Dual-phase multiobjective Bayesian optimization method for estimating hepatocellular carcinoma dynamics parameters from PET/CT scans

Introduction

Liver cancer remains a global health challenge, and its incidence is expected to exceed 1 million cases by 2025. Hepatocellular carcinoma (HCC) is the most common form of liver cancer, accounting for approximately 90% of cases (1). Despite these alarming statistics, recent advancements have demonstrated notable progress in the development of systemic therapeutic strategies for HCC (2-4) and the optimization of cancer treatment protocols (5). Positron emission tomography (PET), which is described as in vivo autoradiography, measures tracer pharmacokinetics in a single experiment, thereby reflecting biochemical processes (6). As a noninvasive functional imaging method, PET/computed tomography (CT) can provide complementary information to that obtained from standalone magnetic resonance imaging (MRI) or CT.

Metabolic alterations are closely associated with the development of HCC (7). ¹⁸F-fluorodeoxyglucose (¹⁸F-FDG), which is the most commonly used tracer for PET/CT, is transported into cells from the plasma and is phosphorylated, thus providing functional insights into cellular physiology and metabolism. Static PET only provides tracer distributions at specific time points and is unable to capture sufficient physiological changes. In contrast, dynamic PET imaging captures real-time tracer distribution changes in vivo by reconstructing image sequences at multiple time points, thereby resulting in the generation of a time-concentration activity curve (TAC) for the regions of interest (ROIs). The analysis of the TAC data derived from plasma and liver tissues and the construction of kinetic models have been shown to enable the determination of physiological parameters such as blood flow, perfusion, and liver metabolism (8). Thus, the analysis of kinetic parameters from PET imaging offers additional indices for obtaining more accurate tumor characterization, differentiation, and efficacy assessment results (9).

The estimation of kinetic model parameters via optimization algorithms has garnered significant attention. The nonlinear least-squares (NLLS) method, which is commonly used for estimating kinetic parameters (10-12), can often become trapped in local optima, thereby limiting its clinical applicability. In our previous research, the Bayesian method (BM) (13) was first compared with the NLLS method, with the results demonstrating that the BM yielded more accurate kinetic parameter estimates. However, this method is highly dependent on the initial sampling points, thus leading to instability in its results.

In contrast, Bayesian optimization (BO) constructs a surrogate model to effectively estimate the mean and uncertainty (variance) of the target objective function, thereby enabling the approximation of its global optimum. As a result, BO is widely applied in the scientific and engineering domains (14-16). However, its applicability in the field of pharmacokinetic parameter estimation requires further investigation.

Furthermore, a dual-phase (DP) strategy can be effectively employed to increase the accuracy of the utilized algorithm by correcting for suboptimal results (17,18); moreover, a multiobjective (MO) approach can be used to improve the global search ability of the algorithm (19). Thus, we assume that the incorporation of the DP and MO strategies into BO (DPMO-BO) may closely align the resulting model with real physiological characteristics to increase its kinetic estimation effect. Consequently, this study evaluated the performance of the BM, BO, and DPMO-BO in the task of estimating dynamic HCC parameters from PET/CT scans. We present this article in accordance with the STARD reporting checklist (available at https://qims.amegroups.com/article/view/10.21037/qims-2024-2767/rc).

Methods

Patient characteristics

The data used in this study were obtained from the PET/CT Center of the First People’s Hospital of Yunnan Province, and this study was conducted in accordance with the Declaration of Helsinki and its subsequent amendments. The study was approved by the Ethics Committee of the First People’s Hospital of Yunnan Province (No. KHLL2022-KY189) and informed consent was taken from all the patients. A total of 23 HCC patients with 27 pathologically diagnosed HCC lesions were included in the study; among them, 21 patients exhibited only one lesion (including one follow-up patient), one patient exhibited two lesions, and one patient exhibited three lesions, with tumor sizes ranging from 1.6 to 17.0 cm and a mean size of 6.75 cm being observed. All of the patients underwent 5-minute dynamic ¹⁸F-FDG PET/CT scans and whole-body routine static ¹⁸F-FDG PET/CT scans at the PET/CT Center of the First People’s Hospital of Yunnan Province.

All of the examinations were performed via a Philips Ingenuity TF PET/CT scanner (Philips, Columbus, OH, USA). ¹⁸F-FDG was automatically synthesized in a chemical synthesis module (Beijing Pate Biotech Co., Ltd., Beijing, China) with a radiochemical purity level >95%. This study used the ordered subset expectation maximization (OSEM) algorithm for PET image reconstruction. After a patient was intravenously administered ¹⁸F-FDG, the data observed in the first minute were reconstructed as 12 frames with 5-second intervals, and the data obtained in the last 4 minutes were reconstructed as 4 frames with 60-second intervals. Moreover, to better perform kinetic modeling for the analysis of the metabolism of ¹⁸F-FDG in each patient’s body, 1 frame of each 60-minute static PET liver image was selected. Seventeen PET image frames were obtained, fused, and aligned with the CT images.

