Surface-based partial-volume correction for high-resolution PET

NeuroImage 102 (2014) 674–687

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NeuroImage journal homepage: www.elsevier.com/locate/ynimg

Surface-based partial-volume correction for high-resolution PET Thomas Funck a,b, Caroline Paquette b,c, Alan Evans a, Alexander Thiel b,c,⁎ a b c

Montreal Neurological Institute, McGill University, Montreal, Canada Jewish General Hospital, Montreal Canada Department of Neurology and Neurosurgery, Montreal, Canada

a r t i c l e

i n f o

Article history: Accepted 20 August 2014 Available online 29 August 2014 Keywords: Positron emission tomography Partial-volume correction Cortical thickness Magnetic resonance imagery

a b s t r a c t Tissue radioactivity concentrations, measured with positron emission tomography (PET) are subject to partial volume effects (PVE) due to the limited spatial resolution of the scanner. Last generation high-resolution PET cameras with a full width at half maximum (FWHM) of 2–4 mm are less prone to PVEs than previous generations. Corrections for PVEs are still necessary, especially when studying small brain stem nuclei or small variations in cortical neuroreceptor concentrations which may be related to cytoarchitectonic differences. Although several partial-volume correction (PVC) algorithms exist, these are frequently based on a priori assumptions about tracer distribution or only yield corrected values of regional activity concentrations without providing PVE corrected images. We developed a new iterative deconvolution algorithm (idSURF) for PVC of PET images that aims to overcome these limitations by using two innovative techniques: 1) the incorporation of anatomic information from a cortical gray matter surface representation, extracted from magnetic resonance imaging (MRI) and 2) the use of anatomically constrained filtering to attenuate noise. PVE corrected images were generated with idSURF implemented into a non-interactive processing pipeline. idSURF was validated using simulated and clinical PET data sets and compared to a frequently used standard PVC method (Geometric Transfer Matrix: GTM). The results on simulated data sets show that idSURF consistently recovers accurate radiotracer concentrations within 1–5% of true values. Both radiotracer concentrations and non-displaceable binding potential (BPnd) values derived from clinical PET data sets with idSURF were highly correlated with those obtained with the standard PVC method (R2 = 0.99, error = 0%–3.2%). These results suggest that idSURF is a valid and potentially clinically useful PVC method for automatic processing of large numbers of PET data sets. © 2014 Elsevier Inc. All rights reserved.

Introduction Positron emission tomography (PET) is the only imaging modality to directly and quantitatively measure cerebral blood flow, metabolism or neuroreceptor binding in vivo using radiotracers. The accuracy of measuring the regional radioactivity concentration of a radiotracer in a brain region depends on the radioisotope and the intrinsic resolution of the PET camera. New generation dedicated brain PET-scanners (CTI/ Siemens HRRT) with a spatial resolution of approximately 2.4 mm in the center of the field of view (Wienhard et al., 2002) can measure differences in neuroreceptor concentrations related to cytoarchitectonic differences (la Fougere et al., 2011) or metabolic rates in small brainstem nuclei (Heiss et al., 2004). All imaging modalities suffer from the intrinsic finite resolution of the imaging system, resulting in images affected by partial volume effects (PVE). In PET, PVEs consist in multiple tissue types contributing to the radioactivity measured in a certain brain volume (voxel) and in the dispersion of focal radiotracer concentrations (Rousset and Zaidi, ⁎ Corresponding author. E-mail address: [email protected] (A. Thiel).

http://dx.doi.org/10.1016/j.neuroimage.2014.08.037 1053-8119/© 2014 Elsevier Inc. All rights reserved.

2006). Objects, such as the cortical gray matter, that are less than approximately half the size of the full width at half maximum (FWHM) of the scanner's point spread function (PSF) are frequently underestimated because their peak values are lost (Hoffman et al., 1979). Partial-volume correction (PVC) of PET images is thus necessary to accurately represent the radiotracer distribution in the cortical gray matter (GM) and distinguish, for example, true neuronal loss from cortical thinning. Methods for correction of PVE can be classified into methods performing the correction on regions-of-interest (ROIs), thus generating corrected regional tracer concentration values, or voxel-based methods which generate corrected images. Early ROI-based methods estimated the amount of loss from neural tissue and used the ratio of lost to observed signal as a recovery coefficient to estimate the original activity concentration (Chawluk et al., 1987; Herscovitch et al., 1986; Kessler et al., 1984; Mazzioatta et al., 1981). To account for PVEs between multiple ROIs, Rousset et al. (1998) proposed the geometric transfer matrix (GTM) method that calculates the cross-contamination for an arbitrarily large number of distinct ROIs derived from segmented MRI. Whereas the initial GTM method calculated the blurring of the PSF in sinogram space with an analytic PET simulator using 2D acquisition,

