A Geometric Model of Ultrasound Backscatter to Describe Microstructural Anisotropy of Tissue

Abstract

Methods to assess ultrasound backscatter anisotropy from clinical array transducers have recently been developed. However, they do not provide information about the anisotropy of microstructural features of the specimens. This work develops a simple geometric model, referred to as the secant model, of backscatter coefficient anisotropy. Specifically, we evaluate anisotropy of the frequency dependence of the backscatter coefficient parameterized in terms of effective scatterer size. We assess the model in phantoms with known scattering sources and in a skeletal muscle, a well-known anisotropic tissue. We demonstrate that the secant model can determine the orientation of the anisotropic scatterers, as well as accurately determining effective scatterer sizes, and it may classify isotropic versus anisotropic scatterers. The secant model may find utility in monitoring disease progression as well as characterizing normal tissue architectures.

Introduction

Characterization of microstructural anisotropy of biological tissue in optical and magnetic resonance imaging has proven useful for monitoring disease (e.g., breast tumor associated collagen signatures using nonlinear optical microscopy) and describing normal tissue architecture (e.g., diffusion tensor imaging of the uterine cervix).^1,2 Similarly, evaluation of anisotropy with ultrasound imaging has proven useful in describing the organization of tissue microstructure in numerous studies.^3–9 Previous works have demonstrated a link between acoustic anisotropy and spatial features of media.^{4,5,7,8,10,11} This motivates our group’s development of novel quantitative ultrasound parameters which characterize anisotropy.^9,12 This paper focuses on extending our quantitative assessment of anisotropy; specifically, characterizing anisotropy of the acoustic impedance distribution.

Currently, techniques to assess ultrasound anisotropy are based on ex vivo evaluation or anisotropy of echo signal power computed using the Reference Phantom Method (RPM), assuming media may be described as a composition of cylindrical scatter sources.^9,12,13 Ex vivo studies have shown great success in quantifying both the magnitude and frequency dependence of acoustic properties like attenuation and backscatter in tissues like muscle, renal cortex, and trabecular bone to name a few.^4–8,10 However, generalizability to in vivo applications is limited given differences in perfused versus non-perfused tissues and challenges of using lab-based systems. An in vivo alternative is to utilize the backscatter power difference with the RPM, which has demonstrated sensitivity in skeletal muscle and the uterine cervix from empirically derived parameters.^9,11 However, this method only quantifies relative power compared to an isotropic phantom and removes frequency dependent information. As such, a model-based approach has potential to characterize anisotropic tissue microstructure.

In this paper, we develop a geometric model we call the “secant model” and hypothesize that it may quantify characteristics of cylindrical sources of scattering. We show that the secant model may be a promising tool to advance our understanding of ultrasound anisotropy and characterize anisotropic media.

Materials and Methods

Geometric Model of Backscatter Anisotropy

We consider acoustic scattering interactions in random media (i.e., scattering from a random spatial distribution variation in compressibility and density). In the case of incoherent scattering occurring in the far-field, the differential scattering cross section per unit volume ${\sigma }_d$ is given by

$${\sigma }_d(k)=\frac{k^4}{16{\pi }^{2}V}{\int }_V{\int }_V⟨{\gamma }^2⟩b_{\gamma }(\mathrm{\Delta }\mathit{r})e^{-i\mathit{K}\cdot \mathrm{\Delta }\mathit{r}}d{v}_{1}d{v}_2$$ (1)

where $\mathbf{K}$ is the scattering vector with magnitude $k$ (the wavenumber), $V$ is the scattering volume, $\gamma$ is the combined fractional variation in compressibility and density, $b_{\gamma }$ is the spatial autocorrelation function dependent on relative position $\mathrm{\Delta }\mathrm{r}$ between volume elements $dv$.¹⁴ Depending on characteristics of the media (e.g., spatial symmetry), equation (1) may be simplified and models may be developed allowing for parameter estimation. The acoustic backscatter coefficient ${\sigma }_b$ is defined as ${\sigma }_d$ for scattering angles of 180°.^15,16 In practice, acoustic form factors $F$ have been derived assuming isotropic scattering and are functions proportional to the Fourier transform of the correlation function. These models are parameterized by the effective scatterer size $d_{eff}$, and are related to ${\sigma }_b$ as follows

