Patlak-Guided Self-Supervised Learning for Dynamic PET Denoising

Abstract

Quantitative ¹⁸F-FDG Patlak $K_{\text{i}}$ images provide physiologically meaningful measures of glucose influx, but they are substantially noisier than SUV images because they are estimated from short-duration dynamic frames. This study aims to improve the quality and quantitative reliability of $K_{\text{i}}$ images by applying a self-supervised deep learning framework for dynamic PET denoising. We propose the Patlak-Guided Self-Supervised Denoising Network (PG-SSDNet), a multi-frame CycleGAN-based model that incorporates a Patlak consistency loss to preserve tracer kinetics across frames. PG-SSDNet was evaluated on 14-frame and 6-frame whole-body ¹⁸F-FDG dynamic PET datasets from 27 patients. Performance was compared with a supervised U-Net, a single-frame CycleGAN, and a multi-frame CycleGAN without Patlak guidance. Metrics included mean normalized residual sum of squares (MNRSS) of Patlak fits, liver normalized standard deviation (NSTD), and lesion signal-to-noise ratio (SNR). PG-SSDNet consistently reduced noise while maintaining kinetic fidelity. It achieved the lowest Patlak fitting errors (MNRSS: 0.98% ± 0.14% for 14 frames, 0.85% ± 0.11% for 6 frames), the lowest liver NSTD values (0.105 ± 0.037 for 14 frames, 0.211 ± 0.107 for 6 frames), and the highest lesion SNR across all methods, while preserving accurate mean $K_{\text{i}}$ values. PG-SSDNet effectively denoises ¹⁸F-FDG dynamic PET data while preserving Patlak linearity, improving both lesion quantification and image quality. Unlike supervised methods, it requires no paired noisy/clean datasets, making it practical for clinical deployment. Future work will extend PG-SSDNet to other tracers with different kinetic properties and to larger multi-center datasets.

I. Introduction

Images of the parametric tracer influx rate, $K_{\text{i}}$ images, provide a fully quantitative measurement of metabolic activity in ¹⁸F-fluorodeoxyglucose (FDG) Positron Emission Tomography (PET). Comparing to conventional Standard Uptake Value (SUV)-based imaging, $K_{\text{i}}$ images offer improved lesion detectability [1] and lower false-positive diagnosis rates [2]. However, generating $K_{\text{i}}$ images requires dynamic PET acquisitions with short frame durations to accurately capture tracer kinetics, resulting in lower signal-to-noise ratios (SNR) [3] and, consequently, reduced $K_{\text{i}}$ image quality. Therefore, effective denoising of $K_{\text{i}}$ images is essential to enhance their clinical utility and support the advancement of fully quantitative parametric imaging.

Parametric images are often generated using an indirect approach, which involves first reconstructing dynamic PET frames and then applying kinetic modeling with an input function [3], [4]. Since most regions in FDG PET can be described by a two-tissue irreversible compartment model, the Patlak graphical model is commonly used as a simplified method to generate $K_{\text{i}}$ images [5], [6]. Denoising techniques can be applied to the dynamic frames prior to Patlak modeling, producing enhanced $K_{\text{i}}$ images that preserve quantitative accuracy while effectively reducing noise.

Deep learning-based denoising methods have recently proven more effective than traditional techniques [7], [8], [9], [10], such as nonlocal mean filtering (NLM) [11], [12], Block Matching and 3D filtering (BM3D) [13], [14], and HighlY constrained back PRojections to a local region of interest (HYPR) [15], [16], [17], which avoided excessive smoothing of the dynamic PET images but achieved only limited noise reduction. Deep learning models such as U-Net [18], [7], Residual Network (ResNet) [19], [20], Generate Adversarial Network (GAN) [21], [22], and Diffusion models [23], [24], [25] can significantly reduce noise while preserving critical image details. However, they typically rely on paired noisyclean datasets, which are difficult to obtain in dynamic imaging due to the short frame durations required to capture tracer kinetics.

To address this challenge, we previously developed two approaches. The first approach involves creating a noise level-matched training dataset using clinical static images: standard-count static images were downsampled to match the noise level of dynamic images by aligning prompt counts, and the resulting trained network was applied to denoise dynamic frames [26]. The supervised model relies on the matched training data, and the denoising results are limited to the label image quality [27]. The second approach leverages deep image prior (DIP) [28], [29], an unsupervised technique that iteratively refines noisy images. In our Population-based DIP (PDIP) method [30], the denoised images from the supervised prompt-matched model serve as prior images, while the target images are the dynamic frames themselves. Although PDIP showed improved performance in preserving lesion values compared to the supervised model, it required approximately 4 hours of inference time per dynamic study. Additionally, due to GPU memory limitations, whole-body images were too large to be processed directly with the DIP model.

Image generation models such as CycleGAN [31], which enable image translation between domains without the need for paired training data, offer a promising approach for dynamic PET denoising. Previous research has utilized CycleGAN to enhance PET imaging by transforming images from lower-resolution scanners to approximate the quality of higher-resolution systems [32]. For dynamic PET denoising, a similar strategy could involve using the mean frame—representing a longer acquisition with improved image quality—as the target domain, while treating individual dynamic frames as the input domain.

