Abstract
Purpose
This study aims to develop and validate a dual-time-window (DTW) Patlak plot method that eliminates the need for invasive blood sampling and reduces scan duration. We seek to improve the accuracy of the net influx constant (
) estimation, addressing the inaccuracies inherent in traditional DTW and single-time-window methods, which often introduce bias and hinder comparability across different cohorts.
Method
We developed an unsupervised, multi-branch neural network (NN) to assist in estimating missing data intervals within the DTW protocol, thereby facilitating accurate Patlak analysis. The model fits the mapping from time to the time-activity curve (TAC), generating multiple pseudo input functions (IFs). A correlation coefficient is then computed between each pseudo IF and the voxel-level measured data, extracting statistical information guided by the kinetic process. These correlation scores were used to construct a weighted statistic, serving as the final IF (NNIF). Our approach was validated using both simulation and clinical data, including 
-FDG PET scans from 67 lung cancer subjects. Additionally, we compared the performance of our method with other simplified quantification techniques to demonstrate its efficacy in achieving high-quality parametric imaging and reliable quantitative analysis within abbreviated scanning protocols.
Result
Our proposed method achieved high accuracy in the estimation of IF, with a maximum mean absolute deviation (MAD) of 0.04 in a real patient study. The regressed 
derived from different DTW scan protocols exhibited good consistency. In simulation studies , the best relative absolute error (RAE) was 0.0302. In real patient study, the optimal average peak signal-to-noise ratio (PSNR) of parametric imaging reached 97.40 dB, while the best average R-squared (
) in ROI-based quantitative analysis reached 0.991.
Conclusions
We demonstrate the feasibility of using a weighted statistic, constructed from a multi-branch neural network, to accurately estimate the complete IF. This approach enables the generation of high-quality parametric images with shortened scan protocols, effectively reducing scanning time while ensuring accurate Patlak analysis.
Keywords: Dynamic PET, Patlak plot, Dual-window acquisition, Neural network, Unsupervised learning
Introduction
Patlak analysis [1, 2] is a widely utilized graphical method in dynamic positron emission tomography (PET) imaging [3] that quantifies the 
of tracers into tissue. The net influx constant (
) is clinically valuable due to its enhanced specificity in assessing metabolic processes [4–6]. Implementing Patlak analysis necessitates the accurate determination of the arterial IF, which traditionally relies on invasive blood sampling [7]. This method poses significant discomfort and inconvenience to patients, limiting its widespread clinical application. To mitigate these challenges, the image-derived input function (IDIF) has emerged as a non-invasive alternative, leveraging imaging data to estimate the arterial tracer concentration [8, 9]. However, a complete IF is time-consuming and negatively impacts patient throughput.
Flow Motion technique [10, 11] and population-based input functions (PBIFs) have been developed to eliminate the need for continuous blood pool scanning. Notable PBIF approaches include scaled PBIF (sPBIF) [12], dual-time-window PBIF (dPBIF) [13], and hybrid IF [14, 15]. A further option is the relative Patlak plot, which fits a subject-specific late-time IF tail and thus avoids population priors. However, it still requires an additional measurement to recover the global scale factor 
, which may vary depending on the tracer used and the disease type [16, 17]. Among these, the DTW Patlak plot stands out for its ability to capture both the initial peak and the later equilibrium of the plasma input function, effectively balancing scan efficiency with quantitative precision. However, traditional DTW methods rely on nonlinear fitting with rational functions to interpolate missing data [15], which may not accurately represent personalized plasma input functions. This can introduce biases that propagate into Patlak analyses, resulting in kinetic parameter estimates that deviate from true values. Such deviations undermine the comparability of 
measurements across different patient cohorts and reduce the interpretability of results compared to more straightforward metrics like the standardized uptake value (SUV). Additionally, PBIFs may fail to account for individual physiological variations, further compromising the accuracy of 
estimations. These limitations highlight the need for improved PBIF methodologies that preserve 
accuracy while reducing scan durations and maintaining non-invasive procedures.
