A scatter correction method for contrast-enhanced dual-energy digital breast tomosynthesis.

Contrast-enhanced dual energy digital breast tomosynthesis (CE-DE-DBT) is designed to image iodinated masses while suppressing breast anatomical background. Scatter is a problem, especially for high energy acquisition, in that it causes severe cupping artifact and iodine quantitation errors. We propose a patient specific scatter correction (SC) algorithm for CE-DE-DBT. The empirical algorithm works by interpolating scatter data outside the breast shadow into an estimate within the breast shadow. The interpolated estimate is further improved by operations that use an easily obtainable (from phantoms) table of scatter-to-primary-ratios (SPR)--a single SPR value for each breast thickness and acquisition angle. We validated our SC algorithm for two breast emulating phantoms by comparing SPR from our SC algorithm to that measured using a beam-passing pinhole array plate. The error in our SC computed SPR, averaged over acquisition angle and image location, was about 5%, with slightly worse errors for thicker phantoms. The SC projection data, reconstructed using OS-SART, showed a large degree of decupping. We also observed that SC removed the dependence of iodine quantitation on phantom thickness. We applied the SC algorithm to a CE-DE-mammographic patient image with a biopsy confirmed tumor at the breast periphery. In the image without SC, the contrast enhanced tumor was masked by the cupping artifact. With our SC, the tumor was easily visible. An interpolation-based SC was proposed by (Siewerdsen et al 2006 Med. Phys. 33 187-97) for cone-beam CT (CBCT), but our algorithm and application differ in several respects. Other relevant SC techniques include Monte-Carlo and convolution-based methods for CBCT, storage of a precomputed library of scatter maps for DBT, and patient acquisition with a beam-passing pinhole array for breast CT. Our SC algorithm can be accomplished in clinically acceptable times, requires no additional imaging hardware or extra patient dose and is easily transportable between sites.

Collimator performance evaluation for In-111 SPECT using a detection/localization task.

In SPECT, the collimator is a crucial element in controlling image quality. We take a task performance approach to collimator performance evaluation in which an ideal observer is applied to the raw camera data without regard to the subsequent reconstruction stage. The clinical context of our collimator study is one of searching for and detecting neuroendocrine tumor metastases in the liver as seen in In-111 Octreotide SPECT. Our task involves detection and localization of a signal and thus differs from the conventionally used detection-only task. The scalar task performance metric is ALROC, the area under the localization receiver operating characteristic curve. Since In-111 emits photons at both 171 and 245 keV, the higher energy emissions can contribute significant septal scatter and penetration. Our collimator evaluations address a question previously considered by Mähler et al (2012 IEEE Trans. Nucl. Sci. 59 47­53) who used a different methodology: does allowing a limited amount of septal scatter and penetration yield improved task performance? We used simulation methods to evaluate five parallel-hole collimators. The collimators had roughly equal geometric sensitivity and resolution but a range of contributions from septal effects leading to variations in total sensitivity and resolution. We found that the best performance was obtained with a collimator that allowed a moderate amount of septal scatter and penetration.

Ideal AFROC and FROC observers.

Detection of multiple lesions in images is a medically important task and free-response receiver operating characteristic (FROC) analyses and its variants, such as alternative FROC (AFROC) analyses, are commonly used to quantify performance in such tasks. However, ideal observers that optimize FROC or AFROC performance metrics have not yet been formulated in the general case. If available, such ideal observers may turn out to be valuable for imaging system optimization and in the design of computer aided diagnosis techniques for lesion detection in medical images. In this paper, we derive ideal AFROC and FROC observers. They are ideal in that they maximize, amongst all decision strategies, the area, or any partial area, under the associated AFROC or FROC curve. Calculation of observer performance for these ideal observers is computationally quite complex. We can reduce this complexity by considering forms of these observers that use false positive reports derived from signal-absent images only. We also consider a Bayes risk analysis for the multiple-signal detection task with an appropriate definition of costs. A general decision strategy that minimizes Bayes risk is derived. With particular cost constraints, this general decision strategy reduces to the decision strategy associated with the ideal AFROC or FROC observer.

