Combined Fan/Parallel Beam Reconstruction in Cardiac SPECT Imaging

Manoj V. Nuayanad, Michael A. King' and Charles L. B p e 2 'University of Massachusetts Medical School, Worcester, MA, aUniversity of Massachusetts, Lowell, MA

Abstract With three-headed SPECT systems, two heads

can acquire truncation-free emission images with parallel-beam (PB) collimators, while the third head acquires transmission images through a fan-beam (FB) collimator. The FB collimator can also be used for emission imaging. Combining FB and PB emission data from the 3 heads, offers the potential for improved sensitivity with the trade-off being truncation artifacts. In this work, we investiga.te 3 methods to combine fan and parallel beam (F&PB) reconstructions. The first approach uses a PB estimate as a prior for the regularized AB-EMML algorithm to reduce truncation in the FB reconstruction. The second approach combines the parallel and resorted FB projection8 and reconstructs a composite image. The third method is based on the work of Jaszczak et al., [4] where the intermediate F&PB reconstructions are combined to reduce the truncation artifacts inherent in FB imaging. MCAT simulation results indicate that the combined F&PB reconstructions provide decreased image noise while greatly limiting the truncation artifacts inherent in FB imaging.

I. INTRODUCTION Three headed SPECT systems offer the capability

for simultaneously acquiring transmission and emission tomographic data in cardiac imaging. Two heads fitted with parallel-beam (PB) collimators can acquire truncation-free cardiac projections. Simultaneously, the third head with a fan-beam (FB) collimator can acquire transmission projections from an opposed line source and emission projections from the patient [l]. This simultaneous transmission-emission acquisition offers the advantage of attenuation correction for cardiac images with no additional patient imaging time.

Currently at our clinic, attenuation-corrected cardiac images are reconstructed using data solely from the two PB colliiators. Due to the different collimator geometry (FB) on the remaining head, we currently discard the additional FB counts that are available for reconstruction. The main drawback with use of the FB emission projection data is the reconstruction artifacts which occur in the slices as a result of the diminishing field-of-view (FOV) with distance from the face of the FB collimators. These artifacts are especially deleterious in cardiac imaging if the heart is truncated in some projections. The presence of these reconstruction artifacts complicates the task of utilizing the additional FB counts to reduce the noiselevels in the resulting reconstruction.

Electrocardiographic (ECG) gating or fespiratory gating [2] of conventional SPECT images results in a reduced number of counts in the image being reconstructed. Under such conditions, utilizing the additional FB counts from the third camer&head would help alleviate to some extent, the poor count-statistics. Therefore, the motivation for this research was to develop combined fan/parallel beam reconstruction strategies that offer the potential for improved sensitivity as well as reduce or eliminate truncation artifacts. Previously, algorithms have been derived to reconstruct a combination of parallel and cone beam data [3, 41 well as cone and fan beam data [5]. In this work, we investigate three iterative approaches to combine FB and PB reconstruction to improve the noise characteristics as well as avoid artifacts due to truncation. These methods are presented in section II. Details of simulations undertaken to validate these methods are described in section III and results demonstrating the artifact-free combined fan/parallel beam reconstructions are detailed in section IV. Finally in section V, we present our conclusions.

11. METHODS

A . Method I In this approach, a PB-only reconstruction is obtained

first. Next, the FB emission projections are reconstructed using a regularized form of the AB-EMML [6] algorithm. The AB-EMML algorithm is an extension of the rescaled block-iterative version of the expectation-maximization maximum-likelihood method (RBI-EM) [7] to include prior upper and lower bound information. In the following, we first give the mathematical details of the regularized AB-EMML algorithm and then detail how we can obtain an artifact-free reconstruction using the PB reconstruction as a prior estimate.

