EMISSION tomography, which includes positron emission

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1 1248 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 Aligning Emission Tomography and MRI Images by Optimizing the Emission-Tomography Image Reconstruction Objective Function James E. Bowsher, Member, IEEE, David M. DeLong, Timothy G. Turkington, Member, IEEE, and Ronald J. Jaszczak, Fellow, IEEE Abstract An important approach to reconstructing PET and SPECT (PET/SPECT) radiotracer images is to utilize high-resolution information from registered MRI or CT (MRI/CT) images. These methods depend on accurate registration of PET/SPECT and MRI/CT images. Herein, we consider registration via optimization of a PET/SPECT image-reconstruction objective function which includes the registration parameters. Potential benefits of this approach include 1) modeling, within the registration process, of PET/SPECT noise and PET/SPECT acquisition effects such as limited spatial resolution, perhaps resulting in more accurate registration and 2) a natural framework for calculating joint uncertainties in registration parameters and radiotracer activity. In cases where the structures imaged by MRI/CT (e.g., gray matter and white matter in the brain) strongly influence the radiotracer distribution, the relatively small number of variables comprised of alignment parameters and regional radiotracer mean activities may account for, not all, but much of the estimable radiotracer distribution, and it may be useful to develop methods for rapid, highly accurate estimation of these few parameters, ultimately embedding such estimation within more general estimation of the full radiotracer distribution. Herein we develop Levenberg Marquardt simultaneous estimation of regional radiotracer mean activities and the six 3D rigid body translation and rotation alignment parameters. The method is tested by a computer-simulation study. With a PET/SPECT scanner spatial resolution of 0.2 cm FWHM, this study shows translational registration errors of about cm and rotational errors which are fractions of a degree. For small regions, estimates of regional mean activities are much closer to true values than are estimates obtained by OSEM using PET/SPECT projection data only. Index Terms CT, image reconstruction, MRI, PET, registration, SPECT. I. INTRODUCTION EMISSION tomography, which includes positron emission tomography (PET) and single photon emission computed tomography (SPECT), is important for its ability to image biological and molecular processes. However, PET/SPECT imaging has limited spatial resolution and is noisy. In order to Manuscript received January 16, 2004; revised July 9, This work was supported in part by NIH/NCI under Grants 5R24 CA and 5R01 CA and in part by NIH/NCRR under Grant P41 RR J. E. Bowsher is with the Departments of Radiation Oncology and Radiology, Duke University Medical Center, Durham, NC USA ( james.bowsher@duke.edu). D. M. DeLong is with the Department of Biostatistics and Bioinformatics, Duke University Medical Center, Durham, NC USA. T. G. Turkington, and R. J. Jaszczak are with the Departments of Radiology and Biomedical Engineering, Duke University, Durham, NC USA. Digital Object Identifier /TNS improve the estimation of radiotracer distribution over distance scales comparable to or smaller than the spatial resolution of PET/SPECT scanners, methods have been developed for including high-resolution, low-noise magnetic resonance (MRI) or x-ray computed tomography (CT) information within the estimation of radiotracer distribution, e.g., [1] [15] and references therein. In these methods, the MRI/CT image enters into the radiotracer-image-reconstruction objective function. For example, the objective function may smooth estimated radiotracer activities between nearby voxels only if those voxels are in the same MRI/CT region (e.g., [3], [10]), or it may smooth estimated voxel activities toward the mean activity of each MRI/CT region (e.g., [7], [8], [10]). Such methods require accurate alignment of MRI/CT and PET/SPECT image coordinates. Given the rapid growth of multi-modality imaging and multimodality systems (e.g., PET/CT), image registration is highly important and is an area of extensive research, e.g., [16]. Even with sequential imaging systems such as PET/CT, patient motion is possible between the two scans. Here we aim toward methods which may usefully correct even minor mis-registrations, as well as the greater mis-alignments associated with separated systems such as PET and MRI. One characteristic of most work on registering MRI/CT with PET/SPECT is that the registration procedures are based on the PET/SPECT radiotracer image, rather than the PET/SPECT projection data, and these registration procedures do not have knowledge of the spatial resolution and variance and covariance characteristics of the PET/ SPECT image. Consequently, these registration procedures are relatively limited in their capacity to determine whether a structure in the PET/SPECT image is strongly supported by the PET/ SPECT projection data and thus likely to correspond to structure in the real radiotracer distribution, or if instead, the PET/SPECT image structure is likely to be due to noise. It may be that substantially improved registration can be obtained by registration procedures which do incorporate resolution and noise information. Within the framework of methods such as [1] [15] this additional information can be introduced into the registration procedure by including MRI/CT PET/SPECT registration parameters within the radiotracer-image-reconstruction objective function. This approach to cross-modality image registration has several potential advantages over more-widespread registration methods. First, iterative statistical reconstruction is well-suited to modeling PET and SPECT projection acquisition /$ IEEE

