2.1.

Forward Model

In our simulations, the frequency domain diffusion approximation and the Robin-type (type III) boundary condition have been adopted for forward computation.¹⁹ A commercial software package using finite element method (FEM), COMSOL, has been employed to solve the forward diffusion equation. A 3-D cylindrical mesh is generated for forward calculation. The optical source is modeled as an isotropic point source placed approximately one reduced scattering distance, $(1∕{\mu }_{s1}^{\prime }+{\mu }_{a1})$ , underneath the boundary, where ${\mu }_{s1}^{\prime }$ and ${\mu }_{a1}$ are the reduced scattering and absorption coefficients of the first layer. The radius and the height of the cylinder are large enough to approximate the semi-infinite geometry. A smooth surface is used to model the tissue and the chest-wall interface. The chest-wall tilting angle with respect to the probe and the tissue-chest interface location in depth $h$ are determined by the co-registered ultrasound images. Figure 2 gives an example of a two-layer model configuration. Figure 2a shows a B-scan ultrasound image of a normal breast with the tilted chest wall marked, and Fig. 2b is the corresponding two-layer model generated.

2.2.

Estimation of Two-Layer Bulk Absorption and Reduced Scattering Coefficients

FEM has been used to relate the bulk absorption and reduced scattering coefficients of the first and second tissue layers to the photon density wave calculated at the surface as

2.3.

Weight Matrix Calculation Using the Two-Layer Model

The forward Jacobian weight matrix $W_{ij}=[\partial {\varphi }_{ij}∕\partial {\mu }_{aj}]$ , which relates the photon density wave perturbation at detector $i$ and imaging voxel $j$ with absorption coefficient change $\Delta {\mu }_{aj}$ , is calculated as

Using a larger ROI helps visualize the effects of mismatches; however, it reduces the reconstructed target absorption coefficients. With the ultrasound spatial location guidance, we typically use a tighter ROI of two times the target size to improve the accuracy of reconstructed absorption coefficient. For a $1\text{-}\mathrm{cm}$ -diam high-contrast absorber, we can achieve about 80% accuracy when the maximum value of reconstructed absorption distribution is used.²¹ For a low-contrast absorber of the same size, about 100 to 120% of true values can be achieved. The reconstructed absorption values presented in this manuscript correspond to results with only target depth guidance, and they are lower than those with additional spatial guidance. Therefore, we have used the percentage of the reconstructed target absorption normalized to the no-background mismatch case to highlight the relative changes in simulation and phantom studies.

2.4.

Inversion

Last, a dual mesh model and a conjugate gradient method are used for reconstruction of optical absorption properties of lesion with the location information provided by the co-registered ultrasound measurement.

2.5.

Experimental System

Our experimental system consists of three laser diodes of 690, 780, and $830\mathrm{nm}$ and 10 parallel detectors. Each laser diode is sequentially switched to nine source positions on the probe, and 10 parallel detection channels acquire backscattered light simultaneously for each source position. More details on our hand-held imaging probe and the NIR system can be found in Ref. 22.

For phantom experiments, intralipid solution and solid plastisol phantoms have been used to emulate the first and second tissue layers, respectively. A commercial ultrasound probe is located in the middle of the hand-held probe to provide the tilting angle and the depth of the second layer and also the region of the target. Each two-layer phantom is imaged with and without a target. The data without a target is used for estimation of background optical properties, and the difference between the target and background data is used to compute perturbation for imaging reconstruction.

The optical properties of the two layers of the phantom were calibrated separately by using a least-squares fitting procedure detailed in Ref. 23. Hereafter, we will refer to these calibrated values as true background optical properties. Briefly, the gains of different source positions and different detector positions as well as the slopes of amplitude (a logarithmic scale) and phase measurements versus source–detector separation were estimated by fitting the measured data using a semi-infinite model of an absorbing boundary condition. The background absorption and reduced scattering coefficients can be readily obtained from the estimated slopes of amplitude and phase measurements.

We also estimated the two-layer optical properties by fitting the reflection measurements using the nonlinear Nelder-Mead algorithm described in Sec. 2.2. We refer to these background values as fitted optical properties in the following text.

