A Priori Motion Models for Four-Dimensional Reconstruction in Gated Cardiac SPECT

David S. Lalush‘,2, Lin Cui’, and Benjamin M. W. Tsui‘,* ’Department of Biomedical Engineering and 2Department of Radiology

The University of North Carolina at Chapel Hill

Abstract ‘We investigate the benefit of incorporating a priori

assumptions about cardiac motion in a fully four-dimensional (4D) reconstruction algorithm for gated cardiac SPECT. Previous work has shown that non-motion-specific 4D Gibbs priors enforcing smoothing in time and space can control noise while preserving resolution. In this paper, we evaluate metlhods for incorporating known heart motion in the Gibbs prior model. The new model is derived by assigning motion vectors to each 4D voxel, defining the movement of that volume of activity into the neighboring time frames. Weights for the Gibbs cliques are computed based on these “most likely” motion vectors. To evaluate, we employ the mathematical cardiac-torso (MCAT) phantom with a new dynamic heart model that simulates the beating and twisting motion of the heart. Sixteen realistically-simulated gated datacsets were generated, with noise simulated to emulate a real TI-20 1 gated SPECT study. Reconstructions were performed using several different reconstruction algorithms, all modeling nonuniform attenuation and three-dimensional detector response. These include ML-EM with 4D filtering, 4D MAP-EM without prior motion assumption, and 4D MAP-EM with prior motion assumptions. The prior motion assumptions included both the correct motion model and incorrect models. Results show that reconstructions using the 4D prior model can smooth noise and preserve time-domain resolution more effectively than 4D linear filters. We conclude that modeling of motion in 4D reconstruction algorithms can be a powerful tool for smoothing noise and preserving temporal resolution in gated cardiac studies.

I. INTRODUCTION Single-photon emission computed tomography (SPECT) is

widely used in cardiac perfusion studies. Conventional cardiac SPECT suffers from the problem of motion blur, in that the heart moves during the entire time the data is collected. Thus, the reconstructed image estimate only represents an average of the myocardial activity over the entire cardiac cycle.

Gated SPECT seeks to correct this problem by synchronizing data acquisition with the cardiac cycle. Instead of colllecting a single projection dataset, in gated SPECT, several “time frames” of projection data are acquired, each frame associated with a portion of the cardiac cycle. In

This work was supported by grant number CA39463 from the United States National Cancer Institute. Its contents are solely the responsibility of the authors and do not necessarily represent the official1 views of the National Cancer Institute.

reconstructing these data, we obtain a time series of 3D images representing the myocardial activity distribution as it changes during the cardiac cycle. The 3D images in each time frame offer improved spatial resolution as compared to the composite (ungated) image since the heart moves less within the time aperture of observation. Unfortunately, since significant increases in total counts are not possible, each time frame of the gated study has a fraction of the total counts of the composite study, and the individual gated images are significantly more noisy. ’

In previous work, we have proposed and evaluated the use of “space-time” Gibbs priors in a fully 4D MAP-EM algorithm for smoothing noise while still preserving some resolution [ 11. These priors enforce smoothing not only in the three spatial dimensions but also in the fourth dimension of time. By assuming that voxels of activity would move a relatively small distance from one time frame to the next, we found that it was possible to enforce smoothing constraints on the basis of the similarity in activity between 4D voxels that are close in both space and time. These constraints did not require any assumptions about the direction of motion.

In this paper, we evaluate methods for incorporating some a priori information about the motion of the heart in the Gibbs prior model. In this case, we will seek to enforce smoothing constraints only between space-time voxels that are consistent with the assumed direction of motion. By assigning motion vectors to each voxel in each time frame, we will weight the smoothing constraints according to the direction of motion.

In the next section, we explain some of the concepts behind the use of the space-time Gibbs model. We follow with an evaluation of the new prior model in comparison with previously-proposed methods.

11. THE SPACE-TIME GIBBS MODEL

A. Gibbs Priors Gibbs distributions [2] provide a framework for modeling

Markov random fields on multidimensional image lattices. They are particularly useful for applying smoothing constraints in iterative reconstruction algorithms using a Bayesian approach [3, 41. A class of Gibbs distributions can be defined on an image lattice whose voxel values are represented by the vector x:

P(x) represents the probability associated with a particular set of values for x, i.e., a particular image. The symbol 2

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represents a normalization factor that can be used to make the expression integrate to unity. it is not necessary for our purposes to know the value of 2. The parameter p controls the peaking of the distribution and ultimately determines the overall weight of smoothing applied in a MAP reconstruction algorithm [5]. The total potential, U@), is equal to the weighted sum of the potentials of individual cliques, or pairs (in our case) of voxels:

U(x) = c wij V(Xi - ” j ) iJuV

in Eq. (2), the summation is taken over all pairs (iJ) of voxels that have been defined to form cliques. Each clique has its own weight in the summation, represented by w,. The clique potential function, V( ), determines the relative penalty associated with two voxels that share a clique being different in value, and it has a significant effect on the image properties obtained in a MAP reconstruction [6].

