Patlak-PBNT: a simple population-based Patlak model to generate [⁶⁸Ga]Ga-PSMA-11 K_i parametric images for shortened total-body PET scan

Abstract

Purpose

The objective of this study is to generate reliable K_i parametric images from short-duration total-body PET scan for clinical applications using a simple population-based Patlak model.

Methods

We proposed a population-based Patlak model named Patlak-PBNT, which incorporates a population-based “normalized time” (PBNT) into the traditional Patlak plot. This model does not require long-duration PET scans to obtain an image-derived input function (IDIF) of full measured kinetics, thereby generating reliable K_i images from short-duration total-body PET scans. We evaluated the effectiveness of Patlak-PBNT based on 60-minute full-dynamic total-body [⁶⁸Ga]Ga-PSMA-11 PET data collected from 20 subjects PET scans. For comparison, the K_i images generated by the traditional Patlak plot (t* = 20 min post-injection) with measured IDIF was used as the gold standard. The differences between Patlak-PBNT-generated K_i images and the gold standard were evaluated at both the voxel and volume of interest (VOI) levels.

Results

Compared to the traditional Patlak, Patlak-PBNT effectively reduces PET scan duration while maintaining the quality of generated K_i images. For [⁶⁸Ga]Ga-PSMA-11, K_i images generated by Patlak-PBNT using only 40 minute dynamic PET images (20–60 min post-injection) show negligible differences compared to those generated by traditional Patlak with the same 40 minute dynamic PET images and a 60 minute full-duration IDIF, with a normalized mean squared error (NMSE) of 0.01, a Pearson correlation coefficient (Pearson’s r) of 0.99, and a peak signal-to-noise ratio (PSNR) of 75.27 dB. It is important to note that K_i images generated by Patlak-PBNT, using only 20 minute dynamic PET images (40–60 min post-injection), exhibit a high correlation with the gold standard in predefined VOIs, achieving a coefficient of determination (R²) of 0.92.

Conclusion

The proposed Patlak-PBNT model reduces the dependency on a complete input function, thereby avoiding the need for long-duration PET scans typically required to obtain a full input function. When utilizing dynamic PET images of identical scan durations (20–60 min post-injection), the K_i images generated by Patlak-PBNT and traditional Patlak are essentially identical. Furthermore, even when the scan duration is further reduced, the Patlak-PBNT method is capable of quantifying 20 minute dynamic [⁶⁸Ga]Ga-PSMA-11 total-body PET images.

Graphical abstract

Supplementary Information

The online version contains supplementary material available at 10.1186/s40658-026-00840-0.

Introduction

Prostate cancer (PCa) ranks first in incidence and second in mortality among male malignancies worldwide [1, 2]. The detection and localization of prostate cancer lesions are of great significance in guiding clinical diagnosis and treatment. Compared to structural imaging techniques such as computed tomography (CT) and magnetic resonance imaging (MRI), positron emission tomography (PET) offers a highly sensitive, non-invasive approach for in vivo investigation of human physiology, metabolism, and molecular pathways [3, 4]. Although [¹⁸F]F-FDG, commonly used in PET imaging, specifically labels high-metabolism tissues by mimicking glucose metabolism, it demonstrates significant diagnostic value in oncology, neurology, and cardiology. However, for slow-growing tumors, such as thyroid cancer, neuroendocrine tumors, and particularly prostate cancer, it shows markedly lower sensitivity [5]. Meanwhile, research on PSMA continues to grow, demonstrating excellent sensitivity and specificity, and has become a fundamental tool for staging prostate cancer [6–13]. More importantly, the integration of diagnosis and therapy is a key advantage that distinguishes PSMA from FDG. PSMA-targeting molecules can carry diagnostic radionuclides (such as [⁶⁸Ga]Ga-PSMA-11) for precise imaging, while also linking to therapeutic radionuclides (such as [¹⁷⁷Lu]Lu-PSMA-617) for targeted radionuclide therapy, thereby establishing a complete “diagnosis followed by therapy” treatment cycle [14–16].

