Tomographic Reconstruction of Dynamic Cardiac Image ... - IEEE Xplore

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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 4, APRIL 2007

Tomographic Reconstruction of Dynamic Cardiac Image Sequences Erwan Gravier, Member, IEEE, Yongyi Yang, Senior Member, IEEE, and Mingwu Jin, Student Member, IEEE

Abstract—In this paper, we propose an approach for the reconstruction of dynamic images from a gated cardiac data acquisition. The goal is to obtain an image sequence that can show simultaneously both cardiac motion and time-varying image activities. To account for the cardiac motion, the cardiac cycle is divided into a number of gate intervals, and a time-varying image function is reconstructed for each gate. In addition, to cope with the under-determined nature of the problem, the time evolution at each pixel is modeled by a B-spline function. The dynamic images for the different gates are then jointly determined using maximum a posteriori estimation, in which a motion-compensated smoothing prior is introduced to exploit the similarity among the different gates. The proposed algorithm is evaluated using a dynamic version of the 4-D gated mathematical cardiac torso phantom simulating a gated single photon emission computed tomography perfusion acquisition with Technitium-99m labeled Teboroxime. We thoroughly evaluated the performance of the proposed algorithm using several quantitative measures, including signal-to-noise ratio analysis, bias-variance plot, and time activity curves. Our results demonstrate that the proposed joint reconstruction approach can improve significantly the accuracy of the reconstruction. Index Terms—Dynamic cardiac image reconstruction, incremental gradient algorithm, motion compensation, single photon emission computed tomography (SPECT).

I. INTRODUCTION N this paper, we propose a joint estimation approach for reconstruction of dynamic images using gated cardiac data acquisition. In a gated study, the cardiac cycle is divided into a number of gate intervals by using the recorded electrocardiogram (ECG) signal, and the image is reconstructed for each gate interval. This effectively alleviates the blur caused by cardiac motion, and, consequently, the resulting gated images can provide valuable information not only about the distribution of the imaging radiotracer in the myocardium but also about the ventricular function [1]. Traditionally, the tracer distribution in a gated study is treated as constant over the data acquisition period, and a single static image is reconstructed for each gate. However, for certain radiotracers with fast update

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Manuscript received April 13, 2006; revised October 9, 2006. This work was supported in part by the National Institutes of Health under Grant HL65425. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Attila Kuba. The authors are with the Department of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, IL 60616 USA (e-mail: graverw@iit. edu; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2006.891328

and washout dynamics which are related to myocardial blood flow (e.g., Technitium-99m labeled Teboroxime), it is more desirable to reconstruct the time-varying tracer distribution in the myocardium, because it could potentially provide valuable extra information for differentiation of myocardial flow abnormalities [2], [3]. In this paper, our goal is to reconstruct from gated acquisition a dynamic image sequence that can show both the cardiac motion and the time-varying tracer distribution simultaneously. In the literature, there have been a number of techniques developed for reconstruction of dynamic tracer distributions. However, most of these techniques, if not all, treat the organ of interest as motionless. For example, in [4], a ML approach is developed based on a prior that the time activity at each pixel is either constant, increasing, decreasing, or first decreasing and then decreasing. In [5], the KL transformation of the data along the dynamic time axis is used to reconstruct the sequence. In [6], cubic B-splines are used to model the time activities at images pixels for reconstruction using list-mode data in positron emission tomography (PET). B-splines were also used in [7], [8] for temporal models in dynamic single photon emission computed tomography (SPECT), where no motion was included. Alternatively, there are also methods developed for directly estimating kinetic parameters from the dynamic projection data (e.g., see [9]–[12]). In the rest of this paper, we will develop our proposed reconstruction approach in the context of gated dynamic SPECT. This will facilitate our presentation of the material without introducing overly complex notation. However, the developed algorithm should also be equally applicable to other imaging modalities, such as PET. Gated SPECT perfusion imaging is currently one of the most important techniques for detection and evaluation of coronary artery disease. In a routine SPECT study, a radio-pharmaceutical is first introduced into the patient, and a rotating camera is used to image the emitted photons afterward. A challenging problem in gated SPECT is the amount of noise due to low data counts. In recent years there have been considerable interests in development of image processing methods for noise reduction in gated cardiac imaging. These methods aim to exploit the correlation among the signal components of the different gated frames [13]–[16]. In our own previous work [17], [18], we developed spatiotemporal reconstruction methods where estimated image motion was used to enforce the correlation among the gated frames. The work presented here is much different in that a dynamic sequence of images (rather than a single static image) is to be reconstructed for each gate. We point out that reconstructing dynamic images can be particularly challenging in gated cardiac SPECT on several aspects. First, during data acquisition, the projection data of the

