2.1.

PVA Phantom Preparation

The recipe for making PVA gel-based phantoms has been described in Kharine ⁷ and Manohar ¹⁸, here we briefly summarize the procedure. PVA with a hydrolysis degree greater than 99% and an average molecular weight of 85, 000 to 140, 000 (Sigma-Aldrich, catalog no. 36 314-6) is used. An aqueous solution of 20 wt. % PVA is obtained by dissolving PVA powder in demineralized water at 95 °C with continuous gentle stirring for 2 h. After it is allowed to stand for a few hours to let air bubbles migrate to the surface, the solution is cast in a Perspex mold and kept for 12 h at −20 °C in a refrigerator and subsequently for 12 h at room temperature. This constitutes one F–T cycle. The final sample is prepared undergoing four F–T cycles according to Kharine ⁷

A large PVA phantom prepared with dimensions of 15 × 10 × 6 cm³, is divided into three blocks (Fig. 1). Block I is used for measuring the [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ (Sec. 2.3). Block II is used for acoustic property measurements (Sec. 2.4). Block III is used for the microstructure study with SEM (Sec. 2.5).

Fig. 1

Schematic of poly(vinyl alcohol) gel prepared for speed of sound, acoustic attenuation, optical reduced scattering coefficient, and microstructure measurements.

2.2.

Temperature Measurements

The temperature at two different locations (at the surface and 30 mm under the surface) inside the PVA gel during F–T cycles is simultaneously recorded using two K-type thermocouples (R [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mathrm{\ddot{o}}$\end{document} $\ddot{\mathrm{o}}$ ssel Messtechnik, Dresden, Germany).

2.3.

Reduced Scattering Coefficient Assessment

A custom-made setup based on the principle of oblique-incidence diffuse reflectance (ODR) is used to measure the reduced scattering coefficient ( [TeX:] \documentclass[12pt]{minimal}\begin{document}${ \mu _{s}^{\prime}}$\end{document} ${\mu }_s^{\prime }$ ). The method is discussed in detail by Wang and Jaques,²¹ for a semi-infinite sample with [TeX:] \documentclass[12pt]{minimal}\begin{document}${\it \mu _{s}^{\prime}}$\end{document} ${\mu }_{\mathit{s}}^{\prime }$ much larger than the absorption coefficient μ_a. Under these assumptions, a narrow beam incident obliquely on the surface can be approximated by a buried isotropic point source that is 1 [TeX:] \documentclass[12pt]{minimal}\begin{document}${\it /\mu _{s}^{\prime }}$\end{document} $/{\mu }_{\mathit{s}}^{\prime }$ away from the incident point [see Fig. 2a]. The center of the diffuse reflectance is shifted from the center of the incident beam by Δx. From the diffuse reflectance intensity map [Fig. 2b], a profile is selected along the major axis of the elliptical spot [Fig 2c] from the edges of the reflectance profile at each intensity level. The spatial difference between the light entry point and the midpoint curve at saturation indicate the shift Δx. Considering also the refraction, Δx is expressed as:

Fig. 2

(a) Schematic of the oblique–incidence diffuse reflectance principle and setup. (b) Diffuse reflectance intensity map. (c) Diffuse reflectance profile (solid curve), calculated midpoint curve (dotted), and light entry point (dashed-dotted) at the position indicated in dashed line in (b).

A calibrated sample with known [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ and refractive index n is measured to validate our ODR system. This sample is based on epoxy resin with the optical scattering provided by adding a proper amount of TiO₂ particles.²² Good agreement between the measured [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ and reported value is obtained (Table 1). Further, 20% Intralipid (batch no. WI15893, Fresnius Kabi Nederland BV, The Netherlands) samples with concentration in water varying from 0.1% to 0.5% in volume were measured. As shown in Fig. 3, our measurements show good agreement with the reported values calculated from Van Staveren ²³

Fig. 3

Reduced scattering coefficient ( [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ ) at 784 nm for varying Intralipid 20% concentrations in water measured using the oblique-incidence diffuse reflectance setup. Our results are compared with [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ values calculated from Van Staveren ²³ Each sample is measured 5 times, error bars represent standard deviations.

Fig. 10

(a) Schematic of block III in Fig. 5. (b) Pore sizes in the locations specified in (a), the error bars represent the standard deviations. (c) Pore density (black stars, left axis) and estimate of wall thickness (gray dots, right axis) for all specimens. (d) Pore diameter and wall thickness for grouped sample in depth. The error bars represent the standard deviations.

Table 1

Validation of the oblique-incidence diffuse reflectance system. The sample is calibrated at 800 nm and measured at 784 nm. Five measurements are taken for the sample.