The ROI was outlined after the PET/CT scanning process was completed for all of the patients. Standard uptake values (SUVs) were extracted from each image via the Philips IntelliSpace Portal (v7.0.4.20175). SUV interference caused by the blood vessels inside of the liver was avoided to the greatest extent possible during the outlining process. Moreover, TACs consisting of the maximum SUV for each frame were obtained from the ROIs of the PET/CT images. Figure 1 shows the ROIs delineated on a single frame of CT and PET/CT images for an HCC patient.

Figure 1 ROIs constructed in dynamic PET/CT. (A) Different slices of the CT image; (B) different slices of the PET/CT fusion image. The HCC is marked by a blue circle, the background liver tissue is marked by a black circle, the aorta is marked by a red circle, and the portal vein is marked by a green circle. CT, computed tomography; HCC, hepatocellular carcinoma; PET, positron emission tomography; ROI, region of interest.

Double-input three-compartment model

The compartmental model used in this study is the reversible (k4 ≥0) double-input three-compartment model (r-DI-3CM) (20), as shown in Figure 2.

Figure 2 r-DI-3CM. Red and blue dashed boxes indicate blood and tissue compartments, respectively. r-DI-3CM, reversible double-input three-compartment model.

In Figure 2, fa represents the percentage of hepatic arterial blood flow. Ci(t) represents the total blood ¹⁸F-FDG concentration as a function of time. Ca(t) and Cv(t) represent the ¹⁸F-FDG concentrations in the hepatic artery and portal vein blood, respectively, as a function of time. The concentration of free ¹⁸F-FDG in the liver tissue as a function of time is indicated by Cf(t); additionally, the concentration of ¹⁸F-FDG-6-phosphate in the liver tissue as a function of time is indicated by Cp(t). The total blood input function of the model was obtained by weighing and summing the hepatic artery and portal vein blood supply functions:

Ci(t)=fa×Ca(t)+(1−fa)×Cv(t)[1]

K1 (mL/min/mL) represents the rate constant for the transport of ¹⁸F-FDG from the blood to the liver tissue, and k2 represents the rate constant for the return of ¹⁸F-FDG from the liver tissue to the blood. k3 represents the rate constant for the phosphorylation of ¹⁸F-FDG to ¹⁸F-FDG-6-phosphate via hexose phosphate kinase in the liver tissue, and k4 represents the rate constant for the conversion of phosphokinase to ¹⁸F-FDG with a rate constant.

Differential equations can be obtained based on the two-input arterial model of the liver shown in Figure 2:

dCf(t)dt=K1Ci(t)−(k2+k3)Cf(t)+k4Cp(t)[2]

dCp(t)dt=k3Cf(t)−k4Cp(t)[3]

The solving of these differential equations yields CT(t), which represents the curve of the tracer concentration in the tissue measured from the input PET image over time; additionally, it indicates the output function of the kinetic model:

CT(t)=(1−vb)K1α2−α1[(k3+k4−α1)e−α1t+(α2−k3−k4)e−α2t]⊗Ci(t)+vb×Ci(t)[4]

where vb is the fractional blood volume, and α1 and α2 can be described as follows:

α1=k2+k3+k4−(k2+k3+k4)2−4k2k42[5]

α2=k2+k3+k4+(k2+k3+k4)2−4k2k42[6]

Estimation of the kinetic parameters

This study proposes an improved algorithm (DPMO-BO) for the estimation of liver kinetic parameters. Our objective is as follows:

argminθRMSE(θ)=argminθ1n∑i=1n(CT(θ,ti)−ci)2[7]

where θ is the set of all of the kinetic parameters to be estimated (K1,k2,k3,k4,fa,vb), ci denotes the TACs obtained from the actual measurements acquired from the PET/CT image frames (n=17), and CT(θ,ti) denotes the TACs obtained from the compartment model. In BO, the initial solution space for the objective function is obtained by conducting uniformly distributed random sampling; moreover, the objective function is approximated by constructing a Gaussian process (GP) as a surrogate model, and the upper confidence bound (UCB) acquisition function is a noninformative method (21) that is used to generate the next candidate points in each iteration and add one to the solution space (22). Finally, the optimal solution is searched in the solution space. The UCB is strongly dependent on its own fixed parameters for weighing the exploitation and exploration processes and cannot dynamically change its exploration or exploitation strategy based on the strengths and weaknesses of candidate points.

The regret-to-sigma ratio (RSR) and pharmacokinetic parameter Ki are introduced as weights to measure the quality of the candidate points during the optimization process. Based on the quality of these candidate points, different parameter estimation strategies are employed. These strategies constitute the first phase of the DPMO-BO method. This approach improves the ability of the constructed algorithm to explore and mine optimal pharmacokinetic parameter points, thus offering a more focused and efficient optimization process while capturing the physiological relevance and optimization potential of the parameters. Overfitting issues are commonly encountered during parameter optimization, and no clear criterion is available for detecting overfitting in this dual-input model. In the second phase, we establish overfitting detection criteria based on the optimization results acquired from the previous phase. Additionally, we integrate pharmacokinetic characteristics to perform targeted optimization on the overfitted points. The overall flowchart describing the DPMO-BO method is shown in Figure 3. DPMO-BO can be divided into three parts based on the GPs: weight calculation and sampling; high- and low-weight optimization; and overfitting detection and resolution.

