EXIRAD-HE: multi-pinhole high-resolution ex vivo imaging of high-energy isotopes

We recently developed a dedicated focusing multi-pinhole collimator for a stationary SPECT system that offers down to 120 µm (or 1.7 nL) spatial resolution SPECT images of cryo-cooled tissue samples (EXIRAD-3D). This collimator is suitable for imaging isotopes that are often used in small animal and diagnostic SPECT such as 125I (27 keV), 201Tl (71 keV), 99mTc (140 keV), and 111In (171 and 245 keV). The goal of the present work is to develop high-resolution pinhole imaging of tissue samples containing isotopes with high-energy photon emissions, for example, therapeutic alpha and beta emitters that co-emit high energy gammas (e.g. 213Bi (440 keV) and 131I (364 keV)) or 511 keV annihilation photons from PET isotopes. To this end, we optimise and evaluate a new high energy small-bore multi-pinhole collimator through simulations. The collimator-geometry was first optimised by simulating a Derenzo phantom scan with a biologically realistic activity concentration of 18F at two system sensitivities (0.30% and 0.60%) by varying pinhole placements. Subsequently, the wall thickness was selected based on reconstructions of a Derenzo phantom and a uniform phantom. The obtained collimators were then evaluated for 131I (364 keV), 213Bi (440 keV), 64Cu (511 keV), and 124I (511 + 603 keV) with biologically realistic activity concentrations, and also for some high activity concentrations of 18F, using digital resolution, mouse knee joint, and xenograft phantoms. Our results show that placing pinhole centres at a distance of 8 mm from the collimator inner wall yields good image quality, while a wall thickness of 43 mm resulted in sufficient shielding. The collimators offer resolutions down to 0.35 mm, 0.6 mm, 0.5 mm, 0.6 mm, and 0.5 mm when imaging 131I, 213Bi, 18F, 64Cu, and 124I, respectively, contained in tissue samples at biologically achievable activity concentrations.


Introduction
Preclinical imaging of radiolabeled molecule distributions in small animals plays an important role in studying the mechanisms of disease and developing diagnostic tracers (Rowland and Cherry 2008). A wide and steadily increased variety of tracers with different radioisotopes is available. While conventional SPECT scanners are designed to image isotopes emitting gammas with a typical energy of around 140 keV, many available and emerging isotopes emit gammas with higher energies. These include PET tracers that emit 511 keV annihilation photons (e.g. 18 F, 124 I, and 64 Cu), and alpha or beta emitters for theranostic applications that co-emit high-energy gammas (e.g. 131 I, 213 Bi, and 209 At). Many commercial preclinical PET scanners for imaging positron emitters have been launched. Recent ones include Mediso NanoPET/CT (Herrmann et al 2013), Bruker Albira (Gonzalez et al 2016), Clairvivo (Sato et al 2016), and Molecubes β-CUBE (Krishnamoorthy et al 2018), which achieve resolutions down to approximately 0.9 mm. With one small field-of-view (FOV) prototype coincidence system, a resolution of 0.6 mm was obtained (Yang et al 2016).
In another approach, PET isotopes are successfully imaged using dedicated clustered multi-pinhole collimators in a triangular detector set up (VECTor) at an initial resolution of < 0.75 mm in vivo for 18 F (Goorden et al 2013, Walker et al 2014, Gonzalez et al 2016. Today resolutions of commercial VECTor scanners are < 0.6 mm for 18 F simultaneously with 0.4 mm resolution 99m Tc imaging. Compared to coincidence PET, multi-pinhole PET effectively removes or reduces several image degrading effects that significantly limit the image quality in small animals. These include limited detector resolution and depth-of-interaction effects whose influence can be strongly diminished by pinhole magnification, as well as non-collinearity of annihilation photons, and random coincidences which do not play a role in pinhole PET. As a result, the resolution of VECTor can outperform coincidence PET in a subset of imaging situations (Walker et al 2014). In addition, with VECTor, the prompt gammas that are emitted by some PET isotopes (e.g. 603 keV for 124 I) can be employed to exclude the effect of positron range that sets a fundamental limit on the resolution in coincidence PET. This enables for example 0.75 mm resolution PET of 124 I and 89 Zr (Beekman et al 2018 as well as multi-isotope PET. At the same time, VECTor has the capability of sub-mm resolution imaging of isotopes that emit single gammas above the conventional SPECT energy range like 131 I (364 keV), 213 Bi (440 keV), and 209 At (545 keV) (de Swart et al 2016, Crawford et al 2018. Therefore, multi-pinhole imaging is a single-system solution for high-resolution imaging of gamma-and positron-emitting isotopes over a broad energy range. One way to improve the resolution-sensitivity trade-off of tissue samples in pinhole SPECT and PET is to increase the pinhole magnification factor by decreasing the collimator's FOV. This concept was used for the ex vivo imaging technique EXIRAD-3D (MILabs B.V.), which was designed for low and medium energy isotopes such as 125 I (27 keV), 201 Tl (71 keV), 99m Tc (140 keV), and 111 In (171 keV and 245 keV) (Nguyen et al 2019(Nguyen et al , 2020. With this technique, 120 µm (or 1.7 nL) resolution was obtained for 99m Tc, i.e. about a ten times better volumetric resolution than state-of-the-art in vivo preclinical SPECT (Ivashchenko et al 2014). EXIRAD-3D consists of a dedicated small FOV multi-pinhole collimator, a tissue holder connected directly to a refillable chamber with dry ice for cryo-cooling the tissue throughout the scan, and dedicated acquisition and reconstruction software.
Today, EXIRAD-3D coupled with in vivo imaging modules as is done in the commercial MILabs' nuclear scanners offers both longitudinal information of biological and physiological functions and extra-fine three-dimensional image details, with a direct link between the two datasets. That is achieved with little more effort than the conventional in vivo imaging workflow without the need for whole new equipment and a complicated tissue sectioning and handling process as in a histological study. However, note that EXIRAD-3D is only designed for imaging gamma-emitters but not pure alpha-and beta-emitters.
The aim of the present paper is to design an EXIRAD option (EXIRAD-HE) that is also suited for imaging high-energy isotopes. The approach is to use narrow pinhole opening angles to suppress pinhole edge penetration and scatter, with thick collimators walls. Full system simulations of a Derenzo phantom scan and a cylindrical phantom scan with a biologically realistic activity concentration of 18 F were performed for various collimator designs at two system sensitivities (0.30% and 0.60%). Best performing collimators were subsequently evaluated for 18 F scans with several activity concentrations and for 64 Cu, 124 I, 213 Bi, and 131 I with digital joint and xenograft phantoms.

