System Modeling and Evaluation of a Prototype Inverted-Compound Eye Gamma Camera for the Second Generation MR Compatible SPECT

Abstract

We have reported the design of the MRC-SPECT-II system based on the inverted-compound-eye (ICE) gamma camera to offer a > 1% detection efficiency while maintaining a sub-500 μm imaging resolution [1]. One of the key challenges of using the ICE camera for SPECT imaging is whether one could develop an accurate point response function (PRF), given its complex aperture design and low fractionation accuracy of 3D printing. In this work, we will discuss (I) a combined experimental and analytical approach for deriving the precise PRF, and (II) an experimental imaging study to demonstrate the feasibility of using the ICE-camera for acquiring high-quality SPECT images with a sub-500 μm resolution. These studies would help to overcome one of the major hurdles for implement ICE-cameras for practical SPECT imaging.

1. Introduction

Our community is motivated by great potential of combining SPECT and MRI for both pre-clinical and clinical applications and have devoted intensive effort in developing this hybrid system in recent years. Most notably, Wagenaar et al. have reported the design considerations of an MRI compatible SPECT system based on CZT detectors [2]. Tsui et al. have presented a prototype of a MRI compatible SPECT system based on similar CZT detectors [3] and proposed a whole body pre-clincal SPECT/MRI system [4]. A consortium consisting of several groups in EU proposed and developed both pre-clinical and clinical SPECT inserts using SiPM based detector [5, 6, 7]. Karel Deprez et al. have constructed an MRI compatible aperture using rapid additive manufacturing with selective laser melting of tungsten powder [8]. And J. Zajicek et al. have presented a MR-SPECT insert using Timepix detectors [9].

We have reported an ultrahigh resolution stationary MR compatible SPECT system (MRC-SPEC-I) [10] and demonstrated its ability to achieve a sub-500 μm resolution in simultaneous SPECT/MRI mode by integrating the SPECT system with the Siemens Trio 3T clinical scanner [11]. Despite the success in developing the MRC-SPECT-I system, the preliminary evaluation revealed several limitations of the system, including the relatively low geometrical sensitivity, limited angular sampling, and a large system dimension.

To overcome these limitations, we proposed an Inverted Compound Eye (ICE) camera and the second-generation MRI compatible SPECT system (MRC-SPECT-II) based on ultra-high-resolution detector technologies [1]. In the simulation study, the system, which has 1536 Micro-Pinhole-Camera-Elements (MPCEs) in 24 ICE cameras, offers a > 1% detection efficiency while maintaining a sub-500 μm imaging resolution. The system performance in experimental studies may degrade compared to the simulation ones because of the potential mismatch between the calibrated and actual imaging geometry, including pinhole position, pinhole shape, ect. This issue will potentially become more critical in ICE and MRC-SPECT-II, because of the large number of the micro-pinholes and the limited mechanical precision of current 3-D printing techniques. In this paper, we developed a hybrid modeling method by combining an analytical calibration and experimentally measured PRF using a point source scanning across a 2D space. This would allow us to incorporate the mechanical imperfections into the system model and therefore minimize the impact of related systematic errors on imaging quality.

2. Method

2.1. Prototype Inverted Compound Eye Camera

The prototype Inverted-Compound-Eye camera system (Fig. 1) consists of an ICE camera module and 4D sub-micron resolution scanning stages. The prototype ICE module has 64 Micro-Pinhole-Camera-Elements (MPCEs), by coupling a high resolution detector with sensor area of 2.56 × 2.56 cm (Fig. 1-D) and an aperture with 64 pinholes (Fig. 1-B and C). The 4D sub-micron resolution scanning system consists of 3 Newport linear stages and 1 Micro ration stage (Fig. 1-A). It allows the system to scan and rotate the object, to mimic a full-ring stationary system with eight detector modules. The design logic of the aperture and distributing scheme of each individual MPCE are the same as ones for the full system with 24 ICE camera modules [1], i.e., the pinholes of MPCEs are uniformly distributed along the axial direction (x-axis shown in Fig. 1) and in the transverse direction, the FOV edge of each individual MPCE is tangent to the system FOV boundary and the adjacent rows of MPCEs are pointing forward opposite direction, i.e., if the first row MPCE in the aperture points to left of the FOV and the second row will point to the right of FOV. With the aperture parameters, A STL file was produced by a home-made 3D drawing package and 3D rapid-prototyping technology [12] was used to print this complicated aperture using platinum (95%) and Iridium (5%). The FOV of the camera is around 10 mm in diameter.

