Statistics of photon travel spread and coincidence timing
Briefly, the detection of γ-photon within scintillation detectors is a two-stage process. Firstly, the incident 511-keV photon is absorbed and optical photons are emitted within the scintillation crystal. Consecutively, a photo-detector converts these photons to electrical pulses [31].
When a narrow beam of 511-keV photons hits a scintillation crystal of thickness L, the original beam intensity (I0) is primarily attenuated due to photoelectric absorption or Compton scattering. In order to calculate the number of events absorbed in each depth layer inside the crystal, the following exponential model can be used:
$$ I(x) = I_{0}\exp(-\beta x)\,\text{,}\quad\text{for}\ 0\le x \le L $$
(1)
where β approximates the material absorption coefficient.
Formula (1) can also be expressed in terms of time:
$$ I(t) = I_{0} \exp(- \lambda t)\, {,}\quad\text{for}\ 0\le t \le T $$
(2)
where λ=βc, c≈0.2998 mm/ps is the speed of light and T=L/c (ps) is the maximum time duration of a γ-photon travelling perpendicularly to the entrance surface (Fig. 1).
The conditional probability density function (PDF) for events that are absorbed is obtained by normalising the above Eq. 2:
$$ g(t;\lambda) = \frac{\lambda\exp(- \lambda t)}{1 - \exp(-\lambda T)}\,\text{,}\quad\text{for}\ 0\le t \le T $$
(3)
Let two photons A and B from a single annihilation event be independently detected. The continuous joint probability distribution is therefore gAB(tA,tB;λ)=gA(tA;λ)gB(tB;λ). Making the substitutions d=tA−tB, which is the time difference between the two photons, and s=tA+tB, we denote the joint distribution depending on d and s as:
$$ f_{DS}(d, s; \lambda) = g_{AB} (h_{1}(d,s), h_{2}(d,s); \lambda) \left| J \right| $$
(4)
where h is the inverse transformation (h1(d,s)=(s+d)/2 and h2(d,s)=(s−d)/2) and J is the Jacobian determinant of h. Then, the CTR kernel can be obtained by integrating fDS over s:
$$ f_{D}(d;\lambda) = \left(\lambda \sinh{\left(\lambda(T - \left|d \right|)\right)}\, \text{csch}\left(\frac{T\lambda}{2}\right)^{2}\right) \Big/ 4 \,\text{,}\quad\text{for} -T\le d \le T $$
(5)
where csch is the hyperbolic cosecant function.
The corresponding cumulative distribution function (CDF) is given by:
$$ F_{D}(d;\lambda) = \frac{1 + \text{sgn}(d)}{2} - \text{sgn}(d) (\cosh{(\lambda(T - \left|d \right|))} - 1)\, \text{csch}\left(\frac{T\lambda}{2}\right)^{2} \Big/ 4 $$
(6)
where sgn is the sign function.
When T→∞, Eq. 5 becomes the Laplace distribution (fL) and FL its CDF, given by:
$$\begin{array}{*{20}l} f_{L}(d; \lambda) &= \frac{\lambda}{2}\exp(-|d\lambda|) \end{array} $$
(7)
$$\begin{array}{*{20}l} F_{L}(d; \lambda) &= \frac{1}{2}\,\left(1 + \text{sgn}(d)\left(1-\exp(-\left|d\lambda\right|)\right)\right) \end{array} $$
(8)
In practice, other system effects will decrease the timing resolution of the system. This non-idealised case can be modelled by using a TOF kernel which is the convolution between the fD(d,λ) and a normal distribution fadd(d,σ), with σ the standard deviation of the normal distribution. See the Appendix A.
Scanner model
The geometry of a cylindrical PET scanner was simulated using the GATE simulation toolkit (v.8.1) [29, 32]. The scanner was comprised of 24 rings with 666 detectors each. No gaps between blocks were considered. The gap between the crystals was 0.2 mm. The inner ring radius was 424.5 mm (with field of view (FOV) radius 297 mm), and the total axial length was 110 mm. The crystals were made of Lu2Y2SiO5:Ce (LYSO) with density (ρ) equal to 7.105 g cm −3. This crystal configuration provided realistic performance similar to the PreLude 420 by Saint Gobain [33]. The scanner’s geometry is illustrated at Fig. 1.
The surface of each scintillation crystal was 4×4 mm2. Different crystal lengths were used, which are defined in each section. The energy resolution was set to 11.4%, and the applied energy window was 435–650 keV. The coincidence timing window was set to 4.1 ns. Each crystal measurement was read out individually without summing up the energy from neighbouring crystals. The minimum allowed radial detector difference (rsector difference in GATE) was 83 detectors. The emstandardemstandard_opt3 physics list was used.
System’s timing resolution
Although various methods for the evaluation of the timing resolution of a PET scanner have been proposed [34–37], in this study, a simpler approach was considered.
To investigate the effect of the PTS on the CTR, a thin (0.05 mm radius) back-to-back γ-photon rod source was simulated until approximately 70×106 total events were recorded.
In GATE, the “macro” command setTimeResolution applies an additional normal blurring to the detection time for each detector, i.e. following the original detection time defined as the time of occurrence of the photoelectric effect (PE) in the crystal.
Using the aforementioned “macro” command, two detector configurations were considered:
-
1
First case, the additional detector time resolution was set to 0 ps which simulated a detector with idealised timing properties. In this case, the CTR kernel was tested for three different crystal thicknesses, namely 10, 20 and 40 mm. For comparison, a normal distribution was fitted to the simulated timing responses using maximum likelihood estimate (MLE) (via the Distribution Fitter App in Mathworks Matlab). With regard to the data boundaries, two cases were considered. In the first case, data boundaries were placed such that we obtained the minimum Kullback–Leibler (K-L) distance (optimum kernel) between the simulated distribution and the timing kernel (fN). On the other hand, on the second fitting (\(f^{\prime }_{N}\)), no data boundaries were placed.
