Monte Carlo-based quantitative pinhole SPECT reconstruction using a ray-tracing back-projector

An attractive, but seldom clinically used, aspect of single-photon emission computed tomography (SPECT) is that it can be utilized for activity quantification in vivo, with applications in, e.g. radionuclide therapy dosimetry [1, 2] and myocardial perfusion studies [3]. The accuracy of SPECT-based activity quantification depends on the accuracy with which radiation transport in the patient and the detector are modelled in the tomographic reconstruction algorithm. The level of accuracy achievable varies with the complexity of the emission photon spectrum of the radionuclide used. Typical photon energies in SPECT imaging range from 50 to 400 keV. Within this energy range, there is a considerable probability that an emitted photon will interact within the patient before reaching the detector, which is difficult to model accurately. Furthermore, the finite hole diameter of the collimator causes substantial degradation of the spatial resolution of the image. High-energy photons, which have non-negligible probabilities of penetrating the collimator septa, further degrade the image quality. For imaging using radionuclides that emit photons of multiple energies, high-energy photons can be down-scattered into lower energy windows. Hence, all of these phenomena must be considered to achieve reliable activity quantification. The accuracy is further limited by image noise, uncertainties associated with system calibration, and patient motion. The absolute quantitative accuracy of gamma camera SPECT quantification has typically been reported to be within 20% for a range of radionuclides [2, 4].

The collimators typically employed in clinical SPECT are parallel-hole collimators, which is also the case for clinical quantitative SPECT. Pinhole collimators also have great potential. Historically, pinhole imaging has been focused on either superficial small organs (e.g. the thyroid gland) or pre-clinical imaging of small animals (e.g. rats and mice). However, pinhole collimation offers several advantages over parallel-hole collimation, such as increased system sensitivity and superior spatial resolution, but at the cost of a smaller field of view (FOV) [5–7]. During the last two decades, renewed interest in pinhole imaging techniques has developed in the nuclear medicine community, as well as in their clinical applications, following extensive hardware developments driven primarily by preclinical research [8, 9]. These developments include new detectors and detector materials [9], advanced pinhole and multi-pinhole geometries [10, 11], and the implementation of iterative reconstruction methods [12, 13]. In the clinical realm, recent pinhole camera-system research has focused on myocardial perfusion imaging. In the commercially available system GE Discovery NM 530c (NM 530c) [14], a pinhole collimator and cadmium zinc telluride (CZT) semiconductor detector are employed. These compact detectors have made it possible to mount multiple detector-pinhole units in a static geometry to acquire projections simultaneously with an angular coverage sufficient for tomographic reconstruction. The NM 530c camera has a sensitivity three times higher than those of conventional dual-headed SPECT cameras equipped with parallel-hole collimators and NaI crystals [14]. However, the small FOV, a spherical volume 19 cm in diameter, limits its application to specific organs of interest. Since all of the projections are acquired simultaneously, NM 530c enables dynamic myocardial blood flow and perfusion studies to be conducted [15, 16], which are otherwise only feasible using positron emission tomography (PET) [17]. An accurate quantitative reconstruction method is a prerequisite for model-based kinetic analysis.

Quantitative SPECT requires attenuation and scatter correction. Typically, SPECT attenuation correction is performed using a co-registered attenuation map based on computed tomography (CT) images. For scatter correction, several correction methods exist [18]. Approximate window-based methods are often employed for gamma cameras. For CZT detectors, the presence of un-scattered photons with energies below the primary energy window will introduce bias when utilizing window-based methods [19]. This bias has been addressed by developing new correction methods for CZT-based systems, including modified window-based methods [20] and iterative deconvolution approaches [21].

In terms of radiation transport modelling, one of the most trusted strategies is to use the Monte Carlo (MC) method, since it enables inclusion of the details of the measurement geometry and physics governing photon interactions. The potential benefits of MC-based SPECT reconstruction was discussed already in the 1980s by Floyd et al. [25] but the procedure was limited to reconstructing a single slice. Even though the computational burden of MC simulations still poses a challenge for full Monte Carlo-based reconstruction (MCR), the continuous improvements in computer hardware and the possibility of parallel processing have made this method increasingly interesting for practical use.

An MC simulation can be included in an ML-EM algorithm in a variety of ways. For example, individual components, such as the scatter contribution, can be modelled using MC methods while deriving the remaining components of the projection model by other means [26–29]. Alternatively, the forward-projection can be fully MC-based, possibly with some approximations to increase the computational efficiency [30, 31], or an MC-based pre-calculated system matrix can be utilized in both the forward projection and the back projection [32–35].

One MC program that is optimized for the simulation of gamma cameras is SIMIND [36]. In the SIMIND code, several variance reduction techniques are implemented to reduce the simulation time, as described in [37]. Recently, a model of CZT photon detection was incorporated into the SIMIND code [38]. In short, the CZT charge collection following a photon interaction is modelled using the Hecht equation [39] and lateral charge diffuses according to a distance-dependent Gaussian distribution sampling. The CZT model in the SIMIND code has been validated for the NM 530c system based on the known geometry of the system [38]. Furthermore, the SIMIND pinhole modelling has been verified using analytical expressions and published data [40].

