 Original research
 Open Access
 Published:
Influence of sampling schedules on [^{177}Lu]LuPSMA dosimetry
EJNMMI Physics volume 7, Article number: 41 (2020)
Abstract
Background
Individualized dosimetry is recommended for [^{177}Lu]LuPSMA radioligand therapy (RLT) which is resourceintensive and protocols are often not optimized. Therefore, a simulation study was performed focusing on the determination of efficient optimal sampling schedules (OSS) for renal and tumour dosimetry by investigating different numbers of time points (TPs).
Methods
Sampling schedules with 1–4 TPs were investigated. Timeactivity curves of the kidneys and two tumour lesions were generated based on a physiologically based pharmacokinetic (PBPK) model and biokinetic data of 13 patients who have undergone [^{177}Lu]LuPSMA I&T therapy. Systematic and stochastic noise of different ratios was considered when modelling timeactivity data sets. Timeintegrated activity coefficients (TIACs) were estimated by simulating the hybrid planar/SPECT method for schedules comprising at least two TPs. TIACs based on one single SPECT/CT measurement were estimated using an approximation for reducing the number of fitted parameters. For each sampling schedule, the rootmeansquared error (RMSE) of the deviations of the simulated TIACs from the ground truths for 1000 replications was used as a measure for accuracy and precision.
Results
All determined OSS included a late measurement at 192 h p.i., which was necessary for accurate and precise tumour TIACs. OSS with three TPs were identified to be 3–4, 96–100 and 192 h with an additional SPECT/CT measurement at the penultimate TP. Kidney and tumour RMSE of 6.4 to 7.7% and 6.3 to 7.8% were obtained, respectively. Shortening the total time for dosimetry to e.g. 96 h resulted in kidney and tumour RMSE of 6.8 to 8.3% and 9.1 to 11%, respectively. OSS with four TPs showed similar results as with three TPs. Planar images at 4 and 68 h and a SPECT/CT shortly after the 68 h measurement led to kidney and tumour RMSE of 8.4 to 12% and 12 to 16%, respectively. One single SPECT/CT measurement at 52 h yielded good approximations for the kidney TIACs (RMSE of 7.0%), but led to biased tumour TIACs.
Conclusion
OSS allow improvements in accuracy and precision of renal and tumour dosimetry for [^{177}Lu]LuPSMA therapy with potentially less effort. A late TP is important regarding accurate tumour TIACs.
Introduction
In recent years, radioligands labelled with ^{177}Lu targeting the prostatespecific membrane antigen (PSMA), such as [^{177}Lu]LuPSMA617 [1] and [^{177}Lu]LuPSMA I&T [2], were established as promising treatment options for patients with metastasized castrationresistant prostate cancer (mCRPC) after exhaustion of approved treatments [3, 4].
Renal dosimetry should be applied for therapy monitoring as kidneys have been identified as a potential doselimiting organ [2, 5]. First data indicate that additional dosimetry of tumour lesions might predict therapy effect [6]. Interpatient variabilities in anatomy and (patho‑)physiology can lead to large differences in absorbed dose coefficients. For example, Okamoto et al. reported renal absorbed dose coefficients in the range of 0.33–1.22 Gy/GBq in a cohort of 15 patients with mCRPC treated with [^{177}Lu]LuPSMA I&T [1]. Thus, individualized treatments are expected to lead to better outcomes than populationbased treatments.
The MIRD Pamphlet No. 16 gives suggestions on sampling schedules for individualized dosimetry [7]. At least three time points (TPs) per exponential clearance should be used [7]. Furthermore, one/two TPs at some fraction of the effective halflife, one near the effective halflife and one or two at three and five times the effective halflife were suggested [7]. Thus, individualized dosimetry meeting high accuracy and precision is in general very resourceintensive. Additionally, clinical demand for [^{177}Lu]LuPSMA radioligand therapy (RLT) is growing. Several studies have already been performed to reduce the number of measurements while maintaining the reliability of the results of dosimetry [8,9,10,11,12,13,14,15,16,17,18]. For example, Merrill et al. investigated sampling schedules comprising 1–3 TPs regarding the timeintegrated activity coefficients (TIACs) [19] for patients with Graves’ disease treated with ^{131}I [8]. They showed that increasing the number of measurements from two to three only led to marginal improvements in accuracy of the determined TIACs [8]. Dosimetry with a single TP and an assumed effective halflife yielded promising results [8]. In general, dosimetry based on a single measurement was of particular interest (especially for [^{177}Lu]LuDOTATATE/DOTATOC [13,14,15,16,17,18]). Therefore, a priori knowledge of the biokinetics, i.e. effective halflives of the clearance rates, is needed. An elegant singletimepoint approach was introduced by Hänscheid et al. [13]. They approximated the TIACs based on the activity value of a single TP. The underlying approximation is exact if the ground truth is monoexponential and the chosen TP matches the effective halflife. Hänscheid et al. reported that a single measurement at 96 h p.i. led to reliable results of organ and tumour dosimetry for patients with neuroendocrine tumours (NETs) treated with [^{177}Lu]LuDOTATATE/DOTATOC [13]. Sundlöv et al. and Del Prete et al. used (among others) this singletimepoint approach for renal dosimetry and achieved similar accuracies and precisions [14, 15].
