MRI signal origin
Magnetic Resonance Imaging
Magnetic Resonance Imaging (MRI) is an non-ionizing and non-invasive imaging technology that uses magnetic fields, generated by a magnet and radio-waves, to view internal organs and produce diagnostic images of the (human) body. The detailed anatomic images are generated by state-of-the-art electronics and computers.
Example images from: www.wikipedia.org
Examples of MRI functional MRI micro MRI
Diffusion Weighted MRI MR Angiography
Examples of MRI
Volume
Dynamic
MRI system
MRI system
Brief history of Magnetic Resonance (MR)
• The MR phenomenon was first discovered simultaneously and independently by Felix Bloch and Edward Purcell in 1946 (Nobel Prize in 1952).
• MR was & still is a very important technique for non-destructive spectroscopic analyses of chemical composition of samples (i.e. signal based).
• Everything changed in 1970ies when Paul Lauterbur and Sir Peter Mansfield employed magnetic field gradients to spatially localize the resonating nuclei and produce first Magnetic Resonance (MR) images.
• Both scientists received the Nobel Prize in Physiology and Medicine in 2003 for their discoveries concerning Magnetic Resonance Imaging (MRI).
Composition of nuclei
All nuclei are composed of two types of particles: protons and neutrons. The following quantities characterize an atomic nucleus:
• Z number of protons in the nucleus (also called atomic or charge number). • N number of neutrons in the nucleus.
• A number of neutrons and protons (also called mass number)
The symbol used: ZA X
where X represents the chemical symbol of the element.
N can be calculated as follows: N=A-Z
Above: Illustration of a nucleus, where protons and neutrons are shown in red and blue respectively.
Isotopes
• The nuclei of all atoms can contain a different number of neutrons. They are referred to as isotopes. The isotopes of an element thus have the same Z value but different N and A values. For example, carbon has four isotopes:
11 12 13 and 14 6C 6C 6C 6C
• The natural abundances of isotopes differ greatly. For example 12C is 98.9% 12 6
abundant naturally. As shall be seen later, 6 Cdoes not give rise to an MR signal. 13C is used in spectroscopy, but with a natural abundance of 1.1%, it is rarely used
6
in MRI.
Nuclear spin
• Another fundamental quantity of the nucleus is spin. Both protons and neutrons have spin within the nucleus. The nucleus is spinning and therefore has an inherent angular momentum.
• The nuclear spin is designated by a quantum number, I, which is proportional to the angular momentum, J:
J = hI 2p
where h is the Planck’s Constant: h = 6.625 x 10-34 Js
• Experimentally the values of I are: integers (if A is even) or half-integers (if A is odd) ranging from zero in 12C up to seven in 176Lu . It has been noted practically that all even-even nuclei (even number of neutrons and protons, N = Z) have I = 0.
Nuclear magnetic moment
• The nuclear angular momentum has a magnetic moment μ, associated with it. • This magnetic moment is given by:
æ hI ö
μ = g çè 2 p ÷ø = g J
• Here g is the nuclear gyromagnetic ratio. This is a characterization constant for each nucleus.
• For the water hydrogen proton (most MRI applications), g = 2.675 x 108 rad/s/T.
• As rad/s can be converted to Hz by dividing by 2p, as we will see later the constant
‘gamma-bar’ is convenient for relating frequency to magnetic field strength, so:
g=g =42.58MHz/T 2p
The net magnetic moment M0
• Where more than one nucleus is present, the net direction and magnitude of the magnetic moment is obtained by summing all the individual cases as three dimensional vectors.
• The components of the net magnetization M0 can be expressed as follows: Mx =åμx
My =åμy Mz =åμz
• The net magnetization is then given by:
M=M =(M)+(M)+(M) 0 tot x2 y2 z2
Magnetic field
• In physics, a magnetic field is a field that permeates space and which exerts a magnetic force on moving electric charges. Magnetic fields surround electric currents and changing electric fields.
• In classical physics, the magnetic field B is a vector field (that is, some vector at every point of space and time), with SI unit of Tesla, T, where 1T=1Ns/(Cm). Being a vector quantity, the magnetic field has both magnitude and direction.
Application of static magnetic field to the spin system
• In the absence of the static magnetic field B0, the net magnetic moment M0 has no preferred orientation as the individual magnetic moments point in different directions. In that case the net magnetic moment is zero.
