Bias Evaluation and Reduction in 3D OP-OSEM Reconstruction in Dynamic Equilibrium PET Studies

Abstract


Background
Iterative image reconstruction algorithms based on the known maximum likelihood -expectation maximization (ML-EM) have been widely used in positron emission tomography (PET) for the past three decades, with the order-subset (OS) variants being particularly prevalent (1)(2)(3)(4).
However, these methods have been shown to cause bias for applications involving low count levels (5)(6)(7)(8)(9).This effect is often problematic in dynamic PET studies with 11 C-labelled radioligands and especially in the case of neuroreceptor binding studies that usually use reference regions, e.g.cerebellum or pons/brainstem, which frequently present low neuroreceptor concentrations levels and thus low counts of the radioligand per frame.
Other groups using ML-EM based reconstruction methods have reported different levels of bias (7) and overestimation, as well as underestimation in volumes of interest (VOIs) with either low or high activity concentrations (10).In the case of 3D OSEM based reconstruction algorithms, the source of the bias has been attributed to the introduction of a positive bias in the reconstructed images due to the non-negativity constraint in the data prior to correction (11).To avoid bias from the non-negativity constraint, 3D ordinary Poisson OSEM (OP-OSEM) can be used as an alternative iterative reconstruction method as it implements all required data corrections in a way that preserves non-negativity during the reconstruction (6,12).However, 3D OP-OSEM uses observed random and scattered coincidence events without updating them for each iteration step during the image reconstruction (13).The accuracy of this method has been studied and a bias of 10% or more was reported (11) in regions of a homogeneous phantom.Byars and colleagues (14) showed that the bias can be reduced when a variance reduction algorithm (VRR) is implemented to reduce the variance in estimated random counts.However, it is important to consider that other factors can also contribute to the bias in image reconstruction at low count rates, including scatter correction implementations (e.g.frame-based) or the framing scheme (10,15,16).
The aim of this study was to evaluate the impact of different framing schemes and reconstruction procedures on the quantitative accuracy of PET images using a 11 C filled phantom decay study and a bolus plus infusion (BI) human neuroreceptor study.Particular focus was given to the bias effects on the values of non-displaceable binding potential (BPND) at count rate situations normally found in dynamic [ 11 C]ABP688 (ABP) equilibrium PET studies.An alternative method for gathering the PET coincidences into a framing scheme is also proposed in which the counts per frame are kept at the same value for all reconstructed frames of the dynamic acquisition.As anticipated in (12,16), the bias is dependent, in part, on the number of counts in the frame.This leads to the conclusion that the bias may be constant throughout the entire TAC when maintaining constant counts per frame over the entire time interval of the dynamic PET acquisition.Based on this premise, different framing schemes were also compared with respect to bias size and the limits of the bias range that a task must overcome in order to be effectively identified (how much change a task must induce to exceed the bias limits) were evaluated.Two spherical and one background VOIs from an adapted NEMA phantom were analyzed.The phantom had different dimensions, reflecting our BI protocol with ABP a glutamatergic receptor ligand with high and low neuroreceptor density regions to evaluate the limitations in binding quantification due to the bias.
In addition, the bias obtained with the vendor's 3D OP-OSEM reconstruction was compared with the bias obtained with an in-house reconstruction (PRESTO (17)) both in the phantom study and with human data with ABP.

A. PET Acquisitions
All PET data (phantom and human brain studies) were acquired using a 3T hybrid MR-BrainPET insert system (SIEMENS, Erlangen, Germany) in list mode (18).The coincidences were corrected for random events using the delayed window technique with VRR, dead time, attenuation and scattered coincidences (single scatter simulation -SSS method), and physical decay.The image reconstruction was performed with the vendor-supplied 3D OP-OSEM (19) and in-house developed PRESTO implementations (2 subsets and 32 iterations, and 80 iterations respectively), with an isotropic voxel of 1.25 mm into a volume consisting of 153 transverse slices of 256 × 256 pixels.The vendor's reconstruction uses sinograms with span 9 axial data compression, whereas PRESTO is realized as a direct line of response (LOR) reconstruction without LOR data compression.Post-processing was performed with a 2.5 mm 3D Gaussian filter.Pmod software 3.9 was used to define the VOIs and to extract activity concentration (kBq/cm 3 ) for the analysis.

