Non-Gaussian space-variant resolution modelling for ...

C Cloquet1 , F C Sureau2 , M Defrise2 , G Van Simaeys1 , N Trotta1 , S Goldman1 1

Dept. Nuclear Medicine, Universit´ e Libre de Bruxelles, B-1070 Brussels, Belgium 2 Dept. Nuclear Medicine, Vrije Universiteit Brussel E-mail: [email protected]

Non-Gaussian space-variant resolution modelling for list-mode reconstruction Abstract. Partial volume effect is an important source of bias in PET images, that can be lowered by accounting for the point spread function (PSF) of the scanner. We measured such a PSF in various points of a clinical PET scanner and modelled it as a product of matrices acting in image space, taking the asymmetrical, shift-varying and non-Gaussian character of the PSF into account (AMP modelling), and we integrated this accurate image space modelling into a conventional list-mode OSEM algorithm (EM-AMP reconstruction). We showed on the one hand, that when a sufficiently high number of iterations is considered, the AMP modelling lead to better recovery coefficients at reduced background noise compared to reconstruction where no or only partial resolution modelling is performed, and on the other hand, that for a small number of iterations, a Gaussian modelling gave the best recovery coefficients. Moreover, we have demonstrated that a deconvolution based on the AMP system response model lead to the same recovery coefficients as the corresponding EM-AMP reconstruction, but at the expense of an increased background noise.

PACS numbers: 87.57.nf

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1. Introduction Positron Emission Tomography (PET) is an in vivo functional imaging modality that provides quantitative measurements of biochemical and biological processes at work in living bodies. Several sources of bias, such as attenuation and scatter, are now commonly taken into account to enhance the quantification. Most clinical systems however do not model the point spread function (PSF) of the scanner during image reconstruction, leading to partial volume effects (PVE) (Hoffman et al., 1979). Depending on the application this bias can be as important as the bias caused by attenuation (Soret et al., 2007). Although advanced detector technologies, such as the measurement of the depth-of-interaction (DOI) or decreasing the crystal size may partially alleviate the PVE problem, an accurate modelling of the system PSF during reconstruction is always warranted to better exploit the potential performance of a scanner. As the most important components of this PVE take their physical origin in the data space, most efforts to tackle this problem have been spent by modelling the distorsion of the data, either before or during image reconstruction. However, the reconstructed image may also be seen as the blurring of the true image by a non-stationary and non-Gaussian PSF, suggesting an alternative approach to PVE correction: modelling the PSF in image space during image reconstruction. Interest for this alternative approach stems from two attractive properties. The first property is a reduced computational complexity compared to data space PSF modelling, as the image space is three-dimensional whereas the data space is fourdimensional unless simplifications are introduced. The second attractive property of image space PSF modelling is that it can be applied efficiently with the list-mode maximum-likelihood expectation-maximisation (MLEM) reconstruction algorithm. This is important for the low count rate PET studies that we are considering. Only a limited number of studies have investigated image space PSF modelling during reconstruction and, to our knowledge, no comparison has ever been made between the Gaussian and non-Gaussian modelling. These studies are reviewed in detail in section 2. For clinical scanners without DOI capability, the measurements of Surti and Karp (2004) suggest that a non-stationary model might be relevant. The purpose of this paper is therefore to analyse the performance of this technique for list-mode reconstruction of data acquired with a large field-of-view (FOV) clinical PET scanner, such as the Philips Gemini 16 Power. In particular, this study aims at evaluating the impact of using a sophisticated non-stationary and non-Gaussian model of the PSF, as opposed to the stationary Gaussian model in most previous works. In addition the method will be compared to an alternative approach in which an initial image reconstructed without resolution modelling is deconvolved using the same non-stationary PSF model. In all these cases, the PSF will be estimated by measuring point sources in various locations inside the FOV. We are thereby adding our contribution to previous studies performed by other groups. The rest of this paper is organized as follows: section 2 reviews the theory of resolution modelling and its use in reconstruction and deconvolution. Point source measurements and fitting, as well as the practical experiments that have been conducted are described in section 3 and our results are described in section 4.

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2. Theory 2.1. Origin of the PVE Modern PET scanners usually store the detected coincident events into a list mode data file. Each event in this file is characterized by a line of response (LOR) defined by the indices of the two crystals in coincidence, together with other information like the time of occurence and the estimated energy of the two photons. The list mode data file can be histogrammed on the fly or off-line into a sinogram, with bins containing the number of events measured in each LOR. List-mode data or sinogram data are then fed to a reconstruction algorithm, eventually producing an image of the spatial distribution of the tracer in the FOV of the scanner. The simplest model of the data assumes that the mean number of events measured in each LOR is equal to the integral of the tracer distribution along the line linking the centres of the two corresponding crystals. This is the model used by the Radon-based analytical reconstruction algorithms but also by many existing implementations of iterative algorithms. This line-integral approximation neglects a number of physical effects which occur at each stage of the detection chain, each contributing, if not modelled, to the blurring of the reconstructed image. These well-known contributions to the PVE include: the positron range, acollinearity, finite crystal size, parallax and intercrystal scatter (Leahy and Qi, 2000). The first effect is best described as acting on the image, since the detector actually sees the true image as blurred by the positron range ; its extent depends on the positron energy and on the medium through which it travels. In contrast the three last effects above are best described as acting on the localization of the LORs ; each causes a loss of resolution that is not uniform in the FOV. Finally the characteristics of the image digitization (voxels, blobs, ...) also contribute to the PVE, but this need not be considered during reconstruction, the goal of which is to recover the best possible estimate of the digitized image of the tracer distribution. 2.2. Modelling In the following, a line over a symbol denotes a real – hence unknown – effect and a tilde indicates a matrix modelling an effect. Let S = RD be the sinogram space and I = RV the space of the reconstructed image, where D is the number of LORs in the scanner and V the number of voxels. In the particular case of the Gemini 16 Power D ' 2 · 108 and V ' 7.5 · 106 . Let freal (x) be the true radioactive tracer concentration in each point x of the FOV, freal ∈ I its discretization on a voxel grid and g , {gj } ∈ S the data, with gj the number of events measured in the LOR j. The mean number of events measured in the LOR j is given by the j th element of the vector gtheo = Hfreal ,