Reconstructing a discrete non-negative image from its linear projections is equivalent to solving for a nonnegative J x 1 vector, z = [xi] that satisfies a linear system of equations, y = Px, with projection matrix P = [fi i] , Pi, 2 0 and projection data y = [yi], y i > 0,

for i = 1, ..., I a n d j = 1 ,..., J. The AB-EMML algorithm converges to a solution x , satisfying the vector inequalities a 5 x 5 b, where a and b are the lower and upper bounds, whenever such a solution exists [SI. To avoid overfitting the image to noisy data, regularization is commonly employed. Using the Bayesian gamma-distributed prior regularization of Lange et ab, (81, a regularized version of the AB-EMML algorithm is

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the following: we assume that Pai < gi < Pbi for d i and aj < x , < bj for all j . Here a, and bj represent the lower and upper bounds (for q), respectively. With Pai and Pbi denoting the ith entries of the vectors Pa and Pb respectively, we have

where

and

(4)

Max over j

where the projection index {i = l , . . . , I} has been partitioned into disjoint subsets {S,,, n = 1,. . . , N), with CiEs, denoting summation over those i in the nth subset S,,. Here, p is a prior estimate of the desired solution, satisfying the constraints ai 5 pj 5 bj and /3 is the regularization parameter. The iteration proceeds until convergence to a solution of g = Px, satisfying the constraints aj 5 zj 5 bj , if one exists.

By setting the prior estimate, p to be equal to the previously reconstructed PB-only reconstruction, scaled appropriately for differences in sensitivity, we shall show that it is possible to greatly reduce the truncation artifacts in the resultant FB reconstruction.

B. Method 11 First, the FB projection data is re-sampled to

equivalent PB data [9]. Next, a composite projection set (PB geometry) is formed by adding this converted FB data to the PB data obtained from the parallel heads. The composite projection set is formed as follows: within the field of view (FOV), both the PB as well as the re-sampled FB projections are averaged, while outside the FOV, the projection set is made up of only PB data. This composite emission projection set is then reconstructed with the ordered-subset expectation maximization (OS-EM) algorithm for a PB geometry.

C. Method III In this appro&, we implement for fan/parallel

reconstruction, a method proposed by Jaszczak et

al., [4] for the combined cone/pardel beam problem. We have modified the original maximum-likelihood expectation-maximization (MLEM) based algorithm to an OS-EM based one. Parallel and fan beam reconstruction estimates are obtained separately, using one complete iteration of the OS-EM algorithm. For the remaining iterations, the FB estimate at m y given sub-iteration k is modified in the following way

FBk = (FB’ + a x PBk) /2 (5)

where, a is a scaling factor that accounts for the differences in sensitivity between FB and PB data. No such modification is made to the PB estimate during the iterative process, which is carried out till a stopping criteria such as a fixed number of iterations is met.

111. SIMULATION DETAILS The mathematical cardiac torso (MCAT) phantom

was used to generate activity and attenuation maps. To consider the impact of truncation on the cardiac regions when using a FB geometry, we simulated two cases: a) apex of the heart extending beyond the FOV for a 65 cm FB collimator, and b) cardiax: regions completely within the FOV of the above mentioned FB geometry. The maps were sampled on grid of 64 x 64 x 64 with a pixel size of 0.634 cm. Noise-free emission projections over 64 steps were generated for both PB and FB (65 cm focal length) geometries for a circular camera rotation of 25 cm. These projections included the effects of distancedependent detector response (corresponding to the low-energy high-resolution collimators used in our clinic) as well as non-uniform attenuation. Poisson noise at a level of 0.5 million total counts corresponding to 2 PB collimators acquiring emission data simultaneously, was added to the noisefree PB projections. This is the average count-level in one frame of an 8-frame gated TcgBm sestamibi acquisition in our clinic. Similarly, Poisson noise at a level of 0.365 million total counts was added to the noise-free FB data (FB sensitivity is approximately 1.45 times that of PB).

