An improved optimization algorithm of the three-compartment model with spillover and partial volume corrections for dynamic FDG PET images of small animal hearts in vivo

Abstract

The three-compartment model with spillover (SP) and partial volume (PV) corrections has been widely used for noninvasive kinetic parameter studies of dynamic 2-[18F] fluoro-2deoxy-D-glucose (FDG) positron emission tomography images of small animal hearts in vivo. However, the approach still suffers from estimation uncertainty or slow convergence caused by the commonly used optimization algorithms. The aim of this study was to develop an improved optimization algorithm with better estimation performance. Femoral artery blood samples, image-derived input functions from heart ventricles and myocardial time-activity curves (TACs) were derived from data on 16 C57BL/6 mice obtained from the UCLA Mouse Quantitation Program. Parametric equations of the average myocardium and the blood pool TACs with SP and PV corrections in a three-compartment tracer kinetic model were formulated. A hybrid method integrating artificial immune-system and interior-reflective Newton methods were developed to solve the equations. Two penalty functions and one late time-point tail vein blood sample were used to constrain the objective function. The estimation accuracy of the method was validated by comparing results with experimental values using the errors in the areas under curves (AUCs) of the model corrected input function (MCIF) and the 18F-FDG influx constant K _i . Moreover, the elapsed time was used to measure the convergence speed. The overall AUC error of MCIF for the 16 mice averaged -1.4 ± 8.2%, with correlation coefficients of 0.9706. Similar results can be seen in the overall K _i error percentage, which was 0.4 ± 5.8% with a correlation coefficient of 0.9912. The t-test P value for both showed no significant difference. The mean and standard deviation of the MCIF AUC and K _i percentage errors have lower values compared to the previously published methods. The computation time of the hybrid method is also several times lower than using just a stochastic algorithm. The proposed method significantly improved the model estimation performance in terms of the accuracy of the MCIF and K _i , as well as the convergence speed.

Figure1.

Illustration of representative PET data distribution around the peak and typical model fitting curve with bias. It also shows that the circles with dense distribution has better model curve fitting.

Figure 2.

Flow chart of the proposed hybrid method including Artificial Immune System and Interior Reflective Newton algorithms. Steps 4–6 indicate the improved algorithm with ‘fmincon’, using initial values optimized by AIS, which removed the uncertainty of ‘fmincon’ and enhanced the convergence performance of normal AIS.

Figure 3.

Representative plot of MCIF upper limit for a combination of the parameters A₁, A₂, A₃, L₁, L₂, L₃, τ with values within the range indicated in Table 1. This results in a MCIF with a peak value 80 MBq/mL, indicating the selected bounds can cover the maximal peak of MCIF based on the injected dose for mice.

Figure 4.

Representative plot of estimated result. (a) Measured blood samples (circles with dash line) and estimated input (solid line) for the first 9 minutes. (b) Measured blood samples (circles) and estimated input (solid line) for the entire scan duration. (c) The PET image derived and model estimated time activity curves of blood and myocardium.

Figure 5.

Representative plots showing the effects of constraints. (a) Peak fitting accuracies of model blood with constraint O₂(p) (solid line) and without constraint O₂(p) (dotted line). (b) Ki error drop at iteration 100 when O₃(p) is introduced.

Figure 6.

(a) Correlation plot of the AUCs of gold standard input function (IF) versus the estimated IF (r = 0.9706). (b) Correlation plot of the Ki using gold standard IF versus Ki using estimated IF (r = 0.9912)