2.1.

Monte Carlo Simulation

The MC method²² was adopted to simulate the light propagation in a scattering medium and to visualize the absorption distribution inside the medium. Briefly, the turbid medium was modeled as having layers with finite thickness (along the depth, $z$ axis) of specified optical properties in each layer. If the medium is homogeneous, all layers will have the same optical properties. In the simulation, millions of photons were generated at each source location. Each photon was incident at $90\mathrm{deg}$ to the surface of the turbid medium and was assigned a unity weight $\mathbf{W}$ , which is analogous to the light intensity. Each photon went through many steps of absorption and scattering processes. After each step, part of the weight $\Delta W$ was absorbed by the medium, and the weight of the photon was decreased. The photon was scattered following the Henyey-Greenstein function. The medium anisotropy, $g$ , was chosen to be 0.9 in this simulation. The roulette technique was used to terminate the photon when $W⩽W_{\mathit{th}}$ , where $W_{\mathit{th}}$ was a threshold value. Each photon either was absorbed in the medium, was detected at the reflecting surface, or exited the transmitting surface. After the migration of one particular photon halted, a new photon was launched into the medium at the source location. The MC simulation was performed in the time domain,²³ and the resulting temporal data were Fourier transformed to provide frequency-domain amplitude and phase shift as a function of distance. In this simulation, the boundary condition can easily be controlled through the use of refractive indices of the surface layer and the medium layer.²⁴ The absorption boundary was used between the scattering medium and the outer surface. The photon energy was recorded when it reached the outer surface (boundary) without further reflection.

The MC program was extended to include a spherical target embedded inside the medium. The size, position, and optical properties of the target were controlled from an input file. The complex photon reflection and refraction at the interfaces of the target and the medium were computed. It is important to decide whether a photon was reflected or refracted at the interface between the target and the medium. The angle of incidence, the angle of reflection, and the angle of refraction were calculated in three dimensions. As shown in Fig. 1 , a plane $P$ was established based on three points: center of the target (O), the original position of the photon (E), and the expected position of the photon after one-step propagation (F), assuming no reflection/refraction. The interception of the photon propagation direction with the target is G. Plane $P$ can be expressed as

Fig. 1

Illustration of the MC simulation when a target is embedded in the medium.

If a photon experienced total internal reflection, the photon was first propagated to the point where its trajectory intersected the boundary between the target and the medium via a shortened step size $\left(s_1\right)$ . Then, the photon’s traveling direction was changed to the reflection direction and propagated the remaining step $(s-s_1)$ in the same medium. If the photon experienced partial internal reflection, we simplify the computation by assuming that the photon was either all reflected or all refracted. To determine that, the reflection coefficient from Fresnel’s equations²⁵ was computed and compared with a random number generated from a uniform distribution. If the random number was less than the reflection coefficient, the photon experienced reflection; otherwise, it was refracted. In the latter case, the photon propagated to the boundary first and then continued by following the refraction direction. The final spatial location of the photon was calculated by propagating the photon with a distance of $s-s_1$ , which was adjusted in length according to the difference in total interaction coefficients between the medium and the target. The total interaction coefficient is the summation of the absorption coefficient and the scattering coefficient.

2.2.

Photon Tracking Method

The MC method has the advantage of tracking each generated photon.²⁶ The detected photon can be separated into several groups according to the propagation path of the photon. To compute the contributions of each target or top portion or bottom portion of the target to the measured reflectance, we categorize each received photon at the surface into different groups. The first group consists of photons that do not enter the target (BG), and the second group consists of photons that enter the target (TG). Figure 2 illustrates example paths of these two photon groups. The TG group can be further categorized into many subgroups based on the deepest depth that each photon has reached. For example, as shown in Fig. 2, if the target is separated into the upper half and the bottom half, the photons that entered the target can be further categorized into two subgroups: the photons entering the upper half target only (UTG) and the photons reaching the bottom half (UBTG). By summing the weights of detected photons from each group and then dividing the total weight, we obtained the intensity ratio as

Fig. 2

Illustration of different photon groups. (a) Illustration of photon groups when a single target is embedded in the medium. Target group (TG) consists of photons that enter the target; background group (BG) consists of photons that propagate in the background medium only. (b) Illustration of photon groups when the target is segmented into top and bottom layers. Upper target group (UTG) consists of photons that enter the upper half target only; upper and bottom target group (UBTG) consists of photons that enter both the upper and bottom parts of the target.

