A prior information-based multi-population multi-objective optimization for estimating 18F-FDG PET/CT pharmacokinetics of hepatocellular carcinoma

Abstract

Background

¹⁸F fluoro-D-glucose (¹⁸F-FDG) positron emission tomography/computed tomography (PET/CT) pharmacokinetics is an approach for efficiently quantifying perfusion and metabolic processes in the liver, but the conventional single-individual optimization algorithms and single-population optimization algorithms have difficulty obtaining reasonable physiological characteristics from estimated parameters. A prior-based multi-population multi-objective optimization (p-MPMOO) approach using two sub-populations based on two categories of prior information was preliminarily proposed for estimating the ¹⁸F-FDG PET/CT pharmacokinetics of patients with hepatocellular carcinoma.

Methods

PET data from 24 hepatocellular carcinoma (HCC) tumors of 5-min dynamic PET/CT supplemented with 1-min static PET at 60 min were prospectively collected. A reversible double-input three-compartment model and kinetic parameters (K₁, k₂, k₃, k₄, f_a, and \(\:{v}_{b}\)) were used to quantify the metabolic information. The single-individual Levenberg–Marquardt (LM) algorithm, single-population algorithms (Particle Swarm Optimization (PSO), Differential Evolution (DE), and Genetic Algorithm (GA)) and p-MPMO optimization algorithms (p-MPMOPSO, p-MPMODE, and p-MPMOGA) were used to estimate the parameters.

Results

The areas under the curve (AUCs) of the three p-MPMO methods were significantly higher than other methods in K₁ and k₄ (P < 0.05 in the DeLong test) and the single population optimization in k₂ and k₃ (P < 0.05), and did not differ from other methods in f_a and v_b (P > 0.05). Compared with single-population optimization, the three p-MPMO methods improved the significant differences between K₁, k₂, k₃, and k₄. The p-MPMOPSO showed significant differences (P < 0.05) in the parameter estimation of k₂, k₃, k₄, and f_a. The p-MPMODE is implemented on K₁, k₂, k₃, k₄, and f_a; The p-MPMOGA does it on all six parameters.

Conclusions

The p-MPMOO approach proposed in this paper performs well for distinguishing HCC tumors from normal liver tissue.

Introduction

Dynamic 18F-FDG PET/CT has been widely applied in clinical studies for liver cancer diagnosis. Using tracer kinetic modeling (TKM) of time-activity curve (TAC) data, the HCC can be diagnosed according to quantified physiological information [1].

Pharmacokinetic-based compartmental modeling stands as a widely utilized approach for acquiring quantitative information on tracer metabolism within biological tissues [2, 3]. Sokoloff et al. [4] proposed an irreversible three-compartment model with the parameters \(\:{K}_{1}\), \(\:{k}_{2}\) and \(\:{k}_{3}\). Graham et al. [5] investigated the effect of the parameter \(\:{k}_{4}\), and the results indicated that neglecting the dephosphorylation process led to modeling bias. Given the dual blood supply characteristics of the liver, utilizing the dual blood input from the hepatic artery and portal vein has emerged as the optimal approach for hepatic kinetic modeling [6, 7]. In addition, image-derived blood input acquisition has replaced blood sampling as a result of its noninvasiveness [8].

