Ultrasonic attenuation in trabecular bone: Theoretical approach and experimental measurement
Abderrazek Bennamane, Tarek Boutkedjirt
Université des Sciences et de la Technologie Houari Boumédiène, USTHB, Faculté de Physique, Laboratoire de Physique des Matériaux,
BP 32, ElAlia, DZ-16111 Algiers, Algeria
Abstract— The aim of this paper is to determine how different parameters such as porosity, scatter size and frequency affect the ultrasonic attenuation process during the acoustic wave propagation through the trabecular bone. In this study, a theoretical model combining absorption and scattering is proposed to predict attenuation in trabecular bones. The total theoretical attenuation coefficient is computed as the sum of the contributions of scattering and absorption. The obtained results have been compared with experimental data achieved on bovine cancellous bone samples filled by water. The porosity of the used samples varies between 40 % and 75 %. A transmission technique is used. This method only requires the measurement of the specimen’s thickness and recording of two pulses: one without and one with the specimen inserted between the transmitting and receiving transducers. The theoretical model used in this paper appears to give appropriate results compared with experimental measurements. The obtained results show that viscous losses are not sufficient to describe the attenuation in trabecular bone. This study indicates that scattering is the primary mechanism responsible for attenuation in trabecular bone and confirms the strong dependence of the attenuation of the size of the scatterer, the bone porosity and of the ultrasound frequency.
Keywords— Cancellous bone; Ultrasound; Attenuation;
Scattering; Propagation modeling.
I. INTRODUCTION
Osteoporosis is a disease that results in a decrease of the mineral density of trabecular bones. Numerous studies have focused on developing ultrasonic techniques for trabecular bone characterization and osteoporosis diagnosis [1, 2]. From the quantitative evaluation of the ultrasonic parameters, these techniques allow predicting the loss of bone mass and the deterioration of the micro architecture. The loss of bone mass is due to ageing process or osteoporosis. It involves an increase of porosity, thinning or even a total disappearance of some trabecular elements and a disturbance of continuity of the structure. The porosity of trabecular bone has been considered as a criterion for the prediction of osteoporosis [2].
The diagnosis methods of bone tissue using ultrasound are generally based on the assessment of the velocities and attenuation of the propagating ultrasonic waves, in order to determine the acoustic or geometrical parameters of the bone by in vivo or in vitro measurements [3]. The interaction of the incident ultrasonic wave with the porous medium, which is
composed of the bone frame saturated with fluid, causes essentially two mechanisms of attenuation: absorption and scattering [3]. These mechanisms are not well understood and are still investigated by many researchers. It is difficult to separate these two contributions starting from ultrasonic measurements, but it is of interest to understand the relative contributions of each. The absorption results from the energy dissipation by viscous friction at the solid-fluid interface, in the solid and in the fluid themselves. The solid trabeculae are likely the main cause for waves scattering in trabecular bone.
This is due to the disparity in the acoustic properties between the mineralized trabeculae and the saturating fluid. The scattering is related to the characteristics of the scatterer (elastic properties, size, number and spacing between solid trabeculae). It reflects the micro architecture parameters of the material [4]. We propose, in this study, to use theoretical approaches to model the ultrasonic attenuation by the trabecular structure of the bone and to compare the variation of the attenuation coefficient versus frequency or porosity with that evaluated from experiments on samples of bovine bone. We aim to assess the validity of these approaches and to obtain predictions of the influence of structure on the acoustic properties of the trabecular bone.
II. ABSORPTIONMODEL
The Biot theory was developed to predict the acoustical properties of fluid saturated porous rocks in the context of geophysical testing but has been extensively used to describe the wave motion in the trabecular bone [5]. Biot model describes the acoustic wave propagation in the porous medium. It consists of a superposition of two coupled continuous media (fluid and solid). This is a poroelastic medium, i.e. a solid with connected cells in which a fluid can freely circulating. The trabecular bone is considered as a porous biphasic medium, consisting of two parts: a solid phase (the skeleton) and a fluid phase (the bone marrow or water).
This model is based on several assumptions (the skeleton is a viscoelastic medium, the fluid phase is continuous and the material is considered as homogeneous and isotropic). A biphasic porous material submitted to a mechanical constraint is deformed elastically. According to this theory, the equations that reflect the interaction dynamics by coupling both inertial and viscous behavior of the fluid and the solid phases, and
which characterize the dispersion and the attenuation of the waves propagating in a porous medium, lead to two longitudinal waves and one transversal wave . For a harmonic plane wave of angular frequency , propagating through the trabecular bone and at normal incidence, the following expressions of the longitudinal velocities are obtained [5]:
2 4( 2)(11 22 122)
0.52) ( 2 2
,
Q PR
Q fast PR
Vslow
(1) whereP.22R.112Q.12
The signs ± in the denominator mean that V2fast will be obtained when "-" is selected, and V2slowwhen"+" is selected.
