The effect of pore size and density on ultrasonic attenuation in porous structures with mono-disperse random pore distribution: A two-dimensional in-silico study

Ultrasound Characterization of Cortical Bone Using Shannon Entropy

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.

The influence of intra-cortical microstructure on the contrast in ultrasound images of the cortex of long bones: A 2D simulation study

Decreased thickness of the bone cortex due to bone loss in the course of ageing and osteoporosis is associated with reduced bone strength. Cortical thickness measurement from ultrasound images was recently demonstrated in young adults. This requires the identification of both the outer (periosteum) and inner (endosteum) surfaces of the bone cortex. However, with bone loss, the cortical porosity and the size of the vascular pores increase resulting in enhanced ultrasound scattering which may prevent the detection of the endosteum. The aim of this work was to study the influence of cortical bone microstructure variables, such as porosity and pore size, on the contrast of the endosteum in ultrasound images. We wanted to estimate the range of these variables for which ultrasound imaging of the endosteum is feasible. We generated synthetic data using a two-dimensional time-domain code to simulate the propagation of elastodynamic waves. A synthetic aperture imaging sequence with an array transducer operating at a center frequency of 2.5 MHz was used. The numerical simulations were conducted for 105 cortical microstructures obtained from high resolution X-ray computed tomography images of ex vivo bone samples with a porosity ranging from 2% to 24 %. Images were reconstructed using a delay-and-sum (DAS) algorithm with optimized f-number, correction of refraction at the periosteum, and sample-specific wave-speed. We observed a range variation of 18 dB of endosteum contrast in our data set depending on the bone microstructure. We found that as porosity increases, speckle intensity inside the bone cortex increases whereas the intensity of the signal from the endosteum decreases. Also, a microstructure with large pores (diameter >250 μm) was associated with poor endosteum visibility, compared with a microstructure with equal porosity but a more narrow distribution of pore sizes. These findings suggest that ultrasound imaging of the bone cortex with a probe operating at a central frequency of 2.5 MHz using refraction-corrected DAS is capable of detecting the endosteum of a cortex with moderate porosity (less than about 10%) if the largest pores remain smaller than about 200 μm.

Study on freezing characteristics of the surface water film over glaze ice by using an ultrasonic pulse-echo technique

This study proposes an ultrasonic pulse-echo technique to examine the freezing characteristics of a thin film of water over ice, and uses it to develop a method to measure the thickness of glaze ice. A multilayer model is first introduced to simulate ultrasonic transmission through multiple media. A transition layer is then inserted between the layers of ice and water, and its properties were in gradient form along the direction of thickness. Following this, a high-frequency ultrasonic experimental device is developed to dynamically measure variations in the thickness of the layers of ice and water. The accuracy of the proposed model of the transition layer was validated by showing that its numerical results agreed well with those of experiments. The results show that the amplitude of echo from the top of the ice layer was at its minimum when the thickness of the film of water was in the range of [40, 45] μm, and increased when the film of water was thinner than 40 μm. A delay in echo from the top of the layer of ice was observed when measuring its thickness because the film of water froze, which yielded a relative error of 3.34%. The proposed numerical model can thus efficiently measure the thickness of glaze ice.

Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus

In this paper, the influence of the transverse wave on sound propagation in a porous medium with a flexible structure is considered. The study is carried out in the time domain using the modified Biot theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The viscous exchanges between the fluid and the structure are described by fractional calculus. When a sound pulse arrives at normal incidence on a porous material with a flexible structure, the transverse waves interfere with the longitudinal waves during propagation because of the viscous interactions that appear between the fluid and the structure. By performing a calculation in the Laplace domain, the reflection and transmission operators are derived. Their time domain expressions depend on the Green functions of the longitudinal and transverse waves. In order to study the effects of the transverse wave on the transmitted longitudinal waves, numerical simulations of the transmitted waves in the time domain by varying the characteristic parameters of the medium are realized whether the transverse wave is considered or not.

Using ultrasonic attenuation in cortical bone to infer distributions on pore size

In this work we infer the underlying distribution on pore radius in human cortical bone samples using ultrasonic attenuation data. We first discuss how to formulate polydisperse attenuation models using a probabilistic approach and the Waterman Truell model for scattering attenuation. We then compare the Independent Scattering Approximation and the higher-order Waterman Truell models' forward predictions for total attenuation in polydisperse samples. Following this, we formulate an inverse problem under the Prohorov Metric Framework coupled with variational regularization to stabilize this inverse problem. We then use experimental attenuation data taken from human cadaver samples and solve inverse problems resulting in nonparametric estimates of the probability density function on pore radius. We compare these estimates to the "true" microstructure of the bone samples determined via microCT imaging. We find that our methodology allows us to reliably estimate the underlying microstructure of the bone from attenuation data.