Our data acquired from patients suggest that typical optical absorption coefficients of cancers were in the range of $0.2{\mathrm{cm}}^{-1}\text{to}0.3{\mathrm{cm}}^{-1}$ .(Ref. 14) For typical benign lesions, the absorption coefficients were in the range of $0.03{\mathrm{cm}}^{-1}\text{to}0.16{\mathrm{cm}}^{-1}$ . Scattering coefficient of lesions was not consistent. For larger tumors, scattering could be quite low compared with the background; while for smaller tumors, scattering coefficient could be higher or lower than that of the background. Therefore, we have chosen an absorption coefficient of $0.20{\mathrm{cm}}^{-1}$ to represent high-contrast tumors, and $0.07{\mathrm{cm}}^{-1}$ to represent benign lesions. We have used similar reduced scattering coefficients for background and targets.

The same setup has been used for clinical studies. The study protocol has been approved by a local Institutional Review Board (IRB) committee. Patients have been scanned in a supine position, and multiple sets of optical measurements were made simultaneously with co-registered ultrasound images of the lesion breast and the normal contralateral breast in the same quadrant as the lesion. Data at the contralateral breast have been used for the estimation of background optical properties as well as the calculation of perturbation for image reconstruction.

3.1.

Effect of Layer Depth Mismatch between Lesion and Reference

Figure 3 shows simulation results of two-layer interface depth mismatch between the reference and target sites. In the simulation, the second layer of the target site was kept at $1.5\mathrm{cm}$ depth, and the second layer of the reference site was varied from $1.5\mathrm{cm}$ , to $1.75\mathrm{cm}$ , and to $2\mathrm{cm}$ in Figs. 3a, 3b, 3c, and to $1.4\mathrm{cm}$ and $1.3\mathrm{cm}$ in Figs. 3d and 3e, respectively. The optical properties of two layers were chosen as ${\mu }_{a1}=0.02$ , ${\mu }_{s1}^{\prime }=7.0$ , ${\mu }_{a2}=0.1$ , and ${\mu }_{s2}^{\prime }=7.0{\mathrm{cm}}^{-1}$ . The target optical properties were chosen as ${\mu }_a=0.20$ and ${\mu }_s^{\prime }=7.0{\mathrm{cm}}^{-1}$ . When the second-layer depth of the reference site matches the second-layer depth of the target site, the reconstructed target appears at the correct location with no artifacts [Figs. 3(a-1)]. The reconstructed maximum absorption coefficient was only 39% of the true value because of the use of a larger ROI (8 times the target size) in spatial dimension. With a tighter ROI (two times the target size) in spatial dimension, the reconstructed maximum absorption coefficient reached 78% of the true value [Figs. 3(a-2)]. The scale of figures (a-2) was adjusted to show a better quantification of the target. To highlight the relative changes when the second-layer depth and titling angle as well as the bulk optical properties of the two layers were varied, we have used 8 times target size as the ROI in the rest of the simulation and phantom studies. Therefore, the results are pertinent to optical tomography with target depth guidance only.

Fig. 3

Effect of second-layer depth mismatch between the reference site and the target site. The second layer at the reference site was located at (a) $1.5\mathrm{cm}$ , (b) $1.75\mathrm{cm}$ , (c) $2\mathrm{cm}$ , (d) $1.4\mathrm{cm}$ , and (e) $1.3\mathrm{cm}$ in depth. The second layer of the target site was located at $1.5\mathrm{cm}$ depth. Background optical properties at both reference and target sites are ${\mu }_{a1}=0.02$ , ${\mu }_{s1}^{\prime }=7.0$ , ${\mu }_{a2}=0.1$ , and ${\mu }_{s2}^{\prime }=7.0{\mathrm{cm}}^{-1}$ . The top picture of each part shows the position of the second layer and the target, and the bottom pictures [(a-1) and (b) to (e)] show the reconstructed absorption maps using an ROI of eight times the target size, while the bottom picture (a-2) shows the absorption maps reconstructed with an ROI of two times the target size. The color bar is the absorption coefficient in units of ${\mathrm{cm}}^{-1}$ , and color bars in (a-2) was adjusted to $0.2{\mathrm{cm}}^{-1}$ for better visualization of the target. In the absorption map, each slice presents a spatial image of $8\mathrm{cm}\times 8\mathrm{cm}$ obtained from $0.4\mathrm{cm}$ underneath the probe surface to $2.9\mathrm{cm}$ in depth, with $0.5\text{-}\mathrm{cm}$ spacing between slices. The spatial image dimensions for each slice in the absorption map and the spacing between the slices were kept the same as in Fig. 3 in the following figures except in Fig. 14. (Color online only.)

