Voxel-based partial volume correction of PET images via subtle MRI guided non-local means regularization

Abstract

Purpose:

Positron emission tomography (PET) images tend to be significantly degraded by the partial volume effect (PVE) resulting from the limited spatial resolution of the reconstructed images. Our purpose is to propose a partial volume correction (PVC) method to tackle this issue.

Methods:

In the present work, we explore a voxel-based PVC method under the least squares framework (LS) employing anatomical non-local means (NLMA) regularization. The well-known non-local means (NLM) filter utilizes the high degree of information redundancy that typically exists in images, and is typically used to directly reduce image noise by replacing each voxel intensity with a weighted average of its non-local neighbors. Here we explore NLM as a regularization term within iterative-deconvolution model to perform PVC. Further, an anatomical-guided version of NLM was proposed that incorporates MRI information into NLM to improve resolution and suppress image noise. The proposed approach makes subtle usage of the accompanying MRI information to define a more appropriate search space within the prior model. To optimize the regularized LS objective function, we used the Gauss-Seidel (GS) algorithm with the one-step-late (OSL) technique.

Results:

After the import of NLMA, the visual and quality results are all improved. With a visual check, we notice that NLMA reduce the noise compared to other PVC methods. This is also validated in bias-noise curve compared to non-MRI-guided PVC framework. We can see NLMA gives better bias-noise trade-off compared to other PVC methods.

Conclusions:

Our efforts were evaluated in the base of amyloid brain PET imaging using the BrainWeb phantom and in vivo human data. We also compared our method with other PVC methods. Overall, we demonstrated the value of introducing subtle MRI-guidance in the regularization process, the proposed NLMA method resulting in promising visual as well as quantitative performance improvements.

1. Introduction

Positron emission tomography (PET) imaging is a highly sensitive molecular imaging tool that is able to quantify functional processes in vivo. However, due to a number of resolution degrading factors, the so-called partial volume effect (PVE) occurs in PET images [1,2], resulting in underestimation of radiotracer uptake in oncological lesions or plaque aggregation measures. Moreover, statistical noise in the data further impacts the quality of PET images, and can be amplified in partial volume correction (PVC) methods [3,4].

A range of PVC algorithms have been explored to improve the spatial resolution of PET images. One general category involves reconstruction-based methods, and is referred to as point-spread-function (PSF) modeling (or resolution modeling). This has been used to improve the quality in PET images by modeling resolution-degrading phenomena within the system matrix of the reconstruction algorithms, including object-domain (e.g. positron range) and detection-domain (e.g. inter-crystal blurring) resolution degrading factors [5]. PSF modeling has attracted considerable interest in PET over the past decade, and has been adopted by major PET vendors in their state-of-the-art PET scanners. At the same time, PSF modeling requires access to the original listmode or sinogram data, which is not easily achievable, especially when exploring application of PVC methods to longitudinal data (including possibly from multiple centers and scanners) obtained over many years [6].

The other category of PVC algorithms involves post-reconstruction processing, as applied to reconstructed images (whether or not they have been generated with PSF modeling). This category itself consists of region-based vs. voxel-based methods. The most commonly used region-based method is the geometric transfer matrix (GTM) method, which performs PVC correction on mean regional uptake values [7]. The GTM approach assumes that the activity within a region is uniform and can be accurately described by its mean value. Du et al. [8] introduced a more refined so-called perturbation-based GTM method for more accurate PVC estimates. Sattarivand et al. [3] also investigated a symmetric GTM (sGTM) method (equivalent to Labbe’s method [4] but avoiding handling of excessively large matrices), showing similar accuracy as GTM but improved precision. In these methods, volumes of interest (VOIs) can be propagated from the Hammers atlas [9], or other sources, and their performance depends on precise segmentation results.

Voxel-based post-reconstruction methods include the Muller-Gartner (MG) method [10], region-based voxel-wise correction (RBV) which is an extension of the GTM method to the voxel domain [11], and the iterative Yang (IY) technique [12]. At the same time, these methods all make strong assumptions about uniformity of PET uptake within MRI regions. This is not the case for the Van Cittert (VC) and Richardson–Lucy (RL) iterative deconvolution techniques that do not use MRI information at all [13]. For instance, VC method or its more statistically sound version ‘reblurred’ VC (RVC) only requires system point spread function (PSF) information and has the advantage of easy implementation. However, it does not use any regularization, and simple application of RVC method will amplify noise levels [13]. As such, regularization may have to be considered.

