Comparison of basis functions for 3D PET reconstruction using a Monte Carlo system matrix

Monte Carlo Simulation and Reconstruction: Assessment of Myocardial Perfusion Imaging of Tracer Dynamics With Cardiac Motion Due to Deformation and Respiration Using Gamma Camera With Continuous Acquisition

Purpose

Myocardial perfusion imaging (MPI) with single photon emission computed tomography (SPECT) is routinely used for stress testing in nuclear medicine. Recently, our group extended its potential going from 3D visual qualitative image analysis to 4D spatiotemporal reconstruction of dynamically acquired data to capture the time variation of the radiotracer concentration and the estimated myocardial blood flow (MBF) and coronary flow reserve (CFR). However, the quality of reconstructed image is compromised due to cardiac deformation and respiration. The work presented here develops an algorithm that reconstructs the dynamic sequence of separate respiratory and cardiac phases and evaluates the algorithm with data simulated with a Monte Carlo simulation for the continuous image acquisition and processing with a slowly rotating SPECT camera.

Methods

A clinically realistic Monte Carlo (MC) simulation is developed using the 4D Extended Cardiac Torso (XCAT) digital phantom with respiratory and cardiac motion to model continuous data acquisition of dynamic cardiac SPECT with slowly rotating gamma cameras by incorporating deformation and displacement of the myocardium due to cardiac and respiratory motion. We extended our previously developed 4D maximum-likelihood expectation-maximization (MLEM) reconstruction algorithm for a data set binned from a continuous list mode (LM) simulation with cardiac and respiratory information. Our spatiotemporal image reconstruction uses splines to explicitly model the temporal change of the tracer for each cardiac and respiratory gate that delineates the myocardial spatial position as the tracer washes in and out. Unlike in a fully list-mode data acquisition and reconstruction the accumulated photons are binned over a specific but very short time interval corresponding to each cardiac and respiratory gate. Reconstruction results are presented showing the dynamics of the tracer in the myocardium as it continuously deforms. These results are then compared with the conventional 4D spatiotemporal reconstruction method that models only the temporal changes of the tracer activity. Mean Stabilized Activity (MSA), signal to noise ratio (SNR) and Bias for the myocardium activities for three different target-to-background ratios (TBRs) are evaluated. Dynamic quantitative indices such as wash-in (K₁) and wash-out (k₂) rates at each gate were also estimated.

Results

The MSA and SNR are higher with higher TBRs while biases were improved with higher TBRs to less than 10%. The correlation between exhalation-inhalation sequence with the ground truth during respiratory cycle was excellent. Our reconstruction method showed better resolved myocardial walls during diastole to systole as compared to the ungated 4D image. Estimated values of K₁ and k₂ were also consistent with the ground truth.

Conclusion

The continuous image acquisition for dynamic scan using conventional two-head gamma cameras can provide valuable information for MPI. Our study demonstrated the viability of using a continuous image acquisition method on a widely used clinical two-head SPECT system. Our reconstruction method showed better resolved myocardial walls during diastole to systole as compared to the ungated 4D image. Precise implementation of reconstruction algorithms, better segmentation techniques by generating images of different tissue types and background activity would improve the feasibility of the method in real clinical environment.

High-throughput, accurate Monte Carlo simulation on CPU hardware for PET applications

Monte Carlo simulations (MCS) represent a fundamental approach to modelling the photon interactions in Positron Emission Tomography (PET). A variety of PET-dedicated MCS tools are available to assist and improve PET imaging applications. Of these, GATE has evolved into one of the most popular software for PET MCS because of its accuracy and flexibility. However, simulations are extremely time-consuming. The use of graphics processing units (GPU) has been proposed as a solution to this, with reported acceleration factors about 400-800. These factors refer to GATE benchmarks performed on a single CPU core. Consequently, CPU-based MCS can also be easily accelerated by one order of magnitude or beyond when exploiting multi-threading on powerful CPUs. Thus, CPU-based implementations become competitive when further optimisations can be achieved. In this context, we have developed a novel, CPU-based software called the PET Physics Simulator (PPS), which combines several efficient methods to significantly boost the performance. PPS flexibly applies GEANT4 cross-sections as a pre-calculated database, thus obtaining results equivalent to GATE. This is demonstrated for an elaborated PET scanner with 3-layer block detectors. All code optimisations yield an acceleration factor of  20 (single core). Multi-threading on a high-end CPU workstation (96 cores) further accelerates the PPS by a factor of 80. This results in a total speed-up factor of 1600, which outperforms comparable GPU-based MCS by a factor of  2. Optionally, the proposed method of coincidence multiplexing can further enhance the throughput by an additonal factor of 15. The combination of all optimisations corresponds to an acceleration factor of  24000. In this way, the PPS can simulate complex PET detector systems with an effective throughput of photon pairs in less than 10 milliseconds.

