Torque Precession Precession
B
Bioengineering 280A µ Principles of Biomedical Imaging For a non-spinning magnetic moment, the torque will try to align the moment with magnetic field (e.g. compass needle) Fall Quarter 2012 MRI Lecture 2 N
N = µ x B Torque
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
Precession Precession Analogous to motion of a gyroscope dµ Torque = µ x γB dt Precesses at an angular frequency of N = µ x B B dS ω = γ Β = µ x B dS dt = N dµ This is known as the Larmor frequency. dt = µ x γB dt dµ µ Change in µ = γ S Angular momentum
Relation between magnetic moment and angular momentum
http://www.astrophysik.uni-kiel.de/~hhaertełmpg_e/gyros_free.mpg TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
1 Magnetization Vector RF Excitation Vector sum of the magnetic moments over a volume. 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. 1
M = ∑µi V protons in V dM = γM × B dt Hansen 2009 http://www.drcmr.dk/main/content/view/213/74/ € TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
€
Free precession about static field Free precession about static field
dM B = M × γB " % " dMx dt% Bz My − By Mz dt $ ' $ ' dM dt = γ B M − B M iˆ ˆj kˆ $ y ' $ x z z x ' dΜ $ ' #$ dMz dt&' #B y Mx − Bx My& = γ Mx My Mz Μ " % B B B 0 Bz −By " Mx% x y z $ '$ ' = γ$− Bz 0 Bx ' My % iˆ B M − B M ( $ ' ' ( z y y z ) * $ B −B 0 '$ M ' ˆ # y x &# z & = γ' − j (Bz Mx − Bx Mz )* ' ˆ * & k (By Mx − Bx My ) ) € TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012 €
2 Precession Gyromagnetic Ratios
" dMx dt% " 0 B0 0%" Mx% $ ' $ '$ ' dM dt B 0 0 M $ y ' = γ$− 0 '$ y' Nucleus Spin Magnetic γ/(2π) Abundance #$ dMz dt&' #$ 0 0 0&'#$ Mz &' Moment (MHz/Tesla)
1H 1/2 2.793 42.58 88 M Useful to define M ≡ Mx + jMy jMy € Mx 23 dM dt = d dt(Mx + iMy ) Na 3/2 2.216 11.27 80 mM
€ = − jγB0 M 31P 1/2 1.131 17.25 75 mM Solution is a time-varying phasor M(t) = M(0)e− jγB0t = M(0)e− jω 0t €
Question: which way does this rotate with time? Source: Haacke et al., p. 27 TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012 €
Larmor Frequency Notation and Units
ω = γ Β Angular frequency in rad/sec 1 Tesla = 10,000 Gauss Earth's field is about 0.5 Gauss f = γ Β / (2 π) Frequency in cycles/sec or Hertz, Abbreviated Hz 0.5 Gauss = 0.5x10-4 T = 50 µT
For a 1.5 T system, the Larmor frequency is 63.86 MHz which is 63.86 million cycles per second. For comparison, γ = 26752 radians/second/Gauss KPBS-FM transmits at 89.5 MHz. γ = γ /2π = 4258 Hz/Gauss Note that the earth’s magnetic field is about 50 µΤ, so that a 1.5T system is about 30,000 times stronger. = 42.58 MHz/Tesla
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012 €
3 Gradients MRI Gradients
Spins precess at the Larmor frequency, which is proportional to the local magnetic field. In a constant
magnetic field Bz=B0, all the spins precess at the same frequency (ignoring chemical shift).
Gradient coils are used to add a spatial variation to Bz such that Bz(x,y,z) = B0+Δ Bz(x,y,z) . Thus, spins at different physical locations will precess at different frequencies.
