Inference of hysteretic respiratory tumor motion from external surrogates: a state augmentation approach

It is important to monitor tumor movement during radiotherapy. Respiration-induced motion affects tumors in the thorax and abdomen (in particular, those located in the lung region). For image-guided radiotherapy (IGRT) systems, it is desirable to minimize imaging dose, so external surrogates are used to infer the internal tumor motion between image acquisitions. This process relies on consistent correspondence between the external surrogate signal and the internal tumor motion. Respiratory hysteresis complicates the external/internal correspondence because two distinct tumor positions during different breathing phases can yield the same external observation. Previous attempts to resolve this ambiguity often subdivided the data into inhale/exhale stages and restricted the estimation to only one of these directions. In this study, we propose a new approach to infer the internal tumor motion from external surrogate signal using state augmentation. This method resolves the hysteresis ambiguity by incorporating higher-order system dynamics. It circumvents the segmentation of the internal/external trajectory into different phases, and estimates the inference map based on all the available external/internal correspondence pairs. Optimization of the state augmentation is investigated. This method generalizes naturally to adaptive on-line algorithms.

Real-time prediction of respiratory motion based on local regression methods

Recent developments in modulation techniques enable conformal delivery of radiation doses to small, localized target volumes. One of the challenges in using these techniques is real-time tracking and predicting target motion, which is necessary to accommodate system latencies. For image-guided-radiotherapy systems, it is also desirable to minimize sampling rates to reduce imaging dose. This study focuses on predicting respiratory motion, which can significantly affect lung tumours. Predicting respiratory motion in real-time is challenging, due to the complexity of breathing patterns and the many sources of variability. We propose a prediction method based on local regression. There are three major ingredients of this approach: (1) forming an augmented state space to capture system dynamics, (2) local regression in the augmented space to train the predictor from previous observation data using semi-periodicity of respiratory motion, (3) local weighting adjustment to incorporate fading temporal correlations. To evaluate prediction accuracy, we computed the root mean square error between predicted tumor motion and its observed location for ten patients. For comparison, we also investigated commonly used predictive methods, namely linear prediction, neural networks and Kalman filtering to the same data. The proposed method reduced the prediction error for all imaging rates and latency lengths, particularly for long prediction lengths.

Diffeomorphic nonlinear transformations: a local parametric approach for image registration

Many types of transformations are used to model deformations in medical image registration. While some focus on modeling local changes, some on continuity and invertibility, there is no closed-form nonlinear parametric approach that addresses all these properties. This paper presents a class of nonlinear transformations that are local, continuous and invertible under certain conditions. They are straightforward to implement, fast to compute and can be used particularly in cases where locally affine deformations need to be recovered. We use our new transformation model to demonstrate some results on synthetic images using a multi-scale approach to multi-modality mutual information based image registration. The original images were deformed using B-splines at three levels of scale. The results show that the proposed method can recover these deformations almost completely with very few iterations of a gradient based optimizer.

A comparison of rotation- and blob-based system models for 3D SPECT with depth-dependent detector response

We compare two different implementations of a 3D SPECT system model for iterative reconstruction, both of which compensate for non-uniform photon attenuation and depth-dependent system response. One implementation performs fast rotation of images represented using a basis of rectangular voxels, whereas the other represents images using a basis of rotationally symmetric volume elements. In our simulations the blob-based approach was found to slightly outperform the rotation-based one in terms of the bias-variance tradeoff in the reconstructed images. Their difference can be significant, however, in terms of computational load. The rotation-based method is faster for many typical SPECT reconstruction problems, but the blob-based one can be better-suited to cases where the reconstruction algorithm needs to process one volume element at a time.