Considerations in collimator design and optimisation
The EXIRAD-HE collimator (see figure 1) is designed to have the same central field-of-view (CFOV) size (4 mm diameter) and inner diameter (21 mm) as the available low-energy collimator meaning that the same size tissue volumes can be scanned by means of the scanning focus method (Vastenhouw and Beekman 2007). With this method, the tissue is moved to multiple positions inside the collimator such that the whole tissue volume is viewed by the CFOV, and acquisitions from all of the scanned positions are used together in image reconstruction. The collimator is mounted in a U-SPECT or VECTor system having a fixed triangular detector setup with three large-field-of-view gamma cameras. In this study, we assume a 9.5 mm thick NaI(Tl) crystal with a usable area of 497.4 * 410.6 mm 2 for each detector. The crystal surface is placed at a distance of 210 mm from the collimator's centre. The EXIRAD-HE collimator contains a large number of round knife-edge pinholes placed manually in such a way that overlapping pinhole projections are limited without using a shielding layer (see section 2.4.3). All pinholes' axes point towards the collimator's centre, and the pinholes' centres are placed at the same distance from the collimator's longitudinal axis. Collimator material assumed is a tungsten alloy consisting of 97% W, 1.5% Ni, and 1.5% Fe.
With the above-mentioned constraints, we restricted the parameters for optimisation to be the distance from the pinhole's centre to the collimator's inner surface (h), the pinhole diameter (d), and the wall thickness (T) (see figure 1). Other parameters as listed in table 1, if not fixed, can be expressed by these parameters. For example, pinhole opening angle only depends on h because the size of the CFOV (4 mm diameter sphere) and the distance from the collimator's longitudinal axis to the collimator's inner surface (10.5 mm) were fixed. Also, the total number of pinholes were calculated from the detector size (3 * 497.4 * 410.6 mm 2 ) and the size of pinhole projections (again only depending on h) (see equation (3)). For the three parameters to be optimised, the following considerations were taken into account: -For h, one may expect that the closer the pinhole is to the object (i.e. h approaches 0), the higher the system resolution that can be achieved for a fixed sensitivity (Rentmeester et al 2007). This suggests putting the pinhole apertures as close to the inner surface of the collimator as possible. However, putting pinholes closer gives more sensitivity per pinhole but fewer pinholes that can be placed for the same detector area, which in some cases, limits the resolution-sensitivity trade-off (Goorden et al 2009). Besides, for the high energy photons (e.g. 511 keV), putting pinholes closer to the inner collimator wall would result in a significant increase of pinhole edge scatter and penetration because of shorter paths that photons travel through the pinhole edges, which deteriorates image quality. Thus an intermediate value for h may be most beneficial. -The pinhole diameter d directly decides the resolution-sensitivity trade-off of the system: reducing the pinhole diameter improves the resolution at the expense of decreasing the sensitivity and vice versa. -For T, we would expect that the larger the wall thickness, the better prevention of direct collimator wall penetration which is important for image quality. However, the thicker the collimator, the heavier and more expensive the collimator is which limits its practical use as a final product.
It is not straightforward to find an optimal set for all of these parameters analytically. We, therefore, utilised an analytical model for the initial estimate of the parameter sets, and then our optimisation was based on evaluating the quality of the final reconstructed images obtained from Monte Carlo Simulation (MCS) data. The following sections will explain this process in more detail.