http://www.magnet.fsu.edu/education/tutorials/magnetacademy/mri/fullarticle.html
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
Gradient Fields Interpretation
∂Bz ∂Bz ∂Bz B (x,y,z) = B + x + y + z ∆Bz(x)=Gxx z 0 ∂x ∂y ∂z = B + G x + G y + G z z 0 x y z x
y
€ Spins Precess Spins Precess at γB0 Spins Precess at at γB0- γGxx M(t) = M(0)e− jγB0t at γB0+ γGxx (slower) (faster) − jω0t ∂Bz ∂B = M(0)e G 0 z − jγ(B −G x)t − jγ(B +G x)t z = > G 0 0 x 0 x ∂z y = > M(t) = M(0)e M (t) = M(0)e ∂y − j(ω −Δω )t − j(ω +Δω )t = M(0)e 0 = M(0)e 0 TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
€ € € € 4 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
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
Phasor Diagram Interpretation ∞ Imaginary G(kx ) = ∫ g(x)exp(− j2πkx x)dx -2∆x -∆x 0 ∆x 2∆x −∞ % % 0 ( ( θ = −2πkx x exp' − j2π' * x* & & 8Δx ) ) Real
€ θ = −2πkx x € % % 1 ( ( exp' − j2π' * x* & & 8Δx ) ) € kx =1; x = 0 x =1/4 x =1/2 x = 3/2
2πk x = 0 2πkx x = π /2 2πkx x = π 2πkx x = 3π /4 € % % 2 ( ( x € exp' − j2π' * x* & & 8Δx ) )
€ € € € € ∆B (x)=G x Faster θ = 0 θ = −π /2 θ = π θ = π /2 Slower z x TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
€ € € €
5 kx=0; ky=0 kx=0; ky≠0
Fig 3.12 from Nishimura TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012 Hanson 2009
Phase with time-varying gradient
Hanson 2009 TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
6 K-space trajectory Gx(t)
ky
t ky (t4 ) t1 t2 Gy(t) ky (t3 ) €
kx t3 t4 k (t ) k (t ) x €1 x 2
€ €
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012 Nishimura 1996
K-space trajectory Gx(t)
ky t t1 t2 Gy(t)
kx
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
7 Spin-Warp Gx(t) k-space
ky Image space k-space
t1
Gy(t)
k x y ky
x k Fourier Transform x
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
k-space
ky
kx
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
8 RF Spin-Warp Pulse Sequence Spin-Warp Gx(t)
Gx(t) t1 ky
Gy(t) Gy(t)
ky Gy(t)
kx
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
(a) (b) y 1 0 1
0 1 0 1 0 1
x
(c) (d) (e)
Hanson 2009 TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
9 Gradient Fields Static Gradient Fields Define In a uniform magnetic field, the transverse magnetization is given by: ˆ ˆ − jω t −t /T G ≡ G iˆ + G ˆj + G k r ≡ xiˆ + yˆj + zk 0 2 x y z M(t) = M(0)e e So that In the presence of non time-varying gradients we have G x + G y + G z = G ⋅ r x y z € € € − jγB ( r )t Also, let the gradient fields be a function of time. Then M(r ) = M(r ,0)e z e− t /T2 ( r ) the z-directed magnetic field at each point in the M(r ,0)e− jγ ( B0+G ⋅r )te− t /T2 ( r ) = − jω t − jγG ⋅r t − t /T ( r ) € volume is given by : = M(r ,0)e 0 e e 2 B (r ,t) B G (t) r z = 0 + ⋅
TT. Liu, BE280A, UCSD Fall 2012 €TT. Liu, BE280A, UCSD Fall 2012
€
Time-Varying Gradient Fields Phasors In the presence of time-varying gradients the frequency as a function of space and time is:
ω(r ,t) = γBz (r ,t) = γB0 + γG (t)⋅ r = ω0 + Δω(r ,t)
θ = 0 θ = −π /2 θ = π θ = π /2 €
€ € € €
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
10 Phase Phase with constant gradient Phase = angle of the magnetization phasor Frequency = rate of change of angle (e.g. radians/sec) Phase = time integral of frequency t ϕ r ,t = − ω(r ,τ)dτ ( ) ∫0 t r ,t = −ω0 + Δϕ( ) Where the incremental phase due to the gradients is
t t3 t 1 r ,t (r , )d € Δϕ(r ,t1) = − Δω(r ,τ)dτ Δϕ( 3 ) = − ∫ Δω τ τ Δϕ(r ,t) = − Δω(r ,τ )dτ ∫0 0 ∫0 t2 t Δϕ(r ,t2 ) = − Δω(r ,τ)dτ = − γG (r ,τ ) ⋅ r dτ ∫0 ∫0 = −Δω(r )t2 € € if Δω is non - time varying.
