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

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 [111In]In-DOTATATE activity in kidneys at T1 = (2.9 ± 0.6) h, T2 = (4.6 ± 0.4) h, T3 = (22.8 ± 1.6) h, T4 = (46.7 ± 1.7) h, and T5 = (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 T3 = (22.8 ± 1.6) h and T4 = (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 TIANLME–PBMS and TIA3-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 111In-DOTATATE data, showing that four-parameter functions perform better than three- and five-parameter functions.

Mathematical model to calculate the total number of decays in Peptide Receptor Radionuclide Therapy using nonlinear mixed effect modelling

“…A study on the development of a simplified dosimetry method using NLME was also conducted by Hardiansyah et al in 2023 on 111 In-DOTATATE pharmacokinetic data [10]. As a result, the method developed by Hardiansyah et al by applying model selection to NLME modeling has better TIAs accuracy than the method of Devasia et al [10]. These results also provide an understanding of the importance of model selection in determining the best model capable of describing pharmacokinetic data.…”

“…The Sum of Exponentials (SOE) was developed with the idea that the pharmacokinetics of a radiopharmaceutical in the body decays exponentially due to physical and biological decay. The SOE functions with 3, 4, and 5 parameters were employed in this study to fit the pharmacokinetics of 177 Lu-DOTATATE in kidneys Equation (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11).…”

“…where fi is a fit function, i is the total number of the fitted parameters, Aj and 𝜆 𝑗 (j = 1,2,3) are the fit function prefactors and the radiopharmaceutical biological decay constants with values ≥ 0. 𝜆 𝑝ℎ𝑦𝑠 is the physical decay constant calculated from the half-life (T1/2) of 177 Lu (𝜆 𝑝ℎ𝑦𝑠 = ln(2)/ T1/2= 7.24 x 10-5 min -1 ), 𝜆 𝑏𝑐 is the blood circulation rate at the time of uptake by the kidney with a half-life of 1 min (𝜆 𝑏𝑐 = ln(2)/ 1 min), and 𝛼 is the fractional contribution value of the corresponding exponential term with a value between 0 and 1 [10]. Each function has a varied physical meaning depending on its constituent parameters and was used for fitting pharmacokinetic data in units of percent injected activity (%IA).…”

“…where σ 2 is the variance in the logarithmic transformation [15]. Furthermore, the AICc and the weighting of each function that met the goodness of fit criteria were calculated using the following equation [10,12]:…”

“…As a result, this method has better accuracy in calculating TIAs compared to other simplified dosimetry studies [9]. A study on the development of a simplified dosimetry method using NLME was also conducted by Hardiansyah et al in 2023 on 111 In-DOTATATE pharmacokinetic data [10]. As a result, the method developed by Hardiansyah et al by applying model selection to NLME modeling has better TIAs accuracy than the method of Devasia et al [10].…”