Ultrasound Characterization of Cortical Bone Using Shannon Entropy

Abstract

Objective:

Ultrasound backscattered signals encompass information on the microstructure of heterogeneous media such as cortical bone, in which pores act as scatterers and result in the scattering and multiple scattering of ultrasound waves. The objective of this study was to investigate whether Shannon entropy can be exploited to characterize cortical porosity.

Methods:

In the study described here, to demonstrate proof of concept, Shannon entropy was used as a quantitative ultrasound parameter to experimentally evaluate microstructural changes in samples with controlled scatterer concentrations made of a highly absorbing polydimethylsiloxane matrix (PDMS). Similar assessment was then performed using numerical simulations on cortical bone structures with varying average pore diameter (Ct.Po.Dm.), density (Ct.Po.Dn.) and porosity (Ct.Po.).

Results:

The results suggest that an increase in pore diameter and porosity lead to an increase in entropy, indicating increased levels of randomness in the signals as a result of increased scattering. The entropy-versus-scatterer volume fraction in PDMS samples indicates an initial increasing trend that slows down as the scatterer concentration increases. High levels of attenuation cause the signal amplitudes and corresponding entropy values to decrease drastically. The same trend is observed when porosity of the bone samples is increased above 15%.

Conclusion:

Sensitivity of entropy to microstructural changes in highly scattering and absorbing media can potentially be exploited to diagnose and monitor osteoporosis.

Introduction

Osteoporosis is the result of unbalanced bone turnover and occurs when there is a malfunction in the process of bone formation and resorption [1]. It is associated with an overall loss in bone mass and manifests as the deterioration of bone micro- and macrostructure, leading to increased porosity, trabecular separation and Haversian canal diameter, as well as decreased cortical thickness [2–8]. Nearly half of women and 20% of men older than 50 experience a fracture caused by osteoporosis in their lifetime [9]. These fractures are associated with high costs and high morbidity. Therefore, it is crucial to improve the methods used to diagnose and monitor this prevalent bone disease.

The most common technique used to assess bone loss is dual-energy X-ray absorptiometry (DXA), which gives a measure of bone mineral density (BMD); however, it suffers from several limitations [10]. Although BMD has proven to be effective in diagnosis of osteoporosis, it relies mainly on bone mass measurements and fails to provide information on bone microstructure, which also affects overall bone quality and further development of osteoporotic fractures [11]. Additionally, X-ray techniques are associated with ionizing radiation. This raises concerns for more vulnerable individuals or in cases in which regular monitoring routines are required [12].

Quantitative ultrasound (QUS) has been used as a safer modality to evaluate trabecular bone through measurements of speed of sound (SOS) and broadband ultrasound attenuation (BUA) in accessible sites such as the calcaneus [13–15]. Despite the fact that cortical bone has high load-bearing capacity and strong correlations between its micro-structural and mechanical parameters [16–20], it has not been studied as extensively as trabecular bone [21]. Cortical bone is a transversely isotropic structure in which cylinder-shaped units (osteons) are embedded in the mineralized matrix. Ultrasound wave propagation in cortical bone is affected by its structure because of the acoustic impedance difference between the vascular canals and the solid matrix, forcing the wave to take random paths as scattering occurs. Scattering of ultrasound is associated with loss of energy and, in the case of multiple scattering, with a diffusive regime, which have previously been used to characterize cortical bone through measurements of the attenuation coefficient [22–25] and the diffusion constant [26].

In addition, multiple scattering affects the statistics of the signal amplitude. Several studies have investigated the potential of statistical models to characterize scattering media. On the basis of its simplicity and convenience, the Nakagami distribution [27] has been used to characterize the structure of liver, breast masses and tissue-mimicking phantoms [28–33]. Another method for the statistical analysis of ultrasound signals is based on evaluation of the Shannon entropy, which quantifies the uncertainties in the system [34]. The definition of Shannon entropy in information theory encompasses the probabilistic features of ensembles of messages, as the definition of entropy would apply to statistical ensembles of microstates in statistical mechanics [35]. In a QUS imaging study, Tsui [36] determined that the concentration of scatterers in phantoms affects the Shannon entropy. In another study, Wang et al. [37] measured entropy for trabecular bone phantoms with different densities, as well as clinically on lumbar vertebrae. They found a high negative correlation between phantom density (r = −0.83) and entropy at 3.5 MHz. In addition, measurements from the lumbar vertebrae of patients with a moderate to high risk of osteoporosis indicated higher entropy values corresponding to higher signal complexity.

