On the correction for radioactive decay in pharmacokinetic modeling

Population-based model selection for an accurate estimation of time-integrated activity using non-linear mixed-effects modelling

PURPOSE

Personalized treatment planning in Molecular Radiotherapy (MRT) with accurately determining the absorbed dose is highly desirable. The absorbed dose is calculated based on the Time-Integrated Activity (TIA) and the dose conversion factor. A crucial unresolved issue in MRT dosimetry is which fit function to use for the TIA calculation. A data-driven population-based fitting function selection could help solve this problem. Therefore, this project aims to develop and evaluate a method for accurately determining TIAs in MRT, which performs a Population-Based Model Selection within the framework of the Non-Linear Mixed-Effects (NLME-PBMS) model.

METHODS

Biokinetic data of a radioligand for the Prostate-Specific Membrane Antigen (PSMA) for cancer treatment were used. Eleven fit functions were derived from various parameterisations of mono-, bi-, and tri-exponential functions. The functions' fixed and random effects parameters were fitted (in the NLME framework) to the biokinetic data of all patients. The goodness of fit was assumed acceptable based on the visual inspection of the fitted curves and the coefficients of variation of the fitted fixed effects. The Akaike weight, the probability that the model is the best among the whole set of considered models, was used to select the fit function most supported by the data from the set of functions with acceptable goodness of fit. NLME-PBMS Model Averaging (MA) was performed with all functions having acceptable goodness of fit. The Root-Mean-Square Error (RMSE) of the calculated TIAs from individual-based model selection (IBMS), a shared-parameter population-based model selection (SP-PBMS) reported in the literature, and the functions from NLME-PBMS method to the TIAs from MA were calculated and analysed. The NLME-PBMS (MA) model was used as the reference as this model considers all relevant functions with corresponding Akaike weights.

RESULTS

The function [Formula: see text] was selected as the function most supported by the data with an Akaike weight of (54 ± 11) %. Visual inspection of the fitted graphs and the RMSE values show that the NLME model selection method has a relatively better or equivalent performance than the IBMS or SP-PBMS methods. The RMSEs of the IBMS, SP-PBMS, and NLME-PBMS (f3a) methods are 7.4%, 8.8%, and 2.4%, respectively.

CONCLUSION

A procedure including fitting function selection in a population-based method was developed to determine the best fit function for calculating TIAs in MRT for a given radiopharmaceutical, organ and set of biokinetic data. The technique combines standard practice approaches in pharmacokinetics, i.e. an Akaike-weight-based model selection and the NLME model framework.

Single-Time-Point Renal Dosimetry Using Nonlinear Mixed-Effects Modeling and Population-Based Model Selection in [177Lu]Lu-PSMA-617 Therapy

The aim of this study was to investigate the accuracy of single-time-point (STP) renal dosimetry imaging using SPECT/CT data, a nonlinear mixed-effects (NLME) model, and a population-based model selection (PBMS) in a large population for 177Lu-labeled prostate-specific membrane antigen therapy. Methods: Biokinetic data (mean ± SD) of [177Lu]Lu-PSMA-617 in kidneys at time points 1 (1.8 ± 0.8 h), 2 (18.7 ± 0.9 h), 3 (42.6 ± 1.0 h), 4 (66.3 ± 0.9 h), and 5 (160.3 ± 24.2 h) after injection were obtained from 63 patients with metastatic castration-resistant prostate cancer using SPECT/CT. Thirteen functions were derived from various parameterizations of 1- to 5-exponential functions. The function's parameters were fitted in the NLME framework to the all-time-point (ATP) data. The PBMS NLME method was performed using the goodness-of-fit test and Akaike weight to select the best function fitting the data. The best function from ATP fitting was used to calculate the reference time-integrated activity and absorbed doses. In STP dosimetry, the parameters of a particular patient with STP data were fitted simultaneously to the STP data at different time points of that patient with ATP data of all other patients. The parameters from STP fitting were used to calculate the STP time-integrated activity and absorbed doses. Relative deviations (RDs) and root-mean-square errors (RMSEs) were used to analyze the accuracy of the calculated STP absorbed dose compared with the reference absorbed dose obtained from the best-fit ATP function. The performance of STP dosimetry using PBMS NLME modeling was compared with the Hänscheid and Madsen methods. Results: The function [Formula: see text] was selected as the best-fit ATP function, with an Akaike weight of 100%. For STP dosimetry, the STP measurement by SPECT/CT at time point 3 (42.6 ± 1.0 h) showed a relatively low mean RD of -4.4% ± 9.4% and median RD of -0.7%. Time point 3 had the lowest RMSE value compared with those at the other 4 time points. The RMSEs of the absorbed dose RDs for time points 1-5 were 23%, 16%, 10%, 20%, and 53%, respectively. The STP dosimetry using the PBMS NLME method outperformed the Hänscheid and Madsen methods for all investigated time points. Conclusion: Our results show that a single measurement of SPECT/CT at 2 d after injection might be used to calculate accurate kidney-absorbed doses using the NLME method and PBMS.