Deep Learning-Based Photoacoustic Imaging of Vascular Network Through Thick Porous Media

Photoacoustic imaging is a promising approach used to realize in vivo transcranial cerebral vascular imaging. However, the strong attenuation and distortion of the photoacoustic wave caused by the thick porous skull greatly affect the imaging quality. In this study, we developed a convolutional neural network based on U-Net to extract the effective photoacoustic information hidden in the speckle patterns obtained from vascular network images datasets under porous media. Our simulation and experimental results show that the proposed neural network can learn the mapping relationship between the speckle pattern and the target, and extract the photoacoustic signals of the vessels submerged in noise to reconstruct high-quality images of the vessels with a sharp outline and a clean background. Compared with the traditional photoacoustic reconstruction methods, the proposed deep learning-based reconstruction algorithm has a better performance with a lower mean absolute error, higher structural similarity, and higher peak signal-to-noise ratio of reconstructed images. In conclusion, the proposed neural network can effectively extract valid information from highly blurred speckle patterns for the rapid reconstruction of target images, which offers promising applications in transcranial photoacoustic imaging.

Ultrasound Scattering in Cortical Bone

Recent advances in imaging of bone microstructure have led to a growing recognition of the role of cortical microstructure in osteoporosis. It is now accepted that the assessment of the microstructure of cortical porosity is essential to assess bone mechanical competence and predict fracture risk. Cortical porosity affects the propagation of ultrasound waves because pores act as ultrasound scatterers. Scattering by the porosity is an opportunity that should be leveraged to extract quantitative information about cortical microstructure. Scattering by the pores affects a number of ultrasound parameters that should be quantified, including attenuation, backscatter coefficient, ultrasound diffusivity, and their frequency dependence. Measuring these ultrasound parameters and developing models that describe their dependence upon parameters of cortical microstructure is the key to solve inverse problems that will allow the quantitative assessment of cortical porosity and ultimately will improve the non-invasive ultrasound-based evaluation of bone mechanical competence and fracture risk. In this chapter, we present recent advances in measuring and modeling those parameters in cortical bone.

In vivopulse-echo measurement of apparent broadband attenuation andQfactor in cortical bone: a preliminary study

Quantitative ultrasound (QUS) methods have been introduced to assess cortical bone health at the radius and tibia through the assessment of cortical thickness (Ct.Th), cortical porosity and bulk wave velocities. Ultrasonic attenuation is another QUS parameter which is not currently used. We assessed the feasibility ofin vivomeasurement of ultrasonic attenuation in cortical bone with a broadband transducer with 3.5 MHz center frequency. Echoes from the periosteal and endosteal interfaces were fitted with Gaussian pulses using sparse signal processing. Then, the slope of the broadband ultrasonic attenuation (Ct.nBUA) in cortical bone and quality factorQ11-1were calculated with a parametric approach based on the center-frequency shift. Five human subjects were measured at the one-third distal radius with pulse-echo ultrasound, and reference data was obtained with high-resolution x-ray peripheral computed tomography (Ct.Th and cortical volumetric bone mineral density (Ct.vBMD)). Ct.Th was used in the calculation of Ct.nBUA whileQ11-1is obtained solely from ultrasound data. The values of Ct.nBUA (6.7 ± 2.2 dB MHz^-1.cm^-1) andQ11-1(8.6 ± 3.1%) were consistent with the literature data and were correlated to Ct.vBMD (R2=0.92,p<0.01, RMSE = 0.56 dB.MHz^-1.cm^-1, andR2=0.93,p<0.01, RMSE = 0.76%). This preliminary study suggests that the attenuation of an ultrasound signal propagating in cortical bone can be measuredin vivoat the one-third distal radius and that it provides an information on bone quality as attenuation values were correlated to Ct.vBMD. It remains to ascertain that Ct.nBUA andQ11-1measured here exactly reflect the true (intrinsic) ultrasonic attenuation in cortical bone. Measurement of attenuation may be considered useful for assessing bone health combined with the measurement of Ct.Th, porosity and bulk wave velocities in multimodal cortical bone QUS methods.