The well-known non-local means (NLM) method, introduced by Buades et al. in 2005 [14], employs the high degree of redundant information that typically contains in images and reduces image noise by replacing each pixel intensity with a weighted average of its nonlocal neighbors. The use of NLM filter for medical imaging has shown promising results, yet the search window is normally defined as a fixed sub-dimension of the image to constrain the search space [15–17]. Ideally, the entire image should be explored as searching window to enhance the performance of the NLM filter, which comes at a high computation cost. Motivated by the success of the NLM filter, a number of NLM based regularization models have been used for image restoration and reconstruction; e.g. in computed tomography (CT) [18–21]. NLM can also be used as a denoising method [17,22,23]. A number of methods combine deblurring and denoising steps by solving an optimization problem (inverse problem) [24,25]. Commonly used least square (LS) objective function only accounts for white Gaussian noise, which oversimplifies the noise property in reconstructed PET image [26,27]. Minimizing the LS cost function usually leads to higher noise results with increasing iterations, similar to the maximum likelihood (ML) approach. A penalty is desired for a penalized ML or MAP solution. In the present work, we specifically focus on regularization utilizing the nonlocal means (NLM) method [14], in order to enable PVC while suppressing image noise amplification. This NLM framework employs a high degree of information redundancy (similar structures) that typically exists in images and reduces image noise by replacing each pixel intensity with a weighted average of its nonlocal neighbors. Chan et al. [28] proposed a NLM filter method with anatomical prior, their results suggest promise in clinical practice for whole-body PET/CT imaging.

A philosophy we pursue in our work is that the ‘subtle’ usage of MRI information in PET reconstruction, as opposed to intense or no usage of MRI information. As additional motivation, clinical PET/MRI systems have been designed and produced in the past decade. Several correction methods based on anatomic images have been developed in the past, although these are not being used routinely in clinical practice [29,30]. In the present work, we use the anatomy (MRI information) to restrict the search window region without further intervention from MRI. Our method introduces a voxel-based PVC method under the least squares framework (LS) employing anatomical non-local means (NLMA) for PET image processing, and considers the Multivariate Gaussian distribution through the data fidelity term and the choice of the regularization parameter. We evaluated our proposed method with NLM as well as RVC and RBV methods. The paper is organized as follows. Sec. 2 introduces the mathematical model and the optimization algorithm. The experimental design as applied to recovery of amyloid PET images is described in Sec. 3, along with the results, which are discussed in Sec. 4. Discussion and conclusion are presented in Sec. 5. Parts of this work were presented at the Society of Nuclear Medicine and Molecular imaging (SNMMI) 2018 Annual Meeting [31] and the 2018 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS-MIC) [32].

2. Methods

2.1. Mathematical model

2.1.1. LS image processing framework

Our aim is to effectively tackle PVE in PET images as deteriorated by a known point spread function (PSF). A simple model for the measured PET image is [33]:

$$g=Hf+\eta$$ (1)

where g is the measured image, f stands for the true image, Hf denotes the result of the convolution operation with the system PSF, and η stands for the noise. The task is to estimate f given g and H. This problem is, in general, ill-posed. In order to effectively reduce noise in the recovered images, a common method is to add a regularization term. Image after PVC can be obtained by minimizing the following regularize function.

$$\widehat{f}=\text{arg}\underset{f}{\text{min}}\parallel g-Hf{\parallel }_2^2+\beta R(f)$$ (2)

where R(f) is the regularization term which uses the MRI information, β is a regularization parameter that balances the strength of regularization and deconvolution, $\parallel g-Hf{\parallel }_2^2$ is the fidelity term which controls the degree of agreement between the estimated and the measured image, ‖X‖² = X^TX, X^T denotes transpose of X. In this work, we set regularization term R(f) as NLM/NLMA functions, which we will describe in detail in the following section.

2.1.2. NLM-based regularization term

The NLM algorithm was originally proposed as a non-iterative edge-preserving filter to denoise natural images corrupted by additive white Gaussian noise [26]. Essentially, it is one of the neighbourhood filters, which denoises each pixel with a weighted average of its neighbouring pixels according to similarity (though the neighbourhoods tend to be large; i.e. non-local). Different from other neighbourhood filters, NLM filter calculates the similarity based on patches instead of pixels.