The aim of our study was to investigate best accuracy and precision of renal and tumour dosimetry for [^{177}Lu]LuPSMA I&T therapy with 1–4 TPs. Biokinetic patient data and a physiologically based pharmacokinetic (PBPK) model are used to create timeactivity data sets used as ground truths. We seek to identify easy and straightforward sampling schedules optimizing both renal and tumour dosimetry for the hybrid planar/SPECT method using 2–4 TPs and for the singletimepoint dosimetry. Additionally, the effects of shortening the time duration for dosimetry on accuracy and precision are systematically investigated.
Methods
Patient data and virtual patients
Biokinetic data of 13 patients with mCRPC were obtained by planar wholebody scans at 30–120 min, 24 h and 7 days post injection (additional measurements at 48 h and 72 h p.i. for several patients) for the first cycle [2, 6, 20, 21]. The patient cohort had a median age of 73 years (range: 58–77 years), a median prostatespecific antigen (PSA) level of 133 ng/l (range: 0.23–2905 ng/l) and median kidney and tumour lesion volumes of 297 ml (range: 233–400 ml) and 12 ml (range: 0.33–92 ml), respectively. Activities of 7.3 ± 0.3 GBq [^{177}Lu]LuPSMA I&T using a peptide amount of 91 ± 5 nmol were applied. Additionally, a pretherapeutic PET/CT scan with [^{68}Ga]GaPSMAHBEDCC (115 ± 16 MBq, 1.6 ± 0.3 nmol) was performed [21].
PBPK modelling was used to create virtual patients. The wholebody PBPK model is described in detail elsewhere [6, 21,22,23]. In brief, the kidneys, the tumour, the liver and the gastrointestinal tract were modelled as PSMApositive tissues. The tumour was analysed selecting two tumour lesions (high uptake, no overlap with other PSMApositive tissue) and a tumour rest. Relevant physically and physiologically mechanisms were included in the PBPK model as e.g. physical decay, blood flows to organs/tumour lesions, specific and unspecific binding, internalisation and release, excretion and plasma protein binding.
The virtual patients were created by individually fitting the PBPK model parameters to the pretherapeutic PET/CT and the planar biokinetic patient data. Furthermore, individual demographic data were included. Time activity curves (TACs) of the kidneys and the two tumour lesions were generated.
The Ethics Committee of the Technical University Munich approved the retrospective analysis (permit 115/18 S), and the requirement to obtain informed consent was waived.
Sampling schedules
The simulation routine introduced by Rinscheid et al. [20, 24] was used, which was implemented in MATLAB (release R2019b, The MathWorks, Inc., Natick, MA, USA). The investigated sampling schedules depended on the simulated dosimetric approach. For the hybrid planar/SPECT method, sampling schedules comprising 2–4 planar images and one SPECT/CT measurement were investigated [14, 20, 25,26,27,28]. Considering working hours [24], following 24 TPs for planar images were used: 1, 2, 3, 4, 20, 22, 24, 26, 28, 44, 48, 52, 68, 72, 76, 92, 96, 100, 116, 120, 124, 144, 168 and 192 h p.i. For sampling schedules comprising four TPs, the following additional constraint was applied: One or two TPs were within the first 4 h p.i. There were no additional constraints for sampling schedules with two and three TPs. Thus, 276, 2024 and 5700 different sampling schedules for the planar images were investigated comprising of 2, 3 and 4 TPs, respectively. For the hybrid planar/SPECT method, the quantitative SPECT/CT measurement was assumed to be 0.5 h after one of the planar images of the investigated sampling schedules. This resulted in 2 × 276, 3 × 2024 and 4 × 5700 sampling schedules for the hybrid planar/SPECT method comprising of 2, 3 and 4 planar images, respectively. For the singletimepoint approach [13], each TP considered for planar images were investigated for the time of the single SPECT/CT scan (i.e. 24 cases).
Timeactivity data sets
