Optimal Design with Automatic Differentiation:Exploring Unbiased Steady-State Relaxometry

Jan 13, 2014Jason Su

Outline

• Concept and theory• Practical usage with automatic differentiation

(AD)• Exploring relaxometry with SPGR and bSSFP

Motivation

• I started with the goal of trying to optimize mcDESPOT and the estimate of MWF– Lankford et al. criticized the theoretical accuracy of mcDESPOT

based on CRLB analysis– Interestingly however, this criticism inspired the approach to

finding the solution

• Since then, I’ve discovered that this project can have greater possibilities– The following provides a framework that is suitable for the

analysis, comparison, and optimization of a broad range of parameter mapping/estimation methods• T1, T21, B1+, B0, MT, diffusion coefficient2, etc.

[1] Jones et al. JMR 1996 Oct;113(1):25-34. [2] Brihuega-Moreno et al. MRM 2003 Nov;50(5):1069-76.

Idea

• The Cramer-Rao Lower Bound (CRLB) provides a “best case” limit on the variance of an estimator of parameter– In quantitative MR, the estimator is defined by the applied

pulse sequence and protocol details

• Typically this is used as an analysis tool, but what if we instead use it as a cost function?– We want to solve the problem:

• Find the protocol that gives the lowest variance estimate of θ

– This is hard to solve for non-linear equations, relax it to:• Find the protocol that gives the best lower bound on the variance

of the estimate of θ

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 formulation1. 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

Automatic Differentiation

• An algorithm for generating computer programs that calculate derivatives of other functions based on their source code/graph– Uses repeated application of the chain rule– Fast and accurate to machine precision– Can differentiate extremely complicated functions– Many packages exist for Matlab, C, and Python

• Typically you will write the function that computes your signal equation– Then your AD package provides the derivatives essentially for free,

i.e. no analysis or coding effort– YMMV depending on the language or package

What does this buy us?

AD gives us a way to compute CRLB or Fisher information cheaply

• Both in terms of coding effort and CPU time

We can use this framework to find the optimal protocol design that provides the best estimate for a given θ• This is usually the first question when someone invents a new mapping method

Or we can explore variety of scenarios and design cases

• What is the optimal protocol for a range of T1s?• What is the protocol that is least affected by B1+/B0 effects?• What precision can be gained by using complex data instead of magnitude data?• Which of these different mapping methods is the most efficient at estimating θ?

Application

• To vet this approach and my code, which is ultimately intended for open-source use– Went back to analyze a number of classic designs

to see if I could reproduce them• Diffusion coefficient, VFA/DESPOT1, DESPOT2• In all cases, the answer was yes

• The DESPOT2-cast was particularly interesting

Steady-State Relaxometry with bSSFP and SPGR (aka DESPOT2)

• DESPOT2 seeks to estimate T1 and T21. First do VFA/DESPOT1 with a pair of SPGR

images at ~0.71 signal level to get a T1 map2. Then acquire a pair of π-phase-cycled bSSFP

images at ~0.71 signal level and back out T2, knowing T1 from step 1

– Constraints:• Phase-cycling is π• Acquisition time is equal between the angles in a pair,

only optimize the relative time between the SPGR and bSSFP parts of the experiment

Methods

• Find α’s and acquisition time fraction• Minimize the sum of the coefficient of variations (CoV)

of T1 and T2: σT1/T1+σT2/T2– Where σ was provided by the CRLB which uses the Jacobians

of the SPGR and bSSFP signal equations– Exhaustively search for the # SPGR and bSSFP images– Fix the total acquisition time so we find the best protocol

per unit time

• Optimize for representative T1 and T2 values of WM and GM at 3T– WM (T1=1100ms, T2=60ms) and GM (T1=1645ms, T2=85ms)

Solver

• Used a general solver for this problem– This is the weakest part of the framework as the

solver sometimes fails to converge or encounters inversion errors

– May be improved with insights from optimal design theory, e.g. using determinant of F instead of actual CRLB

Results

• To within 5 decimal places, framework says to acquire the pairs of flip angles at 1/√2 of the max signal for SPGR and π-phase-cycled bSSFP– Compared to the original literature estimate of 0.71

(Deoni 2003)– I later found that 1/√2 was shown to be the analytically

correct solution (Schabel and Morell 2009)• ~75% of time should be spent on SPGRs and T1 map• Optimal DESPOT2 protocols achieve a sum of T1

and T2 CoVs of 45.7 and 53.0 for WM and GM

Unrestricted Problem

• Can we do better?• What if we remove the constraints from the

DESPOT2 acquisition?– Allow any combination of SPGRs and bSSFPs to be

used, forget about the 2-stage estimate of T1 then T2 and reconstruct with a joint nonlinear fit

– Allow any phase-cycle to be used– Allow the acquisition time for any image to be

freely adjustable

PCVFA Solution

• The 0 and π phase cycles should be collected, each with a pair of flip angles that attain 1/√2 of the max signal for that phase-cycle

• Equal time fraction should be devoted to each of the 4 images

• Achieves a sum of CoVs of 20.8 and 21.6 for WM and GM– More than 2.1x improvement over DESPOT2

Performance of Optimal GM Protocol over a Range of Tissues

DESPOT2 PCVFA

Experimental Validation

• In progress with silicone oil phantom at 3T to mitigate B1 inhomogeneity– Harder to acquire DESPOT2 optimal sequence due to

time difference between SPGR and bSSFP parts– Fractional angles by adjusting ia_rf1, <1 deg. Needed

for PCVFA– Had difficulties doing linear and nonlinear fit for

DESPOT2 and PCVFA with acquired data• Maybe made an error in the acquisition, should

try again

One step further

• What if we used complex images for the reconstruction?

• 21.6 -> 10.81– Another 2x gain?

Complex PCVFA vs. Magn. PCVFA

Summary

• This framework gives the ability to study complex pulse sequences and their efficacy analytically– A way to formally compare and explore potential

mapping methods

• Gets closer to answering the question of, what is the best way of mapping X?– But may not include considerations like immunity to

B1+ inhomogeneity or ease of fitting

Next Steps

• Robust protocol design– Have been looking at B0 robustness, where cost function

evaluates CRLB at many B0 values and try to optimize the worst case estimate of θ among them

– B1+ robustness• Broad comparison of efficiency of mapping methods– T1 or B1+ mapping is probably a good candidate

• mcDESPOT and MWF• MPnRAGE, what should n be?• Analyzing bloch simulations, MRF?