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Pakula M. What kind of waves are measured in trabecular bone? ULTRASONICS 2022; 123:106692. [PMID: 35176689 DOI: 10.1016/j.ultras.2022.106692] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2021] [Revised: 01/06/2022] [Accepted: 01/17/2022] [Indexed: 06/14/2023]
Abstract
The paper discusses the fundamental mechanisms underlying the interaction between ultrasound and trabecular bone, which is considered a two-phase material. When fluid-saturated cancellous bone is interrogated by ultrasound, in some cases, one or two wave modes are observed. Many authors claim that these waves correspond to the fast and slow waves predicted by Biot's theory of elastic wave propagation in fluid-saturated porous media. Within our analysis of the physical conditions, predictions of the existing two-phase models of the propagation of ultrasonic waves in the material as well as numerical simulations for fluid-saturated trabecular bone were performed. On the basis of the theoretical results (from numerical studies) and arguments presented in this paper, we aimed to answer the question of whether two waves observed in ultrasonic wave transmission studies can be interpreted as the fast and slow waves predicted by Biot's theory.
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Affiliation(s)
- Michal Pakula
- Faculty of Mechatronics, Kazimierz Wielki University in Bydgoszcz, Poland.
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Wear KA. Mechanisms of Interaction of Ultrasound With Cancellous Bone: A Review. IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL 2020; 67:454-482. [PMID: 31634127 PMCID: PMC7050438 DOI: 10.1109/tuffc.2019.2947755] [Citation(s) in RCA: 41] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/17/2023]
Abstract
Ultrasound is now a clinically accepted modality in the management of osteoporosis. The most common commercial clinical devices assess fracture risk from measurements of attenuation and sound speed in cancellous bone. This review discusses fundamental mechanisms underlying the interaction between ultrasound and cancellous bone. Because of its two-phase structure (mineralized trabecular network embedded in soft tissue-marrow), its anisotropy, and its inhomogeneity, cancellous bone is more difficult to characterize than most soft tissues. Experimental data for the dependencies of attenuation, sound speed, dispersion, and scattering on ultrasound frequency, bone mineral density, composition, microstructure, and mechanical properties are presented. The relative roles of absorption, scattering, and phase cancellation in determining attenuation measurements in vitro and in vivo are delineated. Common speed of sound metrics, which entail measurements of transit times of pulse leading edges (to avoid multipath interference), are greatly influenced by attenuation, dispersion, and system properties, including center frequency and bandwidth. However, a theoretical model has been shown to be effective for correction for these confounding factors in vitro and in vivo. Theoretical and phantom models are presented to elucidate why cancellous bone exhibits negative dispersion, unlike soft tissue, which exhibits positive dispersion. Signal processing methods are presented for separating "fast" and "slow" waves (predicted by poroelasticity theory and supported in cancellous bone) even when the two waves overlap in time and frequency domains. Models to explain dependencies of scattering on frequency and mean trabecular thickness are presented and compared with measurements. Anisotropy, the effect of the fluid filler medium (marrow in vivo or water in vitro), phantoms, computational modeling of ultrasound propagation, acoustic microscopy, and nonlinear properties in cancellous bone are also discussed.
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The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J Biomech 2018; 79:54-57. [PMID: 30122518 DOI: 10.1016/j.jbiomech.2018.07.048] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2018] [Revised: 07/19/2018] [Accepted: 07/31/2018] [Indexed: 11/23/2022]
Abstract
Cancellous bone is a highly porous material, and two types of waves, fast and slow, are observed when ultrasound is used for detecting bone diseases. There are several possible stimuli for bone remodelling processes, including bone fluid flow, streaming potential, and piezoelectricity. Poroelasticity has been widely used for elucidating the bone fluid flow phenomenon, but the combination of poroelasticity with charge density has not been introduced. Theoretically, general poroelasticity with a varying charge density is employed for determining the relationship between wave velocity and attenuation with charge density. Fast wave velocity and attenuation are affected by porosity as well as charge density; however, for a slow wave, both slow wave velocity and attenuation are not as sensitive to the effect of charge density as they are for a fast wave. Thus, employing human femoral data, we conclude that charged ions gather on trabecular struts, and the fast wave, which moves along the trabecular struts, is significantly affected by charge density.