B. Phantom Study
An adapted NEMA phantom (20) with sphere inserts filled with 11 C was used and coincidence data were acquired during seven isotope half-lives, thus giving ~163 minutes total acquisition time for 11 C with its half-life T1/2 = 20.38 minutes.An interval of 3T1/2, from 62 minutes to 122 minutes of the acquisition time was used for data analysis.This acquisition time interval was chosen according to the typical count rates measured in a human brain study with ABP alongside a BI protocol (see Section C) and a scan time of 65 minutes (~3T1/2 of 11 C).The ratio of the activity concentration between the two spheres (Hot1 with 28 mm of internal diameter and Hot2 with 22 mm of internal diameter) and the background region (BG) was ~2:1.This value is frequently found for the ratio of activity concentrations in the grey matter cortex (GM) and cerebellar grey matter (CER, reference region) during the steady-state condition in ABP studies.The activity concentrations for each time interval during the 11 C decay study with the phantom are given in Table 1.

2.06
The half-life here is expressed as the number of elapsed 11 C half-lives during the decay experiment, where T4 is during the fourth T1/2 ( 11 C) and so forth.The chosen T4, T5 and T6 were selected to represent typical activity concentration in human [ 11 C]ABP688 acquisitions.
A cold transmission scan of the adapted NEMA phantom using 68 Ge sources was acquired in a Siemens ECAT Exact HR+ PET scanner.This acquisition (2 bed positions with 20 minutes of transmission in a 256 × 256 matrix and reconstructed with OSEM 2D -6 iterations and 16 subsets) was used to create the attenuation map for the phantom used in the 11 C decay study.
To obtain the ground truth data for the phantom study, the activity concentrations in the different phantom compartments were measured with a gamma counter (Wizard counter) repeating for 3 probes in a solution of 0.5 ml.Decay correction, counter calibration factor and volumes/weights of the probes were considered and standard corrections applied.
Figure 1 shows a PET image from ABP overlaid with an MR T1 image acquisition and a comparison to the image of an adapted NEMA phantom.Based on the known neuroreceptor distribution of ABP, the background VOI (BG) was considered to be representative of CER as a reference region and the two hot sphere's VOIs (Hot1 and Hot2) were considered to be representative of the GM regions, i.e. parts of the cingulate cortex (neuroreceptor rich region in ABP, see Section C) and the nucleus accumbens.The latter is important for the analysis of psychiatric conditions, such as schizophrenia (21).The effective volumes of the three VOIs were: 112.86 ml (BG, 57782 voxels), 3.42 ml (Hot1, 1751 voxels), and 1.13 ml (Hot2, 578 voxels) respectively.To reduce partial volume effects in the analysis, the VOIs were drawn with a distance of 4 mm to the inner borders of the Hot1 and Hot2 spheres (effective internal diameters of 20 mm and 14 mm, respectively).Moreover, in the 11 C filled phantom decay study, following the application of all data corrections, a constant activity concentration was expected during the entire acquisition time (T1 to T7) in all phantom compartments since a BI study at equilibrium conditions aims to maintain constant activity concentrations in the brain compartments.

