Ex vivo cortical porosity and thickness predictions at the tibia using full-spectrum ultrasonic guided-wave analysis

The axial transmission (AT) system (Azalée, Paris, France) included a custom-made probe (Vermon, Tours, France), driving electronics (Althais, Tours, France), and a human-machine interface (BleuSolid, Paris, France). The multi-channel probe consisted of a central 24-receiver array (pitch = 1.2 mm) and two lateral 5-emitter arrays (pitch = 1.5 mm). The dimensions of each rectangular receiver and emitter element were 1.2 × 13 mm² and 1.5 × 13 mm², respectively. A distance of 8 mm separated the receiver from each emitter array. This configuration enabled the propagation of ultrasound in two opposite directions, a technique used to correct errors induced by small inclination angles between the probe and the bone surface [24]. The excitation signal consisted of a wideband pulse with a center frequency of 500 kHz (− 6 dB frequency bandwidth from 0.3 to 0.7 MHz). The five multi-element transmitters were used successively with time delays ranging from 0 to 0.8 μs. After 16 averages performed by the hardware, a set of 120 radio-frequency (RF) signals corresponding to all possible transmission-receiver pairs were digitized (12 bits, 20 MHz, 1024 samples) for each propagation direction.

Measurements were performed in water at room temperature (21 °C) (Fig. 1a). The probe was placed in contact with the specimens at the medial surface of the tibia (facies medialis) above the medullary cavity. The edge of the probe was aligned with (i) the distal cut plane and (ii) the long main axis of the bone. The protocol required the acquisition of three cycles of 400 successive measurements. During each cycle, the probe was slowly tilted in both circumferential directions (arrow in Fig. 1a) to scan a wide region above the medullary cavity. Between the cycles, the probe was removed from the water bath and repositioned. At each measurement, 120 RF signals (5 × 24) were acquired from each propagation direction. The scan time per cycle was about 3 min.

Cortical thickness (Ct.Th) and porosity (Ct.Po) were estimated by fitting a transverse isotropic free plate model (Fig. 2) to the measured guided wave dispersion curves in line with Minonzio et al. [23]. Briefly, the recorded time signals were transformed to the frequency-wavenumber (f-k) space using a singular value decomposition (SVD) enhanced two-dimensional spatiotemporal Fourier transform. This signal processing step provided the so-called Norm function of which each pixel (f, k) reflects the presence rate of a guided wave mode in a 0 to 1 scale [4]. Subsequently, a transverse isotropic free plate model was fitted to the maxima of the Norm function (Fig. 3c). A plate model was used since effects from the bone’s curvature can be neglected [25]. The model required four elastic coefficients, the mass density, and the thickness of the waveguide. Granke et al. suggested that in aged women, changes in Ct.Po account for most of the bone’s mesoscopic elasticity variations [26]. We used thus a micro-mechanical model [9] to calculate a set of effective mesoscale stiffness tensors for a set of porosity values assuming that transverse isotropic elastic coefficients (c₁₁ = 26.8 GPa, c₁₃ = 15.3 GPa, c₃₃ = 35.1 GPa, and c₅₅ = 7. 3 GPa) and mass density (ρ = 1.91 g^.cm⁻³) for the tissue matrix are invariant [26]. Then, the predicted mesoscale stiffness tensors were used to create a database of dispersion curves for a combination of porosity and thickness values. The thickness ranged from 2.5 to 5.5 mm with intervals of 0.1 mm and the porosity from 1 to 25% with intervals of 1%. Figure 2 shows the effect of changes in Ct.Th and Ct.Po on the modeled dispersion curves. Ct.Po mainly modifies the slope of the curves in the f-k space (a), whereas the curves shift towards lower frequencies with increasing Ct.Th (b).

