The effects of boundary constrained LM fitting methods on compartmental model analysis

Introduction: Compartmental modeling is widely used in PET dynamic quantitative analysis. The standard kinetic parameters, such as V_b, K₁, k₂, k₃, and k₄ (k₄ = 0 for FDG) of two-tissue compartmental model (2TCM), can be derived from nonlinear regression of a time activity curve (TAC). The Levenberg-Marquardt (LM) method is commonly applied to derive the least square problem in the nonlinear compartmental model analysis. However, the LM method is unconstrained in fitting parameters boundary, which may cause meaningless results such as a negative V_b. This study proposed a boundary constrained method based on bounded function and compared with the LM method. Methods: LM algorithm is a nonlinear iterative fitting method. A common boundary constrained method (CBC) when parameters exceed the boundary is preserve the last iteration parameters and change update conditions. This method is easy to fall into local optimum. The proposed boundary constrained method was based on bounded function. Absolute value function (ABS), square function (SQE), exponential function (EXP) and arctangent function (ACT) was tested in this work. Parameter will be replaced by a bounded function if it is nonnegative. Total-body dynamic patient scans using uEXPLORER were used to evaluate this approach. The dataset of dynamic scan was divided into 100 frames: 60×1 sec, 30×5 sec, 10×12 sec, 5×60 sec and 25×60 sec. Dynamic data were reconstructed using a 3D TOF-OSEM algorithm, and the number of iterations was set to be 3 with 20 subsets. The image size was 150×150×673 with 4×4×2.89-mm3 voxel size. Simulated TACs were generated by 2TCM. Six different parameter groups were chosen with the same framing of clinical dataset. Gaussian noise with zero mean and uptake-dependent variance was added to simulate noisy TACs. Results: Two-tissue irreversible compartmental model (2TiCM) was used for parametric estimation. In unconstrained LM method test, all estimated parameters were positive under noise free condition. When testing noisy TACs, V_b and k₃ were negative in two cases. With the proposed non-negative constraints, using bounded functions. The estimated V_b and k₃ were positive while most of other parameters retain the same or closer to true value compare with LM method. There was an exception that the k₃ of liver TAC had a larger relative bias from true value, but the absolute bias was small. Moreover, CBC method fell into local optimum in noise free TACs of liver and spleen, which had larger bias compare to LM method. In clinical 2TiCM parametric imaging calculation, absolute value function was used to replace V_b and k₃. V_b, K₁, k₂, k₃ image was estimated. There were fewer outliers and reduced salt-and-pepper noise in K₁ and K_i with bounded function. Conclusions: This study demonstrated that bounded function method can restrict fitting boundary. The absolute value function can retain a same or lower fitting bias compare to LM method and limiting boundary in iterations. It also shows that this method can achieve better image quality for kinetic image modeling compare to LM fitting method.