2.1.1 Picker Nuclear Thyroid Phantom

Embedded into a sheet of Perspex material with dimensions 100 mm× 100 mm× 18.4 mm, the Picker phantom contains four nodules (three called "cold" and one called "hot") forming two lobes, as shown in Figure 1. It is filled with a solution of water and a ^99mTc radioisotope such that the right-to-left uptake ratio is 2:1. The three cold nodules present no radioactivity, whereas the hot one has the same activity as the other lobe background, meaning it has double the lobe activity of a cold one. In the same figure, cold and hot nodules are colored with black and white, respectively. Moreover, the right and left lobes are colored with black and gray for clarity purposes [21, 22]. To incorporate such irregular shape into the Monte Carlo simulation, we converted the phantom into voxels of dimension 0.25× 0.25× 18.4 mm.

Fig. 1.Full-screen View

Voxelized simulated phantoms. (a) The Picker thyroid phantom consists of a left lobe (9 mm diameter cold rod and 12 mm diameter hot rod) and right lobe (6 mm and 12 mm diameter cold rods) with a left-to-right uptake ratio of 2:1. (b) Three breast phantoms, where SL, MM, and LH refer to small size and low glandularity, medium size and medium glandularity, and large size and high glandularity, respectively.

2.1.2 Breast phantom

Based on the 3D mathematical models of the prone breast provided by Bliznakova et al. [23], we were able to generate nine of them. Each model, including skin, duct system, adipose, and glandular tissues, was adjustable in terms of size (small (S), medium (M), or large (L)) and glandularity percentage (low (L), medium (M), or high (H)). As shown in Figure 1(b), the global dimensions for S, M, and L are 40 mm× 80 mm× 80 mm, 55 mm× 110 mm× 130 mm and 80×140× 180 mm, respectively, and the glandularities for L, M, and H are 25%, 50% and 75%, respectively. Hereafter, we use the above abbreviations to refer to each model; for example, the small breast model with low glandularity is referred to as SL. Each model was voxelized with a 1 mm× 1 mm× 1 mm voxel size. The densities and elemental compositions of the materials used are listed in Table 1.

2.1.3 Mura-mask

In this work, we used a Mura-mask with a 41× 41 array pattern. It has a fixed hole distribution and adjustable thickness (from 0.25 to 2.5 mm), holder material (W or Pb), and hole shape (box or cylinder). Each box shaped hole has a facet size of 2 mm × 2 mm and each cylinder shaped one has a diameter of 2 mm. Similarly to the desired mask, as described in the literature, it contains 50% open/close area, i.e., the same surface area of gamma-ray-opaque and gamma-ray-transparent zones (Fig. 2(b)). The overall shape has dimensions of 82 mm× 82 mm, embedded into a 600 mm × 600 mm opaque holder with adjustable thickness. A more detailed description was given by Alnafea et al. [17].

Fig. 2.Full-screen View

(a) Visualization of the simulated setup using Geant4 and HepRAPP software [24], including breast phantom, Mura-mask and its holder, detector crystal, and the backscatter compartment. (b) schematic view of the Mura-mask with a 41×41 array pattern (white and black pixels represent opaque and transparent holes, respectively).

2.1.4 Lehr collimator

During this study, the Lehr collimator was simulated for comparison purposes. It consisted of a regular distribution of hexagonal holes along a lead sheet with dimensions 328 mm × 328 mm × 35 mm. Each hexagonal hole had an inner diameter of 1.5 mm and a septa thickness of 0.2 mm, forming a collimator structure with a matrix array of 222× 192 holes. Cylindrical hole radii were adjusted to have the same area as the hexagonal ones. More detailed descriptions were provided by Alfuraih et al. [9] and Vieira et al.[25].