2.3.

Weight Matrix Calculation and Image Reconstruction

The Born approximation is used to relate the scattered field $U_{sd}(r_{si},r_{di})$ measured at the source $\left(s\right)$ and detector $\left(d\right)$ pair $i$ to absorption variations $\Delta {\mu }_a\left(r^{\prime }\right)$ in each volume element $r^{\prime }$ within the sample. A dual-zone mesh scheme introduced by us earlier¹² was used to segment the imaging volume into lesion region $\left(L\right)$ and background region $\left(B\right)$ with finer and coarse voxel sizes, respectively. This scheme significantly reduced the total number of voxels with unknown optical properties and improved the convergence. In general, the inversion converged in three to four iterations. The measured scattered field, weight matrix, and total absorption distributions were related in the following equation as

The scattered wave or perturbation received at the surface due to the presence of an absorbing target can be approximated as $U_{sd}=U_{\mathit{\text{top}}}+U_{\mathit{\text{bottom}}}+U_{\mathit{\text{background}}}$ , where $U_{\mathit{\text{top}}}$ and $U_{\mathit{\text{bottom}}}$ represent the perturbation generated from the top part and bottom part of the target, respectively, and $U_{\mathit{\text{background}}}$ is from the background. The reconstructed total absorption distribution has three components of $\Delta {\mu }_{a\text{\_}\mathit{\text{top}}}$ , $\Delta {\mu }_{a\text{\_}\mathit{\text{bottom}}}$ , and $\Delta {\mu }_{a\text{\_}\mathit{\text{background}}}$ if the medium scattering heterogeneities are neglected. We then have the following equation derived from Eq. 4 as

2.4.

Quantifying Light Shadowing Effect Due to Target Optical Contrast Only

Based on the perturbation approach, the $R(s,d)$ is quantitatively related to the $P(s,d)=W_{\mathit{UTG}}∕W_{\mathit{UBTG}}$ in the MC simulation, which is the intensity ratio of the received photons entering the upper-half target only, UTG, and photons reaching the bottom-half target, UBTG. For each pair of high-contrast and low-contrast targets of the same size located at the same depth, $R_{\mathit{\text{high}}}(s,d)$ and $R_{\mathit{\text{low}}}(s,d)$ can be calculated. The ratio of $R_{\mathit{\text{high}}}(s,d)$ and $R_{\mathit{\text{low}}}(s,d)$ given in the following equation cancels the weighting effect due to $\kappa (s,d,{\overline{\mu }}_a,\overline{{\mu }_s^{\prime }},\mathit{\text{location}},\mathit{\text{volume}})$ and characterizes the light shadowing effect due to target contrast only:

The estimation derived earlier attempts to explain the approximate relationship between $\left(\Delta {\mu }_{a\text{\_}\mathit{\text{top}}}\right)∕\left(\Delta {\mu }_{a\text{\_}\mathit{\text{bottom}}}\right)$ measured from absorption maps and intensity ratio $P(s,d)=W_{\mathit{UTG}}∕W_{\mathit{UBTG}}$ computed from the MC simulations. In experiments, it is impossible to separate the perturbation measurements into contributions from top and bottom parts of the target. However, the reconstructed $\Delta {\mu }_{a\text{\_}\mathit{\text{top}}}$ and $\Delta {\mu }_{a\text{\_}\mathit{\text{bottom}}}$ can be measured from absorption images and can be quantitatively related to average percentage of weight $P=W_{\mathit{UTG}}∕W_{\mathit{UBTG}}$ . In the computation, the maximum ${\mu }_{a}s$ measured at the top and bottom parts of the target were used to approximate the $\left(\Delta {\mu }_{a\text{\_}\mathit{\text{top}}}\right)∕\left(\Delta {\mu }_{a\text{\_}\mathit{\text{bottom}}}\right)$ as $\left(\max {\mu }_{a\text{\_}\mathit{\text{top}}}\right)∕\left(\max {\mu }_{a\text{\_}\mathit{\text{bottom}}}\right)$ . This approximation avoids the computation problem when the $\Delta {\mu }_{a\text{\_}\mathit{\text{bottom}}}$ is too small.