C. Human Study with [ 11 C]ABP688
PET Protocol with [ 11 C]ABP688: An analysis similar to that used in the phantom study was applied to already existing data from a dynamic PET acquisition on a subject with 437 MBq of totally administered activity of ABP.Details of this study can be found in (22).The radiosynthesis of ABP was performed according to (23).
Applied BI protocol: The bolus injection had 50% of the total injected activity, followed by 65 minutes of infusion with 92 ml/h.The bolus injection was applied after positioning the subject in the scanner and began simultaneously with the start of PET data acquisition.A distribution equilibrium was observed at around 30 minutes after the bolus injection, and PET data acquisition stopped at 65 minutes with the end of the infusion.The emission data were corrected for attenuation using template-based methods (24).Head motion was corrected with a frame-by-frame realignment to a reference image (frame length 5 minutes post-injection) and was performed with Pmod 3.9.Head motion was lower than 1 mm, which is less than the PET spatial resolution of 3 mm at the center of the FOV.VOIs were selected in CER, anterior cingulate cortex (ACC) and posterior temporal cortex (Post-Tl), as the last two regions have a high density of metabotropic glutamatergic receptors type 5 (mGluR5) (25).

D. Reconstruction -Frame Schemes
The different framing schemes used to test bias in the time activity curves (TACs) and binding potential values were defined by either constant or increasing frame lengths, as well as an alternative framing scheme with variable frame lengths that takes the decreasing count rate during the dynamic PET into account.The framing schemes for the phantom study and the human brain study were defined as follows: Constant Frame Length Schemes (Const): PET list-mode data of the entire acquisition was sorted into time frames with constant frame lengths of 2, 3 or 5 minutes respectively (Const 2 min, Const 3 min and Const 5 min).
Increasing Frame Length Scheme (Incr): PET list-mode data was sorted into time frames with increasing frame length, i.e. during T4 the frame length was set to 2 minutes and during T5 and T6 the frame length was set to 3 minutes and 5 minutes (Incr 2-3-5 min), respectively.In this way, the lower counts for later frames caused by the radioactive decay of 11 C was compensated to some extent.
Increasing Frame Length Scheme with Constant True Counts (Const Trues): True events versus time curves were extracted from the acquisition and the frame lengths were adjusted to values which yielded the same total counts per frame for all frames of the dynamic PET data.The number of counts in the final frame of the acquisition was taken as a reference for counts per frame.Earlier frames were accordingly shorter.A duration of 5 minutes was chosen for the final frame since this is a typical setting for applications with cognitive tasks in simultaneous PET/MR applications and our BI protocol.The three framing schemes were evaluated with both reconstructions.
We would like to emphasize here that the advantage of the BI protocol in our case is the simplicity over the dual-bolus injection approach, because it allows us to measure the baseline and challenge effects in BPND in a single acquisition session.Another important point is the simultaneous use of other imaging or monitoring modalities, such as magnetic resonance (MR) or electroencephalography (EEG) at the same brain state condition.Our presented approach can also be applied to pure bolus or pure infusion protocols since the bias in estimated binding values, is not caused by the activity application protocol, but the image reconstruction itself.However, the bias reduction for parameters estimated with kinetic modeling still needs to be evaluated in an follow up study.

E. Bias Analysis
Bias and variability were analyzed as follows: Activity Concentration Accuracy: The bias of the measured activity concentration (Ameasured) was computed by (1): where Atrue is the true activity concentration from the probes measured in the gamma counter Binding Potential Accuracy: The procedure described above in (1) was also used to estimate the bias and variability of the binding potential values (BP), by considering BP instead of the activity concentration A. The BPtrue (for Hot1 and Hot2) values were estimated as presented in (2).Equation 3was used to calculate BPmeasured.
where AVOI1 is the mean activity concentration in the VOI1 (high activity concentration region -

Standard error (SE):
Statistical SE was computed according to (4) and Gaussian error propagation was used to estimate the SE for the BP values.
where σ is the standard deviation in the VOI regions and  is the number of pixels from the VOI.
This analysis was performed for both reconstructions.

