega ;r)\right)}^{1/2}\right],$$ (6)

which, provided absorption is neglected, gives the following ISA attenuation model:

$$\alpha (\omega ;n_s,r)=\frac{1}{2}{n}_s{\gamma }^{\text{scatt}}(\omega ;r).$$ (7)

Furthermore, from Punurai (2006), we have that the scattering cross section is given by

$${\gamma }^{\text{scatt}}(\omega ;r)=4\pi \sum _{m=0}^{\infty }\frac{1}{2m+1}\left[{\left|A_m\right|}^2+m(m+1)\frac{k_l}{k_s}{\left|B_m\right|}^2\right],$$

where k_l and k_s are the longitudinal and shear wavenumbers, respectively. The unknown coefficients, A_m and B_m, are determined as

$$\left[\begin{array}{c}{A}_m\\ B_m\end{array}\right]=H^{-1}\left[\frac{-1}{k_l}{(-i)}^{m+1}(2m+1)\left[\begin{array}{c}{J}_{11}\\ J_{12}\end{array}\right]\right],$$ (8)

where

$$\begin{array}{c}H=\left[\begin{array}{cc}{H}_{11}& H_{12}\\ H_{21}& H_{22}\end{array}\right],\\ H_{11}=-\left(m^2-m-\frac{{(k_{s}r)}^2}{2}\right)h_m(k_{l}r)-2(k_{l}r)h_{m+1}(k_l),\\ H_{12}=m(m+1)\left[(m-1)h_m(k_{s}r)-(k_{s}r)h_{m+1}(k_{s}r)\right],\\ H_{21}=(m-1)h_m(k_{l}r)-(k_{l}r)h_{m+1}(k_{l}r),\\ H_{22}=-\left(m^2-m-\frac{{(k_{s}r)}^2}{2}\right)h_m(k_{s}r)-(k_{s}r)h_{m+1}(k_{s}r),\end{array}$$

and

$$\begin{array}{c}{J}_{11}=-\left(m^2-m-\frac{{(k_{l}r)}^2}{2}\right)j_m(k_{l}r)-2(k_{l}r)j_{m+1}(k_{l}r),\\ J_{21}=(m-1)j_m(k_{l}r)-(k_{l}r)j_{m+1}(k_{l}r).\end{array}$$

Note that $j_m(·)$ is the spherical Bessel function of the first kind of order m, and $h_m(·)$ is the spherical Bessel function of the third kind of order m.

B. The WT model

Recall that the WT approximation is referred to as second order due to the truncation of the Taylor series at the second-order terms. Thus, the effective wavenumber is approximated as

$$k_{\text{eff}}^2\approx k_0^2+4\pi n_s{f}_0(\omega ;r)+\frac{4{\pi }^2{n}_s^2}{k_0^2}[f_0^2(\omega ;r)-f_{\pi }^2(\omega ;r)].$$

Notice, here, we allow for multiple scattering where a wave can revisit a scatter more than once. In the far-field, the scattering amplitude of the longitudinal wave $f_{\theta }$ is calculated as (Punurai, 2006)

$$f_{\theta }(\omega ;r)=\sum _{n=0}^{\infty }-A_m(i^m)P_m(\text{cos}\hspace{0.17em}(\theta )),$$ (9)

where A_m is defined in Eq. (8), and P_m refers to the Legendre polynomial of degree m.

III. FDTD SIMULATION OF ULTRASONIC ATTENUATION DATA

In the present work, we consider numerically simulated frequency dependent attenuation data. This allows fine-tuned control of the microstructural parameters of interest as well as direct comparison with the monodisperse ISA and WT models' predictions for attenuation. Furthermore, it allows us to consider the attenuation only resulting from scattering. Having nominal or true pore radii and densities that we use to generate the data is essential for determining the efficacy of solving the inverse problem to infer these. We now discuss the specifics of this process.

A. The heterogeneous structure

To generate the random heterogenous structures for a given pore radius (r) and density (n_s), a Monte Carlo method (Mohanty et al., 2019) is implemented to randomly distribute pores throughout the media until the desired number of pores is achieved. The algorithm used to generate structures does not allow overlap between pores. The slab dimensions are $200\hspace{0.17em}\hspace{0.17em}\text{mm}\times 200\hspace{0.17em}\hspace{0.17em}\text{mm}\times 10\hspace{0.17em}\hspace{0.17em}\text{mm}$. Material properties of cortical bone and water are assigned to the solid and fluid phases, respectively, of the binary structures. The pore radius ranges from 50 to $100\hspace{0.17em}\hspace{0.17em}\mu \mathrm{m}$ and the pore density ranges from 20 to 50 pores/mm³.

B. Emitting pulse

To solve the inverse problem, a frequency sweep is needed to capture the frequency dependent attenuation. To do so, Gaussian pulses with a central frequency within the spectroscopy range of 1–8 MHz with 1 MHz intervals and −6 dB bandwidth of 20% are transmitted through the medium. We assume 30 receivers are placed throughout the depth of structure to record the signal as it propagates. The emitter and all of the receivers are large enough so that they cover the width of the structure and the pores are homogeneously distributed. Hence, the transmitted wave is a plane wave, and the recorded signals are averaged over the whole width of the structure.