Ground truths, i.e. timeactivity curves A_{true}(t) of the kidneys and of two tumour lesions, were generated from the virtual patients. Thus, the true activity values for each sampling schedule are known. Random noise was taken into account for each activity value. The used noise model is described in detail in the supplement. In brief, the simulated activity values A_{planar}(t_{i}) and A_{SPECT}(t_{SPECT}) were randomly drawn from lognormal distributions [29]. The standard deviations of the distributions depended on the imaging modality (planar: 20 %; SPECT/CT: 5 %) [20]. The noise of activity values attributed to planar images was subdivided into a systematic and a stochastic part [30], i.e. a superposition of two lognormal distributions was used. The amount of systematic noise (f_{syst}) in the total noise (20 %) is an unknown parameter, which depends e.g. on the anatomy of the patient, the measurement device and the quantification process [30]. Thus, different proportions f_{syst} = 25%, 50% and 75% were investigated for the hybrid planar/SPECT method [20].
Timeintegrated activity coefficients with the hybrid planar/SPECT approach
For determining the TIACs with the planar/SPECT approach, the simulated planar activity values A_{planar}(t_{i}) were firstly fitted with a monoexponential function:
with the prefactor A_{1}, the biological clearance rate λ_{1} and the physical decay constant for ^{177}Lu λ_{phys} = ln(2)/(6.647 ∙ 24) h^{−1} [31]. The TIACs based on the planar images (TIAC_{planar}) were determined by analytical integration of f_{planar} from zero to infinity and subsequent normalization as
where A_{0} is the injected activity for the investigated patient. The TIACs estimated with the hybrid planar/SPECT method (TIAC_{hybrid}) were calculated according to
where A_{SPECT}(t_{SPECT}) is the simulated activity value assuming the SPECT/CT measurement and f_{planar}(t_{SPECT}) is the activity value according to the fit function used for fitting the planar activity data set at TP t_{SPECT} [14, 20, 28]. The relative differences Δ of the simulated TIACs and the ground truth were determined. The values of ΔTIAC also correspond to the relative differences in selfdoses.
Timeintegrated activity coefficients with single time point approach
The dosimetry method introduced by Hänscheid et al. [13] with just one single quantitative SPECT/CT measurement was investigated. The TIACs can be approximated as
where A_{SPECT}(t_{ref}) is the simulated activity value for the SPECT/CT measurement at TP t_{ref} [13, 14].
Optimal sampling schedules
In total, 1000 replications were performed for each sampling schedule and patient [24]. Thus, 13000 ΔTIAC values for the kidneys and 26000 ΔTIAC values for the tumours were simulated for each sampling schedule. The mean (μ_{ΔTIAC}) and standard deviation (σ_{ΔTIAC}) of the ΔTIAC values were used to estimate the rootmeansquared error RMSE for the kidneys (RMSE_{K}) and tumours (RMSE_{T}) individually according to
where the index j represents the number of the sampling schedule. Lower RMSE values represent better sampling schemes for the kidneys or tumour lesions. A joint RMSE_{joint} value was introduced to sort the sampling schedules with respect to accurate and precise results for the kidneys and tumours:
where w_{K} is a weighting factor for the kidney RMSE. A weighting of w_{K} = 2 was used for the simulations to ensure a higher priority of accurate and precise kidney dosimetry than tumour dosimetry. The schedule with the lowest RMSE_{joint} values was defined as the optimal sampling schedule (OSS).
The effect of varying the last two TPs of the determined OSS on the kidney RMSE_{K} and on the tumour RMSE_{T} was investigated for the hybrid planar/SPECT method. Additionally, the best achievable RMSE by limiting the time of the last measurement to 48 h, 72 h, 96 h,… and 192 h were estimated.
Results
Optimal sampling schedules
The determined OSS for estimating renal and tumour TIACs using the hybrid planar/SPECT method were independent on the investigated fraction of systematic error f_{syst} to the total error with one exception (Table 1). The RMSE values decrease with increasing f_{syst} for a fixed total error. This shows that the systematic error due to planar images can be mostly corrected using the hybrid planar/SPECT method. Higher RMSE values were found for tumours than for the kidneys. This is particularly evident for the sampling schedules comprising two TPs. Best achievable RMSE values of either only the kidneys or the tumours resulted in lower RMSE values of maximal 0.6 percentage points (data not shown) than joint optimization. Thus, a joint optimization of the kidneys and tumours was possible using 2–4 TPs and the hybrid planar/SPECT method. The method with a single SPECT/CT resulted in kidney RMSE_{K} values similar to those estimated with 2–4 TPs and the hybrid planar/SPECT method. The tumour TIACs were considerably underestimated.
