Biokinetic data of [177Lu]Lu-PSMA-I&T in kidneys
Thirteen patients with metastatic castration-resistant prostate cancer were included in this retrospective analysis [13, 14]. All patients underwent [177Lu]Lu-PSMA-I&T radioligand therapy (RLT) and post-therapeutic planar whole-body scintigraphies. The biokinetic data (the time-activity data) of [177Lu]Lu-PSMA-I&T RLT in kidneys were calculated from the kidneys regions of interest using the geometric mean of anterior and posterior counts with background corrections. From thirteen patients, 3 patients had 5 time points data, 1 patient had 4 time points data and 9 patients had 3 time points data. The biokinetic data were obtained at (1.1 ± 0.7) h, (20.7 ± 2.3) h, (51.0 ± 10.1) h, (92.3 ± 47.2) h, (163.8 ± 2.1) h p.i..
Investigated set of exponential functions
Sums of exponential functions with increasing complexity were used in the investigated model set, as such mathematical functions are commonly used to describe biological processes [6,7,8,9]:
$$f_{2a} \left( t \right) = 100 e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} - 100 e^{{ - \left( {\lambda_{2} + \lambda_{{{\text{phys}}}} } \right)t}}$$
(1)
$$f_{2b} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}}$$
(2)
$$f_{2c} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} - A_{1} e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}}$$
(3)
$$f_{2d} \left( t \right) = - A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{1} e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}}$$
(4)
$$f_{2e} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + (100 - A_{1} ) e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}}$$
(5)
$$f_{3a} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{2} e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}}$$
(6)
$$f_{3b} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} - A_{1} e^{{ - \left( {\lambda_{2} + \lambda_{{{\text{phys}}}} } \right)t}}$$
(7)
$$f_{3c} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + \left( {100 - A_{1} } \right) e^{{ - \left( {\lambda_{2} + \lambda_{{{\text{phys}}}} } \right)t}}$$
(8)
where \(f_{ia}\) is a fit function with \(i\) parameters, the \(A_{i} \ge 0\) are the prefactors, \(\lambda_{{{\text{phys}}}}\) is the physical decay constant of the radionuclide calculated from the half-life \(T_{1/2}\) of 177Lu \(\left( {\lambda_{{{\text{phys}}}} = \ln \left( 2 \right)/T_{1/2} } \right)\) and \(\lambda_{1}\) and \(\lambda_{2}\) describe the biological clearance rates of the radiopharmaceutical. In addition, the following functions were also used which were defined in analogy to the case of degenerate eigenvalues for a damped oscillator (note the additional factor t):
$$f_{3d} \left( t \right) = A_{1} t e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{2} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}}$$
(9)
$$f_{2a,3d} \left( t \right) = A_{1} t e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}} + A_{2} e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}}$$
(10)
$$f_{2b,3d} \left( t \right) = A_{1} t e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}}$$
(11)
$$f_{2c,3d} \left( t \right) = A_{1} t e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + 100 e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}}$$
(12)
The three functions (10)–(12) are derived from Eq. (9) by reducing the number of fit parameters. In addition to the functions in Eqs. (1)–(12), we examined the functions below using all biokinetic data of the patient population and a shared parameter approach. The shared parameters are assumed to be the same for all patients and are estimated for all data in the patient population together. The other parameters were individually estimated from the data. All the following functions are derived from function \(f_{3a}\) (Eq. (6)) with different shared parameters (Eqs. (13)–(15)) and different parameterizations (Eqs. (16)–(18)):
$$f_{3aS1} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{2} e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}} \;{\text{with}}\,{\text{shared}}\,{\text{parameter}}\;A_{1}$$
(13)
$$f_{3aS2} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{2} e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}} \;{\text{with}}\,{\text{shared}}\,{\text{parameter}}\;\lambda_{1}$$
(14)
$$f_{3aS3} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{2} e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}} \;{\text{with}}\,{\text{shared}}\,{\text{parameter}}\;A_{2}$$
(15)
$$f_{3aS4} \left( t \right) = A_{1} \beta e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{1} \left( {1 - \beta } \right) e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}} \;{\text{with}}\,{\text{shared}}\,{\text{parameter}}\;\beta$$
(16)
$$f_{3aS5} \left( t \right) = A_{1} \beta e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{1} \left( {1 - \beta } \right) e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}} \;{\text{with}}\,{\text{shared}}\,{\text{parameter}}\;A_{1}$$
(17)
$$f_{3aS6} \left( t \right) = A_{1} \beta e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{1} \left( {1 - \beta } \right) e^{{ - \left( {\lambda_{{{\text{phys}}}} } \right)t}} \;{\text{with}}\,{\text{shared}}\,{\text{parameter}}\;\lambda_{1}$$
(18)
where parameters \(\beta\) are the fractional contributions of the corresponding exponentials with values constrained between 0 and 1. The index \(S\) refers to a shared parameter. For completeness, the following exponential functions with one and four estimated parameters were also analysed:
$$f_{1} \left( t \right) = A_{1} e^{{ - \lambda_{{{\text{phys}}}} { }t}}$$
(19)
$$f_{4} \left( t \right) = A_{1} e^{{ - \left( {\lambda_{1} + \lambda_{{{\text{phys}}}} } \right)t}} + A_{2} e^{{ - \left( {\lambda_{2} + \lambda_{{{\text{phys}}}} } \right)t}}$$
(20)
Data fitting
All functions (Eqs. (1)–(20)) were fitted to the biokinetic data of kidneys using the IBMS and the PBMS approaches with all parameters being constrained to positive values. The fittings were performed using the simulation analysis and modelling software SAAMII v.2.3 (The Epsilon Group, Charlottesville, VA, USA) [15]. The following computational settings were used for the fittings: Rosenbrock algorithm, convergence criterion 10–4, and absolute-based variance model with a fractional standard deviation of 0.15 [15].
