A population‑based method to determine the time‑integrated activity in molecular radiotherapy

Abstract

Background The calculation of time-integrated activities (TIAs) for tumours and organs is required for dosimetry in molecular radiotherapy. The accuracy of the calculated TIAs is highly dependent on the chosen fit function. Selection of an adequate function is therefore of high importance. However, model (i.e. function) selection works more accurately when more biokinetic data are available than are usually obtained in a single patient. In this retrospective analysis, we therefore developed a method for population-based model selection that can be used for the determination of individual time-integrated activities (TIAs). The method is demonstrated at an example of [177Lu]Lu-PSMA-I&T kidneys biokinetics. It is based on population fitting and is specifically advantageous for cases with a low number of available biokinetic data per patient.

Methods Renal biokinetics of [177Lu]Lu-PSMA-I&T from thirteen patients with metastatic castration-resistant prostate cancer acquired by planar imaging were used. Twenty exponential functions were derived from various parameterizations of mono- and bi-exponential functions. The parameters of the functions were fitted (with different combinations of shared and individual parameters) to the biokinetic data of all patients. The goodness of fits were assumed as acceptable based on visual inspection of the fitted curves and coefficients of variation CVs < 50%. The Akaike weight (based on the corrected Akaike Information Criterion) 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 A1βe−(λ1+λphys)t+A1(1−β)e−(λphys)t with shared parameter β was selected as the function most supported by the data with an Akaike weight of 97%. Parameters A1 and λ1 were fitted individually for every patient while parameter β was fitted as a shared parameter in the population yielding a value of 0.9632 ± 0.0037.

Conclusions The presented population-based model selection allows for a higher number of parameters of investigated fit functions which leads to better fits. It also reduces the uncertainty of the obtained Akaike weights and the selected best fit function based on them. The use of the population-determined shared parameter for future patients allows the fitting of more appropriate functions also for patients for whom only a low number of individual data are available.

Background Individual treatment planning is desirable for radionuclide therapy to maximize tumour absorbed dose while sparing organs at risk [1,2,3]. The absorbed doses are determined for the largest part by the time-integrated activities (TIAs) [4, 5]. The TIAs are equal to the number of disintegrations of the used radionuclide in the considered organ. To calculate the TIAs, a mathematical function is first fitted to the measured biokinetic data obtained from 2D or 3D imaging at multiple time points [6,7,8,9], and this function is then integrated from time zero to infinity. The calculated TIA values based on this fitting method depend on the chosen fit function [10]. Therefore, using the “optimal” fit function [11] is crucial for the accurate and precise determination of the TIAs and subsequently the absorbed doses. Relevant criteria for an optimal fit function are that.

(1) the investigated function fits the data, i.e. the goodness of fit is satisfactory, and

(2) the function is most supported by the observed data. "Most" here refers to a set of reasonable functions defined by the investigator.

While item (1) can be easily checked by applying standard criteria such as visual inspection of the fitted graph, quantitative assessment using coefficient of variations of the fitted parameters (< 50%) and the constraints for the correlation matrix elements (absolute values being lower than 0.8) [8], item (2) requires model (or function) selection based on quantitative analysis of the corrected Akaike information criterion (AICc) [11, 12].

Model selection has two inputs: On the one hand the set of models and on the other hand the underlying observed data. The former, however, depends on the latter, as few data only allow the use of models (or corresponding functions) with few parameters.

In nuclear medicine, the measurement of biokinetics is often only carried out at a few time points. Therefore, instead of using the data of only a single patient, i.e. individual-based model selection (IBMS), including the data of additional patients with the same disease treated with the same radiopharmaceutical might be important to determine an optimal fit function (item (2) above). Such a population-based model selection (PBMS) increases the ratio of number of observed data used as input to the number of estimated parameters and thus reduces the uncertainty in the model selection. Moreover, it allows to use an expanded model set, as functions with a higher number of parameters become possible. In addition, information about the functional shape of the time-activity curve of previous patients might be used for future patients.

In this work, we therefore present a general method to improve the calculation of TIAs using biokinetic data of a population instead of a single patient only. The method performs the required model selection based on a PBMS approach and is presented at the example of kidneys biokinetics in [177Lu]Lu-PSMA-I&T radioligand therapy. For this purpose, a set of mathematical models or functions is defined, a population-based fit is performed and the function most supported by the data is selected using the Akaike weights method. The developed method can be used to determine individual TIAs of future patients using the best function obtained from a previously measured population