Predictions Of Phase Velocity

Published Date: 02 Nov 2017

1Unit of Medical Technology and Intelligent Information Systems

University of Ioannina

45110 Ioannina, Greece

e-mail: [email protected], web page: http://medlab.cc.uoi.gr

2Department of Mechanical and Aeronautical Engineering

University of Patras

26500 Patras, Greece

e-mail: [email protected]

Keywords: Bone healing, Ultrasound, Scanning acoustic microscopy, Scattering, Attenuation, Phase velocity.

Abstract. The quantitative determination of wave dispersion and attenuation in nonhomogeneous media such as healing bones is a quite complicated procedure due to the complex nature of the callus tissue. In this study, phase velocity and attenuation predictions are conducted in the callus region of a healing bone based on an iterative effective medium approximation (IEMA)[1]. The geometry and material properties were derived from a previous animal study[2] using scanning acoustic microscopy (SAM) and represent an early healing stage after three weeks of the performance of an artificial fracture. The callus region was considered as a composite medium consisting of a matrix with the material properties of blood and a random distribution of spherical inclusions with material properties equal to those of the osseous tissue. We investigated the dependence of wave dispersion and attenuation on frequency and particles’ volume concentration and size. The results indicate that the evaluation of the phase velocity and attenuation coefficient could provide supplementary information for bone assessment during the healing process.

1 introduction

The monitoring role of quantitative ultrasound in the identification of the bone healing progress is a challenging research area in terms of clinical diagnosis. Several studies[3-8] have investigated ultrasound propagation in healing bones experimentally and numerically, using measurements of the velocity of the first arriving signal (FAS) and attenuation estimations as the main indicators of the healing process. In our previous three-dimensional computational study[4], a FAS velocity decrease was found at the first stages of the healing process, followed by a gradual increase as bone restores its original integrity at the final weeks of healing. In addition, an in vitro study of Dodd et al.[6] showed that the attenuation of the FAS propagating in two-dimensional numerical models of healing long bones could be a good quantitative indicator for the fracture healing process. More recently, Machado et al. [7] in a simulation study estimated the attenuation for the first 140 days of healing by calculating the FAS peak amplitude. However, these studies deal with the evaluation of the FAS attenuation and velocity, which behaves as a lateral wave for wavelengths comparable to or smaller than the cortical thickness and reflect only the periosteal region of bone. Later, guided waves were introduced as an advanced method of bone characterization which can capture changes in bone and callus properties occurring at deeper layers[3-4]. Nevertheless, computational studies[3-7] on healing bones are mainly based on simple and homogeneous geometries, thereby ignoring the complex callus microstructure which induces multiple scattering, material dispersion and absorption phenomena.

Thus, when a plane wave propagates in nonhomogeneous media as healing bones the amplitude of the wave decays with a frequency dependent rate, which is expressed via a frequency dependent exponential coefficient called the attenuation coefficient. In addition, a composite medium is said to be dispersive when the phase velocity is a function of the frequency or wavelength. Several theories have been proposed referring to the quantitative determination of wave dispersion and attenuation, due to the presence of a random distribution of particles in a composite medium[9-13]. However, these theories are not able to provide accurate phase velocity and attenuation estimations for all types of inclusions and for a wide range of volume concentrations and wavenumbers.

To this end, an iterative effective medium approximation (IEMA) was presented by Aggelis et al. [1] and Tsinopoulos et al. [14], which first uses the frequency dependent complex density as the main parameter in order to control the whole iteration procedure and then calculates the complex wavenumber for each frequency. The real part of the complex wavenumber is a function of the phase velocity and the circular frequency, while the imaginary part is the attenuation coefficient. In these studies, the numerical results of IEMA were compared to experimental data taken from the literature to demonstrate the methodology effectiveness to predict wave dispersion and attenuation in composite materials including particles with volume concentrations as high as 50%.