Figure 3 Flowchart of the DPMO-BO method. The first phase comprises weight calculation and sampling, followed by separate optimizations for high- and low-weight subsets. The second phase applies overfitting detection and resolution strategies based on the results from the first phase. BO, Bayesian optimization; DP, dual-phase; DPMO-BO, DP and MO strategies into BO; MO, multiobjective; RMSE, root mean square error; RSR, regret-to-sigma ratio.

GPs

We use GPs (23,24) to build a probabilistic model of the black-box objective function that needs to be optimized. A GP(μ,k) can be fully specified by a mean function μ(x) and a covariance (kernel) function k(x,x'). In this process, Xt:={x1,x2,…,xt} denotes t points evaluated by the algorithm after t iterations, where one point is evaluated per iteration. These t points collectively constitute the solution space explored by the algorithm. For any x∈Xt, we note the objective function f|Ft∼GP(μt(x),kt(x,x′)), where:

μt(x)=kt(x)⊤(Kt+σn2I)−1yt[8]

kt(x,x′)=k(x,x′)−kt(x)⊤(Kt+σn2I)−1kt(x′)[9]

Here, Kt:=[k(x′,x″)]x′,x″∈Xt denotes the empirical kernel matrix known as kt(x):=[k(x′,x)]x′∈Xt, and yt denotes {f(x′)+ϵ′}x′∈Xt, where we recall that ϵ′∼N(0,σn2).

Acquisition functions form a tradeoff between exploration and exploitation by utilizing statistics derived from the posterior p(α(⋅)|D), with D denoting the data evaluated in the solution space thus far. The most widely used method in BO is the UCB.

αUCB(x;β)=μ(x)+βσ(x)[10]

The UCB is typically used to maximize the target objective function (rather than minimizing it); moreover, for our minimization problem, we need to consider the objective function as negative. In each iteration, the action that maximizes the UCB selects the optimal evaluation point according to xtUCB∈argmaxμt(x)+βtσt(x). This selected point is then added to the existing solution space. In the first phase of DPMO-BO, we aim to minimize the fitting error between the curve fitted by the kinetic model and the a priori information concerning the kinetic parameters; thus, we use the lower confidence bound (LCB) for the acquisition function.

αLCB(x;β)=μ(x)−βσ(x)[11]

Here, σ(x)=k(x,x′) is the marginal standard deviation of f(x). β>0 is a tradeoff parameter between the exploration term σ(x) and the exploitation term μ(x). Practical implementations typically require the β parameter to be heuristically tuned. In contrast, the DPMO-BO method does not require such β parameter tuning. The β parameter can be adaptively adjusted by the designed weights.

Weight calculation and sampling

The weight design process was based on the methodology by Ren et al. (25), who optimized weights based on the RSR, which is a robust criterion for guiding the exploration process in the parameter space. However, we recognized that solely focusing on the optimization objective may be insufficient for capturing the full physiological significance of the estimated parameters. Therefore, we integrate the net influx rate (Ki) derived from dynamic imaging into our optimization framework, as it is considered the gold-standard quantification index for FDG PET (26). To account for both the RSR and Ki, we calculate the weight for each point as follows:

Wi=|ymin−μ(x)σ(x)|×K1×k3k2+k3[12]

where ymin is the minimum observed value of the target function in the current solution space, and Ki is calculated based on the pharmacokinetic parameters K1, k2, and k3. The weights are subsequently standardized.

During the initial iteration, we utilize Latin hypercube sampling (LHS) to generate preliminary sample points. Based on the calculated weights, we select the m points with the highest weights and the m points with the lowest weights, thereby forming two subsets known as Shigh and Slow, respectively. These 2 × m points, which are referred to as preliminary points, are then optimized. The optimized points form the candidate set for the next iteration. In subsequent iterations i, we directly select the m points with the highest weights and the m points with the lowest weights from the candidate set to from Shigh,i and Slow,i, respectively. This process is repeated for t iterations, and the search space is refined in each iteration.

High- and low-weight optimization

In each iteration, we use different optimization methods for each of the Shigh and Slow subsets obtained from sampling. The high-weight points, which are characterized by elevated Ki values and high RSRs, indicate regions with high metabolic activity and significant potential for improving the current model solutions. Consequently, these points necessitate intensified exploration. For the m points contained within the Shigh subset, we minimize the LCB to guide our exploration process, as follows:

argminx∈ShighαLCB(x;β|D)[13]

Conversely, for low-weight points, we place greater emphasis on exploitation during the experimental process. Low weights indicate lower Ki and RSR values, thus highlighting areas that are likely close to the current optimal solution or that represent clinically normal tissues or regions with low metabolic activity. These points are more likely to yield optimal solutions via further localized fine-tuning.

We propose the application of MO acquisition functions to seek Pareto-optimal solutions. This approach aids in balancing different acquisition schemes, thereby ensuring that no single acquisition function overly influences the optimization process (27). Formally, our objective is to simultaneously minimize both the LCB and RSR, which is structured as follows:

minx(αLCB(x;β|D),αRSR(x|D))[14]

where x∈Slow. In our experiments, we utilize the nondominated sorting genetic algorithm II (NSGA-II) (28), which supports mixed-variable crossover and mutation, thus enabling the optimization of both real-valued and integer-valued inputs. This algorithm is particularly suited for our needs, as it effectively handles MO optimization tasks, thereby driving the search process toward a set of Pareto-optimal solutions.

The parameter β in Eq. [13] and Eq. [14] controls the balance between exploration and exploitation, with larger β values favoring exploration and smaller values prioritizing exploitation. For each x, the associated β value is derived via linear interpolation based on the corresponding weight, thus allowing for flexible and adaptive adjustments. This approach enhances the optimization process by dynamically tailoring the balance between exploration and exploitation to the specific characteristics of each point.

After both the high- and low-weight points are optimized, we obtain all of the candidate points. We select the point with the minimal LCB as the evaluation point to add to the solution space D, thus ensuring the systematic integration of the most promising points and iterative model refinement.

Overfitting detection and resolution

During the parameter estimation process, increasing model complexity can frequently lead to reduced generalization performance. Although parameter estimation results may yield low root mean square error (RMSE) values, they may exhibit a lack of reliability (29). We refer to these results as overfitting points. The importance of Ki in tumor differentiation and FDG PET uptake quantification is well- documented; specifically, it is more reliable compared to SUVs, as it is proportional to tissue glucose metabolism and independent of the measurement time or input function (30,31). To mitigate the impact of overfitting on the robustness of the algorithm and improve the accuracy of kinetic parameter estimation, we introduce Ki in the second phase of DPMO-BO to detect overfitting points.

Combined with actual clinical measurement information, we define overfitting points as follows:

{(type=tumorandKi>kiavgnormal)or(type=tumorandKi<kiavgnormal)}andRMSE<θ[15]

where type∈{tumor,normal}. Additionally, the threshold of the RMSE θ is set to 0.3 in our experiments.

DPMO-BO automatically detects these overfitting points and applies targeted optimization. Similar to the first phase, we employ the NSGA-II optimizer for the overfitted points and incorporate Ki into Eq. [14] to guide our search in the parameter space.

For the normal type,

minx(αLCB(x;β|D),Ki⋅λ1⋅αRSR(x|D)⋅(1−λ1))[16]

For the tumor type,

minx(αLCB(x;β|D),Ki⋅αRSR(x|D)⋅λ2−Ki⋅(1−λ2))[17]

where λ1 and λ2 are different weights for the optimization objectives, which are set to 0.8 and 0.6 in this study, respectively.

Statistical metrics

An independent-sample t-test was conducted to assess the differences between the kinetic parameters of HCC and background liver tissues. Receiver operating characteristic (ROC) analysis and the DeLong test were performed to evaluate the diagnostic performance of Ki in terms of distinguishing between background liver tissues and HCC. If P<0.05, the difference was considered to be statistically significant.

Results

Comparison of the kinetic parameters

The results of the kinetic parameter (K1,k2,k3,k4,fa,vb,Ki) estimates obtained via the three methods are presented in Table 1.

For the BM, the k2, k3, fa, and vb values obtained for the background liver tissues were significantly lower than those obtained for the HCC (P=0.003, P=0.023, P<0.001, and P=0.007, respectively), whereas the K1, k4 and Ki values did not significantly differ between the HCC and background liver tissues (P=0.106, P=0.277, and P=0.280, respectively).

In the BO results, fa and vb were significantly greater for the HCC than for the background liver tissues (both P<0.001); however, K1, k2, k3, k4 and Ki were not significantly different (P=0.147, P=0.171, P=0.081, P=0.123, and P=0.105, respectively).

The DPMO-BO method demonstrated a more significant ability to distinguish between HCC and background liver tissues. Specifically, the P values produced for all six parameters were less than 0.05, thus indicating significance. Notably, k2 and fa achieved even higher significance levels (both P<0.001).

Diagnostic performance

Figure 4 shows the ROC curves of the Ki values estimated by the three methods for differentiating HCC from background liver tissues. The proposed DPMO-BO method exhibited a greater area under the ROC curve (AUC) value compared with the BO method and the BM for Ki. Furthermore, the results of the DeLong test demonstrated that the AUC values produced by DPMO-BO and BO were significantly different (P<0.001). Similarly, the AUC values of DPMO-BO and the BM were also significantly different (P<0.001). When BO was compared with the BM, the DeLong test results revealed that their Ki values were not significantly different (P=0.291).

Figure 4 Comparison of the ROC curves of Ki. AUC, area under the ROC curve; BO, Bayesian optimization; BM, Bayesian method; DP, dual-phase; DPMO-BO, DP and MO strategies into BO; MO, multiobjective; ROC, receiver operating characteristic.

