Peak density is an ultrasound measurement, which has been found to vary according to microstructure, and is defined as the number of local extrema within the resulting power spectrum of an ultrasound measurement. However, the physical factors which influence peak density are not fully understood. This work studies the microstructural characteristics which affect peak density through experimental, computationa,l and analytical means for high-frequency ultrasound of 22–41 MHz. Experiments are conducted using gelatin-based phantoms with glass microsphere scatterers with diameters of 5, 9, 34, and 69 μm and number densities of 1, 25, 50, 75, and 100 mm−3. The experiments show the peak density to vary according to the configuration. For example, for phantoms with a number density of 50 mm−3, the peak density has values of 3, 5, 9, and 12 for each sphere diameter. Finite element simulations are developed and analytical methods are discussed to investigate the underlying physics. Simulated results showed similar trends in the response to microstructure as the experiment. When comparing scattering cross section, peak density was found to vary similarly, implying a correlation between the total scattering and the peak density. Peak density and total scattering increased predominately with increased particle size but increased with scatterer number as well. Simulations comparing glass and polystyrene scatterers showed dependence on the material properties. Twenty-four of the 56 test cases showed peak density to be statistically different between the materials. These values behaved analogously to the scattering cross section.

## Introduction

Quantitative ultrasound (QUS) has become a widely studied field and has seen significant growth since its early beginnings. By and far, the most commonly used QUS methods for tissues make use of the backscatter from pulse-echo measurements. Early studies in the 1980s were able to characterize liver and ocular tissue which helped lay the ground for future QUS research [13]. These methods began to look at the information about the structure hidden in the frequency domain. Particular features of the frequency spectrum such as the midband fit, spectral slope, and spectral intercept were identified to correlate with specific structure properties such as the scatterer size and concentration. These parameters continue to see use [18]. Perhaps, the most popular measurement for QUS in tissues is the backscatter coefficient. This parameter has several formulations and has been developed to account for transducer geometries and attenuation effects [919].

Through-transmission methods, in which separate transducers are used to generate and receive the sound, are less broad than their pulse-echo counterparts. An important technique in industrial applications, this method is most often used in flaw detection by measuring the transmitted amplitude and locating shadows created by the defect [20]. Quantitative measurements commonly conducted for this configuration are the determination of the sound velocity as well as the attenuation. These measurements have been widely used in tissue and material characterization [12,2028]. Through-transmission techniques do have the advantages of stronger received signals which result in higher signal-to-noise ratios and offer rapid measurement.

A study by Doyle et al. in 2011 examined the response of assorted breast tissue pathologies to high-frequency ultrasound [29]. A particular parameter found in the study to vary according to the microstructure of the tissue was the so-called density of peaks, or peak density. This parameter measures the response of through-transmission ultrasound and is defined as the number of peaks and valleys present in the frequency spectrum of the received pulse. Peak density has also been investigated as a means for imaging structure and showed improvement over conventional through-transmission methods [29,30]. These initial results which show its reliance on material characteristics and its ability for very fast calculation and acquisition make it an ideal candidate for future use. However, the precise structural features which affect the value of peak density have not been fully investigated. In addition, this parameter is still in its early stages and is lacking the extensive foundation as found for some of the previously mentioned parameters.

This paper serves to gain insight into the underlying material characteristics that influence peak density. Gelatin-based phantoms with varied sizes and distributions of glass microsphere scatterers are studied. The setup of the experiment and acquisition of peak density is described in Sec. 2. A finite element model simulating the scattering is developed in Sec. 3. The results of the experiment and simulation are given and discussed in Sec. 4. Section 4 also presents an extended model which takes into account glass scatterers of larger sizes and numbers. Section 5 compares the experimental peak density measurements with amplitude measurements for each of the phantoms. The amount of scattering present in the system is considered in Sec. 6 using analytical formulations for the scattering cross section and is compared to the simulated peak density measurements. The model is then applied to scatterers of different material properties in Sec. 7. Section 8 examines the effect of individual material properties on peak density and the scattering present in the system. We then conclude the paper in Sec. 9.