Figure 1:

Prototype ICE camera system. (A) the system consists of an ultrahigh resolution photon accounting detector, rapid prototyped aperture made of platinum (95%) and iridium (5%), and 4D sub micron resolution scanning stages that have 3D linear stages and one rotation stage. (B) and (C) are front-view and back-view of the aperture. (D) the photon counting detector.

2.2. Ultra-high Resolution CdTe detector

The prototype system uses a customized-design prototype detector, PDT50 (Fig. 1-D) provided by XCounter. The PDT50 is an upgraded version of PDT25-DE photon counting detector [13]. The CdTe sensor has an active area of 25.6 × 25.6 mm² and is pixelated to 256 × 256 pixels with a size of 100 μm. The sensor thickness is 1 mm and the detection efficiency is 0.30 for a 140 keV photon. The detector thickness will be increased to 2 mm, or 5 mm to have enough stopping power for the full ring system in the future. The photon counting ASIC attached to the anodes implements a charge sharing correction technique to integrate the charge induced on neighboring (say 2 × 2) pixels. These schemes could significantly reduce the impact of charge sharing effects on the effective spatial resolution for small pixel detectors (Fig.6 and Fig.7 in [13]) and also help to alleviate the impact of charge-sharing (and incomplete charge collection) on the effective detection efficiency of the detectors.

Figure 6:

Projection at the first view angle.

Figure 7:

Reconstructed images of resolution phantom. (A),(C) are reconstructed images using hybrid modeling and they are corresponding to sagittal and axial views respectively. (C) and (D) are images using analytical model

3. Hybrid System Modeling

3.1. Analytical Calibration

A gamma camera could be described as a linear system, mapping discretized counts distribution, i.e., f = [f₁, f₂, ⋯ f_N] in the object space to the sensor detected counts, which is represented by an M dimension vector, p = [p₁, p₂, ⋯p_M]) as:

$$\mathbf{p}=\mathbf{H}\mathbf{f},$$ (1)

where H denotes the system response matrix with M × N dimension. A element, H_mn of the system response matrix H can be calculated using a widely used analytical method [14, 15, 16]:

$$H_{mn}=\underset{{\mathbf{r}}_d\in V_m}{\int }\left\{\frac{{\mu }_d}{4\pi \mid {\mathbf{r}}_d-{\mathbf{r}}_n{\mid }^2}\text{exp}[-{\mu }_d{l}_d({\mathbf{r}}_d,{\mathbf{r}}_n)-{\mu }_p{l}_p({\mathbf{r}}_d,{\mathbf{r}}_n)]\right\}\text{dV},$$ (2)

where H_mn is the probability of a photon emitted from the n^th object voxel detected by the m^th detector pixel; we calculated the probability at r_d for the object voxel at r_n and then integrated over the sensing volume belonging to the m^th detector pixel, i.e., r_d ∈ V_m. μ_d and μ_p are the linear attenuation coefficients of CdTe and the aperture at 140 KeV respectively; l_d and l_p represent the distance that the photon travels in the aperture and the CdTe crystal before it is detected at r_d.

As we can see in Eq. 2, the system response matrix depends on the imaging system geometry, including the position relations between an object, pinholes, and detectors. These parameters were estimated by minimizing difference between weighted centers of simulated point response functions (PRFs) and the experimental ones [17, 18]. The simulated PRFs were calculated based on Eq. 2 and the designed pinhole profile was used when we calculated pinholes’ attenuation.

The experimental PRFs were acquired through following procedures: A 100 μCi Co-57 point source with a diameter of 250 μm was scanned in the object space using the 4D sub-micron resolution stages. We scanned 4 × 4 × 4 grid points with a step size of 1 mm and 6 rings uniformly distributed along the axial direction with a gap of 1.5 mm. Each ring had 16 points uniformly distributed in a circle (radius around 3.0 mm). We acquired 1 min data at each position.

3.2. Experimentally Measured PRFs in a 2D Plane and Interpolation for PRFs in 3D

Besides the geometry relations, the system matrix is also affected by the pinhole profile. In the conventional analytical model, the attenuation effect of the pinhole is modeled based on the design geometry due to high accuracy of the conventional fabrication technique. However, accuracy of 3D printing technology are significantly worse. We have developed a method to compensate the discrepancy between our design and aperture fabrication:

1) As illustrated in Fig. 2, a point source was scanned through a 2D plane, which was almost parallel to the detector surface and located around the FOV center. There were 41 × 49 points covering the 2D planes and the distance between adjacent points was 0.3 mm. Each scanning point acquisition time was 150 seconds and the detection probability in each detector pixel was calculated by dividing the corresponding detected counts with total emitted counts, i.e., product of source activity (100 μCi) and the acquisition time.