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Wear KA. Nonlinear attenuation and dispersion in human calcaneus in vitro: statistical validation and relationships to microarchitecture. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2015; 137:1126-1133. [PMID: 25786928 PMCID: PMC9204557 DOI: 10.1121/1.4908310] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
Through-transmission measurements were performed on 30 human calcaneus samples in vitro. Nonlinear attenuation and dispersion measurements were investigated by estimating 95% confidence intervals of coefficients of polynomial expansions of log magnitude and phase of transmission coefficients. Bone mineral density (BMD) was measured with dual x-ray absorptiometry. Microarchitecture was measured with microcomputed tomography. Statistically significant nonlinear attenuation and nonzero dispersion were confirmed for a clinical bandwidth of 300-750 kHz in 40%-43% of bone samples. The mean linear coefficient for attenuation was 10.3 dB/cm MHz [95% confidence interval (CI): 9.0-11.6 dB/cm MHz]. The mean quadratic coefficient for attenuation was 1.6 dB/cm MHz(2) (95% CI: 0.4-2.8 dB/cm MHz(2)). Nonlinear attenuation provided little information regarding BMD or microarchitecture. The quadratic coefficient for phase (which is related to dispersion) showed moderate correlations with BMD (r = -0.65; 95% CI: -0.82 to -0.36), bone surface-to-volume ratio (r = 0.47; 95% CI: 0.12-0.72) and trabecular thickness (r = -0.40; 95% CI: -0.67 to -0.03). Dispersion was proportional to bone volume fraction raised to an exponent of 2.1 ± 0.2, which is similar to the value for parallel nylon-wire phantoms (2.4 ± 0.2) and supports a multiple-scattering model for dispersion.
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Affiliation(s)
- Keith A Wear
- United States Food and Drug Administration, Center for Devices and Radiological Health, Silver Spring, Maryland 20993
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Nagatani Y, Tachibana RO. Multichannel instantaneous frequency analysis of ultrasound propagating in cancellous bone. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2014; 135:1197-206. [PMID: 24606262 DOI: 10.1121/1.4864464] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
An ultrasonic pulse propagating in cancellous bone can be separated into two waves depending on the condition of the specimen. These two waves, which are called the fast wave and the slow wave, provide important information for the diagnosis of osteoporosis. The present study proposes to utilize a signal processing method that extracts the instantaneous frequency (IF) of waveforms from multiple spectral channels. The instantaneous frequency was expected to be able to show detailed time-frequency properties of ultrasonic waves being transmitted through cancellous bone. The employed method, termed the multichannel instantaneous frequency (MCIF) method, showed robustness against background noise as compared to the IF that was directly derived from the original waveform. The extracted IF revealed that the frequency of the fast wave was affected by both the propagation distance within the specimen and the bone density, independently. On the other hand, the alternation of the center frequency of the originally transmitted wave did not produce proportional changes in the extracted IF values of the fast waves, suggesting that the fast wave IF mainly reflected the thickness of the specimens. These findings may provide the possibility of obtaining a more precise diagnosis of osteoporosis.