Real-Time 3D PET Image with Pseudoinverse Reconstruction

Real-time positron emission tomography (PET) may provide information from first-shot images, enable PET-guided biopsies, and allow awake animal studies. Fully-3D iterative reconstructions yield the best images in PET, but they are too slow for real-time imaging. Analytical methods such as Fourier back projection (FBP) are very fast, but yield images of poor quality with artifacts due to noise or data incompleteness. In this work, an image reconstruction based on the pseudoinverse of the system response matrix (SRM) is presented. w. To implement the pseudoinverse method, the reconstruction problem is separated into two stages. First, the axial part of the SRM is pseudo-inverted (PINV) to rebin the 3D data into 2D datasets. Then, the resulting 2D slices can be reconstructed with analytical methods or by applying the pseudoinverse algorithm again. The proposed two-step PINV reconstruction yielded good-quality images at a rate of several frames per second, compatible with real time applications. Furthermore, extremely fast direct PINV reconstruction of projections of the 3D image collapsed along specific directions can be implemented.

Real-Time 3D PET Image with Pseudoinverse Reconstruction

Real-time positron emission tomography (PET) may provide information from first-shot images, enable PET-guided biopsies, and allow awake animal studies. Fully-3D iterative reconstructions yield the best images in PET, but they are too slow for real-time imaging. Analytical methods such as Fourier back projection (FBP) are very fast, but yield images of poor quality with artifacts due to noise or data incompleteness. In this work, an image reconstruction based on the pseudoinverse of the system response matrix (SRM) is presented. w. To implement the pseudoinverse method, the reconstruction problem is separated into two stages. First, the axial part of the SRM is pseudo-inverted (PINV) to rebin the 3D data into 2D datasets. Then, the resulting 2D slices can be reconstructed with analytical methods or by applying the pseudoinverse algorithm again. The proposed two-step PINV reconstruction yielded good-quality images at a rate of several frames per second, compatible with real time applications. Furthermore, extremely fast direct PINV reconstruction of projections of the 3D image collapsed along specific directions can be implemented.

An automatic algorithm to exploit the symmetries of the system response matrix in positron emission tomography iterative reconstruction

Positron emission tomography (PET) iterative 3D reconstruction is a very computational demanding task. One of the main issues of the iterative reconstruction concerns the management of the system response matrix (SRM). The SRM models the relationship between the projection and the voxel space and its memory footprint can easily exceed hundreds of GB. Moreover, in order to make the reconstruction fast enough not to hinder its practical application, the SRM must be stored in the random access memory of the workstation used for the reconstruction. This issue is normally solved by implementing efficient storage schemes and by reducing the number of redundant patterns in the SRM through symmetries. However, finding a sufficient number of symmetries is often non-trivial and is typically performed using dedicated solutions that cannot be exported to different detectors and geometries. In this paper, an automatic approach to reduce the memory footprint of a pre-computed SRM is described. The proposed approach was named symmetry search algorithm (SSA) and consists in an algorithm that searches for some of the redundant patterns present in the SRM, leading to its lossy compression. This approach was built to detect translations, reflections and coordinates swap in voxel space. Therefore, it is particularly well suited for those scanners where some of the rotational symmetries are broken, e.g. small animal scanner where the modules are arranged in a polygonal ring made of few elements, and dual head planar PET systems. In order to validate this approach, the SSA is applied to the SRM of a preclinical scanner (the IRIS PET/CT). The data acquired by the scanner were reconstructed with a dedicated maximum likelihood estimation maximization algorithm with both the uncompressed and the compressed SRMs. The results achieved show that the information lost due to the SSA compression is negligible. Compression factors up to 52 when using the SSA together with manually inserted symmetries and up to 204 when using the SSA alone, can be obtained for the IRIS SRM. These results come without significant differences in the values and in the main quality metrics of the reconstructed images, i.e. spatial resolution and noise. Although the compression factors depend on the system considered, the SSA is applicable to any SRM and therefore it can be considered a general tool to reduce the footprint of a pre-computed SRM.

