A regret-alpha ratio based bayesian optimization method for pet / ct kinetic parameter estimation

Abstract

The present application relates to a kind of regret-alpha ratio-based PET / CT kinetics parameter estimation bayesian optimization method.The present application utilizes reversible double-input three-chamber model to model the metabolic activity of liver tissue, and determines optimization target in combination with real TAC data.By introducing regret-alpha ratio and net inflow rate as weight, high and low weight of initial sampling point is distinguished, and different optimization strategies are used respectively, these strategies are integrated into bayesian optimization framework, and the best parameters are modeled and predicted by gaussian process to target function.After preliminary optimization is completed, whether overfitting exists is judged in combination with target function value to parameter estimation result that does not satisfy physiological reasonableness, and multi-objective optimization is carried out to overfitting result.The present application combines mathematical optimization model with physiological parameters, aims at accurately estimating kinetic parameters, more effectively distinguishes hepatocellular carcinoma and background liver tissue, and provides reliable basis for the analysis of hepatocellular carcinoma.

Description

TECHNICAL FIELD

[0001] The application relates to a regret-alpha ratio-based PET / CT kinetic parameter estimation Bayesian optimization method and belongs to the technical field of machine learning and kinetic parameter optimization. BACKGROUND

[0002] Liver cancer is a global health challenge, with an estimated incidence of over 1 million cases by 2025. Hepatocellular carcinoma (HCC) is the most common form of liver cancer, accounting for approximately 90% of cases. Although malignant and benign tumors can be distinguished by traditional imaging techniques such as computed tomography (CT) and magnetic resonance imaging (MRI), the pathological diagnosis and analysis of hepatocellular carcinoma in the clinic still presents considerable challenges. Dynamic PET / CT technology, which reconstructs image sequences at multiple time points to capture real-time tracer distribution changes in the body, provides more accurate and dynamic information for the diagnosis and treatment of hepatocellular carcinoma.

[0003] Estimating kinetic model parameters through optimization algorithms to explore pharmacokinetic models in PET / CT imaging has attracted widespread attention. However, traditional parameter estimation methods mainly focus on finding a fitting optimal point, lack comprehensive exploration of the parameter space, and rely on initial points and fixed sampling strategies, which are prone to local optimal solutions. Therefore, although these methods may be ideal in terms of fitting results, they still have certain limitations in accurately estimating pharmacokinetic parameters that conform to physiological characteristics.

[0004] In terms of technology, the application uses a Bayesian optimization algorithm for pharmacokinetic modeling and parameter estimation of PET / CT data. Currently, there is no research that applies Bayesian optimization to the field of pharmacokinetics and improves it on this basis. SUMMARY

[0005] The application provides a regret-alpha ratio-based PET / CT kinetic parameter estimation Bayesian optimization method for parameter estimation in liver PET / CT kinetic modeling. The application applies different optimization strategies to search the solution space according to the weight, detects and optimizes parameters that do not conform to physiological reasonableness, significantly improves the global search ability of the algorithm, and obtains accurate parameter estimation results that conform to physiological characteristics. The obtained parameter estimation results can significantly assist in distinguishing hepatocellular carcinoma from background liver tissue.

[0006] The technical solution of the application is as follows: a regret-alpha ratio-based PET / CT kinetic parameter estimation Bayesian optimization method, the specific steps of the method are as follows:

[0007] Step 1: Use a reversible two-input three-compartment model to model the metabolic activity of liver tissue and, combined with real TACs data, construct an objective function for optimization.

[0008] Step 2: Weight the initial sampling points to determine the corresponding optimization strategy. The weights are calculated based on the regret-alpha ratio and the net tracer inflow rate at each point to determine whether the parameter space is more prone to exploration or exploitation during the optimization process.

[0009] Step 3: For sampling points with high weights, the lower confidence limit (LCB) is used as the acquisition function to predict the current optimal point and add it to the parameter space, thus favoring exploration. For sampling points with low weights, a multi-objective optimization method combining LCB and regret-α ratio is used to guide the prediction of the optimal point. While minimizing both, more attention is paid to developing the area near the current optimal solution.

[0010] Step 4: Perform overfitting detection on the parameter estimation results of all patients and extract all overfitting parameter results;

[0011] Step 5. For patient data that were detected to be overfitted in the preliminary optimization results, the net inflow rate and regret-α ratio were combined with the lower confidence limit (LCB) into the Bayesian optimization framework, and the multi-objective optimization was re-executed to ultimately obtain more accurate kinetic parameter estimation results.