2 BOWSHER et al.: ALIGNING EMISSION TOMOGRAPHY AND MRI IMAGES 1249 effects such as Poisson noise, spatially varying spatial resolution, and non-uniform attenuation. These effects are included in the likelihood component of the image-reconstruction objective function. When registration is performed by also including alignment parameters in the objective function, the registration process inherently utilizes this modeling information. For example, it factors in the spatial resolution model when considering whether two different alignments provide significantly different likelihoods. Use of this modeling information may improve registration accuracy and precision. Second, inclusion of alignment parameters in the objective function provides a natural framework for calculating joint uncertainties in alignment and radiotracer-activity parameters. For example,at the maximum likelihood (ML) estimate of these parameters, their variances and covariances may be well approximated by the inverse of the Fisher information matrix, where the Fisher information matrix is comprised of the negatives of the expected values of the second derivatives of the log likelihood function [17]. With regard to the first point above, note that since these variances and covariances are implicit in the shape of the objective function, and since the alignment is estimated by searching for the peak in this objective function, it follows that the registration process inherently and implicitly utilizes these variances and covariances. In contrast, alignment methods that work directly with an estimated PET/SPECT image generally do not utilize knowledge of these variances and covariances. Previous work on estimating the alignment of high-resolution information directly from projection data includes [5], [6], [18] [21]. Estimation methods have included gradient ascent [18], [21], Levenberg Marquardt [5], [6], and variable metric methods using adjoint differentiation [19][20]. The computational efficiency of the methods in [19][20] is notable. Here we focus on Levenberg Marquardt estimation. The consideration of Levenberg Marquardt methodology is motivated by the direct calculation, which it enables, of exact second-derivatives of the log likelihood with respect to regional radiotracer activities, as well as approximate second-derivatives with respect to alignment parameters, where this approximation may be quite accurate as the parameter estimates approach convergence. Levenberg Marquardt optimization was also investigated in [5], [6] for two 2D translation parameters, along with parameters for angle-dependent radial location of inner and outer heart walls, in a surface-based model of cardiac boundaries. Here we consider the six 3D rigid-body translation and rotation parameters and a more general, segmentation-based model for the high-resolution information. The method provides accurate simultaneous estimation of registration parameters and of the mean activities of multiple radiotracer regions. As in [5], [6], [19] [21], the present work assumes uniform radiotracer concentration within each tissue type. This assumption is unrealistic for most applications, and the methods developed here are not intended as stand-alone PET/SPECT image reconstruction methods. Instead, these methods are being developed and evaluated as one stage in a program that will ultimately embed these techniques within PET/SPECT image reconstruction which does allow for within-region variation in radiotracer activity. The motivation for this approach and a plan for estimating within-region variation are discussed in Section IV. II. MODEL AND ESTIMATION A. Model and Alignment Parameters A general framework for estimating a radiotracer activity distribution from PET/SPECT projection data is to optimize a log likelihood or a penalized log likelihood. Methods for limiting noise in the estimate of include use of a penalty function and choice of the parameterization of. Here, we consider a model for the radiotracer distribution in which each point in three-dimensional space is assumed to belong to a single MRI/CT region and radiotracer activity is assumed to be constant within each MRI/CT region. We do not utilize a penalty function. The objective function to be optimized is thus the log likelihood, considered as a function of regional radiotracer activities and the alignment parameters. This radiotracer model does not imply piecewise flat radiotracer images, since the MRI/CT image may be specified on much smaller voxels. For example, in the studies presented here, at least 64 MRI/CT voxels overlap each PET/SPECT voxel, allowing variable mixing of MRI/CT regions within each PET/SPECT voxel. For most realistic applications, a more general model is needed which allows for continual variation of radiotracer activity throughout each MRI/CT region. The present, more-restricted radiotracer model serves in part as a test bench for investigating MRI/CT PET/SPECT alignment via optimization of the objective function. However, the larger purpose in developing this model and the corresponding estimation procedures is for their use within more general models and estimation methods which do account for within-region variations in radiotracer activity, as discussed in Section IV. The Poisson log likelihood is where is the number of detector bins, and and are respectively the expected and measured numbers of events at projection bin. Let be coordinates of a point in a frame that is fixed relative to the PET/SPECT scanner. Also, let be coordinates of a point in a frame that is fixed with respect to the MRI/CT image, and let be the radiotracer activity per unit volume, specified as a function of. In the present image model, where is the radiotracer activity per unit volume within MRI/CT region, and is the region classification at location. Coordinates for a given point in space can be generally expressed as or,so. The expected number of events is where is the probability that a photon emitted at location will be detected at bin. A discretized (voxelized) version of (1) (2)

