Joint estimation of dynamic PET images and temporal basis functions using fully 4D ML-EM

Deep learning-based PET image denoising and reconstruction: a review

This review focuses on positron emission tomography (PET) imaging algorithms and traces the evolution of PET image reconstruction methods. First, we provide an overview of conventional PET image reconstruction methods from filtered backprojection through to recent iterative PET image reconstruction algorithms, and then review deep learning methods for PET data up to the latest innovations within three main categories. The first category involves post-processing methods for PET image denoising. The second category comprises direct image reconstruction methods that learn mappings from sinograms to the reconstructed images in an end-to-end manner. The third category comprises iterative reconstruction methods that combine conventional iterative image reconstruction with neural-network enhancement. We discuss future perspectives on PET imaging and deep learning technology.

Direct List Mode Parametric Reconstruction for Dynamic Cardiac SPECT

Recently introduced stationary dedicated cardiac SPECT scanners provide new opportunities to quantify myocardial blood flow (MBF) using dynamic SPECT. However, comparing to PET, the low sensitivity of SPECT scanners affects MBF quantification due to the high noise level, especially for 201 Thallium (201Tl) due to its typically low injected dose. The conventional indirect method for generating parametric images typically starts by reconstructing a time series of frame images followed by fitting the time-activity curve (TAC) for each voxel or segment with an appropriate kinetic model. The indirect method is simple and easy to implement; however, it usually suffers from substantial image noise that could also lead to bias. In this paper, we developed a list mode direct parametric image reconstruction algorithm to substantially reduce noise in MBF quantification using dynamic SPECT and allow for patient radiation dose reduction. GPU-based parallel computing was used to achieve more than 2000-fold acceleration. The proposed method was evaluated in both simulation and in vivo canine studies. Compared with the indirect method, the proposed direct method achieved substantially lower image noise and variability, particularly at large number of iterations and at low-count levels.

Simultaneous estimation and segmentation from projection data in dynamic PET

PURPOSE

Dynamic positron emission tomography (PET) is known for its ability to extract spatiotemporal information of a radio tracer in living tissue. Information of different functional regions based on an accurate reconstruction of the activity images and kinetic parametric images has been widely studied and can be useful in research and clinical setting for diagnosis and other quantitative tasks. In this paper, our purpose is to present a novel framework for estimating the kinetic parametric images directly from the raw measurement data together with a simultaneous segmentation accomplished through kinetic parameters clustering.

METHOD

An iterative framework is proposed to estimate the kinetic parameter image, activity map and do the segmentation simultaneously from the complete dynamic PET projection data. The clustering process is applied to the kinetic parameter variable rather than to the traditional activity distribution so as to achieve accurate discrimination between different functional areas. Prior information such as total variation regularization is incorporated to reduce the noise in the PET images and a sparseness constraint is integrated to guarantee the solution for kinetic parameters due to the over complete dictionary. Alternating direction method of multipliers (ADMM) method is used to solve the optimization problem. The proposed algorithm was validated with experiments on Monte Carlo-simulated phantoms and real patient data. Symbol error rate (SER) was defined to evaluate the performance of clustering. Bias and variance of the reconstruction activity images were calculated based on ground truth. Relative mean square error (MSE) was used to evaluate parametric results quantitatively.

RESULT

In brain phantom experiment, when counting rate is 1 × 10⁶ , the bias (variance) of our method is 0.1270 (0.0281), which is lower than maximum likelihood expectation maximization (MLEM) 0.1637 (0.0410) and direct estimation without segmentation (DE) 0.1511 (0.0326). In the Zubal phantom experiment, our method has the lowest bias (variance) 0.1559 (0.0354) with 1 × 10⁵ counting rate, compared with DE 0.1820 (0.0435) and MLEM 0.3043 (0.0644). As for classification, the SER of our method is 18.87% which is the lowest among MLEM + k-means, DE + k-means, and kinetic spectral clustering (KSC). Brain data with MR reference and real patient results also show that the proposed method can get images with clear structure by visual inspection.

CONCLUSION

In this paper, we presented a joint reconstruction framework for simultaneously estimating the activity distribution, parametric images, and parameter-based segmentation of the ROIs into different functional areas. Total variation regularization is performed on the activity distribution domain to suppress noise and preserve the edges between ROIs. An over complete dictionary for time activity curve basis is constructed. SER, bias, variance, and MSE were calculated to show the effectiveness of the proposed method.

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.

History and future technical innovation in positron emission tomography

Instrumentation for positron emission tomography (PET) imaging has experienced tremendous improvements in performance over the past 60 years since it was first conceived as a medical imaging modality. Spatial resolution has improved by a factor of 10 and sensitivity by a factor of 40 from the early designs in the 1970s to the high-performance scanners of today. Multimodality configurations have emerged that combine PET with computed tomography (CT) and, more recently, with MR. Whole-body scans for clinical purposes can now be acquired in under 10 min on a state-of-the-art PET/CT. This paper will review the history of these technical developments over 40 years and summarize the important clinical research and healthcare applications that have been made possible by these technical advances. Some perspectives for the future of this technology will also be presented that promise to bring about new applications of this imaging modality in clinical research and healthcare.

Investigation of optimization-based reconstruction with an image-total-variation constraint in PET

Interest remains in reconstruction-algorithm research and development for possible improvement of image quality in current PET imaging and for enabling innovative PET systems to enhance existing, and facilitate new, preclinical and clinical applications. Optimization-based image reconstruction has been demonstrated in recent years of potential utility for CT imaging applications. In this work, we investigate tailoring the optimization-based techniques to image reconstruction for PET systems with standard and non-standard scan configurations. Specifically, given an image-total-variation (TV) constraint, we investigated how the selection of different data divergences and associated parameters impacts the optimization-based reconstruction of PET images. The reconstruction robustness was explored also with respect to different data conditions and activity up-takes of practical relevance. A study was conducted particularly for image reconstruction from data collected by use of a PET configuration with sparsely populated detectors. Overall, the study demonstrates the robustness of the TV-constrained, optimization-based reconstruction for considerably different data conditions in PET imaging, as well as its potential to enable PET configurations with reduced numbers of detectors. Insights gained in the study may be exploited for developing algorithms for PET-image reconstruction and for enabling PET-configuration design of practical usefulness in preclinical and clinical applications.