By manipulating the weighting parameter p, the clique weights, and the potential function, it is possible to emphasize different conditions in a MAP reconstruction. Generally, such priors are used to promote intensity-specific smoothing between spatially-close voxels [7-91 by requiring that neighboring voxels should not differ too much in intensity, unless perhaps they are on a physical boundary. This is achieved by associating a relatively large potential, via the function V, with undesirable differences in voxel values. This translates to a high total potential, and thus a low posterior probability for a given solution image. In this way, smoother images are more likely, and thus have higher posterior probabilities.

B. Space-Time Gibbs Priors In our previous work in this area [l], we developed a

technique for enforcing smoothing between voxels not only in the three spatial dimensions, but also in the time dimension. if we consider that the reconstruction of gated SPECT data results in a time series of 3D images, we observe that this is simply a four-dimensional dataset. Thus, the image lattice can be defined in four dimensions, and we will refer to the elements of this lattice as space-time voxels.

The Gibbs cliques allow us to define smoothing relationships between neighboring voxels: two neighbors should not differ much in value. We usually think of such relationships in the three spatial dimensions, but it is also reasonable to assume that a voxel at a given spatial position in time frame one should not differ much from a voxel at the same spatial position in time frame two, provided the temporal sampling is sufficient. Thus, we can extend the smoothing concept to a four-dimensional space-time lattice.

We have previously used the edge-preserving properties of the potential function to apply four dimensional smoothing to gated SPECT images [1]. In this “no-motion’’ prior model, the potential function is designed to emphasize smoothing if the difference between neighboring voxels is of a moderate

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value, and to reduce smoothing if the difference is large. This promotes little smoothing between space-time voxels that clearly should not be associated since motion cannot have gone in that direction. It may, however, promote smoothing between voxels that coincidentally have similar intensities, even without motion in that direction.

C. Motion Modeling To carry the space-time model to the next logical step, we

need to incorporate assumptions about the motion into the Gibbs model. There are at least two ways to approach this. One is to modify the individual clique potential functions on the basis of the assumed motion. While we have implied in Eq. (2) that the potential function V is the same for all cliques, we may have different functions for each clique.

The second, somewhat easier possibility is to modify the clique weights (w in Eq. (2)) in accordance with the assumed motion. In this approach, we apply larger weights to those cliques that follow the assumed direction of motion, and smaller weights to those that do not (Fig. 1). This forces smoothing between a volume element in one time frame and the voxels to which it is most likely to have moved in the next time frame. As with any prior information, if the motion assumption is in error, we run the risk of associating voxels that should never be associated on the basis of motion.

D. Implementation In this section, we present the details of our

implementation of the motion model. We begin by defining a 3D motion vector for each space-time voxel to represent the expected motion of that voxel into the neighboring time frame. The motion vectors are computed by modeling the left ventricular inner and outer walls as ellipsoids that undergo affine transformations (rotation, scaling, and translation) with each frame. These motion vectors are defined in both the forward and reverse time dimensions. Thus, our prior information consists of two 3D motion vectors for each voxel in the 4D space. This is a great deal of information required in the most general sense. Yet, it is not difficult to reduce it by excluding locations that do not experience motion.

frame 1 frame 2 Figure 1: Illustration of the Gibbs model with assumed motion. The shaded voxel in frame 1 is assumed to have the motion vector as shown. The cliques involving the shaded voxel in frame 1 are weighted as shown in frame 2 . The cliques involving kame 2 voxels that are closer to the expected motion are weighted more heavily, as shown by the darker shading.

Once the motion vectors have been established, they are used to compute weights for individual cliques in the Gibbs model. The clique structure in the time domain is explained below. We will refer to a particular voxel in time frame two as the “source voxel.” The source voxel is a member of a clique with the voxel at the same spatial location in frame three, as well as all of the voxels that share sides, edges, or comers with that voxel. The same is true for the reverse direction, associating the frame two source voxel with those in frame one. This results in the source voxel in frame two sharing cliques with the twenty-seven nearest voxels in frame three and the twenty-seven nearest voxels in frame one. These fifty-four voxels will be referred to as the “target voxels.” Since cardiac SPECT results in a cyclical dataset, we wrap the last time frame around to associate with the first.