The spatial distribution of [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ within the PVA phantom is investigated by applying the ODR approach at different locations on three planes of block I as indicated in Fig. 1. Plane [TeX:] \documentclass[12pt]{minimal}\begin{document}$\textit {a}$\end{document} $\mathit{a}$ is the surface of the block, and plane [TeX:] \documentclass[12pt]{minimal}\begin{document}$\textit {b}$\end{document} $\mathit{b}$ is obtained by removing a 5 mm thick layer after measurements at plane [TeX:] \documentclass[12pt]{minimal}\begin{document}$\textit {a}$\end{document} $\mathit{a}$ . Plane [TeX:] \documentclass[12pt]{minimal}\begin{document}$\textit {c}$\end{document} $\mathit{c}$ is the surface of the block opposite to plane [TeX:] \documentclass[12pt]{minimal}\begin{document}$\textit {a}$\end{document} $\mathit{a}$ . Therefore, these three planes represent the depths of 0, 5, and 50 mm along the Y axis. Each plane is investigated at 5 depths from 1 to 5 cm along the Z axis (5 times averaging). Care was taken to enable that the diffuse reflectance beam lay well within the sample without interference from edges. A refractive index of 1.36 for the PVA specimens is used according to Kharine ⁷

2.4.

Speed of Sound and Acoustic Attenuation Assessment

The acoustic transmission properties are measured using the insertion technique.²⁴ A 5-MHz unfocused single element transducer (V309 Panametrics) and a broadband needle hydrophone (BLLMCX074 Precision Acoustic Ltd., Dorchester) are mounted in a demineralized water bath at room temperature (Fig. 4). The transducer is driven by a broadband ultrasonic pulser/receiver (Panametrics 5077PR) to emit an ultrasonic pulse whose frequency content is determined by the transducer center frequency and bandwidth (2.25 to 7.8 MHz). The measurement data are recorded with a data acquisition card (U1067A Acqiris, 8 bit, 500 MS/s) while the water temperature is logged using a USB thermocouple measurement device (NI-USB TC01) with the K-type thermocouple. The temperature remained stable within ±0.05 °C during the measurement time of 5 min.

Fig. 4

Schematic of the setup for speed of sound and acoustic attenuation measurements.

A reference signal is obtained when no sample is inserted between the transmitter and the receiver. The ultrasound pulse from the sample measurement has a time-of-arrival shift ΔT and amplitude decrease with respect to the reference signal. The thickness Δd of each sample is measured several times around the position where the sample is insonified. For every sample the Δd used in the estimation of the acoustic parameters is the mean ± standard deviation of 10 measurements. Knowing the speed of sound in water (C _w) at the measurement temperature,²⁵ the frequency dependent SOS function C _s(ω) can be expressed in the frequency domain as:

The KK-based solver has been validated using two calibrated samples based on evaporated milk and agarose^{30, 31, 32} provided by E. L. Madsen (University of Wisconsin, Wisconsin). The sample thicknesses were measured to be 22.95 ± 0.24 mm and 20.85 ± 0.15 mm for samples I and II, respectively. Good agreement is achieved between the measured SOS and AA with the calibrated values (Table 2).

Table 2

Calibrated (22 °C) and measured (22.7 °C) speed of sound and acoustic attenuation values for the two calibrated samples. Temperatures were stable within ±0.05 °C during the measurement time of 5 minutes. The mean and dispersions in measured thicknesses were used in the estimation.

Block II in Fig. 1 is used for the measurements. It is divided into five pieces with the same size of approximately 2 × 2 × 1 cm³. Therefore, these five samples are representative of depths from 1 to 5 cm.

2.5.

Microstructure of PVA Gels

Several authors^{33, 34, 35} have studied PVA hydrogel microstructure. However, their specific applications did not require the cast sample to exceed 3-mm thickness. For breast mimicking phantoms, characterization needs to be done on samples with thickness on the order of a few centimeters.

Trials were carried out to choose the optimum procedure for pore analysis. In samples prepared by critical point drying (CPD)³³ and liquid nitrogen (LN2) treatment,^{34, 36} SEM (FEI/Philips XL30 ESEM^TM FEG, FEI Company, Hillsboro, Oregon) inspection showed 10 to 20 μm cavities on the surface of the specimens (Fig. 5). However, under higher magnification of CPD treated samples [Fig. 5a inset], the structures appeared collapsed (see also Ref. 34). In LN2 treated samples [Fig. 5b inset], smaller cavities are further distinguishable. LN2 treatment is thus preferred since the pores are preserved, but also since it is fast. Further, the samples undergo spontaneous cracking into 2 or more fragments when quenched. This exposes the interiors avoiding the need for further cutting.

Fig. 5

Scanning electron microscopy of (a) outer surface of critical point during processed sample and (b) liquid nitrogen processed sample.

A 2-cm thick slice of the phantom, block III, is cut into four main blocks (Fig. 1, groups 1 to 4). Three specimens from each block are obtained (Fig. 1). Every specimen is inspected with SEM at different locations; average pore size is determined from 150 pores. The number of pores in a 5 × 5 μm² region of interest (A _ROI) is counted for every specimen and is used to calculate the average wall thickness. From the number of pores ν and the average radius ρ, the total area occupied by assumed circular pores is:

3.1.