[0012] Furthermore, in Step 1, the objective function is defined by minimizing the root mean square error (RMSE) between the TAC value predicted by the calculation model and the measured TAC value.

[0013] Furthermore, in Step 2, the regret-α ratio evaluates the contribution of the current sampling point to the optimization result at the current stage by measuring the relative difference between the current sampling point and the optimal solution; the regret-α ratio is related to the tracer net inflow rate K i By multiplying this physiological parameter, the weight value of each sampling point is finally calculated. The optimization strategy adjusts the weight of exploration and development according to the weight value, thereby guiding the search direction of the parameter space and gradually optimizing the accuracy of the solution.

[0014] Furthermore, the calculation formula of the weight W is: W = α RSR (x)×K i ; Among them, α RSR (x) is the regret-α ratio, and Ki represents the net influx rate of the tracer.

[0015] Furthermore, the calculation formula of the regret-α ratio is: Among them, y minμ(x) is the mean function of the Gaussian process probability model of the black-box objective function to be optimized, σ is the standard deviation, k(x, x') is the covariance function.

[0016] Further, in Step 4, those parameter points that obtain excellent RMSE values but the parameter estimation results deviate from the physiological reasonableness are defined as overfitting points.

[0017] Further, in Step 4, the definition formula of the overfitting points is:

[0018]

[0019] Wherein, combined with the clinical actual measurement information type ∈ {tumor, normal}, And K i is the mean value of all patients in the tumor tissue and the normal tissue, respectively, and θ is the threshold value of the RMSE.

[0020] Further, in Step 5, the physiological parameter K i represents 18 The net influx rate of the F-FDG tracer, the uptake of the FDG PET is proportional to the glucose metabolism of the tissue, and is not affected by the measurement time or the input function; therefore, K i is introduced into the Bayesian optimization framework to guide the search of the solution space and effectively correct the parameter estimation results.

[0021] Further, in Step 5, the objective function formula of the multi-objective optimization is:

[0022] For the patient data of the type normal, the objective function formula of the multi-objective optimization is:

[0023]

[0024] For the patient data of the type tumor, the objective function formula of the multi-objective optimization is:

[0025]

[0026] Wherein, D is all the parameter points evaluated in the solution space, α LCB (x; β|D) and α RSR (x|D) are the lower confidence bound LCB and the regret-α ratio value RSR of the parameter points calculated on the basis of the current D, respectively, and λ1 and λ2 represent the weights of the multiple optimization objectives.

[0027] The beneficial effects of the present application are:

[0028] 1. The application uses a reversible double-input three-compartment model to model the metabolic activity of liver tissue, and determines the optimization target in combination with real TAC data;

[0029] 2. The application distinguishes between high and low weight initial sampling points by introducing regret-alpha ratio and net inflow rate as weights, and respectively adopts different optimization strategies, integrates these strategies into the Bayesian optimization framework, and uses Gaussian process to model and predict the optimal parameters of the objective function;

[0030] 3. After preliminary optimization, the application judges whether there is overfitting by combining the parameter estimation results that do not meet the physiological rationality with the objective function value, and performs multi-objective optimization for the overfitting results;

[0031] 4. The application can combine mathematical optimization model with physiological parameters to more accurately estimate pharmacokinetic parameters, significantly distinguish hepatocellular carcinoma from background liver tissue, and provide reliable data support for the evaluation, grading and prognosis analysis of hepatocellular tumor characteristics after resection. BRIEF DESCRIPTION OF DRAWINGS

[0032] Figure 1 is a flowchart of the application;

[0033] Figure 2 is a reversible double-input three-compartment model of the application;

[0034] Figure 3 is the TAC curve fitting result of normal liver tissue of the application and traditional nonlinear least squares method;

[0035] Figure 4 is the TAC curve fitting result of hepatocellular carcinoma lesion of the application and traditional nonlinear least squares method. DETAILED DESCRIPTION

[0036] Example 1: A PET / CT kinetic parameter estimation Bayesian optimization method based on regret-alpha ratio, real PET data of hepatocellular carcinoma patients were used in the experiment, a total of 23 HCC patients were included, among which 21 patients had 1 tumor (including 1 follow-up patient), 1 patient had 2 tumors, and 1 patient had 3 tumors, the tumor size was 1.6cm-17.0cm, and the average was 6.75cm.