(1)

where the matrix element Hji represents the probability that an event occuring at voxel i is detected in the LOR j. Following Qi et al. (1998) and the seminal idea of Shepp and Vardi (1982) H can be factorized as: H'H where

sens

H

sino

H

attn

H

geom

positron

H

,

(2)

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positron

• H (I → I) represents the matrix modelling the effect of the positron range, geom • H (I → S) represents the probability that a photon pair emitted in the voxel i crosses the faces of the crystals of the LOR j, in the absence of attenuation and scatter; this term also accounts for the acollinearity, attn

• H (S → S) is a diagonal matrix accounting for the attenuation in the medium, sens • H (S → S) accounts for the sensitivity of the detectors, sino

• H (S → S) represents the blurring between the line of response, due to parallax, and intercrystal scatter. This last matrix is called the sinogram space ray spread function (SI-RSF). Note that even this representation is approximate as the effect of scatter and random coincidences are neglected, both of which are usually estimated separately and incorporated as an additive background in (1). However, it has the advantage to hold the matrices as sparse as possible, thereby speeding up the algorithms. A detailed and pedagogical description of the factors in (2) can be found in Qi et al. (1998) or Leahy and Qi (2000). Iterative reconstruction algorithms approximate the real probability matrix H by e that more or less accurately models the various physical effects a simplified matrix H e only accounts for the geometrical effects in the detection chain. In many cases, H and for the attenuation and detector sensitivity. The reconstructed image is defined e † g, with H e † being for instance the generalized by some mapping S → I, frecon = H † e inverse. If the mapping H is linear, the mean reconstructed image is equal to the reconstruction from noise-free data e † gtheo = H e † Hfreal . frecon = H (3) so that the reconstructed image is blurred by a V × V matrix reso e † H, H ,H

(4)

which is, in general, different from the identity. However most of the following e Hreso will be called the image-space discussion does not rely on the linearity of H. point spread functions (IM-PSF). Ideally an iterative algorithm should model all physical effects described above. e globally by means of a This can be done either by estimating the system matrix H Monte-Carlo simulation (Mumcuoglu et al., 1996, Qi et al., 1998, Alessio et al., 2004, 2005, 2006) or by building it as a product of matrices, each approximating one of the factors in (2): e positron , e=H e sens H e sino H e attn H e geom H H

(5)

e geom is an efficient numerical model of the geometrical projector. A where H computationally more efficient alternative attempts to incorporate all non-geometric effects in a single matrix. This can be done either in data space with a SI-RSF model e sino as H e=H e sens H e sino H e attn H e geom H (6) e reso as or in image space with an IM-PSF model H e reso . e=H e sens H e attn H e geom H H

(7)

e sino or IM-PSF To get a noise-free estimation of the non-stationary SI-RSF H reso e H in every point of the FOV, two methods have previously been explored, in

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addition to Monte-Carlo simulations of the system matrix. The first possibility is to analytically model some effects, leading to SI-RSFs (Liang, 1994, Schmitt et al., 1988, Rahmim et al., 2008). In particular, Schmitt et al. (1988) developed a method combining analytic modelling and simulations or measurements, and Rahmim et al. positron sino and H , but for 2D reconstructions only. The (2008) separately modelled H second approach consists in performing direct measurements: Brix et al. (1997), Bernardi et al. (2003) and Rahmim et al. (2003) acquired and reconstructed line sources, whereas Panin et al. (2006), Alessio and Kinahan (2008), Alessio et al. (2010) and Sureau et al. (2008) acquired and reconstructed point sources. 2.3. State of the art of PVE correction Approaches based on equations (6) and (7) have quite different implications. In the following, we describe the benefits and drawbacks of each method and how they have been previously used to address the PVE problem in PET. 2.3.1. Sinogram space modelling The main argument for the use of the sinogram space modelling is related to the physics. With 18 F, the positron range in soft tissues is about 0.54 mm (S´ anchez-Crespo et al., 2004), so that, in clinical applications with 18 F, the major factors limiting the spatial resolution are the parallax and inter crystal e geom ). The parallax is scatter as well as the finite crystal size (if not modelled in H particularly important as it is responsible for the asymmetry and non-stationarity of the resolution. Therefore it seems a priori more logical to adopt the data space model (6) rather than the image space model (7). e sino : S → S in (6) should be built by Ideally a reliable model of the matrix H using a collimated source to measure the probability that a 511 keV photon incident with a given angle onto a crystal a is detected in a crystal b. These probabilities e sino . This measurement is however can then be combined to calculate the elements H i,j untractable in practice. Moreover, the SI-RSF is a non-stationary, asymmetric, and non-Gaussian 4D kernel depending on 4 variables. Therefore, due to the large size of this kernel, the SI-RSF method is impractical unless significant simplifications are introduced, such as approximating the 4D matrix by a 1D or 2D shift variant convolution in the radial and azimuthal sinogram variables. Remarkable improvements of image quality have been demonstrated using such simplified models (Mumcuoglu et al., 1996, Brix et al., 1997, Qi et al., 1998, Lee et al., 2004, Alessio et al., 2006, Panin et al., 2006, D’Ambrosio et al., 2008, Tohme and Qi, 2009, Lee et al., 2009, Alessio et al., 2010, Tong et al., 2010) but little is known on potential further gain that could be achieved by fully exploiting the potential of sinogram based approach. Furthermore, independently of its computational complexity (if implemented in 4D) an additional difficulty is that the sinogram based modelling appears incompatible with the list-mode MLEM algorithm. The knowledge of the SI-RSF can also be exploited to deblur the sinograms, prior to any reconstruction (Derenzo, 1986, Huesman et al., 1989). 2.3.2. Image space modelling of the resolution The IM-PSF is also a non-stationary, asymmetric and non-Gaussian kernel, but in contrast to SI-RSF, it is a 3D operator. The image space approach is therefore attractive because it requires a 3D rather than a 4D convolution (i.e. V  D) and because it can easily be applied to list-mode MLEM reconstruction (Reader et al., 2002).