Figure 2:

Illustration of experimental based correction for PRF.

2) We fitted the experiment PRFs with asymmetric Gaussian:

$$G=A\text{exp}\left[-\frac{1}{2}(\mathbf{x}-\overline{\mathbf{x}}{)}^T{\Sigma }^{-1}(\mathbf{x}-\overline{\mathbf{x}})\right]$$ (3)

where the A is the amplitude of the PRFs and x = [x_r, x_c]^T is the detector pixel position and $\overline{\mathbf{x}}$ is the peak position. The co-variance matrix Σ could be decomposed by Singular Value Decomposition (SVD):

$$\Sigma ={\mathbf{V}}^T\Delta \mathbf{V}=\left[\begin{array}{cc}\hfill \cos \theta ,\hfill & \hfill -\sin \theta \hfill \\ \hfill \sin \theta ,\hfill & \hfill \cos \theta \hfill \end{array}\right]\left[\begin{array}{cc}\hfill {\delta }_r^2,\hfill & \hfill 0\hfill \\ \hfill 0,\hfill & \hfill {\delta }_c^2\hfill \end{array}\right]\left[\begin{array}{cc}\hfill \cos \theta ,\hfill & \hfill \sin \theta \hfill \\ \hfill -\sin \theta ,\hfill & \hfill \cos \theta \hfill \end{array}\right]$$ (4)

Reorganize Eq. 3 with SVD form of Σ in Eq. 4:

$$\begin{gathered} G=A\text{exp}\left\{-\frac{1}{2}[{\mathbf{V}}^T(\mathbf{x}-\overline{\mathbf{x}}){]}^T\Delta [{\mathbf{V}}^T(\mathbf{x}-\overline{\mathbf{x}})]\right\} \\ =A\text{exp}\left\{-\frac{1}{2}\left[\frac{(x_r^{\prime }{)}^2}{{\delta }_r^2}+\frac{(x_c^{\prime }{)}^2}{{\delta }_c^2}\right]\right\} \end{gathered}$$ (5)

where ${\mathbf{x}}^{\prime }=[x_r^{\prime },x_c^{\prime }{]}^T$ are transfered pixel position coordinate, where $x_r^{\prime }$ and $x_c^{\prime }$ are decoupled with each other:

$$\begin{gathered} {\mathbf{x}}^{\prime }={\mathbf{V}}^T(\mathbf{x}-\overline{\mathbf{x}}) \\ \left[\begin{array}{c}\hfill x_r^{\prime }\hfill \\ \hfill x_c^{\prime }\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \cos \theta ,\hfill & \hfill -\sin \theta \hfill \\ \hfill \sin \theta ,\hfill & \hfill \cos \theta \hfill \end{array}\right]\left[\begin{array}{c}\hfill x_r-{\overline{x}}_r\hfill \\ \hfill x_c-{\overline{x}}_c\hfill \end{array}\right], \end{gathered}$$ (6)

where the angle θ allows us to control the orientation of the PRFs. δ_r and δ_c are given by:

$${\delta }_r∕c=\{\begin{array}{cc}{\lambda }_{r∕c}{\delta }_{r∕c0}\hfill & x_{r∕c^{\prime }}<0\hfill \\ {\delta }_{r∕c0}\hfill & x_{r∕c^{\prime }}>=0\hfill \end{array}\phantom{\}}$$ (7)

The coefficients, λ_r and λ_c, give the freedom to model those asymmetric PRFs in the row and column direction respectively.

3) Interpolate the system response function in the 3D object space based on the system geometry derived from the analytical calibration and PRFs from 2D fine scanning as illustrated in Fig. 2:

a) Calculated intersection (Voxel I in the figure) between the fine scanning plan and the line between target voxel (voxel J in Fig. 2) and center of a pinhole.

b) Linearly interpolated asymmetric Gaussian parameters of voxel I by using the fitted parameters of four closest adjacent scanning points (marked as 1, 2, 3, and 4 in Fig. 2 and corresponding PRFs are marked as PRF1, PRF2, PRF3, and PRF4).

c) Calculated the Gaussian parameters for target voxel i using:

$${\delta }_{r∕cJ}={\delta }_{r∕cI}\frac{M_I}{M_J}$$ (8)

$${\lambda }_{r∕cJ}={\lambda }_{r∕cI}$$ (9)

where parameters with subscript J are asymmetric Gaussian parameters for voxel J and with I for voxel I. M_I/J are magnification factors for voxels I/J.

