The SNR of time-of-flight positron emission tomography data for joint reconstruction of the activity and attenuation images.

Objective.Measurement of the time-of-flight (TOF) difference of each coincident pair of photons increases the effective sensitivity of positron emission tomography (PET). Many authors have analyzed the benefit of TOF for quantification and hot spot detection in the reconstructed activity images. However, TOF not only improves the effective sensitivity, it also enables the joint reconstruction of the tracer concentration and attenuation images. This can be used to correct for errors in CT- or MR-derived attenuation maps, or to apply attenuation correction without the help of a second modality. This paper presents an analysis of the effect of TOF on the variance of the jointly reconstructed attenuation and (attenuation corrected) tracer concentration images.Approach.The analysis is performed for PET systems that have a distribution of possibly non-Gaussian TOF-kernels, and includes the conventional Gaussian TOF-kernel as a special case. Non-Gaussian TOF-kernels are often observed in novel detector designs, which make use of two (or more) different mechanisms to convert the incoming 511 keV photon to optical photons. The analytical result is validated with a simple 2D simulation.Main results.We show that if two different TOF-kernels are equivalent for image reconstruction with known attenuation, then they are also equivalent for joint reconstruction of the activity and the attenuation images. The variance increase in the activity, caused by also jointly reconstructing the attenuation image, vanishes when the TOF-resolution approaches perfection.Significance.These results are of interest for PET detector development and for the development of stand-alone PET systems.

The SNR of Positron Emission Data With Gaussian and Non-Gaussian Time-of-Flight Kernels, With Application to Prompt Photon Coincidence.

It is well known that measurement of the time-of-flight (TOF) increases the information provided by coincident events in positron emission tomography (PET). This information increase propagates through the reconstruction and improves the signal-to-noise ratio in the reconstructed images. Takehiro Tomitani has analytically computed the gain in variance in the reconstructed image, provided by a particular TOF resolution, for the center of a uniform disk and for a Gaussian TOF kernel. In this paper we extend this result, by computing the signal-to-noise ratio (SNR) contributed by individual coincidence events for two different tasks. One task is the detection of a hot spot in the center of a uniform cylinder. The second one is the same as that considered by Tomitani, i.e. the reconstruction of the central voxel in the image of a uniform cylinder. In addition, we extend the computation to non-Gaussian TOF kernels. It is found that a modification of the TOF-kernel changes the SNR for both tasks in almost exactly the same way. The proposed method can be used to compare TOF-systems with different and possibly event-dependent TOF-kernels, as encountered when prompt photons, such as Cherenkov photons are present, or when the detector is composed of different scintillators. The method is validated with simple 2D simulations and illustrated by applying it to PET detectors producing optical photons with event-dependent timing characteristics.

Rigid motion tracking using moments of inertia in TOF-PET brain studies.

A data-driven method is proposed for rigid motion estimation directly from time-of-flight (TOF)-positron emission tomography (PET) emission data. Rigid motion parameters (translations and rotations) are estimated from the first and second moments of the emission data masked in a spherical volume. The accuracy of the method is analyzed on 3D analytical simulations of the PET-SORTEO brain phantom, and subsequently tested on18F-FDG as well as11C-PIB brain datasets acquired on a TOF-PET/CT scanner. The estimated inertia-based motion is later compared to rigid motion parameters obtained by directly registering the short frame backprojections. We find that the method provides sub mm/degree accuracies for the estimated rigid motion parameters for counts corresponding to typical 0.5 s, 1 s, and 2 s18F-FDG brain scans, with the current TOF resolutions clinically available. The method provides robust motion estimation for different types of patient motion, most notably for a continuous patient motion case where conventional frame-based approaches which rely on little to no intra-frame motion of short time intervals could fail. The method relies on the detection of stable eigenvectors for accurate motion estimation, and a monitoring of this condition can reveal time-frames where the motion estimation is less accurate, such as in dynamic PET studies.