2.5.

Phantom and Clinical Experiments

The experimental validation was performed using our existing frequency domain NIR system.²⁷ It consisted of laser diodes of $780\text{-}\mathrm{nm}$ and $830\text{-}\mathrm{nm}$ wavelengths. The outputs of the laser diodes were amplitude modulated at $140\mathrm{MHz}$ and optically switched to 9 positions on a handheld probe. Fourteen optical fibers of $3\text{-}\mathrm{mm}$ diameter were coupled to 14 parallel detection channels. For phantom experiments, Intralipid solution was used to emulate background breast tissue optical properties. Polyester resin spheres of calibrated values ${\mu }_a=0.23{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=5.45{\mathrm{cm}}^{-1}$ , and ${\mu }_a=0.07{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=5.50{\mathrm{cm}}^{-1}$ were used to emulate high-contrast tumors and low-contrast benign lesions. Both high- and low-contrast targets have three sizes of 1.0-, 2.0-, and $3.0\text{-}\mathrm{cm}$ diameter to emulate small, medium, and larger lesions. In each experiment, the measurements were first made from a simple Intralipid background that was used as the reference data. The measurements were repeated for selected solid target phantoms immersed in the Intralipid background at known depths. A commercial ultrasound probe was located in the middle of the handheld probe to provide the depth and the location of the target.

The same setup has been used for clinical studies. The study protocol was approved by the local Institution Review Board. Signed informed consent was obtained from all patients who agreed to participate in the study. Patients were scanned in a supine position while multiple sets of optical reflectance measurements were made with co-registered ultrasound images at the lesion location and the normal contralateral location of the same quadrant as the lesion. The measurements obtained at the normal location were used to estimate background optical properties, which were used to compute the weight matrix for imaging.

3.1.

Example of Absorption Distributions of Different Photon Groups

In the first set of MC simulations, photons have propagated in a $12\mathrm{cm}\times 12\mathrm{cm}\times 7.5\mathrm{cm}$ volume of background ${\mu }_a=0.03{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=6.0{\mathrm{cm}}^{-1}$ . A simulated lesion of $3\text{-}\mathrm{cm}$ diameter and ${\mu }_a=0.3{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=6.0{\mathrm{cm}}^{-1}$ was located at $(0,0,2.5\mathrm{cm})$ . The simulated lesion has the same refractive index of $n=1.33$ as that of the background medium. The source was located at $(-2.0,0,0\mathrm{cm})$ , and the detector was located at $(0,0,0\mathrm{cm})$ . In MC simulation, each photon was assigned an initial weight of unity. As the photon propagated in the medium, part of the weight $\Delta W$ was absorbed after each step. The absorbed $\Delta W$ inside the 3-D volume was summed in the $y$ direction and projected into the $x\text{-}z$ plane, as shown in Fig. 3 . In each figure, the vertical axis is the propagation depth $z$ in centimeters, and the horizontal axis is the lateral dimension $x$ in centimeters. Each figure is normalized to the total number of photons and displayed in logarithmic scale. The target position is marked with a white dashed circle. Figures 3 and 3 show absorption distributions of the photons that have propagated in the background medium only (BG group) and entered the target (TG group), respectively. Figure 3 is the distribution of the photons that have entered the upper part of the target only (UTG group), and Fig. 3 is the distribution of photons that have entered both the upper and bottom parts of the target (UBTG group).

