2.1.

Computational Algorithms

To estimate two-layer optical properties with an arbitrary breast-tissue and chest-wall interface, the FEM has been used to generate a 3-D mesh consisting of breast tissue and the chest wall. The breast-tissue and chest-wall interface depth and tilting angle are extracted from the coregistered ultrasound images measured at a contralateral reference site. For a semi-infinite model, the depth of the medium is chosen as $6.5\mathrm{cm}$ . The frequency domain diffusion approximation and the Robin-type (type 3) boundary condition are adopted in the forward computation.²⁶

A nonlinear regression algorithm based on the Nelder-Mead method²⁷ has been adopted to estimate the two-layer optical parameters. This algorithm is a multidimensional unconstrained minimization method, and has been used to estimate tissue optical properties from two-layer tissue structures in our earlier studies.^{21, 23, 26} The convergence rate of the algorithm is in the range of 150 to 230 iterations, which is slow. However, fitted first-layer background optical properties can reach 30% of the true value in simulations and phantom experiments. The fitting error of the second-layer optical properties depends on the depth of the two-layer interface and increases as the depth of the second layer increases. As expected, when the second layer is deeper, it affects closer source-detector pair measurements less and distant source-detector pairs more. In general, distant measurements have lower signal-to-noise ratio.

The fitted absorption and reduced scattering coefficients of the first and second layers are used to estimate the forward photon density wave as follows:

For image reconstruction, the two-layer geometry information of the target (lesion) site obtained from ultrasound images, and the optical properties obtained from the nonlinear regression fitting algorithm, have been used to form a 3-D FEM model of the breast tissue and chest wall. A 3-D Jacobian weight matrix

Fig. 1

Flow chart of the image reconstruction procedures. The reference data obtained from the reference site were used to fit the background properties of a two-layer medium. The layer depth and tilting angle measured from coregistered ultrasound was used to segment the medium as breast tissue and chest wall. The fitted background optical properties were input into the FEM model to compute the weight matrix for inversion. The depth at the target site was also estimated from ultrasound to segment the medium with different background optical properties obtained from fitting for weight matrix computation.

2.2.

Experimental Methods

3.1.

Simulations

3.1.1.

Effects of second-layer location on estimation of background optical properties and target absorption

In many clinical cases, the chest wall does not have a major effect on the optical measurements, and the background can be considered as a homogeneous and semi-infinite medium. However, for patients who have a small amount of breast tissue, i.e., the chest wall is located $2.0\mathrm{cm}$ or less from the probe, the effect of the chest wall on the measurement data cannot be neglected and should be considered in the image reconstruction model. The following examples show the necessity of adapting a two-layer model for image reconstruction. Figures 2, 2, 2, 2 show the fitted optical properties of the two-layer medium using the two-layer and semi-infinite models, respectively. The different interface depth and tilting angle is presented in the $x$ axis. The results show that the error of the fitted optical properties increases when the depth of the two-layer interface is closer to the surface if the semi-infinite model is used. However, the two-layer model can estimate the optical properties of the two-layer medium within 13% of the true value when the chest-wall interface is less than $2.0\mathrm{cm}$ deep.

Fig. 2

Fitted optical properties of two-layer media with true values of ${\mu }_{a1}=0.02$ , ${\mu }_{s1}^{\prime }=7.0$ , ${\mu }_{a2}=0.1$ , and ${\mu }_{s2}^{\prime }=7.0{\mathrm{cm}}^{-1}$ . Dashed lines show the true values. Blue squares show the fitted values obtained by the two-layer model, and red dots show the fitted values obtained from the semi-infinite model. The $x$ axis shows the position and tilting angle of the second layer from the probe surface. (Color online only.)

As the depth of the second layer increases, the measurements contain less information from the second layer, as most of the optical paths are confined in the first layer. Therefore, the accuracy of the estimated optical properties of the second layer decreases. Under this condition, the measurement data from the lesion are affected more by the first layer optical properties than by the second layer. Therefore, the performance of the semi-infinite model is similar to that of the two-layer model.

