1 Introduction

To be able to photograph an object, there should be a one-to-one correspondence between an object and its image. The lens performs this function in an ordinary photographic camera. However, a lens cannot create such a correspondence for high-energy radiation, since lenses ordinarily are not able to bend high-energy radiation [1]. Therefore, nuclear imaging (SPECT and gamma camera) is conducted with the help of a radioactive uses aperture (pinhole) and parallel hole collimator (PHC) to direct output radiation from the body. In this way, a one-to-one correspondence is created between radiopharmaceutical distribution in the body and its image [2].

Many studies [3–6] have been conducted in the effort to find appropriate methods for design and construction of collimators of gamma-ray imaging systems. Three such influential studies are the Jaszczak et al. [4] study assessing cone beam collimators; the Tsui et al. [5] research on the assessment of fan beam collimators; and that of Mueller et al. [6], which selected high-resolution low-energy collimators in brain imaging.

One type of collimator is the pinhole collimator (PC), which is comprised of a small pinhole aperture embedded in a piece of lead or tungsten. In PCs, where there is a small object next to the collimator aperture, there usually is good spatial resolution and detection efficiency, but the field of view (FOV) becomes smaller [7]. Such collimators are used in humans to image some body parts such as the wrist and thyroid [7, 8] and are only rarely used for heart and breast [9, 10]. The greater the distance between the object and the PC’s aperture is, the greater the FOV is, and the less the spatial resolution and sensitivity is. On the other hand, the sensitivity will be increased by using a pinhole aperture of large diameter that leads to a decrease in resolution. The more the apertures focus on different points in the FOV, the larger the FOV. Additionally, the higher the sensitivity, the better the spatial resolution [11]. Many studies select the proper parameters to improve the above-mentioned quantities [12, 13]. However, using single aperture eliminates most photons containing information. The sensitivity, FOV, and spatial resolution will rise if apertures are increased [14–16].

Therefore, one of the methods employed to decrease the number of removed photons containing information is to use a thick lead plate with holes (PHC), in which all beams parallel to its holes are allowed to cross [2].

Designing the proper PHC depends on the application, sensitivity, and necessary resolution. Hence, it is constructed in different forms and designs, and the real performance of the best design will usually be assessed empirically [17, 18]. The major problem with PHCs remains the elimination of useful information-containing beams, i.e., of every 10,000 photons, just one is allowed to cross the collimator [19]. This leads to an increase in imaging time and patient-injected radiopharmaceutical dose, and a decrease in image quality.

Since the number of emitted photons is generally very low, collimators have the most important restrictions in gamma camera and SPECT. Consequently, a great step forward will be taken if a radiopharmaceutical spatial distribution in the body is obtained by using a gamma camera without a collimator.

The aim of this study was to present the idea of a gamma camera without collimator (GCWC) based on the specific arrangement of detectors in zigzag form. Ideally, this theory will be assessed through Monte Carlo simulation by MCNP code.

2 Materials and methods

2.1 Theory

In GCWC, instead of using a series of detectors on a flat plate together with a large crystal, gamma detectors are distributed in zigzag form to create a row of detectors (ROD) (Fig. 1). Detector rows are separated with lead plates, called septa.

Fig. 1
figure 1

GCWC in three dimensions: A, distance between two lead septa, detector width, and thickness of image slice; B, row of detectors (ROD); C, septa between detector rows

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In other words, instead of bending the beams (like a lens) or directing the beams (by collimators), some changes are created in the detectors’ FOV in such a way that the FOV of each detector will be different from those of the other detectors (Fig. 2). In this way, the observation of radiopharmaceutical distribution in space will be unique for different detectors. The image can be produced by using the difference in observed radiopharmaceutical distribution and image reconstruction through the filtered back projection (FBP) method, without the necessity of a collimator.