Analytical model
The analytical model was used for 18 F (511 keV), and we only optimised the collimator for this energy, which lies inside the studied energy range (364-603 keV). For the initial optimisation, we used an analytical model which is based on the models in (Metzler et al 2001, Rentmeester et al 2007 that can efficiently predict the performance of multi-pinhole SPECT and has been applied in optimising a small-animal SPECT system, namely U-SPECT-I (Beekman et al 2005), as well as in a theoretical analysis of a human brain SPECT (Goorden et al 2009). It models collimator and detectors as spherical layers surrounding the scanned object (figure 2(a)). The parameters used in this model are included in table 1.
To approximate the cylindrical collimator and the triangularly-placed detectors in EXIRAD-HE by the spherical layers in the analytical model, the average distance from the pinholes' centres and the detector surface to the collimator's centre were set as the radius of the collimator layer (r c ) and the radius of the detector layer (r d ), respectively. To this end, we sampled uniformly on each detector's surface a set of 50 * 50 = 2500 points (D) from each of which a line was drawn through the collimator's centre. Intersections (C) of these lines with the cylindrical surface that goes through all pinholes' centres were calculated. Then, r d (or r c ) was set as the average distance from D (or C) to the collimator's centre. Consequently, r d was 278 mm as the detectors were fixed, and r c depended on the real distance from pinholes' centres to the collimator centre (which is different for the various collimators we considered).
With the analytical model, we calculated the sensitivity over the whole CFOV instead of the peak sensitivity calculated at the collimator's centre as was done in (Rentmeester et al 2007). To this end, we took  the average of the sensitivities over n = 8820 points uniformly distributed within the spherical CFOV. The sensitivity at each point reads (Metzler et al 2001): where S i is the sensitivity at photon source position i;θ i and m i define the location of the photon source (figure 2(b)); d is pinhole diameter; α is pinhole opening angle; µ is attenuation coefficient of the pinhole material. This average sensitivity was multiplied by the number of pinholes (N), detector efficiency (ϵ), and then multiplied by two because of the gamma pair production, to obtain the system sensitivity over the CFOV: The number of pinholes was calculated as: This took into account that a fraction of π/4 of the total detector surface (3 * D X * D Z ) is covered with non-overlapping pinhole projections. A detector photopeak efficiency ϵ of 14.3% for 511 keV was obtained from MCS, as explained in section 2.3.3. The radius of the CFOV (r f ) is 2 mm. Table 2. Studied isotopes and corresponding simulation settings. The attenuation coefficient in collimator was obtained from NIST database (Seltzer 1993, Hubbell andSeltzer 1995), and the attenuation coefficient in the detector was calculated from MCS (section 2.3.4).  (Jan et al 2004, 2011, Staelens et al 2006, Chen et al 2008 was used for all MCS in this research to simulate photon transport inside the scanner. Our simulation method was validated with experimental data from EXIRAD-3D with low-energy collimators in a previous publication (Nguyen et al 2019) where detailed settings can be found. In this work, to simulate the high-energy collimators, we adapted the following. Firstly, instead of manually defining the physics processes, here the 'physic list builder' mechanism was used as recommended in the recent GATE documentation, and the general 'emstandard' builder provided by the Geant4 community was set. This included, but was not limited to, photoelectric effect, Compton scattering, Rayleigh scattering, electron ionisation, bremsstrahlung, multiple scatter, and positron annihilation for positron emitters. Secondly, GATE's 'ion source' was defined by setting the isotope's atomic number (Z) and atomic weight (A) (e.g. Z = 9 and A = 18 for a 18 F source). This way, both the radioactive decay and the atomic de-excitation were incorporated, and it was straightforward to change the radioactive isotopes for different scan simulations. Finally, the detector's spatial resolution and energy resolution were set in a different way than in (Nguyen et al 2019). The detector's spatial resolution was fixed to 3.5 mm for all studied isotopes, instead of scaling it by the reciprocal of the square root of the photon's energy which would result in a too good resolution for high energies because this way of scaling would not appropriately consider the large amount of scatter of the high-energy photons. The detector energy resolution was modelled using the inverse square law (Jan et al 2004), R energy = R 0 √ E 0 /E, where R 0 = 9% was assumed as the full-width-at-half-maximum (FWHM) energy resolution at E 0 = 511 keV.

