2.1.

Component Material Selection

The optical phantoms consist of three components: polyurethane, absorbing chromophores, and the scattering agent. Selection of the three primary components were driven by our choice of absorbing dyes.

We used a phthalocyanine dye (Epolight 6084, Epolin Incorporated, Newark, New Jersey) as the absorber at $690\mathrm{nm}$ . The absorption at $830\mathrm{nm}$ was achieved using a palladium dye (Epolight 4148). The dye manufacturer lists these dyes for use in optical filters and laser goggle applications and suggested polyurethane as a casting material. We verified that both dyes lost their absorption during the curing process in epoxy resin (Marine-grade epoxy resin, Side A 314 resin and Side B 109 hardener, Tap Plastics, Incorporated, Dublin, California). Both dyes exhibit narrow absorption bands [full width at half maximum (FWHM) 32 and $76\mathrm{nm}$ for Epolight 6084 and 4148, respectively] that allow nearly independent absorption at the wavelengths $690\mathrm{nm}$ by Epolight 6084 and $830\mathrm{nm}$ by Epolight 4148 when cured in polyurethane (Fig. 1 ). Titanium dioxide (Ti-Pure R-900, Dupont Chemicals, Wilmington, Delaware) was assumed to behave as a pure scatterer with negligible absorption between 650 and $850\mathrm{nm}$ . Finally, we used a two part polyurethane system (WC-781 A/B, BJB Enterprises Incorporated, Tustin, California) that had a $30\text{-}\min$ pot life to suspend the titanium dioxide and absorbing dyes. This polyurethane is a clear rigid casting resin that is designed to be tintable and pigmentable. This polyurethane is formed by mixing part A with part B in a ratio of 100/85 by weight (100/88 by volume). The pot life for the polyurethane was a convenient duration, because it allowed sufficient time to mix and degas but still cured quickly enough that no apparent settling of the scatterers took place. The viscosity of the mixed polyurethane is $650\mathrm{cps}$ . The demolding time is $24\mathrm{h}$ . Optical measurements could be made at $24\mathrm{h}$ . However, complete curing of the polyurethane requires 5 to 7 days at room temperature or $16\mathrm{h}$ at 71 to $82°\mathrm{C}$ .

Fig. 1

The absorption coefficient spectra of $10\mathrm{\mu }\mathrm{g}∕\mathrm{mL}$ Epolight 6084 in polyurethane and $17\mathrm{\mu }\mathrm{g}∕\mathrm{mL}$ Epolight 4148 in polyurethane.

2.2.

Initial Studies

2.3.

Phantom Design

Our two wavelength phantoms had four variables to specify( ${\mu }_{a,{\lambda }_1},{\mu }_{a,{\lambda }_2},{\mu }_{s,{\lambda }_1}^{\prime }$ , and ${\mu }_{s,{\lambda }_2}^{\prime }$ ) but only three free parameters, since the reduced scattering coefficients at the two wavelengths are related by the use of a single scattering agent. In other words, we could select the optical properties at the first wavelength, but the reduced scattering at the second wavelength has a constant proportional relationship to the reduced scattering at the first wavelength with respect to scatterer concentration. To compensate, we condensed the desired optical properties into three parameters ${\mu }_{a,{\lambda }_1},{\mu }_{a,{\lambda }_2}$ , and the reduced albedo at the second wavelength $\left(a_{{\lambda }_2}^{\prime }\right)$ . The reduced albedo was defined by

Fig. 2

Process for maintaining the four desired optical properties with only three independent variables. The reduced scattering of the first wavelength establishes the reduced scattering at the second wavelength. A new absorption coefficient is calculated at the second wavelength that holds the desired reduced albedo at the second wavelength, constant relative to the new optical properties.

2.4.

Fabrication Method of Tissue Phantoms

The optical properties of the tissue phantoms were chosen to represent a range of different optical properties at the 690- and 830-nm wavelengths. The absorption coefficient at these two wavelengths would be varied to correspond to physiologic relevant values for the sum total of contributions by fractions of water, oxygenated and deoxyhemoglobin, and a wavelength constant background absorption. For all the phantoms, the scattering coefficient would be held constant. However, the wavelength dependence of the scattering from ${\mathrm{TiO}}_2$ differs from tissue in that the ratio of scattering coefficients at 690 to $830\mathrm{nm}$ for ${\mathrm{TiO}}_2$ does not equal the target ratio for tissue. To account for this effect, the target absorption coefficients at $830\mathrm{nm}$ were adjusted as described in the previous section. Finally, a single set of optical properties was used to make 20 identical phantoms.