However, directly aligning each dynamic frame to the mean frame poses the risk of distorting tracer distribution and losing critical tracer kinetic information [33]. To mitigate this issue, Patlak-based constraints can be employed. Traditionally, these constraints have been integrated into iterative reconstruction methods, such as the Nested Direct Reconstruction approach [34]. This technique incorporates Patlak fitting into the reconstruction process, enabling iterative enhancement of image quality while suppressing noise. Additionally, Patlak constraints have been applied in dynamic PET motion correction to improve frame alignment while preserving kinetic information [35]. However, when used as a loss function in network training, the Patlak fitting error alone provides insufficient constraints and must be supplemented with additional loss terms to achieve optimal performance.

It is also important to note that most current clinical dynamic PET is performed on whole-body continuous bed motion (CBM) scanners, where each bed pass is relatively short and thus per-frame counts are low, making denoising especially critical. In contrast, recently developed total-body PET scanners (e.g., uEXPLORER, PennPET Explorer) provide substantially higher sensitivity and larger axial coverage [36], [37], [38], enabling either shorter scans or finer temporal sampling with higher SNR. While these systems may reduce the need for aggressive denoising, they also introduce challenges such as handling more dynamic frames and increased memory demands for deep models. Our method is therefore designed to be practical under the noisier CBM setting most widely adopted clinically, while remaining scalable to future total-body applications.

Here we introduced the Patlak-Guided Self-Supervised Denoising Network (PG-SSDNet), an innovative CycleGAN-based framework tailored for dynamic PET denoising. Unlike traditional CycleGAN, PG-SSDNet processed all dynamic frames simultaneously, leveraging a long short-term memory (LSTM) mechanism [39], [40] to enable efficient inter-frame information transfer. Furthermore, the Patlak model was applied to the denoised frames to generate Patlak-fitted dynamic frames, which served as a direct Patlak model constraint in the model training. This design allows PG-SSDNet to effectively preserve tracer kinetics, improving denoising performance while maintaining critical kinetic information across frames.

We compared PG-SSDNet to several benchmark models, including a supervised U-Net (trained on noise-matched static images), a single-frame CycleGAN, and a variant of the proposed method without the Patlak constraint. Performance evaluations were conducted under both standard-count (14-frame) and low-count (6-frame) settings to assess the models’ capabilities in handling varying temporal resolutions and dynamic frame complexities. Our results demonstrated the effectiveness of the multi-frame CycleGAN approach and underscore the importance of the Patlak constraint in self-supervised training for optimal dynamic PET denoising.

II. Materials and Methods

A. Human dataset

With Yale Institutional Review Board approval, 28 subjects (23 cancer patients and 5 healthy individuals) participated in a 90-minute whole-body FDG Continuous Bed Motion (CBM) PET imaging protocol obtained using a Siemens Biograph mCT scanner. The subjects, aged 24 to 77 years (mean age 56 ± 14 years), had body mass indices (BMI) from 21.6 to 37.2 kg/m² (mean BMI 28.2 ± 3.9 kg/m²) and fasting plasma glucose levels from 75 to 142 mg/dL (mean 101 ± 17 mg/dL). Each subject received a bolus injection of 334.48 ± 32.56 MBq of FDG, followed by a non-contrast CT scan for attenuation correction. An initial 6-minute single-bed scan was acquired over the heart to estimate the early-phase input function from the segmentations of the left ventricle or aorta. This early-phase acquisition was divided into 9 frames (6 frames of 0.5 min each, followed by 3 frames of 1 min each).

Subsequent whole-body CBM PET imaging was performed in 19 passes, with frame durations of 2 minutes for the first 4 passes and 5 minutes for the remaining 15 passes. The scanner operated in a unidirectional mode, moving from the superior to the inferior end of the body during each CBM frame and resetting to the head between each frame. Taller subjects required an increased bed movement speed due to the larger coverage area. Arterial blood samples were obtained from 23 out of the 28 subjects, allowing plasma activity measurements using a calibrated well counter.

All dynamic frames underwent reconstruction using the ordered subsets expectation maximization (OSEM) algorithm (2 iterations, 21 subsets) with point spread function and time-of-flight information. The reconstruction incorporated corrections for attenuation, scatter, random coincidences, normalization, and radioactive decay, followed by a 5-mm Gaussian post-filtering applied to each image. Voxel dimensions were standardized at 2.04 × 2.04 × 2.03 mm³, resulting in a transverse image matrix of 400 × 400 voxels. The total number of axial slices per whole-body scan varied from 489 to 859, depending on the height of each subject.

B. Network architecture: Patlak-Guided Self-Supervised Denoising Network (PG-SSDNet)

The proposed Patlak-Guided Self-Supervised Denoising Network (PG-SSDNet) architecture is shown in Figure 1. The two image domains consist of (1) the noisy dynamic PET frames and (2) the corresponding mean clean frame C. The network utilized two primary generators: the Noise-to-Clean Generator $\left(G_{N2C}\right)$, which transformed noisy dynamic PET frames into synthetic clean mean frames, and the Clean-to-Noise Generator $\left(G_{C2N}\right)$, which performed the reverse operation by introducing synthetic noise into clean mean frames. Specifically, the mean clean frame $C$ is defined as the voxel-wise temporal average of the SUV images from frames 6–19 (20–90 min). In the 6-frame setting, $C$ is computed as the average of the selected six frames. This mean frame serves as a higher-SNR proxy target domain for the CycleGAN mappings.

Fig. 1:

Overview of the Patlak-Guided Self-Supervised Denoising Network (PG-SSDNet) architecture for dynamic PET image denoising. The network comprises two main generators: the Noise-to-Clean generator (N2C) and the Clean-to-Noise generator (C2N), both structured as U-Net architectures with ConvLSTM bottlenecks to facilitate temporal information transfer across frames. The N2C generator denoises dynamic frames, while the C2N generator reintroduces noise to clean mean frames, supporting bidirectional domain translation. The discriminators provide adversarial feedback, ensuring the realism of generated clean mean frames and noisy frames, respectively. The network is guided by a Patlak model constraint, applied as an additional L1 loss to enforce physiologically consistent outputs. Cycle consistency loss further ensures input-output fidelity across transformations, enabling effective self-supervised training for dynamic PET image denoising.

Both generators were designed with a U-Net-like encoder-decoder structure with a 3D convolutional LSTM (ConvLSTM) bottleneck. Each downsampling step included two 3×3×3 convolutional layers, followed by the LeakyReLU activation function. A 2×2×2 max-pooling layer was utilized at the end of each step, except for the last. The channels were set as 64, 256, and 512, respectively. The upsampling step consisted of an up-convolutional layer with a scale factor of 2, and two 3×3×3 convolutional layers followed by the LeakyReLU activation function. The bottleneck featured a ConvLSTM [40] mechanism, consisting of two ConvLSTM layers that take the 14 encoder outputs as input, with 512 channels and 256 hidden channels, enabling the network to capture temporal dependencies. A dropout rate of 0.2 was applied to prevent overfitting.

The network employed two patch-based discriminators to distinguish between real and synthetic frames in each domain: the Clean Discriminator $\left(D_C\right)$ and the Noise Discriminator $\left(D_N\right)$. Each discriminator consisted of three 4×4×4 convolutional layers. The first two convolutional layers had 32 and 64 channels, with a stride of 2, followed by instance normalization and Leaky ReLU activations. The final 4×4×4 convolutional layer had 128 channels, with a stride of 1, and its outputs were reshaped into a 1D tensor.

C. Model training

In the generator training process, the loss was composed of three components: cycle-consistency loss, discriminator loss, and Patlak loss. The cycle consistency loss consists of mean absolute difference (L1) loss designed to ensure that the input and output images remain consistent after passing through two generators. Specifically, L1 loss was calculated between the original noisy dynamic frames $N$ and the synthetic noisy dynamic frames $G_{C2N}\left(G_{N2C}(N)\right)$, which were generated by sequentially applying $G_{N2C}$ and $G_{C2N}$. Similarly, there was an L1 loss between the original clean mean frame $C$ and the synthetic clean mean frame $G_{N2C}\left(G_{C2N}\left(C\right)\right)$, produced by passing $C$ through $G_{C2N}$ and $G_{N2C}$.

We integrated the Patlak model fitting error (described in Section 2.D) as an additional generator loss component to constrain tracer kinetics. By applying a matrix formulation of the Patlak model [41] to perform linear regression on the input function and dynamic PET frames, we generated Patlak-fitted dynamic frames $P\left(G_{N2C}(N)\right)$. The detailed implementation is introduced in Section 2.D. An L1 loss enforced consistency between the generator outputs $G_{N2C}(N)$ and these Patlak-predicted frames $P\left(G_{N2C}(N)\right)$, guiding the network to maintain accurate tracer kinetics. The final generator loss was given by:

$$\begin{gathered} {ℒ}_{\text{G}}\left(G_{N2C},G_{C2N},D_N,D_C\right) \\ =\alpha \left[{ℒ}_{\text{cycle}}\left(C,G_{N2C}\left(G_{C2N}\left(C\right)\right)\right)\right. \\ \left.+{ℒ}_{\text{cycle}}\left(N,G_{C2N}\left(G_{N2C}(N)\right)\right)\right] \\ +\beta \left[{ℒ}_{\text{Patlak}}\left(P\left(G_{N2C}\right),G_{N2C}\right)\right] \\ +\gamma \left[{ℒ}_{\text{GAN}}\left(D_C\left(G_{N2C}\left(G_{C2N}\left(C\right)\right)\right),1\right)\right. \\ \left.+{ℒ}_{\text{GAN}}\left(D_C\left(G_{C2N}\left(G_{N2C}\left(C\right)\right)\right),1\right)\right]. \end{gathered}$$ (1)

The hyper-parameters $\alpha$ and $\beta$ were set to 5 and 0.25, respectively, consistent with previous cycleGAN works [42], [43]. For the first 10 epochs of training, we set $\gamma$ to 0, effectively disabling the Patlak constraint. After these initial epochs, $\gamma$ was updated to 10⁻³ so that its scale was consistent with the other loss terms, thereby ensuring a balanced influence of the Patlak constraint during training.