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Affiliation(s)
- Yoshiki Nagatani
- Department of Electronics, Kobe City College of Technology, Gakuen-higashi-machi 8-3, Nishi-ku, Kobe 651-2194, Japan
| | - Ryosuke O Tachibana
- Department of Life Sciences, Graduate School of Arts and Sciences, The University of Tokyo, Tokyo 153-8902, Japan
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Wear KA. Estimation of fast and slow wave properties in cancellous bone using Prony's method and curve fitting. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2013; 133:2490-501. [PMID: 23556613 PMCID: PMC8243208 DOI: 10.1121/1.4792935] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
The presence of two longitudinal waves in poroelastic media is predicted by Biot's theory and has been confirmed experimentally in through-transmission measurements in cancellous bone. Estimation of attenuation coefficients and velocities of the two waves is challenging when the two waves overlap in time. The modified least squares Prony's (MLSP) method in conjuction with curve-fitting (MLSP + CF) is tested using simulations based on published values for fast and slow wave attenuation coefficients and velocities in cancellous bone from several studies in bovine femur, human femur, and human calcaneus. The search algorithm is accelerated by exploiting correlations among search parameters. The performance of the algorithm is evaluated as a function of signal-to-noise ratio (SNR). For a typical experimental SNR (40 dB), the root-mean-square errors (RMSEs) for one example (human femur) with fast and slow waves separated by approximately half of a pulse duration were 1 m/s (slow wave velocity), 4 m/s (fast wave velocity), 0.4 dB/cm MHz (slow wave attenuation slope), and 1.7 dB/cm MHz (fast wave attenuation slope). The MLSP + CF method is fast (requiring less than 2 s at SNR = 40 dB on a consumer-grade notebook computer) and is flexible with respect to the functional form of the parametric model for the transmission coefficient. The MLSP + CF method provides sufficient accuracy and precision for many applications such that experimental error is a greater limiting factor than estimation error.
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Affiliation(s)
- Keith A Wear
- U.S. Food and Drug Administration, 10903 New Hampshire Avenue, Building 62, Room 3108, Silver Spring, Maryland 20993-0002, USA.
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HOSOKAWA ATSUSHI. NUMERICAL SIMULATIONS OF CHANGE IN TRABECULAR STRUCTURE DUE TO BONE REMODELING UNDER ULTRASOUND PROPAGATION. J MECH MED BIOL 2013. [DOI: 10.1142/s0219519413500036] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Bone remodeling is defined as the coupling of bone formation and resorption on the bone surface. Numerical simulations of the remodeling in cancellous bone were performed to reproduce the change in the trabecular structure. Assuming that the formation/resorption in cancellous bone could be generated on the trabecular surface, where the local stress under the mechanical load was larger/smaller than the averaged stress on the surrounding surface, voxel trabecular elements in a numerical model of bovine cancellous bone were added/removed. An ultrasound continuous wave in the frequency range 0.1–1.0 MHz was applied as the mechanical load, and then, the local stress was analyzed using a finite-difference time-domain (FDTD) method. Using the remodeling simulations, both changes in the trabecular structure could be reproduced with decreasing and increasing porosity. In changes, the trabecular elements and the pore spaces became strongly oriented in the direction of ultrasound propagation. In addition, the remodeling simulations indicated that both bone formation and resorption lessened as the frequency increased.
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Affiliation(s)
- ATSUSHI HOSOKAWA
- Department of Electrical and Computer Engineering, Akashi National College of Technology, 679-3 Nishioka, Uozumi, Akashi 674-8501, Japan
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Wear KA, Nagaraja S, Dreher ML, Gibson SL. Relationships of quantitative ultrasound parameters with cancellous bone microstructure in human calcaneus in vitro. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2012; 131:1605-12. [PMID: 22352530 PMCID: PMC6931152 DOI: 10.1121/1.3672701] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
Ultrasound parameters (attenuation, phase velocity, and backscatter), bone mineral density (BMD), and microarchitectural features were measured on 29 human cancellous calcaneus samples in vitro. Regression analysis was performed to predict ultrasound parameters from BMD and microarchitectural features. The best univariate predictors of the ultrasound parameters were the indexes of bone quantity: BMD and bone volume fraction (BV/TV). The most predictive univariate models for attenuation, phase velocity, and backscatter coefficient yielded adjusted squared correlation coefficients of 0.69-0.73. Multiple regression models yielded adjusted correlation coefficients of 0.74-0.83. Therefore attenuation, phase velocity, and backscatter are primarily determined by bone quantity, but multiple regression models based on bone quantity plus microarchitectural features achieve slightly better predictive performance than models based on bone quantity alone.
