### Journal Club - ARLO

```JOURNAL CLUB:
M. Pei et al., Shanghai Key Lab of MRI, East China
Normal University and Weill Cornell Medical College
“Algorithm for Fast Monoexponential Fitting Based
on Auto-Regression on Linear Operations (ARLO) of
Data.”
Aug 18, 2014
Jason Su
Motivation
• Traditional fitting methods for exponentials
have pros and cons
– Nonlinear LS (Levenberg-Marquardt) – slow, may
converge to local minimum
– Log-Linear – fast but sensitive to noise
• Can we improve upon them?
– Surprisingly, yes!
Background: Numerical Integration
• Approximating the value of a
definite integral
• Trapezoidal Rule: the area
under a 2-pt linear
interpolation of the interval
• Simpson’s Rule: the area under
of the interval
• Newton-Cotes formulas:
Theory
• Log-Linear: linearize the signal equation with a
nonlinear transformation to fit a line
• ARLO: integrate the signal equation to fit a linear
approximation (Simpson’s rule)
−/2∗
= 0
=
+2

= 2∗ [( ) − (+2 )]
∆
≅  =
[  + 4  + ∆ + ( + 2∆)
3
• Assuming decay curve sampled linearly at ∆
intervals
Theory
• An auto-regressive time-series
• Find T2* to minimize the error between model
and data,  −
Methods
• Rician noise compensation
– Data truncation, only keep points with high SNR
• Values > μ + 2σnoise in background
– Apply a bias correction based on a Bayesian model
table look-up depending on the number of coils
Methods
• Simulation to assess bias and variance
– Fitting method vs T2* range, # channels, SNR
– 10,000 trials with Rician noise
• In vivo
– 1.5T, 8ch, 15 patients, 2D GRE, TR=27.4, α=20deg, TE = 1.323.3ms (16 linearly sampled), liver
– 3T, 8ch?, 2 volunteers, 3D GRE, α=20deg, 7/12 echoes with
6.5/4.1ms spacing, brain
– 1.5T, 2D GRE, TR=19ms, α=35deg, TE=2.8-16.8ms (8
– Manual segmentation of liver and brain structures
• Statistical
– Linear regression, Bland-Altman, and t-tests
Results:
Simulation
• LM and ARLO
are effectively
equivalent
• ARLO is
generally
equivalent to
LM except at
T2*=1.5ms
• Log-linear is
sensitive to T2*,
SNR, and
channels
Results: In Vivo, Liver ROI
• Computation time per voxel
– 8.81 ± 1.00ms for LM
– 0.57 ± 0.04ms for LL
– 0.07 ± 0.02ms for ARLO
Results: In Vivo, Whole Liver
Results: In Vivo, Whole Liver
Results: In Vivo, Brain
Results: In Vivo, Brain
Results: In Vivo, Heart
Discussion
• ARLO is more robust than LL to noise with
accuracy as good as LM at 10x the speed of LL
–
–
–
–
Noise is amplified by log-transform
ARLO is a single-variable linear regression, O(N)
LL is a two-variable linear regression, O(6N)
LM is nonlinear LS, O(N3)
• ARLO provides an effective linearization of the
nonlinear estimation problem
– Does not require an initial guess, immune to
convergence issues like in LM
Discussion
• Simpson’s rule much better approximation than
Trapezoidal
– Higher order gave little improvement
• Could also use differentiation but not as good as
integration in low SNR and need finer sampling
• Other applications:
– Other exponential decay models like diffusion, T2, offresonance and T2*
– T1 recovery “from data measured at various timing
parameters such as TR or TI”
• Can also be adapted to multi-exponential fitting
Discussion
• Limitations
– Requires at least 3 data points vs 2 for LM and LL
– Linear sampling of echo times
– Results in minimum T2* of 1.5ms by ARLO
• Probably due to poor protocol
Thoughts
• Nonlinear sampling
– Generally linear sampling is not ideal for experimental
design, are there approximations that don’t require this?
with unequally spaced points (clustered at the endpoints
of the integration interval) are stable and much more
accurate”
• For protocols varying multiple parameters, we would
integrate over multiple dimensions?
– Higher-dimensional integral approximations?
– Simpson’s in each dimension would be a lot of sample
points
Thoughts
• Seems important to have an operation that is
equivalent to a linear combination of the
acquired data
– e.g. integral of exponential is difference of
exponentials
• Consider SPGR:
+2

(1 − 1 ) sin

1 − 1 cos
1 − 1 log 1 cos  − 1
=
1
+2

```