Fig. 2
figure 2

One detector row: Shaded bar is the difference between two FOVs of detectors, so that radioactive source S is observed only by detector D 1 ; the face of ROD: the slice that ROD and some radioactive material laid on it; θ: angle between the incident radiation and detector level

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GCWC is designed in such a manner that distribution of detectors is shaped in a zigzag manner. As a result, the FOV of each detector on the right side of each triangle in Fig. 2 is different from the adjacent detector by merely a strip (similarly, this is true for the detectors on the left side of the triangle). In other words, detectors D 1 and D 2 observe radioactive source S′, but only detector D 1 (out of the detectors on the right side) detects source S.

Therefore, if a radioactive material is laid in front of this detector distribution (each ROD), N D1–N D2 equals the absorbed radiation resulting from radioactive materials in the shaded bar (N D1 and N D2 are, respectively, the number of counts read by detectors D 1 and D 2). In other words, a narrow strip containing radioactive material will be separated without the necessity of eliminating beams through using hardware similar to a PHC.

The imaging is staged in two perpendicular projections; point S is shaped from the intersection of two bands through (D 0i  − D 0i+1) and (D 90j  − D 90j+1) (Fig. 3). Therefore, point S can be determined through image reconstruction by FBP. D φα indicates the αth detector in the projection angle of φ°.

Fig. 3
figure 3

Point S is shaped from the intersection of two bands through two perpendicular projections, S \( \in \) (D 0i − D 0i + 1) ∩ (D 90j − D 90j + 1), where i and j are number of detectors

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Apparently the proposed idea can be implemented theoretically, but its accuracy and verification must be proven through simulation. Therefore, the following sections will investigate the feasibility of this idea, ideally through simulation based on Monte Carlo with MCNP code.

2.2 Computer simulation

Monte Carlo method is based on random sampling to find solutions for physics or math problems according to certain probabilities and laws of statistics. In addition, it is highly accurate in predicting the outcomes of processes that are repeated many times. Monte Carlo simulation (MCS) using MCNP code is a common method in gamma camera imaging and designing proper collimator experimentation [18, 20].

Ninety-eight detectors were arranged in a zigzag manner on a line 1 m in length, which formed a ROD for MCS. Dimensions of each detector’s surface were 2 cm × 1.41 cm. Therefore, the thickness of the imaging slice that was taken by ROD was 2 cm. Technetium-99m (which has an energy of 140 keV) radiopharmaceuticals are in widespread use for imaging and functional studies (over 80 % of all diagnostic procedures) of major organs such as the heart, brain, liver, kidney, and thyroid [21]. Therefore, a radioactive source with energy of 140 keV was situated on the face of the ROD, i.e., the slice on which the ROD and some radioactive materials have been placed (Figs. 2 and 3). Among the active detectors, 49 detectors were laid on the right side of the triangle in ROD. Sixty projections at a distance of 3° (0°–179°) from each other were simulated by MCNP-4C code. In other words, simulation was performed 60 separate times, with 60 different projections at a distance of 3°, and the photons passing through each detector were counted. It was verified that in each simulation period, 100,000,000 photons were released from the radioactive source and were traced by MCS.

3 Results

After simulation and data collection, each detector’s counts were estimated in different projections. For example, Fig. 4 shows the first projection (angle of 0°) in which the vertical axis is the gamma-ray counts and the horizontal axis is the number of detectors on the right side of the zigzag arrangement.

Fig. 4
figure 4

Horizontal axis is each number of detectors. Vertical axis is the count of radiations passing through each detector

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In Fig. 2, the number of counts necessary for the relevant strip will greatly decrease with the angle between the incident radiation and detector surface. Therefore, Eq. (1) observes the number of counts resulting from the difference counts of two consecutive detectors. D φα indicates counts of the αth detector in the projection angle of φ°. Two consecutive detectors determine the bandwidth value of each band (d φα ), calculated by Eq. (1); in addition, ρ and μ are arbitrary values for reinforcing these differences.

$$ d_{\varphi \alpha } = \rho^{*} \left( {\frac{{D_{\varphi \alpha + 1} - D_{\varphi \alpha } }}{{D_{\varphi \alpha + 1} + D_{\varphi \alpha } }}} \right)^{\mu } $$
(1)

Figure 5 is the result of applying Eq. (1) on the collected counts obtained from different detectors in the first projection (at 0°). The band containing the radioactive source is clearly visible.