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Affiliation(s)
- Keith A Wear
- U.S. Food and Drug Administration, Center for Devices and Radiological Health, 10903 New Hampshire Boulevard, Silver Spring, Maryland 20993, USA.
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Cardoso L, Cowin SC. Role of structural anisotropy of biological tissues in poroelastic wave propagation. MECHANICS OF MATERIALS : AN INTERNATIONAL JOURNAL 2012; 44:174-188. [PMID: 22162897 PMCID: PMC3233242 DOI: 10.1016/j.mechmat.2011.08.007] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
Ultrasound waves have a broad range of clinical applications as a non-destructive testing approach in imaging and in the diagnoses of medical conditions. Generally, biological tissues are modeled as an homogenized equivalent medium with an apparent density through which a single wave propagates. Only the first wave arriving at the ultrasound probe is used for the measurement of the speed of sound. However, the existence of a second wave in tissues such as cancellous bone has been reported and its existence is an unequivocal signature of Biot type poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as density, a fabric-dependent anisotropic poroelastic ultrasound (PEU) propagation theory was recently developed. Key to this development was the inclusion of the fabric tensor - a quantitative stereological measure of the degree of structural anisotropy of bone - into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of waves in several soft and hard tissues. It was found that collagen fibers in soft tissues and the mineralized matrix in hard tissues are responsible for the anisotropy of the solid tissue constituent through the fabric tensor in the model.
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Affiliation(s)
- Luis Cardoso
- The Department of Biomedical Engineering, The City University of New York, New York, NY 10031
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Cardoso L, Cowin SC. Fabric dependence of quasi-waves in anisotropic porous media. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2011; 129:3302-16. [PMID: 21568431 PMCID: PMC3115277 DOI: 10.1121/1.3557032] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2010] [Revised: 01/14/2011] [Accepted: 01/19/2011] [Indexed: 05/20/2023]
Abstract
Assessment of bone loss and osteoporosis by ultrasound systems is based on the speed of sound and broadband ultrasound attenuation of a single wave. However, the existence of a second wave in cancellous bone has been reported and its existence is an unequivocal signature of poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as bone mineral density (BMD), a fabric-dependent anisotropic poroelastic wave propagation theory was recently developed for pure wave modes propagating along a plane of symmetry in an anisotropic medium. Key to this development was the inclusion of the fabric tensor--a quantitative stereological measure of the degree of structural anisotropy of bone--into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of mixed wave modes along an arbitrary direction in anisotropic porous media called quasi-waves. It was found that differences between phase and group velocities are due to the anisotropy of the bone microarchitecture, and that the experimental wave velocities are more accurately predicted by the poroelastic model when the fabric tensor variable is taken into account. This poroelastic wave propagation theory represents an alternative for bone quality assessment beyond BMD.
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Affiliation(s)
- Luis Cardoso
- The Department of Biomedical Engineering, The City University of New York, New York, New York 10031, USA.
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Cowin SC, Cardoso L. Fabric dependence of wave propagation in anisotropic porous media. Biomech Model Mechanobiol 2011; 10:39-65. [PMID: 20461539 PMCID: PMC3393603 DOI: 10.1007/s10237-010-0217-7] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2010] [Accepted: 04/08/2010] [Indexed: 10/19/2022]
Abstract
Current diagnosis of bone loss and osteoporosis is based on the measurement of the bone mineral density (BMD) or the apparent mass density. Unfortunately, in most clinical ultrasound densitometers: 1) measurements are often performed in a single anatomical direction, 2) only the first wave arriving to the ultrasound probe is characterized, and 3) the analysis of bone status is based on empirical relationships between measurable quantities such as speed of sound (SOS) and broadband ultrasound attenuation (BUA) and the density of the porous medium. However, the existence of a second wave in cancellous bone has been reported, which is an unequivocal signature of poroelastic media, as predicted by Biot's poroelastic wave propagation theory. In this paper, the governing equations for wave motion in the linear theory of anisotropic poroelastic materials are developed and extended to include the dependence of the constitutive relations upon fabric-a quantitative stereological measure of the degree of structural anisotropy in the pore architecture of a porous medium. This fabric-dependent anisotropic poroelastic approach is a theoretical framework to describe the microarchitectural-dependent relationship between measurable wave properties and the elastic constants of trabecular bone, and thus represents an alternative for bone quality assessment beyond BMD alone.
