Strain-dependence of ultrasound speed in cancellous bone was determined by applying a range of uniaxial compressive strains in the elasticity region, in a single direction, parallel or perpendicular to the propagating wave. Compressive strain modulated the ultrasound speed significantly. The decrease of ultrasound speed was found to change linearly as a function of strain. The changes of broadband ultrasound attenuation were also determined for the two dilatational waves (parallel or perpendicular to the strain). They do not follow linear relation or constant sign of change with strain for the examined specimens. Considerable possibilities open up for using developments in acoustoelasticity for nondestructive ultrasonic techniques.
1. Introduction
Acoustoelasticity addresses the propagation of elastic waves in initially stressed bodies and includes practical issues—determination of the physicomechanical properties and the stress–strain state of solids. Considerable possibilities open up for using developments in acoustoelasticity to create nondestructive ultrasonic techniques.
Ultrasound elastography is a method to assess the mechanical properties of tissues, by applying stress and detecting tissue displacement using ultrasound. As a noninvasive method, it has been successfully used in detecting lesions and pathological changes in a variety of tissues or organs, including skeletal muscle, cardiac muscle, liver, prostate, breast, and thyroid (Drakonaki et al., 2012). In elastography, ultrasound speed is traditionally assumed to be independent of the mechanical deformation of the tissue. This is generally a valid assumption in soft tissues as their composition, density, and modulus of elasticity may not change significantly during the mechanical compression. These assumptions are not valid with bone.
In this work, the change of the velocity of ultrasonic wave propagation through cancellous bone, subjected to uniaxial compression, was studied. The expressions of Hughes and Kelly (1953) were used to calculate the elastic constants of the third order of cancellous bone. The change of broadband ultrasonic attenuation (BUA) during compression was also determined.
2. Materials and methods
The expressions of Hughes and Kelly (1953) for the relative changes in wave speed with respect to the unstressed state, propagating in a solid stressed by a uniaxial stress, in a direction parallel or perpendicular to the propagating wave, using Murnaghan's theory of finite deformations, are the following [Egle and Bray (1976)]:
where Vax, Vlat are speeds of dilatational waves propagating in directions parallel or perpendicular to the applied strain, respectively; are Lamé elastic constants (or second order); l, m are Murnaghan's elastic constants of third order; and ν is Poisson's ratio.
A number of 15 cubic specimens of tibial bovine cancellous bone with a side length of 22–25 mm were prepared (Deligianni et al., 2007). Measurement of ultrasound propagation speed and BUA through the specimens was performed before and after application of compressive strain. Unfocused immersion transducers (Panametrics, V303, d = 0.5 in., center frequency 1 MHz) have been used, connected to an ultrasonic pulser receiver (USD 10NF, Krautkraemer, Germany). Radio frequency (RF) signals were digitized at 35 MHz. The −20 dB frequency bandwidths were 0.24–1.24 MHz. A standard through-transmission substitution method was used to measure ultrasonic speed of sound and BUA (Deligianni et al., 2007).
For the determination of ultrasonic parameters in the stressed state, the specimens, after two cycles of preconditioning well below the elastic limit (stress 70 MPa), were loaded uniaxially in compression up to 2% strain (elastic area). The strain was applied incrementally with a custom made device. When the specimens were compressed in a direction parallel to the propagating wave, the ultrasound transducers were mounted on the jigs of the compression testing device. The strain was measured with a linear variable displacement transformer extensometer. Ultrasound was propagated along or perpendicularly to the applied strain (Fig. 1).
(Color online) Directions of propagation of ultrasound waves through cancellous bone specimens and of compressive loading (εax, compressive strain; εlat, tensile strain).
(Color online) Directions of propagation of ultrasound waves through cancellous bone specimens and of compressive loading (εax, compressive strain; εlat, tensile strain).
From the experiments, the experimental values of (dVax/V0ax)/dεax and (dVlat/V0lat)/dεlat [Eqs. (1a), (1b)] were calculated. The Lamé elastic constants for each specimen were measured in standard compression tests, after the ultrasound propagation tests. The Murnaghan elastic constants m and l were calculated from Eqs. (1a), (1b). The ultrasound speed and BUA were measured again after the unloading of each specimen, and their values were compared to the corresponding properties of the unloaded specimen, in order to ensure that the elastic limit has not been exceeded.
3. Results
A statistically significant (p < 0.05) strain-dependent variation in sound speed was observed during uniaxial straining, in both directions of ultrasound propagation. Compression decreases sound speed, whereas tension increases it. The change of ultrasound speed was found to be linear with strain, either compressive or tensile. The mean decrease of SoS with compression was 20% and mean increase with tension 11%, at the maximum strain applied. A plot of the relative changes in the speed of sound as a function of axial strain (compressive) and lateral strain (tensile) for the waves propagating parallel or perpendicular to the load axis is shown in Fig. 2. As predicted by the theoretical model of Hughes and Kelly (1953), the wave speed changes are linear functions of the strain. From the experimental curves, the third order Murnaghan elastic material constants were determined (Table 1).
Relative changes in the speed of sound as a function of axial strain (compressive) and lateral strain (tensile).
Relative changes in the speed of sound as a function of axial strain (compressive) and lateral strain (tensile).
Lamé and third order Murnaghan elastic material constants determined from the experimental data, with Poisson ratio ν = 0.4.