Temperature Measurements

Temperatures measured at the surface and the inside of the large PVA phantom during F–T procedure are shown in Fig. 6. The aqueous PVA solution was allowed to stabilize for 2 h before submitting it to the F–T procedure. At the instant when the exterior had cooled to room temperature, the interior was still warm. The thermocouple positioned close to the surface experiences the typical saw tooth temperature oscillations in the refrigerator produced by simple ON–OFF control exerted by the thermostat. The thawing cycle on the other hand shows a smooth temperature rise since this is not a thermostat controlled effect. The interior temperature lags the surface cooling, and at any instant in time is 5 to 10 °C higher than the surface temperature. When the surface temperature reaches the freezing temperature of water, the bulk temperature is still about 15 °C. The outer layer of the PVA phantom thus starts freezing earlier than the inside. The same phenomenon occurs during thawing. Both the surface temperature and bulk temperature measured do not reach −20 °C when undergoing 12 h freezing.

Fig. 6

Temperature recorded at the surface (gray line) and 30 mm under the surface (black line) of a large PVA phantom during F–T cycles.

3.2.

Reduced Scattering Coefficient

The [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ distribution inside the large PVA phantom is shown in Fig. 7. Three sets of measurements are performed for the three planes of block I (Fig. 1). The measured [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ on the PVA surface is generally lower than that measured in the bulk. Both plane [TeX:] \documentclass[12pt]{minimal}\begin{document}$\textit {a}$\end{document} $\mathit{a}$ and plane [TeX:] \documentclass[12pt]{minimal}\begin{document}$\textit {c}$\end{document} $\mathit{c}$ measurements show that [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ increases approaching the center of the phantom and the maximum [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ inside the phantom is approximately 3 times larger than that on the surface. The [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ from plane [TeX:] \documentclass[12pt]{minimal}\begin{document}$\textit {c}$\end{document} $\mathit{c}$ at depths from 1 to 5 cm has a maximum variation of less than 30%. However, the largest variation is within the first few millimeters from the surface layer. Measurements from plane b and plane c show similar behavior.

Fig. 7

Measured [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ distribution at 784 nm inside the large PVA phantom. Data points represent average [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ value of 5 measurements from the plane a, plane b and plane c of the phantom shown in Fig. 5. The error bars represent the standard deviations.

The [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ values of the large PVA phantom at different locations measured using the ODR technique are dependent on the refractive index n of the sample. According to Kharine,⁷ a refractive index of 1.36 for PVA is used for [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ calculation. Since the temperature distribution might affect the refractive index, we recalculated the [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ using a refractive index in the range between 1.33 (water) and 1.5 (most polymers). The corresponding variations are less than 10%. This ensures a weak dependence of the determination of [TeX:] \documentclass[12pt]{minimal}\begin{document}$\mu _{s}^{\prime}$\end{document} ${\mu }_s^{\prime }$ on refractive index variation.

3.3.

Speed of Sound and Acoustic Attenuation Measurements

The SOS and AA measured in specimens located at different depths from the surface down into the bulk of the large PVA phantom are shown in Fig. 8. Though their standard deviations are relatively high due to roughness of the sample surfaces caused by cutting leading to dispersions in thickness estimation (standard deviations ± 0.5 mm), in general the SOS values obtained are comparable to the values reported for biological tissue. SOS is usually quoted as between 1425 and 1575 m s⁻¹, (Ref. 24) and AA between 0.5 and 1.1 dB cm⁻¹ at 1.76 MHz.³⁷ Both SOS and AA decrease slightly from the surface toward the center of the phantom. The spatial variation in the acoustic properties, however, is within 1% for SOS and less than around 13% for AA.

Fig. 8

Speed of sound and acoustic attenuation (at 5 MHz) at different depths from the surface to the bulk of large PVA phantom at 22.5 °C. The data points represent the average values from 10 measurements, error bars represent the standard deviations.

3.4.

Microstructures of PVA Gels

Besides 10 to 20 μm large cavities on the surface (Fig. 5), the SEM micrograph of a deep cross-section and of the surface show similar small scale pores (less than 1 μm), almost uniformly distributed throughout the entire specimen area (Fig. 9). Figure 10b summarizes the average pore size at the locations specified in Figure 10a with their standard deviation. It also compares these dimensions with the small scale pores which are located at the outer surface [sample 0 in Figs. 10a and 10b). Within the same specimen, the pore size variability is considerable, but the overall distribution does not exhibit any particular trend. The average pore size in the bulk is sub-micrometer and smaller than in Hyon ³⁸ if the large 10 to 20 μm structures present at the outer surfaces are discarded.

Fig. 9

Scanning electron microscope imaging of liquid nitrogen preprocessed specimens situated at (a) surface and (b) a deep lying cross-sections.

Estimates of the wall thickness are shown in Fig. 10c, along with the number of pores in the specified ROI. The pore size does not change, but a difference is found in density and wall thickness. Bulk specimens (groups 3 and 4) exhibit lower wall thickness (approximately −25%) than side specimens (groups 1 and 2) as shown in Fig. 10d. Thus, pore density increases toward the center of the phantom.