In this work, IEMA is used to conduct wave dispersion and attenuation predictions in the callus region of a healing bone. Callus was assumed to be a porous, composite medium consisting of a blood matrix with spherical particles of osseous tissues. The material properties and the geometry of the osseous tissues were derived using a scanning acoustic microscopy (SAM) image. SAM is a micro elastic imaging technique that has been extensively used to investigate the microstructural and elastic alterations of mineralized callus and cortical tissues[2,15,16]. First, the particles’ volume concentration and diameter were measured as 44.75% and 350 μm derived from the SAM callus region. Numerical calculations of the phase velocity and attenuation are performed in the frequency range from 24 – 1200 kHz. Then, the frequency was set at 500 kHz in order to estimate wave dispersion and attenuation variation for different inclusions’ volume concentrations and diameters. The numerical results indicate that IEMA could provide supplementary information for the assessment of healing bones.

2 MATERIALS AND METHODS

2.1 Scanning acoustic microscopy image

The SAM image illustrated in Fig. 1 was obtained from the right tibia of a female Merino sheep and depicts an embedded longitudinal section of a 3-mm osteotomy. The bone specimen examined here, corresponds to the third postoperative week. SAM measurements were taken in[2], using a spherically focused 50 MHz transducer with spatial resolution of 23 μm and scan increment of 16 μm.

Figure 1. SAM image from an animal study[2] representing the 3rd postoperative week.

2.2 The IEMA for particle suspensions

In this section we briefly describe the iterative methodology proposed by Aggelis et al.[1]. A plane wave propagating in a composite medium can be considered as a sum of: a) a mean wave travelling in the medium with the dynamic effective properties of the composite, and b) fluctuating waves obtained from the multiple scattering of the mean wave. This consideration implies a complicated self-consistent multiple scattering condition so to estimate the dynamic effective properties of the composite. In order to simplify the calculations, Kim et al.[13] proposed a simple self-consistent condition, which for nonhomogeneous media is expressed as:

(1)

Figure 2. The single scattering problems of the self-consistent condition (Eq. (1)).

whererepresent the volume fraction of the inclusions and the matrix, respectively, is the direction in which a -polarized plane mean wave propagates and are the forward scattering amplitudes derived from the solution of the scattering problems 1 and 2, respectively, presented in Fig. 2.

The mean wave is both dispersive and attenuated and has a complex wavenumber defined as:

(2)

with denoting the effective and frequency dependent phase velocity and attenuation

coefficient, respectively, of a longitudinal (d ≡ P) or transverse (d ≡ S) mean wave propagating with circular frequency ω. Under these assumptions the steps of IEMA for the determination of and are described in detail below.

First, the composite medium is replaced by an elastic homogeneous and isotropic material with bulk and shear moduli and , respectively, calculated using the static mixture model of Christensen[11] via the equations:

(3)

(4)

where are functions of given in the study of Christensen [11] and the indices 1, 2 correspond to the material properties of the inclusion and matrix, respectively. Considering the effective density of the composite to be defined as:

(5)

the real effective wave number of the mean wave can be estimated through the material properties (4) and (5) and the relations:

(6)

(7)

where Cp and Cs are the longitudinal and the shear velocity of the propagated wave, respectively. This is the first step of the IEM approximation.

Next, utilizing the material properties calculated in the first step, we proceed to the second step where the scattering problems 1 and 2, illustrated in Fig. 2, are solved in order to evaluate the forward scattering amplitudes Subsequently, as in Eq. (1) the scattering amplitudes can be estimated according to the following equation:

(8)

and making use of the dispersion relation proposed in [9], we can estimate the new effective wavenumber of the mean wave as:

(9)

where a is the radius of a volume equivalent to the particle sphere.

Then, the new complex density is calculated based on the and the Eqs. (4), (6) and (7). The second step is repeated with the material properties (4) and the new density (ρeff)step2 until the self-consistent condition (1) is satisfied. Thus, applying the Eqs. (3) and (9) the frequency dependent, effective phase velocity and attenuation coefficient of the mean wave can be derived.

2.3 Material Properties

The material properties of each pixel composing the osseous tissues in Fig. 1 were derived using empirical equations. The density ρ was calculated using the equation[15]:

(10)

where Z is the acoustic impedance.

Then, the elastic constant in the axial direction c33 is calculated via the equation [15]:

(11)

The Young modulus E is finally defined as [2]:

(12)

where ν is the Poisson’s ratio. The osseous tissues are considered isotropic with a Poisson’s ratio ν = 0.3. The calculated average values of the callus material properties, as well as the material properties of blood derived from a previous study[3] are shown in Table I.