Gamma detector dead time correction using Lambert W function.

BACKGROUND: For therapeutic applications of several isotopes (e.g., 131I, 153Sm, 177Lu) in nuclear medicine, the high activities typically applied require accurate dead time correction in early time point imaging. We present a novel, straightforward dead time correction method using the Lambert W function, which is in principle exact for the paralyzable detector model with a single parameter τ (i.e., dead time). RESULTS: As a proof of concept, the method is validated with a simple model: a commonly used isotope, 99mTc, with a single photopeak. We measured count rates of a gamma camera both intrinsically and extrinsically (i.e., with collimators) with point sources in air and in a scatter phantom (extrinsic only). τ was estimated for both open window (τOW) and a 99mTc photopeak window (τTc), using a "graphical" method for fitting the count rate of decaying sources. These values for τ were subsequently used for dead time correction. τ varied significantly between the different geometries for both energy windows, but τOW was more reproducible than τTc, particularly for the scatter phantom measurements. τOW measured from the phantom measurements was approximately 30% lower than τOW from the intrinsic measurement but corresponded within 15% with the extrinsic point source measurements. Accordingly, using the intrinsic τOW led to an overcorrection of 8% at high count rates; τOW from the extrinsic point source measurements corrected the phantom measurement to within 2%. However, significant differences were observed between τTc values. All measured τTc values underestimated dead time losses in a second independent phantom measurement, with even τTc from the first phantom measurement underestimating activity with 5-10% at the highest count rates. Based on measurements of the effect of energy window settings and geometry, we tentatively attribute the added dead time losses to pulse pile-up. CONCLUSIONS: Analytic dead time correction based on the Lambert W function is accurate for the range in which gamma detectors behave as paralyzable systems. However, further investigation indicated measured τ values to be variable with geometry as well as window fraction. We propose that dead time correction should be based on the open window value, τOW, corrected for window fraction.

Size and affinity kinetics of nanobodies influence targeting and penetration of solid tumours.

A compound's intratumoural distribution is an important determinant for the effectiveness of molecular therapy or imaging. Antibodies (Abs), though often used in the design of targeted compounds, struggle to achieve a homogenous distribution due to their large size and bivalent binding mechanism. In contrast, smaller compounds like nanobodies (Nbs) are expected to distribute more homogenously, though this has yet to be demonstrated in vivo at the microscopic level. We propose an intravital approach to evaluate the intratumoural distribution of different fluorescently labeled monomeric and dimeric Nb tracers and compare this with a monoclonal antibody (mAb). Monomeric and dimeric formats of the anti-HER2 (2Rb17c and 2Rb17c-2Rb17c) and control (R3B23 and R3B23-R3B23) Nb, as well as the dimeric monovalent Nb 2Rb17c-R3B23 were generated and fluorescently labeled with a Cy5 fluorophore. The mAb trastuzumab-Cy5 was also prepared. Whole-body biodistribution of all constructs was investigated in mice bearing subcutaneous xenografts (HER2+ SKOV3) using in vivo epi-fluorescence imaging. Next, for intravital experiments, GFP-expressing SKOV3 cells were grown under dorsal window chambers on athymic nude mice (n = 3/group), and imaged under a fluorescence stereo microscope immediately after intravenous injection of the tracers. Consecutive fluorescence images within the tumour were acquired over the initial 20 min after injection and later, single images were taken at 1, 3 and 24 h post-injection. Additionally, two-photon microscopy was used to investigate the colocalization of GFP (tumour cells) and Cy5 fluorescence (tracers) at higher resolution. Whole-body images showed rapid renal clearance of all Nbs, and fast tumour targeting for the specific Nbs. Specific tumour uptake of the mAb could only be clearly distinguished from background after several hours. Intravital imaging revealed that monomeric Nb tracers accumulated rapidly and distributed homogenously in the tumour mere minutes after intravenous injection. The dimeric compounds initially achieved lower fluorescence intensities than the monomeric. Furthermore, whereas the HER2-specific dimeric bivalent compound remained closely associated to the blood vessels over 24 h, the HER2-specific dimeric monovalent tracer achieved a more homogenous tumour distribution from 1 h post-injection onwards. Non-specific tracers were not retained in the tumour. Trastuzumab had the most heterogenous intratumoural distribution of all evaluated compounds, while -due to the long blood retention- achieving the highest overall tumour uptake at 24 h post-injection. In conclusion, monomeric Nbs very quickly and homogenously distribute through tumour tissue, at a rate significantly greater than dimeric Nbs and mAbs. This underlines the potential of monomeric Nb tracers and therapeutics in molecular imaging and targeted therapies.

