Acoustic diffusion constant of cortical bone: Numerical simulation study of the effect of pore size and pore density on multiple scattering

Abstract

While osteoporosis assessment has long focused on the characterization of trabecular bone, the cortical bone micro-structure also provides relevant information on bone strength. This numerical study takes advantage of ultrasound multiple scattering in cortical bone to investigate the effect of pore size and pore density on the acoustic diffusion constant. Finite-difference time-domain simulations were conducted in cortical microstructures that were derived from acoustic microscopy images of human proximal femur cross sections and modified by controlling the density (Ct.Po.Dn) ∈[5-25] pore/mm² and size (Ct.Po.Dm) ∈[30-100] μm of the pores. Gaussian pulses were transmitted through the medium and the backscattered signals were recorded to obtain the backscattered intensity. The incoherent contribution of the backscattered intensity was extracted to give access to the diffusion constant D. At 8 MHz, significant differences in the diffusion constant were observed in media with different porous micro-architectures. The diffusion constant was monotonously influenced by either pore diameter or pore density. An increase in pore size and pore density resulted in a decrease in the diffusion constant (D =285.9Ct.Po.Dm^-1.49, R²=0.989 , p=4.96×10^-5,RMSE=0.06; D=6.91Ct.Po.Dn^-1.01, R²=0.94, p=2.8×10^-3 , RMSE=0.09), suggesting the potential of the proposed technique for the characterization of the cortical microarchitecture.

FIG. 1.

(a) Binarized SAM image of a human femur shaft cross-section with Ct.Po.Dn of 10 $\mathit{pore}/m{m}^2$, average Ct.Po.Dm of 60.6 $\mu m$ (min = 11.3 $\mu m$; max = 300 $\mu m$; mean = 60.6 $\mu m$; std = 36.2 $\mu m$) and porosity of 4%. The left edge of the frame indicates the position and length of the 64-element array transducer with respect to the bone image; (b) Detail of the region in the frame in (a). To manipulate the Ct.Po.Dm the cortical bone is divided in four regions based on the distance to the endosteum: the increase in pore size is smaller for the outer bands since trabecularization of the bone starts from the endosteal region; (c) The average Ct.Po.Dm is artificially increased while keeping the Ct.Po.Dn constant. Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$; Average Ct.Po.Dm = 80.6 $\mu m$ (min = 11.3 $\mu m$; max = 354.5 $\mu m$; mean = 80.6 $\mu m$; std = 41.4 $\mu m$); Porosity = 6.9%; (d) Ct.Po.Dn is artificially increased while keeping the average Ct.Po.Dm constant; Ct.Po.Dn = 25.3 $\mathit{pore}/m{m}^2$; average Ct.Po.Dm = $60.1\hspace{0.17em}\mu m$ (min = 11.3 $\mu m$; max = 259 $\mu m$; mean = 60.1 $\mu m$; std = 35 $\mu m$); Porosity = 9.6%. Both an increase of Ct.Po.Dm and of Ct.Po.Dn lead to an increase of the sample Ct.Po.

FIG. 2.

Schematic representation of data acquisition. (a) An 8 MHz Gaussian pulse is transmitted from one transducer element to the multiple scattering medium; (b) The response is recorded on all the elements of the array transducer. Repeating the process for all the elements results in an inter-element response matrix.

FIG. 3.

(a) Signal emitted by element 1 and received by element 10 for one of the bone structures. The start of the signal is detected using a threshold equal 0.02 times the maximum of the signal (black circle) (b) All 64 received signals for the first emission. (c) The starts of the signals are detected and shifted to time T = 0 μs. (d) The initially emitted signals are cut out and the backscattered signals from bone are time-shifted with the threshold equal 0.02 times the maximum of the signals.

FIG. 4.

The first two time-windows are shown for one of the time-shifted signals.

FIG. 5.

Normalized backscattered intensity vs emitter–receiver distance for a given time window T. A sharp peak corresponds to the coherent contribution and wider pedestal corresponds to the incoherent contribution of intensity.

FIG. 6.

The normalized incoherent intensity for a given time window T is fitted with a Gaussian curve.

FIG. 7.

Incoherent intensities corresponding to bone samples with pore density of 10 $\mathit{pore}/m{m}^2$ and mean pore diameter of 50 $\mu m$ (a) and 90 $\mu m$ (b) are shown. The growth of the incoherent intensity is more pronounced for the sample with smaller pore size.

FIG. 8.

Variance of the Gaussian fit of the incoherent intensity versus time for bone samples with 8.95% and 2.63% porosity (Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$, $\text{Ct}.\text{Po}{.\text{Dm}}_{\mathit{circles}}$= 90 μm, $\text{Ct}.\text{Po}{.\text{Dm}}_{\mathit{stars}}$= 50 μm). The variance increases at a higher rate for the sample with lower porosity.

FIG. 9.

Diffusion constant for samples with Ct.Po.Dn of 10 $\mathit{pore}/m{m}^2$ and different pore sizes. Error bars are associated with 3 different generated geometries for each size-density combination. Fitted Curve: $D=285.9Ct.\mathit{Po}.D{m}^{-1.49}$; $R^2=0.989$, $p=4.96\times {10}^{-5}$, RMSE = 0.06. Results from modified femur SAM images of other donors are shown by crosses (mod2: Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$, Ct.Po.Dm = 55.34 $\mu m$. mod3: Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$, Ct.Po.Dm= 60.48 $\mu m$. mod4: Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$, Ct.Po.Dm = 72.70 $\mu m$. mod5: Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$, Ct.Po.Dm = 72.29 $\mu m$).

FIG. 10.

Diffusion constant for samples with average Ct.Po.Dm of 60 $\mu m$ and different pore densities. Error bars are associated with 3 different generated geometries for each size-density combination. Fitted Curve: $D=6.91Ct.\mathit{Po}.D{n}^{-1.01}$; $R^2=0.94$ $p=2.8\times {10}^{-3}$, RMSE = 0.09. Results from modified femur SAM images of other donors are shown by crosses (mod3: Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$, Ct.Po.Dm = 60.48 $\mu m$. mod4: Ct.Po.Dn = 15 $\mathit{pore}/m{m}^2$, Ct.Po.Dm = 59.84 $\mu m$. mod5: Ct.Po.Dn = 15.34 $\mathit{pore}/m{m}^2$, Ct.Po.Dm = 60.19 $\mu m$. s3: Ct.Po.Dn= 21 $\mathit{pore}/m{m}^2$, Ct.Po.Dm = 59.16 $\mu m$).