A feasibility study of mutual information based setup error estimation for radiotherapy

We have investigated a fully automatic setup error estimation method that aligns DRRs (digitally reconstructed radiographs) from a three-dimensional planning computed tomography image onto two-dimensional radiographs that are acquired in a treatment room. We have chosen a MI (mutual information)-based image registration method, hoping for robustness to intensity differences between the DRRs and the radiographs. The MI-based estimator is fully automatic since it is based on the image intensity values without segmentation. Using 10 repeated scans of an anthropomorphic chest phantom in one position and two single scans in two different positions, we evaluated the performance of the proposed method and a correlation-based method against the setup error determined by fiducial marker-based method. The mean differences between the proposed method and the fiducial marker-based method were smaller than 1 mm for translational parameters and 0.8 degree for rotational parameters. The standard deviations of estimates from the proposed method due to detector noise were smaller than 0.3 mm and 0.07 degree for the translational parameters and rotational parameters, respectively.

Maximum-likelihood transmission image reconstruction for overlapping transmission beams

In many transmission imaging geometries, the transmitted "beams" of photons overlap on the detector, such that a detector element may record photons that originated in different sources or source locations and thus traversed different paths through the object. Examples include systems based on scanning line sources or on multiple parallel rod sources. The overlap of these beams has been disregarded by both conventional analytical reconstruction methods as well as by previous statistical reconstruction methods. We propose a new algorithm for statistical image reconstruction of attenuation maps that explicitly accounts for overlapping beams in transmission scans. The algorithm is guaranteed to monotonically increase the objective function at each iteration. The availability of this algorithm enables the possibility of deliberately increasing the beam overlap so as to increase count rates. Simulated single photon emission tomography transmission scans based on a multiple line source array demonstrate that the proposed method yields improved resolution/noise tradeoffs relative to "conventional" reconstruction algorithms, both statistical and nonstatistical.

Mean and variance of single photon counting with deadtime

The statistics of photon counting by systems affected by deadtime are potentially important for statistical image reconstruction methods. We present a new way of analysing the moments of the counting process for a counter system affected by various models of deadtime related to PET and SPECT imaging. We derive simple and exact expressions for the first and second moments of the number of recorded events under various models. From our mean expression for a SPECT deadtime model, we derive a simple estimator for the actual intensity of the underlying Poisson process; simulations show that our estimator is unbiased even for extremely high count rates. From this analysis, we study the suitability of the Poisson statistical model assumed in most statistical image reconstruction algorithms. For systems containing 'modules' with several detector elements, where each element can cause deadtime losses for the entire module, such as block PET detectors or Anger cameras, the Poisson statistical model appears to be adequate even in the presence of deadtime losses.

Regularization for uniform spatial resolution properties in penalized-likelihood image reconstruction

Traditional space-invariant regularization methods in tomographic image reconstruction using penalized-likelihood estimators produce images with nonuniform spatial resolution properties. The local point spread functions that quantify the smoothing properties of such estimators are space-variant, asymmetric, and object-dependent even for space-invariant imaging systems. We propose a new quadratic regularization scheme for tomographic imaging systems that yields increased spatial uniformity and is motivated by the least-squares fitting of a parameterized local impulse response to a desired global response. We have developed computationally efficient methods for PET systems with shift-invariant geometric responses. We demonstrate the increased spatial uniformity of this new method versus conventional quadratic regularization schemes in simulated PET thorax scans.

Tomographic reconstruction of tracer gas concentration profiles in a room with the use of a single OP-FTIR and two iterative algorithms: ART and PWLS