A Convergence Proof of MLEM and MLEM-3 With Fixed Background.

Maximum likelihood expectation-maximization (MLEM) is a popular algorithm to reconstruct the activity image in positron emission tomography. This paper introduces a "fundamental equality" for the MLEM complete data from which two key properties easily follow that allows us to: 1) prove in an elegant and compact way the convergence of MLEM for a forward model with fixed background (i.e., counts such as random and scatter coincidences) and 2) generalize this proof for the MLEM-3 algorithm. Moreover, we give necessary and sufficient conditions for the solution to be unique.

Time-of-flight PET time calibration using data consistency.

This paper presents new data driven methods for the time of flight (TOF) calibration of positron emission tomography (PET) scanners. These methods are derived from the consistency condition for TOF PET, they can be applied to data measured with an arbitrary tracer distribution and are numerically efficient because they do not require a preliminary image reconstruction from the non-TOF data. Two-dimensional simulations are presented for one of the methods, which only involves the two first moments of the data with respect to the TOF variable. The numerical results show that this method estimates the detector timing offsets with errors that are larger than those obtained via an initial non-TOF reconstruction, but remain smaller than [Formula: see text] of the TOF resolution and thereby have a limited impact on the quantitative accuracy of the activity image estimated with standard maximum likelihood reconstruction algorithms.

sMLACF: a generalized expectation-maximization algorithm for TOF-PET to reconstruct the activity and attenuation simultaneously.

The 'simultaneous maximum-likelihood attenuation correction factors' (sMLACF) algorithm presented here, is an iterative algorithm to calculate the maximum-likelihood estimate of the activity λ and the attenuation factors a in time-of-flight positron emission tomography, and this from emission data only. Hence sMLACF is an alternative to the MLACF algorithm. sMLACF is derived using the generalized expectation-maximization principle by introducing an appropriate set of complete data. The resulting iteration step yields a simultaneous update of λ and a which, in addition, enforces in a natural way the constraints [Formula: see text] where [Formula: see text] is a fixed lower bound that ensures the boundedness of the reconstructed activities. Some properties-like the monotonic increase of the likelihood and the asymptotic regularity of the estimated [Formula: see text]-of sMLACF are proven. Comparison of sMLACF with MLACF for two data sets reveals that both algorithms show very similar results, although sMLACF converges slower.

Plane-dependent ML scatter scaling: 3D extension of the 2D simulated single scatter (SSS) estimate.

Scatter correction is typically done using a simulation of the single scatter, which is then scaled to account for multiple scatters and other possible model mismatches. This scaling factor is determined by fitting the simulated scatter sinogram to the measured sinogram, using only counts measured along LORs that do not intersect the patient body, i.e. 'scatter-tails'. Extending previous work, we propose to scale the scatter with a plane dependent factor, which is determined as an additional unknown in the maximum likelihood (ML) reconstructions, using counts in the entire sinogram rather than only the 'scatter-tails'. The ML-scaled scatter estimates are validated using a Monte-Carlo simulation of a NEMA-like phantom, a phantom scan with typical contrast ratios of a 68Ga-PSMA scan, and 23 whole-body 18F-FDG patient scans. On average, we observe a 12.2% change in the total amount of tracer activity of the MLEM reconstructions of our whole-body patient database when the proposed ML scatter scales are used. Furthermore, reconstructions using the ML-scaled scatter estimates are found to eliminate the typical 'halo' artifacts that are often observed in the vicinity of high focal uptake regions.

