This study evaluated the effect of the choice of imaging time points on the accuracy and precision of Patlak linearization for 89Zr-immuno-PET, considering different conditions. Simulations showed that inclusion of a PET scan and blood sample at 24 h p.i. improves accuracy and precision of Patlak results. Different combinations of later time points did not change the accuracy and precision in most cases. Moreover, increase in rKi, rVT and noise level decreased accuracy and precision of Patlak results. Additionally, IFs with smaller AUCp showed decreased accuracy and precision of Patlak results as compared to IFs with larger AUCp.
Underestimation of K
Bias in Ki was negative in all simulations. This can be explained by the shape of the IF in combination with the calculation of AUCp in the Patlak equation . In case the IF is fully described, for instance with a bi-exponential equation, determining the AUCp by integration will result in the true value for AUCp. However, when only a finite set of points is known from the IF, determining the AUCp will be based on trapezoidal numerical integration. For the simulations in this study, the latter applies, because data sampling is always finite. Since the activity concentration in plasma decreases over time in an exponential manner, the shape of the IF is curved downwards, leading to an overestimation of the AUCp with trapezoidal numerical integration. The overestimated AUCp increases the x-coordinates of the Patlak plot, which is AUCp/ACp, while the y-coordinates remain the same, because the ratio ACt/ACp does not change. This results in a decreased positive slope of the Patlak plot, e.g., negative bias of Ki.
24 h time point
Inclusion of time point 24 h p.i. showed to improve accuracy and precision of Patlak linearization. This is also due to the better assessment of the shape of the IF and the calculation of AUCp as detailed before. The better the curve of the IF is described, by adding a time point in the most curved part of the IF, the more accurate the determination of AUCp and Patlak parameters. One assumption for Patlak linearization is that equilibrium is reached between the 89Zr-mAb concentration in plasma and in the reversible tissue compartment, meaning that all fluxes are constant with respect to time . In this study, activity concentrations in tissue were simulated by means of Patlak linearization and therefore were directly in equilibrium with activity concentrations in plasma. However, mAbs are relatively large proteins, therefore distribution inside the body takes relatively long, so tissue is not in rapid equilibrium with plasma . Therapeutic antibodies cetuximab and trastuzumab showed approximately homogeneous distributions after 24 h p.i. in tumor-bearing mice . For this reason, a period of 24 h was estimated to reach equilibrium between tissue and plasma. Additionally, from a practical point of view, it would not be possible to include time points after approximately 12 h, because PET scans should then be obtained outside working hours. Hence, time points before 24 h p.i. were not included in the simulations. This moment of equilibrium may differ between 89Zr-mAbs, and inclusion of a slightly earlier or later time point may be better depending on the mAb pharmacokinetics.
Time point combinations
After inclusion of the 24 h p.i. time point, different time point combinations barely influenced Patlak results, which is advantageous from a practical perspective. Postponing a late imaging time point to a different day would not influence Patlak results. This is in contrast with obtaining the SUV, for which differences in the uptake time between injection and PET scan does influence the result, because SUV changes as a function of time . In case the assumption of equal clearance between patients is true, comparisons of SUVs between patients would only be possible for PET scans that are obtained at the same uptake time post-injection . Therefore, postponing a PET scan, resulting in different scan days for patients accompanied by different plasma activity concentrations, will influence SUV results. Apart from the ability to distinguish between reversible and irreversible, and potentially between non-specific and specific uptake of 89Zr-mAbs , the option to postpone a PET scan is another advantage of using Patlak linearization over using SUV in the quantification of 89Zr-immuno-PET.
Simulations showed that increasing rKi and rVT resulted in similar or increased bias and variability in both Ki and VT. As Patlak linearization is only applied when the assumption of irreversible uptake is met, Ki is never zero. Additionally, Jauw et al.  showed that organs without target expression have Ki values higher than zero, representing the catabolic rate of 89Zr-mAbs in healthy tissue. Values for Ki in this study are therefore all above zero.