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Affiliation(s)
- Stephen C Cowin
- The New York Center for Biomedical Engineering, Departments of Biomedical and Mechanical Engineering, School of Engineering of The City College and Graduate School of The City University of New York, New York, NY 10031, USA.
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Wear KA. Cancellous bone analysis with modified least squares Prony's method and chirp filter: phantom experiments and simulation. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2010; 128:2191-203. [PMID: 20968389 PMCID: PMC9130964 DOI: 10.1121/1.3478779] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
The presence of two longitudinal waves in porous media is predicted by Biot's theory and has been confirmed experimentally in cancellous bone. When cancellous bone samples are interrogated in through-transmission, these two waves can overlap in time. Previously, the Modified Least-Squares Prony's (MLSP) method was validated for estimation of amplitudes, attenuation coefficients, and phase velocities of fast and slow waves, but tended to overestimate phase velocities by up to about 5%. In the present paper, a pre-processing chirp filter to mitigate the phase velocity bias is derived. The MLSP/chirp filter (MLSPCF) method was tested for decomposition of a 500 kHz-center-frequency signal containing two overlapping components: one passing through a low-density-polyethylene plate (fast wave) and another passing through a cancellous-bone-mimicking phantom material (slow wave). The chirp filter reduced phase velocity bias from 100 m/s (5.1%) to 69 m/s (3.5%) (fast wave) and from 29 m/s (1.9%) to 10 m/s (0.7%) (slow wave). Similar improvements were found for 1) measurements in polycarbonate (fast wave) and a cancellous-bone-mimicking phantom (slow wave), and 2) a simulation based on parameters mimicking bovine cancellous bone. The MLSPCF method did not offer consistent improvement in estimates of attenuation coefficient or amplitude.
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Affiliation(s)
- Keith A Wear
- Center for Devices and Radiological Health, U.S. Food and Drug Administration, Room 3108, Building 62, 10903 New Hampshire Boulevard, Silver Spring, Maryland 20993, USA.
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Wear KA. Frequency dependence of average phase shift from human calcaneus in vitro. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2009; 126:3291-300. [PMID: 20000943 DOI: 10.1121/1.3257550] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
If dispersion in a medium is weak and approximately linear with frequency (over the experimental band of frequencies), then it can be shown that the constant term in a polynomial representation of phase shift as a function of frequency can produce errors in measurements of phase-velocity differences in through-transmission, substitution experiments. A method for suppressing the effects of the constant phase shift in the context of the single-wave-model was tested on measurements from 30 cancellous human calcaneus samples in vitro. Without adjustment for constant phase shifts, the estimated phase velocity at 500 kHz was 1516+/-6 m/s (mean+/-standard error), and the estimated dispersion was -24+/-4 m/s MHz (mean+/-standard error). With adjustment for constant phase shifts, the estimated mean velocity decreased by 4-9 m/s, and the estimated magnitude of mean dispersion decreased by 50%-100%. The average correlation coefficient between the measured attenuation coefficient and frequency was 0.997+/-0.0026 (mean+/-standard deviation), suggesting that the signal for each sample was dominated by one wave. A single-wave, linearly dispersive model conformed to measured complex transfer functions from the 30 cancellous-bone samples with an average root-mean-square error of 1.9%+/-1.0%.
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Affiliation(s)
- Keith A Wear
- Center for Devices and Radiological Health, U. S. Food and Drug Administration, Silver Spring, Maryland 20993, USA.
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