Simulation of a gamma-ray imaging technique using detector response patterns

We introduce a novel gamma-ray imaging technique that uses detector response patterns. This method employs multiple shielding cubes randomly positioned in a three-dimensional configuration. Within the volume defined by these cubes, a unique gamma-ray flux pattern is formed based on the incidence direction of the gamma rays. This pattern can be measured using the responses of several scintillator cubes. By pre-measuring the detector response pattern and incidence direction of the gamma rays, the incidence direction can be estimated using an unfolding technique. Simulations were performed using a 137Cs point source. Our results show that a 10 MBq 137Cs source, located 3 m away from the imager, can be imaged with an angular resolution close to 10°. These findings suggest that our new method is comparable to existing gamma-ray imaging techniques. Potential applications of this imaging method include nuclear power plant decommissioning, nuclear medicine, security, and astronomy.


Introduction
][10][11] Common gamma-ray imagers, such as pinholes, [12][13][14][15] coded-masks, [16][17][18][19] and Compton cameras [20][21][22][23][24][25][26] serve diverse needs; however, they have their own set of challenges. Fr example, pinhole cameras, which are effective in high-dose-rate environments and are easier to use for measuring radioactivity, 27) require a heavy shield around the detector, often weighing a few tens of kilograms.Their efficiency is also limited because gamma rays are detected only through a tiny pinhole. Whilecoded-mask cameras are portable, and do not require heavy shielding, their image quality decreases for gamma rays in directions away from their field of view (FOV), and they offer a narrower FOV than the other methods.Compton cameras, although compact and offering a wide FOV, struggle in highdose-rate environments as they require time coincidence between the scatterer and absorber detectors.They are suitable for detecting photons that undergo Compton scattering as the main interaction.
Given these challenges, several scholars have investigated new gamma-ray imaging methods.][32] In the medical field, there is a trend toward replacing metal collimators with self-shielding collimators. 33)][36][37][38][39][40][41][42][43][44] We introduce a novel gamma-ray imaging approach.The proposed method uses multiple shielding cubes randomly spread in three-dimensional (3D) space to create unique gammaray flux patterns.By utilizing these shielding cubes, the proposed technique can more efficiently modulate gamma-ray flux compared to active coded-mask cameras.The ability to modulate flux effectively with shielding cubes is especially advantageous for gamma rays in the high-energy range from several hundred keV to several MeV.This is because the imager can be miniaturized beyond what is possible with active codedmask cameras that only uses self-shielding.Additionally, this approach allows for a significant reduction in the number of readout channels.The reduction in the number of readout channels not only cuts operational costs, but also simplifies the entire signal processing system.A simpler signal processing system does not need complex, application-specific systems like application specific integrated circuits (ASICs), enabling the creation of more flexible and versatile imager systems.
We aim to develop a gamma-ray imager that can flexibly adapt to various applications, including the decommissioning of nuclear power plants or in restoration following nuclear accidents.We designed a new gamma-ray imager, called the coded cube camera for gamma rays (C3G) that achieves these objectives, and we tested our novel method through simulations.We aimed for an angular resolution of 20 degrees in C3G, which corresponds to the spatial resolution that distinguishes between two sources that are 1 m apart, from a distance of 3 m.