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A practical and intuitive method (Reader et al., 2002) to construct an image e reso in (7) is to acquire data gi = Hfi for a point source in air located space model H 0 0 in voxel i0 and described by an image fi0 ,i = δi0 ,i . The i0 th column of the matrix e reso is the image pi obtained by solving H 0 e sens H e geom pi gi0 = H 0

(8)

i.e. by reconstructing the point source data without any resolution modelling, using e geom . Equation (8) can only be solved in as system matrix the geometrical model H e sens H e geom )† gi . an approximate way, using for instance a generalized inverse p†i0 = (H 0 † th reso e This generates the i0 column of the matrix, as (H )j,i0 = (pi0 )j . The full matrix is built by repeating this procedure (measurement of gi0 then reconstruction) for each voxel i0 in the FOV. A practical implementation is described in section 3. e reso has been calculated, the data are reconstructed using OSEM, now Once H e reso . In practice, the IM-PSF has a limited support e=H e sens H e geom H with the model H e reso can be implemented efficiently as a convolution and multiplication of an image by H with a finite impulse response, spatially variant, kernel. Let us stress that resolution modelling in image space is more than only incorporating the effect of the positron range because the above procedure also automatically incorporates the effect of the detector blurring. In fact, if the condition e sino H e geom } ⊆ Range{H e geom } Range{H (9) e reso that results in exactly the is satisfied then there exists an image space matrix H e sino , i.e. such that same system matrix model as the data space model defined by H e reso . e sino H e geom = H e geom H H (10) In general the range condition (9) cannot be satisfied because V  D, and in that e sino is a 1D radial stationary case the equivalence (10) can only be approximate. If H convolution model, analytically exact equivalence is still guaranteed because this convolution is equivalent by the convolution theorem for the x-ray transform to a 2D stationary convolution of the image (Natterer and W¨ ubbeling, 2001, p.13). The image space methods have not been extensively explored up to now. As for the sinogram space modelling, the IM-PSF can be used in two ways. On the one hand, as pointed out in the introduction, the whole system (acquisition + reconstruction algorithm) can be viewed as a non ideal imaging system characterized by its IM-PSF. The reconstructed image can therefore be deconvolved. Teo et al. (2007) used a stationary Van Cittert deconvolution technique without addressing the problem of increased noise intensity. The authors tried several (stationary Gaussian) IM-PSF widths, and found that, for some sizes of structures, the exact IM-PSF tended to overestimate the recovery coefficient by about 5 %, whereas a narrower one gave the right recovery coefficient. Kirov et al. (2008) used a Richardson-Lucy deconvolution with a one-step-late regularization and a deconvolution function surprisingly slightly larger than the measured IM-PSF. Barbee et al. (2010) explored a shift-variant postreconstruction deconvolution, but restricted to a region of interest. One advantage of the post-reconstruction deconvolution is that existing reconstruction code should not be modified. A potential drawback is the fact that the noise in the reconstructed image is correlated, for which no simple model is known. On the other hand, the IM-PSF can be used inside the reconstruction algorithm. Reader et al. (2002, 2003), Fazendeiro et al. (2004) and Antich et al. (2005) implemented stationary Gaussian resolution modelling, whereas the latter also

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compared to post-reconstruction deconvolution. Li et al. (2009) used a blind deconvolution procedure where the IM-PSF is estimated between each iteration of an OSEM reconstruction scheme including resolution modelling. The implementation of Sureau et al. (2008) for the brain scanner HRRT used a spatially invariant but non-Gaussian model for the IM-PSF. Rahmim et al. (2003) described an image space method that used the transaxial component of a symmetric non-stationary Gaussian IM-PSF. These authors indirectly estimated its parameters by reconstructing line sources at several positions in the FOV. For each position, the best parameters were those which minimized the width of the reconstructed line source. Finally, a recent work of Rahmim et al. (2008) strictly applied the idea of equation (4), putting the positron range correction in image space and the other corrections in sinogram space. Other works and applications on clinical data include Rahmim, Cheng and Sossi (2005), Rizzo et al. (2007), Varrone et al. (2009), Mourik et al. (2010) and Hoetjes et al. (2010). 2.4. PET reconstruction PET images can be reconstructed using the well-known MLEM algorithm (Shepp and Vardi, 1982). Using the notations of Barrett et al. (1994), we can write   g 1 t (k+1) (k) e ·H , (11) f =f · et a e (k) + b H Hf e is the system matrix, g are the data where f (k) is the image at iteration k, H in LOR-sinogram format, a is a data vector containing for each LOR the product of the attenuation and normalization factors and b is an estimate of the random and scattered coincidences, corrected for the attenuation. The multiplications e and H e t – corresponding respectively to forward- and backprojection – are by H matrix multiplications, whereas the other operations are component-wise operations. Equation (11) is straightforwardly translated to the list-mode case (Rahmim, Cheng, Blinder, Camborde and Sossi, 2005, Parra and Barrett, 1998). The convergent algorithm MLEM can be accelerated by sub-iterating on subsets of the data, as originally proposed by Hudson and Larkin (1994) with the (nonconvergent) OSEM algorithm. In this work, subsets have been defined by assigning consecutive events of the list-mode file to different subsets. 2.5. Post-reconstruction deconvolution The PVE in the reconstructed image can be reduced by several ways. For the sake of clarity, we will not consider other methods than deconvolution, and focus on Richardson-Lucy and Landweber algorithms. The Richardson-Lucy algorithm (RL) (Bertero and Boccacci, 1998) assumes Poisson noise on h, the image to be deconvolved, and automatically enforces the non-negativity of the deconvolved image. It reads h e reso,t f (k+1) = f (k) · H , (12) reso e H f (k) e reso is a matrix multiplication, and all the other where the multiplication by H operations are componentwise. The images can alternatively be deconvolved using the Landweber algorithm (LW) (Bertero and Boccacci, 1998). The (k + 1)th iteration is computed as :   e reso,t h − H e reso f (k) , f (k+1) = f (k) + λ H (13)

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e reso ||−2 . where λ is a relaxation parameter lower than 2 · ||H 3. Materials & Methods 3.1. Instrumentation Our acquisitions have been done with the Philips Gemini 16 Power PET scanner, whose characteristics are summarized on figure 1 (Surti and Karp, 2004). As this scanner does not have DOI capability, the resolution varies largely in the radial direction along the FOV. Surti and Karp (2004) reported a degradation of the radial resolution from 5.49 mm FWHM at 1 cm to 7.69 mm FWHM at 20 cm from the centre of the FOV.

XX

Figure 2: Point source acquisition setup.

XX

XX

Figure 1. Physical characteristics of the Philips Gemini 16P PET scanner. The scanner is represented with its block structure, at scale 1:16.

Figure 3. phantom.