Fig. 3

Simulated absorbed weight distribution of a single target inside the 3-D volume was summed in the $y$ direction and projected to the $x\text{-}z$ plane. (a) The background photon group (BG) consists of photons that do not enter the target. (b) The target group (TG) consists of photons that enter the target. (c) The upper target group (UTG) includes the photons entering the upper half target only, when the target is separated into the upper half and the bottom half. (d) The upper and bottom target group (UBTG) includes the photons that reach the bottom half target when the target is separated into two parts.

3.2.

Computed Intensity Ratios of Two-Photon Groups

The MC simulations were performed for small, medium, and large spherical targets of 1.0-, 2.0-, and $3.0\text{-}\mathrm{cm}$ diameters located at different depths, respectively. The background optical properties were ${\mu }_a=0.03{\mathrm{cm}}^{-1}$ , ${\mu }_s^{\prime }=6.0{\mathrm{cm}}^{-1}$ , and the target optical properties were ${\mu }_a=0.3{\mathrm{cm}}^{-1}$ , ${\mu }_s^{\prime }=6.0{\mathrm{cm}}^{-1}$ for the high-contrast case and ${\mu }_a=0.1{\mathrm{cm}}^{-1}$ , ${\mu }_s^{\prime }=6.0{\mathrm{cm}}^{-1}$ for the low-contrast case. All targets were located at $(0,0,z)\mathrm{cm}$ , and center depth, $z$ , was varied from $1.0\text{to}3.0\mathrm{cm}$ . Figure 4 is the imaging probe with the source and detector positions marked. Using this probe, the intensity ratio $(P=W_{\mathit{UTG}}∕W_{\mathit{UBTG}})$ was calculated for both high- and low-contrast targets at all source–detector pairs. The average $P=W_{\mathit{UTG}}∕W_{\mathit{UBTG}}$ was 3.04, 4.34, and 6.62 for the high-contrast $3\text{-}\mathrm{cm}$ target located at 2.0, 2.5, and $3.0\mathrm{cm}$ , respectively; and the corresponding $P$ was 1.86, 2.68, and 4.06 for the low-contrast target of the same size. The intensity ratios under other imaging conditions are given in Table 1 . As predicted by Eq. 7, $P=W_{\mathit{UTG}}∕W_{\mathit{UBTG}}$ increases with the target depth. However, the ratio of $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ according to Eq. 8 is independent of the source–detector separation and the target location and dependent on target contrast only when the target size is the same. As shown in Table 2 , the average values of $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ were 1.62, 1.65, and 1.65 when the center of the high- and low-contrast targets was located at 2.0, 2.5, and $3.0\mathrm{cm}$ , respectively. The average is fairly constant and independent of depth. This indicates that the ratio of the total number of received photons from the top portion of the target to the bottom portion of the target is 1.6 higher for the high-contrast target compared with that of the low-contrast target of the same size. In other words, the high-contrast target experiences 1.6 times higher light shadowing effect independent of the target location.

Fig. 4

Probe geometry used for simulation and phantom experiments. The red stars are the locations of sources, and the blue circles are the locations of detectors. The middle slot is used for a commercial ultrasound transducer. (Color online only.)

Table 1

Simulation results for high-contrast targets and low-contrast targets at different center depths.

Table 2

Simulation results for the ratio Phigh∕Plow when a pair of high- and low-contrast targets located at different center depths.

A similar comparison can be made for the medium-size and small-size targets. Figure 5 shows an example of $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ versus source–detector separation for the pair of high- and low-contrast targets of $2.0\text{-}\mathrm{cm}$ diameter located at $2.0\text{-}\mathrm{cm}$ depth. The ratio $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ remains fairly constant over the entire range of source–detector separation. The mean value of $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ is 1.34, and the standard deviation is 0.57. Note that when the source–detector separation is larger, the total number of photons received at each detector is smaller, and the signal-to-noise ratio (SNR) is lower. The mean values and standard deviations of $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ under other imaging conditions are given in Table 2.

