### Patient data and virtual patients

Biokinetic data of 13 patients with mCRPC were obtained by planar whole-body 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 prostate-specific 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]Lu-PSMA I&T using a peptide amount of 91 ± 5 nmol were applied. Additionally, a pre-therapeutic PET/CT scan with [^{68}Ga]Ga-PSMA-HBED-CC (115 ± 16 MBq, 1.6 ± 0.3 nmol) was performed [21].

PBPK modelling was used to create virtual patients. The whole-body 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 PSMA-positive tissues. The tumour was analysed selecting two tumour lesions (high uptake, no overlap with other PSMA-positive 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 pre-therapeutic 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 single-time-point approach [13], each TP considered for planar images were investigated for the time of the single SPECT/CT scan (i.e. 24 cases).

### Time-activity data sets

Ground truths, i.e. time-activity 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 log-normal 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 log-normal 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].

### Time-integrated 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 mono-exponential function:

$$ {f}_{\mathrm{planar}}(t)={A}_1\cdot {\mathrm{e}}^{-\left({\lambda}_1+{\lambda}_{\mathrm{phys}}\right)\cdot t} $$

(1)

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

$$ {TIAC}_{\mathrm{planar}}=\frac{1}{A_0}\cdot {\int}_0^{\infty }{f}_{\mathrm{planar}}(t)\mathrm{d}t=\frac{1}{A_0}\cdot \frac{A_1}{\lambda_1+{\lambda}_{\mathrm{phys}}} $$

(2)

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

$$ {TIAC}_{\mathrm{hybrid}}=\frac{A_{\mathrm{SPECT}}\left({t}_{\mathrm{SPECT}}\right)}{f_{\mathrm{planar}}\left({t}_{\mathrm{SPECT}}\right)}\cdot {TIAC}_{\mathrm{planar}} $$

(3)

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 Δ*TIAC* also correspond to the relative differences in self-doses.

### Time-integrated 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

$$ {TIAC}_{1\mathrm{TP}}=\frac{1}{A_0}\cdot \frac{2}{\mathrm{In}(2)}{A}_{\mathrm{SPECT}}\left({t}_{\mathrm{ref}}\right)\cdot {t}_{\mathrm{ref}} $$

(4)

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 Δ*TIAC* values for the kidneys and 26000 Δ*TIAC* values for the tumours were simulated for each sampling schedule. The mean (*μ*_{ΔTIAC}) and standard deviation (*σ*_{ΔTIAC}) of the Δ*TIAC* values were used to estimate the root-mean-squared error RMSE for the kidneys (*RMSE*_{K}) and tumours (*RMSE*_{T}) individually according to

$$ {RMSE}_j=\sqrt{{\left({\sigma}_{\Delta TIAC,j}\right)}^2+{\left({\mu}_{\Delta TIAC,j}\right)}^2} $$

(5)

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:

$$ {RMSE}_{\mathrm{joint},j}={w}_K\cdot {RMSE}_{\mathrm{K},j}+{RMSE}_{\mathrm{T},j\cdot } $$

(6)

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.