Dosimetry with 2–4 planar images only, i.e. without a SPECT/CT measurement, led to kidney and tumour RMSE values about two to three times higher than with the hybrid planar/SPECT method (exceptions for the tumour RMSE_{T} values using two TPs and f_{syst} = 25 %, 50 %; 1.4 and 1.7fold). The determined OSS for dosimetry based on planar images only are listed in Additional file 1: Table S1.
The frequency distributions of the relative deviations between the simulated TIACs and the ground truths for the hybrid planar/SPECT method and for the planar images only are presented in Fig. 1. The broad frequency distribution of the simulated kidney TIACs based on the planar images e.g. led to about 25% of the TIACs deviating more than 20% from the ground truth. This percentage was reduced below 1% by using the hybrid planar/SPECT method instead.
The frequency distributions of the kidneys and tumours using 1–4 TPs are depicted in Fig. 2. One single SPECT/CT measurement at 52 h resulted in a high number of underestimations of the tumour TIACs (Fig. 2a). A more appropriate TP regarding only tumour dosimetry was determined to be 72 h leading to the kidney and tumour RMSE values of 11 % (mean: − 6.1 ± 9.1%) and 12% (mean: − 7.8 ± 8.8%), respectively. Using two TPs led to a skewed distribution of the tumour TIACs (Fig. 2b). Thus, the standard deviations of the kidneys σ_{K} and the tumours σ_{T} differed (Table 1) by about 30–60%. Using three and four time points resulted in similar frequency distributions for kidney and tumour TIACs, which can also be seen from the similar means μ and standard deviations σ given in Table 1.
Variation of the last two time points from the determined optimal sampling schedule
The dependence of the RMSE on variations of the last two TPs from the determined OSS for the hybrid planar SPECT/CT method (Table 1) was investigated (Fig. 3).
The dependence of the RMSE on the sampling schedule comprising two TPs are depicted in Fig. 3a, b. A reduction of the total time for dosimetry with acceptable accuracy and precision (e.g. kidney RMSE_{K} < 10% and tumour RMSE_{T} < 15%) could be achieved with sampling schedules in the range of 20–24, 144h (t_{SPECT} = t_{1} + 0.5 h). Furthermore, the schedule of 4 and 68 h (t_{SPECT} = t_{2} + 0.5 h) seemed promising. However, the kidney RMSE was slightly above 10% (RMSE_{K} = 11%, RMSE_{T} = 14%). Here, the RMSE_{T} value increased with decreasing t_{1} from 4 to 1 h.
Investigations of variations from the determined sampling schedule comprising three TPs, i.e. 3, 4–168 and 20–192 h (t_{SPECT} = t_{2,3} + 0.5 h), are depicted in Fig. 3c, d. RMSE values below 10 % for both kidneys and tumours could be reached within 120 h using e.g. 3, 72 and 120 h (t_{SPECT} = t_{2} + 0.5 h). Here, the kidney RMSE_{K} = 6.9 % was even slightly improved in comparison to the joint OSS. Dosimetry within 72 h by e.g. using 3, 20 and 72 h (t_{SPECT} = t_{3} + 0.5 h) as sampling schedule led to kidney RMSE_{K} = 9.3% and tumour RMSE_{T} = 13%.
Figure 3e, f shows the RMSE in dependence of the used sampling schedules comprising four TPs within 3, 4, 20–168 and 22–192 h (t_{SPECT} = t_{3,4} + 0.5 h). The time duration for dosimetry could also be shortened to 120 h with both RMSE values still below 10%. Dosimetry within 72 h was possible with kidney RMSE_{K} = 8.2% and tumour RMSE_{T} = 13% using e.g. 3, 4, 68 and 72 h (t_{SPECT} = t_{3} + 0.5 h). These RMSE values could be further reduced to RMSE_{K} = 8.0% and RMSE_{T} = 11% by e.g. using 4, 20, 68 and 72 h (t_{SPECT} = t_{3} + 0.5 h; data not shown).
The effects on the RMSE by varying the last two TPs of the determined OSS for the simulations with f_{syst} = 25 % and f_{syst} = 75 % are given in the supplement (Additional file 1: Figures S1 and S2).
Reductions of the time duration for dosimetry
The best achievable RMSE by using the hybrid planar/SPECT method with limiting the time for the last TP t_{last} of the sampling schedules are depicted in Fig. 4. Only slight changes (≤ 1.0 percentage points) of the kidney RMSE was observed for OSS comprising three or four TPs with t_{last} = 96 h…192 h. For tumours and investigated schedules with three and four TPs, the RMSE steadily increased with shortening time duration for dosimetry, i.e. with decreasing t_{last}. Using four instead of three TPs resulted in lower RMSE values of less than 0.8 percentage points for t_{last} = 96 h…192 h. The schedules 4, 68–72 and 96 h (t_{SPECT} = t_{2} + 0.5 h) were best suited for dosimetry within 96 h p.i. Dosimetry within 72 h with kidney RMSE_{K} ≤ 10% and tumour RMSE_{T} ≤ 15% was possible using schedules with four TPs for f_{syst} = 25%, with at least three TPs for f_{syst} = 50% and even with two TPs for f_{syst} = 75%. Dosimetry with RMSE_{K} ≤ 10% and RMSE_{T} ≤ 15% within 48 h was not possible.
Discussion