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GRAVIER et al.: TOMOGRAPHIC RECONSTRUCTION OF DYNAMIC CARDIAC IMAGE SEQUENCES

time-varying tracer distribution are collected at only one angular position of the rotating camera for a given time instance, and, therefore, are far from being sufficient for reconstructing the dynamic images. This is especially troublesome when a slow rotating camera is used (as is the case in our experiments in Section III, where only three projection angles are available at any time point), because the fast-changing tracer dynamics can lead to significant inconsistencies in the projection data over time among the different angular positions. Second, the data counts will now be divided not only among the different gate intervals, as in traditional gated SPECT, but also throughout the dynamic time period. Therefore, the imaging noise will be even more significant compared to gated SPECT. The work in this paper was motivated by the early work in [19] and [20], which, to our knowledge, is the first to treat gated SPECT and dynamic SPECT jointly. In [21], a dynamic image sequence was first reconstructed by using the dynamic expectation–maximization (EM) algorithm of [4] for each gate frame, then smoothed by using Wiener filtering across the gated frames to further reduce the noise (without taking into account the image motion). In this work, we propose a joint reconstruction approach for dynamic gated image sequences, where the dynamic images of the different gates are treated collectively as a single signal, and determined from all the available data by using maximum a posteriori (MAP) estimation. Specifically, we will use spline functions to regularize the dynamic image activities along the time axis at the individual pixels, as in the B-spline modeling method in [6] and [22] for dynamic PET reconstruction. This will yield a compact representation of the dynamic images for each gate, and thereby can help cope with the incompleteness of the data. Then, we will introduce a motion-compensated smoothness prior as in gated SPECT [17] to exploit the similarity among the different gates. In our experiments, we tested the proposed reconstruction approach using the 4-D gated mathematical cardiac-torso (gMCAT) D1.01 phantom [23] with a tracer-kinetic model simulating imaging with Technitium-99m labeled Teboroxime [24]. The use of phantom data allowed us to thoroughly evaluate the reconstructed images using several quantitative measures where the ground truth was known. The results demonstrate that the proposed joint reconstruction approach can greatly improve the accuracy of dynamic images. The rest of the paper is organized as follows. In Section II, we present the imaging model used and derive the reconstruction algorithm. Evaluation study and implementation issues are discussed in Section III. Experimental results and discussions are given in Section IV, and, finally, conclusions are presented in Section V.

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sample time (postinjection). Then, the data are assumed to obey the following imaging model: (1) and are vectors representing, respectively, the where image activity and projection data during time interval for gate , denotes the expected value, and is the system matrix which is time-varying because of the rotation of the SPECT represent the data counts at the system. The elements of correspond individual projection bins, and the elements of to the tracer distribution at individual pixels. Our goal is to estifrom the data for all gates mate the image function and time . While the imaging model in (1) bears some similarity in form to that of gated SPECT, they are fundamentally different. In is treated as constant gated SPECT, the tracer distribution over during the course of data acquisition, and consequently, only a single static image needs to be reconstructed for each gate. With the dynamic imaging model, however, we need to refor each time . construct In this paper, we propose a joint estimation approach based on maximum MAP for reconstructing the dynamic images simultaneously for all the gates and time . To address the severely underdetermined nature of the problem, we will use B-splines to regulate the temporal activities at the individual pixels. Moreover, we will exploit the by enforcing similarity among the different gate frames smoothing along the motion trajectories. As will be demonstrated by our experiments, such an approach can lead to significant improvement in the reconstructed images. B. Joint Maximum A Posteriori Reconstruction 1) B-Spline Model for Dynamic Frames: Let denote that corresponds to pixel , . the element of For notational simplicity, here, the image pixels are denoted by over time using a single index. We model the activity cubic B-splines [25] as (2) where is the spline basis function, is the at pixel , weight (called control point) associated with is the total number of basis functions used. and Let be a vector formed by the control points at . Then (2) can be rewritten in a all the pixels in gate frame more compact form as

