1. A Time-Averaged Projection Matrix Approach to Motion Correction in ET by Charles Byrne,Department of Mathematical Sciences, University of Massachusetts Lowell

2. Basic Problem Determine the spatial distribution of radionuclide intensity; Discretize the body into J voxels; r(j) denotes the emission element of radionuclide within jth voxel at beginning of scan; Estimate the amount of r(j).

3. The Motion-Free Case For j=1,…,J, x(j) is the expected emissions from emission element r(j). For i=1,…,I, P(i,j) is the probability that a photon emitted from r(j) will be detected at detector i. y(i) is the count at detector i. Solve y=Px (approximately) for x=(x(j)).

4. When There Is Patient Motion Emissions from emission element r(j) are detected in “the wrong place”, due to the motion of the patient. Can we determine which detector location is the “right place” and move the data accordingly? For rigid motion, yes. Even for known non-rigid motion, we would need to know the r(j) to correct by moving the data.

5. Discretized Motion We discretize the motion into M poses, with d(m) the duration of the mth pose. Scale in time so that total duration is one.

6. Iterative Reconstruction The EM algorithm: for k=0,1,… x(j,k+1)=x(j,k) S i P(i,j) y(i)/(Px(k))(i). sum on i is the back-projection step. forming (Px(k))(i)=  j P(i,j) x(j,k) is the forward-projection step.

7. The Case of Rigid Motion determine the motion by external observation; move the counts, to correspond with motion-free emission ; reconstruct the intensities.

8. Non-Rigid Motion- Possible Approaches Use gated or time-stamped data to reconstruct M images, then combine. Move the estimated intensities prior to forward projection (Fulton and Meikle). Calculate time-averaged probability matrix to replace the matrix P (Byrne, Reyes, et al.)

9. Non-Rigid Motion -Gated or Time-Stamped Data Observe external motion; Infer internal motion; Time-stamp or gate data; Reconstruct image for each pose. Is there sufficient data to obtain good image for each pose?

10. Fulton and Meikle Approach- MIC-2005 Only a single image is reconstructed, but during the forward projection step the current image is moved to each of the poses before projection into data space. Time-averaging after projection. Probabilities are constant over time. Each voxel moves to one voxel. Efficient, but does it converge?

11. Time-Averaged Probabilities We model the patient-motion problem as one of motionless patient, but time- dependent probabilities connecting locations within the body to the detectors. We need a model of actual motions, to determine these probabilities. We average these probabilities over time.

12. Time-Averaged Probabilities-2 We do not need to know when specific data was recorded, nor the position of the patient when specific data was obtained. The same iterative or non-iterative algorithms are used for reconstruction. Only the matrix of probabilities is different.

13. The Time-Averaged Matrix- Continuous Time For t in [0,T], let P(i,j,t) be the probability that an emission, at time t, from the emission element r(j), will be detected at i. Let P(i,j)=(1/T)  P(i,j,t)dt be the time- averaged projection matrix. Solve y=Px for x.

14. Practical Considerations -1 We assume M distinct positions or poses and replace the integral with a time- weighted sum. We may or may not assume that motion moves each voxel to a new voxel; We may allow for expansion or contraction.

15. Practical Considerations -2 The portion of the body initially in voxel j may later expand to occupy portions of several known voxels. In that case, we must assume that whatever radionclide r(j) is present in that portion is spread uniformly over the several voxels at the later time, so that the new probabilities can be calculated.

16. Voxel-to-Voxel Motion: notation during pose m, emission element r(j) occupies voxel g, with g=g(j,m) during pose m, voxel j is occupied by emission element r(h), for h=h(j,m); therefore, h(g(j,m),m)=j and g(h(j,m),m)=j, for all j and m.

17. Fulton-Meikle Forward Projection Replace (Px(k))(i)=  j P(i,j) x(j,k) with  j P(i,j) (  m d(m) x(h(j,m),k) ). This can be rewritten as  j (  m d(m) P(i,g(j,m)) ) x(j,k), which is (Qx(k))(i).

18. Fulton-Meikle Back-Projection The back-projection uses P, not Q. for the EM example, we have x(j,k+1)=x(j,k)  i P(i,j) y(i)/(Qx(k))(i). Does this converge? Compare with our approach: x(j,k+1)=x(j,k)  i Q(i,j) y(i)/(Qx(k))(i).

19. Back-Projection with $Q$ given x(k) and Qx(k), initialize the vector z=(z(1),…,z(J)) at zero. Then, calculate a(j) =  i P(i,j) y(i)/(Qx(k))(i). For m=1,…,M and j=1,…,J, replace z(h(j,m)) with z(h(j,m) + d(m)a(j). Then x(j,k+1)=x(j,k) z(j).

20. Related Work- Reyes, et al. Fully 3-D 2005 The idea of using the time-averaged projection matrix also occurs in the work of Reyes, et al. Their method is the same as ours. They use the “emission element” terminology and index these elements, not the voxels.

21. Acknowledgments This work was supported in part by grants R01EB001457 and R01EB002798, from the National Institute of Biomedical Imaging and Bioengineering to the University of Massachusetts Medical School, Worcester, MA, Dr. Michael King, PI.

22. References Reyes, M., Maladain, G., Koulibaly, P., and Darcourt, J. (2005) Respiratory motion correction in emission tomography imaging, Eighth Fully 3- D meeting. Fulton, R., and Meikle, S. (2005) Reconstruction of projection data corrupted by rigid or non-rigid motion, MIC-05, Puerto Rico, M10-1. Byrne, C. (2004) A new approach to motion compensation (preprint).