Computed tomographic (CT) reconstructions of air contaminant concentration fields were conducted in a room-sized chamber employing a single open-path Fourier transform infrared (OP-FTIR) instrument and a combination of 52 flat mirrors and 4 retroreflectors. A total of 56 beam path data were repeatedly collected for around 1 hr while maintaining a stable concentration gradient. The plane of the room was divided into 195 pixels (13 x 15) for reconstruction. The algebraic reconstruction technique (ART) failed to reconstruct the original concentration gradient patterns for most cases. These poor results were caused by the "highly underdetermined condition" in which the number of unknown values (156 pixels) exceeds that of known data (56 path integral concentrations) in the experimental setting. A new CT algorithm, called the penalized weighted least-squares (PWLS), was applied to remedy this condition. The peak locations were correctly positioned in the PWLS-CT reconstructions. A notable feature of the PWLS-CT reconstructions was a significant reduction of highly irregular noise peaks found in the ART-CT reconstructions. However, the peak heights were slightly reduced in the PWLS-CT reconstructions due to the nature of the PWLS algorithm. PWLS could converge on the original concentration gradient even when a fairly high error was embedded into some experimentally measured path integral concentrations. It was also found in the simulation tests that the PWLS algorithm was very robust with respect to random errors in the path integral concentrations. This beam geometry and the use of a single OP-FTIR scanning system, in combination with the PWLS algorithm, is a system applicable to both environmental and industrial settings.

Exact distribution of edge-preserving MAP estimators for linear signal models with Gaussian measurement noise

We derive the exact statistical distribution of maximum a posteriori (MAP) estimators having edge-preserving nonGaussian priors. Such estimators have been widely advocated for image restoration and reconstruction problems. Previous investigations of these image recovery methods have been primarily empirical; the distribution we derive enables theoretical analysis. The signal model is linear with Gaussian measurement noise. We assume that the energy function of the prior distribution is chosen to ensure a unimodal posterior distribution (for which convexity of the energy function is sufficient), and that the energy function satisfies a uniform Lipschitz regularity condition. The regularity conditions are sufficiently general to encompass popular priors such as the generalized Gaussian Markov random field prior and the Huber prior, even though those priors are not everywhere twice continuously differentiable.

Ordered subsets algorithms for transmission tomography

The ordered subsets EM (OSEM) algorithm has enjoyed considerable interest for emission image reconstruction due to its acceleration of the original EM algorithm and ease of programming. The transmission EM reconstruction algorithm converges very slowly and is not used in practice. In this paper, we introduce a simultaneous update algorithm called separable paraboloidal surrogates (SPS) that converges much faster than the transmission EM algorithm. Furthermore, unlike the 'convex algorithm' for transmission tomography, the proposed algorithm is monotonic even with nonzero background counts. We demonstrate that the ordered subsets principle can also be applied to the new SPS algorithm for transmission tomography to accelerate 'convergence', albeit with similar sacrifice of global convergence properties as for OSEM. We implemented and evaluated this ordered subsets transmission (OSTR) algorithm. The results indicate that the OSTR algorithm speeds up the increase in the objective function by roughly the number of subsets in the early iterates when compared to the ordinary SPS algorithm. We compute mean square errors and segmentation errors for different methods and show that OSTR is superior to OSEM applied to the logarithm of the transmission data. However, penalized-likelihood reconstructions yield the best quality images among all other methods tested.

Ordered subsets algorithms for transmission tomography

The ordered subsets EM (OSEM) algorithm has enjoyed considerable interest for emission image reconstruction due to its acceleration of the original EM algorithm and ease of programming. The transmission EM reconstruction algorithm converges very slowly and is not used in practice. In this paper, we introduce a simultaneous update algorithm called separable paraboloidal surrogates (SPS) that converges much faster than the transmission EM algorithm. Furthermore, unlike the 'convex algorithm' for transmission tomography, the proposed algorithm is monotonic even with nonzero background counts. We demonstrate that the ordered subsets principle can also be applied to the new SPS algorithm for transmission tomography to accelerate 'convergence', albeit with similar sacrifice of global convergence properties as for OSEM. We implemented and evaluated this ordered subsets transmission (OSTR) algorithm. The results indicate that the OSTR algorithm speeds up the increase in the objective function by roughly the number of subsets in the early iterates when compared to the ordinary SPS algorithm. We compute mean square errors and segmentation errors for different methods and show that OSTR is superior to OSEM applied to the logarithm of the transmission data. However, penalized-likelihood reconstructions yield the best quality images among all other methods tested.

