Physics Of MRI - NYU Tandon School Of Engineering

Physics of MRIYao WangPolytechnic University, Brooklyn, NY 11201Based on J. L. Prince and J. M. Links, Medical Imaging Signals andSystems, and lecture notes by Prince. Figures are from the textbookexcept otherwise noted.

Lecture OutlineEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn2

Lecture Outline Overview of MRINuclear spin propertiesPrecession and Larmor FrequencyRF excitationRelaxationContrast mechanismEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn3

Magnetic Resonance Imaging Provide high resolution anatomic structure (as with X-ray CT)Provide high contrast between different soft tissues (X-ray CT cannot)No exposure to radiation and hence safeMore complicated instrumentationTakes longer to acquire a scan than CT, more susceptible to patient motionPETMRICTEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn4

X-ray projectionMRIEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn5

Basic Principle of MRI The hydrogen (1 H) atom inside body possess “spin” In the absence of external magnetic field, the spin directions of allatoms are random and cancel each other. When placed in an external magnetic field, the spins align with theexternal field. By applying an rotating magnetic field in the direction orthogonal tothe static field, the spins can be pulled away from the z-axis with anangle \alpha The bulk magnetization vector rotates around z at the Larmorfrequency (precess) The precession relaxes gradually, with the xy-component reduces intime, z-component increases The xy component of the magnetization vector produces a voltagesignal, which is the NMR signal we measureEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn6

What is Spin? Spin is a fundamental property of nature like electricalcharge or mass. Spin comes in multiples of 1/2 and canbe or -. Protons, electrons, and neutrons possess spin.Individual unpaired electrons, protons, and neutronseach possesses a spin of ½ or - ½. Two or more particles with spins having opposite signscan pair up to eliminate the observable manifestations ofspin. In nuclear magnetic resonance, it is unpaired nuclearspins that are of importance.EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn7

Nuclear Spin A nucleus consists of protons and neutronsWhen the total number of protons and neutrons ( mass number A) is odd orthe total number of protons is odd, a nucleus has an angular momentum(\phi) and hence spin– Ex. Hydrogen (1 H) (1 proton), 13 C The spin of a nucleus generates a magnetic filed, which has a magneticmoment (\mu)The spin causes the nucleus behave like a tiny magnet with a north andsouth poleEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn8

Angular momentum vs Magnetic Moment ]EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn9

Nuclear Spin System Collection of identical nuclei in a given sample ofmaterial (also known as spin packet, a voxel in theimaged volume) In the absence of external magnetic field, the spinorientations of the nuclei are random and cancel eachother When placed in a magnetic field, the microscopic spinstend to align with the external field, producing a net bulkmagnetization aligned with the external fieldEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn10

In the absence of external magnetic fieldHydrogen Nuclei (Protons)From Graber, Lecture note for BMI F05EL582 MRI PhysicsYao Wang, Polytechnic U., BrooklynAxis of Angular Momentum(Spin), Magnetic Moment11

Nuclear Magnetization(low energy state)(high energy state)N-/N e-E/kTEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn12

PrecessionSpins PRECESS ata single frequency(w0), butincoherently theyare not in phase, sothat the sum of x-ycomponents is 0,with netmagnetizationvector in z directionW0 \gamma B 0:Larmor freq.mzFrom Graber, Lecture note for BMI F05EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn13

Bulk Magnetization at EquilibriumWhich depends on tissue typeEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn14

How to make the spins in phase?Irradiating with arotating magneticfield B 1 offrequency w0,causes spins toprecess coherently,or in phase,generating a xycomponentEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn15

Process Involved in MRI Put patient in a static field B 0 (much stronger than the earth’s field) (step 1) Wait until the nuclear magnitization reaches an equilibrium(align with B 0) Applying a rotating magnetic field B 1 (much weaker than B 0) tobring M to an initial angle \alpha with B 0 (rotating freq Larmorfreq.) M(t) precess around B 0 at Larmor frequency around B 0 axis (zdirection) with angle \alpha The component in z increases in time (longitudinal relaxation) withtime constant T1 The component in x-y plane reduces in time (transverse relaxation)with time constant T2 Measure the transverse component at a certain time after theexcitation (NMR signal) Go back to step 1 By using different excitation pulse sequences, the signal amplitudecan reflect mainly the proton density, T1 or T2 at a given voxelEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn16

Evolution of magnetization when a Timevarying magnetic field is appliedEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn17

M(t) experiences a torque when an external magnetic field B(t) isappliedUsing the right hand rule, M will rotate around z if M is not aligned with zEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn18

Cross Product: ReviewiM B MxBxjMyBykMzBz ( M y Bz M z By ) i ( M z Bx M x Bz ) j ( M x By M y Bx ) kDirection of MxB follows “right hand” ruleEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn19

Solution under a Static Field with anInitial Angle B(t) [0,0,B 0] MxB M y B 0 i - M x B 0 j 0 k dM x/dt M y B 0 dM y/dt - M x B 0 Solving above yields solution in the next slideEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn20

Precession Due to a Static Field with anInitial AnglemzThis is the frequency of the photon which would cause atransition between the two energy levels of the spin.B0 1.5T, \gamma 42.58 MHz/T, v0 63.9 MHzEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn21

