Session S08 - Inverse Problems and Applications

Wednesday, July 14, 17:00 ~ 17:30 UTC-3

Mechanical Characterization of Trabecular Bone

María Gabriela Messineo

INTEMA - Universidad Nacional de Mar del Plata, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Life expectancy worldwide has been increasing during last decades, resulting in a significant proliferation of bone diseases such as osteoporosis [1]. Osteoporosis leads to the loss of bone mass, highly increasing the fracture risk. This entails an enormous cost for the social security system around the world. There is plenty research effort aimed at a better understanding of bone mechanics to improve diagnosis and treatments. A comprehensive characterization of bone structure is relevant to assess some aspects of its properties which are not yet understood. Quantitative ultrasound (QUS) methodologies have proven to be suitable for in vivo bone evaluation, as transmitted waves bring information about its mechanical properties. The most usual QUS techniques measure the speed of sound (SOS) and the broadband ultrasonic attenuation (BUA) [2-6]. In vitro SOS measurements are typically done via transmission tests in which a bone sample is placed in a water tank between a wave source and a wave receiver. Experimental studies have demonstrated that SOS is highly correlated with the bone volume-to-total volume ratio, BVTV, (volume of mineralised bone per unit volume of the sample), the bone mineral density (BMD) and bone strength. This suggests that SOS also might be a good candidate for fracture risk prediction. Cancellous bone is a poroelastic and biphasic medium composed of an elastic skeleton (trabecular network) filled with a viscous fluid (bone marrow). Theoretical models using the Biot theory [7,8] –initially developed in the context of geophysical applications– have been applied to model ultrasound in cancellous bone with some success. Biot's model predicts the existence of two longitudinal waves resulting from the coupling of the fluid and the solid. Experimental and theoretical works provide evidence that the propagation speed of both waves is influenced not only by BVTB, but also by the bone structural anisotropy and the orientation of the ultrasound beam relatively to the direction of trabeculae alignment [9].

The main objective of this work is to develop a procedure to compute the BVTV from SOS data by means of inverse analysis. The procedure is based on the analytical model developed by Nguyen et al. [10], which is a closed-form solution of the equations proposed by Biot and Willis [11-13]. Nguyen’s model allows for the computation of the SOS from the solution of an eigenvalue problem that is posed in terms of the bone elastic constants, the trabecular microstructure tortuosity, permeability, and mass densities of the solid, the fluid and the mixture. It is proposed here to relate the bone sample properties to the BVTV (using either correlations from the bibliography or by developing correlations ad hoc), in such a way that the SOS results a function of the BVTV only. Then, the estimation of the BVTV is found by minimizing the difference between the SOS measured in a transmission test and that calculated with the analytical model.

The talk addresses the work done to assess the performance of Nguyen’s model, to develop the correlations between the bone properties and BVTV, and to set-up the inverse problem.

Computed microtomographies were used to fully characterize 23 samples of bovine trabecular bone: they were processed with BoneJ [14] to measure BVTV, trabecular thickness and trabecular space; tortuosity was obtained with Dijkstra shortest path algorithm [15], and effective elastic tensors and permeabilities in the principal anisotropy directions were computed using micro-finite element models in combination with asymptotic homogenization schemes. The resulting data was used to correlate tortuosity, permeability and elastic tensor components with BVTV.

Three-dimensional ultrasonic transmission tests were simulated using the finite difference time derivative (FDTD) software SimSonic. The model geometries were directly obtained from microtomographies. Ultrasonic propagations were simulated in the three principal anisotropy directions for the 23 samples, and the corresponding SOS were estimated from the amplitude vs time data.

The SOS from the FDTD simulations were compared to those of Nguyen’s model with the elastic and geometric parameters of the bone samples. Very good agreement was found between the two sets of results. In most of the cases, SOS values predicted by Nguyen’s somewhat lower than of the simulations, with relative error around 4% on average. The ratio between the analytical model velocity and the simulated velocity is close to one for most of the samples. Based on these results, it was concluded that Nguyen’s model is accurate enough to perform reliable estimation of SOS of trabecular bone.

Finally, a sequential quadratic programming (SQP) method [16] implemented with fminimax Matlab function was used to solve the inverse problem to estimate BVTV from SOS results. Differences between the estimated and actual BVTV values of the 23 samples were around 18% in average with standard deviation of 25%.

It is concluded that the proposed procedure allows for acceptable estimations of the BVTV. Further developments of the procedure will address the effect of attenuation into the model in order to extend the analysis to incorporate BUA data.

Bibliography

1- https://www.osteoporosis.foundation/facts-statistics

2- Langton C.M., Palmer S.B. & Porter R. W., (1984) The measurement of broadband ultrasonic attenuation in cancellous bone, Eng. Med. 13, 89-91.

3- Fredfelt K. E., (1986) Sound velocity in the middle phalanges of the human hand, Acta Radiol. Diagn. (Stockh), 27, 95-6 8.

4- Otani, T. et al., (2009) Estimation of in vivo cancellous bone elasticity, Jpn. J. Appl. Phys. 48, 0-5.

5- Foldes, A. J., Rimon, A., Keinan, D. D. & Popovtzer, M. M., (1995) Quantitative ultrasound of the tibia: a novel approach for assessment of bone status, Bone 17, 363-367.

6- Barkmann, R. et al., (2008) In vivo measurements of ultrasound transmission through the human proximal femur, Ultrasound Med. Biol. 34, 1186-1190.

7- Biot M.A, (1956) Theory of propagation of elastic waves in fluid-saturated porous solid I. Low-frequency range, Jour. Acoust. Soc. Am., 28(2), 168-178.

8- Biot M.A, (1956) Theory of propagation of elastic waves in fluid-saturated porous solid II. Higher frequency range, Jour. Acoust. Soc. Am., 28(2), 179-191.

9- Nguyen, V.H., Naili, S., Sansalone, V., (2009) Simulation of ultrasonic wave propagation in anisotropic cancellous bone immersed in fluid, Wave Motion, 47 (2), 117-129.

10- Nguyen, V.H., Naili, S., Sansalone, V., (2010) A closed-form solution for in vitro transient ultrasonic wave propagation in cancellous bone, Mech. Res. Com., 37 (4), 377-383.

11- M.A. Biot, General theory of three-dimensional consolidation (1941), J. Appl. Phys. 12 (2) 155–164.

12- M.A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid (1955), J. Appl. Phys. 26 (2) 182–185.

13- M.A. Biot, D.G. Willis (1957), The elastic coefficients of the theory of consolidation, J. Appl. Mech. 79 594–601.

14- https://bonej.org/

15- Dijkstra E.W, (1959) A Note on Two Problems in Connexion with Graphs, Num. Math. Vol. 1, 269-271.

16- Brayton, R. K., Director, S. W., Hachtel, G. D. and Vidigal, L., (1979) A New Algorithm for Statistical Circuit Design Based on Quasi-Newton Methods and Function Splitting, IEEE Trans. Circuits and Systems, Vol. CAS-26, pp. 784-794.

Joint work with Ing. Joaquín García Zárate (INTEMA - Universidad Nacional de Mar del Plata) and Prof. Adrián Cisilino (INTEMA - Universidad Nacional de Mar del Plata).

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