Topics. The History of Spin. Spin. The concept of spin Precession of magnetic spin Relaxation

Transcription

1 Topics Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2008 MRI Lecture 1 The concept of spin Precession of magnetic spin Relaation Spin The History of Spin Intrinsic angular momentum of elementary particles -- electrons, protons, neutrons. Spin is quantied. Key concept in Quantum Mechanics Stern and Gerlach observed quantiation 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. 1

2 Classical Magnetic Moment Energy in a Magnetic Field Maimum Energy State I A r µ = IAˆ n B E = " r µ B = "µ B Lorent Force Minimum Energy State Stern-Gerlach Eperiment Force in a Field Gradient F = "#E = µ $B $ Deflected up Increasing vertical B-field. Deflected down Image from 2

3 Stern-Gerlach Eperiment Quantiation of Magnetic Moment The key finding of the Stern- Gerlach eperiment is that the magnetic moment is quantied. That is, it can only take on discrete values. In the eperiment, the finding was that the component of magnetiation along the direction of the applied field was quantied: µ = + µ 0 OR - µ 0 Image from 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. Quantiation of Angular Momentum Because the magnetic moment is quantied, so is the angular momentum. In particular, the -component of the angular momentum Is quantied as follows: S = m s h m s "{#s,#(s #1),...s } s is an integer or half intege 3

4 Nuclear Spin Rules Hydrogen Proton Number of Protons Even Even Odd Odd Number of Neutrons Even Odd Even Odd Spin 0 j/2 j/2 j Eamples 12 C, 16 O 17 O 1 H, 23 Na, 31 P 2 H Spin 1/2 # S = $ +h/2 %& "h/2 # +'h /2 µ = $ %& "'h /2 Magnetic Field Units 1 Tesla = 10,000 Gauss Earth's field is about 0.5 Gauss 0.5 Gauss = T = 50 µt Boltmann Distribution Ε = µ Β 0 B 0 ΔΕ = γhβ 0 Ε = µ 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 4

5 Bigger is better Equilibrium Magnetiation # n (µ ) " n down (µ ) & M 0 = N µ = N % up ( N $ ' e µ B /( kt ) " e" µ B /( kt ) e µ B /( kt ) + e" µ B /( kt ) 2 ) Nµ B /(kt) = Nµ 3T Human imager at UCSD. 7T Human imager at U. Minn. = N* 2 h 2 B /(4kT)! N = number of nuclear spins per unit volume Magnetiation is proportional to applied field. 7T Rodent Imager at UCSD 9.4T Human imager at UIC Gyromagnetic Ratios Torque B Nucleus Spin Magnetic γ/(2π) Abundance Moment (MH/Tesla) µ 1H 1/ M 23Na 3/ mm N 31P 1/ mm N=µB For a non-spinning magnetic moment, the torque will try to align the moment with magnetic field (e.g. compass needle) Torque Source: Haacke et al., p. 27 5

6 Precession Precession Torque N = µ B ds = N Change in Angular momentum ds = µ B µ = γ S dµ = µ γb dµ = µ γb B dµ µ Analogous to motion of a gyroscope Precesses at an angular frequency of ω = γ Β This is known as the Larmor frequency. Relation between magnetic moment and angular momentum Larmor Frequency Notation and Units ω = γ Β f = γ Β / (2 π) Angular frequency in rad/sec Frequency in cycles/sec or Hert, Abbreviated H 1 Tesla = 10,000 Gauss Earth's field is about 0.5 Gauss 0.5 Gauss = T = 50 µt For a 1.5 T system, the Larmor frequency is MH which is million cycles per second. For comparison, KPBS-FM transmits at 89.5 MH. Note that the earth s magnetic field is about 50 µτ, so that a 1.5T system is about 30,000 times stronger. " = radians/second/gauss " = " /2# = 4258 H/Gauss = MH/Tesla 6

7 Recap Spins: angular momentum and magnetic moment are quantied. Spins precess about a static field at the Larmor frequency. In MRI we work with the net magnetic moment. In the presence of a static field and non-ero temperature, the equilibirum net magnetic moment is aligned with the field (longitudinal), since transverse components cancel out. We will use an radiofrequency pulse to tip this longitudinal component into the transverse plane. Vector sum of the magnetic moments over a volume. M = 1 V Magnetiation 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 " protons in V µ i = "M # B RF Ecitation 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 From Levitt, Spin Dynamics,

8 RF Ecitation At equilibrium, net magnetiaion is parallel to the main magnetic field. How do we tip the magnetiation away from equilibrium? RF Ecitation 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 magnetiation vector as it tips away from the -ais. - lab frame of reference RF Ecitation a) Laboratory frame behavior of M b) Rotating frame behavior of M Images & caption: Nishimura, Fig From Buton

9 Free Induction Decay (FID) RF Ecitation M0 y y y Doing nothing Ecitation M 0 (1 e -t/t1 ) Relaation e -t/t2 y T1 recovery T2 decay Credit: Larry Frank Relaation Longitudinal Relaation An ecitation pulse rotates the magnetiation vector away from its equilibrium state (purely longitudinal). The resulting vector has both longitudinal M and tranverse M y components. dm = " M " M 0 T 1 Due to thermal interactions, the magnetiation will return to its equilibrium state with characteristic time constants. After a 90 degree pulse M (t) = M 0 (1" e "t /T 1 ) T 1 spin-lattice time constant, return to equilibrium of M T 2 spin-spin time constant, return to equilibrium of M y Due to echange of energy between nuclei and the lattice (thermal vibrations). Process continues until thermal equilibrium as determined by Boltmann 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. 9

10 T1 Values Transverse Relaation Gray Matter muscle White matter dm y = " M y T 2 y y y kidney liver fat Each spin s local field is affected by the -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 magnetiation, which is characteried 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. Image, caption: Nishimura, Fig. 4.2 T2 Relaation T2 Relaation Runners Net signal After a 90 degree ecitation M y (t) = M 0 e "t /T 2 Credit: Larry Frank 10

11 T2 Values Eample 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. 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? 11