Motivated by the success of the NLM filter, a number of NLM based regularization models have been used for image restoration and reconstruction in computed tomography (CT) [18–21]. It can also be used as a denoising method [17,22,23]. (With specific testing for amyloid PET images), where image recovery (based on knowledge of point-spread function model) is coupled with an NLM-based regularization term to control noise amplification. In order to perform PVC in PET imaging, we bring NLM within our PVC efforts. The NLM prior (regularization) term is given as:

$$Rf=\sum _i\varnothing \left(f_i-\sum _{keS{W}_i}{w}_{ik}(f)\cdot f_k\right)$$ (3)

where ∅ is a positive potential function and here we choose ∅(f) = f²/2, f represents the PET image, and SW_i denotes a search window with i^th voxel as the centre. The weighting coefficient w_ik(f) is determined by how similar local patches around the two voxels (i and k) being compared in the PET image as:

$$w_{ik}(f)=\frac{\text{exp}\left(-{\Vert P\left(f_i\right)-P\left(f_k\right)\Vert }_{2,a}^2/{\sigma }^2\right)}{{\sum }_{k\in S{W}_i}\text{exp}\left(-{\Vert P\left(f_i\right)-P\left(f_k\right)\Vert }_{2,a}^2/{\sigma }^2\right)}$$ (4)

where P(f_m) is the image patch centered at the m^th voxel, and $\parallel \cdot {\parallel }_{2,a}^2$ computes the squared Euclidean distance weighted by a Gaussian kernel with a > 0 denoting the standard deviation of the Gaussian kernel and the parameter σ acts as a degree of filtering to control the decay of the exponential function. In conventional NLM applications, NLM is commonly used as denoising filter. In the present work, this term is presented as regularization, coupled with the resolution recovery model incorporating the PSF of the system, transferring the problem from noise reduction, to PVC with noise suppression.

2.1.3. Incorporating MRI anatomical information into NLM regularization term (NLMA)

In the past, MR images have been taken as priors due to the excellent soft-tissue contrast, and anatomy-guided image processing methods have been used to improve PET image quality. However, these methods often suffer from the potential risk of over-usage of MRI information (e. g., assuming PET images are relatively uniform within anatomical regions, which is a very strong assumption [29,34–38]). By contrast, in our work we seek subtle usage of MRI information without uniformity assumption on PET activity. Specifically, in order to incorporate MRI information into the NLM regularization term, we propose to utilize anatomical region masks to merely restrict the search window, thus restricting usage of non-local information to voxels that belong to the same tissue. As shown in Fig. 1, SW is the search window, f_i, f_j, f_k are from the PET image, while A_i, A_j, A_kare from the MRI segmentation regions (eg. GM, WM, Caudate, Putamen, etc.). When pixels i and j belong to the same tissueT_k, that is A_i ∈ T_kandA_j ∈ T_k, we use the tissue area to replace the search window. This method is called ‘NLMA’, where the additional ‘A’ denotes anatomical guidance. This also improves the searching efficiency. The regularization function of NLMA is:

$$Rf=\sum _i\varnothing \left(f_i-\sum _{k\epsilon S{W}_i}{w}_{ik}(f)\cdot f_k\cdot {\chi }_{T_i}\left(A_k\right)\right)$$ (5)

where ${\chi }_{T_i}$ is the indicator function of tissue T_i and is defined as:

$${\chi }_{T_i}\left(A_k\right)=\{\begin{array}{l}1,if{A}_k\in T_i\hfill \\ 0,if{A}_k\notin T_i\hfill \end{array}$$ (6)

Fig. 1.

Schematic representation of the NLMA method, (A) is the PET image, (B) is the segmented MRI image.

T_i denotes the segmented region which the i^th voxel belongs to, and SW_i denotes a search window with i^th voxel being the centre

2.2. Optimization

Direct optimization of the resulting objective function can be complicated. In practice, an empirical one-step-late implementation is usually employed in iterative approaches to obtain solutions [20,34,39]. In this work, the weighting coefficients are computed on the current image estimate, and are considered constants when the image is updated. Although this strategy inevitably increases the computational load, it may produce a more accurate solution. In our study, we applied the Gauss-Seidel (GS) algorithm combined with the one-step-late scheme to minimize the objective function (2). Following iterative OSEM or MLEM reconstruction, the variance of the image has been shown approximately equal to the activity distribution value of the PET measured image [40–42].

