Introduction au cours

Fund BioImag 20129-19: Relaxation of nuclear magnetizationHow is the MR signal detected ?What is the quantum-mechanical equivalent of the rotating frame ? What is the rotating frame description good for ?How can the return of the magnetization to thermodynamic equilibrium described ?What is the behavior of magnetization in the presence of an external magnetic field ?

After this course youCan describe the principle of MR detection and excitationCan explain how MR excitation is frequency selective (resonance) Understand the principle of relaxation to the equilibrium magnetization Know what are the major relaxation times and how they phenomenologically affect magnetization in biological tissue, in particular that of water.Can explain the elements of the Bloch equations and FIDUnderstand the MR contrast strongly depends on experimental parameters Fund BioImag 20129-2

MRI contrast depends on experimental parametersI. Time after excitation TETE=25 ms50 ms75 ms100 msFund BioImag 20129-3

aII. Flip angle a and time between excitations TR

aa deg pulsemsFund BioImag 20129-4Magnetic resonance so farNeed:Nucleus with non-zero spinMagnetic field B0Get:Nuclear (equilibrium) magnetization M0 (Magnitude dictated by Boltzmann distribution)M0 increases with Number of spins in voxelMagnetic field B0Gyromagnetic ratio gImaging 1H in H2O is most sensitiveAll this does not generate a measurable signalThermodynamic equilibrium magnetization M0 is || B0M0 does not precess=0

Fund BioImag 20129-5

9-1. Detection of MR signal

Magnetic field of dipole decreases with distance : x decreases with distance from magnetizationFaradays Law of Induction

Lenzs Lawinduced voltage x current magnetic field opposes the change in the magnetic flux that produces the current (Completely analogous to power generation!)

Biot-Savart Lawmagnetic field falls off with r-2

Magnetic flux FBInduced voltageBar magnetMagnetizationz(t)dAMR: dA=constant x dB/dt B1I(t)Coil currentNB. Generation of the RF field B1B1 decreases with distance from RF coil (Biot-Savart)

Fund BioImag 20129-6

RF Coils: Examples of MR Detectors

Distance is important: A coil for every organ!

Fund BioImag 20129-7xyz9-2. Rotating frame revisitedRotating frame: reference frame rotating about z at frequency RFLarmor frequency in the rotating frame: = Beff Beff = Bo - RF/

qLtMoMosinqBoRF/ Case II: rotating frame with RF = L magnetization is stationary (precesses in xy with zero frequency)Equation of motion is still valid, i.e. precession frequency gDBeff/2p Beff = 0Equation of motion for M (always valid in any reference fame) in presence of B0Case I: non-rotating reference frame (wRF=0) magnetization precesses in xy plane with frequency gB0/2pmagnetization precesses in xy plane with frequency gDBeff/2pFund BioImag 20129-8Schrdinger representation:

If HS=const in t:NB. Interaction representation (Higher order perturbation theory)Supplement: Rotating frameA brief sketch of quantum-mechanical equivalenciesQuantum mechanics:http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/HigherOrderPT.htmNote: Demodulation of RF to audio frequency is the electrical engineering equivalent

How to determine etc ? Split HS into time-invariant and -dependent terms:

Quantum mechanical equivalencies:Mz, Mx , My For one spin-1/2 (1H), i.e. two energy levels

For spin:

What is VI(t) [wRF=gB0]?

Quantum mechanical equivalencies:B0Iz, B1x,y Ix,yFund BioImag 20129-99-3. Resonance: RF field and flip angle

Rotating frame of reference

Laboratory frame of reference

flip angle a = gB1t [rad]In MRI typically gB1/2p ~ 0.1-1kHz (t~1ms) MxyMxyMzMzB1B1

B1 radiofrequency field at Larmor frequency wL applied in transverse (xy) plane for duration t nutation (at wL) of M as it tips away from the z-axis.RF field rotates M towards xy plane

Amplitude B1 determines how quickly the magnetization is rotated. Fund BioImag 20129-10Quantum mechanical resonance Transition probability highest : hn = hgB0/2pAt Dw=wL-wRF (from wL) magnetization experiences effective field strength Beff

Rotation axis : tilted by q. on resonance: gB1 >> Dw effective field || B1 short RF pulses (t1 T1~ s to min)Boltzmann distribution re-established by energy (photon) transfer from spins to system (lattice).

Fund BioImag 20129-15

along z

along x

along y

B1 : RF field in rotating frameSubstituting W=-gB0+wRF (B0=Bz is not time-dependent) yields:add relaxation terms (T1, T2) to the fundamental Eq of motion of magnetization:Bloch Equations

Rotating reference frame

Felix BlochPhysics1952Fund BioImag 20129-169-5. Free induction decayPrecession and relaxation (after RF pulse)

MxMy

MzMzMxMytttT1T2T2

Transverse magnetizationLongitudinal magnetization(after 900 RF excitation)

Mxy

NB. M can never exceed M0 T2 T1Fund BioImag 20129-17

Measuring T1 changesrepetitive pulsingTR

Converts Mz to Mxy

TRopt = T1

Optimal TR to detect changes in T1 ?Use noise error propagation calculation (Lesson 1)Multipulse experiment with RF pulses applied every TR secondsa=900(signal)

The effect of T1 (and T2) on the signal depends on how it is measured

F=max:t=TRopt:

TR (ms)Fund BioImag 20129-18

20080016006400320030406080120160MR signal intensity depends on relaxation and how it is measuredTE/TR

TE (ms)Gray matterWhite matterFat

TR (ms)Gray matterWhite matterFatFund BioImag 20129-19SummaryMagnetic resonance so farOnly mobile spins (e.g. water) are detectedM0 reflects amount of nuclei and thus water content[Water content varies 70-100ml/100g in body (poor contrast)]Effect of T1 and T2 changes on image contrast depend strongly on experimental parameters (RF pulse timing and flip angle) Magnetic field B0Equilibrium magnetization M0||zproportional tonumber of spins in voxelStatic magnetic field B0gyromagnetic ratio gRF field B1 (applied on-resonance i.e. wL)tilts magnetization M into transverse plane xyPrecession of Mxy is detectedT2 and T1 relaxationexponential decay of Mxy exp. return of Mz to M0reflect molecular environmentsource of contrast Fund BioImag 20129-20Randomization in XY plane: Magnetization DecaysI hNET MagnetizationTransverse T2 RelaxationLongitudinal T1 RelaxationRelaxation Longitudinal and transverseMagnetic fieldA /2 pulse flips the z-magnetization to xy-planeRandomAlignment..tDefocusing; Loss of phase memory; with time constant T2 20