Scattered coincidences degrade image contrast and compromise quantitative accuracy in positron emission tomography (PET). A number of approaches to estimating and correcting scattered coincidences have been proposed, but most of them are based on estimating and subtracting a scatter sinogram from the measured data. We have previously shown that both true and scattered coincidences can be treated similarly by using Compton scattering kinematics to define a locus of scattering which may in turn be used to reconstruct the activity distribution using a generalized scatter maximum-likelihood expectation maximization (GS-MLEM) algorithm. The annihilation position can be further confined by taking advantage of the patient outline (or a geometrical shape that encompasses the patient outline). The proposed method was tested on a phantom generated using GATE. The results have shown that for scatter fractions of 10–60% this algorithm improves the contrast recovery coefficients (CRC) by 4 to 28.6% for a source and 5.1 to 40% for a cold source while the relative standard deviation (RSD) was reduced. Including scattered photons directly into the reconstruction eliminates the need for (often empirical) scatter corrections, and further improvements in the contrast and noise properties of the reconstructed images can be made by including the patient outline in the reconstruction algorithm as a constraint.
Scattered photons are a significant source of image quality degradation and lead to quantitative errors in positron emission tomography (PET) [
The energy resolution of PET detectors has improved in [
In our previous study, we have shown that true coincidences can be considered to be a subset of scattered coincidences and that a GS-MLEM algorithm can use both true and scattered events to reconstruct the source distribution [
A diagram illustrating a Compton scattering event. A positron annihilates at the green dot and generates two 511 keV photons. One is observed by detector A and the other one undergoes a Compton interaction and is finally detected by detector B. The possible scattering positions can be described by the two circular arcs (TCA) defined by the kinematics of Compton scattering.
We hypothesize that the annihilation position can be further confined by making use of the patient outline (or a geometrical shape that encompasses the patient outline) as a further spatial constraint. In this work, we evaluated the contrast and noise properties of the constrained GS-MLEM algorithm with a patient/phantom outline constraint as well as the dependency of the proposed method on the accuracy of the patient/phantom outline constraints employed.
In PET, the three main sources of photon scatter are
In such a scenario, the possible annihilation positions can be further constrained by connecting the intersection points between the TCA and the patient outline, C (the furthest from the unscattered photon detector A) and D (closer to the unscattered photon detector A), with the unscattered photon detector A. The position of annihilation is confined to the area encompassed by the TCA, the patient/phantom outline, and the line AC (area CDE as shown in Figure
In this case, one of the TCA interacts with the patient at point D (closer to A) and C (further from A). If the extent of the scatter volume is known (the patient outline), the possible annihilation area may be further confined to the area C-D-E encompassed by line CA, TCA, and the outline of the patient.
The patient outline can be estimated by a variety of means. One way is to use an external X-ray source or optical system (either laser based or photogrammetric). In PET/CT or PET/MRI systems the anatomical image provided by the CT or MRI could be used. Alternatively it may be possible to estimate the constraints using an approximation of the patient outline based on initial iterations or by using a basic geometric shape (say a circle or ellipse) as shown in Figure
The patient outline is replaced by an ellipse which is slightly larger than the patient outline and intersects with TCA at point C (the furthest from the unscattered photon detector) and point D (closer to the unscattered photon detector). The possible annihilation positions are confined to the area C-D-E encompassed by line CA, TCA, and the ellipse calculated in the same way as the case of patient/phantom outline.
Before building the patient outline constraint into the GS-MLEM algorithm, the expected number of coincidences in which an unscattered photon is observed at A while the other photon is observed at B following a Compton scattering through an angle
In the above expression the total number of photons which could reach point
To reduce the large computational workload, we ignore the attenuation and the electron density and assume that the expected number of detected coincidences is linearly proportional to the product of the differential Klein-Nishina electronic cross section and the total activity within area CDE as given by
where
where
To make the comparison consistent with our previous study [
A cylindrical water phantom with three hot areas (yellow color) and one cold area (black color) was simulated. The radii for the 1 to 4 circles are 1.5 mm, 3 mm, 4.5 mm, and 6 mm, respectively. We set the activity ratio R between the hot and background to 4.
In this initial evaluation of the proposed method, only the coincidences scattered within the phantom were used in the reconstruction. The image quality was evaluated using the contrast recovery coefficient (CRC) and relative standard deviation (RSD) which reflects the contrast and noise properties of the reconstructed images. The local contrast recovery coefficient (CRC) for hot disk was defined by
where
where
An evaluation point on the
The image quality of the image reconstructed with the phantom outline constraint was compared to the results using GS-MLEM without the phantom outline constraint and to the conventional LOR-MLEM method given in [
The improvement in the contrast and noise properties of the reconstructed images was related to the accuracy of the patient/phantom outline used. When the phantom outline constraints chosen are smaller than the actual phantom outline, the confined area may not encompass the annihilation position, which will reduce the image contrast and introduce artifacts. Thus, the minimum but still most accurate outline constraint would be the actual patient/phantom outline. A larger area can be used to constrain the annihilation position but will not be optimal. When the constraint outline approaches that of the detector positions, the outline constraint based GS-MLEM algorithm will approach the nonoutline constraint based algorithms. To evaluate the effect of different constraint sizes on the reconstructed image quality, we characterize the patient/phantom outline sizes as a function of the ratio of difference between the tested outline constraint and the actual phantom outline size. We therefore tested the effect of various phantom outline constraints using a circle with radii of 42 mm, 45 mm, 50 mm, and 60 mm, being 5%, 12.5%, 25%, and 50% larger than the actual phantom outline, respectively. A total of 6 × 105 coincidences with 50% scatter fraction were generated by GATE and reconstructed using the proposed method with the different outline constraints. The images were also reconstructed using the same dataset but with GS-MLEM without the phantom outline constraint and with the conventional LOR-MLEM algorithm, as well as with 3 × 105 true coincidences using conventional LOR-MLEM algorithm as a comparison.
To illustrate the performance of the proposed method, we plot the CRC versus RSD for different methods/or data for the no. 3 (the largest hot) disk in Figure
The CRC curves of no. 3 (the largest hot) source were calculated with GS-MLEM with phantom outline constraints, GS-MLEM method and the conventional LOR-MLEM approach.
The CRC curves of no. 4 (cold) source were calculated with GS-MLEM with phantom outline constraints, GS-MLEM method and the conventional LOR-MLEM approach.
Figure
(a) shows the image using 6 × 105 coincidences with a 50% scattering fraction by the proposed method (GS-MLEM plus patient/phantom constraint method). (b) shows the image using the same data as (a) and by GS-MLEM without phantom outline constraint; (c) shows the image using the same true 3 × 105 coincidences plus 3 × 105 scattered coincidences that fall into the 350 to 511 keV energy window and was reconstructed using the conventional LOR-MLEM algorithm as a comparison. The second row shows profiles of the above images passing through the center of the images in the horizontal and vertical directions, respectively.
Images with different phantom outline constraints were also reconstructed to evaluate the dependency of the proposed algorithm on the accuracy of the phantom outline constraints. The CRC curves of the no. 3 (the largest hot) disk for different phantom outline constraints as a function of the relative background standard deviation were obtained by varying the number of iterations as shown in Figure
The CRC curves of no. 3 (largest hot) disk calculated using GS-MLEM with different phantom outline constraints. The CRC for 3 × 105 true coincidences and 6 × 105 coincidences with 50% scattered fraction using conventional LOR-MLEM were also reconstructed as a comparison.
The CRC curves of no. 4 (cold) disk calculated using GS-MLEM with different phantom outline constraints. The CRC for 3 × 105 true coincidences and 6 × 105 coincidences with 50% scattered fraction using conventional LOR-MLEM were also reconstructed as a comparison.
The CRC and noise properties of no. 3 (largest hot) disk as a function of relative increase of radii of the phantom outline constraints for the evaluation points of Figure
The CRC and noise properties of no. 4 (cold) disk as a function of relative increase of radii of the phantom outline constraints for the evaluation points of Figure
In our previous study [
The results for the evaluation of different outline constraints on image quality have shown that uncertainties in defining the patient/phantom outline are not a significant obstacle when implementing the proposed method. This method is based on the assumption that the patient scattering dominates over detector, gantry, and surrounding environment scattering, which is true in human imaging. In this initial test of the proposed method, we carried out this work in 2D, and only the coincidences scattered within phantom were selected to show the principle of the proposed method and at the same time to reduce the complexity of the mathematics. This approach could be implemented in 3D where it could be of even greater value.
The TCA are calculated using the detector positions and the scattering angle which is closely related to the scattered photon energy using Compton equation. The accuracy with which the locus of the TCA assigns the possible scattering positions and encompasses the annihilation position will therefore depend on the energy resolution of the detectors. The area defined using a scattered photon energy with a large uncertainty may not encompass its annihilation position or may overestimate the confined area, resulting in artifacts or blurring of the reconstructed images. The effects of energy resolution on the proposed method will be investigated in the future work.
Previous work demonstrated that scattered coincidences can be directly included in the reconstruction process. The results of this study show that further improvements in the contrast and noise properties of the reconstructed images can be made by including the patient outline in the reconstruction algorithm as a constraint. While these results are promising, we are currently investigating ways of correcting for attenuation and improving the limited energy resolution of current detectors, both of which will need to be resolved before scatter reconstruction can be used for clinical applications.
This work was supported in part by CancerCare Manitoba Foundation, University of Manitoba, and Natural Sciences and Engineering Research Council of Canada.