Model Selection Based on Population Fitting at an Example of ¹⁷⁷Lu-PSMA Kinetics in Kidneys with a Low Number of Data

Aim: Model selection is important to ensure that the used function could approximate the reality given by the observed data [1,2]. In this study, we develop an algorithm to be used to perform model selection for the determination of the time-integrated activity coefficients (TIACs) and demonstrate it at an example of ¹⁷⁷Lu-PSMA kidneys biokinetics. The algorithm is based on population fitting, i.e. simultaneous fitting of all patients together, and is specifically advantageous for cases with a low number of available biokinetic data.

Methods: Biokinetic data of ¹⁷⁷Lu-PSMA in kidneys were collected from 13 patients with metastatic prostate cancer. Twenty exponential functions derived from various parameterizations of a mono- (A1e^-λ1+λphyst) and a bi-exponential function (A1e^-λ1+λphyst+A2e^-λphyst ) were used as the model set for the model selection. The parameters of the functions with different combinations of shared parameters and individual parameter estimations were fitted to the data simultaneously. The goodness of the fits (visual inspection and coefficient of variation CV<50% [3]) were used to test the quality of the fits. The corrected Akaike Information Criterion (AICc) weight [2] was used to select the fit function most supported by the data from the set of functions with acceptable goodness of fit.

Results: The function A1 α e^-λ1+λphyst+A1 1-α e^-λphyst with a shared parameter α was selected as the model best supported by the data (AICc weight=97%) from those fit functions found to have an acceptable fit based on the goodness of fit criteria. In this function, the A1 and λ1 parameters were fitted individually for each patient while parameter α was fitted as a shared parameter in the population with the estimated value of 0.963±0.004.

Conclusions: An algorithm to define an adequate fit function (to be used for future patients) based on a relatively low number of data was developed. Based on our results, the function A1 α e^-λ1+λphyst+A1 1-α e^-λphyst with a fix parameter α can be used to estimate the TIACs for patients with a low number of data, e.g. three biokinetic data, with an individual estimation of A1 and λ1 and fixing α to 0.963. References: 1. Burnham KP, Anderson DR. Model Selection and Multimodel Inference: Springer-Verlag New York; 2002. 2. Glatting G, Kletting P, Reske SN, Hohl K, Ring C. Choosing the optimal fit function: comparison of the Akaike information criterion and the F-test. Med Phys. 2007;34(11):4285-92. 3. Kletting P, Schimmel S, Kestler HA, Hänscheid H, Luster M, Fernandez M, Bröer JH, Nosske D, Lassmann M, Glatting G. Molecular radiotherapy: the NUKFIT software for calculating the time-integrated activity coefficient. Med Phys. 2013;40(10):102504.