Does the beta regularization parameter of bayesian penalized likelihood reconstruction always affect the quantification accuracy and image quality of positron emission tomography computed tomography?

Abstract Purpose This study aims to provide a detailed investigation on the noise penalization factor in Bayesian penalized likelihood (BPL)‐based algorithm, with the utilization of partial volume effect correction (PVC), so as to offer the suitable beta value and optimum standardized uptake value (SUV) parameters in clinical practice for small pulmonary nodules. Methods A National Electrical Manufacturers Association (NEMA) image‐quality phantom was scanned and images were reconstructed using BPL with beta values ranged from 100 to 1000. The recovery coefficient (RC), contrast recovery (CR), and background variability (BV) were measured to assess the quantification accuracy and image quality. In the clinical assessment, lesions were categorized into sub‐centimeter (<10 mm, n = 7) group and medium size (10–30 mm, n = 16) group. Signal‐to‐noise ratio (SNR) and contrast‐to‐noise ratio (CNR) were measured to evaluate the image quality and lesion detectability. With PVC was performed, the impact of beta values on SUVs (SUVmax, SUVmean, SUVpeak) of small pulmonary nodules was evaluated. Subjective image analysis was performed by two experienced readers. Results With the increasing of beta values, RC, CR, and BV decreased gradually in the phantom work. In the clinical study, SNR and CNR of both groups increased with the beta values (P < 0.001), although the sub‐centimeter group showed increases after the beta value reached over 700. In addition, highly significant negative correlations were observed between SUVs and beta values for both lesion‐size groups before the PVC (P < 0.001 for all). After the PVC, SUVpeak measured from the sub‐centimeter group was no significantly different among different beta values (P = 0.830). Conclusion Our study suggests using SUVpeak as the quantification parameter with PVC performed to mitigate the effects of beta regularization. Beta values between 300 and 400 were preferred for pulmonary nodules smaller than 30 mm.


| INTRODUCTION
Positron emission tomography (PET) is used extensively in clinical oncology for tumor detection, staging and therapy response assessment. It also facilitates theranostic PET-guided therapy protocols. 1 A range of image-derived numerical metrics, such as the standardized uptake value (SUV) or the kinetic attributes are used in PET for quantitative analysis, which is better than visual assessment for distinguishing effective early treatment response from ineffective one in oncotherapy. 2 Recently, progress and innovative developments of new probes targeting different biological features (cell proliferation, amino acid transport/metabolism, integrin receptor expression in angiogenesis and metastasis), 3 as well as the use of artificial intelligence-based techniques (machine learning and deep learning of radiomics from PET imaging), 4 are revolutionizing clinical practice in oncology. As a result, the quantitative accuracy is extremely important when quantitative PET is still challenged by several degrading physical factors related to the physics of PET imaging. The quantification accuracy is inherently compromised by the limited spatial resolution due to partial volume effect (PVE), which results in the underestimation of radiotracer uptake, particularly for lesions smaller than 2 times of the system spatial resolution, 5 such as sub-centimeter pulmonary nodules and lymph nodes PVEs include both spill-in and spill-out of activity to and from a region-or organ-of-interest. Activities from the hot regions may interfere with PET quantification and visualization of nearby lesions, resulting in an overestimation of their SUVs. This effect is often referred to as the "spill-in." 6 On the opposite, activities are usually underestimated due to the "spillout" of counts to neighboring regions with lower activity. For example, in dynamic cardiac images, activities in ventricular chamber are usually underestimated due to the spill-out effect, and overestimation of activity in neighboring regions such as the blood pool is caused by spill-in effect. 7 These errors in the estimated activities can affect quantitative parameters. Therefore, compensating for PVEs should be carefully performed to ensure the accuracy of PET measurements. One of the simple correction methods to overcome the bias caused by PVE is to use the recover coefficient (ratio of observed to true activity, RC). 8 When applying this method, the measured lesion uptake in a ROI is divided by a correction factor RC. Srinivas  fied as a "regional correction" method, which means it does not yield a PVE-corrected image but only corrects the bias in an ROI and obtains a PVE-corrected uptake value. 8 As a crucial aspect, the PET reconstruction algorithm has a huge impact on the accuracy of SUV measurement. [10][11][12] Currently, the most widely used reconstruction algorithm in clinical practice is ordered subsets expectation maximization (OSEM). OSEM is not able to reach full convergence because the noise in the image grows with each iteration and hence there exists a compromise between iteration and noise resulting in partial convergence 13 . Besides, postsmoothing method for noise suppression 14 improves the acceptance of image quality but at a cost of reduced quantitative accuracy and volume distortions in small objects.
A Bayesian penalized likelihood (BPL)-based reconstruction algorithm (Q.Clear, GE Healthcare, Milwaukee) has been recently introduced for clinical routine with distinct advantages over OSEM. The BPL-based algorithm is able to achieve global convergence for all the image voxels by using relative difference penalty and block sequential regularized expectation maximization approaches, [15][16][17] incorporating point-spread-function (PSF) modelling. 18,19 The noise of the PET image is limited by a regularization parameter "beta value," which is the only user-input variable.
Many studies demonstrated the advantages of the BPL reconstruction algorithm for evaluating small pulmonary nodules, 20,21 liver metastasis, 22 and mediastinal nodes in nonsmall cell lung cancer. 23 Teoh et al. examined beta values of 100-1000 on a nondigital PET/CT system with attenuation correction and scatter correction performed, and recommended a beta value of 400 for the optimal clinical use. 22 They also used the beta value of 400 to assess small pulmonary nodules 20 and mediastinal lymph nodes. 23 Other researches applied fixed beta for BPL-based reconstruction, for instance, a beta of 50 was