The standardized uptake value (SUV) is the most commonly employed semi-quantitative metric for quantifying PSMA-PET in clinical practice [17]. However, SUV is influenced by various biological and technical factors, including injected dose, time between injection and scan, and differences in scanners’ spatial resolution [18]. Compared to semi-quantitative analysis, absolute quantification from dynamic PET is regarded as a more reliable method [17, 19–23]. The net uptake rate constant (K_i) derived from dynamic [⁶⁸Ga]Ga-PSMA-11 total-body PET images, provide absolute quantification of tracer uptake [14, 22, 24]. Moreover, the K_i parametric images have high sensitivity, significantly improved likelihood of accurate lesion detection, and could be instrumental for clinical imaging diagnosis and clinical trial design [20–23].

However, K_i parametric imaging requires long-duration full-dynamic PET scans (≥ 60 min) and continuous arterial blood sampling as the input function (IF), potentially causing patient discomfort and increasing the risk of motion-related artifacts in the images [14, 21, 24, 25]. Moreover, arterial blood sampling, regarded as the gold standard, is invasive and carries risks such as radiation exposure, infection, bleeding, and thrombosis [26, 27]. The image-derived IF (IDIF) is a common alternative to arterial blood sampling for deriving the complete IF from dynamic PET images during long-duration scans [28–30]. Notably, a complete IDIF for K_i parametric imaging cannot be derived from short-duration PET scans.

To generate reliable K_i images from short-duration PET scans, the population-based IF (PBIF) is commonly constructed to replace or supplement the incomplete IDIF [27, 31–35]. The PBIF is constructed based on the assumption that the shape of the IF is consistent across all subjects. The method for constructing a PBIF based on the mean IDIF involves first shifting individual blood time-activity curves (TACs) to align their peaks, then calibrating them using parameters (such as injected dose, body surface area, and body weight), and finally averaging the TACs across multiple subjects [27].

Compared to PBIF, the Population-based Normalized Time (PBNT) we previously proposed in the construction of a self-supervised neural network-based parametric imaging algorithm (SN-Patlak) is also an effective method for supplementing missing input function information [4]. The PBNT is constructed based on the phenomenon that the image-derived normalized time (IDNT) in the population exhibits an excellent linear relationship with the real time t (). In both PBNT and IDNT, “NT” refers to the “normalized time” in Patlak plot, defined as the ratio of the blood time integral to the instantaneous concentration. Notably, although in previous studies we explored the feasibility of using PBNT for parametric imaging based on short-duration FDG-PET scans, we did not provide a mathematical derivation for the construction of PBNT and conducted experiments solely based on [¹⁸F]F-FDG data [4]. Theoretically, PBNT is also applicable to parametric imaging for PSMA-PET [4, 17], but its specific feasibility has yet to be explored.

In this study, we derived a mathematical formulation for the construction of PBNT and proposed a simple population-based Patlak model, named Patlak-PBNT. This model reduces the reliance on long-duration PET scans to obtain a complete IDIF, thereby enabling the generation of reliable Patlak-based K_i parametric images from short-duration total-body PET scans. Experimental results demonstrate that our proposed Patlak-PBNT is robust and reliable for the quantification of 20-minute (40–60 min post-injection) dynamic [⁶⁸Ga]Ga-PSMA-11 total-body PET.

Materials and methods

Dynamic PET acquisition, reconstruction, and data preprocessing

Twenty patients with prostate cancer referred to Renji Hospital for [⁶⁸Ga]Ga-PSMA-11 PET/CT from December 2020 to April 2023 were retrospectively analyzed. [⁶⁸Ga]Ga-PSMA-11 was synthesized by Renji Nuclear Medicine Laboratory in accordance with good manufacturing practice quality requirements [36]. The injection dose of [⁶⁸Ga]Ga-PSMA-11 was 2.03 (1.85–2.18) MBq/kg. Low-dose attenuation CT scan (120 kV, 20 mA) was performed first, followed by a 60-minute dynamic total-body PET scan using a long-axial-field-of-view PET scanner (uEXPLORER, Shanghai United Imaging Healthcare Co. Ltd) after intravenous injection of the drug near the ankle, according to the method in the expert consensus [20]. To enhance patient comfort and minimize motion artifacts during the scan, patients were positioned using fixation devices (e.g., foam pads and straps) and instructed to remain as still as possible.

The dynamic PET images of 92 frames (30 × 2 s, 12 × 5 s, 6 × 10 s, 4 × 30 s, 25 × 60 s, 15 × 120 s) were reconstructed with radioactive decay, scatter, attenuation, and random correction using an ordered-subset expectation maximization (OSEM) algorithm (voxel size of 1.667 × 1.667 × 2.886 mm³ in x, y and z direction, 4 iterations, and 20 subsets) incorporating time-of-flight (TOF) and point spread function (PSF) modeling, and without post reconstruction smoothing [4].

The post-processing of the reconstructed dynamic PET data and all volumes of interest (VOIs) delineations were performed manually using PMOD 4.2 software (PMOD Technologies LLC, Zurich, Switzerland). The VOIs encompassed normal organs, blood pool (descending aorta), and lesions, which included primary prostate lesions (PT), lymph node metastases (LNM), and bone metastases (BM). Two experienced nuclear medicine physicians performed the evaluation and pathological lesion confirmation according to the guidelines [37]. The time-activity curves (TACs) of total-body dynamic PET show the average activity concentration of the delineated VOIs during the 60-min acquisition period.

Patlak plot and IDNT implementation

The Patlak plot was implemented as Eq. (1) [14, 22, 24]:

1

where is the tracer concentration in plasma, and is the tissue tracer concentration at time t. This equation is constructed by plotting the normalized tissue tracer concentration against the normalized time (NT), defined as . By performing a linear regression on the plotted points after the starting time t*, K_i is determined as the slope of the linear portion, and v_i as the y-intercept of the fitting line. This computational process requires the full-duration blood time-activity curve (TAC) and dynamic PET images acquired post the t*.

The TAC of the descending aorta, extracted from full dynamic PET images spanning 0–60 min, was utilized as the IDIF [17, 38–40] and employed to generate the image-derived NT (IDNT). Note that, the partial volume effects (PVE) on the image derived input function were minimized by incorporating PSF spatial resolution modeling in the PET reconstruction. Moreover, since is derived from the descending aorta, the PVE on both and are canceled out in the ratio . Consequently, the IDNT is quite independent of PET spatial resolution.

In this study, K_i images generated by the Patlak plot using IDNT with a t* of 20 min post-injection were chosen as the gold standard, denoted as K_i*.

Population-based Patlak plot and PBNT implementation

The PBNT was derived from the linear relationship between the IDNT and time (), based on population data. This linear relationship can be observed in previous research [4, 41] and elucidated through an analytical derivation. For , can be fitted using an exponential function [42]. Therefore, the integral of can be expressed by the following formula:

2

where is a constant representing the integral of from 0 to t*. Based on Eq. (2), the ratio can be expressed as:

3

Given that is a value close to zero, applying a Taylor expansion to the exponential function and substituting the first two terms of this expansion into the Eq. (3) reveals that the ratio exhibits a linear relationship with time , as shown below:

4

Based on the elucidated linear relationship, we conducted linear regression and statistical analysis on the IDNT across all data collected after 20 min post-injection. The statistical results are illustrated in Fig. 1, showcasing a high and uniform fitting quality in various participants by the coefficient of determination (R²) of 1.00 ± 0.00. After averaging these regression lines, it is feasible to model the integral-to-instantaneous blood concentration ratio as a linear function of time, characterized by a slope of 1.84 ± 0.20 and a y-intercept of 0.53 ± 5.22. The specific formula is as the following:

Fig. 1.