C. Boundary conditions

Perfectly matched layers at the two ends of the structure in the wave propagation direction reduce reflections from those boundaries. Symmetric boundary conditions in the direction perpendicular to the wave propagation are implemented to eliminate diffraction and ensure plane wave transmission.

D. Simulation parameters

The grid step of $\Delta x=20\hspace{0.17em}\hspace{0.17em}\mu \mathrm{m}$ exceeds the 20 points per wavelength spatial sampling requirement proposed by Bossy (2012), ensuring the accuracy of the simulation results while keeping the computational costs sufficiently low. Choosing Courant-Friedrichs-Lewy (CFL) $=0.99$, the temporal grid step is defined as

$$\Delta t=0.99\frac{1}{\sqrt{d}}\frac{\Delta x}{c_{\text{max}}},$$ (10)

where $\Delta x$ is the spatial grid step, c_max is the highest speed of sound in the simulation medium, and d is the dimension of space (d = 3 for the 3D simulation). The procedure for measuring the attenuation and spectroscopy are given in Yousefian et al. (2019) and Yousefian et al. (2018). Figure 1 depicts the simulated 3D structure with monodisperse, randomly distributed pores.

FIG. 1.

Schematic of 3D structures with dimensions 10 mm × 20 mm× 20 mm.

IV. INVERSE PROBLEM

Solving the forward problem involves taking the microstructural parameters, pore radius, and density and using the mathematical models (i.e., ISA- or WT-based) to predict attenuation. Here, we are interested in solving the inverse problem, where one takes attenuation data along with a mathematical model and attempts to estimate the pore radius and density of a sample. To do so, we must formulate and then solve the inverse problem. The first step is to model the data observation process. Here, we consider that a realization of the data generation procedure is given by

$$y_j=\alpha ({\omega }_j;n_{s_0},r_0)+{ϵ}_j,\hspace{1em}j=1,\dots ,N,$$ (11)

where ω_j, $j=1,\dots ,N$, are the frequency points, $n_{s_0}$ and r₀ denote the nominal pore density and radius values used to generate the data, respectively, and ϵ_j's are independent identically distributed (IID) error terms. This is referred to as an absolute error model and results in an ordinary least squares (OLS) formulation (Banks et al., 2014; Banks and Tran, 2009) of the inverse problem in which all data observations are treated as equally important. The cost function we wish to minimize is given by

$$J(n_s,r)=\sum _{j=1}^N{[y_j-\alpha ({\omega }_j;n_s,r)]}^2,$$ (12)

where y_j represents the attenuation data collected at frequency points ω_j, $j=1,\dots ,N$, and α represents the corresponding model solution. Solving the inverse problem results in estimates for pore radius and density given by

$$({\widehat{n}}_s,\widehat{r})=\underset{(n_s,r)}{\text{arg}\hspace{0.17em}\text{min}}\hspace{0.17em}J(n_s,r).$$ (13)

V. OPTIMIZATION RESULTS AND DISCUSSION

We now present the results of calibrating both the ISA and WT models to the simulated data by solving the inverse problem laid out in Sec. IV. The optimization is done using MATLAB's fmincon, an interior point algorithm. We considered 11 simulated 3D monodisperse structures with combinations of pore radii (r) ranging from 50 to $100\hspace{0.17em}\hspace{0.17em}\mu \mathrm{m}$ and pore densities (n_s) ranging from 20 to 50 pores/mm³. Figure 2 provides representative results for two specific datasets in which the nominal microstructural parameters are given in the titles and the resulting estimates for each model are given in the inset boxes.

FIG. 2.

(Color online) Optimized ISA- vs WT-based models and resulting parameter estimates for nominal pore density ns = 30 pores/mm³ and radius $r=75\hspace{0.17em}\mu \mathrm{m}$ (a) and nominal pore density ns = 50 pores/mm³ and radius $r=50\hspace{0.17em}\mu \mathrm{m}$ (b).

We see that the ISA- and WT-based models produce similar parameter estimates, which correspond well to the nominal parameter values. Furthermore, both calibrated models correspond to the simulated attenuation data. These results validate the use of both the ISA and WT models for inferring pore radius and density from simulated attenuation data. To fully see how accurately each model predicts the microstructure, Figs. 3 and 4 contain comparisons of the nominal parameter values vs the estimates for each model. These result from optimizing both models to the 11 datasets generated from the combinations of pore radii and densities given above. Notice that we consider each parameter separately, whether pore radius or pore density, and the closer the pattern lies to the line y = x, the more accurate the estimates.

FIG. 3.

(Color online) True pore radius (r₀) vs estimated pore radius ($\widehat{r}$) for the ISA model (a) and WT model (b). The corresponding nominal densities are given in the legend.