While it is theoretically possible to pre-compute the system matrix for a given (patient-specific) geometry, the storage requirements increase dramatically with the image and projection matrix sizes [33]. The typical mode of operation is instead to compute forward- and back-projections during the reconstruction process without reference to a pre-computed system matrix. A problem that is evident in MCR, due to the nature of the simulation procedure, is then that the MC program is only directly applicable as a forward-projector, and its use as a back-projector would be computationally cumbersome. Hence, the back-projector often must be derived in a more simplistic fashion.

where b _ij denotes the elements of the system matrix used in the back-projection.

In this study, we developed and extended the functionality of SIMIND to include tomographic pinhole reconstruction. Our aims were to take advantage of the accurate forward-projection modelling provided by a full MC simulation with the SIMIND program and to combine it with an approximate back-projector, thereby creating an integrated program for MC-based pinhole SPECT reconstruction. We then evaluated the reconstruction method with respect to its quantitative accuracy using phantom measurements. Qualitative comparison with the clinical reconstruction program of the camera was also performed.

In this paper, we have presented a pinhole reconstruction method that is entirely embedded in the well-known SIMIND MC code. The method has been applied for a dedicated clinical myocardial SPECT system, based on CZT technology, and we have evaluated its usefulness for activity quantification.

Generally, the TARCs were slightly overestimated for all of the spheres, but the estimated sphere activities typically remained within approximately 10% of the reference value (defined by the syringe activity measured in an activity meter and the sphere volumes). A noteworthy exception was the 20% overestimation of the TARC in the 0.5-mL sphere in the empty torso. The trend of total activity overestimation is not evident for the cardiac phantom in Table 3. However, the total estimated myocardial activities in Table 3 depend on the VOI definition used. The VOI used to estimate the myocardiac activity was delineated from a density map and included all of the cardiac phantom voxels and any potential signal spill-out. Increasing the VOI even further caused activity overestimation, especially for the water-filled torso, due to the inclusion of more background voxels (data not shown). The fact that the problem was more pronounced for water suggests that the scatter compensation in the reconstruction was not complete, i.e. that the model underestimated the amount of scatter in the projections. For large VOIs (dilation with a 9 × 9 × 9 voxel mask, i.e. a considerable expansion of the volume), the relative errors were up to 5% for the water-filled torso and up to 2% for the empty torso. The relative errors achieved with MCR are comparable to those previously published. Pourmoghaddas et al. reported relative errors of 5% ± 3% [47] when correcting for attenuation and scatter [20] for a similar measurement setup.

It should be noted that the MCR activity estimates does not depend on an explicit calibration of the system. Instead, the relationship between the projection counts and activity estimates is a result of the SIMIND camera model only. Hence, the estimated activities are completely independent of a separate activity meter measurements, as well as any errors associated with such measurements. However, this finding indicates that the accuracy of MCR-estimated activity concentrations is highly dependent on the accuracy of the camera model, not only in a relative sense, but also in absolute terms. Examples of sources of uncertainties are the cross sections used in the simulations, both with respect to the mass attenuation coefficients employed for various materials and the estimated density map obtained using CT. Apart from the uncertainties in the camera model, the reference activity is also associated with an uncertainty that would manifest itself as systematic deviations of the TARC and ARC. The CZT model used in SIMIND has previously been validated for sources in air [38]. In order to validate the handling of scatter by SIMIND for CZT, we compared measured and simulated spectra for the sphere geometry described in the “Sphere phantom” section. The simulated and measured spectra are shown in Fig. 11. Without any torso content, the two spectra agree well down to 60 keV. The same conclusion is drawn for the case of the water-filled torso despite a slight underestimation of the number of counts in the photopeak. Note that the spectra are absolute (number of counts) and not normalized to their maximum values. From the good agreement, we conclude that SIMIND models photon scattering well.

As shown in Table 1, which presents the MCR results for projections simulated with SIMIND, the total activities for the spheres in the empty and water-filled torsos agree well. So, if the forward-projector used in the reconstruction exactly matched the generation of the measured projections, the errors are substantially reduced. Hence, we believe that the majority of the errors in the measured results originated from model uncertainties in the forward-projector and reference values rather than the approximations used in the back-projector model.

As opposed to the TARCs, the ARCs exhibit volume-dependent underestimation, as demonstrated in Fig. 4. Generally, a smaller sphere means lower recovery. The exception to this rule, the 2-mL sphere in water, can probably be explained by the coarse discretization of the source volume, which made it difficult to delineate the VOIs accurately. Additional errors would occur when the VOI centre point does not coincide with the physical centre of the sphere. The fact that compensation for spatial resolution in the iterative reconstruction did not yield perfect recovery is in line with general resolution compensation experiences [48]. Figure 4 illustrates that the activity concentration recovery continued until the user-selected end-point of 100 iterations for all sphere sizes, albeit slowly for the larger spheres at high iteration numbers. This finding suggests that it would be possible to increase the recovery even further, but at the expense of increasing the computational burden. The same trend of improved recovery and image contrast with increasing number of iterations, even up to 100 iterations, can be seen in the line profiles in Figs. 8 and 9, where the central values approach activity concentrations close to the true activity concentration in the myocardium. Hence, to utilize fully the potential of an accurate forward-projector, such as an MC-based one, more iterations than are typically used clinically may be necessary.