The sequence for phantom preparation was as follows using the stock solutions of dye and ${\mathrm{TiO}}_2$ described earlier. We made cylindrically shaped phantoms $7\mathrm{cm}$ in diameter and about $5\mathrm{cm}$ in height. This corresponded to $\sim 200\text{-}\mathrm{mL}$ polyurethane by mixing $113.4\mathrm{g}$ of part A and $97\mathrm{g}$ of part B. In addition, an extra $5.3\mathrm{g}$ of part A and $4.5\mathrm{g}$ of part B were prepared for making thin slabs for optical property characterization. The proportions of each part of polyurethane were determined by weight, and the addition of either dye or ${\mathrm{TiO}}_2$ was neglected since their contribution was less than 1 part in 200 for all phantoms. Absorbers were added to the polyurethane in part A while the scatterers were mixed to part B. Since the scattering of all phantoms were to be identical, $11\mathrm{ml}$ of ${\mathrm{TiO}}_2$ stock was mixed into $2628\mathrm{g}$ of part B polyurethane, which was a sufficient quantity for all phantoms.

Part A of the polyurethane $\left(118.8\mathrm{g}\right)$ was weighed out for a single phantom into a polyethylene container. Using stock of Epolight 4148, the near-infrared dye was first pipetted into part A and stirred in with a glass rod until visibly homogeneous. This dye had a pale violet tint when mixed into the polyurethane but was not noticeable after the addition of the Epolight 6084 stock, which produces a strong turquoise blue tint. When both dyes were homogeneously stirred into part A, about $2\mathrm{ml}$ was placed in a cuvette for an absorbance measurement on the Cary spectrometer using a cuvette of part A polyurethane without dye in the reference arm to verify proper absorption attenuation.

At this point, $5.3\mathrm{g}$ of the part A with dye was set aside in a plastic weighing boat dish. The remaining portion of polyurethane part A with dye was weighed again to calculate the appropriate quantity of the part B with ${\mathrm{TiO}}_2$ to be added by weight. While on the scale, part B was poured into part A. The final amount of part B was added drop by drop using a wooden tongue depressor. The large flat area of the tongue depressor allowed drops to be controlled so that the weight of any drops could be controlled with precision of about $0.03\mathrm{g}$ . The polyurethane was thoroughly mixed using a tongue depressor to scrape the container sides/bottom and stirred for about 2 min. Most of the mixture was then poured into 16-oz HDPE molds (16-oz specimen containers, Fisher Scientific International, Incorporated, Fair Lawn, New Jersey). The remaining material was poured into two plastic weighing boats to cast thin disks of differing thickness and about $6\mathrm{cm}$ in diameter. The set-aside $5.3\mathrm{g}$ of polyurethane part A and dye was treated identical as the previous, except it had $4.5\mathrm{g}$ of transparent part B added to it. Also, these samples were mixed then poured into 1-cm pathlength cuvettes for casting.

It was necessary to remove entrapped bubbles in the polyurethane introduced by the mixing together of components. Up to five phantoms or weighing boat slabs could be degassed at a time. This restricted us to mixing no more than two sets of optical properties together at a single time, since for each set, a phantom and two slabs were made. The second set was mixed while the first set was degassing. Degassing was done by placing the samples in a desiccator/vacuum chamber (Nalgene 5311 desiccator, Nalge Nunc International, Rochester, New York). The chamber was connected to a vacuum pump (Speedivac 2, Edwards High Vacuum International, England) capable of reducing the pressure to less than $1\mathrm{mbar}$ . Samples were degassed for about $8\min$ , at which time bubbles stopped forming.

After degassing, the phantoms were set onto a level surface and covered. The phantoms were solid after $24\mathrm{h}$ and could be removed from the mold. However, at $24\mathrm{h}$ , the polyurethane is relatively soft, such that fingerprints can be embedded on the surface with moderate pressure. In general, we waited $48\mathrm{h}$ before removing the samples from the casting mold. The phantoms in the specimen container molds were removed by flexing the walls of the container and then pushing on the bottom while holding the phantom upside down. The samples cast in weighing boats were removed by peeling the weighing boat, which tore away from the polyurethane.

