Aug 18, 2014 Jason Su

1. Aug 18, 2014 Jason Su

2. 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!

3. 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 a 3-pt. quadratic interpolation of the interval • Newton-Cotes formulas:

4. 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) • Assuming decay curve sampled linearly at intervals

5. Theory • An auto-regressive time-series • Find T2* to minimize the error between model and data,

6. 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

7. 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 = 1.3-23.3ms (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 echoes), heart with iron overload • Manual segmentation of liver and brain structures • Statistical • Linear regression, Bland-Altman, and t-tests

8. 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

9. 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

10. Results: In Vivo, Whole Liver

11. Results: In Vivo, Whole Liver

12. Results: In Vivo, Brain

13. Results: In Vivo, Brain

14. Results: In Vivo, Heart

15. 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

16. 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, off-resonance and T2* • T1 recovery “from data measured at various timing parameters such as TR or TI” • Can also be adapted to multi-exponential fitting

17. 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

18. Thoughts • Nonlinear sampling • Generally linear sampling is not ideal for experimental design, are there approximations that don’t require this? • “Gaussian quadrature and Clenshaw–Curtis quadrature 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

19. 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: