This work has shown the strong variation in PSF with location in FOV for SPECT imaging (Fig. 3). In addition, the assumption of an isotropic PSF is also inappropriate in most positions within the FOV. This indicates that accurate application of any PVC algorithm requiring information on the PSF requires a case-specific measurement.

One factor included in the ‘case-specific’ description is the number of updates used in the reconstruction. The PSF was found to depend on the reconstruction update number (Fig. 3). In each case investigated with the simple phantom, it takes around 100 reconstruction updates (10 OSEM iterations with 10 subsets) for the FWHM values in each direction to become roughly constant with reconstruction update number. Even at 200 updates, the shape of the plot suggests that the FWHM measurements could continue to change with reconstruction update number—in particular the tangential direction at the off-centre position.

Further support to a potential recommendation to ensure sufficient reconstruction updates are used for quantitative images is given in the results of the experiment comparing zero and non-zero activity within the bladder of the XCAT phantom. Here, a difference was seen when comparing otherwise identical images when images were reconstructed with fewer than 50 updates. For higher updates, the perturbation-estimated PSF was virtually independent of the presence of activity within the bladder.

The results presented here indicate that factors such as lesion contrast and intensity can also impact the PSF estimation if a low number of reconstruction updates is used (Fig. 4). The range of reconstruction updates this applies to include reconstruction settings typically used in clinical practice. For example, international guidance recommends 24–50 updates, and notes the need to balance noise and resolution [25]. Lesions may change in size and avidity throughout a course of treatment, and this can confound quantification since the extent of the PVE will vary due to these changes. Reconstructing with fewer than 100 OSEM updates may introduce variability in the PSF measurement which in turn may impact the accuracy of the applied PVC. However, further work would be required to assess the impact of varying PSF dimensions on lesion quantification.

Different FWHM variation behaviour was observed for the anthropomorphic phantoms when compared with the simple cylinder phantom. A possible explanation for this difference in behaviour is the difference in surrounding activity distribution. The lesions of interest within the anthropomorphic phantoms, in particular the XCAT bone phantom lesions, are surrounded by many more low activity voxels compared with the simple phantom lesions. An additional contribution may be the relative magnitudes of the FWHM values which are lower in the anthropomorphic phantom cases. This was explored further by simulating a lesion within a uniform ring, with the same external dimensions as the simple phantom, but with the inner volume set to zero voxel value. Simulation was performed using a circular orbit, as per the filled simple phantom. In this case, the lesion was positioned further from the centre of the FOV and the curve shapes were more similar to those observed in the XCAT data; reaching a relatively unchanging value at an earlier update number compared with the simple phantom. This supports the hypothesis that the difference in shape between the FWHM versus reconstruction update curves is due to the size of the PSF and/or the surrounding activity distribution.

Figure 3 shows that the standard deviation values on FWHM measurements are greater for earlier reconstruction updates. This suggests a higher level of uncertainty in the 2D Gaussian fits for images reconstructed with low numbers of iterations compared with more iterations. For increased confidence in the fit of the PSF, and to reduce the dependence of FWHM measurement on iteration number, this work suggests that at least 100 updates should be used for reconstruction.

Another relevant factor to consider, regarding typical clinical practice, is the matrix or voxel size used for acquisition and reconstruction. The normally accepted limit for sufficient sampling suggests that voxel dimensions should be less than 0.5 times the image resolution [26]. In the cases studied in this work, to reflect typical clinical practice, 4 mm or 4.4 mm voxels were used. Referring to Fig. 3, we observe FWHM values approaching approximately 8 mm. These measurements were based on a position 8 cm from the centre of the FOV, however for objects positioned closer to the collimator face, the resolution is expected to be superior and therefore voxel sizes may exceed the suggested limit.

Changing acquisition and reconstruction parameters will affect image appearance—possibly adversely. To decouple the problem of producing visually good images, for clinical reporting, and quantitatively accurate datasets there is an argument for reconstructing multiple different image sets. For example, reconstructing one set of images for visual interpretation and another with smaller voxels, reconstructed using more iterations which could be used for quantification. However, we must note that an adverse effect of increasing the number of OSEM iterations is an increase in image noise. The optimal reconstruction settings would need to balance the bias and variance for quantification.

Another factor relevant to the reconstruction, as mentioned in the introduction, is the use of Resolution Modelling. The data presented in this work have all been reconstructed without the use of RM. Had RM been used in the reconstruction, we may have expected to see some reduction in the underestimation of regional mean value measurements for some non-PVC data. However, this reduction in bias would depend on the size and intensity of the lesion and on other reconstruction parameters. Due to Gibbs artefacts, for some smaller lesions, data reconstructed with RM may demonstrate a maximum voxel value of greater than 100% of the ground truth [27]. The application of PVC using perturbation to data reconstructed using RM is not guaranteed to be reliably accurate since Gibbs artefacts would also make the assumption of a Gaussian PSF inappropriate. In addition, the assumption of constant PSF across the width of a lesion may no longer be appropriate due to increased enhancement (i.e. improved resolution) at the edges of an object compared with the centre.

The accuracy of corrected images depends on both the accuracy of the PVC method and the accuracy of the resolution measurement. Therefore, residual bias in regional mean values may not be solely due to limitations of the perturbation method—they may be due to limitations of the STC method. However, the RMSE analysis did demonstrate lower error when the image was convolved with a perturbation-estimated PSF versus a non-specific PSF, acting as a check of the suitability of the perturbation method independent of the PVC method.

