Lecture 22

1. Lecture 22 • Nuclear magnetic resonance

2. A very brief history Stern and Gerlach – atomic beam experiments Isidor Rabi – molecular beam exp.; nuclear magnetic moments (angular momentum) Felix Bloch & Edward Purcell – NM resonance Ernst & Anderson – FT NMR (FT=Fourier transform) Ernst, Aue, Jeener, et al – 2D FT NMR Bloch, Purcell and Ernst have been awarded the Nobel Prize for their work Lauterbur & Mansfield – NMR imaging - the Nobel prize 2003 (adding space coordinate) Actually first body images are due to Raymond Damadian - who discovered different spin relaxation times for tumors.

3. In quantum mechanics orbital momentum, L, is quantized in units of , so it takes discreet values of L=(0,1,2...) where =h/2πand h is the Plank constant. In addition an elementary particle can have internal orbital motion - spin, S, which also takes discreet values. It is quantized in half units of .Spin quantum number J - 0,1/2, 1, 3/2. Proton, neutron have spin 1/2, while nuclei can have a wide range of spins. We discussed that most of the stable nuclei are even-even. In such nuclei spins of protons as well as spins of neutrons are oriented in opposite directions resulting in the total spin equal zero: 4He, 14N, 16O.

4. Spin 1/2 charged point-like particles have magnetic moment which can be calculated in the Dirac theory and (more accurately) in quantum electrodynamics. Protons and neutron have internal quark structure leading to modification of the magnetic moment and in particular to non zero magnetic moment for the neutron. Spinning charged particle or charged particle having orbital motion can be considered as a small magnet generated by a closed current.

5. Magnetic fields of two nucleons with spins in opposite directions cancel: Hence only nuclei with unpaired nucleons have magnetic properties. Nuclear magnetic moment is proportional to spin: Strictly speaking, in QM this is the operator relation, and are operators of magnetic moment and spin. is the gyromagnetic ratio

6. Nuclear Spin - Energy in magnetic field Projection of spin to a given direction is also quantized: corresponding to magnetic quantum number, m, changing between -j and j. For the nucleon Spin cannot be precisely directed in say z direction, there is always a bit of wobbling. which is always larger than

7. Magnetic moment /Spin interacts with magnetic field. Energy of interaction is Corresponding term in the Hamiltonian is leading to correction to the energy of the state

8. ℏwith m=-j,-j+1,...j. For a particle like a proton with s=1/2 there are two possible energy values . The energy difference between UP and DOWN states depends both on magnetic field strength B and gyromagnetic ratio

9. TheZeemaneffectforparticleswithspinj = 1/2 . Inthepresenceofatime-independentexternalmagneticfieldBofmagnitudeB0, theparticlecanoccupytwodifferentenergystates, “spinup” ( ) and“spindown” ( ). TheenergydifferencebetweenbothstatesisproportionaltoB0.

10. The frequency of the photon with energy equal to difference of these energies is This is a resonance condition, and is the Larmor (angular frequency) - very important formula which significance will become clear later For proton, for

12. State Population Distribution Boltzmann statistics provides the population distribution for these two states: N-/N+ = e-ΔE/kTwhere: ΔE is the energy difference between the spin states k is Boltzmann's constant (1.3805x10-23 J/Kelvin) T is the temperature in Kelvin. At physiologic temperature approximately only ~3 in 106 excess protons are in the low energy state for one Tesla field.

13. Net Magnetization Nlower/Nhigher=exp(-ΔE/kT) Example: take 1 billion protons at room temperature(37oC) = k=8.62 x 10-5 eV/oK B0(Tesla) Excess spin 0.15 495 0.35 1155 1.0 3295 1.5 4945 4.0 13,200

14. Alignment in an Applied Magnetic Field This is a dynamical equilibrium where individual protons have nearly random orientation. The stronger the field, the larger the net magnetization and the bigger the MR signal !!!!

15. Important theorem: for description of dynamical equilibrium for a larger collection of protons (sufficiently large voxel) the expected behavior of a large number of spin is equivalent to the classical behavior of the net magnetization vector representing the sum of individual spins. where is the number of protons in the voxel. Operator satisfies equation:

16. Due to symmetry around z axis in the dynamical equilibrium expectation value of the vector M has only z component parallel to B. At the same time expectation value of is not equal to zero. Hence it is instructive to consider also classical picture of the interaction of magnetic dipole with magnetic field. Classical consideration. Consider motion of an atom with angular momentum and associated magnetic moment μin the external magnetic field . The vectors are parallel and

17. The potential energy E is E is minimal if are parallel. Since the potential energy depends only on z coordinate it is clear that it corresponds to a torque force acting on the atom. In difference from QM the z projection of J can take any values between -J and J.

