A review of partial volume correction techniques for emission tomography and their applications in neurology, cardiology and oncology

An effective method to enhance the spatial resolution in reconstructed SPECT and PET images is to incorporate the PSF into the system matrix used for forward and back-projection (see e.g. Tsui et al 1994, Liow et al 1997, Hutton and Lau 1998, Pretorius et al 1998, Zeng et al 1998, Reader et al 2003). Inclusion of resolution in the system matrix (model) is now available as an option on most SPECT and PET systems and therefore immediately applicable. It should be noted, however, that the application of this more exact model of the system is mainly used in practice to reduce the noise in the reconstruction, improved noise properties being a by-product of the improved system model. Although in theory the inclusion of the resolution aims to reconstruct the exact activity distribution, there is a limit to the recovery achievable, even at high iterations, due to the loss of high frequency information during data acquisition. Recovery is therefore limited since there will still be residual partial volume effects, that can be hard to estimate, given the nonlinear nature of the reconstruction. At best, partial volume effects are reduced in magnitude provided a sufficient number of iterations are performed, but quantitative recovery is limited.

Better results can be obtained with reconstruction algorithms involving anatomical priors to control noise and enhance edges between functional structures (most often assuming that edges between two functional compartments match the edges between two anatomical compartments). Visually these techniques can produce very striking image contrast. There is, however, rather limited reference to the effectiveness of these techniques in achieving quantitative PVC.

When using penalized iterative reconstruction algorithms (including Bayesian or maximum a posteriori (MAP) algorithms), a smoothness prior is often used. This encourages solutions with a low degree of variability between adjacent image voxels. With a segmented anatomical image, the smoothness constraint can be restricted to each anatomical region, and larger differences can thereby be allowed between adjacent voxels belonging to different regions. This is the basic principle used by most groups in the past. Early work includes Chen et al (1991), Fessler et al (1992), Gindi et al (1993) and Ouyang et al (1994).

Ardekani et al (1996) developed a method based on minimisation of cross-entropy, which utilized multi-spectral MRI data. Bowsher et al (1996) presented a Bayesian method for simultaneous reconstruction and segmentation of PET or SPECT data, in which higher prior probabilities were assigned when segmented regions stayed within anatomical boundaries. Lipinski et al (1997) presented a method that assumes a Gaussian distribution of the image values with the same mean value for all voxels within a given anatomical region. Sastry and Carson (1997) presented a MAP algorithm based on a tissue composition model, in which the image is described as a sum of activities for different tissue types, determined from a segmented MR image. Comtat et al (2002) presented an algorithm for reconstructing whole-body PET/CT data, in which the anatomical labels were blurred in order to take into account the uncertainty associated with the anatomical information. This method was later also applied to brain studies by Bataille et al (2007). Baete et al (2004) presented a MAP reconstruction algorithm for brain PET studies using a tissue composition model, in which WM and CSF were treated as uniform for the purpose of estimating the GM values. Bowsher et al (2004) described an algorithm based on an image model promoting greater smoothing among nearby voxels that have similar MRI signals, which therefore did not require segmentation of the anatomical image. Alternative methods, based on an anatomically driven anisotropic diffusion filter, were presented recently (Yan and Yu 2007, Chan et al 2009, Kazantsev et al 2012). Another novel approach was presented by Pedemonte et al (2010), in which a hidden variable is introduced, denoting tissue composition that conditions a similarity function based on entropy.

More work needs to be undertaken in order to better understand the sensitivity of these methods to mis-matched anatomical information (whether due to mis-registration or simply lack of functional/anatomical correspondence). Application has been largely in neurological studies where registration tends to be straightforward.

The second approach consists of performing an image enhancement by post-processing the image reconstructed without resolution modelling (or in an attempt to reduce residual partial volume effects after reconstruction with resolution modelling). This can be performed either using a deconvolution operation on a reconstructed image, or by incorporating into the reconstructed image some high frequency information taken from a structural image.

In the deconvolution approach, the blurred image is described as follows:

$$\begin{equation} b\left( {\bm x} \right) = \mathop \int a( {\bm y} )h( {{\bm x},{\bm y}} )\,{\rm d}{\bm y} \end{equation} \tag{ 1 }$$

where b( · ) is the uncorrected image, a( · ) is the true image, h( ·, ·) is the PSF, which can be space variant or invariant, and x and y are 3D coordinates. (When no integration limits are specified, it means that the integration is to be performed over the whole space.)

Various iterative techniques have been used for image restoration: Teo et al (2007) and Tohka and Reilhac (2008) used the Van Cittert deconvolution technique, Tohka and Reilhac (2008) also used the Richardson–Lucy algorithm, and Kirov et al (2008) and Barbee et al (2010) used the MLEM algorithm. As an example, the reblurred Van Cittert iterative scheme, can be described as follows (Tohka and Reilhac 2008):

$$\begin{equation} \left\{ {\begin{array}{@{}l@{}} {\tilde a_0 ( {\bm x} ) = b( {\bm x} )} \\ {\tilde a_{k + 1} ( {\bm x} ) = \tilde a_k ( {\bm x} ) + \alpha ( {h \otimes ( {b - h \otimes \tilde a_k } )} )( {\bm x} )} \\ \end{array}} \right. \end{equation} \tag{ 2 }$$

where $\tilde a_k ( \cdot )$ is the corrected activity distribution after k iterations, α is the step-length, in practice usually selected close to 1, and ⊗ represents the convolution operator. Here h( ⋅ ) is assumed to be a position-invariant PSF. The goal of this algorithm is to find the activity distribution (ã_k) such that the blurred distribution (h ⊗ ã_k) matches the observed distribution (b) in a least squares sense. The formula arises since minimizing (b − h ⊗ ã_k)² is achieved when its derivative $( - 2h \otimes (b - h \otimes \tilde a_k ))$ is zero. The basic idea of the iterative scheme is to detect structures in the residual (b − h ⊗ ã_k), and to put these back into the restored image. The process starts with the observed image and finishes when no more structures are detected.

