Basics of Magnetic Resonance Imaging Angular Momentum Orbital Angular Momentum L mr v Principles of Medical Imaging – Shung, Smith.
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Basics of Magnetic Resonance Imaging Angular Momentum Orbital Angular Momentum L mr v Principles of Medical Imaging – Shung, Smith and Tsui Angular Momentum Spin Angular Momentum Spin is an intrinsic property of the nucleons (protons and neutrons) in a nucleus HOWEVER – The name doesn’t mean that spin results from the nucleons rotating about an axis!!! http://svs.gsfc.nasa.gov/vis/a000000/a001300/a001319/ Spin Angular Momentum Spin is quantized – it can only take certain values Lz mI mI I , I 1, I 2,... I Here I is the total spin quantum number of the nucleus. The proton has I = ½. Lz is the angular momentum due to that spin. Spin Angular Momentum To get the total spin of a nucleus we add up (separately) the spins of the protons and neutrons 15N has spin ½ . We do pairwise addition: 7 protons 8 neutrons Only nuclei with an odd number of protons or neutrons will be visible to MRI Note: 14N has spin 1. Alignment of Spins in a Magnetic Field, B0 The spin angular momentum yields a magnetic moment μ m L z Principles of Medical Imaging – Shung, Smith and Tsui Energy Levels of Spins and B0 Principles of Medical Imaging – Shung, Smith and Tsui Energy Levels and Spin E1 = B0 E2 = -B0 E = E1 - E2 = 2 B0 = (h/2) B0 for spin-1/2 particles B0 is the main magnetic field Energy Level Population and Field Strength 2,000,000 999,999 999,995 1,000,002 1,000,005 999,987 1,000,014 B0 = 0 n = 0 B0 = 0.5T n = 3 B0 = 1.5T n = 10 B0 = 4.0T n = 27 Spins are distributed according to the Boltzmann distribution ( Nupper / Nlower ) eE / kT Larmor Frequency B0 Principles of Medical Imaging – Shung, Smith and Tsui Excitation Energy and Frames of Reference B0 Beff B0 = main magnetic field B1 = applied field (pulse) Beff = vector sum B0+B1 B1 z z y y x lab frame x rotating frame Net Magnetization, M, and the Rotating Frame While B1 0, M precesses around B1 z z M M y y B1 x x B0 0 B1 = 0 B0 0 B1 0 Net Magnetization, M, and the Rotating Frame We turn B1 “on” by a applying radiofrequency (RF) to the sample at the Larmor frequency. This is a resonant absorption of energy. If we leave B1 on just long enough for M to rotate into the x - y plane, then we have applied a “90 pulse”. In this case, Nupper = Nlower. If we leave B1 on just long enough for M to rotate along the -z axis, then we have applied a “180 pulse” (inversion). In this case, Nupper = Nlower + M. Free Induction Decay What is the effect of applying a 90 pulse? /2 RF time Free Induction Decay The effect of a 90 pulse is to rotate M into the x - y (transverse) plane. If we place a detection coil (a loop of wire) perpendicular to the transverse place, we will detect an induced current in the loop as M precesses by (in the lab frame). z Minitial y B1 Mfinal x Principles of Medical Imaging – Shung, Smith and Tsui Signal Processing of Free Induction Decay Principles of Medical Imaging – Shung, Smith and Tsui We can characterize the signal by its: Amplitude Phase Fourier Transform Frequency Signal Processing of Free Induction Decay We see that after a 90 pulse, we get a cosinusoidal signal. To quantitatively describe the signal we calculate its Fourier transform. (think: Larmor frequency) Principles of Medical Imaging – Shung, Smith and Tsui Fourier Transform of Time Domain Data http://www.med.harvard.edu/JPNM/physics/didactics/improc/intro/fourier3.html Image Contrast (I) We can detect the signal from water molecules in the body. Can we make an image? Will it be a useful image? Relaxation Processes Fortunately for us, the signal we get from water molecules in the body depends on their local environment. Spins can interact by exchanging or losing energy (or both). As in all spectroscopy methods, we put energy into the system and we then detect the emitted energy to learn about the composition of the sample. We then use some variables to characterize the emission of energy which (indirectly) tell us about the environment of the spins. Image Contrast! Relaxation Processes (T2) 1. Spin-spin relaxation time (T2): when spins interact With each others magnetic field, they can exchange energy (perform a spin flip). They can lose phase coherence, however. Only affects Mxy. Signal without T2 interaction between spins T2* http://irm-francophone.com/htm/signal.htm Signal including T2 interactions between spins T2*??? T2* and T2 T2 is an intrinsic property of the sample. This is what we are interested in to use for contrast generation. T2* is the time constant of the decay of the free induction decay. It is related to the intrinsic T2 in the following way: 1 1 1 1 inhomogeneity susceptibility * T2 T2 T2 T2 sample only depends upon the local magnetic field T2* and T2 1 1 1 1 inhomogeneity susceptibility * T2 T2 T2 T2 random process Not a random process Inhomogeneity term - dephasing due to magnet (B0) imperfections depends upon position Susceptibility term - dephasing due to the interaction of different sample regions with B0 (depends upon position) Relaxation Processes (T2) /2 pulse (delay) Effect of Spin Coherence on Signal