F
After getting the projection images from the GATE output, realistic background radiation (from e.g. cosmic radiation) was added to both the photopeak window and the two side windows. This was based on the count rate from a background radiation measurement with a U-SPECT/CT system at MILabs B.V., Utrecht, the Netherlands, as reported in table 2. Table 2 also specifies the photopeak window, the side windows, and some other settings for each isotope.

Simulation of system sensitivity
The system sensitivity obtained from MCS was used in the collimator optimisation process to obtain pairs of h and d that would provide such sensitivity (section 2.4). Here a uniform spherical 18 F distribution that exactly fits inside the CFOV (4 mm diameter) and emits 10 8 positrons was simulated. The activity distribution was placed at the centre of a PMMA sphere having a diameter of 10 mm that acts as the annihilation environment. Sensitivity was calculated as the number of detected counts within the photopeak window (20%) over the number of emissions. Note that all the sensitivities reported in this work were for 18 F.

Simulation of detector efficiency
The detector efficiency was used in the analytical model. To simulate this, a uniform spherical 18 F source that fits in the CFOV and emits 10 8 positrons was placed at the centre of a full multi-pinhole collimator (h = 8 mm, d = 0.27 mm, and T = 60 mm). The activity distribution was placed at the centre of a PMMA sphere having a diameter of 10 mm that acts as the annihilation environment. By setting the detector's material as NaI with a normal density (3.67 g cm −3 ) and then with a very high density (367 000 g cm -3 ), and taking the ratio between the counts in these two cases, we obtained a detector efficiency of 14.3% for 511 keV.

Simulation of attenuation coefficient in detector crystal
The attenuation coefficient in the detector was simulated for each isotope and used both in the analytical model (section 2.2) and in generating the system matrix for image reconstruction (section 2.5.1). To this end, a point source containing the studied isotope that emits 10 5 photons in a beam perpendicular to one of the 9.5 mm thick NaI(Tl) scintillators was simulated. No collimator was included, and the beam was defined in GATE by the source's emission angle. Only for this simulation, the source was defined with gamma particle type and the corresponding mono-energy instead of the ion type. Subsequently, the attenuation coefficient was calculated with equation (4), and the obtained numbers are listed in table 2. This equation assumed Beer's law that did not fully include the Compton scatter (only the part that ends up being detected in the photopeak was considered); however, it provided an effective attenuation coefficient that we need for the analytical model and the system matrix generation.