18. Analogy - torque for gyroscope and proton. T=-mg

19. If a particle with angular momentum J and magnetic moment μ is suspended without friction in an external magnetic field B , a precession about B occurs. The angular frequency ω0 of this precession is proportional to B0. For positive γ the precession is clockwise.

20. Motion equation: in Classical Mechanics is the net torque acting on the system; Combining with we obtain: Solution of this eqn is which is exactly the same frequency as we obtained in QM!!!

21. The constants are values of components at t=0. Here I use complex variables. i is imaginary unit. Hence for positive γ , the transverse component of rotates clockwise about z’=z axis with Larmor frequency. This motion is called precession.

22. Rotating frame. Further simplification: use of rotating frame with coordinate axes x’,y’,z’ that rotate clockwise with frequency in which stands still. In this frame an effective magnetic field is zero.

23. Disturbing the dynamic equilibrium: The RF field We discussed above that if the system is placed in the field of strength B, the energy splitting of the levels is given by If the photon with the resonance energy is absorbed by the system spin can flip with system being excited to a higher level Eup. For B= 1 T,

24. RF wave can be generated by sending alternative currents in two coils positions along the x- and y-axes of the coordinate system (in electronics - quadrature transmitter) where is time independent. Denoting as net magnetization To solve this equation we switch to the rotating frame which we discussed before.

25. It is the frame which rotates with angular velocity In this frame the field B does not act on M. At the same time is stationary in the rotating frame. Hence the motion of relative to in the rotating frame is the same as relative to in the case we considered before( in the stationary frame). Consequently, processes around with the precession frequency

26. At t=0 the effective magnetic field lies along the x’ axis, and it rotates away from z=z’ axis along the circle in z’,y’ plane to the y’ axis. The angle between z-axis and is called the flip angle : Hence it is possible to rotate M by any flip angle. If the up-time of the RF field is halved, should be doubled, which implies a quadrupling of the delivered power. Due to electric part of EM field substantial part of it is transformed to heat which limits the increase of .

27. Two important flip angles: • The 90o pulse: brings along y’-axis There is no longitudinal magnetization. Both levels are occupied with the same probability. If pulse is stopped when this angle is reached, in the rest frame will rotate clockwise in x-y plane.

28. The 180o or inverse pulse. is rotated to negative z-axis: QM - the majority of spins occupy the highest energy level. The magnetic field interacts independently with different nuclei, hence the rotations of different nuclei are coherent. This phenomenon is called phase coherence. It explains why in nonequilibrium conditions magnetization vector can have transverse component. (see also animation in the folder - images_mri/spinmovie.ppt)

29. After RF is switched off the process of the return to the dynamical equilibrium: relaxation starts. (a) Spin-spin relaxationis the phenomenon which causes the disappearance of the transverse component of the magnetization vector. On microscopic level it is due slight variations in the magnetic field near individual nuclei because of different chemical environment (protons can belong to H2O, -OH, CH3 , ...). As a result spins rotate at slightly different angular velocity. t=T2 t=0 t=∞ Dephasingofmagnetizationwithtimefollowinga 90oRFexcitation. (a) Att=0, allspinsareinphase (phasecoherence). (b) AfteratimeT2, dephasingresultsinadecreaseofthetransversecomponentto 37% ofitsinitialvalue. (c) Ultimately, thespinsareisotropicallydistributedandthereisnonetmagnetizationleft.

30. Dephasing process can be described by a first order decay model: T2 depends on the tissue: T2 =50ms for fat and 1500 ms for water. See next slide. Spin -spin interaction is an entropy phenomenon. The disorder increases, but there is no change in the energy (occupancy of two levels does not change). (b) Spin-Lattice Relaxation causes the longitudinal component of the net magnetization to change from which is the value of the longitudinal (z) component right after the RF pulse to

31. This relaxation is the result of the interaction of the spin with the surrounding macromolecules (lattice). Process involves de-excitation of nuclei from a higher energy level - leading to some heat release (much smaller than in RF). One can again use the first order model with spin-lattice relaxation time (see figure in next slide). 100 ms for fat; 2000 ms for water

32. (a) Thespin-spinrelaxationprocessforwaterandfat. Aftert = T2, thetransversemagnetizationhasdecreasedto 37% ofitsvalueatt = 0. Aftert = 5T2, only 0.67% oftheinitialvalueremains. (b) Thespin-latticerelaxationprocessforwaterandfat. Aftert = T1, thelongitudinalmagnetizationhasreached 63% ofitsequilibriumvalue. Aftert = 5T1, ithasreached 99.3%.