Fig. 5

An example of ratio $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ versus source–detector separation. The $x$ axis is the source–detector distance, and the $y$ axis is the $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ . In the simulation, a pair of high-contrast and low-contrast $2\text{-}\mathrm{cm}$ targets was located at $2.0\text{-}\mathrm{cm}$ depth.

In Fig. 6 , solid lines with circles show the average $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ versus target center depth for the three pairs of large (blue), medium (red), and small (black) targets. The average $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ values were 1.28, 1.34, 1.39, and 1.45 for the medium-size target located at 1.5-, 2.0-, 2.5-, and $3\text{-}\mathrm{cm}$ depths, respectively, while the average values were 1.04, 1.14, 1.16, and 1.17 for the small $1\text{-}\mathrm{cm}$ target located at the corresponding depths. On average, the larger, medium, and small high-contrast targets experience 1.64, 1.36, and 1.14 times higher light shadowing effect due to the target contrast compared with the low-contrast target of the same size. The results suggest that the $1\text{-}\mathrm{cm}$ target is free of the posterior light shadowing effect. The ratio $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ is independent of the target location and depends on the target contrast only.

Fig. 6

Average ratio of $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ (intensity ratio) and $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ (reconstruction ratio) versus target center location. The three groups of curves correspond to target diameter of $1\mathrm{cm}$ , $2\mathrm{cm}$ , and $3\mathrm{cm}$ . Both $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ and $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ are independent of target location, as predicted by Eq. 8.

3.3.

Image Reconstruction Results of the MC Simulations

The image reconstruction was performed on each data set, and the ratio of reconstructed maximum absorption coefficients of top and bottom target layers was calculated for each target location. In the reconstruction, the background region was divided into a relatively coarse mesh of pixel size $1.0\times 1.0{\mathrm{cm}}^2$ in $x$ and $y$ dimensions, and the target region was segmented into a finer mesh of $0.1\times 0.1{\mathrm{cm}}^2$ in $x$ and $y$ dimensions. The target region was chosen three to four times larger than the true target area. To compare the intensity ratio obtained from the MC simulation with the reconstructed maximum absorption ratio measured from the top and bottom target layers in depth, we have reconstructed targets in two layers. The first target layer was located at the center depth of the top half of the target, and the bottom target layer was located at the center depth of the bottom half of the target. The separation of two target layers was the radius of the target. This imaging reconstruction scheme matches the intensity ratio calculation used in simulation and makes the comparison easier.

The reconstruction results are listed in the second row of each category in Table 1. The reconstructed images are not shown here. The ratios of reconstructed maximum ${\mu }_a$ measured from the top-half target to the bottom-half target are listed in the third row of each category. For the high-contrast $3\text{-}\mathrm{cm}$ target, the ratios were 3.30, 4.23, and 5.08 when the target was located at 2.0-, 2.5-, and $3.0\text{-}\mathrm{cm}$ depth, respectively. For the low-contrast $3\text{-}\mathrm{cm}$ target, the reconstructed ratios were 2.08, 2.52, and 3.13, respectively. Similarly, the ratio $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ , which is independent of the target location and depends only on the target contrast, is plotted in Fig. 6 using dashed lines with cross symbols. The ratio of $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ closely follows the $P_{\mathit{\text{high}}}∕P_{\mathit{\text{low}}}$ computed from the MC simulations. The average ratios of $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ for the three high- and low-contrast target pairs of larger, medium, and small were 1.63, 1.35, and 1.06.

3.4.