Evaluation of MLACF based calculated attenuation brain PET imaging for FDG patient studies.

Calculating attenuation correction for brain PET imaging rather than using CT presents opportunities for low radiation dose applications such as pediatric imaging and serial scans to monitor disease progression. Our goal is to evaluate the iterative time-of-flight based maximum-likelihood activity and attenuation correction factors estimation (MLACF) method for clinical FDG brain PET imaging. FDG PET/CT brain studies were performed in 57 patients using the Biograph mCT (Siemens) four-ring scanner. The time-of-flight PET sinograms were acquired using the standard clinical protocol consisting of a CT scan followed by 10 min of single-bed PET acquisition. Images were reconstructed using CT-based attenuation correction (CTAC) and used as a gold standard for comparison. Two methods were compared with respect to CTAC: a calculated brain attenuation correction (CBAC) and MLACF based PET reconstruction. Plane-by-plane scaling was performed for MLACF images in order to fix the variable axial scaling observed. The noise structure of the MLACF images was different compared to those obtained using CTAC and the reconstruction required a higher number of iterations to obtain comparable image quality. To analyze the pooled data, each dataset was registered to a standard template and standard regions of interest were extracted. An SUVr analysis of the brain regions of interest showed that CBAC and MLACF were each well correlated with CTAC SUVrs. A plane-by-plane error analysis indicated that there were local differences for both CBAC and MLACF images with respect to CTAC. Mean relative error in the standard regions of interest was less than 5% for both methods and the mean absolute relative errors for both methods were similar (3.4% ± 3.1% for CBAC and 3.5% ± 3.1% for MLACF). However, the MLACF method recovered activity adjoining the frontal sinus regions more accurately than CBAC method. The use of plane-by-plane scaling of MLACF images was found to be a crucial step in order to obtain improved activity estimates. Presence of local errors in both MLACF and CBAC based reconstructions would require the use of a normal database for clinical assessment. However, further work is required in order to assess the clinical advantage of MLACF over CBAC based method.

Fourier rebinning and consistency equations for time-of-flight PET planograms.

Due to the unique geometry, dual-panel PET scanners have many advantages in dedicated breast imaging and on-board imaging applications since the compact scanners can be combined with other imaging and treatment modalities. The major challenges of dual-panel PET imaging are the limited-angle problem and data truncation, which can cause artifacts due to incomplete data sampling. The time-of-flight (TOF) information can be a promising solution to reduce these artifacts. The TOF planogram is the native data format for dual-panel TOF PET scanners, and the non-TOF planogram is the 3D extension of linogram. The TOF planograms is five-dimensional while the objects are three-dimensional, and there are two degrees of redundancy. In this paper, we derive consistency equations and Fourier-based rebinning algorithms to provide a complete understanding of the rich structure of the fully 3D TOF planograms. We first derive two consistency equations and John's equation for 3D TOF planograms. By taking the Fourier transforms, we obtain two Fourier consistency equations and the Fourier-John equation, which are the duals of the consistency equations and John's equation, respectively. We then solve the Fourier consistency equations and Fourier-John equation using the method of characteristics. The two degrees of entangled redundancy of the 3D TOF data can be explicitly elicited and exploited by the solutions along the characteristic curves. As the special cases of the general solutions, we obtain Fourier rebinning and consistency equations (FORCEs), and thus we obtain a complete scheme to convert among different types of PET planograms: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF planograms. The FORCEs can be used as Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. As a byproduct, we show the two consistency equations are necessary and sufficient for 3D TOF planograms. Finally, we give numerical examples of implementation of a fast 2D TOF planogram projector and Fourier-based rebinning for a 2D TOF planograms using the FORCEs to show the efficacy of the Fourier-based solutions.