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similar singoram-based implementations based on 3D acquisition PET scanners have also been proposed (Reilhac et al., 2000; Rousset et al., 2008). The GTM was also implemented using image-based convolution to model the PET scanner's PSF (Frouin et al., 2002; Labbe et al., 1998). Recently, an extension of the GTM method has been proposed by Sattarivand et al. (2012) that uses probabilistic instead of discrete representations of the ROIs. Least-squares minimization (Aston et al., 2002) and iterative deconvolution methods (Teo et al., 2007) have also been used in the context of ROI-based PVC. Initial voxel-based correction methods defined tissue types on MRI and corrected each voxel for spill-out from that tissue region and for spill-in from neighboring tissue regions (Meltzer et al., 1990; Muller-Gartner et al., 1992; Videen et al., 1988). These methods were fundamentally limited by the assumption of a homogeneous tracer activity within tissue regions. Several voxel-based techniques were developed to provide PVC without either the assumption of regional homogeneity or the use of anatomic information (Boussion et al., 2009; Kirov et al., 2008; Tohka and Reilhac, 2008). This makes the PVE problem significantly more difficult to solve, but has the advantage of being applicable when no high resolution anatomic scan has been performed. Voxel-based approaches that do not assume regional homogeneity but do use high resolution anatomic information are currently the most promising algorithms and provide the best PVC with the least restrictive assumptions. One approach has been to use wavelet-based methods to incorporate anatomic information from MRI into the PET image (Boussion et al., 2006; Le Pogam et al., 2011; Shidahara et al., 2009; Turkheimer et al., 2008). Markov random fields have also been used to incorporate more sophisticated prior constraints on the deconvolution process (Bousse et al., 2012; Wang and Fei, 2012). The earlier methods described above make simplifying a priori assumptions about the radiotracer distribution which limits the range of possible applications for PVC. The most important of these assumptions is that radiotracer concentrations are homogeneous within a brain region under investigation (Aston et al., 2002; Muller-Gartner et al., 1992; Rousset et al., 1998). Contemporary PVC methods have begun to relax this assumption by assuming only that the tracer distributions within a specific region are similar but not necessarily uniform, e.g. regional values are sometimes assumed to follow the same sampling distribution (Bousse et al., 2012). These methods often require user input to adjust the regularization parameters for the deconvolution (e.g., Bousse et al., 2012; Wang and Fei, 2012). This user interaction makes automated PVC of large data sets difficult, especially for nonexpert users unfamiliar with advanced signal processing and probabilistic estimation (e.g., in a clinical context). We therefore developed idSURF, a new and fully automated iterative deconvolution (id) based PVC method that eschews the assumption of homogeneous radiotracer distribution and minimizes user interaction. This method uses mesh-based surface representations of the GM in conjunction with surface-based anatomically constrained filtering (SURF). The mesh-based cortical GM representation is generated from T1-weighted magnetic resonance imaging (MRI) and therefore provides a very accurate model of the anatomical structure of the cortical GM. Mesh-based representations have certain advantages over traditional, volumetric representations because they provide a smooth, continuous representation of the cortical GM. Although a previous attempt was made to use a mesh-based surface for PVC, this method assumed regional homogeneity and was designed for use specifically with [18F]-FDG (Park et al., 2006). The validity of idSURF was assessed using both simulated and real data. We first performed a qualitative comparison of mesh-based representations of the GM compared to traditional segmentation-based representations. idSURF was validated against numerically simulated PET images to demonstrate its performance across a broad range of noise-to-signal values. The performance of idSURF was also evaluated on BPnd values derived from real clinical data sets, where the true

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distribution of radioactivity concentrations in the object was not known, and results of idSURF were compared to the standard GTM method. Method Image acquisition Twenty one subjects were used for the numerical and in vivo validation experiments. Nine subjects were healthy (age 61 ± 10 years, 9 males) without history of focal morphological pathologies other than mild microvascular disease. Twelve subjects (62 ± 4 years old, range: 56–70, 8 males) with a clinical diagnosis of idiopathic Parkinson's disease (PD) and treated with levodopa participated in the study. Subjects had a Hoehn & Yahr score between 1.5 and 3. All subjects provided informed consent in accordance to McGill University's Faculty of Medicine Internal Review Board regulations for human subjects' studies and the Helsinki Declaration. [18F]Flumazenil scans were obtained for all subjects with an ECAT HRRT PET scanner in list mode (Siemens Medical Solutions, Knoxville, TN, USA) (Wienhard et al., 2002). The ECAT HRRT is a dedicated full 3D high resolution brain scanner, with a field view of 25.2 cm (axially) and 31.2 cm (diameter), and has a spatial resolution of between 2.3 and 3.4 mm at FWHM and enables data acquisition with high spatial resolution combined with high sensitivity. In addition, the use of two crystal layers (LSO/LYSO) permits photon detection with depth-of-interaction information. After a transmission scan for attenuation correction (137Cs-source), approximately 370 MBq [18F] FMZ were injected intravenously as a slow bolus over 60 s. The list mode data were acquired for 60 min after injection and were subsequently binned into 2209 sinograms (each of size 256 radial bins × 288 azimuthal bins) using span 9 compression for a total of 17 time frames (40 s, 20 s, 2 × 30 s, 360 s, 4 × 50 s, 3 × 300 s, and 3 × 600 s), resulting in images with a voxel size of 1.22 × 1.22 × 1.22 mm3. Fully 3D FBP by 3D reprojection (3D RP) was carried out with a Hamming windowed Colsher filter (alpha = 0.5, cut off at the Nyquist frequency). MRI sequences were obtained for all subjects. For the healthy subjects, anatomical T1-weighted MRI sequences were obtained on a Siemens Sonata 1.5 Tesla Scanner (Siemens MedicalSolutions, Erlangen, Germany) using a 3D fast-field echo scan sequence (TR, 22 ms; TE of 9.2 ms, and a matrix size of 256 × 256 resulting in images with a voxel size of 1 × 1 × 1 mm). For the remaining 12 subjects, anatomical T1-weighted MRI was obtained on the Siemens TIM Trio 3T MRI scanner with a gradient-echo sequence (TR, 2300 ms; TE, 2.91 ms, and a matrix size of 256 × 240 resulting in images with a voxel size of 1 × 1 × 1 mm). Iterative deconvolution with surface-based anatomically constrained filtering (idSURF) Mesh-based volumetric reconstruction of cortical GM CIVET is an image processing pipeline that generates mesh-based representations of the cortical GM from MR images. CIVET uses the non-parametric N3 method to correct MR field non-uniformity (Sled et al., 1998). The MRI is then transformed to MNI stereotaxic space of the ICBM 152 6th generation non-linear brain atlas (Mazziotta et al., 2001), using a 12 parameter affine transformation (Collins et al., 1994). Spatially normalized images are then segmented into 3 tissue classes: GM, white matter (WM) and cerebral spinal fluid (CSF) using INSECT (Zijdenbos et al., 1998), a discrete classifier, as well as a probabilistic classifier (Tohka et al., 2004). The Constrained Laplacian Anatomic Segmentation using Proximity (CLASP) algorithm generates a meshbased representation of the cortical GM using two deformable mesh models consisting of vertices connected to form triangles. Vertices forming the outer mesh are defined by expanding the inner vertices to the GM/CSF boundary thus generating two meshes with corresponding