Monotonic algorithms for transmission tomography

We present a framework for designing fast and monotonic algorithms for transmission tomography penalized-likelihood image reconstruction. The new algorithms are based on paraboloidal surrogate functions for the log likelihood. Due to the form of the log-likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity. Unlike previous methods, the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events, such as scatter and random coincidences. The gradient and the curvature of the likelihood terms are evaluated only once per iteration. Since the problem is simplified at each iteration, the CPU time is less than that of current algorithms which directly minimize the objective, yet the convergence rate is comparable. The simplicity, monotonicity, and speed of the new algorithms are quite attractive. The convergence rates of the algorithms are demonstrated using real and simulated PET transmission scans.

Penalized-likelihood estimators and noise analysis for randoms-precorrected PET transmission scans

This paper analyzes and compares image reconstruction methods based on practical approximations to the exact log likelihood of randoms-precorrected positron emission tomography (PET) measurements. The methods apply to both emission and transmission tomography, however, in this paper we focus on transmission tomography. The results of experimental PET transmission scans and variance approximations demonstrate that the shifted Poisson (SP) method avoids the systematic bias of the conventional data-weighted least squares (WLS) method and leads to significantly lower variance than conventional statistical methods based on the log likelihood of the ordinary Poisson (OP) model. We develop covariance approximations to analyze the propagation of noise from attenuation maps into emission images via the attenuation correction factors (ACF's). Empirical pixel and region variances from real transmission data agree closely with the analytical predictions. Both the approximations and the empirical results show that the performance differences between the OP model and SP model are even larger, when considering noise propagation from the transmission images into the final emission images, than the differences in the attenuation maps themselves.

Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

Statistical image reconstruction methods for randoms-precorrected PET scans

Positron emission tomography (PET) measurements are usually precorrected for accidental coincidence events by real-time subtraction of the delayed-window coincidences. Randoms subtraction compensates on average for accidental coincidences but destroys the Poisson statistics. We propose and analyze two new approximations to the exact log-likelihood of the precorrected measurements, one based on a 'shifted Poisson' model, the other based on saddle-point approximations to the measurement of probability mass function (PMF). The methods apply to both emission and transmission tomography; however, in this paper we focus on transmission tomography. We compare the new models to conventional data-weighted least-squares (WLS) and conventional maximum-likelihood methods [based on the ordinary Poisson (OP) model] using simulations and analytic approximations. The results demonstrate that the proposed methods avoid the systematic bias of the WLS method, and lead to significantly lower variance than the conventional OP method. The saddle-point method provides a more accurate approximation to the exact log-likelihood than the WLS, OP and shifted Poisson alternatives. However, the simpler shifted Poisson method yielded comparable bias-variance performance to the saddle-point method in the simulations. The new methods offer improved image reconstruction in PET through more realistic statistical modeling, yet with negligible increase in computation time over the conventional OP method.

Preliminary results from intensity-based CT-SPECT fusion in I-131 anti-B1 monoclonal-antibody therapy of lymphoma

BACKGROUND

In treatment of non-Hodgkin's lymphoma patients with predose-plus-I-131-labeled anti-B1 (anti-CD20) monoclonal antibody, an intratherapy single photon emission computed tomography (SPECT) image is an important part of research estimates of tumor dosimetry. For that imaging, a computed tomography (CT)-SPECT fusion is used both to obtain an attenuation map for the space-alternating generalized expectation maximization reconstruction and to provide CT-based volumes of interest (VoI) to determine activity in tumors and organs. Fusion based on external, skin-surface markers has been used but may not correctly superimpose internal structures.

METHODS

A new algorithm, developed and implemented in the Department of Radiology, University of Michigan, and based on the mutual information of grayscale values, was investigated. Results from four anti-B1 therapy patients are presented.