In Method I, the PB reconstructions were obtained separately using 5 iterations of OS-EM. The FB projections were then reconstructed with 5 iterations of the regularized AB-EMML algorithm using the previously reconstructed PB images as the prior p in Eq. 1-4. In Method II, the composite projection data set, made up of both PB and the reoriented FB projections, was reconstructed with 5 iterations of OS-EM. Method I11 reconstructions were implemented with 5 iterations of OS-EM modified to reconstruct both FB as well as PB data simultaneously as mentioned in section 11-C. In addition, three sets of reconstructions were also obtained to contrast against the 3 combined fan/pardel beam reconstruction strategies that we are investigating. First is a PB-only OS-EM reconstruction (5 iterations) using projections counts from 2 PB heads which we denote

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using emission data from 3 PI3 collimators with noise levels corresponding to 0.75 million total counts. This simulation should account for the scenario if all 3 heads of a threeheaded SPECT acquisition system were fitted with PB collimators. Finally, the third was a FB-only

~~ ~

OS-EM reconstruction (5 iterations) obtained to illustrate the truncation artifacts that result from the missing data inherent in FB acquisitions. Note, that all reconstructions compensated for the effects of non-uniform attenuation only.

Twenty five different noise realizations were generated to estimate the reduction in noise when doing a combined fan/paraUel reconstruction. The left ventricular counts as assessed by the bull's-eye polar maps were used to compute quantitative measures of image quality such as bias and variance from these 25 noise realizations. The polar map is divided into 9 regions. These regions axe the apex, basal and mid-ventricle sections of the lateral region (lat-b and lat-mv), basal and mid-ventricle sections of the anterior region (ant-b and ant-mv), basal and mid-ventricle sections of the septal region (sepb and sepmv) and basal and mid- ventricle sections of the inferior region (if-b and inf-mv). Bias and variance measures were computed for each of these 9 Merent region-of-interests (ROIs). Let A; be the uptake value of the j th pixel from the nth noise realization, n = 1,. . . , N = 25. We then define the average uptake value within the ith ROI, i = 1,. . . ,9

where the summation is over Ji pixels-in the ith ROI. If Ofruth represents the ideal values for e:, the bias in the different reconstruction strategies for each of the 9 ROIs is then computed as

. N . N

The sample variance of uptake v d u a 6:(n) within each of the 9 sub-regions of the bull's eye polar maps, computed for every noise realization, n is

Finally, the mean variance @:of uptake values averaged over the N noise realizations for each of the 9 ROIs is then computed as

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(9)

Figure 1: Comparison of (a) transverse slice reconstructions and (b) the corresponding bull's-eye polar maps of the 1'' MCAT study where the apex extends beyond the FOV for a 65 em FB collimator. The following reconstructions am displayed from left to right: PB-only (2 heads), PB-only (3 heads), Method I, Method 11, Method I11 and FB-only.

A . Statistical Tests Since the PB-only (2 heads) reconstruction is our

clinical standard for attenuation corrected "emission images, a series of s ta t i s t id comparisons were made via the 2 sample &test with unequal variances between the PB-only (2 heads) reconstruction and the remaining 5 strategies: Methods I, 11, 111, PB-only (%heads) and FB-only reconstructions. Five sets of c?mparisons were made on the average uptake values, O r (Eq. 6) as well as the sample variances, &:(n) (a. 8). These comparisons were made for each of the 9 different ROIS under consideration. Since multiple paired comparisons were made, a Bonferroni correction [lo] was applied to the level of significance a = 0.05, i.e. monfemni = a / { K = 5) = 0.01. K here represents the number of paired comparisons made. In other wordfi, 5 separate tests of significance were performed at the modified, (YBonfemni = 0.01 level. This corresponds to a level of significance of 0.05 for the entire family of comparisons that are undertaken.

In order to get an overall assessment of performance, series of 5, two-sample paired &tests were conducted on the bias estimates. Once again, the paired comparisons (paired by cardiac region) were made between the PB-only (2 heads) reconstructions and the remaining 5 reconstruction strategies. Finally, a series of 5 sign-tests were also conducted on the mean variance of the uptalre valuea @: to 898e88 the overall reduction in noise with the proposed reconstruction approachea.