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Fig. 1. Example of a sub-volume constructed from vertices in meshes. Green points represent vertex points and red points are locations on the volumetric grid that were found to lie inside the sub-volume defined by the two congruent pentagonal sub-surfaces and the cortical thickness. Dotted lines depict rays cast from grid points to determine whether those points are inside the sub-volume. Case I shows a point inside the sub-volume ray with only 1 intersection, while cases II and III show that points outside the mesh will have 0 or 2 points of intersection.

vertices. These meshes are fitted to the interior and exterior of the cortical GM (Kim et al., 2005). The interior and exterior meshes produced by CIVET were used to reconstruct a discrete volumetric representation of the cortical GM. A grid containing the CIVET meshes was defined at MRI resolution. Each grid point represented the center of a voxel. The CIVET meshes were divided into small sub-surfaces, each centered on a single vertex within the larger meshes. The sub-surfaces consisted of a single central vertex and its five or six neighboring vertices. The volume contained by two corresponding sub-surfaces on the inner and outer meshes, together with the planes formed by joining these inner and outer surfaces, defined a closed sub-volume. These sub-volumes represented a small volume in the overall cortical GM. Classifying the interior grid points inside of the sub-volumes was done using an even–odd ray casting algorithm (Haines, 1994). The algorithm proceeded as follows: for each point near a given sub-volume, a ray was drawn in an arbitrary direction from a grid-point. The number of intersections of this ray with the planes defining the sub-volume around the grid-point was calculated. If the ray was found to intersect 0 or 2 of these planes, then the point was defined as being outside the sub-volume. If the ray intersected a single plane, then the point was considered as being within the sub-volume (Fig. 1). The grid points inside the sub-volume were identified as the centers of voxels within the cortical GM. By defining an arbitrarily fine grid, it is possible to use the information from the CIVET meshes to create a super-resolution volumetric (SRV) image of the cortical GM. SRV GM masks at MRI resolution were used for PVC in the present study. Probabilistic masks for the WM and CSF were derived using CIVET and thresholded at 50% to produce binary WM and CSF masks. The binary WM and CSF masks were interpolated to the same dimensions as the SRV GM mask using the nearest neighbor sampling. Masks for the basal ganglia were defined by segmenting approximate masks manually and removing regions overlapping with the WM and CSF masks.

mean values of voxels within the same anatomic region as the voxel over which the filter kernel was centered (Fig. 2). It thus provided smoothing without blurring across borders of distinct regions, but requires predefined binary masks that define distinct anatomic regions. In this case SRV GM and WM masks were used to define the regions. The PET image was co-registered to the MRI using a 6-parameter affine transformation and then transformed into stereotaxic MNI-space with linear interpolation (Mazziotta et al., 2001). The PET image was then interpolated to SRV mask resolution with the nearest neighbor interpolation. The WM, CSF and GM values of the PET image served as the initial estimate of the true tracer distribution. The initial estimate, or test image ti, of the true tracer distribution was then convolved with the PSF of the PET scanner, h, which was approximated by an isotropic Gaussian filter (Eq. (1), where r is an index value for a voxel location). Division of the observed PET image values, o, by the filtered test image produced a volume of residuals. The initial estimate image was then multiplied by the residuals and denoised with the SURF filter. The new test image, ti + 1, was then used as an improved estimate of the true radiotracer distribution and

Iterative deconvolution and surface-based filtering An iterative deconvolution algorithm was designed to be used in conjunction with the SRV mask and the SURF filtering process. The SURF filter is an anatomically constrained filter that calculated the

Fig. 2. Illustration of the SURF filter. The image represents a small slice from a mask, where voxels of value 1 are considered to be within a particular anatomic region. The SURF filter is centered at a voxel in the mask and averages over the surrounding voxels with a value of 1. In this example, the total number of voxels in the intersection of the mask and kernel is 6, so the voxels in this intersection are divided by 6.

