Partial volume correction of PET image data using geometric transfer matrices based on uniform B-Splines

Abstract

Objective:

Most methods for partial volume correction (PVC) of positron emission tomography (PET) data employ anatomical segmentation of images into regions of interest. This approach is not optimal for exploratory functional imaging beyond regional hypotheses. Here, we describe a novel method for unbiased voxel-wise PVC.

Approach:

B-spline basis functions were combined with geometric transfer matrices to enable a method (bsGTM) that provides PVC or alternatively provides smoothing with minimal regional crosstalk. The efficacy of the proposed method was evaluated using Monte Carlo simulations, human PET data, and murine functional PET data.

Main Results:

In simulations, bsGTM provided recovery of partial volume signal loss comparable to iterative deconvolution, while demonstrating superior resilience to noise. In a real murine PET dataset, bsGTM yielded much higher sensitivity for detecting amphetamine-induced reduction of [¹¹C]raclopride binding potential. In human PET data, bsGTM smoothing enabled increased signal-to-noise ratios with less degradation of binding potentials relative to Gaussian convolution or non-local means.

Significance:

bsGTM offers improved performance for PVC relative to iterative deconvolution, the current method of choice for voxel-wise PVC, especially in the common PET regime of low signal-to-noise ratio. The new method provides an anatomically unbiased way to compensate partial volume errors in cases where anatomical segmentation is unavailable or of questionable relevance or accuracy.

Introduction

Positron emission tomography (PET) often presents twin challenges of low signal-to-noise ratio (SNR) and low spatial resolution, two issues that can be difficult to address simultaneously. Spatial smoothing boosts SNR at the expense of resolution and partial volume contamination, while voxel-wise approaches for partial volume corrections (PVC), such as deconvolution, can amplify noise. Not surprisingly, most PVC methods are based on clustering data into regions of interest (ROIs) based on anatomical delineation (Erlandsson et al., 2012). These methods have the advantage of enabling PVC together with spatial smoothing into regions, subject to the constraint that data conforms to the imposed prior information. Other anatomically based PVC methods include MRI-guided PET reconstruction, an iterative method that encourages MRI-similar solutions during the generation of PET images (Bai et al., 2013; Lalush, 2017), and deep learning methods based upon anatomical priors or anatomically-derived PVC corrections (Matsubara et al., 2022; Sanaat et al., 2023; Song et al., 2020). Because changes in PET functional markers may not conform to anatomical boundaries, the use of anatomical priors to guide PET functional studies can bias results.

Exploratory dynamic functional imaging represents a particularly challenging regime regarding the tradeoff between SNR and temporospatial resolution. Examples include within-scan designs to assess changes in neurotransmitter release (Sander et al., 2020) or metabolism (Greve et al., 2016; Villien et al., 2014). Such studies restrict temporal averaging and are not well served by ROI analyses, which sacrifice spatial information and impose strong biases by the selection of regions. The alternative approach, voxel-wise analyses, suffer from low sensitivity, so some form of spatial smoothing is a virtual requirement in analysis of function studies. Many PET studies could benefit from a method that combines PVC with SNR enhancement without requiring the imposition of a priori information.

While most PVC methods require prior information, iterative deconvolution techniques, especially reblurred Van Cittert deconvolution (RVD) (Lagendijk and Biemond, 1991), have been shown to provide some recovery of partial volume crosstalk between voxels while avoiding the severe noise amplification that compromises deconvolution by conventional inverse filtering (Tohka and Reilhac, 2008). RVD is easily implemented and less subject to the inherent bias imposed by imperfect prior information and segmentation. However, recovery of partial volume effects is imperfect and cannot appreciably increase SNR. A further drawback of iterative methods is that the convergence rate depends upon signal and noise, which change throughout the time-activity function produced by a decaying radioligand. Although PET studies typically report summary parameters like binding potentials, PVC should be applied on a frame-by-frame basis to account for time-dependent regional variations in the signal, so PVC methods ideally should be independent of SNR.

We report a new post-reconstruction method for PET studies that combines spatial smoothing with partial volume correction. The method employs the well-known framework of the geometric transfer matrix (GTM), commonly employed using a set of anatomical regions (Rousset et al., 1998). However, we embed GTM within uniform B-splines that span the space. We show that the new method provides recovery of partial volume spillover comparable to RVD but is less affected by noise. Also, varying degrees of smoothing can be imparted in exchange for some loss of PVC. We evaluate and demonstrate the method using Monte Carlo simulations, human positron emission tomography (PET) data, and a preclinical model that assesses neurotransmitter release using PET.