[0037] The present invention uses the ordered subset expectation maximization (OSEM) algorithm for PET image reconstruction. After the patient is intravenously injected with 18F-FDG, the first minute of data is reconstructed into 12 frames with an interval of 5 seconds, and the last 4 minutes of data is reconstructed into 4 frames with an interval of 60 seconds; in order to better analyze the metabolism of 18F-FDG in the patient's body using a kinetic model, a 60-minute static PET image of the liver is selected. 17 frames of PET images are obtained, fused, and aligned with the CT image. After all patients complete the PET / CT scan, professional doctors manually outline the region of interest. During the outlining process, interference from the standard uptake value (SUV) of the intrahepatic blood vessels is avoided as much as possible. The TAC consisting of the maximum standard uptake value (SUVmax) of each frame is obtained from the region of interest of the PET / CT image.

[0038] like Figure 2 The reversible two-input three-compartment model shown in Figure 1 can simulate the complex distribution and metabolism of tracers in liver tissue. a represents the proportion of hepatic arterial blood flow, C i (t) represents the tracer in the total blood 18 Functional relationship between F-FDG concentration and time, C a (t) and C v (t) represents the blood in the hepatic artery and portal vein, respectively. 18 The functional relationship between F-FDG concentration and time. 18 The relationship between the concentration of F-FDG and time is expressed as C f (t) indicates liver tissue 18 The relationship between the concentration of F-FDG-6P and time was expressed as C p (t) represents the total blood input function of the model. The total blood input function of the model is obtained by weighted summation of the hepatic artery and portal vein blood supply functions:

[0039] C i (t) = f a ×C a (t)+(1-f a )×C v (t)(1)

[0040] K1 (ml / min / ml) represents 18 The rate constant of F-FDG transport from blood to liver tissue, k2 represents 18 The rate constant of F-FDG returning from liver tissue to blood, k3 represents the rate constant of F-FDG returning from liver tissue to blood. 18 F-FDG is phosphorylated to 18 The rate constant of F-FDG-6P, k4 represents the rate constant of F-FDG-6P catalyzed by phosphokinase 18 Rate constant of F-FDG.

[0041] according toFigure 2 The reversible dual-input three-compartment model can be used to obtain the differential equation:

[0042]

[0043] Solving the differential equation can obtain C T (t), which represents the curve of the tracer concentration in the tissue measured from the PET image over time, is the output function of the kinetic model:

[0044]

[0045] where v b is the blood volume fraction, and a1 and a2 can be described as follows:

[0046]

[0047] The parameter estimation of the liver PET activity curve data is modeled based on the above-mentioned dual-input three-compartment model.

[0048] A regret-a ratio-based Bayesian optimization method for PET / CT kinetic parameter estimation, the specific steps of the method are as follows:

[0049] Step 1, model the metabolic activity of liver tissue using a reversible dual-input three-compartment model, and combine real TAC data to construct an objective function for optimization. In Step 1, the objective function is defined by minimizing the root mean square error RMSE between the TAC values predicted by the calculation model and the measured TAC values.

[0050] Specifically, in Step 1, the objective function is defined as:

[0051]

[0052] where θ is the set of all kinetic parameters to be estimated (K1, k2, k3, k4, f a ,v b ), c i is the real TAC data obtained from n = 17 frames of PET / CT images, C T (θ, t i ) is the data of the above-mentioned dual-input three-compartment kinetic model adopted by the liver.

[0053] Step2, weight calculation for initial sampling points to determine the corresponding optimization strategy. The weight is calculated based on the regret-alpha ratio and tracer net influx rate of each point to determine whether the parameter space is more inclined to exploit or explore during optimization. In Step2, the regret-alpha ratio evaluates the contribution of the current sampling point to the optimization result by measuring the relative difference between the current sampling point and the optimal solution; the regret-alpha ratio and the tracer net influx rate K i This physiological parameter is multiplied to finally calculate the weight value of each sampling point. The optimization strategy adjusts the weight of exploration and exploitation according to the weight value, thereby guiding the search direction of the parameter space and gradually optimizing the accuracy of the solution.

[0054] Specifically, in Step2, first construct the Gaussian process probability model GP(μ, k) of the black box objective function to be optimized, μ(x) is the mean function of the Gaussian process probability model of the black box objective function to be optimized, k(x, x') is the covariance (kernel) function. The regret-alpha ratio is the ratio between the interpolation (regret) of the objective function between the current optimal solution and the predicted mean and the predicted standard deviation (alpha) to ensure that the sampling point can both improve the current solution and provide valuable information. The formula of regret-alpha ratio is:

[0055]

[0056] Where y min is the minimum objective function value in the current solution space, is the standard deviation.