## Experiment

### Sample Preparation.

Ultrasound measurements were carried out with the use of tissue-mimicking phantom with soda-lime glass microspheres acting as scatterers. The phantoms were prepared using water and gelatin in a manner similar to Bude and Adler [31]. However, in this study, the psyllium fiber powder used in Bude was omitted. The base water–gelatin mixture was a 12.5:1 ratio of water to gelatin. Phantoms were formed into 3.2 mm tall and 40 mm wide cylinders. Glass microspheres (Cospheric LLC, Santa Barbara, CA) of different nominal diameters were added to the gelatin mixture in varying concentrations and mixed. The mean sphere diameters used were 5, 9, 34, and 69 μm. Exact distributions of the spheres are shown in Table 1.

In order to form the phantoms, the gelatin solution was placed over moderate heat while being gently stirred taking care to not introduce air bubbles. Once the gelatin was fully dissolved the warm solution was slowly poured into the container, and the microspheres were added. The mixture was gently stirred and placed in a refrigerator to cool.

Each phantom contained a different number density of microspheres. The five concentrations used in this study were: 1, 25, 50, 75, and 100 mm−3. Through-transmission optical microscopy was used to image the phantoms to ensure the proper sizes and concentrations of the glass microspheres. A Meiji MC 50 optical microscope (Meiji Technoco, Saitama, Japan) along with a Nikon DS-Fi1 digital camera (Nikon, Tokyo, Japan) were used for the imaging. Images of select phantoms are shown in Fig. 1.

### Experimental Setup.

The ultrasound setup in this study used two small-diameter, single-element, high-frequency transducers operated in a through-transmission configuration. Each transducer had a center frequency of 31.5 MHz (2 mm diameter, 1.5 mm focus, TransducerWorks, Center Hall, PA). Figure 2 shows the through-transmission response of the transducers when separated by a small amount of coupling gel. The -6 dB bandwidth of the transducers is 22–41 MHz.

The transducers were excited by a 100 V, 5 ns square wave pulse by a high-frequency pulser-receiver (UT340; UTEX Scientific Instruments, Inc., Mississauga, ON, Canada). Received pulses were digitized at 1.25 GS/s using a digital storage oscilloscope (DPO 3052, Tektronix, Inc., Beaverton, OR). The transducers were placed in the far field of each other and mounted on a homemade, automated three-axis control stage and interfaced with LabView (National Instruments, Austin, TX). Measurements were taken at 1 mm steps, and the samples were scanned in a standard raster pattern for 100 steps. To ensure sufficient contact between the transducers and the sample, ultrasound gel (Sonotech Clear Image; Magnaflux Inc., now NEXT Medical, Branchburg, NJ) was amply used and carefully monitored.

### Parameter Calculation.

Here, we describe the calculation of the peak density parameter. In order to minimize local variations, each sample was scanned at 100 separate locations. The received waveforms were loaded into Matlab for analysis. Each pulse was windowed by selecting 1500 points about the maximum of the pulse which was found using a Hilbert transform. A Tukey window with an adjustment parameter α = 0.1 was then applied to the pulse. This was performed to eliminate any edge effects introduced by the finite windowing of the original rectangular window. The power spectrum for each pulse was then determined. To negate the specific frequency effects introduced by the experimental setup, the measured power spectrum was divided by a reference spectrum. The reference spectrum was found by measuring the signal received through a very small amount of coupling gel. This is the same spectrum as shown in Fig. 2. This acquisition procedure was previously determined to give optimal peak density measurements [32].

The peak density is defined as the number of local maxima and minima present in the spectrum. This value was determined by calculating the numerical derivative of the normalized spectrum. The number of times the derivative crosses the x-axis then denotes the peak density. Process flow for this calculation can be found in Fig. 3. The peak density was calculated for each individual scan point. The measured peak density for each phantom was then the average of the peak density of the individual scan lines.