II. METHODS (3) A. Dynamic Imaging Model As in conventional gated SPECT, the dynamic projection data are binned into gating intervals within each cardiac cycle by using the recorded ECG signal. We use the angular incremental steps of the rotating SPECT camera to denote the progress of

Substituting (3) into the imaging model in (1), we get

(4)

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Hence, the problem becomes how to determine the control from the projection data. The dynamic images are points subsequently interpolated from the estimated control points by using the B-spline model. As will be clear from our experiments, the B-spline model in (3) provides a compact representation of the time activity curves (TACs) at the individual pixels, i.e., the number of basis is much smaller than that of the time points . functions This can have several desirable effects. First, it can help suppress the noise and cope with the incompleteness of the data. Second, such a compact representation can also reduce both the memory requirement and execution time in the reconstruction was used algorithms. For example, in our experiments, was used for the basis functions. for the time points and This represents more than 13-fold reduction in memory requirement. 2) MAP Estimate: We seek a penalized maximum likelihood (ML) approach for estimating the unknowns from the projection data , , for . For convenience, let , i.e., a vector consisting of all the control points associated with , i.e., a vector gate ; similarly, let consisting of all the projection data for gate . First, we derive the likelihood function of the data pa. Let denote the elrameterized by the control points corresponding to projection bin , . ement of denote the element of at position , Also, let which corresponds to the probability that at time a photon emitted from pixel will be detected in projection bin . Then, the imaging model in (4) can be rewritten as

(5) , which characAssuming a Poisson distribution for terizes emission tomography, we have the likelihood function

We seek a joint estimation approach for estimating all the from all the data unknowns . The MAP estimate of is obtained as

(8) where is the likelihood function of parameterized by , and is a prior distribution of . We note that the prior in (8) is defined on the control points instead of on the dynamic images explicitly. This is because the B-spline basis functions are well conditioned: small variations in the control points also correspond to small variations in the images, and vice versa [25]. Thus, a smoothness constraint imposed on will implicitly have a similar effect on . the reconstructed images. Below we define the prior term , defined in the following We assume a Gibbs prior for form: (9) where , are two potential functions defined in space and gate, respectively, and , are their corresponding scalar weighting parameters. in We choose the following spatial potential function (9) (10) where is the unit-distance neighborhood around pixel . The term is used to enforce local smoothness among neighboring pixels in each gate frame. in (9) as folWe choose the gate potential function lows:

(11)

(6)

Assuming that the detections at different detectors are statistically independent, we can write the overall likelihood function for each gate as

(7)

where the constant term

is omitted.

where

denotes the motion-compensated prediction of from its correspondence in gate , is the temporal distance between gates and , and is a non-negative constant. is For a gated cardiac sequence, the temporal distance replaced by the periodic distance between the two gates. in (11) is used to enforce smoothing along The term the motion trajectories across the different gates. Due to the periodic nature of gated acquisition, the different gate frames would be identical if not for the motion. Thus, in theory, the different gate frames are equally predictable from each other if the motion is known. The gate regularizer in (11) is intended to exploit this fact. However, in practice, the motion is not known in advance and has to be estimated from the noisy image data. With , the compensation coefficients in (11) are so defined that


temporally neighboring gates will have more contribution to the current gate than gates that are further apart. In our experiment, was used. In such a case, the motion compensation term of the current gate is formed primarily by the two nearest neighboring gates. In our implementation, linear interpolation is used for the pre, which is a linear function of the corrediction term in gate . Consequently, as a sponding control points whole the potential function in (11) assumes a quadratic form . in terms of the unknowns With all the terms defined, the MAP estimate in (8) can then be solved from maximizing the following objective function: (12) Such a definition of the objective function allows us to investigate conveniently the respective effects of spatial and gated regularization terms on the reconstructed images in our the optimal solution experiments. Specificically, when simply amounts to that of indeof the objective function pendently reconstructing the different gate frames using MAP; , the solution further reduces to that of in addition, when the ML estimate for individual frames. C. Reconstruction Algorithm The incremental gradient algorithm [22], [26] is used for optimization of the objective function in (12). First, as in a standard is ordered-subset algorithm [27], the set of projection data . Then, the divided into a number of subsets , objective function is partitioned accordingly as