d) Amplitude (A) of Gaussian parameter was scaled so that the detection probability satisfies one-over-square-of-distance relation:

$$\frac{{\sum }_k^N{H}_{kJ}}{{\sum }_k^N{H}_{kI}}=\frac{d_I^2}{d_J^2}$$ (10)

, where H_kJ/J is probability of voxel J/I detected by a detector pixel k and d_I/J is the distance from pinhole to voxels I/J

3.3. Resolution Phantom Study

To evaluate the performance of the prototype system and benefits of hybrid modeling, we carried the resolution phantom system. The home-made resolution phantom had four groups of hot rods with sizes of 0.35 mm, 0.5 mm, 0.75 mm and 1 mm respectively. This distance between the centers of hot rods within a group was two times of their size. The phantom with a diameter of 12 mm was placed in 3D printed tube and injected with total activity of 598μCi (the activity filled in the hot rods were significantly less than this value). The phantom was rotated 8 angles with a step size of 45° to simulate the full ring system consisting of 8 camera modules. The images was reconstructed using ML-EM.

4. Results

4.1. Point Response Function

To evaluate the model accuracy of experimental correction, we compared PRFs from measurement, the analytical model, and the hybrid model. As shown in Fig. 5, although the weighted center of the analytical PRF is close to the measured ones with difference around 0.1 of the detector pixels size, the profile is much wider than the measured one. In comparison, the hybrid modeled PRF matches much better in terms of shape. Please note that this measured PRF was selected from the analytical calibration measurement (Section 3.1) and not from 2D scanning points used for the PRF interpolation in Sect. 3.1.

Figure 5:

Left: detection probability of one of the corner pinhole (marked PH 1 in Fig. 1); right: detection probability of a center pinhole (marked as PH 28 in Fig. 1)

4.2. Detection Accuracy

We also evaluated the detection accuracy, by comparing measurement detection probability with the analytical model and experimental based correction. The 160 scanned points in the analytical calibration (Section. 3.1) was used to derive measured detection probability by summing each individual PRF. Fig. 4 shows the detection probability of one of the corner pinholes (marked PH 1 in Fig. 1) and the other of a center pinhole (marked as PH 28 in Fig. 1). Hybrid modeling can significantly improve detection accuracy with a maximum error < 10 %, while the error of the analytical one could go as high as 40%.

Figure 4:

(A) measured PRF; (B) corresponding PRF derived from analytical modeling; (C) PRF using hybrid modeling

4.3. Resolution Phantom Studies

Fig. 6 shows a projection at the first angle. As we can see, each individual MPCE samples part of the phantom in one specific angle. Combining 64 MPCEs projects in all eight rotation angles, the reconstructed image resolves 0.5 mm hot rods (Fig. 7-A and C) using the hybrid model. As the comparison, using the same data set with analytical modeling, the hot rods larger than 0.5 mm could be resolved but severely distorted (Fig. 7-B and D). The modeling error of the analytical method also causes the boundaries of the reconstructed phantom to be truncated to discontinuous sections and the reconstructed image has artifact dots outside the phantom region, where no events should exist unless scattering is considered.

5. Discussion and Conclusion

A prototype inverted-compound eye camera was built using an ultra-high resolution CdTe detector coupled to a 64 pinhole aperture. To accurately model the response of this complicated system, a hybrid system modeling scheme, combining the analytical geometry calibration and experimental fine scanning 2D data was developed. Comparing PRF and detection accuracy has shown that hybrid modeling scheme can significantly improve modeling accuracy. The resolution phantom study has demonstrated the system could achieve 500 μm resolution and the modeling scheme has significantly improved the imaging qualities. This prototype system has evaluated feasibility and potentials of the ICE concept and shows that combining ultra-high resolution detector technologies, advanced aperture design and fabrication, and novel system modeling methodologies could allow us to develop a high performance SPECT camera.

Our community has already explored how to use sparse 3D experimental measurements to interpolate PRFs [19]. The methodology developed here was unique in terms of using measurements in a 2D space to interpolate PRFs in the full 3D space. 2D scanning allows us to achieve a good trade-off between scanning step size and total number of measurements. The fine step size (0.3 mm) is required for ICE, given that the small open angle of the pinholes in the aperture and each pinhole does not cover the full FOV. While 2D scanning could significantly reduce measurement times, the total acquisition was still long (83 hours with 100 Ĉi Co-57), but it can be reduced by using a stronger activity source.

Figure 3:

Asymmetric Gaussian PRF model. The first row shows PRFs with orientation angles θ equal to −60°, 0° and 30°; The second row shows PRFs with asymmetric coefficients λ equal to 0.75, 1.0, and 1.5.