Low-dose CT of the Jaszczak

3.2. Algorithms In the sequel, a left handed coordinate system has been used and positioned as follows: the origin is at the centre of the FOV. The x-axis is horizontal, and points to the left hand side of a supine patient. The y-axis is vertical, pointing towards the floor. A list-mode OSEM algorithm has been implemented to reconstruct the images e geom has been implemented using a on a 2 x 2 x 2 mm voxel grid. The matrix H 16-ray Siddon projector, where each LOR is obtained by averaging 16 line integrals connecting 4 points randomly distributed on the entry surface of each of the two

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crystals in coincidence. Each line integral through the voxelized image is calculated on-the-fly during the reconstruction using Siddon’s method (Siddon, 1985). e geom This averaging accurately models the finite crystal size, which is needed in H geom reso e is even if some of its effects could be taken into account into H . The better H e reso . Furthermore, with an accurate modelled, the less effects are to be modelled in H model of the tube of response induced by the finite crystal size, we also prevent the e reso . The potential occurence of artefacts, that would be difficult to account for into H ji e t a in (11) has been computed by backprojecting all the LORs. sensitivity image H 3.3. Point sources acquisition and reconstruction To assess the resolution throughout the FOV, we used a 30 µCi 22 Na source, of 300 µm diameter, enclosed in a small acrylic cylinder of 6.1 mm height and 25.4 mm diameter as point source. Alessio et al. (2005) showed that this configuration leads to the same positron range as 18 F in water. A grid has been printed on a sheet of paper and sticked onto a polystyrene support with negligible attenuation. This grid has been placed in a central coronal plane covering the largest part of one quadrant of the FOV, in order to exploit the rotational symmetry of the scanner (see figure 1). The interval between adjacent positions of the grid was 41 mm radially (7 positions), and 15 mm axially (6 positions), resulting in 42 different positions. The acquisitions have been performed over several days. At the beginning of each series of acquisitions, the grid has been centered using the point source, and aligned with the axis by means of the axial laser of the PET gantry. The source has then been displaced by hand at each position on the grid, where compact list-mode acquisitions (Phi, 2005) of 2 million counts (prompts + delayed) have been performed, resulting in about 1.5 to 4 minutes acquisition per position. Data have been reconstructed with 9 iterations and 4 interleaved temporal subsets of list-mode OSEM. The positions of the point source have been approximated by the centre of mass of the reconstructed image, rather than inferred from the physical location of the 22 Na source. Therefore, potential small displacements could not be corrected. 3.4. Point sources fitting As can be seen from the reconstructions of the point sources images (figure 4), the IM-PSFs are space-variant, non-Gaussian and asymmetrical. This was also observed by Alessio et al. (2006) and will be further discussed in the results section. To avoid introducing noise in the modelling of the resolution, a smooth functional model of the IM-PSF is needed, which will also allow to separate the axial and transaxial directions, thereby speeding up the algorithm. IM-PSFs are assumed to be separable in radial, tangential and axial directions. We therefore use a z axis-rotationally invariant parametric model of the IM-PSF, which is the product of one axial, one radial and one tangential function. Let k = [r t z]t be the radial, tangential and axial coordinates of the position of the centre of mass of a point source in the scanner reference frame, mk be the corresponding reconstructed image, centered on k, ∆ = [∆r ∆t ∆z]t the coordinates in that image reference frame, corresponding to k+∆ in the scanner reference frame. The

Non-Gaussian space-variant resolution modelling for list-mode reconstruction (x, z) = (0 mm, 0 mm)

(x, z) = (246 mm, 60 mm) 0.6

AMP fit SVG fit Data

0.4

intensity [a. u.]

intensity [a. u.]

0.6

0.2

0

10

0

10

20 x [mm]

30

AMP fit SVG fit Data

0.4

0.2

0

0

10

20 x [mm]

30

Figure 4. (a) point source + fits at the centre of the FOV. (b) point source + fits at x = 246 mm and z = 60 mm. These figures have been obtained by selecting the central slice (z-coordinate) of the point source and fits, and then summing the profiles along the tangential (y) direction. The three functions (data, SVG fit and AMP fit) have the same 3D-centre of mass. The function in (b) is wider than in (a), shows a slight asymmetry, and a clear non-Gaussianity, demonstrating the superiority of the AMP fit.

reconstructed images mk have been fitted in the least-square sense with a separable space-variant 3D function f3D (Pk ; ∆) = P0,k · f1D (Uk ; ∆r) · f1D (Vk ; ∆t) · f1D (Wk ; ∆z),

(14)

where the f1D are 1D functions described in box 2, and Pk = [P0,k Uk Vk Wk ] are the parameters of the fit, with P0,k a global scaling factor and Uk (resp. Vk and Wk ) a set of parameters for the radial (resp. tangential and axial) direction: t

Pk = arg min ||mk (∆) − f3D (P; ∆)||2 . P

(15)

The non-convex and nonlinear cost function in (15) has been minimized using the Levenberg-Marquardt algorithm (Levenberg, 1944, Marquardt, 1963), implemented in C by Lourakis (2004). To start the minimisation, the parameters have been initialized using the parameters found by a preliminary 1D fit on the 1D profiles. Several separable 3D models have been fitted: a shift variant Gaussian Model (SVG) (6 parameters), a shift variant and asymmetrical Gaussian model (SVGA) (9 parameters) and the Asymmetric Modified Pearson model (AMP) (9 parameters) (see box 2). In each case, the IM-PSF fitted on each acquired point source is rotated (see box 1), finally producing a separable kernel made of a 2D transaxial kernel and a 1D axial kernel in each point of a grid in the FOV. In our case, we used a grid with sampling [10.25, 10.25, 3.75 mm]. From the parameters of the AMP model, we computed the Full Width at Half Maximum (FWHM) and the Full Width at Tenth Maximum (FWTM) with the formula’s of box 2. 3.5. Non Gaussianity measurements The shape of a unidimensional profile can be assessed by computing the skewness, describing the asymmetry, and the kurtosis, describing the flatness (see. box 2). This kurtosis can be separately computed on the left or right side of the profile after symmetrization of the half-profile. This should be done when assessing the nonGaussianity of asymmetric profiles, i.e. when the skewness is different from zero. In e = K − 3, so that, for a Gaussian, this article, we used the shifted kurtosis defined as K e = 0. K

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Box 1. How to produce kernels spanning the whole FOV from the acquisition in a plane.

Box 2. Fit functions and non-Gaussianity Measurements.

The notations are those of section 3.4.