The GS method is a special case of (projected) successive over relaxation (+SOR), which is derived via Taylor expansion and, similar to Newton methods, updates each image parameter individually by minimizing the objective function (2) over that parameter while holding other parameters fixed [18–21]. According to the OSL strategy, the NLMA weighting coefficients (w_ik) are always computed on a current image estimate and then assumed to be constant when updating the image. The updating expression in GS without regularization is given by:

$$f_i^{n+1}=\frac{h_{\mathit{i}}^{\mathit{T}}(g-Hf)}{h_{\mathit{i}}^{\mathit{T}}{h}_i}+f_i^n$$ (7)

where h_i is i^th column of H. The updating expression for GS with NLMA regularization is:

$$f_i^{n+1}=\frac{h_i^T(g-Hf)+\left(h_i^{T}H\right)f_i^n+\beta {\sum }_{k\in S{W}_i}{w}_{ik}{f}_k^n}{h_i^T{h}_i+\beta {\sum }_{k\in S{W}_i}{w}_{ik}}$$ (8)

where β is hyper parameter to control the regularization strength in the method, we set it according to the practical experiment (ranging from 0.04 to 2) and ${\sum }_{k\in S{W}_i}{w}_{ik}{f}_k^n$ is calculated from the NLMA method.

3. Experimental design

3.1. Phantom simulations

We performed simulation studies using the BrainWeb phantom [43], which provides T1-weighted MR image as shown in Fig. 2(A). Fig. 2(B) shows the corresponding segmented tissue map containing 10 different tissue regions. The anatomical model used to generate simulated brain MRI data consist of a set of 3-dimensional “fuzzy” tissue membership volumes. In addition, a discrete anatomical model is provided which consists of a class label (integer) at each voxel, representing the tissue which contributes the most to that voxel [20,43]. We can use this phantom to create PET activity and attenuation images. The simulated PET model consists of a set of non-uniform activity tissues including the volumes (GM, WM, CSF and others). The tissue volumes are defined at a 1 mm isotropic voxel grid in Talairach space [44], with dimensions of 181 × 181 × 217. Realistic PET activities were assigned into different regions based on ¹¹C-PIB patient data: 12500 Bq ml⁻¹ in GM, 3125 Bq ml⁻¹ in WM, 0 Bq ml⁻¹ in air, CSF and bone, 1000 Bq ml⁻¹ in all other tissues including skin, muscle, connective tissue and fat [45]. Furthermore, non-uniform activity was assigned in GM region. To do this, we convolved grey matter with a Gaussian function with a full-width at half-maximum (FWHM) of 10 mm to create non-uniform distribution in GM. The resulting PET phantom is shown in Fig. 2(C). We realize this method may still simplify realistic scenarios [46]. However, in this manuscript, we use such Gaussian convolution method as a simple way to generate non-uniform activity distribution.

Fig. 2.

(A) MRI image, (B) Segmented MRI image (10 tissues map), (C) Ground truth PET image, (D) Attenuation map, (E) Reconstructed PET image, and (F) Segmented MRI image (18 tissues map).

Different degradation factors, including attenuation, detector blurring, and normalization effect, were considered in the projection domain in the context of our analytic simulations [47–49]. To build the attenuation map, the tissues were assigned the same attenuation coefficients as in PET-SORTEO [13], which is 0 cm⁻¹ for air, 0.146 cm⁻¹ for bone and 0.096 cm⁻¹ for other tissues to build the attenuation map, as shown in Fig. 2(D) [50]. The normalization coefficients were obtained using a component-based method [51]. The total number of counts was adjusted to generate a medium noise (10⁹ number of coincidence events) in the projection domain. 20 noise realizations were computed in the simulation to evaluate quantitative performance. Image reconstruction was performed using the maximum likelihood expectation maximum (MLEM) algorithm with 240 iterations. Fig. 2(E) shows the reconstructed PET image.

To simulate more delicate segmentation guidance, we need to get another different segmentation guidance map. We used Freesurfer software to segment the MRI image into 18 tissues [52]. The generated tissue map contains many small regions such as caudate, putamen, frontal lobe, temporal lobe, hippocampus, and amygdala, as shown in Fig. 2(F). With this phantom, we are able to evaluate performance of different PVC methods in more regions.