Transmission-less attenuation estimation from time-of-flight PET histo-images using consistency equations.

In positron emission tomography (PET) imaging, attenuation correction with accurate attenuation estimation is crucial for quantitative patient studies. Recent research showed that the attenuation sinogram can be determined up to a scaling constant utilizing the time-of-flight information. The TOF-PET data can be naturally and efficiently stored in a histo-image without information loss, and the radioactive tracer distribution can be efficiently reconstructed using the DIRECT approaches. In this paper, we explore transmission-less attenuation estimation from TOF-PET histo-images. We first present the TOF-PET histo-image formation and the consistency equations in the histo-image parameterization, then we derive a least-squares solution for estimating the directional derivatives of the attenuation factors from the measured emission histo-images. Finally, we present a fast solver to estimate the attenuation factors from their directional derivatives using the discrete sine transform and fast Fourier transform while considering the boundary conditions. We find that the attenuation histo-images can be uniquely determined from the TOF-PET histo-images by considering boundary conditions. Since the estimate of the attenuation directional derivatives can be inaccurate for LORs tangent to the patient boundary, external sources, e.g. a ring or annulus source, might be needed to give an accurate estimate of the attenuation gradient for such LORs. The attenuation estimation from TOF-PET emission histo-images is demonstrated using simulated 2D TOF-PET data.

ML-reconstruction for TOF-PET with simultaneous estimation of the attenuation factors.

In positron emission tomography (PET), attenuation correction is typically done based on information obtained from transmission tomography. Recent studies show that time-of-flight (TOF) PET emission data allow joint estimation of activity and attenuation images. Mathematical analysis revealed that the joint estimation problem is determined up to a scale factor. In this work, we propose a maximum likelihood reconstruction algorithm that jointly estimates the activity image together with the sinogram of the attenuation factors. The algorithm is evaluated with 2-D and 3-D simulations as well as clinical TOF-PET measurements of a patient scan and compared to reference reconstructions. The robustness of the algorithm to possible imperfect scanner calibration is demonstrated with reconstructions of the patient scan ignoring the varying detector sensitivities.

Transmission-less attenuation correction in time-of-flight PET: analysis of a discrete iterative algorithm.

The maximum likelihood attenuation correction factors (MLACF) algorithm has been developed to calculate the maximum-likelihood estimate of the activity image and the attenuation sinogram in time-of-flight (TOF) positron emission tomography, using only emission data without prior information on the attenuation. We consider the case of a Poisson model of the data, in the absence of scatter or random background. In this case the maximization with respect to the attenuation factors can be achieved in a closed form and the MLACF algorithm works by updating the activity. Despite promising numerical results, the convergence of this algorithm has not been analysed. In this paper we derive the algorithm and demonstrate that the MLACF algorithm monotonically increases the likelihood, is asymptotically regular, and that the limit points of the iteration are stationary points of the likelihood. Because the problem is not convex, however, the limit points might be saddle points or local maxima. To obtain some empirical insight into the latter question, we present data obtained by applying MLACF to 2D simulated TOF data, using a large number of iterations and different initializations.

Simultaneous reconstruction of activity and attenuation in time-of-flight PET.

In positron emission tomography (PET) and single photon emission tomography (SPECT), attenuation correction is necessary for quantitative reconstruction of the tracer distribution. Previously, several attempts have been made to estimate the attenuation coefficients from emission data only. These attempts had limited success, because the problem does not have a unique solution, and severe and persistent "cross-talk" between the estimated activity and attenuation distributions was observed. In this paper, we show that the availability of time-of-flight (TOF) information eliminates the cross-talk problem by destroying symmetries in the associated Fisher information matrix. We propose a maximum-a-posteriori reconstruction algorithm for jointly estimating the attenuation and activity distributions from TOF PET data. The performance of the algorithm is studied with 2-D simulations, and further illustrated with phantom experiments and with a patient scan. The estimated attenuation image is robust to noise, and does not suffer from the cross-talk that was observed in non-TOF PET. However, some constraining is still mandatory, because the TOF data determine the attenuation sinogram only up to a constant offset.