RESULTS

In one patient, the new intensity-based fusion provided total reconstructed counts for kidneys that were higher than those produced by marker-based fusion; therefore, the VoI was probably located more accurately. In a second patient, after an acquisition that did not include any skin markers, the new algorithm produced counts/pixel that were similar for four of five tumors consistent with what is expected from an ideal therapy combined with accurate count density estimates. The fifth tumor was quite small and will have its final activity estimate moved toward consistency with the others after a recovery coefficient multiplication. For four tumors in two patients, direct comparison of the two algorithms yielded count totals that were different by no more than 7.2%.

CONCLUSIONS

The use of CT-SPECT fusion and subsequent transfer of tumor VoI originally drawn in high-resolution CT space offers potential advantages for quantifying tumor uptake of radioactivity. A new, mutual-information-based fusion algorithm is usable without skin markers. Results indicate that the new fusion algorithm gives equal tumor count values within 7.2% compared with fusion based on external markers. It increases estimates of kidney activity by an average of 6.4%.

Grouped-coordinate ascent algorithms for penalized-likelihood transmission image reconstruction

This paper presents a new class of algorithms for penalized-likelihood reconstruction of attenuation maps from low-count transmission scans. We derive the algorithms by applying to the transmission log-likelihood a version of the convexity technique developed by De Pierro for emission tomography. The new class includes the single-coordinate ascent (SCA) algorithm and Lange's convex algorithm for transmission tomography as special cases. The new grouped-coordinate ascent (GCA) algorithms in the class overcome several limitations associated with previous algorithms. 1) Fewer exponentiations are required than in the transmission maximum likelihood-expectation maximization (ML-EM) algorithm or in the SCA algorithm. 2) The algorithms intrinsically accommodate nonnegativity constraints, unlike many gradient-based methods. 3) The algorithms are easily parallelizable, unlike the SCA algorithm and perhaps line-search algorithms. We show that the GCA algorithms converge faster than the SCA algorithm, even on conventional workstations. An example from a low-count positron emission tomography (PET) transmission scan illustrates the method.

In vivo mapping of cholinergic terminals in normal aging, Alzheimer's disease, and Parkinson's disease

To map presynaptic cholinergic terminal densities in normal aging (n = 36), Alzheimer's disease (AD) (n = 22), and Parkinson's disease (PD) (n = 15), we performed single-photon emission computed tomography using [123I]iodobenzovesamicol (IBVM), an in vivo marker of the vesicular acetylcholine transporter. We used coregistered positron emission tomography with [18F]fluorodeoxyglucose for metabolic assessment and coregistered magnetic resonance imaging for atrophy assessment. In controls (age, 22-91 years), cortical IBVM binding declined only 3.7% per decade. In AD, cortical binding correlated inversely with dementia severity. In mild dementia, binding differed according to age of onset, but metabolism did not. With an onset age of less than 65 years, binding was reduced severely throughout the entire cerebral cortex and hippocampus (about 30%), but with an onset age of 65 years or more, binding reductions were restricted to temporal cortex and hippocampus. In PD without dementia, binding was reduced only in parietal and occipital cortex, but demented PD subjects had extensive cortical binding decreases similar to early-onset AD. We conclude that cholinergic neuron integrity can be monitored in living AD and PD patients, and that it is not so devastated in vivo as suggested by postmortem choline acetyltransferase activity (50-80%).

Simultaneous transmission/emission myocardial perfusion tomography. Diagnostic accuracy of attenuation-corrected 99mTc-sestamibi single-photon emission computed tomography

BACKGROUND

The purpose of the present study was to assess the diagnostic performance of attenuation-corrected (AC) stress 99mTc-sestamibi cardiac single-photon emission computed tomography (SPECT) for the identification of coronary heart disease (CHD).