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Fig. 3. Schema of idSURF. The iterative deconvolution proceeded using the difference between a blurred test image and the observed PET image to make iteratively better estimations of the true tracer distribution. Anatomically constrained averaging was used in each iteration to attenuate noise. The process was repeated until convergence to a stable solution was reached.

entered again into the iterative process. The algorithm repeated the previous steps until either a stable solution or a maximum number of iterations were reached (Fig. 3). A solution was considered stable when the average percentage change over three iterations in the rootsquared error between the observed and the filtered test image fell below a predetermined convergence threshold. !

t

iþ1

i

ðr Þ ¼ SURF t ðr Þ 

oðr Þ h  t i ðr Þ

ð1Þ

Implementation The algorithms described above, including both SRV, idSURF and the GTM method, were implemented in C programming language and compiled using the GNU C Compiler (GCC) (version: gcc-4.4.4). The code was written using standard C libraries as well as the open-source libraries MINC (McConnel Brain Imaging Center, Montreal Neurological Institute, version: minc-2.0.18, http://www.bic.mni.mcgill.ca/ServicesSoftware/ MINC) (Montreal Neurological Institute, McConnell Brain Imaging Centre) and the Numerical Recipes library (Press et al., 1992). All codes were run in a Linux environment on the servers of the McConnell Brain Imaging Center (BIC), Montreal Neurological Institute. Validation of idSURF Validation metrics The idSURF algorithm was evaluated for accuracy and precision. Accuracy refers to the average difference between idSURF's estimation of the radioactivity concentrations and the true activity concentrations.

This difference is measured as the normalized mean root squared error (MRSE). This metric is normalized by dividing the absolute value of the error by the true value (Eq. (2)). MRSE thus gives the average percentage error per voxel. Precision is measured as the standard deviation of the MRSE.

MRSE ¼

1X n

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Trueðr Þ−Test ðr Þ Trueðr Þ

ð2Þ

Qualitative comparison of cortical GM representations Reference GM-representations were obtained by segmenting MR images using the standard volume-based probabilistic method implemented in CIVET with a 50% threshold (Tohka et al., 2004). A qualitative comparison was then made between those reference images and the SRV masks which incorporated surface mesh information. SRV masks were produced at 1 mm3 (i.e., at MRI resolution), 0.5 mm3 and 0.333 mm3. The masks were superimposed on the original MRI and visually compared for fidelity to the underlying anatomy and for the presence of noise. Validation with numerical simulation An image-based approach was used to simulate 21 PET images from each subject's segmented MRI. The first step was to intersect the SRV GM mask with the Juelich cytoarchitectonic atlas (Eickhoff et al., 2005), which labels distinct anatomic regions with values ranging from 0 to 121. This intersected image was then multiplied by 0.10 and a uniform value of 30 was added to all voxels within the cortical GM, thus yielding simulated activity distributions ranging from 30 to 42

Fig. 4. Example of a randomly generated initial image for PVC with idSURF. Images were generated by sampling a uniform distribution (with values from 1 to 96) at each voxel within the brain.

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Reference GM Segmentation

(1x1x1mm)

(0.5x0.5x0.5mm)

(0.33x0.33x0.33mm)

Fig. 5. Comparison of reference GM masks with SRV GM masks. Increasing the sampling of the CIVET mesh produces images with smoother, better defined sulci. Blue and red circles indicate sulci that were fused in the reference mask, but became better delineated with SRV GM masks of progressively higher resolution. The green circles show a sulcus that was still fused in the 1 mm isotropic SRV mask, but became delineated with higher resolution. The straight surface near the corpus callosum does not represent the cortical GM because CIVET is not designed to model this anatomic region.

units. The CSF, WM and basal ganglia received values of 5, 15 and 25, respectively. The aim was to produce a physiologically plausible ground truth radiotracer distribution based on cytoarchitechture. This image was then convolved with an approximation of the PET scanner PSF, modeled using an isotropic Gaussian filter with a FWHM of 3 mm. White noise with standard deviations ranging from 0 to 5 was added to each voxel in the image. While the standard deviation of the noise was uniform across the image, the noise-to-signal ratio varied between regions. Thus the noise-to-signal ratio was higher in regions with below average signal, e.g., the WM, and lower in region average signal. The total noise level was calculated from the sum of the absolute value of the noise at each voxel within the brain. It was then divided by the sum of the activity in all brain voxels of the blurred image to give the average noise-to-signal ratio. The range of standard deviations corresponded to noise-to-signal ratios of 0 to 16% of the average whole brain activity. These simulated PET images then underwent PVC with idSURF. To determine the number of iterations needed to reach a stable solution, the convergence threshold was set at 0 and the maximum number of

Reference GM

iterations was set to 13. This number of iterations proved sufficient for the algorithm to reach a stable state at which point further iterations did not improve the PVC image. To assess the rate of convergence of the algorithm, the MRSE was plotted against the number of iterations. To evaluate the effect of using surface-based filtering on the recovered PVC values, 1 × 1 × 1 and 5 × 5 × 5 kernels were used. Note that the 1 × 1 × 1 kernel takes the average over 1 voxel and is therefore equivalent to no smoothing being performed. Corrected images were then compared to the true image in order to calculate the average and standard deviation of the true MRSE between the true image and the PVC corrected image. A linear regression was performed across anatomic regions in each subject. Anatomic regions were defined on the 2009 MNI ICBM 152 atlas and were not the same with those used to define the activity distributions (Fonov et al., 2009). A second numerical simulation was performed to investigate the effectiveness of using SRV masks in idSURF when correcting inhomogeneous tracer distributions defined along the cortical surface. In addition, deconvolution was performed with the GTM to test the accuracy of

Super-Resolution (1x1x1mm)

Super-Resolution (0.5x0.5x0.5mm)

Fig. 6. Overlay of reference mask and SRV masks. The SRV masks are less noisy, smoother and more accurate than the reference GM mask. Increasing resolution produced smoother images with better definition of sulci.

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using a region-based approach in such a situation. A SRV GM mask with 0.333 mm3 voxels was derived from a single subject. The central sulcus for this subject was identified visually and ROIs for the anterior and posterior banks were segmented manually on the SRV GM volume with 0.333 mm3 voxels. The posterior bank was assigned a value of 50 and the anterior bank a value of 45. The remaining GM was given a value of 30. The image was convolved with an isotropic Gaussian filter with a FWHM of 3 mm and white noise was added to produce an image with 4% noise-to-signal. The blurred, noisy image was then deconvolved with idSURF. The PVC with idSURF used two SRV GM masks. The first GM mask was the SRV image generated with 0.333 mm3 voxels, while the second was generated with 1 mm3 voxels that were then upsampled to 0.333 mm3 with the nearest neighbor interpolation. Validation against GTM on real PET images The idSURF method was empirically validated by comparing its performance to the standard GTM method using real PET images. Each of the 10 frames (frames 8–17) from 21 dynamic [18F]Flumazenil PET scans from healthy older adults and subjects with PD was corrected with both the idSURF and the GTM method. For the GTM method, GM ROIs were defined in each hemisphere on the 2009 MNI ICBM 152

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atlas (Eickhoff et al., 2005) for the frontal, parietal, temporal, and occipital lobes. ROIs for the WM and CSF were defined using the probabilistic CIVET masks, thresholded at 50%. For the idSURF PVC, the PSF was an isotropic Gaussian filter with a FWHM of 2.4 mm. A maximum of 10 iterations was used in the deconvolution procedure and a 5 × 5 × 5 kernel was used for the filtering. A convergence threshold of 5% was used because it led to convergence within 5–8 iterations, which was found to be a sufficient number from the numerical simulations. A linear regression of idSURF on GTM values across subjects and GM regions defined on the 2009 MNI ICBM 152 (Fonov et al., 2009) was performed. The CSF contains regions with large PVE, e.g. within the lateral sulcus, that render the problem of PVC significantly more difficult to solve, but is unlikely to have significant [18F]Flumazenil tracer concentrations. The CSF was thus not used in the idSURF deconvolution of the in vivo PET data and CSF voxels were assumed to have a value of 0. Parametric non-displaceable binding (BPnd) was created using the Logan plot method (Logan et al., 1996), with a WM reference region and a start time of 300 s for the linear regression. The GTM was implemented as described in Rousset et al. (1998), but with image-based blurring instead of sinogram-based PSF modeling. Briefly, frontal, parietal, occiptal and temporal GM regions (defined

Fig. 7. The average MRSE is shown for each subject at noise-to-signal ratios ranging from 0 to 16%. Columns represent anatomic regions: frontal lobe, occipital lobe, parietal lobe and temporal lobe. Rows represent the number of voxels in the SURF filter. IdSURF provides accurate PVC (within 1–5%) when a SURF filter of 5 × 5 × 5 voxels is used (bottom row). Note the difference in order of magnitude of the MRSE between the SURF filter sizes.

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according to the 2009 MNI ICBM 152 (Fonov et al., 2009)) as well as WM, CSF and basal ganglia regions were convolved with a 5 × 5 × 5 Gaussian kernel (FWHM = 2.4). Recovery coefficients were extracted from the regional spread functions as described and used to correct the observed regional activity concentrations. Convergence of idSURF algorithm A Monte-Carlo simulation approach was used to test the convergence of idSURF under different starting conditions. The same in vivo [18F]Flumazenil PET image from a single healthy subject was deconvolved 20 times, each time starting with a different randomly generated initial test image. The randomly generated initial images were created by assigning uniformly distributed random values (ranging from 1 to 96) to the GM, WM and CSF voxels (Fig. 4). This range of values was selected to yield an image with an expected value of the activity distribution that would be larger than the PVC values on which idSURF converged. Each deconvolution used a maximum of 20 iterations, a 5 × 5 × 5 SURF filter, and a convergence threshold of 1%.