Figure 3 shows the reconstructed maximum absorption coefficient of a $1\text{-}\mathrm{cm}$ -diam target when the second layer was located at different depths with different tilting angles from the optical probe. The optical properties of the target were ${\mu }_a=0.07$ and ${\mu }_s^{\prime }=7.0{\mathrm{cm}}^{-1}$ , and the target center was located at $0.9\mathrm{cm}$ depth. The results show that when the two-layer interface is closer to the surface, the error of the reconstructed maximum absorption coefficient of the target increases from 20 to 200% in the semi-infinite model when the second-layer depth decreases from $3.0\text{to}1.5\mathrm{cm}$ . This error can cause misclassification of a benign lesion into the malignant category, because the reconstructed absorption is higher than its true value. However, the two-layer model can reconstruct the target absorption coefficient within 10% of the true value.

Fig. 3

Reconstructed maximum absorption coefficient of a $1\text{-}\mathrm{cm}$ -diam spherical target embedded in the same two-layer medium, as given in Fig. 2. The second layer was located at $1.5\text{-}\mathrm{cm}$ depth with 0 and $10\mathrm{deg}$ angled in difference directions, and 2-, 2.5-, and $3\text{-}\mathrm{cm}$ depth with zero-degree tilting angle from the probe surface. The target was positioned at $x=0$ , $y=0$ , and $z=0.9\mathrm{cm}$ . Dashed lines show the true value. Red dots and blue squares show the reconstructed maximum absorption coefficients of the target using the semi-infinite and two-layer models, respectively. The $x$ axis shows the position and tilting angle of the two-layer interface from the probe surface. (Color online only.)

3.1.2.

Effect of second-layer optical properties on estimation of background optical properties and target absorption

In addition to the location of the second layer, the error caused by using the semi-infinite model also depends on the optical properties of the second layer. Figure 4 shows the fitted optical properties of the two-layer medium for different second-layer absorption coefficients, as given in the $x$ axis. The results indicate that the error of the fitted values obtained by the semi-infinite model increases when the absorption coefficient of the second layer increases. However, the two-layer model is capable of estimating optical properties of both layers, (except for ${\mu }_{s2}^{\prime }$ ) within 10% error.

Fig. 4

Fitted optical properties of two-layer media with different optical properties. The two-layer interface was located at $1.5\text{-}\mathrm{cm}$ depth with $10\text{-}\mathrm{deg}$ tilting angle from the probe. Dashed lines show the true optical properties of the two-layer media. Blue squares and red dots show the fitted values obtained from the two-layer and semi-infinite models when the second-layer absorption coefficient increases, as shown in the $x$ axis. In each figure, the $x$ axis shows the optical properties of the two-layer medium, and the $y$ axis shows (a) the fitted first-layer absorption coefficient, (b) first-layer reduced scattering coefficient, (c) second-layer absorption coefficient, and (d) second-layer reduced scattering coefficient. (Color online only.)

The fitting error of the second layer ${\mu }_{s2}^{\prime }$ is larger than that of ${\mu }_{a2}$ due to its smaller contribution to the measurement data as compared with ${\mu }_{a2}$ . Figure 5 demonstrates the effect of ${\mu }_{a2}$ [Figs. 5 and 5] and ${\mu }_{s2}^{\prime }$ [Figs. 5 and 5] on measured amplitude (in logarithmic sale) and phase profiles as a function of the source-detector distance. The simulated two-layer medium had an interface located at $1.5\mathrm{cm}$ depth with a $0\text{-}\mathrm{deg}$ tilting angle. Optical properties of the first layer were ${\mu }_{a1}=0.02$ and ${\mu }_{s1}^{\prime }=7.0{\mathrm{cm}}^{-1}$ . ${\mu }_{s2}^{\prime }$ was fixed at $7.0{\mathrm{cm}}^{-1}$ in Figs. 5 and 5, and ${\mu }_{a2}$ was fixed at $0.1{\mathrm{cm}}^{-1}$ in Figs. 5 and 5. The results show that ${\mu }_{a2}$ has more influence on the measurement data than that of ${\mu }_{s2}^{\prime }$ . As a result, the accuracy of the estimated second-layer absorption coefficient is higher than that of the reduced scattering coefficient.