We evaluated our proposed method (NLM and NLMA) against RVC and RBV methods. For implementation of NLM and NLMA, we set the overall search window size to 21 × 21 × 21 voxels and patch size to 7 × 7 × 7 voxels. Different regularization parameters β were chosen which we will specify later. The step size in RVC method was set as 0.5.

Among the four different methods, NLMA and RBV require additional anatomical guidance. In order to evaluate these two methods with imperfect MR guidance that may occur in practice, we simulated both registration mismatch and segmentation mismatch. Different extents of registration mismatches were simulated by shifting anatomical images. For registration mismatch, we generated misregistration MRI images from 1.0 mm to 4.0 mm offsets (along the × axis). To simulate segmentation mismatch, we also used Freesurfer to generate elaborate tissue map with noisy MRI image (the noise level is 9%) which is shown in Fig. 3(A). The original noise-free MRI image we used in the PET simulation phantom is shown in Fig. 3(B). In Fig. 3(C) and (D), we also showed the segmentation maps of the corresponding two MRI images. Red arrow points out an example where difference exists between the two segmentation maps.

Fig. 3.

(A) 9% noise MRI phantom, (B) noise-free MRI phantom, (C) MRI segmentation map of (A) from Freesurfer, (D) MRI segmentation map of (B) from Freesurfer.

The simulation and experiment works were implemented in Matlab 2017b on a computing system with a 3.30 GHz Intel Core i5-4590 processor. It takes about 15 min for one noise realization of NLMA method and about 10 min for one noise realization of NLM method.

3.2. Quantitative measurements

To quantitatively evaluate the images resulted from different algorithms, we computed noise, namely coefficient of variability (Cov), as well as bias in different regions. Cov measures image variability over different noise realizations for the same region, and is defined as:

$$\text{Cov}=\frac{\frac{1}{M}{\sum }_{i=1}^M\sqrt{\frac{1}{R-1}\sum _{r=1}^R{\left(v_i^r-{\overline{v}}_i\right)}^2}}{\overline{v}}\times 100\text{\%}$$ (9)

where M is the number of voxels in the same region, R is the number of noise realization, ${\overline{v}}_i=\frac{1}{R}{\sum }_{r=1}^R{v}_i^r$ is the mean over different noise realizations, $v_i^r$ is the value of voxel i in r^th noise realization, and $\overline{v}$ is the regional mean value of the ground truth activity image.

Biases in activity uptake quantitation is defined in terms of deviation from the ground truth, calculated as follows:

$$\text{Bias}=\frac{1}{R}\frac{{\sum }_{r=1}^R\left({\overline{v}}^r-\overline{v}\right)}{\overline{v}}\times 100\text{\%}$$ (10)

where ${\overline{v}}^r$ is the regional mean value of the r^th noise realization.

3.3. Clinical patient data

Our clinical data consists of one 85-year-old cognitively normal amyloid negative male and one 85-year-old cognitively normal amyloid positive female collected from Baltimore Longitudinal Study of Aging (BLSA). Two scans were performed for each participant with two different tracers. In the first scan, participants were injected with 370 MBq ¹⁵O-H₂O and 60 s PET scans were performed once the counts reached threshold levels. In the second scan, 555 MBq ¹¹C-PIB was injected PET scans were performed over 70 min. We use the average of motion-corrected time frames corresponding to 50–70 min post PIB injection in this study. These two scans were performed on a GE Advance scanner [45]. Filtered back projection (FBP) algorithm was used for reconstruction to generate 3D images of 128 × 128 × 35 voxels with voxel size of 2mm × 2mm × 4.25mm. The FWHM of PSF was set as 7.5mm × 7.5mm × 7.5mm for GE Advance scanner. Anatomical information was also provided by performing a magnetization-prepared rapid gradient echo (MPRAGE) scan for each participant on a 3 T Philips Achieva scanner (repetition time = 6.8 ms, echo time = 3.2 ms, flip angle=8^◦, image size=256 × 256 × 170, voxel size = 1 × 1 × 1.2mm³). The MPRAGE scans were inhomogeneity-corrected, skull-stripped, co-registered with PET images, and anatomically-labelled using the MUSE algorithm [53]. We assessed our NLMA method, together with RBV, RVC and NLM methods, using these four scans. For NLM and NLMA methods, search window was set as 9 × 9 × 9 and patch size was 5 × 5 × 5. Different regularization parameters were applied for these two methods.

4. Results

4.1. Results for simulation experiments

We evaluated the performance of four different methods, including NLM, RVC, RBV and NLMA. We first show the visual results of these four different methods when perfect MR guidance is provided. In this case, we set the β parameter for NLM as 1.5, and 0.5 for NLMA. For RVC, we set the number of iterations as 30. The visual results are shown in Fig. 4. We can see that NLMA has the lowest noise and depicts more structural details compared with other methods.

Fig. 4.

Results for different PVC methods. (A) Ground truth PET image and (B) Uncorrected PET images, followed by PET images that are PVC-corrected using (C) RVC, (D) RBV, (E) NLM, and (F) NLMA. All images use the same colour scale.

Mean percentage bias and coefficient of variability (Cov) are computed for quantitative comparisons. For NLM and NLMA, bias-noise curves were obtained by changing regularization strength β. Specifically, the scale of the β parameter ranging from 0.04to 2. Fig. 5 plots quantitative performance of these four methods with perfect MR guidance. Among different β values, the best bias-noise balance was obtained whenβ is close to 1.2. For RVC, the curve was plotted by ranging the iteration number from 1 to 30. Overall, we see that our proposed NLMA method outperformed other methods and largely improves quantitative performance, compared to non-MRI-guided NLM framework (e.g. by ~50% reduced bias when matching noise).

Fig. 5.

Bias vs. noise trade-off curves using different algorithms, for GM, WM, Caudate, Putamen, Frontal lobe, Temporal lobe, Hippocampus and Amygdala tissues. Each curve is generated by varying the hyper parameter β.

Fig. 6 (for GM, WM, Caudate and Putamen) shows quantitative performance of different PVC methods when registration mismatches occur. It is seen that with increasing mis-coregistrations, the quantitative performances degrade, as expected; Still, with small errors, NLMA method outperforms others, especial in the small areas. For putamen even to 4 mm errors, the performance of NLMA is better than NLM method.

Fig. 6.

Bias - noise trade-off curves for (1st row) 0 mm, (2nd row) 1 mm, (3rd row) 2 mm and (4th row) 4 mm MR-PET registration errors in GM (1st column), WM (2nd column), Caudate (3rd column) and Putamen (4th column) regions.

As summary of above, Fig. 4 and Fig. 5 depict visual and quantitative results of different PVC methods with perfect MR guidance (no errors in registration and segmentation). With perfect segmentation and no registration mismatch, the two segmentation-based methods, NLMA and RBV show better performance compared to NLM method. RVC method has less bias compared with NLM method. Fig. 6 shows bias-noise curves for RBV, NLM and NLMA by varying co-registration accuracies. Bias-noise trade-off performance assessment is prevalent and valuable in quantitative image analysis. The closer the curve is to the origin, the better its performance. These results demonstrate that NLMA is less influenced by registration error except in putamen. In this region, when the error reaches 4 mm, the performance of NLMA is worse than that of NLM method. We observed faster degradation for RBV compared to other methods. As registration mismatch increased, all MR-guided PVC methods degraded in performance. Within 4 mm registration error is match, NLMA shows stronger resilience to segmentation and registration mismatches.

Fig. 7 plots bias-noise trade-off curves in GM, WM, caudate, putamen, frontal lobe, temporal lobe, hippocampus and amygdala tissues for RBV and the proposed NLMA method, with perfect and mis-segmented MR guidance. We also put the NLM result and input image (measured image) in the plot as the baseline. It is seen that segmentation mismatch can degrade performance of both methods. At the same time, after PVC method, the quality of the images outperforms original non-PVC image in the mismatch segmentation MR guidance. In some tissues, the NLMA still performs better than NLM in both bias-noise. In some tissues, the NLMA still shows better bias.

Fig. 7.

Bias vs. noise trade-off curves of NLMA method, RBV using perfect MRI guidance (NLMA, RBV) and with mis-segmentation MRI guidance (NLMA-M, RBV-M), in the cases of original PET image (Input) and NLM showed results (NLM) as comparation. Results shown are for GM, WM, Caudate, Putamen, Frontal Lobe, Temporal Lobe, Hippocampus and Amygdala tissues.

