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2D Fourier Transform Phasor Diagram 2D Fourier Transform Fourier Transform ∞ ∞ − j 2π (kx x+ky y) G(kx,ky ) = F[g(x,y)] = ∫ ∫ g(x,y)e dxdy Bioengineering 280A" −∞ −∞ Principles of Biomedical Imaging" " Inverse Fourier Transform Fall Quarter 2014" ∞ ∞ MRI Lecture 2" j 2π (kx x+ky y) g(x,y) = ∫ ∫ G(kx,ky )e dkxdky −∞ −∞ € TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 Phasor Diagram ∞ Imaginary G(kx ) = ∫ g(x)exp(− j2πkx x)dx −∞ θ = −2πkx x Real € θ = −2πkx x € x =1/4 x =1/2 x = 3 / 4 kx =1; x = 0 2πkx x = π /2 2πkx x = π 2πkx x = 3π / 2 2πkx x = 0 € € € € 3 / 2 θ = 0 θ = −π /2 θ = −π θ = − π TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 € € 1 Vector Vector sum sum x 25 x 13 Vector Vector sum sum x -5 x -3 TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 Vector Vector sum sum x -1 x -13 Vector Vector sum sum x 3 xx 13 TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 2 Precession Larmor Frequency Analogous to motion of a gyroscope dµ = µ x γB ω = γ Β Angular frequency in rad/sec dt Precesses at an angular frequency of B ω = γ Β f = γ Β / (2 π) Frequency in cycles/sec or Hertz, This is known as the Larmor frequency. Abbreviated Hz d µ For a 1.5 T system, the Larmor frequency is 63.86 MHz µ which is 63.86 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. TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 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 http://gulfnews.com/business/economy/german-factories-race-to-meet-big-surge-in-orders-1.638066 TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 3 Interpretation Rotating Frame of Reference ∆Bz(x)=Gxx Reference everything to the magnetic field at isocenter. x Spins Precess Spins Precess at γB0 Spins Precess at at γB0- γGxx at γB0+ γGxx (slower) (faster) TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 Spins There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. Erwin Hahn kx=0; ky=0 kx=0; ky≠0 Fig 3.12 from Nishimura TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 4 TT. Liu, BE280A, UCSD Fall 2014 Hanson 2009 TT. Liu, BE280A, UCSD Fall 2014 Vector sum x -13 Vector sum xx 13 Hanson 2009 TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 5 (a) (b) y 1 0 1 0 1 0 1 0 1 x (c) (d) (e) TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 Gradient Fields Gradient Fields ∂B ∂B ∂B Define B (x,y,z) = B + z x + z y + z z z 0 G ≡ G iˆ + G ˆj + G kˆ r ≡ xiˆ + yˆj + zkˆ ∂x ∂y ∂z x y z = B + G x + G y + G z So that z 0 x y z Gx x + Gy y + Gzz = G ⋅ r y € € Also, let the gradient fields be a function of time. Then € the z-directed magnetic field at each point in the € volume is given by : ∂Bz G 0 ∂Bz B (r ,t) B G (t) r z = > G = > 0 z = 0 + ⋅ ∂z y ∂y TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 € € € 6 Time-Varying Gradient Fields Phase MY ! ! ! M(r,t) = M(r,0)eϕ (r ,t) In the presence of time-varying gradients the frequency as a function of space and time is: Phase = angle of the magnetization phasor MX Frequency = rate of change of angle (e.g. radians/sec) Phase = time integral of frequency r ,t B (r ,t) ! ω( ) = γ z θ = Δϕ(r,t) t ϕ r ,t = − ω(r ,τ)dτ = γB0 + γG (t)⋅ r ( ) ∫0 (r ,t) = −ω t + Δϕ r ,t = ω0 + Δω 0 ( ) Where the incremental phase due to the gradients is t € € Δϕ r ,t = − Δω(r ,τ )dτ ( ) ∫0 t = − γG (r ,τ ) ⋅ r dτ 0 ∫ TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 € Phase with constant gradient Phase with time-varying gradient ! t1 ! Δϕ r,t = − Δω(r,τ )dτ ! t1 ! ( 1) ∫ 0 ! t3 ! Δϕ r,t = − Δω(r,τ )dτ r,t (r, )d ( 1) ∫ 0 Δϕ ( 3 ) = − Δω τ τ ! t3 ! = −γG xt ∫ 0 x 1 = −γG xt Δϕ (r,t3 ) = − Δω(r,τ )dτ G xt x 1 ∫ 0 ! t2 ! = −γ x 3 ! t2 ! Δϕ r,t = − Δω(r,τ )dτ Δϕ r,t2 = − Δω(r,τ )dτ G x t t t ( 2 ) ∫ 0 ( ) ∫ 0 = −γ x ( 1 +( 3 − 2 )) ! t1 ! = −Δω(r)t2 = − Δω(r,τ )dτ + 0 ∫ 0 = −γGx xt2 = −γGx xt1 TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 7 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 2014 TT. €Liu, BE280A, UCSD Fall 2014 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 2014 TT. Liu, BE280A, UCSD Fall 2014 € 8 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 2014 TT. Liu, BE280A, UCSD Fall 2014 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 kx γ/(2π) has units of Hz/G or Hz/Tesla. 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 2014 TT. Liu, BE280A, UCSD Fall 2014 € 9 Example K-space trajectory G (t) Gx(t) = 1 Gauss/cm x ky t t ky (t4 ) t2 = 0.235ms t1 t2 Gy(t) ky ky (t3 ) € kx kx t3 t4 k (t ) k (t ) x €1 x 2 kx (t1) kx (t2 ) γ t 1 cm kx (t2 ) = Gx (τ )dτ € € 2π ∫0 € € = 4257Hz /G ⋅1G /cm ⋅0.235 ×10−3 s =1 cm−1 TT. Liu, BE280A, UCSD Fall 2014 TT. Liu, BE280A, UCSD Fall 2014 € 10 .
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