Whole-body direct 4D parametric PET imaging employing nested generalized Patlak expectation-maximization reconstruction

Whole-body (WB) dynamic PET has recently demonstrated its potential in translating the quantitative benefits of parametric imaging to the clinic. Post-reconstruction standard Patlak (sPatlak) WB graphical analysis utilizes multi-bed multi-pass PET acquisition to produce quantitative WB images of the tracer influx rate K i as a complimentary metric to the semi-quantitative standardized uptake value (SUV). The resulting K i images may suffer from high noise due to the need for short acquisition frames. Meanwhile, a generalized Patlak (gPatlak) WB post-reconstruction method had been suggested to limit K i bias of sPatlak analysis at regions with non-negligible (18)F-FDG uptake reversibility; however, gPatlak analysis is non-linear and thus can further amplify noise. In the present study, we implemented, within the open-source software for tomographic image reconstruction platform, a clinically adoptable 4D WB reconstruction framework enabling efficient estimation of sPatlak and gPatlak images directly from dynamic multi-bed PET raw data with substantial noise reduction. Furthermore, we employed the optimization transfer methodology to accelerate 4D expectation-maximization (EM) convergence by nesting the fast image-based estimation of Patlak parameters within each iteration cycle of the slower projection-based estimation of dynamic PET images. The novel gPatlak 4D method was initialized from an optimized set of sPatlak ML-EM iterations to facilitate EM convergence. Initially, realistic simulations were conducted utilizing published (18)F-FDG kinetic parameters coupled with the XCAT phantom. Quantitative analyses illustrated enhanced K i target-to-background ratio (TBR) and especially contrast-to-noise ratio (CNR) performance for the 4D versus the indirect methods and static SUV. Furthermore, considerable convergence acceleration was observed for the nested algorithms involving 10-20 sub-iterations. Moreover, systematic reduction in K i % bias and improved TBR were observed for gPatlak versus sPatlak. Finally, validation on clinical WB dynamic data demonstrated the clinical feasibility and superior K i CNR performance for the proposed 4D framework compared to indirect Patlak and SUV imaging.

Federated optimisation of kinetic analysis problems

Positron Emission Tomography (PET) data is intrinsically dynamic, and kinetic analysis of dynamic PET data can substantially augment the information provided by static PET reconstructions. Yet despite the insights into disease that kinetic analysis offers, it is not used clinically and seldom used in research beyond the preclinical stage. The utility of PET kinetic analysis is hampered by several factors including spatial inconsistency within regions of homogeneous tissue and relative computational expense when fitting complex models to individual voxels. Even with sophisticated algorithms inconsistencies can arise because local optima frequently have narrow basins of convergence, are surrounded by relatively flat (uninformative) regions, have relatively low-gradient valley floors, or combinations thereof. Based on the observation that cost functions for individual voxels frequently bear some resemblance to each-other, this paper proposes the federated optimisation of the individual kinetic analysis problems within a given image. This approach shares parameters proposed during optimisation with other, similar voxels. Federated optimisation exploits the redundancy typical of large medical images to improve the optimisation residuals, computational efficiency and, to a limited extent, image consistency. This is achieved without restricting the formulation of the kinetic model, resorting to an explicit regularisation parameter, or limiting the resolution at which parameters are computed.

Dynamic PET reconstruction using temporal patch-based low rank penalty for ROI-based brain kinetic analysis

Dynamic positron emission tomography (PET) is widely used to measure changes in the bio-distribution of radiopharmaceuticals within particular organs of interest over time. However, to retain sufficient temporal resolution, the number of photon counts in each time frame must be limited. Therefore, conventional reconstruction algorithms such as the ordered subset expectation maximization (OSEM) produce noisy reconstruction images, thus degrading the quality of the extracted time activity curves (TACs). To address this issue, many advanced reconstruction algorithms have been developed using various spatio-temporal regularizations. In this paper, we extend earlier results and develop a novel temporal regularization, which exploits the self-similarity of patches that are collected in dynamic images. The main contribution of this paper is to demonstrate that the correlation of patches can be exploited using a low-rank constraint that is insensitive to global intensity variations. The resulting optimization framework is, however, non-Lipschitz and nonconvex due to the Poisson log-likelihood and low-rank penalty terms. Direct application of the conventional Poisson image deconvolution by an augmented Lagrangian (PIDAL) algorithm is, however, problematic due to its large memory requirements, which prevents its parallelization. Thus, we propose a novel optimization framework using the concave-convex procedure (CCCP)

Direct Parametric Image Reconstruction in Reduced Parameter Space for Rapid Multi-Tracer PET Imaging

The separation of multiple PET tracers within an overlapping scan based on intrinsic differences of tracer pharmacokinetics is challenging, due to limited signal-to-noise ratio (SNR) of PET measurements and high complexity of fitting models. In this study, we developed a direct parametric image reconstruction (DPIR) method for estimating kinetic parameters and recovering single tracer information from rapid multi-tracer PET measurements. This is achieved by integrating a multi-tracer model in a reduced parameter space (RPS) into dynamic image reconstruction. This new RPS model is reformulated from an existing multi-tracer model and contains fewer parameters for kinetic fitting. Ordered-subsets expectation-maximization (OSEM) was employed to approximate log-likelihood function with respect to kinetic parameters. To incorporate the multi-tracer model, an iterative weighted nonlinear least square (WNLS) method was employed. The proposed multi-tracer DPIR (MT-DPIR) algorithm was evaluated on dual-tracer PET simulations ([18F]FDG and [11C]MET) as well as on preclinical PET measurements ([18F]FLT and [18F]FDG). The performance of the proposed algorithm was compared to the indirect parameter estimation method with the original dual-tracer model. The respective contributions of the RPS technique and the DPIR method to the performance of the new algorithm were analyzed in detail. For the preclinical evaluation, the tracer separation results were compared with single [18F]FDG scans of the same subjects measured two days before the dual-tracer scan. The results of the simulation and preclinical studies demonstrate that the proposed MT-DPIR method can improve the separation of multiple tracers for PET image quantification and kinetic parameter estimations.