Hoetjes et al (2010) compared post-reconstruction deconvolution with reconstruction-based resolution recovery for oncological PET studies, and found the two approaches had similar accuracy, but the reconstruction-based method had better precision.

The main problem with the deconvolution-based methods is the need to control noise by regularizing the solution (e.g. post-reconstruction smoothing (Boussion et al 2009)). Consequently with realistic noise levels it is very difficult to achieve full resolution recovery (indeed impossible given the high frequency losses that occur during acquisition). As a result full partial volume correction cannot be achieved. In order to address this problem, Bousse et al (2012) recently presented a maximum penalised likelihood deconvolution technique, based on a model that assumes the existence of activity classes that behave like a hidden Markov random field driven by a segmented MRI image.

An alternative to performing deconvolution is an approach that consists of transferring high-frequency information directly from a high-resolution anatomical image to a PET or SPECT image. Calvini et al (2006) proposed a method in which high-frequency information was extracted by subtracting a blurred version of an MRI image from the original one. These data were weighted and then added to a SPECT image. Boussion et al (2006) propose a more sophisticated method, in which the high frequency information was transferred in the wavelet domain from a CT or MR image to a PET or SPECT image. One advantage with this approach is that it does not require segmentation of the anatomical image. A potential disadvantage is that artefacts may be introduced in areas where there is mismatch between the structural and functional images, although this problem can be reduced using a local analysis approach (Le Pogam et al 2011). Shidahara et al (2009) showed that improved performance could be obtained by using a segmented anatomical image instead of the raw CT or MR image. However, if segmentation is required after all, it may be preferable to choose a method based on a more solid theoretical foundation.

In one of the first works on PVC for PET, Hoffman et al (1979) defined the recovery coefficient (RC) as the apparent activity concentration of an object divided by its true concentration. The RC could be calculated based on the system PSF and the geometry of the object. The authors determined the RC for objects of different sizes and shapes scanned on their PET system, and found it to be strongly dependent on object size. The RC concept was introduced in Hoffman et al (1979), but no mathematical expression was provided. Below we derive such an expression. In all equations below, variables with a tilde (e.g. $\tilde A$) represent an estimation of the corresponding variable without tilde (e.g. A).

The mean value of an image b inside a region Ω is defined as follows:

$$\begin{equation} B_{\rm \Omega } = \frac{1}{{\int_{\rm \Omega } {{\rm d}{\bm x}} }}\int_{\rm \Omega } {b( {\bm x} ){\,\rm d}{\bm x}} . \end{equation} \tag{ 3 }$$

Using (1) we get:

$$\begin{equation} B_{\rm \Omega } = \frac{1}{{\int_{\rm \Omega } {{\rm d}{\bm x}} }}\int_{\rm \Omega } {\int {a( {\bm y} )h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}{\,\rm d}{\bm x}} } . \end{equation} \tag{ 4 }$$

Assuming a( · ) to be approximately constant inside Ω, with a mean value of A_Ω, and zero outside leads to:

$$\begin{equation} B_{\rm \Omega } \approx A_{\rm \Omega } \frac{1}{{\int_\Omega {{\rm d}{\bm x}} }}\int_\Omega {\int_\Omega {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}{\,\rm d}{\bm x}} } . \end{equation} \tag{ 5 }$$

The true regional mean value can then be estimated as:

$$\begin{equation} \tilde A_{\rm \Omega } = \frac{1}{R}B_{\rm \Omega } \end{equation} \tag{ 6 }$$

where R is the RC value, defined as:

$$\begin{equation} R = \frac{1}{{\int_{\rm \Omega } {{\rm d}{\bm x}} }}\int_{\rm \Omega } {\int_{\rm \Omega } {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}{\,\rm d}{\bm x}} } . \end{equation} \tag{ 7 }$$

The RC values from Hoffman et al (1979) were used by Wisenberg et al (1981) to quantify myocardial blood flow in dogs with PET. The wall-thickness, needed for selecting the appropriate RC values, was determined from echocardiographic measurements. Kessler et al (1984) determined RC values for objects of various shapes and various dimensions with non-zero background. The RC definition can be based either on the mean apparent activity concentration in the target, or on the maximum activity concentration in the target, e.g. Srinivas et al (2009).

In oncological applications, it is a common practice to experimentally determine recovery coefficients using spherical phantom inserts. In this case, the correction would only be theoretically applicable to spherical lesions. However, the determination of recovery coefficients as described above is valid for any shape of the volume concerned (provided known).

Although the RC-based correction was initially proposed assuming that the target could be delineated using a structural imaging modality (CT or MR), it can be extended to be applicable without such structural information (Avril et al 1997, Gallivanone et al 2011). In that case, the volume of the target is directly estimated from the PET image, using a threshold-based method.