Longitudinal and TransverseComponentsNo changeRapidly rotatingEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn22

Laboratory Frame vs. Rotating FrameCoordinate systemrotated about z axisat the Larmor freq.z, z’x’ yy’xThe rotating M(t) vector appearstationary in the rotating frameEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn23

See animation at mEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn24

NMR Signal The rapidly rotating transverse magnetization (M xy)creates a radio frequency excitation within the sample. If we put a coil of wire outside the sample, the RFexcitation will induce a voltage signal. In MRI, we measure this voltage signal. Voltage produced is (Faraday’s Law of Induction)EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn25

Simplification B r(r) B rV ω0Vs M 0 sin α B rB0γ 2 h 2Recall ω0 γB0 , M 0 PD4κT2Therefore V B0 , PDEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn26

How do we tilt M to an initial angle? Applying a circularly polarized (rotating) magnetic field B 1(t) in thex-y plane with the same Larmor frequency forces the magnetizationvector to tilt down to the x-y plane– B 1(t) has two orthogonal components, in x and y directionsrespectively, and is produced by using quadrature RF coil– Simplest envelop B 1,e is a rectangular pulse Motion of M(t) is spiralEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn27

Animation of spiral motion Laboratory 4-5.htm Rotating frame: EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn28

Circularly Polarized Magnetic FieldSSSB0two more magnets,whose fields areorthogonal to B0, thatrotate, in oppositedirections, at theLarmor frequencyNNNEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn29

Tip Angle If M is parallel to z-axis before the RF excitation pulse, the tip angleafter the excitation (with duration \tau p) is If B 1 e(t) is rectangular Pulse that leads to \alpha \pi/2 is called “\pi over 2 pulse”, whichelicits the largest transverse component M xy, and hence largestNMR signal Pulse that leads to \alpha \pi is called “\pi pulse” or inverse pulse,which is used to induce spin echo (later) The excitation pulse (envelop of B 1(t)) is also called “an alphapulse”EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn30

RelaxationEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn31

Longitudinal Relaxation The magnetization vectors tend to return to equilibriumstate (parallel to B 0) M 0 cos\alpha 0 for \pi/2 pulseEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn32

In the laboratory frame, M takes aspiralling path back to itsequilibrium orientation. But herein the rotating frame, it simplyrotates in the y ׳ -z ׳ plane.z ׳ MzB0My ׳ x ׳ The z component of M, Mz, grows backinto its equlibrium value, exponentially:Mz M (1 - e-t/T1)EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn33

Transverse Relaxation The strength of the magnetic field in the immediate environment of a1H nucleus is not homogeneous due to presence of other nucleus(and their interactions) Hence the Larmor frequencies of nearby nuclides are slightly different(some spins faster, some slower)– Spin-spin interactions This causes dephasing of the xy components of the magnetizationvectors, leading to exponential decay of M xyEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn34

See animation at http://www.cis.rit.edu/htbooks/mri/inside.htm– Under T2 processes Overall effect of both transverse and longitudinalrelaxation: mEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn35

T 2 is called transverse relaxation time, which is the time for M xyto decrease by 1/e. Also called spin-spin relaxation time T2 is much smaller than T1– For tissue in body, T2: 25-250ms, T1: 250-2500 msEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn36

Free Induction Decay The voltage signal (NMR signal) produced by decaying M xy alsodecays This is called free induction decay (FID), and is the signal we measurein MRIEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn37

T2 Star Decay Received signal actually decays faster than T 2 (having a shorter relaxationtime T 2 *)Caused by fixed spatial variation of the static field B 0 due to imperfectionof the magnet– Accelerates the dephasing of magnetization vectors– Note that T2 is caused by spatial variation of the static field due to interactions ofnearby spins The initial decay rate is governed by T 2 * , but the later decay by T 2.EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn38

Formation of Spin Echo By applying a 180 degree pulse, the dephased spins can recover theircoherence, and form an echo signalEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn39

RF Pulse Sequence and CorrespondingNMR SignalEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn40

Spin echo sequence Multiple π pulses create “Carr-Purcell-Meiboom-Gill(CPMG)” sequence Echo Magnitude Decays with time constant T2T R (pulse repetition time)EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn41

Bloch EquationsRelaxationForced precessionStatic fieldEL582 MRI PhysicsYao Wang, Polytechnic U., BrooklynAlpha pulse(RF excitation atLarmor freq.)42

Solving the previous equation in x, y, z direction will yieldthe equations representing the transverse andlongitudinal relaxations, shown previouslyEL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn43

Source of MR Contrast Different tissues vary in T1, T2 and PD (proton density) The pulse sequence parameters can be designed so that thecaptured signal magnitude is mainly influenced by one of theseparameters Pulse sequence parameters– Tip angle \alpha– Echo time T E– Pulse repetition time T REL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn44

Typical Brain Tissue Parameters Table 12.2 in [Prince]P DT 2 (ms)T 1 2802650EL582 MRI PhysicsYao Wang, Polytechnic U., Brooklyn45

P DT 2 (ms)T 1 2802650PD weightedWhite matterT2- weightedCSFT1- weightedGray matter