computation of the cramér-rao lower bound for virtually any pulse sequence an open-source framework...
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![Page 1: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/1.jpg)
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.Rutt2
1Department 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.
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Declaration of Conflict of Interest or Relationship
I 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.Rutt2
1Department of Electrical Engineering, Stanford University, Stanford, CA, United States2Department of Radiology, Stanford University, Stanford, CA, United States
ISMRM 2014 E-POSTER #3206
![Page 3: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/3.jpg)
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
![Page 4: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/4.jpg)
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
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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
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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
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CRLB: How does it work?• A common formulation
1. Unbiased estimator2. A signal equation with normally distributed
noise3. Measurement noise is independent
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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
![Page 9: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/9.jpg)
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
![Page 10: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/10.jpg)
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
![Page 11: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/11.jpg)
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
![Page 12: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/12.jpg)
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
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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
![Page 14: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/14.jpg)
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
![Page 15: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/15.jpg)
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
![Page 16: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/16.jpg)
Optimal Design• Optimality conditions• Applications in other fields• Cite other MR uses of CRLB for optimization
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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
![Page 18: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/18.jpg)
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
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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
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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)
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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
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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
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Results: T1=500-5000ms
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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
![Page 25: Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework in Python: sujason.web.stanford.edu/quantitative/ SPGR](https://reader033.vdocuments.us/reader033/viewer/2022051401/56649e4e5503460f94b446bb/html5/thumbnails/25.jpg)
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.
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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.
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T1 in DESPOT2 vs PCVFA
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T1 in DESPOT2 vs PCVFA
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Discussion & Conclusion