The phantoms were finished by leveling the top surface of the polyurethane block. A lathe was used to mill the meniscus lip from the top of each phantom until the shiny surface was completely removed. Then each phantom was sanded by hand in two steps: rough finishing with 200-grit wet sandpaper and fine finishing with 600-grit wet sandpaper. In each sanding step, the paper was wet with water and placed on a glass surface while the polyurethane block was sanded in a circular motion. Elimination of visible surface scratches were used as a completion condition.

2.5.

Testing of Phantom Optical Properties

Absorption of each phantom was measured using the Cary spectrometer $48\mathrm{h}$ after casting. The clear samples cured in cuvettes for each phantom were measured, referenced to cured polyurethane without dyes in a cuvette. Four absorbance measurements of each sample were recorded. Additional measurements of the same cuvette samples were recorded 13 and 14 months later to establish long-term stability.

Inverse adding doubling was used to find the scattering and absorption of a slab of turbid material using total reflection and total transmission measurements. Two measurements on each of the two thin slabs for each phantom were made. Total diffuse reflection measurements were made using a 203.2-mm-diam integrating sphere (IS-080-SF, Labsphere, Incorporated, North Sutton, New Hampshire) with a 12.7-mm-diam detector port and a 31.75-mm-diam sample port with a baffle between ports. Light was guided to the phantom disk through a $400\text{-}\mathrm{\mu }\mathrm{m}\text{-}\mathrm{diam}$ optical fiber (0.22 numerical aperture) placed about $4\mathrm{mm}$ above the sample centered with the port hole. A 4-mm-diam steel tube coated with multiple coats of flat white paint was used to sleave the fiber through a 6.35-mm top port down to the phantom disk. An Oriel mercury lamp was used for the source and a $600\text{-}\mathrm{\mu }\mathrm{m}\text{-}\mathrm{diam}$ (0.38 numerical aperture) delivered light from the integrating sphere to a scanning grating detector system of a Fluorolog-3 spectrofluorometer (Instruments S.A., Incorporated, Edison, New Jersey). Signals above $850\mathrm{nm}$ were often poor as a consequence of lower power from the lamp at longer wavelengths and lesser efficiency of the photomultiplier tube above $800\mathrm{nm}$ . The detector fiber was connected to the integrating sphere using an SMA connector surrounded by white shielding in the 12.7-mm port. Measurements were referenced to a 99% Spectralon reflectance standard (Labsphere, Incorporated, North Sutton, New Hampshire) and a dark measurement where the sample port was open and the fiber illuminating black felt on the optical table was $254\mathrm{mm}$ below the port. Four measurements were made on each disk; two measurements on each side. The phantom disks were placed flush with the sample port. Figure 3 shows the sphere measurements for total reflection, and Fig. 4 shows the reference sphere calibration measurements.

Fig. 3

Diagrams of various integrating sphere reflectance measurements that are needed.

Fig. 4

Diagrams of various measurements needed to determine the fraction of diffuse illumination and the sphere reflection efficiency.

Total diffuse transmission measurements were made with another identical integrating sphere. In this setup, only two ports were open, the 25.4-mm-diam sample port and 12.7-mm-diam detector port with a baffle between ports. For this arrangement, the phantom disk was placed on the top port, and light was collected with the same detector $600\text{-}\mathrm{\mu }\mathrm{m}\text{-}\mathrm{diam}$ fiber we previously described. Light was delivered to the phantom disk with the $400\text{-}\mathrm{\mu }\mathrm{m}$ fiber. The end of the fiber was positioned just above the phantom disk $(<1\mathrm{mm})$ centered with the port hole. Again, four measurements were recorded on each disk; two measurements on each side. Measurements were referenced to 100% with the fiber illuminating the open port hole, and a dark measurement with an open port but with the fiber removed from the sample porthole. Figure 5 shows the sphere arrangement for total diffuse transmission measurements.

Fig. 5

Diagrams of various integrating sphere transmission measurements that are needed. The empty port on the bottom of the spheres should be identical to the size of the entrance port in the sphere reflection measurements.

Two other values were needed to calculate the scattering properties. The thickness of each phantom disk was measured using a depth gage (ID-C112EB, Mitutoyo Corporation, Japan). The refractive index of the cured polyurethane was measured at $670\mathrm{nm}$ using an Abbé refractometer at 1.468. We assumed negligible change in the refractive index over the spectral range of 500 to $900\mathrm{nm}$ .