Osseous tissue

Blood

ρ (kg/m3)

1425

1055

E (GPa)

13.6

3x10-3

λ (GPa)

7.9

2.6

μ (GPa)

5.2

100x10-9

ν

0.3

0.5

Table I: Material properties of the callus components.

2.4 Wave dispersion and attenuation predictions

In this section, we describe the different cases examined in order to investigate the dependence of the phase velocity and attenuation coefficient on frequency and inclusions’ volume concentration and diameter. Callus was assumed as a porous, two-phase medium with the material properties of Table I. Blood was considered as the material of the matrix of the composite and the osseous tissue as the content of the spherical particles. The average particle diameter and the volume fraction were calculated using the SAM image as 350 μm and 44.75%, respectively. Measurements of the phase velocity and the attenuation coefficient were performed in the frequency range from 24 – 1200 kHz. Then, the frequency and the volume concentration were set at 500 kHz and 44.75%, respectively, in order to examine the behavior of the phase velocity and attenuation when the diameter of the inclusions gradually increases from 200 – 500 μm. This diameter range covers the different particles’ sizes measured in the callus region of Fig. 1. Finally, we applied a constant operating frequency and diameter of 500 kHz and 350 μm, respectively to investigate the predictions’ variation when the particles’ volume concentration increases from 30% to 55%. By examining wave dispersion and attenuation for different particle diameters and volume fractions we approximate the callus composition and conditions occurring at earlier and later healing weeks.

3 results

Figs. 3 and 4 represent phase velocity and attenuation predictions derived from IEMA as a function of frequency for an average particle diameter equal to 350 μm and a volume fraction equal to 44.75%. In Fig. 3, it can been seen that the phase velocity decreases from 1826 m/s down to 1743 m/s with increasing frequency from 24 – 1200 kHz, exhibiting thus a negative dispersion. On the other hand, the attenuation increases from 0.06 – 76.36 m-1 in the examined frequency range (Fig. 4).

Figs. 5 and 6 show the phase velocity and attenuation variation with increasing scatterer volume concentration for a frequency 500 kHz and a scatterer diameter 350 μm. It can be observed that the phase velocity increases from 1713 – 1896 m/s when the inclusions’ volume concentration increases from 30 – 55% (Fig. 5). Moreover, it can be seen in Fig. 6 that the attenuation coefficient slightly increases from 1.97 – 2.68 m-1 with increasing particle volume concentration.

Finally, Figs. 7 and 8 represent the phase velocity and attenuation coefficient estimations for increasing particle diameters (the frequency and the inclusions’ volume concentration are 500 kHz and 44.75%, respectively). It is shown that for diameters from 200 – 500 μm the phase velocity slightly decreases from 1820 – 1795 m/s (Fig. 7). On the other hand, the attenuation coefficient increases from 0.58 – 7.12 m-1 with increasing particle sizes (Fig. 8).

Figure 3. The dependence of the phase velocity of the composite medium on frequency.

Figure 4. The dependence of the attenuation of the composite medium on frequency.

Figure 5. The dependence of the phase velocity of the composite medium on particle volume concentration.

Figure 6. The dependence of the attenuation of the composite medium on particle volume concentration.

Figure 7. The dependence of the phase velocity of the composite medium on particle diameter.

Figure 8. The dependence of the attenuation of the composite medium on particle diameter.

4 discussion

In the present work, we make use of an iterative methodology in order to conduct wave dispersion and attenuation estimations in healing bones. To achieve more realistic conditions, the geometry and the material properties were derived from a SAM image of a healing tibia which corresponds to the third postoperative week. Wave propagation effects are examined at an early healing stage reflecting the complex microstructure of the newly formed tissues of callus. To this end, predictions of the phase velocity and attenuation coefficient were presented for different frequencies, particle volume concentrations and diameters.