Acoustic Attenuation: Multifrequency Measurement and Relationship to CT and MR Imaging

Transcranial magnetic resonance-guided focused ultrasound (tcMRgFUS) is gaining significant acceptance as a noninvasive treatment for motion disorders and shows promise for novel applications such as blood-brain barrier opening for tumor treatment. A typical procedure relies on CT-derived acoustic property maps to simulate the transfer of ultrasound through the skull. Accurate estimates of the acoustic attenuation in the skull are essential to accurate simulations, but there is no consensus about how attenuation should be estimated from CT images and there is interest in exploring MR as a predictor of attenuation in the skull. In this study, we measure the acoustic attenuation at 0.5, 1, and 2.25 MHz in 89 samples taken from two ex vivo human skulls. CT scans acquired with a variety of X-ray energies, reconstruction kernels, and reconstruction algorithms, and MR images acquired with ultrashort and zero echo time sequences are used to estimate the average Hounsfield unit value, MR magnitude, and T2^* value in each sample. The measurements are used to develop a model of attenuation as a function of frequency and each individual imaging parameter.

Frequency-dependent analysis of ultrasound apparent absorption coefficient in multiple scattering porous media: application to cortical bone

The effect of matrix viscoelastic absorption on frequency-dependent attenuation in porous structures mimicking simplified cortical bone is addressed in this numerical study. An apparent absorption is defined to quantify the difference between total attenuation (resulting from both absorption and scattering) and attenuation exclusively due to scattering. A power-law model is then used to describe the frequency-dependent apparent absorption as a function of pore diameter and density. The frequency response of the porous structures to a Gaussian pulse is studied to determine the frequency range over which the system can be considered linear. The results show that for low scattering regimes (normalized frequency [Formula: see text]0.80), the system and its apparent absorption can be considered linear. Hence, the total attenuation coefficient results from the summation of scattering and absorption coefficients. However, for highly scattering regimes, the system can no longer be considered linear, as the apparent absorption vs. frequency deviates from a linear trend. As the pore density increases, the apparent absorption coefficient increases as well.

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

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  μm 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.

Estimation of Thickness and Speed of Sound in Cortical Bone Using Multifocus Pulse-Echo Ultrasound

Most bone loss during the development of osteoporosis occurs in cortical bone at the peripheral skeleton. Decreased cortical thickness (Ct.Th) and the prevalence of large pores at the tibia are associated with reduced bone strength at the hip. Ct.Th and cortical sound velocity, i.e., a surrogate marker for changes of cortical porosity (Ct.Po), are key biomarkers for the identification of patients at high fracture risk. In this study, we have developed a method using a conventional ultrasound array transducer to determine thickness (Ct.Th) and the compressional sound velocity propagating in the radial bone direction (Ct. ν₁₁ ) using a refraction-corrected multifocus imaging approach. The method was validated in-silico on porous bone plate models using a 2-D finite-difference time-domain method and ex vivo on plate-shaped plastic reference materials and on plate-shaped cortical bovine tibia samples. Plane-wave pulse-echo measurements provided reference values to assess precision and accuracy of our method. In-silico results revealed the necessity to account for inclination-dependent transmission losses at the bone surface. Moreover, the dependence of Ct. ν₁₁ on both porosity and pore density was observed. Ct.Th and Ct. ν₁₁ obtained ex vivo showed a high correlation ) with reference values. The ex-vivo accuracy and precision for Ct. ν₁₁ were 29.9 m/s and 0.94%, respectively, and those for Ct.Th were 0.04 mm and 1.09%, respectively. In conclusion, this numerical and experimental study demonstrates an accurate and precise estimation of Ct.Th and Ct. ν₁₁ . The developed multifocus technique may have high clinical potential to improve fracture risk prediction using noninvasive and nonionizing conventional ultrasound technology with image guidance.

Artificial neural network to estimate micro-architectural properties of cortical bone using ultrasonic attenuation: A 2-D numerical study

The goal of this study is to estimate micro-architectural parameters of cortical porosity such as pore diameter (φ), pore density (ρ) and porosity (ν) of cortical bone from ultrasound frequency dependent attenuation using an artificial neural network (ANN). First, heterogeneous structures with controlled pore diameters and pore densities (mono-disperse) were generated, to mimic simplified structure of cortical bone. Then, more realistic structures were obtained from high resolution CT scans of human cortical bone. 2-D finite-difference time-domain simulations were conducted to calculate the frequency-dependent attenuation in the 1-8 MHz range. An ANN was then trained with the ultrasonic attenuation at different frequencies as the input feature vectors while the output was set as the micro-architectural parameters (pore diameter, pore density and porosity). The ANN is composed of three fully connected dense layers with 24, 12 and 6 neurons, connected to the output layer. The dataset was trained over 6000 epochs with a batch size of 16. The trained ANN exhibits the ability to predict the micro-architectural parameters with high accuracy and low losses. ANN approaches could potentially be used as a tool to help inform physics-based modelling of ultrasound propagation in complex media such as cortical bone. This will lead to the solution of inverse-problems to retrieve bone micro-architectural parameters from ultrasound measurements for the non-invasive diagnosis and monitoring osteoporosis.