## Finite Element Analysis Simulation

### Theory.

COMSOL MULTIPHYSICS v5.2 software was used to carry out the finite element simulations in this study. This software was chosen due to its ability to couple various types of physical partial differential equations, in this case, acoustics and solid mechanics. Moreover, while other numerical methods for studying acoustic systems may exist, commercial finite element analysis (FEA) software was chosen for its robust computational ability and its reproducibility among different researchers.

In the fluid region, comsol solves the two-dimensional Helmholtz equation
$∇·−1ρc∇pt−qd−keq2ρcpt=Qm$
(1)
where the equivalent wave number $keq$ is defined for frequency $f$ and the fluid sound speed $cc$ as
$keq2=(2πf)2cc2−kz2=k2−kz2$
(2)
The out-of-plane wave number $kz$ is set to zero in our models. $Qm$ and $qd$ correspond to the pressure fields created by monopole and dipole source terms. These terms are both zero in our system. The total pressure, $pt$, is defined as the sum of the background pressure field and the scattered field: $pt=pb+ps$. The background pressure field in our models is a plane wave propagating in the $x$-direction and is given by $pb=p0e−kx$. Compiling these simplifications and using a constant fluid density, $ρc$ the equation becomes
$∇2pt+k2pt=0$
(3)
In order to fully capture the effects created by having elastic scatterers, the two-dimensional Navier's equation of motion is also solved for the solid scatterers. The microspheres are modeled as a linear elastic material with isotropic material properties. The governing equation is then
$−2πf2ρsu=∇·S+Feiϕ$
(4)

where $f$ is the frequency, $ρs$ is the density of the solid, $S$ is the Cauchy stress tensor, $F$ is the force acting on the solid, and $ϕ$ is the phase component of the force. The displacement field of the solid is given by $u$.

At the boundary of the fluid and the solid interface, the partial differential equations for the fluid acoustics and solid mechanics are coupled with the following boundary conditions:
$n·1ρc∇pt=−n·utt$
(5)

$FA=ptn$
(6)

where $n$ denotes the unit vector normal to the boundary, $utt$ is the structural acceleration, and $FA$ is the load acting on the solid structure.

### Model Setup.

The model is arranged into two simulation domains-fluid and solid. A schematic of the model geometry is shown in Fig. 4. In addition, we make use of a perfectly matched layer surrounding the fluid domain. This layer mimics a nonreflecting boundary condition and allows any outgoing waves to leave the model domain without any reflection. A dense and adaptive mesh was used to ensure proper resolution for the propagating waves, and the maximum element size was $λ/8$. The wavelength of the propagated wave $λ$, varied according to the medium, solid or fluid, as well as the frequency of the incident plane wave. This element size is slightly finer than the size of $λ/6$ recommended by comsol. A mesh sensitivity study was conducted by running the simulation with sizes of $λ/6$, $λ/8$, and $λ/10$ for each diameter for the two and seven scatterer cases. The model frequency spectra and peak densities were then determined. The peak density was consistent for each element size and scatterer diameter except for the 9 μm, high number case. For this instance, the peak density was different for the $λ/6$ mesh but converged for the finer meshes. From this analysis the element size of $λ/8$ was found to be sufficiently fine to give adequate results.

In order to study the scattering due to different densities of scatterers the simulations are conducted with different numbers of scatterers randomly distributed in space using the same diameters as the microspheres in the experiment. The simulations are repeated for four different spatial distributions. The positions of each scatterer were randomly generated and were repeated for each subsequent scatterer diameter. The number of scatters used in the simulation was determined by calculating the expected number of spheres present in a thin layer of the experimental phantom. A layer with an area of 1 mm2 and a height of 69 μm was used as well as the experimental number densities. The resulting number of scatterers used was 1, 2, 3, 5, and 7. Material properties used in the simulations are shown in Table 2.