Results Qualitative comparison of cortical GM representations The SRV GM masks show improved anatomic representation of the GM and less noise than those produced with the reference method. From Figs. 5 and 6, it appears that the SRV representations better identified the convoluted structure of the GM by distinguishing more clearly between sulcal walls than the reference images. Increasing the resolution of the SRV further improved the representation of the GM by clearly differentiating between narrower sulcal walls, e.g. in the cingulate sulcus, that were not clearly delineated at MRI resolution. Accuracy and precision of numerical simulations The results of the numerical simulations (Fig. 7) showed that across a broad range of noise levels, idSURF was consistently able to recover accurate values when the SURF filter was used. All subjects showed a similar positive relationship between noise and accuracy.

Fig. 8. The standard deviation of the MRSE is shown for each subject at noise-to-signal ratios ranging from 0 to 16%. Columns represent anatomic regions: frontal lobe, occipital lobe, parietal lobe and temporal lobe. Rows represent the number of 1 mm3 voxels in each dimension of the SURF filter. Given a slope of 0.002 to 0.02 for linear models of the cortical GM, the precision of the recovery was largely unaffected by noise when the SURF filter was employed.

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Fig. 9. The MRSE of the idSURF PVC decreases rapidly within the first 6 iterations and quickly converges to a stable solution. Convergence is achieved quickly across all noise levels.

In general, the linear regression of MRSE on noise-to-signal showed a strong correlation between noise and error. For images deconvolved without the SURF filter (i.e., a SURF filter of 1 × 1 × 1) the R2 value was consistently above 0.98, while the R2 value for deconvolution with a 5 × 5 × 5 filter was between 0.58 in the occipital lobe and 0.95 in the frontal lobe. This shows a strong linear relationship between the noise-to-signal ratio and MRSE, but exhibits regional variation.

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The use of the SURF filter greatly increases the accuracy of the deconvolution. Without the SURF filter, the slope of the linear regression model for the lobes of the cortical GM was between 7.4 and 7.9. However, when the SURF filter was employed the slope of the linear regression model decreased to between 0.06 and 0.09 for the cortical GM. The precision of idSURF was much more strongly linearly related to the noise-to-signal ratio when the SURF filter was not used (Fig. 8). Without a SURF filter, the slope for the linear regression model was between 10 and 12, with an R2 of 0.99, within the lobes of the cortical GM. This indicated that increasing noise-to-signal ratio greatly decreased the precision of the deconvolution. When a SURF filter was used, there was a very small, but statistically significant, linear relationship between noise-to-signal ratio and the standard deviation of the MRSE. On average the linear regression for the cortical GM had a slope between 0.002 and 0.02. The R2 varied between regions, between 0.006 in the temporal lobe and 0.52 in the occipital lobe. Therefore, the precision of the deconvolution was almost unaffected by the noise level in the image when the SURF filter is used, but was affected by regional differences. The algorithm converged rapidly—within 6 to 8 iterations—across all noise levels (Fig. 9). In practice a 5 × 5 × 5 SURF filter appeared to provide an optimal trade-off between computational efficiency and denoising. Fig. 10 gives an illustrative example of the results from the numerical simulations in a typical subject. In the deconvolution of the numerically simulated PET image with activity distributions defined along the central sulcus, only the idSURF PVC with a SRV GM mask with 0.333 mm3 voxels was able to accurately recover the true tracer distribution. The MRSE for the PVC values produced using a SRV GM mask defined at 0.333 mm3 was 4% ± 4.7% on the posterior bank of the central sulcus and an MRSE of 3% ± 3.6% on the anterior bank of the central sulcus. When performing the deconvolution with a SRV GM mask that was defined at 1 mm3 and then upsampled to 0.333 mm3, the MRSE on the posterior bank was 19.7% ± 30% and 16.4% ± 29%. This can be seen in Fig. 11.

Fig. 10. Effect of idSURF for subject C1. Note the excellent recovery of the original regional activity. The PVC images are visually very similar to the true image, illustrating the effective noise attenuation of the SURF filter. The MRSE image shows the error in the PVC and has a scale from 0 to 1.

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Fig. 11. Coronal slice of PVC images defined on simulated image based on a SRV GM mask with 0.333 mm3 voxels. PVC performed with a SRV mask defined with 0.333 mm3 voxels is very accurate, while the PVC performed with an SRV mask defined at 1 mm3 and subsequently upsampled to 0.333 mm3 did not recover the distinct activity values on the posterior and anterior banks of the central sulcus.

converged to virtually the same solution for regions in the GM and very similar solutions in the WM (Fig. 15).

Correlation between GTM and idSURF The linear regression of PVC radiotracer concentrations across several regions of the cortical GM show that both idSURF and the GTM method produced similar results—within 0–3.2% of one another—when correcting real subject PET data (Table 1, Fig. 12). This finding is consistent both across regions and in individual subjects (Fig. 13). The occipital lobe had the largest discrepancy between idSURF and GTM values (3.2% in the left hemisphere and, 1.7 in the right hemisphere). The BPnd values produced by both methods exhibited greater variability than in the tracer concentrations, specifically in the occipital lobe (Table 1). Whereas the other regions had an approximately 2% difference between idSURF and the GTM, the difference in the occipital lobe was approximately 7–8%. From Fig. 14, it is clear that even when the GTM and idSURF have close regional averages there can be large local variations of estimated BPnd values within a ROI. The 5% convergence threshold was small enough for fast convergence after approximately 4–8 iterations (approximately 1 min per PET frame). For both radiotracer concentrations and BPnd values, the linear regression exhibits an almost perfect correlation between the two methods, with an R2 of 0.99 for each region.