The method initially described in Hoffman et al (1979) corrects only for the spill-out effect. To correct for both spill-out and spill-in, the approach described above can be extended to multiple regions. Assuming a( · ) to be approximately piece-wise constant, with mean value A_i in region Ω_i, equation (5) becomes:

$$\begin{equation} B_i \approx \mathop \sum \limits_{j = 1}^M A_j \frac{1}{{\int_{{\rm \Omega }_{\rm i} } {{\rm d}{\bm x}} }}\int_{{\rm \Omega }_{\rm i} } {\int_{{\rm \Omega }_{\rm j} } {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}{\,\rm d}{\bm x} = \mathop \sum \limits_{j = 1}^M \phi _{ij} A_j ;\quad i = 1, \ldots ,M} } \end{equation} \tag{ 8 }$$

where M is the number of regions and ϕ_ij are the cross-talk factors, defined as:

$$\begin{equation} \phi _{ij} = \frac{1}{{\int_{{\rm \Omega }_i } {{\rm d}{\bm x}} }}\int_{{\rm \Omega }_i } {\int_{{\rm \Omega }_j } {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}{\,\rm d}{\bm x}} } . \end{equation} \tag{ 9 }$$

(Note that R_i = ϕ_ii). With a matrix formulation, the solution to this system of equations is given by:

$$\begin{equation} \tilde {\bm A} = {\bm \phi }^{ - 1} {\bm B} \end{equation} \tag{ 10 }$$

where $\tilde {\bm A}$ and B are the vectors of estimated and uncorrected regional mean values, respectively, and ϕ = [ϕ_ij] is a matrix containing the cross-talk factors. This method was first introduced by Henze et al (1983). They developed the technique to correct for cross-contamination between the myocardium and the left ventricular blood-pool. Although these authors used only two regions (M = 2), the technique they described would be valid for any number of regions. The method from Henze et al (1983) was applied by Herrero et al (1988) in myocardial blood flow PET studies on dogs. This method later became known as the 'geometric transfer matrix' (GTM) method, after it was described by Rousset et al (1998) for application to the brain.

The GTM method is widely applied for regional analysis and has the advantage that it can account for spillover effects between multiple regions. The main disadvantage is that the method only provides mean values for the predetermined regions but does not provide a partial volume corrected image.

A voxel-based PVC method was developed by Videen et al (1988), using anatomical information from either CT or MRI. The method was developed for brain PET studies, and the anatomical image was segmented into two regions; a brain region (including both GM and WM) and a non-brain region (including CSF). The latter was assumed not to contain any tracer. A binary map of the brain region was convolved with the system PSF to generate an image of RC values for each voxel. PVC was then performed by dividing the original PET image with the RC map, considering only voxels within the brain region. From (1) we obtain, with the assumption that $a( {\bm x} ) \approx A_{\rm \Omega } \forall {\bm x} \in {\rm \Omega }$ and zero elsewhere:

$$\begin{equation} b( {\bm x} ) \approx \tilde b( {\bm x} ) = A_{\rm \Omega } \int_\Omega {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}} . \end{equation} \tag{ 11 }$$

The correction can then be described as follows:

$$\begin{equation} \tilde a( {\bm x} ) = \left( {\frac{{A_{\rm \Omega } }}{{\tilde b( {\bm x} )}}} \right)b( {\bm x} ) = \frac{1}{{r( {\bm x} )}}b( {\bm x} )\forall {\bm x} \in {\rm \Omega } \end{equation} \tag{ 12 }$$

where r( ⋅ ) is the distribution of recovery factors for each voxel:

$$\begin{equation} r( {\bm x} ) = \int_\Omega {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}} . \end{equation} \tag{ 13 }$$

Please note that A_Ω does not appear in the final correction formula. This method is essentially an extension of the RC-method (6) (Hoffman et al 1979) from a regional to a voxel basis. In Videen et al (1988) the method was illustrated with 2D simulations, but the authors stated that, with real data, the correction must be applied in 3D. This was later done by Meltzer et al (1990).

Müller-Gärtner et al (1992) extended the voxel-based Videen method (described above) from two to three regions in order to account for WM and to get more accurate GM values. An MRI image of the brain was segmented into GM, WM and CSF regions. The GM region was treated as the target region, and the other two as background regions. CSF was assumed to be devoid of activity. The binary WM map, scaled to an estimate of the true WM mean value, was convolved with the system PSF and subtracted from the original PET image. Finally, corrected values were obtained for GM voxels as described above for the Videen method. We will refer to this method as MGM (short for 'the Müller-Gärtner method'). If indices 1 and 2 represent GM and WM, respectively, we obtain from (1):

$$\begin{equation} b( {\bm x} ) \approx \tilde b( {\bm x} ) = A_1 \int_{{\rm \Omega }_1 } {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y} + A_2 } \int_{{\rm \Omega }_2 } {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}} . \end{equation} \tag{ 14 }$$

If $A_2 \approx \tilde A_2$ where $\tilde A_2$ is an estimate of the mean value in the background region (WM), the correction can be described as follows:

$$\begin{equation} \tilde a( {\bm x} ) = \frac{1}{{r_1 ( {\bm x} )}}[ {b( {\bm x} ) - \tilde A_2 \phi _2 ({\bm x})} ]\,\,\, \forall \, {\bm x} \in {\rm \Omega }_1 \end{equation} \tag{ 15 }$$

where r₁( · ) is the distribution of RC values in Ω₁:

$$\begin{equation} r_1 ( {\bm x} ) = \int_{{\rm \Omega }_1 } {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}\,\,\, \forall \, {\bm x} \in {\rm \Omega }_1 } \end{equation} \tag{ 16 }$$

and ϕ₂( · ) is the distribution of cross-talk factors from Ω₂ in Ω₁:

$$\begin{equation} \phi _2 ({\bm x}) = \int_{{\rm \Omega }_2 } {h( {{\bm x},{\bm y}} ){\,\rm d}{\bm y}\,\,\, \forall \, {\bm x} \in {\rm \Omega }_1 } . \end{equation} \tag{ 17 }$$