Principle
We provide a concise description of the principle of our method by comparing it with that of a coded-mask camera.Our method can be viewed as a 3D evolution of the codedmask camera based on their numerous similarities.Figure 1 shows a shadow projected onto the ground, with light paths visualized as streaks.Our method can be described analogously to the coded-mask camera by considering a gammaray source and gamma rays instead of the Sun and sunlight.
Determining the gamma-ray source direction uses a similar principle as determining the direction of the Sun, as shown in Fig. 1.For the coded-mask camera, the shadow pattern on the ground combined with the shape of clouds was used to estimate the direction of the gamma-ray sources.The codedmask camera consists of a metal plate with multiple holes, termed a coded-mask, and a position-sensitive pixel detector, which correspond to clouds and ground, respectively.A unique shadow pattern of the mask was projected onto the position-sensitive detector, depending on the direction of the gamma-ray source.The projected shadow pattern was measured as the detector response pattern.With prior knowledge of the detector response and gamma-ray irradiation direction, the direction of the gamma-ray source can be estimated from the measured detector response pattern.The advantages of this method include a high spatial resolution and portability.Furthermore, unlike Compton cameras, time coincidence is not required, making it suitable for relatively high-dose-rate environments.However, its limitations include a restricted FOV to the front of the detector because shadows are projected onto the 2D-pixel detector.The image quality decreases when gamma rays come from outside the FOV owing to the reduced shadow contrast.The coded-mask camera performed the best when the gamma ray source was located in front of the camera.
Our approach uses a 3D light streak pattern visualized in space to determine the direction of the Sun.The shield cubes that spread out in space are similar to clouds.These cubes are designated as 3D coded masks, representing the 3D expansion of the coded mask.Inside the space enclosed by the 3D coded mask, a unique 3D gamma-ray flux pattern appears based on the incident direction of the gamma-ray.This 3D flux pattern is measured as the response intensity pattern of multiple gamma-ray detectors positioned within the space enclosed by a 3D coded mask.The gamma-ray source direction can be determined by pre-measuring the response of the detector and irradiation direction of the gamma rays.Unlike coded-mask cameras composed of planar masks and planar detectors, the proposed method comprises a 3D mask and multiple detectors.The setup ensures that gamma rays entering from any direction will pass through the mask, thus extending the FOV to 4π steradians.Consequently, the position of the gamma-ray source is not a limiting factor.

Reconstruction algorithm
We employed the gradient descent method (GDM) to reconstruct the gamma-ray source direction based on the measured detector response pattern.Because GDM operates within an underdetermined system, it cannot yield strictly unique solutions.Consequently, we aim to identify the most plausible solution.
We defined matrix S as the measured detector response, matrix R as the response function pre-measured in advance, and matrix W as the gamma-ray source direction, ideally satisfying the following relationship: In this context, n denotes the number of gamma-ray detectors and m is the number of gamma-ray incidence points.As ( 1) is indeterminate because of its under-determined system and uncertainties in S and R, we aim to identify W that minimizes the squared error, serving as a viable solution Parameter σ is a multidimensional surface.Calculating the gradient at a given position on this surface produces a gradient vector ∇σ in the direction of the increasing slope of the surface.If the present position vector is x, then a position vector closer to the minimum value can be expressed as new By repeated calculations of the gradient vector and updates to the position vector, it becomes possible to approach values near the minimum of σ.In this equation, ò represents the learning rate that determines the extent of position adjustment in a single update.
We used the Monte Carlo simulation toolkit Geant 4 [45][46][47] to calculate matrix R. We considered a count range of 662 ± 28 keV for R. The ±28 keV value corresponds to the experimentally measured full width at half maximum (FWHM).The number of divisions m in the gamma-ray incidence direction was set to 614.This is because the gamma-ray irradiation points were determined by dividing both the polar angle θ and azimuthal angle f into intervals of 10°.From each irradiation point, 10 8 gamma rays of 662 keV were emitted parallelly.
We developed the GDM program in Python 3 using machine-learning frameworks such as TensorFlow 48) and Keras. 49)For our gradient descent approach, the Adam Optimizer 50) from Keras was used.This optimizer automatically adjusts the learning rate ò in (3) to an optimal value.