So, any given voxel is a member of fifty-four cliques in the time domain, not counting any spatial-only cliques. Spatial-only cliques, cliques where both members are in the same time frame, may be used, but we handle them separately [l]. The weight for a given time-domain clique is computed by taking the Euclidean distance between the center of the target voxel and the location of the expected motion of the source voxel (Fig. 2):

r 1-1 r 1-1

where: d,J is the Euclidean distance between the motion vector for source voxel i and the center of target voxel j . The summations are taken with respect to all of the target voxels in a given time frame; i.e., in Eq. (3), all of the elements k are in the same time frame as voxelj, and all the elements I are in the: same time frame as voxel i. For balance, the weight used for a given clique (iJ) must be computed as the average of the weights obtained with each voxel in the clique in the role of source voxel; hence, the two terms of Eq. (3).

The result of Eq. (3) is that cliques with voxels that are far from the expected target location are assigned very small weights while those that are close are assigned relatively

distance djj .

frame 1 frame 2 Figure 2: The method used for computing the clique weighting. The weighit for the clique involving the source voxel i and the target voxel , j is based on the Euclidean distance between the expected location of the movement, indicated by the arrow, and the center of the target voxel. This illustration is two-dimensional for simplicity; our implementation is three-dimensional.

large weights. If one of the dij terms is zero, then that clique receives a weight of one and all others receive a weight of zero. The summations ensure that the total weight applied is uniform for all voxels.

The final clique weights used in our reconstruction routine are then computed by applying a smoothing power equalization factor as discussed in [ 11:

where the c terms are elements of the projection matrix. For example, c k i represents the contribution of voxel i to projection bin k. The equalization factor makes the strength of smoothing more uniform throughout the image space [ 11.

Once the individual ‘clique weights are established, it remains to select the weighting parameter and the potential function. The weighting parameter is critical in determining the overall amount of smoothing applied [5] and generally must be selected through careful study. For the potential function, there are a variety to choose from [3,7-91. As in our previous work [ 11, we may apply different weighting parameters (ps) and potential functions (V) for the time- domain cliques and the spatial-only cliques. The spatial-only cliques are weighted in a uniform fashion based on the distance between voxels [I].

D. The 4 0 MAP-EM Algorithm If we were to reconstruct gated SPECT data without the

use of space-time Gibbs priors, we might employ an iterative algorithm with a fully 3D model such as ML-EM [lo] and reconstruct each time frame separately. With the addition of the Gibbs prior, we must reconstruct all of the time frames simultaneously. Thus, we have developed a fully four- dimensional reconstruction algorithm. This 4D MAP-EM algorithm was reported in [l] , and we include it here for completeness:

-old n

where i”” and x0ld represent the vectors of previous and current image estimates. As before, the c terms are elements of the projection matrix, and pj represents the measured counts in projection bin j . The summations over bins are taken for only the set (ti) of bins that are in the same time frame as space-time voxel i.

The algorithm computes one iteration for a particular time frame given the current image estimates for all time frames. When the estimate for one frame is complete, it continues with computing a single iteration for the next time frame, and so on. While it would be more appropriate to update all time frames at once, our approach saves precious program memory in this large 4D problem.

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111. SIMULATION STUDY

A. Methods To evaluate our motion-modeling technique, we employ

the mathematical cardiac-torso (MCAT) phantom [ 1 11 with a newly-developed dynamic heart model that simulates the beating and twisting motion of the heart. This phantom consists of sixteen frames in which the heart contracts, expands, and rotates. The total rotation used was fifteen degrees. A moving cold lesion at 80% intensity reduction was placed in the inferiolateral region of the left ventricle.

Very low-noise projection data for the sixteen time frames were simulated with Monte Carlo methods, including the effects of attenuation, scatter, and detector response. The acquisition simulated a typical T1-201 gated SPECT study with a LEGP collimator, 4 cm intrinsic camera resolution, and a radius of rotation of 22.5 cm. The acquisition arc was for 64 views over 180 degrees from 45” RA0 to 45” LPO. The bin size and reconstructed voxel size were 6.25 mm. The energy window was 30% centered at 74 kEv.

From the low-noise datasets, Poisson noise was simulated for a count level approximating a patient dose of 4 mCi TI- 201 with a total acquisition time of 27 seconds per view. In reconstructions, the potential function used in both space and time was a nonconvex function we have introduced previously [ 1,9]. Parameters were set by trial and error.

B. Results Images from a single slice of the MCAT data taken over

the sixteen frames are shown in Figs. 3 through 5. The noise- free ML-EM reconstruction after 50 iterations of ML-EM is shown in Figure 3. Reconstructions using our MAP-EM with the 4D motion model prior are shown in Fig. 4, and reconstructions using ML-EM with a 4D Butterworth filter are shown in Fig. 5 for comparison. Both MAP and filtered ML are able to recover the important features of the phantom from noisy data. The edge-enhancing properties of the Butterworth filter create a result with higher contrast, but the 4D filter results in a loss of time-domain resolution, making the defect persist in frames where it should appear only weakly. The 4D MAP technique sacrifices some of that contrast, but maintains the time-domain resolution.