A. Phantom Study
Activity Concentration Accuracy:  In Figure 2 a) it is possible to notice a slightly increased bias in the BG region with decreasing frame length (lower count statistics per frame).In contranst, an opposite behaviour is shown in hot regions in b) and c) where the bias becomes negative with decreasing frame length.Figure 3 compares the relative errors for the three phantom regions, for the three intervals T4, T5 and T6, and for the Const 5 min framing scheme, the Incr and Const Trues schemes (described in Section D).A similar trend for the BG region can be noticed in terms of increased bias with decreasing frame length, but this time with reduced bias in T6 and the opposite behavior for Incr and Const Trues schemes is seen.In the hot regions, the negative bias is maintained, but again there is a reduced bias in the T6 time interval for Incr and Const Trues schemes compared to Const schemes, especially with the Const Trues scheme.
Binding Potential Accuracy:  Note that, for the scheme with Const 5 min framing, the bias became variable within the time interval (from 12% to 5% for Hot1 and from 1% to -10% for Hot2), suggesting a different bias range according to the time intervals during the acquisition.This trend is minimized when Incr 2-3-5 min and Const Trues framing schemes were used and the bias range remained even closer in the all-time interval with the Const Trues scheme (Figure 4 a) bias around 5% in all-time interval and b) bias around -10% in all-time interval).However, in the Hot2 region, the bias became higher negative and with higher variability, especially in the T6 interval for all schemes tested (up to -10%) even with the proposed framing Const Trues.It is important to bear in mind that these data were not corrected for partial volume effects (PVE).However, the maximum estimated effect is around 15%-17% and the estimated correction factor is around 0.84 for the reported activity concentration values (hot regions are 4 cm far away from the FOV center).The PVE can be expected to be the same size for all framing schemes and time intervals.PVE was estimated with a geometric transfer matrix (GTM) using Pmod.

Time Activity Curves and Fit Analysis:
As shown in Figure 5 for four different framing schemes, the TACs of VOIs (BG and Hot1) were approximated separately by individual linear fits for the intervals T4, T5 and T6.Hot2 will no longer be shown since a larger bias and variability for quantification was noticed in this region.
The BP values became particularly unreliable and we do not intend to analyze such small region in our BI protocol in the future due to the complexity of task effects analysis.A similar trend found in Figure 5. can be seen in the box plots (Figure 2 and Figure 3), which show a negative slope for the BG region in T4 and T5 and positive slopes in T6 (more pronounced in Const schemes).Note the negative slopes in the Hot1 region for different framing scheme methods.Slope values ± SE for the decay corrected TACs are outlined in Table 2.

B. Application to a Human Brain Study Data
For human brain studies, the true activity concentrations in brain regions and the binding potential values are not known.Therefore, here, the TACs during the equilibrium phase of the acquisition were directly analyzed using linear fits (Figure 6) to compare the reconstruction bias of the different framing schemes.Table 3 shows the corresponding slope results (%/h ± SE).Trues; all with vendor's 3D OP-OSEM reconstruction.Const Trues 0.6 ± 1.9 -4.9 ± 1.9

TACs Analysis for [ 11 C]ABP688:
*Slope -Linear fits from graphs presented in Figure 6, and SE -Standard Error.
Using the human brain data, it is again possible to confirm the bias behavior for low activity concentration regions as CER and high activity concentration regions as ACC (see Figure 6 and Table 3).Please note the higher slope values in CER when using Const framing methods and its reduction when applying the proposed Const Trues framing scheme method (results using vendor's reconstruction).

Time Activity Curves and Fit Analysis:
Slope values for the different framing schemes with PRESTO reconstruction are presented in Table 4, per half-life and phantom regions as shown in Figure 6 and Table 2 for comparison with vendor's 3D OP-OSEM reconstruction.TACs can be seen in the supplementary file 1 .

Human Study: TACs Fit Analysis with [ 11 C]ABP688:
The evaluation explained in Section D was repeated for the PRESTO reconstruction and the results are given in Figure 9 and Table 5. Trues; all with PRESTO reconstruction.Comparisons between the vendor's 3D OP-OSEM reconstruction (see Figure 6 and Table 3) and PRESTO are possible by looking at the slope values in BG and CER low activity concentration regions.As already shown in the box plots (Figure 8 a)), it is possible to achieve lower slope values in these regions using PRESTO reconstruction.However, higher bias in Hot1 and ACC regions was also shown for PRESTO.