RC ¼
where A M is the measured mean activity concentration(in kBq/mL) within a VOI in each sphere delineated on CT images, A B is the measured mean activity concentration in the background, A K is the known activity concentration (in kBq/mL) in the sphere; C is the ratio of known activity concentration in the sphere and that in the background (that is four in the study), SD B is the standard deviation of the measured background activity concentration and C B is the mean of the corresponding measured background activity concentration.

2.A.3 | Phantom image reconstruction
Both attenuation and scatter corrections were performed and the matrix size of reconstructed images was 256 × 256 with a pixel size (mm) of 2.73 × 2.73. A total of 345 slices was obtained for each scan with a slice thickness of 2.8 mm. All the PET images were reconstructed using the BPL-based algorithm with a range of beta value (β) from 100 to 1000 in an interval of 50, where beta is defined in the BPL objective function as follow [Eq. (4)]: where x is the image estimate, i is the pixel index, y i represents the measured PET coincidence data, P is the system geometry matrix, β is a regularization parameter, and R(x) is the relative difference penalty (RDP) to control noise. 15 The RDP can be expanded as Eq.
(5) shown below: where γ is the parameter controls the edge preservation, w j and w k are relative weights for different components of the function and N j represents a set of voxels surrounding voxel j. By applying the RDP to the objective function in Eq. (4), the reconstruction algorithm has the advantage of providing activity-dependent noise control. 26 With the increase of γ, images with sharper edges will be generated, which results in a more accurate quantification, especially for small lesions. 27 where r is the activity concentration (kBq/ml), a 0 is the decay-corrected dose of injected FDG (kBq), w is the body weight of the patient (g). 28 The background region was selected according to NEMA NU2-2012 guidelines. 29,30 For each sphere, different background regions were selected according to the sphere's diameter. As an example, the background activity concentration for the 37mm sphere was derived from 12 circular ROIs with a diameter of 37 mm were drawn. These 12 ROIs were located at background regions that did not contain any hot sphere and they were not allowed to overlap. The same set of 12 circular ROIs was then drawn on two slices above and two slices below the maximum intensity pixel to obtain a final of 60 background ROIs. The final background SUV reading was taken from the mean value of these 60 ROIs. 9 2.B | Clinical study

2.B.5 | Quantitative imaging analysis
The SUVs of each primary lung tumor were recorded using a standard volume of interest (VOI, segmented by using a 42% threshold of the maximum SUV) tool. 31

2.B.6 | PVC for pulmonary nodules
The PVC method used in this study was a simplified version modified from Kumar et al. 34 With the results from the phantom study, the relationships between RC and sphere diameter reconstructed using different beta values were established using linear regression analysis.