Advances in PET/MR instrumentation and image reconstruction

The combination of positron emission tomography (PET) and MRI has attracted the attention of researchers in the past approximately 20 years in small-animal imaging and more recently in clinical research. The combination of PET/MRI allows researchers to explore clinical and research questions in a wide number of fields, some of which are briefly mentioned here. An important number of groups have developed different concepts to tackle the problems that PET instrumentation poses to the exposition of electromagnetic fields. We have described most of these research developments in preclinical and clinical experiments, including the few commercial scanners available. From the software perspective, an important number of algorithms have been developed to address the attenuation correction issue and to exploit the possibility that MRI provides for motion correction and quantitative image reconstruction, especially parametric modelling of radiopharmaceutical kinetics. In this work, we give an overview of some exemplar applications of simultaneous PET/MRI, together with technological hardware and software developments.

PET performance evaluation of MADPET4: a small animal PET insert for a 7 T MRI scanner

MADPET4 is the first small animal PET insert with two layers of individually read out crystals in combination with silicon photomultiplier technology. It has a novel detector arrangement, in which all crystals face the center of field of view transaxially. In this work, the PET performance of MADPET4 was evaluated and compared to other preclinical PET scanners using the NEMA NU 4 measurements, followed by imaging a mouse-size hot-rod resolution phantom and two in vivo simultaneous PET/MRI scans in a 7 T MRI scanner. The insert had a peak sensitivity of 0.49%, using an energy threshold of 350 keV. A uniform transaxial resolution was obtained up to 15 mm radial offset from the axial center, using filtered back-projection with single-slice rebinning. The measured average radial and tangential resolutions (FWHM) were 1.38 mm and 1.39 mm, respectively. The 1.2 mm rods were separable in the hot-rod phantom using an iterative image reconstruction algorithm. The scatter fraction was 7.3% and peak noise equivalent count rate was 15.5 kcps at 65.1 MBq of activity. The FDG uptake in a mouse heart and brain were visible in the two in vivo simultaneous PET/MRI scans without applying image corrections. In conclusion, the insert demonstrated a good overall performance and can be used for small animal multi-modal research applications.

Development of computational small animal models and their applications in preclinical imaging and therapy research

The development of multimodality preclinical imaging techniques and the rapid growth of realistic computer simulation tools have promoted the construction and application of computational laboratory animal models in preclinical research. Since the early 1990s, over 120 realistic computational animal models have been reported in the literature and used as surrogates to characterize the anatomy of actual animals for the simulation of preclinical studies involving the use of bioluminescence tomography, fluorescence molecular tomography, positron emission tomography, single-photon emission computed tomography, microcomputed tomography, magnetic resonance imaging, and optical imaging. Other applications include electromagnetic field simulation, ionizing and nonionizing radiation dosimetry, and the development and evaluation of new methodologies for multimodality image coregistration, segmentation, and reconstruction of small animal images. This paper provides a comprehensive review of the history and fundamental technologies used for the development of computational small animal models with a particular focus on their application in preclinical imaging as well as nonionizing and ionizing radiation dosimetry calculations. An overview of the overall process involved in the design of these models, including the fundamental elements used for the construction of different types of computational models, the identification of original anatomical data, the simulation tools used for solving various computational problems, and the applications of computational animal models in preclinical research. The authors also analyze the characteristics of categories of computational models (stylized, voxel-based, and boundary representation) and discuss the technical challenges faced at the present time as well as research needs in the future.

