Spin. Nuclear Spin Rules

Transcription

1 Spin Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 203 MRI Lecture Intrinsic angular momentum of elementary particles -- electrons, protons, neutrons. Spin is quantized. Key concept in Quantum Mechanics. Magnetic Moment and Angular Momentum Nuclear Spin Rules A charged sphere spinning about its axis 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. Number of Protons Number of Neutrons Spin Examples Even Even 0 2 C, 6 O Even Odd j/2 7 O Odd Even j/2 H, 23 Na, 3 P Odd Odd j 2 H

2 Classical Magnetic Moment Energy in a Magnetic Field Maximum Energy State I A µ = IAˆ n B E = µ B = µ z B Lorentz Force Minimum Energy State Energy in a Magnetic Field Magnetic Field Units Tesla = 0,000 Gauss Earths field is about 0.5 Gauss 0.5 Gauss = 0.5x0-4 T = 50 µt Earth-magnetic-field.jpg 2

3 Boltzmann Distribution Equilibrium Magnetization M0 = N µ z Nµz2 B /(kt) = Nγ 2 2 B /(4kT) B0 UCSD 3T.7T NIH U. Minn. 7T N = number of nuclear spins per unit volume Magnetization is proportional to applied field. Hansen 2009 UIC 9.4T.7T NeuroSpin France 3

4 MRI System Nova 32 channel Siemens 32 channel MRI Gradients Torque B µ For a non-spinning magnetic moment, the torque will try to align the moment with magnetic field (e.g. compass needle) N N = µ x B Torque 4

5 Precession Precession Torque ds dt N = µ x B = N Change in Angular momentum ds dt = µ x B µ = γ S dµ dt = µ x γb dµ dt = µ x 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 Vector sum of the magnetic moments over a volume. Magnetization Vector RF Excitation 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. M = V protons in V µ i dm dt = γm B Hansen

6 Free precession about static field Free precession about static field dm dt = M γb ˆ i ˆ j ˆ k = γ M x M y M z B x B y B z ( ) % i ˆ B z M y B y M ( z * = γ ˆ j ( B z M x B x M z )* k ˆ & ( B y M x B x M y ) * ) B dμ Μ " dm x $ $ dm y # $ dm z dt% " B z M y B y M z % $ dt = γ$ B x M z B z M x dt& # $ B y M x B x M y & " 0 B z B y %" $ $ = γ$ B z 0 B x $ # $ B y B x 0 & # $ M x M y M z % & " dm x $ $ dm y # $ dm z Precession dt% " 0 B 0 0% " $ $ dt = γ $ B $ dt& # $ 0 0 0& # $ M x M y M z % & Gyromagnetic Ratios Nucleus Spin Magnetic Moment γ/(2π) (MHz/Tesla) Abundance Useful to define M M x + jm y jm y H / M dm dt = d dt( M x + im y ) = jγb 0 M M x 23 Na 3/ mm Solution is a time-varying phasor 3 P / mm M(t) = M(0)e jγb 0t = M(0)e jω 0t Question: which way does this rotate with time? Source: Haacke et al., p. 27 6

7 Larmor Frequency Notation and Units ω = γ Β f = γ Β / (2 π) Angular frequency in rad/sec Frequency in cycles/sec or Hertz, Abbreviated Hz Tesla = 0,000 Gauss Earths field is about 0.5 Gauss 0.5 Gauss = 0.5x0-4 T = 50 µt For a.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.5t system is about 30,000 times stronger. γ = radians/second/gauss γ = γ /2π = 4258 Hz/Gauss = MHz/Tesla Gradients MRI System Spins precess at the Larmor frequency, which is proportional to the local magnetic field. In a constant magnetic field B z =B 0, all the spins precess at the same frequency (ignoring chemical shift). Gradient coils are used to add a spatial variation to B z such that B z (x,y,z) = B 0 +Δ B z (x,y,z). Thus, spins at different physical locations will precess at different frequencies. 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. Image, caption: copyright Nishimura, Fig

8 Interpretation Gradient Fields B z (x)=g x x B z (x, y,z) = B 0 + B z x x + B z y y + B z z z x z = B 0 + G x x + G y y + G z z y Spins Precess at γb 0 - γg x x (slower) Spins Precess at γb 0 Spins Precess at at γb 0 + γg x x (faster) G z = B z z > 0 G y = B z y > 0 Rotating Frame of Reference Spins Reference everything to the magnetic field at isocenter. There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. Erwin Hahn 8

9 Phasors θ = 0 θ = π /2 θ = π θ = π /2 G(k x ) = Phasor Diagram ( ) g(x)exp j2πk x x dx θ = 2πk x x Imaginary Real θ = 2πk x x % % 0 ( ( exp j2π * x* & & 8Δx ) ) % % ( ( exp j2π * x* & & 8Δx ) ) Interpretation -2 x - x 0 x 2 x k x =; x = 0 2πk x x = 0 x =/4 2πk x x = π /2 x =/2 2πk x x = π x = 3/2 2πk x x = 3π /4 % % 2 ( ( exp j2π * x* & & 8Δx ) ) θ = 0 θ = π /2 θ = π θ = π /2 Slower B z (x)=g x x Faster 9

10 k x =0; k y =0 k x =0; k y 0 Fig 3.2 from Nishimura Hanson 2009 k-space Image space k-space y k y x Fourier Transform k x 0

11 Hanson D Fourier Transform Fourier Transform G(k x,k y ) = F[ g( x,y) ] = g(x, y) ( ) dxdy e j 2π k x x +k y y Inverse Fourier Transform g(x, y) = G(k x,k y ) ( ) dkx dk y e j 2π k xx +k y y Hanson 2009

12 Seeley et al, JMR 2004 Seeley et al, JMR 2004 y (a) (b) x (c) (d) (e) PollEv.com/be280a k y k x 2