Methods

B-spline GTM

This method aims to combine smoothing and PVC into a robust post-reconstruction step that does not require prior spatial knowledge. To approach this issue intuitively, note that smoothing relates closely to the creation of ROIs, i.e., smoothing replaces each voxel with a weighted average of surrounding values. If we span the three-dimensional space with basis functions approximating Gaussian shapes, we can provide smoothing comparable to Gaussian convolution. Creating non-binary regions from this type of spatial averaging would enable application of the GTM framework, which provides a simple and robust way to apply PVC corrections to ROI-based data in a non-iterative way independent of frame SNR. Gaussian-shaped regions would be suitable for our task if we can ensure that 1) the regions accurately fit three-dimensional data consistent with the resolution of the basis functions, and 2) the regions are sufficiently separated in space to enable identification of unique coefficients by avoiding collinearity during image fitting.

The ROI characteristics described above closely resemble three-dimensional uniform B-spline basis functions, which are piece-wise polynomials that overlap and sum to unity across the three-dimensional space. Uniform B-splines are often used in image processing for fitting data and for multi-resolution representation of images such as the deformation fields used during nonlinear registration to standardized spaces (Skare and Andersson, 2005). Cubic B-splines have a continuous second derivative and were employed in this work. As shown in Fig 1a using a unit grid spacing, uniform cubic B-splines are shifted copies of a single function that closely approximates a Gaussian form with a grid spacing 30% smaller than the full width at half maximum (FWHM) and with a finite support region. Fig. 1b shows two rows of these functions across a single plane of mouse brain; the full set must span the full three-dimensional field of view (FOV). The explicit algorithm combines B-spline basis functions with standard GTM formalism.

Fig. 1.

a) Uniform cubic B-splines are depicted in one dimension. The red curve shows a matched Gaussian function for comparison. b) A subset of cubic B-splines designed to model a 1 mm PSF in mouse brain. c) The matrix condition number, indicating the stability of inversion of the geometric transfer matrix describing the overlap of PSF-blurred B-spline basis functions onto the original basis set, is shown versus the ratio of basis spacing to the PSF.

1. Choose the grid spacing for three-dimensional B-spline functions based upon the known PSF introduced by the imaging system plus any prior post-reconstruction smoothing. The selection of grid spacing is addressed below. The three-dimensional spacings between function maxima fully determine the b-spline set.

2. Pad the FOV using signal reflection at boundaries to extend the placement of B-spline functions and minimize drop-off near the edge of the FOV. Fill the padded FOV with B-spline functions using the specified grid spacing.

3. Apply smoothing by a Gaussian function with the system PSF to all B-spline functions $X_{vr}$, where subscripts identify region ($\mathrm{r}$) and voxel ($\mathrm{v}$), to create a smoothed set ${\tilde{X}}_{vr}$. The PSF is determined for PET systems using point sources positioned at various locations within the PET field of view (Gsell et al., 2020, Delso et al., 2011). In this work, we used a spatially invariant PSF, but the method can easily accommodate space-varying PSF due to the local nature of basis functions. Create the geometric transfer matrix (G) as projections of smoothed onto unsmoothed functions. The matrix G is a square matrix with N_r² elements, where N_r is the total number of regions needed to span the FOV.

   $$G_{r{r}^{\prime }}=\sum _v{X}_{rv}^T{\tilde{X}}_{v{r}^{\prime }},$$ [1]

4. For each frame of a dynamic scan, convert voxel-based PET data $\left(Y_v\right)$ as a function of time (t) to a set of ROIs: $Y_r^{\mathit{meas}}(t)={\sum }_v{p}_{rv}{Y}_v(t)$, with ${\mathrm{p}}_{\mathrm{r}\mathrm{v}}$ representing a normalized probability across the region.

