### GE discovery NM 530c

The GE Discovery NM 530c [14] consists of 19 detector-pinhole units, and each detector has 32 × 32 individual detector elements in a 78.7 mm × 78.7 mm area. The detectors are arranged in five detector triplets alternating with four single detectors, and the whole detector package is curved around the torso of the patient, providing angular coverage of 180° with a 19-cm-diameter spherical FOV. The lateral detectors in each triplet, of which there are 10 in total, are tilted relative to the pinhole axis.

Clinical image reconstruction is performed using an iterative Bayesian (‘one-step-late’) algorithm [42, 43]. The reconstructed images have a matrix size of 70 × 70 × 50 with 4 mm × 4 mm × 4 mm voxels. It is possible to include an attenuation correction in clinical ML-EM reconstruction, but doing so requires a CT image to be acquired on a separate system, and consequently, the attenuation coefficient map must be co-registered with the SPECT volume prior to reconstruction. A SPECT/CT version of NM 530c, called NM 570c, is commercially available. Without accurate attenuation and scatter correction and a calibration coefficient, reconstructed data cannot be used for activity quantification. In this study, neither attenuation nor scatter correction was included in the NM 530c reconstruction.

### SIMIND ML-EM

The pinhole reconstruction method described in this work was fully incorporated into the SIMIND MC code and operates as a stand-alone program. Thus, the reconstruction program can take advantage of all future improvements and updates in the main SIMIND code.

Three separate files are used to define the system and SPECT acquisition. The first two are parts of the general SIMIND framework for pinhole SPECT simulation and describe (a) the source and detector characteristics and (b) the detector and pinhole positions and orientations. A third file, stored in interfile 3.0 format, provides study-specific information, such as the acquisition time and energy window settings.

The reconstruction starts with loading the measured projection data and an optional co-registered CT. A three-dimensional source volume with a uniform positive value is used as a first estimate. The SIMIND program then simulates projections scaled with the proper acquisition time of the measured projections. In this particular work, the projections were simulated for the NM 530c by including all of the physical effects modelled using SIMIND, such as non-homogeneous photon attenuation, scatter in the object, scatter and penetration in the collimator, interactions in the CZT material, and the subsequent charge transport (including electron- and hole-trapping and charge diffusion).

The description of the system design and detector geometry specifications was provided by GE Healthcare, Haifa, Israel.

#### Back-projector

As opposed to the MC-based forward-projector, the back-projector model includes several approximations. These are that (a) no attenuation or scatter occurs in the object, (b) the pinhole aperture opening is infinitely small without penetration effects (i.e. perfect collimation), and (c) the efficiency is independent of the incident angle. An analytical expression for the pinhole point spread function (PSF), including penetration through the pinhole edges, has been described elsewhere [43] but was not considered in this study due to its complexity in relation to the NM 530c geometry. Instead, we used a more approximate ray-tracing algorithm to derive the individual elements of *b*
_{
ij
}.

The probabilities *b*
_{
ij
} are derived on a voxel-by-voxel basis for an *N* × *N* × *N* source matrix with voxel side lengths *m*. A focal line originating at point **P** will intercept the plane of an individual detector at point **D** by passing through the infinitesimally small pinhole. The line from the source volume origin through the pinhole centre is referred to as the pinhole axis, and the direction of the pinhole axis is specified by a polar angle *θ* and an azimuthal angle *φ*. The coordinates of **D** are calculated from those of **P** using the detector-to-pinhole distance **H** and origin-to-pinhole distance **R**. For this purpose, the coordinate system is rotated so that the *z*
^{′}-axis becomes parallel to the pinhole axis. The new coordinates **P**
^{′} after rotation of the reference system about the *x*-axis and *y*-axis are given by

$$ \left[\begin{array}{c}{x}^{\hbox{'}}\\ {}{y}^{\hbox{'}}\\ {}{z}^{\hbox{'}}\end{array}\right]=\left[\begin{array}{ccc}\cos \varphi & \sin \varphi \sin \theta & \sin \varphi \cos \theta \\ {}0& \cos \theta & -\sin \theta \\ {}-\sin \varphi & \cos \varphi \sin \theta & \cos \varphi \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ {}y\\ {}z\end{array}\right], $$

(3)

The coordinates of **D** are calculated by mirroring and then scaling *x*
^{′} and *y*
^{′} by a magnification factor *M*:

$$ {x}_{\mathrm{D}}=-{Mx}^{\hbox{'}}, $$

(4)

$$ {y}_{\mathrm{D}}=-{My}^{\hbox{'}}, $$

(5)

$$ M=\frac{H}{R-{z}^{\hbox{'}}} $$

(6)

Due to the finite size of the source voxels, photons originating from a single voxel can contribute to several projection bins. This behaviour is modelled by adapting a scalable grid centred on the intercept point, and the width of the grid *G*
_{D} is calculated by scaling *m* by *M*:

$$ {G}_{\mathrm{D}}= mM. $$

(7)

The grid procedure is illustrated in Fig. 1, where the yellow point represents the detector intercept, with *N*
_{grid} × *N*
_{grid} black grid points distributed over the *G*
_{D} × *G*
_{D} area.