The total diffuse reflectance and transmittance measurements in terms of percentage at all wavelengths were input into the IAD program. The total diffuse reflection was calculated using

The batch variability was measured on 12 out of 20 phantoms from a single batch to establish the variability in our fabrication method. A thin slab was cut and sanded flat from 12 of the phantoms. The thickness of each slab was measured with a depth gage. Total diffuse reflection and transmission measurements were recorded for each of the 12 slabs, along with reference measurements as previously described. Inverse adding doubling was used to determine the absorption and reduced scattering coefficient for each slab over the wavelength range of 650 to $850\mathrm{nm}$ .

3.1.

Absorption Characterization

Both dyes exhibit stable absorption properties in polyurethane after a period of 14 months from casting. The visible/near-infrared (NIR) spectrum for the absorption coefficient of Epolight 6084 and 4148 is shown in Fig. 1. The absorption coefficient at $690\mathrm{nm}$ relates only to the concentration of Epolight 6084 $\left(C_{6084}\right)$ by the relation

Fig. 6

The absorption coefficient at $690\mathrm{nm}$ as a function of dye concentration of Epolight 6084 in part A of the polyurethane (circles) and after curing in polyurethane (squares). The fitting line is ${\mu }_a^{690}=a_1{C}_{6084}+{\mu }_{a0}^{690}$ , where $a_1=0.16{\mathrm{cm}}^{-1}\mathrm{mL}∕\mathrm{\mu }\mathrm{g}$ and ${\mu }_{a0}^{690}=0.046{\mathrm{cm}}^{-1}$ is the absorption coefficient of polyurethane at $690\mathrm{nm}$ one week after casting.

Fig. 7

The absorption coefficient at $830\mathrm{nm}$ as a function of dye concentration of Epolight 4148 in part B of the polyurethane (circles) and after curing in polyurethane (squares). The fitting line is ${\mu }_a^{830}=a_1{C}_{4148}+{\mu }_{a0}^{830}$ , where $a_2=0.065{\mathrm{cm}}^{-1}\mathrm{mL}∕\mathrm{\mu }\mathrm{g}$ and ${\mu }_{a0}^{830}=0.042{\mathrm{cm}}^{-1}$ is the absorption coefficient of polyurethane at $830\mathrm{nm}$ one week after casting.

The tissue phantoms are limited by a minimum intrinsic absorption coefficient that is due to the absorption of the binding medium. The absorption coefficient of polyurethane at a time point one week after casting is greater than after 13 months from casting, as shown in Fig. 8 . The change of absorption at these time points is $0.03\pm 0.003{\mathrm{cm}}^{-1}$ over the wavelength range of 500 to $835\mathrm{nm}$ . The polyurethane absorption is stable between 13- and 14-month time points, but it is not known how gradual or when exactly the drop in absorption occurred between the one-week and 13-month time points. The polyurethane exhibited less absorption than two other resins (an epoxy resin and a polymer resin), which showed significant absorption occurring below $500\mathrm{nm}$ , thus giving each a yellow tint as shown in Fig. 8.

Fig. 8

The absorption coefficient spectra of an epoxy resin, a polymer resin, and the polyurethane 13 months after being cast in addition to the polyurethane one week after casting.

3.2.

Scatterer Characterization

The scattering coefficients were determined from diffuse reflection and transmission measurements. A typical spectra is shown in Fig. 9 . This illustrates the wavelength separation in the absorption of the two dyes in the reflected and transmitted light. The reduced scattering coefficient as a function of wavelength (Fig. 10 ) was obtained using inverse adding doubling, using the total diffuse reflection and transmission measurements shown in Fig. 9. For comparison, the absorptioncoefficient obtained from a spectrometer measurement of a sample without titanium dioxide is also shown in Fig. 10 with the absorption coefficient calculated by IAD. The reduced scattering coefficient ${\mu }_s^{\prime }$ relates to the scattering coefficient $\left({\mu }_s\right)$ , and the cosine of the mean angle of scattering $\left(g\right)$ from the incident direction of light by the equation ${\mu }_s^{\prime }=(1-g){\mu }_s$ . The reduced scattering coefficient is an equivalent optical parameter that is one over the mean free path between scattering events when the scattering phase function is isotropic. Since two measurements are recorded, only absorption and the reduced scattering coefficient can be determined. The anisotropy and the scattering coefficient ${\mu }_s$ were not determinable separately without another measurement of the ballistic photons. Our slabs (ranging from 2 to $7\mathrm{mm}$ ) were sufficiently thick that all transmitted light was completely diffuse.