Fig. 5

(a) Amplitude (logarithmic scale) and (b) phase profiles of reflectance measurements for different second-layer absorption coefficients from $0.02\text{to}0.1{\mathrm{cm}}^{-1}$ . The other optical properties of the two-layer medium were ${\mu }_{a1}=0.02$ , ${\mu }_{s1}^{\prime }=7.0$ , and ${\mu }_{s2}^{\prime }=7.0{\mathrm{cm}}^{-1}$ . (c) Amplitude (logarithmic scale) and (d) phase profiles of reflectance measurements for different second-layer reduced scattering coefficients from $5\text{to}15{\mathrm{cm}}^{-1}$ . The other optical properties of the medium were ${\mu }_{a1}=0.02$ , ${\mu }_{a2}=0.1$ , and ${\mu }_{s2}^{\prime }=7.0{\mathrm{cm}}^{-1}$ . The two-layer interface was located at $1.5\mathrm{cm}$ with a $0\text{-}\mathrm{deg}$ tilting angle from the probe. $\rho$ is the source-detector distance.

Figure 6 shows the reconstructed maximum absorption coefficient of a $1\text{-}\mathrm{cm}$ -diam target embedded in a different second-layer background absorption coefficient. The target was located at $0.9\mathrm{cm}$ depth with optical properties of ${\mu }_a=0.07$ and ${\mu }_s^{\prime }=7.0{\mathrm{cm}}^{-1}$ . The background optical properties were the same as those used in Fig. 4. The results show that by increasing the absorption coefficient of the second layer, the error of the reconstructed maximum absorption coefficient of the target increases in the semi-infinite model. Again, this error can cause misclassification of a benign lesion into the malignant category. However, the target absorption coefficient can be reconstructed within 10% of the true value when the two-layer model is used.

Fig. 6

Reconstructed maximum absorption coefficient of a $1\text{-}\mathrm{cm}\text{-}\text{diam}$ spherical target embedded in a two-layer medium with different second-layer absorption coefficients. The two-layer interface was located at $1.5\text{-}\mathrm{cm}$ depth with $10\text{-}\mathrm{deg}$ tilting angle from the probe surface. The target was positioned at $x=0$ , $y=0$ , and $z=0.9\mathrm{cm}$ . The dashed line shows the true values. Red dots and blue squares show the maximum value of the reconstructed absorption coefficient of the target using the semi-infinite and two-layer models, respectively. The $x$ axis shows the optical properties of the two-layer media. (Color online only.)

A two-layer model of optical properties of ${\mu }_{a1}=0.02$ , ${\mu }_{s1}^{\prime }=7.0$ , and ${\mu }_{a2}=0.02{\mathrm{cm}}^{-1}$ was used to compare the performance of a semi-infinite model when the second-layer reduced scattering coefficient was different than its first layer. Figure 7 shows the fitted optical properties of the two-layer medium as ${\mu }_{s2}^{\prime }$ varies from $4\text{to}15{\mathrm{cm}}^{-1}$ . The second layer was located at $1.5\mathrm{cm}$ with a $10\text{-}\mathrm{deg}$ tilting angle from the probe. At the higher reduced scattering coefficient of ${\mu }_{s2}^{\prime }=15{\mathrm{cm}}^{-1}$ , the error of fitted first-layer background absorption is about 1.8 times of the true value, in the semi-infinite model.

