Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR (DESPOT1, VFA) and bSSFP-based (DESPOT2) relaxometry methods are re-optimized under this framework Allowing phase-cycling as a free variable improves the theoretical precision by 2.1x in GM O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE C RAMÉR -R AO L OWER B OUND J.Su 1,2 and B.K.Rutt 2 1 Department of Electrical Engineering, Stanford University, Stanford, CA, United States 2 Department of Radiology, Stanford University, Stanford, CA, United States ISMRM 2014 E-P OSTER #3206C OMPUTER N O. 12 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.

Declaration of Conflict of Interest or Relationship I have no conflicts of interest to disclose with regard to the subject matter of this presentation. O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE C RAMÉR -R AO L OWER B OUND J.Su 1,2 and B.K.Rutt 2 1 Department of Electrical Engineering, Stanford University, Stanford, CA, United States 2 Department of Radiology, Stanford University, Stanford, CA, United States ISMRM 2014 E-P OSTER #3206

Directory O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206 Automatic Differentiation Automatic Differentiation Cramér-Rao Lower Bound Cramér-Rao Lower Bound T1 mapping with SPGR (VFA, DESPOT1) T1 and T2 mapping with SPGR and bSSFP (DESPOT2) Diffusion? Optimal Design Optimal Design

How precisely can I measure something with this pulse sequence? O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206

CRLB: What is it? A lower limit on the variance of an estimator of a parameter. – The best you can do at estimating say T 1 with a given pulse sequence and signal equation: g(T 1 ) Estimators that achieve the bound are called “efficient” – The minimum variance unbiased estimator (MVUE) is efficient

CRLB: Fisher Information Matrix

CRLB: How does it work?

CRLB: Computing the Jacobian Questionable accuracy Numeric differentiation Has limited the application of CRLB Difficult, tedious, and slow for multiple inputs, multiple outputs Symbolic or analytic differentiation Solves all these problems Calculation time comparable to numeric But 10 8 times more accurate Automatic differentiation

Directory O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206 Automatic Differentiation Automatic Differentiation Cramér-Rao Lower Bound Cramér-Rao Lower Bound T1 mapping with SPGR (VFA, DESPOT1) T1 and T2 mapping with SPGR and bSSFP (DESPOT2) Diffusion? Optimal Design Optimal Design

Automatic Differentiation Your 21 st century slope-o-meter engine. O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206

Automatic Differentiation Automatic differentiation IS: – Fast, esp. for many input partial derivatives Symbolic requires substitution of symbolic objects Numeric requires multiple function calls for each partial O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206

Automatic Differentiation Automatic differentiation IS: – Fast, esp. for many input partial derivatives – Effective for computing higher derivatives Symbolic generates huge expressions Numeric becomes even more inaccurate O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206

Automatic Differentiation Automatic differentiation IS: – Fast, esp. for many input partial derivatives – Effective for computing higher derivatives – Adept at analyzing complex algorithms Bloch simulations Loops and conditional statements 1.6 million-line FEM model O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 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 A comparison between automatic and (central) finite differentiation vs. symbolic AD matches symbolic to machine precisionFinite difference doesn’t come close O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206

Directory O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206 Automatic Differentiation Automatic Differentiation Cramér-Rao Lower Bound Cramér-Rao Lower Bound T1 mapping with SPGR (VFA, DESPOT1) T1 and T2 mapping with SPGR and bSSFP (DESPOT2) Diffusion? Optimal Design Optimal Design

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

Directory O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206 Automatic Differentiation Automatic Differentiation Cramér-Rao Lower Bound Cramér-Rao Lower Bound T1 mapping with SPGR (VFA, DESPOT1) T1 and T2 mapping with SPGR and bSSFP (DESPOT2) Diffusion? Optimal Design Optimal Design

T1 Mapping with VFA/DESPOT1 Protocol optimization – What is the acquisition protocol which maximizes our T 1 precision? Christensen 1974, Homer 1984, Wang 1987, Deoni 2003

DESPOT1: Protocol Optimization

Problem Setup

Results: T 1 =1000ms

Results: T 1 = ms

Directory O PTIMAL U NBIASED S TEADY -S TATE R ELAXOMETRY WITH P HASE -C YCLED V ARIABLE F LIP A NGLE (PCVFA) BY A UTOMATIC C OMPUTATION OF THE CRLBISMRM 2014 #3206 Automatic Differentiation Automatic Differentiation Cramér-Rao Lower Bound Cramér-Rao Lower Bound T1 mapping with SPGR (VFA, DESPOT1) T1 and T2 mapping with SPGR and bSSFP (DESPOT2) Diffusion? Optimal Design Optimal Design

DESPOT2 Using SPGR and FIESTA These have been previously paired in the DESPOT2 technique as a two-stage experiment The toolkit creates the CRLB as a callable function and delivers it to standard optimization routines Finds the same optimal choice of flip angles and acquisition times as in literature 1 under the DESPOT2 scheme with 5+ decimal places of precision A pair of SPGR for T1 mapping and a pair of FIESTA for T2 with ~75% time spent on SPGRs A new method with FIESTA only: phase-cycled variable flip angle (PCVFA) 2.1x greater precision per unit time achieved by considering a joint reconstruction and allowing phase-cycling to be free parameter 3 Deoni et al. MRM 2003 Mar;49(3): The optimal PCVFA protocol acquires two phase cycles, each with flip angle pairs that give 1/√2 of the maximum signal.

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.

T1 in DESPOT2 vs PCVFA

Discussion & Conclusion