Fig. 5
figure 5

Subtracting two consecutive detectors based on Eq. (1) in first projection (angle of 0°). The band containing radioactive source is clearly visible

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The counts obtained after simulation for each detector in different projections were reconstructed after being transferred to MATLAB software by the command “iradon,” based on the FBP technique; the final image is illustrated in Fig. 6. An imaginary point source was also placed within a uniform matrix through using MATLAB software, and then 60 projections were provided by the command “radon.” Then, the mentioned point was reconstructed by the command “iradon” (Fig. 6a). Figure 6 shows that the point image emerged as plus (+) after reconstruction from its projections due to the limitation of the command “iradon” in MATLAB software. This software uses the back projection method for image reconstruction based on projections; the main problem with this method is the star artifact that is shown in Fig. 6a. All photons received by the detectors have energy of 140 keV due to the lack of scattering in the environment. Thus, spectrum evaluation of detected photons’ energy was not possible and deemed meaningless.

Fig. 6
figure 6

a The reconstructed image made from 60 projections produced from an ideal point using MATLAB software’s “radon” command. b Image reconstruction by 60 simulated projections using MCNP-4C code with normalization values μ = 2 and ρ = 1. c Image reconstruction of 60 simulated projections using code MCNP-4C with normalization values μ = 1 and ρ = 1

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Sensitivity is defined as the number of counts passing through the detectors per second per megabecquerel of source (cps/MBq). Geometric sensitivity of ROD in each projection was different with respect to other projections, because radioactive source position varied in the face of ROD in each projection. Mean, maximum, and minimum sensitivity in 60 projections are presented in Table 1. Sensitivity was calculated before and after subtracting two consecutive detectors by Eq. (1).

Table 1 Row of detectors of sensitivity of gamma camera without collimator (GCWC), using 108 photons that were released from the radioactive source and were traced by Monte Carlo simulation (MCS), totaling 60 projections presented before and after subtracting two consecutive detectors
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4 Discussion

Sixty simulations were conducted by rotation of detectors around the radioactive point in 60 projections using MCNP-4C code; then data were transferred, differences were counted with Eq. (1) by MATLAB software, and the space was threaded in each projection. Figure 6b shows the outcome of this action. The star pattern around the point source results from the image reconstruction process [22], as a sharp curve appeared after subtracting (Fig. 5), and the band containing radioactive source was clearly visible. Proving this issue, a zero matrix was built in order to determine the contribution of blurring the image in running the command “iradon” in MATLAB software for the image reconstruction process. Digit 1 was placed as one of its elements, similar to radioactive source, in front of the detectors. After that, ideal projections were made by the command “radon” and the point’s image was reconstructed by the command “iradon.” Figure 6a showsthe outcome of this action. The comparison made between this image and the reconstructed image through 60 projections in MCNP-4C code (Figs. 6b, c) indicates that lack of clarity in the image is related to running the command “iradon.” As a result of the comparison made between point T, resulting from the simulation with point + due to the command “radon,” the hypothesis of the elimination of collimators through detectors’ zigzag distribution ideally is verified, even though the radioactivity in the reconstructed images can be achieved without shadow by edge filters (e.g., Roberts filter). Regarding the study’s aim (investigating the possibility of omitting collimator and observing a radioactive source by a special distribution of detectors), the simulation of various sources at different distances from the camera or the distribution of radioactive material has not been dealt with and will be a future research topic. MCS and practical construction of slit–slat collimators have been investigated for resolution and sensitivity along the axis and perpendicular to the axis as a function of aperture [23, 24]. Saad et al. [25] also used MCS to find the best mesh in PHC for breast imaging. Islamian et al. [26] have applied MCS to investigate the suitable thickness of collimators.