3 1250 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 (2) is formed as follows: For a given MRI/CT segmentation and alignment, each radiotracer voxel, i.e. each voxel associated with the coordinate frame, will in general contain a mixture of MRI/CT regions. A vector is associated with each radiotracer voxel, where indicates the volume of voxel that is in region. Thus, where is the volume of voxel. From (2) where the symbol indicates integration over voxel is the average value of over voxel, and (3) makes the usual assumption that voxels are sufficiently small relative to PET/ SPECT scanner spatial resolution, such that the distribution of activity within the voxel has no significant influence on, and consequently the same detection probability can be used for all points within a given voxel. Elaborating (3) (3) (8) and (9) also constitute a tissue-composition model, in that they account for mixtures of tissue type within individual PET/SPECT voxels. (Other tissue composition models include [10][13]). Both the continuous form (2) and discrete form (8) are important for the discussion below. For fixed-scale, rigid-body transformation (10) where specifies displacement of the coordinate frame relative to the coordinate frame, and specifies a rotation of the coordinate frame. is characterized by 3 angles. This paper utilizes the xyz Euler angle convention described in [22] (pp ), in which the first rotation is the yaw angle about the initial axis, the second rotation is the pitch angle about the intermediary axis, and the third rotation is the bank or roll about the final axis. The columns of (rows of ) are where if and 0 otherwise, and For this paper, the quantities are computed by multiple periodically-spaced samples for values of within each voxel. Equation (6) compares to (31) in the Discussion. The expected number of photons emitted from voxel is. Arranging (5) differ- so that (5) is a case of ently gives the dependence of where on (4) (5) (6) (7) where depends on the MRI/CT alignment. It represents the expected number of events at a projection bin per unit radiotracer activity per unit volume in region. Equations (8) (9) (11) (12) (13) and are summarized by. All parameters within the model are indicated by, where. B. Estimation Since the number of registration parameters is six and since the number of MRI/CT regions would typically be well under one hundred, candidate methods for optimizing the objective function with respect to these parameters might include Newton Raphson, Levenberg Marquardt, variable metric, and conjugate-gradient techiques [23]. Newton Raphson techniques would involve the computational burden of forward projecting second derivatives of the radiotracer distribution with respect to alignment parameters. Accordingly, we have chosen instead a Levenberg Marquardt approach with a Gauss Newton approximation to. This approach avoids forward projecting while still utilizing information about the second derivatives, information carried in products of first derivative terms. These products are exact for second derivatives with respect to region mean activities. For second derivatives involving alignment parameters, these products may dominate as the alignment converges and the fit to the measured PET/SPECT projection data becomes quite good.

4 BOWSHER et al.: ALIGNING EMISSION TOMOGRAPHY AND MRI IMAGES 1251 The derivatives of with respect to are (14) are independent of and. Therefore they can be moved outside of the integration over in (18), leaving three forward projections of the form (20) (15) The Gauss Newton approximation to (15) is obtained by ignoring second derivatives of For displacement parameters, (18) can then be computed as (21) (16) thus estimating the first and second derivatives of from the first derivatives. Equation (15) indicates that approximation (16) may be quite accurate when the fit to the measured data is good, e.g., as the estimate of alignment and of regional mean activities approaches convergence. 1) Derivatives With Respect to Alignment: The derivative of with respect to an alignment parameter is where notation such as indicates as a subscript for and 1 as a subscript for. Equation (21) indicates that the weight factors of (19) are applied in the projection space after forward projection. When is a rotation parameter,, the derivatives are (22) which can be expanded as (17) The 3 3 matrices, are independent of and can be moved outside of the integral (18). However the factors remain within the integral (18) and require nine forward projections of the form (18) where the subscript on the brackets indicates that both terms within the brackets are evaluated at. The detection-probability function in (18) includes a model of PET/SPECT spatial resolution and other acquisition effects, and (18) shows explicitly that this model is incorporated into image registration. For (18), is computed using the segmentation which is fixed with respect to the M coordinates. Three derivatives images are computed in the M-coordinate frame. Each derivative is computed numerically as the difference in radiotracer activity between two adjacent voxels and is located midway between those two voxels. For example, the array of derivatives is stored on a voxel grid which is shifted by half a voxel width in the -direction, relative to the grid on which is stored. For purposes of evaluating (18), these derivatives images are then expressed in the E-coordinate frame according to the transformation (10). The terms can be evaluated by differentiating both sides of. When is a displacement parameter,, the derivatives are (23) where and. For rotation parameters, (18) can then be computed as (24) Thus the evaluation of (18) for all six alignment parameters can be accomplished by 12 forward projections, three unweighted (20) and nine weighted (23). Each set of four forward projections specified by a given can be computed simultaneously since all four are on the same shifted grid. This may reduce computation cost. For example, calculation or storage of may constitute a significant portion of the computational burden and need be performed only once for each group of four forward projections. 2) Derivatives With Respect to Regional Radiotracer Mean Activities: From (8), and, so that, via (14) and (15), the first and second derivatives of with respect to regional radiotracer activities are (19) (25) On the right-hand side, when and 1 when, so the vector of derivatives,is equal to the negative of column of. Thus, the derivatives For (26), the Gauss Newton approximation is exact. (26)