METHODS AND RESULTS

With a triple-detector SPECT system with a 241Am transmission line source, simultaneous transmission/emission tomography (TCT/ECT) was performed on 60 patients with angiographic coronary disease and 59 patients with < or = 5% likelihood of CHD. Iteratively reconstructed AC stress 99mTc-sestamibi perfusion images were compared with uncorrected (NC) filtered-backprojection images. Normal database polar maps were constructed from AC and NC images for quantitative analyses. From the low-likelihood patients, the visual and quantitative normalcy rates increased from 0.88 and 0.76 for NC to 0.98 and 0.95 for AC (P < .05). For the detection of CHD, the receiver operating characteristic curves for the AC images demonstrated improved discrimination capacity (P < .05), and sensitivity/specificity values increased from 0.78/0.46 (NC) to 0.84/0.82 (AC) with visual analysis and from 0.84/0.46 (NC) to 0.88/0.82 (AC) with quantitative analysis. For localization of stenosed vessels, visual and quantitative sensitivity values were 0.51 and 0.63 for NC and 0.64 and 0.78 for AC images (P < .05), respectively.

CONCLUSIONS

TCT/ECT myocardial perfusion imaging significantly improves the diagnostic accuracy of cardiac SPECT for the detection and localization of CHD. Clinical use of TCT/ECT imaging deserves serious consideration.

Spatial resolution properties of penalized-likelihood image reconstruction: space-invariant tomographs

This paper examines the spatial resolution properties of penalized-likelihood image reconstruction methods by analyzing the local impulse response. The analysis shows that standard regularization penalties induce space-variant local impulse response functions, even for space-invariant tomographic systems. Paradoxically, for emission image reconstruction, the local resolution is generally poorest in high-count regions. We show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections. This analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution. The modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized-likelihood methods.

Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography

Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore, investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson statistics. We also describe a "plug-in" approximation that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement. The approximations should be useful in a wide range of estimation problems.

Simultaneous transmission-emission thallium-201 cardiac SPECT: effect of attenuation correction on myocardial tracer distribution

This study evaluates the effect of attenuation correction on regional myocardial tracer distributions defined by 201TI cardiac perfusion SPECT images obtained from healthy volunteers and patients with coronary heart disease.

METHODS

A three-detector SPECT system equipped with an 241Am line source and a fanbeam collimator was used for simultaneous transmission/emission (201TI) tomography on 40 patients and 10 normal volunteers. Uncorrected emission images were reconstructed using filtered backprojection (FBP), whereas the attenuation corrected images were iteratively reconstructed with a regularized, least-squares algorithm utilizing the attenuation map computed from the transmission data. Both sets of images were reoriented into short-axis and vertical long-axis slices. Circumferential profile analysis was applied to both datasets of short-axis slices.

RESULTS

The normal volunteers demonstrated improved homogeneity in tracer distribution. For a basal short-axis slice, the lateral-to-posterior activity ratio improved from 1.17 +/- 0.12 for FBP to 1.01 +/- 0.07. Basal attenuation appeared properly compensated as the peak basal-to-apical slice activity gradient along the posterior-inferior wall changed from 1.15 +/- 0.12 for FBP to 1.01 +/- 0.09. The apex of the attenuation corrected images showed a significant decrease in activity relative to the base which appeared consistent with anatomic wall thinning. For the inferior and basal septal regions, the defect severity was slightly less in the attenuation corrected images, but the defects were more sharply defined compared to the FBP image defects.

CONCLUSION

These results indicate that attenuation correction is clinically feasible and accurately corrects for photon attenuation. Clinical validation, however, is necessary to define the diagnostic benefits.

Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms

Most expectation-maximization (EM) type algorithms for penalized maximum-likelihood image reconstruction converge slowly, particularly when one incorporates additive background effects such as scatter, random coincidences, dark current, or cosmic radiation. In addition, regularizing smoothness penalties (or priors) introduce parameter coupling, rendering intractable the M-steps of most EM-type algorithms. This paper presents space-alternating generalized EM (SAGE) algorithms for image reconstruction, which update the parameters sequentially using a sequence of small "hidden" data spaces, rather than simultaneously using one large complete-data space. The sequential update decouples the M-step, so the maximization can typically be performed analytically. We introduce new hidden-data spaces that are less informative than the conventional complete-data space for Poisson data and that yield significant improvements in convergence rate. This acceleration is due to statistical considerations, not numerical overrelaxation methods, so monotonic increases in the objective function are guaranteed. We provide a general global convergence proof for SAGE methods with nonnegativity constraints.