Collimator optimization in SPECT based on a joint detection and localization task.

In SPECT the collimator is a crucial element of the imaging chain and controls the noise-resolution tradeoff of the collected data. Optimizing collimator design has been a long studied topic, with many different criteria used to evaluate the design. One class of criteria is task based, in which the collimator is designed to optimize detection of a signal (lesion). Here we consider a new, more realistic task, the joint detection and localization of a signal. Furthermore, we use an ideal observer-one that attains a theoretically maximum task performance-to optimize collimator design. The ideal observer operates on the sinogram data. We consider a family of parallel-hole low-energy collimators of varying resolution and efficiency and optimize over this set. We observe that for a 2D object characterized by noise due to background variability and a sinogram with photon noise, the optimal collimator tends to be of lower resolution and higher efficiency than equivalent commercial collimators. Furthermore, this optimal design is insensitive to the tolerance radius within which the signal must be localized. So for this scenario, the addition of a localization task does not change the optimal collimator. Optimal collimator resolution gets worse as signal size grows, and improves as the level of background variability noise increases. These latter two trends are also observed when the detection task is signal-known-exactly and background variable.

Impact on reader performance for lesion-detection/ localization tasks of anatomical priors in SPECT reconstruction.

With increasing availability of multimodality imaging systems, high-resolution anatomical images can be used to guide the reconstruction of emission tomography studies. By measuring reader performance on a lesion detection task, this study investigates the improvement in image-quality due to use of prior anatomical knowledge, for example organ or lesion boundaries, during SPECT reconstruction. Simulated (67)Ga -citrate source and attenuation distributions were created from the mathematical cardiac-torso (MCAT) anthropomorphic digital phantom. The SIMIND Monte Carlo software was then used to generate SPECT projection data. The data were reconstructed using the De Pierro maximum a posteriori (MAP) algorithm and the rescaled-block-iterative (RBI) algorithm for comparison. We compared several degrees of prior knowledge about the anatomy: no knowledge about the anatomy; knowledge of organ boundaries; knowledge of organ and lesion boundaries; and knowledge of organ, lesion, and pseudo-lesion (non-emission uptake altering) boundaries. The MAP reconstructions used quadratic smoothing within anatomical regions, but not across any provided region boundaries. The reconstructed images were read by human observers searching for lesions in a localization receiver operating characteristic (LROC) study of the relative detection/localization accuracies of the reconstruction algorithms. Area under the LROC curve was computed for each algorithm as the comparison metric. We also had humans read images reconstructed using different prior strengths to determine the optimal trade-off between data consistency and the anatomical prior. Finally by mixing together images reconstructed with and without the prior, we tested to see if having an anatomical prior only some of the time changes the observer's detection/localization accuracy on lesions where no boundary prior is available. We found that anatomical priors including organ and lesion boundaries improve observer performance on the lesion detection/localization task. Use of just organ boundaries did not provide a statistically significant improvement in performance however. We also found that optimal prior strength depends on the level of anatomical knowledge, with a broad plateau in which observer performance is near optimal. We found no evidence that having anatomical priors use lesion boundaries only when available changes the observer's performance when they are not available. We conclude that use of anatomical priors with organ and lesion boundaries improves reader performance on a lesion-detection/localization task, and that pseudo-lesion boundaries do not hurt reader performance. However, we did not find evidence that a prior using only organ boundaries helps observer performance. Therefore we suggest prior strength should be tuned to the organ-only case, since a prior will likely not be available for all lesions.

The efficiency of the human observer for lesion detection and localization in emission tomography.

