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) $\in [5-25]\hspace{0.17em}p\mathit{ore}/m{m}^2$ and size (Ct.Po.Dm) $\in [30-100]\hspace{0.17em}\mu m$ 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 $=285.9Ct.\mathit{Po}.D{m}^{-1.49},\hspace{0.17em}{R}^2=0.989\hspace{0.17em},\hspace{0.17em}p=4.96\times {10}^{-5},\mathit{RMSE}=0.06;\hspace{0.17em}D=6.91Ct.\mathit{Po}.D{n}^{-1.01},\hspace{0.17em}{R}^2=0.94,\hspace{0.17em}p=2.8\times {10}^{-3}\hspace{0.17em},\hspace{0.17em}\mathit{RMSE}=0.09$), suggesting the potential of the proposed technique for the characterization of the cortical microarchitecture.

I. INTRODUCTION

Long bones consist of a trabecular core and an outer compact cortical bone shell. The latter is highly dense, but is pervaded by elongated pores, which results in a porosity (Ct.Po) ranging between 3% and 28% (Bousson et al., 2001; Eneh et al., 2017). For decades, studies of bone pathologies have focused on the characterization and evaluation of trabecular bone, as it is viewed as metabolically more active and is a common fracture site (Cooper et al., 2016). However, the assessment of cortical bone is also of high interest since it bears considerable loads and is closely linked to bone strength (Cooper et al., 2016; McCalden et al., 1993; Palacio-Mancheno et al., 2014; Rajapakse et al., 2015; Schneider et al., 2007; Zebaze et al., 2010). The load-carrying capacity of cortical bone is considerable at relevant sites such as vertebrae and femoral neck (Augat et al., 2009; Burghardt et al., 2010; Cooper et al., 2016; Rockoef et al., 1969; Holzer et al., 2009; Spadaro et al., 1994). It has been shown that bone elastic modulus, toughness, and impact energy absorption capacity decrease as cortical porosity increases (Augat et al., 2009).

Bone mineral density (BMD) is extensively used to diagnose and monitor osteoporosis as it is inversely related to fracture risk (Marshall et al., 1996; Ross et al., 1990). The most common technique to measure BMD is dual energy x-ray absorptiometry (DXA). Due to its limited resolution and use of ionizing radiation, it is important to find reliable alternatives to DXA. Quantitative ultrasound (QUS) is a potential alternative, as it permits measurements of fracture-associated bone properties at relevant anatomical regions (e.g., the distal radius and the proximal femur).

Despite significant improvements in ultrasound imaging techniques for soft tissues, ultrasound still presents limitations for imaging solid and porous media such as bone, in which the difference in acoustic impedance between bone tissue and marrow-filled pores is large, leading to large amounts of scattering and attenuation. The potential of multiple scattering to reflect microstructural features of heterogeneous media has been applied in fields such as geophysics, optics, and acoustics (Aubry et al., 2008; Aubry and Derode, 2007; Van Der Mark et al., 1988; Sato and Fehler, 2012; Tourin et al., 1997; Tourin et al., 2000; Wiersma et al., 1997; Wolf and Maret, 1985). Aubry et al. have observed the signature of multiple scattering in trabecular bone (Derode et al., 2005). Tourin et al. took advantage of constructive interferences of backscattered signals in multiple scattering media to extract parameters such as the diffusion constant (D) and transport mean free path (Tourin et al., 2000). Aubry and Derode have exploited the same principles to measure the diffusion constant in phantoms of steel rods submerged in water (Aubry and Derode, 2007) and concluded that the diffusion constant could discriminate between samples with different scatterer densities. We hypothesize that a similar approach can be applied to bone, in which the typical pore size and the impedance difference between pores and bone tissue is appropriate to lead to multiple scattering and give rise to a diffusive regime of ultrasound waves.

In the present numerical study, we calculate the diffusion constant D for bone geometries with artificially modified porosities when cortical pore density (Ct.Po.Dn) and pore diameter (Ct.Po.Dm) are varied independently. The aim is to determine the dependence of D to structural features such as pore size and pore density. Compared with experimental data acquisition, the numerical approach proposed here allows the necessary, full control over Ct.Po.Dm and Ct.Po.Dn.