This scheme can easily be extended to include multiple background regions:

$$\begin{equation} \tilde a( {\bm x} ) = \frac{1}{{r_1 ( {\bm x} )}}\left( {b( {\bm x} ) - \mathop \sum \limits_{j = 2}^M \tilde A_j \phi _j ({\bm x})} \right)\,\,\, \forall \, {\bm x} \in {\rm \Omega }_1 \end{equation} \tag{ 18 }$$

where M is the total number of regions. This method has similarities to scatter and attenuation correction used in PET and SPECT: first there is a subtraction of events that are misplaced into the target region (similar to scatter correction), and then a multiplicative correction is applied for events that are lost from the target region (similar to attenuation correction). The disadvantages with this method are that; (1) the correction is valid only for voxels within the target region, and (2) the true mean values in the background regions $( {\tilde A_j ,j \ge 2} )$ have to be initially estimated. These would typically be obtained from regions large enough that the central part can be assumed to be unaffected by PVE. An alternative would be to use the GTM method to determine the mean values of the background regions, as described in Quarantelli et al (2004). This is known as the modified MGM (mMGM). Figure 3 illustrates MGM applied to a 1D phantom profile.

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MGM is one of the most used PVC methods. Originally developed for brain PET studies, it was implemented for brain SPECT studies by Matsuda et al (2003). It was also implemented for cardiac SPECT studies by Da Silva et al (1999). These authors developed a new method for deriving the cross-talk and recovery factors in order to take into account the distance-dependent resolution of the SPECT camera. Instead of convolving regional maps by a Gaussian PSF, the regional maps were forward projected with a model that included attenuation and distance-dependent resolution, and then reconstructed. In Da Silva et al (1999) the method was evaluated with phantom data, and later it was also applied to real data from animal studies (Da Silva et al 2001).

There are thus two methods to calculate the cross-talk and recovery factors: either by convolving the binary maps of the region of interest by the PSF of the imaging system, or by forward projecting and then reconstructing the binary map of the regions using a realistic forward model and the same reconstruction method as the one used for the data to be later corrected. These two methods were compared for brain PET studies by Frouin et al (2002) and for brain SPECT studies by Soret et al (2003). In both cases, no significant difference was found between the two methods. The method based on forward-projection and reconstruction of regional maps is in principle only valid when analytical reconstruction algorithms (e.g. filtered backprojection (FBP)) are used. Iterative algorithms are nonlinear and the result depends on the activity distribution. This means that the reconstruction of a regional map in isolation will produce a different result than if it had been part of a larger object. A solution to this problem was proposed by Du et al (2005) and by Boening et al (2006) using a perturbation technique. Two images with a small difference are forward-projected and reconstructed, and the difference of the two reconstructed images corresponds to the map of correction factors. Pretorius and King (2009) pointed out a limitation with this approach. They observed that when local differences in uptake exist between the emission data and the added forward projected regional map, such as in the case of a cardiac perfusion defect, the map of the correction factors (difference of the two reconstructed images) erroneously takes on some of the characteristics of the emission data.

One of the disadvantages of MGM is (as mentioned before) that the correction is valid only for voxels within the target region. Yang et al presented a method where PVC is applied to the whole image (Yang et al 1996). In this method, a piece-wise constant image is created, where each region is represented by its true relative mean value (A^r_i = A_i/A₁∀i), where A₁ is the mean value in an arbitrary reference region. This image is convolved with the system PSF, and PVC factors were obtained as the ratio of the two images (before and after convolution). The convolved piece-wise constant image can be described as:

$$\begin{equation} \tilde b( {\bm x} ) = \mathop \sum \limits_{i = 1}^M A_i^r \int_{{\rm \Omega }_i } {h({\bm x},{\bm y}){\,\rm d}{\bm y}} \end{equation} \tag{ 19 }$$

and the correction is then given by:

$$\begin{equation} \tilde a( {\bm x} ) = c( {\bm x} )b( {\bm x} ) \end{equation} \tag{ 20 }$$

where

$$\begin{equation} c( {\bm x} ) = \frac{{A_i^r }}{{\tilde b( {\bm x} )}}\,\, \forall \, {\bm x} \in {\rm \Omega }_i ,\quad i = 1, \ldots ,M \end{equation} \tag{ 21a }$$

or

$$\begin{equation} c({\bm x}) = \frac{{a_c ( {\bm x} )}}{{\{ {h \otimes a_c } \}( {\bm x} )}} \end{equation} \tag{ 21b }$$

and a_c( ⋅ ) is a piece-wise constant image (the subscript c indicating a constant distribution within each region):

$$\begin{equation} a_c ( {\bm x} ) = A_i^r \,\, \forall \, {\bm x} \in {\rm \Omega }_i ,\quad i = 1, \ldots ,M. \end{equation} \tag{ 22 }$$

This method is essentially an extension of the Videen method (12) from one to multiple regions. The method requires knowledge of the true relative activity concentrations in all regions, which is a strong prior. Absolute values are not needed, as any scaling factor would just come out in the wash. In Yang et al (1996), the method was applied to brain blood flow studies to correct for cross-talk between the GM, WM and CSF regions, which were assigned relative mean values of 4, 1 and 0, respectively. Figure 4 illustrates the Yang method on a 1D phantom profile.