4D PET iterative deconvolution with spatiotemporal regularization for quantitative dynamic PET imaging

Quantitative measurements in dynamic PET imaging are usually limited by the poor counting statistics particularly in short dynamic frames and by the low spatial resolution of the detection system, resulting in partial volume effects (PVEs). In this work, we present a fast and easy to implement method for the restoration of dynamic PET images that have suffered from both PVE and noise degradation. It is based on a weighted least squares iterative deconvolution approach of the dynamic PET image with spatial and temporal regularization. Using simulated dynamic [(11)C] Raclopride PET data with controlled biological variations in the striata between scans, we showed that the restoration method provides images which exhibit less noise and better contrast between emitting structures than the original images. In addition, the method is able to recover the true time activity curve in the striata region with an error below 3% while it was underestimated by more than 20% without correction. As a result, the method improves the accuracy and reduces the variability of the kinetic parameter estimates calculated from the corrected images. More importantly it increases the accuracy (from less than 66% to more than 95%) of measured biological variations as well as their statistical detectivity.

Single-cell tracking with PET using a novel trajectory reconstruction algorithm

Virtually all biomedical applications of positron emission tomography (PET) use images to represent the distribution of a radiotracer. However, PET is increasingly used in cell tracking applications, for which the "imaging" paradigm may not be optimal. Here, we investigate an alternative approach, which consists in reconstructing the time-varying position of individual radiolabeled cells directly from PET measurements. As a proof of concept, we formulate a new algorithm for reconstructing the trajectory of one single moving cell directly from list-mode PET data. We model the trajectory as a 3-D B-spline function of the temporal variable and use nonlinear optimization to minimize the mean-square distance between the trajectory and the recorded list-mode coincidence events. Using Monte Carlo simulations (GATE), we show that this new algorithm can track a single source moving within a small-animal PET system with 3 mm accuracy provided that the activity of the cell [Bq] is greater than four times its velocity [mm/s]. The algorithm outperforms conventional ML-EM as well as the "minimum distance" method used for positron emission particle tracking (PEPT). The new method was also successfully validated using experimentally acquired PET data. In conclusion, we demonstrated the feasibility of a new method for tracking a single moving cell directly from PET list-mode data, at the whole-body level, for physiologically relevant activities and velocities.

Application of adaptive kinetic modelling for bias propagation reduction in direct 4D image reconstruction

Parametric imaging in thoracic and abdominal PET can provide additional parameters more relevant to the pathophysiology of the system under study. However, dynamic data in the body are noisy due to the limiting counting statistics leading to suboptimal kinetic parameter estimates. Direct 4D image reconstruction algorithms can potentially improve kinetic parameter precision and accuracy in dynamic PET body imaging. However, construction of a common kinetic model is not always feasible and in contrast to post-reconstruction kinetic analysis, errors in poorly modelled regions may spatially propagate to regions which are well modelled. To reduce error propagation from erroneous model fits, we implement and evaluate a new approach to direct parameter estimation by incorporating a recently proposed kinetic modelling strategy within a direct 4D image reconstruction framework. The algorithm uses a secondary more general model to allow a less constrained model fit in regions where the kinetic model does not accurately describe the underlying kinetics. A portion of the residuals then is adaptively included back into the image whilst preserving the primary model characteristics in other well modelled regions using a penalty term that trades off the models. Using fully 4D simulations based on dynamic [(15)O]H2O datasets, we demonstrate reduction in propagation-related bias for all kinetic parameters. Under noisy conditions, reductions in bias due to propagation are obtained at the cost of increased noise, which in turn results in increased bias and variance of the kinetic parameters. This trade-off reflects the challenge of separating the residuals arising from poor kinetic modelling fits from the residuals arising purely from noise. Nonetheless, the overall root mean square error is reduced in most regions and parameters. Using the adaptive 4D image reconstruction improved model fits can be obtained in poorly modelled regions, leading to reduced errors potentially propagating to regions of interest which the primary biologic model accurately describes. The proposed methodology, however, depends on the secondary model and choosing an optimal model on the residual space is critical in improving model fits.

Direct reconstruction of cardiac PET kinetic parametric images using a preconditioned conjugate gradient approach

PURPOSE

Our research goal is to develop an algorithm to reconstruct cardiac positron emission tomography (PET) kinetic parametric images directly from sinograms and compare its performance with the conventional indirect approach.

METHODS

Time activity curves of a NCAT phantom were computed according to a one-tissue compartmental kinetic model with realistic kinetic parameters. The sinograms at each time frame were simulated using the activity distribution for the time frame. The authors reconstructed the parametric images directly from the sinograms by optimizing a cost function, which included the Poisson log-likelihood and a spatial regularization terms, using the preconditioned conjugate gradient (PCG) algorithm with the proposed preconditioner. The proposed preconditioner is a diagonal matrix whose diagonal entries are the ratio of the parameter and the sensitivity of the radioactivity associated with parameter. The authors compared the reconstructed parametric images using the direct approach with those reconstructed using the conventional indirect approach.

RESULTS

At the same bias, the direct approach yielded significant relative reduction in standard deviation by 12%-29% and 32%-70% for 50 × 10(6) and 10 × 10(6) detected coincidences counts, respectively. Also, the PCG method effectively reached a constant value after only 10 iterations (with numerical convergence achieved after 40-50 iterations), while more than 500 iterations were needed for CG.

CONCLUSIONS

The authors have developed a novel approach based on the PCG algorithm to directly reconstruct cardiac PET parametric images from sinograms, and yield better estimation of kinetic parameters than the conventional indirect approach, i.e., curve fitting of reconstructed images. The PCG method increases the convergence rate of reconstruction significantly as compared to the conventional CG method.