This article presents numerical results exploiting ultrasound multiple scattering and the diffusive properties of cortical bone with controlled porous microstructures. For each structure, the diffusion constant D is extracted. The sensitivity of D to both pore size and pore density is then evaluated.

II. MATERIALS AND METHODS

A. Generation of bone geometries

To study the effect of Ct.Po.Dn and Ct.Po.Dm on wave propagation in bone, these two parameters needed to be accurately controlled in the simulated cortical bone geometries. First, a two dimensional (2D) image of a human femoral shaft cross-section was obtained by scanning acoustic microscopy (SAM) through the following processes: The left femur was obtained from a 71-year-old male donor in accordance with the German Law “Gesetz über das Leichen-, Bestattungs- und Friedhofswesen des Landes Schleswig-Holstein - Abschnitt II, §9 (Leichenöffnung, anatomisch)” from 04.02.2005. A 21-mm thick cross section was extracted from the proximal femur shaft and prepared for SAM as described elsewhere (Iori et al., 2018). Briefly, after removal of soft tissues and washing in phosphate buffered saline (PBS), one surface of the sample was polished on a planar grinder (Phoenix 4000, Buehler Ltd. Illinois). After the polishing, samples were washed again and degassed while submerged in 1% buffer solution. The SAM measurement was performed using a custom built quantitative scanning acoustic microscope (Malo et al., 2013) and a 100-MHz spherically focused transducer (KSI 100/60°, KSI, Herborn, Germany), providing a beam diameter of 19.8 μm at a focal depth of 139 μm (Raum, 2008). Device and scanning procedure have been described in detail elsewhere (Lakshmanan et al., 2007). The image was acquired with a pixel size of 12 μm in the x-y plane. Recorded signals were then bandpass-filtered (Chebyshev filter) and the reflected amplitude determined as the maximum of the (Hilbert-transformed) envelope signal. Established procedures were applied for defocus correction, calculation of acoustic impedance (Z) values (Hube et al., 2006; Raum et al., 2006) and thresholding of the bone tissue (Lakshmanan et al., 2007). The obtained image was then resampled to reach the desired pixel size of 10 $\mu m$ for simulations. To resample the image, bicubic interpolation method was used such that the output pixel value would be the weighted average of pixels in the nearest 4 by 4 neighborhood.

To measure the average pore diameter and pore density, every single closed surface within the segmented bone cross-section was labelled. The number of labels indicated the total number of pores, while the ratio between pore number and the entire bone area provided the pore density Ct.Po.Dn, in number of pores/mm². To estimate the mean pore diameter, pores were assumed to have circular shape. By measuring the surface area of each pore and averaging the calculated diameters, the mean pore diameter Ct.Po.Dm was obtained.

The reference image was modified to obtain structures with desired Ct.Po.Dn and Ct.Po.Dm. The modification of Ct.Po.Dm, was performed such that the average pore size would decrease radially from the endosteal towards the periosteal surface. The reason for this is that trabecularization of cortical bone commonly starts from the endosteal surface (Cooper et al., 2016) and larger pores are usually observed close to the endosteum (Bakalova et al., 2018).

To obtain bone images with average pore size varying radially, the total number of pores and subsequently Ct.Po.Dn were kept constant at 10 $\hspace{0.17em}\mathit{pore}/m{m}^2$, according to results of a previous work by Bousson et al. (2001). To obtain the exact Ct.Po.Dn of 10 $\mathit{pore}/m{m}^2$, some pores were randomly removed from the reference SAM image. The cortical bone in the resulting image was then divided into four concentric curved bands with approximate width of 750 $\mu m$ [Fig. 1(b)].

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.

By morphologically dilating or eroding pixels from the pores falling within each band and averaging Ct.Po.Dm over the whole image we created 8 bone structures with varying average Ct.Po.Dm ranging from 30 to 100 $\mu m$. Figures 1(b) and 1(c) show examples of two bone slices with the same Ct.Po.Dn and different average Ct.Po.Dm.

