Retrospective Resolution Adaption

In the following section, the previously described 3D GA radial sequence is applied to retrospectively reconstruct data at resolutions which are adapted to the measured signal time curves of a perfusion phantom.

Process of Resolution Adaption

The principle of retrospective resolution adaption is described in figure 8.7.

First, temporally equidistant images are reconstructed at low spatial/high temporal resolutions, such that the initial signal time curves Sinit(r, t) at voxel positions r are

sampled sufficiently fast for high fitting accuracy. For each voxel within a chosen ROI, the pharmacokinetic model is fitted to Sinit(r, t), resulting in fitted curves Sf it(r, t) and

low spatial resolution PK maps pinit.

From Sf it(r, t) of each voxel, sampling times {ti} are derived which have the highest

possible spatial resolution whilst preserving fitting accuracy. The choice of sampling times will be described in detail in the next section.

Finally, images are reconstructed at the time points {ti} and the PK model is fitted

to generate the adaptive PK map padapt. The resolution of padapt may vary for different

parameters throughout the map.

Choice of Sampling Times and Corresponding Resolutions

It is assumed in this study that for accurate model fitting, the time points {ti} have

to be distributed such that 4 equidistant time points are placed at baseline, upslope (including peak) and wash-out, respectively. This is visualized in figure 8.8.

Intervals of length Δti are defined such that the start and end points of each interval

lie in the middle between adjacent ti. The number of projections Nprwithin each interval

is determined by:

Npr =

Δti

T R, (8.12)

where TR is the repetition time.

To determine the corresponding resolution for Nprgiven projections, it is assumed that

the GA radial profiles are approximately homogeneously distributed in k-space and that data under-sampled by 90% still yield acceptable image quality. The latter assumption is based on a work by Stehning et al [Stehning2004], in which it has been shown that for MR angiography 12.5% under-sampling is feasible.

Using equation 2.36, describing the Nyquist criterion of uniformly spaced 3D radial

k-space data, the matrix size M can be derived, for which 10% of the Nyquist criterion

is met:

M =



10Npr/π. (8.13)

Since an integer value for M is required, the obtained value is rounded up. At a fixed FOV, the isotropic spatial resolution Δx is given by: Δx = F OV /M .

Figure 8.7: Process of resolution adaption: Equidistant low spatial/high temporal res- olution images of voxel signal Sinit(r, t) are reconstructed. The PK model

is fitted to the data, providing fitted curves Sf it(r, t) and initial low spatial

resolution PK maps pinit. From the fitted curves Sf it(r, t), sampling times

{ti} are derived which yield the maximal spatial resolution whilst preserving

fitting accuracy. Data at times {ti} are reconstructed and PK maps padapt

8.2 Methods 133

Figure 8.8: Sampling times and corresponding resolutions: It is assumed that 4 equidis- tant time points {ti} are required on the baseline, upslope and wash-out of

Sf it, respectively. From the length of intervals Δti with start and end point

between adjacent ti, the number of projections Npr and the corresponding

feasible resolutions Δx are derived.

With the perfusion phantom used, some curves with large onset times are only partially acquired, such that large parts of the wash-out are missing. In that case, the number of required time points is adjusted such that the spatial resolution never falls below that of the initial low spatial resolution images.

Cluster Generation

To increase reconstruction speed, all curves Sf it(r, t) within the ROI are divided into

clusters of similar onset times τ and peak times tmax. In increments of Δt=25 s, the time

between 0 and the maximal possible τ -value is divided into intervals. The same is done for tmax. All curves falling into the same τ - and tmax-interval form a cluster. For each

cluster, the mean curve Smean(t) is calculated. The sampling procedure as described

above is done based on Smean(t). The resulting sampling times {ti} are used for PK

map generation for all voxels having curves within the cluster. Two example clusters with mean curves Smean(t) can be seen in figure 8.9.

Figure 8.9: All fitted signal time curves Sf it of two clusters from perfusion phantom

and the resulting mean curves Smean(t) and the resulting sampling schemes

(dotted lines) for each cluster.

Image Scaling

Throughout the signal time curves, images are reconstructed at varying spatial reso- lutions. Since during gridding a resolution-dependent number of profiles Npr is used,

different scaling factors are imposed due to varying density compensation functions. To scale the signal intensity of image I₁ reconstructed with Npr,1 profiles to image I2

reconstructed with Npr,2 profiles, the following rescaling has to be performed:

I_1,sc= I₁·Npr,2

Npr,₁

, (8.14)

where I_1,sc is the scaled image I₁.

This is investigated using phantom data of a water-filled sphere. Images with varying numbers of profiles Npr = 625, 1000, 1250, 5000 at a constant spatial resolution (ma-

trix size M=64 at a FOV of 300 mm) are reconstructed and scaled to the image with

Npr=5000. For each image, signal profiles denoted as S625, S1000, S1250 and S5000, going

through the center of the sphere are shown in figure 8.10 a) and b) without and with scaling.

Since a small shift is detected to remain in the phantom profiles after scaling, a constant shift s is empirically determined by calculation of the mean difference of the signal profiles to S₅₀₀₀ and division by the scaling factors given in equation 8.14. The scaled and shifted phantom profiles are given in figure 8.10 c).

8.2 Methods 135

Figure 8.10: Signal profiles of images of a water-filled sphere for varying numbers of pro- files Npr= 625, 1000, 1250 and 5000 a) without scaling, b) scaled according

to equation 8.14, c) scaled and shifted by the constant shift s.

8.2.8 Comparison of Adaptive and Equidistant Schemes Using a Perfusion