Fig. 7

Fitted optical properties of two-layer media with different second-layer reduced scattering coefficients. The two-layer interface was located at $1.5\text{-}\mathrm{cm}$ depth with $10\text{-}\mathrm{deg}$ tilting angle from the probe. Dashed lines show the true optical properties of the two-layer media. Blue squares and red dots show the fitted values obtained from the two-layer and semi-infinite models when the second-layer reduced scattering coefficient increases, as shown in the $x$ axis. In each figure, the $x$ axis shows the optical properties of the two-layer medium, and the $y$ axis shows (a) the fitted first-layer absorption coefficient, (b) first-layer reduced scattering coefficient, (c) second-layer absorption coefficient, and (d) second-layer reduced scattering coefficient. (Color online only.)

Figure 8 compares the reconstructed maximum absorption coefficient of the target when the second-layer reduced scattering coefficient varies from $4\text{to}15{\mathrm{cm}}^{-1}$ . The target location and optical properties were the same as those used in Fig. 6. The optical properties of the two-layer medium were the same as those used in Fig. 7. At the higher scattering coefficient of ${\mu }_{s2}^{\prime }=15{\mathrm{cm}}^{-1}$ , the reconstructed maximum absorption coefficient of the target is about 1.5 times the true value, in the semi-infinite model..

Fig. 8

Reconstructed maximum absorption coefficient of a $1\text{-}\mathrm{cm}\text{-}\text{diam}$ spherical target embedded in a two-layer medium with different second-layer reduced scattering coefficients. The two-layer interface was located at $1.5\text{-}\mathrm{cm}$ depth with $10\text{-}\mathrm{deg}$ tilting angle from the probe surface. The target was positioned at $x=0$ , $y=0$ , and $z=0.9\mathrm{cm}$ . The dashed line shows the true values. Red dots and blue squares show the maximum value of the reconstructed absorption coefficient of the target using the semi-infinite and two-layer models, respectively. The $x$ axis shows the optical properties of the two-layer media. (Color online only.)

3.2.

Phantom Experiments

To validate the results of the simulation, phantom experiments with various two-layer interface depths and tilting angles were performed. The top layer of the medium was made of 0.8% intralipid solution of calibrated absorption and reduced scattering coefficients of ${\mu }_{a1}=0.025$ and ${\mu }_{s1}^{\prime }=9.9{\mathrm{cm}}^{-1}$ . The bottom layer was made of solid plastisol phantom with calibrated optical properties of ${\mu }_{a2}=0.08$ and ${\mu }_{s2}^{\prime }=7.0{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ .

Figures 9 and 9 show the B-scan ultrasound images of the $x\text{-}z$ and $y\text{-}z$ cross section of the two-layer medium with and without the target, respectively. The plastisol phantom, which was used as a chest-wall layer, was placed at $1.4\mathrm{cm}$ with a $7\text{-}\mathrm{deg}$ tilting angle from the probe surface. The target was a $1\text{-}\mathrm{cm}$ -diam spherical solid absorber with calibrated optical properties of ${\mu }_a=0.07$ and ${\mu }_s^{\prime }=5.5{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . The target was located at $(x,y,z)=\left(\mathrm{0,0,0.9}\right)\mathrm{cm}$ inside the intralipid solution. The fitted absorption and reduced scattering coefficients of the two-layer medium obtained by the two-layer model were ${\mu }_{a1}=0.031$ , ${\mu }_{a2}=0.097$ , ${\mu }_{s1}^{\prime }=10.0$ , and ${\mu }_{s2}^{\prime }=7.2{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ . The fitted optical properties as estimated by the semi-infinite model were ${\mu }_a=0.069$ and ${\mu }_s^{\prime }=7.1{\mathrm{cm}}^{-1}$ . Figures 9 and 9 show the reconstructed absorption coefficient of the target obtained by the two-layer and semi-infinite models, respectively. In the absorption map, each slice presents a spatial image of $8\times 8\mathrm{cm}$ obtained from $0.4\mathrm{cm}$ underneath the probe surface to $2.9\mathrm{cm}$ in depth with $0.5\text{-}\mathrm{cm}$ increments between slices. Table 1 shows the fitted background optical properties and reconstructed absorption coefficient of a target located at different depths. The results show that the reconstructed maximum absorption coefficient of the target improves from 207% using the semi-infinite model to 114% using the two-layer model.