Time-Activity data fitting in molecular Radiotherapy: Methodology and pitfalls

Absorbed radiation doses are essential in assessing the effects, e.g. safety and efficacy, of radiopharmaceutical therapy (RPT). Patient-specific absorbed dose calculations in the target or the organ at risk require multiple inputs. These include the number of disintegrations in the organ, i.e. the time-integrated activities (TIAs) of the organs, as well as other parameters describing the process of radiation energy deposition in the target tissue (i.e. mean energy per disintegration, radiation dose constants, etc). TIAs are then estimated by incorporating the area under the radiopharmaceutical's time-activity curve (TAC), which can be obtained by quantitative measurements of the biokinetics in the patient (typically based on imaging data such as planar scintigraphy, SPECT/CT, PET/CT, or blood and urine samples). The process of TAC determination/calculation for RPT generally depends on the user, e.g., the chosen number and schedule of measured time points, the selection of the fit function, the error model for the data and the fit algorithm. These decisions can strongly affect the final TIA values and thus the accuracy of calculated absorbed doses. Despite the high clinical importance of the TIA values, there is currently no consensus on processing time-activity data or even a clear understanding of the influence of uncertainties and variations in personalised RPT dosimetry related to user-dependent TAC calculation. As a first step towards minimising site-dependent variability in RPT dosimetry, this work provides an overview of quality assurance and uncertainty management considerations of the TIA estimation.

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

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 [Formula: see text] with shared parameter [Formula: see text] was selected as the function most supported by the data with an Akaike weight of 97%. Parameters [Formula: see text] and [Formula: see text] were fitted individually for every patient while parameter [Formula: see text] 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.

An automated voxelized dosimetry tool for radionuclide therapy based on serial quantitative SPECT/CT imaging

PURPOSE

To create an accurate map of the distribution of radiation dose deposition in healthy and target tissues during radionuclide therapy.

METHODS

Serial quantitative SPECT∕CT images were acquired at 4, 24, and 72 h for 28 (177)Lu-octreotate peptide receptor radionuclide therapy (PRRT) administrations in 17 patients with advanced neuroendocrine tumors. Deformable image registration was combined with an in-house programming algorithm to interpolate pharmacokinetic uptake and clearance at a voxel level. The resultant cumulated activity image series are comprised of values representing the total number of decays within each voxel's volume. For PRRT, cumulated activity was translated to absorbed dose based on Monte Carlo-determined voxel S-values at a combination of long and short ranges. These dosimetric image sets were compared for mean radiation absorbed dose to at-risk organs using a conventional MIRD protocol (OLINDA 1.1).

RESULTS

Absorbed dose values to solid organs (liver, kidneys, and spleen) were within 10% using both techniques. Dose estimates to marrow were greater using the voxelized protocol, attributed to the software incorporating crossfire effect from nearby tumor volumes.

CONCLUSIONS

The technique presented offers an efficient, automated tool for PRRT dosimetry based on serial post-therapy imaging. Following retrospective analysis, this method of high-resolution dosimetry may allow physicians to prescribe activity based on required dose to tumor volume or radiation limits to healthy tissue in individual patients.