These points are illustrated quantitatively in Fig. 6, where we plot the activity in a region of interest in the inferiolateral wall of the left ventricle. The defect starts outside of this location and rotates into the region of interest in frames 2 through 5 and then back out again. Results are shown for ML-EM with both 3D and 4D Butterworth filters (order 8, cutoff .2 cycles/pixel), and for MAP-EM with true motion, erroneous motion, and the no motion assumption model [I].

From Fig. 6, we find that MAP-EM is able to preserve the sharpness of the transition with the defect rotating into and out of the region much more effectively than the 4D Butterworth filter, yet MAP smoothes noise more effectively than the 3D Butterworth filter. The model with true motion

achieves the lowest mean-squared error, but all of the MAP techniques achieve lower mean-squared error than the filtered ML results. This indicates that perhaps the method with prior motion modeling will not be too sensitive to small errors in motion modeling. Of course, all of these results are dependent on the parameters chosen and on one instance of noise, so they should be interpreted only as indications of the potential of the smoothing techniques.

IV. CONCLUSION We have introduced techniques for modeling motion as a

prior constraint in reconstruction of gated cardiac SPECT studies. The technique permits noise smoothing in four dimensions with less degradation of time-domain resolution than linear 4D filtering. If the motion model is reasonably correct, the prior that models motion outperforms a previously-proposed technique that requires no motion model.

V. REFERENCES [I] D. S. Lalush and B. M. W. Tsui, “Space-time Gibbs

priors applied to gated SPECT myocardial perfusion studies,” in Three-dimensional Image Reconstruction in Radiology and Nuclear Medicine. Dordrecht, Netherlands: Kluwer Academic Publishers, 1996, pp.

[2] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans Putt Anal Mach Int, vol. PAMI-6, pp. 721- 741,1984.

[3] P. J. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm,’’ IEEE Trans Med Im, vol. MI-9, pp. 84-93, 1990.

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Figure 3: One slice taken kom sixteen frames of MCAT phantom reconstructions from noise-free data using 50 iterations of ML-EM, considered the best result possible with the given model.

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Figure 4: One slice taken from sixteen frames of MCAT phantom reconstructions from noisy data using 50 iterations of MAP-EM with the itrue motion modeled in the prior.

Figure 5: One slice takenom sixteen frames of MCAT phantom reconstructions from noisy data using 50 iterations of ML-EM followed by a 4D Butterworth filter (order 8 cutoff .2 cycles/pixel).

D. S . Lalush and B. M. W. Tsui, “A fast and stable maximum a posteriori conjugate gradient reconstruction algorithm,” Med Phys, vol. 22, pp. 1273-1284, 1995. D. S . Lalush and B. M. W. Tsui, “Simulation evaluation of Gibbs prior distributions for use in maximum a posteriori SPECT reconstructions,” IEEE Trans Med Im,

D. S. Lalush, “The application of a Gibbs prior with a generalized potential function to maximum a posteriori SPECT reconstruction,” PhD Dissertation, The University of North Carolina at Chapel Hill, 1992.

vol. MI-1 1, pp. 267-275, 1992.

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Figure 6: Activity from a region of interest in the inferiolateral region of the heart plotted versus time frame for ML-EM (top) followed by 3D and 4D Butterworth filters and MAP-EM reconstructions (bottom) using the true model, the erroneous model, and the model requiring no motion assumption. Mean-squared- errors with respect to the noise-free reconstruction are given in parentheses.

[7] S. Geman and D. McClure, “Bayesian image analysis: an application to single photon emission tomography,” Proceedings of the American Statistical Society, Statistical Computing Section, Washington, DC, 1985.

[8] T. J. Hebert and R. Leahy, “A generalized EM algorithm for 3D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans Med Im, vol. MI-8, pp. 194- 202, 1989.

[9] D. S. Lalush and B. M. W. Tsui, “A generalized Gibbs prior for maximum a posteriori reconstruction in SPECT,”Phys Med Biol, vol. 38, pp. 729-741, 1993.

[ 101 K. Lange and R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J Comp Assist Tomogr, vol. 8, pp. 306-316, 1984.

[11]B. M. W. Tsui, J. A. Terry, and G. T. Gullberg, “Evaluation of cardiac cone-beam Single Photon Emission Computed Tomography using observer performance experiments and receiver operating characteristic analysis,” Invest Radiol, vol. 28, pp. 1101- 1112, 1993.

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