Fitting quality analysis

Example TAC curve fitting results produced for normal and tumor tissues by the three methods are shown in Figure 5. In the first minute of the HCC fitting curve plot, the DPMO-BO and BM curves were closely aligned. However, the DPMO-BO method demonstrated superior global fitting ability in comparison with the BM (yellow line). Additionally, in the normal tissue fitting curve plot, the fitting performance of BO was superior to that of the BM. In both plots, the DPMO-BO curve (blue line) provided a more accurate fit for the actual measured data points (black dots). The means and variances of the RMSE values yielded by the three methods are presented in Table 2. Compared with the BM and BO, DPMO-BO achieved the lowest RMSEs for both background liver tissues and HCC.

Figure 5 Comparison of the TAC fitting results obtained by the three methods. BO, Bayesian optimization; BM, Bayesian method; DP, dual-phase; DPMO-BO, DP and MO strategies into BO; HCC, hepatocellular carcinoma; MO, multiobjective; TAC, time-concentration activity curve.

Discussion

The combination of dynamic ¹⁸F-FDG PET/CT imaging with pharmacokinetic modeling offers a robust analytical pathway for assessing tumor biochemical processes. However, conventional dynamic ¹⁸F-FDG PET/CT imaging necessitates a time period of up to 60 minutes to scan a specific organ, thereby requiring the subject to remain still; thus, this technique is impractical for clinical settings. This study employed a 5-minute dynamic PET/CT protocol supplemented by a 60-minute postinjection static PET/CT scan. Hepatic artery and portal vein blood were acquired from the image ROIs as input functions, which were combined with a r-DI-3CM for kinetic parameter estimation. This protocol maintains a reliable signal-to-noise ratio (10) and is more suitable for patients compared with techniques using shorter scanning intervals and invasive blood collection methods.

The NLLS method for estimating kinetic parameters struggles with complex and irregular search spaces; moreover, it is sensitive to noise and uncertainty and is often trapped by local optima. Although the BM improves the accuracy of parameter estimation, it is limited by its reliance on prior results and the initial parameters. In this study, we applied the BO strategy to kinetic parameter estimation for the first time, compared its performance with that of the BM, and introduced an enhanced DPMO-BO approach to more effectively address these limitations. BO constructs a surrogate model to estimate the objective function and uses an acquisition function to guide the search process, thereby efficiently exploring the search space to identify the global optimum. Furthermore, it does not rely on initial parameter values and is able to handle complex and irregular functions; moreover, it can manage uncertainty with high sample efficiency, thus making it suitable for automated parameter tuning.

The hepatic organ requires a dual-input function that includes both the hepatic artery and portal vein to accurately characterize its blood supply. Healthy liver tissue receives 70–80% of its blood from the portal vein, whereas HCC is primarily supplied by the hepatic artery because of its vasoproliferative nature (11). The fa parameter results derived from all of the methods in this study revealed a statistically greater distribution in HCC tissues than in normal liver tissues. Specifically, the fa (%) values for HCC were 77.2±21.3 for the BM, 81.6±23.2 for BO, and 82.4±20.6 for DPMO-BO, all of which significantly differentiated HCC from background liver tissues. These findings preliminarily indicated that BO was effective for estimating pharmacokinetic parameters.

Based on BO, this study used the net inflow rate Ki and the RSR as weights to partition the search space into high- and low-weight regions. For the high-weight region, increased exploration of the uncertainty domain was needed. Conversely, for the low-weight region, increased exploitation around the current optimal solution was necessary to prevent the model from being trapped by local optima and being unable to escape. The abovementioned design describes only the first phase of DPMO-BO. We observed that the kinetic parameter estimates obtained at this stage could differentiate HCC from background liver tissues, with P values for six of the parameter estimates being significant (P<0.05), with the exception of k3 (P=0.05). However, a more in-depth examination revealed that the optimal points resulted in very low RMSE values and did not conform to physiological norms, thus limiting their clinical applicability.

As a quantitative marker of glucose metabolism, Ki effectively reflects tumor glucose uptake rates and is primarily used in clinical settings to assess the metabolic activity of tumors (32). The quantification of FDG uptake by Ki helps in standardizing diagnostic criteria for FDG PET oncology (30), and dynamic changes in Ki can be used to assess the tumor response to therapy (33), evaluate the tumor proliferative capacity (34), and predict pathological outcomes (35). Previous studies have indicated that Ki should be greater in HCC tissues than in background liver tissues (36,37). In the second phase of DPMO-BO, we targeted the optimization of results exhibiting significantly abnormal Ki values and low objective function values. The final estimated results (Ki) were analyzed via ROC analyses and DeLong tests to evaluate the performance of the three methods in terms of distinguishing HCC from background liver tissues. Our experimental results demonstrated that the DPMO-BO method significantly outperformed BO and the BM with respect to differentiating HCC from liver tissues.

Hepatocytes possess numerous glucose transporter proteins on their plasma membranes, thus facilitating the transport of ¹⁸F-FDG from the blood and into the cells, where it is phosphorylated to ¹⁸F-FDG-6-phosphate by hexokinase. Cancer cells typically exhibit increased hexokinase expression and activity levels, thus leading to elevated K1 and k3 values. Zuo et al. (38) demonstrated that K1 can be used to assess human liver inflammation. Our experimental results demonstrated that only the k3 parameter (estimated via the BO method) deviated from the clinical findings described above, whereas the K1 and k3 parameters (estimated via the other methods) aligned well with clinical expectations.