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is a stepsize, is the gradient operator, and finally, diagonal normalization matrix (defined below). for the update on In (15), the diagonal entry of is given by for for where

(16)

is chosen as [22] (17)

has been chosen to mimic the EM Note that in the above is the algorithm. This is because the term detector to the activity associated with the sensitivity of the basis function over the scan duration. The diagonal eleis used to keep the update within the interval . ment becomes very small when the update gets close to Indeed, either 0 or . The algorithm in (15) is known to be globally convergent [22], [26]. For each full iteration , the update in (15) is carried . Note that all the iterates out for every subiteration will stay inside the interval . This will guarantee that the resulting dynamic images will always be nonnegative. and , and the In our experiments, we set was chosen as . stepsize III. EVALUATION STUDY AND IMPLEMENTATION ISSUES A. Image Data

(13) where is the total number of subsets, and likelihood function corresponding to the subset

is the . That is

(14) The control points

are solved iteratively as [22], [26] (15)

where is the iteration index, is the subiteration index, is with a projection operator onto being a small positive number and is a constant, is the

To test the proposed reconstruction method, we used the 4-D gated mathematical cardiac-torso (gMCAT) phantom [23] with a tracer-kinetic model simulating imaging with Technitium-99m (Tc m) labeled Teboroxime [24]. The use of phantom data provides the ground truth for our quantitative evaluation. As an example, a slice of the phantom (no. 64) is shown in Fig. 1(a) (at 3.8 min). The introduced TACs are shown in Fig. 1(b) for the different organs, including a defect introduced at the base of the septal wall of the left ventricular myocardium. We simulated a clinical Tc m Teboroxime study with a Philips Prism 3000 SPECT system, which is a three-headed system. The field of view was 40.5 cm, which results in a pixel size of 0.634 cm for a 64 64 pixel acquisition matrix. In our simulation, a total of eight gates were used, and the total data acquisition time was 12 min, during which the camera underwent 480 of continuous rotation. We used a total of 60 angular positions for each full camera rotation (i.e., 360 by each head). Thus, each rotation step corresponds to approximately 9 s in time. Poisson noise was introduced such that the total number of counts for the entire scan was eight millions. Specifically, the heart had a total of about 360 000 counts, whereas the liver had about five million counts. The acquisition was simulated as including distance-dependent spatial resolution, and no attenuation was included. The average spatial resolution at the location of the heart in the reconstructed slices was approximately

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Fig. 2. Illustration of the different ROIs used in quantitative evaluation of the reconstructed images: ROI5D-LV for SNR, ROI5D-BVH, and ROI5D-BVD for bias-variance and TACs.

the entire left ventricle. This ROI will be referred to as ROI5D-LV, as shown in Fig. 2. The SNR is defined as (18)

Fig. 1. (a) Slice #64 of the gMCAT Phantom and (b) the introduced dynamics in the different organs of the phantom.

1.3-cm full width at half-maximum (FWHM). Our evaluation results below were based on the slice shown in Fig. 1(a). In this study, a corrupting effect, such as attenuation, was not included in the simulated acquisition in the development stage of the methods. In practice, such an effect will have to be estimated, and is subject to errors; thus, inclusion of it would greatly complicate our simulation study. By isolating it from the simulations, we can focus on investigating the effect solely due to the proposed reconstruction approach. While this may lead to biased results in performance assessment in the absolute sense, it is reasonable to expect that it would not affect the relative ordering of the reconstruction methods in performance, as it would equally corrupt the various reconstruction methods. B. Performance Evaluation Methods To quantify the accuracy of the reconstruction results, we used a number of measures, including: 1) signal to noise ratio (SNR), 2) bias-variance plot, and 3) TACs for different regions of interest (ROI) selected in the myocardium. These measures are defined as follows. 1) To calculate the overall reconstruction accuracy of our algorithm, we use SNR analysis for a ROI containing

where and represent the original and reconstructed images of the ROI, respectively. 2) To measure the local reconstruction accuracy, we use biasvariance plot for two 2 2 ROIs in the left ventricle (the first gate). The first ROI is a healthy region in the heart wall, labeled as as ROI5D-BVH in Fig. 2, whereas the second ROI is positioned at the perfusion defect, labeled as ROI5D-BVD in Fig. 2. In our experiments, we calculated the average intensity for each of these two ROIs from 20 different noise realizations. Then, we calculated the bias (from the original) and variance of this estimator. 3) We also calculate the TACs for the two ROIs: ROI5D-BVH and ROI5D-BVD. This will demonstrate whether the reconstruction can effectively differentiate healthy and abnormal temporal activities in the myocardium.

