Fisher Information Matrix of DESPOT

1. Fisher Information Matrix of DESPOT Dec 17, 2012

2. Theory • The Fisher Information Matrix (FIM) is a key part of the Cramer-Rao Lower Bound (CRLB) • It is a measure of how sensitive the output signals (SPGR and SSFP images) are to the input parameters (tissue characteristics: T1s, T1f, MWF, etc.) • For an unbiased estimator, as is assumed in Lankford with a genetic algorithm, the FIM completely defines the CRLB

3. Theory • gi= the signal equation of the ith image • θj = the jth tissue parameter • Σ = the added noise correlation matrixscaled as per Lankford, 1e-4*M0 for SPGR and sqrt(3)*e-4*M0 for SSFP, assumed diagonal/independent • σθj = CRLB precision bound on jth parameter for an unbiased estimator

4. The Jacobian • Numerical estimation of the Jacobian matrix is performed for one of the tissues specified in Lankford (which comes from Gleaning) • Sample tissue = X • {'T1s','T1f','T2s','T2f','fF','kFS','offResonance','M0'}; • {800 450 100 20 20 10 0 100e3} • With 7 (SPGR) + 9 (SSFP phase 0) + 9 (SSFP phase 180) = 25 images, the resulting size is 25x8 • Looking at the Jacobian itself seems like it would be useful because it gives a feeling for both the sensitivity and specificity of a sequence to a tissue parameter • Intrinsic signal level difference between sequences makes it difficult to compare, tried to account for this by normalizing by the signal value

5. Numerical Estimation of Partial Derivatives • There are a few ways to compute partial derivatives numerically • Usually a constant step size, h, is assumed • 2-point, 3-point, 5-point methods are then defined by calculating gi(X;θ) systematically over a discrete range of θ values • For example for the 5-point method, θj±nk*h, where nk = 0,1,2 • I compute the two simplest forms • Forward difference = g(X; θj+h) - g(X; θj)/h • Central difference = g(X; θj+h) - g(X; θj-h)/2h • Forward difference error ∝ h, central difference error ∝ h2 • Choosing h is a balancing act • We want it as small as possible • But don’t want to run into machine precision rounding errors

6. Step Size Importance h = 1e-4 h = 1e-9

7. Step Size Importance h = 1e-12 If you pick too small of a step size, you can run into precision errors This is an obvious case of that, at h = 1e-11, it’s more subtle In general, should use “good” region of h where values are accurate (small h) and converge between estimation methods Should see similar J across a good range of h

8. Step Size Importance – Rates h = 1e-4 (Lankford) h = 1e-9

9. Methods • h = 1e-9 for all following slides • Several protocols are compared: • mcDESPOT 2pc (7 SPGR angles and 9+9 SSFP angles with phase 0 and phase 180 cycling) • mcDESPOT 2pc mean norm – mean normalized version of the above within each subtype (SPGR, SSFP ph0, SSFP ph180) • mcDESPOT1pc – no phase 0 images, relevant because Lankford noticed that adding phase 0 significantly improved the precision • 3pc – add 9 more SSFP phase 90 images

10. 2pc: Jacobian

11. 2pc: J/S = Jacobian/Signal

12. 2pc: log(abs( J/S ))

13. 2pc mean norm: log(abs( J/S ))

14. 3pc: log(abs( J/S ))

15. Unbiased Estimator, CRLB

16. eig(F)

17. Rate – 2pc: log(abs( J/S ))

18. Rate – 2pc: log(abs( J/S ))

19. Rate – 2pc mean norm: log(abs( J/S ))

20. Rate – Estimator, CRLB