Linear relationship between normalized time and real time in [⁶⁸Ga]Ga-PSMA-11 PET. The dashed lines indicate the individual fitting results for each participant, while the solid line delineates the mean of these fitting results

5

By integrating Eqs. (1), (2), (3), (4) and (5), the population-based Patlak plot for [⁶⁸Ga]Ga-PSMA-11 is implemented as Eq. (6):

6

Evaluation setting

We employed normalized mean square error (NMSE), Pearson’s correlation coefficient (Pearson’s r), and the peak signal-to-noise ratio (PSNR) to evaluate the K_i images generated by Patlak with IDNT (Patlak-IDNT) and PBNT (Patlak-PBNT). Notably, PSNR is a common metric used to quantify the noise level in the images. PSNR is computed from the mean squared error between the evaluated K_i image and the gold-standard K_i* image (Patlak plot, t*=20 min), with higher PSNR indicating lower noise and better image fidelity [22]. Scatter plots and Bland-Altman plots were utilized to analyze the correlation and discrepancies between the gold standard and the K_i images, which were generated from different duration PET scans, based on previously defined VOIs.

Results

Figure 2 presents an example of K_i images generated by Patlak-IDNT and Patlak-PBNT with varying scan durations: 40 min (20–60 min post-injection), 30 min (30–60 min post-injection), 20 min (40–60 min post-injection), and 10 min (50–60 min post-injection). When the scan duration is 40 min, the K_i images generated by Patlak-IDNT and Patlak-PBNT are virtually identical, with both clearly revealing the lesions. As the scan duration decreases from 40 min to 10 min, the K_i images generated by both methods exhibit consistent changes, including a progressive increase in noise and a reduction in lesion visibility. Especially when the scan duration is reduced to 20 min, the primary tumor can still be discerned in the fusion images (comprising CT and K_i images generated by Patlak-PBNT), but when further reduced to 10 min, this tumor becomes obscured by noise in the K_i images.

Fig. 2.

Comparison of K_i images generated by Patlak-IDNT and Patlak-PBNT. The upper panel displays the maximum intensity projection (MIP) of K_i images (ml/g/min) derived from Patlak-IDNT and Patlak-PBNT across various scan durations, with red arrows highlighting the primary tumor. The lower panel presents axial fusion diagrams that combine CT and K_i images across different scan durations, with red arrows highlighting the same primary tumor. Notably, the apparent increase in K_i values with shorter scan durations, derived from two methods, can be attributed to the use of MIP images, which inherently capture all upward noise biases by selecting maximum values

Figure 3 presents the quantitative evaluation results based on total-body K_i images from all 20 subjects. When the scan duration is 40 min, the K_i images generated by Patlak-PBNT closely align with the gold standard, those generated by Patlak-IDNT. This is demonstrated by a low average NMSE of 0.01, indicating minimal deviation from the standard; a high Pearson’s r of 0.99, reflecting a strong correlation with the standard; and a PSNR of 75.27 dB, signifying excellent image quality with low noise.

Fig. 3.

Variations in the accuracy of K_i images generated from dynamic PET scans of different durations using the Patlak-IDNT and Patlak-PBNT. Specifically, A, B, and C represent NMSE, Pearson’s r, and PSNR, respectively. Since two identical images would result in an infinite PSNR value, the PSNR for K_i images generated by the Patlak-IDNT method with a 40-minute scan duration is not displayed in panel C

Consistent with observations from Figs. 2 and 3 also demonstrates that the accuracy of the K_i images generated by both methods decreases as the scan duration is reduced. Moreover, the sharpest decline in accuracy for both methods occurs when the scan duration is reduced from 20 min to 10 min. Overall, based on the results in Figs. 2 and 3, there are no obvious differences between the K_i images generated by the two methods. As the scan time window is shortened, the accuracy of the K_i images generated by Patlak-PBNT decreases slightly, but observable (Fig. 3).