Modeling ultrasound attenuation in porous structures with mono-disperse random pore distributions using the independent scattering approximation: a 2D simulation study

The validity of the independent scattering approximation (ISA) to predict the frequency dependent attenuation in 2D models of simplified structures of cortical bone is studied. Attenuation of plane waves at central frequencies ranging from 1 to 8 MHz propagating in structures with mono-disperse random pore distributions with pore diameter and pore density in the range of those of cortical bone are evaluated by finite difference time domain numerical simulations. An approach to assess the multiple scattering of waves in random media is discussed to determine the pore diameter ranges at which the ISA is applicable. A modified version of the ISA is proposed to more accurately predict the attenuation in porosity ranges where it would traditionally fail. The results show that the modified ISA can model the frequency-dependent attenuation of ultrasonic wave with pore diameter and density ranges comparable to those of cortical bone.

Predicting Structural Properties of Cortical Bone by Combining Ultrasonic Attenuation and an Artificial Neural Network (ANN): 2-D FDTD Study

The goal of this paper is to predict the micro-architectural parameters of cortical bone such as pore diameter ( ϕ ) and porosity ( v ) from ultrasound attenuation measurements using an artificial neural network (ANN). Slices from a 3-D CT scan of human femur are obtained. The micro-architectural parameters of porosity such as average pore size and porosity are calculated using image processing. When ultrasound waves propagate in porous structures, attenuation is observed due to scattering. Two-dimensional finite-difference time-domain simulations are carried out to obtain frequency dependent attenuation in those 2D structures. An artificial neural network (ANN) is then trained with the input feature vector as the frequency dependent attenuation and output as pore diameter ( ϕ ) and porosity ( v ) . The ANN is composed of one input layer, 3 hidden layers and one output layer, all of which are fully connected. 340 attenuation data sets were acquired and trained over 2000 epochs with a batch size of 32. Data was split into train, validation and test. It was observed that the ANN predicted the micro-architectural parameters of the cortical bone with high accuracies and low losses with a minimum R2 (goodness of fit) value of 0.95. ANN approaches could potentially help inform the solution of inverse-problems to retrieve bone porosity from ultrasound measurements. Ultimately, those inverse-problems could be used for the non-invasive diagnosis and monitoring of osteoporosis.

Inferring porosity from frequency dependent attenuation in cortical bone mimicking porous media

Osteoporosis affects porosity in cortical bone. Quantifying levels of osteoporosis by inferring the micro-architectural properties from ultrasonic wave attenuation in cortical bone has yet to be done. In this work we use a phenomenological, power law model to describe the frequency dependent attenuation in non-absorbing porous media mimicking a simplified cortical bone structure. We optimize this model to fit data generated using a finite-difference, time domain (FDTD) numerical simulation. Model parameters are estimated using an ordinary least squares (OLS) formulation of the inverse problem. With these we determine linear, functional relationships between the model parameter estimates and the micro-architectural parameters, pore density and pore diameter. These relationships allow us to infer ranges of porosity from simulated attenuation data. Repeating this process for attenuation data collected from cortical bone samples could allow one to characterize the micro-architectural properties of bone.

Influence of micro-structural parameters on apparent absorption coefficient in porous structures mimicking cortical bone

Ultrasound wave propagation in porous media is associated with energy loss and attenuation. Attenuation is caused by both scattering and absorption, and is influenced by the microstructure as well as the operating frequency. In the present simulation study, we calculate the attenuation coefficient in porous structures mimicking cortical bone both in presence and absence of absorption to isolate the effects of scattering and absorption on the total attenuation. A parameter called apparent absorptionα a p p . a b s is defined as the difference between the total attenuationα total and the attenuation exclusively due to scatteringα scat .α total , α scat andα app.abs are estimated in porous structures with varying pore diameters ( ϕ ∈ [ 40, 120 ] μ m ) and pore densities ( ρ ∈ [ 5, 25 ] pore / m m 2 ) at 5MHz and 8MHz. Results show that both scattering and absorption contribute to the total attenuation. They also illustrate that, although absorption only occurs in the solid matrix, the apparent absorptionα a p p . a b s is a function of porosity, presumably due to the presence of multiple scattering. For large values of k ϕ , an increase in pore size or density does not lead to increase inα s c a t and only results in an increase of the total attenuation as a result of increase inα app.abs. . On the other hand, in low/intermediate scattering regimes ( k ϕ ≤ 1 ) , an increase in either pore size or pore density results in increase inα scat whileα app.abs remains constant.