The structure of cortical bone changes with age, and its porosity increases because of an increase in pore diameter and pore density [38]. These microstructural changes affect the scattering of ultrasound waves, and we hypothesize that Shannon entropy would be an effective tissue characterization tool to evaluate these changes associated with osteoporosis. Shannon entropy had not been evaluated in cortical bone. The non-invasive and quantitative ultrasound evaluation of cortical bone is especially relevant because skeletal sites such as the tibia are particularly accessible to ultrasound backscattering data acquisition.

Such entropy imaging techniques have been implemented to evaluate fatty liver disease, cataract lenses and breast tumors [39–43]. Entropy as a measure of disorder in a medium is used in the present study to characterize heterogeneous structures through experiments on polydimethylsiloxane (PDMS) (Dow Sylgard 184, Dow Corning, Midland, MI, USA) samples with varying concentrations of barium titanate (BaTiO₃) acting as scatterers and simulations in cortical bone based on X-ray computed tomography (CT) images of human femur. The multiple scattering that takes place in cortical bone is a result of the interference of ultrasound waves that have been scattered. Entropy measurement of PDMS–BaTiO3 samples is done as a proof of concept to assess its applicability in characterizing scattering media with different microstructures. These phantoms not only allow for control over the scatterer concentration, but also help us design the signal processing method to assess acoustical parameters such as entropy in a highly absorbing material with an attenuation coefficient of about 17dB/cm at 5 MHz [44].

Implementation of the proposed QUS approach to characterize cortical bone structures with respect to microstructural parameters such as porosity, pore diameter and density makes entropy a potentially effective tool for clinical evaluation of cortical bone.

Methods

PDMS phantom preparation and ultrasound data acquisition

To make the ceramic–silicon composite samples, the two parts of PDMS are measured so that the weight ratio of the base to the curing agent is 25 to 1. These are mixed in a plastic beaker and BaTiO₃ powder is added to the mixture at ratios of 4:1, 2:1, 4:3, 1:1 and 1:2, corresponding volume fraction (VF) percentages of 4%, 7%, 11%, 14% and 24%, respectively [45,46]. The BaTiO₃ particles act as scatterers because of the acoustical impedance difference with the surrounding PDMS matrix. The resulting ceramic–silicone mixture is stirred consistently, de-gassed in a vacuum chamber, cured at 100°C for 2 h and cut into approximately 12 mm thick cylinders with a 100 mm diameter. A total of five PDMS phantoms with the approximate speed of sound of 0.7 mm/μs were used in this study.

A 128 element L7–4 array transducer was used to acquire data at a central frequency of 5 MHz, from a Verasonics Vantange (Verasonics Inc., Kirkland, WA, USA) 128 system. The data for each phantom submerged in water were acquired three times, at slightly different locations to account for possible variations in the spatial distribution of scatterers. The transducer was positioned at close proximity to the phantom and parallel to its surface. The focus was adjusted to be at 6 mm below the surface of the phantom.

Acquisition of CT images in cortical bone samples

We used a collection of images from a previous study [47]. Briefly, we obtained 43 rectangular prism specimens with nominal size of 3 × 4 × 5 mm³ from the central part of the diaphysis of the femur of elderly human donors. For each specimen, an X-ray synchrotron radiation micro-computed tomography (SRμCT) 3-D scan (pixel size: 6.5 μm) was performed. After binarization as discussed in Cai et al. [47], we calculated the structural parameters following Mohanty et al. [48]: average pore diameter (Ct.Po.Dm.), pore density (Ct.Po.Dn.) and volume fraction of pore, or porosity (Ct.Po.).