C. Implementation Issues 1) B-Spline Knot Placement: To apply the cubic B-spline model in (2), we need to specify both the number of knots and their temporal positions. First, to allow discontinuities at both the start and end of the acquisition period (from 0 to 12 min), we introduce four knots to clamp the B-spline curves at each of the two end-points. Next, we have to decide how many internal knots to use and where to place them. Toward a compact spline representation, we would like to keep the number of knots as small as possible. This is because with a compact model we can not only avoid over-fitting the data by the B-splines (hence, more robust to noise), but also achieve both memory saving and faster computation. On the other hand, we also want to have enough knots to obtain an accurate representation of the radiotracer dynamics.


Fig. 3. Dynamic activity curve of the entire phantom estimated directly from the projection data.

In our experiments, we found that using two internal knots can lead to sufficiently accurate representation of the tracer dynamics for reconstruction. Of these two internal knots, one is placed during the initial uptake period of the heart, where there is fast change in the tracer activity, whereas the other is placed at the start of the washout period. Specifically, these two knots were placed in the experiments as follows. First, by summing up the acquired counts over all the projection bins at each camera step, we obtain an estimate of the overall dynamic curve of the entire phantom. For noise reduction, the resulting curve is further smoothed by an approximate cubic spline. As an example, Fig. 3 shows an overall dynamic curve obtained from one noise realization using this process. The first internal knot is then placed at the first peak location of the curve. Next, the second derivative of the overall dynamic curve is computed, which is further smoothed by a cubic spline approximation. The second internal knot is placed afterward at the first zero-crossing location of the resulting curve. In our experiments, we tested the robustness of the knot placement procedure above. As to be discussed later, the results demonstrate that the reconstruction algorithm is rather insensitive to pertubation of the knot locations. It was found that, with 20 different noise realizations, the first knot was placed on average at 0.5 min, and the second knot was placed on average at 1.5 min. In Fig. 4, we show the six , , with the knot placement at basis splines [0,0,0,0,0.5,1.5,12,12,12,12] min. The locations of the knots are indicated by the vertical lines. Mathematically, these basis splines have the following form: (19) where

and if otherwise where

1, 2, 3 and

is the location of the

knot.

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Fig. 4. Illustration of the six basis B-spline functions. Vertical lines indicate the knot locations.

Fig. 5. Illustration of the partitioning of the projection data into subsets. The three angular sectors (in bold) form a single subset.

D. Partitioning of the Data into Subsets We need to partition the data for the incremental gradient algorithm in (15). Since the camera we used has three heads, it is natural to partition the data accordingly. As specified above, a total of 60 angular positions were used during a full rotation of the camera. The projection data were accordingly partitioned into 20 subsets, of which each consisted of projections obtained at three equally-spaced angular positions. This is illustrated in Fig. 5, where each angular sector of the disk represents the portion of the data acquired at that angular position; the three sectors in shade form one subset, and the whole dataset can be accessed by successively turning the selection wheel. To ensure that the consecutive subsets in the update algorithm contain as much different information as possible [27], we need to process the subsets according to a specific order. Let the 20 subsets be numbered sequentially from 1 to 20 when the selection wheel was advanced one angular position at a time. Then the following order of subsets was used in our experiments: 1, 11, 6, 16, 2, 12, 7, 17, 3, 13, 8, 18, 4, 14, 9, 19, 5, 15, 10, 20.

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Fig. 6. (a) Reconstructed six control frames for gate one; (b) the third control frame for gates one to eight.