The discriminator loss ensured that the generated frames closely matched their respective target domains. During the generation step, this loss encouraged the synthetic clean frames $G_{N2C}\left(G_{C2N}\left(C\right)\right)$ and synthetic noisy frames $G_{C2N}\left(G_{N2C}(N)\right)$ to closely resemble real clean and real noisy frames, respectively.

In the discriminator training step, the discriminators $D_C$ and $D_N$ aimed to distinguish real clean samples from fake clean samples and real noisy samples from fake noisy samples, respectively. The discriminator loss function was defined as:

$$\begin{gathered} {ℒ}_{\text{D}}\left(D_N,D_C\right)={ℒ}_{\text{GAN}}\left(D_C\left(G_{N2C}\left(G_{C2N}\left(C\right)\right)\right),0\right) \\ +{ℒ}_{\text{GAN}}(C,1) \\ +{ℒ}_{\text{GAN}}\left(D_C\left(G_{C2N}\left(G_{N2C}\left(C\right)\right)\right),0\right) \\ +{ℒ}_{\text{GAN}}(N,1) \end{gathered}$$ (2)

The training was conducted in a self-supervised, patch-based manner, using one patient’s data per epoch to allow the input function to be passed for computing each patient’s Patlak loss individually. Each batch, with a patch size of 64×64×64, required approximately 22 GB of memory, and training was performed on an NVIDIA H100 GPU and PyTorch version 1.10.2.

To evaluate generalizability and mitigate overfitting, we employed a three-fold cross-validation strategy, splitting the 28 subjects into groups of 9, 9, and 10 patients, respectively. Models were trained on two folds and evaluated on the remaining fold in rotation. In addition to the standard 14-frame setting, we also investigated a reduced 6-frame protocol to simulate lower-count acquisitions. We selected frames 6, 8, 10, 12, 15, and 19, corresponding to time intervals between 20–90 minutes post-injection. These frames were chosen to (i) cover the full Patlak linear phase, (ii) maintain temporal spacing across early, mid, and late windows, and (iii) ensure sufficient counts per frame for stable Patlak fitting. The six-frame subset therefore provides a reduced-duration protocol while still preserving the kinetic information needed for $K_{\text{i}}$ estimation.

D. Parametric image generation

The parametric $K_{\text{i}}$ images were computed from dynamic PET frames using the standard Patlak model. For 23 out of 28 patients, an arterial input function (AIF) was directly measured through continuous blood sampling throughout the scan. For the remaining 5 patients, a personalized population-based input function (PBIF) was generated [44]. This PBIF was created by averaging the AIFs from the 23 patients with available data, resulting in a population-based input function template. To tailor this template to each patient, an image-derived input function (IDIF) was extracted from a 4×4×8 mm³ volume in the center of the descending aorta, using the last 14 dynamic frames. The area under the curve (AUC) of each patient’s IDIF was then used to scale the population-based template, producing a personalized PBIF for each of the 5 patients without direct AIF data. The dynamic frames included in the Patlak model are from 20 minutes (start time $t^{*}$) to 90 minutes, totaling 14 frames.

The Patlak model is represented in matrix form as:

$$\mathbf{Y}=\mathbf{X}\mathit{\beta }+\mathit{ϵ},$$ (3)

where $Y$ is the vector of measured tissue concentrations at different time points, $X$ contains the integrated plasma concentrations and the plasma concentration itself as separate columns, and $\mathit{\beta }=\left[\begin{array}{c}{K}_{\text{i}}\\ V_{\text{b}}\end{array}\right]$ represents the parameters to be estimated, where $K_{\text{i}}$ is the net uptake rate constant and $V_{\text{b}}$ is the fitted constant.

$$\mathbf{Y}=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_k\end{array}\right],\mathbf{X}=\left[\begin{array}{cc}\text{Sp}\left(1\right)& \text{Tp}\left(1\right)\\ \text{Sp}\left(2\right)& \text{Tp}\left(2\right)\\ ⋮& ⋮\\ \text{Sp}\left(k\right)& \text{Tp}\left(k\right)\end{array}\right],$$ (4)

where $k$ corresponded to the frame number, $S_p\left(k\right)$ represented the integrated input function up to time $t_k$ and $T_p\left(k\right)$ was the instantaneous plasma concentration at the frame middle time $t_k$. The regression weights for weighted least squares (WLS) fitting were computed to enhance parametric estimation accuracy, given by:

$$\mathit{\beta }={\left({\mathbf{X}}^{\top }\mathbf{W}\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\top }\mathbf{W}\mathbf{Y},$$ (5)

where $W$ was a diagonal weight matrix, defined as:

$$\mathbf{W}=\left[\begin{array}{cccc}\frac{1}{{\sigma }_1^2}& 0& \dots & 0\\ 0& \frac{1}{{\sigma }_2^2}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & \frac{1}{{\sigma }_k^2}\end{array}\right].$$ (6)

Each element ${\sigma }_k^2$ in $W$ was defined as:

$${\sigma }_k^2=\frac{{\Delta }_t^2}{N_k*DC{F}_k^2},$$ (7)

where ${\Delta }_t$ was the frame duration, $N_k$ was the total counts, and $DC{F}_k^2$ represented the decay correction factor.