Design principles of the coded-cube camera for gamma-ray (C3G)
The proposed approach can be applied to gamma-ray imagers with various geometries.However, to achieve higher Fig. 1.General principles of the coded-mask camera (left) and our method (right).The coded-mask camera measures a two-dimensional shadow pattern on a pixel detector, while our method measures the spatial gamma-ray intensity distribution.
032005-2 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd performance, certain design considerations should be considered.Primarily, there should be a clear link between the direction in which the gamma rays arrive and the detector response pattern.Figure 2(a) illustrates the suboptimal gamma-ray imager design.For incidence directions 1 and 2, the direction of the gamma-ray source can be estimated because the detector response patterns are distinct.However, for Directions 3 and 4, the patterns are identical, making it impossible to identify the source direction.The design shown in Fig. 2(b) illustrates the shield and gamma-ray detector cubes placed randomly in the 3D configuration.This layout is more suitable for our method because it ensures unique patterns for different gamma-ray incidence directions.
The choice of material for both the shield and gamma-ray detector cubes is another crucial factor.For instance, if the goal is to measure 662 keV gamma rays emitted from 137 Cs, dense materials such as lead and tungsten can enhance the pattern contrast while requiring less shielding volume.A 22 mm thick layer of lead can reduce direct exposure to 662 keV gamma rays by 90%.For gamma rays or X-rays in the tens of keV range, lighter materials offer sufficient contrast.
We designed a new gamma-ray imager, C3G, to evaluate the feasibility of the proposed method.The target gamma ray energy was 662 keV, originating from the 137 Cs sources.In this feasibility study, the imager design should be simple to ensure that the results are easily understood.Furthermore, considering future prototype creation, the design should be easy to implement.Considering these factors, the C3G was designed as a 3 × 3 × 3 cube arrangement, as shown in Fig. 3.Each individual cube measures 10 mm × 10 mm × 10 mm, resulting in the overall dimensions of the C3G being 30 mm × 30 mm × 30 mm.The configuration included 18 shielding cubes, eight gamma-ray detector cubes, and one vacant cube arranged randomly.Consequently, the number of detectors n as shown in (1), is 8. Lead was selected as the shield cube material, and a gadolinium aluminum gallium garnet (GAGG) scintillator was employed for the gamma-ray detector.

Simulation condition
We evaluated the feasibility of our method through simulations using Geant 4. Imaging tests were conducted with three different gamma-ray source configurations, which included both single-and double-source setups, as described below.For the simulations, we assumed a 10 MBq 137 Cs source located 3 m from the C3G.Considering this distance, the incidence of gamma rays on C3G from the source was almost parallel, and we used a parallel beam for our test data.The energy resolution of the GAGG (Ce) scintillator, which was experimentally determined to be 8.5% at 662 keV, was accounted for during the simulation.To determine the necessary measurement time for stable results, simulations were performed for 10, 30, and 100 min for each configuration.Ten separate simulations were performed for each time setting to examine the consistency of the imaging outcomes.

Results and discussion
Figure 4 presents the temporal progression results for each source distribution.Under all conditions, imaging was centered around the true source position.It was evident that the imaging accuracy improved with increasing measurement time.A comparison of the results from the 10 min measurement with those from the 30 min measurement revealed significant differences.However, a comparison of the results from the 30 min measurement to those from the 100 min measurement did not show a pronounced difference.Based on these observations, it can be inferred that, under the simulation conditions presented, a measurement time of approximately 30 min is sufficient to obtain results with adequate accuracy.
Focusing on the results of the source configuration (a), in addition to the imaging centered around the true source position, false imaging was observed.This is believed to arise from the characteristics of the reconstruction algorithm in an underdetermined system, and such false imaging can be reduced by choosing an appropriate reconstruction algorithm  and optimizing its parameters.This is an issue we wish to address in future research.Table I lists the changes in the image formation position and angular resolution with respect to measurement time for source distribution patterns (a) and (b), respectively.The image formation position was defined as the point of maximum intensity in the reconstructed image, and the average of ten measurements determined its value.The standard deviation, denoted by ±σ, represents the variability in the image formation position across the ten measurements.The angular resolution was derived from the normalized outcomes of ten measurements.Specifically, it was characterized by the FWHM of the intensity distribution curve.We calculated the FWHM for cross Sects.along the x-and y-axes intersecting the image formation position, and the mean of these measurements provided the angular resolution.
For the source distribution patterns (a) and (b), the precision of the image formation position and its reliability improved with increasing measurement time.Importantly, the angular resolution showed a significant improvement between the 10-and 30 min measurements.In the case of pattern (b), peaks related to false imaging were excluded when determining the angular resolution.
Table II presents how the image formation position and angular resolution for source configuration (c) change with the measurement time.For the double-source case, similar to single-source patterns (a) and (b), the precision increased with increasing measurement times.However, the results were more accurate for a single source.
The final angular resolution, between 10°and 15°, is approximately equivalent to conventional Compton cameras but does not reach the performance of coded-mask type cameras.Notably, such imaging with a resolution in this range could be achieved with only 8 channels, covering a full 4π FOV.This feature is one of the notable characteristics of our imaging system, as illustrated in Table III, where it is compared with other gamma-ray imagers.
Under these conditions, a measurement time of approximately 30 min is sufficient to achieve the desired accuracy.However, the required measurement time can vary with factors such as the source intensity or distance, and guidelines are necessary to determine the degree of convergence under different conditions.The statistical error in the counts can be used as an indicator of convergence.For a 30 min measurement, the count for each detector ranged from approximately 200-2000, with a combined total of Table I.Time evolution of the image formation position and angular resolution for single source case.approximately 9000 for all detectors.The statistical errors for the individual detectors ranged between 2% and 7%.In this context, the relative error for count N, represented by N N was considered as the statistical error.For the C3G design discussed in this study, this level of statistical error suggests that the imaging results converged sufficiently.As observed in Fig. 4, the imaging distribution does not form a round shape around the true source position but shows asymmetric distortions.It could be caused by the inherent asymmetry of the 3D coded mask.Additionally, such asymmetry also affects the uniformity of detection sensitivity, as depicted in Fig. 5.The detection sensitivity, η m , in direction m is defined as follows: h = å = r m n nm 1