One advantage of the STC method for partial volume correction is the production of a corrected image, rather than assuming uniformity across the lesion and assigning a single value across the region. Since some stages of the STC correction include voxelwise correction, the heterogenity of the lesion could be retained. This is demonstrated in Fig. 9, in particular the bottom row which shows the STC method applied to a non-uniform lesion with uptake pattern typical of a lesion with a necrotic core.

An activity distribution based on Tektrotyd studies was investigated as this may be an area of clinical practice where the accurate application of PVC could be important. The semi-quantitative Krenning score which assesses the uptake in lesions relative to the uptake in the liver and in the spleen [28] has been shown to correlate with outcome [29]. In general, lesions are assessed visually, however there is interest in reviewing these studies with quantitative SPECT [18]. To illustrate the impact of the PVE, and the importance of accurate application of PVC, we included a lesion in the Tektrotyd dataset with Ground Truth uptake higher than the spleen (corresponding to a Krenning score of 4). However due to the PVE, as show in Fig. 10, the uncorrected reconstructed image demonstrates a regional mean value higher than the liver but lower than the spleen (corresponding to a Krenning score of 3). The extent of the PVE depends on multiple factors including the size, contrast and position within the FOV. Accurate application of PVC would improve lesion and position independence when quantifying uptake. This is especially important as Tektrotyd studies can be used to monitor response to treatment where size and avidity of the same lesion is likely to change which, as noted above, can confound quantification. Accurate application of PVC could reduce the impact of this effect.

### Limitations

One limitation of this work is the fact that system models and software used for generating sinograms and for reconstructing were identical. This does not replicate the “real life” situation accurately. However, we performed some simulations with non-matched system models. From these simulations, the estimated PSF was found to depend on the system model used to simulate imaging of the point source (i.e. used to generate the sinogram of the perturbation source). A possible option to pursue in clinical practice may be to use a measured point source—ensuring that the geometry and conditions for generating the sinogram of the perturbation source exactly matched that of the acquisition of patient data. However, a significant disadvantage of this is the requirement to make a measurement at every possible position in the transaxial plane. In addition, the measurement would not match the patient-specific attenuation and scatter situation. There would be experimental errors also in the production of a point source. Therefore, work to assess the required accuracy of the system model would be beneficial to guide the assessment of the suitability of using a simulated perturbation source.

This digital phantom simulation study did not investigate other factors which may limit accuracy in the practical application of SPECT quantification. For example, in this work segmentation was performed directly from the generated image and was therefore accurate. While noisy datasets were produced, scatter was not modelled and so results here assume perfect scatter correction. Scatter correction methods are commonly accepted to be reasonably accurate for (^{99{mathrm{m}}})Tc SPECT, however may not be perfect as assumed in the current simulation. While an imperfect attenuation correction was applied in the reconstruction, we acknowledge that other factors which limit the accuracy of attenuation correction in real clinical data are not replicated here. In reality, the accuracy of attenuation correction based on CT images may be limited by issues related to scaling Hounsfield Unit values to the energy of the SPECT photons, mis-registration and image noise. Real-life scatter and attenuation conditions may result in a reduction in the accuracy of the perturbation measurement and PVC applied to clinical data. Further work should involve testing the perturbation method and PVC algorithm on clinical datasets—potentially utilising Monte Carlo simulation with synthetically added lesions to provide a ground truth for the lesion. For translation into clinical practice, in addition to testing the suitability of a simulated perturbation source as mentioned above, it would be important to include the effect of segmentation, registration, scatter and attenuation correction into the overall assessment of quantitative accuracy. Again, further work on error propagation within SPECT quantification, and estimation of the resulting uncertainty, would be important for the evaluation of an optimal PVC method. For clinical data, uncertainty analysis could be performed by following the EANM guidelines [30].

Note that some error may be introduced due to variation in the PSF across the distance of the lesion. This work assumes that, while the PSF may change significantly across the FOV, it does not change significantly across the local region around the lesion or object of interest. “Significantly” in this case refers to variations in the PSF that would impact the accurate application of PVC such that the result of quantification would be different. This is likely to be a reasonable assumption for small lesions, but may not be for larger objects, e.g. organs at risk. However, since larger objects will be less affected by the PVE in the first instance, there may be a trade-off in accuracy. Further assessment of this would be useful.

Referring to Fig. 9c, while the quantitative accuracy of the STC correction was good, and the corrected image had improved edge definition, the uniformity of the ground truth lesion was not reproduced in the corrected image. This merits further exploration.

Areas of further investigation around the subject of this work include, but are not limited to, an assessment of precision (i.e. evaluating uncertainties), and error propagation—including an assessment of the required accuracy of the system model used for sinogram generation/reconstruction. Data presented here were limited to the relatively simple imaging radionuclide, (^{99{mathrm{m}}})Tc. Further investigation would be required to test whether the application of the perturbation method can be generalised to other radionuclides such as those used for imaging in the context of theranostics (for example (^{123})I or (^{111})In), or those used with therapeutic intent (for example (^{90})Y, (^{131})I or (^{177})Lu). The PSF of these radionuclides is likely to be wider and may include star artefacts due to high energy emissions and septal penetration. The perturbation method could potentially account for this, provided that the implementation does not make the assumption of a Gaussian PSF. Further investigation towards implementing perturbation on radionuclides other than (^{99{mathrm{m}}})Tc would be informative, perhaps utilising Monte-Carlo modelling to explore shapes and position-dependence of the PSF.

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