Fig. 9

(a) and (b) show the $x\text{-}z$ and $y\text{-}z$ cross section of ultrasound images of the two-layer medium. The two-layer interface was located at $1.4\text{-}\mathrm{cm}$ depth with a $-7\text{-}\mathrm{deg}$ tilting angle from the probe. The target was located at $x=0$ , $y=0$ , and $z=0.9\mathrm{cm}$ . The reconstructed target absorption map using the two-layer model is shown in (c), and the semi-infinite model in (d). In the absorption map, each slice presents a spatial image of $8\times 8\mathrm{cm}$ 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.

Table 1

Fitted optical properties of a two-layer media and reconstructed absorption coefficient of a 1-cm -diam sphere target located 0.9cm from the optical probe using two-layer and semi-infinite models.

3.3.

Clinical Results

Two clinical examples are given here to demonstrate that using the two-layer model can potentially improve the lesion quantification. Figure 10 shows an example of a $49\text{-year}$ -old woman who had a $1.2\text{-}\mathrm{cm}$ suspicious lesion seen by ultrasound. The ultrasound-guided biopsy confirmed that the lesion was a benign fibroadenoma. The center of the breast-tissue and chest-wall interface at the reference site, as shown in Fig. 10, was located at $2.2\mathrm{cm}$ depth. The interface was best modeled as a piece-wise linear curve as shown in the figure. The estimated optical properties of the breast tissue and chest wall were ${\mu }_{a1}=0.036$ , ${\mu }_{a2}=0.061$ , ${\mu }_{s1}^{\prime }=6.5$ , and ${\mu }_{s2}^{\prime }=1.5{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ , and ${\mu }_{a1}=0.043$ , ${\mu }_{a2}=0.087$ , ${\mu }_{s1}^{\prime }=5.9$ , and ${\mu }_{s2}^{\prime }=2.1{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . The fitted background optical properties using the semi-infinite model were ${\mu }_a=0.055$ and ${\mu }_s^{\prime }=4.47{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ , and ${\mu }_a=0.077$ and ${\mu }_s^{\prime }=4.66{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . The breast-tissue and chest-wall interface at the lesion site as shown in Fig. 10 was located at $2.3\mathrm{cm}$ depth with $-2.2\text{-}\mathrm{deg}$ tilting angle, and the center of the lesion was located at $x=0$ , $y=0$ , and $z=1.2\mathrm{cm}$ from the probe. Figures 10, 10, 10 were reconstructed absorption maps obtained at $780\mathrm{nm}$ , $830\mathrm{nm}$ , and the computed total hemoglobin concentration using the semi-infinite model, while Figs. 10, 10, 10 were corresponding results using the two-layer model. The reconstructed maximum absorption coefficients of the lesion acquired by the two-layer model were $0.11{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ and $0.13{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . However, the corresponding maximum values obtained by the semi-infinite model were $0.15{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ and $0.18{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . The computed total hemoglobin concentration showed a decrease from $79.1\mu \mathrm{mol}∕\mathrm{L}$ obtained from the semi-infinite model to $57.9\mu \mathrm{mol}∕\mathrm{L}$ obtained from the two-layer model.

Fig. 10

(a) and (b) are ultrasound images of the lesion and reference sites. (c)– through (e) show reconstructed optical absorption maps of a benign fibroadenoma at (c) $780\mathrm{nm}$ , (d) $830\mathrm{nm}$ , and (e) computed total hemoglobin map using the semi-infinite model. Reconstructed maximum absorption coefficients were $0.15{\mathrm{cm}}^{-1}$ $\left(780\mathrm{nm}\right)$ , $0.18{\mathrm{cm}}^{-1}$ $\left(830\mathrm{nm}\right)$ , and maximum total hemoglobin concentration was $79.1\mu \mathrm{mol}∕\mathrm{L}$ . (f) through (h) show reconstructed optical absorption maps at (f) $780\mathrm{nm}$ , (g) $830\mathrm{nm}$ , and (h) computed total hemoglobin map using the two-layer model. Reconstructed maximum absorption coefficients were $0.11{\mathrm{cm}}^{-1}$ $\left(780\mathrm{nm}\right)$ , $0.13{\mathrm{cm}}^{-1}$ $\left(830\mathrm{nm}\right)$ , and the computed maximum total hemoglobin concentration was $57.9\mu \mathrm{mol}∕\mathrm{L}$ .