Asymmetric Shift-Variant Gaussian

1 For each point x = [xi yi zi ]t of a Cartesian grid subset of the 3D FOV, q (a) compute ρi = x2i + yi2 and

f1D (Q; w) = 2   − 1 w−µ if χµ > 0    σ1 e 2 σL L (external) 2  √1 · 1 w−µ 2π  −2 if χµ ≤ 0 1  σ R  σ e R (internal) where χµ = (w − µ)µ, Q = [µ, σL , σR ].

θi = tan−1 (yi /xi ) (b) define • ki , [ρi  0 zi ] t  cos θ sin θ 0 • R(θ) ,  − sin θ cos θ 0  0 0 1 • Ωi , [∆x ∆y ∆z]t are coordinate centered on x and • Ω0i , [∆x0 ∆y 0 ∆z 0 ]t = R(θi )Ωi . (c) compute Uki , Vki and Wki by 3D linear interpolation of the nearest parameters (d) the IM-PSF centered at x now reads h3D (Pki ; Ω) = P0 · f1D (Uki , ∆x0 ) · f1D (Vki , ∆y 0 ) · f1D (Wki , ∆z 0 ). 2 for implementation reasons, as the radial and tangential directions do not necessarily coincide with the x and y directions, we stored the profiles as a 2D transaxial function and a 1D axial function. ftrans ([UV]; [∆x∆y]) = f1D (U, ∆x0 ) · f1D (V, ∆y 0 ), and faxial (W ; ∆z) = f1D (W, ∆z 0 ). 3 during reconstruction: linear interpolation of the profiles.

Asymmetric Modified Pearson (AMP) f1D (Q; w) =    2 −σL if χ > 0  µ w−µ   1 + 2σ −3 L (external)    2 −σR if χ ≤ 0  µ   1 + w−µ 2σR −3 (internal)

Width measurements with the parameters of AMP: p ∆L (m) = |2σL − 3| pm1/σL − 1 ∆R (m) = |2σR − 3| m1/σR − 1 FWHM = ∆L (2) + ∆R (2) FWTM = ∆L (10) + ∆R (10) Non-Gaussianity measurements of a unidimensional profile p(x) of mean µ and standard deviation σ. R +∞  x−µ 3 p(x) dx, Skewness: S = −∞ σ R +∞  x−µ 4 p(x) dx Kurtosis: K = −∞ σ

3.6. Phantom acquisition, reconstruction and deconvolution A Jaszczak Deluxe phantom with hot inserts, consisting of 6 sextants containing rods of different sizes (4.8, 6.4, 7.9, 9.5, 11.1 and 12.7 mm) has been filled with a [18 F]FDG solution with a specific activity of 177.23 kBq/ml. The phantom has been placed horizontally, and only the rods have been filled to avoid scatter from the upper part. To test the off-axis resolution modelling, the centre of the phantom has been placed at 13.6 cm from the axis of the scanner. The edge was thereby at 24.1 cm off-centre. A reference volume large enough to avoid PVE, filled with 75 ml of the solution has been placed in the FOV (see figure 3). This reference volume provides an activity reference as well as a noise reference, as described in the sequel. A listmode file of 2 · 109 counts has been acquired (prompts + delayed), and then divided into 10 independent statistical realizations of 2 · 108 counts (replicas), by selecting one event every ten. The ten realizations have been reconstructed with standard list-mode OSEM (denoted by EM-noRM in the sequel) and list-mode OSEM with resolution modelling, with the following kernels: stationary Gaussian with FWHM=4.7 mm, which corresponds roughly to the resolution of the scanner

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in the external half of the FOV (EM-GA), symmetric shift-variant Gaussian fitted on the data (EM-GA-SV), asymmetric shift-variant Gaussian (EM-GA-SV-A) and shift-variant asymmetric modified Pearson (EM-AMP), each time with 20 iterations and 8 interleaved temporal subsets. Eighty-seven iterations of EM-noRM, EMGA-SV-A and EM-AMP for 2 realization have also been performed, and a postsmoothed reconstruction (EM-noRM-PS) has been obtained by applying a 3-mm FWHM Gaussian kernel to EM-noRM. Finally, the Richardson-Lucy and Landweber deconvolution algorithms have been applied on each of the five first iterations of EMnoRM. Attenuation has been corrected by means of a low dose CT-scan. Scatter coincidences have been estimated with the single scatter simulation algorithm provided by the manufacturer. The randoms have been estimated by means of a variance reduction procedure (Hogg et al., 2001) applied on the delayed coincidences, and included together with the scatter in the denominator of the OSEM iteration. Data have been normalized with a normalization matrix received from Philips (Wang et al., 2007), and based on measurements done on our scanner. The reconstruction algorithm has been parallelized with OpenMP into 4 processes, which have been found to be the optimal number of parallel threads on our architecture, though it has not been further optimized. On our platform‡, each iteration of EM-noRM took about 9 hours, while each iteration of OSEM with resolution modelling took about 10.5 hours, hence a 16.6% increase in computation time per iteration. 3.7. Region of interest drawing and recovery coefficients We estimated the position of the centre of the phantom based on the low-dose CT image. The orientation has been assumed to be known, and visually checked on the CT image. The centres of the rods have then been defined as the theoretical positions relative to the estimated centre of the phantom. For each 12.7 mm rod p, we defined several cylindrical ROIs of 10 mm height and varying diameter d, centered on these positions. Let B , B(sch, ite, rep, v ox) be the intensity for the voxel v ox of the reference volume, the reconstruction scheme sch, the replica rep and the iteration ite, and A , A(sch, dia, ite, rep, rod) is the mean activity in the ROI of diameter dia around the centre of the rod rod. The reference intensity µB is computed on the last iteration of the EM-noRM reconstruction scheme as follows: µB = hBivox,rep , where h. . .ia,...,b denotes the averaging on the variables a, . . . , b. For nine of the 12.7 mm-rods, we computed the Recovery Coefficient (RC) as: RC(sch, dia, ite) = µ−1 B hAir ep,r od . 3.8. Noise evaluation The noise level of the reference volume is computed as σB (sch, ite) =

µ−1 B

!1/2

 2  B − hBivox,rep

. v ox,r ep

‡ http://www.vub.ac.be/BFUCC/hydra/about.html, update of 8 March 2010

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We also computed σM (sch, dia, ite), the standard deviation of the estimator of the mean activity in a rod, as follows: σM (sch, dia, ite) =

D

A − hAirep

1/2


2 E

.

r ep,r od

This way of computing prevents the variance to be corrupted by any bias on the rods mean intensities. 4. Results 4.1. Point sources Let F (psce, v ox) be the fit of the image of the point source psce at the voxel v ox. Table 1 summarizes the average relative mean squared error of the fits, computed as ARMSE =

DD

(I − F )

2

E v ox

 E / I 2 vox

1/2 ,

psce

and shows that the non-Gaussian function better fits the data. ARMSE

SVG 0.2925

SVGA 0.2644

AMP 0.1747

Table 1. Average relative mean squared error of the fits. The AMP function is the function that best fits the data.