5. Compute the PVC-corrected image $\left(Y_r^{PVC}(t)\right)$ from the inverse of the GTM; alternatively, solve for the inverse coefficients by lower-upper decomposition with back-substitution. Note that this step corrects for crosstalk between regions due to smoothing, as in the typical GTM method, while correcting for overlapping regions in the absence of smoothing. When smoothing is omitted in step 3, multiplication by the GTM inverse produces the coefficients for a fit to the image

   $$Y^{PVC}(t)=\sum _r{Y}_r^{PVC}(t)=\sum _{r,}{G}_{r{r}^{\prime }}^{-1}{Y}_{r^{\prime }}^{\mathit{meas}}(t)$$

6. Repeat step 5 with maxima shifted to all possible grid positions $\overrightarrow{g}$, then average the resulting set of image volumes to produce a final volume with reduced dependence upon basis function locations.

In step 1 above, the selection of B-spline grid spacing is not arbitrary because basis functions that are packed more closely than the PSF lead to an ill-conditioned GTM matrix. In other words, if smoothing of the B-spline basis functions in step 3 blurs boundaries between adjacent functions so that they are no longer clearly separable, the inversion process in step 5 will become unstable or fail. To investigate the relationship between PSF and grid spacing, we computed the matrix condition number for three different PSF values corresponding to three real PET systems, including two systems used in this report. Figure 1b shows that the matrix is well-conditioned if the basis spacing exceeds the PSF by about 25% or more. This means that the FWHM of the B-spline functions should be roughly twice as large as the PSF or larger.

The square GTM matrices are large, with dimensions along each side equal to the number total of basis functions (N_r) across the FOV. For applications discussed in this paper, matrix dimensions range between 800 (mouse brain with 1 mm PSF) and 26656 (human brain with 4.5 mm PSF). Because the matrix dimensions scale with FOV, we choose to apply the bsGTM method in a normalized template space rather than native space. To reduce computation time, we employed parallel threads on a 36-core Linux workstation with 1 GHz processors and 96 Gb of random-access memory, and we saved the decomposed GTM matrix to avoid repeated computation for each subject. For the largest matrices spanning a human FOV, lower-upper decomposition took about 48 min. To perform PVC on typical four-dimensional murine datasets (800 basis functions, 56 frames) required only about 3 minutes using bsGTM, whereas PVC using large human matrices required about 3 min per frame or about 80 min per four-dimensional dataset using multi-threaded processing.

As a comparison method for bsGTM, we employed RVD, which was reported to be the best iterative deconvolution method (Tohka and Reilhac, 2008) and is the PVC method incorporated into the online software by our small-bore PET vendor. For convergence criteria, we used the tolerance (1%), maximum number of iterations (30), and the value of “alpha” (1) within the range suggested by prior reports (Thomas et al., 2016; Thomas et al., 2011).

Monte Carlo Simulation of a Derenzo-type phantom

We employed GATE (Geant4 Application for Tomographic Emission, version 9.1) software (Sarrut et al., 2021) to perform a Monte Carlo simulation of the physics of positron emission and photon detection to generate data for the reconstruction of images containing realistic PSF and noise. We used a validated Siemens Biograph mMR scanner based on (Aklan et al., 2015). In the simulation, we included the effects of the 3 Tesla static magnetic field.

We simulated the geometry of a digital Derenzo-like phantom (Derenzo et al., 1997). The cylindrical phantom was 220 mm in diameter, and the rod sources had radii of 4.0-, 4.5-, 5.0-, 5.5-, and 6.0 mm in typical Derenzo-like arrangements (Fig.2). The contrast ratio between the rods and the background was 2.87. Simulated cylindrical sources were not separated from the background activity with a cold cylinder that, in a real-world experiment, would be the cylinder’s wall. We used a [¹¹C] positron emitter source and the emstandard_opt4 physics list.

Fig. 2.

[¹¹C] radioactive decay within a digital Derenzo phantom was simulated using GATE Monte Carlo software that incorporated a validated model of the Siemens Biograph mMR scanner (Aklan et al., 2015), and data were reconstructed to produce phantom images with a realistic scanner PSF of 4.5 mm. a) Images without PVC (original) are compared to different voxel-based PVC methods at high (top) and low (bottom) counts. b) Contrast Recovery Coefficient (%) as a function of rod diameter without PVC and with different PVC methods. c) Percent change in SNR averaged across all rods using RVD or different instances of bsGTM.