Figure 11 shows an example of a $62\text{-year}$ -old woman who had a $7\text{-}\mathrm{mm}$ suspicious lesion seen by ultrasound. The ultrasound-guided biopsy confirmed that the lesion was a ductal carcinoma. The breast-tissue and chest-wall interface at the reference site was located at $1.4\mathrm{cm}$ with a zero-degree tilting angle from the probe [Fig. 11]. The estimated optical properties of the breast tissue and chest wall were ${\mu }_{a1}=0.037$ , ${\mu }_{a2}=0.075$ , ${\mu }_{s1}^{\prime }=7.76$ , and ${\mu }_{s2}^{\prime }=8.98{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ , and ${\mu }_{a1}=0.065$ , ${\mu }_{a2}=0.083$ , ${\mu }_{s1}^{\prime }=4.92$ , and ${\mu }_{s2}^{\prime }=7.27{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . The fitted background optical properties using the semi-infinite model were ${\mu }_a=0.068$ and ${\mu }_s^{\prime }=4.9{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ , and ${\mu }_a=0.087$ and ${\mu }_s^{\prime }=4.86{\mathrm{cm}}^{-1}$ at $830\text{-}\mathrm{nm}$ wavelengths. The breast-tissue and chest-wall interface at the lesion site as shown in Fig. 11 was located at $z=1.5\mathrm{cm}$ depth with $5.5\text{-}\mathrm{deg}$ tilting angle, and the center of the lesion was located at $x=-1.4$ , $y=1.2$ , and $z=1.3\mathrm{cm}$ from the probe. The reconstructed maximum absorption coefficients obtained by the two-layer model were $0.19{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ and $0.24{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . The reconstructed maximum absorption coefficients obtained by the semi-infinite model were $0.25{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ and $0.26{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . The computed total hemoglobin concentration obtained from the two-layer and semi-infinite models was 103 and $122\mu \mathrm{mol}∕\mathrm{L}$ , respectively. In both benign and malignant cases, the lesion total hemoglobin contrast has been reduced; however, the total hemoglobin ratio of malignant over benign lesions was improved from 1.54 obtained from the semi-infinite model, to 1.78 obtained from the two-layer model. Based on the results obtained from simulations and phantoms, the two-layer model should be more accurate in lesion quantification by reducing errors in estimated background optical properties. Currently, we do not have enough benign and malignant cases in this subpopulation to statistically evaluate the improvement of using the two-layer model versus the semi-infinite model in lesion characterization. More patients are being recruited and more cases in this subpopulation will be pooled together to evaluate the performance of the two-layer-model-based reconstruction.

Fig. 11

(a) and (b) show ultrasound images of the lesion and reference sites. (c) through (e) reconstructed optical absorption maps of a $7\text{-}\mathrm{mm}$ malignant lesion at (c) $780\mathrm{nm}$ , (d) $830\mathrm{nm}$ , and (e) computed total hemoglobin map using the semi-infinite model. Reconstructed maximum absorption coefficients were $0.25{\mathrm{cm}}^{-1}$ $\left(780\mathrm{nm}\right)$ , $0.26{\mathrm{cm}}^{-1}$ $\left(830\mathrm{nm}\right)$ , and maximum total hemoglobin concentration was $122\mu \mathrm{mol}∕\mathrm{L}$ . (f) through (h) show reconstructed optical absorption maps at (f) $780\mathrm{nm}$ , (g) $830\mathrm{nm}$ , and (h) computed total hemoglobin map using the two-layer model. Reconstructed maximum absorption coefficients were $0.19{\mathrm{cm}}^{-1}$ $\left(780\mathrm{nm}\right)$ , and $0.24{\mathrm{cm}}^{-1}$ $\left(830\mathrm{nm}\right)$ . The computed maximum total hemoglobin concentration was $103\mu \mathrm{mol}∕\mathrm{L}$ .