The parameters P,Q and R are the elastic moduli. They are expressed as functions of the coefficients Kf, Ks, Kb andN, which are the bulk moduli of the fluid, of the elastic solid and of the porous skeletal frame, and the shear modulus of the skeletal frame respectively [6,7].
11
and 22 are the apparent densities of the solid and of the liquid respectively. 12 is the inertial coupling density. The viscous coupling is integrated in the expression of these complexes densities [8]. The wavenumber associated to each wave mode is given by:
fast fast slow
slow V
k f
, 2, (2)
where f is the frequency. The attenuation caused by the absorption phenomenon can be deduced from the imaginary part of the wavenumberkslow,fast . It is given by:
, ) , Im(kslow fast a fast
slow
(3)
The simulation parameters of the Biot model for a bovine trabecular bone are listed in Table (I).
The variations of the absorption coefficient αaof the fast wave and of the slow wave predicted by the Biot theory, as a function of porosity, when the pore spaces are filled by water at the frequencies 0.2, 0.4, 0.6, 0.8 and 1 MHz are represented in Fig.1 and Fig.2 respectively. For the fast wave, the absorption determined using Biot’s theory in the frequency range [0.2 -1.0 MHz] has peak values at a porosity of about 33%-45%. The absorption increases with porosity for the pores of low size such as for cortical bone porosity. It decreases with porosity for pores of great size. For the slow wave, the absorption decreases as the porosity increases. It is greater than that of the fast wave.
TABLE I. . STRUCTURAL AND ACOUSTIC PARAMETERS OF BOVINE TRABECULAR BONE[9].
Young’s modulus of solid bone :Es 22 GPa Poisson’s ratio of solid bone:υs 0.32 Poisson’s ratio of skeletal frame:υb 0.32 Compressibility modulus of the solid:κs 20.37GPa
Solid density:ρs 1960 kg/m3
Compressibility modulus of the fluid:κf
marrow
water 2.0GPa
2.28GPa Fluid viscosity :ηf
marrow
water 1.5 Pa.s
0.001 Pa.s Fluid density:ρf
marrow water Pore size
930 kg/m3 103kg/m3 0 to 3.5 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5 2 2.5
Porosity
Absorption (dB/cm)
f=0.2 MHz f=0.4 MHz f=0.6 MHz f=0.8 MHz 1MHz
Fig. 1. Absorption coefficientαa,fastof the fast wave predicted by Biot’s theory, versus porosity, for pores space filled by water at the frequencies [0.2, 0.4, 0.6, 0.8, 1.0] MHz.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10 15 20 25 30 35 40
Porosity
Absorption (dB/cm)
f=0.2 Mhz f=0.4 MHz f=0.6 MHz f=0.8 MHz 1.0 MHz
Fig. 2. Absorption coefficientαa,slowof the slow wave predicted by Biot’s theory, versus porosity, for pores space filled by water at frequencies [0.2, 0.4, 0.6, 0.8, 1.0] MHz.
The attenuation of the slow longitudinal wave is much higher than that of the fast wave. This is in contrast with the experimental results published by Hosokawa and Otani [9].
Therefore, the prediction of attenuation by the Biot model is
not very accurate. Haire and Langton [10] have indicated in their review of the application of the Biot’s model to trabecular bone that inaccurate prediction of the attenuation coefficients was partially due to incomplete evaluation of the attenuation by the model. Despite its success in predicting propagation velocities in trabecular bone [9], the Biot analytical model underestimates the attenuation phenomenon.
III. SCATTERINGMODEL
The trabecular bone consists of various components such as fat, minerals (hydroxyapatite) and collagen. These elements contribute significantly to the ultrasound scattering through the porous bone. In this study, we assume that the structure of the trabecular bone is limited to two components: marrow or water (the fluid phase) and the bone matrix (the solid phase).
The trabecular bone is considered as a weak scattering medium; the scattering is due to velocity and density fluctuations.
Applying the equation of Sehgal [11] for a two-component mixture, the square of the velocity fluctuationμ2is calculated from the sound speeds in solid bone (cs) and fluid (cf), and the porosity ().μ2can be written in the form:
22 2
1 ) .