Sample No. . | E (MPa) . | λ (MPa) . | μ (MPa) . | l (MPa) . | m (MPa) . |
---|---|---|---|---|---|
1 | 118 | 33 | 49.5 | −406.7 | −84.8 |
2 | 558 | 155 | 232.5 | −1912.1 | −398.5 |
3 | 383 | 106.5 | 159.8 | −1314.3 | −273.9 |
4 | 1113 | 309.1 | 463.6 | −3813.4 | −794.8 |
5 | 1452 | 403.4 | 605.1 | −4977.3 | −1037.4 |
6 | 977 | 271.5 | 407.3 | −3350.0 | −698.2 |
7 | 224 | 62.2 | 93.3 | −767.7 | −160.5 |
8 | 708 | 196.7 | 295.5 | −2426.5 | −505.7 |
9 | 1205 | 334.7 | 502.1 | −4129.8 | −860.7 |
10 | 886 | 246.1 | 369.2 | −3036.5 | −632.9 |
11 | 986 | 273.9 | 410.8 | −3379.2 | −704.3 |
12 | 607 | 168.6 | 252.9 | −2080.3 | −433.6 |
13 | 905 | 251.4 | 377.1 | −3101.6 | −646.4 |
14 | 1315 | 365.3 | 547.9 | −4506.8 | −939.3 |
15 | 541 | 150.3 | 225.4 | −1854.1 | −386.4 |
Sample No. . | E (MPa) . | λ (MPa) . | μ (MPa) . | l (MPa) . | m (MPa) . |
---|---|---|---|---|---|
1 | 118 | 33 | 49.5 | −406.7 | −84.8 |
2 | 558 | 155 | 232.5 | −1912.1 | −398.5 |
3 | 383 | 106.5 | 159.8 | −1314.3 | −273.9 |
4 | 1113 | 309.1 | 463.6 | −3813.4 | −794.8 |
5 | 1452 | 403.4 | 605.1 | −4977.3 | −1037.4 |
6 | 977 | 271.5 | 407.3 | −3350.0 | −698.2 |
7 | 224 | 62.2 | 93.3 | −767.7 | −160.5 |
8 | 708 | 196.7 | 295.5 | −2426.5 | −505.7 |
9 | 1205 | 334.7 | 502.1 | −4129.8 | −860.7 |
10 | 886 | 246.1 | 369.2 | −3036.5 | −632.9 |
11 | 986 | 273.9 | 410.8 | −3379.2 | −704.3 |
12 | 607 | 168.6 | 252.9 | −2080.3 | −433.6 |
13 | 905 | 251.4 | 377.1 | −3101.6 | −646.4 |
14 | 1315 | 365.3 | 547.9 | −4506.8 | −939.3 |
15 | 541 | 150.3 | 225.4 | −1854.1 | −386.4 |
The changes of BUA were also determined for two waves measured. They do not follow a linear relation or constant sign of change with strain for the tested specimens (Fig. 3).
Relative changes in the BUA as a function of axial strain (compressive) and lateral strain (tensile).
Relative changes in the BUA as a function of axial strain (compressive) and lateral strain (tensile).
4. Discussion
To our best knowledge, this study represents the first investigation of the ultrasonic wave propagation in prestressed bone. Similar studies have been reported in soft tissues, articular cartilage (Lötjönen et al., 2009), and technical materials, e.g., steel (Egle and Bray, 1976) and rocks (Bourbié et al., 1986). Most of this work was concentrated on the effect of stress on the velocities of different types of ultrasonic waves in homogeneous materials.
Biot (1940), in a plane strain problem with initial tension, concluded that the dilatational waves are not affected by the initial tension, but only the rotational ones. Subsequent findings contradicted this result. A number of models have been developed to study ultrasonic wave propagation in prestressed materials. Coulson (1955) showed that the wave velocity of a longitudinal vibration of a bar in tension is given by
where V0 is the velocity under no tension, T0 is the initial tension, and E is Young's modulus. Tang (1967), assuming homogeneous and isotropic material when it is stress free, gave
as the velocities of the dilatational wave in the directions parallel or perpendicular to the direction of strain, respectively; T is tension and ρ is the bone density. When it is in the deformed state of equilibrium the material is, strictly speaking, no longer isotropic. The induced anisotropy has not been taken into account in the above results. Both of the above models underestimated the change of the ultrasound speed in cancellous bone. They predicted decrease of the speed in the stressed material at 0.8%–2% in comparison to that in the unstressed material, whereas the experimental results of the current study displayed much higher decrease.
In articular cartilage, unlike soft biological tissues, ultrasound speed was also found to be strain-dependent and to decrease with strain, but nonlinearly (Lötjönen et al., 2009). The origin of load-related variation in ultrasound speed in articular cartilage was attributed to compositional changes and, particularly, to collagen orientation. In cancellous bone the main factor that influences ultrasound speed is the tissue density. Further investigation is necessary to determine the factors responsible with the variation in ultrasound speed under mechanical loading and explain the corresponding mechanisms.
In this work, two assumptions were made. The first is that possible longitudinal/transversal mode conversions at the solid-fluid interfaces were neglected (Deligianni et al., 2007). However, the shear waves should be considered and their measured values used in Hughes and Kelly (1953) expressions, in order to calculate third order elasticities of bone. The second assumption, that tibial bovine bone can be considered as an isotropic material with respect to ultrasonic properties or when its homogeneities are distributed isotropically, is justified by determination of ultrasonic properties in Deligianni et al. (2007). Values of yield strain for bovine bone, found in the literature, are 1.0%–1.5%. Our measured strain values were higher, but the specimens were checked for remaining strains after unloading.
This study addressed how straining affects ultrasonic properties of bovine cancellous bone. Although the results of the study for bovine bone are not directly applicable to human bone, the usefulness of the experimental results for the understanding of the ultrasonic behavior of cancellous bone may be significant. Understanding of the ultrasound propagation through bone can improve ultrasonic methods of estimation. Moreover, the study of variation of ultrasound speed with strain can possibly be applicable for diagnostic purposes, by determining higher order elastic constants and estimating bone fracture risk directly and not through the bone density estimation (Destrade et al., 2010).