4.2. Results for clinical dataset

Fig. 8 shows the anatomical clinical images of two MRI scans, which were used as anatomical reference in the anatomy-based methods in Fig. 9. The left image is the Subject0′s and the right is the Subject1′s. Standard uptake value ratio (SUVR) is a very important semi-quantitative indicator commonly in PET tumour diagnosis. It refers to the radioactivity of the imaging agent. In this paper, we obtained it by dividing the PET image by the mean value of the reference region. The reference region of ¹⁵O-H₂O is the whole brain, and the ¹¹C-PIB is the Cerebellar GM. In Fig. 9, we show transverse slices of SUVR for ¹⁵O-H₂O and ¹¹C-PIB clinical data for different PVC methods (the reference region for ¹⁵O is the whole brain and for ¹¹C is cerebellar GM). Fig. 10 shows the NLM and NLMA results with different regularization strength (β = 0.1,0.2,0.5,0.8,1.2) for ¹⁵O-H₂O clinical data and Fig. 11 shows the similar results for ¹¹C-PIB clinical data.

Fig. 8.

Anatomical clinical reference images for Subject0 (left) and Subject1 (right) MRI scan. (a) is the transverse, (b) is the coronal and (c) is the sagittal.

Fig. 9.

PVC results for clinical participants with Subject0 (left) and Subject1 (right) ¹⁵O-H₂O and ¹¹C-PIB PET scans.

Fig. 10.

Five different β parameters including 0.1, 0.2, 0.5, 0.8 and 1.2 from left to right for NLM and NLMA results of clinical participants with amyloid negative (left) and amyloid positive (right) ¹⁵O-H₂O PET scan.

Fig. 11.

Five different β parameters including 0.1, 0.2, 0.5, 0.8 and 1.2 from left to right for NLM and NLMA results of clinical participants with amyloid negative (left) and amyloid positive (right) ¹¹C-PIB PET scan.

We then plotted SUVR in different ROIs. Fig. 12 plots ROI SUVR from different PVC methods for amyloid negative (left) and amyloid positive (middle) participants. We choose β parameters as 0.1 and 0.8 for NLM and NLMA methods, respectively.

Fig. 12.

SUVR in different ROIs for different PVC methods for amyloid negative (left), amyloid positive (right) participants, for ¹¹C-PIB (1st row) and ¹⁵O-H₂O (2nd row).

5. Discussion

In this study, we proposed a PVC approach based on subtle MRI information inducing nonlocal regularization under the LS framework in order to recover PET activity loss due to PVE and suppress image noise. Specifically, the presented NLMA method utilizes the geometrical information from the MRI data to improve the measured PET data. The method was evaluated with the digital PET/MRI phantom and in vivo human data. The results presented in Sec. 4 demonstrated that the gains from the presented NLMA PVC algorithm are notable compared with those from the NLM and RBV PVCs in terms of quantitative measurements. From the RBV results, we can see that the noise is much higher than NLMA results. We did experiments using different tissues like 10, 12, 18 tissues, and found that as the number of tissues increasing the results of NLMA will be better.

We also compared different MR-based PVC methods in the context of varying MR information mismatches including mis-segmentation and misregistration. The results showed that our proposed PVC methods with subtle MR guidance, demonstrate stronger resilience to mismatches between PET and MR images compared to methods with stronger assumptions invoked in MR-guided PVC. As registration mismatch increased, all MR-guided PVC methods degraded in performance. We observed faster degradation for RBV relative to other methods. With 4 mm registration mismatch, NLMA shows stronger resilience to registration mismatches. We also experimented with larger (>4 mm) registration mismatches (not shown here), and differences between different methods with subtle MR guidance were no longer obvious, but they all outperformed RBV. We compared MRI-guided PVC methods with both perfect and mis-segmented tissue maps. As shown in Fig. 7, with imperfect segmentation, performances of NLMA and RBV degraded. While RBV with segmentation error was inferior compared to NLMA, which still outperformed other methods. We can see NLMA with mis-segmentation case has lower noise and lower bias level in these eight regions. Under the same noise level, NLMA has the lowest bias among these methods. This shows that the proposed methods are less affected by mis-segmentation compared with other methods. In all cases, NLMA with perfect segmentation always provided best performance, though we noticed good segmentation was still important to guarantee its performance.