Figures 4 and 5 present detailed quantitative comparisons based on VOIs of the K_i images generated by Patlak-IDNT and Patlak-PBNT with varying scan durations. The K_i images generated by Patlak-PBNT (20–60 min post-injection) exhibit a strong correlation with the gold standard K_i*, in major organs and lesions, as demonstrated by the regression equation K_i* = 0.96K_i(Patlak-PBNT) + 0.00, with an R² of 0.98 (Fig. 4B₁). As the scan duration decreases, the R² values for both methods progressively decline, especially when the duration is reduced from 20 min (40–60 min post-injection) to 10 min (50–60 min post-injection) with the R² for Patlak-IDNT dropping from 0.93 to 0.74 and for Patlak-PBNT from 0.92 to 0.77. The results of the difference analysis, as depicted in Fig. 5, also show that the bias between K_i values derived from the two methods and the gold standard increases as the scan duration decreases. When the scan duration is reduced from 20 to 10 min in the later phase, the mean difference line for Patlak-PBNT increases from well below 0.01 to reaching 0.01, and the 95% limits of agreement double.

Fig. 4.

Scatter plots comparing mean values within delineated VOIs from K_i images (ml/g/min) to the gold standard (K_i*), derived from Patlak-IDNT with t* set at 20 min. Subplots A (Patlak-IDNT comparison) and B (Patlak-PBNT comparison) show the comparisons over various scan durations

Fig. 5.

Bland-Altman plots comparing mean values within delineated VOIs from K_iimages (ml/g/min) to the gold standard (K_i*), derived from Patlak-IDNT with t* set at 20 min. Subplots A (Patlak-IDNT comparison) and B (Patlak-PBNT comparison) show the comparisons over various scan durations

Discussion

This study proposed a simple population-based Patlak model (Patlak-PBNT) for K_i parametric imaging. The evaluation results indicate that, compared to the conventional Patlak (Patlak-IDNT), Patlak-PBNT effectively reduces PET scan duration while maintaining the quality of generated K_i images. For [⁶⁸Ga]Ga-PSMA-11, K_i images generated by Patlak-PBNT just using 40 minute dynamic PET images (20–60 min post-injection) show almost no difference from those generated by Patlak-IDNT using the same 40 minute dynamic PET images with a 60-minute full-duration IDIF. Importantly, Patlak-PBNT can generate reliable K_i images using only 20 minute dynamic PET images (40–60 min post-injection). This reduction in scan duration has important clinical implications. From the patient’s perspective, shortended scan duratuation is likely to reduce discomfort and body motion, resulting in higher image quality. For the hospital, this improvement in patient throughput allows for more efficient use of equipment and resources, enabling a greater number of patients to be scanned in the same amount of time. This could ultimately enhance the overall efficiency of clinical workflows in hospital, improve comfortability and reduce financial burden for patients.

The proposed Patlak-PBNT can also be used for v_i parametric imaging from short-duration dynamic PET images (Figure S1). It can be observed that as the scan duration decreases, the accuracy of v_i images declines more rapidly. This is evidenced by the average Pearson’s r for v_i images generated by Patlak-PBNT falling around 0.62 when the duration is reduced to 20 min. Additionally, the composition of v_i images is inherently complex [24], as a detailed analysis of v_i images was not conducted in this study.

In Patlak-PBNT, we replaced the IDNT of the traditional Patlak method with a linear PBNT, thus avoiding the need for longer-duration PET scans to obtain a complete IDIF. According to Eq. (3), PBNT can be derived in the form of an exponential function: . Based on the [⁶⁸Ga]Ga-PSMA-11 PET dataset in this study, the fitted average values for parameters ‘a’ and ‘b’ are 102.87 and 0.01, respectively (Fig. 6).

Fig. 6.

