Topics. 2D Image. a b. c d. 1. Representing Images 2. 2D Fourier Transform 3. MRI Basics 4. MRI Applications 5. fmri

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1 Topics Neuroscience 200C Spring Quarter 2005 Imaging/MRI Lecture 1. Representing Images 2. 2D Fourier Transform 3. MRI Basics 4. MRI Applications 5. fmri Signals and Images Discrete-time/space signal/image: continuous valued function with a discrete time/space inde, denoted as s[n] for 1D, s[m,n] for 2D, etc. Continuous-time/space signal/image: continuous valued function with a continuous time/space inde, denoted as s(t) or s() for 1D, s(,y) for 2D, etc. n n m 2D Image a 0 0 b a b = + c d c 0 0 d t y 1

2 Image Decomposition 1 0 a b 0 1 a b = + c d 0 0 c 0 0 d Basis Functions 1/2 1/2 Coefficients = 1/2-1/2 Sum g[m,n] = aδ[m,n] + bδ[m,n 1] + cδ[m 1,n] + dδ[m 1,n 1] 1 1 = g[k,l] δ[m k,n l] k= 0 l= 0 = c k,l b k,l [m,n] k= 0 l= /2 1/2 1/2 1/2-1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2 Object Topics 1D Fourier Transform 1. Representing Images 2. 2D Fourier Transform 3. MRI Basics 4. MRI Applications 5. fmri KPBS KIFM Fourier Transform KIOZ 2

3 2D Plane Waves 2D Fourier Transform e j 2π (k +k y) y = cos( 2π(k + k y y) ) + j sin( 2π(k + k y y) ) 1 k 2 + k y 2 1/k y 1/k cos(2πk ) cos(2πk y y) cos(2πk +2πk y y) k-space Eamples Image space k-space y k y Fourier Transform k 3

4 Eamples Eamples Eamples Eamples 4

5 Fourier Transform 2D Fourier Transform G(k,k y ) = F[ g(, y) ] = e j 2π k +k y y Inverse Fourier Transform g(, y) = G(k,k y ) e j 2π k +k y y ( ),g = g(, y) ( ) dk dk y ( ) ddy e j 2π k +k y y Phasor Diagram Recall that a comple number has the form z = a + jb = z ep( jθ) = z ( cosθ + j sinθ) where z = a 2 + b 2 and θ = tan 1 ( b / a) e j2πk = cos( 2πk ) j sin( 2πk ) Imaginary Real θ = 2πk Interpretation Interpretation ep j2π 8Δ 1 ep j2π 8Δ 2 ep j2π 8Δ k =0; k y =0 k =0; k y 0 Fig 3.12 from Nishimura 5

6 1. Representing Images 2. 2D Fourier Transform 3. MRI Basics 4. MRI Applications 5. fmri Topics History of MRI 1946: Feli Bloch (Stanford) and Edward Purcell (Harvard) demonstrate nuclear magnetic resonance (NMR) 1973: Paul Lauterbur (SUNY) published first MRI image in Nature. History of MRI Spin Late 1970 s: First human MRI images Early 1980 s: First commercial MRI systems 1993: functional MRI in humans demonstrated Intrinsic angular momentum of elementary particles -- electrons, protons, neutrons. Spin is quantized. Key concept in Quantum Mechanics. 6

7 Classical Magnetic Moment Energy in a Magnetic Field Maimum Energy State I A r µ = IAˆ n B E = r µ B = µ z B Lorentz Force Minimum Energy State 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. 7

8 Boltzmann Distribution Equilibrium Magnetization Ε = µ z Β 0 B 0 ΔΕ = γhβ 0 Number Spins Up Number Spins Down Ε = µ z Β 0 = ep(-δe/kt) Ratio = at 1.5T!!! Corresponds to an ecess of about 10 up spins per million M 0 = N µ z = N n up µ z N = Nµ eµ z B / kt e µ z B / kt e µ z B / kt + e µ z B / kt Nµ z 2 B /(kt) = Nγ 2 h 2 B /(4kT) ( ) + n down ( µ z ) N = number of nuclear spins per unit volume Magnetization is proportional to applied field. Bigger is better Torque B 3T Human imager at UCSD. 7T Human imager at U. Minn. 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 B Torque 7T Rodent Imager at UCSD 9.4T Human imager at UIC 8

9 Precession Larmor Frequency dµ dt = 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. ω = γ Β f = γ Β / (2 π) Angular frequency in rad/sec Frequency in cycles/sec or Hertz, Abbreviated Hz dµ µ 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. Vector sum of the magnetic moments over a volume. 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. M = 1 V dm dt protons in V µ i = γm B RF Ecitation 9

