1. Fund BioImag 2013 9-1 9: Relaxation of nuclear magnetization 1.How is the MR signal detected ? 2.What is the quantum-mechanical equivalent of the rotating frame ? 3.What is the rotating frame description good for ? 4.How can the return of the magnetization to thermodynamic equilibrium described ? 5.What is the behavior of magnetization in the presence of an external magnetic field ? After this course you 1.Can describe the principle of MR detection and excitation 2.Can explain how MR excitation is frequency selective (resonance) 3.Understand the principle of relaxation to the equilibrium magnetization 4.Know what are the major relaxation times and how they phenomenologically affect magnetization in biological tissue, in particular that of water. 5.Can explain the elements of the Bloch equations and FID 6.Understand the MR contrast strongly depends on experimental parameters

2. Fund BioImag 2013 9-2 MRI contrast depends on experimental parameters I. Time after excitation TE TE=25 ms50 ms75 ms 100 ms

3. Fund BioImag 2013 9-3  II. Flip angle  and time between excitations TR   deg pulse ms

4. Fund BioImag 2013 9-4 What do we know about MR so far ? Need: Nucleus with non-zero spin Magnetic field B 0 Get:Nuclear (equilibrium) magnetization M 0 (Magnitude dictated by Boltzmann distribution)M 0 increases with 1. Number of spins in voxel 2. Magnetic field B 0 3. Gyromagnetic ratio  Imaging 1 H in H 2 O is most sensitive All this does not generate a measurable signal Thermodynamic equilibrium magnetization M 0 is || B 0 M 0 does not precess =0

5. Fund BioImag 2013 9-5 9-1. How is the MR signal detected ? Magnetic field of dipole decreases with distance :  decreases with distance from magnetization Faraday’s Law of Induction Lenz’s Law induced voltage   current → magnetic field opposes the change in the magnetic flux that produces the current (Completely analogous to power generation!) Biot-Savart Law magnetic field falls off with r -2 Magnetic flux  B Induced voltage Bar magnet Magnetization  (t) dA MR: dA=constant →   dB/dt B1B1 I(t) Coil current NB. Generation of the RF field B 1 B 1 decreases with distance from RF coil (Biot-Savart)

6. Fund BioImag 2013 9-6 What does an MR detector look like ? RF Coil examples Distance is important: A coil for every organ!

7. Fund BioImag 2013 9-7 x y z 9-2. Rotating frame revisited Rotating frame: reference frame rotating about z at frequency  RF Larmor frequency  in the rotating frame:  =  B eff  B eff = B o -  RF /   LtLt MoMo M o sin  BoBo  RF /  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  B eff /2    B eff = 0 Equation of motion for M (always valid in any reference fame) in presence of B 0 Case I: non-rotating reference frame (  RF =0)  magnetization precesses in xy plane with frequency  B 0 /2  magnetization precesses in xy plane with frequency  B eff /2 

8. Fund BioImag 2013 9-8 Schrödinger representation: If H S =const in t: NB. Interaction representation (Higher order perturbation theory) Supplement: Rotating frame What are the quantum-mechanical equivalencies ? Quantum mechanics: http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/HigherOrderPT.htm Note: Demodulation of RF to audio frequency is the electrical engineering equivalent How to determine etc ?  Split H S into time-invariant and -dependent terms: Quantum mechanical equivalencies: M z , M x , M y  Quantum mechanical equivalencies: M z , M x , M y  For one spin-1/2 ( 1 H), i.e. two energy levels For spin: What is V I (t) [  RF =  B 0 ]? Quantum mechanical equivalencies: B 0  I z, B 1x,y  I x,y Quantum mechanical equivalencies: B 0  I z, B 1x,y  I x,y

9. Fund BioImag 2013 9-9 9-3. What is the motion of magnetization when an RF field induces a flip angle ? Rotating frame of reference Laboratory frame of reference flip angle  =  B 1  [rad] In MRI typically  B 1 /2  ~ 0.1-1kHz (  ~1ms) M xy MzMz MzMz B1B1 B1B1 B 1 radiofrequency field at Larmor frequency  L applied in transverse (xy) plane for duration   nutation (at  L ) of M as it tips away from the z-axis. RF field rotates M towards xy plane Amplitude B 1 determines how quickly the magnetization is rotated.

10. Fund BioImag 2013 9-10 Quantum mechanical “resonance” Transition probability highest : h = h  B 0 /2  At  =  L -  RF ( from  L ) magnetization experiences effective field strength B eff Rotation axis : tilted by . “on resonance”:  B 1 >>  → effective field || B 1  short RF pulses (  <1ms) B1B1   B eff  “off resonance”: B 1 <<  B → B eff || z What range of frequencies can be excited with a given RF pulse? What is “resonance” ? RF field with amplitude B 1 can excite a range of frequencies on the order of ±  B 1 B eff Magnetization (tip) B eff B1B1  /  B eff  / 

11. Fund BioImag 2013 9-11 9-4. How is the return to equilibrium M 0 governed ? Relaxation Transverse magnetization: (along x and y-axis, on resonance ) x y z M xy B1B1 x y z B1B1   RF pulse(s) B 1 Relaxation T 1, T 2 Thermodynamic equilibriumAfter excitation Exponential decay of M xy Equations formally equivalent to linear attenuation coefficient (x-ray) (same solution) M0M0

