MR Results

Cardiac images were reconstructed from the volunteer data acquired during free-breathing.

Images are able to be reconstructed in any cardiac and respiratory phase. One also has the ability to select a specific cardiac phase (such as end-diastole) and reconstruct car-diac images at different respiratory phases. This capability is highly useful for studying the influence of respiration on left ventricular filling for example.

Table 6.1: Comparison of exact source images and reconstructed images using the sinc RKHS method, the splines RKHS method, the Sobolev RKHS method, and sin-exp interpolation in computer simulated data.

Reconstruction method

Correlation coefficient Normalized mutual information Mean Standard deviation Mean Standard deviation

Sinc RKHS 0.94 0.043 0.20 0.041

Splines RKHS 0.96 0.020 0.23 0.063

Sobolev RKHS 0.97 0.008 0.33 0.060

Sinc-exp 0.96 0.019 0.29 0.096

Breath-hold vs reconstruction from free-breathing data

Of all the possible motion states existing in the range of s and t, two in particular have been selected for comparison with standard clinical images acquired in breath-hold. Two cardiac timepoints corresponding to diastole and systole were first selected manually (the same as for the free-breathing and breath-hold reconstructions). The respiratory phase used for comparison was the closest to the breath-hold reference, as defined by the correlation coefficient, of the 30 respiratory phases reconstructed at uniformly distributed values of s between 0 and 1. For the two cardiac timepoints, images are shown in Fig. 6.5. The correlation coefficient was then computed between the reference images (breath-hold) and the reconstructions from free-breathing data (see Table 6.2) for all the 30 cardiac phases. As in the phantom experiment, imaging results obtained using the Sobolev RKHS exhibited a statistical advantage over the other RKHS methods and sinc-exp interpolation.

Table 6.2: The mean of correlation coefficient between images (the respiratory phase is s=0.55 fixed, and 30 different cardiac phases) reconstructed from free-breathing using sinc RKHS, splines RKHS, Sobolev RKHS, sinc-exp interpolation and standard clinical images obtained in breath-hold for a healthy volunteer.

Reconstruction Method Correlation coefficient Standard deviation

Sinc RKHS 0.433 0.035

Splines RKHS 0.433 0.030

Sobolev RKHS 0.448 0.028

Sinc-exp interpolation 0.447 0.029

Comparison of the different RKHS

Data from each of the five healthy volunteers were used to reconstruct 30 cardiac phases (at a fixed respiratory phase), as is usually done in standard protocols. Fig. 6.6 shows examples of such reconstructions using the different methods to be tested. The figure

Comparison of images obtained using different RKHS

Diastole Systole

Diastole Systole

(a)

(b)

Figure 6.5: Cardiac images from a healthy volunteer, in two cardiac phases : diastole and systole respectively. (a) Images reconstructed using Sobolev RKHS from data acquired in free-breathing (for two cardiac phases and one respiratory phase corresponding to breath-hold: s = 0.55 ). (b) Standard clinical images obtained from data acquired during a breath-hold.

also shows a temporal profile accounting for cardiac motion along a line section drawn in the left ventricle. These profiles show less blurring using the Sobolev RKHS and sinc-exp interpolation compared with the sinc and splines RKHS. However a few residual aliasing artifacts can be seen.

This typical example illustrates the inherent compromise which has to be made between resolution (or equally, the amount of blurring) and aliasing artifacts, as the temporal sampling density is nonuniform. The results in Fig. 6.7 aim to give a better understanding of this compromise. Shannon entropy of images was used in order to assess the ”effective“ image resolution (in terms of the size of structures that can be detected). Entropy is a good choice for that as it has been used as an objective criterion

0 0.81

Figure 6.6: Images reconstructed for a fixed cardiac-respiratory phase from data acquired in free breathing (left column). The specific cardiac phase chosen for display is illustrated at the top of the left column with red line, and the respiratory phase is end-inspiration.

Temporal profiles (midle column) correspond to the line marked in the image at the top of the middle column. The one dimensional profile of the interpolation kernel (right column) centered in 0.5 and for p= 5. (a) Sinc RKHS. (c) Splines RKHS. (c) Sobolev RKHS (d) Sinc-exp interpolation.

for such tasks as image restoration [53] or motion-compensated MRI reconstruction [5].

Generally speaking, entropy increases as the effective resolution is improved. This can be seen by varying the kernel opening parameter p (see Fig. 6.7). On the one hand, increasing p, results in narrower kernels. In the case of the sinc RKHS, increasing p, the interpolation kernel became very localized and we record a loss of signal that produce artifacts. On the other hand, significantly decreasing the magnitude of p will result in a loss of resolution (i.e. blurring). Results from Fig. 6.7 suggest that, compared to sinc

Comparison of images obtained using different RKHS

or splines RKHS, the Sobolev RKHS method is an efficient and robust choice, as its results exhibit a relatively weak dependence upon the opening parameter. The results obtained with Sobolev RKHS method are comparable with the results obtained with sinc-exp interpolation, even though for small values ofp the entropy of images obtained with sinc-exp have lower entropy. These better results obtained using Sobolev RKHS or sinc-exp interpolation can be attributed to the exponential decay of both kernels. The exponential function represents a more rapid decay than any polynomial function, and has a more desirable effect than that obtained by varying p.

p = 1

p = 1

p = 10

p = 10

p = 20

p = 20

0 5 10 15 20

4.6 4.8 5 5.2 5.4 5.6 5.8 6

Kernel Opening Parameter (p)

Image Entropy

Sinc RKHS Spline RKHS Sobolev RKHS sinc−exp interpolation

p0

(a)

(b)

(c)

Figure 6.7: (a) The entropy of cardiac images from a healthy volunteer reconstructed with different kernel opening parameters. The parameterp0 = 11.34 marked in the figure is the kernel opening parameter given by (6.7). (b) Cardiac images reconstructed using Sobolev RKHS, for different values of the kernel opening parameterp. (c) Cardiac images reconstructed using sinc RKHS, for different values of the kernel opening parameter p.

The analyses of the cine sequences produced for three subjects in inspiration and expiration by the two cardiologists blinded observers were consistent. Two of the meth-ods, sinc RKHS and splines RKHS, produced images of very poor quality with many artifacts, and were of little diagnostic value. The two other methods, Sobolev RKHS and sinc-exp interpolation, were comparable with overall good image quality, much less artifacts and good diagnostic accuracy. Both observers agreed that the images

pro-duced with Sobolev RKHS contained slightly less artifacts than those using the sinc-exp interpolation.

The mean of the image entropy from further results is shown in Fig. 6.8 over the full range of cardiac phases from images from five healty volunteers. Here the kernel opening parameter was set as in (6.7). The graph shows again that the Sobolev RKHS and sinc-exp interpolation give more stable values compared with other kernels.

0 0.2 0.4 0.6 0.8 1

5.4 5.5 5.6 5.7 5.8 5.9 6 6.1

Cardiac Phase

Image Entropy

sinc RKHS splines RKHS Sobolev RKHS sinc−exp interpolation

Figure 6.8: The mean of image entropy, over the images reconstructed for five healthy volunteers, for 30 cardiac phases uniformly distributed between 0 and 1.

Extension to three-dimensional RKHS