[0057] And K i is composed of a set of kinetic parameters (K1, k2, k3), which is used to reflect 18 the physiological parameters of F-FDG tracer net influx rate, K i The formula of K

[0058]

[0059] The weight of the initial sampling point is calculated as the basis for using different optimization strategies. The formula for calculating the weight is:

[0060] W = α RSR (x) × K i (10)

[0061] Step3, for sampling points with higher weight, use lower confidence limit (LCB) as the acquisition function to predict the current optimal point and add it to the parameter space, so as to be more inclined to explore; for sampling points with lower weight, combine LCB with the multi-objective optimization method of regret-alpha ratio to guide the prediction of the optimal point, and pay more attention to the development of the area near the current optimal solution while minimizing both. The formula of LCB is:

[0062] α LCB (x; β) = μ(x) - βσ(x) (11)

[0063] where the parameter β needs to be heuristically adjusted, in the present invention it is automatically interpolated by the calculated weight size.

[0064] Specifically, in Step 3, the Gaussian process probability model constructed in Step 2 will explore the parameter space according to the high-low weight region strategy and predict the optimal parameters. The sampling points with high weight not only have the potential to significantly improve the current model solution, but also may represent areas with high metabolic activity in physiology, so they are worth further exploration, i.e., minimizing the LCB. In contrast, sampling points with low weight indicate that the current objective function value is already relatively satisfactory, and the improvement potential is limited, so they are more suitable for local fine development. A multi-objective optimization algorithm is used to balance the LCB and the regret-α ratio.

[0065] Step 4, perform overfitting detection on the parameter estimation results of all patients, and extract all overfitting parameter results.

[0066] Specifically, in Step 4, those parameter points that obtain excellent RMSE values but deviate significantly from the physiological reasonableness are defined as overfitting points. The definition of overfitting points is:

[0067]

[0068] where type ∈ {tumor, normal} combines the clinical actual measurement information, and are the mean values of K i in tumor tissue and normal tissue of all patients, respectively, and θ is the threshold value of RMSE.

[0069] Step 5, for the patient data detected as overfitting in the preliminary optimization results, combine the net influx rate and the regret-α ratio with the lower confidence limit (LCB) into the Bayesian optimization framework, and re-execute the multi-objective optimization to finally obtain more accurate kinetic parameter estimation results.

[0070] Specifically, in Step 5, the physiological parameter K i represents 18 the net influx rate of F-FDG tracer. The uptake of FDG PET is proportional to the glucose metabolism of the tissue and is not affected by the measurement time or the input function. Therefore, K i is introduced into the Bayesian optimization framework to guide the search of the solution space, which can effectively correct the parameter estimation results. The objective function formula of multi-objective optimization is:

[0071] For patient data of type normal:

[0072]

[0073] For patient data of type tumor:

[0074]

[0075] where D is all the parameter points evaluated in the solution space so far, a LCB (x; β | D) and a RSR (x | D) are the lower confidence bound (LCB) and the regret-a ratio value (RSR) of the parameter point calculated based on the current D, respectively, and λ1 and λ2 represent the weights of multiple optimization objectives.

[0076] According to the experimental procedure, the TAC curve fitting results of the present application and the conventional nonlinear least squares method are shown in Figs. Figure 3 and Figure 4 Compared with the conventional method, the method described in the present application can be closer to the real measurement data at each time point, thereby achieving more accurate fitting. More specific parameter estimation results and experimental evaluation indexes are shown in Table 1:

[0077] Table 1 Experimental results of the present application

[0078] Kinetic parameters [K1] <k2> [ k3 ] [ k4 ] fa (%) vb RMSE HCCs 1.307±0.293 1.535±0.232 0.04±0.069 0.11±0.088 82.4±20.6 0.062±0.059 1.226±0.581 Normal liver tissue 1.062±0.418 1.103±0.447 0.005±0.016 0.168±0.065 29.3±18.1 0.024±0.012 1.051±1.026 P 0.016 <0.001 0.014 0.008 <0.001 0.002 /

[0079] Independent sample T test is a statistical test method for comparing whether there is a significant difference in the mean values of two independent samples. The test outputs a P value for two sample data, and when P < 0.05, it represents a significant difference, indicating that the liver cancer tissue and the normal liver tissue can be well distinguished by the parameter result value. The table shows that all the kinetic parameter estimation results of the present application have significant differences. Among them 18 F-FDG-6P can be dephosphorylated to 18 F-FDG, and this metabolic process occurs more frequently in normal liver tissue, so the k4 value in normal liver tissue will be greater than that in hepatocellular carcinoma tissue. While the K1, k2, k3 and v b parameter values in normal liver tissue will be less than those in hepatocellular carcinoma tissue.

