Multi-exponential decomposition for MR imaging of HCC and fibrosis

1. Multi-exponential decomposition for MR imaging of HCC and fibrosis Giovanni Motta Jan 7, 2005

2. Sequences • UTE • Fat saturation • 4 echoes • 20 sequences 256x256 (4) or 320x320 (16) • TE = 0.08, 3.25, 6.42 and 9.59ms (2) • TE = 0.08, 4.53, 8.98 and 13.5ms (11) • TE = 0.08, 5.81, 11.6 and 17.4ms (4) • TE = 0.08, 6.90, 13.8 and 19.6ms (3) • One slice from each sequence

3. Example UTE_0015: TE = 0.08, 5.81, 11.6 and 17.4ms, 320x320 pixels

4. Model • Voxel value is proportional to the transverse magnetization of the corresponding volume • Transverse magnetization decays exponentially with TE • The time behavior of a voxel can be described by a linear combination of exponentials (plus a residual error)

5. Model • Exponentials are the basis functions used in the decomposition

6. Method • Given the four echoes • We want to solve • With respect to and

7. Method • An exact solution is not always possible, so we look for an approximation that minimizes the error • Where

8. Example UTE_0015: voxel of coordinates (208, 63) Best fit with T2A=13 and T2B=6:

9. Advantages • Short Term • Allows generation of synthetic images for arbitrary TE • Exponentials and reconstruction error can be isolated and imaged individually • Subtracting the reconstruction error from the image provides a form of denoising • Long Term • The parameters of this representation can be used in the classification of the voxels

10. Main Assumption The parameters of this decomposition are an advantageous way of representing all the information necessary for the classification

11. Experiments with two exponentials • In experiment sets 1 and 3 the system is non linear • Unknowns can only assume non negative values

12. Experiments with four exponentials • T2 =100, 20, 10 and 5ms. • System of equations is linear • Non negativity constraints on M

13. Supervised Classification • Rudimentary nearest neighborhood classification • Voxels are represented by the parameters of the decomposition • Fixed a set of parameters (target), find the voxels that that have parameters closer than a predetermined amount (threshold) • Distance is measured by the squared error between the two sets of parameters

14. Example • Matlab program • Experiment sets 2 and 3

15. Unupervised Classification • Voxels are represented by the parameters of the decomposition • Fixed the number of classes, partition the voxels into classes so that voxels belonging to the same class have similar parameters • Distance is measured by the squared error

16. Unsupervised Classification • Two exponentials • T2=20 and 5ms. • 8 classes (left) and 16 classes (right)

17. Unsupervised Classification • Four exponentials • T2=100, 20, 10 and 5ms. • 8 classes (left) and 16 classes (right)

18. What’s Next? • Speed up decomposition • Verify assumptions (linearity, for ex.) • More echoes • Sub pixel operations • Registration of different sequences • Classification • Integration with anatomic info • …