¹⁸F-FDG-6-phosphate can be dephosphorylated back to ¹⁸F-FDG by phosphatase (39,40). This metabolic activity varies between normal cells and tumor cells, with greater dephosphorylation being observed in normal liver tissues and lower phosphorylation being observed in tumor cells, thereby resulting in greater k4 values in normal liver tissues. Our findings demonstrated that all three parameter estimation methods met clinical expectations, with the DPMO-BO method achieving the lowest RMSE values.

To assess the feasibility of the DPMO-BO method, we evaluated its runtime performance in 23 HCC patients encompassing 27 pathologically confirmed lesions. The entire analysis took approximately 4 hours to be performed, and the average runtime was 8.23 minutes per lesion. When considering that our imaging protocol comprises only a 5-minute dynamic acquisition phase followed by a static scan at the 60-minute mark, this computational overhead appears to be feasible for clinical practice.

In summary, ¹⁸F-FDG kinetics offer detailed insights into physiological systems and disease pathogenesis. The DPMO-BO method more accurately estimates kinetic parameters compared to the BM, thus improving the clinical applicability of dynamic ¹⁸F-FDG PET/CT imaging; moreover, the BO method is crucial for assessing tumor characteristics, grading tumors, and conducting postresection prognostic analyses. Nonetheless, the complexity of these optimization techniques and their practical integration into routine clinical workflows remain key challenges. To bridge this gap, researchers should develop user-friendly interfaces and conduct larger prospective studies for clinical validation. Over the next 5 years, ongoing refinements in machine learning–based algorithms are expected to broaden the application of complex kinetic modeling in personalized oncology, thereby enhancing our capacity to accurately characterize tumor metabolism.

However, there are several limitations in this study. First, the relatively small sample size could limit the generalizability of our findings, thus emphasizing the need to expand the number of available clinical samples or incorporate high-quality simulated data in future studies. Second, the algorithm focuses on high- and low-weight regions to improve its convergence speed, thereby potentially missing new optimal solutions in unsearched areas. The design of multiregion independent search strategies or the introduction of perturbations in the parameter space may mitigate this limitation; however, the implementation of such approaches remains challenging. Therefore, in the future, we plan to explore more comprehensive optimization strategies from these perspectives, thus aiming to achieve low-error pharmacokinetic parameter estimations and thereby ensuring robust, accurate results.

Conclusions

This study demonstrated the feasibility of using the BO method for kinetic parameter estimation. Additionally, the proposed DPMO-BO method more effectively and intuitively highlights the physiological differences between HCC and background liver tissues compared to the previously developed BM, thus providing a reliable basis for HCC analysis tasks.

Acknowledgments

None.

Footnote

Reporting Checklist: The authors have completed the STARD reporting checklist. Available at https://qims.amegroups.com/article/view/10.21037/qims-2024-2767/rc

Data Sharing Statement: Available at https://qims.amegroups.com/article/view/10.21037/qims-2024-2767/dss

Funding: This work was supported by the National Natural Science Foundation of China (Nos. 82160347, 82260355, and 82060329), the Yunnan Key Laboratory of Smart City in Cyberspace Security (No. 202102AE090031), the Ten Thousand People Plan in Yunnan Province (No. YNWR-QNBJ2018-243), the Basic Research on Application of Joint Special Funding of Science and Technology Department of Yunnan Province-Kunming Medical University (No. 202301AY070001-211), and the Yunnan Provincial Science and Technology Department Social Development Special Project (No. 202403AC100018).

Conflicts of Interest: All authors have completed the ICMJE uniform disclosure form (available at https://qims.amegroups.com/article/view/10.21037/qims-2024-2767/coif). The authors have no conflicts of interest to declare.

Ethical Statement: 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. This study was conducted in accordance with the Declaration of Helsinki and its subsequent amendments. The study was approved by the Ethics Committee of the First People’s Hospital of Yunnan Province (No. KHLL2022-KY189) and informed consent was taken from all the patients.

Open Access Statement: This is an Open Access article distributed in accordance with the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International License (CC BY-NC-ND 4.0), which permits the non-commercial replication and distribution of the article with the strict proviso that no changes or edits are made and the original work is properly cited (including links to both the formal publication through the relevant DOI and the license). See: https://creativecommons.org/licenses/by-nc-nd/4.0/.