For the medically relevant task of joint detection and localization of a signal (lesion) in an emission computed tomographic (ECT) image, it is of interest to measure the efficiency, defined as the relative task performance of a human observer versus that of an ideal observer. Efficiency studies can be used in system optimization, improving postprocessing (e.g., reconstruction) algorithms, deriving human-emulating model observers and computer-aided detection methods. Calculation of ideal observer performance for ECT is highly computationally complex. We can, however, compute ideal observer performance exactly using a simplified 'filtered-noise' model of ECT. This model results in images whose correlation structure, due to quantum noise, background variability and regularization, is similar to that of real ECT reconstructed images. A two-alternative forced choice test is used to obtain the performance of the human observers. We compare the efficiency of our joint detection-localization task with that of a corresponding signal-known-exactly (SKE) detection task. For the joint task, efficiency is low when the search tolerance is stringent. Efficiency for the joint task rises with signal intensity but is flat for the SKE task. For both tasks, efficiency peaks at a mid-range level of regularization corresponding to a particular noise-resolution tradeoff.

Aperture optimization in emission imaging using ideal observers for joint detection and localization.

For the familiar 2-class detection problem (signal present/absent), ideal observers have been applied to optimization of pinhole and collimator parameters in planar emission imaging. Given photon noise and background and signal variabilities, such experiments show how to optimize an aperture to maximize detectability of the signal. Here, we consider a fundamentally different, more realistic task in which the observer is required to both detect and localize a signal. The signal is embedded in a variable background and is known except for location. We inquire whether the addition of a localization requirement changes conclusions on aperture optimization. We have previously formulated an ideal observer for this joint detection/localization task, and here apply it to the classic problem of determining an optimal pinhole diameter in a planar emission imaging system. We conclude that as search tolerance on localization decreases, the optimal pinhole diameter shrinks from that required by detection alone, and, in addition, task performance becomes more sensitive to fluctuations about the optimal pinhole diameter. As in the case for detection only, the optimal pinhole diameter shrinks as the amount of background variability grows and, in addition, conspicuity limits can be observed. Unlike the case for detection only, our task leads to a finite aperture size in the absence of background variability. For both tasks, the inclusion of background variability yields a finite aperture size.

Rapid Optimization of SPECT Scatter Correction Using Model LROC Observers.

The problem we address is the optimization and comparison of window-based scatter correction (SC) methods in SPECT for maximum a posteriori reconstructions. While sophisticated reconstruction-based SC methods are available, the commonly used window-based SC methods are fast, easy to use, and perform reasonably well. Rather than subtracting a scatter estimate from the measured sinogram and then reconstructing, we use an ensemble approach and model the mean scatter sinogram in the likelihood function. This mean scatter sinogram estimate, computed from satellite window data, is itself inexact (noisy). Therefore two sources of noise, that due to Poisson noise of unscattered photons and that due to the model error in the scatter estimate, are propagated into the reconstruction. The optimization and comparison is driven by a figure of merit, the area under the LROC curve (ALROC) that gauges performance in a signal detection plus localization task. We use model observers to perform the task. This usually entails laborious generation of many sample reconstructions, but in this work, we instead develop a theoretical approach that allows one to rapidly compute ALROC given known information about the imaging system and the scatter correction scheme. A critical step in the theory approach is to predict additional (above that due to to the propagated Poisson noise of the primary photons) contributions to the reconstructed image covariance due to scatter (model error) noise. Simulations show that our theory method yields, for a range of search tolerances, LROC curves and ALROC values in close agreement to that obtained using model observer responses obtained from sample reconstruction methods. This opens the door to rapid comparison of different window-based SC methods and to optimizing the parameters (including window placement and size, scatter sinogram smoothing kernel) of the SC method.

An overview of fast convergent ordered-subsets reconstruction methods for emission tomography based on the incremental EM algorithm.

Statistical reconstruction has become popular in emission computed tomography but suffers slow convergence (to the MAP or ML solution). Methods proposed to address this problem include the fast but non-convergent OSEM and the convergent RAMLA [1] for the ML case, and the convergent BSREM [2], relaxed OS-SPS and modified BSREM [3] for the MAP case. The convergent algorithms required a user-determined relaxation schedule. We proposed fast convergent OS reconstruction algorithms for both ML and MAP cases, called COSEM (Complete-data OSEM), which avoid the use of a relaxation schedule while maintaining convergence. COSEM is a form of incremental EM algorithm. Here, we provide a derivation of our COSEM algorithms and demonstrate COSEM using simulations. At early iterations, COSEM-ML is typically slower than RAMLA, and COSEM-MAP is typically slower than optimized BSREM while remaining much faster than conventional MAP-EM. We discuss how COSEM may be modified to overcome these limitations.