Breast tissue optical properties have been reported by several groups^{28, 29, 30, 31, 32, 33, 34, 35} and a summary can be found in a recent paper by Mo ³⁶ The reported background absorption coefficient of breast tissue from nine different groups was in the range of $0.04\text{to}0.05{\mathrm{cm}}^{-1}$ measured in the neighborhood of $780\mathrm{nm}$ . The reduced scattering coefficient was in the range of $8\text{to}12{\mathrm{cm}}^{-1}$ . Literature data on optical properties of chest walls are not available. The only available data are optical properties of muscles and bones. Muscle was reported to have large absorption and medium scattering coefficients, with ${\mu }_a$ in the range of $0.08\text{to}0.27{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }$ in the range of $5\text{to}10{\mathrm{cm}}^{-1}$ .^{37, 38, 39} Bone was a highly absorbing and scattering medium, and was reported to have large absorption and scattering coefficients, with ${\mu }_a$ in the range of $0.05\text{to}0.15{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }$ in the range of $20\text{to}30{\mathrm{cm}}^{-1}$ .^{40, 41} To estimate breast-tissue and chest-wall optical properties, we have applied the fitting algorithm to a group of ten patients. The average age of this group of patients is $50\text{years}$ . These cases were selected from about 100 patients’ data from our on-going clinical studies and represented about 10% of the study population. The coregistered B-scan ultrasound images of these cases showed that the patients’ chest walls were less than $2\mathrm{cm}$ deep from the surface of skins. Our fitting results (see Fig. 12 ) show the average ${\mu }_{a1}=0.044{\mathrm{cm}}^{-1}$ $(\pm 0.025)$ and ${\mu }_{a2}=0.08{\mathrm{cm}}^{-1}$ $(\pm 0.023)$ at $780\mathrm{nm}$ , and the average ${\mu }_{a1}=0.045(\pm 0.017)$ and ${\mu }_{a2}=0.11(\pm 0.058)$ at $830\mathrm{nm}$ . The average ${\mu }_{s1}=7.4{\mathrm{cm}}^{-1}$ $(\pm 3.8)$ and ${\mu }_{s2}=6.7{\mathrm{cm}}^{-1}$ $(\pm 3.5)$ at $780\mathrm{nm}$ , and the average ${\mu }_{s1}=7.5(\pm 3.7)$ and ${\mu }_{s2}=4.9(\pm 2.2)$ at $830\mathrm{nm}$ . If the semi-infinite model was used, the fitted average background absorptions were $0.06{\mathrm{cm}}^{-1}$ $(\pm 0.024)$ at $780\mathrm{nm}$ and $0.072{\mathrm{cm}}^{-1}$ $(\pm 0.023)$ at $830\mathrm{nm}$ . The corresponding reduced scattered coefficients were $5.4{\mathrm{cm}}^{-1}$ at $780\mathrm{nm}$ and $5.1{\mathrm{cm}}^{-1}$ at $830\mathrm{nm}$ . The fitted first-layer ${\mu }_{a1}$ is lower and the second-layer ${\mu }_{a2}$ is higher than the average ${\mu }_a$ obtained from the semi-infinite fitting at both 780- and $830\text{-}\mathrm{nm}$ wavelengths. The fitted first-layer ${\mu }_{s1}^{\prime }$ is higher and the second-layer ${\mu }_{s2}^{\prime }$ is lower than the average obtained from the semi-infinite fitting. The fitted first-layer breast-tissue optical properties reported in this study are in good agreement with the literature data. The fitted second-layer data are within the range of optical properties reported for muscles.

Fig. 12

The estimated mean background optical properties of ten clinical cases obtained by the semi-infinite and two-layer models. The units are in ${\mathrm{cm}}^{-1}$ .