$${\sigma }_b(f)={\sigma }_{o}F\left(f,d_{eff}\right)$$ (2)

where $f$ is the temporal frequency and ${\sigma }_o$ is the backscatter coefficient in the long-wavelength limit (i.e., $f^4$ Rayleigh scattering).¹⁵ In cases where the frequency-dependence of ${\sigma }_o$ deviates from $f^4$ (i.e., when the product of the wavenumber and the scatterer radius, referred to as $ka$, are greater than 0.5), one may estimate $d_{\mathit{eff}}$ and extract information about the media’s microstructure.¹⁵

We now consider the case of a cylindrical scatterer whose dominant axis is located at some constant orientation angle $\varphi$ relative to the transducer face (see Figure 1). To examine anisotropic behavior, we electronically steer the beam lines to arbitrary beam steering angles $\theta$. At this steering angle, the variation of the major axis of the cylindric section as a function of beam steering angle and orientation angle may be determined via geometry, as described by Hilbert,¹⁷ as

$$a\left(\theta \right)=d\mathrm{s}\mathrm{e}\mathrm{c}(\theta -\varphi )$$ (3)

where $a$ is the major axis of the cylindric section created by $\theta ,$ $d$ is the cylinder’s diameter, and sec is the secant function. Now, if we assume that $d_{\mathit{eff}}$ is primarily determined by the major axis of the cylindric section, and the cylindrical scattering does not violate any of the assumptions necessary to estimate $d_{\mathit{eff}}$, we may write

$$d_{eff}\left(\theta \right)\approx a\left(\theta \right)=d\mathrm{s}\mathrm{e}\mathrm{c}(\theta -\varphi )$$ (4)

Figure 1.

A diagram showing the orientation of the cylinder and transducer. As the beam steering angle $\theta$ of the transducer is varied, the major axis of elliptical cross-sections varies according to equation (1) and is minimized upon perpendicular incidence.

Equation (3) suggests that one may estimate the diameter and orientation angle of a cylindrical scatterer based on the angle-dependence of $d_{\mathit{eff}}$. In practice, we found that it was necessary to add an empirically motivated “anisotropy factor” $\xi$ to our geometric model, resulting in the following equation

$$d_{eff}\left(\theta \right)=d\mathrm{s}\mathrm{e}\mathrm{c}(\xi (\theta -\varphi ))$$ (5)

In essence, $\xi$ quantifies the deviation of the angle-dependence of $d_{\mathit{eff}}(\theta )$ from the angle-dependence of the diameter of the major axis using geometry as given in equation (2). The effect of different values of $\xi$ on equation (5) are shown in Figure 2. An interpretation of $\xi$ is given in the Discussion below.

Figure 2.

A plot showing the variation of effective scatterer size $d_{\mathit{eff}}$ (as given in equation (5)) as a function of the “anisotropy factor” $\xi$, with an orientation angle $\varphi =0^{\circ }.d_{\mathit{eff}}$ is plotted as a function of beam steering angle u and is normalized to the cylinder’s diameter $d$. In the case where $\theta$ is equal to unity, $d_{\mathit{eff}}$ represents the major axis of the cylindric section, as shown in equation (2). As $\theta$ increases beyond unity, we can appreciate that $d_{\mathit{eff}}$ is larger than the length of the major axis predicted from the geometric model. As $\xi$ decreases below unity, we find that $d_{\mathit{eff}}$ becomes less sensitive to beam steering angle, until reaching $\xi =0$ when $d_{\mathit{eff}}$ is independent of $\mathrm{\theta }$.

