Problem Set #6 BioE 326B/Rad 226B

Transcription

1 . Chemical shift anisotropy Problem Set #6 BioE 26B/Rad 226B 2. Scalar relaxation of the 2 nd kind. 0 imaging 4. NMRD curves

2 Chemical Shift Anisotropy The Hamiltonian a single-spin system in a magnetic field subject to both the isotropic part of the chemical shift shielding tensor, σ, and an anisotropic component, Δσ, is given by: Ĥ = Ĥ0 + Ĥ ( t ) where Ĥ 0 = γb 0 ( σ )Îz Ĥ ( t) = γb 0 Δσ 2 and ( F 0 ( t)îz F t 6 ( )Î+ F t 6 ( )Î ) with F 0 (t) = 2 ( cos 2 θ ) and F ± (t) = sinθ cosθe iφ Find: T =? T 2 =?

3 Scalar relaxation of the 2 nd kind Consider a system of J-coupled spins. a. Show: Ĥ = ω I Î z ω S Ŝ z + 2π J = 2 2π J T ( ) 2 S( S +) ) ( ÎzŜz + Î xŝx + Î yŝy) In this case, the T relaxation time of the S spin is very short (T S << /J). One way of analyzing this system is to assume the S spin is in continuous equilibrium with the lattice because of its short relaxation time. By assuming the S spin is part of the lattice, the perturbing Hamiltonian can be rewritten as: Ĥ = S z ( t)îz + S x ( t)î + S x y ( t)î y where S z (t), S x (t), and S y (t) are well modeled as stochastic functions with the following correlation functions: S z ( S x ( t) + is y ( t) ) S x t +τ ( ( t)s z ( t +τ ) = 2π J ) 2 S( S +) ) ( ( ) is y ( t +τ )) = S + t ( ) 2 S( S +) ) = 2π J T 2 ( )S t +τ T 2,S + ( ω I ω s ) 2 2 T 2,S " $ T,S + # e τ T,S ( ) = 2 ( 2π J ) 2 S( S +) ) T 2,S + ( ω I ω s ) 2 2 T 2,S ( ) = % ' & e iω sτ e τ T 2,S Note: the S(S+)/ factor comes from Tr Ŝp 2 S(S +), p = product operator where S = spin of the unpaired electron system or nucleus.

4 T ρ Relaxation by scalar coupling of the 2 nd kind (cont.) We now wish to perform a spin lock experiment and measure relaxation in the rotating frame (T ρ ). D. Spielman H # π & % ( $ 2 ' x On resonance with amplitude γb = ω acquire Spin lock t b) What is the Hamiltonian in the laboratory frame with the spinlock pulse on? c) What is the Hamiltonian in a frame of reference rotating around the z axis at a frequency ω 0? d) What is the Hamiltonian in the doubly rotating frame (i.e. also rotating around the x axis at a frequency ω )? e) Find an expression for the relaxation superoperator in the doubly rotating frame. Hint: the following transformation may help simplify your result: z! = x, y! = z, and x! = y. f) Assuming an exponential correlation time for the perturbation fields of τ c, find an expression for /T ρ ( longitudinal relaxation rate in the doubly rotating frame). g) What is /T ρ in the limit of ω = 0?

5 O imaging. A research team decides to measure in vivo levels of H 2 O ( O is a spin 5/2 nuclei with a natural abundance of 0.07%) using an indirect detection approach in which serial T 2 -weighted echo planar images are acquired with the decoupler power alternately on and off every eighth image. The pulse sequence diagram is shown below. H O

6 O imaging (cont.) The following in vitro and in vivo data are obtained. In particular, the data show: (a) signal time courses during a serial decoupling experiment for four tubes of water and one tube of acetone. Baseline T 2 -weighted EPI images of the tubes are shown on the left. Next to these are maps of the correlation coefficient between the signal timecourse for each pixel and the decoupler power waveform. The plots are from retangular ROIs indicated on the T 2 -weighted images. (b) in vivo data from normal and ischemic rat brain. a acetone

7 O imaging (cont.) b Rat brain Give a theoretical explaination for the observed data. In particular: a) Why does image signal intensity increase when the decoupler in on and why does the effect increase with increasing O fractional enrichment? b) Why does the effect dissapear at low ph (ph = 4) or with the use of acetone instead of water? c) Give an explaination for the differential response between normal and ischemic brain. d) Discuss potential applications for H 2 O imaging. e) A graduate student proposes to improve the sensitivity of the method by turning on the decoupler prior to the 90 o excitation (see pulse sequence diagram on next page) in order to exploit the Nuclear Overhauser Effect (NOE). Will this change the sensitivity and by what factor?

8 O imaging (cont.) O Rf dec. H

9 NMRD Curves D. Spielman Plot T relaxivity NMRD curves for Gd-DTPA and Gd-DTPA bound to serum albumin.