1) Motion Estimation: The definition of the prior term in (11) requires the correspondence of the image pixels among the different gate frames. To be realistic, in our experiments, we estimated the image motion from the image data. First, we from the image data using reconstruct the control points our algorithm with spatial only regularization ( , ). As an example, Fig. 6(a) shows the resulting control for the first gate from one particular noise realizaframes tion, where the first row shows frames one to three (left to right), whereas the second row shows frames four to six. As can be seen, the heart is most visible in the third frame, while the liver is most visible in the fourth control frame. This is because the shape of the third spline basis function closely matches that of the TACs of the pixels in the left ventricle. In a way the third control frame effectively separates most of the heart signals from the rest of the control frames. Therefore, this control frame is used to estimate the image motion between the different gates, as our primary interest is to reconstruct the cardiac signals. This can help reduce the effect of other undesired signal components such as imaging noise and activities of other organs. In Fig. 6(b), we show the reconstructed third control frame for all the gates from one particular noise realization, where the first row shows gates one to four (left to right), whereas the second row shows gates five to eight. As can be seen, the heart wall motion is clearly manifested in these gate frames. In our experiments, we applied the optical flow method in [17] to estimate the image motion between the different gates from these control frame images. The resulting motion vectors are then used in the reconstruction algorithm for all the control frames. As an

Fig. 7. (a) Motion field estimated from noiseless data (from gate #2 to gate #3); (b) the motion field estimated from noisy data (from gate #2 to gate #3).

Fig. 8. SNR plot of the reconstruction for different parameter settings at 1.5 min.

example, in Fig. 7, we show the estimated motion field between gates #2 and #3 from one noise realization; for comparison the


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Fig. 10. Reconstructed dynamic images with no spatial and gate regularization ( = 0 and = 0). The horizonal axis shows the images at different gate intervals (1–8), whereas the vertical axis shows the progression over time (1 min 20 s, 2 min 40 s, 4 min, etc.).

Fig. 9. SNR plot of the reconstruction for different parameter settings at: (a) 1.5 min; (b) 5 min; and (c) 9 min.

motion field estimated from the noiseless phantom data of these two gates is also shown. IV. EXPERIMENTAL RESULTS AND DISCUSSIONS To investigate the effect of the spatial and gate smoothing terms in the prior, we tested the reconstruction algorithm over and a wide range of values of the regularization parameters . We show in Fig. 8 a plot of the SNR of the reconstructed and . At each setting, the images for different values of SNR was calculated for the first gate at time 1.5 min, and the same noise realization for the projection data was used. Note

that the best SNR result (8.8 dB) was obtained when and , i.e., when no spatial smoothing was used. Moreover, in Fig. 9, we show the SNR results at three different time points: 1.5, 5.0, and 9 min. Each curve in Fig. 9 was while holding the obtained by varying the spatial parameter constant. The SNR results were calculated gate parameter for all the eight gates, and then averaged. To mitigate the effect of random realizations, we used 20 different noise realizations in our experiments, and the results in Fig. 9 are the average from these realizations. The results in Figs. 8 and 9 reveal that use of gate regularization term offers the most benefit in improving the reconstruction accuracy of the myocardium. In particular, in the early stage of the dynamics (1.5 min) the best result was obtained with gate , but no spatial smoothing at all. Howparameter ever, in the later stage (5 and 9 min), the best results were obtained with increased spatial smoothing. We believe that this is due to the fact that the heart wall is very thin (only a few pixels thick) and, thus, use of spatial smoothing will lead to undesirable blurring (i.e., loss of spatial resolution). However, in the later washout period of the tracer dynamics, the noise level of the data increases with time (due to reduced counts), and use of additional spatial smoothing will be of benefit. Another observation that can be made from the results in Figs. 8 and 9 is that when no gated regularization was used, , the optimal result was achieved with a much i.e., larger value of the spatial parameter (hence, more spatial

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Fig. 11. Reconstructed dynamic images with spatial only regularization ( = 2 10 and = 0). The horizonal axis shows the images at different gate intervals (1– 8), whereas the vertical axis shows the progression over time (1 min 20 s, 2 min 40 s, 4 min, etc.).

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smoothing) than when gate regularization was used (e.g., ). Therefore, it is reasonable to expect that the use of gate regularization can help improve the spatial resolution of the reconstruction. We show in Figs. 10–12 a set of reconstructed images from our algorithm with the following three different settings: 1) no , ), 2) spatial spatial and gate regularization, i.e., ( only regularization ( , ), and 3) gate regu, ). In each figure, the horizonal larization ( axis shows the cardiac images at different gate intervals (1– 8), whereas the vertical axis shows the progression over time (1 min 20 s, 2 min 40 s, 4 min, etc.). Thus, these images allow us to examine not only the effect of wall motion (horizontally), but also the uptake and washout of the tracer over time (vertically). For comparison, we show in Fig. 13 a magnified view of the images of the myocardium reconstructed by the three methods for gate #3 and gate #8 at time 1 min 30 s. Here we also show the corresponding noiseless reconstruction of the phantom, which was reconstructed by assuming all projection angles available at a particular time instance. For brevity, the noiseless reconstruction is shown only for the gates and time point in Fig. 13. As can be seen, the impact of imaging noise is clearly visible in the reconstructed images when no regularization was used. The noise level is notably mitigated when spatial regularization was used, but the heart wall also suffers from notable spatial blurring (which also exhibits as blurred wall motion). The images from the gate regularization method suffers from the least of these artifacts.


