MR Fundamentals. 26 October Mitglied der Helmholtz-Gemeinschaft

Transcription

1 MR Fundamentals 26 October 2010 Mitglied der Helmholtz-Gemeinschaft

2 Mitglied der Helmholtz-Gemeinschaft Nuclear Spin

3 Nuclear Spin Nuclear magnetic resonance is observed in atoms with odd number of protons and/or neutrons (possess spin angular momentum) Spin angular momentum S is an intrinsic vector property of elementary particles S Associated with S is a magnetic dipole moment μ = I I = spin operator μ = γ S= γ I γ = gyromagnetic ratio Classical picture: charged sphere spinning about its axis like a planet Slide 3

4 Spin Dynamics Classical Description Mitglied der Helmholtz-Gemeinschaft

5 Spins at B 0 = 0 Spins have random orientation Vectorsum over spins is zero Hanson 2008 No net magnetisation Slide 5

6 Spins in a Static Field Instantaneous precession of spins around axis of external field at the Larmor frequency ω = γ B 0 0 Spin distribution skews slightly towards magnetic north pole through relaxation net magnetisation along direction of magnetic field is formed Hanson 2008 Slide 6

7 Hanson 2008 Equilibrium Magnetisation How large is the equilibrium magnetisation? Energy dependent on angle, between field and magnetic moment μ: E( θ ) = μb0 cosθ Proton magnetic moment: μ Angular spin distribution: P( θ ) = E( θ ) kt Z-component of magnetisation Slide 7 pi 0 π 0 e e E kt ( θ ) sin( θ ) dθ μz = P( θ)( μcosθ)sinθdθ μbu 0 1 udu kt e 1 μz = μ μbu 0 1 du kt e γ B 4kT 0 θ, = 3 4 γ

8 Interaction with a Radio Frequency Field Hanson 2008 A homogeneous magnetic field B 1 in the transverse direction rotating at the Larmor frequency is applied The B 1 field induces torque on the magnetisation Leads to coherent (no spin position change relative to one another) rotation of the spin distribution Net magnetisation is rotated into the transverse plane Slide 8

9 Bloch Equation Without Relaxation The magnetic field causes a torque, T, on a magnetic moment, μ. T = μ B The torque is the 1 st derivative of the angular momentum, I, with respect to time. di 1 dμ T = = dt γ dt Ensemble averaging from the microscopic magnetic moments to the macroscopic magnetisation: dm dt = M γ B = ω M Equation of motion* solved by precessional motion with frequency ω=-γb. (clockwise). * Bloch-Equation without relaxation Slide 9

10 Spin Dynamics Quantum Mechanics Mitglied der Helmholtz-Gemeinschaft

11 Zeeman Effect Stern Gerlach Experiment Classical prediction 0 F = ( μ B ges ) = 0 B μz z Observed: two discrete levels J z =± Slide

12 Zeeman Effect In absence of field: degenerate states (multiple quantum states with identical energy) External magnetic field breaks degeneracy (each sublevel at different energy) Atomic level Spin dynamics, Levitt, p. 6 & 14 Nuclear Zeeman effect Slide 12

13 Zeeman Effect H = μ B = γ B0I z Eigenvalues are give by interaction energy of the nucleus E = γ B m with m= I, I 1,..., I 0 For Protons I= 1/2, thus two energy levels E E -½ = γ B0 = ω E ½ = γ B0 = ω0 m = -½ m = ½ ΔE Wikipedia Slide 13

14 P = e e E kt Equilibrium Magnetisation Reminder: using the density operator ρ the expectation value <A> of an operator A can be expressed as A = Tr( ρ A) Diagonal elements P, P of the density operator express the relative spin populations In equilibrium they are given by the Boltzmann distribution γ B 2kT E E γb γb kt kt 2kT 2kT + e = e e e P = e e E kt γ B 2kT E E γb γb kt kt 2kT 2kT The net longitudinal magnetisation is the observable population difference between Zeeman eigenstates given by 2 2 γ γ B ( ) 0 μz = P P 2 4kT + e = e e e Slide 14