The simulation durations of 5 and 30 minutes were employed to produce images with low and high signal-to-noise ratios, respectively. The simulated activity restricted the random events to less than one percent. We reconstructed using the ordered subset expectation maximization (OSEM) algorithm, as implemented in the e7tools, with 21 subsets, 4 iterations, and 4 mm Gaussian post-filtering. With a 45 mm off-center line source and filtered back-projection, we determined the PSF to be about 4.5 mm, consistent with real measured results (Delso et al., 2011).

The reconstructed images were resampled from 2.1 × 2.1 × 2 mm into isotropic 1.5 mm voxels and subjected to PVC using either RVD or bsGTM with varying b-spline widths. For each method, the signal across all true locations of rods (circles in the cross-sectional image of Fig. 2a) was computed as a function of rod diameter and normalized relative to the true signal.

The Contrast Recovery Coefficient (CRC) was calculated similarly to the NEMA protocol suggestion (Daube-Witherspoon et al., 2002):

$$CRC=\frac{\raisebox{1ex}{${\mu }_r$}\!\left/ \!\raisebox{-1ex}{${\mu }_b$}\right.-1}{2.87-1}$$

where the ${\mu }_r$ is the mean activity all rods of each size, and ${\mu }_b$ is the mean activity of 24 same-sized ROIs in the background. The background ROIs were radially placed in the gaps between the rod arrangements. We must note that placing the 5.5- and 6 mm background ROIs in this region was challenging, and some underestimation of the CRC in smoother bsGTM images can be expected. The SNR was calculated as the average rod signal divided by the standard deviation of the phantom background outside rods and then normalized by the result obtained without PVC.

Temporal Simulation

We performed simplified simulations using a synthetic phantom to assess the efficacy of PVC and the performance for identifying a small change in signal in the presence of noise. A three-dimensional letter combination with a font stroke width of 5 mm and additive, time-dependent signal, and Gaussian noise (Teymurazyan et al., 2013) was blurred by a Gaussian kernel with an isotropic 3 mm FWHM; the noise level was selected to approximate the relative level of noise in the human basal ganglia data of Fig. 5. Prior to the addition of noise and blurring, signal during the baseline period (0–45 min) was set to a uniform value of 100, and the level of signal increased by 10% uniformly within the font at 45 minutes to simulate a change in binding potential. As in real studies that are corrected for time-dependent decay of radioactive tracer, the signal-to-noise ratio decreased versus time in the simulation by amplifying noise proportional to $\sqrt{e^{t/\tau }}$ using a time-constant of 29.4 minutes, corresponding to the ¹¹C isotope.

Fig. 5.

a) Representative maps of [11C]raclopride BP_ND (a) and SNR (b) in human subjects on a PET system with 4.5 mm resolution: no smoothing or PVC (left), bsGTM with a grid spacings 1.3 times larger than the PSF (middle), or bsGTM with a grid spacing twice the PSF (right). c) Normalized BP_ND and SNR (mean ± SD, n=20) using bsGTM (red, relative grid spacings of 1.3, 1.8, 2.2, and 2.7 in left to right order), Gaussian smoothing (black, with widths of 0, 3, 6, or 9 mm), or nonlocal means with varying signal variance.

Ten simulated scans were generated and analyzed by a linear model, including a constant term and a step function to fit the signal increase. Scans were preprocessed by RVD or bsGTM with varying grid spacings for the basis function. Basis function separations for bsGTM were selected to be 4, 5, or 6 mm, and these choices are denoted by referencing the ratio of the grid spacing to the PSF; e.g., “bsGTM (2.0)” has a grid spacing equal to 6 mm, which is twice the PSF in this case. For each processing method, we analyzed the percentage increase in baseline and challenge signal for the ten simulated scans, and we quantified the contrast-to-noise ratio (CNR) for detection of the step change in signal (“challenge”) using the linear-model T statistic.

Biological Systems

Two biological models were used to evaluate the new method: 1) Human [¹¹C]raclopride PET data were used to test the relative merits of bsGTM for smoothing data to boost SNR but minimally affect measurements of non-displaceable binding potentials (BP_ND), and 2) mouse [¹¹C]raclopride data were used as a positive in vivo control to test whether bsGTM could recover functional changes in BP_ND in striatum that are corrupted due to spillover of activity from the Harderian glands. In each case, only the form of smoothing or partial volume correction changed across comparisons. Smoothing and/or partial volume correction were applied on each frame of the time series, and then tracer kinetic analysis determined binding potentials according to published methods as described below.