Currently, acoustic form factors used to estimate $d_{\mathit{eff}}$ are derived from form factors that assume spherically symmetric scatterers and assume deviations occur from $f^4$ Rayleigh scattering.¹⁵ Subsequently, our simplified geometric model may have errors in terms of the absolute value of the $d_{\mathit{eff}}$ masked by the addition of $\xi$. An ongoing area of investigation by our group is the development of form factors based on scattering from cylindrically symmetric scatterers ($f^3$ in the long-wavelength limit).¹⁸ However, this is outside the scope of this work. The proceeding sections highlight a validation study of the secant model.

Phantom Compositions

Two phantoms were constructed to determine if the secant model appropriately characterizes the angular dependence of scattering from cylindrical structures: an isotropic and anisotropic phantom. The isotropic phantom was composed of a uniform distribution of 75–90 μm diameter glass beads (6 g/L, Potter’s Industries, Malvern, PA, USA) randomly distributed in a water-based gel (4.5 g/L; BD Bacto Dehydrated Agar, Becton, Dickinson and Company, Franklin Lakes, NJ, USA) with graphite powder added (50 g/L; 9039 Desulco, Superior Graphite, Chicago, IL, USA). A 25-μm thick Saran^™ film (Dow Chemical, Midland, Michigan, USA) was used as a scanning window. The anisotropic phantom was created using a hemodialysis filter (Baxter Healthcare Corporation, Deerfield, IL, USA) composed of straight, hollow cellulose triacetate rods, with an inner diameter of 200 μm and wall thicknesses of 15 μm. The filter was filled with an agar mixture, removed from its mold, and then suspended in deionized water. Photographs of the dialysis filter and corresponding B-mode image of the resultant anisotropic phantom can be seen in Figure 3. A reference phantom was used to remove system transfer functions through the RPM, described below.¹³ This was made of a mixture of water-based gel (4.5 g/L; BD Bacto Dehydrated Agar, Becton, Dickinson and Company, Franklin Lakes, NJ, USA), graphite powder (50 g/L; 9039 Desulco, Superior Graphite, Chicago, IL, USA), and glass beads (4 g/L, 3000 E beads; ~5–20 μm, Potter’s Industries, Malvern, PA, USA). A 25-μm thick Saran^™ film was used as a scanning window for reference phantom.

Figure 3.

The dialysis filter composed of straight, hollow cellulose triacetate tubes: (a) prior to casting in gel and (b) a corresponding B-mode of the anisotropic phantom. The black region above the surface of the phantom corresponds to deionized water.

In Vivo Skeletal Muscle

Skeletal muscle is a well-documented anisotropic tissue in terms of acoustic properties.^3,5 Thus we chose to apply the secant model to the in vivo human rectus femoris, one of the muscle groups comprising the quadriceps muscle. A diagrammatic representation of the imaging plane and a B-mode image of the rectus femoris and underlying tissues are provided in Figure 4.

Figure 4.

(a) A diagrammatic representation of the bipennate rectus femoris. The black box represents the area over which imaging planes were acquired. (b) B-mode image of different structures in the field of view, which include the: 1. rectus femoris, 2. vastus intermedius, and 3. surface of the femur.

Data Acquisition

For the phantom experiments, backscattered radiofrequency (RF) echo data were collected using an 18L6 linear array transducer operating at a nominal frequency of 8 MHz using a Siemens Acuson S3000 ultrasound system (Siemens Healthcare, Ultrasound Business Unit, Mountain View, CA, USA). The transducer was secured to a positioning stage and coupling gel was used to establish contact with the Saran scanning window of the isotropic phantom, whereas the transducer was placed in an intervening water layer for the anisotropic phantom (see Figure 3). The Axius Direct Ultrasound Research Interface was used to obtain electronically beam steered and focused RF data from −40° to 40° in steps of 4°.^19,20 Due to power spectral signal-to-noise ratio (SNR) considerations (analysis restricted to SNR 10 dB above the noise floor), the steering angle domains were restricted to −24° to 24°. Five independent planes of data were collected.