Globally convergent algorithms for maximum a posteriori transmission tomography

This paper reviews and compares three maximum likelihood algorithms for transmission tomography. One of these algorithms is the EM algorithm, one is based on a convexity argument devised by De Pierro (see IEEE Trans. Med. Imaging, vol.12, p.328-333, 1993) in the context of emission tomography, and one is an ad hoc gradient algorithm. The algorithms enjoy desirable local and global convergence properties and combine gracefully with Bayesian smoothing priors. Preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the EM algorithm. This superiority stems from the larger number of exponentiations required by the EM algorithm. The convex and gradient algorithms are well adapted to parallel computing.

Hybrid Poisson/polynomial objective functions for tomographic image reconstruction from transmission scans

This paper describes rapidly converging algorithms for computing attenuation maps from Poisson transmission measurements using penalized-likelihood objective functions. We demonstrate that an under-relaxed cyclic coordinate-ascent algorithm converges faster than the convex algorithm of Lange (see ibid., vol.4, no.10, p.1430-1438, 1995), which in turn converges faster than the expectation-maximization (EM) algorithm for transmission tomography. To further reduce computation, one could replace the log-likelihood objective with a quadratic approximation. However, we show with simulations and analysis that the quadratic objective function leads to biased estimates for low-count measurements. Therefore we introduce hybrid Poisson/polynomial objective functions that use the exact Poisson log-likelihood for detector measurements with low counts, but use computationally efficient quadratic or cubic approximations for the high-count detector measurements. We demonstrate that the hybrid objective functions reduce computation time without increasing estimation bias.

Comparison of americium-241 and technetium-99m as transmission sources for attenuation correction of thallium-201 SPECT imaging of the heart

This study compares the ability of 241Am and 99mTc to estimate 201Tl attenuation maps while minimizing the loss in the precision of the emission data.

METHODS

A triple-head SPECT system with either an 241Am or 99mTc line source opposite a fan-beam collimator was used to estimate attenuation maps of the thorax of an anthropomorphic phantom. Linear attenuation values at 75 keV for 201Tl were obtained by linear extrapolation of the measured values from 241Am and 99mTc.

RESULTS

Lung and soft-tissue estimates from both isotopes showed excellent agreement to within 3% of the measured values for 201Tl. Linear extrapolation did not yield satisfactory estimates for bone from either 241Am (+11.7%) or 99mTc (-15.3%). Patient data were used to estimate the dependence of crosstalk on patient size. Contamination from 201Tl in the transmission window was 5-6 times greater for 241Am compared to 99mTc, while the contamination in the 201Tl data in the transmission-emission detector head (head 1) was 4-5 times greater for 99mTc compared to 241Am. No contamination was detected in the 201Tl emission data of heads 2 and 3 from 241Am, whereas the 99mTc produced a small crosstalk component giving a signal-to-crosstalk ratio near 20:1. Measurements with a fillable chest phantom estimated the mean error introduced into the data from the removal of the crosstalk.

CONCLUSION

Based on the measured data, 241Am is a suitable transmission source for simultaneous transmission-emission tomography for 201Tl cardiac studies.

In vivo mapping of cholinergic neurons in the human brain using SPECT and IBVM

In the search for an in vivo marker of cholinergic neuronal integrity, we extended to human use the tracer (-)-5-[123I]iodobenzovesamicol (IBVM).