A similar workflow was followed to generate bone images with varying Ct.Po.Dn. A reference image with average Ct.Po.Dm of 60 $\mu m$ was modified by randomly adding/removing pores, all the while maintaining a higher Ct.Po.Dn in the endosteal region and a lower Ct.Po.Dn near the periosteal surface. This provided 8 different structures with average Ct.Po.Dm of 60 $\mu m$ and Ct.Po.Dn ranging from 5 to 25 $\mathit{pore}/m{m}^2$.

For each combination of Ct.Po.Dn and Ct.Po.Dm, three structures were created with different random pore distributions resulting in a total of 45 structures. The ranges for Ct.Po.Dn and Ct.Po.Dm were chosen in agreement with those reported in previous works (Cowin and Telega, 2003; Evans and Bang, 1967; Rajapakse et al., 2015; Thomas et al., 2006).

B. Data acquisition and processing

The SimSonic open-source software (www.simsonic.fr) (Bossy et al., 2004) was used to compute a numerical solution to the elastic wave propagation in the 2D cortical bone structures described above. The approach was based on a finite-difference time-domain algorithm and could model the wave propagation in heterogeneous media with a combination of solid and fluid materials. A 64-element linear array transducer with central frequency of 8 MHz and element width of 0.3 mm was simulated. Simulations were carried out in the near field, resembling potential future in vivo applications of ultrasound, where cortical bone is accessible; e.g., at the tibia shaft. Despite being derived from a femoral bone image, the present simulations remain informative considering the structural and acoustical similarities between the cortical bone of the tibia and femur (Tatarinov and Sarvazyan, 2008; Wang and Norman, 1997). The linear array transducer was placed at a distance of 3 mm from the periosteal surface. Homogeneous material properties (Table I) for bone and water were assigned to each pixel (bone tissue and pores, respectively) of the binary SAM image.

TABLE I.

Values of stiffness constants for isotropic cortical bone (Bossy et al., 2004) and water used in the simulations.

	Bone	Water
Density	1.85 $\left(\frac{\mathrm{g}}{{\text{cm}}^3}\right)$	1 $\left(\frac{\mathrm{g}}{{\text{cm}}^3}\right)$
${\mathit{C}}_{11}={\mathit{C}}_{22}={\mathit{C}}_{33}$	29.6 (GPa)	2.37 (GPa)
${\mathit{C}}_{12}={\mathit{C}}_{23}={\mathit{C}}_{31}$	17.6 (GPa)	2.37 (GPa)
${\mathit{C}}_{44}={\mathit{C}}_{55}={\mathit{C}}_{66}$	6.0 (GPa)	0 (GPa)

In addition to the structures modified from one reference SAM image (45 structures), simulations were run for four other femur images obtained by scanning acoustic microscopy from different donors to account for the effect of individual differences. To have a wider range of average pore diameter (Ct.Po.Dm ∈ [55.34-71.42 μm]) and pore density (Ct.Po.Dn ∈ [10–21 $\mathit{pore}/m{m}^2$]), simulations were also run on modified versions of these images where either average Ct.Po.Dm or Ct.Po.Dn were reduced to 60 $\mu m$ and 10 $\mathit{pore}/m{m}^2,$ respectively.

Perfectly matched layers with a thickness of 7.5 mm were added at the four boundaries of the simulation map, to ensure minimal reflections while not increasing the simulation time drastically. A grid step of 10 $\mu m$ was chosen for the simulations to meet 20 points per wavelength requirement (Bossy, 2012).

Each element of the array was excited with a Gaussian pulse with a central frequency of 8 MHz (3DB bandwidth= 3.33 MHz). For each single-element transmit, the reflected signals were recorded by all 64 elements. Figure 2 illustrates the data acquisition process for one transmit. The process was repeated for all 64 transducer elements providing an inter-element response matrix $H$, containing $64\times 64$ time traces. Due to reciprocity, the inter-element response matrix $H$ is symmetric such that $h_{\mathit{ij}}=h_{\mathit{ji}}$ where $h$ is the backscattered signal in the time domain, and the indices i and j indicate the number of emitter and receiver elements, respectively.

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.