-
2
Second case, a range of values were set to the setTimeResolution, simulating a non-ideal detector. The crystal size was fixed to 20 mm, as this is the one of most common thicknesses for 176Lu-based crystals [38–40]. Seven additional detector timing resolutions were considered FWHMadd = 0, 5, 10, 20, 40, 60, 80 and 100 ps. This additional timestamp smearing represents other factors affecting the timing spread, such as fluctuation in the detection of optical photons, pulse integration and electronic noise. According to the central limit theorem, the additional timing uncertainty due to all these effects can be described by the normal distribution. The two kernels (i.e. the additional timing kernel and the CTR) are then convolved. As such, the shape of the final kernel depends on both kernels.
In addition, the histograms of the timing differences corresponding to the non-idealised scanners were compared with the convolution kernel given in the Appendix A (Eq. 11).
Image reconstruction
Average depth of interaction
STIR takes into account an average depth-of-interaction (DOI) effect in the crystal for the calculation of the line of response (LOR)’s position. In order to find a good approximation for the average DOI, the detected γ-photons were binned into histograms based on the depth where they were absorbed. Then, the mean absorption depth was found by summing the bin values until the mean value was found.
The average DOI values were found to be 3.6, 5.8 and 7.4 mm for the 10-, 20- and 40-mm crystals, respectively.
Calculation of the TOF projection matrix
The TOF kernel, as implemented in STIR, is applied on top of the non-TOF LOR (pij) as [41]:
$$ \begin{aligned} p_{it;j} &= p_{ij} K_{it;j},\\ K_{it;j} &= \text{cdf}(k_{t+1} - v'_{cj}) - \text{cdf}(k_{t} - v'_{cj}) \end{aligned} $$
(9)
where Kit;j is the time response for the tth TOF position of the ith bin and jth image element, cdf is the CDF corresponding to the timing kernel used, [kt,kt+1) is the timing interval for the tth TOF bin and \(v^{\prime }_{cj}\) is the projection of the voxel’s centre on the TOF line.
Reconstruction algorithm
STIR [42, 43] supports a wide range of algorithms for the determination of the maximum likelihood estimate (MLE), including ordered subset expectation maximization (OSEM), median root prior (MRP) and quadratic prior (QP) Bayesian one step late methods [44, 45], and the ordered subset separable paraboloidal surrogates algorithm [46].
In this paper, listmode (LM)-maximum likelihood-expectation maximisation (MLEM) was used [47, 48] as it is the simplest option, and is guaranteed to converge (even slowly) to a solution. The TOF version of LM-MLEM in the STIR library was previously presented with simulated data [41] and recently validated using measured PET data [49]. The size of the TOF bins was 1 ps (numbering 4101 in total). No TOF mashing, view mashing or axial compression was used for the data.
The voxel size of the reconstructed images was 1×1×2.08 mm3. In order to reduce the reconstruction duration, the number of voxels was adjusted to fit the size of the phantom in each case. No post-reconstruction smoothing filters were applied to the images.
Attenuation correction factors were calculated with an analytical simulation, of the phantom, having the appropriate linear attenuation values for 511-keV γ-photons, as found in NIST [50]. Normalisation factors were not used. The scattered and random events were omitted from the reconstructions; all datasets had 40×106 true events.
The iterative process was performed for up to 105 iterations for the contrast recovery coefficient (CRC) and 150 iterations for the spatial resolution. However, all results are discussed for the 60th iteration, as it ensures that region of interest (ROI) values have almost converged, without introducing noise amplification and reduction in signal to noise ratio [51].
Further, expansion of the software allowed us to parallelise the TOF LM-MLEM reconstruction using OPEN-MP. STIR supported the options for OPEN-MP and MPI for reconstruction of sinograms only. The new code reduced the amount of time needed for a single iteration 10× running with 25 threads on 28 processor on the University’s cluster.
Simulated phantoms
NEMA image quality: contrast recovery coefficients
A NEMA image quality phantom [52] was designed and simulated for all scanner geometries under consideration.
The CRC of a hot sphere with inner diameter d was calculated as:
$$ \textrm{CRC}_{r} = \left. \left(\frac{\mu_{H,d}}{\mu_{B,d}} - 1\right) \middle/ (\alpha - 1) \right. $$
(10)
where α=4.5 is the actual contrast ratio of the sphere, μH,d is the mean value of the ROI and μB,d is the mean value of the background in the reconstructed images. The inner diameters of the hot spheres were 10,13,17 and 22 mm. In order to reduce the statistical error, the simulations were repeated 7 times; μH,d and μB,d values were averaged over all datasets.
Spatial resolution
In order to evaluate the effect of the different kernels on the spatial resolution, a computational Derenzo-style phantom was simulated. The phantom material was set to plastic (as defined in the GATE materials database) with a 5-cm radius and a 7-cm height. The hot rods were subdivided into six sections, with diameters of (A) 7.0 mm, (B) 5.0 mm, (C) 4.0 mm, (D) 3.5 mm, (E) 3.0 mm and (F) 2.5 mm (the letters denote the name-ID of each section). The separation distance between the rods was set to the double of their diameter [53]. In total, 40×106 true events were used to reconstruct the images.
As previously discussed [54], the assessment of the actual image resolution with statistical image reconstruction is not trivial as spatial resolution depends on the iteration number and activity distribution. In order to limit the effect of the non-negativity constraint, a high activity background source was used.
Furthermore, in order to evaluate whether the reconstructed sources contain edge artefacts, the ratio between the pixel value on the centre of gravity (COG) of the source and the average ROI value was recorded for the sources of the largest section.