When the second layer of the reference site is deeper than that of the target site, image artifacts appear. The artifact is more pronounced when the second layer is deeper [Figs. 3b and 3c]. When the second layer of the reference site was placed at a lower depth than that of the target site, for example, at 1.4 or $1.3\mathrm{cm}$ , the reconstructed absorption coefficient and the size of the target became lower and smaller [Figs. 3d and 3e]. The decrease in reconstructed absorption coefficient of the target depends on the optical properties of the two-layer media at the target and reference sites and the depth mismatch.

Table 1 shows the fitted optical properties of the two layer media used in obtaining Fig. 3. The nonlinear Nelder-Mead algorithm discussed in Sec. 2.2 was used for fitting and 1% noise was added to the simulated data. The fitting error of ${\mu }_{s2}^{\prime }$ is larger when the second layer is deeper.^{24, 25} Using the fitted background optical properties, we reconstructed the target again, and the percentage of maximum reconstructed absorption coefficient versus the reference interface depth is shown in Fig. 4 . Compared to those obtained using true background values, the results of using fitted background values are essentially the same.

Fig. 4

Percentage of maximum reconstructed absorption coefficient of the target normalized to the no-background mismatch case as a function of the two-layer interface depth at the reference site using both calibrated background and fitted background optical properties (simulation data). The two-layer interface at the target side is located at $1.5\mathrm{cm}$ from the probe.

Table 1

Fitted optical properties of two-layer media used in Fig. 3.

To validate the simulation results, we performed phantom experiments. An example of phantom experiments is given in Fig. 5 . Figure 5a shows a B-scan ultrasound image of the two-layer medium with a low-contrast absorber located at $(x,y,z)=(0,0,0.9\mathrm{cm})$ inside the intralipid solution (target site). The top layer of the phantom was made of a homogeneous 0.8% intralipid solution of calibrated values ${\mu }_a=0.025{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=7.53{\mathrm{cm}}^{-1}$ , and the bottom layer was a solid phantom made of plastisol of calibrated values ${\mu }_a=0.08{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=6.5{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . The target was a $1.0\text{-}\mathrm{cm}$ -diam spherical absorber of calibrated optical properties ${\mu }_a=0.07{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=5.5{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . The plasitsol phantom was used as a chest-wall layer and placed at $1.4\mathrm{cm}$ underneath the intralipid with zero-degree tilt. Figure 5b is a B-scan ultrasound image of the phantom without target (reference site). Figure 5c shows the reconstructed absorption map at $780\mathrm{nm}$ , which is an ideal image obtained without any mismatch. Figure 5d shows a B-scan ultrasound image of the two-layer medium with the target. The image is the same as Fig. 5a. Figure 5e is a B-scan ultrasound image of the reference site with the plastisol phantom placed at $2\mathrm{cm}$ depth underneath the intralipid with zero-degree tilt. Figure 5f shows the reconstructed absorption map at $780\mathrm{nm}$ . The image artifacts pattern is similar to the one obtained from simulations under the same conditions.

Fig. 5

Reconstructed absorption map of a $1\text{-}\mathrm{cm}$ -diam spherical target with calibrated optical properties ${\mu }_a=0.07$ and ${\mu }_s^{\prime }=5.5{\mathrm{cm}}^{-1}$ , located at $(x,y,z)=(0,0,0.9\mathrm{cm})$ . (a) B-scan ultrasound image of a two-layer medium with the target. A plastisol phantom was located at $1.4\mathrm{cm}$ underneath the probe surface with a zero-degree tilting angle. (b) B-scan ultrasound image of the phantom without the target. (c) Reconstructed target absorption map at $780\mathrm{nm}$ . (d) B-scan ultrasound image of the two-layer medium with the target [same as (a)]. (e) The two-layer medium with the plastisol phantom located at $2\mathrm{cm}$ with a zero-degree tilting angle. (f) Reconstructed target absorption map at $780\mathrm{nm}$ .