15 Equation of Motion of the Expectation Value In QM coherent evolution can be described using the density operator ρ and the Liouville equation ρ 1 = [ H, ρ] t i For the spin-1/2 system we get the following density matrix ρ 2 * c c c = * 2 c c c Expectation value of the spins x component is given by μ x γ μx = Trace( μρ x ) = ( ρ + ρ ) 2 Slide 15

16 Equation of Motion of the Expectation Value Liouville equation gives us μx γ ρ ρ γ = i + i = ([ H, ρ] + [ H, ρ] ) t 2i t t 2i γ = (( H H )( ρ ρ ) + ( ρ ρ )( H H )) 2i Remember we have the dipolar Hamiltonian H = μ B μ ( ) 2 y z ( ) x B ρ ρ B ρ ρ = γ + = γby μz + γbz μy t 2 2i Slide 16

17 Equation of Motion of the Expectation Value Cyclic permutation gives us all three components μx = γ By μz + γbz μy t μ y = γ Bz μx + γb x μz t μz = γ Bx μy + γby μx t μ = γ μ B t This is remarkably similar to the classical Bloch equation without relaxation dm = γ μ B dt Slide 17

18 Equation of Motion of the Expectation Value For large populations of non interacting spins QM and classical mechanics give identical results Magnetic resonance is a good example of the correspondence principle Quantum effects are visible when individual spins are considered or spin-spin interactions occur Slide 18

19 Mitglied der Helmholtz-Gemeinschaft Relaxation

20 Relaxation Following an excitation pulse the magnetisation returns to its equilibrium state The longitudinal magnetisation regains its equilibrium value M 0 The transverse magnetisation (the measurable magnetisation component) returns to zero Stirnberg Slide 20

21 Longitudinal Relaxation Upon perturbation the longitudinal magnetisation returns to equilibrium by exchanging energy with the surrounding tissue (lattice) Time evolution is given by Solution: dm dt = z M z M T M = M + ( M (0) M ) e z 0 z t T 1 M 0 = longitudinal equilibrium T 1 magnetisation = spin-lattice relaxation time constant Assuming a 90 excitation pulse at t=0; M z (0)=0: M z = M0 1 e t T 1 Stirnberg Slide 21

22 Longitudinal Relaxation Energy exchange with surrounding tissue (mostly water and fat) requires interaction at or near resonance condition Energy transfer via oscillating field resulting from movement of the spins molecule (molecular tumbling) Possible motions: rotation, vibration and translation With respect to T 1 three states of water are of interest: Free water; unbound structured water; bound to macromolecule with single bond (retains rotational freedom) bound water; bound to macromolecule with two bonds BPP Theory (Bloembergen et al. 1948) Slide 22

23 Transverse Relaxation Transient and random local field fluctuations occur on atomic level through spin-spin interactions Leads to irreversible loss of phase coherence between the spins The time evolution of the transverse magnetisation is described by dm xy M xy = dt T2 Assuming a 90 pulse at t=0 and M xy (0)=M 0 the solution is T 2 = spin-spin relaxation time constant M xy = Me o t T 2 Stirnberg Slide 23

24 Bloch Equation with Relaxation Adding the relaxation terms to our equation of motion we arrive at the phenomenological Bloch equation which completely describes our spin system d M Mx ( Mz M0) = M γ B i k dt T T 2 1 precession transversal (spin-spin) relaxation γ T1 T2 M 0 ijk,, longitudinal (spin-lattice) relaxation = gyromagnetic ratio = spin-lattice relaxation time = spin-spin relaxation time = equilibrium magnetisation = unit vectors in x,y,z Slide 24