Collimator optimisation and evaluation procedure
We here optimised the collimator for 18 F because 511 keV lies near the middle of the studied energy range (364-603 keV) and we aimed to attain a collimator that can be used for a wide energy range. The optimisation procedure was as follows: i. First, we assumed T = 60 mm which is significantly larger than the thickness of the available lowenergy collimator for EXIRAD-3D (25.5 mm) to prevent collimator wall penetration by the high-energy photons. The probability that 511 keV photons pass through a 60 mm thick collimator material of 97% W, 1.5% Ni, and 1.5% Fe is 3.1 * 10 -5 %. ii. For a fixed sensitivity averaged over the CFOV (S CFOV ) of either 0.30% (sensitivity of an available in vivo high-energy collimator in VECTor (Goorden et al 2013)) or double that, 0.60%, h was varied from 0 mm to 10 mm with a step size of 2 mm and the corresponding d to achieve the desired sensitivity was roughly calculated with an analytical model as explained in section 2.2. Subsequently, d was fine-tuned with a step size of 5 µm using MCS (see section 2.3) to match the sensitivity exactly. iii. The combinations of h and d, each defining a unique collimator, were used to create complete multipinhole collimators with the pinhole projections illustrated in figure 3. The pinholes were distributed into rings, and in each ring, the axes between adjacent pinholes made an angle equal to the pinhole opening angle. The angular shift between adjacent rings was half the pinhole opening angle. The distances between the rings were set manually. iv. The collimators were then evaluated at fixed S CFOV based on full image reconstruction of a Derenzo phantom with a realistic activity concentration of 18 F (described in section 2.5.2). For each S CFOV , the value of h that results in the reconstructed Derenzo image with the highest contrast-to-noise ratio (CNR, equation (5)) was selected. v. For each S CFOV with the chosen set of h and d, the wall thickness T was reduced from 60 mm to 50 mm, 43 mm, and 30 mm to see which thickness still maintains good reconstructed image quality. To this end, we simulated both a Derenzo phantom and a uniform phantom (described in section 2.5.3) with a realistic 18 F activity concentration. The image quality was based on visual assessment, and quantitative metrics (CNR calculation for the Derenzo phantom, and portion of reconstructed activity present outside the uniform phantom). vi. Finally, we obtained an optimised collimator for each S CFOV , defined by a set of h, d, and T, and the estimated collimator weights, as well as the updated sensitivities for the finally chosen wall thickness, were recorded.
Subsequently, two collimator designs selected for the two considered sensitivities were evaluated with high concentrations of 18 F and biologically realistic activities of some other high-energy isotopes, namely 64 Cu, 124 I, 213 Bi, and 131 I, to see which spatial resolution can be achieved.  . Geometric projection on one of the detectors with the manually placed pinholes for varying distance from pinhole centres to the collimator inner surface (h). The indicated opening angle for each h is for the pinholes in middle rings. The opening angle of the pinholes in the two outermost rings is two degrees smaller than that to prevent projection overlapping.
Furthermore, a digital mouse knee joint phantom and a digital tumour xenograft phantom (described in section 2.5.4) were simulated to evaluate the collimators' performance in tissue scans. The reconstructed images with the new collimators were also compared with those obtained with the VECTor collimator and 1 mm blurred images of the original activity distribution that represents typical images of commercial coincidence PET.

Image reconstruction 2.5.1. General settings
All studied scans were simulated with MCS. In each scan, the object was placed at multiple bed positions and the projections acquired from all bed positions were used together in image reconstruction (Vastenhouw and Beekman 2007). For image reconstruction, a system matrix was generated for each of the investigated collimators, using the ray-tracing method as described in . In this method, the system matrix was calculated given the position and orientation of the collimators and detectors, as well as the energy-dependent linear attenuation coefficients of the collimator and detector materials. The system matrix for each isotope was generated at the energy of the chosen photopeak (table 2). To keep a reasonable matrix size and reconstruction time, only the gamma photon paths that have a probability of at least 1% to pass through the collimator material are included into the system matrix. Reconstructions were performed using the similarity-regulated ordered-subset expectation maximisation (SROSEM) algorithm (Vaissier et al 2016). Post-filtering was applied with the filter size indicated for each phantom in sections 2.5.2-2.5.4. Images were reconstructed on an isotropic 0.25 mm voxel grid. Scatter and background radiation were corrected using the triple-energy window method (Ogawa et al 1991) with two side windows adjacent to the photopeak, each having a width of 25% of the photopeak window's width. Positron range was corrected by including it in the forward step of the reconstruction using pre-calculated kernels . Note that positron range correction was not applicable for 124 I, 213 Bi, and 131 I for the corresponding acquired photopeaks.