Human PET: Effects of smoothing on SNR and BP_ND

Traditional volumetric smoothing can increase SNR at the cost of mixing tissue types and degrading binding estimates, whereas bsGTM offers the potential to perform smoothing with less mixing of adjacent tissues. We evaluated several alternative smoothing methods across human striatum, a region with relatively high and uniform [¹¹C]raclopride binding compared to surrounding tissue.

All human PET studies were conducted at the Athinoula A. Martinos Center for Biomedical Imaging at Massachusetts General Hospital, Boston, MA. A cohort of human volunteers (n=9 males, n=11 females) received [¹¹C]raclopride by a bolus injection (7.8 ± 1.7 mCi) plus continuous infusion (5.8 ± 1.4 mCi) in a Siemens Biograph mMR PET/MR scanner. Four-dimensional time-activity volumes were reconstructed using the OSEM algorithm with 21 subsets and 3 iterations into 47 timeframes with progressively larger durations from 1 to 4 minutes. The vendor-supplied reconstruction algorithm incorporates PSF modeling to improve SNR and resolution, achieving a nominal PSF of 4.5 mm (Delso et al., 2011).

Data were analyzed to produce parametric maps of BP_ND (Mandeville et al., 2022) relative to the reference region (cerebellum gray matter), and voxel-wise time-averaged SNR, calculated as the signal mean divided by the temporal standard deviation of the data with respect to the fit. Within each subject, a striatal mask above a BP_ND value of unity was defined based on data without smoothing or PVC. Within the defined striatal mask, we evaluated SNR and degradation of BP_ND versus the amount of smoothing for three different methods.

1. Isotropic three-dimensional Gaussian smoothing is expected to increase SNR but mix tissue types, reducing striatal BP_ND.

2. Non-local means (NLM) operates in signal space rather than real space, and so smoothing occurs preferably along isocontours of PET signal. NLM will smooth data preferentially within striatum, producing less effect on regional BP_ND than isotropic smoothing in real space. We employed NLM based on the least-square fitting of neighborhood fields (Tristan-Vega et al., 2012), which was determined from the summed image and applied consistently for each frame.

3. bsGTM combines smoothing and PVC and thus may produce a good tradeoff between SNR and BP_ND without resorting to the highly anisotropic smoothing neighborhoods of NLM. The scanner PSF of 4.5 mm requires a minimal bsGTM grid spacing of about 6 mm; we employed spacings of 6 to 12 mm.

Murine PET: PVC for functional imaging of dopamine release

Mouse PET imaging can inform mechanisms of image contrast (Skinbjerg et al., 2010) and disease (de Paula Faria et al., 2014), but partial volume spillover due to the small size of brain, together with high uptake of most PET radioligands from the Harderian glands (HG), can produce large partial volume effects (Thanos et al., 2002). Spillover error for FDG studies in frontal cortex can be significant, leading some investigators to advocate removal of HG prior to imaging (Brammer et al., 2007; Kim et al., 2014).

Wild-type mice (n=8 mice, male, C57BL/6J, 27 ± 2 g weight) were ordered from Jackson Laboratories (Bar Harbor, ME) and scanned using a PET/MR scanner (Bruker Biospin Corp., Billerica MA) with a nominal PSF of about 1.0 mm (Gsell et al., 2020). Two tail vein catheters per mouse enabled infusion of [¹¹C]raclopride (261 ± 76 uCi) at the start of the scan and amphetamine (1 mg/kg IV) at 35–37 minutes later. All procedures were approved by and complied with the Institutional Animal Care and Use Committees (IACUC) regulations at Massachusetts General Hospital and the Association for Assessment and Accreditation of Laboratory Animal Care (accreditation number 000809).

PET images were reconstructed using an iterative vendor-supplied algorithm, the maximum-likelihood expectation-maximization method, with 12 iterations, a final voxel size of 0.5 mm in each dimension, and 56 time frames with increasing durations from 1 to 2 min. Data were analyzed using a pseudo-linear variant of the simplified reference tissue model (Alpert et al., 2003; Ichise et al., 2003), including a constant challenge term. Analyses produced maps of BP_ND and the amphetamine-induced change DBP_ND using 1) no PVC, 2) PVC by RVD, or 3) PVC by bsGTM. Cross-subject data were analyzed by a mixed-effects linear model based upon a summary statistic (Worsley et al., 2002). Voxel-wise averages for BP_ND, ΔBP_ND, and CNR (computed as a T statistic) within caudate-putamen (CPu) were computed for different processing methods.