A sequence of phantom experiments was performed to further validate the simulation results. The top layer of the phantom was made of a homogeneous 0.8% intralipid solution of calibrated optical properties ${\mu }_a=0.024{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=8.25{\mathrm{cm}}^{-1}$ , and the bottom layer was a solid phantom made of plastisol of calibrated values ${\mu }_a=0.08{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=6.5{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . The target was a $1.0\text{-}\mathrm{cm}$ -diam spherical absorber of calibrated optical properties ${\mu }_a=0.23{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=5.45{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . The two-layer interface was located at $1.5\mathrm{cm}$ from the probe at the target site, and it varied from $1.2\text{to}2\mathrm{cm}$ at the reference site. The target and reference measurements were made from the two-layer medium with and without the target, respectively. The percentage of maximum reconstructed absorption coefficient of the target normalized to the no-background mismatch case using both calibrated and fitted background optical properties is shown in Fig. 6 . The fitted optical properties of the two-layer media for the phantom experiments are given in Table 2 . The error of the fitted second layer ${\mu }_{s2}^{\prime }$ is larger when the second layer is deeper.^{24, 25} However, the reconstructed target ${\mu }_a$ is reasonably robust to the fitting error.

Fig. 6

Percentage of the maximum reconstructed ${\mu }_a$ normalized to the no-background mismatch case as a function of the two-layer interface depth at the reference site (phantom data) using both calibrated background and fitted background optical properties values. The two-layer interface at the target site is located at $1.5\mathrm{cm}$ depth from the probe.

Table 2

Fitted optical properties of two-layer phantoms.

The preceding reported simulation and experimental results can be explained intuitively. As the second layer at the reference site becomes shallower, more photons are absorbed by the second layer, and therefore fever photons are detected at the surface. As a result, the perturbation, which is the normalized difference between lesion data and reference data, is reduced. This reduction in perturbation degrades the target reconstruction accuracy and reduces the target contrast. Certainly, the reduction in the reconstructed target absorption with depth mismatch depends on the optical properties of the two layers and target contrast. On the other hand, as the second layer at the reference site becomes deeper, fever photons are absorbed by the second layer and more are detected at the surface of the reference site. As a result, the perturbation consists of both portions caused by lesion and lesion-reference background mismatch. The perturbation caused by lesion-reference background mismatch produces image artifacts in background regions and increases the reconstructed absorption coefficient of the target.

3.2.

Effect of Mismatch Tilting Angle between Lesion and Reference Sites on Image Reconstruction

As shown in Fig. 1, when the second layer is tilted with respect to the imaging probe, the amplitude and phase (not shown) measurements spread over a larger angular region. Figure 7 demonstrates the effect of mismatch between the interface tilting angle in reference and lesion measurements. In this set of simulations, the second-layer depth in both the target and reference measurements was kept at $1.5\mathrm{cm}$ , and the interface angle at the target site was zero deg. The mismatch at the second layer of the reference site was introduced as $-10$ , $-5$ , 0, 5, or $10\mathrm{deg}$ in the $y$ or $x$ direction. The optical properties of the two layers were chosen as ${\mu }_{a1}=0.02$ , ${\mu }_{s1}^{\prime }=7.0$ , ${\mu }_{a2}=0.1$ , and ${\mu }_{s2}^{\prime }=7.0{\mathrm{cm}}^{-1}$ . The target optical properties were chosen as ${\mu }_a=0.20$ and ${\mu }_s^{\prime }=7.0{\mathrm{cm}}^{-1}$ . In our system, the ultrasound probe is located at the center of the optical probe. Therefore, with the assistance of the real-time ultrasound image, the lesion is always positioned in the center of the optical probe. The results show that the mismatch angle introduces a target position shift and reduces the accuracy of the reconstructed absorption coefficient. Figure 8 shows the percentage of maximum reconstructed ${\mu }_a$ relative to the values obtained with the flat interface (no-background mismatch) as a function of the tilting angle difference between the target and the reference sites. As a result of this lesion-reference tilting angle mismatch, the reconstructed target degrades in accuracy and shifts in position.

Fig. 7

Reconstructed absorption maps of a target located at $(x,y,z)=(0,0,0.9\mathrm{cm})$ . The first-and second-layer interfaces of both the target and the reference site were positioned at $1.5\mathrm{cm}$ depth. The interface at the target site has a zero-degree tilting angle. (a) to (i) sequentially show the absorption maps of the target with a target–reference layer tilting angle mismatch of (a) $0\mathrm{deg}$ , (b) $10\mathrm{deg}$ , (c) $5\mathrm{deg}$ , (d) $-10\mathrm{deg}$ , and (e) $-5\mathrm{deg}$ in the $x$ direction, and of (f) $10\mathrm{deg}$ , (g) $5\mathrm{deg}$ , (h) $-10\mathrm{deg}$ , and (i) $-5\mathrm{deg}$ mismatch in the $y$ direction. Optical properties of both the reference and target sites are ${\mu }_{a1}=0.02$ , ${\mu }_{s1}^{\prime }=7.0$ , ${\mu }_{a2}=0.1$ , and ${\mu }_{s2}^{\prime }=7.0{\mathrm{cm}}^{-1}$ The top picture of each part shows the position of the second layer and the target, and bottom picture shows the reconstructed absorption map.