BPND values with [ 11 C]ABP688:
Figure 10 presents BPND values using PRESTO reconstruction and Table 6 shows the slope from the BPND curves for the different framing schemes and for both reconstruction procedures during the equilibrium phase (from 1800 seconds to the acquisition end).*Slope -Linear fits from graphs presented in Figure 7 and Figure 10, SE -Standard Error.

Discussion
In this study, the quantification bias given by different framing schemes was investigated.In addition, an alternative framing scheme (Const Trues) that kept constant true counts per frame in all of the PET dynamics was proposed.The propagation of the bias into the binding was also evaluated by using simple ratio methods at equilibrium.Moreover, the analysis was additionally performed with the PRESTO reconstruction and compared to the vendor's 3D OP-OSEM.
Quantification bias as a result of low counts in PET images reconstructed with the 3D OP-OSEM algorithm has been studied by several groups with special emphasis being placed on the impact of the bias on estimated activity concentrations and binding potentials in neuroreceptor studies (8-10, 16, 19, 26, 27).At low count rates, the distribution of the reconstructed events per voxel is asymmetric, leading to a bias in the mean of the voxel activity concentration values (9).MLEM and OP-OSEM reconstructions tend to be biased in regions with low activity concentrations, particularly if these regions are surrounded by regions of high activity concentrations (16).
Slambrouck et al., also demonstrated that regions with low activity concentrations will converge much more slowly and a very high number of iterations would be required to eliminate the positive bias (16).However, it is not usual to have a high enough number of iterations to avoid this convergence bias.Moreover, since there are other sources for bias apart from iteration numbers, a completely bias-free image is not possible (16) and at low count statistical levels, other factors, such as random estimation, can be critical.This is because single rates are not constant and/or the activity distribution is not static during the frame (at this point framing schemes should be carefully chosen).Previous works have reduced the random events estimation bias considerably by applying a reduction variance given by VRR algorithms (9,14,19).However, other factors, such as scatter overcorrection, global dead time correction instead of block-wise correction, inconsistences in the attenuation map, etc., can still contribute to the reconstruction bias.These factors are not within the scope of this work.
Other methods to reduce bias at low count rates have been proposed, e.g.Hong et al. (28) used a method called complementary frame reconstruction.This method involves the indirect formation of a low count image (short time frame) through the subtraction of two frames with longer acquisition times (high statistics).The short time frame is then excluded from the second, long frame data prior to reconstruction.The method was tested with a phantom and with clinical data using HRRT and Biograph mCT scanners and with OP-OSEM reconstruction.In contrast to this study, the authors focused their work on applications relating to estimation of the arterial input function, although they also commented on potential future applications.While, our alternative framing scheme, (Const Trues), is intended to mitigate the bias in any group of reconstructed images, so far it was validated for equilibrium methods at low count rates and particularly for BI protocols when used together with ABP and simultaneously MR sequences.The goal here is to maximize the possibility to identify pre and post task effects together with other modalities and during the same brain conditions.In addition, it increases the possibility to detect binding differences by reducing the bias and keeping at the same levels during the pre-challenge and postchallenge time periods for a possible comparison.Since the bias cannot be avoided entirely, even when using 3D OP-OSEM + VRR reconstructions, we hypothesize that, if the bias is mainly due to the low count rates, the bias can be mitigated and maintained at a constant level by keeping the counts per frame constant during all PET dynamics (keeping bias constant for the all acquisition time interval).