METHODS

IBVM, an analog of vesamicol that binds to the acetylcholine transporter on presynaptic vesicles, was prepared with specific activity greater than 1.11 x 10(9) MBq mmole-1. After intravenous injection of [123I]IBVM, body distribution studies (n = 5) and brain SPECT studies (n = 5) were performed on normal human subjects (n = 10). SPECT images of the brain were collected sequentially over the first 4.5 hr following injection, and again 18 hr later. Data were realigned and transformed to stereotaxic coordinates, and localized activities were extracted for tracer kinetic analysis. The cerebral tracer input function was determined from metabolite-corrected radial arterial blood samples. The best data fit was obtained using a three-compartment model, including terms reflecting cerebral blood volume, exchange of free tracer between plasma and brain and specific binding.

RESULTS

Dissociation of bound tracer was negligible for up to 4 hr. For the fitted parameters reflecting transport (K1) and binding site density index (k3), coefficients of variation were approximately 8% in cortical regions of interest. Relative distributions corresponded well with postmortem immunohistochemical values reported for the acetylcholine-synthesizing enzyme choline acetyltransferase, k3 (IBVM binding site density index), and tracer activity distribution at 22 hr, but not at 4 hr after injection.

CONCLUSION

SPECT imaging of [123I]IBVM succeeds as an in vivo measure of cholinergic neuronal integrity and should be useful for the study of cerebral degenerative processes such as Alzheimer's disease.

Model-based estimation with boundary side information or boundary regularization [cardiac emission CT]

The authors have previously developed a model-based strategy for joint estimation of myocardial perfusion and boundaries using ECT (emission computed tomography). They have also reported difficulties with boundary estimation in low contrast and low count rate situations. Here they propose using boundary side information (obtainable from high resolution MRI and CT images) or boundary regularization to improve both perfusion and boundary estimation in these situations. To fuse boundary side information into the emission measurements, the authors formulate a joint log-likelihood function to include auxiliary boundary measurements as well as ECT projection measurements. In addition, they introduce registration parameters to align auxiliary boundary measurements with ECT measurements and jointly estimate these parameters with other parameters of interest from the composite measurements. In simulated PET O-15 water myocardial perfusion studies using a simplified model, the authors show that the joint estimation improves perfusion estimation performance and gives boundary alignment accuracy of <0.5 mm even at 0.2 million counts. They implement boundary regularization through formulating a penalized log-likelihood function. They also demonstrate in simulations that simultaneous regularization of the epicardial boundary and myocardial thickness gives comparable perfusion estimation accuracy with the use of boundary side information.

Model-based estimation for dynamic cardiac studies using ECT

The authors develop a strategy for joint estimation of physiological parameters and myocardial boundaries using ECT (emission computed tomography). They construct an observation model to relate parameters of interest to the projection data and to account for limited ECT system resolution and measurement noise. The authors then use a maximum likelihood (ML) estimator to jointly estimate all the parameters directly from the projection data without reconstruction of intermediate images. They also simulate myocardial perfusion studies based on a simplified heart model to evaluate the performance of the model-based joint ML estimator and compare this performance to the Cramer-Rao lower bound. Finally, the authors discuss model assumptions and potential uses of the joint estimation strategy.

Penalized weighted least-squares image reconstruction for positron emission tomography

Presents an image reconstruction method for positron-emission tomography (PET) based on a penalized, weighted least-squares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, the author argues statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. The author proposes a simple data-based method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive over-relaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood expectation-maximization (ML-EM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP.

Object-based 3-D reconstruction of arterial trees from magnetic resonance angiograms

By exploiting a priori knowledge of arterial shape and smoothness, subpixel accuracy reconstructions are achieved from only four noisy projection images. The method incorporates a priori knowledge of the structure of branching arteries into a natural optimality criterion that encompasses the entire arterial tree. An efficient optimization algorithm for object estimation is presented, and its performance on simulated, phantom, and in vivo magnetic resonance angiograms is demonstrated. It is shown that accurate reconstruction of bifurcations is achievable with parametric models.