Convergence of idSURF algorithm to a unique solution The 20 randomly generated initial test images for idSURF produced virtually identical values to one another. Whereas the initial images had values of approximately 66.6 ± 37.2 in all regions, the PVC images

Table 1 Slopes for linear regressions of radiotracer concentrations and BPnd values produced by regressing idSURF on GTM PVC values. For radiotracer concentrations, the slopes are between 0–3.2% points of unity. There is greater variability in the slopes for the linear regression of regional BPnd values (1.2–7.8% points). The WM was used as the reference region for the Logan plot and so by definition has a BPnd value of 0 in both the GTM and idSURF. Region

Radiotracer concentrations

BPnd

Frontal (L) Frontal (R) Parietal (L) Parietal (R) Occipital (L) Occipital (R) Temporal (L) Temporal (R) Basal ganglia White matter

1.009 1.009 1.007 1.006 0.968 0.983 1.014 1.01 0.999 1.032

0.987 0.985 0.983 0.980 0.920 0.925 0.980 0.971 0.984 NA

Discussion The results from the PVC of the numerically simulated and clinical images show that idSURF is able to recover accurate values across a broad noise range and shows excellent correlation with the standard GTM method. IdSURF is an iterative algorithm that convolves an estimate of the true tracer distribution with a model of the PET scanner PSF. This convolution results in a blurred version of the estimate that is compared to the actual observed PET image. Discrepancies between the blurred estimate and the observed PET image are used to correct the initial estimate. To prevent noise amplification, the corrected estimate is smoothed with surface-based filtering to give a new estimate of the true tracer distribution. The process begins again until the average change in the residuals over several iterations falls below a predetermined threshold. Advantages of mesh-based cortical gray matter mask The aim of using SRV was to reconstruct an improved volumetric GM representation based on the more accurate mesh representation produced by CIVET. The increased accuracy was due to the more precise modeling of the GM topology with CLASP. CLASP uses the topology of the WM–GM surface, where PVEs are minimal, to constrain the surface representation of the exterior GM–CSF surface where PVEs are more prevalent. This implicit correction results in improved GM representations where sulci that were fused in the volume-based segmentation are now separated. In addition, the SRV GM representations exhibited less noise because the noise in the probabilistic GM mask was insufficient to distort the surface fitting process. Accurate partial volume correction with numerical simulation The validation of idSURF using numerical simulations demonstrated that idSURF was able to recover accurate values for real cortical surfaces convolved with a simulated PSF, across a broad noise range. The results also demonstrated the importance of using the SURF filter to denoise the data. Failing to include such denoising led to greatly decreased accuracy and precision in the recovered values. When the SURF filter was used during the iterative deconvolution, the MRSE in the PVC images was reduced below the noise level of the observed image. This suggests that instead of amplifying noise, which is common in iterative algorithms, idSURF is able to attenuate noise.

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Fig. 12. Linear regression through the origin of idSURF on GTM across regional averages. Each point represents a regional average for a subject at a given time frame. The close correlation between the values recovered by idSURF and the GTM indicates that idSURF consistently recovers very similar values to the standard method over a wide variety of brain regions. The red line is the line of unity and the blue line is the line of regression.

While the accuracy of the PVC in the numerical simulations decreased linearly with increasing noise, this was much less pronounced with the precision of the PVC. A 1% increase in noise corresponded to a 0.002% to 0.02% increase in the standard deviation of the MRSE. The effect of noise on precision was not identical across lobes of the cortical GM. This is because the efficacy of the SURF filter for noise attenuation depends on the number of voxels over which it averages at any particular location. Thinner portions of the cortical GM will thus have less precise PVC values. The variance around the solution on which idSURF converges is thus only very slightly affected by the noise in the observed image. IdSURF is therefore robust in the presence of high noise-to-signal levels. Overall, in simulated images of the cerebral cortex idSURF was able to recover activity concentrations within 1–5% of the true values. The idSURF has similar accuracy to the leading voxel-based methods that incorporate anatomic information. On images derived from analytic simulations of the HR + with (Videen et al., 1988) F-FDG PET, the algorithm of Le Pogam, et al. (Shidahara et al., 2009) produced PVC

values with a mean error of 8.9% ± 2.7%. With the numerically simulated cylindrical images, the iterative Bayesian method of Wang and Fei (2012) produced PVC values with less than 5% recovery error with 5% noise-tosignal. Meanwhile, the Markov Random Field approach of Bousse et al. (2012) found approximately 90–95% recovery of the true tracer distribution in the cortex for numerically simulated images. In the deconvolution of the simulated image with activity defined on the central sulcus, only the use of idSURF with a SRV GM mask defined at 0.333 mm3 voxel dimensions was able to recover accurate values. Thus upsampling a GM mask that was defined at 1 mm3 to 0.333 mm3 was not equivalent to using a GM mask that is originally defined at 0.333 mm3. Specifically, in the GM mask defined at 0.333 mm3, the posterior and anterior banks of the precentral sulcus were clearly distinguished, whereas they were often fused in the upsampled image. Insofar as surface-based models of the cortical surface are able to accurately represent the cortical GM, volumetric images derived from these surface-based models are better able to model the complex topology of the cortical GM than GM segmentations based only on volumetric MRI.