List-mode image reconstruction for positron emission tomography using tetrahedral voxels

Image space decomposition based on tetrahedral voxels are interesting candidates for use in emission tomography. Tetrahedral voxels provide many of the advantages of point clouds with irregular spacing, such as being intrinsically multi-resolution, yet they also serve as a volumetric partition of the image space and so are comparable to more standard cubic voxels. Additionally, non-rigid displacement fields can be applied to the tetrahedral mesh in a straight-forward manner. So far studies incorporating tetrahedral decomposition of the image space have concentrated on pre-calculated, node-based, system matrix elements which reduces the flexibility of the tetrahedral approach and the capacity to accurately define regions of interest. Here, a list-mode on-the-fly calculation of the system matrix elements is described using a tetrahedral decomposition of the image space and volumetric elements-voxels. The algorithm is demonstrated in the context of awake animal PET which may require both rigid and non-rigid motion compensation, as well as quantification within small regions of the brain. This approach allows accurate, event based, motion compensation including non-rigid deformations.

Image reconstruction in higher dimensions: myocardial perfusion imaging of tracer dynamics with cardiac motion due to deformation and respiration

Myocardial perfusion imaging (MPI) using slow rotating large field of view cameras requires spatiotemporal reconstruction of dynamically acquired data to capture the time variation of the radiotracer concentration. In vivo, MPI contains additional degrees of freedom involving unavoidable motion of the heart due to quasiperiodic beating and the effects of respiration, which can severely degrade the quality of the images. This work develops a technique for a single photon emission computed tomography (SPECT) that reconstructs the distribution of the radiotracer concentration in the myocardium using a tensor product of different sets of basis functions that approximately describe the spatiotemporal variation of the radiotracer concentration and the motion of the heart. In this study the temporal B-spline basis functions are chosen to reflect the dynamics of the radiotracer, while the intrinsic deformation and the extrinsic motion of the heart are described by a product of a discrete set of Gaussian basis functions. Reconstruction results are presented showing the dynamics of the tracer in the myocardium as it deforms due to cardiac beating, and is displaced due to respiratory motion. These results are compared with the conventional 4D-spatiotemporal reconstruction method that models only the temporal changes of the tracer activity. The higher dimensional reconstruction method proposed here improves bias, yet the signal-to-noise ratio (SNR) decreases slightly due to redistribution of the counts over the cardiac-respiratory gates. Additionally, there is a trade-off between the number of gates and the number of projections per gate to achieve high contrast images.

Blob parameter selection for image representation

A technique for optimizing parameters for image representation using blob basis functions is presented and demonstrated. The exact choice of the basis functions significantly influences the quality of the image representation. It has been previously established that using spherically symmetric volume elements (blobs) as basis functions, instead of the more traditional voxels, yields superior representations of real objects, provided that the parameters that occur in the definition of the family of blobs are appropriately tuned. The technique presented in this paper makes use of an extra degree of freedom, which has been previously ignored, in the blob parameter space. The efficacy of the resulting parameters is illustrated.

Assessment of a fully 3D Monte Carlo reconstruction method for preclinical PET with iodine-124

Iodine-124 is a radionuclide well suited to the labeling of intact monoclonal antibodies. Yet, accurate quantification in preclinical imaging with I-124 is challenging due to the large positron range and a complex decay scheme including high-energy gammas. The aim of this work was to assess the quantitative performance of a fully 3D Monte Carlo (MC) reconstruction for preclinical I-124 PET. The high-resolution small animal PET Inveon (Siemens) was simulated using GATE 6.1. Three system matrices (SM) of different complexity were calculated in addition to a Siddon-based ray tracing approach for comparison purpose. Each system matrix accounted for a more or less complete description of the physics processes both in the scanned object and in the PET scanner. One homogeneous water phantom and three heterogeneous phantoms including water, lungs and bones were simulated, where hot and cold regions were used to assess activity recovery as well as the trade-off between contrast recovery and noise in different regions. The benefit of accounting for scatter, attenuation, positron range and spurious coincidences occurring in the object when calculating the system matrix used to reconstruct I-124 PET images was highlighted. We found that the use of an MC SM including a thorough modelling of the detector response and physical effects in a uniform water-equivalent phantom was efficient to get reasonable quantitative accuracy in homogeneous and heterogeneous phantoms. Modelling the phantom heterogeneities in the SM did not necessarily yield the most accurate estimate of the activity distribution, due to the high variance affecting many SM elements in the most sophisticated SM.

