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.Su1,2 and B.K.Rutt2
1Department
2Department
of Electrical Engineering, Stanford University, Stanford, CA, United States
of Radiology, Stanford University, Stanford, CA, United States
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
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.
Allowing phase-cycling as a free variable improves
the theoretical precision by 2.1x in GM
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
2Department of
Electrical Engineering, Stanford University, Stanford, CA, United States
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.
OPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB
ISMRM 2014 #3206
Directory
T1 mapping with
SPGR (VFA,
DESPOT1)
Cramér-Rao
Lower Bound
Optimal
Design
Automatic
Differentiation
T1 and T2 mapping
with SPGR and
bSSFP (DESPOT2)
Diffusion?
OPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB
ISMRM 2014 #3206
How precisely can I
measure something
with this pulse
sequence?
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
𝛿𝑔𝑖 𝜃
𝐽𝑖,𝑗 =
𝛿𝜃𝑖
−1
𝐹 = 𝐽𝑇 Σ𝑛𝑜𝑖𝑠𝑒
𝐽
• 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 estimator
2. A signal equation with normally distributed noise
3. Measurement noise is independent
−1
𝐹 = 𝐽𝑇 Σ𝑛𝑜𝑖𝑠𝑒
𝐽
Σ𝜃 ≥ 𝐹 −1
𝜎 𝜃𝑖 ≥
Σ𝜃,𝑖𝑖
CRLB: Computing the Jacobian
Numeric
differentiation
Symbolic or analytic
differentiation
Automatic
differentiation
• Questionable accuracy
• Has limited the application of CRLB
• Difficult, tedious, and slow for multiple inputs, multiple
outputs
• Solves all these problems
• Calculation time comparable to numeric
• But 108 times more accurate
OPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB
ISMRM 2014 #3206
Directory
T1 mapping with
SPGR (VFA,
DESPOT1)
Cramér-Rao
Lower Bound
Optimal
Design
Automatic
Differentiation
T1 and T2 mapping
with SPGR and
bSSFP (DESPOT2)
Diffusion?
OPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB
Automatic Differentiation
Your 21st century
slope-o-meter engine.
ISMRM 2014 #3206
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
Directory
T1 mapping with
SPGR (VFA,
DESPOT1)
Cramér-Rao
Lower Bound
Optimal
Design
Automatic
Differentiation
T1 and T2 mapping
with SPGR and
bSSFP (DESPOT2)
Diffusion?
Optimal Design
• Optimality conditions
• Applications in other fields
• Cite other MR uses of CRLB for optimization
OPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB
ISMRM 2014 #3206
Directory
T1 mapping with
SPGR (VFA,
DESPOT1)
Cramér-Rao
Lower Bound
Optimal
Design
Automatic
Differentiation
T1 and T2 mapping
with SPGR and
bSSFP (DESPOT2)
Diffusion?
T1 Mapping with VFA/DESPOT1
𝑆𝑆𝑃𝐺𝑅
1 − 𝑒 −𝑇𝑅/𝑇1
𝛼, 𝑇𝑅 ∝ 𝑀0
1 − cos 𝛼 𝑒 −𝑇𝑅/𝑇1
• Protocol optimization
– What is the acquisition protocol which maximizes
our T1 precision?
Christensen 1974, Homer 1984, Wang 1987, Deoni 2003
DESPOT1: Protocol Optimization
𝑔1 𝑀0 , 𝑇1 = 𝑆𝑆𝑃𝐺𝑅 𝛼1 , 𝑇𝑅 , 𝜎12
⋮
𝑔𝑁 𝑀0 , 𝑇1 = 𝑆𝑆𝑃𝐺𝑅 𝛼𝑁 , 𝑇𝑅 , 𝜎𝑁2
• Acquiring N images: with what flip angles and how long
should we scan each?
• Cost function, λ = 0 for M0
– Coefficient of variation (CoV = 𝜎𝑇1 /𝑇1 ) 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
– 𝛼 > 0, 𝜎𝑖2 > 0
– 𝛼 < 90°
• Solver
– Sequential least squares programming
with multiple start points
(scipy.optimize.fmin_slsqp)
Results: T1=1000ms
•
•
•
•
N=2
𝜎12 = 𝜎22
α = [2.373 13.766]°
This agrees with Deoni 2003
– Corresponds to the pair of flip angles producing signal at
– Previously approximated as 0.71
1
2
of the Ernst angle
Results: T1=500-5000ms
• N=2
• 𝜎12 = 𝜎22
• 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
OPTIMAL UNBIASED STEADY-STATE RELAXOMETRY WITH PHASE-CYCLED VARIABLE FLIP ANGLE (PCVFA) BY AUTOMATIC COMPUTATION OF THE CRLB
ISMRM 2014 #3206
Directory
T1 mapping with
SPGR (VFA,
DESPOT1)
Cramér-Rao
Lower Bound
Optimal
Design
Automatic
Differentiation
T1 and T2 mapping
with SPGR and
bSSFP (DESPOT2)
Diffusion?
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 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
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.
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
T1 in DESPOT2 vs PCVFA
Discussion & Conclusion