8
. Here, r nm represents the response function defined in (3), with m = 614 and the number of detectors n = 8.Data points are linearly interpolated between measured points.The detection sensitivity, η, varied almost by a factor of two at its maximum.

Conclusions
In this study, we introduce a novel gamma-ray imaging technique that departs from traditional methodologies.Using unique gamma-ray flux patterns generated using multiple shielding cubes placed randomly in a 3D space, our method provides an innovative approach to gamma-ray imaging.The key findings of this research are as follows: • Simulations using a 137 Cs source demonstrated that the proposed method could achieve an angular resolution close to 10°.This value satisfies the target angular resolution of 20°.While the angular resolution is comparable to that of Compton cameras and inferior to coded mask cameras, achieving this angular resolution with only eight readout channels is significant.• While the method shows promise in single-source configurations, the precision in double-source configurations remains slightly inferior.• The statistical error, derived from the counts, serves as a reliable measure to evaluate the stability of the imaging technique.For our C3G design, a statistical error ranging between 2%-7% suggests that the imaging results are consistent and reliable.• It was observed that certain source configurations can lead to the formation of false images.The decline in imaging accuracy with two sources compared to that with a single source suggests that the accuracy could further decrease with three or more sources.For instance, for sources spread over an area, there is the possibility of a  [20]  Compton 20°121 ch 4π Compton 1 [38]  Compton 11°1440 ch 4π Compton 2 [22]  Compton Better than 10°256 ch 180°i PIX [17]  Coded-mask 6.0°65536 ch 45°E PSILON-G [11]  Coded-mask no data 144 ch 45°P inhole [12]  Pinhole 9.5°256 ch 44°N uVISION [51]  Coded-mask 3.5°15°256 ch 45°4π Compton Fig. 5. Non-uniformity of detection sensitivity η.

032005-5
© 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd significant drop in imaging precision.Further, false imaging can be improved by optimizing the reconstruction algorithm.In addition to the reconstruction algorithm, this study prioritized verifying the feasibility of the method; therefore, the placement of the shielding and scintillator cubes was not optimized.We aim to optimize both the software and hardware aspects to achieve further performance enhancement.Furthermore, multi-nuclide simultaneous imaging, such as with 137 Cs and 60 Co, remains a challenge for future work.Currently, our focus is on the photoelectric absorption peaks of target energies.However, during multi-nuclide simultaneous imaging, the photoelectric peaks are influenced by the scattering components of other nuclides.Using training data that includes scattering components can solve this problem.
Even with these aforementioned challenges, the results strongly suggest the feasibility of the proposed method.We anticipate that further refinement of this technique will bring significant advancements in the gamma-ray imaging industry.In the future, we plan to develop a prototype and experimentally evaluate its characteristics.

Fig. 2 .
Fig. 2. Examples of gamma imager designs.In (a), the design can only discriminate between directions 1 and 2, whereas in (b), the design can estimate the gamma-ray incidence direction for all directions.

Fig. 4 .
Fig. 4. Temporal progression of imaging outcomes based on source configuration (a), (b), and (c).These results depict a representative example from one of the 10 measurements performed.The white arrow indicates the true source position, and the red arrow indicates the fake image formation.

Table II .
Time evolution of the image formation position and angular resolution for double source case.Comparison of characteristics with other gamma-ray imagers.