Optimizing Phases of CPMG Pulse Sequence and Applying Exact Solution to Measure Relaxation Time

Transcription

1 Optimizing Phases of CPMG Pulse Sequence and Applying Exact Solution to Measure Relaxation Time Alex Bain 1, Christopher Anand 2, Zhenghua Nie 3 1 Department of Chemistry & Chemical Biology 2 Department of Computing and Software 3 School of Computational Engineering and Science McMaster University May 31, 2011 CORS

2 Outline Motivation Simulating NMR Solution of Single Spin System CPMG Experiments Conclusion

3 Measuring Relaxation Time 2D spectroscopy of the Iα of Protein Kinase A (PKA). (Provided by Dr. Melacini s Lab)

4 Measuring Relaxation Time 2D spectroscopy of the Iα of Protein Kinase A (PKA). (Provided by Dr. Melacini s Lab)

5 Measuring Relaxation Time 2D spectroscopy of the Iα of Protein Kinase A (PKA). (Provided by Dr. Melacini s Lab)

6 Effect of Offset The power cannot be infinity. Some probes require low power; Maximum power output by the amplifier is limited; Sample may boil if the power is too high; If sample is a large molecule, the range of offset frequencies may be wide. For example, ω = γb 1. Offset will significantly affect the measurements.

7 Ideal Experiments

8 Real Experiments

9 Ideal Experiments Spectroscopy (Phase Correction) [SquarePulseGoalMagnet.txt] frequencies (Hz)

10 Real Experiments Spectroscopy (Phase Correction) [SquarePulseOriMagnet.txt] frequencies (Hz)

11 Measuring Relaxation Time Objective: Can we measure the relaxation time of all nuclei at one carrier frequency?

12 Calculation of NMR Using Liouville-von Neumann Equation Vector of observables (ρ) in Liouville space method Evolution of the spin system in Liouville space method d ρ d t d ρ d t = i(l + B)ρ R(ρ ρ eq ) (Inhomogeneous) = ( i(l + B) R)ρ (Homogeneous)

13 Solution of the Liouville Space Method Precession: dρ = ( i(l + B) R)ρ dt Solution of a rectangular pulse: ρ(t) = e ( i(l+b) R)t ρ(0) Solution of a shaped pulse or pulse sequence: ρ(t p ) = ˆL N ˆL 1 ρ(0) with ˆL j = e ( i(l+b j ) R) t j

14 Bloch Equations and Its Solution d dt M x M y M z M e = R 2 ω γb 1 sin φ 0 ω R 2 γb 1 cos φ 0 γb 1 sin φ γb 1 cos φ R 1 R (1) A is used to represent the coefficient matrix. When A is constant, the solution is M(t) = e At M(0). M x M y M z M e

15 CPMG and Its Solution M n = E n M n 1 E n = E fid E π (ϕ n )E fid M n = E n M 0 (2)

16 CPMG and Its Solution (cont.) f (n) = n k=1 ) 2 ( Mx,k 2 + M2 y,k Mx,k M2 y,k 1 f (φ, ω, γ B 1, τ, t p )

17 Mixed Integer Problem to Measure Oscillations min φ {0,1,2,3} max f (φ, ω, γ B 1, τ, t p ) ω/γ B 1 [ 1/2,1/2],τ/t p [0,4] Solution: Maximum difference is of phase 0 or 2. Please note that the solution will depend on the initial magnetization. Here, we suppose the initial magnetization is along with the x axis.

18 Phase Variations of CPMG and Its Solution E = E fid E π,φ4 E fid E fid E π,φ3 E fid E fid E π,φ2 E fid E fid E π,φ1 E fid M n = E n M 0

19 Mixed Integer Problem to Measure Oscillations for Phase Variations min φ 1,,φ K {0,1,2,3} max f (φ 1,, φ K, ω, γ B 1, τ, t p ) ω/γ B 1 [ 1/2,1/2],τ/t p [0,4] Solution: Maximum difference is of using {0, 0, 1, 3} for a group of 4 pulses.

20 Experiments ω/γb 1 = ω/γb 1 = ω/γb 1 = ω/γb 1 =

21 Effective Relaxation Rates of Using {0, 0, 1, 3} for a group of 4 pulses R2 eff 0013 = R 2 + (R 1 R 2 )t p + 1 e (2τ+tp)(R1 R2) µ 2 + O(µ 4 ) 8τ + 4t p 2τ + t p where µ = ω/γ B 1. R 2 = R 2 eff t 4 p(r1 R 2 eff ) 8τ + 3 t p ( 1 e ) 4(2τ+tp) 2 (R1 R 2 eff ) 8τ+3 tp 8τ + 3 t p µ 2 +O(µ 4 )

22 Fitting Problem Subject to: M x d M y dt M z M e min = n M meas (i) I 0 Mx,i 2 + M2 y,i i=1 Unknown Variable: R 2 R 2 ω γb 1 sin φ 0 ω R 2 γb 1 cos φ 0 γb 1 sin φ γb 1 cos φ R 1 R M x M y M z M e

23 Reformed Fitting Problem min Subject to Eq. (2). n M meas (i) I 0 Mx,i 2 + M2 y,i i=1 2

24 Comparison of Second-Order Formula and Exact Fitting R 2 (s -1 ) ω/γb 1

25 Conclution An optimization model is proposed to quantify oscillations of measured intensities of CPMG experiments. Repeating one group of four pulses with different phases can significantly smooth the dependence of measured intensities on frequency offset in the range of ± 1 2 γ B 1 A second-order expression with respect to the ratio of offset to π-pulse amplitude is developed to describe the effective R 2 of CPMG experiments when using a group phase variation scheme. The second-order expression of the effective decay rate with phase variation is able to provide reliable estimates of R 2 when offsets are roughly within ± 1 2 γ B 1. Most significantly, the more sophisticated optimization model using an exact solution of the discretized CPMG experiment extends, to ±γ B 1, the range of offsets for which reliable estimates of R 2 can be obtained when using the preferred phase variation scheme.

26 Conclution (cont.) Objective: Can we measure the relaxation time of all nuclei ( 15 N) of a protein at one carrier frequency? Answer: Yes, we can!

27 Reference 1. Alex D. Bain, Christopher Kumar Anand, Zhenghua Nie, Exact Solution of the CPMG Pulse Sequence with Phase Variation Down the Echo Train: Application to R 2 Measurements, Journal of Magnetic Resonance, 209 (2011) (doi: /j.jmr ) 2. Alex D. Bain, Christopher Anand, Zhenghua Nie, Exact Solution to the Bloch Equations and Application to the Hahn Echo, Journal of Magnetic Resonance, 206 (2010) (doi: /j.jmr )

28 Thanks!

29 Abstract Accurately measuring relaxation time is a key process to investigate the dynamics of molecules. In this presentation, we demonstrate grouped phase variation in CPMG which removes oscillation and field-inhomogeneity effects and the exact CPMG simulation to fit the experiment data which provides reliable relaxation times in a wide range.