TT. Liu,€ BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012 €
Time-Varying Gradient Fields Signal Equation The transverse magnetization is then given by Signal from a volume s (t) = M(r ,t)dV r ∫V t −t /T2 (r ) − jω 0t M(r ,t) M(r ,0)e−t /T2 (r )eϕ(r ,t ) = ∫ ∫ ∫ M(x,y,z,0)e e exp − jγ ∫ G (τ)⋅ r dτ dxdydz = x y z ( o ) t = M(r ,0)e−t /T2 (r )e− jω 0t exp − j Δω(r ,t)dτ ( ∫o ) For now, consider signal from a slice along z and drop € z z / 2 t 0 +Δ −t /T2 (r ) − jω 0t the T2 term. Define m(x,y) ≡ M(r ,t)dz = M(r ,0)e e exp − jγ G (τ)⋅ r dτ ∫z z / 2 ∫ 0 −Δ ( o ) To obtain t − jω 0t sr (t) = ∫€ ∫ m(x,y)e exp − jγ ∫ G (τ)⋅ r dτ dxdy x y ( o ) €
TT. Liu, BE280A, UCSD Fall 2012 €TT. Liu, BE280A, UCSD Fall 2012
11 Signal Equation MR signal is Fourier Transform Demodulate the signal to obtain
jω 0t s(t) = e sr (t) t = ∫ ∫ m(x,y)exp − jγ ∫ G (τ)⋅ r dτ dxdy s(t) = m(x,y)exp − j2π k (t)x + k (t)y dxdy x y ( o ) ∫x ∫y ( ( x y )) t = m(x,y)exp − jγ G (τ)x + G (τ)y dτ dxdy ∫x ∫y ( ∫o [ x y ] ) = M(kx (t),ky (t)) = m(x,y)exp − j2π k (t)x + k (t)y dxdy ∫x ∫y ( ( x y )) = F[m(x,y)] kx (t ),ky (t ) Where γ t k (t) = G (τ)dτ € x 2π ∫0 x γ t € k (t) = G (τ)dτ y 2π ∫0 y
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012 €
Recap K-space At each point in time, the received signal is the Fourier • Frequency = rate of change of phase. transform of the object • Higher magnetic field -> higher Larmor frequency -> s(t) = M kx (t),ky (t) = F[m(x,y)] phase changes more rapidly with time. ( ) kx (t ),ky (t )
• With a constant gradient Gx, spins at different x locations precess at different frequencies -> spins at greater x-values evaluated at the spatial frequencies: change phase more rapidly. γ t € k (t) = G (τ)dτ • With a constant gradient, distribution of phases across x x 2π ∫0 x locations changes with time. (phase modulation) γ t k (t) = G (τ)dτ • More rapid change of phase with x -> higher spatial y 2π ∫0 y frequency kx Thus, the gradients control our position in k-space. The design€ of an MRI pulse sequence requires us to efficiently cover enough of k-space to form our image.
TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
12 K-space trajectory Units Gx(t) Spatial frequencies (kx, ky) have units of 1/distance. Most commonly, 1/cm t
t1 t2 Gradient strengths have units of (magnetic field)/ k y distance. Most commonly G/cm or mT/m
/(2 ) has units of Hz/G or Hz/Tesla. kx γ π k (t ) k (t ) x 1 x 2 γ t kx (t) = Gx (τ )dτ γ t ∫0 k (t) = G (τ)dτ 2π x 2π ∫0 x € € = [Hz /Gauss][Gauss /cm][sec] = [1/cm]
€ TT. Liu, BE280A, UCSD Fall 2012 TT. Liu, BE280A, UCSD Fall 2012
€ Example Gx(t) = 1 Gauss/cm
t t2 = 0.235ms
ky
kx
kx (t1) kx (t2 )
γ t k (t ) = G (τ )dτ 1 cm x 2 2π ∫0 x € € = 4257Hz /G ⋅1G /cm ⋅0.235 ×10−3 s =1 cm−1
TT. Liu, BE280A, UCSD Fall 2012
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