Recent machine-learning studies have further illustrated the feasibility of non-invasive input-function recovery. For example, a convolutional convolutional neural network (CNN) trained on small-animal -FDG sequences reproduced arterial curves and preserved Patlak accuracy, but required image-derived reference labels for supervision [18]. Another approach employed a long short-term memory (LSTM) framework with kinetic-model–regularized loss, trained on total-body brain -FDG data using the ascending aorta as reference [19]. A third method combined depth-wise separable convolutions with an arterial-curve template to simultaneously predict whole-blood and metabolite-corrected 11C-PBR28 functions, though its performance relied heavily on tracer-specific shape priors [20]. Taken together, these approaches confirm that deep neural networks, when guided by substantial prior information and labelled arterial curves, can yield accurate IFs. By dispensing with external labels and tracer-specific templates, the present unsupervised dual-time-window strategy seeks to achieve comparable accuracy while maximizing applicability.
To address these challenges, we propose an unsupervised, robust IF estimation method that eliminates the need for prior population information and introduces an optimal DTW scan protocol for Patlak analysis, effectively shortening scan duration. Our approach employs a neural network-assisted sampling method based on an abbreviated DTW protocol, comprising an early short scan and a longer scan post-equilibrium. A multi-branch neural network (NN) generates diverse pseudo IFs from the incomplete DTW IF, and a nonlinear estimator constructs the weighted IF. The main innovations of this work are twofold: first, we theoretically derive how sampling bias in DTW can disrupt the linear assumption of the Patlak plot, leading to inaccurate kinetic parameter estimates; second, the multi-branch network architecture enhances IF estimation by capturing a wide range of pseudo IF variations, enhancing the robustness and accuracy of IF estimation. Unlike previous data-driven PBIF or deep-learning methods that rely on training across separate subjects, our network is trained for each subject. All voxel-level TACs act as internal constraints, enabling IF to be fully individualized without the need for an external database or population prior.
Materials and methods
Patlak analysis in dynamic PET
In the Patlak plot, the tracer concentration in the tissue, denoted as 
, is modeled as the sum of two components: the product of fractional blood volume (
) and plasma concentration (
), and an irreversible trapping term, which represents the accumulation of tracer over time:
 |
1 |
Here, 
is the net FDG influx rate, and 
represents the equilibration time after which the linear relationship holds. The dynamic process of tracer uptake is thus simplified into a linear regression model of the form: 
, where 
, 
.
Neural network-assisted IF estimation
As illustrated in Fig. 1, after DTW acquisition, time points (t) were fed into a multi-branch NN to fit incomplete IF measurements. Each branch is randomly initialized and operates with independent parameters to generate diverse pseudo IF values 
, which were further used to construct NNIF and to perform integration in Patlak analysis. We define 
to be 
, assuming that the integral of 
includes a bias term 
from 
. The bias propagated to X from 
is denoted as 
, i.e., 
. According to the Cauchy-Schwarz inequality in Eq. 4, sampling bias in DTW disrupts the linear relationship between Y and 
, causing the Pearson correlation coefficient 
to fall below 1 and thereby violating the linear assumption of the Patlak plot.
 |
2 |
 |
3 |
 |
4 |
To mitigate this issue, we construct an estimator from randomly sampled pseudo IFs by evaluating the linear correlation between Y and 
, and then assigning weights to different branches based on their correlation, defined as:
 |
5 |
where the dynamic image contains N voxels, where 
represents the left-hand-side dependent variable from the k-th voxel, 
(set to 
) avoids singularities when 
. To construct the statistic precisely, 
is the weight for i-th branch, and is computed according to the correlation coefficients of multiple 
s and 
derived from i-th branch, which involving performing M shuffles on the N voxels, with each branch randomly assigned to the shuffled voxels covering 
voxels on average.