The findings of our study support a previous bias investigation undertaken by Planeta-Wilson and colleagues using HRRT and OP-OSEM (MOLAR) reconstruction.Here, a bias of -4 ± 2% was shown for GM (high activity concentration region) and 4 ± 5% for white matter (low activity concentration) regions (7).In terms of bias range, for low and high activity concentration regions, our results showed agreement, since the bias range values were closer for BG (with ± 3%) and Hot1 (between -3% and 5%) with respect to activity concentration accuracy.In addition, based on data obtained from a 11 C filled decay phantom study and a human data (slopes), we also observed a negative bias for high activity concentration regions when the frame length was shortened (low counts).Our study differs from previous studies reported in the analysis approach since we additionally evaluated the slope changes from a complete dynamic PET acquisition rather than just a single frame with different count statistics.Furthermore, our results completely diverged from others in terms of the amount of bias found.Johansson et al., reported a significantly higher bias which was in the range of -16% to -18% in high activity concentration regions (1M and 200k counts, respectively) using HRRT and 3D OP-OSEM.van Velden et al., reported a bias of around -9% to -14% in GM (Hoffman phantom in 5 seconds frame) using the same scanner and reconstruction (5,19).Moreover, Reilhac and colleagues reported a bias of up to 80% in the CER region for the end of the activity time course in simulations using 3D UW-OSEM (8).
It is important to interpreter these results carefully since our study differs in some aspects, for example, in terms of the activity concentration and high to low activity concentration ratios used, the radioisotope and radiopharmaceuticals, and the range of analyzed statistics.Moreover, there are some configuration differences between scanners and the 3D OSEM reconstruction procedures presented (number of iterations and subsets), post-processing smoothing, etc.However, in the interest of comparability, we tried to compare our results with the closest 3D OP-OSEM procedures and statistics as possible.
In terms of bias propagation into binding values, our study is in agreement with van Velden et al (19), where a bias of -14% BPND was reported when using reference tissue models.In our study, a BPND bias of up to -40% ± 6% per hour (Const 2 min) was found and when mitigated with the Const Trues framing scheme, the bias was -7% ± 5% per hour in ACC (observed as slopes in the human data example, see in Table 6).It is important to emphasize that our approach is based on BI studies and the BPND values in other studies were obtained from bolus only analysis applying kinetic modeling instead of simple ratio methods.We hypothesize, that pure bolus and pure infusion studies may suffer from bias issues similar to those found in our study and that it might be possible to mitigate the bias with the proposed Const Trues framing approach.As the bias is introduced during iterative image reconstruction, parameters estimated via kinetic modeling will potentially be affected as well.However, further studies for evaluating this are needed.
The slopes approach was used in this study because we expected a more similar bias between time intervals (T4, T5 and T6) and along the TACs when applying the Const Trues framing scheme.In this case, the slope should not be changed by the bias.Nevertheless, bias between subjects can.We noticed the same trend as found in the 11 C filled phantom decay study, where the bias became negative along the TAC, especially in time interval T6.Furthermore, it was also possible to identify a stronger bias in a smaller region with high activity concentration, represented by Hot2 (simulating nucleus accumbens volume).Unfortunately, this region showed consistently larger bias values, especially for BP.The negative bias for activity concentration was in the 1% to -8% range, and the variability was higher between the different time intervals (T4, T5 and T6) leading to larger BP mean bias values of up to -10%.The high BP bias values and its high variability for regions comparable to Hot2 in size could cause spurious results during neuroreceptor study analysis.Walker and colleagues (9) found -13% without VRR and -5.5% with VRR bias (for 1.7M counts) in the caudate head (1.3 ml) which is of similar size to Hot2 (1.13 ml).