The main drawback of MCR is the long reconstruction time. The overall reconstruction time scaled with the number of iterations and varied with the amount of scattering material, and typical reconstruction times for a heart phantom (50 iterations) were roughly 10 and 20 h for empty and water-filled torsos, respectively. The majority of the reconstruction time is spent in the MC forward-projection simulation step, and any attempts to speed up the back-projection step would have very little effect on the overall reconstruction time. Several variance reduction techniques have been implemented in SIMIND code to improve the simulation efficiency [37]. A specific variance reduction technique, applicable to pinhole simulations, is that photons exiting the source/phantom volume are directed into a solid angle that, for each separate emission point, covers a circular area around the centre of the pinhole having a diameter equal to four times the pinhole diameter. This technique speeds up simulations significantly compared to those in which the emission is isotropic, while still modelling the penetration and scatter effects at the pin-hole edges.

Currently, research is in progress on potential methods of decreasing the calculation time further using parallel-processing message-passing interfaces in the forward-projection, where the simulation can be performed in parallel on multiple central processing units (CPUs). The reduction in simulation time scales roughly with the number of cores used. Since the reconstruction times reported here can be achieved using a single CPU, we believe that this algorithm can be applied in cases in which expedience is not of primary importance, such as for research and post-study evaluations. As the cost of hardware continues to decrease and with the possibility of parallelization, the time aspect is also expected to decrease in importance in the years to come. The reconstruction time could also be reduced by implementing the OS-EM algorithm. In this work, however, we did not utilize OS-EM due to the difficulties in defining subsets from 19 projections.

From an image-quality perspective, the MCR and NM 530c reconstruction results differ somewhat, as shown in Fig. 5. The spheres reconstructed using NM 530c are more uniform due to the noise regularization applied during the NM 530c reconstruction and the Butterworth post-filtering, but are similar in shape to those obtained using MCR. The difference in noise characteristics between MCR and NM 530c can be counteracted by post-filtering the MCR images with a Butterworth filter. For the largest sphere, a ringing artefact is visible in the NM 530c reconstruction but not in the MCR. The relative sphere intensities in the water-filled and empty torsos are similar in the MCR since photon attenuation is included in the forward-projector model, whereas the absence of attenuation compensation in the NM 530c reconstruction led to non-homogeneous sphere intensity suppression. The clinical reconstruction software does allow for attenuation correction, but it is seldom used clinically in our department. Since there are no means of correcting scatter in the clinical program, thereby making a “fair” comparison impossible, we used the clinically employed settings (i.e. no attenuation correction). The myocardium phantom profiles in Fig. 10 for MCR and NM 530c are similar, but the presence of noise regularization produced a more uniform myocardial distribution for NM 530c, as is evident in the individual slices. However, noise regularization could in principle be applied to MCR as well, but its implementation was beyond the scope of this study. Without attenuation correction in the NM 530c reconstruction there is as a negative gradient in the reconstructed activity towards the myocardiac base when the torso is water filled. For MCR, which includes attenuation in the forward-projector, there is no gradient.

One area of potential further development is improvement of the model of the pinhole in the back-projector by accounting for its finite size. This objective could be achieved by analytical determination of the pinhole PSF [43, 47], which would still be an approximation with regard to the model in the forward-projector but would be an improvement over the current ray-tracing model. As an example, the signal spreads due to the finite voxel and pinhole sizes for a point source located 15 cm from the pinhole are 0.14 and 0.68 cm respectively, so for this specific position the total spread is underestimated when ignoring the finite pinhole aperture. Incomplete modelling of the spatially variant PSF affects the noise characteristics in the reconstructed image. In this paper, we determined total activity in VOIs and the cardiac phantom mean activity profiles. In that regard, our results display a good accuracy, but is should be noted that both these measures are fairly resistant to noise. An alternative to calculating the system matrix during reconstruction is to pre-calculate the entire system matrix [33]. Referencing an MC-calculated system matrix in both the forward- and back-projections is optimal from an image quality standpoint, but with the downside of requiring storage of the full or a compressed system matrix.

In general, the full potential of MC-based image reconstruction is not necessarily best demonstrated with a radionuclide-like ^99mTc, but rather with radionuclides with more complex emission spectra, such as ¹²³I and ¹³¹I. These radionuclides emit several high-energy photons during their decay that, despite low abundance, can scatter in and penetrate through the collimator edges as well as penetrate though the detector and scatter in the material behind the detector and back into the detector in a way that is not easy to predict using analytical models. Realistic projections can, however, be obtained with full modelling of the radiation transport in the object, collimator, and detector.

We developed an MC-based reconstruction method for pinhole SPECT and evaluated it using phantom measurements. The combination of an MC-based forward-projector and a simplified analytical ray-tracing back-projector produced quantitative images of acceptable image quality. No explicit calibration is necessary in this method since the forward-projector model maintains a consistent relationship between count and activity.