Fig. 8

Percentage of maximum reconstructed ${\mu }_a$ relative to the values obtained with the flat interface (no-background mismatch) as a function of the difference between the tilting angle of the two-layer interface at the target and reference sites.

We validated the simulation results with phantom experiments using the same setup and the target as described in the previous section. Figure 9a shows a B-scan ultrasound image of a two-layer medium with target located at $(x,y,z)=(0,0,0.9\mathrm{cm})$ . The solid phantom was used as a chest-wall layer and placed underneath the intralipid at $1.4\mathrm{cm}$ with a zero-degree tilting angle. Figure 9b is the B-scan ultrasound image of the same phantom without target and $-8\mathrm{deg}$ titling angle. Figure 9c shows the reconstructed absorption map at $780\mathrm{nm}$ . Similar distortion effects are observed as seen in simulations.

Fig. 9

Reconstructed absorption map of $1\text{-}\mathrm{cm}$ -diam spherical target located at (0, 0, $0.9\mathrm{cm}$ ) with calibrated optical properties ${\mu }_a=0.07{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=5.5{\mathrm{cm}}^{-1}$ . (a) B-scan ultrasound image of a two-layer medium with target. The plastisol phantom was located at $1.4\mathrm{cm}$ with a zero-degree tilting angle. (b) B-scan ultrasound image of the same phantom without the target. The layer was located at $1.4\mathrm{cm}$ with $-8\text{-}\mathrm{deg}$ tilting angle. (c) Reconstructed target absorption map at $780\mathrm{nm}$ .

3.3.

Effect of Optical Property Mismatch between Lesion and Reference Media

In this section, we investigate the effect of background optical properties on image reconstruction. In our clinical experiments, background optical properties were estimated from the normal contralateral breast of the same quadrant with the assumption that both breasts at the symmetric locations have similar bulk optical properties. This is not a problem in general (see Sec. 4). However, for patients who have reconstructed contralateral breasts due to prior surgery, the estimated bulk optical properties at the reference site can be different from the lesion site regardless of the absorption properties of the lesion. For patients with contralateral disease, the optical properties of the reference site may also defer from that at the lesion site.

Figure 10 demonstrates the effect of mismatch between the first-layer absorption coefficient of reference site and the lesion site. The first-layer absorption coefficient has been chosen as $0.023{\mathrm{cm}}^{-1}$ (intralipid) and $0.04{\mathrm{cm}}^{-1}$ (intralipid with ink). Other optical properties have calibrated values of ${\mu }_{s1}^{\prime }=7.5$ , ${\mu }_{a2}=0.08$ , and ${\mu }_{s2}^{\prime }=6.5{\mathrm{cm}}^{-1}$ . The target is the $1\text{-}\mathrm{cm}$ solid phantom with calibrated optical properties of ${\mu }_a=0.23$ and ${\mu }_s^{\prime }=5.45{\mathrm{cm}}^{-1}$ . Figure 10 shows the reconstructed image, and Fig. 11 shows the changes in the reconstructed absorption coefficient for cases when the target site has higher, equal, and lower first-layer absorption coefficient compared to the reference site. The comparison has been made using both the true and fitted background optical properties, and the results are essentially the same. The fitted optical properties of the two-layer media were ${\mu }_{a1}=0.043$ , ${\mu }_{s1}^{\prime }=8.07$ , ${\mu }_{a2}=0.072$ , and ${\mu }_{s2}^{\prime }=13.22{\mathrm{cm}}^{-1}$ for the case of higher first-layer ${\mu }_{a1}$ at the target site and ${\mu }_{a1}=0.023$ , ${\mu }_{s1}^{\prime }=7.5$ , ${\mu }_{a2}=0.098$ , and ${\mu }_{s2}^{\prime }=19.45{\mathrm{cm}}^{-1}$ for the case of lower ${\mu }_{a1}$ at the target site.