References

1. Llovet JM, Kelley RK, Villanueva A, Singal AG, Pikarsky E, Roayaie S, Lencioni R, Koike K, Zucman-Rossi J, Finn RS. Hepatocellular carcinoma. Nat Rev Dis Primers 2021;7:6. [Crossref] [PubMed]
2. Guven DC, Erul E, Kaygusuz Y, Akagunduz B, Kilickap S, De Luca R, Rizzo A. Immune checkpoint inhibitor-related hearing loss: a systematic review and analysis of individual patient data. Support Care Cancer 2023;31:624. [Crossref] [PubMed]
3. Rizzo A, Ricci AD, Brandi G. Trans-Arterial Chemoembolization Plus Systemic Treatments for Hepatocellular Carcinoma: An Update. J Pers Med 2022;12:1788. [Crossref] [PubMed]
4. Rizzo A, Ricci AD. Challenges and Future Trends of Hepatocellular Carcinoma Immunotherapy. Int J Mol Sci 2022;23:11363. [Crossref] [PubMed]
5. Sahin TK, Ayasun R, Rizzo A, Guven DC. Prognostic Value of Neutrophil-to-Eosinophil Ratio (NER) in Cancer: A Systematic Review and Meta-Analysis. Cancers (Basel) 2024;16:3689. [Crossref] [PubMed]
6. Lancelot S, Zimmer L. Small-animal positron emission tomography as a tool for neuropharmacology. Trends Pharmacol Sci 2010;31:411-7. [Crossref] [PubMed]
7. Vitale E, Rizzo A, Santa K, Jirillo E. Associations between "Cancer Risk", "Inflammation" and "Metabolic Syndrome": A Scoping Review. Biology (Basel) 2024;13:352. [Crossref] [PubMed]
8. Morris ED, Endres CJ, Schmidt KC, Christian BT, Muzic RF, Fisher RE. Kinetic modeling in positron emission tomography. In: Wernick MN, Aarsvold JN. editors. Emission tomography: the fundamentals of PET and SPECT. San Diego: Academic Press; 2004:499-540.
9. Huo L, Guo J, Dang Y, Lv J, Zheng Y, Li F, Xie Q, Chen X. Kinetic analysis of dynamic (11)C-acetate PET/CT imaging as a potential method for differentiation of hepatocellular carcinoma and benign liver lesions. Theranostics 2015;5:371-7. [Crossref] [PubMed]
10. Samimi R, Kamali-Asl A, Geramifar P, van den Hoff J, Rahmim A. Short-duration dynamic FDG PET imaging: Optimization and clinical application. Phys Med 2020;80:193-200. [Crossref] [PubMed]
11. Wang S, Li B, Li P, Xie R, Wang Q, Shi H, He J. Feasibility of perfusion and early-uptake (18)F-FDG PET/CT in primary hepatocellular carcinoma: a dual-input dual-compartment uptake model. Jpn J Radiol 2021;39:1086-96. [Crossref] [PubMed]
12. Geist BK, Xing H, Wang J, Shi X, Zhao H, Hacker M, Sang X, Huo L, Li X. A methodological investigation of healthy tissue, hepatocellular carcinoma, and other lesions with dynamic (68)Ga-FAPI-04 PET/CT imaging. EJNMMI Phys 2021;8:8. [Crossref] [PubMed]
13. He J, Li Y, Wang T, Deng Y, Wang S. Kinetic parameter estimation of hepatocellular carcinoma on (18) F-FDG PET/CT based on Bayesian method. Med Phys 2023;50:2860-71. [Crossref] [PubMed]
14. Calandra R, Seyfarth A, Peters J, Deisenroth MP. An experimental comparison of Bayesian optimization for bipedal locomotion. Proceedings of the 2014 IEEE International Conference on Robotics and Automation (ICRA). Hong Kong; 2014:1951-8.
15. Wang Z, Jegelka S, Kaelbling LP, Lozano-Pérez T. Focused model-learning and planning for non-Gaussian continuous state-action systems. In: Proceedings of the 2017 IEEE International Conference on Robotics and Automation (ICRA). Singapore, 2017:3754-61.
16. Bi W, Kwok JT. Bayes-optimal hierarchical multilabel classification. IEEE Trans Knowl Data Eng 2015;27:2907-18.
17. Xu X, Li J, Zhou MC, Xu J, Cao J. Accelerated two-stage particle swarm optimization for clustering not-well-separated data. IEEE Trans Syst Man Cybern Syst 2018;50:4212-23.
18. Uğuz H. A two-stage feature selection method for text categorization by using information gain, principal component analysis and genetic algorithm. Knowledge-Based Systems 2011;24:1024-32.
19. Ye SQ, Zhou KQ, Zhang CX, Mohd Zain A, Ou Y. An improved multi-objective cuckoo search approach by exploring the balance between development and exploration. Electronics 2022;11:704.
20. Huang SC, Phelps ME, Hoffman EJ, Sideris K, Selin CJ, Kuhl DE. Noninvasive determination of local cerebral metabolic rate of glucose in man. Am J Physiol 1980;238:E69-82. [Crossref] [PubMed]
21. Claeys E, Gancarski P, Maumy-Bertrand M, Wassner H. Dynamic allocation optimization in A/B-tests using classification-based preprocessing. IEEE Trans Knowl Data Eng 2023;35:335-49.