25 Free Induction Decay (FID) A 90 excitation pulse is applied at time t=0 The received MR signal decays exponentially with the time constant T 2 At the same time the longitudinal magnetisation (M z ) recovers Stirnberg Slide 25

26 T 2* Decay Spin-spin interaction lead to irreversible loss of phase coherence between the spins (T 2 -decay) B 0 inhomogeneities and susceptibility effects cause additional shortening with time constant T 2 (reversible with spin-echo) Additional shortening of decay constant to T 2 * given by = + * ' T T T Slide 26 Stirnberg

27 Mitglied der Helmholtz-Gemeinschaft Important Effects

28 Spin Echo τ τ irreversible e 2τ T 2 reversible Slide 28

29 Spin Echo Equilibrium 90 RF Dephasing More Dephasing 180 RF Rephasing Echo! Slide 29

30 Chemical Shift Small displacement of the resonance frequency due to shielding by local electronic environment Effective field experienced by the nucleus is given by B (1 ) eff = B0 B0 σ = B0 σ ω (1 ) eff = ω0 ω0 σ = ω0 σ Defined relative to reference frequency ω ω σ σ δ = 10 = 10 ( σ σ ) 10 ωs S R 6 R S 6 6 R S ωr 1 σr σ = environment dependent shielding constant = reference frequency = resonance frequency of sample δ = chemical shift in parts per million (ppm) ω R Slide 30

31 FID Chemical Shift NMR spectroscopy: acquire FID and Fourier transform Fourier Transform Spectrum σ time ω R 1 H spectrum of Methanol (CH 3 OH) Frequency ω S Peak area ratio 1:3 Lorentzian line shape G( ) due to T 2 decay, centered at frequency G( ω) ω 0 = T2 1 + ( ω ω ) T ω Magnete Spins und Resonanzen, p.75, Siemens Slide 31

32 Chemical Shift 1 H spectrum of the brain Faria et al. 2004; doi: /s x Slide 32

33 Chemical Shift 31 P spectrum ph Dependence of the Inorganic Phosphate Peak p.p.m. relative to PCr ph Gadian 1982, p.32 Noninvasive ph measurement! Friebolin Ein- und zweidimensionale NMR-Spketroskopie, p. 358, 1999 Slide 33

34 Chemical Shift Chemical shift artefact in MRI (fat/water) Frequency is used for spatial encoding in MRI Chemical shift changes frequency -> erroneous encoding CS good for spectroscopy Not so good for imaging -> fat suppression Slide 34

35 J-Coupling Indirect coupling between two nuclear spins of a molecule through involvement of bonding electrons Observable in NMR spectroscopy Leads to splitting of lines into multiplets in molecular spectra Independent of magnetic field (opposed to chemical shift) Quantum effect (not explainable within classical mechanics) Coupling between nuclear pins I j and I k described by 3x3 J- coupling tensor J jk (reduces to scalar coupling in isotropic liquids) H = 2π I j J jk Ik Magnete Spins und Resonanzen, p.158, Siemens Slide 35

36 Additional References Textbooks Principles of Magnetic Resonance Imaging, D.G. Nishimura, Stanford University, 2010 Magnetic Resonance Imaging: Phsical Principles and Sequence Design, Haacke, Brown and Vankatesan, Wiley, 1999 Principles of Magnetic Resonance, C.P. Slichter, Springer-Verlag Berlin heidelberg New York, 1978 Papers Is Quantum Mechanics Necessary for Understanding Magnetic Resonance?, L.G. Hanson, Conc. Magn. Res. A, Vol.32A(5), pp , 2008 Nuclear Induction, F. Bloch, Phys. Rev., Vol. 70(7,8), 1946 Spin Echoes, E.L. Hahn, Phys. Rev., Vol. 80(4), pp , 1950 Relaxation Effects in Nuclear Magnetic Resonance Absorption, Bloembergen, Pound, Purcel, Phys. Rev., Vol. 73, p. 679, 1948 Slide 36