Fig. 1.
The flow chart of proposed method: from incomplete IF to neural network-Assisted input function (NNIF). (1) Dual-time-window PET acquisition: collects an early and late windows data of plasma TAC 
and tissue TACs 
. (2) Multi-branch fitting: an 
-branch neural network is trained to fit the incomplete 
, generating pseudo IFs 
. (3) Random uniform allocation for each branch: voxel-level tissue curves 
are randomly distributed to these branches. (4) Correlation-based branch scoring: iterative Pearson-correlation updates and normalizes the branch weights 
for fusing the pseudo IFs into a subject-specific NNIF. (5) Patlak regression & 
estimation: the NNIF drives Patlak analysis to obtain the 
for voxels or ROI. EQM = equilibrium.
The final estimator 
is given by:
 |
6 |
Each candidate branch is weighted proportionally to its Pearson correlation coefficient with the voxel-wise TACs. Branches with large estimation errors tend to have low 
values, and thus receive small weights. As a result, their influence on the fused input function (NNIF) and consequently on the Patlak regression is negligible. 
is then used in the integral step of further Patlak analysis, employing numerical integration with a mesh size matching the smallest frame in the scan protocol.
Implementation details
In this study, each branch of the multi-branch NN is structured identically, comprising three layers with 16 neurons each and utilizing a Sigmoid activation function. The final layer consists of a single neuron for dimensional adjustment. Each branch takes time (in minutes) as input and outputs a pseudo IF value. The number of branches (n) is set to 2048, ensuring a sufficient number of pseudo IF samples while keeping GPU memory usage and runtime within acceptable limits. Given that computing resources vary across medical centers, this parameter should be viewed as a practical trade-off rather than a universal optimum. During the training phase, the neural network is optimized using the Adam optimizer with an initial learning rate of 0.01 and no decay, running for a maximum of 1000 iterations. Training was conducted on an AMD EPYC 9754 CPUs and NVIDIA RTX L40s GPUs, utilizing PyTorch version 2.1.0.
Dataset
Simulation study
Simulations of TACs were developed to evaluate the bias, measured through relative absolute error (RAE), and the standard deviation (SD) of Ki estimates derived from various methods. 
was simulated using typical kinetic parameters from lesions (
) [21] based on Eq. 1. The simulation model for IF is defined as follows:
 |
7 |
The PBIF method requires population information. We generate 5000 realizations of IF using random parameters, where the parameters were randomly generated with uniform distribution in the following range: 
, 
, 
, 
, 
, 
, and 
[22] Correspondingly, 5000 noise realizations of were generated. For the proposed NNIF method, which does not require population information, we used a set of typical plasma parameters from preceding range for generating IF: 
, 
, 
, 
, 
, 
, and 
[23], and conduct 5000 noise realizations according to this IF.
The noise samples were generated based on time-varying Gaussian noise model [24] with four noise levels (0.05, 0.1, 0.2, 0.3) to match different statistical characteristics. The 65 min framing protocol used for simulation consisted of: 
s, 
, 
s, 
s, 
s. The realizations are shown in Fig. 2.
Fig. 2.
Representative Time-Activity Curves (TACs) generated from the simulation study. Left: Ten realizations of the input function (IF). Middle: IF realizations alongside their corresponding noisy tissue TACs (
) at a noise level of 0.3. Right: Patlak linear regression analysis performed on a noisy realization when equilibration time (
) was set as 35 min
Real patient study
This study includes 
-FDG PET data of 67 subjects (Sex:38 females, 39 males; Weight: 63.18 ± 10.82 kg; Inject dose: 285.50 ± 50.57 MBq). The framing protocol involved 
s, 
, 
s, 
s, 
s. The slice thickness was 2.97 mm and reconstructed as a matrix of 
. The complete IDIF was derived from the descending aorta. The discrepancy between blood and plasma uptake was not considered. ROI-based analysis was performed based on 28 pathologically verified lesions. In order to assess the reliability of the constructed statistic, we use various acquisition protocols: early windows were set as 0-3, 0-6, and 0-9 min, and late windows were set as 35-65, 40-65, 45-65, and 50-65 min, respectively.