[0080] The liver organ needs double input including the hepatic artery and the portal vein to accurately characterize its blood supply. Normal liver tissue receives 70%-80% of blood supply from the portal vein, while hepatocellular carcinoma tissue is mainly supplied by the hepatic artery. Compared with normal liver tissue, the f a parameter value in hepatocellular carcinoma tissue will be greater.

[0081] All parameter estimation results in the application can accurately reflect physiological characteristics while ensuring low RMSE, and can significantly distinguish hepatocellular carcinoma tissue from normal liver tissue. The method provides reliable data support for evaluation, grading and prognosis analysis after resection of hepatocellular tumor characteristics.

[0082] The specific embodiments of the application are described in detail above, but the application is not limited to the above-mentioned embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the purpose of the application.

Claims

1. A regret-a-ratio based PET / CT kinetic parameter estimation Bayesian optimization method, characterized in that, The specific steps of the method are as follows: Step 1, the metabolic activity of liver tissue is modeled using a reversible double-input three-compartment model, and a real TAC data is combined to construct an objective function for optimization; Step 2, weight calculation is performed on the initial sampling points to determine the corresponding optimization strategy; The weight is calculated based on the regret-alpha ratio and the tracer net inflow rate of each point to determine whether the parameter space is more inclined to develop or explore during the optimization process; Step 3, for sampling points with high weight, lower confidence limit LCB is used as the acquisition function to predict the current optimal point and add parameter space, so as to be more inclined to explore; for sampling points with low weight, a multi-objective optimization method combining LCB and regret-alpha ratio is used to guide the prediction of the optimal point, and in the process of minimizing both, more attention is paid to the area near the current optimal solution; Step 4, overfitting detection is performed on the parameter estimation results of all patients, and all overfitting parameter results are extracted; Step 5, for patient data detected as overfitting in the preliminary optimization results, the net inflow rate and regret-alpha ratio are combined with the lower confidence limit LCB to be integrated into the Bayesian optimization framework, and multi-objective optimization is performed again to obtain more accurate kinetic parameter estimation results; The formula for calculating the regret-a ratio is: wherein is the minimum objective function value in the current solution space, is the mean function of the Gaussian process probability model of the black-box objective function to be optimized, is the standard deviation, is the covariance function; In Step 5, the objective function formula of multi-objective optimization is: For patient data of type normal, the objective function formula of multi-objective optimization is: ； For patient data of type tumor, the objective function formula of multi-objective optimization is: ； wherein, is the set of all evaluated parameter points in the solution space, and are the lower confidence bound LCB and the regret-α ratio value RSR of the parameter point calculated on the basis of the current respectively, and represent the weights of the plurality of optimization objectives.

2. The regret-a-ratio based PET / CT kinetic parameter estimation Bayesian optimization method of claim 1, wherein: In Step 1, the objective function is defined by minimizing the root mean square error RMSE between the TAC value predicted by the calculation model and the measured TAC value.

3. The regret-a-ratio based PET / CT kinetic parameter estimation Bayesian optimization method of claim 1, wherein: In Step 2, the regret-a ratio evaluates the contribution of the current sampling point to the optimization result by measuring the relative difference between the current sampling point and the optimal solution; the regret-a ratio is related to the tracer net inflow rate K i This physiological parameter is multiplied to finally calculate the weight value of each sampling point, and the optimization strategy adjusts the weight of exploration and development according to the weight value, thereby guiding the search direction of the parameter space and gradually optimizing the accuracy of the solution.

4. The regret-a-ratio based PET / CT kinetic parameter estimation Bayesian optimization method of claim 1, wherein: The formula for calculating the weight W is: ; wherein, is the regret-α ratio, and Ki represents the tracer net inflow rate.

5. The regret-a-ratio based PET / CT kinetic parameter estimation Bayesian optimization method of claim 1, wherein: In Step 4, those parameter points that obtain excellent RMSE values but whose parameter estimation results deviate significantly from physiological reasonableness are defined as overfitting points.

6. The regret-a-ratio based PET / CT kinetic parameter estimation Bayesian optimization method of claim 5, wherein: In Step 4, the definition formula of overfitting points is: ； Wherein, combine with clinical actual measurement information , and are the mean of K i of all patients in tumor tissue and normal tissue respectively, is the threshold of RMSE.

7. The regret-a-ratio based PET / CT kinetic parameter estimation Bayesian optimization method of claim 1, wherein: The physiological parameter K i represents 18 The net influx rate of the F-FDG tracer, FDG PET uptake is proportional to the glucose metabolism of the tissue, and is not affected by the measurement time or the input function; therefore, K i A Bayesian optimization framework is introduced to guide the search of the solution space and effectively correct the parameter estimation results.

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