For all structures, the diffusion constant D was determined according to a method proposed by Aubry and Derode (2007). The first step was to time-shift the $h_{\mathit{ij}}$ signals such that all the response signals started at the same time. This time-shifting step was necessary to compensate for differences in arrival times due to differences in transmitter–receiver distances and to the curvature of the bone surface. The received signals have two main components: (1) the signal that initially has been transmitted and has travelled the exact distance between the emitter and receiver, (2) the signal that has been reflected by the medium. Figure 3(a) shows the signal emitted by element 1 and received by element 10 and Fig. 3(b) shows all the 64 received signals for the first emission. By setting a threshold of 0.02 times the maximum of the signal amplitude, the start of each signal was detected and shifted to time T = 0 μs [Fig. 3(c)]. Then the initial part of the signal (that has just traveled the emitter–receiver distance and was shifted to time T = 0 $\mu s$) was cut out and the shifting process with the same threshold is repeated for the backscattered signals from bone [Fig. 3(d)].

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.

After the time-shifting process, each signal was truncated into $0.5\hspace{0.17em}\mu s$ half-overlapping time-windows (Fig. 4).

FIG. 4.

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

By integrating the squared value of signals over each time window, the backscattered intensity $I_{\mathit{ij}}(T)$ was obtained:

$$I_{\mathit{ij}}\left(T\right)={\int }_{T-\Delta /2}^{T+\Delta /2}{h}_{\mathit{ij}}^2(t)\mathit{dt},$$ (1)

where $T$ is the time at the center of the time window and $\Delta$ is the width of the time window. The backscattered intensity $I(X,T)$ was calculated by averaging $I_{\mathit{ij}}(T)$ over the emitter–receiver pairs separated by the same distance $X=\left|X_{\mathit{emitter}}-X_{\mathit{receiver}}\right|$ where $X_{\mathit{emitter}}$ and $X_{\mathit{receiver}}$ indicated the location of the emitter and receiver elements, respectively. The initial part of the reflected intensity corresponds to the reflection (Derode et al., 2005) that takes place mainly at the first water–bone interface. Accordingly, data associated with the first time window ($0.5\mu s$) is cut out of the intensity matrix $I(X,T)$ to remove the initial part caused by reflection. At each time-window a typical backscattered intensity versus emitter–receiver distance (Fig. 5) exhibits a steep peak (coherent contribution) over a wider pedestal (incoherent contribution), as described in Tourin et al. (2000). The growth of the incoherent contribution over time provided access to the diffusion constant (Tourin et al., 2000). To separate the two (coherent and incoherent) contributions, an antisymmetrization method was used (Aubry and Derode, 2007). Signals were first rewritten as

$$\begin{array}{c}{h}_{\mathit{ij}}^A=h_{\mathit{ij}},\hspace{1em}\text{for}\hspace{0.17em}i>j\hfill \\ h_{\mathit{ij}}^A=-h_{\mathit{ij}},\hspace{1em}\text{for}\hspace{0.17em}i<j\hfill \\ h_{\mathit{ij}}^A=0,\hspace{1em}\text{for}\hspace{0.17em}i=j\hfill \end{array}.$$ (2)

It can be shown (Aubry and Derode, 2007) that the incoherent contribution of the backscattered intensity is

$$I_{\mathit{inc}}=\frac{1}{2}\left(I+I^A\right),$$ (3)

where $I^A$ is the backscattered intensity derived from $H^A$. For highly scattering media, the incoherent intensity can be approximated as follows (Aubry and Derode, 2007):

$$I_{\mathit{inc}}(X,T)\simeq {\left(4\pi \mathit{DT}\right)}^{-(1/2)}I\left(X,T\right)\text{exp}\hspace{0.17em}\left(-\frac{X^2}{4DT}\right),$$ (4)

where D represents the diffusion constant and the term $\text{exp}(-X^2/4DT)$ describes the temporal evolution of the incoherent intensity. To retrieve the constant D, a Gaussian curve was fitted to the function I_inc (X,T) at each time window (Fig. 6). It was then normalized with respect to its maximum and its standard deviation was obtained. According to Eq. (4), the variance or squared standard deviation of the Gaussian curves over time follows a linear trend with slope equal to 2D. Identifying this slope with a linear fit allowed to measure D for all Ct.Po.Dm–Ct.Po.Dn combinations. All data processing was performed using Matlab 2016 (Mathworks, Inc., Natick, MA).

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.