The Patlak loss is then defined as the weighted residual sum of squares (WRSS):

$${ℒ}_{\text{Patlak}}=\frac{1}{K}\sum _{k=1}^K{w}_k{\left(Y_k-{\hat{Y}}_k\right)}^2,{\hat{Y}}_k={\mathbf{X}}_k\stackrel{\mathit{ˆ}}{\mathit{\beta }}.$$

This term is added to the overall training loss, encouraging the denoised frames to be consistent with Patlak linearity while suppressing frame-wise noise.

E. Comparing models

We evaluated the PG-SSDNet against several baseline approaches to assess its effectiveness in denoising $K_{\text{i}}$ images. These approaches included: (1) a supervised U-Net model trained on 100 downsampled static images, described as the prompts-matched method [26], (2) the CycleGAN trained using only a single dynamic frame, (3) the proposed multi-frame denoising method without additional Patlak constraints, and (4) the proposed multi-frame self-supervised denoising method with the Patlak loss. To assess model performance in a low-count setting, all methods used six specific dynamic frames (frames 6, 8, 10, 12, 15, and 19) to generate $K_{\text{i}}$ images as well. In this setting, the multi-frame network used only six frames as input.

F. Noise evaluation

To evaluate the denoising performance of each method, we computed two primary metrics: the mean normalized residual sum of squares (MNRSS) and the normalized standard deviation (NSTD) in the liver region to assess local noise levels in the $K_{\text{i}}$ images.

The MNRSS of the Patlak error was calculated for each voxel by comparing the Patlak-fitted dynamic frames, $P\left(G_{N2C}(N)\right)$, with the denoised dynamic frames, $G_{N2C}(N)$, as shown in Equation 8:

$$\text{MNRSS}=\frac{1}{K}\left(\frac{\sum _{j=1}^K{\left[P{\left(G_{N2C}(N)\right)}_k-G_{N2C}(N{)}_k\right]}^2}{{\left(\sum _{j=1}^K{G}_{N2C}(N{)}_k\right)}^2}\right)$$ (8)

In this equation, $K$ represented the total number of frames (14 for the full dataset and 6 for the low-count setting), and $k$ was the frame index. The mean NRSS across the body region, referred to as the MNRSS of the Patlak error, was used as a summary measure of noise.

To quantify the local noise level specifically in the liver, we computed the liver NSTD within a defined volume of interest (VOI) on dynamic images, as the liver represents a relatively uniform physiological region where measured variability predominantly reflects image noise, whereas other organs may exhibit true uptake heterogeneity. The liver NSTD was given in Equation 9, where ${\bar{I}}_i$ represented the mean SUV value within the liver VOI, and $N_{\text{VOI}}$ denoted the total number of voxels within the VOI:

$$\text{NSTD}=\frac{\sqrt{\sum _{i\in \text{VOI}}{\left(I_i-{\bar{I}}_i\right)}^2/N_{\text{VOI}}}}{{\bar{I}}_i}$$ (9)

G. Lesion SNR

An experienced nuclear medicine physician manually identified and delineated 54 lesion VOIs on the $K_{\text{i}}$ images, with an average lesion size of 8.90 ± 21.00 cm³. For each VOI, the mean $K_{\text{i}}$ values, $\overline{I_{\text{i}}^{\text{lesion}}}$, were measured both before and after denoising. To compare these values, we generated both a correlation plot and a Bland-Altman plot, using the 14-frame fitted $K_{\text{i}}$ image as the gold standard for both the 14-frame and 6-frame settings.

To further assess lesion detectability, we computed the signal-to-noise ratio (SNR) for each lesion. A background VOI was defined in the surrounding tissue adjacent to each lesion to serve as a noise reference. The SNR was calculated using the following equation:

$$\text{SNR}=\frac{\overline{I_{\text{i}}^{\text{lesion}}}}{\text{STD}\left(\text{background}\right)}$$ (10)

where STD(background) is the standard deviation of the background VOI.

III. Results

A. Dynamic frames and Ki images

Figure 2 presents denoised frames and $K_{\text{i}}$ images for a lung cancer patient, comparing various methods: input, supervised U-Net, CycleGAN, the SSDNet, and the PG-SSDNet. The supervised U-Net effectively maintained SUV values across frames but showed slight blurring in lesion regions, which could impact lesion visibility. In contrast, CycleGAN, trained in a self-supervised manner with a smaller cohort (20 dynamic studies), resulted in lower SUV values and reduced lesion contrast. The SSDNet improved noise reduction but still underestimated lesion intensity. Notably, the PG-SSDNet achieved the highest SUV preservation, comparable to the supervised U-Net. In the $K_{\text{i}}$ images, the PG-SSDNet provided consistent $K_{\text{i}}$ values $\left(K_{\text{i}}=0.0143\right)$ with the supervised U-Net, and it performed better than CycleGAN and the non-Patlak model, which yielded lower $K_{\text{i}}$ values.