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Fig. 13. Linear regression through the origin for idSURF on GTM PVC values across subjects. Each point represents a regional average for a subject at a given time frame. This shows that idSURF is consistently able to recover similar values to the GTM across subjects. The red line is the line of unity and the blue line is the line of regression.

Although radioactive decay follows a Poisson distribution (Vardi et al., 1985), Teymurazyan et al. (2013) showed that noise in PET images is poorly modeled by the Poisson distribution. Instead, when using FBP, PET image noise is well modeled by a normal distribution and is thus an appropriate noise model for image-based PET simulations. While noise levels are known to vary within the PET scanner's field of view (Pajevic et al., 1998), the SURF filter assumes only that the noise follows a sampling distribution with a mean of 0. Thus even if noise in the PET image varies spatially in its standard deviation, this will not impact the performance of the SURF filter. What must be demonstrated is that, given a predetermined filter size, the SURF filter is able to attenuate noise even at the largest plausible noise-to-signal ratio. We thus evaluated the SURF filter's performance across a broad range of noise-tosignal ratios to demonstrate its robustness. Close correlation between gold standard and iterative deconvolution The empirical validation reveals a close correlation between the average restored values of idSURF and the GTM method. The GTM

method is a ROI-based and is known to produce accurate results. IdSURF consistently produces very similar regional averages across all regions, with an excellent correlation between the two methods. The slope of the regression is between 0–3.2% greater than unity indicating a small but systematic and linear difference between idSURF and the GTM method. The difference between the GTM and idSURF PVC values was particularly large in the occipital lobe. In the visual comparison in Fig. 14, the uncorrected PET image appears highly heterogeneous. Given that the GTM assumes tracer homogeneity within a ROI, the violation of this assumption will produce biased results that will not conform to the PVC values produced by idSURF. In PET phantom studies, the GTM produces recovered tracer distributions within approximately 0–5% of the true tracer distribution (Frouin et al., 2002; Rousset et al., 1998). Therefore the difference between the idSURF and GTM is within the error range of the GTM. Overall, the data show that the idSURF produces PVC images that are closely linearly related, but with approximately 0–3.2% difference, to the PVC values obtained with the standard method.

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BPnd

Fig. 14. BPnd image from a typical subject. The first image is the uncorrected PET image, followed by the GTM corrected and idSURF corrected PET images. Although the regional averages between the GTM and idSURF images are similar, the GTM may conceal focal areas of high activity within ROIs.

The difference in BPnd values between idSURF and the GTM was greater than that observed in the radiotracer concentrations. The reason for this increased difference is that the Logan plot is a non-linear function of the radiotracer concentrations in a target region and a reference region. Hence the BPnd values are affected by discrepancies between idSURF and GTM in both the target and reference region. This suggests that while both methods produce similar PVC estimates of radiotracer concentrations, small differences in these estimates can have a significant impact on estimates of physiological parameters, like BPnd. Convergence to unique solution independent of initial conditions Whereas some iterative deconvolution algorithms can be proven to converge to a unique and correct solution, this was not possible in the case of idSURF. The SURF filter cannot be easily represented mathematically because it does not have a closed form representation and is spatially variant. By generating a series of random images and using these as initial guesses of the true underlying distribution, the iterative deconvolution algorithm was empirically tested for potential local minima that would lead to divergent PVC solutions. From this experiment, idSURF does not appear to be very sensitive to starting conditions and consistently converges to a common solution despite highly variable start conditions.

Limitations The current implementation of idSURF has two limitations: a spatially invariant PSF and the assumption of white noise. Modeling spatially variant PSFs has been widely adopted in reconstruction algorithms (Rahmim et al., 2013). Alessio et al. (2010) evaluated the effect of an empirically derived, spatially variant PSF versus a spatially invariant PET on 3D full-body PET images and found similar levels of performance between the two methods. Similarly, D'Ambrosio et al. (2010) evaluated the effect of invariant versus variant PSF in reconstructing images with an expectation maximization algorithm for a small animal PET scanner and reported only a small improvement in recovered values with the invariant PSF versus the spatially invariant PSF. The majority of post-reconstruction PVC methods assume a spatially invariant PSF (Erlandsson et al., 2012). A post-reconstruction PVC method has been proposed by Barbee et al. (2010) that uses an empirical PSF derived from a point source. The empirical PSF produced similar results to using a spatially invariant PSF. These studies seem to suggest that only a very moderate improvement could be expected from implementing a spatially variant PSF into idSURF. Another key assumption of most PVC methods is that the noise is spatially independent with a mean centered at 0. Aston et al. (Sattarivand et al., 2012) modeled both spatially correlated and uncorrelated noises but found very similar results for the GTM method,

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Fig. 15. The bottom row shows the average values and standard deviations between the 20 randomly generated initial images. The top row represents the average value and the standard deviation between the PVC images produced using the random initial images. The high variability seen between the initial images is virtually eliminated after PVC with idSURF, showing that idSURF is not sensitive to the start conditions of the iterative deconvolution.

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