Digital derenzo phantoms
Two Derenzo phantoms simulated in this work are shown in figure 4. Derenzo phantom 1 was used most of the time, and Derenzo phantom 2 was only for the 131 I scan in figure 7 to show smaller rods. Each phantom has six sectors, in each of which, the distance between centres of two adjacent rods was twice the rod diameter. The whole phantom was placed at the centre of a PMMA cylinder having a diameter of 14 mm and a length of 8 mm.
Various activities in the phantoms and scan times were assumed for the collimator optimisation and evaluation. We simulated biologically realistic activity concentration in mice of 18 F (4.39 MBq ml -1 ), 64 Cu (9 MBq ml -1 ), 124 I (9.6 MBq ml -1 ), 213 Bi (12.4 MBq ml -1 ), 131 I (54 MBq ml −1 ) that were retrieved from (Laitinen et 2). Data acquisition with sixteen bed positions was simulated for this phantom. The images that were shown had the iteration number and the 3D Gaussian post-filter that maximised the CNR. The CNR was calculated in the same way as in (Walker et al 2014). For this purpose, we drew the activity regions (red circle) and the background regions (blue circles) as indicated in figure 4 that span a depth of 1.5 mm (or six slices). CNR was defined as: where I s and B s are the mean intensity over the activity regions and the background regions of sector s, respectively. σ Is,p and σ Bs,p are standard deviations over these regions, calculated on a subset of planes p, p ∈ {1, 3, 5} to reduce covariance between planes. IB s is mean intensity over all regions of interest in sector s, and n s is the number of rod sectors. For EXIRAD-HE, all sectors were included, while for VECTor, only the three largest sectors were used in CNR calculation because the smaller sectors were not well resolved. The CNR's mean and standard deviation reported in tables 3 and 4 were calculated with three noise realisations, while the CNR for figure 7 was from one realisation.

Digital uniform phantom
To study the wall thickness, a uniformly filled syringe (6.5 mm inner diameter and 10 mm length) was simulated. The whole phantom was placed at the centre of a PMMA cylinder having a diameter of 14 mm and a length of 14 mm. The phantom was assumed to contain 4.39 MBq ml −1 18 F and to have been scanned for 4 h. Forty-five bed positions were used for this phantom. All the uniform phantom images were shown at the 10th iteration with the same post-reconstruction filters, a 0.7 mm FWHM 3D Gaussian filter followed by a 7 * 7 * 7-voxels Median filter. Reconstructed image quality was evaluated based on visual assessment and the percentage of activity reconstructed outside the phantom. The optimal image for each sensitivity is enclosed in a square box. Slice thickness was 1.5 mm.