Figure 4 shows two summed profiles and the corresponding fits. Figure 4-(a) displays a point source at the centre of the FOV. The profile is symmetric, and the agreement between the fits and the data is good. Figure 4-(b), at x = 246 mm and z = 60 mm, shows that the IM-PSF is wider than at the centre of the FOV, slightly asymmetrical and that the Gaussian function does not adequately fit the data anymore, whereas the AMP function does. These properties can be further understood by looking at figure 5. The first row shows the FWHM (left scale) and FWTM (right scale) computed with the parameters of the AMP fit. The second row presents the skewness (left scale) and the left and right shifted kurtosis (right scale) based on the data. All values are averaged over the point sources located along the direction orthogonal to the abscissa of each figure. Figure 5 (upper row) displays that the FWHM of the radial profiles varies from 3.5 mm at the centre of the FOV to 6 mm at 24.6 cm, which is compatible with the observations of Surti and Karp (2004). The same behaviour is observed for the FWTM. Figure 5 (lower row, left scale) illustrates that the data are skewed towards the centre in the radial direction, and the absolute value of this skewness increases from 0 to 0.4 from the centre to the edge of the FOV. The data in the tangential and axial direction are essentially not skewed. Figure 5 (lower row, right scale) also shows that the data are non-Gaussian and mostly heavy tailed (shifted kurtosis higher than 0) in the three directions, except for the left kurtosis of the radial profiles, which becomes short-tailed after r = 164 mm.

Non-Gaussian space-variant resolution modelling for list-mode reconstruction Radial

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Figure 5. Shift variance and non-Gaussianity measurements of the point sources. (upper row, left scale) FWHM and (upper row, right scale) FWTM, computed on the basis of the parameters of the AMP fit. (lower row, left scale) Skewness computed on the data for the radial profiles. (lower row, right scale) Shifted kurtosis computed on the data. All curves present values that are averaged on the point sources observations made in the direction orthogonal to the abscissa of the figure. The first column present results for radial profiles, the second column for tangential profiles, and the third for axial profiles.

4.2. Resolution modelling phantom evaluation Figure 6 shows the reconstruction of one realization of the Jaszczak phantom. The gray scale of the four images has been adjusted to the maximum intensity of the EMAMP image. There are line-shaped artefacts in the transaxial direction, mostly visible on the smallest rods; these are believed to come from the faster convergence in the axial direction. There also remain some non-uniformities presumably due to the non perfect correction of the detector efficiencies. Comparison of subfigures (a) and (d-e) illustrate the well-known property that RM-methods lead to a lowered noise compared to EM-noRM for a fixed number of iterations. On subfigures (b-e) we fixed the number of iterations so that the noise level is the same (max 2 % of difference). The figures show that the RM-methods lead to a visually better contrast compared to EM-noRM and EM-noRM-PS. A profile through one rod (figure 7) shows that, at approximately the same noise level, the reconstruction with EM-AMP gives the highest value at the centre of the profile, while having the steepest flanks. EM-noRM-PS gives the smallest peak value and the largest profile, while EM-noRM and EM-GA provide intermediate results. To further characterize this trend, we evaluated the intensity of the reference volume µB , the recovery coefficients RC(sch, dia, ite), the reference volume noise σB (sch, ite) and the estimator noise σM (sch, dia, ite), as described in section 3.

Non-Gaussian space-variant resolution modelling for list-mode reconstruction EM-noRM (4 it.)

15

EM-noRM-PS (20 it.)

EM-noRM (20 it.)

EM-AMP (19 it.)

EM-GA (17 it.) (b)

(c)

(a)

Figure 6. OSEM, 8 subsets. The vertical direction is parallel to (e) the axis of the (d) scanner. The horizontal direction is transaxial to the scanner. (a) EM-noRM 20 iterations, (b) EM-noRM 4 iterations, (c) EM-noRM-PS 20 iterations, (d) EMGA 17 iterations, (e) EM-AMP 19 iterations. Subfigures (b-e) all correspond to roughly the same noise level, with a tolerance of 2%. Comparison of subfigures (a) and (d-e) illustrates that RM-methods lead to a lowered noise compared to EM-noRM for a fixed number of iterations. Subfigures (b-e) show that, at roughly the same noise level, the RM-methods lead to a visually better contrast compared to EM-noRM and EM-noRM-PS.

intensity [a. u.]

400 300 200 100 0

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Figure 7. Profiles through a 12.7 mm rod for EM-noRM (4 iterations) (4), EM-noRM-PS (20 iterations) (+), EM-GA (17 iterations) () and EM-AMP (19 iterations) (o), corresponding approximately to the same noise level, averaged on 5 successive coronal slices. The EM-AMP profile shows straighter flanks and a total activity inside the volume of the rod larger than the other reconstruction schemes.

Figure 8 shows σB (sch, ite), the reference volume noise, versus RC(sch, dia, ite) for the different reconstruction schemes, and for several diameters of the ROIs (12.7, 7.7 and 5.7 mm). Curves illustrating the RC vs noise compromise for 20 iterations/10 realizations, and 87 iterations/2 realizations are presented. The figures reveal that at fixed noise level, RM-methods lead to a higher RC than noRM-methods. Conversely,

Non-Gaussian space-variant resolution modelling for list-mode reconstruction 12.7 mm - 10 realiz.

