Single-time-point dosimetry using model selection and nonlinear mixed-effects modelling: a proof of concept

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Abstract Purpose This project aims to develop and evaluate a method for accurately determining time-integrated activities (TIAs) in single-time-point (STP) dosimetry for molecular radiotherapy. It performs a model selection (MS) within the framework of the nonlinear mixed-effects (NLME) model (MS–NLME). Methods Biokinetic data of [ 111 In]In-DOTATATE activity in kidneys at T 1 = (2.9 ± 0.6) h, T 2 = (4.6 ± 0.4) h, T 3 = (22.8 ± 1.6) h, T 4 = (46.7 ± 1.7) h, and T 5 = (70.9 ± 1.0) h post injection were obtained from eight patients using planar imaging. Eleven functions were derived from various parameterisations of mono-, bi-, and tri-exponential functions. The functions’ fixed and random effects parameters were fitted simultaneously (in the NLME framework) to the biokinetic data of all patients. The Akaike weights were used to select the fit function most supported by the data. The relative deviations (RD) and the root-mean-square error (RMSE) of the calculated TIAs for the STP dosimetry at T 3 = (22.8 ± 1.6) h and T 4 = (46.7 ± 1.7) h p.i. were determined for all functions passing the goodness-of-fit test. Results The function $$f_{4d} \left( t \right) = A_{1} /\left\{ {\left( {\frac{1 - \alpha }{{\lambda_{1} + \lambda_{{{\text{phys}}}} }}} \right) - \left( {\frac{\alpha }{{\lambda_{2} + \lambda_{{{\text{phys}}}} }}} \right) - \left( {\frac{1 - 2\alpha }{{\lambda_{bc} + \lambda_{{{\text{phys}}}} }}} \right)} \right\} \cdot e^{{ - \lambda_{{{\text{phys}}}} t}} \cdot \left\{ {\left( {1 - \alpha } \right) \cdot e^{{ - \lambda_{1} t}} - \alpha \cdot e^{{ - \lambda_{2} t}} - \left( {1 - 2\alpha } \right) \cdot e^{{ - \lambda_{bc} t}} } \right\}$$ f 4 d t = A 1 / 1 - α λ 1 + λ phys - α λ 2 + λ phys - 1 - 2 α λ bc + λ phys · e - λ phys t · 1 - α · e - λ 1 t - α · e - λ 2 t - 1 - 2 α · e - λ bc t with four adjustable parameters and $$\lambda_{bc} = \frac{{{\text{ln}}\left( 2 \right)}}{{1\;{\text{ min}}}}$$ λ bc = ln 2 1 min was selected as the function most supported by the data with an Akaike weight of (45 ± 6) %. RD and RMSE values show that the MS–NLME method performs better than functions with three or five adjustable parameters. The RMSEs of TIA NLME–PBMS and TIA 3-parameters were 7.8% and 10.9% (for STP at T3), and 4.9% and 10.7% (for STP at T4), respectively. Conclusion An MS–NLME method was developed to determine the best fit function for calculating TIAs in STP dosimetry for a given radiopharmaceutical, organ, and patient population. The proof of concept was demonstrated for biokinetic 111 In-DOTATATE data, showing that four-parameter functions perform better than three- and five-parameter functions.