Human-observer LROC study of lesion detection in Ga-67 SPECT images reconstructed using MAP with anatomical priors.

We compare the image quality of SPECT reconstruction with and without an anatomical prior. Area under the localization-response operating characteristic (LROC) curve is our figure of merit. Simulated Ga-67 citrate images, a SPECT lymph-nodule imaging agent, were generated using the MCAT digital phantom. Reconstructed images were read by human observers.Several reconstruction strategies are compared, including rescaled block iterative (RBI) and maximum-a-posteriori (MAP) with various priors. We find that MAP reconstruction using prior knowledge of organ and lesion boundaries significantly improves lesion-detection performance (p < 0.05). Pseudo-lesion boundaries, regions without increased uptake which are incorrectly treated as prior knowledge of lesion boundaries, do not decrease performance.

Decision strategies that maximize the area under the LROC curve.

For the 2-class detection problem (signal absent/present), the likelihood ratio is an ideal observer in that it minimizes Bayes risk for arbitrary costs and it maximizes the area under the receiver operating characteristic (ROC) curve [AUC]. The AUC-optimizing property makes it a valuable tool in imaging system optimization. If one considered a different task, namely, joint detection and localization of the signal, then it would be similarly valuable to have a decision strategy that optimized a relevant scalar figure of merit. We are interested in quantifying performance on decision tasks involving location uncertainty using the localization ROC (LROC) methodology. Therefore, we derive decision strategies that maximize the area under the LROC curve, A(LROC). We show that these decision strategies minimize Bayes risk under certain reasonable cost constraints. The detection-localization task is modeled as a decision problem in three increasingly realistic ways. In the first two models, we treat location as a discrete parameter having finitely many values resulting in an (L + 1) class classification problem. In our first simple model, we do not include search tolerance effects and in the second, more general, model, we do. In the third and most general model, we treat location as a continuous parameter and also include search tolerance effects. In all cases, the essential proof that the observer maximizes A(LROC) is obtained with a modified version of the Neyman-Pearson lemma. A separate form of proof is used to show that in all three cases, the decision strategy minimizes the Bayes risk under certain reasonable cost constraints.

Fast LROC analysis of Bayesian reconstructed emission tomographic images using model observers.

Lesion detection and localization is an important task in emission computed tomography. Detection and localization performance with signal location uncertainty may be summarized by a scalar figure of merit, the area under the localization receiver operating characteristic (LROC) curve, A(LROC). We consider model observers to compute A(LROC) for two-dimensional maximum a posteriori (MAP) reconstructions. Model observers may be used to rapidly prototype studies that use human observers. We address the case background-known-exactly (BKE) and signal known except for location. Our A(LROC) calculation makes use of theoretical expressions for the mean and covariance of the reconstruction and, unlike conventional methods that also use model observers, does not require computation of a large number of sample reconstructions. We validate the results of the procedure by comparison to A(LROC) obtained using a gold-standard Monte Carlo method employing a large set of reconstructed noise samples. Under reasonable simulation conditions, our theoretical calculation is about one to two orders of magnitude faster than the conventional Monte Carlo method.

An accelerated convergent ordered subsets algorithm for emission tomography.

We propose an algorithm, E-COSEM (enhanced complete-data ordered subsets expectation-maximization), for fast maximum likelihood (ML) reconstruction in emission tomography. E-COSEM is founded on an incremental EM approach. Unlike the familiar OSEM (ordered subsets EM) algorithm which is not convergent, we show that E-COSEM converges to the ML solution. Alternatives to the OSEM include RAMLA, and for the related maximum a posteriori (MAP) problem, the BSREM and OS-SPS algorithms. These are fast and convergent, but require ajudicious choice of a user-specified relaxation schedule. E-COSEM itself uses a sequence of iteration-dependent parameters (very roughly akin to relaxation parameters) to control a tradeoff between a greedy, fast but non-convergent update and a slower but convergent update. These parameters are computed automatically at each iteration and require no user specification. For the ML case, our simulations show that E-COSEM is nearly as fast as RAMLA.