Fig. 12. Reconstructed dynamic images with gate regularization ( = 0 and = 2 10 ). The horizonal axis shows the images at different gate intervals (1–8), whereas the vertical axis shows the progression over time (1 min 20 s, 2 min 40 s, 4 min, etc.).

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Fig. 13. Magnified view of the reconstructed images (gates #3 and #8 at time 1.5 min) by the different methods: 1) noise-free dynamic gMCAT phantom (Noise-free), 2) no spatial and gate regularization (likelihood), 3) spatial only regularization (spatial), and 4) gate regularization (5D).

Furthermore, while the images in Fig. 12 show that the heart wall is most visible during the uptake stage, which gradually becomes dimmer during the washout period, the defect at the base of the heart wall is notably differentiated from the rest of the healthy myocardium in gates two through five. That is, it is initially darker than the rest of the heart wall, but becomes brighter as the rest of the heart wall becomes dimmer. To quantify the differentiation in dynamics between the defect and healthy myocardium, in Fig. 14, we show the average TACs of ROI5D-BVH and ROI5D-BVD, respectively, of the reconstructed images for the case of spatial only regularization


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Fig. 14. Reconstructed TACs of the defect and healthy ROIs in the myocardium for 1) spatial only regularization (spatial), and 2) gate regularization (5D). The noise-free reconstruction TACs are also shown for comparison (noise-free).

( , ) and the case of gate regularization , ). For comparison, the noise-free recon( struction TACs of these two ROIs are also shown. These results show that gate regularization yields more accurate reconstruction of the TACs for both the defect and healthy ROIs, and that the TAC of the defect ROI is better separated from that of the healthy ROI (thus, better differentiation). This is consistent with the earlier observed image difference between the defect and the healthy myocardium. To further quantify the reconstruction accuracy, in Fig. 15, we show the bias-variance plots of the health ROI5D-BVH and defect ROI5D-BVD respectively at time 5 min. Clearly, the use of gate regularization leads to more accurate reconstruction for both the healthy and defect regions of the myocardium. We also tested the effect of the knot locations on the reconstruction results. For example, when two internal knots were deliberately perturbed from their original positions of [0.5,1.5] to [0.1,2.5], our results show that the reconstructed TACs of ROI5D-BVH and ROI5D-BVD still show good differentiation between the defect and healthy ROIs. Due to space limitation, these results are not shown. In summary, our test results indicate that the reconstruction algorithm is rather robust to the knot placement in the B-spline model. Finally, we note that for demonstration of the proposed reconstruction approach the optical flow method was used in our experiments, where the motion was estimated from reconstruction without temporal regularization. This is similar in spirit to the work in [28] where the motion and image parameters were estimated jointly. It is reasonable to expect that some additional improvement could be achieved if a more elaborate motion model is used. For example, one could re-estimate the motion from the newly reconstructed images (at the expense of additional computation). This would be similar to the approach of [15] and [16], where the image values and motion were alternately estimated from the data. Thus, the results achieved in this study can be viewed as a lower bound on the performance of the proposed algorithm.

Fig. 15. Bias-variance plots of the mean intensity of (a) ROI5D-BVH and (b) ROI5D-BVD, respectively, estimated from reconstruction with different parametric settings.

V. CONCLUSION In this paper, we presented a new imaging technique for reconstructing a dynamic image sequence which shows not only heart motion but also the time-varying distribution of the radiotracer. To deal with the high level of noise in the data, we regularized the dynamic images both along the temporal direction and across the different gate intervals. We used B-splines to model the dynamic evolution of the tracer, and introduced both spatial and gate regularization to exploit the signal correlations. The use of B-splines also enabled a compact representation of the data, reducing the storage of the data. The evaluation results are both illuminating and promising: use of gate regularization can improve significantly the accuracy of the reconstruction without at the expense of spatial resolution; in addition, it also allows better differentiation between healthy and abnormal dynamics in the myocardium. Encouraged by this success, we plan to further test the proposed algorithm using clinical acquisitions in an ensuing study.