It should also be noted, that the negative trend of bias in hot regions and the positive trend of bias in the BG region leads to an amplified underestimation of BP (Equation 3).This is an important point of consideration for neuroreceptor studies using equilibrium conditions, especially if the aim of the study is to evaluate the binding prior, during, and post a specific task.This holds especially for cognitive tasks where the potentially induced effects are caused by endogenous release and are thus expected to be smaller than pharmacological tasks (e.g.video game playing tasks with [ 11 C]raclopride, 13% BP decrease versus psychostimulants 10% to 20% BP decrease (29)).In such a case, the bias could mask the effect in the Hot1 region, since it would be larger than 10% during the time interval T4 and 5% during the time interval T6, when a scheme with Const frames is used (see in Figure 4).This effect can also be observed for ABP in-vivo TACs (see Figure 6, Figure 7 and Table 3).Here, we observed a relative change of up to -40% per hour in the ACC region with the vendor's 3D OP-OSEM reconstruction and with the Const 2 min framing scheme.This could lead to misinterpretation of the effects caused by the task and give rise to incorrect conclusions.With the proposed, alternative Const Trues framing scheme, the average BP bias could be reduced/mitigated to mean values around 5% in the phantom study (see Figure 8) which is constant for all intervals T4, T5, and T6.A drawback of the proposed framing method is the use of the very short time frames at the beginning of the acquisition, which result from the reference counts number in the last frame.This also leads to an increased variability (Figure 8).
Nevertheless, the time required to reach equilibrium in neuroreceptor studies, e.g. for ABP usually starts at around 30 minutes p.i., so this drawback is tolerable for our studies, since the relevant part of the PET study is during the equilibrium phase interval (see Table 6).Note that the SE values remain constant with our Const Trues framing scheme (Figure 7 and Figure 10).
For the Incr 2-3-5 min framing, a reduction in the BP mean bias values was also observed.
However, regarding the difference between time intervals T4, T5 and T6, a higher bias (around 10%) can be seen in T4 for Incr schemes compared to Const Trues (around 5% from T4 to T6).
From these findings, framing schemes with constant frame lengths are less suitable for the evaluation of BPND when is required for the analysis prior and post effects in neuroreceptor studies as the bias range has high variability between the time intervals (from T4 = 15% to T6 = 5%).In Figure 5, the falling and rising trends of the bias for the BG region and the falling behavior of the bias for the hot regions can be clearly observed for all studied framing schemes.The similar results were found for the TACs in the human brain study with [ 11 C]ABP688 (especially the rising behavior in the last time interval for CER and falling behavior for ACC, see Figure 6).This finding was also noticed by van Velden et.al. (19) using reference region (pons) approaches with In summary, all framing schemes methods, even with the minimized bias range due to our proposed framing scheme Const Trues, produce a bias of at least 5%, which should be taken into account for the conclusions drawn from the observed BPND values.Moreover, when task effects are evaluated with the mitigated bias framing scheme (Const Trues), the task must induce a change sufficiently larger than 5% in order to be observable in equilibrium studies with ABP during the analyzed time interval.
Further studies are planned to investigate the influence of scatter correction at low count rates, as it has already been pointed out by Jian and colleagues (10) that scatter correction is a potential bias source.