Time-of-flight PET data determine the attenuation sinogram up to a constant.

In positron emission tomography (PET), a quantitative reconstruction of the tracer distribution requires accurate attenuation correction. We consider situations where a direct measurement of the attenuation coefficient of the tissues is not available or is unreliable, and where one attempts to estimate the attenuation sinogram directly from the emission data by exploiting the consistency conditions that must be satisfied by the non-attenuated data. We show that in time-of-flight PET, the attenuation sinogram is determined by the emission data except for a constant and that its gradient can be estimated efficiently using a simple analytic algorithm. The stability of the method is illustrated numerically by means of a 2D simulation.

Improved quantification in multiple-pinhole SPECT by anatomy-based reconstruction using microCT information.

PURPOSE: The aim of this study was to evaluate the potential of anatomy-based reconstruction, using microCT information, to improve quantitative accuracy in multiple-pinhole SPECT. METHODS: Multiple-pinhole SPECT and microCT images were acquired with the Micro Deluxe Phantom using both hot and cold rod inserts. The phantoms were filled with 3.7 MBq/ml of (99m)Tc. To improve microCT contrast, the phantoms were also filled with contrast agent. Emission images were reconstructed using a one-step-late (OSL) modification of the ordered subsets expectation maximization (OSEM) algorithm for incorporation of microCT information, to encourage smoothing within but not across boundaries. To allow quantification, the OSL OSEM algorithm takes into account imperfect camera motion, collimator response, angular variation of the sensitivity, intrinsic camera resolution, attenuation and scatter. For comparison, the emission images were also reconstructed by OSEM using post-reconstruction filtering and by OSL OSEM using a quadratic prior and an edge-preserving prior. In each rod of the phantoms the recovery coefficient (RC), defined as measured divided by the true activity concentration, was expressed as a function of the noise. Different noise levels were obtained by varying the amount of spatial filtering during or after reconstruction and by the use of binominal deviates. RESULTS: Compared to conventional OSEM using post-reconstruction filtering and compared to OSL OSEM using a quadratic prior, our study demonstrated that the use of anatomical information during reconstruction significantly improved the quantitative accuracy in both cold and hot rods with a diameter larger than or equal to 2.4 mm. When compared to the edge-preserving prior, the anatomical prior performs significantly better for hot rods with a diameter ≥ 2.4 mm. For the 4.0-mm hot rods for example, the RC averaged over the different noise levels was 0.67 ± 0.02 when multiple-pinhole SPECT images were reconstructed using anatomical information, compared to 0.54 ± 0.08, 0.60 ± 0.04 and 0.64 ± 0.02 when OSEM in combination with a post-reconstruction filter, OSL OSEM using a quadratic prior and OSL OSEM using a median root prior was used, respectively. For the 4.0-mm cold rods, the RC averaged over the different noise levels was 0.61 ± 0.03 when the multiple-pinhole SPECT images were reconstructed using anatomical information, compared to 0.54 ± 0.07, 0.53 ± 0.08 and 0.60 ± 0.03 when OSEM in combination with a post-reconstruction filter, OSL OSEM using a quadratic prior and OSL OSEM using a median root prior was used, respectively. CONCLUSION: Anatomy-based reconstruction using microCT information has the potential to improve quantitative accuracy in multiple-pinhole SPECT.

Comparing implementations of penalized weighted least-squares sinogram restoration.

