Lecture 21 Nuclear magnetic resonance
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.
In quantum mechanics orbital momentum, L, is quantized in units of so it takes discreet values of L=(0,1,2...) where and h is the Plank h 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 h h h 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: 4 He, 14 N, 16 O.
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.
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: ˆµ = γŝ Strictly speaking, in QM this is the operator relation, ˆµ and Ŝ are operators of magnetic moment and spin. γ is the gyromagnetic ratio
Nuclear Spin - Energy in magnetic field Projection of spin to a given direction is also quantized: s h,(s 1) h,...( s + 1) h, s h corresponding to magnetic quantum number, m, changing between -j and j. For the nucleon s = 1 2 h,s z = ± 1 2 h Spin cannot be precisely directed in say z direction, there is always a bit of wobbling. S 2 = S 2 z + S 2 x + S 2 y = S(S + 1) h 2 which is always larger than < S 2 z > S 2 h 2
Magnetic moment /Spin interacts with magnetic field. Energy of interaction is E = µ B Corresponding term in the Hamiltonian is H = ˆ µ ˆ B leading to correction to the energy of the state
ΔE = µb 0 = mγb 0 with m=-j,-j+1,...j. For a particle like a proton with s=1/2 there are two possible energy values. E up = 1 2 hγb 0 E down = 1 2 hγb 0 The energy difference between UP and DOWN states depends both on magnetic field strength B and gyromagnetic ratio γ
The Zeeman effect for particles with spin j = 1/2. In the presence of a timeindependent external magnetic field B of magnitude B0, the particle can occupy two different energy states, spin up ( ) and spin down ( ). The energy difference between both states is proportional to B0.
E up E down = hγb 0 The frequency of the photon with energy hω 0 equal to difference of these energies is ω 0 = γb 0 This is a resonance condition, and ω 0 is the Larmor (angular frequency) ω 0 = γb 0 - very important formula which significance will become clear later For proton, for B 0 = 1Tesla,ω 0 = 42.6MHz
ΔE = hf L Protons moving from low to high energy state require radiofrequency. Protons moving from high to low energy release radiofrequency.
State Population Distribution Boltzmann statistics provides the population distribution for these two states: N - /N + = e -ΔE/kT where: 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 10 6 excess protons are in the low energy state for one Tesla field.
Net Magnetization Nlower/Nhigher=exp(-ΔE/kT) Example: take 1 billion protons at room temperature(37 o C) = k=8.62 x 10-5 ev/ o K B0(Tesla) Excess spin 0.15 495 0.35 1155 1.0 3295 1.5 4945 4.0 13,200
Alignment in an Applied Magnetic Field Bo 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!!!!
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. M 0 = n s µ i, i=1 where n s is the number of protons in the voxel. Operator M 0 satisfies equation: d M 0 dt = M 0 γ B
Due to symmetry around z axis in the dynamical equilibrium expectation value of the vector M has only z component 2 parallel to B. At the same time expectation value of < M 0 > 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 μ in the external magnetic field B =(0,0,B 0 ). The vectors J and µ are parallel and µ = γ J J
The potential energy E is E = µ B = µb 0 cosθ = γjb 0 cosθ E is minimal if B and µ 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.
Analogy - torque for gyroscope and proton. In a Static Field Gyroscope in a gravitational field Proton in a magnetic field M B 0 mg T=-mg
If a particle with angular momentum J and magnetic moment is suspended without friction in an external magnetic field B, a precession about B occurs. The angular frequency of this precession is proportional to B. For positive the precession is clockwise.
where τ Motion equation: in Classical Mechanics d J dt = τ is the net torque acting on the system; in our case τ = µ B. Combining with µ = γ J d µ dt = µ γ B. we obtain: Solution of this eqn is µ x (t)=µ x (0)cos(ω 0 t)+µ y (0)sin(ω 0 t) with ω 0 = γb 0 obtained in QM!!! µ y (t)=µ y (0)cos(ω 0 t) µ x (0)sin(ω 0 t) µ z (t)=µ z (0) which is exactly the same frequency as we
The constants µ x (0), µ y (0), µ z (0) are values of components at t=0. Let µ xy (t)=µ x (t)+iµ y (t), and µ xy (0)=µ x (0)+iµ y (0). Here I use complex variables. i is imaginary unit. µ xy (t)=µ xy (0)e iω 0t Hence for positive γ, the transverse component of µ rotates clockwise about z =z axis with Larmor frequency. This motion is called precession.
Rotating frame. Further simplification: use of rotating frame with coordinate axes x,y,z that rotate clockwise with frequency ω 0 in which µ stands still. In this frame an effective magnetic field is zero.
