A probabilistic study of the ultrasonic reflection coefficient from cortical bones A. Abdoulatufa, V-H. Nguyena, C. Desceliersb and S. Nailia* Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS a 61, avenue du Général de Gaulle, 94010, Créteil Cedex, France b 5, boulevard Descartes, 77454, Marne-la-Vallée Cedex 2, France Keywords: Ultrasound; cortical bone; reflection coefficient; stochastic model; FEM
1. Introduction Different metabolic diseases such as osteoporosis may affect bone quality, resulting in a decrease in bone mass and micro-architectural deterioration of bone tissue, which implies an increase in bone fragility. The diaphysis of long bones such as radius and femur is mainly constituted of cortical bone. Investigating cortical bone quality is of interest because it accounts for about 80% of the skeleton, supports most of the load of the body, and is mainly involved in osteoporotic fractures. Among the methods used in diagnosis of bones, ultrasound techniques, which are based on the evaluation of mechanical properties of bones, have been shown particularly suitable for cortical bone evaluation. Basically, mechanical properties of a bone plate may be derived by determining the reflection/transmission coefficients, or the velocity of bulk/guided waves. From a mechanical point of view, cortical bone is a highly complex composite material formed by a hierarchical heterogeneous structure of different multiscale constituents. The heterogeneity of cortical bone at the vascular scale is due to the pore distribution and physical properties of mineralization of bone tissues which depend of many factors (genetic, environmental, physiological, and pathological). These factors may be taken into account by introducing a probabilistic framework to describe these properties via the random variables which are used in a model to obtain of the ultrasonic response of cortical bone plate.
for the bone diagnostic using the so-called axial transmission test, has been studied as the quantity of interest. This work presents a probabilistic study of the reflection coefficient from a cortical bone plate due to a random fluctuation of elastic properties with respect to the bone’s thickness direction
2. Model and method Figure 1 presents the configuration of the model in consideration. The bone is modeled by a constantthickness solid plate immersed in two acoustic fluid halfspaces representing the soft tissue and the marrow, respectively. Here, a bone plate with 4mm thickness is studied. Both fluids are assumed to be homogeneous; theirs mechanical properties will be considered as deterministic fields and are given by: the mass densities ρ1 = ρ2 = 1000 kg.m−3, the wave velocities c1 = c2 = 1500 m.s−1. The bone plate is assumed to be an anisotropic elastic medium. It is homogeneous along its longitudinal direction ( ) but randomly heterogeneous along its thickness direction ( ). By using a probabilistic model (Desceliers et al., 2012), the elasticity tensors at each point in the thickness direction may be generated from a mean-valued elasticity tensor and two scalar parameters and which control the dispersion and the correlation length of the random elasticity field, respectively.
Some studies have recently been carried out to investigate the ultrasonic response from random cortical bone plates. Different models have been used. The bone plate was described as randomly homogeneous media (Macocco et al., 2006), or randomly heterogeneous media in thickness direction (Desceliers et al., 2012), or randomly heterogeneous media both space directions (Naili et al., 2015). However, only the FAS (First Arriving Signal) Figure 1 Model description velocity, which has been shown to be a key parameter ______________________________________________________________________________________________ * Corresponding author. Email:
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For this study, the mean model of bone material is assumed to behave as a transversely isotropic elastic one. The components of the mean elasticity tensor c are: c11 = 23.05 GPa, c22 = 15.10 GPa, c12 = 8.71 GPa, c66 = 9.40 GPa, c16 = c26 = 0, and the mass density ρ = 1722 kg.m−3. Consider a harmonic plane wave (amplitude and frequency ) arriving to the upper surface of the bone plate from an incident angle wave . The reflection coefficient is defined by the ratio between the amplitude of reflected wave and one of the incident wave. In this study, the solution of reflected wave from the heterogeneous plate in consideration has been computed by using the semi-analytical finite element method (Nguyen and Naili, 2013).
In this figure, two similar solid lines represents the upper and lower envelope delimiting the confidence region of the reflection coefficient obtained with the stochastic model when δ = 0.1251 (black region), δ = 0.2431 (dark grey region) and δ = 0.3647 (light grey region). The dotted red line curve represents the reflection coefficient obtained with the deterministic model. It may be noticed that the confident region of strongly depend to and becomes wider when increases. However, the difference between the mean value of this quantity (figure is not shown here) and the one obtained by the mean model is not significant.
4. Conclusions
A statistic study has been carried out based on the Monte Carlo simulation. For each realization of the random elastic matrix, the reflection coefficient is calculated. Convergence analysis based on the number of realizations was performed. The quantile method was applied to construct the confidence interval associated to a probability level for the random variable .
The paper presents a framework for studying the reflection coefficient of ultrasonic waves from a random bone plate. It has shown the effects of the dispersion of the material, which represents the fluctuation level of mechanical properties, on the measured reflection coefficient. This effect may be significant, especially in frequency range higher than 1MHz, and would be taken into account for estimation of bone quantity.
3. Results and discussion
Acknowledgements
The statistic study has been performed for different frequencies in the ultrasonic frequency range (250MHz - 2MHz) and incident angles. The obtained results shown that reflection coefficient is highly sensible to the dispersion parameter . For illustration purposes, Figure 2 shows the confidence regions associated with a probability level = 0.95 of the reflection coefficient with respect to the incident angle .
A. Abdoulatuf would like to acknowledge the Comoros Ministry of Education and Research for his PhD scholarship.
Figure 2 Reflection coefficients versus the incident angle for different dispersions δ
References Desceliers, C., Soize, C., Naili, S., Haiat, G., 2012. Probabilistic model of the human cortical bone with mechanical alterations in ultrasonic range. Mech Syst Signal Pr 32 (0), 170 – 177. Macocco, K., Grimal, Q., Naili, S., Soize, C., 2006. Elastoacoustic model with uncertain mechanical properties for ultrasonic wave velocity prediction: Application to cortical bone evaluation. J Acoust Soc Am 119 (2), 729–740. Naili, S., Nguyen, V.-H., Vu, M. B, Desceliers, C., Soize, C., 2015. Modeling of transient ultrasound wave propagation in a random heterogeneous long bone coupled with fluid. J Acoust Soc Am, 137(2), 668-678 Nguyen, V.-H., Naili, S., 2013. Ultrasonic wave propagation in viscoelastic cortical bone plate coupled with fluids: a spectral finite element study. Comput Methods Biomech Biomed Engin 16 (9), 963–974.