Iv. RESULTS AND DISCUSSION Fig. 1 compares the noisy reconstructions 89 well tu

the polar maps for PB-only (2 heads), PB-only (3 heads), Methods I, 11 and 111 as well as a FBonly reconstruction for the first simulation study, where the cardiac walls do extend beyond the FOV of a 65 cm FB collimator. Note that all reconstructions have been post-filtered with a three-dimensional (3D) Butterworth filter of order 5 and cut-off frequency 0.25 cycles/pixel as used in our clinic. In the transverse slice as well as the polar map for the FB-only reconstruction, a large defect that

1 1 4 4

Average I 0.36 (0.181) I 0.341 (0.166)J. I 0.36 (0.164)l I 0.36 (0.166) 1 I 0.37 (0.163)J. I 0.37 (0.263)t I

Figure 2: Comparison of (a) transverse slice reconstructions and (b) the corresponding bull's-eye polar maps of the Znd MCAT study. The following reconstructions are displayed from left to right: PB-only (2 heads), PB-only (3 heads), Method I, Method 11, Method I11 and FB-only.

extends from the apex to the anterior mid-ventricle regions of heart is quite evident. This result illustrates the truncation artifacts that we should be concerned with when the cardiac regions do extend beyond the FOV of the FB collimator. On the other hand, the 3 combmed faulparallel beam reconstruction strategies provide images that are relatively artifact-free. The other aspect to be noted from Fig. 1 is that the PB-only (3 heads), Method I and Method 11 reconstructions appear to be less noisy than the remaining reconstructions.

These observations are reiterated by the bias and variance estimates shown in Table 1, computed as explained in section 111-A. Note that the variance values tabulated represent the average d u e s 8; (Eq. 9) from 25 noise-realizations. Upon conducting the 2 sample Btest (with unequal variances) comparisons, any significant differences at the aBonfwmnj = 0.01 significance level for either 0; (q. 6) or 6;(n) (&. 8 ) are highlighted in bold-face. Note that t and J. are used to indicate whether the bias or variance for any reconstruction strategy is significantly higher or lower, respectively, than the corresponding values for PB (%heads). Bias estimates at the apex as well as the anterior mid-ventricle regions for the FB-only reconstructions, reiterate the visual

impressions of the severe truncation defect. Furthermore, Methods I, 11 and III as well as the PB-only (3 heads) reconstructions have smaller variances in general than the conventional PB-only (2 heads) reconstruction, highlighting the noise-reduction that is available by including data from the third FB collimator.

The overall average bias and variance (i.e., biasi and $;averaged over the 9 different ROTS) are presented in the last row of II&ble 1. Results in terms of significant differences from the global assessment of both bias (paired &test) and Variance (8igWteSt) at the (XBonfmoni = 0.01 level are again highlighted in bold-face. The PB-only (3 heads) reconstruction has a significantly lower overall bias than the conventional PB-only (2 heads) reconstruction. Additionally, Methods I, I1 and 111 as well as PB-only (3 heads) reconstructions provide significantly less noisy images than PB-only (2 heads) reconstructions. Also, comparing PB-only (2 heads) with FB-only reconstructions, we see that FB-only reconstructions are significantly more noisy as one would expect.

2 compares the noisy reconstructions as well as the polar maps for PB-only (2 heads), PB-only (3 heads), Methods I, II and 111 as well as the FB-only reconstructions for the second MCAT simulation study, where all the cardiac walls were within the FOV of a 65 cm FB collimator. Note that all reconstructions have been post-filtered with a three-dimensional (3D) Butterworth filter of order 5 and cut-off frequency 0.25 cycles/pixel. For this particular simulation, truncation artifacts do not affect visualization of the cardiac walls.

Bias and variance estimates for the second simulation are shown in Table 2. The statistical analysis outlined for the first simulation was again repeated. All significant differences at the OBonfemnj = 0.01 significance level for either 0: or 6?(n) are highlighted in bold-face. Methods I, 11 and III as well as the PB-only (3 heads) reconstructions have smaller variances in general than the conventional PB-only (2 heads) reconstruction, again highlighting the

Fig.