Results

Monte Carlo simulations, Derenzo-like phantom

Monte Carlo simulations of a digital Derenzo-like phantom and subsequent image reconstruction produced three-dimensional volumes with high or low counts depending upon simulation duration (Fig. 2a, “original” images). Applying PVC using RVD increased the average signal across the real location of the rods (Fig. 2b) but decreased SNR, with a larger relative drop in SNR for the low-count simulation (Fig. 2c). bsGTM using the smallest grid spacing increased the average rod similarly to RVD and proved more noise resistant, with a small SNR reduction for the smallest b-spline grid spacing and increases at the two larger spacings (7.5 mm and 9 mm, corresponding to 1.7 and 2 times the width of PSF) (Fig. 2c).

Temporal Simulation

Figures 3 and 4 illustrate PVC results on a digital phantom with time-varying signal and noise. When a 5 mm font width was subjected to a PSF of 3 mm, the uniform signal level during the baseline period was reduced (Fig. 2b, blue line) to 78% of the actual value when averaged across the full binary letter mask of the original unsmoothed image. RVD recovered signal recovery to 89% of the true value (Fig. 2b, green line). When using a grid spacing near the minimum value, bsGTM achieved a slightly higher recovery to 93% (Fig. 2b, red solid line). However, some ringing artifacts were evident due to using lower-resolution basis functions.

Fig. 3.

a) The true (unsmoothed) 3-letter synthetic phantom used an input to the simulations (inset), with a cyan line showing the projection in the right panel. The graph shows a representative time series from a single voxel (black) with fit (red). b) A profile along the cyan line in the letter phantom of Fig. 2 showing shapes with and without PVC using RVD or bsGTM.

Fig. 4.

a) Parametric maps of fitted challenge signal (top) and challenge CNR (bottom) with no processing, RVD, or bsGTM using grid spacings that are 1.3, 1.7, and 2.0 times larger than the PSF. Consistent scales were used for signal (0–15) and CNR (0–8) for all methods. b) The percentage change in challenge signal and CNR across the full three-dimensional letter phantom relative to the value with no PVC.

Fig. 4 depicts PVC recovery of baseline and challenge signal and effects on CNR, which determines the ability to identify the challenge-induced signal during the late noisy portion of the time-activity curve. The top row of Fig. 4a shows the challenge signal, and the bottom row depicts CNR; all maps were created using a second-stage random effects model to analyze the ten simulated runs. The most pronounced difference between RVD and bsGTM was seen in the CNR maps, where RVD exhibited poor performance relative to bsGTM or unprocessed data. Figure 4b quantifies results, demonstrating that RVD suffered a 37% loss in CNR relative to data without PVC and that bsGTM significantly increased CNR while offering similar benefits in signal recovery. Basis functions with larger widths offered higher voxel-wise CNR due to more smoothing but suffered losses in PVC. Using a grid spacing of 4 mm, which is 33% larger than the PSF, bsGTM offered 8% better signal recovery and 35% better CNR than RVD. Using a grid spacing of 5 mm (66% larger than the PSF), bsGTM increased CNR relative to RVD by a factor of 2.3 while sacrificing only a 5% loss in signal recovery relative to that method.

Human PET: SNR and BP_ND

To assess tradeoffs between smoothing and PVC, we analyzed data in 20 human subjects who received [¹¹C]raclopride and were scanned by a PET system with a PSF of about 4.5 mm. Fig. 5a shows representative maps of BP_ND and SNR in a single subject. Fig. 5b summarizes the relationships between these two quantities averaged across all subjects' high-BPND region within the striatum. All three methods progressively increased SNR by smoothing. Of the three evaluated methods, isotropic smoothing produced the largest decrease in BP_ND, as expected. NLM produced less loss of BP_ND by smoothing preferentially within the striatum, although smoothing neighborhoods were spatially anisotropic with nonuniform volumes.