Accurate and efficient modeling of the detector response in small animal multi-head PET systems

In fully three-dimensional PET imaging, iterative image reconstruction techniques usually outperform analytical algorithms in terms of image quality provided that an appropriate system model is used. In this study we concentrate on the calculation of an accurate system model for the YAP-(S)PET II small animal scanner, with the aim to obtain fully resolution- and contrast-recovered images at low levels of image roughness. For this purpose we calculate the system model by decomposing it into a product of five matrices: (1) a detector response component obtained via Monte Carlo simulations, (2) a geometric component which describes the scanner geometry and which is calculated via a multi-ray method, (3) a detector normalization component derived from the acquisition of a planar source, (4) a photon attenuation component calculated from x-ray computed tomography data, and finally, (5) a positron range component is formally included. This system model factorization allows the optimization of each component in terms of computation time, storage requirements and accuracy. The main contribution of this work is a new, efficient way to calculate the detector response component for rotating, planar detectors, that consists of a GEANT4 based simulation of a subset of lines of flight (LOFs) for a single detector head whereas the missing LOFs are obtained by using intrinsic detector symmetries. Additionally, we introduce and analyze a probability threshold for matrix elements of the detector component to optimize the trade-off between the matrix size in terms of non-zero elements and the resulting quality of the reconstructed images. In order to evaluate our proposed system model we reconstructed various images of objects, acquired according to the NEMA NU 4-2008 standard, and we compared them to the images reconstructed with two other system models: a model that does not include any detector response component and a model that approximates analytically the depth of interaction as detector response component. The comparisons confirm previous research results, showing that the usage of an accurate system model with a realistic detector response leads to reconstructed images with better resolution and contrast recovery at low levels of image roughness.

Averaging and Metropolis iterations for positron emission tomography

Iterative positron emission tomography (PET) reconstruction computes projections between the voxel space and the lines of response (LOR) space, which are mathematically equivalent to the evaluation of multi-dimensional integrals. The dimension of the integration domain can be very high if scattering needs to be compensated. Monte Carlo (MC) quadrature is a straightforward method to approximate high-dimensional integrals. As the numbers of voxels and LORs can be in the order of hundred millions and the projection also depends on the measured object, the quadratures cannot be precomputed, but Monte Carlo simulation should take place on-the-fly during the iterative reconstruction process. This paper presents modifications of the maximum likelihood, expectation maximization (ML-EM) iteration scheme to reduce the reconstruction error due to the on-the-fly MC approximations of forward and back projections. If the MC sample locations are the same in every iteration step of the ML-EM scheme, then the approximation error will lead to a modified reconstruction result. However, when random estimates are statistically independent in different iteration steps, then the iteration may either diverge or fluctuate around the solution. Our goal is to increase the accuracy and the stability of the iterative solution while keeping the number of random samples and therefore the reconstruction time low. We first analyze the error behavior of ML-EM iteration with on-the-fly MC projections, then propose two solutions: averaging iteration and Metropolis iteration. Averaging iteration averages forward projection estimates during the iteration sequence. Metropolis iteration rejects those forward projection estimates that would compromise the reconstruction and also guarantees the unbiasedness of the tracer density estimate. We demonstrate that these techniques allow a significant reduction of the required number of samples and thus the reconstruction time. The proposed methods are built into the Teratomo system.

High Performance 3D PET Reconstruction Using Spherical Basis Functions on a Polar Grid

Statistical iterative methods are a widely used method of image reconstruction in emission tomography. Traditionally, the image space is modelled as a combination of cubic voxels as a matter of simplicity. After reconstruction, images are routinely filtered to reduce statistical noise at the cost of spatial resolution degradation. An alternative to produce lower noise during reconstruction is to model the image space with spherical basis functions. These basis functions overlap in space producing a significantly large number of non-zero elements in the system response matrix (SRM) to store, which additionally leads to long reconstruction times. These two problems are partly overcome by exploiting spherical symmetries, although computation time is still slower compared to non-overlapping basis functions. In this work, we have implemented the reconstruction algorithm using Graphical Processing Unit (GPU) technology for speed and a precomputed Monte-Carlo-calculated SRM for accuracy. The reconstruction time achieved using spherical basis functions on a GPU was 4.3 times faster than the Central Processing Unit (CPU) and 2.5 times faster than a CPU-multi-core parallel implementation using eight cores. Overwriting hazards are minimized by combining a random line of response ordering and constrained atomic writing. Small differences in image quality were observed between implementations.