Evaluation metrics
We used 65 mins dynamic scan as the ground truth reference. The evaluation was conducted using three key metrics: (1) the distribution of bias between the estimated complementary TACs and the complete IFs to assess the accuracy of the IF estimations; second, (2) the PSNR to evaluate the quality of the 
parametric images, ensuring that the kinetic parameter mappings were precise and reliable; (3) the R-squared (
) value to measure the accuracy of the linear regression in region-of-interest (ROI)-based quantitative analyses, thereby determining the fidelity of the Patlak plot assumptions.
During evaluation, different acquisition protocols were applied for PBIF-based reference methods in both simulation and real patient studies. For dPBIF method [13], the midpoints of the early and late windows were selected from the 20–25 min and 60–65 min intervals, respectively. In the sPBIF approach, the scaling factor was determined by calculating the area under the curve (AUC) of the IF from 50–65 min, which was found to achieve the best accuracy in ROI-based analyses [12]. Furthermore, the sPBIF was employed as the population-based information for 
in the hybrid IF method [14, 15].
Result
We validated our method using both simulations and real patient studies, comparing it with several existing simplified quantification methods. Parametric images and quantitative analyses demonstrate reliable accuracy compared to ground truth derived from complete IFs.
Simulation study
As shown in Fig. 3, boxplots summarize the estimated 
using various methods. The ground truth 
is marked as a dashed gray line for comparison. HybridIF (RAE: 0.0371
0.0912, SD: 0.0290
0.0715) outperforms sPBIF (RAE: 0.2386
0.2583, SD: 0.1536
0.1791) and dPBIF (RAE: 0.5313, SD: 0.3791) in both accuracy and variability across various equilibration times (
). However, hybrid IF exhibits more and wider outliers compared to the NNIF methods. NNIF methods achieve the best performance with high accuracy low variability: 
(RAE: 0.0314
0.0927, SD: 0.0239
0.0702), 
(RAE: 0.0301
0.0936, SD: 0.0230
0.0712), and 
(RAE: 0.0302
0.0932, SD: 0.0230
0.0709).
Fig. 3.
Estimated 
for simulation data across different methods at a noise level of 0.3 [21]. Blue line within each box represents the median value, while the upper and lower edges of the box correspond to the 75th and 25th percentiles, respectively. Whiskers extend from the box to show the range of the data, up to 1.5 times the interquartile range. Red dots indicate outliers that fall outside this range. The mean RAE of the estimated 
is marked above the blue median line
Real patient study
Bias analysis of estimated NNIFs
We analysed the error distribution of each frame with the maximum mean absolute deviation (MAD) and median to explore which window provides the best control over bias. As shown in Fig. 4, when comparing protocols with the same early or late window, increasing the length of either early or late window consistently improves the accuracy of the estimated NNIFs and reduces the bias range, median error, and mean error. This indicates that a longer scan provides more reliable estimation.
Fig. 4.
Bias analysis of estimated NNIFs on pseudo duration across various DTW scan protocols
When the early window was fixed at 0–3 min and the late window was gradually extended, the maximum MAD remained consistently greater than 0.1 (
= 35 min: MAD = 0.11; 
= 40 min: MAD = 0.12; 
= 45 min, MAD = 0.13; 
= 50 min, MAD = 0.16). In contrast, when the late window was fixed, for example, at 35-65 min, and the early window was extended from 6 min to 9 min, the the maximum MAD significantly decreased from 0.11 to 0.04. Similar results were observed in other DTW protocols. Although shorter early and late window led to bias, these bias derived from proposed method are generally tolerable and the method remains robust, as will be discussed further in the ROI-based analysis (Sec.3.2).