Finite-difference time-domain simulations

SimSonic [49], an open-source finite-difference time domain (FDTD) simulation software, was used to model ultrasound propagation in bone structures obtained from CT scanning. FDTD simulations were carried out on 7 cross-sectional images for each cortical bone specimen, resulting in 301 total simulations. Bone and water material properties [49] were assigned to solid and liquid phases of binarized images, respectively. According to previous studies on cortical bone [24], the total attenuation, which accounts for both scattering and absorption losses, is observed to be approximately 10 dB/cm·MHz. To the best of our knowledge, scattering and absorption contributions of attenuation in cortical bone have not been reported separately, and their relative contributions are complex [50]. Hence, in our simulations, the absorption coefficient for the bone matrix was arbitrarily set to 3 dB/cm·MHz, which is below the observed total attenuation value of 10 dB/cm·MHz. As illustrated in Figure 1, for each acquisition, a plane wave at 5 MHz is transmitted in the medium, and the backscattered signals are received on the transducer array. The speeds of sound in bone matrix and water-filled pores are assumed to be 4 and 1.54 mm/μs, respectively. The densities for bone and pores are modeled at 1.85 and 1 g/mL, respectively [49]. The spacing between transducer elements is kept at 0.3 mm, which meets the half-wavelength requirement and is extensively used in the fabrication of ultrasonic transducers in this frequency range [51]. Considering the small size of the structures (2.11 × 3.18 mm²), the number of elements is limited to 11. Perfectly matched layer (PML) boundary conditions are used to minimize reflections from the edges of the sample. The gridstep for the simulations is set to 6.5 μm, which meets the 20 points per wavelength simulation requirement [49]. The duration of acquisition is 3.2 μs, which is longer than the time required for the wave to travel the whole structure back and forth. This allows us to extract longer backscattered signals to take multiple scattering into account.

Figure 1.

Left: Configuration used for the numerical simulation. A plane wave is transmitted at 5 MHz, and the backscattered signals are received on the array transducer (located on the top). Black: Material properties of water. White: Material properties of bone. Right: Three snapshots of the propagation at respectively 0.8, 1.2 and 2.4 μs.

Shannon entropy

To obtain the entropy map for each of the PDMS samples and bone structures, a sliding window is used to obtain the signals locally. The size of the window for PDMS samples is 0.6 × 0.6 mm², with 50% overlap in both the lateral and axial directions. Each window comprises two lateral lines as transducer elements are 0.3 mm wide. The length of the window (0.6 mm) is approximately 4λ (wavelength) which is sufficient to capture the characteristics of the medium.

The probability density function (PDF) of the amplitudes of the RF signals in each time window are obtained, and Shannon entropy is calculated as [34]

$$H_c=\sum _{y_{\text{min}}}^{y_{\text{max}}}w(y){\text{log}}_2[w(y)]dy$$ (1)

where H_c is the entropy, w(y) is the amplitude distribution, y_min and y_max are the minimal and maximal values of the backscatter envelope y = f(t). To obtain w(y), a histogram of the signals with a bin width of 0.004 is used. By assigning the calculated entropy value to each window, the entropy map is formed.

Figure 2 illustrates the process of acquiring the entropy image of a PDMS sample with a 4% scatterer VF. For each sample the entropy is averaged over a 2.4 × 21.3 mm² region of interest (ROI). The large width of the ROI is justified by the large size of the sample and the fact that averaging over a larger area would provide a better representation of the overall structure of the sample. However, the depth is kept on the order of a few wavelengths as the samples are highly attenuative and the penetration of ultrasound waves is limited.

Figure 2.

Algorithmic scheme to obtain the entropy image using the PDF of the signals in the sliding window. The frame on the entropy image of a polydimethylsiloxane matrix sample (scatterer volume fraction = 4%) indicates the region of interest. PDF, probability density function; RF, radiofrequency.

