5.2 Algorithm development
5.2.2 Surface-based SMPVC
The noise sensitivity of the voxel-based approach led to the reformulation of SMPVC which modifies boundary surfaces rather than voxels. The core concept of iteratively fitting regions according to the observed PET data remains, although the surface-based approach contains notable differences. The key changes between the voxel- and surface-based approaches are the PVC technique used internally by the algorithm and how the boundary changes are evaluated.
5.2.2.1 Iterative Yang partial volume correction
Voxel-based SMPVC relies on the RBV correction, that in itself is not particularly compu- tationally demanding and can be performed in a few minutes. However, when evaluating thousands of voxels, the overhead quickly becomes very large. This led to the approximation of voxel changes, which while considerably faster to calculate, are less accurate than complete RBV correction.
The computationally intensive part of RBV is the calculation of corrected mean values using GTM. Whereas, the voxel-wise correction step of Yang et al.[1996] is quick to calculate. The previously discussed projection-based PVC method of Erlandsson and Hutton[2010] (section 2.3.2.3, page 36) uses an iterative PVC approach based on the Yang method. This correction is referred to as iterative Yang (iY). The iY correction was applied in the projection domain, although later h-PVC was performed using iY in the image domain. Due to the lower computational cost of iY, this technique was selected for PVC inside the surface-based
5.2. Algorithm development 129 SMPVC. iY is described by the equation below:
fk+1(x) = fP ET(x) sk(x) sk(x) ⊗ h(x) , (5.7) sk(x) = X i=1..N [Tipi(x)], Ti = 1 ni X j ∈ pi fk j , f0 = fP ET.
where fP ET is the observed PET image, pi is the binary mask of region i, ni is the total number of voxels in i and Ti is the mean value of region i at iteration k. sk is a piece-wise constant image of the regional mean values and fk is the PV-corrected image at iterationk. The iY method typically converges in 3 - 5 iterations, producing a PV-corrected image faster than if the GTM were used for the calculation of the mean values. Not needing to calculate the GTM also makes iY arguably easier to implement than RBV.
5.2.2.2 Surface evolution
The surface-based SMPVC uses a fast marching level set method[Sethian, 1996] to evolve region boundaries. The purpose of fast marching methods is to track a moving boundary which moves according to a knownspeed function. This approach has advantages over other methods that rely on control points to define surface meshes in that there are fewer problems in terms of the preservation of topology.
Fast marching methods are a special case of the more generallevel sets[Sethian, 2001], in that fast marching methods are monotonic and therefore only move ‘forward’. This makes fast marching methods very quick compared to other numerical methods, however their sim- plicity may make them unsuitable for some applications. For the purposes of SMPVC, where the aim is to evolve a region boundary a short, constrained distance, fast marching methods were deemed acceptable.
The fast marching method operates by evolving a surface according to the speed function and recording the amount of ‘time’ it takes the surface boundary to propagate to a particular point. The method is initialised with a set of seed points which define the surface to be evolved. In this instance the output is an image where each voxel represents the time taken for the surface to reach it.
The output image consists of zeros at the seed points, with larger values at voxels where the boundary took longer to reach them. The voxel value is referred to as thecrossing time.
5.2. Algorithm development 130 A new region can then be defined by applying a binary threshold for a desired crossing time. The crossing time at a particular voxel depends on the speed function. In this instance, the speed function is an image and commonly referred to as aspeed image.
T0 T1 T2
Figure 5.2: An illustrative example of how the initial boundary of an object (crossing time T0) could propagate at crossing times T1 and T2.
In the case of SMPVC, the boundary to be propagated is that of a particular brain region. The speed image is defined according to the regional image statistics from the PV-corrected PET data. More specifically, when evolving the surface of regioni, the mean of region i (µi) is used as input to a modified sigmoid function. The function is linear transformation that is necessary, first because the fast marching filter of the Insight Segmentation and Registration Toolkit (ITK) expects the speed image to contain values of between 0 and 1 and second to encourage faster surface propagation in regions that are more similar to each other. The surface moves faster in areas of the speed image that are equal to 1 and slows down where the image tends to 0. The function (φ) used to calculate the speed image is:
φ(x) = 1 − |Ψ(x) − 0.5|, (5.8) Ψ(x) = 1 1+ e−x−βα , α = 2 × µi 6 , β = µi.
whereµi is the iY-corrected PET mean value in region i andφ is the value assigned in the speed image. The value ofφ as a function of the percentage difference from µi can be seen in figure 5.3.
In the current implementation, the surface evolution is constrained to GM regions only. Additionally, GM regions are restricted to their respective hemispheres. During initial tests
5.2. Algorithm development 131 −100 −75 −50 −25 0 25 50 75 100 0.5 0.6 0.7 0.8 0.9 1
Percentage difference from region mean (µi)
V ox el value of speed imag e (φ )
Figure 5.3: The modified sigmoid function (equation 5.8) used to compute the speed image.
of the voxel-based approach, regions such as the cingulate could spread from one hemisphere to another. While it is feasible that inter-hemispheric region definitions may reduce the global variance, they may not accurately describe the PET distribution. This behaviour is therefore prevented by checking the voxel labels in the parcellated brain mask image.
5.2.2.3 Iterative scheme
The surface-based SMPVC algorithm iterates in a different manner to the voxel-based ap- proach. The previous implementation would evaluate voxel changes from all regions during an iteration. However, the surface-based method evolves the surface of a single region and then calculates whether the changes to the region surface improves the PVC. The algorithm iterates over the set of regions, evolving each surface individually. For a region change to be accepted the following three criteria must be met:
1. The acceptance criterion for an iteration is calculated in the same way the voxel-based approach (section 5.2.1.3).∆GV must be less than 0 for the change to be accepted. 2. The CoVr in the region must not increase after modification.
3. The size of the region must be within ±20% of the original region size.
The criterion based on the CoVr is applied to prevent a modification that degrades a region but is not reflected by the∆GV . As ∆GV is a global measure, it is possible for this situation to occur. Conversely, it is entirely possible that the CoVr in a region should increase. In this
5.2. Algorithm development 132 instance, the change should occur when the neighbouring region is modified as the criterion would be satisfied.
The size restriction was added for two reasons. Firstly, it prevents small regions from disappearing and secondly it limits the problem where two neighbouring regions may be very similar to each other and can therefore change dramatically in terms of their size, while having little effect on the mean or CoVr of those regions. The latter effect could make any parameter based on volume change quite unstable.
As discussed earlier, the surface-based algorithm iterates over the set of regions. This is performed several times as changes to one region may prompt changes in another. The al- gorithm therefore has two loops, an inner loop controlling iterations over the set of regions and an outer loop controlling the number of times to evaluate the sets. The surface-based SMPVC algorithm is described in pseudocode in section 5.2.2.4. Once both loops have ter- minated, the modified mask image is used to perform a RBV correction on the original PET data to produce the final PV-corrected image. RBV is applied because, while results of iY correction is very similar to RBV after 5 iterations, RBV is still considered to be the most ac- curate correction when assuming regional uniformity. The surface-based SMPVC approach was selected over the voxel-based technique and all subsequent discussion of SMPVC refers to the surface-based approach unless specifically stated.