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Figure 8. Reference volume noise versus relative mean volumic intensity in the ROIs of various diameters, all placed within 12.7-mm rods (1=reference volumic intensity), for the different reconstruction schemes (EM-noRM (4), EM-noRMPS (+), EM-GA (), EM-GA-SV (B), EM-GA-SV-A (C), EM-AMP (o), and for different diameters of the ROIs: (a) d = 12.7 mm (10 realizations), (b) d = 9.7 mm (10 realizations), (c) d = 12.7 mm (2 realizations), (d) d = 5.7 mm (2 realizations). The numbers besides the points refers to the iteration during reconstruction. Inside a subfigure, the same iterations are displayed for the different reconstruction schemes.

at fixed RC level, RM-methods lead to a reduced noise compared to noRM-methods. This strong noise in noRM can be lowered by applying a post-smoothing (EM-noRMPS), but at the price of a degradation of the RC. In short, RM-methods lead to a better compromise between RC and noise. We also see that RM-methods converge slower than noRM-methods, and that, interestingly, when the RC becomes stable, the noise level of the RM-methods seems to be higher than noRM-methods. Finally, it should be noted that, in contrast to the other methods, the RC for EMnoRM and EM-noRM-PS methods stay strictly lower than 1 for each ROI-diameter and iteration number. Figure 8-(a) shows that for a 12.7 mm diameter-ROI (size of the rod), EMAMP leads to the highest RC. Among the reconstruction methods with Gaussian modelling, we also see that even if EM-GA shows an improvement over the methods without resolution modelling, the recovery is better when the shift-variance is taken into account (EM-GA-SV). Nevertheless, the asymmetry modelling (EM-GA-SV-A) does not bring any significant improvement, suggesting that the most important effects

Non-Gaussian space-variant resolution modelling for list-mode reconstruction

17

to model are the shift-variance and the non-Gaussianity. Figure 8-(b) reveals that, for a ROI slightly smaller than the rod size, the RC of EM-AMP rapidly reaches 1 and remains relatively stable. Results for iterations above 20 (not shown) confirm this stability. Figure 8-(c) confirms that for the 12.7 mm-rods the RC is stable from the 20th iteration on. This figure only includes 2 realizations, however, the comparison between figures 8-(a) and 8-(c) shows that, for the 20 first iterations, the evolution of the RC and the noise in function of the iterations are similar. Figure 8-(d) shows that, for the RM methods, after the 2nd iteration the RC in the 5.7-mm ROI decreases, as the iterations increases, suggesting that the bell-shaped profile is progressively becoming flatter. Beyond the 60th iteration, the RC becomes smaller in the 5.7-mm ROI than in larger ROIs, which may reveal the onset of a Gibbs effect. When iterated until iteration 87, EM-AMP leads to the RC which is both the highest and the closest to the reference value of 1 (see also the online supplementary figure 12). However, if a small number of iterations is considered (eg. below 20), EM-AMP leads to the RC the closest to 1 for ROI-diameters ranging from 9.7 mm to 12.7-mm (and larger – see online supplementary figure 12) only. For ROIs diameters ranging from 5.7 mm to 9.7 mm excluded, the EM-GA-SV and EM-GA-SV-A curves are the closest to 1, and EM-AMP overestimates the RC up to 10 %. This means that, although EM-AMP leads to the best recovery coefficients at high iteration numbers, for to date routinely achievable reconstructions, EM-GA-SV and EM-GA-SV-A best recover the maximum activity of the rods, while EM-AMP best recovers the mean activity inside the true volume of the rods. The estimator noise (σM (sch, dia, ite)) dependance on the number of iterations is pointed out in figure 9. Because the number of replicas is small, these observations should be taken with care. However, this figure suggests that, at fixed RC, the estimator noise level is lower for the RM-methods than that for the noRM methods. The GA methods give intermediate results. The same observation is made for the RC at fixed noise level. 4.3. Post-reconstruction deconvolution phantom evaluation We compared resolution modelling inside the reconstruction with a Richardson-Lucy (RL) post-deconvolution based on the same AMP IM-PSF-model. Figure 10 illustrates the noise properties of the RL deconvolution, for 12.7 mm-ROIs, computed on ten replicas. As in figure 8, the curves depict the behaviour of the reference volume noise versus the RC. The three black curves illustrate these properties for deconvolutions starting respectively from the iterations 1, 3 and 5 of EM-noRM. Each point of the black curves corresponds to a different deconvolution iteration, as indicated by the roman numbers. This figure shows that the asymptotic RC for the deconvolution is close to the RC obtained with EM-AMP. At this intensity level, the reference noise level is higher for the deconvolved images, and the higher the noise in the original image, the higher the noise in the deconvolved image. Comparable behaviours have been observed for the other kernels, and with a projected Landweber deconvolution (not shown).

Non-Gaussian space-variant resolution modelling for list-mode reconstruction

18

12.7 mm - 10 realiz. 1.8 20

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Figure 9. Estimator noise versus relative mean volumic intensity in the ROIs (1=reference volumic intensity), for the different reconstruction schemes : EM-noRM (4), EM-noRM-PS (+), EM-GA (), EM-GA-SV (B), EM-GASV-A (C), EM-AMP (o). The numbers besides the points indicate the reconstruction iteration number. The same iterations are displayed for the different reconstruction schemes.

12.7 mm - 10 realiz. XIII

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EM-noRM-DE (3→)

EM-GA

EM-noRM-DE (1→)

EM-noRM-DE (5→)

Figure 10. Reference volume noise versus relative mean volumic intensity in the ROIs (1=reference volumic intensity). The iterations of three post-reconstruction deconvolutions (EM-noRM-DE) starting respectively from iterations 1 (×), 3 (?) and 5 (O) (see arrows) of EM-noRM are compared to other methods EM-noRM (4), EM-GA (), EM-AMP (o), presented in figure 8. Numbers from 1 to 20 stand for the reconstruction iteration number. The same iterations are displayed for each reconstruction scheme. Numbers from I to XIII indicate the deconvolution iteration numbers. The same iterations are displayed for each deconvolution.

