In this section, we show the recovered images for healthy breasts using the above described algorithm and compare the output with the results from unconstrained image reconstructions. We also estimated the optical properties for adipose and fibroglandular tissue and compare the findings with the previous results computed based on expert segmentation.

First, using a set of 31 DBT images of healthy breast as a learning set, we study the X-ray contrasts of different tissue types and seek possibilities for an automatic compositional segmentation approach. The intensity values in all DBT images were first rescaled to the range of 0-255. For each DBT image, we manually selected two regions-of-interest (ROI) corresponding to “pure” adipose and fibroglandular tissue. Each ROI is a 2D rectangular region about 1cm x 1cm in size from a selected image slice. From these ROIs, we estimated *f*(*P*_{f}) and *f*(*P*_{a}) values by averaging the DBT image intensities inside the ROIs. We plot the estimated *f*(*P*_{f}) and *f*(*P*_{a}) pairs for all selected breasts in
.

From , we assume a linear relationship between *f*(*P*_{f}) and *f*(*P*_{a}), i.e.

and the coefficients

*a* = 1.226 and

*b* = 13.76 are determined by linear regression from the plot in . The

*R*^{2} of the linear fitting is 0.92. Thus, to compute the compositional map from

Eq. (3), we only need to select a single ROI for adipose (or fibroglandular) tissue and compute the other from

Eq. (10).

Using the above correlation, we apply the compositional segmentation for another 29 pairs of bilateral DOT/DBT measurements. In this case, we only manually select a single ROI corresponding to the “pure” adipose tissue, estimate

*f*(

*P*_{a}) from the ROI and calculate the value for

*f*(

*P*_{f}) from

Eq. (10). The compositional maps of the 58 breasts are then obtained by

Eq. (3).

To use the compositional maps in our FE-based image reconstructions, we first generated forward/parameter meshes using the “iso2mesh” mesh generator [

34]. In all reconstructions, we first estimated a set of bulk optical properties which served as the homogeneous initial guess for the full image reconstruction. We used both RF and CW measurements at 685 nm and 830 nm in all the reconstructions. A multi-spectral algorithm [

35,

21] was used to estimate HbO, HbR and scattering properties from multi-wavelength measurements simultaneously. All image reconstructions ran for 5 Gauss-Newton iterations. All calculations were performed on a single core of an Intel Xeon E5530 (2.4 GHz) CPU.

In
, we show sample reconstructed HbT images (in μM) from the right breast of a 45-year-old healthy volunteer. The node numbers for forward and parameter meshes are 14824 and 1934, respectively. We solved for the forward solutions at 13/6 RF source/detector and 26/19 CW source/detector locations at two wavelengths and built the Jacobian matrix using the adjoint method [

36]. The DBT image and the compositional map slice for fibroglandular tissue are shown in . From (g) to (j), we show the image slices reconstructed with the compositional-prior-guided algorithm with different

*λ* values, (

*λ* = 2.5 to 0.039 with a multiplicative step factor of 0.25). In (c), we show the reconstruction output from the “binary soft-prior” algorithm (the binary segmentation of fibroglandular tissue was achieved by thresholding at C

_{f} >0.5) and (d) the result without structural prior (i.e.

*L* =

*I*). All images were extracted at a horizontal plane

*z* = 2.6 cm from the bottom of the compressed breast. The prior-guided image reconstructions took about 102 seconds per iteration, while that for the prior-free reconstruction is around 73 seconds. The relative residuals (normalized by the residual from a common initial guess) for the prior-guided reconstruction range from 0.27 (

*λ* = 0.039) to 0.34 (

*λ* = 2.5); that for the prior-free reconstruction is 0.32.

Notice that the compositional map in contains an edge artifact due to the application of an edge-enhancing algorithm in the DBT image processing. We deliberately choose this case as we want to see how the algorithm performs when the compositional map contains errors.

Based on , we chose *λ* around 1 as the default regularization parameter. We show SO_{2} and *μ*’_{s} (at 830nm) for the same breast using compositional priors (*λ* = 1); in comparison, those without priors are shown in and , respectively. Then we ran reconstructions for 58 healthy breasts. Similar to the previous cases, we ran 5 Gauss-Newton iterations to recover 3D volumetric images for HbO, HbR, scattering amplitude and power, respectively. Images from 4 additional healthy subjects are shown in
.

From the recovered images, we estimate the optical properties by

Eq. (4) for adipose and fibroglandular tissue. In
, we plot the estimated adipose HbT, SO

_{2} and reduced scattering coefficient (

*μ*_{s}’) at 830nm for left vs. right breasts (circles). We also overlapped the results estimated by the prior-free algorithm using manually created ROIs [

22] (dots). We also used these same manually generated ROIs to extract the optical properties for adipose and fibroglandular tissue from the prior-guided images. These were plotted as “pluses” in . In
, we summarize the correlation coefficients between the optical properties of the adipose tissue between left and right breasts, as well as the parameter values recovered from the 3 approaches. Those for the fibroglandular tissue are similar (not shown).

| **Table 1**Correlation coefficients and the corresponding P-values between the bilateral adipose tissue properties extracted from 29 pairs of bilateral measurements are reported in columns entitled “R (P)”. We also summarize the means and standard (more ...) |