Image Reconstruction Results of Phantom Experiments

Phantom experiments were performed at similar conditions as the simulation. The Intralipid solution was used to emulate typical breast tissue optical properties of ${\mu }_a=0.027{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=7.20{\mathrm{cm}}^{-1}$ calibrated at $780\text{-}\mathrm{nm}$ wavelength. The high-contrast and low-contrast targets of three sizes were used to emulate the small, medium, and large lesions of 1.0-, 2.0-, and $3\text{-}\mathrm{cm}$ diameter. Each target was located at different depths from $1.5\mathrm{cm}\text{to}3.0\mathrm{cm}$ with $0.5\text{-}\mathrm{cm}$ increment in depth.

The reconstructed absorption maps were obtained following the same procedure described in Sec. 2.3. In the reconstruction, the coarse mesh was the same as the simulation in $x$ and $y$ spatial dimensions. The target region was segmented into a finer mesh of $0.25\times 0.25{\mathrm{cm}}^2$ in $x$ and $y$ dimensions. The choice of fine-mesh size was based on considerations of the system signal-to-noise ratio and the total number of voxels with unknown optical proprieties. The layer or depth separation was chosen as $1.0\mathrm{cm}$ for coarse mesh and $0.5\mathrm{cm}$ for fine mesh, which were the same as those used in the clinical experiments. The fine-mesh target region was chosen three to four times larger than the true target area. In the reconstructed absorption map, each slice is the spatial $x\text{-}y$ image of $8\mathrm{cm}\times 8\mathrm{cm}$ . The depth of each slice is marked in the figure. In order to compare with the clinical experiments, the reconstruction was performed in $0.5\text{-}\mathrm{cm}$ spacing in depth. Thus, the $3\text{-}\mathrm{cm}$ -diam target and the $2\text{-}\mathrm{cm}$ -diam target occupied five target layers and three target layers, respectively. The other layers in depth were the background.

The reconstructed images of the $3\text{-}\mathrm{cm}$ high-contrast target located at 2.0-, 2.5-, and $3\text{-}\mathrm{cm}$ depth are shown in Figs. 7, 7, 7 , respectively. The images of the $3\text{-}\mathrm{cm}$ low-contrast target located at the corresponding depths are shown in Figs. 8, 8, 8 , respectively. For each set of images, the measured maximum ${\mu }_a$ at the target layers are given in Table 3 . If the ratios $R$ of measured maximum ${\mu }_a$ obtained from the top target layer over that obtained from the second target layer are calculated, the ratio $R$ depends on the target location and increases with the target depth, as predicted by Eq. 7.

Fig. 7

Experimental results of $3\text{-}\mathrm{cm}$ -diam high-contrast target. The target center was located at (a) $2.0\mathrm{cm}$ , (b) $2.5\mathrm{cm}$ , and (c) $3.0\mathrm{cm}$ , respectively. Each slice is the tomographic display of reconstructed absorption distribution in $8\mathrm{cm}\times 8\mathrm{cm}$ spatial $x\text{-}y$ dimensions. The depth of each slice is marked in the figure.

Fig. 8

Experimental results of $3\text{-}\mathrm{cm}$ -diam low-contrast target. The target center was located at (a) $2.0\mathrm{cm}$ , (b) $2.5\mathrm{cm}$ , and (c) $3.0\mathrm{cm}$ , respectively. Each slice is the spatial $x\text{-}y$ image of $8\mathrm{cm}\times 8\mathrm{cm}$ , and the depth of each slice is marked in the figure.

Table 3

Reconstruction results for phantom experiments.