### Parameter Estimation.

To simulate the resulting experimental power spectra the simulation is repeated, and the Helmholtz equation is solved for each different $k$. In order to detect small changes in the resulting scattered pressure field for each frequency, the frequency was incremented in steps of 200 kHz and spanned the bandwidth of the transducer from 22 to 41 MHz. Since we are interested in all of the forward scattering, the scattered pressure field is determined on the back wall (Fig. 4) of the fluid modeling domain to obtain the overall forward response. After the simulation has been conducted for each wavenumber the results are compiled into a model frequency spectrum. The derivative of the model spectrum is calculated, and the number of x-axis zero crossings are counted to calculate the simulated peak density. The values are then averaged for each trial distribution. This process is illustrated in Fig. 5. Examples of the resulting simulated spectra for a single scatterer are shown in Fig. 6.

## Experiment and Simulation Comparison

Simulated peak densities were calculated for each spatial distribution of scatterers as well as for each scatterer diameter. The peak density values were averaged for each diameter and configuration over the four trials. The peak densities for the experiment and model can be found in Figs. 7(a) and 7(b), respectively. Upon inspecting the graphs, we see the overall trend in which peak density tends to rise for increasing scatterer size as well as increasing numbers of scatterers.

Additionally, in the experimental data, we see a minimum value for peak density of around four. This suggests that the phantom material provides a baseline value for peak density. For these low concentrations and sizes, the microspheres do not scatter enough of the field to increase the value of peak density beyond this background level.

Until the amount of scattering by the microspheres reaches a certain level this background value of peak density is the major contributor. Once the amount of scattering is sufficient we begin to see more peaks.

We note that our model considers a fluid medium. This means the model captures the entire response of the scattered pressure and incorporates even minimal scattering contributions. Ultimately, by comparing the model with the experiment, they both show the same overall trend we are interested in.

In both the model and experiment, it appears that the peak density may be beginning to level off at large scatter size and high numbers of scatterers and there seem to be local variations in the overarching trend. In order to explore this further, the simulations were repeated using both larger scatterers and higher numbers. These expanded simulations included scatterer diameters of 100 and 150 μm and also considered scatterer numbers of 10, 15, 20, and 25. The resulting simulated peak density values are shown in Fig. 8.

From the additional simulations, we see that the same general trend noted earlier still holds—peak density tends to increase with scatterer size and number. We do note, however, that this is not necessarily absolute and in a few cases we may see a small dip in the trend. In addition, it is seen at the edges of the data the trend begins to decay. The peak density does not continue to increase with diameter at low numbers of scatterers and the peak density flattens out for the smallest scatterer at high numbers.

## Comparison of Peak Density With Traditional Methods

In order to demonstrate the purposefulness of peak density measurement, we also compare the experimental peak density values gathered for the phantoms with more conventional through-transmission, amplitude-based measurements. The amplitude measurements were found by finding the maximum of the envelope of the pulse received through the phantom. The envelope was obtained through the Hilbert transform of the waveform. As with the previous peak density measurements, the amplitudes were determined for each individual scan line and then averaged. Figure 9 gives a side-by-side comparison of these results. From Fig. 9(b) it is apparent that the amplitude measurements do not really discriminate among the different phantoms, whereas peak density shows a much more drastic difference.

## Scattering Theory

From the results, we have found that peak density increases with the size and number of scatterers present. This suggests that the total amount of scatter in the system may be the driving force behind the value of peak density. In this section, we briefly discuss relevant scattering theory and consider the total amount of scattering present in our system by looking at the scattering cross section of the system.

First, we consider the acoustic scattering by an elastic sphere as derived by Faran [33]. Moreover, we determine the predicted scattered field in the forward direction for each size of scatterer. This is accomplished by calculating the magnitude of the scattered pressure according to Faran Eq. (31) for θ = 0 over our frequency bandwidth.

Figure 10 shows the resulting Faran scattering in the forward direction for each of the different size of spheres. Inspection quickly tells us that for increasing microsphere diameter the number of peaks in the frequency spectrum for the scattered field also increases. Since Faran theory shows a change in the number of expected peaks in the scattered spectra akin to our peak density measurement this suggests that peak density may be a measure of the variability in the scattered field.