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Shcherbinin and Celler (2010) implemented this method for SPECT using forward projection and reconstruction to generate $\tilde b( {\bm x} )$ in the calculation of the PVC factors instead of convolving with a PSF. The correction was applied to phantom data using the known activity concentrations in the different compartments for A^r_i.

Both MGM and the Yang method suffer from the drawback of requiring initial information in terms of mean or relative mean values in various regions. Novel approaches have been proposed which avoid this problem by combining MGM and the Yang method, respectively, with the GTM method.

An extension of MGM was proposed by Erlandsson et al (2006). The first step of this 'multi-target correction' (MTC) method is to use the GTM method to obtain estimates of the true mean values in all regions $( {\tilde A_j \forall j} )$. Next, the MGM formula with multiple background regions (18) is applied repeatedly, so that each region in turn is treated as the target region while all other regions are considered as background regions. Thereby, all regions would be considered alternatively as a target or background regions. The final image is obtained as a mosaic of all the corrected regions. The correction of each region is independent of that of all other regions.

Thomas et al (2011a) presented a modified version of the Yang method called 'region-based voxel-wise correction' (RBV). The regional mean values $( {\tilde A_j \forall j} )$ are initially obtained using GTM, and the Yang correction (20) is then applied. The results obtained are similar to MTC, but RBV is simpler to implement. Instead of using GTM, the mean values in all regions can be estimated with an iterative procedure as follows. The correction is first applied with $\tilde A_i = B_i \forall i$, then new regional mean values are calculated and a new correction applied, etc. As only a few iterations are needed (typically 3–5), this procedure is faster than RBV and simpler as well. This approach is presented here for the first time. We call it the 'iterative Yang' (iY) method. These methods produce a voxel-by-voxel correction of the whole image, do not require any prior information about the activity distribution and can be used with any number of regions. A comparison of the MGM, GTM and RBV methods is presented in the example below (section 2.7).

Shcherbinin and Celler (2011a) also presented a new algorithm, based on the Yang method (Yang et al 1996), which can be described as follows: a template image is calculated based on an initial image reconstructed without PVC and on a structural image. The voxel values in the target region of the template are set to a constant equal to the mean of the top ten percentile voxels in the corresponding region of the original image while the background values are set to their values in the original image. The template is projected and reconstructed to mimic the detection/reconstruction process. The voxel values in the target region are then corrected by multiplying them by correction factors, as in (20). These factors are equal to the ratio of the template values to the values in the images reconstructed from the template projections, similar to (21). The corrected target region is then projected again and the result is used to update the background estimate. Two iterations are performed to get the PVC image. This method has been called the 'iterative partial volume effect correction' (itPVEC) method.

Huesman (1984) developed an analytic algorithm to obtain ROI mean values directly from projection data, without reconstructing the complete image. Carson (1986) developed an iterative ML-EM algorithm for the same purpose. By incorporating the resolution of the scanner into the algorithm, PV corrected ROI mean values were automatically obtained. An alternative approach, based on least-squares algorithm was developed by Formiconi (1993). This algorithm was evaluated with brain SPECT studies on phantoms and patients by Vanzi et al (2007). These methods require segmentation of the entire object. In order to avoid this, Moore et al (2012) developed a method, which only requires segmentation of few tissue types (2 to 4) within a small sub-VOI surrounding the lesion seen on the CT or MR structural image. Only the projection rays traversing the VOI are used to estimate the mean activity concentration in each tissue type. These activity concentrations are estimated simultaneously by maximizing iteratively the log likelihood of measuring a given projection dataset. The contribution of the background outside the VOI is accounted for by reprojecting through the reconstructed image outside the VOI. One advantage of this method is that, being based on maximum-likelihood fitting in projection space, it utilizes the knowledge that the projection noise is white (spatially uncorrelated) and Poisson distributed.

In all the anatomically based PVC methods discussed above, the correction is performed in the image domain. The correction can actually also be performed in the projection domain, as described in Erlandsson and Hutton (2010) and Erlandsson et al (2010, 2011) for SPECT, still requiring the segmentation of the different tissue types from a structural CT or MR image. The original method by Erlandsson and Hutton (2010) was developed in combination with iterative FBP reconstruction. A second version that worked in combination with OSEM reconstruction (Hudson and Larkin 1994) was later presented by Erlandsson et al (2010, 2011). In these approaches, the projection domain is related to the image domain by the forward-projection operator:

$$\begin{equation} p( {\bm \theta } ) = F\{ {a({\bm x})} \} \end{equation} \tag{ 23a }$$

$$\begin{equation} p_R ( {\bm \theta } ) = F_R \{ {a({\bm x})} \} \end{equation} \tag{ 23b }$$

where p(θ) and p_R(θ) are projection data sets without and with resolution blurring, respectively, F{ ⋅ } and F_R{ ⋅ } represent the forward-projection operator without and with resolution modelling, respectively, and θ is a coordinate in the projection domain. The projection space correction is based on the Yang approach (Yang et al 1996), and can be described as follows:

$$\begin{equation} \tilde p({\bm \theta }) = \left( {\frac{{F\{ {a_c ( {\bm x} )} \}}}{{F_R \{ {a_c ( {\bm x} )} \}}}} \right)p_R ({\bm \theta }) \end{equation} \tag{ 24 }$$

where a_c( ⋅ ) is a piece-wise constant image:

$$\begin{equation} a_c ( {\bm x} ) = A_i \,\, \forall \, {\bm x} \in {\rm \Omega }_i ,\quad i = 1, \ldots ,M. \end{equation} \tag{ 25 }$$

New mean value estimates $( {\tilde A_i \forall i} )$ are calculated after each iteration of the reconstruction algorithm. The main advantage of this method is that it takes into account the distance dependent resolution in SPECT.