Considering the small size of the bone structures, the lateral resolution for bone entropy maps is limited to 11 elements with the element size of 0.3 mm. For each element, the received signals are truncated into 0.21 μs half-overlapping time windows for which the entropy is calculated and assigned to obtain the entropy map (Fig. 3a).

Figure 3.

Simulation results. (a) An entropy map is formed using the backscattered signals from the bone, for two different bone samples: (i) Ct.Po.Dm. = 52.45 μm, Ct.Po.Dn. = 12.54 pores/mm², porosity = 3.71%, entropy = 3.96; (ii) Ct.Po.Dm. = 108.07 μm, Ct.Po.Dn. = 12.27 pores/mm², porosity = 17.13%, entropy = 4.93. (b–d) Entropy versus (b) Ct.Po. (R² = 0.75, RMSE = 0.17), (c) Ct.Po.Dm. (R² = 0.61, RMSE = 0.21) and (d) Ct.Po.Dn. for structures with 52.06 < Ct.Po.Dm. < 61.29 μm (R² = 0.17, RMSE = 0.18). Ct.Po., cortical bone porosity; Ct.Po.Dm., cortical bone pore diameter; Ct.Po.Dn., cortical bone pore density; RMSE, root mean square error.

All data post-processing was performed using MATLAB 2018a (The MathWorks, Natick, MA, USA).

Results

Figure 4b indicates the entropy average over the 2.4 × 21.3 mm² ROI (Fig. 4a) with respect to the scatterer VF percentages of the PDMS samples. To account for deviations in scatterer distributions, three data sets were obtained and processed for each sample. The entropy indicates an increasing trend with increasing particle VF from 4% to 14%. The trend is interrupted as VF is increased to 24%.

Figure 4.

(a) Entropy image of a polydimethylsiloxane sample with scatterer VF of 4%. The entropy is average over the 2.4 × 21.3 mm² region of interest. (b) Average entropy versus scatterer VF. VF, volume fraction.

The entropy map of a bone structure is illustrated in Figure 3a. Considering that the scattered signals convey microstructural information from a heterogeneous structure like cortical bone, the entropy is averaged over the whole map over a 3.2-μs time span. The Ct.Po.Dm. for the 301 structures used in this study ranges from 36.02 to 133.75 μm, and the Ct.Po.Dn. ranges from 8.41 to 23.85 pores/mm². Minimum and maximum Ct.Po. values are 2.17% and 28.07%, respectively. Figure 3b illustrates entropy as a function of Ct.Po. An increase in Ct.Po. results in an increase in entropy; however, for highly scattering structures with porosities >15%, the power law fit seems to flatten, indicating a relatively constant entropy range. This is quite similar to the observation from the PDMS samples with the highest scatterer VF (Fig. 4b). Figure 3c depicts a positive correlation between Ct.Po.Dm. and entropy (R² = 0.61, p ≪ 0.001). Because the average pore diameter (Ct.Po.Dm = 66.28 ± 19.17 μm) has a relatively large standard deviation among the sample population, only the structures within the first (52.06 μm) and second (61.29 μm) quartiles are used to evaluate entropy versus Ct.Po.Dn. to limit the effect of the change in diameter. Results (Fig. 3d) suggest a weak correlation between entropy and Ct.Po.Dn. (R² = 0.17, p = 0.0002), indicating that the increase in pore density has a less significant effect on entropy compared with changes in pore diameter or porosity.

Discussion

In the past, statistical parameters such as entropy have been successfully exploited in QUS studies to evaluate structural changes in biological tissues such as liver and breast [39,41,42]. In the present study, we use entropy as a measure of randomness in ultrasound backscattered signals to assess PDMS samples with different concentrations of BaTiO₃, as well as cortical bone structures. The control over the scatterer concentration of the PDMS–BaTiO₃ samples and the resemblance between PDMS matrix and bone tissue in terms of high ultrasound absorption make the phantoms relevant models for preliminary tissue characterization studies such as the present work.