The smallest smoothing-related loss of SNR occurred using bsGTM, which smooths data isotropically and then partially corrects for spillover between regions. Closely packed basis functions, with a grid spacing 1.3 times larger than the PSF, provided the best PVC but did not benefit SNR. More widely spaced basis functions increased SNR two-fold before leading to a drop in BP_ND relative to unsmoothed data. Note that RVD increased BP_ND without loss of SNR in this case when averaged across whole striatum; however, RVD offers no way to increase SNR.

Murine PET: PVC for functional imaging of dopamine release

Fig 6 illustrates the application of bsGTM to a murine dataset (n=17) acquired using bolus infusion of [¹¹C]raclopride followed by a within-scan amphetamine challenge to evaluate the effects of dopamine release. All parametric maps correspond to a random-effects linear model analysis across subjects. During PET scans, injection of amphetamine caused high uptake in D2-rich CPu and in HG near eyes (Fig. 6a). Amphetamine, administered midway through each scan, produced a reduction in BP_ND in CPu, consistent with increased D2 receptor occupancy due to dopamine release. However, BP_ND in HG exhibited a large paradoxical increase, indicated by blue-green colors in Fig. 6b. Note that the absolute change in binding potential, DBP_ND in Fig. 6b, has a positive sign when BP_ND decreases according to the standard PET convention (Innis et al., 2007).

Fig 6.

Comparisons of parametric maps created using a within-scan amphetamine challenge in mice (n=8) when [¹¹C]raclopride time-activity curves were processed without PVC, with PVC using RVD, or with PVC using bsGTM. Use of RVD or bsGTM increased both baseline BP_ND (a) and amphetamine-induced ΔBP_ND (b) relative to no PVC, but RVD significantly reduced the CNR for detecting ΔBP_ND, whereas bsGTM significantly increased CNR (c).

Both RVD and bsGTM enhanced BP_ND within CPu (Fig. 6a). Relative to results obtained without PVC, RVD and bsGTM increased average voxel-wise values of BP_ND within CPu by 29 ± 5 and 28 ± 2 percent, respectively; these results were not significantly different (p>0.05, pair T test across subjects). Similarly, RVD and bsGTM increased amphetamine-induced DBP_ND by relative amounts (74 ± 19 %, 78 ± 17%) that were not significantly different. However, statistical scores differed greatly (p<10⁻⁴). Relative to CNR values obtained without PVC, RVD decreased CNR (−22 ± 4 %), whereas bsGTM more than doubled CNR (137 ± 33%).

Discussion

PVC is a necessary correction for some PET studies. examples include degeneration (Thomas et al., 2011) or aging (Greve et al., 2016; Smith et al., 2019), where limited PET resolution and the progressive nature of the underlying biological processes conspire to cloud the interpretation of observed group differences due to uncertainties in disentangling the mixing of tissue types. Additionally, opposite changes in binding potentials can occur in adjacent tissue regions due to drug or electrical stimulation; our murine data presents an example of the former case. Opposing functional changes in PET signals can mask adjacent responses, but PVC can be applied on a frame-by-frame basis to help disentangle competing signals.

While many PET studies benefit from PVC in quantifying radiotracer concentrations, the suitability of the applied method depends upon the type of PET application. Some studies have well-defined anatomical targets, so PVC can usefully employ ROI methods, like the traditional GTM approach based upon anatomical segmentation, to disentangle regional crosstalk with better recovery than voxel-wise methods (Erlandsson et al., 2012). In principle, anatomically based PVC can fully recover PVC errors (Erlandsson et al., 2012), but of course, this interpretation depends upon the accuracy of prior knowledge. Anatomical GTM assumes uniform concentrations of radioligand within all ROIs at all times and also requires avoidance of experimental errors, such as misregistration between anatomy and PET data due to motion, distortion, or other causes. The murine data presented here represent a good application for traditional ROI-based GTM because the functional target of interest (caudate-putamen) is known, and the MR images facilitate template-based segmentation (Mandeville et al., 2023). In this study, we used the murine model as a suitable in vivo control case because a drug challenge is expected to affect receptor occupancy irrespective of region similarly, so we know that the largest changes in binding potential should occur in the highest binding regions.