Simulation-based optimisation of the PET data processing for partial saturation approach protocols

Positron emission tomography (PET) with [(11)C]Raclopride is an important tool for studying dopamine D2 receptor expression in vivo. [(11)C]Raclopride PET binding experiments conducted using the Partial Saturation Approach (PSA) allow the estimation of receptor density (B(avail)) and the in vivo affinity appK(D). The PSA is a simple, single injection, single scan experimental protocol that does not require blood sampling, making it ideal for use in longitudinal studies. In this work, we generated a complete Monte Carlo simulated PET study involving two groups of scans, in between which a biological phenomenon was inferred (a 30% decrease of B(avail)), and used it in order to design an optimal data processing chain for the parameter estimation from PSA data. The impact of spatial smoothing, noise removal and image resolution recovery technique on the statistical detection was investigated in depth. We found that image resolution recovery using iterative deconvolution of the image with the system point spread function associated with temporal data denoising greatly improves the accuracy and the statistical reliability of detecting the imposed phenomenon. Before optimisation, the inferred B(avail) variation between the two groups was underestimated by 42% and detected in 66% of cases, while a false decrease of appK(D) by 13% was detected in more than 11% of cases. After optimisation, the calculated B(avail) variation was underestimated by only 3.7% and detected in 89% of cases, while a false slight increase of appK(D) by 3.7% was detected in only 2% of cases. We found during this investigation that it was essential to adjust a factor that accounts for difference in magnitude between the non-displaceable ligand concentrations measured in the target and in the reference regions, for different data processing pathways as this ratio was affected by different image resolutions.

Patlak image estimation from dual time-point list-mode PET data

We investigate using dual time-point PET data to perform Patlak modeling. This approach can be used for whole body dynamic PET studies in which we compute voxel-wise estimates of Patlak parameters using two frames of data for each bed position. Our approach directly uses list-mode arrival times for each event to estimate the Patlak parametric image. We use a penalized likelihood method in which the penalty function uses spatially variant weighting to ensure a count independent local impulse response. We evaluate performance of the method in comparison to fractional changes in SUV values (%DSUV) between the two frames using Cramer Rao analysis and Monte Carlo simulation. Receiver operating characteristic (ROC) curves are used to compare performance in differentiating tumors relative to background based on the dynamic data sets. Using area under the ROC curve as a performance metric, we show superior performance of Patlak relative to %DSUV over a range of dynamic data sets and parameters. These results suggest that Patlak analysis may be appropriate for analysis of dual time-point whole body PET data and could lead to superior detection of tumors relative to %DSUV metrics.

Direct estimation of kinetic parametric images for dynamic PET

Dynamic positron emission tomography (PET) can monitor spatiotemporal distribution of radiotracer in vivo. The spatiotemporal information can be used to estimate parametric images of radiotracer kinetics that are of physiological and biochemical interests. Direct estimation of parametric images from raw projection data allows accurate noise modeling and has been shown to offer better image quality than conventional indirect methods, which reconstruct a sequence of PET images first and then perform tracer kinetic modeling pixel-by-pixel. Direct reconstruction of parametric images has gained increasing interests with the advances in computing hardware. Many direct reconstruction algorithms have been developed for different kinetic models. In this paper we review the recent progress in the development of direct reconstruction algorithms for parametric image estimation. Algorithms for linear and nonlinear kinetic models are described and their properties are discussed.

Nonlinear spatio-temporal filtering of dynamic PET data using a four-dimensional Gaussian filter and expectation-maximization deconvolution

We introduce a method for denoising dynamic PET data, spatio-temporal expectation-maximization (STEM) filtering, that combines four-dimensional Gaussian filtering withEMdeconvolution. The initial Gaussian filter suppresses noise at a broad range of spatial and temporal frequencies and EM deconvolution quickly restores the frequencies most important to the signal. We aim to demonstrate that STEM filtering can improve variance in both individual time frames and in parametric images without introducing significant bias. We evaluate STEM filtering with a dynamic phantom study, and with simulated and human dynamic PET studies of a tracer with reversible binding behaviour, [C-11]raclopride, and a tracer with irreversible binding behaviour, [F-18]FDOPA. STEM filtering is compared to a number of established three and four-dimensional denoising methods. STEM filtering provides substantial improvements in variance in both individual time frames and in parametric images generated with a number of kinetic analysis techniques while introducing little bias. STEM filtering does bias early frames, but this does not affect quantitative parameter estimates. STEM filtering is shown to be superior to the other simple denoising methods studied. STEM filtering is a simple and effective denoising method that could be valuable for a wide range of dynamic PET applications.

Direct 4-D PET list mode parametric reconstruction with a novel EM algorithm

The production of images of kinetic parameters is often the ultimate goal of positron emission tomography (PET) imaging. The indirect method of PET parametric imaging, also called the frame-based method (FM), is performed by fitting the time-activity curve (TAC) for each voxel with an appropriate compartment model after image reconstruction. The indirect method is simple and easily implemented, however, it usually leads to some loss of accuracy or precision, due to the use of two separate steps. This paper presents a direct 4-D method for producing 3-D images of kinetic parameters from list mode PET data. In this application, the TAC for each voxel is described by a one-tissue compartment model (1T). Extending previous EM algorithms, a new spatiotemporal complete data space was introduced to optimize the maximum likelihood function. This leads to a straightforward closed-form parametric image update equation. This method was implemented by extending the current list mode platform MOLAR to produce a parametric algorithm PMOLAR-1T. Using an ordered subset approach, qualitative and quantitative evaluations were performed using 2-D (x, t) and 4-D (x, y, z, t) simulated list mode data based on brain receptor tracers and also with a human brain study. Comparisons with the indirect method showed that the proposed direct method can lead to accurate estimation of the parametric image values with reduced variance, especially at low count levels. In the 2-D test, the direct method showed similar bias to the frame-based method but with variance reduction of 23%-60%. In the 4-D test, bias values of both methods were no more than 4% and the direct method had lower variability (coefficient of variation reduction of 0%-64% compared to the frame-based method) at the normal count level. The direct method had a larger reduction in variability (27%-81%) and lower bias (1%-5% for 4-D and 1%-19% for FM) at low count levels. The results in the human brain study are similar with PMOLAR-1T showing lower noise than FM.