The anisotropic phantom was tilted from −12° to 12° in steps of 4° relative to the transducer aperture to quantify rod orientation with the secant model in terms of $\varphi$. Five independent planes of data were collected for each experiment by translating the transducer elevationally by at least one elevational aperture after each data collection.

For the rectus femoris, a 14L5 SP linear array transducer was used to acquire five independent elevational planes of beam steered RF data with imaging planes assumed to be oriented with muscle fibers parallel to the face of the transducer based on visualization from B-mode imaging. The transducer operated at a nominal frequency of 6 MHz. The scan was performed with the muscle relaxed.

With the same system settings, 20 independent planes of data were collected from the reference phantom. Both reference phantom sound speed and attenuation had been previously estimated over 2.25 to 10 MHz using a narrow-band through transmission technique, with test samples manufactured at the same time as the phantom.²¹ A broadband planar interface technique derived by Chen et al. was used to estimate ${\sigma }_b$ for the reference phantom.²²

Data Processing

All data processing was performed offline using MATLAB (Mathworks, Natick, MA, USA). The RPM was used to compensate for system effects.¹³ Using the methods described by Thijssen (Section 4.1, Gray level statistics), axial and lateral RF echo signal correlation lengths were estimated to be 282 and 378 μm, respectively, in the reference phantom for the 18L6 transducer.²³ The same methods were applied to determine the RF echo signal correlation lengths for the 14L5 SP transducer, yielding 295 and 321 μm for the axial and lateral correlation lengths, respectively.²³ Following the recommendations of Rosado-Mendez et al. 4 × 4 mm² power spectral estimation regions were determined to be optimal for estimation of backscatter parameters (i.e., approximately 15 pulse-echo correlation lengths).²⁴ Power spectra were calculated on RF echo data using non-overlapping regions of interest (ROIs) using a multi-taper method with four tapers for specimens and the reference phantoms at the same spatial locations over the angular domain.²⁵ For both phantoms, a spectral difference method was used to estimate local attenuation coefficients.²⁶ The spectral difference attenuation compensation approach has been shown to be successful at estimating both frequency dependence and magnitude of scattering in phantoms when compared to Faran scattering theory.²⁷ These local attenuation coefficients were then used to compensate for total attenuation to the depth of the power spectral estimation region. For the anisotropic phantom, attenuation compensation through the water layer was accomplished using the published attenuation coefficient of pure water at 22°C.²⁸ As a result of the heterogeneity of in vivo tissue, we performed attenuation compensation for the rectus femoris by measuring skin thickness and then compensating for skin and muscle attenuation using values of 2.1 and 1.1 dB cm⁻¹ MHz⁻¹, respectively.^29,30 Attenuation anisotropy was not considered, as it was assumed to be relatively small due to the limited beam steering angles used in this work.^3,5

It is well documented that $d_{\mathit{eff}}$ model fitting requires operating at an appropriate $ka$.^15,31 Given prior, and assumed, knowledge of scattering sources, different frequency bandwidths were used for the isotropic phantom (4–8 MHz), anisotropic phantom (4–6 MHz), and rectus femoris (3–6 MHz). Stationary 4 × 4 mm² ROIs were used to estimate $d_{\mathit{eff}}$ to apply the secant model. The spherical shell form factor model was used to estimate $d_{\mathit{eff}}$ for the isotropic phantom given its prior use and established accuracy for rigid spherical scatterers.^15,31 A Gaussian model was used to estimate $d_{\mathit{eff}}$ for the anisotropic phantom and the rectus femoris due to the narrow size distributions and more complex boundary conditions of the underlying acoustic impedance distribution.³² This was accomplished using a least squares estimator.³³ While choosing a form factor model affects the accuracy of $d_{\mathit{eff}}$ (a simple parameter that quantifies the slope of ${\sigma }_b$), all models assume spherical scatterer symmetry and isotropic scattering. As such, deviations from these conditions were anticipated to be captured by the secant model.