Simultaneous extraction of plasma TAC and tissue TAC data serves as the basis for kinetic modeling, allowing the derivation of physiological information through parameter estimation. The nonlinear least squares (NLLS) method is commonly used for parameter estimation. In previous studies, single-individual optimization algorithms such as the Levenberg–Marquardt (LM) algorithm [9], which is generally considered prone to convergence to local optima [10,11,12,13], have been applied. Moreover, metaheuristic-based population optimization algorithms such as PSO [14], DE [15], and GA [16] perform well in global optimization. Nevertheless, challenges persist in ensuring that these parametric solutions accurately represent physiological processes. Although studies employing population optimization have shown the potential of achieving smaller fitting errors, these parameters might unexpectedly fall outside the anticipated range, thereby adversely affecting diagnostic outcomes. In one study, the identifiability of the three-compartment model was analyzed, revealing the possibility of multiple local solutions within the solution space and implying that modeling and solving of TAC data may be more challenging in practice than theoretically anticipated [10]. Thus, estimating parameters based on finding the solution with the smallest error may not truly reflect physiological characteristics, potentially leading to nonsignificant statistical differences in the parameter results [17, 18]. Applying prior information is one solution, and straightforward alternatives include setting physiologically reasonable fixed boundaries for the parameters [19] or using a specific reliable value to determine a certain kinetic parameter [20]; however, the actual parameter may not almost conform to these fixed restriction rules. He et al. [13] combined prior knowledge to the Bayesian method and obtained reliable parameters. Lin et al. [21] used prior information to guide the optimization process for parametric imaging of hybrid models, obtaining more stable results in terms of the Cramér‒Rao lower bounds metric. Ghovvati et al. [22] used a penalty function in a hybrid GA and PSO algorithm to avoid infeasible points. Kanga et al. [23] used PSO for infinity-norm regularization-based parameter estimation of three-compartment models and used prior information to accelerate the optimization process. However, the utilization of prior information within population optimization algorithms has not been extensively explored in the literature. In this work, we innovatively integrated prior information into the framework of population optimization algorithms, aiming to enhance the physiological rationality of parameter estimation outcomes in the dynamic 18F-FDG PET/CT pharmacokinetic modeling.

Materials and methods

Dynamic PET/CT data (patient characteristics)

This study received approval from the Institutional Review Committee of the First People’s Hospital of Yunnan Province (No. KHLL2022-KY189). The patients all underwent 5-min short-term dynamic 18F-FDG PET/CT scans and 1-min whole-body conventional static scans prior to receiving any treatment. Twenty-one patients, all of whom had confirmed diagnoses of HCC, participated in the study, contributing data. Among the patients, there were 20 males and 1 female, with ages ranging from 31 to 78 years. Nineteen patients had one tumor, one patient had two tumors, and one patient had three tumors, resulting in a total of 24 pathologically diagnosed HCC tumors. These tumors varied in size, with the long axis ranging from 1.9 to 15.0 cm (mean 6.5 ± 3.6). In terms of differentiation grade, 7 tumors were classified as well-differentiated, 10 as moderately differentiated, and 7 as poorly differentiated. Informed consent was acquired from all patients, and all methods adhered to the principles outlined in the Declaration of Helsinki.

PET imaging was performed using a Philips Ingenuity TF PET/CT scanner (Cleveland, OH, USA), while Philips IntelliSpace Portal v7.0.4.20175 was used for imaging post-processing. 18F-FDG synthesis was carried out using a chemical synthesis module (PET Biotechnology Co., Ltd., Beijing, China), ensuring a radiochemical purity exceeded 95%. The PET/CT scanning procedure for each patient was as follows: at least 6 h of fasting before injection and a bedside low-dose liver CT scan (120 kV, 100 mAs) for attenuation correction and image fusion was performed; 18F-FDG injection was then performed, followed by rapid manual application of 18F-FDG (5.5 MBq/kg) in 2 mL of 0.9% saline and at a 2 mL/s. The flow was flushed with 2 mL of 0.9% saline, and after injection into the vein, a 5-min dynamic PET scan was performed. To observe the time course of tracer uptake, the PET axial field of view was centered on the liver during the scan; Subsequently, a conventional static PET scan was conducted around the 60th minute after the injection, complemented by whole-body CT scans spanning from the apex of the skull to the proximal thighs (120 kV, 200 mAs). Following the CT scans, a 1-min PET scan was conducted at each scanning position. Twelve frames of 5s and four frames of 60s were reconstructed from the 5-min dynamic PET data. In addition, one frame of static PET scan data at 60 min to form a total of 17 frames of PET data. The reconstruction algorithm employed adhered to the standard ordered subsets expectation maximization (OSEM).

Regions of interest (ROIs) of the artery, portal vein, HCC, and background liver tissues, were primarily manually drawn on PET images or CT images and adjusted slice-by-slice. When the delineation was challenging, CT images were used as an aid to refine the boundaries. For arteries and portal veins, ROIs were drawn to cover approximately two-thirds of the vascular cross-section, ensuring the exclusion of adjacent structures. For HCC tumors and normal liver tissues, blood vessels were carefully excluded from the ROIs to avoid interference. The maximum standardized uptake values (SUVmax) were extracted from each frame of the PET/CT images in the ROIs and comprised the time-activity curves (TACs) of the tissues.