Advances in 4D radiation therapy for managing respiration: part I - 4D imaging

Techniques for managing respiration during imaging and planning of radiation therapy are reviewed, concentrating on free-breathing (4D) approaches. First, we focus on detailing the historical development and basic operational principles of currently-available "first generation" 4D imaging modalities: 4D computed tomography, 4D cone beam computed tomography, 4D magnetic resonance imaging, and 4D positron emission tomography. Features and limitations of these first generation systems are described, including necessity of breathing surrogates for 4D image reconstruction, assumptions made in acquisition and reconstruction about the breathing pattern, and commonly-observed artifacts. Both established and developmental methods to deal with these limitations are detailed. Finally, strategies to construct 4D targets and images and, alternatively, to compress 4D information into static targets and images for radiation therapy planning are described.

Fourier domain closed-form formulas for estimation of kinetic parameters in reversible multi-compartment models

Background

Compared with static imaging, dynamic emission computed tomographic imaging with compartment modeling can quantify in vivo physiologic processes, providing useful information about molecular disease processes. Dynamic imaging involves estimation of kinetic rate parameters. For multi-compartment models, kinetic parameter estimation can be computationally demanding and problematic with local minima.

Methods

This paper offers a new perspective to the compartment model fitting problem where Fourier linear system theory is applied to derive closed-form formulas for estimating kinetic parameters for the two-compartment model. The proposed Fourier domain estimation method provides a unique solution, and offers very different noise response as compared to traditional non-linear chi-squared minimization techniques.

Results

The unique feature of the proposed Fourier domain method is that only low frequency components are used for kinetic parameter estimation, where the DC (i.e., the zero frequency) component in the data is treated as the most important information, and high frequency components that tend to be corrupted by statistical noise are discarded. Computer simulations show that the proposed method is robust without having to specify the initial condition. The resultant solution can be fine tuned using the traditional iterative method.

Conclusions

The proposed Fourier-domain estimation method has closed-form formulas. The proposed Fourier-domain curve-fitting method does not require an initial condition, it minimizes a quadratic objective function and a closed-form solution can be obtained. The noise is easier to control, simply by discarding the high frequency components, and emphasizing the DC component.

Kinetic parameter estimation using a closed-form expression via integration by parts

Dynamic emission computed tomographic imaging with compartment modeling can quantify in vivo physiologic processes, eliciting more information regarding underlying molecular disease processes than is obtained from static imaging. However, estimation of kinetic rate parameters for multi-compartment models can be computationally demanding and problematic due to local minima. A number of techniques for kinetic parameter estimation have been studied and are in use today, generally offering a tradeoff between computation time, robustness of fit and flexibility with differing sets of assumptions. This paper presents a means to eliminate all differential operations by using the integration-by-parts method to provide closed-form formulas, so that the mathematical model is less sensitive to data sampling and noise. A family of closed-form formulas are obtained. Computer simulations show that the proposed method is robust without having to specify the initial condition.

The development, past achievements, and future directions of brain PET

The early developments of brain positron emission tomography (PET), including the methodological advances that have driven progress, are outlined. The considerable past achievements of brain PET have been summarized in collaboration with contributing experts in specific clinical applications including cerebrovascular disease, movement disorders, dementia, epilepsy, schizophrenia, addiction, depression and anxiety, brain tumors, drug development, and the normal healthy brain. Despite a history of improving methodology and considerable achievements, brain PET research activity is not growing and appears to have diminished. Assessments of the reasons for decline are presented and strategies proposed for reinvigorating brain PET research. Central to this is widening the access to advanced PET procedures through the introduction of lower cost cyclotron and radiochemistry technologies. The support and expertize of the existing major PET centers, and the recruitment of new biologists, bio-mathematicians and chemists to the field would be important for such a revival. New future applications need to be identified, the scope of targets imaged broadened, and the developed expertize exploited in other areas of medical research. Such reinvigoration of the field would enable PET to continue making significant contributions to advance the understanding of the normal and diseased brain and support the development of advanced treatments.

Quantitative accuracy of MAP reconstruction for dynamic PET imaging in small animals

PURPOSE

Iterative reconstruction algorithms are becoming more commonly employed in positron emission tomography (PET) imaging; however, the quantitative accuracy of the reconstructed images still requires validation for various levels of contrast and counting statistics.

METHODS

The authors present an evaluation of the quantitative accuracy of the 3D maximum a posteriori (3D-MAP) image reconstruction algorithm for dynamic PET imaging with comparisons to two of the most widely used reconstruction algorithms: the 2D filtered-backprojection (2D-FBP) and 2D-ordered subsets expectation maximization (2D-OSEM) on the Siemens microPET scanners. The study was performed for various levels of count density encountered in typical dynamic scanning as well as the imaging of cardiac activity concentration in small animal studies on the Focus 120. Specially designed phantoms were used for evaluation of the spatial resolution, image quality, and quantitative accuracy. A normal mouse was employed to evaluate the accuracy of the blood time activity concentration extracted from left ventricle regions of interest (ROIs) within the images as compared to the actual blood activity concentration measured from arterial blood sampling.

RESULTS

For MAP reconstructions, the spatial resolution and contrast have been found to reach a stable value after 20 iterations independent of the β values (i.e., hyper parameter which controls the weight of the penalty term) and count density within the frame. The spatial resolution obtained with 3D-MAP reaches values of ∼1.0 mm with a β of 0.01 while the 2D-FBP has value of 1.8 mm and 2D-OSEM has a value of 1.6 mm. It has been observed that the lower the hyper parameter β used in MAP, more iterations are needed to reach the stable noise level (i.e., image roughness). The spatial resolution is improved by using a lower β value at the expense of higher image noise. However, with similar noise level the spatial resolution achieved by 3D-MAP was observed to be better than that by 2D-FBP or 2D-OSEM. Using an image quality phantom containing hot spheres, the estimated activity concentration in the largest sphere has the expected concentration relative to the background area for all the MAP images. The obtained recovery coefficients have been also shown to be almost independent of the count density. 2D-FBP and 2D-OSEM do not perform as well, yielding recovery coefficients lower than those observed with 3D-MAP (approximately 33% lower for the smallest sphere). However, a small positive bias was observed in MAP reconstructed images for frames of very low count density. This bias is present in the uniform area for count density of less than 0.05 × 10(6) counts/ml. For the dynamic mouse study, it was observed that 3D-MAP (even gated at diastole) cannot predict accurately the blood activity concentration due to residual spill-over activity from the myocardium into the left ventricle (approximately 15%). However, 3D-MAP predicts blood activity concentration closer to blood sampling than 2D-FBP.