Dosimetric effectiveness of targeted radionuclide therapy based on a pharmacokinetic landscape

Assessment of targeted radionuclide therapy (TRT) agent effectiveness based on its pharmacokinetic (PK) properties could provide means to expedited agent development or its rejection. A broad PK model that predicts the relative effectiveness of TRT agents based on the relationship between their normal body (k(12), k(21)) and tumor (k(34), k(43)) PK parameters has been developed. A classic two-compartment open model decoupled from a tumor was used to represent the body. Analytically solved differential equations were used to develop a relationship that predicts TRT effectiveness. Various PK scenarios were created by pairing normal body PK parameters of 38 pharmaceuticals found in the literature with estimated tumor PK parameters. Each PK scenario resulted in a maximum permissible injected activity that limited the whole-body dose to 2 Gy and yielded a maximum delivered tumor dose. The model suggests that a k(34):k(43) ratio greater than 5 and a k(12):k(21) ratio less than 1 is effective at delivering doses that ensure sufficient solid tumor control. It was also shown that there is no direct relationship between tumor dose and acid dissociation constant (pK(a)), lipophilicity (log P), and fraction unbound (fu), which are important physicochemical properties. This study suggests that although effective TRT may be difficult to achieve for solid tumors, good TRT agents must have extremely desirable normal body PKs in conjunction with very high tumor retention. The developed PK TRT model could serve as a tool to compare the relative dosimetric effectiveness of existing TRT agents and novel TRT agents early in the developmental phase to potentially reject those that possess unfavorable PKs.

Model selection for time-activity curves: the corrected Akaike information criterion and the F-test

Data analysis often requires a multi model approach, i.e. the best model or models are selected from a well chosen set of candidate models and subsequent parameter inference is conducted. The selection of the model or models which are best supported by the data can be accomplished using various criteria. The present work focuses on the comparison of two approaches namely the corrected Akaike information criterion (AICc) and the F-test for sparse data sets, which are common in medical research. The selection of the true model and the determination of relevant pharmacokinetic parameters as the clearance, the volume of distribution and the mean residence time are examined using Monte Carlo simulations with 10000 replications. The data (N = 10 per replication) are generated from a sum of two exponentials, which parameters were determined by fitting to time-concentration data of 111In labelled anti-CD66 antibody in blood serum. Four different normal distributed multiplicative statistical errors (0.05, 0.1, 0.15, 0.2) were examined. The set of candidate models consists of sums of up to 3 exponentials. Comparisons with two different model set sizes were conducted. All candidate models are fitted to the generated data and selected according to the AICc and the F-test. Both selection criteria perform well for our data. The selection frequency of functions of lower dimension increases proportionally to the statistical error for both criteria, while for higher errors, the AICc tends to choose a model of lower dimension more frequently than the F-test. In addition, the overfitted fraction decreases proportionally to the statistical error for both methods but selection frequency of function of higher dimension is larger using the F-test. The choice of the adequate model set is important for the positive effect of model averaging concerning the bias and the variability of the estimated parameters. It is in general assumed and has been confirmed in this study that parameter estimation using the AICc has clear advantages over the F-test.

Validity of model approximations for receptor-ligand kinetics in nuclear medicine

An appropriate mathematical model is required for quantitative analysis of high affinity radioligands as direct or surrogate probes to measure receptor distribution, affinity, concentration, binding potential, and endogenous or exogenous ligand occupancy levels. For studies with positron emission tomography (PET) or single photon emission computed tomography (SPECT), the receptor-ligand compartment model has been well established and widely used. This pharmacokinetic model is represented mathematically by a set of nonlinear ordinary differential equations. Variations of models for PET and SPECT account for radioactive decay differently. These are not equivalent and entail assumptions or approximations that may be not appreciated. In this study, a general form of the model is presented and compared with others with various approximations, which are valid only under specific conditions. The various approximate formulations were analytically compared to the exact model to identify the terms that were neglected in the approximate formulations. The extent to which the approximations impact the model solutions was assessed by computer simulations based on numerical solutions to each set of equations. Specifically, each model formulation was tested using three simulated injection protocols representing a typical PET experiment, a typical SPECT experiment, and an extreme experiment where both the injected activity and the specific activity were very high. No significant differences were found among the output of the three model formulations when the PET and SPECT injection protocols were tested. The only conditions that produced significant differences occurred when the specific activity and the administered activity were simultaneously very high. These conditions, however, have little practical relevance to experimentally achievable conditions due to radiation dose and specific activity of radiopharmaceuticals

Rapid dual-tracer PTSM+ATSM PET imaging of tumour blood flow and hypoxia: a simulation study