Estimates of $d_{\mathit{eff}}$ at each beam steering angle were averaged over the five elevational planes and nonlinear regression was performed to fit the secant model to compute $d,$ $\xi$, and $\varphi$ using the “nlinfit” function in MATLAB (Mathworks, Natick, MA, USA), which implements the Levenberg-Marquardt algorithm.³⁴ This algorithm and its variations balances tradeoffs between convergence and computational speed and has extensive use in nonlinear curve fitting.³⁵ Parameters were left unconstrained using starting points of $d=0,$ $\xi =1$, and $\varphi =0$.

Statistical tests were performed to evaluate both $\varphi$ and $\xi$. The Pearson correlation coefficient $r$ was computed between the estimated $\varphi$ and the true tilt angle for the anisotropic phantom to determine the model’s ability to predict alignment of underlying rod-like scatterers. The 95% confidence interval for $\varphi$ was assessed to see if it contained the tilt angle. Differences in $\xi$ amongst specimens (e.g., isotropic and anisotropic phantom) were assessed by evaluating the overlap in the 95% CI, as estimated by nonlinear regression, using the three pairs of samples: the isotropic versus anisotropic phantom, the isotropic phantom versus rectus femoris, and the anisotropic phantom versus rectus femoris.

Results

Plots of $d_{\mathit{eff}}$ as a function of beam steering angle $\theta$ are shown for the isotropic and anisotropic phantom in Figure 5. Qualitatively, one may appreciate the larger angle-dependence of $d_{\mathit{eff}}$ in the anisotropic phantom compared to isotropic phantom. For the isotropic phantom, the estimated $d$ from the secant model was $d=81.8\mathrm{\mu }\mathrm{m}$ (95% confidence interval: 80.5–83.1 μm), near the median of the uniform glass bead distribution. For the anisotropic phantom, the estimated $d$ from the secant model was $d=175.0\mathrm{\mu }\mathrm{m}$ (95% confidence interval: 168.0–182.0 μm), near the manufacturer’s stated inner diameter of the cellulose triacetate rods. Application of the secant model to the rectus femoris yielded an estimated diameter of the secant model $d=104.7\mathrm{\mu }\mathrm{m}$ (95% confidence interval: 102.1–107.7 μm) and $\varphi =-{9.35}^{\circ }$.

Figure 5.

Effective scatterer size $d_{\mathit{eff}}$ as a function of beam steering angle $\theta$ for the (a) isotropic and (b) anisotropic phantoms, respectively. The anisotropic phantom data was acquired at a tilt angle of 0° (e.g., rods parallel with the transducer face). The blue curve represents the mean $d_{\mathit{eff}}$ while the blue shaded regions represent a standard deviation of $I\sigma$. Red dashed curves correspond to the secant model fit to the mean $d_{\mathit{eff}}$, yielding $d=177.5\mathrm{\mu }\mathrm{m}$ (95% confidence interval: 171.6–183.5 μm) and $\varphi =0^{\circ }$ (95% confidence interval: −1.6°–1.6°) for the anisotropic phantom.

Variation in $\varphi$ as a function of the tilt angle in the anisotropic phantom can be seen in Figure 6. The estimated orientation angle $\varphi$ mirrored the tilt angle for the anisotropic phantom, producing $r>0.99$. Across every estimate, the 95% confidence interval of $\varphi$ contained the tilt angle. Overlap in $\xi$ was not observed between the isotropic and anisotropic phantom (0.52–0.98 vs. 1.36–1.78) nor in the isotropic phantom and the rectus femoris (0.52–0.98 vs. 1.51–2.73). Overlap in $\xi$ between the anisotropic phantom and the rectus femoris was observed (1.36–1.78 vs. 1.51–2.73). Key findings from the experiments are provided in Table 1. In the case of the anisotropic phantom, $d$ and $\xi$ represent the average value across all true tilt angles.

Figure 6.

Estimates of the orientation angle $\varphi$ versus anisotropic phantom’s true tilt angle. Error bars represent the 95% confidence interval of the parameter estimate.