Fig. 10

Effect of first-layer ${\mu }_a$ mismatch between the reference site and the target site on the reconstructed image. First-layer optical properties at the target and reference sites are (a) ${\mu }_{a1̱\mathit{tar}}=0.023$ , ${\mu }_{a1̱\mathit{ref}}=0.023$ ; (b) ${\mu }_{a1̱\mathit{tar}}=0.04$ , ${\mu }_{a1̱\mathit{ref}}=0.023$ ; (c) ${\mu }_{a1̱\mathit{tar}}=0.04$ , ${\mu }_{a1̱\mathit{ref}}=0.04$ ; and (d) ${\mu }_{a1̱\mathit{tar}}=0.023$ , ${\mu }_{a1̱\mathit{ref}}=0.04$ . The other background optical properties of both the reference and target sites are ${\mu }_{s1}^{\prime }=7.5$ , ${\mu }_{a2}=0.08$ , and ${\mu }_{s2}^{\prime }=6.5{\mathrm{cm}}^{-1}$ . The optical properties of the target are ${\mu }_a=0.23$ and ${\mu }_s^{\prime }=5.45{\mathrm{cm}}^{-1}$ .

Fig. 11

Percentage of maximum reconstructed ${\mu }_a$ normalized to the no-background mismatch case using both calibrated and fitted background values. Lower ${\mu }_{a1̱\mathit{tar}}$ shows the results when ${\mu }_{a1̱\mathit{tar}}=0.023$ and ${\mu }_{a1̱\mathit{ref}}=0.04{\mathrm{cm}}^{-1}$ , and higher ${\mu }_{a1̱\mathit{tar}}$ shows the results when ${\mu }_{a1̱\mathit{tar}}=0.04$ , and ${\mu }_{a1̱\mathit{ref}}=0.023{\mathrm{cm}}^{-1}$ . The other optical properties of both the target and reference sites are ${\mu }_{s1}^{\prime }=7.5$ , ${\mu }_{a2}=0.08$ , and ${\mu }_{s2}^{\prime }=6.5{\mathrm{cm}}^{-1}$ .

The preceding experiments can be explained intuitively. As the first-layer background absorption of the reference site increases, more photons are absorbed and fewer photons are reflected. As a result, the perturbation is reduced as compared with no mismatch, and the reconstructed target contrast is lower and the target mass becomes smaller. On the other hand, lower ${\mu }_{a1}$ of the reference site or higher ${\mu }_{a1}$ of the target site increases the perturbation caused by non-target-related background mismatch, which contributes to image artifacts in the background regions.

The effect of the first layer ${\mu }_{s1}^{\prime }$ mismatch on the reconstructed absorption coefficient of the target is shown in Fig. 12 . The top layer of the phantom was made of a homogeneous intralipid solution of different concentrations. The calibrated ${\mu }_{s1}^{\prime }$ varied from $5.8\text{to}8.2{\mathrm{cm}}^{-1}$ with ${\mu }_{a1}=0.024{\mathrm{cm}}^{-1}$ , and the bottom layer was a solid plastisol phantom with calibrated values of ${\mu }_a=0.08{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=6.5{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . The target was the $1.0\text{-}\mathrm{cm}$ -diam spherical absorber of calibrated optical properties ${\mu }_a=0.23{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=5.45{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . While ${\mu }_{s1}^{\prime }$ was changed from $5.8\text{to}8.2{\mathrm{cm}}^{-1}$ at the target site, it was fixed to $7.0{\mathrm{cm}}^{-1}$ at the reference site. Other optical properties of both reference and target sites were ${\mu }_{a1}=0.024$ , ${\mu }_{a2}=0.08$ , and ${\mu }_{s2}^{\prime }=6.5{\mathrm{cm}}^{-1}$ . Figure 13 shows the percentage of reconstructed absorption coefficient of the target normalized to the no-background mismatch case using both true and fitted values of two-layer optical properties. The fitted optical properties of the two-layer medium were ${\mu }_{a1}=0.031$ , ${\mu }_{s1}^{\prime }=10.49$ , ${\mu }_{a2}=0.097$ , and ${\mu }_{s2}^{\prime }=7.25{\mathrm{cm}}^{-1}$ .