Fig. 9

Typical reflectance and transmission measurements used to derive optical properties using the inverse adding-doubling method.

Fig. 10

The reduced scattering and absorption coefficient spectra for the reflection and transmission data in Fig. 9 are shown in comparison to the spectrometer measurement of absorption for the same phantom made without added scatter. The error bars are the standard deviation of three spectrometer measurements through the clear phantom. Above $830\mathrm{nm}$ , the sensitivity of the detector drops precipitously, causing error in the resultant optical properties. The reduced scattering is fit by the relation ${\mu }_s^{\prime }=\left(5.2{\mathrm{cm}}^{-1}\right)(\lambda ∕1000\mathrm{nm}{)}^{-0.8}$ .

The derived scattering properties for the titanium dioxide follow the relations

Fig. 11

The size distribution of the diameters of 112 titanium dioxide particles has a mean of $340\pm 90\mathrm{nm}$ .

3.3.

Phantom Optical Properties

A set of ten optical phantoms were made with differing levels of absorption but with the same scattering properties. The variation among the ten phantoms in the scattering coefficient at 690 and $830\mathrm{nm}$ is shown in Fig. 12 . The measured absorption coefficients at 690 and $830\mathrm{nm}$ of the ten samples are shown relative to the predicted absorption from Eqs. 1, 2 in Fig. 13 . The difference between predicted and the measured absorption coefficients for the ten phantoms was plus or minus 3% at $690\mathrm{nm}$ and 6% at $830\mathrm{nm}$ .

Fig. 12

The variation in reduced scattering coefficient for ten different phantoms at $690\mathrm{nm}$ (circles) and $830\mathrm{nm}$ (squares) measured by the IAD method. All samples were designed to have the same reduced scattering properties. The error bars are the standard deviation of four sets of reflection and transmission measurements for each sample.

Fig. 13

The predicted absorption coefficient using Eqs. 1, 2 are shown relative to the spectrometer measured absorption coefficient for the ten different phantoms.

The reproducibility of a phantom with a single set of absorption and scattering properties is shown in Fig. 14 for 12 phantoms made from a single batch. Both absorption and scattering varied by plus or minus 3% between 600 and $800\mathrm{nm}$ . Between 500 and $600\mathrm{nm}$ , the percentage absorption variation increases but the absolute error remains constant. At wavelengths longer than $800\mathrm{nm}$ , the error increased (as previously stated) due to lamp power and decreased detector efficiency. Since both the Epolight 6084 and 4148 were mixed together in one part of the polyurethane and titanium dioxide to the other part before combining the two parts, the actual percentage variation in absorption at $830\mathrm{nm}$ should be comparable to that at $690\mathrm{nm}$ . Since all 20 phantoms come from a single large batch where all the dye was added to part A of the polyurethane and the ${\mathrm{TiO}}_2$ was added to part B for all 20 samples, the variation in the samples can only come from two sources: weighing errors for the combination to the two parts of polyurethane, or inhomogeneities in our large solutions of components. The largest weighing error was less than 2% for all phantoms, including the ten with different absorption properties.

Fig. 14

The absorption and reduced scattering coefficient spectra for 12 of the 20 identical phantoms. The error bars show the standard deviation for each optical property among the 12 phantoms and are approximately $\pm 3\%$ between 600 and $800\mathrm{nm}$ .

To minimize error due to inhomogeneities, the part of the polyurethane with ${\mathrm{TiO}}_2$ was regularly stirred to prevent settling of the scatterers before mixing with the part with added absorbers. However, after combining the two parts of the polyurethane together, the phantom cannot be stirred once the degassing step begins. Samples were held under vacuum at a minimum of eight minutes, but can be degassed for 10 to $12\min$ . Though the pot life of the polyurethane was a half-hour, by 30 to $45\min$ the polyurethane was still liquid but warming and noticeably beginning to gel. It is doubtful that full degassing of phantoms can be performed with a pot life less than half an hour, since gelling of the polyurethane will inhibit gas from escaping. It took about $5\mathrm{h}$ to make all 20 phantoms with two people working. The timeliest part of the process was degassing, because we were limited to only four or five phantoms in a vacuum chamber due to space constraint.