The impact of PVC on tumour assessment has been studied in simulated data, phantoms and in patients.

Using simulations and phantom data, it has been shown that tumour uptake significantly increased with PVC compared to without PVC, both in SPECT (e.g. Boening et al 2006, Shcherbinin and Celler 2010) and in PET (e.g. Tylski et al 2010, Hoetjes et al 2010, Gallivanone et al 2011). In Gallivanone et al (2011), the uptake values obtained using an RC-based correction were significantly more accurate than uncorrected values for lesion diameter > 1 cm with error in activity estimates <13%. Below that size, the method failed as the RCs were estimated from the PET images and the tumour delineation was not accurate enough for small spheres. In Hoetjes et al (2010), PVC methods were shown to recover true SUV within 10% for spheres equal to or larger than 1 mL (i.e. about 1.2 cm in diameter). Also, in simulations, PVC using a deconvolution approach was reported to be useful to improve the characterization of the heterogeneity in tumour uptake (van Velden et al 2011).

In patients, PVC also significantly increased the tumour SUV in PET, although the accuracy of the measurement could not be determined due to the lack of gold-standard value (Hoetjes et al 2010, Gallivanone et al 2011, Hatt et al 2012). Still, PVC was found to increase the correlation between FDG uptake and Ki-67 proliferation index and to decrease the positive correlation between FDG uptake and tumour size (Vesselle et al 2008) in non-small cell lung cancer. It was also found to significantly increase the accuracy of differentiation between benign and malignant tissue in breast lesions (Avril et al 1997) and in lung nodules less than 2 cm in diameter (Hickeson et al 2002). Yet, another study reported a significantly poorer diagnosis of malignant lung nodules versus benign masses with RC-based PVC SUV than with uncorrected SUV (Menda et al 2001). The authors observed that this was because the SUV increase resulting from PVC led to the wrong classification of benign lesions that reached the threshold above which the lesion was considered as malignant.

Regarding the role of PVC upon tumour staging, in non-small cell lung cancer, although the maximum SUV in the tumour was found to be associated with the TNM stage, PVC SUV was not (Vesselle et al 2004). In this study, the tumour size estimated using a CT scan also appeared to be significantly different as a function of the T status, N status and M status. The fact that uncorrected SUV bears some information about the tumour size due to PVE might therefore explain why removing size-information through PVC actually made the correlation between SUV and stage disappear.

PVC based on an iterative deconvolution method did not improve the prediction of therapy response nor the prognostic value based on SUV at baseline in patients with oesophageal cancer (Hatt et al 2012), although this result should be interpreted with caution given the relatively large volume of the tumours included in this study (40 ± 30 cm³). In line with the explanation given above regarding the lack of benefit brought by PVC for tumour staging, a reason for the absence of benefit of PVC for predicting therapy response or for prognosis might be that prognostic value depends on both the lesion volume and the lesion metabolic activity. When PVE is not compensated for, the measured SUV reflects both tumour volume and metabolic uptake, so that for an actual metabolic activity, observed SUV will be lower in small tumours than in large tumours. With ideal PVC, the measured SUV reflects only the metabolic activity of the tumour. Hence a potentially prognostic factor, namely the tumour volume, is no longer present in the SUV measurement, making it even less relevant as a prognostic factor. In van Heijl et al (2010), the lack of predictive values of various SUV indices, be they corrected for partial volume or not, was also reported for oesophageal cancer patients.

Accurate quantification might be crucial for patient monitoring, both in PET and in SPECT. In PET, the question is mostly to determine whether the tumour is responding to therapy (Weber 2009), while in SPECT, an important oncological application is the monitoring of the dose to the tumour and organs-at-risk in patients treated by targeted radionuclide therapy (TRT) (Bardies and Buvat 2011). The feasibility and impact of PVC have been studied in these two applications.