Associations between D and the micro-structural parameters Ct.Po.Dm and Ct.Po.Dn were characterized in terms of the coefficient of determination $R^2$ and the root mean square error with respect to a power-law fit of the data. A Spearman rank correlation test was also performed to evaluate the monotonic relationship between D and those parameters.

III. RESULTS

The reference SAM image of a femur of a 71 yr old male donor had average Ct.Po.Dm = 60.7 $\mu m$ (min = 11.28 $\mu m$; max= 525.76 $\mu m$; std = 44.31 $\mu m$), Ct.Po.Dn = 19.83 $\mathit{pore}/m{m}^2$ and porosity = 8.79%, before porosity manipulation. To confirm the choice of pixel size, one structure (modified from the original reference SAM image) with Ct.Po.Dm = 30 μm, Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$ and the pixel size of 10 μm was resampled down to 5 μm. Results for D were compared for the two pixel sizes of 10 and 5 μm. A 2% difference in D was found between simulations run with a 5 $\hspace{0.17em}\mu m$ pizel size and simulations run with a 10 $\hspace{0.17em}\mu m$ pixel size, while the computational cost for the smaller pixel size indicated a 700% increase. In addition, the signals corresponding to emitter 32 and receiver 32 were compared for that image with the two mentioned pixel sizes. The root mean square error of the difference was 2%. Results suggest that 10 $\mu m$ was a reasonable pixel size that allowed to maintain accuracy while mitigating computational costs.

The selected time-window length for all the simulations was 0.5 $\mu s$. The choice of time-window length was evaluated by comparing the obtained diffusion constant values for seven different time-windows ($\Delta \in [0.25,\hspace{0.17em}0.4,\hspace{0.17em}0.5,\hspace{0.17em}0.6,\hspace{0.17em}0.7,\hspace{0.17em}0.8,0.9,\hspace{0.17em}1]\mu s$) for a structure with average Ct.Po.Dm = 60 $\mu m$ and Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$. The coefficient of variation [CV $=100\times (\mathit{std}/\mathit{mean})$] of D for those given time-windows, was 1.59% suggesting that a variation of the time window has a minor effect on D. Hence $\Delta =0.5\mu s$ was an acceptable time-window.

In Fig. 7, the incoherent intensity (in grey shade) is plotted as a function of time and receiver–transmitter distance, for two images modified to have different average Ct.Po.Dm. The growth of the diffusive halo over time is larger for the sample with smaller pore size (and lower porosity). The variance of the Gaussian fit of the incoherent intensity was obtained at each time widow, for two different structures (Fig. 8). As explained in Sec. II, this variance is expected to increase linearly with time. The slope of its linear fit, which is proportional to the diffusion constant, was smaller for the structure with higher porosity (Fig. 8). When pore density was kept constant (Fig. 9), the diffusion constant decreased as the average pore size increased. A power law was fitted to model the D-Ct.Po.Dm relation: D $=285.9Ct.\mathit{Po}.D{m}^{-1.49}$ ($R^2=0.99,\hspace{0.17em}p=4.96\times {10}^{-5},\hspace{0.17em}\mathit{RMSE}=0.06$). In Fig. 10, the diffusion constant for samples with constant Ct.Po.Dm of 60 $\mu m$ is plotted for different pore densities. A power fit of this plot provided the relation $D=6.91Ct.\mathit{Po}.D{n}^{-1.01}\hspace{0.17em}(R^2=0.94,\hspace{0.17em}p=2.8\times {10}^{-3},\hspace{0.17em}\mathit{RMSE}=0.09)$, between D and Ct.Po.Dn.

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 $\mu m$).

The average and standard deviation of the diffusion constant D for all the structures were 0.69 and 0.47 $m{m}^2/\mu s,$ respectively. The p values for the Spearman correlation between D and parameters Ct.Po.Dm and Ct.Po.Dn were $4.96\times {10}^{-5}$ and $2.8\times {10}^{-3},$ respectively; suggesting significant correlations and monotonous relationships between D and those micro-structural parameters.

To account for individual differences, additional femur SAM images from other donors were used in simulations to obtain the diffusion constant values. Table II summarizes the results. Results from the modified versions of those samples at Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$ and average Ct.Po.Dm = 60 $\mu m$ are presented in Figs. 9 and 10, respectively, and summarized in Table II.