Fig. 2:

Sample denoised frames, $K_{\text{i}}$ images, and standard deviation (STD) images of Patlak error for a representative lung cancer patient, comparing results from the input, supervised U-Net model, CycleGAN, proposed model without Patlak, and the proposed Patlak-guided self-supervised multi-frame denoising network. The first three rows display frames from different time points: early (frame 6), middle (frame 12), and late (frame 18). The fourth row shows the $K_{\text{i}}$ images generated from 14 dynamic frames, and the bottom row presents the STD of Patlak error images. For each frame, the mean standardized uptake values (SUV) of the indicated lesion are displayed below. Similarly, the $K_{\text{i}}$ mean values for the same lung lesion are provided under each $K_{\text{i}}$ image.

Figure 3 shows similar results for a myeloma patient in a low-count 6-frame setting, showcasing three lesions in the leg region. Here, the PG-SSDNet demonstrated effective denoising, offering high lesion clarity and reduced noise in the surrounding muscle tissue, outperforming both CycleGAN and the non-Patlak model, which exhibited some blurring and reduced lesion contrast. While the supervised U-Net maintained slightly higher lesion values, it introduced more noise compared to the PG-SSDNet. The $K_{\text{i}}$ images from the PG-SSDNet preserved accurate lesion quantification $\left(K_{\text{i}}=\right.0.0140)$, closely matching the Input $\left(K_{\text{i}}=0.0151\right)$, while outperforming CycleGAN and the non-Patlak model, which yielded lower $K_{\text{i}}$ values. Together, these results illustrated that the proposed Patlak-guided model achieves a favorable balance of noise reduction, lesion clarity, and quantification accuracy across different low-count scenarios.

Fig. 3:

Results from the 6-frame experiment for a myeloma patient, demonstrating denoised frames (Leg region coronal view) and $K_{\text{i}}$ images from different methods: input, supervised U-Net model, CycleGAN, proposed model without Patlak guidance, and the proposed Patlak-guided self-supervised multi-frame denoising network. The first three rows show frames at various time points—early (frame 6), mid (frame 12), and late (frame 19). The fourth row presents $K_{\text{i}}$ images derived from these 6 frames. The mean SUV for the selected lesion are provided below each frame, with corresponding $K_{\text{i}}$ mean values shown beneath each $K_{\text{i}}$ image.

B. Noise evaluation

Table I compares the MNRSS of Patlak fitting errors across different methods for 14-frame and 6-frame $K_{\text{i}}$ predictions over the entire 28 studies. In both settings, the PG-SSDNet achieved the lowest Patlak error MNRSS.

TABLE I:

Comparison of Patlak Error MNRSS across Methods for 14-Frame and 6-Frame Settings (mean ± standard deviation). P-value was calculated under the Wilcoxon signed-rank test.

Method	MNRSS	P-value
14 Frames
100%-count Input	0.0844 ± 0.0132	< 0.001
Supervised U-Net	0.0523 ± 0.0089	< 0.01
CycleGAN	0.0418 ± 0.0069	< 0. 01
SSDNet	0.0131 ± 0.0016	< 0.05
PG-SSDNet	0.0098 ± 0.0014	-
6 Frames
100%-count Input	0.0916 ± 0.0154	< 0. 01
Supervised U-Net	0.0469 ± 0.0101	< 0. 01
CycleGAN	0.0414 ± 0.0091	< 0. 01
SSDNet	0.0133 ± 0.0029	0.25
PG-SSDNet	0.0085 ± 0.0011	-

Figure 4 illustrates the mean liver NSTD values for each frame, with shaded areas representing ± one standard deviation, across different methods in both dynamic acquisition settings. Across both tables, the Input method exhibited relatively high liver NSTD values, with measurements increasing from approximately 0.177 ± 0.051 at frame 6 to 0.248 ± 0.068 at frame 19. The CycleGAN method lowered these values, ranging from 0.112 ± 0.041 to 0.137 ± 0.049. In contrast, the Supervised U-Net method consistently reduces the liver NSTD, achieving values as low as 0.076 ± 0.033 at frame 6 and around 0.085 ± 0.040 by frame 19. Notably, the denoising performance improves even more with SSDNet and the Patlak-guided PG-SSDNet: SSDNet achieves liver NSTD values of approximately 0.063 ± 0.033 at frame 6 and 0.073 ± 0.041 at frame 19, while PG-SSDNet yields the lowest values—around 0.054 ± 0.037 at frame 6 and 0.062 ± 0.044 at frame 19. This consistent reduction in liver NSTD, particularly with PG-SSDNet, indicates that incorporating the Patlak loss not only improves model adherence but also enhances denoising performance.

Fig. 4:

Liver NSTD values for frames 6 to 19 in the 14-frame and 6-frame settings. Comparisons include input, supervised U-Net, CycleGAN, SSDNet, and PG-SSDNet. The PG-SSDNet shows the lowest liver NSTD in both settings.