z y x z T2* y x http://irm-francophone.com/htm/signal.htm Irreversible versus Reversible http://www.cchem.berkeley.edu/demolab/images/HahnEchoSpinRes.htm Start Reverse Hahn Spin Echo Pulse Sequence http://www.esr.ethz.ch/intro/spinecho.html http://www.chem.queensu.ca/FACILITIES/NMR/nmr/webcourse/t2.htm Hahn Spin Echo and T2 We can calculate T2 by changing the echo spacing, , and recording the signal at 2. http://spiff.rit.edu/classes/phys273/exponential/exponential.html S (2 ) e Signal Echo spacing , 2 / T2 Spin-Lattice Relaxation, T1 To look at the behavior of the longitudinal component of M (Mz), we start by putting M along the -z axis and then read it out with a 90 pulse. M z M 0 (1 2e TI / T1 ) Spin-Lattice Relaxation, T1 Energy levels and Inversion Equilibrium Net Magnetization: = After Inversion = Relaxation Processes (T1) pulse short TI long TI /2 pulse /2 pulse Image Contrast and T1 In (a) the TI is chosen to null the signal from curve [ii], while the TI in (b) nulls out [i] http://www.fmrib.ox.ac.uk/~stuart/thesis/chapter_2/section2_4.html Now Can We Make A Useful Image? Magnetic Resonance Imaging Magnetic Resonance Image Formation What do we need? 1. Ability to image thin slices 2. No projections - image slices with arbitrary orientation 3. Way to control the spatial resolution 4. Way to carry over the spectroscopic contrast mechanisms to imaging. Magnetic Resonance Image Formation B0 signal A sin(t ) How can we spatially encode this signal? Principles of Medical Imaging – Shung, Smith and Tsui Magnetic Resonance Image Formation - Slice Selection Magnetic field gradients: B = B(position) Beff B0 Gz z eff (B0 Gz z) Gz z B0 precessional frequency is now a function of position Magnetic Resonance Image Formation - Slice Selection 2 ( ) z Gz = frequency bandwidth z = slice thickness MRI: Basic Principles and Applications - Brown, Semelka Magnetic Resonance Image Formation - Slice Selection How do we excite only a slice of spins? Fourier Pairs! sinc rect FT time frequency Magnetic Resonance Image Formation - Phase Encoding So far we have a slab of tissue whose spins are excited. The next step is to place a grid over the slab and define pixels. Magnetic Resonance Image Formation - Phase Encoding Beff B0 Gy y Gy eff ( B0 G y y ) 2 eff < 0 eff > 0 Center of magnet Magnetic Resonance Image Formation - Phase Encoding Apply gradient for a finite duration = (y) (the phase of M over each region of the sample depends upon it’s position) This is because the gradient makes each spin precess with an angular frequency that depends on it’s position. For the duration of the gradient, t, spins move faster or slower than 0 depending upon where they are. After the gradient is turned off, all spins again precess at 0. The phase accumulated during time, t, is: Gy yt Magnetic Resonance Image Formation - Phase Encoding z’ M after a 90 pulse y’ x’ Turn on the phase encoding gradient z’ z’ y’ x’ Gy = 0 z’ y’ x’ Gy < 0 y’ x’ Gy > 0 Magnetic Resonance Image Formation Frequency Encoding (Readout) Now we need to encode the x-direction... How do we spatially encode the frequency of the signal? Can we turn on another gradient? When? And for how long? Magnetic Resonance Image Formation Frequency Encoding (Readout) We apply a gradient while the signal is being acquired as the spin-echo is being formed. /2 RF Signal Gx time Therefore, the precessional frequency is a function of position Magnetic Resonance Image Formation Spin-echo Pulse Sequence ?(3) ? (1) ?(2) http://spl.harvard.edu:8000/pages/papers/zientara/fast/fastimaging.html (1) Phase Encoding: Each acquisition is separately encoded with a different phase. The sum total of the N acquisitions is called the ‘k-space’ data. The Effect of Frequency Encoding on the Signal (dephasing gradient) (2) MRI: Basic Principles and Applications - Brown, Semelka (3) z y Slice-select refocusing gradient x The effect of having the gradient on during the time that the magnetization is moving from the z-axis to the y-axis is to curve the path. In general, one needs to apply a slice refocussing gradient of opposite magnitude after the RF pulse so that the spins are in phase at the end of the pulse. The area of the negative gradient must be one half the area of the slice selection gradient pulse. Structure of MRI Data: k-space k-space 256 x 256 points A/D, 256 points row 40 row -55 view 40 view -55 view -128 row -128 kx = frequency ky = phase http://www.indyrad.iupui.edu/public/lectures/mri/iu_lectures/mri_homepage.htm Structure of MRI Data: k-space What is k-space? The time-domain signal that we collect from each spatially encoded spin echo gets put in a matrix. This data is called k-space data and is a space of spatial frequencies in an image. To get from spatial frequencies to image space we perform a 2-D Fourier Transform of the k-space data. General intensity level is represented by low spatial frequencies; detail is represented by high spatial freq’s. low spatial frequencies high spatial frequencies all frequencies http://www.indyrad.iupui.edu/public/lectures/mri/iu_lectures/mri_homepage.htm Structure of MRI Data: k-space FT k-space data image data http://www.jsdi.or.jp/~fumipon/mri/K-space.htm