Scatter plots illustrating the exponential relationship between NT and time, based on the data from a cohort of 20 patients. The dashed lines indicate the individual fitting results for each participant. The dashed lines indicate the individual fitting results for each participant, while the solid line delineates the mean of these fitting results

In this study, PBNT was constructed using IDIF extracted from the descending aorta. It is important to note that PBNT could also be constructed based on IDIF from other blood pools, such as the ascending aorta, left ventricle, and right ventricle. The consistency of IDIF across these regions has been demonstrated in previous research [17]. Additionally, when performing K_i parametric imaging with data obtained from short-axial-field-of-view PET scanners, PBNT can also be constructed using IDNT from the common carotid artery, abdominal aorta, and iliac artery [43].

Based on the experiments in this study and our previous work (SN-Patlak method) [4], the proposed PBNT should also be applicable to other tracers with irreversible kinetics (for which the conventional Patlak plot is applicable for quantification), including [¹⁸F]FDG. Therefore, a careful validation procedure is suggested to use Patlak-PBNT for other irreversible tracers. Moreover, based on the experimental results from SN-Patlak, controlling noise in the K_i images generated by Patlak-PBNT can further reduce the required scanning time. For example, combining PBNT with spatially constrained parametric imaging methods [44, 45], or employing denoising post-processing on K_i images generated by Patlak-PBNT [31], could also involve using machine learning algorithms (random forests [46] or neural networks [4, 47]) to control noise.

It should be noted that a key theoretical basis of this method is that the input function () can be fitted by an exponential function for irreversible tissue tracer kinetics. Although we have validated this property for the Patlak Plot-suitable tracers [⁶⁸Ga]Ga-PSMA-11 (Fig. 1) and [¹⁸F]FDG [4], there may still be cases in which the input function cannot be well fitted by an exponential function. Therefore, when applying the proposed Patlak-PBNT method to new tracers, simple validation with a small dataset is necessary. As an example, the early phase (0–30 min) after [¹¹C]methionine injection is suitable for quantitative analysis using the Patlak plot method [48]. Based on our observations, its input function () in the early phase can be fitted with an exponential function, and therefore Patlak-PBNT method is only suitable for this early-phase quantitative analysis using Patlak plot.

In future work, we will further evaluate the clinical value of K_i images generated by Patlak-PBNT. Additionally, we will validate this approach on PET data from other tracers and incorporate the previously mentioned denoising techniques to further reduce the scan duration required for K_i parametric imaging.

Conclusion

The proposed Patlak-PBNT model reduces the dependency on a complete input function, thereby avoiding the need for long-duration PET scans typically required to obtain a full input function. When utilizing dynamic PET images of identical scan durations (20–60 min post-injection), the K_i images generated by Patlak-PBNT and traditional Patlak are essentially identical. Furthermore, even when the scan duration is further reduced, the Patlak-PBNT method is capable of quantifying 20-minute dynamic [⁶⁸Ga]Ga-PSMA-11 total-body PET images. This model significantly reduces scanning time, enhances patient comfort, and increases clinical throughput, thereby expanding its potential for broader clinical applications.

Supplementary Information

Below is the link to the electronic supplementary material.

Authors’ contributions

Study conception and design: JG, GL, SX and YZ. Material preparation and data collection: LL, WG and WY. Data analysis: LL, WG, WW, JL, GH, SX, GL and YZ. Drafting of the manuscript: LL, WG and WW. All authors commented on previous versions of the manuscript and read and approved the final manuscript.

Data availability

Data are available on request to the corresponding author.

Declarations

Ethics approval and consent to participate

The study was conducted in accordance with the Declaration of Helsinki (as revised in 2013). The study was approved by ethics board of Renji Hospital, School of Medicine, Shanghai Jiao Tong University (approval number: 2018 − 104), and informed consent was taken from all the patients. The authors are accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Consent for publication

All participants have provided written consents to the publication of the data.

Competing interests

All authors have completed the ICMJE uniform disclosure form. The authors have no conflicts of interest to declare.

This work was partially supported by the National Natural Science Foundation of China under Grant 62272135, 62372135, 82127807, and 62001144; the construction project of Shanghai Key Laboratory of Molecular Imaging(18DZ2260400); the National Key Research and Development Program of China (2020YFA0909000).