Kinetic modeling

The compartmental model used in this paper is the reversible ( k4 ≥ 0 ) double-input three-compartment model (r-DI-3CM) [24], as shown in Fig. 1.

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\(\:{C}_{a}\left(t\right)\) is the hepatic arterial input concentration, \(\:{C}_{v}\left(t\right)\) is the portal vein input concentration. The total model input function (the blood tracer concentration \(\:{C}_{i}\left(t\right)\)) is obtained by weighted summation of the two blood input functions according to the hepatic artery blood supply fraction (\(\:{f}_{a}\)):

$$\:{C}_{i}\left(t\right)={f}_{a}\times\:{C}_{a}\left(t\right)+(1-{f}_{a})\times\:{C}_{v}\left(t\right)$$

(1)

\(\:{C}_{f}\left(t\right)\) and \(\:{C}_{p}\left(t\right)\) represent the tracer concentrations of the free 18F-FDG compartment and phosphorylated FDG compartment, respectively. The kinetic parameter \(\:{K}_{1}\) (ml/min/ml) in the figure represents the rate constant of 18F-FDG transport from the blood to the hepatocyte, \(\:{k}_{2}\) (1/min) is the clearance rate of 18F-FDG transport back to the blood, \(\:{k}_{3}\) represents the rate constant of phosphorylation of 18F-FDG to 18F-FDG-6-phosphate, and \(\:{k}_{4}\) represents the dephosphorylation rate of phosphatase.

The pharmacokinetic process of r-DI-3CM can be modeled by the following ordinary differential equations:

$$\:\frac{d}{dt}C\left(t\right)=M\cdot\:C\left(t\right)+{k}_{1}{C}_{i}\left(t\right)\cdot\:e,\hspace{1em}C\left(0\right)=0$$

(2)

$$\:M=\left[\begin{array}{cc}-({k}_{2}+{k}_{3})&\:{k}_{4}\\\:{k}_{3}&\:-{k}_{4}\end{array}\right],\hspace{1em}C=\left[\begin{array}{c}{C}_{f}\\\:{C}_{p}\end{array}\right],\hspace{1em}e=\left[\begin{array}{c}1\\\:0\end{array}\right]$$

(3)

where \(\:t\) is time, and \(\:C\left(t\right)\) is the total output tracer concentration function, the expression of the concentration function is \(\:\left[c\right({t}_{1}),c({t}_{2}),\cdots\:,c({t}_{k}){]}^{T}\), and \(\:k\) is the total number of PET scanning protocol frames. The matrix form of the system of equations is as follows:

$$\:\left[\begin{array}{c}\frac{d}{dt}{C}_{f}\left(t\right)\\\:\frac{d}{dt}{C}_{p}\left(t\right)\end{array}\right]=\left[\begin{array}{cc}-({k}_{2}+{k}_{3})&\:{k}_{4}\\\:{k}_{3}&\:-{k}_{4}\end{array}\right]\times\:\left[\begin{array}{c}{C}_{f}\left(t\right)\\\:{C}_{p}\left(t\right)\end{array}\right]+\left[\begin{array}{c}{K}_{1}{C}_{i}\left(t\right)\\\:0\end{array}\right]$$

(4)

The ordinary differential equation is solved to obtain:

$$\:C(t;k,{C}_{i})={k}_{1}{\int\:}_{0}^{t}{e}^{M\cdot\:(t-\tau\:)}\cdot\:{C}_{i}\left(\tau\:\right)ed\tau\:$$

(5)

The compartmental model also includes the parameter \(\:{v}_{b}\), which is the fractional blood volume. The tissue concentration is calculated by the following equation with \(\:{v}_{b}\):

$$\:{C}_{t}\left(t\right)={v}_{b}\times\:{C}_{i}\left(t\right)+(1-{v}_{b})\times\:C\left(t\right)$$

(6)

where \(\:{C}_{t}\left(t\right)\) is the total concentration, \(\:{C}_{i}\left(t\right)\) is the blood input concentration; and \(\:C\left(t\right)={C}_{f}\left(t\right)+{C}_{p}\left(t\right)\) is the tissue concentration.