Digital mouse tissue phantoms
A mouse knee joint phantom was derived from a real 3D image of a mouse knee joint containing 99m Tc-methylene diphosphonate ( 99m Tc-MDP) scanned with the low-energy EXIRAD collimator, and here assumed to contain 18 F-Na as these isotopes label the same target in the bone. The mouse knee joint contained 56 MBq ml −1 18 F (obtained from experimental data with 18 F) and has been scanned for 4 h. This phantom volume had a size of 4.45 * 4.45 * 6.45 mm 3, and three bed positions were used. Furthermore, a tumour xenograft phantom consisting of a large tumour with a necrotic core, and two small spherical tumours next to each other was simulated to contain 131 I-NaI. The mouse was assumed to be injected with 20 MBq 131 I and 10.2% ID/g was on the tumour at the time of termination (Tijink et al 2009). The xenograft was presumed to be snap-frozen and imaged for 16 h. This phantom volume had a size of 6.25 * 6.25 * 6.25 mm 3, and five bed positions were used.
In both cases, the reconstructed images were displayed at the iteration number and the 3D Gaussian filter size that yielded the lowest normalised mean-squared error (NMSE) and evaluated based on both visual assessment and NMSE. Figure 5 compares reconstructed images of Derenzo phantom 1 when varying h. The pinhole diameters d that result in the desired system sensitivities and the image CNRs are reported in table 3. 'N/A' represents the case in which no non-negative value of d can be found to achieve the target S CFOV . This was also the case when setting h = 0 mm. For both sensitivity levels, h = 8 mm offered the images with the highest CNR and was hence selected. Figure 6 shows the reconstructed images of Derenzo phantom 1 and the uniform phantom when varying wall thickness, and table 4 presents the corresponding CNRs of the Derenzo images as well as the percentage of activity reconstructed outside the uniform. Visual assessment on the Derenzo phantom's rods and on the activity within the uniform phantom volume does not show significant image degradation as the wall thickness reduces from 60 mm to 30 mm. However, the artefacts outside the uniform phantom do increase, especially for the 30 mm thick collimator. Table 4 indicates that as the collimator wall gets thinner, the CNR decreases and the activity reconstructed outside the uniform phantom increases with the most degradation when going from 43 mm to 30 mm. Therefore, we chose T = 43 mm to keep a reasonable collimator weight of 16 kg for the core. The weight variation between the collimators for the two sensitivities is negligible because only pinhole diameters change (0.270 mm versus 0.595 mm). Table 5 presents the final parameters of the two optimal collimators, EXIRAD-HE-1 and EXIRAD-HE-2, with reference to the VECTor collimator. It turns out that EXIRAD-HE-1 and EXIRAD-HE-2 are only different in the pinhole diameter, which leads to the different system sensitivities. The listed sensitivities of the EXIRAD-HE collimators here were calculated with 18 F and the chosen wall thickness (T = 43 mm), which are now slightly higher than the initially targeted sensitivities set at the beginning of the optimisation process. Here the value of h was already fixed to 8 mm. Note that the listed sensitivities are only for T = 60 mm and we would expect slightly higher sensitivities for the smaller wall thicknesses. The Derenzo images were displayed with the iteration and the 3D Gaussian filter size that maximises the CNR. All the uniform phantom images were shown at the 10th iteration with the same post-reconstruction filters, 0.7 mm FWHM 3D Gaussian filter + 7 * 7 * 7-voxels Median filter. Slice thickness was 1.5 mm. Table 5. Parameters of the selected EXIRAD-HE collimators and reference to the HE-UHR-M collimator in VECTor which was also simulated in this paper for comparison. For each collimator, two angles are listed: the larger one is for the inner rings, and the smaller one is for the two outermost rings near two ends of the collimator.  Figure 7 displays simulated images of the Derenzo phantoms filled with 18 F at several activity concentrations as well as some other high-energy isotopes using the optimal collimators. These reconstructed images are placed next to the images obtained with the in vivo collimator in VECTor to see the benefit of the new ex vivo collimator. Each column corresponds to a scan with a fixed setup, and only the collimator is changed. The image that shows the clearest rods in each scan is enclosed in a box. Here only the scans with 131 I used the smaller phantom, Derenzo phantom 2, to visualise rods with smaller sizes. In all of the investigated scans, the two EXIRAD-HE collimators clearly offer better images than the VECTor collimator. Comparing between EXIRAD-HE-1 and EXIRAD-HE-2, in some scans, the higher-sensitivity collimator performs better, while in the other scans the lower-sensitivity design offers superior images. That is understandable since, with relatively low radioactivity, we would choose a high-sensitivity collimator to acquire more counts that benefit image quality, and if the radioactivity is already relatively high, a high-resolution collimator will offer a better image resolution.

Evaluation with digital derenzo phantoms
The achievable spatial resolutions, defined by the diameters of the smallest visible rods, are 0.5 mm, 0.6 mm, 0.5 mm, 0.6 mm, and 0.35 mm for 18 F, 64 Cu, 124 I, 213 Bi, and 131 I imaging, respectively. . Simulated images of the Derenzo phantoms with 18 F at several activity levels as well as with some other high-energy isotopes using the optimal collimators in table 5. Only the 131 I scan (rightmost column) was simulated with a smaller Derenzo phantom to visualise rods with smaller diameters. The images were displayed with the iteration and the 3D Gaussian filter size that maximises CNR. However, maximising CNR was visually suboptimal for 213 Bi and 131 I scans with EXIRAD-HE, so the iteration number and the filter size were chosen manually for these cases. In each column, the performance of the three collimators is compared, and the image that shows the clearest rods is enclosed in a square box. Slice thickness was 1.5 mm. Figures 8 and 9 demonstrate the performance of the new collimators in imaging a mouse knee joint containing 18 F-NaF, and a tumour xenograft containing 131 I-NaI, respectively. In both cases, EXIRAD-HE resolves more details than VECTor. The xenograft image obtained with the VECTor collimator shows a smaller necrotic core of the central tumour than the actual shape, an unclear separation between the central tumour and the smaller tumours, and the invisibility of the smallest tumour. With the EXIRAD-HE collimators, the whole reconstructed shape of the xenograft is very close to that of the true phantom, and each of the tumours can be recovered with great detail. EXIRAD-HE-1 performs better than EXIRAD-HE-2 in the knee joint scan, while EXIRAD-HE-2 yields an image with clearer shape of the tumours than EXIRAD-HE-1 does in the xenograft scan. These results agree with the NMSE as reported in figures 8 and 9's legends. In both cases, the 1 mm blurred images of the phantom offer significantly less detail than the images with EXIRAD-HE and VECTor.