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 E up E down = hγb 0 If the photon with the resonance energy ω 0 = γb 0 is absorbed by the system spin can flip with system being excited to a higher level E up. For B= 1 T, ω 0 = 43.57 MHz
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) B 1 (t)=b 1 (cos(ω 0 t), sin(ω 0 t),0) = B 1 cos(ω 0 t) ib 1 sin(ω 0 t),0)=b 1 e iω 0t. =( where B 1 is time independent. Denoting M as net magnetization d M dt = M γ( B + B 1 (t). ) To solve this equation we switch to the rotating frame which we discussed before.
It is the frame which rotates with angular velocity In this frame the field B does not act on M. ω 0 At the same time B 1 is stationary in the rotating frame. Hence the motion of M relative to B 1 in the rotating frame is the same as M relative to B in the case we considered before( in the stationary frame). Consequently, M processes around B 1 with the precession frequency ω 1 = γb 1
At t=0 the effective magnetic field lies along the x axis, and it rotates M away from z=z axis along the circle in z,y plane to the y axis. The angle between z-axis and M is called the flip angle : α = Z t 0 α γb 1 dτ = γb 1 τ = ω 1 t. Hence it is possible to rotate M by any flip angle. If the uptime of the RF field is halved, B 1 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 B 1.
Two important flip angles: The 90 o pulse: brings M along y -axis M =(0,M 0,0) There is no longitudinal magnetization. Both levels are occupied with the same probability. If pulse is stopped when this angle is reached, M in the rest frame will rotate clockwise in x-y plane.
The 180 o or inverse pulse. M is rotated to negative z-axis: M =(0,0, M 0 ) 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)
After RF is switched off the process of the return to the dynamical equilibrium: relaxation starts. (a) Spin-spin relaxation is 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 H 2 O, -OH, CH 3,...). As a result spins rotate at slightly different angular velocity. t=0 t=t 2 Dephasing of magnetization with time following a 90 o t= RF excitation. (a) At t=0, all spins are in phase (phase coherence). (b) After a time T 2, dephasing results in a decrease of the transverse component to 37% of its initial value. (c) Ultimately, the spins are isotropically distributed and there is no net magnetization left.
Dephasing process can be described by a first order decay model: M tr (t)= M tr (0)e t/t 2 T 2 depends on the tissue: T 2 =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 M 0 cosθ which is the value of the longitudinal (z) component right after the RF pulse to M 0
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). M l (t)=m 0 cosαe t/t 1 + M 0 (1 e t/t 1 ) T 1 = T 1 = 100 ms for fat; 2000 ms for water
(a) The spin-spin relaxation process for water and fat. After t = T 2, the transverse magnetization has decreased to 37% of its value at t = 0. After t = 5T 2, only 0.67% of the initial value remains. (b) The spin-lattice relaxation process for water and fat. After t = T 1, the longitudinal magnetization has reached 63% of its equilibrium value. After t = 5T 1, it has reached 99.3%.
Schematic overview of a NMR experiment. The RF pulse creates a net transverse magnetization, due to energy absorption and phase coherence. After the RF pulse, two distinct relaxation phenomena ensure that the dynamic (thermal) equilibrium is reached again.
Signal detection and detector Consider 90 0 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.
(a) The coil along the horizontal axis measures a cosine (b) the coil along the vertical axis measures a sine. s x (t)=ae t/t 2 cos( ω 0 t) s y (t)=ae t/t 2 sin( ω 0 t) s(t)=ae t/t 2 e iω 0t This is for stationary frame. In moving frame s(t)=ae t/t 2
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. Slices in any direction can be selected by applying an appropriate linear magnetic field gradient. This dynamic sequence shows the four cardiac chambers together with the heart valves in a plane parallel to the cardiac axis
Let us consider an example of slices perpendicular to the z-axis (though any direction can be used). Linear gradient field is characterized by G =(G x,g y,g z )= ( 0,0, B z z Gradients on millitesla/meter are used. ω(z)=γ(b 0 + G z z) Hence a slice/ slab of thickness contains a well defined range of processing frequencies Δω = γg Z Δz Δz )
Let us consider an example of slices perpendicular to the z-axis (though any direction can be used). Linear gradient field is characterized by G =(G x,g y,g z )= ( 0,0, B z z Gradients on millitesla/meter are used. ω(z)=γ(b 0 + G z z) Hence a slice/ slab of thickness Δz contains a well defined range of processing frequencies Δω = γg Z Δz )
Δz = BW γg Z Principle of slice-selection. A narrow-banded RF pulse with bandwidth BW = ω is applied in the presence of a slice-selection gradient. The same principle applies to slab-selection, but the bandwidth of the RF pulse is then much larger. Slabs are used in 3D imaging.
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 k 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.
In rotating frame this generates rotation of magnetization with a frequency which depends on x: leading to t-depend M: ω(x)=γg x x M tr (x,y,z)=m tr (x,y,0)e iγg xxt Signal in receiver: s(t)= Z Z ρ(x,y)e iγg xxt dxdy where ρ(x, y) Define magnetization density k x = γ 2π G xt s(t)=s(k x )= Z Z ρ(x,y)e i2πk xx dxdy
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. When a positive gradient in the x-direction is applied (a), the spatial frequency k x increases (b).