In this study, noise was approximated based on counting statistics, which resulted in noise increasing over time. This was similar to results from a study about noise-induced variability in PET imaging for 89Zr-immuno-PET, where recovery coefficients (RC) also increased over time from day 0 to day 6 . RC was defined as 1.96*SD(%). RCs found for Kidney, lung, spleen and liver combined ranged from 2 to 11 , resulting in a maximum SD of approximately 5%. Similarly, SD derived from the RCs of tumor SUVpeak results in 15%. Simulations including TACs with a 5% noise level may therefore represent biodistribution and TACs with a 15% noise level may represent tumor uptake. Increasing the noise level from 5 to 15% only increased the variability, biases remained the same. Additionally, results of simulations with a noise level of 15% showed the same pattern as simulations with a 5% noise level and were chosen not to be presented.
The literature search provided five different 89Zr-mAb plasma IFs in patients, of which three were used for the simulations, while there are currently 119 therapeutic antibodies approved by the FDA . However, these three 89Zr-mAb plasma IFs used in this study provide a wide range of clearances, covering substantial variability in IFs.
Simulations showed a dependency of Patlak results on the IF. For high rKi, accuracy and precision in Patlak results decreased with AUC of the IF (i.e., faster clearance), in the following order: 89Zr-pertuzumab, 89Zr-trastuzumab and 89Zr-huJ591. A decrease in AUCp will result in lower x-coordinates of the Patlak plot, thereby bringing the datapoints closer together resulting in higher contribution of noise. The AUCp is the integral of the activity concentration in plasma, which is the total 89Zr-mAbs present in the plasma cumulated over time from injection to moment of PET scan. For the simulations, the IF and rKi were regarded as two independent variables; however, they are physiologically related. For IFs with lower AUCp, so faster clearance, higher irreversible uptake in tissue (rKi) is expected. However, simulations showed that a higher rKi for the 89Zr-huJ591 IF resulted in decreased accuracy of Ki (− 16%) and precision of VT as compared to the other IFs. This indicates that accuracy and precision of Patlak results are worse for 89Zr-mAbs with faster clearance combined with higher irreversible uptake. However, for volumes of interest showing high irreversible uptake, a bias in Ki of − 16% would not change the (clinical) decision-making based on the data, because the observed irreversible uptake would still be high.
This study considers input functions with binding of targets on cells that do not redistribute during the course of the PET studies (HER2 for trastuzumab and pertuzumab, and PSMA for huJ591). However, the usefulness of Patlak linearization may be limited in case of 89Zr-mAbs that bind to mobile immune cells, such as the PD-1 receptors on T-cells. In order to apply Patlak linearization, an equilibrium between reversible processes is assumed as well as a constant density of specific targets or receptors. Changes in receptor availability during the course of the study may introduce inaccuracies in Patlak linearization. Yet, Patlak analysis also has several advantages over SUV. Patlak linearization can also be applied with higher mass dose. However, there are two phenomena that need to be considered. First of all, higher mass doses will result in slower plasma clearance. Patlak linearization takes into account the mAb concentration in plasma (or input function) and no assumptions are required with regard to (changes in) plasma clearance as the measured plasma kinetics are used. Secondly, a higher administered mass dose will result in lower uptake in tissue of interest. Patlak linearization is still valid with higher mass doses; however, lower Ki values are expected because of the reduced receptor availability/higher receptor or target occupancy. Also, Menke-van der Houven van Oordt et al.  showed in their study that Patlak linearization applied to PET imaging data with different administered mass doses allows evaluation of the optimal therapeutic dose. By plotting the Patlak Ki values against increasing mass doses a S-curve can be obtained. Ki values decrease because of target binding competition between labeled and unlabeled mAbs. This curve allows evaluation of the 50% inhibitory mass dose (ID50). The ID50, the dose at which 50% of the targets are occupied, can be used in establishing the optimal therapeutic dose .
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