CONCLUSIONS

The authors observed that 3D-MAP produces more accurate activity concentration estimates than 2D-FBP or 2D-OSEM at all practical levels of statistics and contrasts due to improved spatial resolution leading to lesser partial volume effect.

Development and experimental medicine applications of PET in oncology: a historical perspective

Nearly 90 years of scientific research have led to the use of PET and the ability to forge advances in the field of oncology. In this Historial Review we outline the key developments made with this imaging technique, including the evolution of cyclotrons and scanners, together with the associated advances made in image reconstruction, presentation, analysis of data, and radiochemistry. The applications of PET to experimental medicine are summarised, and we cover how these are related to the use and development of PET, especially in the assessment of tumour biology and pharmacology. The use of PET in clinical oncology and for tissue pharmacokinetic and pharmacodynamic studies as a means of supporting drug development is detailed. The current limitations of the technology are also discussed.

Direct 4D parametric imaging for linearized models of reversibly binding PET tracers using generalized AB-EM reconstruction

Due to high noise levels in the voxel kinetics, development of reliable parametric imaging algorithms remains one of most active areas in dynamic brain PET imaging, which in the vast majority of cases involves receptor/transporter studies with reversibly binding tracers. As such, the focus of this work has been to develop a novel direct 4D parametric image reconstruction scheme for such tracers. Based on a relative equilibrium (RE) graphical analysis formulation (Zhou et al 2009b Neuroimage 44 661-70), we developed a closed-form 4D EM algorithm to directly reconstruct distribution volume (DV) parametric images within a plasma input model, as well as DV ratio (DVR) images within a reference tissue model scheme (wherein an initial reconstruction was used to estimate the reference tissue time-activity curves). A particular challenge with the direct 4D EM formulation is that the intercept parameters in graphical (linearized) analysis of reversible tracers (e.g. Logan or RE analysis) are commonly negative (unlike for irreversible tracers, e.g. using Patlak analysis). Subsequently, we focused our attention on the AB-EM algorithm, derived by Byrne (1998, Inverse Problems 14 1455-67) to allow inclusion of prior information about the lower (A) and upper (B) bounds for image values. We then generalized this algorithm to the 4D EM framework, thus allowing negative intercept parameters. Furthermore, our 4D AB-EM algorithm incorporated and emphasized the use of spatially varying lower bounds to achieve enhanced performance. As validation, the means of parameters estimated from 55 human (11)C-raclopride dynamic PET studies were used for extensive simulations using a mathematical brain phantom. Images were reconstructed using conventional indirect as well as proposed direct parametric imaging methods. Noise versus bias quantitative measurements were performed in various regions of the brain. Direct 4D EM reconstruction resulted in notable qualitative and quantitative accuracy improvements (over 35% noise reduction, with matched bias, in both plasma and reference-tissue input models). Similar improvements were also observed in the coefficient of variation of the estimated DV and DVR values even for relatively low uptake cortical regions, suggesting the enhanced ability for robust parameter estimation. The method was also tested on a 90 min (11)C-raclopride patient study performed on the high-resolution research tomograph wherein the proposed method was shown across a variety of regions to outperform the conventional method in the sense that for a given DVR value, improved noise levels were observed.

Spatio-temporal diffusion of dynamic PET images

Positron emission tomography (PET) images are corrupted by noise. This is especially true in dynamic PET imaging where short frames are required to capture the peak of activity concentration after the radiotracer injection. High noise results in a possible bias in quantification, as the compartmental models used to estimate the kinetic parameters are sensitive to noise. This paper describes a new post-reconstruction filter to increase the signal-to-noise ratio in dynamic PET imaging. It consists in a spatio-temporal robust diffusion of the 4D image based on the time activity curve (TAC) in each voxel. It reduces the noise in homogeneous areas while preserving the distinct kinetics in regions of interest corresponding to different underlying physiological processes. Neither anatomical priors nor the kinetic model are required. We propose an automatic selection of the scale parameter involved in the diffusion process based on a robust statistical analysis of the distances between TACs. The method is evaluated using Monte Carlo simulations of brain activity distributions. We demonstrate the usefulness of the method and its superior performance over two other post-reconstruction spatial and temporal filters. Our simulations suggest that the proposed method can be used to significantly increase the signal-to-noise ratio in dynamic PET imaging.

Error-corrected estimation of regional kinetic parameter histograms directly from pet projections

Positron emission tomography (PET) is a unique method to investigate physiology in the living body. Kinetic models with kinetic rate constants describe the dynamic radioactive tracer uptake in living tissue. If the variation of the kinetic parameter values within a specific tissue region could be determined accurately, it would give valuable quantitative information about the tissue heterogeneity. In this study we developed a unique method to estimate the variation from the regional kinetic parameter histograms. To determine the kinetic parameter values, we chose non-penalized maximum likelihood (ML) estimation due to the specific statistical error properties of the ML estimates. The parameter values were estimated directly from the time series of PET projections. The choice of the estimation method enabled us to utilize the ML theory in error correction. We developed a Monte Carlo approach to determine the regional error distributions. The true variation of the kinetic parameters could then be revealed by correcting the regional ML estimate histograms with the estimated error distributions. The method was tested with simulated data. In simulations both the average and the deviation of the kinetic parameters were determined from the error-corrected histograms with good numerical accuracy for the selected region of interest.

