Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework...

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.

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

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






bSSFP (DESPOT2)

Diffusion?

Optimal Design


Your 21st centuryslope-o-meter engine.


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




partial derivatives– Effective for computing

higher derivatives

Symbolic generates huge expressions

Numeric becomes even more inaccurate





higher derivatives– Adept at analyzing

complex algorithms

Bloch simulations

Loops and conditional statements

1.6 million-line FEM model


Automatic Differentiation• Automatic differentiation IS:– Fast, esp. for many input


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







bSSFP (DESPOT2)

Diffusion?

Optimal Design

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






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

Results: T1=500-5000ms






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.

T1 in DESPOT2 vs PCVFA

Discussion & Conclusion

Computation of the Cramér-Rao Lower Bound for virtually any pulse sequence An open-source framework...

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