Table 1.

Key parameters in the secant model: diameter d and anisotropy factor ξ.

	Sample
Parameter	Isotropic phantom	Anisotropic phantom	Rectus femoris
d (μm)	81.8 (80.5, 83.1)	175.0 (168.0, 182.0)	104.7(102.1, 107.7)
ξ	0.75 (0.52, 0.98)	1.57 (1.36, 1.78)	2.10 (1.51, 2.73)

The isotropic phantom was composed of a random spatial distribution of glass spheres in a uniform size distribution from 75 to 90 μm. The anisotropic phantom was composed of hollow cellulose rods. Parameters were obtained by fitting acquired beam steered data from −24° to 24° using the Levenberg-Marquardt algorithm in MATLAB (Mathworks, Natick, MA, USA). Values in parentheses represent the 95% confidence interval of the parameter. In the case of the anisotropic phantom, d and ξ represent the average values obtained from tilting the anisotropic phantom from −12° to 12°.

Discussion

The primary hypothesis in this manuscript is that the anisotropy of the acoustic impedance distribution is consistent with geometrical anisotropy of the cross-sectional major axis created by an acoustic plane intersecting a cylinder along its axis. Results shown in Figure 5(b) indicate that this assertion is appropriate: $d_{\mathit{eff}}$ increases as the beam steering angle moves away from perpendicular incidence with the anisotropic phantom. Our hypothesis also results in equation (4), which indicates that the diameter, as well as the orientation angle, of a cylindrical scatterer may be estimated from the anisotropy of $d_{\mathit{eff}}$. Reassuringly, the diameters estimated from the secant model closely matched the diameter of the cylindrical scatterers which composed the anisotropic phantom. Further, the orientation angle of the anisotropic phantom was correctly described by $\varphi$, as there was no significant difference between $\varphi$ and the anisotropic phantom’s true tilt angle, as shown in Figure 6.

Previous methods quantifying anisotropy were limited to lab-based equipment or evaluated the relative magnitude of backscatter. In contrast, $d_{\mathit{eff}}$ directly quantifies the impedance distribution, can be estimated from clinical systems, and via the secant model is sensitive to anisotropic tissue microstructure. It should be noted this anisotropic phantom’s aligned dominant orientation angle of cellulose rods is a simplified representation of anisotropic tissue. Future works to evaluate an ensemble of cylindrical scatterers with a distribution of orientation angles may be more generalizable. Nonetheless, these results suggest that the anisotropy of $d_{\mathit{eff}}$ may be primarily determined by the geometrical anisotropy of the cross-sectional major axis created by an acoustic plane intersecting a cylinder and may be exploited through our geometric model of $d_{\mathit{eff}}$ anisotropy.

We show in vivo applicability of the secant model using our analysis of skeletal muscle tissue. The secant model computed a diameter of skeletal muscle tissue consistent with skeletal muscle fiber diameters; $d$ was within 1 standard deviation of published muscle fiber diameter measured using optical techniques and biopsy samples from the vastus lateralis: 98 ± 20 μm and 93 ± 18 μm, for type I and type IIa fibers, respectively.³⁶ These results suggest that it is possible to apply the secant model in vivo, and that it, again, provides estimates of the size and orientation angle of cylindrical scatterers. However, in vivo generalizability is challenged by appropriate selection of bandwidth and potential bias in $d_{\mathit{eff}}$ estimates based on model selection. As highlighted by Zhu et al. combining a priori knowledge of histology investigated and pulse frequencies near $ka=1$ may serve as a starting point for bandwidth and model selection.³⁷ Additionally, Mamou et al. demonstrated direct evaluation of acoustic impedance distributions via generation of ex vivo impedance maps may also serve as a starting point to determine appropriateness of model selection.³⁸ Caution must be placed to not over-interrupt results by acknowledging limitations in the characterized acoustic impedance distributions, both in terms of $d_{\mathit{eff}}$ and boundary conditions which may bias parameter estimates as a result of choice of form factor models.^32,37 Despite these challenges, most models have been demonstrated to converge in media not supporting shear waves for $ka<1$.¹⁵ While absolute size estimates linked to underlying dominant scatterers and the corresponding size distributions remain a challenge, the secant model provides a simple means of quantifying deviation from isotropic scattering conditions.