5 1252 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE ) Cross Derivatives: By (16), Gauss Newton approximations for the cross derivatives are (27) where is obtained by (18). 4) Levenberg Marquardt Estimation: The derivatives described above allow for Levenberg Marquardt estimation [23] of local maxima in the objective function with respect to alignment parameters and regional radiotracer activities. At a maximum (that is not on the boundary of the parameter space), and, where is specified by (14) and (17) and is specified by (25). More concisely,. Consider current parameter values. A single Levenberg Marquardt update (e.g., a single subset in Section III) involves the following steps. Step 1) Compute by (9) using to determine. Step 2) Compute by (8) using, and compute by (1). Step 3) Compute using, and compute by (18) using. Step 4) Compute a Hessian matrix according to (15) and (16), using the Gauss Newton approximations provided by (16) and (18) for alignment parameters and (27) for cross derivatives, and using the exact form (26) for region means. Step 5) Compute by (14), (18), and (25). Step 6) The equation expresses the notion that for second derivatives (15) which are constant in, the maximum value of would be located at. Solve this system of linear equations for (we used the CLAPACK routine DSYSV), thereby obtaining. Step 7) If any region mean activities are less than zero, reset these mean activities to, thereby assuring non-negative values for all components of. Step 8) Compute by (7) using and obtaining from. Step 9) Compute. Step 10) Compute the log likelihood by (1). Step 11) If, this Levenberg Marquardt iteration is complete. Set, and return to Step 1 to perform another Levenberg Marquardt update. Step 12) If, loop over the ridging procedure described in Steps 13 and 14 below, starting with and incrementing by 1 until. Step 13) Subtract a positive constant (e.g., the results presented here use ) from the diagonal of, providing a new system of equations. Solve these equations to obtain a new parameter update. Enforce non-negativity as in Step 7 above. Step 14) Using the new parameter values, compute, as described in Steps 8 through 10 above. Step 15) Once, the Levenberg Marquardt iteration is complete. Reset, and return to Step 1 to perform another Levenberg Marquardt update. Regarding the ridging procedure, with sufficiently high powers of, is approximately diagonal, and. Hence the Levenberg Marquardt updates range between Gauss-Newton updates, (i.e. Newton Raphson estimates using the approximation (16) to ) and steepest ascent updates [23]. 5) Computational Burden: The computation cost of the proposed method is determined largely by the number of forward projections. In the absence of ridging, each Levenberg Marquardt update involves about 14 forward projections: The calculation of in Step 1 of Section II-B-4 has a computation cost somewhat greater than that of a single forward projection. Step 3 of Section II-B-4 utilizes 12 forward-projections in evaluating by (18), as described in Section II-B-1. The calculation of in Step 9 of Section II-B-4 is one forward projection. In addition, Step 14 of Section II-B-4 involves one forward projection for each ridging, this occuring during the calculation of. For the example in Section III, there were only two ridgings over 24 updates, so that the computational expense of ridging was minimal. Iterative reconstruction of PET/SPECT images generally involves one forward projection and one equally burdensome backprojection for each image update. Hence the computation burden of forward projection in one Levenberg Marquardt update (with 14 forward projections) may approximately equal the burden of seven image updates in a typical iterative reconstruction algorithm. This burden may be mitigated somewhat by the simultaneous implementation of each group of four forward-projections in Step 3 of Section II-B-4. There can also be significant cost in estimating if the number of samples-per-voxel is large. These points concern the per-update computation burden. It is possible that overall computation cost for image reconstruction is less when MRI/CT information is incorporated and simultaneously aligned, due to convergence in fewer iterations. The computer-simulation studies of Section III illustrate this point. III. EVALUATIONS Two phantom distributions were generated an MRI/CT segmentation and a radiotracer phantom. Fig. 1 shows transaxial, sagittal, and coronal slices through the radiotracer phantom. The figure includes a numerical labeling of the regions underlying this phantom. The regions were generated from ellipsoids, and Table I indicates the displacements and radii of these ellipsoids in the E-coordinate frame. The Fig. 1 radiotracer phantom was constructed in three steps. First, a segmentation of the radiotracer distribution was implemented on a grid of cm-wide voxels. The region assignment of each cm-wide voxel was determined by the one or