FIG. 4.

(Color online) True pore density ($n_{s_0}$) vs estimated pore density (${\widehat{n}}_s$) for the ISA model (a) and WT model (b). The corresponding nominal radii are given in the legend.

We see from the patterns in Fig. 3 that the estimates for both models not only align well with each other but also with the nominal pore radii. Figure 4 shows a less linear pattern, implying we are less accurate in inferring the nominal pore density. This difficulty in accurately estimating both pore density and pore radius simultaneously results from a high correlation between the two parameters. Correlation coefficients are calculated using the OLS estimate for the covariance matrix following Banks et al. (2014) and, for all the datasets, fall in the interval [–0.9986, –0.9584], implying a strong negative linear relationship. This is somewhat expected; notice that when calculating the attenuation due to scattering, ${\alpha }_{\text{scatt}}$ in Eq. (3), we have pore density (n_s) multiplied by forward scattering pressure ($f_{\theta }$), which is explicitly a function of the pore radius (r). This indicates there could be some trade-off between these parameter values that result in the same overall attenuation value. However, studies show that the pore density is less relevant in predicting fracture risk because for advanced bone porosity, merging of large pores leads to a decrease in the pore density (Iori et al., 2019). Overall, these results still show acceptable estimates that are informative regarding the microstructure of the simulated samples.

As noted above, both the ISA and WT models produce similar parameter estimates. This is, in part, due to the fact that their forward model attenuation predictions are similar for the radii, pore density levels, and frequencies considered in this work. Specifically, we see little effects as a result of second-order scattering, and similar forward model predictions result in similar microstructural parameter estimates. It is worth noting that this may not be true in different regimes. Namely, if one expects large amounts of second-order scattering, as is the case for high porosity mediums, there may be a more significant difference between the ISA and WT model predictions.

A. Consistency across multiple realizations

As mentioned in Sec. III, the data are numerically simulated using a Monte Carlo approach to arranging pores within the geometry. To ensure that the results presented in Sec. V are not dependent on the random geometry, we generate multiple data realizations for a given pore radius and density combination. We can then compare, across realizations, the calibrated models' attenuation predictions and the resulting parameter estimates. We consider five realizations and provide representative results in Fig. 5 for samples with a pore radius of $100\hspace{0.17em}\mu \mathrm{m}$ and density of 40 pores/mm³.

FIG. 5.

(Color online) Comparison of the model calibration across five random geometries with pore density n_s = 40 pores/mm³ and radius $r=100\hspace{0.17em}\mu \mathrm{m}$.

We see that the model fit to the data, as well as the accuracy of the resulting parameter estimates, is not dependent upon the random geometry of the sample as there is little variation across realizations. This further justifies the use of both models for inferring the microstructure of media mimicking cortical bone by showing the results are not an artifact of the numerical simulation process but rather dependent on the microstructure itself.

VI. CONCLUSIONS

Previously used phenomenological modeling of attenuation is not capable of providing detailed information regarding the microstructure, such as the distribution of pore radius, and, thus, cannot be applied to experimental samples. Therefore, physics-based models for attenuation are essential. As shown in the present work, such models can be used to infer porosity in structures mimicking cortical bone.

We proposed the use of two physics-based models for frequency dependent attenuation, the ISA and WT models, to describe heterogeneous media mimicking cortical bone. Both models are based on approximation schemes for predicting effective wavenumbers with the ISA describing independent scattering modes and the higher order WT model describing higher order multiple scattering. We then generated attenuation data using a numerical FDTD package, allowing us to control the nominal microstructural parameters, pore sizes, and densities. The simulated structures contained pores arranged randomly within the 3D media mimicking cortical bone. Based on the data generation process, we formulated an inverse problem to infer the microstructure of samples from attenuation data.

We demonstrated that both calibrated models predicted attenuation values in line with data as well as parameter estimates that closely matched the nominal values. Despite the similar predictions given by the ISA and WT models, investigating both is still of interest as we may not have this behavior in the next phase of our research where bone samples with varying pore sizes (i.e., polydisperse) are examined. Monodisperse models were chosen here as a first step. This also enabled us to investigate the contributions of pore radius and pore density to attenuation. This is critical because the evolution of these parameters with osteoporosis is not monotonous. In osteopenic bone and for early stages of osteoporosis, both the radius and pore density begin to increase. However, at later stages of osteoporosis, pores start to merge into larger pores, which reduces pore density (Andreasen et al., 2020). One of the results of the present study is to show that changes in pore density do not affect ultrasound attenuation as much as do the changes in pore radius.

We also verified that our results were consistent across realizations of the random geometry of the data simulation process. This validated the use of both models in predicting ultrasonic attenuation in cortical bone-like structures as well as in inferring the microstructure of these samples solely from ultrasound data. In doing so, this work provides a necessary step toward solving more complex, real-world problems in which experimental data is used. In our future work, we will consider such experimental cortical bone samples in addition to addressing the effects of absorption.