33. SchematicoverviewofaNMRexperiment. TheRFpulsecreatesanettransversemagnetization, duetoenergyabsorptionandphasecoherence. AftertheRFpulse, twodistinctrelaxationphenomenaensurethatthedynamic (thermal) equilibriumisreachedagain.

35. Signal detection and detector Consider 900 pulse. Right after RF each voxel has a net magnetization vector which rotates clockwise ( in the rest frame). This leads to an induced current in the antenna (coil) placed around the sample. To increase signal to noise ratio (SNR) two coils in quadrature are used.

36. (a) Thecoilalongthehorizontal axismeasuresacosine (b) thecoilalongthe verticalaxismeasuresasine. This is for stationary frame. In moving frame

37. Imaging The detected signal in the case described above does not carry spacial information. New idea: superimpose linear gradient (x-, y-, z- directions) magnetic fields onto z-direction main field. Nobel prize 2003. Allows a slice or volume selection. Slicesinanydirectioncanbeselectedby applyinganappropriatelinearmagnetic fieldgradient. Thisdynamicsequenceshows thefourcardiacchamberstogetherwiththe heartvalvesinaplaneparallelto thecardiacaxis

38. Let us consider an example of slices perpendicular to the z-axis (though any direction can be used). Linear gradient field is characterized by Gradients on millitesla/meter are used. Hence a slice/ slab of thickness contains a well defined range of processing frequencies

39. Let us consider an example of slices perpendicular to the z-axis (though any direction can be used). Linear gradient field is characterized by Gradients on millitesla/meter are used. Hence a slice/ slab of thickness contains a well defined range of processing frequencies

40. Principleofslice-selection. Anarrow-bandedRFpulsewithbandwidthBW = ∆ωisappliedinthepresenceofaslice-selectiongradient. Thesameprincipleappliestoslab-selection, butthebandwidthoftheRFpulseisthenmuchlarger. Slabsareusedin 3Dimaging.

41. Constrains: Gradient cannot be larger than (10-40) mT/m. Hard to generate narrow RF pulse. A very thin slice - too few protons - too weak signal. Small Signal /Noise ratio. Position encoding: the theorem After RF pulse there is a transverse component of magnetization in every point in (x,y). To encode position on the slice additional gradient is applied in x,y plane. For simplicity consider x direction only.

42. In rotating frame this generates rotation of magnetization with a frequency which depends on x: leading to t-depend M: Signal in receiver: where magnetization density Define

43. Can generalize to the case of field depending on x, y,z dependent gradient. Signal in different moments t measures FT for different k’s. Whenapositivegradientinthex-directionisapplied (a), thespatialfrequencykxincreases (b).

44. Relaxation effects can be included in the expression of S(k). Using inverse FT one can restore the density. Illustrationofthek-theorem: (a) Modulusoftherawdata measuredbytheMRimagingsystem (fordisplaypurposes, thelogarithmofthemodulusisshown) (b) Modulusoftheimageobtainedfroma 2DInverse Fourier Transform oftherawdatain (a).

45. Basic pulse sequences What k range is necessary? Let us consider the object of length X and a measurement involving taking N slices. When doing Fourier transform we need several waves in the object. Hence kminX<1, or kmin< 1/X. Also we need to resolve all slices kmax>N/X. So the condition is Many strategies (>100) for scanning k space. I will discuss few of them briefly.

46. The Spin Echo Pulse sequence ky 90o 180o RF Gz Gy kx Gx Signal (a) (b) Figure (b) shows trajectory of k. (a) describes G, RF, Signal Slice selection gradient is applied Gz together with 90 and 180 degree pulses. It leads to compensation of dephasing at t=2TE. Gz leads to dephasing which can be compensated by change of sign of Gz - instead one extend a bit the second Gz pulse. Ladder represents phase-encoding gradient Gy

47. It leads to a y-dependent phase shift of s(t) which depends on time, t. is the constant time when Gy is on. In practical imaging one changes Gy in integer steps: leading to the ladder in the plot. (a) The 2DFLASHpulsesequenceisaGE (gradient - echo)sequenceinwhichaspoilergradientisappliedimmediatelyafterthedatacollectioninordertodephasetheremainingtransversemagnetization. (b) Thetrajectoryofthek-vectorobtainedwiththepulseschemein (a)

48. In 3D imaging one has to use two phase encoding gradient ladders: where is the on - time of the gradient in the slab selection direction. mprage_cor.avi, mprage_sag.avi, mprage_trans.avi, 3D GE image of the brain, shown as a coronal, sagittal and transaxial sequence.

50. Alternate strategy – spiral, much more efficient k-space coverage Gx Gy