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ACKNOWLEDGMENT The authors would like to thank Dr. M. A. King in the Department of Radiology, University of Massachusetts Medical School, for generously providing them with the simulated imaging data used in this study.

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[20] T. H. Farncombe, B. Feng, M. V. Narayanan, M. N. Wernick, A. M. Celler, M. A. King, and J. A. Leppo, “Toward 5 dimensional SPECT reconstruction: Determining myocardial blood flow and wall motion in a single study,” presented at the 7th Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, 2003. [21] T. H. Farncombe, M. A. King, A. Celler, and S. Blinder, “A fully 4D expectation maximization algorithm using gaussian diffusion based detector response for slow camera rotation dynamic SPECT,” in Proc. Int. Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, 2001, pp. 129–132. [22] Q. Li, E. Asma, and R. M. Leahy, “A fast fully 4D incremental gradient reconstruction algorithm for list mode PET data,” in Proc. IEEE Int. Symp. Biomedical Imaging: Macro to Nano, 2004, pp. 555–558. [23] P. H. Pretorius, M. A. King, W. Xia, B. M. W. Tsui, T. S. Pan, and B. J. Villegas, “Evaluation of right and left ventricular volume and ejection fraction using a mathematical cardiac torso phantom for gated blood pool SPECT,” J. Nucl. Med., vol. 38, no. 10, pp. 1528–1534, 1997. [24] A. Celler, S. Blinder, D. Noll, T. Tyler, F. Duclercq, and R. Harrop, “Investigation of Tc99m-teboroxime myocardial perfusion using dynamic SPECT and a 4D kinetic thorax model dmCAT,” in Proc. Int. Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, 2002, pp. 1–34. [25] C. D. Boor, Ed., A Practical Guide to Splines. New York: SpringerVerlag, 1978. [26] S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms,” IEEE Trans. Med. Imag., vol. 22, no. 5, pp. 613–626, May 2003. [27] G. T. Herman and L. B. Meyer, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imag., vol. 12, no. 3, pp. 600–609, Mar. 1993. [28] M. W. Jacobson and J. A. Fessler, “Joint estimation of image and deformation parameters in motion-corrected PET,” in Proc. IEEE Nuclear Science Symp. Medical Imaging Conf., 2003, vol. 5, pp. 3290–3294. Erwan Gravier (M’00) received the diploma of electrical engineering from the Ecole Nationale Superieure de l’Electronique et de ses Applications (ENSEA), Paris, France, in 2001, and the M.S. and Ph.D. degrees in electrical engineering from the Illinois Institute of Technology (IIT), Chicago, in 2001 and 2005, respectively. From 2001 to 2005, he was with the Medical Imaging Research Center, IIT. His main projects were 4-D and dynamic cardiac SPECT image reconstruction. He is currently a Consultant for Altran Automotive Infrastructure and Transportation (Altran AIT), Paris. His current projects deal with on-board software for diesel engine management. Yongyi Yang (M’97–SM’03) received the B.S.E.E. and M.S.E.E. degrees from Northern Jiaotong University, Beijing, China, in 1985 and 1988, respectively, the M.S. degree in applied mathematics, and the Ph.D. degree in electrical engineering from Illinois Institute of Technology (IIT), Chicago, in 1992 and 1994, respectively. He is currently on the faculty of the Department of Electrical and Computer Engineering, IIT, where he is an Associate Professor. Prior to this position, he was a faculty member with the Institute of Information Science, Northern Jiaotong University. He is a coauthor of Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, 1998). His research interests are in signal and image processing, medical imaging, machine learning, pattern recognition, applied mathematical and statistical methods, real-time signal processing systems, and biomedical applications. Mingwu Jin (S’05) received the B.S. degree in space physics and the M.S. degree in electrical engineering from Peking University, Beijing, China, in 1997 and 2001, respectively. He is currently pursuing the Ph.D. degree in electrical engineering at Illinois Institute of Technology, Chicago. His current research is focused on signal and image processing, medical imaging, pattern recognition, and data mining.