Since the real structure of the detector was not considered in the simulations, the crossing of each photon from the small plate—considered a detector—was recorded, and the beam was assumed to have been completely absorbed. Therefore, the scattering due to photons’ interaction with the detector components was not taken into account; instead, it was assumed that each beam crossing from the detector surface was recorded by the detector, put aside, and could not penetrate adjacent detectors. In fact, penetration of beam to adjacent detectors occurs and should be considered. Penetration and scattered photons in the collimator have a major impact on nuclear medicine imaging [27–29].

The goal behind using collimators is to thread the space in each projection and to determine the bands including radioactive points, and then to obtain the distribution of radioactive material through intersection of stripped spaces by different projections.

The main problem in using collimators is that most of the radiation containing information is eliminated for threading. However, in the above-mentioned method, the detectors receive the majority of information-containing radiations. As a result, the sensitivity of GCWC (Table 1) is higher than of PHC and PCs [30–32]. Therefore, according to Eq. (1), through the difference between the two detectors’ data, a string can be created that can be seen by one detector and not by the other detectors (Fig. 2).

Apparently there is no difference between the presence of collimators or the above-mentioned method, and sensitivity after subtracting (Table 1) is approximately similar to conventional gamma camera [16, 17]. For example, sensitivity was obtained with 10 cameras used for Tc-99m thyroid scintigraphy by Seret et al. [31]. Measured minimum–maximum values of sensitivity were 62–131 cps/MBq for six pinhole cameras and 55–85 cps/MBq for four non-pinhole cameras. Additionally, Fleming et al. [32] reported sensitivity of two kinds of SPECT, using 99mTc at 120.2 and 153 cps/MBq. Nevertheless, it is worth mentioning that the strings made up of two adjacent detectors produce the maximum spatial resolution. The image quality can be improved by making strings at the distance of two detectors or more; besides, it is possible to correct the final image through combining several images with various strings’ widths.

When two adjacent detectors observe the radioactive material clearly, the counts of these two detectors will rise. Therefore, their difference may be much greater than the difference between two border detectors (one discerns radioactive source but the other does not) with just a few counts. Each difference will be divided by the sum of counts of two detectors in order to reinforce the difference between the two small values relative to that of the two large values. The point will be observed vividly in this stage, although it is possible to affect border reinforcement or final image clarification by changing the coefficient or power in Eq. (1). For example, in Fig. 6, the effect of selecting (ρ = 1, μ = 2) is noted compared with (ρ = 1, μ = 1).

Clearly, if several RODs are separated by septa, detectors will be distributed in two dimensions (Fig. 1). Therefore, a volumetric image will be formed instead of a sectional one when the camera rotates around the patient. If two parallel heads are used instead of a one-head system, increasing image-making potential, the applied corrections, and imaging speed during recording will be observed. As a result, some corrections can be imagined for GCWC, which could inspire future research. The topics could well include optimizing the number and size of detectors, suitable angle of detectors, taking account of the corrections due to the influence of the radiations from adjacent detectors, and absorption of scattered radiations.

5 Conclusion

Simulation of gamma camera with arrangement of zigzag detectors using MCNP-4C code in 60 projections has revealed that the space can be threaded and the radioactive source can be determined through image reconstruction without using collimator, and can instead be accomplished through a special distribution of the detectors. Simulation was conducted ideally to prove the main idea; issues, such as scattering of beam due to striking detector components, the beam passing through a detector and recording in another detector, are related to photons’ collision with the material in a real location, which can affect image quality and have not been considered in the simulation. These issues should be taken into account for simulation of real GCWC in the future.