Prior-based multi-population optimization algorithm

In this paper, we propose to introduce physiological information from prior kinetic parameters into the population optimization process. As a crucial element of the proposed approach, the prior information is obtained through parameter estimation on true TAC data and conducting probability statistics, and the final representation is a statistical distribution over each parameter dimension. The prior parameter samples should be categorized into two groups (“normal” and “tumor”) based on the diagnostic classification of the corresponding TACs. As illustrated in Fig. 2, the proposed p-MPMO optimization delineates two independent subpopulations, highlighted by the blue dotted lines in the figure, wherein independent population optimization is conducted. The actual optimization algorithm applied within the two subpopulations can be freely chosen. According to the differences in the statistical distributions of the kinetic parameters between the “normal” and “tumor” categories of the TAC data, two distinctions between optimizations of subpopulations are: First, the parameters of individuals in the subpopulations are initialized with different probabilities of prior information, enabling each subpopulation to present its category’s prior probability distribution. Second, the objective function of the optimization applied within the subpopulation differs. The method involved utilizing the corresponding category’s prior information to perform a weighted sum multi-objective optimization, with the two objectives being the root-mean-square error (RMSE) between the measured and fitted curves and the prior probability scores. After both subpopulations were optimized, a classification judgment was performed to select one of the results as the final result, where RMSE was used as a metric for judgment.

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In the p-MPMO optimization, each subpopulation performed a multi-objective optimization of the RMSE objective and the prior probability scores, and a weighted sum of the two objectives formed a prior weighted objective function. The probability values in the prior information statistical distribution (histogram distribution in this work) of each parameter solution were \(\:{p}_{{K}_{1}}\), \(\:{p}_{{k}_{2}}\), \(\:{p}_{{k}_{3}}\), \(\:{p}_{{k}_{4}}\), \(\:{p}_{{f}_{a}}\) and \(\:{p}_{{v}_{b}}\). The values were normalized by a [0, 1] normal distribution, and a weighted sum was used to to obtain the prior probability score \(\:{s}_{p}\):

$$\:{s}_{p}={\sum\:}_{i}{w}_{i}\times\:\frac{{p}_{i}-{\mu\:}_{i}}{{\sigma\:}_{i}}$$

(7)

where \(\:{\mu\:}_{i}\) and \(\:{\sigma\:}_{i}\) were the probability mean and standard deviation, respectively, in the prior distribution of the \(\:i\) th kinetic parameter, and \(\:{w}_{i}\) represented the prior probability weights set for the \(\:i\) th kinetic parameter. Finally, the prior weighted RMSE

$$\:pRMSE=RMSE-{s}_{p}$$

(8)

is defined as the actual objective function in each subpopulation optimization.

Parameter estimation and metrics

The optimization algorithm and parameter estimation were implemented using Python 3.8. Statistical analyses and Receiver Operating Characteristic (ROC) analyses were performed using scipy 1.6.2 and sklearn 0.24.1.

The results of parameter estimation were evaluated from the aspects of fitting effect, physiological characteristics, and diagnostic significance. RMSE and the fitting curves were used to objectively and subjectively assess the fitting effect of the parameter estimates, respectively; The mean and standard deviation of the parameter estimation were used to present the quantified physiological characteristics of the parameters. Student’s t-test was used to test for statistical differences between HCCs and background liver tissue (p < 0.05) is considered statistically significant), i.e., the diagnostic significance of the parameters.

Results

Prior information acquisition

First, this study used the single-population GA to estimate the parameters and generate prior information. The source data from 24 cases were used to perform twenty parameter estimations for kinetic modeling, and the results were presented as the statistical histogram distribution shown in Fig. 3:

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The difference between the statistical distributions of HCC and normal liver tissues in Fig. 3. is significant, and the parameter values are in the correct range and magnitude relationship, i.e., presenting a physiological rationality of the kinetic parameters.

Parameter estimation and statistical analysis

This section demonstrates the fitting effects and statistically significant differences among the kinetic parameters estimated by the p-MPMOPSO, p-MPMODE, and p-MPMOGA. The average value of all repeated parameter estimations was used as the final result for each parameter, and the mean and standard deviation were obtained from the sample of all cases’ parameter results. The results are compared with those of the corresponding single-population algorithm as well as the single-individual LM algorithm. The kinetic parameters results of all methods are shown below.

