Topics. Spin. The concept of spin Precession of magnetic spin Relaxation Bloch Equation

Transcription

1 Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2005 MRI Lecture 1 Topics The concept of spin Precession of magnetic spin Relaation Bloch Equation Spin Intrinsic angular momentum of elementary particles -- electrons, protons, neutrons. Spin is quantized. Key concept in Quantum Mechanics. 1

2 The History of Spin 1921 Stern and Gerlach observed quantization of magnetic moments of silver atoms 1925 Uhlenbeck and Goudsmit introduce the concept of spin for electrons Stern and Gerlach measure the effect of nuclear spin Rabi predicts and observes nuclear magnetic resonance. Classical Magnetic Moment I A r µ = IAˆ n Energy in a Magnetic Field Maimum Energy State B E = r µ B = µ z B Lorentz Force Minimum Energy State 2

3 Stern-Gerlach Eperiment Image from Force in a Field Gradient F = E = µ z B z z Deflected up Increasing vertical B-field. Deflected down Stern-Gerlach Eperiment Image from 3

4 Quantization of Magnetic Moment The key finding of the Stern- Gerlach eperiment is that the magnetic moment is quantized. That is, it can only take on discrete values. In the eperiment, the finding was that the component of magnetization along the direction of the applied field was quantized: µ z = + µ 0 OR - µ 0 Magnetic Moment and Angular Momentum A charged sphere spinning about its ais has angular momentum and a magnetic moment. This is a classical analogy that is useful for understanding quantum spin, but remember that it is only an analogy! Relation: µ = γ S where γ is the gyromagnetic ratio and S is the spin angular momentum. Quantization of Angular Momentum Because the magnetic moment is quantized, so is the angular momentum. In particular, the z-component of the angular momentum Is quantized as follows: S z = m s h m s { s, (s 1),...s } s is an integer or half intege 4

5 Nuclear Spin Rules Number of Protons Number of Neutrons Spin Eamples Even Even 0 12 C, 16 O Even Odd j/2 17 O Odd Even j/2 1 H, 23 Na, 31 P Odd Odd j 2 H Hydrogen Proton Spin 1/2 S z = +h/2 h/2 +γh /2 µ z = γh /2 Boltzmann Distribution Ε = µ z Β 0 B 0 ΔΕ = γhβ 0 Ε = µ z Β 0 Number Spins Up Number Spins Down = ep(-δe/kt) Ratio = at 1.5T!!! Corresponds to an ecess of about 10 up spins per million 5

6 Equilibrium Magnetization n ( µz ) + n down (µz ) M 0 = N µ z = N up N e µ z B / kt e µ z B / kt e µ z B / kt + e µ z B / kt Nµz2 B /(kt) = Nµ = Nγ 2 h 2 B /(4kT) N = number of nuclear spins per unit volume Magnetization is proportional to applied field. Bigger is better 3T Human imager at UCSD. 7T Rodent Imager at UCSD 7T Human imager at U. Minn. 9.4T Human imager at UIC Gyromagnetic Ratios Nucleus Spin Magnetic Moment γ/(2π) Abundance (MHz/ Tesla) 1H 1/ M 23 Na 3/ mm 31 P 1/ mm Source: Haacke et al., p. 27 6

7 Torque B µ N For a non-spinning magnetic moment, the torque will try to align the moment with magnetic field (e.g. compass needle) Torque N = µ B Precession Torque ds N = µ B = N ds = µ B dµ = µ γb Change in Angular momentum µ = γ S Relation between magnetic moment and angular momentum Precession dµ = µ γb B Analogous to motion of a gyroscope Precesses at an angular frequency of ω = γ Β This is known as the Larmor frequency. dµ µ 7

8 Larmor Frequency ω = γ Β f = γ Β / (2 π) Angular frequency in rad/sec Frequency in cycles/sec or Hertz, Abbreviated Hz For a 1.5 T system, the Larmor frequency is MHz which is million cycles per second. For comparison, KPBS-FM transmits at 89.5 MHz. Note that the earth s magnetic field is about 50 µτ, so that a 1.5T system is about 30,000 times stronger. Notation and Units 1 Tesla = 10,000 Gauss Earth's field is about 0.5 Gauss 0.5 Gauss = T = 50 µt γ = radians/second/gauss γ = γ /2π = 4258 Hz/Gauss = MHz/Tesla Vector sum of the magnetic moments over a volume. M = 1 V Magnetization Vector For a sample at equilibrium in a magnetic field, the transverse components of the moments cancel out, so that there is only a longitudinal component. Equation of motion is the same form as for individual moments. dm µ i protons in V = γm B 8