Blood flow and hypoxia are interrelated aspects of physiology that affect cancer treatment and response. Cu-PTSM and Cu-ATSM are related PET tracers for blood flow and hypoxia, and the ability to rapidly image both tracers in a single scan would bring several advantages over conventional single-tracer techniques. Using dynamic imaging with staggered injections, overlapping signals for multiple PET tracers may be recovered utilizing information from kinetics and radioactive decay. In this work, rapid dual-tracer PTSM+ATSM PET was simulated and tested as a function of injection delay, order and relative dose for several copper isotopes, and the results were compared relative to separate single-tracer data. Time-activity curves representing a broad range of tumour blood flow and hypoxia levels were simulated, and parallel dual-tracer compartment modelling was used to recover the signals for each tracer. The main results were tested further using a torso phantom simulation of PET tumour imaging. Using scans as short as 30 minutes, the dual-tracer method provided measures of blood flow and hypoxia similar to single-tracer imaging. The best performance was obtained by injecting PTSM first and using a somewhat higher dose for ATSM. Comparable results for different copper isotopes suggest that tracer kinetics with staggered injections play a more important role than radioactive decay in the signal separation process. Rapid PTSM+ATSM PET has excellent potential for characterizing both tumour blood flow and hypoxia in a single, fast scan, provided that technological hurdles related to algorithm development and routine use can be overcome.

Intrapatient consistency of imaging biodistributions and their application to predicting therapeutic doses in a phase I clinical study of 90Y-based radioimmunotherapy

Intrapatient variation in the biodistribution of the chimeric monoclonal antibody cT84.66 was assessed in 19 patients having a variety of carcinoembryonic antigen (CEA) positive tumors. The two studies, including whole-body imaging and blood and urine specimen collections, were conducted within 14 days of each other using (111)In-cT84.66 at a fixed total protein dose of 5 mg per patient per study. An initial pretherapy infusion of (111)In-cT84.66 was administered followed by a therapy coinfusion of (111)In-ct84.66 and 90Y-cT84.66 A closed five-compartment model was used to integrate source organ activity curves as residence time inputs into the MIRDOSE3 program. Normal organ absorbed doses were estimated for 90Y-cT84.66, the corresponding radiotherapeutic agent. For the two (111)In-cT84.66 biodistributions, all data were modeled with a R2 value of between 0.72 and 1.00 with the exception of the urine data taken during therapy. This was due to the need of diethylenetriaminepentaacetic acid during the therapy phase because of the possibility that yttrium might escape from the chelator attached to the antibody. With the assurance that the biodistributions were reproducible, we were able to estimate the 90Y-cT84.66 absorbed doses on a per-patient basis. Concordance coefficients showing the agreement between the imaging and therapy phase dose estimates were between the 0.60 and 0.99 levels for liver, spleen, red marrow, total body, and other organ systems. Median results were: 27, 17, and 2.7 rad/mCi of 90Y-cT84.66 for liver, spleen, and red marrow, respectively. Because of decreases in platelets and white cells as the amount of 90Y was increased, dose-limiting toxicity was found at 22 mCi/m2. We conclude that patient biodistributions were consistent over time to 14 days so as to allow absorbed dose estimation in a radioimmunotherapy trial involving the cT84.66 anti-CEA antibody.

Treatment of radioactive decay in pharmacokinetic modeling: influence on parameter estimation in cardiac 13N-PET

Radioactive decay during measurement can be accounted for by either a decay correction of the measured data before modeling (DbM) or by direct implementation of decay into the pharmacokinetic model (DiM). The purpose of this study was to quantify the influence of the type of decay correction on the calculated parameters for the example of a three-compartment model used for the calculation of myocardial perfusion with 13N ammonia and positron emission tomography (PET). For a given input function [Ca(t)infinity t exp(-kt), k= 1.72/min] the tissue uptake for two parameter sets of K1, k2, k3, TBV were calculated for 20 frames (12 x 10 s, 4 x 30 s, 3 x 120 s, 1 x 300 s). These values were mathematically deteriorated by various noise levels according to Poisson statistics and fitted by a Levenberg-Marquardt algorithm. Estimated parameter means and coefficients of variation of the fitted parameters were calculated for the DbM and DiM case. The estimated parameter means for both decay correction methods were of comparable quality. The important measure for a single fit is the relative variability of the fitted parameters. This value is up to a factor 1.15 smaller for K1 obtained with DiM and a reasonable noise level of 10%. Therefore, decay correction should be taken into account during modeling to reduce the variability in the fitted parameters.