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Table 1 shows the results of the single-individual LM optimization algorithm. Since the optimization of the LM algorithm is very sensitive to the initial parameter points, ten repetitions were performed, and the initial parameter values for each optimization were randomly valued within the upper and lower bounds. The LM algorithm showed statistically significant differences among the parameters \(\:{k}_{2}\), \(\:{k}_{3}\), and \(\:{f}_{a}\) (p < 0.05) between HCC tumors and healthy liver tissue, while it did not among the \(\:{K}_{1}\), \(\:{k}_{4}\), and \(\:{v}_{b}\).

Three population optimization algorithms PSO, DE, and GA, and that of their prior-based multi-population optimization algorithms were compared. For the single-population algorithms, the number of individuals was 100, and the number of iterations was 20. However, for the multi-population methods, each subpopulation included 60 individuals, and the number of iterations was 12 because two sub-populations were considered. As for the prior probability weights, all p-MPMO methods used a setting of \(\:w=\left[\text{0.08,0.02,0.06,0.03,0.05,0.02}\right]\).

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Table 2 shows a comparison of the results obtained by the SPPSO and p-MPMOPSO. The SPPSO showed statistically significant differences in \(\:{k}_{4}\) and \(\:{f}_{a}\) (p < 0.05), while it did not in the other four parameters. In comparison, the p-MPMOPSO had lower p values for \(\:{k}_{2}\), \(\:{k}_{4}\), and \(\:{f}_{a}\) (p < 0.001). Furthermore, the statistical distribution of \(\:{K}_{1}\) values is larger in HCC compared to normal liver tissue, which indicates the correct characterization of the pharmacokinetics process, however, that is not presented in the results of SPPSO.

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Table 3 shows a comparison of the results obtained by the SPDE and p-MPMODE. The SPDE showed statistically significant differences in \(\:{k}_{2}\), \(\:{k}_{4}\), and \(\:{f}_{a}\) (p < 0.05), while the p-MPMODE showed significant differences in \(\:{K}_{1}\), \(\:{k}_{2}\), \(\:{k}_{3}\), \(\:{k}_{4}\), and \(\:{f}_{a}\).

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Table 4 shows a comparison of the SPGA and p-MPMOGA results. The SPGA made three parameters (\(\:{k}_{2}\), \(\:{k}_{4}\), and \(\:{f}_{a}\)) showed statistically significant differences, while the p-MPMOGA did that among all parameters.

ROC analysis

As shown in Fig. 4, the three p-MPMO optimization algorithms (blue line) had higher AUC values than the corresponding single population algorithms (orange line) on most parameters.

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To determine whether the diagnostic performance of the parameter from one algorithm was significantly better than the others, the following DeLong tests [25] were performed.

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The results in Fig. 5; Table 5 indicated that: In \(\:{K}_{1}\) and \(\:{k}_{4}\), the three p-MPMO optimization algorithms achieved significantly better diagnostic performance than the corresponding single population algorithms and the LM algorithm. In \(\:{k}_{2}\) and \(\:{k}_{3}\), the three p-MPMO methods were better than the corresponding single population algorithms. However, compared with the LM algorithm in \(\:{k}_{2}\), the p-MPMOPSO was significantly better, the p-MPMOGA was no significant difference, and the p-MPMODE was significantly worse. Moreover, compared with the LM algorithm in \(\:{k}_{2}\), the three p-MPMO optimization algorithms were significantly worse. In \(\:{f}_{a}\) and \(\:{v}_{b}\), there was no statistically significant difference in diagnostic performance among the seven algorithms, except that p-MPMODE and p-MPMOGA were significantly worse than that of the LM algorithm in \(\:{f}_{a}\).

TAC curve fitting

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Figure 5 demonstrates the comparison of TAC curve fitted by the SPPSO, SPDE, SPGA, p-MPMOPSO, p-MPMODE, and p-MPMOGA. The LM algorithm was not included due to it being a single individual optimization algorithm. The agreement between the measured data (started point) and the fitted data curve represents the fitting effect. The 60 min point was not shown due to it being too far from the previous point at 5 min. As shown, the three p-MPMO optimization algorithms showed ideal agreement with the original curve data.