Relaxation effects can be included in the expression of S(k). Using inverse FT one can restore the density. S(k x,k y ) Illustration of the k-theorem: (a) Modulus of the raw data measured by the MR imaging system (for display purposes, the logarithm of the modulus is shown) (b) Modulus of the image obtained from a 2D Inverse Fourier Transform of the raw data in (a).
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 k min X<1, or k min < 1/X. Also we need to resolve all slices k max >N/X. So the condition is k max > 1/ΔX, k min < 1/X Many strategies (>100) for scanning k space. I will discuss few of them briefly.
The Spin Echo Pulse sequence RF G z G y Gx 90 o 180 o k y k x Signal (a) (b) Figure (b) shows trajectory of k. (a) describes G, RF, Signal Slice selection gradient is applied G z 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
It leads to a y-dependent phase shift time, t. T ph φ(y)=γg y T ph φ(y) of s(t) which depends on is the constant time when Gy is on. In practical imaging one changes Gy in integer steps: G y = mg y leading to the ladder in the plot. (a) The 2D FLASH pulse sequence is a GE (gradient - echo) sequence in which a spoiler gradient is applied immediately after the data collection in order to dephase the remaining transverse magnetization. (b) The trajectory of the k-vector obtained with the pulse scheme in (a)
In 3D imaging one has to use two phase encoding gradient ladders: φ(y,z)=γ(mg y yt ph + ng z zt ss ) where T ss 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.
Alternate strategy spiral, much more efficient k-space coverage G x G y
Imaging of moving spins. Previous discussion assumed that atoms do not move. Problem in case of blood, breathing,... as during the pulse atoms moves relative to the magnetic field and hence the resonance frequency will change. The total phase shift at time TE is Φ m = Expanding in Taylor series Z TE 0 γ G(t) r(t)dt r(t)= r(0)+ d r dt (0)t +... = r(0)+ v(0)t + a(0)t2 /2 +... Φ m = r 0 Z TE 0 γ G(t)dt + v 0 Z TE 0 γ G(t)tdt + a 0 Z TE 0 γ G(t) t2 dt +... 2 or Φ m = r 0 m(0)+ v(0) m(1)+ a(0) m(2)+... where m l = Z TE 0 γ G(t) tl l! dt
This motion induced dephasing is another source of dephasing - it leads to suppression of signal from blood vessels and to artifacts like ghosting. Ghosting is a characteristic artifact caused by periodic motion. In this T1- weighted SE image of the heart, breathing, heartbeats and pulsating blood vessels yield ghosting and blurring. Note that blood flow yields a total dephasing and thus dark blood vessels.
Vessel Signal Voids Early multi-slice spin echo images depicted vessels in the neck as signal voids
Magnetic Resonance Angiography. Main idea: choose the pulses which will put the lowest moments m l = Z TE 0 γ G(t) tl l! dt to zero: m 0 = 0, m 1 = 0 For example: use (b) instead of (a).
An example of practical sequence Schematic illustration of a 3D FLASH based Time Of Flight sequence. First order flow rephasing gradients are applied in the volume-selection and frequency-encoding directions to prevent the dephasing that otherwise would be caused by the corresponding original gradients.
Bonus of the procedure - all stationary components loose coherence and so the signal from them becomes wicker. Bright Blood Images Using gradient (field) echo images with partial flip angles allowed blood which flowed through the 2D image plane to be depicted as being brighter than stationary tissue.
MRI allows to study various types of motion: diffusion 0.01-0.1 mm/s, perfusion 0.1-1 mm/s, cerebrospinal fluid flow 1mm/s -1cm/s, venous flow 1-10 cm/s, arterial flow 10-100 cm/s, stenotic flow 1-10 m/s. - SIX orders of magnitude!!!
To visualize the results one often uses maximum intensity projection (MIP) (a) (b) (a) Illustration of the MIP algorithm. A projection view of a 3D data set is obtained by taking the maximum signal intensity along each ray perpendicular to the image. (b) MIP of a 3D MRA data set of part of the brain.
Willis_mip.ave MIP projections of a high resolution 3D time-of-flight acquisition of the intracranial arteries (without contrast). A 3D impression is obtained by calculating MIPs from subsequent directions around the vascular tree and quickly displaying them one after the other.
mra_ax mra_sag mra_cor mra_mip To study a complex flow of blood a contrast with enriched concentration of protons is injected. In the videos contrast enhanced 3D MR angiography of the thoracic vessels. (a) Axial, (b) sagittal (c) coronal view and (d) maximum intensity projection are presented.
Normal runoff MRA in a 30 year old male Image of tissue surrounding vessel can be manually striped off http://www.uth.tmc.edu/radiology/publish/mra/55_male.html