Genetic algorithms-based inversion of multimode guided waves for cortical bone characterization

Abstract

Recent progress in quantitative ultrasound has exploited the multimode waveguide response of long bones. Measurements of the guided modes, along with suitable waveguide modeling, have the potential to infer strength-related factors such as stiffness (mainly determined by cortical porosity) and cortical thickness. However, the development of such model-based approaches is challenging, in particular because of the multiparametric nature of the inverse problem. Current estimation methods in the bone field rely on a number of assumptions for pairing the incomplete experimental data with the theoretical guided modes (e.g. semi-automatic selection and classification of the data). The availability of an alternative inversion scheme that is user-independent is highly desirable. Thus, this paper introduces an efficient inversion method based on genetic algorithms using multimode guided waves, in which the mode-order is kept blind. Prior to its evaluation on bone, our proposal is validated using laboratory-controlled measurements on isotropic plates and bone-mimicking phantoms. The results show that the model parameters (i.e. cortical thickness and porosity) estimated from measurements on a few ex vivo human radii are in good agreement with the reference values derived from x-ray micro-computed tomography. Further, the cortical thickness estimated from in vivo measurements at the third from the distal end of the radius is in good agreement with the values delivered by site-matched high-resolution x-ray peripheral computed tomography.

Bone cortical thickness and porosity assessment using ultrasound guided waves: An ex vivo validation study

“…Note that the model parametrization in terms of porosity Ct.Po does not modify the dispersion equations Eq. (1), but can simply be regarded as a sampling of the complete elasticity domain spanned by the four elastic coefficients c ij [22]. Therefore, the dispersion equations can now be written as:…”

“…2, for two typical thickness values (1 mm and 2 mm), and two values of porosity (5 and 15%). The number of modes increases with thickness, whereas an increase in porosity is associated with an increase in wave number k (or a decrease in phase velocity c(f ) = 2πf /k) [22].…”

“…Moreover, only the cortical thickness could be validated as no reference values for the elastic coefficients were available. To face these limitations, our group recently introduced an automated inversion procedure to provide concurrent estimates of cortical thickness and porosity [22]. This procedure, applied in vivo at the distal radius of healthy subjects, yielded thickness estimates in good agreement with ground truth values derived from site-matched HR-pQCT measurements [23].…”

Ultrasound-Based Estimates of Cortical Bone Thickness and Porosity Are Associated With Nontraumatic Fractures in Postmenopausal Women: A Pilot Study

“…The problem is that, in the full waveform, distinguishing modes or their dispersion curves in recorded signals requires a specific acquisition scheme or signal processing. Recently, the determination of guided modes and the identification of cortical bone waveguide characteristics have sparked increased discussions of signal‐processing approaches, modeling, and inverse problem‐solving …”

“…Recently, the determination of guided modes and the identification of cortical bone waveguide characteristics have sparked increased discussions of signal-processing approaches, (29)(30)(31)(32) modeling, (33)(34)(35) and inverse problem-solving. (28,(36)(37)(38) Our group has developed the bidirectional axial transmission technique (BDAT) to this goal. (39) We recently showed that BDAT combined with an appropriate waveguide model allows the concurrent identification of Ct.Th and Ct.Po of ex vivo human specimens.…”