The goodness of the fits were checked by visual inspection of the fitted graphs, the coefficient of variation CV of the fitted parameters (< 0.5) and the off-diagonal values of the correlation matrix (-0.8 < CM < 0.8 for most elements) according to the compilation in Table 1 in Ref. [8].
Model selection
To select which function is most supported by the data, the corrected Akaike Information Criterion \(AICc\), which is corrected for a low ratio of the number of data \(N\) to the number of parameters \(K\), i.e. N/K < 40 [11], and the corresponding Akaike weights [11] were calculated as follows:
$$AICc = - 2\ln \left( P \right) + 2K + \frac{{2K\left( {K + 1} \right)}}{N - K - 1}$$
(21)
$$\Delta _{i} = AICc_{i} - AICc_{\min }$$
(22)
$$w_{{AICc_{i} }} = {{e^{{ - \frac{{\Delta _{i} }}{2}}} } \mathord{\left/ {\vphantom {{e^{{ - \frac{{\Delta _{i} }}{2}}} } { \mathop \sum \limits_{i = i}^{F} e^{{ - \frac{{\Delta _{i} }}{2}}} }}} \right. \kern-\nulldelimiterspace} { \mathop \sum \limits_{i = i}^{F} e^{{ - \frac{{\Delta _{i} }}{2}}} }}$$
(23)
where \(P\) is the estimated objective function minimized for the fitting, \(AICc_{\min }\) is the lowest \(AICc\) value of all fitted functions, \(\Delta _{i}\) is the difference between the \(AICc_{i}\) of function \(i\) and \(AICc_{\min }\), \(F\) is the total number of investigated functions and \(w_{{AICc_{i} }}\) is the Akaike weight of function \(i\). The Akaike weights indicate the probability that the model is the best among the whole set of considered models [11].
From those functions which passed the goodness-of-fit test (“Data fitting” section), the functions with an Akaike weight > 0.05 were selected as the functions most supported by the data. These were used to determine the area under the curve of the time-activity curve of [177Lu]Lu-PSMA-I&T RLT in kidneys.
Workflow
In the proposed PBMS method, the parameters of Eqs. (1)–(12) were fitted to the kidneys biokinetic data of the population (13 patients). To investigate if the data of the patients could be described by shared parameters, the population fitting was performed to estimate the parameters of functions in Eqs. (13) to (18) with shared parameter estimation. Model selections were performed using the Akaike weights (“Data fitting” section).
In addition to the PBMS method, we also performed the IBMS method [8, 9] using the functions in Eqs. (1)–(12) for patients P1, P3 and P4, for who five biokinetic measurement data points are available. The minimum number of data points for AICc-based model selection is equal to the number of adjustable parameters Kmax + 2 as seen from Eq. (21). Therefore, only for these 3 patients all functions with up to 3 parameters could be used. The best model obtained from the IBMS method of these patients was then used to calculate the TIAs of the [177Lu]Lu-PSMA-I&T in all thirteen patients. The performance of the functions selected as most supported by the data using the PBMS and IBMS approach, respectively, was evaluated based on the visual inspection of the fitted graphs. In addition, the relative deviation RD between the TIAs from both approaches was also compared and analysed. The Jackknife method was used to analyse the stability of the best model selected through model selection [11, 16]: For this purpose, the leave-one-out method was applied 13 times with only 12 patients for the calculation of the Akaike weights. The Jackknife was applied to check if the output of the model selection from both PBMS and IBMS would change for different set of data (i.e. leaving one patient out 13 times) used in the analysis.