Joint reconstruction of image and motion in gated positron emission tomography

We present a novel intrinsic method for joint reconstruction of both image and motion in positron emission tomography (PET). Intrinsic motion compensation methods exclusively work on the measured data, without any external motion measurements. Most of these methods separate image from motion estimation: They use deformable image registration/optical flow techniques in order to estimate the motion from individually reconstructed gates. Then, the image is estimated based on this motion information. With these methods, a main problem lies in the motion estimation step, which is based on the noisy gated frames. The more noise is present, the more inaccurate the image registration becomes. As we show both visually and quantitatively, joint reconstruction using a simple deformation field motion model can compete with state-of-the-art image registration methods which use robust multilevel B-spline motion models.

Image reconstruction for PET/CT scanners: past achievements and future challenges

PET is a medical imaging modality with proven clinical value for disease diagnosis and treatment monitoring. The integration of PET and CT on modern scanners provides a synergy of the two imaging modalities. Through different mathematical algorithms, PET data can be reconstructed into the spatial distribution of the injected radiotracer. With dynamic imaging, kinetic parameters of specific biological processes can also be determined. Numerous efforts have been devoted to the development of PET image reconstruction methods over the last four decades, encompassing analytic and iterative reconstruction methods. This article provides an overview of the commonly used methods. Current challenges in PET image reconstruction include more accurate quantitation, TOF imaging, system modeling, motion correction and dynamic reconstruction. Advances in these aspects could enhance the use of PET/CT imaging in patient care and in clinical research studies of pathophysiology and therapeutic interventions.

Direct 4D reconstruction of parametric images incorporating anato-functional joint entropy

We developed an anatomy-guided 4D closed-form algorithm to directly reconstruct parametric images from projection data for (nearly) irreversible tracers. Conventional methods consist of individually reconstructing 2D/3D PET data, followed by graphical analysis on the sequence of reconstructed image frames. The proposed direct reconstruction approach maintains the simplicity and accuracy of the expectation-maximization (EM) algorithm by extending the system matrix to include the relation between the parametric images and the measured data. A closed-form solution was achieved using a different hidden complete-data formulation within the EM framework. Furthermore, the proposed method was extended to maximum a posterior reconstruction via incorporation of MR image information, taking the joint entropy between MR and parametric PET features as the prior. Using realistic simulated noisy [(11)C]-naltrindole PET and MR brain images/data, the quantitative performance of the proposed methods was investigated. Significant improvements in terms of noise versus bias performance were demonstrated when performing direct parametric reconstruction, and additionally upon extending the algorithm to its Bayesian counterpart using the MR-PET joint entropy measure.

Motion compensation for fully 4D PET reconstruction using PET superset data

Fully 4D PET image reconstruction is receiving increasing research interest due to its ability to significantly reduce spatiotemporal noise in dynamic PET imaging. However, thus far in the literature, the important issue of correcting for subject head motion has not been considered. Specifically, as a direct consequence of using temporally extensive basis functions, a single instance of movement propagates to impair the reconstruction of multiple time frames, even if no further movement occurs in those frames. Existing 3D motion compensation strategies have not yet been adapted to 4D reconstruction, and as such the benefits of 4D algorithms have not yet been reaped in a clinical setting where head movement undoubtedly occurs. This work addresses this need, developing a motion compensation method suitable for fully 4D reconstruction methods which exploits an optical tracking system to measure the head motion along with PET superset data to store the motion compensated data. List-mode events are histogrammed as PET superset data according to the measured motion, and a specially devised normalization scheme for motion compensated reconstruction from the superset data is required. This work proceeds to propose the corresponding time-dependent normalization modifications which are required for a major class of fully 4D image reconstruction algorithms (those which use linear combinations of temporal basis functions). Using realistically simulated as well as real high-resolution PET data from the HRRT, we demonstrate both the detrimental impact of subject head motion in fully 4D PET reconstruction and the efficacy of our proposed modifications to 4D algorithms. Benefits are shown both for the individual PET image frames as well as for parametric images of tracer uptake and volume of distribution for (18)F-FDG obtained from Patlak analysis.

Acceleration of the direct reconstruction of linear parametric images using nested algorithms

Parametric imaging using dynamic positron emission tomography (PET) provides important information for biological research and clinical diagnosis. Indirect and direct methods have been developed for reconstructing linear parametric images from dynamic PET data. Indirect methods are relatively simple and easy to implement because the image reconstruction and kinetic modeling are performed in two separate steps. Direct methods estimate parametric images directly from raw PET data and are statistically more efficient. However, the convergence rate of direct algorithms can be slow due to the coupling between the reconstruction and kinetic modeling. Here we present two fast gradient-type algorithms for direct reconstruction of linear parametric images. The new algorithms decouple the reconstruction and linear parametric modeling at each iteration by employing the principle of optimization transfer. Convergence speed is accelerated by running more sub-iterations of linear parametric estimation because the computation cost of the linear parametric modeling is much less than that of the image reconstruction. Computer simulation studies demonstrated that the new algorithms converge much faster than the traditional expectation maximization (EM) and the preconditioned conjugate gradient algorithms for dynamic PET.

DIRECT RECONSTRUCTION OF DYNAMIC PET PARAMETRIC IMAGES USING SPARSE SPECTRAL REPRESENTATION

To generate parametric images for dynamic PET, direct reconstruction from projection data is statistically more efficient than conventional indirect methods that perform image reconstruction and kinetic modeling in two separate steps. Existing direct reconstruction methods often use nonlinear compartmental models, which require the knowledge of model order. This paper presents a direct reconstruction approach using a linear spectral representation and does not require model order assumption. A Laplacian prior is used to ensure sparsity in the spectral representation. The resultant maximum a posteriori (MAP) formulation is solved by an expectation maximization shrinkage algorithm. A bias correction step is developed to improve the MAP estimate. Computer simulations show that the proposed method achieves better bias-variance tradeoff than a conventional indirect method for estimating parametric images from dynamic PET data.