Perhaps the most surprising result of the secant model was the need to include our empirically motivated “anisotropy factor”$\xi$ in the model. Without $\xi$ we found that estimated diameters for the cylindrical scatterers were inconsistent with the known size of scatterers in our phantom. Beyond the symmetry of the acoustic form factor, this may be due to the cellulose rods having $ka=2.4$ (calculated using the inner diameter of the rods prior to filling with agar) and there being roughly one cylinder per pulse-echo volume, representing conditions differing from diffuse scattering which are necessary for $d_{\mathit{eff}}$ estimation. Figure 5(b) suggests the reason for this inconsistency is the result of a larger deviation of $d_{\mathit{eff}}$ with beam steering angle than that predicted by geometry. From this, we believe $\xi$ may quantify deviations in the anisotropy of $d_{\mathit{eff}}$ from our geometric model. The larger values of $\xi$ in anisotropic materials and tissues suggest that the angle-dependence of $d_{\mathit{eff}}$ deviates more from the geometric model than the deviation of angle-dependence of the $d_{\mathit{eff}}$ in isotropic media. This result further reinforces the idea that these materials and tissues are correctly classified as isotropic and anisotropic. Further, it appears that the magnitude of $\xi$ can differentiate isotropic from anisotropic scatterers: the isotropic phantom had significantly smaller values of $\xi$ compared to the anisotropic phantom (at all tilt angles) or rectus femoris. Interestingly, $\xi$ was higher in skeletal muscle fibers than in the anisotropic phantom. This finding suggests that $\xi$ may be describing properties of anisotropy which differ between the muscle fibers and the cellulose rods, which compose the rectus femoris and anisotropic phantom respectively. However, deviations in $\xi$ from the purely geometric model may be due to incorrectly compensating for acoustic attenuation. Mottley and Miller demonstrated in a phantom composed of oriented graphite fibers that anisotropy in acoustic attenuation is 90° out of phase with backscatter anisotropy and the magnitude of the variations were ⩽ 0.1 dB cm⁻¹ MHz⁻¹ for the angular range investigated in this study.^6,7 While it is incorrect to assume an angle-independent attenuation, potentially resulting in undercompensating for its effects, its impact is expected to be minimal. Furthermore, this is not anticipated to affect the location of the minimum value, and subsequently $\varphi$, given the previously noted lag between backscatter and attenuation.^6,7 Further work will focus on investigating the properties of anisotropy which may give rise to differing values of $\xi$.

The secant model may find utility in characterizing tissues with highly aligned microstructures composed of cylindrical scatterers. For example, studies in rodent models have found that collagen fibers (which have similar anisotropy properties to those of rod-like scatterers) at the boundaries of malignant breast tumors transition from being wrapped around the tumor to a radial orientation.^1,9 It has also been shown that aligned collagen is a prognostic indicator of survival in human breast carcinoma patients.³⁹ Given that collagen and the extracellular matrix have been shown to contribute to ultrasound backscatter in a variety of tissues, this simple geometric model may find utility in novel tissue characterizations.^40,41

Conclusions

We explored the use of a geometric model on parameters of the acoustic impedance distribution based on the assumption of cylindrical scatterers. The geometric model correctly estimated diameters and the orientation angle of cylindrical scatterers. Furthermore, we found that an empirically motivated “anisotropy factor”” could quantify deviations of the anisotropy of the acoustic impedance distribution, suggesting this factor describes a novel property of acoustic anisotropy. The secant model appears to be an interesting extension of our quantitative analysis of acoustic anisotropy and holds promise for characterizing microstructural organization for monitoring disease and describing normal tissue architectures.