Individualized dosimetry for PSMA targeting agents labelled with ^{177}Lu is demanding high resources especially when high accuracy and precision are required. Simplified dosimetric approaches leading to reliable results are therefore needed. In this study, the achievable accuracy and precision (combined in the RMSE) for the kidney and tumour TIACs in [^{177}Lu]LuPSMA I&T therapy were investigated. The hybrid planar/SPECT method and the method introduced by Hänscheid et al. using one single SPECT/CT scan [13] were used. OSS for joint renal and tumour dosimetry comprising four TPs (3, 4, 92, 192 h), three TPs (3–4, 96–100, 192 h), two TPs (20, 192 h) and one single TP (52 h) were identified. For the hybrid planar/SPECT method (2–4 TPs), the SPECT/CT was assumed to be 0.5 h after the penultimate planar measurement in all cases. As all these OSS have a very late TP, the effects of shortening the time duration for dosimetry on the RMSE was additionally investigated. Dosimetry with one single SPECT/CT at 52 h p.i. yielded promising results for kidney TIACs, but biased tumour TIACs.
The renal and tumour RMSE values were similar considering three and four optimized TPs with at least one TPs ≥ 96 h (Figs. 3c–f and 4). Thus, three TPs may be sufficient for accurate and precise renal and tumour dosimetry using a monoexponential fit function. To account for practicability in clinical routine and patient comfort, the sampling schedule of 4, 68–72 and 96 h (t_{SPECT} = t_{2} + 0.5 h) can be proposed as a suitably shortened OSS.
OSS with three TPs for renal dosimetry alone were already determined earlier [20]. There, we showed that using a schedule of 3–4, 72–76 and 124–144 h p.i. with a SPECT/CT at t_{2} + 0.5 h led to renal RMSE of 6.2 –7.2 %. These results were reproduced within this study as shown in Fig. 3c and Additional file 1: Figures S1c and S2c. Thus, TPs later than about 144 h p.i. were not necessary for renal dosimetry alone. All determined optimal sampling schedules for joint renal and tumour dosimetry comprised a late TP at 192 h. This late TP was therefore important for additional accurate and precise tumour TIACs as shown in Fig. 3d.
Using planar images at 4 h and 68 h with a SPECT/CT following the last measurement or a single SPECT/CT measurement at 52 h p.i. yielded good results for the estimation of renal TIACs. These approaches are expected to be sufficient if additional accurate tumour dosimetry is not required. In our study, dosimetry based on the hybrid planar/SPECT method seems to outperform dosimetry based on planar images only, even if fewer time points were used.
The single SPECT/CT measurement for treatment control might be predefined in nuclear medicine departments based on their individual logistics. Therefore, the time point of the SPECT/CT scan may be chosen different to the determined optimal time point. Assuming the SPECT/CT scan defined at e.g. 24 h p.i., a final planar image at about 144–168 h p.i. should be considered if accurate and precise tumour dosimetry is of interest (Fig. 3). In any case, our simulations allow estimating the loss of accuracy and precision due to a predefined SPECT/CT measurement.
Aiming at dosimetry with a single TP, inclusion of a priori knowledge is essential. The here used approximation of the TIACs with Eq. 4 would be exact if the ground truth is a monoexponential function, and the time of the single measurement matches the effective halflife [13] (i.e. the used a priori knowledge is the effective halflife). Since the kidneys and the tumour lesions have different effective halflives, the joint optimisation with this dosimetric approach did not lead to satisfactory tumour dosimetry. A more suitable approach might e.g. be the usage of averaged population values of the effective halflives depending on the investigated organs and tumours [12]. This procedure was not investigated in this study.
Several groups have already investigated a singletimepoint approach on patients with neuroendocrine tumours (NETs) and meningioma injected with [^{177}Lu]LuDOTATATE/DOTATOC [13,14,15]. The used TPs and the deviations from the respective ground truths are listed in Table 2. In this study, using a single TP at 96 h p.i. mostly underestimated the renal absorbed dose. This has not been observed that drastically in the literature [13,14,15]. These differences can have several causes. Firstly, different tumour entities and radiopharmaceuticals have been investigated (mCRPC vs. NETs/meningioma; [^{177}Lu]LuPSMA I&T vs. [^{177}Lu]LuDOTATATE/DOTATOC). Secondly, the kidneys showed different effective halflives. For NETs and meningioma patients treated with [^{177}Lu]LuDOTATATE/DOTATOC, effective halflives of 47–52 h are given for the kidneys in the literature [13,14,15]. In this study, a median effective halflife of the kidneys of 40 h (range: 30–62 h) and of the tumours of 50 h (range: 34–94 h), respectively, from 24 h onwards was determined. Thirdly, the ground truths from the literature based on mono/biexponential functions fitted to the fulltime activity data sets. In contrast, a wholebody PBPK model was used to create the ground truth in this study. The results provided by Hänscheid et al. [13] using a single measurement at 48 h p.i. were more consistent with our results considering the determined OSS at 52 h p.i.