2.1.5 Imaging Modality

The detection system, including the Mura-mask/Lehr collimator, crystal detector, and backscatter compartment, formed the main part of the simulation procedure in this work, as shown in Fig. 2(a). The adjustable material of the crystal detector, NaI(Tl) or CdZnTe, has a surface of 328× 328 mm and a thickness of 9.5 and 5 mm, respectively, as provided elsewhere [17]. The detector was pixelated in with a 1 mm× 1 mm pixel size. NaI(Tl) material has a density of 3.7 g/cm³ and three elemental compounds of Na (15.2%), I (83.8%), and Tl (1%) according to Garcia et al. [26]. Cadmium zinc telluride was modeled as three elemental compounds of Cd (36.81%), Zn (21.41%), and Te (41.78%) with a density of 5.78 g/cm³ [27]. The backscatter compartment was made of Pyrex material and had dimensions of 328 mm × 328 mm × 68 mm.

2.1.6 Geant4-based program

As shown in Fig. 2(a), the major axis of the breast phantom was set parallel to the XY plane, and the object-to-mask distance was measured between their centers. The Lehr collimator was set in front of the crystal detector. Three spherical tumors characterized by the same elemental compositions and densities of the breast tissues with diameters of 8, 5, and 2 mm were included at three different depths. The positions were chosen to simulate a deeper scenario, where tumors are located on the X axis (Z=0), a superficial scenario, where they are located close to the phantom surface, and a third scenario that lies between the two. Note that for all scenarios, Z=0, and the X and Y coordinates change. We placed the tumor with a diameter of 8 mm at the base of the breast phantom, whereas the 2 mm diameter tumor was placed close to the top of the breast phantom (nipple), and the 5 mm diameter tumor was placed between both locations. During all simulations, we activated the Bremsstrahlung, ionization, and multiple scattering models for electrons with an energy threshold of 990 eV, allowing almost full tracking of particle history. For the photon particles, we activated the Compton effect, the Photoelectric effect, and Rayleigh scattering with a cutoff energy of 990 eV. The standard electromagnetic physics options governed by the specific physics builder "G4EmStandardPhysics" were used to fulfill our purposes. Thanks to the multithreading option provided by Geant4 [28, 29], all the simulations had a reasonable run time, despite the complexity of the experimental setup. Moreover, we utilized another tool for the simulation of the large number of voxels (breast phantoms, Mura-mask, and Lehr collimator) called nested parameterization owned by the class "G4VNestedParameterization", which allows the memory allocated by hardware to be minimized during the execution phase of the simulation. For all the cases, we fixed the mask-to-detector distance to be three times the source-to-mask distance in order to have a magnification factor of 3.0 for better image resolution of this near-field and medium phantom size application. Energy deposition of particles in the detector crystal was used to form coded images. We carried out our simulations on laptops (for testing) and two Dell Precision T7610 workstations, each with a 40-core Intel Xeon E5-2680v2 CPU at 2.80 GHz and 256 GB RAM. The operating system used for the management of all workflow was Ubuntu 14.04. The number of generated primary particles exceeded 2× 10⁹, as we focused the generation towards the Mura-mask or the Lehr collimator only. In our case, the number of detected counts (primary particles) through the Mura-mask was half of the generated primaries, as the open area (holes) formed 50% of the field of view. This gave us a statistical uncertainty of no more than 1% for all studied cases, with run time ranging from 20 min to more than 7 h.

2.2.1 Image deconvolution

Basically, the coded aperture imaging procedure involved the construction of the decoding matrix (Gij) corresponding to the mask-array pattern (Aij), in order to reconstruct the image M from a projected one, P, in the following way [30, 31]:

2.2.2 Statistical parameters

For a meaningful comparison of two given images with X and Y signals and xi and yi pixel values, we used an ImageJ plugin, published by the Biomedical Imaging Group [32] to calculate the following statistical parameters [33]:

where SNR, PSNR, MAE, and RMSE represent signal-to-noise ratio, peak of SNR, mean absolute error, and root mean square error. N refers to the total number of pixels per image; here, it is equal to 328× 328. We also introduced a parameter called quality factor (QF) to select the configuration providing the best PSF. For a reconstructed image M, as the PSF is proportional to contrast (C), SNR, maximum pixel value (Max), and the inverse of the full width at half maximum (FWHM), we calculated the parameter as follows:

For a given region of interest, the contrast was calculated according to the same area (of the same image) considered as background.