Rapid calculation of detectability in Bayesian single photon emission computed tomography.

We consider the calculation of lesion detectability using a mathematical model observer, the channelized Hotelling observer (CHO), in a signal-known-exactly/background-known-exactly detection task for single photon emission computed tomography (SPECT). We focus on SPECT images reconstructed with Bayesian maximum a posteriori methods. While model observers are designed to replace time-consuming studies using human observers, the calculation of CHO detectability is usually accomplished using a large number of sample images, which is still time consuming. We develop theoretical expressions for a measure of detectability, the signal-to-noise-ratio (SNR) of a CHO observer, that can be very rapidly evaluated. Key to our expressions are approximations to the reconstructed image covariance. In these approximations, we use methods developed in the PET literature, but modify them to reflect the different nature of attenuation and distance-dependent blur in SPECT. We validate our expressions with Monte Carlo methods. We show that reasonably accurate estimates of the SNR can be obtained at a computational expense equivalent to approximately two projection operations, and that evaluating SNR for subsequent lesion locations requires negligible additional computation.

A new convex edge-preserving median prior with applications to tomography.

In a Bayesian tomographic maximum a posteriori (MAP) reconstruction, an estimate of the object f is computed by iteratively minimizing an objective function that typically comprises the sum of a log-likelihood (data consistency) term and prior (or penalty) term. The prior can be used to stabilize the solution and to also impose spatial properties on the solution. One such property, preservation of edges and locally monotonic regions, is captured by the well-known median root prior (MRP), an empirical method that has been applied to emission and transmission tomography. We propose an entirely new class of convex priors that depends on f and also on m, an auxiliary field in register with f. We specialize this class to our median prior (MP). The approximate action of the median prior is to draw, at each iteration, an object voxel toward its own local median. This action is similar to that of MRP and results in solutions that impose the same sorts of object properties as does MRP. Our MAP method is not empirical, since the problem is stated completely as the minimization of a joint (on f and m) objective. We propose an alternating algorithm to compute the joint MAP solution and apply this to emission tomography, showing that the reconstructions are qualitatively similar to those obtained using MRP.

Rapid Computation of LROC Figures of Merit Using Numerical Observers (for SPECT/PET Reconstruction).

The assessment of PET and SPECT image reconstructions by image quality metrics is typically time consuming, even if methods employing model observers and samples of reconstructions are used to replace human testing. We consider a detection task where the background is known exactly and the signal is known except for location. We develop theoretical formulae to rapidly evaluate two relevant figures of merit, the area under the LROC curve and the probability of correct localization. The formulae can accommodate different forms of model observer. The theory hinges on the fact that we are able to rapidly compute the mean and covariance of the reconstruction. For four forms of model observer, the theoretical expressions are validated by Monte Carlo studies for the case of MAP (maximum a posteriori) reconstruction. The theory method affords a 10(2) - 10(3) speedup relative to methods in which model observers are applied to sample reconstructions.

Joint-MAP Bayesian tomographic reconstruction with a gamma-mixture prior.

We address the problem of Bayesian image reconstruction with a prior that captures the notion of a clustered intensity histogram. The problem is formulated in the framework of a joint-MAP (maximum a posteriori) estimation with the prior PDF modeled as a mixture-of-gammas density. This prior PDF has appealing properties, including positivity enforcement. The joint MAP optimization is carried out as an iterative alternating descent wherein a regularized likelihood estimate is followed by a mixture decomposition of the histogram of the current tomographic image estimate. The mixture decomposition step estimates the hyperparameters of the prior PDF. The objective functions associated with the joint MAP estimation are complicated and difficult to optimize, but we show how they may be transformed to allow for much easier optimization while preserving the fixed point of the iterations. We demonstrate the method in the context of medical emission and transmission tomography.