A weighting factor of w_{k} = 2 was used for the kidney RMSE values (Eq. 6). Using weightings w_{k} of e.g. 1 and 4 only had minor effects on single time points of the OSS with 2–4 time points (± 1 h for t_{i} ≤ 4 h and ± 4 h for t_{i} ≥ 20 h). For the singletimepoint approach, a weighting factor of w_{k} = 1 led to an OSS of 68 h, which was more favourable for tumour dosimetry. Factors with w_{k} ≥ 2 did not further change the OSS of 52 h for the single time point approach, i.e. it was already optimized for renal dosimetry.
The simulation routine used a monoexponential fit function for alltime activity data. A fit function neglecting an initial uptake phase seems an acceptable simplification for the kidneys and for the tumour lesions [7]. Regarding kidney kinetics, the median maximum TAC value was at 2 h (range: 0.6–3 h). Furthermore, at least 96.8% of the maximum kidney’s activity value was reached already 1 h post injection in all virtual patients. The tumour lesions showed slower uptake kinetics compared to the kidneys. Here, the median maximal activity uptake value was reached after 2.5 h (range: 0.5–9 h). Three hours post injection an uptake value of at least 95% was reached in almost all investigated tumour lesions (two exceptions with 89% and 86%). Nevertheless, using a set of appropriate fit functions and selection criteria [32, 33] could further improve accuracy and precision. Clearly, different optimal sampling schedules are expected for other fit functions.
Noise levels of 5% for SPECT/CT and 20% for planar measurements seem reasonable for kidney activity values [20]. For simplicity, the same noise levels were used for simulated activity values in tumour lesions. However, higher noise might be in general more realistic for small tumour lesions [24]. Furthermore, the noise levels were assumed to be constant over time. Obviously, this is an approximation as e.g. Poisson noise increases for later time points. Assuming a minimal kidney activity at 192 h p.i. of 3.1 MBq, a sensitivity of 9.4 cps/MBq [34], a fieldofview in zdirection of 38.7 cm and a bed speed of 10 cm/min, a maximal Poisson noise of 1.2% can be expected for the measured counts within the kidney ROIs for planar imaging. Thus, increases in Poisson noise over time could be neglected for the kidneys. For the tumour lesions, the total uptake of activity and thus the Poisson noise is size dependent. Here, in analogy a median Poisson noise of 1.1% (range: 0.3–9.2%) can be estimated at 192 h p.i. for planar imaging, where the maximal Poisson noise came from the smallest investigated tumour lesion of 0.33 ml volume. Thus, higher noise levels of about 25–30 % could have been more realistic for small tumour lesions with low activity uptake for late time points. We expect that this would generally lead to higher RMSE values for the tumour lesions. Furthermore, sampling schedules with a last time point earlier than 192 h might be more favourable for tumour dosimetry.
Individual pretherapeutic PET/CT and planar imaging data were used to estimate the PBPK model parameters. Clearly, quantitative SPECT/CT data instead of planar images would additionally improve the estimations of the parameters. Thus, differences between the virtual patients’ biokinetics used as ground truth and the true patients’ biokinetics (which is unknown) may exist. However, since a population of virtual patients with different uptake and washout kinetics was investigated, we expect only minor changes of the results with SPECT/CT input data. Nevertheless, the determined OSS have to be validated in future prospective studies.
Conclusion
The used simulation routine is ideally suited to determine optimal sampling schedules for combined renal and tumour dosimetry in [^{177}Lu]LuPSMA I&T therapy. Considering 2–4 time points, best results are achieved with a last time point at 192 h p.i. The difference in accuracy and precision between optimal sampling schedules with three and four TPs is marginal. Thus, dosimetry based on not more than three time points seems to be sufficient. Focusing on renal dosimetry only, the overall time duration of dosimetry can be safely shortened to e.g. 96 h p.i. when using three time points. Dosimetry based on one single time point at 52 h p.i. led to reliable renal TIACs but biased tumour TIACs.
Availability of data and materials
The data underlying the analyses in this manuscript are available on demand from the author.
Abbreviations
 CT:

