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J.Su 1,2 and B.K.Rutt 2

Computer No. 12. ISMRM 2014 E-Poster #3206. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the Cramér -Rao Lower Bound . J.Su 1,2 and B.K.Rutt 2

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J.Su 1,2 and B.K.Rutt 2

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  1. Computer No. 12 ISMRM 2014 E-Poster #3206 Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the Cramér-Rao Lower Bound J.Su1,2and B.K.Rutt2 1Department of Electrical Engineering, Stanford University, Stanford, CA, United States 2Department of Radiology, Stanford University, Stanford, CA, United States Coefficient of variation for T1 (top) and T2 (bottom) of the optimal protocols over a tissue range with columns (left) DESPOT2 and (right) PCVFA. «denotes the target gray matter tissue.

  2. ISMRM 2014 E-Poster #3206 Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the Cramér-Rao Lower Bound J.Su1,2 and B.K.Rutt2 1Department of Electrical Engineering, Stanford University, Stanford, CA, United States 2Department of Radiology, Stanford University, Stanford, CA, United States Declaration of Conflict of Interest or Relationship I have no conflicts of interest to disclose with regard to the subject matter of this presentation.

  3. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Directory • Cramér-Rao Lower Bound Optimal Design • Automatic Differentiation

  4. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Cramér-Rao Lower Bound How precisely can I measure something with this pulse sequence?

  5. CRLB: What is it? • A lower limit on the variance of an estimator of a parameter. • The best you can do at estimating say T1 with a given pulse sequence and signal equation: g(T1) • Estimators that achieve the bound are called “efficient” • The minimum variance unbiased estimator (MVUE) is efficient

  6. CRLB: Fisher Information Matrix • Typically calculated for a given tissue, θ • Interpretation • J captures the sensitivity of the signal equation to changes in a parameter • Its “invertibility” or conditioning is how separable parameters are from each other, i.e. the specificity of the measurement

  7. CRLB: How does it work? • A common formulation • Unbiased estimator • A signal equation with normally distributed noise • Measurement noise is independent

  8. CRLB: Computing the Jacobian

  9. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Directory • Cramér-Rao Lower Bound Optimal Design • Automatic Differentiation

  10. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Automatic Differentiation Your 21st centuryslope-o-meter engine.

  11. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Automatic Differentiation • Automatic differentiation IS: • Fast, esp. for many input partial derivatives

  12. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Automatic Differentiation • Automatic differentiation IS: • Fast, esp. for many input partial derivatives • Effective for computing higher derivatives

  13. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Automatic Differentiation • Automatic differentiation IS: • Fast, esp. for many input partial derivatives • Effective for computing higher derivatives • Adept at analyzing complex algorithms

  14. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Automatic Differentiation • Automatic differentiation IS: • Fast, esp. for many input partial derivatives • Effective for computing higher derivatives • Adept at analyzing complex algorithms • Accurate to machine precision

  15. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Directory • Cramér-Rao Lower Bound Optimal Design • Automatic Differentiation

  16. Optimal Design • Optimality conditions • Applications in other fields • Cite other MR uses of CRLB for optimization

  17. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Directory • Cramér-Rao Lower Bound Optimal Design • Automatic Differentiation

  18. T1 Mapping with VFA/DESPOT1 • Protocol optimization • What is the acquisition protocol which maximizes our T1 precision? Christensen 1974, Homer 1984, Wang 1987, Deoni 2003

  19. DESPOT1: Protocol Optimization • Acquiring N images: with what flip angles and how long should we scan each? • Cost function, λ = 0 for M0 • Coefficient of variation (CoV = ) for a single T1 • The sum of CoVs for a range of T1s

  20. Problem Setup • Minimize the CoV of T1 with Jacobiansimplemented by AD • Constraints • TR = TRmin = 5ms • Solver • Sequential least squares programming with multiple start points(scipy.optimize.fmin_slsqp)

  21. Results: T1=1000ms • N = 2 • α = [2.373 13.766]° • This agrees with Deoni 2003 • Corresponds to the pair of flip angles producing signal at of the Ernst angle • Previously approximated as 0.71

  22. Results: T1=500-5000ms • N = 2 • α = [1.4318 8.6643]° • Compare for single T1 = 2750ms, optimalα = [1.4311 8.3266]° • Contradicts Deoni 2004, which suggests to collect a range of flip angles to cover more T1s

  23. Results: T1=500-5000ms

  24. Optimal Unbiased Steady-State Relaxometry with Phase-Cycled Variable Flip Angle (PCVFA) by Automatic Computation of the CRLB ISMRM 2014 #3206 Directory • Cramér-Rao Lower Bound Optimal Design • Automatic Differentiation

  25. DESPOT2 The optimal PCVFA protocol acquires two phase cycles, each with flip angle pairs that give 1/√2 of the maximum signal. 3Deoni et al. MRM 2003 Mar;49(3):515-26.

  26. DESPOT2 vs. PCVFA Coefficient of variation for T1 (top) and T2 (bottom) of the optimal protocols over a tissue range with columns (left) DESPOT2 and (right) PCVFA. «denotes the target gray matter tissue.

  27. T1 in DESPOT2 vs PCVFA

  28. T1 in DESPOT2 vs PCVFA

  29. Discussion & Conclusion

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