It was shown that the phase velocity decreases with increasing frequency, exhibiting a negative dispersion. However, this behavior is opposite to the Kramers-Kroning relations which are causality dependent mathematical relations that connect the real and imaginary part of any complex function having a positive imaginary part[17]. When the Kramers-Kroning relations are used to estimate the dispersion of a medium with an increasing attenuation coefficient, the expected result is a phase velocity which increases logarithmically with frequency, known as positive dispersion. Nevertheless, a negative dispersion has been also reported in various experimental and numerical studies investigating wave dispersion in cancellous bones[18-25]. Although, several attempts have been made in order to explain this abnormal behavior of the phase velocity, no conclusion has been drawn yet. Chakraborty et al.[23], presented a non-local extension of the Biot theory, that can result to a negative dispersion under some circumstances. In the same study, it was also shown that the modes of Lamb waves show similar negative dispersion when predicted by the nonlocal poroelastic theory. Moreover, Haiat et al.[24] suggested that the coupling of multiple scattering and absorption may contribute to negative dispersion. A similar phase velocity trend was presented in the study of Bauer et al.[18] in a cancellous bone phantom suggesting that when a negative dispersion is observed at specific bone regions, the attenuation coefficient increases almost linearly with frequency. According to another study[19] the Kramers-Kroning relations are not able to provide accurate phase velocity predictions due to the fact that they use integrals over an infinite bandwidth, while the experimental bandwidths are finite.

In addition, the phase velocity was found to increase with increasing scatterer volume concentration. Although, the dependence of the phase velocity on frequency and volume concentration was in agreement with previous experimental and numerical findings[20, 25], this was not the case when the particle size was increasing. In particular, a slight decrease of the phase velocity was observed with increasing particle diameter, which is opposite to the phase velocity behavior reported in previous studies on cancellous bones[20,21,22,25]. However, further numerical research is needed in order to explain the scattering effects of the complex microarchitecture of cortical bone.

On the other hand, the attenuation coefficient was found to increase with increasing frequency. In addition, an attenuation coefficient increase was also observed when the particles’ volume concentration and diameter increased. A similar attenuation variation has been also reported in a previous experimental study[22] on a cancellous bone phantom performing velocity and attenuation measurements over a range of porosity values from 10– 50% for frequencies from 500 – 900 kHz. The gradual increase of the attenuation coefficient in all the examined cases is attributed to the significant impact of wave absorption, scattering and reflection phenomena as larger diameters and higher particle volume concentrations enhance bone heterogeneity.

Nevertheless, IEMA introduces the assumption that the geometry of the inclusions is spherical which is not realistic as a better representation for the newly formed callus tissues would be a cylindrical shape. The incorporation of cylindrical scatterers into IEMA constitutes our future research. Another assumption is that the random distributed particles all have the same size. This condition is not realistic as different callus sub-regions have different particle diameters derived even from the same healing stage. This observation is more evident at later healing stages where callus tissues gradually fill the osteotomy gap. The size and the volume concentration of the inclusions inside the gap tissues are different in comparison to the callus tissues surrounding the osteotomy. Also, in this study we present wave dispersion and attenuation estimations corresponding to a single healing stage. Our ongoing research takes into consideration successive healing stages derived using serial SAM images in order to verify that IEMA can provide supplementary information for bone healing assessment. Therefore, the complex SAM geometries in combination with the effective material properties calculated using IEMA can contribute to the development of more realistic, nonhomogeneous and anisotropic computational models of the callus tissue during the healing process. Finally, this methodology could be extended in order to examine wave propagation effects in computational models of osteoporotic bones.

5 CONCLUSIONS

In the present work, we used an iterative methodology for the quantitative determination of attenuation and wave dispersion in the callus region of a healing bone corresponding to the third postoperative week. Realistic conditions were incorporated as the material properties and the examined geometry were derived from a scanning acoustic microscopy image. The phase velocity and attenuation coefficient predictions indicate that scattering and absorption effects are enhanced by the heterogeneity of bones at the early healing stages. Therefore, this study could be regarded as a starting point for the development of more realistic, nonhomogeneous and anisotropic computational models of healing bones taking into account the complex callus microstructure. Our future research will make use of IEMA in order to investigate phase velocity and attenuation evolution in callus sub-regions derived from SAM images corresponding to successive healing stages.