For clinical data, we notice our NLMA method generates lower noise compared to RVC, RBV and NLM methods for these two tracers, as shown in Fig. 9. Similar as we observed in simulation experiments, RVC gives improved contrast but also higher noise. If we compare RBV and NLMA, they both keep structure detail of images but NLMA is able to further reduce image noise. Between these two tracers, we observed our method shows better structure detail for ¹¹C-PIB tracer. The beta value affects the results of NLM and NLMA. From the Fig. 10 and Fig. 11, we can see that with the increase of beta value (from 0.1 to 1.2), the results of NLM method are getting worse, but the results of NLMA are getting better. Best NLMA results are obtained near 0.5, which coincides with the conclusion of phantom experiment. In Fig. 9, we choose β for NLM method as 0.1 in ¹⁵O-H₂O and 0.2 in ¹¹C-PIB. We choose β as 0.5 for both ¹⁵O-H₂O and ¹¹C-PIB images of NLMA method. To ensure that both methods have the best visual effect. In Fig. 12, we use 0.1 and 0.8 beta parameters for NLM and NLMA to draw these two graphs. ROI SUVR (including Caudate, Putamen, Frontal GM, WM, Frontal lobe, and Temporal lobe) shown in Fig. 12 are all better than no-PVC method. For Amygdala, ROI SUVRs of NLMA method is quite close to those from the uncorrected image for ¹¹C-PIB, while in other ROIs, NLMA method is better than uncorrected image and most of other methods. For WM, Frontal lobe, and Temporal lobe SUVR values are quite close to the uncorrected image for ¹⁵O-H₂O. In other ROIs, NLMA method is better than uncorrected image. From Fig. 12, it can be observed that RBV method performs better in some ROIs, e.g. Caudate and Frontal GM, compared to other PVC methods.

A number of existing MRI-based PET reconstruction or PVC methods have employed more MRI information in the processing stage [29]. Such approaches risk significantly biasing PET images. As such, our framework attempts to make ‘subtle’ usage of MRI information, meaning that MRI information is only used to define the NLMA search window range. Our approach shows promise, as demonstrated in above results. At the same time, ongoing efforts include application to real clinical longitudinal studies to assess impact on assessment of disease. The process using anatomical images to guide functional image formation typically requires registration of the images from different image modalities. The recent advent of integrated PET/MRI systems apparently brings more opportunities for this task and other forms of synergistic analysis by obtaining anatomical and functional images simultaneously, allowing easier spatial alignment with minimal error.

Some existing image-based PVC methods make significant simplifying assumptions. The geometric transfer matrix (GTM) method, a gold-reference PVC method, only produces quantitative values at the region-of-interest level and does not produce PVC images. In our proposed work, we do assume activity distribution is non-uniform in PET. In the present work, by contrast, we explore a voxel-based PVC framework which performs subtle usage of MRI information, and moreover generates corrected images, while controlling for noise amplification. GTM and RBV make assumptions of uniformity of PET values within anatomical segments of MRI, whereas our proposed methods have the different assumption, aiming to quantify image details more accurately.

There are ways in which the performance of the presented NLMA algorithm could be improved. One of these ways, as we have discussed previous in the paper, is to optimize the algorithm parameters, namely the search-window, the patch-window, the control parameter and the hyper-parameter. In some other studies [54,55], the size of search-window and patch-window that are set in a reasonable range do not have noticeable effect on the resultant PET images quality, and the control parameter is set according to the scalar parameter that is determined by experiments. In addition, the determination of hyper-parameter is tuned manually to achieve a good trade-off between noise and resolution in all the cases. It should be noted that the question, what a “good trade-off” is, depends strongly on the specific tasks at hand, and the preference of the observer. Thus, optimization of parameters would be an interesting topic in our future research efforts.

6. Conclusion

In this work, we have demonstrated that subtle MR guidance under the LS framework can result in improved PET images. Our method specifically involves anatomically guided NLM (NLMA) regularization. Visually, NLMA is superior compared to other investigated methods, and quantitatively, it provides improved bias-noise trade-off performance. We also demonstrated that NLMA is not significantly affected by mis-segmentation and mis-registration. The proposed method depicts stronger resilience to mismatches between PET and MR images compared to other methods. In patient data, visual observations indicated that NLMA method had lower noise relative to other algorithms. From the quantitative results, we observed that PVC methods resulted in higher SUVR values compared to performing no PVC.