The number of points within the *G*
_{D} × *G*
_{D} grid is calculated using the heuristic equation,

$$ {N}_{\mathrm{grid}}=\left\lfloor 2+\frac{60}{1+{\left(R-{z}^{\hbox{'}}\right)}^{0.8}}+0.5\right\rfloor, $$

(8)

where ⌊⋅⌋ denotes truncation to the nearest lower integer. The total number of grid points (*N*
_{grid} × *N*
_{grid}) should be inversely proportional to the squared voxel-to-pinhole distance (*R* − *z*
^{′})^{2} to model the distance-dependent sensitivity of the pinhole. However, accurate modelling of this sensitivity would require a large number of grid points for voxels located close to the pinhole, which would result in long calculation times. Instead, Eq. 8 was adopted to reduce the back-projection computation time. The constant 2 in Eq. 8 is included to ensure that the total number of grid points is greater or equal to 2^{2} for all voxel positions, whereas the maximum number of grid points is never greater than 62^{2}. The constant 0.5 in Eq. 8 assures rounding to nearest integer. It should be noted that Eq. 8 does not account for the decreased sensitivity for oblique angles.

The probability *b*
_{
ij
} is finally obtained by normalizing the number of sample points in a pixel *n*
_{
ij
} to the total number of sample points on all projections for a given *j*, i.e. \( {\sum}_{k=0}^{K-1}{N}_{\mathrm{grid},j,k}^2 \) where the sum goes over the all *K* projections

$$ {b}_{ij}=\frac{n_{ij}}{\sum_{k=0}^{K-1}{N}_{\mathrm{grid},j,k}^2}. $$

(9)

#### Source estimate update

The new source estimate is the product of the current source estimate and correction factors calculated from the back-projected ratios, according to Eq. 2, and is stored as two datasets. One dataset is stored as floating-point single-precision values, with each voxel value equalling the activity in MBq in the volume represented by that voxel, while the other dataset is stored as integers, where the sum of the integer values is the predefined number of histories for the MC simulation. The SIMIND program is then restarted using the latter of the two source files as the input. In this way, the SIMIND program continuously maintains an accurate relation between the number of histories generating counts in the projection images and the underlying activity in a particular voxel in the source volume. Hence, no additional calibration factor is needed to obtain absolute values of the activity in each reconstructed source voxel. The procedure is repeated until the predefined number of iterations has been conducted.

### Simulations and measurements

For MCR, measured projections were reconstructed into a 70 × 70 × 70 matrix with 4-mm cubic voxels. This voxel size is identical to that in the reconstructions produced by NM 530c systems (70 × 70 × 50 and 4 mm). For qualitative comparison, the measured projections were also reconstructed using the clinical reconstruction algorithm, which involves 40 iterations of a regularized (*α* = 0.51 and *β* = 0.3) one-step-late Green ML-EM algorithm [44] and post-filtered using a Butterworth filter (cut-off: 0.37, order: 7).

#### Simulated data

To ensure consistency in the reconstruction algorithm as such, SPECT projections were simulated using SIMIND and then reconstructed using SIMIND MCR, meaning that the forward-projector utilized in the reconstruction was identical to the projector employed to create the projections. The purpose of reconstructing simulated data is to isolate any causes of inaccuracy to either the forward projector or back projector. If the estimated activity from a simulated reconstructed sphere is close to the reference value it means that the back projector is properly normalized. On the other hand, if the relative difference is off by a factor similar to that of measured spheres, this would indicate that the back-projector is not correctly normalized since the forward projector in this case can be considered “perfect”. In this study, a 9-mL sphere was centred at seven different positions, namely, at the reconstruction volume centre and at six positions with 2 cm positive or negative translations along the *x*-, *y*-, and *z*-axes. The activity was defined as 1 MBq, and simulated projections were calculated for each position with the sphere both in a non-attenuating medium, subsequently called ‘air’, and in a cylindrical water phantom (21.4 cm in the axial direction, 11.0 cm radius) with a density of 1 g/cm^{3}. Sufficiently many histories were simulated to attain essentially noise-free projections. No Poisson distributed noise was added to the projections.