TABLE II.

Diffusion constant values for bone samples from different donors.

	Average Ct.Po.Dm ($\mu \mathrm{m}$)	Ct.Po.Dn ($\frac{\mathrm{pore}}{m{m}^2}$)	D ($\frac{{\text{mm}}^2}{\mu \mathrm{s}}$)
Sample 1 (original sample modified to obtain Figs. 9 and 10)	60.7	19.83	0.09
Sample 2	55.43	16.52	0.24
Modified sample 2 (Ct.Po.Dn artificially reduced to 10$\hspace{0.17em}\frac{\mathit{pore}}{m{m}^2}$)	55.34	10.01	0.44
Sample 3	59.16	21.00	0.15
Modified sample 3 (Ct.Po.Dn artificially reduced to 10$\hspace{0.17em}\frac{\mathit{pore}}{m{m}^2}$)	60.48	10.00	0.49
Sample 4	72.92	15.53	0.25
Modified sample 4 (Ct.Po.Dn artificially reduced to 10$\hspace{0.17em}\frac{\mathit{pore}}{m{m}^2}$)	72.70	10.02	0.31
Modified sample 4 (average Ct.Po.Dm artificially reduced to 60$\mathrm{\mu m}$	59.84	15.39	0.55
Sample 5	71.42	15.74	0.22
Modified sample 5 (Ct.Po.Dn artificially reduced to 10$\hspace{0.17em}\frac{\mathit{pore}}{m{m}^2}$)	72.29	10.02	0.35
Modified sample 5 (average Ct.Po.Dm artificially reduced to 60$\mathrm{\mu m}$	60.19	15.34	0.40

IV. DISCUSSION

High-resolution scanning acoustic microscopy images of human cortical bone were processed to model the ultrasound backscatter from cortical bones with different pore densities and pore diameters.

The diffusion constant was calculated for different cortical microstructures by measuring the growth rate of the incoherent portion of the backscattered intensity over time. The effects of pore size and pore density on the diffusivity of the medium were studied independently. Increasing average Ct.Po.Dm and Ct.Po.Dn both resulted in a significant and consistent decrease in the diffusion constant. This result suggested that the diffusion constant as assessed by quantitative ultrasound could be used to characterize the cortical bone microstructure, with potential applications to diagnosis and monitoring of osteoporosis. Our results showed that an increase in either pore size or pore density contributes to higher number of scattering events taking place in the medium, leading to lower diffusion constants.

In healthy bone, the average diameter of Haversian canals ranges from about 40 to 100 $\mu m$ (Cowin and Telega, 2003; Rajapakse et al., 2015). Results from a study on femur neck biopsies suggest that osteonal canals with diameters as large as 172 $\hspace{0.17em}\mu m$ exist in cortical bone. However, such large pores contribute to only about 2.5% of the total pore population (Bell et al., 2000). For this reason, the mean Ct.Po.Dm did not exceed 100 $\hspace{0.17em}\mu m$ in our simulations. Our findings demonstrated that the acoustic diffusion constant of cortical bone was strongly related to the average pore diameter, in the healthy Ct.Po.Dm range (30–100 $\mu m$; Fig. 9; $R_{\mathit{powerlaw}}^2=0.98$). Similarly, the acoustic diffusion constant was highly dependent (Fig. 10; $R_{\mathit{powerlaw}}^2=0.94$) on variations of the pore density in the 5–25 $\mathit{pore}/m{m}^2$ range which was close to the ranges reported in the literature (6–30 $\mathit{pore}/m{m}^2$) (Nalla et al., 2005; Thomas et al., 2006).

The present study, evaluates the diffusion constant as a tool to evaluate changes in the porous microstructure of cortical bone. Ct.Po.Dm and Ct.Po.Dn combined are related to porosity (i.e., the ratio of pore volume over total volume) which is a major determinant of the tissue mechanical properties and resistance. However, the specific focus of this study was to evaluate the independent effects of Ct.Po.Dm and Ct.Po.Dn on D. The numerical procedure utilized here has the advantage of providing full control over the cortical porosity by allowing the manipulation of pore density and pore size independently.

