The effect of boundary conditions on guided wave propagation in two-dimensional models of healing bone

Vavva, Maria G.; Protopappas, Vasilios C.; Gergidis, Leonidas N.; Charalambopoulos, A.; Fotiadis, Dimitrios I.; Polyzos, Demos

doi:10.1016/j.ultras.2008.04.013

Cited by 38 publications

(13 citation statements)

References 30 publications

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“…While these simplifications were made purposefully to follow the systematic approach of investigating effects caused by individual properties, we expect the observed effects of changing cortical thickness to hold for real bone—even though they could possibly be overshadowed by other effects. For example, pores and bone marrow have substantially different acoustic properties compared to air and are known to modify wave propagation [15,24,27,43]. These effects, as well as the effects arising from anisotropy [44,45] or a more realistic shape [15,46], should be investigated in future work.…”

Section: Resultsmentioning

confidence: 99%

Sensitivity of low-frequency axial transmission acoustics to axially and azimuthally varying cortical thickness: A phantom-based study

2019

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Purpose Cortical thickness (cTh) is one of the main factors determining a bone’s mechanical properties, and its quantification is therefore critical for understanding and monitoring bone pathologies such as osteoporosis. Axial quantitative acoustics (ax-QA) offers a non-radiative, non-invasive method to measure cTh. Even though previous works have ascertained ax-QA’s ability to measure azimuthally varying cTh, the effect of axially varying cTh remains unclear. Furthermore, previous experiments and theoretical predictions indicate that measurement of the fundamental flexural mode at low frequencies in the kHz range could increase sensitivity to cTh. However, due to the associated long wavelengths, the approximation of bone geometry as a tube could break down at such frequencies. The presented study therefore investigates a) the sensitivity of ax-QA measurements to cTh in the kHz-regime, b) the applicability of tube theory in this regime, and c) the effect of varying cTh along the long axis on the bone. Materials and methods Axial-transmission acoustic measurements were performed at 3kHz on 14 bone phantoms with a femur-like cross-section and a) axially varying cortical thickness or b) axially and azimuthally varying cortical thickness (cTh-range: 1.5mm-7.5mm). Experimental results were compared to theoretical predictions based on an exact elastic tube theory. Results and discussion Phase velocity measurements using low-frequency ax-QA exhibited a high sensitivity to local cTh less than 4mm, albeit with a complex, not yet understood pattern. Tube theory failed to predict the wave’s behavior in the kHz range, indicating that due to the corresponding long wavelengths the bone can no longer be approximated by a tube, thus requiring more faithful modelling of the bone geometry. The fact that results from both types of phantoms were similar (Pearson correlation coefficient: 0.94) further indicates that the slowly varying cTh along the bone’s long axis did not strongly affect wave propagation as measured by ax-QA measurements.

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Section: Resultsmentioning

confidence: 99%

Sensitivity of low-frequency axial transmission acoustics to axially and azimuthally varying cortical thickness: A phantom-based study

2019

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“…In recent years, gradient elastic theory has gained significant interest in deriving analytical and numerical solutions of ultrasound propagation through cortical bone [23], [24]. Although classical elastic theory has been extensively used in bone studies it cannot provide a complete description of dynamic mechanical behavior since it is associated with concepts of homogeneity and locality of stresses.…”

Section: Gradient Elasticity and Cortical Bonementioning

confidence: 99%

“…From the application point of view, the investigation of guided waves in elastic structures mimicking -although via a simple primitive 2-D model -the cortical bones, have attracted the interest of the authors in [23] and [24], where it has been shown that gradient elasticity can provide supplementary information to better understand guided waves in porous media physically mimicking the bone structure. These works have been suggested as alternative approaches to a series of primitive studies ( [25], [26], [27]) aiming at investigating non destructive monitoring of biomedical systems within the framework of classical elastic wave propagation.…”

Section: Introductionmentioning

confidence: 99%

On the gradient elastic wave propagation in cylindrical waveguides with microstructure

Charalambopoulos

Gergidis

Kartalos

2012

Composites Part B: Engineering

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The present work describes the development of a complete theoretical framework of wave propagation in cylindrical waveguides possessing microstructure. In parallel, a thorough investigation of the full 3-D model of wave propagation in cylinders is presented. The first step is the spectral decomposition of the boundary value problem emerging via wave propagation analysis. The spectral representation of the specific gradient elasticity problem reflects the ability to construct all the possible propagating modes in cylindrical geometry. Several byproducts arise along the present work, which constitute generalizations of well known important features of classical elasticity and are indispensable for modeling the gradient elasticity problem. We note the construction of the set of dyadic Navier eigenfunctions which constitute the generalization of the Navier eigenvectors. The restriction of the Navier eigendyadics on cylindrical surfaces gives birth to the dyadic cylindrical harmonics, which constitute the generalization of the well known vector harmonics. This set is also a basis in the sense that the trace of every dyadic field on a cylindrical surface can be represented as a countable superposition of dyadic cylindrical harmonics. The method aims at providing the necessary theoretical establishment for the determination of the dispersion curves emerging in cortical bones.

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“…Vibration analysis utilizes an impact hammer to generate vibrational waves and measure the resonant frequencies to assess the mechanical properties of long bones (Sonstegard and Matthews 1976) (Jurist 1970, Steele, et al 1988, Van der Perre, et al 1983). In QUS analysis, guided waves from axial transmission measurement along the bone surface has been used to determine the material properties of long bones (Gerlanc, et al 1975, Siegel, et al 1958) (Bossy, et al 2004, Lee and Yoon 2004, Lowet and Van der Perre 1996, Moilanen 2008, Mole and Ganesan 2010, Rose 2004, Ta, et al 2009, Vavva, et al 2008, Viktorov 1970). Recent progress in QUS, has focused on the extraction of the dispersion curves and to exploit multimodal waveguide of long bones to extract cortical thickness and stiffness (Xu, et al 2016, Vallet, et al 2016).…”

Section: Introductionmentioning

confidence: 99%

Ultrasound Characterization of Bone Demineralization Using a Support Vector Machine

Denis

Wan

Fatemi

et al. 2018

Ultrasound in Medicine & Biology

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We propose an ultrasound-guided remote measurement technique, utilizing an acoustic radiation force beam as our excitation source and a receiving hydrophone, to assess non-invasively a bone's mechanical properties. Features, such as velocity, were extracted from the acoustic pressure received from the bone surface. The typical velocity of an intact bone (3540 m/s) was higher in comparison to that of a demineralized bone (2231 m/s). According to the receiver operating characteristic curve, the optimal velocity cutoff value of ≥3096 m/s yields 80% sensitivity and 82.61% specificity between intact and demineralized bone. Utilizing a support vector machine, the hours of bone demineralization were successfully classified with maximum accuracy >80% using 18% training data. The results indicate the potential application of our proposed technique and support vector machine for monitoring bone mechanical properties.

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The effect of boundary conditions on guided wave propagation in two-dimensional models of healing bone

Cited by 38 publications

References 30 publications

Sensitivity of low-frequency axial transmission acoustics to axially and azimuthally varying cortical thickness: A phantom-based study

Sensitivity of low-frequency axial transmission acoustics to axially and azimuthally varying cortical thickness: A phantom-based study

On the gradient elastic wave propagation in cylindrical waveguides with microstructure

Ultrasound Characterization of Bone Demineralization Using a Support Vector Machine

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