10 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 a) Laboratory frame behavior of M b) Rotating frame behavior of M Images & caption: Nishimura, Fig RF Ecitation Free Induction Decay (FID) a) Laboratory frame behavior of M b) Rotating frame behavior of M B 1 induces rotation of magnetization towards the transverse plane. Strength and duration of B 1 can be set for a 90 degree rotation, leaving M entirely in the y plane. Images & caption: Nishimura, Fig

11 M0 z y RF Ecitation Doing nothing Ecitation M 0 (1 e -t/t1 ) z y Relaation z y z e -t/t2 y 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 T1 recovery T2 decay Credit: Larry Frank Relaation 1) Longitudinal component recovers eponentially. 2) Transverse component precesses and decays eponentially. dm z dt Longitudinal Relaation = 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. 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. 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 Transverse Relaation Gray Matter muscle White matter dm y dt = M y T 2 z y z y z y kidney liver fat 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. Image, caption: Nishimura, Fig. 4.2 T2 Relaation There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. Erwin Hahn After a 90 degree ecitation M y (t) = M 0 e t /T 2 12

13 T2 Relaation T2 Values Runners Net signal 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. Credit: Larry Frank Static Inhomogeneities In the ideal situation, the static magnetic field is totally uniform and the reconstructed object is determined solely by the applied gradient fields. In reality, the magnet is not perfect and will not be totally uniform. Part of this can be addressed by additional coils called shim coils, and the process of making the field more uniform is called shimming. In the old days this was done manually, but modern magnets can do this automatically. In addition to magnet imperfections, most biological samples are inhomogeneous and this will lead to inhomogeneity in the field. This is because, each tissue has different magnetic properties and will distort the field. Buton

14 The overall decay has the form. where T 2 * decay ep t /T 2 * v r ( ( )) 1 T 2 * = 1 T T 2 Due to random motions of spins. Not reversible. Due to static inhomogeneities. Reversible with a spin-echo sequence. T 2 * decay Gradient echo sequences ehibit T 2 * decay. RF G z (t) G (t) G y (t) ADC Slice select gradient Slice refocusing gradient TE = echo time Gradient echo has ep(-te/t 2 *) weighting Spin Echo Discovered by Erwin Hahn in RF Spin Echo Pulse Sequence τ 180º τ G z (t) τ τ G (t) The spin-echo can refocus the dephasing of spins due to static inhomogeneities. However, there will still be T 2 dephasing due to random motion of spins. G y (t) ADC ep( t TE / T 2 ) ep( t /T 2 ) Image: Larry Frank TE = echo time 14

15 Image Contrast Saturation Recovery Sequence 90 TE 90 TE 90 Different tissues ehibit different relaation rates, T 1, T 2, and T 2*. In addition different tissues can have different densities of protons. By adjusting the pulse sequence, we can create contrast between the tissues. The most basic way of creating contrast is adjusting the two sequence parameters: TE (echo time) and TR (repetition time). TR Gradient Echo TR [ ]e TE /T 2 * (,y ) I(, y) = ρ(, y) 1 e TR /T 1 (,y) TE Spin Echo I(, y) = ρ(, y) 1 e TR /T 1 (,y) [ ]e TE /T 2 (,y ) TR T1-Weighted Scans Make TE very short compared to either T 2 or T 2 *. The resultant image has both proton and T 1 weighting. T2-Weighted Scans Make TR very long compared to T 1 and use a spin-echo pulse sequence. The resultant image has both proton and T 2 weighting. I(, y) ρ(, y) 1 e TR /T 1 (,y) [ ] I(, y) ρ(, y)e TE /T 2 15

16 Proton Density Weighted Scans Eample Make TR very long compared to T 1 and use a very short TE. The resultant image is proton density weighted. I(, y) ρ(, y) T 1 -weighted Density-weighted T 2 -weighted 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 (,y,z) = B 0 +Δ B z (,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 1. Image, caption: copyright Nishimura, Fig

17 Z Gradient Coil Gradient Fields B(mT) z B z (, y,z) = B 0 + B z + B z y y + B z z z = B 0 + G + G y y + G z z y L G z = B z z > 0 G y = B z y > 0 Credit: Buton 2002 Precession with Gradient Field Interpretation ep j2π 8Δ 1 ep j2π 8Δ t ( ) = Δω( r 1 Δϕ r,t 1 0 r,τ) Δϕ r t (,t 3 ) = Δω( r 3 0 t ( ) = Δω( r 2,τ) dτ dτ Δϕ r,t 2 0 = Δω( r )t 2 if Δω is non - time varying. r,τ) dτ 2 ep j2π 8Δ Slower B z ()=G Faster 17