To further understand this, we now further investigate this variability by quantifying the amount of scattering present in the system. We begin with the standard representation for the scattered far-field pressure for a sphere in a fluid medium isonified by a plane wave of unit amplitude with wavenumber $k$ [3336]
$ps,far=e−ikrrΦθ$
(7)

$Φθ=−1k∑l=0∞2l+1−il+1sinηleiηlPl(cosθ)$
(8)
where the associated phase shift $ηl$ is found from the boundary conditions applied to the fluid–solid interface on the surface of the sphere. This phase shift is dependent on the material properties of the sphere, and since we are concerned with elastic scatterers, we will use the expression for $ηl$ as derived by Faran [33]. The total amount of the scattered field can be represented using the scattering cross section, $σs$ and can be calculated using the angular dependence of the scattered field as [36,37]
(9)
Using the form of the angular dependence previously shown we obtain, as shown in the Appendix
$σs=4πk2∑l=0∞2l+1sin2ηl$
(10)
Let us next consider $N$ identical scatterers. Assuming no multiple scattering, it is reasonable to take the scattering cross section of all of the scatterers to be [37,38]
$σs,N=Nσs$
(11)
For our peak density measurements, we conduct measurements over a frequency band. Let us now take into account the scattering cross section across multiple frequencies. We define $σsk=σs$ at a given frequency. Then, for a given scatterer, the total scattering cross section across the frequency band is given by
$σsk=∑kσsk$
(12)
From Eqs. (11) and (12) it follows that the total scattering cross section for $N$ identical spheres across a given frequency band is:
$σs,total=N∑kσsk=Nσsk$
(13)

This value then corresponds to the entire scattering taking place over multiple frequencies. Figure 11 shows this value for numbers and sizes of scatterers considered earlier. We see that these results suggest a similar general trend as shown by our model in Fig. 8. This implies that peak density, which is essentially a measure of the complexity present in the frequency spectrum, is influenced by the increase in the complexity of the scattered pressure field caused by increasing the total amount scattering present in the system. Obviously, at some point, the number and size of scatterers becomes high enough that multiple scattering would become a factor. While this simple scattering representation may not incorporate multiple scattering, what it does show is that as multiple scattering becomes more likely, peak density increases.

## Application of Peak Density Measurements for Scatterers of Different Materials

To further examine the effect of the material properties of the scatterers on the total scattering within in the system as well as the peak density, we next present the results for another type of microsphere often used in experiment—polystyrene. The simulations were rerun in the same manner as done previously using the same sizes, numbers, and configurations. The material properties used can be found in Table 3.

We expect that the difference in the material properties of polystyrene should give rise to different amounts of total scattering and in turn different values of peak density. Figure 12 plots the peak density and $σs,total$ for each different diameter and number. From these results we notice that glass scatterers only have higher amounts of scattering for the 5 and 9 μm cases. Incidentally, we also notice these are the only two cases in which the peak density for glass is consistently greater than that of polystyrene. For the 150 μm cases, the peak density of polystyrene is noticeably larger than for glass. Table 4 summarizes the particular cases in which the mean peak density is statistically significantly different between glass and polystyrene. In general, it is seen that the scatter size affects the peak density and cross section differently depending on the material. Ultimately, these results suggest that peak density may discern among scatterers based upon the size, number and material. It is also observed that total amount of scattering in the systems affects the resulting peak density values.

## Effects of Material Properties on Peak Density

We have seen the material properties of the scatterers do influence the peak density. This section explores the effect on peak density due to three fundamental material properties of the scatterers: Young's modulus, Poisson's ratio, and density. To study this, the simulations were conducted for the 69 μm, seven scatterer case. This one was chosen to capture the overall trend since it is in the middle of our possible configurations. The values of the Young's modulus considered were 0.05, 0.5, 5, and 50 GPa. Values of 0.22, 0.33, and 0.49 were used for the Poisson's ratio, and the densities were 1000, 1500, 2000, and 2500 kg/m3. The speed of sound, $c$, was calculated using $c=Ey(1−ν)/ρ1+ν1−2ν$, where $Ey$ is the Young's modulus, $ν$ is the Poisson's ratio, and $ρ$ is the density.