3.A | Phantom studies
The results for the phantom study are illustrated in Figs. 1-3. As beta value increased, the CR, RC, and BV decreased for all spheres.
By considering the size of different spheres throughout the whole beta value range, both CR and RC values (Fig. 1) increased with the sphere's diameter. In particular, small spheres (10 and 13 mm) were observed having steeper gradient as beta value went up compared with large spheres. Curves for BV values of all spheres (Fig. 2) dropped to a plateau when the beta value was higher than 650 and the figure also showed that large size spheres gave lower BV values compared with small spheres.

3.C | Subjective image quality evaluation
The results of the subjective image assessment including all study subjects are given in Table 1 23 and Linstrom et al. 35 On top of this, we also reported the effect of beta values on RC values. By using the relationship between RCs and the sphere diameter, we established a phantomderived linear regression model at the contrast of 4:1 to estimate the clinical RC and to perform PVC.
In clinical studies, SNR results showed that noise level decreased with high beta values which indicated that images reconstructed with a high beta value might have better quality compared to those using low beta values. However, unlike the medium-size group, SNR of the <10 mm group showed that such noise reduction became less obvious when beta values reached over 700, which agrees with our phantom study where BV reached a plateau when beta value was 650. In addition, the SNR values for small lesions (<10 mm) were consistently lower than the medium-size group (10-30 mm) and this was caused by the PVE, which affected more on small lesions. The effect of beta on SNR was also reported by Linstrom et al. 36 and their results were similar to this study although the subjects they enrolled were mixed with different diseases and they did not categorize lesions into different sizes. The clinical result of SNR suggested that the increase in the regularization parameter can reduce the image noise, but for small lesions, such improvement will become less obvious. The results for CNR agreed with SNR, the increase in beta value led to a higher CNR and this was contributed by the reduction in noise. At this stage, CNR results suggest that a better lesion detectability could be achieved by having a high beta value.
In the clinical assessment of 18 F-FDG PET/CT quantification, we applied PVC on all pulmonary nodules as the partial volume effect has a great impact on the quantification accuracy of small lesions.
Before the PVC, regardless of nodule sizes, the highly significant negative correlation found between beta values and SUVs implied that a choice of very high beta value will lead to the loss of quantifi- The values of contrast-to-noise ratio (CNR) of the same small pulmonary nodules. Highly significant differences in SNR and CNR were found between those measured at beta = 100 to beta = 1000 for both groups (P < 0.001, for all).
explained by recalling the definitions of different SUV parameters.
SUVmean is highly dependent on the lesion delineation as it measures the uptake of ROI around the maximum pixel, and a change in beta value may affect the ROI definition which consequently affects the quantification accuracy. 35 In addition, SUVmax represents uptake in the highest metabolic region which has the maximum pixel value, and it tends to be affected by the different noise levels. 37  One of the major limitations of this study is the accuracy of the PVC method. Correction factors were estimated from the NEMA phantom study for which hot spheres in the uniform background were considered. In reality, nodules could be surrounded by a region of high uptake. In addition, spill-in of activity from large surrounding structures will most likely occur. For such cases, the proposed PVC will lead to inaccuracy and misinterpretation (overestimation of activity and false positive). In order to avoid such scenario, patients with nodules located close to the heart or pleura were not considered. In addition, this study assumed that the tumor-to-background ratio was approximately 4:1 for all nodules when applying the PVC. Although this contrast ratio was close for those small pulmonary nodules, PVC models with more comprehensive contrasts are needed in the future study to achieve better accuracy. Another challenge with the proposed PVC is that the lesion in this study was defined on the CT images, which might not correspond to that on the PET image due to motion between/during scans. Normally, the solution was to apply the respiratory gating technology. However, respiratory gating was not widely used in our hospital by considering the convenience.
Moreover, the smallest sphere diameter of the NEMA phantom used in this study was 10 mm. This means the phantom-derived PVC model did not cover small lesions with the diameter smaller than 10 mm, and accuracy of PVE-corrected uptake values for those small lesions might be affected. Patients enrolled in this study were not histopathologically confirmed, which prevented our study from investigating the impact of beta values on benign and malignant tumors. Finally, the number of patients enrolled was considerably low compared to past studies. Future studies should employ a more appropriate PVC method considering any potential spill-in effects to ensure more accurate quantification for a wider range of lesions.
Moreover, the effects of beta parameter values on a larger set of independent factors should be investigated, including patient body T A B L E 1 Results of subjective positron emission tomography (PET) image quality ratings for different beta value.

Beta value
General image quality Image sharpness Lesion conspicuity Overall score