Discussion

In this work, kinetic parameters were estimated within the reversible double-input three-compartment model based on PET TAC data, and the hepatic physiological information is presented for the diagnostic study of HCC. The primary objective of this study was to introduce an enhanced approach that offers a more precise quantification and qualification of this information, which may be difficult due to the incorrect physiological characteristics of the kinetic parameters result. Our findings demonstrate that the incorporation of prior information into multi-population optimization algorithms may help parameter estimation more representative of reasonable pharmacokinetic activity. In the results, the p-MPMOPSO, p-MPMODE, and p-MPMOGA demonstrated statistical differences in parameters between normal liver and HCC TACs. The individuals in the population tend to conform to the statistical distribution of the physiological characteristic after applying the prior information, without completely negating the potential of other locations in the solution space.

In this work, a scanning protocol that 5-min dynamic PET supplemented by 1-min static PET at 60 min post-injection [26, 27] was used, and this scheme could maintain a reliable signal-to-noise ratio compared to the scheme that uses a shorter scan interval [28, 29]. The image ROI-derived input function acquisition method was used to obtain hepatic artery and portal vein blood input functions, which is more patient-friendly and easier to perform compared to invasive blood sampling methods [30].

Intelligent population optimization algorithms are a category of optimization algorithms that are gaining popularity across diverse fields. These algorithms have more powerful high-dimensional global optimization capabilities than the single-individual optimization algorithms employed in previous studies. Several studies have utilized the computational advantages to reduce fitting errors. However, in practice, certain positions within the solution space may lead to parameter values that, despite minor errors, lack physiological interpretability. This circumstance may lead to statistically insignificant parameter differences, then failing to yield sufficient clinical diagnostic significance [22]. To address this problem, some studies have used prior information to improve the algorithmic optimization by setting bounds on parameters based on the prior information [19] or setting a parameter with a specific estimated parameter [20]. Nevertheless, these approaches impose excessive restrictions on the parameters, potentially neglecting parameters in other value ranges that could provide an accurate fit. Thus, further research to utilize the prior information based on soft constraints, guiding the algorithm to obtain parameters that are more physiologically reasonable [13, 21,22,23]. In our study, the guiding role of the prior information is imposed on each individual (parameter point) of the population optimization, while reward mechanisms for physiological characteristics are implemented in the population initialization and objective function. Ultimately, a framework for the multi-population multi-objective optimization algorithm based on the idea of the prior classification is proposed.

In the utilization of prior information, p-MPMO optimization has the following three features: Independent, the subpopulations are independent without information exchange. Each subpopulation works independently, so the mechanism of the method is easy to implement and modify; Adaptive, the number of subpopulations is determined by the number of data categories so that the optimization problem with a wide range of categories can be adapted to the design of multiple subpopulations; Prior-based, the prior information in our approach is the prior result of the parameter estimation, and is involved in the computation of the optimization process in the form of a probability distribution. The prior probability distribution may help to ensure the population retains the parameters’ physiological reasonableness while optimizing, however, the fitting remains the main objective in practice. We recommend that the prior probability weights be set to a small value (\(\:{\sum\:}_{i}{w}_{i}\le\:0.3\)) and the specific prior probability weight of each kinetic parameter needs to be determined according to the prior probability distribution used and the physiological rationality preference on this parameter.

The p-MPMO optimization was applied with three population algorithms, PSO, DE, and GA. A comparison was made with the LM algorithm and the SPPSO, SPDE, and SPGA. The optimization problem in this study was constructed around tracer kinetic modeling, and each parameter in r-DI3CM was expected to present the physiological significance of Pharmacokinetics in the result. The infusion of liver tissue includes both hepatic artery and portal vein, and the proportion of arterial supply in lesion tissue was higher (70 ~ 80%), while that in healthy liver tissue is lower (20%~30%) [28], the parameter results for all methods in this paper correctly demonstrate this difference. The cellular uptake of 18F-FDG involves its transport across the cell membrane via glucose transporters (Gluts), followed by phosphorylation mediated by hexokinase to produce 18F-FDG-6-phosphate. This metabolic conversion renders it non-metabolizable and amenable to cellular retention. Moreover, phosphatase dephosphorylates 18F-FDG-6-phosphate back to 18F-FDG. Elevated levels of glucose-6-phosphatase are characteristic of the normal liver, leading to the dephosphorylation of 18F-FDG. Consequently, this results in its diminished accumulation within cells and its re-entry into the metabolic cycle [31,32,33,34]. Previous studies have shown higher \(\:{K}_{1}\) and \(\:{k}_{3}\) values in kinetic modeling due to a substantial increase in hexokinase activity in malignancies [35], p-MPMOPSO, p-MPMODE, and p-MPMOGA both correctly presented such results and were statistically significant differences in our experimental results. In contrast, the \(\:{k}_{4}\) of dephosphating is higher than HCC due to greater G6P activity in healthy liver tissue [1], as demonstrated by the three p-MPMO methods. In addition, the differentiation grade of the tumor affects the presentation of some physiological characteristics, such as a well-differentiated tumor may have a similar level of FDG uptake to normal liver tissue [36]. It is not clear but worth to study whether kinetic parmeters function to distinguish the degree of HCC differentiation; unfortunately, this study was not performed because of a small size of the sample.