Conclusion
This work aimed to analyze the bias in activity concentrations estimated from PET images and its propagation into binding potential values for equilibrium studies using a 11 C filled decay phantom study and a human data set from a BI study with ABP.Taking into account all framing schemes tested, one can conclude that PRESTO showed lower bias values for the low activity concentration region, particularly when using the Const Trues framing scheme method proposed.
Therefore, in order to be observable, the size of any potentially endogenous response to tasks or challenges in neuroreceptor studies with ABP using the BI protocol should be sufficiently larger than 5% for our proposed framing scheme and when using vendor's reconstruction, since the bias

Supplementary Files
This is a list of supplementary les associated with this preprint.Click to download. Supplement.docx

Fig. 1 .
Fig. 1. a) Sagittal PET image from the ABP study overlaid with MR T1 image and b) Sagittal PET image of the NEMA phantom overlaid with an MR T1 image during the 11 C filled phantom decay study.The high activity concentration region (pink VOI) and low activity concentration region (blue VOI) is highlighted for bias analysis.

(
ground-truth value) and Ameasured is the mean activity concentration in the VOIs representing the cerebellum (blue VOI in Figure1(b)) or hot spheres (pink Hot1VOI as an example in Figure1(b)) in each frame.PRESTO was cross-calibrated to the vendor's 3D OP-OSEM reconstruction with respect to the activity concentration accuracy.Box plots were used to represent the bias variability of the measurements for each VOI, for the different acquisition intervals T4, T5, T6, and for the different framing schemes.
scheme, we hypothesized that the same bias would always be obtained throughout the TACs.Thus, Figure 2 shows the relative bias for the three regions in the phantom, for the intervals T4, T5 and T6, and for the three different Const framing schemes explained in Section D.

Fig. 2 .
Fig. 2. Bias and variability in activity concentration for a) BG region (low activity concentration), b) Hot1 region (high activity concentration) and c) Hot2 region (high activity concentration); all with vendor's reconstruction.

Fig. 3 .
Fig. 3. Bias and variability in activity concentration for different framing schemes a) BG region (low activity concentration), b) Hot1 region (high activity concentration) and c) Hot2 region (high activity concentration); all with vendor's 3D OP-OSEM reconstruction.

Figure 4
Figure 4 presents the relative errors of BP values in the two hot phantom VOIs for different framing

Fig. 6 .
Fig. 6.TACs and linear fits during equilibrium in CER (left side) and ACC (right side) regions from a human brain study for a) Const 2 min, b) Const 5 min, c) Incr 2-3-5 min and d) Const effects of the different framing schemes on BPND values are shown in Figure 7.Here the effects of bias propagation into BPND values is clearly noticeable, especially after 3000 seconds for the Cont 2 min framing schemes.Note the reduction in the bias when Incr 2-3-5 min and Const Trues time length schemes were used.

Fig. 8 .
Fig. 8. Bias in activity concentrations and BP values for different framing schemes comparing PRESTO (light blue, light red) and vendor's 3D OP-OSEM (dark blue, dark red) in a) BG, b) Hot1 and c) Hot1 BP.

Fig. 9 .
Fig. 9. TACs and linear fits during equilibrium in CER (left side) and ACC (right side) regions from a human brain study for a) Const 2 min, b) Const 5 min, c) Incr 2-3-5 min and d) Const

Fig. 10 .
Fig. 10.Values of BPND ± SE values for ACC (left side) and Post-Tl (right side) regions from a human brain study for a) Const 2 min, b) Const 5 min, c) Incr 2-3-5 min and d) Const Trues; all with PRESTO reconstruction.
propagates into the biding potential values.Further studies are required to estimate the bias introduced by other sources, such as scatter correction.Declaration Ethics Approval and Consent to ParticipateOur work has been carried out in accordance to The Code of Ethics of the World Medical Association (Declaration of Helsinki) for experiments.All participants (human example) signed written informed consents in addition to verbal consent.The privacy rights of human subjects were observed.The ICMJE Recommendations for the Conduct, Reporting, Editing, and Publication of Scholarly Work in Medical Journals have been followed.

Figures
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Figure 1 a
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Figure 10 Values
Figure 10

TABLE 1 11
C DECAY EXPERIMENT WITH A PHANTOM (SELECTED TIME INTERVAL)

TABLE 2 TIME
ACTIVITY CURVES -SLOPE* RESULTS -PHANTOM DATA *Slope -Linear fits from graphs presented in Figure5, and SE -Standard Error.

TABLE 3 TIME
ACTIVITY CURVES -SLOPE * RESULTS -HUMAN DATA

TABLE 4 TIME
ACTIVITY CURVES -SLOPE * RESULTS -PHANTOM DATA -PRESTO *Slope -Linear fits from PRESTO reconstructed data, and SE -Standard Error.

TABLE 5 TIME
ACTIVITY CURVES -SLOPE * RESULTS -HUMAN DATA -PRESTO Linear fits from graphs presented in Figure9, and SE -Standard Error.