22. Srinivas N, Krause A, Kakade SM, Seeger M. Gaussian process optimization in the bandit setting: No regret and experimental design. arXiv:0912.3995 [Preprint]. 2009. Available online: https://arxiv.org/abs/0912.3995
23. Rasmussen CE, Nickisch H. Gaussian processes for machine learning (GPML) toolbox. J Mach Learn Res 2010;11:3011-5.
24. Li R, Shi Y, Han Y, Shao Y, Qi M, Li B. Active and compact entropy search for high-dimensional Bayesian optimization. IEEE Trans Knowl Data Eng 2023;35:59-70.
25. Ren Z, Li N. Minimizing the Thompson Sampling Regret-to-Sigma Ratio (TS-RSR): a provably efficient algorithm for batch Bayesian Optimization. arXiv:2403.04764v1 [Preprint]. 2024. Available online: https://arxiv.org/html/2403.04764v1
26. Wu J, Liu H, Ye Q, Gallezot JD, Naganawa M, Miao T, Lu Y, Chen MK, Esserman DA, Kyriakides TC, Carson RE, Liu C. Generation of parametric K(i) images for FDG PET using two 5-min scans. Med Phys 2021;48:5219-31. [Crossref] [PubMed]
27. Lyu W, Yang F, Yan C, Zhou D, Zeng X. Batch Bayesian optimization via multi-objective acquisition ensemble for automated analog circuit design. In: Proceedings of the 35th International Conference on Machine Learning. PMLR; 2018:3306-14.
28. D Deb K. Pratap A, Agarwal S, Meyarivan T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 2002;6:182-97.
29. Gábor A, Banga JR. Robust and efficient parameter estimation in dynamic models of biological systems. BMC Syst Biol 2015;9:74. [Crossref] [PubMed]
30. Sadato N, Tsuchida T, Nakaumra S, Waki A, Uematsu H, Takahashi N, Hayashi N, Yonekura Y, Ishii Y. Non-invasive estimation of the net influx constant using the standardized uptake value for quantification of FDG uptake of tumours. Eur J Nucl Med 1998;25:559-64. [Crossref] [PubMed]
31. Basset-Sagarminaga J, van de Weijer T, Iozzo P, Schrauwen P, Schrauwen-Hinderling V. Advances and challenges in measuring hepatic glucose uptake with FDG PET: implications for diabetes research. Diabetologia 2024;67:407-19. [Crossref] [PubMed]
32. Geist BK, Wang J, Wang X, Lin J, Yang X, Zhang H, Li F, Zhao H, Hacker M, Huo L, Li X. Comparison of different kinetic models for dynamic (18)F-FDG PET/CT imaging of hepatocellular carcinoma with various, also dual-blood input function. Phys Med Biol 2020;65:045001. [Crossref] [PubMed]
33. Doot RK, Dunnwald LK, Schubert EK, Muzi M, Peterson LM, Kinahan PE, Kurland BF, Mankoff DA. Dynamic and static approaches to quantifying 18F-FDG uptake for measuring cancer response to therapy, including the effect of granulocyte CSF. J Nucl Med 2007;48:920-5. [Crossref] [PubMed]
34. Kajáry K, Lengyel Z, Tőkés AM, Kulka J, Dank M, Tőkés T. Dynamic FDG-PET/CT in the Initial Staging of Primary Breast Cancer: Clinicopathological Correlations. Pathol Oncol Res 2020;26:997-1006. [Crossref] [PubMed]
35. Dunnwald LK, Doot RK, Specht JM, Gralow JR, Ellis GK, Livingston RB, Linden HM, Gadi VK, Kurland BF, Schubert EK, Muzi M, Mankoff DA. PET tumor metabolism in locally advanced breast cancer patients undergoing neoadjuvant chemotherapy: value of static versus kinetic measures of fluorodeoxyglucose uptake. Clin Cancer Res 2011;17:2400-9. [Crossref] [PubMed]
36. Fahrni G, Karakatsanis NA, Di Domenicantonio G, Garibotto V, Zaidi H. Does whole-body Patlak (18)F-FDG PET imaging improve lesion detectability in clinical oncology? Eur Radiol 2019;29:4812-21. [Crossref] [PubMed]
37. Wang J, Shao Y, Liu B, Wang X, Geist BK, Li X, Li F, Zhao H, Hacker M, Ding H, Zhang H, Huo L. Dynamic (18)F-FDG PET imaging of liver lesions: evaluation of a two-tissue compartment model with dual blood input function. BMC Med Imaging 2021;21:90. [Crossref] [PubMed]
38. Zuo Y, Sarkar S, Corwin MT, Olson K, Badawi RD, Wang G. Structural and practical identifiability of dual-input kinetic modeling in dynamic PET of liver inflammation. Phys Med Biol 2019;64:175023. [Crossref] [PubMed]
39. Amann T, Maegdefrau U, Hartmann A, Agaimy A, Marienhagen J, Weiss TS, Stoeltzing O, Warnecke C, Schölmerich J, Oefner PJ, Kreutz M, Bosserhoff AK, Hellerbrand C. GLUT1 expression is increased in hepatocellular carcinoma and promotes tumorigenesis. Am J Pathol 2009;174:1544-52. [Crossref] [PubMed]
40. He YX, Guo QY. Clinical applications and advances of positron emission tomography with fluorine-18-fluorodeoxyglucose (18F-FDG) in the diagnosis of liver neoplasms. Postgrad Med J 2008;84:246-51. [Crossref] [PubMed]