The entropy results indicate an increasing trend as the BaTiO₃ VF increases from 4% to 15%. This parameter is significantly correlated (p < 0.05) to changes in the heterogeneity of the medium (Fig. 4). Although for lower VFs the entropy proves effective, with an increase to 15% VF, entropy decreases. This observation can be justified by the fact that signals in highly scattering media are attenuated at higher rates because of both scattering and longer paths in the absorbing matrix. The attenuation in the signals results in a reduction in the amplitudes and subsequent calculated entropy values because variance in the amplitudes decreases as well. This is why we used a narrow ROI (depth = 2.4 mm ~ 8λ) for PDMS samples considering the facts that PDMS is a highly absorbing material and the penetration of ultrasound waves is limited. The large width of the ROI (21.3 mm) is justified by the large size of the sample and the fact that averaging over a larger area would provide a better representation of the overall structure of the sample. While compensating for attenuation might be a plausible solution, we believe that at high VFs, it would result in normalization of the noise, which not only does not convey structural information from the deeper parts of the sample, but also introduces more errors into the overall QUS evaluation.

On the basis of these findings on the applicability of entropy measurement in PDMS samples with different levels of heterogeneity, we applied the same approach to evaluate cortical bone structures. We obtained the entropy maps in time and did not convert to the spatial domain (Fig. 3a). Although a power-law fit indicates a positive correlation between Ct.Po. and entropy for porosities <15%, the entropy plateaus at higher porosities (Fig. 3b), suggesting that attenuation is the dominant phenomenon affecting the entropy at higher porosities. This agrees with the observations from Figure 4b, with entropy and BiTO3 volume fraction depicting a similar trend.

On the other hand, Ct.Po.Dm. exhibits monotonous behavior and is highly correlated (R² = 0.61, p ≪ 0.001) with entropy (Fig. 3c). To observe the effect of Ct.Po.Dn. on entropy, the other structural variable, Ct.Po.Dm., needs to be kept constant. Because of large variations in Ct.Po.Dm. (mean = 66.28 μm, standard deviation = 19.17 μm), only the structures with Ct.Po.Dm. ranging from 52.06 μm (first quartile) to 61.29 μm (second quartile) are used to independently investigate the effect of changes in Ct.Po.Dn. on entropy. Hence, only 75 of the total of 301 bone images are used in Figure 3d where entropy and Ct.Po.Dn. do not indicate a strong correlation (R² = 0.17, p = 0.0002). Our results suggest that there is a more significant association between either pore diameter or porosity and entropy. To better understand the independent effect of changes in the microstructure on entropy, a larger data set would need to be exploited in the future where Ct.Po.Dn. and Ct.Po.Dm. are kept constant independently. The mean and standard deviation of the entropy measured for 301 structures are 4.44 and 0.34, respectively. It is worth noting that this parameter is affected by other factors such as signal amplitude and gain, which need to be accounted for during the acquisition and analysis processes especially when data sets from different ultrasound systems are being compared.

Conclusion

The aim of the present study was to evaluate effectiveness of a statistical parameter such as Shannon entropy as a measure of randomness in the backscattered signals with respect to microstructural changes in cortical bone. On the basis of our results, pore diameter and porosity are positively correlated with entropy, while changes in pore density do not indicate a strong association with entropy. The PDMS sample models different scatterer concentrations in a highly absorptive matrix (similar to bone tissue) and is used as a proof of concept for the proposed methodology while providing control over the changes in the microstructure. On the basis of the preliminary results and effectiveness of entropy in cortical bone characterization, future in vivo experiments can be conducted on accessible cortical bone sites (e.g., tibia) to evaluate osteoporosis in patients. Considering the recent studies on cortical bone, a combination of several parameters such as entropy, diffusion constant [26] and backscatter coefficient [38] can be exploited to assess bone microstructure and enhance the prediction of bone fracture non-invasively.

Data availability statement

The data (bone maps, experimental and simulation ultrasound data) are available on request.