The $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ ratios for $1\text{-}\mathrm{cm}$ targets that characterize the light shadowing effect due to contrast only were 0.97, 0.91, 0.97, and 0.99 for this pair of high- and low-contrast targets located at 1.5-, 2-, 2.5-, and $3\text{-}\mathrm{cm}$ depths. The ratio $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ was computed using $R_{\mathit{\text{high}}}$ and $R_{\mathit{\text{low}}}$ obtained from the first target layer over the second target layer. Compared with the simulation, the $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ increases slightly with the target depth. When the target depth increases beyond $2.5\mathrm{cm}$ , the source–detector pairs greater than $5\text{to}6\text{-}\mathrm{cm}$ separation mainly receive photons from the bottom layers and the signal-to-noise ratios of these data are lower compared with those obtained from shorter pairs. This problem is more pronounced when a larger absorber is embedded in the medium. As a result, the reconstructed maximum ${\mu }_a$ values at the bottom layers were slightly above the background values and were the same for both high- and low-contrast targets. Under this condition, the $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ is mainly the ratio of reconstructed maximum ${\mu }_a$ values of high- and low-contrast targets at the top layer, and it is higher than that predicted by Eq. 8. However, the mean value is 1.54, which is very close to the mean value of 1.6 obtained from the MC simulation. Thus, on average, the larger high-contrast target of $3\mathrm{cm}$ in size has experienced 1.6 times higher light shadowing effect due to the target contrast compared with the low-contrast target of the same size.

For the $1\text{-}\mathrm{cm}$ target, the reconstructed images of the high-contrast target located at 1.5, 2, 2.5, and $3\mathrm{cm}$ are shown in Figs. 9, 9, 9, 9 , respectively, and the images of the low-contrast target located at the corresponding depths are shown in Figs. 10, 10, 10, 10 , respectively. For each set of images, the measured maximum ${\mu }_a$ at target layers is given in Table 3. The ratio $R$ depends on the target location and slightly increases with the target depth.

Fig. 9

Experimental results of $1\text{-}\mathrm{cm}$ -diam high-contrast target. From (a) to (d), the target center was located at $1.5\mathrm{cm}$ , $2.0\mathrm{cm}$ , $2.5\mathrm{cm}$ , and $3.0\mathrm{cm}$ , respectively. Each slice is the spatial $x\text{-}y$ image of $8\mathrm{cm}\times 8\mathrm{cm}$ , and the depth of each slice is marked in the figure.

Fig. 10

Experimental results of $1\text{-}\mathrm{cm}$ -diam low-contrast target. The target center was located at (a) $1.5\mathrm{cm}$ , (b) $2.0\mathrm{cm}$ , (c) $2.5\mathrm{cm}$ , and (d) $3.0\mathrm{cm}$ , respectively. The depth of each slice is marked in the figure.

Note that in our reconstruction procedure with $0.5\text{-}\mathrm{cm}$ increment in depth, the small target with diameter of $1\mathrm{cm}$ is typically reconstructed in one target layer. However, to compare results obtained from large and medium-size targets, the small targets were reconstructed in two target layers with the top layer located at the middle of the top-half of the target and the bottom layer located at the middle of the bottom-half of the target.

The $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ ratios for $1\text{-}\mathrm{cm}$ targets that characterize the light shadowing effect due to contrast only were 0.97, 0.91, 0.97, and 0.99 for this pair of high- and low-contrast targets located at 1.5-, 2-, 2.5-, and $3\text{-}\mathrm{cm}$ depths. These values were constant over the target depths studied. This result indicates that on average, the small high-contrast target of $1\mathrm{cm}$ in size did not experience the light shadowing effect due to target optical contrast.

The phantom experiments were also performed on the medium-size $2\text{-}\mathrm{cm}$ diameter high- and low-contrast targets. The reconstruction results are given in Table 3, and the reconstructed images are not shown here. The $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ ratios for the $2\text{-}\mathrm{cm}$ target, which characterize the light shadowing effect due to contrast only, were 1.00, 1.20, 1.25, and 1.39 for this pair of high- and low-contrast targets located at 1.5-, 2-, 2.5-, and $3\text{-}\mathrm{cm}$ depths. The ratio $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ was computed using $R_{\mathit{\text{high}}}$ and $R_{\mathit{\text{low}}}$ obtained from the first target layer over that obtained from the second target layer. The $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ increases slightly with the target depth for the same reason discussed earlier. However, the mean value, 1.22, is very close to the mean value of 1.36 obtained from the MC simulation.