18 k (t) = γ 2π k y (t) = γ 2π K-space At each point in time, the received signal is the Fourier transform of the object s(t) = M( k (t),k y (t)) = F[ m(, y) ] k (t ),k y (t ) evaluated at the spatial frequencies: t 0 t 0 G (τ)dτ G y (τ)dτ 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. G (t) G y (t) K-space trajectory t 1 t 2 t k y (t 4 ) k y (t 3 ) k t 3 t 4 k (t 1 ) k (t 2 ) k y G (t) K-space trajectory G (t) Spin-Warp t k y k y G y (t) t 1 t 2 G y (t) t 1 k k 18

19 G (t) Spin-Warp Spin-Warp Pulse Sequence RF G y (t) t 1 k y G (t) G y (t) k y k k 1. Representing Images 2. 2D Fourier Transform 3. MRI Basics 4. MRI Applications 5. fmri Topics Phase of Moving Spin ΔB z () ΔB z () time 19

20 PCA eample Inflow Effect Prior to imaging -G 0 time Relaed spins flowing in Saturated spins Time of Flight Angiography Arterial Spin Labeling (ASL) 1: Wait Tag by Magnetic Inversion Acquire image 2: Wait Control Control - Tag CBF Acquire image Credit: Wen-Ming Luh 20

21 Multislice CASL and PICORE 2D random walk Diffusion 100 steps CASL 400 steps PICORE QUIPSS II N random steps of length d <Δ 2 >= Nd 2 = 2DT D = diffusivity In brain: D mm 2 /s For T=100 msec, Δ 15 µ Δ Credit: E. Wong Credit: Larry Frank Diffusion Weighted Images T2 weighted Diffusion Weighted Angiogram Restricted Diffusion D depends on direction z After a stroke, normal water movement is restricted in the region of damage. Diffusivity decreases, so the signal intensity increases. y Diffusion tensor: 3 values of D 3 angles Credit: Larry Frank 21

22 Diffusion Imaging Eample Q-ball imaging Tuch et al, Neuron 2003 Fiber Tract Mapping Topics 1. Representing Images 2. 2D Fourier Transform 3. MRI Basics 4. MRI Applications 5. fmri Mori et al., MRM

23 MRI studies brain anatomy. fmri Functional MRI (fmri) studies brain function. fmri Setup High spatial resolution fmri Acquisition High temporal resolution Visual Activation Flickering Checkerboard OFF (60 s) - ON (60 s) -OFF (60 s) - ON (60 s) - OFF (60 s) Brain Activity MP-RAGE Voel volume: 1 mm 3 Imaging time: 6 min EPI Voel volume: 45 mm 3 Imaging time: 60 msec Buton 2002 Time Source: Kwong et al.,

24 Memory Encoding Hemoglobin 10 Familiar scenes 10 Novel Scenes 10 Familiar scenes 10 Novel Scenes 25 seconds The same 4 familiar scenes are used throughout the eperiment Oygen binds to the iron atoms to form oyhemoglobin HbO 2 Release of O 2 to tissue results in deoyhemoglobin dhbo 2 Effect of dhbo 2 dhbo 2 is paramagnetic due to the iron atoms. As it becomes oygenated, it becomes less paramagnetic. dhbo 2 perturbs the local magnetic fields. As blood becomes more deoygenated, the amount of perturbation increases and there is more dephasing of the spins. Thus as dhbo 2 increases we find that T 2 * decreases and the amplitude ep(-te/ T 2 *) image of a T 2 * weighted image will decrease. Conversely as dhbo 2 decreases, T 2 * increases and we epect the signal amplitude to go up. Buton

25 BOLD Effect Blood Oygen Level Dependent signal neural activity blood flow oyhemoglobin T2* MR signal BOLD Dynamics M y Signal M o sinθ S task S control T 2 * task T 2 * control TE optimum time ΔS Stimulus Neural Activity ATP ADP ADP CMRO 2 CO 2 NO, K + CBF ADP Lac CMRGlc CBV BOLD Signal Source: fmrib Brief Introduction to fmri Source: Jorge Jovicich Credit: Rick Buton BOLD Dynamics BOLD and Buton