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C d b C PB PB Method Method Method Region (2 heads) (3 heads) I I1 111

noise-redudion that is available by including data from VII. REFERENCES the third FB collimator. The overall average bias and variance (i.e., biasi a d 8: averaged over the 9 different ROIs) are presented in the last row of Table 2. Results in terms of sipnificant differences from the dobal assessment

111 M, V, N ~ ~ ~ ~ , M. A, mg, T-S. pm and s. Daberg, '(Investigation of appro&= to Educe truncation of attenuation maps with simultaneous transmission and emission SPECT imaging", IEEE %ns. on NUC. Sci., vol.

FB O d Y

of both b&s (paired &test) and variance (saptest) at the aBonfemni = 0.01 level are again highlighted in bold-face. Method I reconstruction has a significantly lower overall bias than the conventional PB-only (2 heads) reconstruction. Additionally, Methods I, 11 and III reconstructions provide significantly less noisy images than PB-only (2 heads) reconstructions. This suggests that when all the cardiac walls are within the FOV of the FB collimator, as is the case with a substantial f r a i o n of the patient studies done at our clinic, the combined fan/parallel beam reconstruction strategies tend to perform better in terms of noisereduction.

V. CONCLUSIONS Three different combined fan/parallel beam

reconstruction strategies have been presented and analyzed via simulations as well as clinical patient studies. The 3 strategies: Methods I, 11 and 111 result in reconstructions that are relatively free of truncation artifacts. Simulation studies indicate that in the law-count realm, using the additional FB counts from the third head does help in reducing the noise-levels in the reconstructions. This aspect would especially be significant with this 3-headed setup (i.e., 2 PB and 1 FB collimators) when acquiring ECG gated or respiratory gated cardiac SPECT data with its inherent low-count statistics.

VI. ACKNOWLEDGMENTS

- - 45, no. 3, June 1998, pp. 1200-1206 M. M. Ter-Pogossian, S. R. Bergmann and B. E. Sobel, "Influence of cardiac and respiratory motion on tomographic reconstructions of the heart: Implications for quantitative nuclear cardiology", J. Comp. Assisted Tomo., vol. 6, no. 6, pp. 1148-1155, Dec., 1982 M. Defiise and R. Clack, i'Filtered-backprojectio~i reconstruction of combined parallel beam and cone-beam SPECT data" Phys. Med. Biol., vol. 40, pp. 1517-1537, 199.5 R. J. Jaszczak, J. Li, H. Wang and R. E. Coleman, iLThree- dimensional SPECT reconstruction of combined cone-bean1 and parallel beam data", Phys. Med. Biol., vol. 37, pp. 535- 548, 1992 G. T. Gullberg, G. L. Zeng, "Three-dimensional SPECT reconstruction of combined cone-beam and fan-bean1 data acquired using a threedetector SPECT system", Proceedings of the 1995 International Meeting on f i l l y Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, pp.329-32, 1995, Grenoble, France. C. L. Byme, "Iterative algorithm for deconvolution and deblurring with constraints", Inverse Problems, 14, pp. 1455-1467, 1999. C. L. Byme, "Accelerating the EMML algorithm a n d related iterative algorithms by rescaled block-iterative methods", IEEE f i n . Image Process., vol. 7(1), pp. 100- 109, 1998. K. Lange, M. Bahn, and R. Little, I'A theoretical study d some miuimum likelihood algorithms for emission and trnnnminsion tomography", IEEE f i n s . Med. h a g . , vol. 6, pp. 106-114, June 1987 A. C. Kak and M. Slaney, "Principles of computerized tomographic imaging", ZEEE Press, pp. 75-93, 1988

This work supported by the National Heart, Lung [lo] w- L* Hays,"Statistiw 5'h editionn, fIafCOUrt BmM

and Blood grant HL 50349. The contents are solely the responsibility of the authors and do not necessarily represent the official views of the National Institutes of Health.

Co''ege Pub'isherai pp. 451, lgg4

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