1
(
F S
S F
c c
c
c .
(4) The square of the density fluctuation, δ2, is calculated from the densities of the solid (ρs) and fluid (ρf) and the porosity ().δ2can be written as:
2 2
2 (1 )1 2
F F S S
F
. (5)
The attenuation of the ultrasonic wave (during its propagation through the porous bone), which is caused by the scattering phenomenon, is calculated according to the procedure used by Sehgal and Greenleaf [12]. Because of the difference in velocity and wave number associated to each type of wave (fast wave or slow wave) for a given frequency, the attenuation caused by the scattering phenomenon will be different according to the type of wave. The attenuation coefficient due to scattering, s, is the sum of two termsα1s
and α2s; the first one is related to the scattering caused by the velocity fluctuation and the second one by the density fluctuation. For an exponential autocorrelation function and assuming a medium composed of identical scatterers, we obtain [13]:
(Neper/m) 2)
9 1 2)(
1 (
3 4 2
1 p p
p k
s
(6)
163
2)) 1 ln(
2) 9 1 )(ln(
2 1 3 ( 8 2
) 13 1 tan tan 3 )(
2 3 ( 2) 1 2)(
9 1 ( 2
2) 1 2( 2) 9 1 (
2 2 p
p p p p
p p p
p p p
p p k p
s
(6)
wherekrepresents the wave number associated to each type of wave. k=kslowfor the slow wave andk=kfastfor the fast wave p=ka, with a is the mean diameter of the scatterers. For the computations, the values of the velocities and the densities of
the fluid and the solid media have been taken from the literature [14] and summarized in Table II.
TABLE II. : VELOCITY AND DENSITY VALUES USED IN MODELLING THE TRABECULAR BONE[14].
Material Velocity (m/s) Density (kg/m3)
marrow 1470 930
water 1500 1000
Cortical
bone 3300 1960
To observe the influence of the porosity on the attenuation coefficientαs, which is due to the scattering of the fast wave, we plotted in Fig. 3 the variations of this coefficient according to porosity, at a frequency of 0.5 MHz and for scatterers of respective size: 0.3, 0.6 and 1.2 mm by taking into account the velocity and the density fluctuations. The qualitative trends for the variation of the attenuation with porosity are substantially similar for all considered scatterers.
An increase of the attenuation with porosity up to a maximum for porosities between 40% and 45%, followed by a decrease beyond, can be noticed. A slight shift in the maximum attenuation towards high porosities, when the scatterer size decreases, is also noticeable. A nonlinear trend of the theoretical variations of the coefficientαsversus porosity can also be observed.
To observe the influence of the frequency on the attenuation coefficient due to the scattering of the fast wave, αs, we plotted in Fig. 4 the variations of this coefficient versus frequencyf, for respective porosities [20%, 40%, 50%, 60%, 70%, 80% and 90%] and for a scatterer of specific size 0.6 mm, by taking into account the velocity and density fluctuations. For this scatterer size, the variations of αs are approximately linear for higher frequencies (above 500 kHz for= 30%, for example). In the studied frequency range, αs
is more important for a porosity of about 40%.
0 0.2 0.4 0.6 0.8 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Porosity
s (dB/cm)
a=0.3mm a=0.6mm a=1.2mm
Fig. 3. Attenuation coefficient αs due to scattering of the fast wave in trabecular bone versus porosity,for scatterers of respective size [0.3, 0.6, 1.2] mm and at a frequency of 0.5 MHz.
0 2 4 6 8 10 x 105 0
1 2 3 4 5 6 7 8
9 Scatter size 0.6mm
Frequency (Hz)
s (dB/cm)
=0.3
=0.4
=0.5
=0.6
=0.7
=0.8
=0.9
Fig. 4. Attenuation coefficient αsdue to scattering of the fast wave versus frequency for respective porosities [20%, 40%, 50%, 60%, 70%, 80%, 90%] and for a scatterer of specific size of 0.6 mm.
IV. GLOBALULTRASOUNDATTENUATION
The global attenuation coefficient α can be estimated by summing the absorption coefficient αa, which has been calculated by using the Biot’s model,and the coefficient αs, which has been estimated from the scattering model of the binary texture. The figures (5) and (6) show the attenuation coefficient α of the fast longitudinal wave as a function of porosity and frequency respectively. The attenuation coefficient increases approximately linearly with increasing frequency for high frequencies (superior to 500 kHz for a porosity= 40%, for example). This result is consistent with that found by Langton and Njeh[15] for calcaneus trabecular bone in the diagnostic frequency range from 0.2 to 0.6 MHz.