Parametric imaging
Figure 5 shows the Patlak 
images generated using NNIFs with various DTW scan protocol for a representative subject with lung cancer. The transverse slices illustrate the high qualitative resemblance with the images derived using complete IDIF (79.03 - 97.40 dB). Parametric images generated using NNIFs provided excellent image quality with high PSNR, as shown in Table 1.
Fig. 5.
images derived from complete IF and NNIFs
Table 1.
The PSNR mean and SD comparing with respective reference across various 
 |
35-65 min | 40-65 min | 45-65 min | 50-65 min |
|---|---|---|---|---|
| IF | ||||
| dPBIF | 57.38(3.68) | 56.03(3.69) | 54.47(3.54) | 51.69(3.19) |
| sPBIF | 61.69(6.81) | 61.19(6.24) | 60.99(6.10) | 60.13(6.18) |
| hybridIF | 47.04(2.80) | 47.14(2.80) | 47.57(2.76) | 48.82(2.76) |
 |
87.62(10.14) | 86.40(10.07) | 83.54(8.89) | 79.40(8.73) |
 |
94.19(8.61) | 90.29(9.80) | 84.89(8.37) | 79.03(7.20) |
 |
97.40(9.26) | 95.19(9.53) | 92.99(9.41) | 88.43(9.39) |
The midpoints of early and late window of dPBIF were selected from the 20-25 and 60-65 min, respectively. The scaling factor for sPBIF is determined by the IF from 50-65 min, achieving optimal accuracy in ROI-based analysis, and the population-based information for the hybridIF is derived from 4-50 min interval
Quantitative analysis of ROI-based 
As shown in Fig. 6, we compared the accuracy of estimated 
among the proposed method and other PBIFs across various DTW protocols. Considering all early windows, when 
= 35 and 40 min, the linear correlation between 
generated by NNIFs and the reference values exceeds 0.98, indicating robustness and high consistency across early window. As shown in Fig. 4, although some bias was introduced during estimating, it did not appear to significantly propagate to the ROI-based 
analysis, indicating that the errors in estimating the IF are still tolerable. The extension of the early window did not significantly improve the accuracy of parameter estimation or the linear correlation. When the late window was shortened, regression slope and 
exhibited a significant downward trend, the mean deviation in the slope increased from 0.037 to 0.093, and mean 
decreased from 0.984 to 0.936, this suggests that the last window is more important in the accuracy of 
estimation.
Fig. 6.
ROI-based 
analysis, where 
represents the 
reference values derived from the complete IF. The midpoints of early and late window of dPBIF were selected from the 20-25 and 60-65 min, respectively. The scaling factor for sPBIF is determined by the IF from 50-65 min, achieving optimal accuracy in ROI-based analysis, and the population-based information for the hybridIF is derived from 4-50 min interval
Discussion
The simulation and real patient studies demonstrate that 
estimates from the NNIF method consistently outperform those from PBIF methods applied to the same data. As shown in Table 1, all 12 DTW protocols achieved PSNR values exceeding 79 dB, indicating high reliability. Similar to previous studies [13–15, 25], the duration of the early and late windows plays different roles in the accuracy of 
estimation. As shown in Sec.3.2, we observed that errors introduced by shortening early scans do not significantly impact Patlak regression accuracy, whereas later scans have a greater effect on the results’ accuracy. From the perspective of shortening scanning duration, Fig. 4 indicates that selecting an early window of 0–3 min and a late window of 40–65 min offers the most cost-effective solution, with biases introduced by the proposed method generally within 1%, evidenced by in our ROI-based analysis.
Moreover, the proposed method operates in an unsupervised manner without requiring population information, which is particularly crucial for multi-center applications. In practice, data collected across different centers often exhibit significant shifts, making it difficult for new centers to transfer existing population data for dynamic PET studies. Our method addresses these challenges by constructing an accurate statistic through the NNIF approach. The 2048 branches strike a balance between computational cost and experimental accuracy. Normally, the number of voxels is always much larger than the number of branches. Therefore, by utilizing a large volume of voxel data to select sampled pseudo IFs, we effectively evaluate the deviation of each sampled pseudo IF under the constraints of the underlying dynamic regression process, thereby performing accurate weighting.