Discussion
We have introduced two multi-pinhole collimators for ex vivo imaging of high-energy isotopes. Both collimators have shown, in various simulated scans, superior image resolution to the in vivo multi-pinhole SPECT/PET. The new collimators yielded a resolution of 0.5 mm, 0.6 mm, 0.5 mm, 0.6 mm, and 0.35 mm for 18 F, 64 Cu, 124 I, 213 Bi, and 131 I, respectively at realistic activity concentration levels. We believe that these resolution gains are important in many applications. For example, in a mouse scan with Fluorodeoxyglucose to detect the reduction in posterior cingulate activity-a sign for Alzheimer's disease, a resolution of 0.5 mm is on the verge of being possible to detect this effect, and a resolution of 0.75 mm or worse fails to detect significantly abnormal activity (Valla et al 2002). Besides, the 1 mm resolution images presented in figures 8 and 9 emphasise that sub-mm resolutions can reveal a lot of extra details.
These achievable resolutions with EXIRAD-HE for the PET isotopes 18 F and 64 Cu are probably already near the fundamental limit because of the positron range effect (Levin and Hoffman 1999). The average positron ranges of 18 F and 64 Cu in water are 0.64 mm and 0.56 mm, respectively (Cal-González et al 2013). This effect is not present if we utilise prompt gammas from non-pure positron emitters or single gammas from isotopes like 213 Bi or 131 I for imaging; therefore, EXIRAD-HE is most suitable for such isotopes.  (table 5) in comparison with the HE-UHR-M collimator in VECTor and the 1 mm blurred image of the phantom. The images are shown at the iteration number that gives the lowest NMSE. The optimal NMSE is 0.17, 0.18, and 0.25 for EXIRAD-HE-1, EXIRAD-HE-2, and VECTor, respectively. Slice thickness was 0.5 mm. Figure 9. Simulation of a mouse tumour xenograft phantom with the optimal collimators (table 5) in comparison with the HE-UHR-M collimator in VECTor and the 1 mm blurred image of the phantom. Images are shown at several iteration numbers, and the images with the lowest NMSE are enclosed in boxes. The optimal NMSE is 0.05, 0.03, and 0.12 for EXIRAD-HE-1, EXIRAD-HE-2, and VECTor, respectively. Slice thickness was 0.5 mm.
Besides, for sufficiently long half-life isotopes, the scan duration could be extended as the tissue sample is kept frozen over the course of the scan for good image quality.
Considering the manufacturing feasibility, the proposed EXIRAD-HE collimators only consist of round pinholes with no shielding outside as is present for the low-energy EXIRAD collimator (Nguyen et al 2019). Adding a trapezium-hole shielding, we would be able to place the pinholes of the same opening angle slightly tighter, which translates into more pinholes without the cost of projection overlapping. Even though each pinhole will be truncated, the overall system sensitivity is expected to slightly increase for the same system resolution. However, the manufacturing complexity and cost will increase significantly because of the large number of narrow trapezium holes in this shielding.
As seen in figures 7-9, each of the two proposed collimators has certain advantages, and the choice between them depends on the imaging situation. For relatively low activity levels, one may prefer the higher-sensitivity collimator for higher counts whereas when the count level is sufficient, the lower-sensitivity collimator would be selected for optimal image resolution.

Conclusion
This paper presented the design, optimisation, and evaluation of new multi-pinhole collimators for ex vivo 3D tomographic imaging of high-energy isotopes. We found that placing pinhole centres at a distance of 8 mm from the collimator inner surface yielded optimal image resolution for fixed system sensitivity over the CFOV. A wall thickness of 43 mm that keeps a reasonable weight (16 kg for the core) still maintains excellent image quality compared to a wall thickness of 60 mm (25 kg for the core). With the two optimal high-energy collimators, the system offers a resolution of 0.35 mm, 0.6 mm, 0.5 mm, 0.6 mm, and 0.5 mm when imaging 131 I, 213 Bi, 18 F, 64 Cu, and 124 I, respectively, contained in tissue samples.