6 BOWSHER et al.: ALIGNING EMISSION TOMOGRAPHY AND MRI IMAGES 1253 TABLE II TRUE AND ESTIMATED REGION ACTIVITIES. COLUMNS! AND TRUE INDICATE THE REGION NUMBER (FIG. 1)AND THE TRUE,PHANTOM ACTIVITY.THE TWO COLUMNS UNDER REGION-BASED ARE ESTIMATED ACTIVITIES FROM THE RECONSTRUCTION OF FIG. 3(E). PLEASE SEE TEXT FOR TC, ROI, AND N. THE THREE COLUMNS UNDER VOXEL-BASED ARE ESTIMATED ACTIVITIES FROM THE RECONSTRUCTIONS OF FIG. 2(A,B,D), RESPECTIVELY. THE NUMBERS IN PARENTHESES ARE PERCENT ERRORS, DEFINED AS (estimated 0 true)=true. ACTIVE REGIONS (! = 9...3) ARE LISTED FIRST, FOLLOWED BY COLD REGIONS (! = ).WITHIN EACH GROUP, REGIONS ARE ORDERED BY ERROR IN COLUMN ML Fig. 1. Transaxial (left), coronal (top), and sagittal (bottom) slices through the radiotracer-activity phantom from which projection data were simulated. Voxels are cm wide. Numerical labels indicate regions!. The lines T indicate the approximate location of the transverse slice. The line S ( C ) indicates the approximate location of the sagittal (coronal) slice. Regions 17 and 18 do not intersect the planes shown. TABLE I DISPLACEMENTS AND RADII FOR THE ELLIPSOIDS WHICH SPECIFY THE MRI/CT AND RADIOTRACER PHANTOMS. (PLEASE SEE TEXT.) THE FINAL COLUMN GIVES THE VOLUME OF EACH REGION more ellipsoids which enclosed the center of the voxel. Centers which were within more than one ellipsoid were assigned to the region corresponding to the highest-numbered of these ellipsoids. Second, a radiotracer phantom was implemented on cm-wide voxels by assigning the phantom regional radiotracer activities given in Table II to each voxel in accord with the region classification of that voxel. Finally, a phantom with cm-wide voxels the phantom shown in Fig. 1 was formed by averaging the activities over the corresponding cm-wide voxels. The phantom MRI/CT segmentation was implemented in a manner similar to step-1 of the construction of the radiotracer phantom, with the difference being that the ellipsoid dimensions of Table I were evaluated in a coordinate frame which was translated and rotated by relative to the coordinate frame in which the radiotracer phantom was constructed. Thus, the entries in Table I are the coordinates for ellipsoids of the radiotracer phantom, and the coordinates for the ellipsoids of the MRI/CT phantom are obtained by (10). Parallel-beam SPECT projections of the Fig. 1 phantom were analytically simulated for 120 angles evenly spaced over 360 degrees. Projection bins were 0.05 cm wide. Scatter and attenuation were not modeled. The projection of each voxel was blurred by a 0.2 cm-fwhm Gaussian kernel. Although the specific geometry is more characteristic of SPECT acquisition, the width of the kernel was chosen to represent approximately the spatial resolution of our small-animal PET scanner (Concorde Microsystems, MicroPET R4, Knoxville, TN, USA), in order to obtain some indication of the degree of partial volume correction that can be provided to small-animal PET imaging through the use of high-resolution MRI/CT images. The blurring was applied by the methods of [24][25], in which integrals of the blurring kernel over the detector bins are pre-computed and stored as a function of voxel-to-collimator distance and as a function of the location of a voxel s projection within the detector bin that contains that projection. Here 32 within-bin locations were considered across the transaxial dimension of the bin, providing for an upper limit of cm on errors in the transaxial placement of the blurring kernel. In the axial direction, two within-bin locations were utilized, thereby exactly modeling the axial projection-locations of the two cm-wide image slices that overlap each 0.05 cm-wide projection slice. This method accurately represents the generally-assumed invariance of SPECT collimator and detector-crystal response to translations of a radiotracer source within any given plane

7 1254 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 Fig. 2. Voxel-based OSEM reconstruction without high-resolution region information. (a) Iteration 500 (200 iterations of 8 subsets followed by 100 iterations each of 4, 2, 1 subsets). (b-f) Results of Hann filter applied to iteration 500. Filter cut-off frequencies are 6, 5, 4, 3, and 2 cycles per cm in b-f, respectively. The transaxial slice shown occupies the volume 0 <= e approximately indicated by the lines T in Fig. 1. < 0:05 cm which is Fig. 3. Joint estimation of alignment and activity using high-resolution region information, for the same projection data as in Fig. 2. Shown are subsets (a) 1 of 8 (1=8) in iteration 1, (b) 1=8, (c) 3=8, and (d) 4=8 in iteration 2, and (e) 1=1 in iteration 6. Image (f) is the true, phantom radiotracer distribution implemented on a grid of 0.05 cm-wide voxels. The transaxial slice shown is the same as in Fig. 2, i.e. it occupies the volume 0 <= e < 0:05 cm. For both figures, voxel intensities above 1500 are set to parallel to the plane of the collimator surface. (Although an invariant 0.2 cm-fwhm kernel is utilized here, the methods just as readily model a dependence of the kernel on voxel-to-collimator distance.) Noisy projection data were generated, at the level of 2 million total events over all angles and slices, by pseudo-random Poisson sampling of the analytically-simulated projections. Images were reconstructed by OSEM [26] with post-filtering and by the region-based methods developed here. Both reconstructions were onto a grid of 0.05 cm-wide voxels, with the image-reconstruction support given by the union of regions 1 18 (Table I). Spatial resolution effects were included in the detection probabilities by the same technique [24][25] as described above for simulating projection data, in this case using 32 within-bin locations transaxially and one axially. The blurring kernel was obtained by convolving the aforementioned 0.2 cm-fwhm Gaussian with the 0.05 cm-wide voxel size, thus modeling the larger voxel size used in reconstruction as compared to projection data simulation. For OSEM reconstruction, the initial activity estimate was 400 for all voxels in the support. In order to allow for contrast recovery in small regions, OSEM was run for 500 iterations [27], with 200 iterations of 8 subsets followed by 100 iterations each of 4, 2, and 1 subsets. Fig. 2(a) shows the approximation, provided by iteration 500, to the voxel-based ML estimate of radiotracer activity. Fig. 2(b f) shows the results of