Figure 11 shows the $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ versus target depth obtained from 1.0-, 2.0-, and $3.0\text{-}\mathrm{cm}$ diameter targets. The $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ was computed using $R_{\mathit{\text{high}}}$ and $R_{\mathit{\text{low}}}$ obtained from the first target layer over the second target layer. The $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ is fairly constant over the depth studied; however, it increases slightly at deeper depth for the medium and larger target data because the signal-to-noise ratio decreases with depth.

Fig. 11

The ratio $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ versus target center location obtained from phantom experiments. The solid lines are the ratios computed using $R_{\mathit{\text{high}}}$ and $R_{\mathit{\text{low}}}$ obtained from the first target layer over the second target layer.

3.5.

Clinical Examples

Two examples were obtained from our ongoing clinical study conducted at the University of Connecticut Health Center. Figure 12 shows a co-registered ultrasound B-scan of a suspicious lesion. The center of the lesion was located at $2.75\text{-}\mathrm{cm}$ depth, and the size of the lesion was about $2.7\mathrm{cm}$ . The fitted background tissue optical properties were ${\mu }_a=0.031{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=3.10{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . Using the same reconstruction scheme as the phantom experiment, the lesion was reconstructed with a $0.5\text{-}\mathrm{cm}$ increment, and the depth of each slice is marked in the figure. Considering the higher noise level in clinical data, the target region was segmented into a fine mesh of voxel size $0.5\times 0.5\times 0.5{\mathrm{cm}}^3$ and the coarse mesh of $1.0\times 1.0\times 1.0{\mathrm{cm}}^3$ . The fine-mesh region chosen was 4.5 times larger than the target area estimated by ultrasound. The reconstructed absorption map is shown in Fig. 12. At the first target layer, the reconstructed maximum ${\mu }_a$ was $0.244{\mathrm{cm}}^{-1}$ , the next layer was $0.116{\mathrm{cm}}^{-1}$ , and the last layer was only $0.041{\mathrm{cm}}^{-1}$ . The ratio between the first and second layer was 2.11. Ultrasound-guided core biopsy revealed that the lesion was an invasive lobular carcinoma.

Fig. 12

An example of a $2.7\text{-}\mathrm{cm}$ malignant lesion. (a) A co-registered ultrasound B-scan image of the cancer. (b) Reconstructed absorption map at $780\mathrm{nm}$ .

Figure 13 shows a co-registered ultrasound B-scan of a suspicious lesion. The lesion center was located at $1.8\text{-}\mathrm{cm}$ depth, and the diameter of the lesion was about $3.0\mathrm{cm}$ . The fitted tissue background optical properties were ${\mu }_a=0.012{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=3.16{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . The lesion was reconstructed with a $0.5\text{-}\mathrm{cm}$ increment, and the reconstructed absorption map is shown in Fig. 13. In this example, the reconstructed maximum ${\mu }_a$ was $0.087{\mathrm{cm}}^{-1}$ at the first target layer and $0.068{\mathrm{cm}}^{-1}$ and $0.049{\mathrm{cm}}^{-1}$ at the second and third target layers, respectively. The ratio between the first layer and the second layer was 1.28, and the ratio between the first layer and the third layer was 1.77. Ultrasound-guided core biopsy revealed that the lesion was a benign mass with fibrocystic changes.

Fig. 13

An example of a $3\text{-}\mathrm{cm}$ benign lesion. (a) A co-registered ultrasound B-scan image of the lesion. (b) Reconstructed absorption maps at $780\mathrm{nm}$ .

The $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ due to optical contrast only for this pair of malignant and benign lesions of $3\text{-}\mathrm{cm}$ size is 1.6. It is interesting to note that the location and size of this pair of malignant and benign lesions are slightly different; however, the ratio $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ is very close to that obtained from the pair of $3\text{-}\mathrm{cm}$ high- and low-contrast phantom targets. This result is promising and suggests that it is possible to use the simple measure $R_{\mathit{\text{high}}∕\mathit{\text{low}}}$ derived from this paper to compare the malignant versus benign lesions.