Currently, the main limitation of using a DTW is the need for repeated segmentation of the descending aorta region at both two windows. This indicates that future work should incorporate more advanced automatic segment algorithms [26] for clinical convenience. Fig. 4 shows that the regions where the IFs estimate deviates the most are typically in the middle segment of the missed duration, which corresponds to the frames farthest from the DTW. Currently, we do not have a specific method to control the deviation of all missed frames within a consistent range. Future research in this direction will allow us to obtain more accurate estimates.
Conclusion
This study proposes an approach for dynamic PET Patlak analysis by utilizing a multi-branch NN to estimate the complete IF from incomplete data obtained through DTW protocols. By leveraging a weighted statistic derived from the dynamic process of the Patlak model, we demonstrated that the NN-based method effectively reconstructs the missing IF data with high accuracy. Crucially, the abbreviated scan protocol not only shortens the total scanning time but also preserves the precision and reliability of the resulting parametric images and quantitative analysis. This advancement holds significant potential for improving the efficiency and accessibility of dynamic PET imaging in clinical practice. Overall, this work offers a robust and practical solution for enhancing Patlak plot analysis, making dynamic PET imaging both faster and more accurate.
Acknowledgements
Not applicable.
Author contributions
All authors contributed significantly to the manuscript, and their individual contributions are summarized as follows: Wenrui Shao designed and performed the experiments, analyzed the data, and wrote the manuscript. Yinxin Chen provided segmentation support for ROI-based analysis. Fangxiao Cheng offered critical feedback during manuscript preparation. Fen Du and Ying Liang provided valuable insights during manuscript preparation. Xiangxi Meng finalized the interpretation of the results and made substantial revisions to the manuscript. Zhaoheng Xie conceived the original idea, guided the overall research framework. Wenrui Shao and Yarong Zhang are co-first authors; Ying Liang and Zhaoheng Xie are corresponding authors.
Funding
This work was supported in part by the Natural Science Foundation of China under Grant 62394311 and Grant 62394310; and in part by Beijing Natural Science Foundation under Grant Z210008. The work of Zhaoheng Xie was supported by the Start-Up Funds of Peking University Health Science Center. The work of Zhaoheng Xie and Xiangxi Meng was supported by the 2024 China Industrial Technology Infrastructure Public Service Platform Project (GN2024-31-4700). The work of Xiangxi Meng was supported by the Clinical Medicine Plus X - Young Scholars Project, Peking University.
Data availability
The datasets used or analyzed during the current study are available from the corresponding author on reasonable request. The code has been made publicly available at https://github.com/PKU-MIPET/Network-Aided-Unsupervised-Input-Function-Estimation.git
Declarations
Ethics approval and consent to participate
This prospective study was approved by the ethics committee of Cancer Hospital & Shenzhen Hospital, Chinese Academy of Medical Sciences (Clinical Trial Number: KYLH2022-1), which followed the 1964 Helsinki Declaration ethical standards and its subsequent amendments. All patients were provided written informed consent.
Consent for publication
Not applicable.
Competing interests
No potential Conflict of interest relevant to this article exist.
Footnotes
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Wenrui Shao and Yarong Zhang contributed equally to this work.
Contributor Information
Ying Liang, Email: liangying_473@163.com.
Zhaoheng Xie, Email: xiezhaoheng@pku.edu.cn.
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Associated Data
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Data Availability Statement
The datasets used or analyzed during the current study are available from the corresponding author on reasonable request. The code has been made publicly available at https://github.com/PKU-MIPET/Network-Aided-Unsupervised-Input-Function-Estimation.git