1. Optimal SSFP Pulse-Sequence Design for Tissue Density Estimation Zhuo Zheng Advanced Optimization Lab McMaster University Joint Work with C. Anand, R. Sotirov, T. Terlaky

2. Overview Motivation Model Optimization Problem Numerical Results

3. Motivation MRI is widely used in diagnosis, treatment monitoring and research. Quantitatively determining different tissue types is crucial. Exploring the applicability of optimization in biomedical engineering research.

4. MRI Basics (Step-by-step Illustration)

5. The Dynamic System Magnetization is dependent on several parameters and. The dynamic system satisfies: The system can be built up from several components.

6. SSFP Pulse-Sequence Fast scanning and good signal-to-noise ratio. Steady-state is achieved if Denoted as, we have with and.

7. Model Components Based on the physical mechanisms, we have

8. Imaging For simplicity, we write the results of n experiments as a real 2n vector and m tissue densities as a real m vector: MPPI is an unbiased estimator for tissue densities if has full rank.

9. Objective and Formulation Objective: choose pulse-sequence design variables such that the error in the reconstructed densities is minimized. Error given by in which  is the white measurement noise.

10. SDO Problem Exerting SVD

11. Relaxation We replace the sines and cosines in the components by unit vectors and and add the constraints: Then relax the constraints to:

12. Complete System Adding upper and lower bounds for the repetition times we have now the system: s.t.

13. where

14. Trust Region Algorithm for NL-SDO How to deal with and semidefinite constraint: Defining a linear SDO-SOCO subproblem by linearizing the nonlinear constraints around the current point. Linearizing : and its partial derivatives for information.

15. A Clinical Application Carotid artery tissue densities estimation We reconstruct the densities based on the optimal solutions obtained by our formulation.

16. Comparison Reconstructed gray-scale images obtained by optimal solutions and grid-search.

17. Numerical Results

18. Concluding Remarks Innovative method for tissue densities estimation by taking into account many parameters using optimization methods. Iteratively solving the problem with semi- definite and highly-nonlinear constraints. Many interesting applications of our method, such as brain development studies in infants.

19. Future Work Formulating the mixed imaging pulse- sequence selection problems. Making the robust formulation possible. Developing an embedded solver to improve performance.