applying varying degrees of 3D Hann filtering to the Fig. 2(a) 3D image, where the Hann filter is defined as for and zero otherwise. Cut-off frequencies were, and cycles/cm for Fig. 2(b f) respectively. For region-based reconstruction, there were 2 iterations with 8 subsets, followed by 2 iterations with 2 subsets, followed by 4 iterations with 1 subset. The simultaneous use of all projection data during iterations 5-8 allows for maximization of the objective function with respect to. Voxel-composition vectors were calculated using samples within each voxel during iterations 1 and 2. There were 8 and 16 samples along each dimension during iterations 3 4 and 5 8, respectively. During each update, non-zero activities were allowed only within the intersection of the aforementioned image-reconstruction support and the volume occupied by regions 2 18 of the translated and rotated MRI/CT segmentation. Among support voxels, initial activities were 400 for those in regions 2 18 of the translated and rotated MRI/CT segmentation and zero for other voxels. Our results to date suggest that initial convergence may be faster when the possibly-active regions of the MRI/CT segmentation (e.g., 2 18) are primarily within the image-reconstruction support. Fig. 3(a e) shows estimated images after updates 1, 9, 11, 12, and 22 of the 24 updates to, where each subset is an update. The alignment parameters do not change much after iteration 5 (update 21), suggesting that the estimation had very nearly converged by iteration 5. The radiotracer distribution appears broader in Fig. 3(a) (update 1) because a high activity is estimated for region 2. By update 12 in Fig. 3(d), estimated activity for region 2 is near its true value of zero. Errors in estimated alignment parameters are shown in Table III. The initial alignment vector is, corresponding to an initial assumption of no mis-alignment between the PET/SPECT image and MRI/CT segmentation. The true alignment vector is, representing the transformation of the M coordinate frame

8 BOWSHER et al.: ALIGNING EMISSION TOMOGRAPHY AND MRI IMAGES 1255 TABLE III ERRORS IN ALIGNMENT PARAMETERS. COLUMN I INDICATES ITERATION. THERE IS ONE ROW FOR EACH SUBSET OF EACH ITERATION. relative to the E frame. The alignment errors in the row of Table III represent the difference between these initial and true alignments. By iteration 6 (update 22), displacement errors are reduced to 8, 14, and 9 micrometers, and the highest rotation error is 0.23 degrees. Region 10 appears to be slightly more narrow in Fig. 3(e) than in the true, phantom image of Fig. 3(f). There is the opposite appearance in an adjacent slice (not shown). These results are apparently due to the slight alignment errors still remaining after iteration 6. Even with this alignment error, the region-based reconstruction still estimates the correct value for the activity of region 10 (column TC of Table II). Various estimates of region activity are shown in Table II. The two estimates under Region-Based are from iteration 6 of region-based reconstruction [Fig. 3(e)]. The column labeled TC (for tissue composition) gives the direct estimate of, obtained on Step 7 of Section II-B-4 in the case of no ridging and on Step 13 in the case of ridging. These estimates utilize the tissue composition model of (8) and (9), and consequently these estimates exactly account for mixtures of tissue type within individual voxels. The next four columns ( ROI H-4 ) of Table II give estimates of obtained from images. For these four columns, each estimate of is computed as the average value of over voxels that have at least 80% of their volume in region. The final column, N, of Table II indicates the number of such voxels in each region. The estimates ROI H-4 are at least somewhat biased by the % contribution of nearby regions. The estimates in column ROI are computed over voxels in the region-based reconstruction of Fig. 3(e). The estimates in columns ML, H-6, and H-4 are computed over voxels in the voxel-based reconstructions of Fig. 2(a,b,d), respectively. The labels H-6 and H-4 reflect Hann filtering with cut-off frequencies of 6 and 4 cycles per cm. The region-based ROI estimate utilized the translated and rotated MRI/CT segmentation obtained on iteration 6 (Table III). Voxelbased estimates were obtained using the (unrealistically) exactly correctly aligned segmentation. The present study involving only a single image does not allow for experimental separation of bias and noise. However, one expects the approximate-ml voxel-based estimate to have lower bias and higher noise as compared to the Hann-filtered estimates [27] [29], and the results in Table II are consistent with this expectation. For regions 2, 4, and 5, which are large and relatively wide, voxel-based OSEM estimates of region activity, both ML and Hann-filtered, are reasonably close to true values, though not as close as region-based estimates. For region 3, which is large but narrow, the voxel-based approximate-ml estimate shows little error. Presumably this approximate-ml estimate has minimal bias, and the large size of region 3 limits noise in the estimated average activity across the region. In contrast, the Hann-filtered images underestimate region-3 activity by 16% and 27%, presumably reflecting significant bias. For smaller regions (6 18), errors are consistently large with the Hann-filtered images, again presumably reflecting significant bias. Among the smaller regions with non-zero true activity, Hann-filter underestimations range from 25% (region 6) to 91% (region 18) of the true activity, and overestimations range from 85% (region 8) to 140% (region 9). For smaller regions with zero true activity, Hann-filter estimates are not far below the activities of surrounding background regions, i.e. Hann-filter estimates for regions 12 and 15 are only slightly below the activity (400) of the surrounding region 4, and Hann-filter estimates for regions 16 and 17 are fairly near the activity (1000) of the surrounding region 3. For small regions with non-zero activity, the voxel-based approximate-ml estimates generally show less error than the Hann-filter estimates. ML estimates are expected to have less bias, but also significantly more noise, and noise may for example underlie the relatively poor approximate-ml estimate of activity for region 9. Noise may also partially explain the only-slightly-better performance of approximate-ml, as compared to Hann-filtering, for small regions with zero activity, since the positivity constraint on implies that random fluctuations will bias activity estimates upward from zero, e.g., [28]. (Stated more generally, the ML estimator can be significantly biased near the boundary of the parameter space [17].) Table II shows that, as compared to voxel-based image reconstruction, the region-based method of Section II provides significantly better estimates of activity in small regions. The region-based method has at least two advantages the tissuecomposition model and the information that activity is constant within each region. A third, possible advantage is that even 500 iterations of OSEM may not have fully converged to the voxel-based ML estimate, so that the full bias-reducing potential of voxel-based ML estimation is not realized, whereas the 6 iterations of region-based Levenberg Marquardt estimation may very nearly have reached the region-based ML estimate and thus very nearly have inverted fully the spatial blur of the PET/SPECT acquisition. Full elucidation of these issues will require studies more extensive than those presented here. Comparison of columns TC and ROI gives some information on the specific contribution of the tissue-composition model. The region-based ROI estimates are overall significantly better than

9 1256 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 the voxel-based estimates, but significant further improvement is provided by the tissue-composition estimate. For region 10, which is the most narrow region, there were no voxels with 80% (or even 50%) of their volume in region 10, so that a tissue-composition approach was essential. Reconstructions were performed on a 3.2 GHz Intel P4 Xeon processor. The 500 iterations of OSEM required 682 minutes. Reconstruction times for region-based reconstruction were 21 minutes for the 2 iterations 1 2 (10.5 minutes per iteration), 33 minutes for the 2 iterations 3 4, and 107 minutes for the 2 iterations 5 6. Increased times at later iterations are presumably due to finer sampling when estimating. There are specific coding steps which would significantly reduce the time required for sampling in this initial software implementation of the Section II methods. Also, the time required for estimating might be reduced by interpolation methods [16]. IV. DISCUSSION All of the radiotracer image reconstructions presented here included accurate modeling of the spatial resolution kernel used to simulate the PET/SPECT projection data. This modeling enables the inversion of blurring effects within image reconstruction [27] [30], as in the voxel-based [Fig. 2(a)] and region-based [Fig. 3(e)] approximate-ml estimates of Table II. Accurate spatial resolution modeling is thus one key to accurate quantification and partial volume correction. A second factor which contributes to image quality is the model of uniform activity within each region. Since activity is truely uniform within each region of the Section III phantom, introducing that uniformity during image reconstruction reduces noise without losing any true signal. Also, the recovery of contrasts between regions is not hampered. For more realistic activity distributions, it will generally be necessary for image reconstruction to allow for low- and some-mid-frequency within-region variations in estimated activity. Mid- and high-frequency within-region variations both from true signal and from noise will, as usual, be limited, e.g., by penalties on within-region variation. Thus noise reduction may cause some loss of true within-region signal variation. Yet, analogous to the results obtained here, these penalties on within-region variation presumably will not hamper the recovery of contrasts between regions. The results of Section III show improved recovery of contrasts between regions, and presumably such improvements will persist when low- to mid-frequency within-region variations are allowed. Furthermore, a general notion underlying the use of MRI/CT within radiotracer image reconstruction is that variations of radiotracer activity are relatively gradual within MRI/CT regions, and when this is case, within-region penalties may have relatively little effect on true within-region variations. A third factor which may improve quantification is that ML estimation of directly from PET/SPECT projection data is much faster computationally than estimation of via regional averages over an ML estimate of. The results of Section III suggest that the voxel-based ML estimates may often not be practical. Estimation of from PET/SPECT projection data may be much better conditioned than estimation of, and this may partly explain its better performance. Accelerated image reconstruction through direct estimation of is also discussed in Section III-C of [8] for the case of penalties that act within regions. A fourth factor which may contribute, at least to computational speed, is the specific algorithm for obtaining ML estimates of, and the rapid convergence of Levenberg Marquardt methods may be relevant. The third and fourth points are part of the motivation for developing methods that estimate, and, directly from PET/SPECT projection data. In this paper, Levenberg Marquardt estimation has been developed based on a model which involves a small number of parameters alignment parameters and mean regional radiotracer activities. More realistic models which allow for within-region activity variation will typically increase the number of parameters beyond what is practical