Diffuse optical tomography (DOT) is another emerging technique that can provide functional assessment about tissue status [9,10]. By measuring the scattered light at multiple wavelengths within the near-infrared frequency range, one can recover hemoglobin (oxygenated, HbO, or deoxygenated, HbR), water and lipid concentrations as well as the tissue scattering properties. These measured quantities were found to be clinically relevant and are particularly promising as biomarkers for breast cancer diagnosis [11–15]. However, similar to PET, standalone DOT images suffer from low resolution due to the diffusive nature of the photons inside the tissue and the subsequent ill-posedness in solving the inverse model. Combining DOT with a high-resolution structural-oriented imaging modality can be highly beneficial [2]. Various dual-modality breast imaging systems involving DOT have been developed in the past decade and yielded different levels of success. Zhu et al. [16] built a combined DOT/Ultrasound hand-held probe for breast tumor diagnosis. Ntziachristos et al. [17], Brooksby et al. [18] and Carpenter et al. [19] have reported efforts combining DOT with MRI by either simultaneous measurement or post-registration. At Massachusetts General Hospital, we have developed a combined DOT and digital breast tomosynthesis (DBT) system for breast cancer screening [20,21]. In the past 4 years, we have been focusing on the clinical evaluation of the system by measuring healthy or tumor-bearing breasts from nearly 200 volunteers; we have shown promising diagnostic efficiency for malignant tumors using the reconstructed optical parameters from this pilot clinical study [22].

In most multi-modality DOT studies as mentioned above, the structural imaging modalities were only used to provide the exterior boundary of the breast, or overlaid on the reconstructed DOT images and used for image interpretation. This appears to be an under-use of the available information as the internal tissue structures are not fully exploited. In fact, the under-used internal structure is the key ingredient that is missing to produce DOT images of higher spatial resolution. To incorporate this ingredient into the optical image construction, novel algorithms were proposed and tested by combining a spatial prior to “inform” the optical image recovery. The simplest approach to bind a spatial structure to the DOT reconstruction is to segment the structural image and represent nodes within the same tissue segment by a single set of optical parameters, i.e. the so-called “hard-prior” approach [18,23]. Several limitations were found for this approach. The implied piecewise-constant optical properties within each tissue type is likely inaccurate in realistic biological tissue. The highly over-determined reconstruction equation effectively cancels the information from most optical measurements, which are usually precious to get. The resulting images are also intolerable to segmentation error, as it is essentially identical to the segmentation itself. To balance the contributions from the structural prior and optical measurements, “softer” constraints, in the form of regularizations, have been investigated. Li et al. has applied different regularization coefficients for different tissue types based on a segmented structural image [24]. Brooksby et al. [23] and Yalavarthy et al. [25] used a regularization matrix, defined by a Laplacian operator for the nodes sharing the same tissue type. Although these algorithms allow certain variations from the structural prior, the underlying piecewise-constant assumption remains to be arguable in realistic tissues. Only recently, researchers begin to exploit more general prior-guided DOT reconstruction by treating a target domain as statistical mixtures of different tissue types [26,27].

Here, we present a structural-prior guided DOT reconstruction algorithm using a compositional segmentation. Different from the binary approach, in a compositional segmentation we assume each pixel/node in the structural image is a combination of two or more tissue components. Using a weighted-graph representation in the compositional space, we are able to construct a regularization matrix, being the modified Laplacian operator of the weighted-graph, and use it in the DOT image reconstruction. We investigate the control of the “softness” of the structural constraints by changing the weight of the regularization. To test this algorithm, we use a clinical data set collected in the past few years with both DOT and DBT measurements. With these clinical data, we are able to identify the potential improvement in image resolution and quality using the proposed algorithm. We also calculate the estimates for the optical properties for each component of the breast tissue, i.e. adipose and fibroglandular tissue, using the compositional priors and compare them to our previous results using prior-free reconstruction and 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 Fig. 2 .

From Fig. 2, 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 Fig. 2. The R² 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 Fig. 3 , 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 Figs. 3 (a) and (b). 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 Fig. 3(b) 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 Fig. 3, we chose λ around 1 as the default regularization parameter. We show SO₂ and μ’_s (at 830nm) for the same breast using compositional priors (λ = 1); in comparison, those without priors are shown in Figs. 3(e) and 3(f), 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 Fig. 4 .

As we discussed in [22], the bilateral correlations serve as a metric to test the robustness of the system. From Fig. 5 and Table 1, we found that the optical properties estimated by the compositional-map-guided reconstructions generally match those estimated from prior-free reconstruction with expert segmentations. Particularly, the least-square estimates using the compositional map (circles) show a great similarity to those using the same image but with ROIs (pluses). These results render the prior-guided approach a very attractive choice as it is almost entirely automated, while the conventional method requires a manual segmentation to create the adipose and fibroglandular ROIs. Moreover, the compositional-prior-guided approach offers more objectivity than expert segmentation as the latter approach depends on the experiences of the operator. We also noticed that the standard deviations for HbT and SO2 values became larger when using priors in the reconstructions. This may be potentially a result of different regularization mechanisms in these methods and deserves further investigation. The optical properties estimated using priors are generally consistent with literature findings [11,18,21–23].