Non-Gaussian space-variant resolution modelling for list-mode reconstruction

19

5. Discussion Our key observation is that image space resolution modelling during PET image reconstruction improves simultaneously the contrast recovery, the reference volume noise, and the estimator noise. The IM-PSF asymmetry modelling does not play an important role, but modelling the space-variance, and more importantly, the non-Gaussianity do. This latter fully space-variant, non-Gaussian and asymmetrical modelling provides the best contrast recovery for all ROIs sizes when iterated to 87 iterations. When iterated to a small number of iterations (eg. below 20), corresponding to a currently routinely achievable number of iterations, the spacevariant Gaussian modelling leads to the best recovery coefficients for ROIs much smaller than the size of the rod, such as when quantification is done using the maximum standard uptake value. Nonetheless, when recovery coefficients for ROIs of the size of the rod are considered, such as in the case of the more useful mean standard uptake value, the reconstruction based on the full modelling leads again to the best recovery coefficients. Furthermore, a post-reconstruction deconvolution based on the same IM-PSF recovers the same level of intensities as resolution modelling, but at the price of a higher reference volume noise. It is known that, with ML-algorithms, a point source in the air has a sharper response than in a radioactive background. This leads to an underestimation of the IMPSF, and an undercorrection of the recovery coefficient and therefore makes difficult to ensure an optimal PVE correction. Moreover, as the response to a point is different for different intensities of the background (Nuyts and Bequ´e, 2006), an improved procedure would perhaps be to estimate the IM-PSF in function of the background, and incorporate this knowledge in the algorithm. Concerning the deconvolution, no regularisation term has been added to the cost functions. In this work, we actually follow the recovery coefficient and noise evolution with the number of iterations to illustrate the different trade-offs obtained by stopping the OSEM algorithm before convergence. An additional penalty term would have lowered the noise at the expense of a diminution of the RC. An interesting issue would be to compare reconstructions with resolution modelling and deconvolution, using an identical spatial penalty term. We expect that this would lower both the noise and the RC in each case, but moderating the difference between the noise in the deconvolution case and in the RM case. As far as noise is concerned, the deconvolution performed worse than the resolution modelling. This can be attributed to two factors. The first is the fact that, according to Nuyts (2008), an ”accurate modelling of the physics [results in the fact that a] larger fraction of the data becomes consistent [and that a] larger fraction of the noise becomes inconsistent”. The second factor is that the deconvolutions should ideally rely on a model of the correlations within the image, which is not available, as pointed out in the introduction. As one can see in figures 8 and 11 (online), the RC is dependent on the size of the ROI. We therefore stress that a correct evaluation of such algorithms should rely on measurements with various ROI sizes. Other way of defining the ROI, for instance by thresholding, might be relevant and have not been studied here. One should also note that this work has focused on quantification, and not on lesion detection or delineation. We expect, however, that the enhancements in quantification reported here would be transferrable to the detection and contour definition of lesions as well.

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This study only adressed the case where the positron range is negligible. The method should be extended to higher energy positron emitters, such as 15 O, 68 Ga and 82 Rb, whose positron range is not negligible. All potential improvements here described should best be tested by direct comparisons between image space and sinogram space methods. The outcome of such comparisons might however be strongly dependent on the specific approximations made when implementing the two methods. Defining a reliable methodology to compare these two approaches remains an open problem. 6. Conclusion In this work, we measured the system response of our scanner in various points, and modelled it as a product of matrices acting in image space, taking the asymmetrical, shift-varying and non-Gaussian character of the PSF into account (AMP modelling), and we integrated this accurate image space modelling into a conventional list-mode OSEM algorithm (EM-AMP reconstruction). We showed on the one hand, that when a sufficiently high number of iterations is considered, the AMP modelling lead to better recovery coefficients at reduced background noise compared to reconstruction where no or only partial resolution modelling is performed, and on the other hand, that for a small number of iterations, a Gaussian modelling gave the best recovery coefficients. Moreover, we have demonstrated that a deconvolution based on the AMP system response model lead to the same recovery coefficients as the corresponding EM-AMP reconstruction, but at the expense of an increased background noise. Acknowledgments This work has been supported in part by a grant of the Universit´e Libre de Bruxelles (funded by Philips Belgium), in part by a grant of the Fonds pour la Recherche dans l’Industrie et l’Agriculture (F.R.I.A., Belgium), and by a F.W.O. grant G.0569.08. The point source has been provided by S. Tavernier and the Jaszack Phantom by C. Vanhove (VUB). Our gratitude goes to G. Destree and the team of the ULB computing centre who provide a very professionnal support. We would like to acknowledge the support of M. Guerchaft, S. Rassel and P. George from Philips Belgium, as well as P. Olivier, D. Gagnon, C.-H. Tung, and P. Khurd from Philips Medical Systems, Cleveland.

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Supplementary online figures In order to refine our analysis of the behaviour of the RC in function of the size of the ROI, and to compare more accurately the efficiencies of the different schemes, we plotted the RC vs ROI size around the centres of the 12.7 mm-rods, for the 20th iteration (figure 11) and for the 87th iteration (figure 12). In both figures, we again observe a different behaviour 1.2 1.1

RC

1.0 0.9 0.8 0.7 0.6 0.5

5.0

7.5

10.0

12.5

15.0

width of the ROI [mm] EM-noRM EM-noRM-PS

XX

EM-GA EM-GA-SV

EM-GA-SV-A EM-AMP

Figure 11. Recovery coefficient at the 20th iteration versus size of the ROI for the different reconstruction schemes EM-noRM-PS (green), EM-noRM (blue), EM-GA (cyan), EM-GA-SV (magenta), EM-GA-SV-A (brown), EM-AMP (red).

for the methods including resolution modelling compared to the others. Indeed, the recovery coefficients are higher for the RM-methods. At the 20th iteration, they plateau at small ROI sizes, while the RC at small ROI sizes is decreasing more rapidly than at higher iterations. 1.2 1.1

RC

1.0 0.9 0.8 0.7 0.6 0.5

5.0

7.5

10.0

12.5

15.0

width of the ROI [mm] XX

EM-noRM EM-noRM-PS

EM-GA-SV-A EM-AMP

Figure 12. Recovery coefficient at the 87th iteration versus size of the ROI for the different reconstruction schemes EM-noRM-PS (green), EM-noRM (blue), EM-GA (cyan), EM-GA-SV (magenta), EM-GA-SV-A (brown), EM-AMP (red).

When iterated until the 87th iteration, the RC of EM-AMP is both the highest and the

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closest to the ideal value of 1. Animated figure 13 displays the evolution of the RC in function of the iteration number, for different ROI-sizes.

recovery coefficient of the VOLUMIC activity iter=87 1.2 1.1

RC

1 0.9 0.8 0.7 0.6 0.5

XX

5

7.5 10 12.5 width of the ROI [mm] EM-noRM EM-noRM-PS

15

EM-GA-SV-A EM-AMP

Figure 13. Evolution of the RC in function of the iteration number, for different ROI-sizes in the 12.7-mm rods. Click on the figure to see the animation.