PURPOSE: A CT scanner measures the energy that is deposited in each channel of a detector array by x rays that have been partially absorbed on their way through the object. The measurement process is complex and quantitative measurements are always and inevitably associated with errors, so CT data must be preprocessed prior to reconstruction. In recent years, the authors have formulated CT sinogram preprocessing as a statistical restoration problem in which the goal is to obtain the best estimate of the line integrals needed for reconstruction from the set of noisy, degraded measurements. The authors have explored both penalized Poisson likelihood (PL) and penalized weighted least-squares (PWLS) objective functions. At low doses, the authors found that the PL approach outperforms PWLS in terms of resolution-noise tradeoffs, but at standard doses they perform similarly. The PWLS objective function, being quadratic, is more amenable to computational acceleration than the PL objective. In this work, the authors develop and compare two different methods for implementing PWLS sinogram restoration with the hope of improving computational performance relative to PL in the standard-dose regime. Sinogram restoration is still significant in the standard-dose regime since it can still outperform standard approaches and it allows for correction of effects that are not usually modeled in standard CT preprocessing. METHODS: The authors have explored and compared two implementation strategies for PWLS sinogram restoration: (1) A direct matrix-inversion strategy based on the closed-form solution to the PWLS optimization problem and (2) an iterative approach based on the conjugate-gradient algorithm. Obtaining optimal performance from each strategy required modifying the naive off-the-shelf implementations of the algorithms to exploit the particular symmetry and sparseness of the sinogram-restoration problem. For the closed-form approach, the authors subdivided the large matrix inversion into smaller coupled problems and exploited sparseness to minimize matrix operations. For the conjugate-gradient approach, the authors exploited sparseness and preconditioned the problem to speed up convergence. RESULTS: All methods produced qualitatively and quantitatively similar images as measured by resolution-variance tradeoffs and difference images. Despite the acceleration strategies, the direct matrix-inversion approach was found to be uncompetitive with iterative approaches, with a computational burden higher by an order of magnitude or more. The iterative conjugate-gradient approach, however, does appear promising, with computation times half that of the authors' previous penalized-likelihood implementation. CONCLUSIONS: Iterative conjugate-gradient based PWLS sinogram restoration with careful matrix optimizations has computational advantages over direct matrix PWLS inversion and over penalized-likelihood sinogram restoration and can be considered a good alternative in standard-dose regimes.

Comparison between parallel hole and rotating slat collimation: analytical noise propagation models.

We have previously proposed a method to compare tomographic systems. It is assumed that each system acquires a tomographic scan of a certain tracer distribution in the same acquisition time. From this scan, each system is forced to reconstruct an image with a predefined spatial resolution. The system that can perform this task with the "most favorable" noise propagation is considered as the best system. The variance on pixel values or region-of-interest (ROI) values is used to assess the noise in the reconstructed image. In this paper, we extend this idea to compare the performance of parallel hole (PH) and rotating slat (RS) collimations. Two different analytical approaches were used to analyze the variance of the reconstructed pixel/ROI values. The first method is based on the filtered-backprojection (FBP) theory, and was applied to the central point of a uniform symmetrical phantom. It yields analytical expressions for the optimal collimator aperture and the corresponding variance of the reconstructed pixel values, but it can only be applied to highly symmetrical configurations. The second method is based on approximations for the Fisher information matrix. It provides numerical results, and it is more general and can be applied to nonsymmetrical objects and shift-variant tomographic systems. The collimations were compared for both planar imaging and volume imaging. The main results are as follows. 1) For cases where both methods are valid, they are in excellent agreement. 2a) The optimal collimator aperture varies linearly with the target resolution. 2b) For a fixed target resolution, the optimal collimator aperture depends on the collimator type and the imaging mode (planar or volume). 2c) The optimal aperture of PH is a factor of √2 larger than that of RS. 3a) The relative performance of the two collimators is determined by both the object size and the object-to-detector distance. 3b) Pixel variance and variances of ROIs with varying sizes yield very similar relative performance for RS versus PH.