2.1

CBBCT system

The projections were acquired by a CBBCT system that is a Pre-FDA approval prototype (KBCT1000, Koning Corp., West Henrietta, NY). The operated x-ray tube was RAD71-SP (Varex Imaging, Salt Lake City, UT), and the x-ray was operated at 49 kV with a pulse-width of 8 milliseconds. 300 projections with uniform 1.2 deg/view angular sampling in a full scan (360 deg) were performed. A CsI:Tl scintillator coupled, amorphous silicon-based flat-panel detector (PaxScan 4030CB, Varex Imaging, Salt Lake City, UT) used in the CBBCT system. The operating pixel size of this detector is 0.388 mm, and the dimension of the detector is 1024×768. The distance between the source and the AOR is 650 mm, and the source-to-detector distance is 898 mm.

2.2

Angular DgN^CT computation

The breast images were all first reconstructed by our developed deep learning-based algorithm, multi-slice residual dense network (MS-RDN) [13], that reduces image noise. Then all MS-RDN reconstructed images were segmented into air, skin, adipose, and fibroglandular tissues (Fig. 1) using a previously reported method [15]. The CNL, D_eff, and f_g can be estimated from segmented images for a semi-ellipsoidal homogeneous breast model.

Figure 1.

Segmentation of the image. The images were reconstructed by MS-RDN deep learning algorithm. Each image was then segmented into air, skin, adipose, and fibroglandular tissues.

The DgN^CT (mGy/mGy) of the patient-specific breast model can be calculated as [5], [6]

where the subscript, hete represents heterogeneous tissue distribution, E_g,dep is the total energy deposited in all fibroglandular tissue voxels, n_g indicates the total number of fibroglandular tissue voxels, m_g is the mass of a fibroglandular tissue voxel, and AK(E) is the air kerma at the breast center with the energy, E, of the incident photons. The same computational method was used for calculating the angular DgN^CT. In MC simulations, all photons were radiated to the breast from the assigned x-ray source position (angle). The MC simulations were performed using the MC code (GATE 8.0) validated in our previous study [10]. The number of photons was 10⁶ as suggested by literature [7], [16], and resulted in a variation of less than 0.7%.

The angular DgN^CT of the homogeneous breast of a semi-elliptical shape is the same as its DgN^CT at any angle because of the rotational symmetry. The angular DgN^CT of this model can be expressed as a fitting function (standard deviation is 1.13%) [11]

2.3

Simplified Math Model

The angular DgN^CT in Eq. (1) is related to the energy deposited on the fibroglandular tissues, which is proportional to the pathlengths (PL) of photons through the fibroglandular tissues. To easily demonstrate this concept, a 2D circle of a finite size presents the fibroglandular tissues here. The center of the circle is the center of the geometry of the fibroglandular tissues (COG_f). The PL can be analytically solved as

where L, R, r, and β, α, and θ are the distance between the x-ray source and AOR, the radius of the orbit of the 2D circle, the radius of the circle, the polar angle of the circle in the orbit, the angle between the central line of the circle and the central line of the fan-beam, the angle deviated from the central line of the circular object, respectively. The figure is depicted in Fig. 2. Similar to the concept of the Radon transform, the curve of the PL is a sine wave varying with projection angle.

Figure 2.

Simplified mathematical model. A 2D circular object in a fan-beam geometry.

3.1

Total Simulation Time

All MC simulations were performed on a Dell workstation 7810 with Intel Xeon CPU (3.20 GHz) and 32 GB RAM. For each angle, the MC simulation took approximately 40 minutes, resulting in 40×10×75=30000 minutes for all 75 samples.

3.2

Numerical results of the simplified math model

Numerical results of two particular examples of our simplified math model are when the COG_f is located at the center and above the AOR (i.e. β=0 in Eq. (3) and Fig. 2). Let the radius of the COG_f be 15 mm with (1) 0 mm (reference) and (2) 50 mm distance away from the AOR. Without losing any generality, θ = 1° (or −1°) was considered here. The results (Fig. 3) show that the PL is a sine wave as predicted and in this particular example, the minimum of the curve occurs when the x-ray source is at ϕ=180 deg (feet position). The average PL of the sine wave is 0.1296 mm less than the reference. The sine wave has the same PL as that of the reference at ϕ=92.2042 deg and 267.7958 deg, which can be analytically solved by Eq. (3).

Figure 3.

The pathlength of the object off positioned above from the AOR. ϕ is the angle of the x-ray source.

3.3

Center of the geometry of fibroglandular tissues

In MC simulations, the center of the geometry of the entire breast COG_b is at the AOR. The deviation of COG_f from COG_b for each sample was shown in Fig. 4. The breast laterality is factored with 90 deg representing medial and 270 deg representing lateral aspects of the breast. It was found that 62.67% of samples have COG_f between 36 deg and 324 deg. From the previous section, the numerical results show that the minimum of the PL curve would happen at 180 deg if the COG_f is exactly at 0 deg. Thus, the angular DgN^CT in this particular dataset should be a sine wave with a minimum at approximately within the range of 144 to 216 deg.

Figure 4.

Center of the geometry of the fibroglandular tissues (COG_f) deviated from the center of the geometry of the entire breast (COG_b). ϕ is the angle of the x-ray source.

3.4

AngularDgN^CT

For each breast volume, the angular DgN_CT was normalized by the DgN_CT from the homogeneous semi-ellipsoidal model (reference) to characterize its deviation. The average of this normalized angular DgN^CT from the 75 breast CT volumes is shown in Fig. 5. Consistent with the prediction from section III. B and III. C above, the curve of the angular DgN^CT is approximately a sine wave with a minimum between 144 deg and 216 deg. The curve can be fitted to sine wave of the form 0.0376 sin(Ø + 83°), with a root mean square error of 0.0106.

Figure 5.

The angular DgN^CTof the patient-specific model is normalized by the reference (homogeneous breast of semi-elliptical shape).