C. Quantitative evaluations

In the 14-frame setting, the supervised U-Net consistently demonstrated the highest correlation with the 100%-count input in $K_{\text{i}}$ values, followed closely by the PG-SSDNet as shown in Figure 5. Although CycleGAN and the SSDNet showed some correlation, their correlation was notably lower than the supervised U-Net and PG-SSDNet. The Bland-Altman plots reinforced these findings, with the supervised U-Net and PG-SSDNet showing the smallest mean differences, indicating more accurate $K_{\text{i}}$ quantification.

Fig. 5:

Correlation and Bland-Altman plots for $K_{\text{i}}$ values using 14-frame across different methods: supervised U-Net, CycleGAN, SSDNet, and PG-SSDNet. The correlation plots (top row) display the relationship between predicted $K_{\text{i}}$ values and the standard 100%-count input, with the supervised U-Net showing the highest correlation $\left(R^2=0.9591\right)$ to the input. The PG-SSDNet achieves a comparable similarity with $R^2=0.9321$, while CycleGAN and the SSD-Net exhibit lower correlations. The Bland-Altman plots (bottom row) show the mean differences and limits of agreement for each method. The PG-SSDNet has a mean difference of −0.0017, comparable to the supervised U-Net (−0.0015), indicating that it closely aligns with the reference $K_{\text{i}}$ values.

As shown in Figure 6, under the 6-frame setting, the supervised U-Net remained the most closely aligned with the 100%-count input, followed by the PG-SSDNet. CycleGAN and the SSDNet showed reduced correlations in comparison. The Bland-Altman analysis further highlights that both the supervised U-Net and Patlak-guided method maintain small mean differences, underscoring their stability and accuracy in $K_{\text{i}}$ quantification even in low-count settings.

Fig. 6:

Correlation and Bland-Altman plots for $K_{\text{i}}$ values using 6-frame settings across different methods: supervised U-Net, CycleGAN, SSDNet, and the PG-SSDNet. In the correlation plots (top row), the supervised U-Net once again achieves the highest similarity with the standard 100%-count input, with the PG-SSDNet showing a comparable performance. The PG-SSDNet achieves a stronger correlation than CycleGAN and the non-Patlak method, suggesting its robustness in low-count conditions. In the Bland-Altman plots (bottom row), the supervised U-Net and PG-SSDNet demonstrate smaller mean differences, suggesting their better alignment with reference $K_{\text{i}}$ values.

The results in Table II demonstrated that the PG-SSDNet consistently achieved the highest SNR across all lesion sizes and frame settings, showing robust performance even under low-count and small-lesion scenarios. In the 6-frame setting, the PG-SSDNet achieved an SNR of 15.9 ± 14.3 for lesions smaller than 1 cm³, outperforming supervised UNet (14.5 ± 12.7), CycleGAN (11.6 ± 7.6), and SSDNet (13.1 ± 10.1). This performance advantage was even more pronounced with larger lesions, where the PG-SSDNet reached an SNR of 56.8 ± 29.9, significantly exceeding the SNR achieved by Supervised UNet (44.8 ± 21.6). In the 14-frame setting, the PG-SSDNet maintained superior performance across all lesion sizes, achieving an SNR of 20.1 ± 16.7 for small lesions, 51.8 ± 40.4 for medium lesions, and 72.3 ± 42.5 for large lesions.

TABLE II:

Comparison of SNR across methods for 14-frame and 6-frame settings (mean ± standard deviation). All p-values for comparisons with the PG-SSDNet were less than 0.01.

14 Frames
Lesion Size Category	Input	Supervised UNet	CycleGAN	SSDNet	PG-SSDNet
< 1 cm3	11.2 ± 9.6	16.2 ± 15.3	16.0 ± 9.5	19.2 ± 14.5	20.1 ± 16.7
[1, 10] cm3	21.4 ± 11.5	35.9 ± 23.6	30.9 ± 17.0	43.7 ± 30.0	51.8 ± 40.4
> 10 cm3	30.7 ± 14.7	52.9 ± 26.0	48.1 ± 22.5	65.6 ± 34.4	72.3 ± 42.5
6 Frames
< 1 cm3	11.6 ± 8.6	14.5 ± 12.7	11.6 ± 7.6	13.1 ± 10.1	15.9 ± 14.3
[1, 10] cm3	18.3 ± 9.2	32.1 ± 18.6	23.9 ± 12.9	31.6 ± 18.6	43.0 ± 31.0
> 10 cm3	25.2 ± 12.0	44.8 ± 21.6	36.5 ± 16.8	46.2 ± 21.9	56.8 ± 29.9

Without physiologic constraints, the performance of multi-frame CycleGAN (SSDNet) varied with lesion size and frame count, showing occasional gains but also residual bias, particularly for small lesions in the 6-frame setting. By integrating the Patlak constraint, PG-SSDNet stabilized multi-frame learning and consistently improved lesion SNR across lesion sizes and frame settings. The constraint ensured that denoised frames preserved linear tracer kinetics, reducing lesion underestimation and enabling reliable $K_{\text{i}}$ quantification even under reduced-frame acquisitions.

IV. Discussion

This study introduces the Patlak-Guided Self-Supervised Denoising Network (PG-SSDNet), a specialized deep learning framework for denoising dynamic PET images. By integrating a Patlak model constraint into a multi-frame CycleGAN-based architecture with long short-term memory (LSTM) components, PG-SSDNet achieves effective noise reduction while preserving essential tracer kinetics across frames.