Each projection set was reconstructed 10 times to determine the variance in the recovered total activity due to the finite number of photon histories in the SIMIND forward-projection step during MCR. The reconstructed sphere activity was determined using a spherical volume of interest (VOI) with twice the diameter of the sphere used in the simulations, and from this VOI, the relative differences between the estimated activities and those defined in the simulations were calculated.

#### Point source measurements

The correct placement of the activity in the reconstructed images was investigated using a point-like source, in the form of a 600-μm-diameter ion resin bead soaked in ^{99m}Tc-pertechnetate and placed on the tip of a plastic rod. The rod was in turn mounted on a motorized 3D translational table, and projections of 125 source positions, defined as a 5 × 5 × 5 cubic grid with 20 mm spaces between the points, were obtained.

The acquired projections were reconstructed using 5, 10, 15, and 20 iterations, and the source volume was filtered using a 3D Gaussian filter with a standard deviation equal to the side length of a voxel in a 7 × 7 × 7 kernel. To investigate the spatial integrity of the reconstruction, we (a) determined the voxel coordinates for the maximal value in each reconstructed volume for each source position, (b) calculated the mean coordinate position of a plane along the coordinate system axis orthogonal to that plane as the mean of the 25 voxel coordinates (repeated for all planes orthogonal to one of the coordinate system axes), and (c) determined the distance between adjacent planes.

#### Sphere phantom

The quantitative accuracy of the MCR was investigated using six spheres with volumes ranging from 0.5 to 16 mL (NEMA standard) that were filled with ^{99m}Tc and each of which had the same activity concentration. The spheres were mounted in an elliptical torso phantom and measured with and without water present in the torso, the activity concentrations being 4.1 and 3.8 MBq/mL, respectively. The sphere-mounting disk was designed so that the spheres could be mounted in the torso phantom using the original cardiac mounting bracket. The angulation of the mounting disk was such that a plane through the centre of the spheres was the same as the cardiac short-axis plane.

The MC simulations of photon interactions in SIMIND rely on a combination of tabulated material and energy-specific mass-attenuation coefficients and a density distribution. A CT study of the phantom was performed, and the results were co-registered to the clinically reconstructed SPECT images using the GE Xeleris™ workstation. The results were then exported by GE Xeleris™ as attenuation images and rescaled to density distributions.

In this study, we defined the activity recovery coefficient (ARC) as the ratio between the estimated and reference sphere activities. The ARC indicates how well the MCR recovers the activity within a sphere and was determined for each sphere using

$$ \mathrm{ARC}=\frac{A_E}{A_R}, $$

(10)

where *A*
_{
E
} and *A*
_{
R
} are the estimated and reference activities, respectively. Two sets of spherical VOIs were used. In one set, the VOI diameter was equal to the physical diameter of the sphere, and in the second set, an additional 8 mm (the size of two voxels) was added to the diameter of each VOI to collect counts otherwise lost due to the spatial resolution. The coefficients obtained from the second type of VOIs are called total activity recovery coefficients (TARCs). Each of the two types of VOIs was generated by manually defining the sphere centre from the images and then producing a spherical VOI mask based on the sphere radius.

#### Cardiac phantom

Measured projections of a myocardial phantom (Data Spectrum Corp, Hillsborough, NC, USA) were used to enable quantitative MCR evaluation and to compare the MCR to the NM 530c reconstruction. The 120-mL myocardium cavity was filled with a ^{99m}Tc solution and imaged twice. For the first acquisition, the torso was filled with water, and the activity in the myocardium at the start of acquisition was 25.2 MBq. For the second acquisition, there was no water in the torso cavity except in the heart insert, and the activity at the start of acquisition was 21.1 MBq. The ventricle cavity was water-filled during both measurements. The thickness of the myocardium compartment was 10.3 mm, and the width of the ventricle compartment was 38.0 mm.

Reconstructed images (40 iterations) were re-oriented and overlaid on the CT image to enable visual confirmation of the spatial registration between the two modalities. Line profiles calculated as the average of seven rows over the myocardium was plotted for different iterations.

The MCR reconstruction was compared to the clinical NM 530c reconstruction of these measurements by plotting a line profile (an average of seven rows) in a long-axis slice. Prior to this comparison, the MCR images were filtered using a Butterworth low-pass filter (cut-off frequency 0.5 cm^{−1}, order 2).