j.su 1,2 and b.k.rutt 2
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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 - PowerPoint PPT PresentationTRANSCRIPT
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
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.Rutt21Department of Electrical Engineering, Stanford University, Stanford, CA, United States2Department of Radiology, Stanford University, Stanford, CA, United States
ISMRM 2014 E-POSTER #3206COMPUTER NO. 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 RelationshipI have no conflicts of interest to disclose with regard to the subject matter of this presentation.
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.Rutt21Department of Electrical Engineering, Stanford University, Stanford, CA, United States2Department of Radiology, Stanford University, Stanford, CA, United States
ISMRM 2014 E-POSTER #3206
DirectoryOPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB ISMRM 2014 #3206
Automatic Differentiation
Cramér-Rao Lower Bound
T1 mapping with SPGR (VFA, DESPOT1)
T1 and T2 mapping with SPGR and
bSSFP (DESPOT2)
Diffusion?
Optimal Design
Cramér-Rao Lower BoundHow precisely can I measure something with this pulse sequence?
OPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB ISMRM 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 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
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
CRLB: How does it work?• A common formulation
1. Unbiased estimator2. A signal equation with normally distributed
noise3. Measurement noise is independent
CRLB: Computing the Jacobian
• Questionable accuracyNumeric
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 108 times more accurate
Automatic differentiation
DirectoryOPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB ISMRM 2014 #3206
Automatic Differentiation
Cramér-Rao Lower Bound
T1 mapping with SPGR (VFA, DESPOT1)
T1 and T2 mapping with SPGR and
bSSFP (DESPOT2)
Diffusion?
Optimal Design
Automatic Differentiation
Your 21st centuryslope-o-meter engine.
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
Symbolic requires substitution of symbolic objects
Numeric requires multiple function calls for each partial
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
Symbolic generates huge expressions
Numeric becomes even more inaccurate
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
Bloch simulations
Loops and conditional statements
1.6 million-line FEM model
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
A comparison between automatic and (central) finite differentiation vs. symbolic
AD matches symbolic to machine precision
Finite difference doesn’t come close
OPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB ISMRM 2014 #3206
DirectoryOPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB ISMRM 2014 #3206
Automatic Differentiation
Cramér-Rao Lower Bound
T1 mapping with SPGR (VFA, DESPOT1)
T1 and T2 mapping with SPGR and
bSSFP (DESPOT2)
Diffusion?
Optimal Design
Optimal Design• Optimality conditions• Applications in other fields• Cite other MR uses of CRLB for optimization
DirectoryOPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB ISMRM 2014 #3206
Automatic Differentiation
Cramér-Rao Lower Bound
T1 mapping with SPGR (VFA, DESPOT1)
T1 and T2 mapping with SPGR and
bSSFP (DESPOT2)
Diffusion?
Optimal Design
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
𝑆𝑆𝑃𝐺𝑅 (𝛼 ,𝑇𝑅 )∝𝑀 01−𝑒−𝑇𝑅 /𝑇 1
1−cos (𝛼 )𝑒−𝑇𝑅/𝑇1
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
Problem Setup• Minimize the CoV of T1 with Jacobians implemented by AD• Constraints
– TR = TRmin = 5ms
• Solver– Sequential least squares programming
with multiple start points(scipy.optimize.fmin_slsqp)
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
Results: T1=500-5000ms• N = 2
• α = [1.4318 8.6643]°– Compare for single T1 = 2750ms, optimal
α = [1.4311 8.3266]°
• Contradicts Deoni 2004, which suggests to collect a range of flip angles to cover more T1s
DirectoryOPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB ISMRM 2014 #3206
Automatic Differentiation
Cramér-Rao Lower Bound
T1 mapping with SPGR (VFA, DESPOT1)
T1 and T2 mapping with SPGR and
bSSFP (DESPOT2)
Diffusion?
Optimal Design
DESPOT2Using 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 literature1 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
3Deoni et al. MRM 2003 Mar;49(3):515-26.
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.