Fully 4D list-mode reconstruction applied to respiratory-gated PET scans

(18)F-fluoro-deoxy-glucose ((18)F-FDG) positron emission tomography (PET) is one of the most sensitive and specific imaging modalities for the diagnosis of non-small cell lung cancer. A drawback of PET is that it requires several minutes of acquisition per bed position, which results in images being affected by respiratory blur. Respiratory gating techniques have been developed to deal with respiratory motion in the PET images. However, these techniques considerably increase the level of noise in the reconstructed images unless the acquisition time is increased. The aim of this paper is to evaluate a four-dimensional (4D) image reconstruction algorithm that combines the acquired events in all the gates whilst preserving the motion deblurring. This algorithm was compared to classic ordered subset expectation maximization (OSEM) reconstruction of gated and non-gated images, and to temporal filtering of gated images reconstructed with OSEM. Two datasets were used for comparing the different reconstruction approaches: one involving the NEMA IEC/2001 body phantom in motion, the other obtained using Monte-Carlo simulations of the NCAT breathing phantom. Results show that 4D reconstruction reaches a similar performance in terms of the signal-to-noise ratio (SNR) as non-gated reconstruction whilst preserving the motion deblurring. In particular, 4D reconstruction improves the SNR compared to respiratory-gated images reconstructed with the OSEM algorithm. Temporal filtering of the OSEM-reconstructed images helps improve the SNR, but does not achieve the same performance as 4D reconstruction. 4D reconstruction of respiratory-gated images thus appears as a promising tool to reach the same performance in terms of the SNR as non-gated acquisitions while reducing the motion blur, without increasing the acquisition time.

A survey of approaches for direct parametric image reconstruction in emission tomography

The quantitative data obtained by emission tomography are decoded using a number of techniques and methods in sequence to provide physiological information. Conventionally, the data are reconstructed to produce a series of static images. Then, pharmacokinetic modeling techniques are applied, and kinetic parameters that have physiological or functional significance are derived. Although it is possible to optimize each estimation step in this process, many simplifying assumptions have to be introduced to make the methods that are used practicable. Published research has shown that if the kinetic parameters are estimated directly from the measured data, the parametric images will have higher quality and lower mean-squared error than if this was done indirectly. This review highlights some aspects of the methods that have been proposed for such direct estimation of pharmacokinetic information from raw emission data.

Direct 4D List Mode Parametric Reconstruction for PET with a Novel EM Algorithm

We present a direct method for producing images of kinetic parameters from list mode PET data. The time-activity curve for each voxel is described by a one-tissue compartment, 2-parameter model. Extending previous EM algorithms, a new spatiotemporal complete data space was introduced to optimize the maximum likelihood function. This leads to a straightforward parametric image update equation with moderate additional computation requirements compared to the conventional algorithm. Qualitative and quantitative evaluations were performed using 2D (x,t) and 4D (x,y,z,t) simulated list mode data for a brain receptor study. Comparisons with the two-step approach (frame-based reconstruction followed by voxel-by-voxel parameter estimation) show that the proposed method can lead to accurate estimation of the parametric image values with reduced variance, especially for the volume of distribution (V(T)).

Advances in PET Image Reconstruction

Until recently, the most widely used methods for image reconstruction were direct analytic techniques. Iterative techniques, although computationally much more intensive, produce improved images (principally arising from more accurate modeling of the acquired projection data), enabling these techniques to replace analytic techniques not only in research settings but also in the clinic. This article offers an overview of image reconstruction theory and algorithms for PET, with a particular emphasis on statistical iterative reconstruction techniques. Future directions for image reconstruction in PET are considered, which concern mainly improving the modeling of the data acquisition process and task-specific specification of the parameters to be estimated in image reconstruction.

Cutting-edge brain imaging with positron emission tomography

A review of recent advances in brain imaging using positron emission tomography (PET) is presented in this article. Some properties of the high-resolution research tomograph are described as examples of state-of-the-art PET instrumentation. A summary of current research topics in image reconstruction and quantification is given, with emphasis on the requirements of brain dynamic imaging. A brief overview of image analysis methods is presented, together with some examples of the contributions of quantitative PET imaging to the current understanding of brain function and disease. PET findings must be evaluated in the context of clinical observations and complemented by other imaging modalities whenever possible to ensure a proper interpretation of the data.

Dynamic PET reconstruction using wavelet regularization with adapted basis functions

Tomographic reconstruction from positron emission tomography (PET) data is an ill-posed problem that requires regularization. An attractive approach is to impose an l(1) -regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al., 2004. In this paper, we apply this technique and extend it for the reconstruction of dynamic (spatio-temporal) PET data. Moreover, instead of using classical wavelets in the temporal dimension, we introduce exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time activity curves (TACs) in PET. We show that the exponential-spline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution. We address the issue of the selection of the "optimal" E-spline parameters (poles and zeros) and we investigate their effect on reconstruction quality. We demonstrate the usefulness of spatio-temporal regularization and the superior performance of E-spline wavelets over conventional Battle-LemariE wavelets in a series of experiments: the 1-D fitting of TACs, and the tomographic reconstruction of both simulated and clinical data. We find that the E-spline wavelets outperform the conventional wavelets in terms of the reconstructed signal-to-noise ratio (SNR) and the sparsity of the wavelet coefficients. Based on our simulations, we conclude that replacing the conventional wavelets with E-spline wavelets leads to equal reconstruction quality for a 40% reduction of detected coincidences, meaning an improved image quality for the same number of counts or equivalently a reduced exposure to the patient for the same image quality.

Spatiotemporal decomposition in object-space along reconstruction in emission tomography

Emission tomography has provided a new insight in brain mechanisms past years. Although reconstructions are nowadays mostly static, trend is going toward dynamic acquisitions and reconstructions. This opens a new range of investigations, for instance for drugs discovery. Indeed new drugs are studied through the dynamic ability of tissues to catch them. However, it is required to know radiotracer concentration of blood that irrigates tissues in order to draw conclusions on potentials of these drugs. This concentration is called 'input function' and this paper presents a new method for measuring it in a non-invasive way. Our new method relies on simultaneous estimations of vessels kinetics and vessels spatial distribution. These estimations are performed during the reconstruction process and take into account the statistical nature of measured signals. Indeed, this method is based on the maximisation of the likelihood of counts in detectors. It takes advantages of a non-negative matrix factorisation which separate spatial and temporal components. Results are very promising, since it estimates arterial input function accurately although object emits just a limited amount of photons, especially within the first minutes.