To find the best fit between the plate model and the experimental dispersion curves (Fig. 3c), the model database was projected onto the singular vector basis U(f) of the Norm function. Accordingly, the objective function is denoted aswhere f_min and f_max correspond to the frequency bandwidth limits, M denotes the number of theoretical guided modes, and e^test the testing vector being a normalized attenuated plane wave. Figure 3b shows contour plot representations of the objective function with the global maxima corresponding to the fitted models in Fig. 3c. Due to ill conditioning of the objective function, i.e., incomplete experimental dispersion curves, often more than one local maxima was obtained. To remove this model ambiguity, we compared the two highest local maxima for each of the 400 successive measurements: when the highest maximum (global) exceeded the second highest of at least 3%, this was considered to be a valid solution to the problem. The threshold was empirically chosen based on a tradeoff between the standard deviation of the Ct.Th/Ct.Po estimates and the total number of valid measurements per measurement cycle. If at least 10 of the 400 successive measurements produced a valid parameter pair, the medians of the Ct.Th/Ct.Po estimates were calculated. Otherwise, the entire measurement cycle would have been rejected. Finally, if at least two out of three cycles were valid, the medians of these valid cycles were averaged for every specimen. Otherwise, the entire measurement series for that sample would have been rejected.

The velocity of the first arriving signal (υ_FAS) was calculated based on a bi-directional measurement [27]. Briefly, the time of flight was determined for each emitter-receiver distance using the first extremum of the signal in the time domain. The sound velocity was derived from the inverse slope of a linear fit through these time points plotted against the known emitter-receiver distances. This procedure was performed for each of the five transmissions and from both directions to account for small inclination angles between the probe and the bone surface. It was previously shown that larger probe inclination angles increase the relative measurement error of υ_FAS [28]. Accordingly, bi-directional measurements for which the absolute difference between the two opposite velocities exceeded 50 m s⁻¹ were eliminated [29]. Ideally, a measurement provided five corrected velocities, corresponding to the five bi-directional ultrasound transmissions per measurement, which were then averaged. Histograms of the corrected velocities were obtained for each measurement cycle. For each specimen, the υ_FAS was calculated as the average of the three histogram peaks.

The A₀ mode phase velocity (υ_A0) was calculated in the frequency-domain based on SVD-enhanced 2D Fourier transforms of the acquired multi-dimensional radiofrequency signals. The principal signal processing steps are illustrated in Online Resource 1. First, the Norm function was converted from the frequency-wavenumber (f-k) into the frequency-phase velocity (f-cp) domain (cp = 2π f/k) [30]. Afterwards, the A₀ mode was extracted using a fixed frequency (0.5 to 0.8 MHz) and c_p range (1400 to 1900 m s⁻¹). Inside this window, the amplitudes of the Norm function were averaged over frequency generating a characteristic single-peaked function of which the maximum was defined as uni-directional velocity υ_A0. For each individual measurement of a cycle, the harmonic mean of the two bi-directional velocities was calculated to correct for inclination angles between the probe and bone surface [28]. Unstable measurements for which the absolute difference between two opposite velocities was larger than 50 m s⁻¹ were eliminated [15]. The final velocity υ_A0 of a tibia specimen was calculated by averaging the peak values of the velocity histograms obtained for each cycle.

Micro-computed tomography (μCT) with 39 μm isotropic voxel size was used to measure cortical thickness (Ct.Th_μCT) and vBMD. The proximal epiphyses were removed with a hand saw to fit the frozen shaft specimens into a custom-made thermo-isolated plastic tube. The tube containing the specimen was filled with dry ice and scanned with the μCT system (VivaCT 80, Scanco Medical, Brüttisellen, Switzerland). Before scanning, the bone’s longitudinal axis was aligned with the rotation axis of the sample holder. The diameter of the field of view was 50 mm, allowing the imaging of the entire shaft cross-section. Source voltage and current were set at 70 kV and 114 μA, respectively. Five hundred projections were acquired over a range of 360° using an integration time of 200 m s. A filtered back-projection reconstruction was used to obtain stacks of 1850 slices with an isotropic voxel size of 39 μm. The gray values of the images were transformed into mgHA^.cm⁻³ based on a calibration procedure provided by the scanner vendor. The bone region insonified by AT was extracted from the μCT stack (approximately 30 mm, equivalent to 795 slices) and first binarized using Otsu’s method [31]. After this, the cortical bone compartment was automatically segmented applying the algorithm proposed by Burghardt et al. [32]. The radius of the structuring element for morphological closing of the mask was set to 0.03 mm. A manual correction was needed for one sample which had the highest Ct.Po_μCT (22%). vBMD was defined as the mean mineralization value for all voxels in the cortical compartment at the medial portion of the tibia and above the medullary canal. Ct.Th was calculated in that region as the minimum distance between endosteal and periosteal surfaces [33].