To account for the effect of individual differences, results from four femur images in addition to the original reference image (sample 1) are presented in Table II. For each sample, a reduction of Ct.Po.Dn, caused an increase in D. For samples 4 and 5, a reduction of average Ct.Po.Dm, increased the diffusion constant value, indicating that higher pore sizes are associated with less diffusive regimes. Comparison between samples 2, 4, 5, and Ct.Po.Dm-reduced versions of samples 4 and 5, indicated that for relatively high pore densities (Ct.Po.Dn $\approx 16\hspace{0.17em}\mathit{pore}/m{m}^2$), the diffusion constant did not present a monotonous behavior. An increase in Ct.Po.Dn for samples from different donors at Ct.Po.Dn = 10 $\mathit{pore}/m{m}^2$, resulted in a decrease in D (Fig. 10) which was in accordance with the results from modified versions of sample 1. As shown in Fig. 9, an increase in Ct.Po.Dm resulted in an overall decrease in D for samples from other donors (crosses) as well as the modified versions of the reference SAM image (sample 1). However, results from modified femur images of other donors indicated slightly smaller values for D compared to the results from sample 1. Different factors such as the shape of the bone, local concentration of the pores close to the periosteal region, and presence of relatively large pores (Ct.Po.Dm > $300\mu m$) may have resulted in slightly different diffusion constant values for different samples with the same average Ct.Po.Dm and Ct.Po.Dn.

Our approach is affected by several limitations. First, our simulations did not model absorption and the decay of the backscattered signals was only due to scattering. We expect absorption to affect the correlations between acoustic diffusion and cortical bone microstructure reported here, to an extent that is yet to be determined. The main reason for neglecting absorption in our simulations of ultrasound backscatter was the lack of a reliable numerical model for absorption of shear waves. The influence of absorption should be addressed in future studies. The effect of cortical thickness [ranging from 0.25 to 6 mm (Treece et al., 2010)] on the diffusion constant could also be investigated in the future, considering the fact that thinning of the cortical shell is one of the signs of osteoporosis. However, it is worth noting that in our simulations, no second echo (from the inner cortical wall) was observed. This is attributed to high amounts of multiple scattering at the operating frequency of the simulations. Another point to notice is that the size of the time-window for integration of the backscattered intensity must be small enough to allow for analysis of the intensity while having the adequate length to include at least one scattering event. Thus, pre-existing knowledge regarding the length of the scattering mean free path, and therefore of the range of pore diameter and density, is beneficial when choosing the length of the time window. The time window used in processing the data corresponds to four wavelengths in the solid phase. However, changing its length for one of the structures from 0.25 to 1 $\mu s$ did not have a significant effect on D (CV = 1.59%). The material properties of the two phases (bone and water) were assumed to be homogeneous within the same bone and for all simulations. This is not the case in real bone. 2D simulations like those performed in this study represent a strong simplification of the highly heterogeneous 3D bone structure, mainly because they fail to model out-of-plane diffraction and out-of-plane scattering (Cooper et al., 2006). While 3D simulations could provide more accurate models, they could not be done because of their excessive computational cost. Supported by the translational symmetry of cortical bone pores in the diaphysis of long bones, we assumed 2D models to hold validity. Finally, adding a layer of soft tissue surrounding the bone would also improve the simulations in terms of mimicking in vivo experiments. Considering all the simplifying assumptions, it is worth noting that the main purpose of this study is to reveal the existence of a monotonous relation between micro-structural parameters and D in a qualitative context rather than a quantitative one. The power-law fits are exclusively used to show the decreasing, monotonous trend that D manifests as Ct.Po.Dm or Ct.Po.Dn increase, rather than to predict an exact quantitative relationship between D and the pore size-density combination. This will be the subject of future studies, which will combine numerical, experimental, and analytical methods.

V. CONCLUSION

The current numerical study takes advantage of multiple scattering of ultrasound in diffusive media to measure the diffusion constant in cross-sections of cortical bone. Scanning acoustic microscopy images of human proximal femur shaft cross sections were modified to control the average cortical pore diameter and density. The diffusion constant was found to be monotonously influenced by both scatterer size and density. These results support the use of ultrasound multiple scattering for the assessment of cortical bone, and proposes the acoustic diffusion constant as a potential tool for the characterization of the cortical bone microstructure.