The simulation was repeated for every possible material configuration and corresponded to 48 different cases for a full factorial design. We do note that these are idealized materials and some of the cases may not exist in nature. To determine which property most affects peak density we present the results in the form of a main effects plot as shown in Fig. 13(a). Upon inspection, it is seen that varying the Young's modulus has the greatest effect on peak density. Meanwhile, varying the density or Poisson's ratio has minimal consequences for the resulting peak density. Figure 13(b) shows the main effects plot for the analytical cross section. We see a similar trend as for the simulated peak densities where the Young's modulus has the largest influence. It is observed however, that $σs,total$ is more sensitive to the Poisson's ratio than peak density. Nevertheless, the stiffness or Young's modulus of the scatterers appears to be the main driving force behind the increased scattering and peak density.

## Conclusion

The work presented here investigated the response of peak density to varied microstructure within tissue-mimicking phantoms. Experiments showed that peak density was most receptive to larger scatterer sizes and high number densities. Finite element simulations were also carried out and showed comparable trends to the experiment. To gain a deeper understanding of the peak density measurement, the simulations were extended to include additional scatterer sizes and numbers. These expanded simulations reinforced the trend noted earlier and showed that the peak density is capable of differentiating glass scatterers of different sizes for a given number of scatterers. We then briefly discussed the total amount of scattering present in the system through analytical techniques. We calculated $σs,total$ for our different configurations and noted the correlation between scattering strength and peak density. The analytical and finite element methods were then applied to systems of scatterers with different material properties. The results were compared for the scatterers, showed that $σs,total$ was dependent upon the material properties and that peak density tended to vary in a similar manner. It was also found that the stiffness of the scatterers had the largest impact on the peak density and scattering within the system. Finally, we also demonstrated the actual usefulness of peak density by showing that peak density showed higher sensitivity to the microstructure of the phantoms than amplitude-based measurements.

## Acknowledgment

The authors thank TransducerWorks for providing the transducers and technical assistance.

## Funding Data

• The National Science Foundation (NSF) GK-12 Program, Grant No. 0947869.

### Appendix: Derivation of the Total Scattering Cross Section

Consider a generalization of the standard representation for the scattered far-field pressure for a sphere at the origin for an incident plane wave of unit amplitude [3336]
$ps,far=e−ikrrΦθ$
(A1)

$Φθ=−1k∑l=0∞2l+1−il+1sinηleiηlPl(cosθ)$
(A2)
where the associated phase shift $ηn$ is dependent on the appropriate boundary conditions. The amount of the total scattered field is known as the scattering cross section, $σs,$ and can be calculated as [34,35]
(A3)
Using the expression for the angular dependence, we then have
$σs=2π∫0π|1k∑l=0∞2l+1−il+1sinηleiηlPl(cosθ)|2sinθdθ$
(A4)

$σs=2πk2∫0π∑l,m2l+1(2m+1)sinηlsinηmei(ηl−ηm)Pl(cosθ)Pm(cosθ)sinθdθ$
(A5)

$σs=2πk2∑l,m2l+1(2m+1)sinηlsinηmei(ηl−ηm)∫0πPlcosθPmcosθsinθdθ$
(A6)
We then use the orthogonality of the Legendre polynomials [39]
$∫0πPlcosθPmcosθsinθdθ=22l+1δl,m$
(A7)
where $δl,m$ is the Dirac-delta function. This yields
$σs=4πk2∑l,m2m+1sinηlsinηmeiηl−ηmδl,m$
(A8)
This collapses the double summation, and we obtain
$σs=4πk2∑l=0∞2l+1sin2ηl$
(A9)

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