N

There is no assumption here on the noise characteristics of the image. Although Poissonian in essence, the final noise h(r) in the image includes also components from attenuation, scatter, or other type of corrections during data acquisition and reconstruction. Since the partial volume solutions proposed here do not call for the explicit characterization of a noise model, we will omit the noise component in the subsequent equations.

6.3.1 Pixel-guided Approach

If one seeks to recover the entire true activity distribution f (r) within tissue component Di, there are N unknowns, the true image distribution of activity of each tissue component of the tracer distribution model, but only one equation, the emission image g(r). Several authors proposed to solve this equation, first with N=1 to compensate for signal dilution in spaces void of activity.43 Compensating for dilution of radioactivity in non-active tissues such as cerebro-spinal fluid is more important in the case of tissue atrophy, as the decrease of metabolism seen with increased tissue atrophy might be confounded by the loss of signal consequent to increased partial volume effect. For example, the decline in blood flow observed with PET has been shown not to decline with age after correcting for tissue atrophy using this method.44 These techniques make use of an anatomical mask defined from MRI or CT, and by assigning pixels corresponding to the cerebral tissue (i.e., grey and white matter) a value of 1, and the space occupied by non-cerebral tissue being kept at 0. This is equivalent to defining an anatomical mask f1(r) as follows:

r = cerebral tissue .0 r = non — cerebral f (r) = Í 1 r = cerebral tissue (29)

As discussed earlier (Eq. 5), if we consider the point-spread function h of the imaging system as being a spatially invariant function, the equation becomes a simple convolution. The next step of this correction method hence consists in convolving the binarized volume fi(r) at the scanner's resolution. The PET image in divided by this "low resolution'' mask in order to obtain images corrected for dilution of radioactivity in non-active spaces. Hence, the very approximate of the true activity distribution is given by:

This equation being derived from the approximation:

It can be seen that the approximation proposed in Eq. (31) is only justified when f(r) possesses the same spatial characteristics as those defined by the anatomical mask f (r). This approach ignores the difference in tracer uptake between grey and white matter.

An extension to a more heterogeneous distribution of the tracer was later proposed, this time with a more realistic mask that makes the distinction between grey and white matter to account for white matter activity contribution to measurements of grey matter activity concentration.22 We have this time the problem that the number of unknowns is now equal to N = 2, while the number of equations has not changed (only one). The way these authors overcame this problem is by transforming one of the unknowns into a known variable. They assume that the true white matter activity can be accurately measured from a large remote white matter area in the emission image (Figure 6). This new mask can be defined as follows:

(f1 (r) r = Grey matter f2(r) r = White matter (32)

Where e.g., f1(r) = 1 for pixels identified as Grey matter, and f1(r) = 0 elsewhere. The tissue components Grey, White, and BKG + CSF are obtained by segmentation of the MR images realigned with the PET image volume. Figure 6 illustrates the general principle of MR-guided PVC in functional brain imaging.

By considering that the radioactivity concentrations in white matter and CSF spaces are constant, and that those components do not suffer themselves from partial volume thanks to their important dimensions with respect to the imaging system resolution, the PET image can be written as:

where T2 and T3 are known constants representing the true activity of the white matter and background plus CSF, respectively. After rearranging the previous equation, one can write:

White matter activity T2 is considered as being uniform throughout the brain whose activity concentration is considered as being accurately estimated from a measurement in the pons, assumed to elude partial volume due to its large cross-sectional size. This value is assigned to the low-resolution white matter image f2(r) ® h(r), and then subtracted from the PET image g(r). These authors proposed to do the same with background + CSF compartment, although one might wonder whether the measured value for that compartment is not the result of spillover from adjacent tissue plus noise. However, for increased realism, and to make a distinction in grey matter tracer uptake, these authors extended this elimination-substitution

Figure 6. Schematic representation of the different steps required for MRI-guided correction for partial volume effect illustrating the original MR (A) and SPECT (B) images, binary mask for whole brain (C); MRI after scalp editing (D); original SPECT image coregistered to MRI (E); MR image segmented into white matter (F) and grey matter (G); White matter SPECT image (H) simulated from convolved white matter MR image; convolved grey matter MR image (I); white matter MR images (J); grey matter SPECT images (K) obtained by subtraction of simulated white matter SPECT image from original SPECT image coregistered to MRI; binary mask for grey matter (L) applied to finally obtain grey matter SPECT image corrected for partial volume effect (M). (Reprinted from ref.45 with permission).

Figure 6. Schematic representation of the different steps required for MRI-guided correction for partial volume effect illustrating the original MR (A) and SPECT (B) images, binary mask for whole brain (C); MRI after scalp editing (D); original SPECT image coregistered to MRI (E); MR image segmented into white matter (F) and grey matter (G); White matter SPECT image (H) simulated from convolved white matter MR image; convolved grey matter MR image (I); white matter MR images (J); grey matter SPECT images (K) obtained by subtraction of simulated white matter SPECT image from original SPECT image coregistered to MRI; binary mask for grey matter (L) applied to finally obtain grey matter SPECT image corrected for partial volume effect (M). (Reprinted from ref.45 with permission).

scheme one step further by incorporating a distinct volume-of-interest (Vol) such as the amygdale.46 Their method consists first in solving the problem of white matter contribution by using Eq. (33). Subsequently, true cortical activity concentration is measured from the corrected image given in Eq. (34). The corrected image for the amygdala is then given by:

- ,, g(r) - Tfi(r) 0 h(r) - ^(r) 0 h(r) - T3/3 0 h(r)

The "true" cortical value Ti derived from the "white matter and CSF-corrected image" given in Eq. (34), must satisfy the following criteria:

(i) it must be representative of the true cortical grey activity that actually contaminates the VoI;

(ii) it must be sufficiently far from the VoI to avoid integrating some of its signal.

We can see that those two criteria might rapidly introduce some conflict with increasing tracer heterogeneity, and would be for example violated if one seeks to account for cross-contamination between e.g., the caudate nucleus and the Putamen since they are lying so close to each other.

6.3.2 Model-based Approach

Another type of post-reconstruction correction methods is the model-based optimization method developed by Chen et al.42 to simultaneously recover the size and the activity concentration of small spheroids thus improving estimates of lesion activity in clinical oncology when object size is unknown (Figure 7). The algorithm is based on a 3D spatially varying object size- and contrast-dependent Gaussian model of the system PSF. A match index is then used to estimate the best model parameters. The authors report a reduction in the activity error by 11%-63% compared to the error obtained without correction. Moreover, the accuracy of the simple RC method is dependent on object-to-background ratio (OBR) and the data used for estimating fitting parameters. Large errors were reported for small spheroids, which are obviously very sensitive to OBR variation and noise. A modified version of the algorithm described above combined with an extension to non-spherical objects was recently proposed.47 The method is being improved currently allowing the quantification of lung tumours with smallest radii with improved convergence properties towards the true model.

System PSF

initial estimate

Transformation matrix te

0 0

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