In ROC analysis, the three p-MPMO optimization algorithms achieved better diagnostic performance than the corresponding single population algorithms. In \(\:{K}_{1}\) and \(\:{k}_{4}\), the three p-MPMO methods achieved significantly better diagnostic performance than the corresponding single population algorithms and the LM algorithm. In \(\:{k}_{2}\) and \(\:{k}_{3}\), the three p-MPMO methods were better than the corresponding single population algorithms, but compared to the LM algorithm in \(\:{k}_{2}\), the p-MPMOPSO was significantly better, the p-MPMOGA was no significant difference, the p-MPMODE was significantly worse, and compared to the LM algorithm in \(\:{k}_{3}\), the three p-MPMO optimization algorithms were significantly worse. This is most likely due to the lack of significant difference in \(\:{k}_{2}\) and \(\:{k}_{3}\) between HCC and normal liver tissue in the distribution of prior parameters shown in Fig. 3. In \(\:{f}_{a}\) and \(\:{v}_{b}\), there was no statistically significant difference in diagnostic performance between the seven algorithms, except that p-MPMODE and p-MPMOGA were significantly worse than that of the LM algorithm in \(\:{f}_{a}\). It can be seen that the resulting trend of these two parameters is relatively fixed in parameter estimation practice, and the changes in the algorithm rarely have an impact on estimation.

This study has limitations. First, the acquisition method and quality of the prior information and hepatic dynamic PET data are critical problems. These factors lead to research being hampered by experimental source data. Second, the relationship between kinetic parameters and tumor differentiation grade was not analyzed in a small sample size in this study, and a large sample size research should be further carried out. Another limitation is that the method presented in this paper involves many hyperparameters, and the setting of these parameters relies on subjective judgments, especially the prior weights of each kinetic parameter. Finally, the approach employed in this research adopts a simplistic weighted sum method to perform the multi-objective optimization between the RMSE and prior score. However, this approach may not assure an optimal equilibrium between the two objectives, thus a more effective alternative should be developed in future research.

Conclusions

In this paper, we propose a multi-population multi-objective approach based on the prior information for kinetic parameter estimation in a reversible double-input three-compartment model. The approach achieved ideal performance when combined with the PSO, DE, and GA, and the experimental results demonstrated that these algorithms may present more physiological differences between normal liver tissue and HCC especially according to K1 and k4 than conventional methods with the help of the kinetic parameter prior information.

Data availability

The datasets generated and analyzed during the present study are not accessible to the public due to data security concerns. However, interested parties may request access to these datasets from the corresponding author, subject to reasonable conditions.

Abbreviations

18F-FDG:

18 F fluoro-D-glucose

PET:

Positron emission tomography

CT:

Computed tomography

p-MPMOO:

Prior-based multi-population multi-objective optimization

HCC:

Hepatocellular carcinoma

LM algorithm:

Levenberg–marquardt algorithm

PSO:

Particle swarm optimization

DE:

Differential evolution

GA:

Genetic algorithm

AUC:

Areas under the curve

TKM:

Tracer kinetic modeling

TAC:

Time-activity curve

NLLS:

Nonlinear least squares

OSEM:

Ordered subsets expectation maximization

ROIs:

Regions of interest

SUVmax:

Maximum standardized uptake value

r-DI-3CM:

Reversible double-input three-compartment model

RMSE:

Root-mean-square error

ROC:

Receiver operating characteristic

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