Fig. 12

Effect of the first-layer ${\mu }_{s1}^{\prime }$ mismatch on the reconstructed absorption map of the target. The first-layer reduced scattering coefficient at the target site is (a) ${\mu }_{s1}^{\prime }=5.8{\mathrm{cm}}^{-1}$ , (b) ${\mu }_{s1}^{\prime }=6.5{\mathrm{cm}}^{-1}$ , (c) ${\mu }_{s1}^{\prime }=7{\mathrm{cm}}^{-1}$ , (d) ${\mu }_{s1}^{\prime }=7.3{\mathrm{cm}}^{-1}$ , (e) ${\mu }_{s1}^{\prime }=7.7{\mathrm{cm}}^{-1}$ , and (f) ${\mu }_{s1}^{\prime }=8.15{\mathrm{cm}}^{-1}$ , respectively. ${\mu }_{s1}^{\prime }$ at the reference site is $7.0{\mathrm{cm}}^{-1}$ . Other optical properties of both the reference and lesion sites are ${\mu }_{a1}=0.024$ , ${\mu }_{a2}=0.08$ , and ${\mu }_{s2}^{\prime }=6.5{\mathrm{cm}}^{-1}$ .

Fig. 13

Percentage of maximum reconstructed ${\mu }_a$ normalized to the no-background mismatch case as a function of first-layer reduced scattering coefficient at the target site using both calibrated and fitted background optical properties. The reduced scattering coefficient is $7.0{\mathrm{cm}}^{-1}$ at the reference site.

As shown in Figs. 12 and 13, a higher background scattering at the target site produces image artifacts, and a lower background scattering causes reduction in target contrast. This is because the higher background scattering at the target site increases the perturbation due to decreased signal strength measured at the target site. The extra perturbation due to background scattering mismatch contributes to the image artifacts in the background region, while the lower background scattering at the lesion site causes the reduction in perturbation due to the increased signal strength measured at the target site. As a result, the reduction in perturbation due to scattering mismatch causes the decrease in reconstructed target absorption.

3.4.

Clinical Examples

To evaluate the mismatch in breast imaging, we reconstructed the absorption map of a benign fibroadenoma using three different reference sites acquired at the contralateral breast. Figure 14a shows a co-registered ultrasound B-scan of the lesion. The center of the lesion was at $1.2\mathrm{cm}$ depth, and the chest-wall layer was located approximately at $1.5\mathrm{cm}$ with an $8\text{-}\mathrm{deg}$ tilting angle. The fitted optical properties of the background were ${\mu }_{a1}=0.049$ , ${\mu }_{s1}^{\prime }=9.87$ , ${\mu }_{a2}=0.11$ , ${\mu }_{s2}^{\prime }=2.9{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . Figure 14b shows the US B-scan of the first reference and the corresponding reconstructed absorption map at $830\mathrm{nm}$ . The chest-wall interface at the reference is almost flat and located at $1.5\mathrm{cm}$ depth. Figure 14c shows the second reference used and the reconstructed absorption map. This reference is deeper and tilted compared to the lesion site; therefore, the lesion mass was more spread out compared to part (a). Figure 14d shows the third reference and the reconstructed absorption map. This reference is almost flat and deeper than the reference in part (a); therefore, the lesion mass was more spread out compared to part (a). Due to the depth mismatch, some artifacts appeared in the figure; however, since the chest-wall and breast tissue of clinical cases are not homogenous and their interference is not well-defined as in the phantom and simulation studies, the artifacts are not as pronounced as the phantom and simulation. The reconstructed maximum and average ${\mu }_a$ of lesion for different reference sites are listed in Table 3 .

Fig. 14

Co-registered ultrasound and reconstructed absorption maps of a benign fibroadenoma. (a) B-scan ultrasound image of the lesion site. The chest-wall layer was located at $1.5\mathrm{cm}$ with $8\mathrm{deg}$ tilt. (b) B-scan ultrasound of the first reference site and the reconstructed absorption maps at $830\mathrm{nm}$ . (c) B-scan ultrasound of the tilted reference site and the reconstructed absorption maps at $830\mathrm{nm}$ . (d) B-scan ultrasound of the third reference site and the reconstructed absorption maps at $830\mathrm{nm}$ . Each slice presents a spatial image of $8\mathrm{cm}\times 8\mathrm{cm}$ obtained from $0.18\mathrm{cm}$ underneath the probe surface to $2.68\mathrm{cm}$ in depth, with $0.5\text{-}\mathrm{cm}$ spacing between slices.

Table 3

Comparison of four parameters for the clinical case shown in Fig. 14: interface depth and tilting angle of the two-layer interface at the reference site and maximum and mean values of the reconstructed target absorption coefficient.