Xray computed tomography
 mCRPC:

Metastasized castrationresistant prostate cancer
 n/a:

Not available
 NET:

Neuroendocrine tumour
 OSS:

Optimal sampling schedule
 p.i.:

Post injection
 PBPK:

Physiologically based pharmacokinetic
 RLT:

Radioligand therapy
 RMSE:

Rootmeansquared error
 ROI:

Region of interest
 SPECT:

Singlephoton emission computed tomography
 TAC:

Timeactivity curve
 TIAC:

Timeintegrated activity coefficient
 TPs:

Time points
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This work was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) (KL2742/21, BE4393/11 and GL236/111).
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AR, PK and GG designed the study. AR performed and evaluated the simulations and wrote the manuscript. PK created the PBPK model. ME helped with patient recruitment and the patients’ biokinetic data analysis. AJB and GG supervised the project. AR, PK, GG and AJB contributed with helpful discussions. All authors edited, reviewed and agreed to the manuscript content. The authors read and approved the final manuscript.
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The Ethics Committee of the Technical University Munich approved the retrospective analysis (permit 115/18 S), and the requirement to obtain informed consent was waived.
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Supplementary information
Additional file 1.
The supplementary information includes a complete description of the used noise model. Table S1. Optimal sampling schedules for dosimetry based on planar images. Figure S1. Variation of the last two time points for f_{syst} = 25%. Figure S2. Variation of the last two time points for f_{syst} = 75%.
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Rinscheid, A., Kletting, P., Eiber, M. et al. Influence of sampling schedules on [^{177}Lu]LuPSMA dosimetry. EJNMMI Phys 7, 41 (2020). https://doi.org/10.1186/s40658020003110
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Keywords
 Optimal sampling schedules
 Individualized dosimetry
 mCRPC
 ^{177}LuPSMA I&T
 Single time point
 Radioligand therapy