9 Simplified Drawing of Basic Instrumentation. Body lies on table encompassed by coils for static field B o, gradient fields (two of three shown), and radiofrequency field B 1. Image, caption: copyright Nishimura, Fig RF Ecitation From Levitt, Spin Dynamics, 2001 RF Ecitation At equilibrium, net magnetizaion is parallel to the main magnetic field. How do we tip the magnetization away from equilibrium? Image & caption: Nishimura, Fig. 3.2 B 1 radiofrequency field tuned to Larmor frequency and applied in transverse (y) plane induces nutation (at Larmor frequency) of magnetization vector as it tips away from the z-ais. - lab frame of reference 9

10 a) Laboratory frame behavior of M b) Rotating frame behavior of M Images & caption: Nishimura, Fig RF Ecitation From Buton 2002 Free Induction Decay (FID) 10

11 z RF Ecitation z z M0 y y y Doing nothing Ecitation M 0 (1 e -t/t1 ) z Relaation e -t/t2 y T1 recovery T2 decay Credit: Larry Frank Relaation An ecitation pulse rotates the magnetization vector away from its equilibrium state (purely longitudinal). The resulting vector has both longitudinal M z and tranverse M y components. Due to thermal interactions, the magnetization will return to its equilibrium state with characteristic time constants. T 1 spin-lattice time constant, return to equilibrium of M z T 2 spin-spin time constant, return to equilibrium of M y Longitudinal Relaation dm z = M z M 0 T 1 After a 90 degree pulse M z (t) = M 0 (1 e t /T 1 ) Due to echange of energy between nuclei and the lattice (thermal vibrations). Process continues until thermal equilibrium as determined by Boltzmann statistics is obtained. The energy ΔE required for transitions between down to up spins, increases with field strength, so that T 1 increases with B. 11

12 T1 Values Gray Matter muscle White matter kidney liver fat Image, caption: Nishimura, Fig. 4.2 Transverse Relaation dm y = M y T 2 z y z y z y Each spin s local field is affected by the z-component of the field due to other spins. Thus, the Larmor frequency of each spin will be slightly different. This leads to a dephasing of the transverse magnetization, which is characterized by an eponential decay. T 2 is largely independent of field. T 2 is short for low frequency fluctuations, such as those associated with slowly tumbling macromolecules. T2 Relaation After a 90 degree ecitation M y (t) = M 0 e t /T 2 12

13 T2 Relaation Runners Net signal Credit: Larry Frank T2 Values Tissue T 2 (ms) gray matter 100 white matter 92 muscle 47 fat 85 kidney 58 liver 43 CSF 4000 Table: adapted from Nishimura, Table 4.2 Solids ehibit very short T 2 relaation times because there are many low frequency interactions between the immobile spins. On the other hand, liquids show relatively long T 2 values, because the spins are highly mobile and net fields average out. Eample T 1 -weighted Density-weighted T 2 -weighted Questions: How can one achieve T2 weighting? What are the relative T2 s of the various tissues? 13

14 Bloch Equation dm = M γb M i + M j y ( M z M 0 )k T 2 T 1 Precession Transverse Relaation Longitudinal Relaation i, j, k are unit vectors in the,y,z directions. Free precession about static field dm = M γb ˆ i ˆ j ˆ k = γ M M y M z B B y B z ( ) i ˆ B z M y B y M z = γ ˆ j ( B z M B M z ) k ˆ ( B y M B M y ) Free precession about static field dm dm y dm z B z M y B y M z = γ B M z B z M B y M B M y 0 B z B y = γ B z 0 B B y B 0 M M y M z 14

15 dm dm y dm z Precession 0 B 0 0 = γ B M M y M z Useful to define M M + jm y dm = d ( M + im y ) = jγb 0 M Solution is a time-varying phasor M(t) = M(0)e jγb 0 t = M(0)e jω 0 t M jm y Question: which way does this rotate with time? Matri Form with B=B 0 dm dm y dm z 1/T 2 γb 0 0 = γb 0 1/T /T 1 M M y M z M 0 /T 1 Z-component solution M z (t) = M 0 + (M z (0) M 0 )e t /T 1 Saturation Recovery If M z (0) = 0 then M z (t) = M 0 (1 e t /T 1 ) Inversion Recovery If M z (0) = M 0 then M z (t) = M 0 (1 2e t /T 1 ) 15

16 M M + jm y Transverse Component ( ) dm = d M + im y = j( ω 0 +1/T 2 )M M(t) = M(0)e jω 0 t e t /T 2 Summary 1) Longitudinal component recovers eponentially. 2) Transverse component precesses and decays eponentially. Fact: Can show that T 2 < T 1 in order for M(t) M 0 Physically, the mechanisms that give rise to T 1 relaation also contribute to transverse T 2 relaation. 16