12. Fund BioImag 2013 9-12 What are the mechanisms of relaxation ? Tumbling of Molecule (Brownian motion) Creates local oscillating/fluctuating magnetic field in solution, rapid motion averages the dipolar interaction –Brownian motion in crystals, positions are fixed for single molecule, but vary between molecules leading to range of frequencies and broad lines Fluctuating magnetic field depends on orientation of the whole molecule & correlation time  c (=time for molecule to rotate 1 rad) Sources of fluctuating magnetic field: Dipolar coupling between nuclei and solvent interaction between nuclear magnetic dipoles Correlation function G Describes degree of correlation of motion t sec apart Correlation time  c :  : viscosity k: Boltzmann constant r: size of molecule

13. Fund BioImag 2013 9-13 What is the cause of loss of transverse Magnetization ? fluctuating microscopic magnetic fields  B Rule of thumb for tissue water: The less “tissue” (bone, solutes, proteins, membranes) is in contact with bulk water, the longer bulk water T 2 Rule of thumb for tissue water: The less “tissue” (bone, solutes, proteins, membranes) is in contact with bulk water, the longer bulk water T 2 T 2 : phenomenological time constant Range 10µs (bone)… several s (water) “transverse relaxation”, “T2 relaxation” Cause: Molecular dynamics and spin-spin interactions Historically : “spin-spin” relaxation  loss of signal in xy plane “Memory” relaxation M xy x y z After excitation random precession of nuclei  dephasing of spins with time constant ‘T 2 ’ Phase  accrued over  c :  =  B  c  c large (immobile spins): large phase differences  short T 2 bone, membranes, proteins are MR-”invisible”

14. Fund BioImag 2013 9-14 How does M z return to equilibrium ? Longitudinal relaxation T 1 Longitudinal Relaxation (along z-axis) Rule of thumb for water: The less “tissue” is in contact with bulk water (bone, solutes, proteins, membranes), the longer bulk water T 1 Rule of thumb for water: The less “tissue” is in contact with bulk water (bone, solutes, proteins, membranes), the longer bulk water T 1 Mechanisms: Incoherent molecular fluctuations on the order of the Larmor frequency  L possibility of energy transfer  matching frequency x y z After T 2 relaxation After decay of M xy by T 2 : Historically : spin-lattice relaxation (heat lost to the surroundings) T 1 ~ 0.5-5s (water) M z → M 0 m=-1/2 m=1/2 After 90 0 excitation: M z =0  population distribution corresponds to T=  : Most efficient when energy levels of system and nuclear spins match, i.e.  c ~1  T 1 minimal (bone:  c >>1  T 1 ~ s to min) Boltzmann distribution re-established by energy (photon) transfer from spins to system (lattice).

15. Fund BioImag 2013 9-15 along z along x along y B 1 : RF field in rotating frame Substituting  =-  B 0 +  RF (B 0 =B z is not time-dependent) yields: add relaxation terms (T 1, T 2 ) to the fundamental Eq of motion of magnetization: What equations describe the change in magnetization ? Bloch Equations Rotating reference frame Felix Bloch Physics 1952

16. Fund BioImag 2013 9-16 9-5. What characterizes the basic MR signal ? Free induction decay: Precession and relaxation (after RF pulse) MxMx MyMy MzMz MzMz MxMx MyMy t t t T1T1 T2T2 T2T2 Transverse magnetization Longitudinal magnetization (after 90 0 RF excitation) M xy NB. M can never exceed M 0  T 2  T 1

17. Fund BioImag 2013 9-17 How can T 1 changes be measured ? repetitive pulsing TR Converts M z to M xy TR opt = T 1 Optimal TR to detect changes in T 1 ? Use noise error propagation calculation (Lesson 1) Multipulse experiment with RF pulses applied every TR seconds  =90 0 (signal) The effect of T 1 (and T 2 ) on the signal depends on how it is measured F=max: t=TR opt : TR (ms)

18. Fund BioImag 2013 9-18 200800160064003200 30 40 60 80 120 160 MR signal intensity depends on relaxation and how it is measured TE/TR TE (ms) Gray matter White matter Fat TR (ms) Gray matter White matter Fat

19. Fund BioImag 2013 9-19 Summary Magnetic resonance so far 1.Only mobile spins (e.g. water) are detected 2.M 0 reflects amount of nuclei and thus water content [Water content varies 70-100ml/100g in body (poor contrast)] 3.Effect of T 1 and T 2 changes on image contrast depend strongly on experimental parameters (RF pulse timing and flip angle) Magnetic field B 0 Equilibrium magnetization M 0 ||z proportional to 1.number of spins in voxel 2.Static magnetic field B 0 3.gyromagnetic ratio  1.number of spins in voxel 2.Static magnetic field B 0 3.gyromagnetic ratio  RF field B 1 (applied on-resonance i.e.  L ) tilts magnetization M into transverse plane xy Precession of M xy is detected T 2 and T 1 relaxation exponential decay of M xy exp. return of M z to M 0 reflect molecular environment source of contrast reflect molecular environment source of contrast