The NMR Inverse Imaging Problem

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1 The NMR Inverse Imaging Problem Nuclear Magnetic Resonance Protons and Neutrons have intrinsic angular momentum Atoms with an odd number of proton and/or odd number of neutrons have a net magnetic moment=> MR active For example, 1 H exhibit nuclear magnetic resonance Neutrons + protons 1 H 1

2 Nuclear Spins Rotating charges correspond to an electrical current Current gives rise to a magnetic moment Proton Nuclear spins are like magnetic dipoles. Electron Moments align with external magnetic field Polarization Spins are normally oriented randomly In an applied magnetic field, the spins align with the applied field in their equilibrium state Excess along B 0 results in net magnetization vector M M No Applied Field Applied Field B 0 2

3 Static Magnetic Field Longitudinal z x, y Transverse B 0 Spin Precession Magnetic Spinning Top Bloch equation with no relaxation Precession frequency For protons (H), 3

4 NMR Nuclei Nucleus 1 H(1/2) 23 Na(3/2) 31 P(1/2) 17 O(5/2) (MHz/T) (M) MR: Intro: RF Magnetic field This is all well and good, but how do we get M r away from B r? The RF Magnetic Field, also known as the B 1 field To excite nuclei M = Mokˆ, apply rotating field at ω o in x-y plane. (transverse plane) B 1 radiofrequency field tuned to Larmor frequency and applied in transverse (xy) plane induces nutation (at Larmor frequency) of magnetization vector as it tips away from the z-axis. - lab frame of reference 4

5 MR: Intro: RF Magnetic field (2) 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 xy plane. MR: Intro: RF excitation r r By design, B 1 B o In the rotating frame, the frame rotates about z axis at ω o radians/sec z B r 1 x M r y 1) B 1 applies torque on M 2) M rotates away from z. (screwdriver analogy) This process is referred to as RF excitation. Strength: B 1 ~.1 G What happens if we leave B 1 on? 5

6 MR: Intro: RF excitation What happens as we leave B 1 on? - B 1 at approximately.1g for 1ms will induce 90º excitation. p =? B1 τ where τ = duration 2 B r 1 x To set rotation for 90º, Turn B 1 on long enough to get 90º rotation, then turn it off. - This is referred to as 90º excitation. z M r y In Laboratory frame, B r oscillates at ω 1 o MR: Intro: RF excitation z Then what happens? B r 1 M r y z M r will process at ω o B r o x M r y 6

7 MR: Intro: Detection Switch RF coil to receive mode. z x y M Precession of M r induces EMF in the RF coil. (Faraday s Law) EMF time signal - Lab frame Voltage t (free induction decay) dφ EMF= - dt Putting it all together: The Bloch equation with Relaxation Sums of the phenomena r dm r r ( mxî + my ĵ) ( M )kˆ?b z M = M o dt T2 T 1 precession, RF excitation Changes the direction of M r, but not the length. transverse magnetization B r includes B o, B 1, and G r longitudinal magnetization These change the length of only, not the direction. M r 7

8 Challenges 1. The spins are not phase coherent - Synchronization is required. 2. All spins are precessing at the same larmour frequency regardless of location ω =γb From NMR to the Imaging Equation RF Spin Phasing If the spins are brought into phase, a net transverse magnetization is created. 8

9 Gradient Fields Gradient Fields G y changes field strength of B field in z direction as a function of y G z changes field strength of B field in z direction as a function of z Similarly for G x For example, if G x =1.0 G/cm, then the frequencies vary as khz/cm for 1 H, amounting to a frequency bandwidth of khz for a 10 cm-wide object. 9

10 Gradient Coils Field always along z Gradient along x, y, or z I Gz Usually, only one gradient active at a time, but can combine to give any arbitrary direction I Gy I Spatial Localization and Frequency Encoding Larmor Equation: Generalize to: ω =γb ω ( x) = γb( x) 10

11 Spin Encoding Spin Encoding of with position dependent frequency shifts Image, caption: copyright Proruk & Sawyer, GE Medical Systems Applications Guide, Fig. 11 Spin Encoding-Example 11

12 Selective Excitation Selective Excitation B 1 applied in presence of G z excites a plane perpendicular to z. The Fourier Transform of B 1 (t) approximates the slice profile 12

13 Selective Excitation Slice Encoding Gradient field B = ( B + G z)ˆ z 0 ω = γ ( B + G z) Slice Location & Width: 0 z z Position Magnitude Gradient strength Center frequency Bandwidth (BW) of RF pulse Slope = (a) (c) 1 γg Frequency Frequency RF Amplitude (b) (d) Time Slice Selection Convention y transverse x z coronal sagittal Patient Table The orientation of the orthogonal slice planes correlated with the standard axes of a whole body MRI magnet system 13

14 Generalized Slice Selection Problem z G z G O Φ Θ selected slice G y y G x x Slice selection at an arbitrary direction Refocusing 14

15 Uniqueness and the Problem of Spatial Encoding 3D Slice Selection 2D??? To encode information in a sine wave: Frequency Phase Amplitude Concept of Phase Encoding 15

16 Phase Encoding Step Phase Encoding The local signal under the influence of this gradient is: The received signal is the sum of all the local phase-encoded signals and is given by: 16

17 Frequency Encoding Apply magnetic field gradient along x-axis during data read-out Frequency of signal depends on the x-position of the spins from which it is generated Fourier transform from (spatial) frequency space to image space Frequency Encoding The Larmor frequency at position at x is: The FID signal generated locally from spins in interval dx at point x is: The signal from the entire object in the presence of this gradient is: 17

18 Frequency and Phase Encoding Fourier Interpretation of Imaging Equation 18

19 Fourier Interpretation of Imaging Equation K-Space Trajectories 19

20 K-space Trajectories Object Domain Perspective 20

21 Central Section Theorem (Fourier Slice Theorem) 21

22 K-space Trajectories K-space Trajectory 22

23 Object Domain The signal equation for this one-sided sequence is: Full Sequence with Data Acquisition 23

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