In PET, it was observed that RC-based PVC could either reduce or increase the change in SUV between two consecutive scans (Gallivanone et al 2011) as a function of the lesion. These significant changes could even alter the classification of the metabolic response of the tumour based on the EORTC criteria (Young et al 1999). In another study in patients with non-small cell lung cancer (Hoetjes et al 2010), PVC resulted in slightly smaller SUV decreases between two scans than without PVC, and this difference in SUV response magnitude was significant. Whether PVC actually improves the early evaluation of treatment response (at 2 weeks of a first course of chemotherapy) has been recently investigated in patients with metastatic colorectal cancer treated by chemotherapy by considering that the actual response was given by the RECIST criteria (Therasse et al 2000) as measured with contrast enhanced CT at baseline and 6–8 weeks after (Maisonobe et al 2011). In this study, PVC SUV did not appear to be useful to improve the identification of the tumour response later observed by RECIST. Again, the reason might be identical to those explaining the lack of improvement brought by PVC for tumour staging, for predicting the tumour response to therapy, and for prognosis. Indeed, it is expected that when a tumour responds to therapy, this yields both a decrease in the metabolic activity of the tumour cells and in the number of metabolically active cells, i.e. in the metabolically active volume. When SUV is not corrected for PVE, it is correlated both with the metabolic activity and with the metabolically active volume, so it is sensitive to two features affected by therapy response. When correcting SUV for PVE, it becomes insensitive to the metabolically active volume, and changes in SUV then only reflect the change in metabolic activity of the tumour. A piece of information potentially useful to assess the tumour response, namely the change in metabolically active volume, is thus lost. These findings suggest that there might be an opportunity to consider both SUV corrected for PVE and metabolic volume when assessing a lesion, either separately or in combination (e.g. PVC SUV × volume = PVC total lesion glycolysis (pTLG)). Yet, large variability in metabolic volume estimates, due to the difficulty in accurately delineating tumour, actually reduces the informative value of the metabolic volume (Maisonobe et al 2011). Still, as the accuracy of tumour delineation methods increases, indices such as pTLG might be preferable to the previously defined TLG (Larson et al 1999), usually based on uncorrected mean SUV × volume.

In SPECT and in the context of TRT, PVC was found to significantly improve the estimate of the total activity in each organ in simulated and phantom data (He et al 2005). The same group investigated the accuracy and precision of activity estimates in PVC corrected images when the VOIs were erroneously defined or mis-registered. They concluded that poor definition and mis-registration of the VOI could introduce significant quantitative errors especially in small organs or tumours and in organs with low uptakes with respect to surrounding activity (He and Frey 2010, Song et al 2010). Tang et al (2001) reported that PVC using the MGM method significantly changed the activity estimates in different organs delineated using a CT scan at a given time point, although they could not demonstrate that the PVC values were more accurate than the uncorrected values in these patients due to the lack of gold standard.

PET images tend to be increasingly incorporated into the definition of a biological target volume for treatment planning in radiotherapy (MacManus et al 2009). Yet, due to the poor spatial resolution of PET compared to CT or MR images conventionally used in treatment planning, accurate delineation of the tumour contour is extremely challenging. Any improvement in spatial resolution of PET images would therefore be beneficial to tumour delineation.

Among PVC methods, those belonging to the image enhancement family have potential to improve spatial resolution and enhance edges between different functional regions in the PET images. They might then contribute to a more accurate definition of the biological target volume. Yet, only few results have been reported regarding the influence of PVC upon tumour delineation. In the study of Hatt et al (2012) including rather large tumours (40 ± 30 mL without partial volume correction), the tumour volumes estimated using an automatic delineation algorithm were systematically lower by 4 ± 3 mL in PVC images compared to uncorrected images, where PVC correction consisted of an iterative deconvolution. This difference was within the reproducibility limits of confidence intervals measured using this delineation algorithm in repeated scans so the authors did not find a significant effect of PVC upon metabolically active tumour volume estimates. Additional insight into the potential benefit of image enhancement for tumour delineation can be obtained by interpreting the results of a comparison of tumour volume estimates as a function of the imaging parameters (resolution, noise and contrast) (Cheebsumon et al 2011). In particular, this study shows that tumour metabolic volume estimates tend to be smaller when image resolution is improved or when contrast increased. It also shows that the impact of image resolution is greater on the metabolic volume estimates than on the SUV. Similarly, the test-retest variability of tumour volume measurements depends on the spatial resolution in the images more than the test-retest variability of the SUV. As a result, for optimal use of PET images in the context of treatment planning in radiotherapy, the impact of PVC based on image enhancement on the delineation of tumour contours, in small tumours, is certainly worth further investigation.

For illustration of the application of PVC in neurology we used data provided by courtesy of Susan Landau and William Jagust (Jagust Lab, University of California, Berkeley). An AD subject underwent T1-weighted MRI, ¹⁸F-FDG PET and ¹⁸F-florbetapir (AV45) PET (an amyloid radioligand). The MRI data was segmented into anatomical regions using FreeSurfer (Fischl et al 2002, 2004), a popular suite of tools for MRI brain analysis. Forty-two regions were defined for the purposes of PVC. The ¹⁸F-FDG and ¹⁸F-florbetapir were independently registered to the MRI data using the block matching technique of Ourselin et al (2001). RBV correction was performed assuming a resolution of 8 mm FWHM. The data were then normalized by mean cerebellar grey matter uptake to generate Standardised Uptake Value Ratio (SUVR) images.

The ¹⁸F-FDG and ¹⁸F-florbetapir PET images with and without PVC are shown in figure 7. Regional mean values for selected regions with and without PVC are shown in figure 8 for both tracers. In the ¹⁸F-FDG data, large increases in SUVR were observed in cortical GM regions after PVC, while a decrease was seen in sub-cortical WM. It is interesting to note that after PVC, the SUVR in GM is 3–4 times higher than in WM, corresponding to typical relative blood flow values. Also with ¹⁸F-florbetapir the higher SUVR values were observed after PVC in cortical GM regions, while the sub-cortical WM SUVR remains similar to the uncorrected data, suggesting that it is relatively unaffected by PVEs. Before correction, sub-cortical WM tends to be higher than cortical GM, while the opposite is true after PVC. Large increases in cortical SUVR values are to be expected when applying PV-correction to AD subjects as they often have extensive atrophy which leads to larger PVEs and a greater under-estimation of the true signal.