for pure Levenberg Marquardt methods, since the Hessian matrix would be too large. However, we have developed the Levenberg Marquardt approach for estimating the relatively few parameters because in many realistic problems, such as brain imaging for example, these parameters may represent, not all, but much of the information that can be obtained regarding radiotracer distribution, and the procedures developed here provide accurate and rapid estimation of these important parameters. That noted, it will be essential in future work also to estimate within-region variations in radiotracer activity. Our basic notion for accomplishing this will be to interleave estimates of and with estimates of only, where the simultaneous estimates of and allow only for a region-dependent scaling of. The purpose of the scaling procedure is to conserve the current within-region shape of the radiotracer distribution during updates of. (More specifically, the scaling is needed to conserve within-region shape during updates of. Updates of alone, by the methods proposed here, already conserve within-region shape.) Between updates of, within-region shape can then be updated by interleaved estimates of only, accomplished by one of more iterations of a voxel-based PET/SPECT reconstruction algorithm, e.g., [26]. Given current estimates and, region-dependent scaling of can be expressed as The expected number of events at detector bin where is (28) (29) (30) (31) (32)

10 BOWSHER et al.: ALIGNING EMISSION TOMOGRAPHY AND MRI IMAGES 1257 An expression similar to that for of (32) was suggested in (15) of [5], for purposes of modeling heterogeneity of radiotracer concentration within the large background region surrounding the ventricle and myocardium in the model of [5], [6]. Reference [5] also notes a related concept in [31]. Equation (32) is an analog to (9) and is a basis for estimating along with simultaneous region-dependent scaling of, while interleaving less restricted updates of so as to adjust the within-region shape of the radiotracer distribution. The less-restricted, high-dimensional search along is intended to extract whatever additional components of that can be estimated from the PET/SPECT projection data. These additional components of may involve only the usual relatively low frequencies estimable from PET/SPECT projection data alone, and numerous algorithms are available for reasonably fast estimation of these relatively low frequencies. However, such algorithms may be slow for those higher-frequency components that can be estimated with the benefit of well-aligned high-resolution MRI/CT. The purpose of this paper has been to develop methods which converge rapidly to accurate and precise estimates of the few parameters which underlie these higher-frequency components. Previous studies using statistical ensembles of reconstructed images have demonstrated the ability of high-resolution, low-noise non-radiotracer images to improve the detection and quantification of radiotracer structure [10] [15], provided that similarity in radiotracer concentrations at different locations is correlated with similarity in MRI/CT image amplitudes (or segmentation labels) at those locations. The anecdotal example presented here in Section III is consistent with those previous results, again illustrating the potential for markedly improved quantification and detection of small radiotracer regions. Here this potential is suggested even for the case in which the true alignment of the high-resolution image is unknown and must be estimated. The region-based results obtained here benefitted from the particularly strong and known correlation between radiotracer and non-radiotracer structure: Radiotracer concentration was uniform within each high-resolution region, and exact knowledge of this was included in the reconstruction. Other scenarios involving less correlation e.g., significant variation of radiotracer concentration within each high-resolution region would presumably result in radiotracer image quality somewhere between that obtainable from PET/SPECT projection data only and that obtainable with very strong information such as has been employed here. Regarding alignment, as with any registration method, the effectiveness of the present method presumably also depends on the nature of the relation between radiotracer distribution and MRI/CT image. The discussion of the previous paragraph suggests the importance of establishing the nature of the high-resolution, low-noise information that may be available. The use of MRI/CT images within radiotracer image reconstruction can be thought of as involving image-reconstruction, biological, and high-resolution-imaging principles and methods. Biological principles establish the relation between radiotracer distribution and non-radiotracer structure. High-resolution-imaging methods are then needed in order to provide high-resolution, low-noise images of that non-radiotracer structure. For example, a biological principle underlying the use of MRI/CT images in brain radiotracer imaging is that variations of radiotracer concentration within gray matter, white matter, and cerebrospinal fluid are much less, or are much more gradual, than differences in radiotracer concentration between these tissue types [2] [4], [7] [11], [13], [15], [32] [34]. Recent work [35] has utilized small-animal studies to investigate biological and high-resolution-imaging principles and methods for utilizing MRI/CT within radiotracer imaging of tumors. Image reconstruction principles and methods are needed for accurately and effectively incorporating the biological principles and the high-resolution, low-noise images into radiotracer image estimation. The present paper develops an aspect of the image-reconstruction principles and methods, addressing unified estimation of alignment parameters and radiotracer activities. ACKNOWLEDGMENT The authors would like to thank K. 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