Scanning acoustic microscopy (SAM) provided the acoustic impedance (Z_SAM) of the cortical bone matrix. Cross sections of approximately 20 mm thickness were extracted from the diaphysis, site-matched with the region of the AT receiver array (Fig. 1b). Of each section, the proximal surface was polished using a planar grinder (Phoenix 4000, Buehler Ltd., Illinois) at constant speed (50 rpm) and decreasing grain size (ISO/FEPA grit: P80, P600, P1200, P2500, and P4000, Buehler Ltd., Illinois). Subsequently, the samples were washed and degassed for approximately 30 min. The custom-made microscope and scanning procedure have been described in detail elsewhere [34]. Briefly, a 100-MHz spherically focused transducer was used (KSI 100/60°, Krämer Scientic Instruments, Herborn, Germany) which had a − 6-dB bandwidth at the confocal pulse echo between 84.4 and 100.7 MHz. The − 6-dB depth of focus and lateral beam diameter in the focal plane were 139 and 19.8 μm, respectively [35]. The samples were immersed in a temperature-controlled tank with a 25 °C degassed 1% PBS solution. Images were acquired by moving the transducer along the x-y-plane with a scan increment of 12 μm. The scan time was up to 5 h. A defocus correction was applied before the images were converted into acoustic impedance maps (Fig. 3a) using calibration materials (PMMA and titanium). The cortical compartment was obtained by drawing the endosteal boundary manually (following the rules proposed by Malo et al. [10]) whereas the periosteal boundary was detected automatically by morphological region filling and tracing of the contour on the segmented image. Segmentation was performed using an adaptive threshold as described by Lakshmanan et al. [36]. The acoustic impedance (Z_SAM) was calculated as the mean impedance value of all bone tissue pixels within the cortical compartment at the medial portion of the tibia and above the medullary canal. Z_SAM was converted into the stiffness coefficient c₃₃ using a non-linear regression function [37].

Rectangular parallelepiped samples of cortical bone were harvested from the cross sections, previously scanned with SAM (Fig. 3a), for the characterization of cortical porosity (Ct.Po_μCT). The typical sample size was 2 × 3 × 4 mm³. Cutting was performed using a precision linear saw (Isomet 4000, Buehler GmbH, Düsseldorf, Germany). In the desktop μCT system (Skyscan 1172, Bruker MicroCT, Kontich, Belgium), the samples were positioned so that the anatomical vertical axis was aligned with the rotation axis of the sample holder. A source voltage of 80 kV, a current of 100 μA, and steps of 0.3° over 180° rotation were used. The exposure time for each frame was 320 ms. Twenty frames were averaged leading to a total scan time of 60 min per sample. A 0.5-mm-thick aluminum filter reduced beam hardening artifacts. Images were saved as 16-bit TIFF files and reconstructed using a filtered back-projection algorithm (NRecon, V1.6.10.4, Skyscan NV, Kontich, Belgium) with 20% ring artifact correction. For each parallelepiped sample, a stack of 650 sections was reconstructed with a 1968 × 1968 pixel field of view and 7.4 μm isotropic voxel size. Further post-processing was performed using the software CTan (V1.16.1.0, Skyscan NV, Kontich, Belgium). To separate the sample from the background, a semi-automatic procedure was performed, based on manual contouring on a selected number of slices, and followed by interpolation. A Gaussian 2D filter (R = 1) was applied to the images which were then segmented using the Otsu’s method [31]. Finally, in a 3D analysis, the tissue volume (TV), pore volume (PV), and cortical porosity (Ct.Po = PV / TV * 100%) were calculated.