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As an example of the possible clinical value of PVC, an NCAT (Segars and Tsui 2002) body phantom was used in combination with a high-resolution contrast-enhanced gated cardiac CT image set to create both a cardiac perfusion SPECT study through Monte Carlo simulation (Ljungberg and Strand 1989) and co-registered templates for implementation of the modified template-reprojection method described in Pretorius and King (2009). The high-resolution contrast-enhanced gated cardiac CT was segmented as described by Pretorius et al (2002), using a greyscale histogram technique in combination with some thresholding. These image sets were averaged to form the basis of a non-gated cardiac perfusion study and the myocardium template respectively. The liver, lungs, and background (majority of other structures) from the NCAT phantom were inserted around the CT-based heart. Noise was added to the Monte Carlo simulated projections such that heart counts totalled 500 k for a 360° acquisition. Only 204 degrees worth of data were used in the reconstruction to simulate the IRIX gamma camera with Beacon transmission (Philips, Cleveland, OH) cardiac perfusion protocol. Emission data were reconstructed using the rescaled-block-iterative (RBI) method, a perturbation of ordered subsets expectation maximization (OSEM) with 17 subsets and 5/10 iterations with attenuation, triple energy window scatter and distance dependent resolution compensation included. The reprojected templates were similarly reconstructed without scatter compensation. Reconstructed sets of data were filtered using a three-dimensional Gaussian filter with a cutoff taking the voxel size into account (Pretorius and King 2009). PVC was performed, data reoriented to short axis slices, and polar map quantification performed using 17 myocardial segments according to the ASNC guidelines (Holly et al 2010). Results are given in figures 9, 10 and 11. The short axis slices in figures 9 and 10 show the importance of voxel size. PVC data in figure 9 (bottom row) are clearly a better representation of the true perfusion compared to the bottom row of figure 10. In figure 11, the uptake values together with the standard deviations are shown compared to the absolute truth. Only the template and data obtained using ten iterations are given. From the plots in figure 11 it is evident that even the heart template at the reconstruction voxel size, does not fully represent the true absolute perfusion value. The reason is twofold. First, the voxel size (3.12 mm) is still too coarse to describe the myocardium accurately at its thinnest parts, and second, the averaging due to cardiac gating adds additional blurring to the folded down heart template. It should be noted that the emission projections were acquired (simulated) using finer sampled source and attenuation maps (256 cube) of the NCAT phantom ensuring a more accurate description of the myocardium. Further study of all the data reveals that the average error compared to the template (best possible outcome) is an under estimation 11.1% ± 2.3% when five iterations are employed, improving to 9.6% ± 2.3% with ten iterations. Compared to the expected absolute perfusion value, these values deteriorate to 23.7% ± 6.8% and 22.4% ± 7.0% respectively. Results for the larger voxel size of 4.68 mm (not tabulated) are similar when compared to the template at that voxel size (12.7% ± 3.9% and 10.6% ± 3.9%), but worse compared to the expected absolute perfusion value (35.5% ± 7.6% and 33.8% ± 8.5%). Without PVC the average error for the two sets of voxel sizes ranges from 44.2% to 49.9%.

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To illustrate the impact of PVC in PET tumour imaging, it is useful to examine how SUV varies when PVC is used. Images from patients with metastatic colon cancer are shown in figure 12. In the first patient, two rather large tumours can be seen in the selected slice (figure 12(A)). The tumour volumes estimated using an isocontour set to 40% of the maximum SUV in the tumours were 7.1 mL (tumour 1) and 10.3 mL (tumour 2). For the second patient (figure 12(B)), the tumour (tumour 3) was smaller (estimated volume of 1.4 mL). The images enhanced using a deconvolution approach (Lucy Richardson deconvolution assuming a 7 mm spatial resolution in the reconstructed images) are also shown in figure 12. For each tumour, four SUVs were calculated: SUVmax (SUV in the brightest voxel of the tumour), SUV40% (mean SUV in the tumour volume estimated using an isocontour set to 40% of SUVmax), SUV40% corrected for PVE using a RC (calculated based on the tumour volume estimated using the isocontour method), and SUVdecon defined as the mean SUV in the image enhanced by deconvolution (where the mean was calculated in a tumour volume estimated as an isocontour set to 70% of the maximum SUV in the original image (Teo et al 2007)). The corresponding values are shown in figure 13. These values illustrate the complexity of PVC in tumour imaging. The increase in tumour SUV due to PVC does not only depend on the tumour volume: although tumour 2 had a larger volume (10.3 mL) than tumour 1 (7.1 mL), the increase between SUV40% and SUV40%RC was greater for tumour 2 compared to tumour 1. This is because tumour 2 was less compact than tumour 1 due the presence of a large necrotic core. PVE was therefore more severe in tumour 2 than in tumour 1, as confirmed with the greater SUV increase in tumour 2 than in tumour 1 when comparing SUVdecon with SUVmax. The results also show that the SUV in the very small tumour 3 was greatly affected by PVC. In addition, figure 13 suggests that the PVC SUV still highly depends on which PVC method is used (substantial differences between SUV40%RC and SUVdecon). Still, the differences between PVC SUV tend to be smaller than the differences between uncorrected SUV (SUVmax and SUV40%). Obviously, the true SUVs for these tumours are unknown. Better understanding of the performance of the various PVC methods on real data and of the PVC impact on tumour assessment in the context of diagnosis and patient monitoring is strongly needed. PVC should now be offered as an option for tumour assessment, so as to facilitate investigations regarding the role of PVC in the context of tumour imaging.

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