As regards the porosity dependence, similar nonlinear trends of the attenuation coefficient versus porosity were found in trabecular bone-mimicking phantoms [16]. This suggests that this nonlinear behavior is a general feature of fluid-saturated porous media like trabecular bone filled with fluid (marrow or water).
This theoretical approach shows that the values of the attenuation coefficient due to scattering αs are significantly higher than those due to absorption and calculated by the Biot theory, αa. It suggests that the scattering is the principal mechanism responsible for the attenuation in trabecular bone.
It also shows that the density fluctuation contribute significantly to the attenuation phenomenon.
In Fig. 7, the variations of the measured attenuation coefficient of the fast longitudinal wave, α in dB/cm at 0.5 MHz versus porosity, which has been otained with various bone samples saturated with water, are represented. The error bars represent the standard deviations. For comparison, we added on this figure the curve representing the variation of the attenuation coefficient obtained with the theoretical model of the global attenuation and for scatterer sizes of 0.9, 1.2 and 1.5 mm respectively. The measured values are significantly higher than those predicted by this model. One can first notice that the model taking into account both absorption and scattering still underestimates the attenuation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1 2 3 4 5 6
Porosity
(dB/cm)
f = 0.5MHz a=0.3mm
a=0.6mm a=0.9mm a=1.2mm a=1.5mm a)
Fig. 5. Global attenuation coefficient α of the fast wave versus porosity
0 1 2 3 4 5 6 7 8 9 10
x 105 0
2 4 6 8 10 12
frequency (Hz)
(dB/cm)
a = 1.2mm
=0.3
=0.4
=0.5
=0.6
=0.7
=0.8
=0.9 b)
Fig. 6. Global attenuation coefficient α of the fast wave versus frequency
0 0 .2 0 .4 0 .6 0 .8 1
0 5 1 0 1 5 2 0 2 5 3 0
p o r o s it y
theoretical and experimental attenuation (dB/cm)
a = 0 . 9 m m a = 1 . 2 m m a = 1 . 5 m m E xp .
T he o re t ic a l
Fig. 7. Experimental and theoretical attenuation coefficient of the fast longitudinal wave versus porosity, at a frequency of 0.5 MHz.
Fig. 8 is an illustration of the attenuation coefficient evolution of the bone samples with different porosities in the frequency range from 0.2 to 0.7 MHz. The experimental curves of the attenuation have been juxtaposed to the theoretical ones. An increase in the attenuation with frequency and porosity of the insonified samples should be noted. The evolution of the experimental attenuation coefficient versus frequency is quasi linear in the considered frequency range. It also remains higher than its theoretical values predicted by the presented model. Nevertheless, there is a similarity in the shape of the experimental and the theoretical curves. In addition, the theoretical curves for the three scatterer sizes are certainly not
very representative of real bone. The differences in these values are probably due to a growth of the attenuation coefficient due multiple scattering and mode conversions in the porous material which have not been considered by this model.
3 3.5 4 4.5 5 5.5 6 6.5 7
x 105 0
5 10 15 20 25 30 35 40
F requency (Hz)
Attenuation coefficient (dB/cm)
=50%
= 60%
= 70%
E xperim ental
T heoretical
Fig. 8. Experimental and theoretical attenuation coefficient of the fast wave versus frequency, for a porosity of (50%, 60% and 70%) respectively.
V. CONCLUSION
The prediction of the attenuation coefficient by using only the Biot’s model is not very sufficient. The calculated attenuation coefficient derived from this model is far below that measured. In the Biot’s model the slow wave is more attenuated than the fast wave. In addition, the experimental results show that the fast wave attenuation increases rapidly with the frequency, which is not predicted by Biot’s theoretical results. A strong ultrasonic scattering by trabeculae occurs during the propagation of the ultrasonic wave through the trabecular bone, due to the large structure of the bone in addition to the difference in acoustic impedance between marrow or water and solid tissue bone. The theoretical approach presented in this paper combines the two contributions predicted by two models to calculate the global attenuation coefficient. This combination of the two models predicts a nonlinear variation of the attenuation according to the porosity which is very similar to those observed experimentally. However this approach doesn’t take into account neither multiple scattering nor attenuation due to mode conversion and assumes that the scatterers are identical.
This may explain the difference between the theoretical values and the experimental ones. Furthermore, the study reveals that the density fluctuation in the trabecular bone contributes
appreciably to scattering and can be an important additional source of ultrasonic attenuation and thus should not be neglected as suggested by other studies.
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