Our method explicitly leverages the linear Patlak relationship that holds after tracer equilibration with reversible compartments, where tissue activity is well modeled as a linear function of the plasma integral and contemporaneous plasma concentration [41], [45]. In our dataset, the dynamic frames used for modeling span 20–90 minutes (frames 6–19), a time window in which Patlak linearity is expected for most tissues and lesions [3], [4]. PG-SSDNet imposes a Patlak-based loss computed from the denoised frames, encouraging the network to (i) reduce frame-wise noise, yet (ii) preserve the inter-frame temporal structure required for accurate linear fits. This directly addresses a key limitation of generic image-to-image denoisers (including unsupervised CycleGAN) that may inadvertently distort time-activity curves and bias $K_{\text{i}}$ values.

Our experimental results indicated that PG-SSDNet achieved superior noise reduction compared to traditional denoising models. PG-SSDNet recorded the lowest MNRSS of Patlak fitting errors, outperforming U-Net, single-frame CycleGAN, and SSDNet without Patlak constraint. Additionally, PG-SSDNet achieved the lowest liver NSTD values across both 14-frame and 6-frame settings, highlighting its denoising efficacy in dynamic images.

In lesion quantification, PG-SSDNet provided $K_{\text{i}}$ values closely aligned with reference standards and showed minimal bias across both 14-and 6-frame settings (mean difference of −0.0017 for 14 frames and −0.0016 for 6 frames). While Bland–Altman and correlation analyses showed that PG-SSDNet did not always outperform the supervised U-Net in global voxel-wise correlation, this reflects an inherent limitation of self-supervised approaches, which do not benefit from clean paired targets. PG-SSDNet may therefore introduce slight residual bias in $K_{\text{i}}$ estimation compared with supervised models. Additionally, our dataset of 28 subjects limits the statistical power to detect subtle differences, underscoring the need for larger multi-center validations. This small training cohort indicates that our findings should be viewed as proof-of-concept rather than definitive evidence of generalizability. While the results are promising, this acknowledgment sets realistic expectations for PG-SSDNet’s performance on independent datasets, particularly in terms of clinical applicability. Additionally, the single-frame CycleGAN, which aligns individual frames with the mean frame, showed higher underestimation likely due to tracer kinetics distortions. Nonetheless, PG-SSDNet demonstrates high lesion SNR values across all sizes, highlighting its robustness in preserving lesion clarity and quantification accuracy.

It should also be emphasized that this study focused exclusively on ¹⁸F-FDG, an irreversible tracer well suited to Patlak modeling. Extending PG-SSDNet to reversible tracers (e.g., ¹¹C-raclopride, ⁸²Rb) will require incorporating different kinetic constraints such as Logan graphical analysis or compartmental modeling [46], [47]. Moreover, FDG uptake after 20 minutes is relatively stable, whereas tracers like ⁸²Rb show rapid washout and frame-to-frame variability [48], [49], which pose additional challenges for joint multi-frame denoising. Additionally, the Patlak model applied here is less suitable for regions with unique tracer kinetics, such as the liver [50], and inter-frame motion introduces challenges for maintaining Patlak-based information accuracy.

PG-SSDNet was developed with an emphasis on clinical practicality. The patch-based architecture overcomes the GPU memory limitations of Deep Image Prior (DIP) methods, enabling efficient processing of whole-body datasets. Unlike diffusion models, which remain computationally intensive even for 2D data [23], PG-SSDNet retains the full 4D structure while offering inference in 2 to 3 minutes. The framework is implemented as a post-reconstruction step and requires no modification to scanner protocols, making it compatible with standard AIF- or PBIF-based dynamic acquisitions [44]. Compared with optimization-based DIP/PDIP methods that denoise each frame iteratively and require hours per study [30], PG-SSDNet jointly denoises all frames in a single forward pass, substantially reducing inference time. Furthermore, unlike supervised denoisers, it does not require paired noisy/clean datasets for training. This makes it feasible for real-world deployment, where paired dynamic PET data are virtually impossible to obtain. By leveraging Patlak constraints, PG-SSDNet maintains physiologic fidelity without paired supervision, highlighting its translational potential.

Future work will broaden PG-SSDNet’s adaptability by extending it to support additional PET tracers and incorporating segmentation-based approaches to address organ-specific kinetics. We also plan to tackle the challenges posed by motion artifacts by integrating motion correction with denoising. This study demonstrates the potential of physiologically informed deep learning models like PG-SSDNet to advance dynamic PET imaging, improving lesion quantification, noise suppression, and the overall reliability of quantitative parametric imaging for clinical applications.

V. Conclusions

This study introduces PG-SSDNet, a Patlak-Guided Self-Supervised Denoising Network designed to enhance $K_{\text{i}}$ images for dynamic PET by achieving superior noise reduction and lesion quantification compared to both supervised U-Net and self-supervised CycleGAN models. PG-SSDNet’s architecture enables effective denoising without requiring paired training data, thus improving the signal-to-noise ratio by leveraging the Patlak constraint to maintain accurate lesion quantification.