Although standard anatomically based methods like GTM or symmetric GTM (Sattarivand et al., 2012) could be applied to the murine data in this study given the well-established target region of striatum, for many other applications, prior information is lacking. The goal of exploratory functional PET studies is to reveal the magnitude and location of neurotransmitter release rather than constraining solutions a priori. This paradigm includes human task designs (Egerton et al., 2009; Hansen et al., 2020) or deep brain stimulation (Mandeville et al., 2022). Constraining exploratory functional studies based on prior assumptions is a fraught task. Moreover, some important regions cannot be segmented using MRI contrast, while using PET to segment regions can be difficult due to SNR and resolution limitations.

While voxel-based methods for PVC are unbiased by prior assumptions, these methods are challenged by SNR limitations. Low SNR is particularly problematic in human subjects due to regulatory limitations on injected radioligand doses, and functional responses using a within-scan challenge paradigm occur during the latter stages of a time-activity sequence when voxel-wise SNR generally is in the single digits. These considerations motivate PVC methods that robustly control noise and enable smoothing with minimal tissue mixing. Among anatomically unconstrained methods, regularized deconvolution was the only such method reported in the most recent comprehensive review of PVC methodologies (Erlandsson et al., 2012), with RVD offering the best performance (Tohka and Reilhac, 2008). RVD is easy to apply, fast, and provided by the small-bore PET scanner vendor used in our studies. The general problems with deconvolution-based methods are well-known: correction is imperfect, and noise amplification is destructive without regularization (Karaoglanis K. et al., 2015). However, to our knowledge, no functional PET studies have explicitly reported problematic noise amplification with regularized deconvolution or evaluated alternative voxel-based comparison methods. Our simulation results show that degradation of SNR by RVD is particularly pronounced in the regime of low SNR (Fig. 2) and the latter stages of a dynamic scan (Fig. 4). Our results caution against the use of RVD for functional PET studies, such as detection of neurotransmitter release because application of this method degrades the CNR for detecting changes in binding potential.

Unlike corrections applied to static summary images for visualization, PVC of dynamic functional studies prior to quantitative parameter estimation must avoid bias due to temporally variant methods. From this perspective, GTM is attractive because it yields a linear and temporally invariant correction that satisfies a fundamental tenet of functional image processing, which is that all time points should be treated identically. In contrast, iterative methods, including RVD or MR-guided PET reconstruction, depend on signal and noise within each PET time frame and risk biasing dynamic data. A potential approach to this problem for iterative methods is to extend the correction through the time domain (Novosad and Reader, 2016; Reilhac et al., 2015), which forces a model-specific temporal regularization on the data. Similarly, deep learning neural networks risk biasing temporal data due to the intrinsically nonlinear relationship between input images and neural outputs; to date, neural networks have not been applied for partial volume correction of dynamic PET data.

The new method reported here, which combines the commonly used methods of B-spline interpolation and GTM-based PVC correction, enables signal recovery comparable to RVD but shows better resilience to noise. In both simulations and in vivo murine data, bsGTM increased CNR about 2-fold relative to RVD. For similar statistical scores, four time as many subjects would be required if data were processed by RVD rather than bsGTM. The essential difference between regularized deconvolution and bsGTM resides in noise treatment. The new bsGTM method can reduce noise significantly relative to RVD or even no PVC correction, although robust noise reduction comes at the expense of PVC.

Finally, we should note that three-dimensional bsGTM is generally appropriate for lissencephaly species or subcortical structures. When applied to brain data in gyrencephalic for interpretation cortical data, the method should be reduced to two dimensions and applied to the cortical sheet in a surface-based analysis to avoid smoothing across nonadjacent cortical areas (Greve et al., 2014).

Conclusions

A new method for partial volume correction, bsGTM, enables a flexible tradeoff of smoothing and PVC for processing noisy PET data. bsGTM exhibited favorable characteristics relative to iterative deconvolution for correction and smoothing of noisy PET data, focusing on applications using within-scan challenges to measure changes in binding potentials. In simulation and in vivo, bsGTM exhibited PVC recovery comparable to iterative deconvolution for low noise regimes, such as the determination of basal BP_ND. However, bsGTM outperformed iterative deconvolution for noisy data, including extraction of a functional challenge late in a time series or regions with inherently low SNR. This new method offers an anatomically unbiased post-reconstruction method for smoothing and mitigating partial volume errors for modalities including PET and single photon emission tomography (SPECT) and regions including the brain and other organs.