• When a strong static magnetic field B0 is applied to the spin system, the magnetic moments orient either parallel or anti- parallel to the direction of the applied magnetic field. The net magnetic moment is non-zero and pointing in the direction of the magnetic field B0.
Application of static magnetic field
• When a static magnetic field B0 is applied, the magnetic moments align in the field with certain allowed orientations, called mI.
• The number of orientations of a nucleus with spin I, is 2I+1. For spin 1⁄2 nuclei, the number of orientations is (2×1/2+1=2) and these orientations are called parallel and anti-parallel, refereed to as +1/2 and –1/2.
• Initially, in the absence of the magnetic field, there is no energy difference between the parallel and anti-parallel state.
• As the strength of B0 is increased, the energy difference between the parallel and anti-parallel states increases
• For small magnetic fields, the energy level splitting increases linearly with the size of the applied magnetic field B0.
Application of static magnetic field
• The Zeeman energies of the water hydrogen proton spins in each quantum
state are:
E parallel = – 12 !gB0 1 Eanti- parallel = 12 !gB0 2
• Therefore the energy difference between the two states is given by:
1æ1ö DE=Eanti-parallel-Eparallel =2!gB0-çè-2!gB0÷ø=!gB0 3
• Using Planck’s well-known relation: Applying 3 = 4 we get:
!w0 = !gB0
•orequivalently: 2pf =gB
DE=hf = h ×2pf =!w 2p
4 5
6
w0 =gB0
so f = g B0 =gB0 0 2p
Application of static magnetic field
• The effect of a magnetic field on an ensemble of nuclei having nuclear spin of 1⁄2 is summarized in the following figure.
MRI coordinate system
• By convention we specify that the magnetic field, B0, is applied in the z-axis.
• It is not applied along the x or y axes.
Coordinate system used in MRI For open magnets, B0 is in the For a solenoid (cylindrical) magnet, B0 vertical direction. is along the axis of the magnet and so the coordinate system must be rotated
MRI coordinate system
Precession of M0
• When the net magnetization M0 is not perfectly aligned with the external B0 – field, a torque is exerted on the magnetization, which is perpendicular to both the magnetization and B0 – field. This torque attempts to align M0 with the B0 field.
• However, the magnetic moment also has an angular momentum, which causes the nucleus to precess (rotate around B0). The frequency of precession can be shown quantum-mechanicallytobetheLarmorfrequency(w0 =gB0).
z
Precession of M0 w0 =gB0
M0
z
Laboratory vs. rotating frame of reference
• Another important concept regularly employed in MRI is the rotating frame of reference (x’, y’, z), which is assumed to rotate around the z-axis at the Larmor precessional frequency. As long as the applied static magnetic field is exactly equal to B0, the magnetization vector will be perceived as stationary in the rotating frame of reference.
z
(a) Magnetization precession around the static magnetic field in the laboratory frame of reference and (b) absence of rotation in the rotating frame of reference.
• Here as animation:
Laboratory vs. rotating frame of reference
laboratory frame of reference
Courtesy Shaihan Malik: http://mriphysics.github.io
B0
• Here as animation:
Laboratory vs. rotating frame of reference
Courtesy Shaihan Malik: http://mriphysics.github.io
B0
rotating frame of reference
Radio-frequency (RF) pulse application
• The act of applying a radio-frequency (RF) pulse for a fixed period of time causes the magnetization to tip.
• The magnetization initially aligned along the z-axis tips to some other direction. • For a 90o pulse the magnetization tips into the x – y (transverse) plane.
Rotating frame of reference
B0
Radio-frequency (RF) pulse application
• An RF field is a combination of an oscillating electric and an oscillating magnetic field perpendicular to each other.
• The magnetic field, B1, of the RF pulse must be perpendicular to the external magnetic field, B0. In this way B1 can exert a torque on the magnetization which is aligned with static field B0 . The B1 field is only able to exert a constant direction torque if it is at the same frequency as the Larmor precessional frequency, w0.
The net magnetization undergoes a spiralling motion under the influence of the oscillating B1 (here: 90o pulse) and static B0 field: (a) in the laboratory frame of reference; (b) in the rotating frame of reference.
Pulse angles
• The angle, q, by which the magnetization is tipped away from the z-axis, is dependent on the power of the RF pulse and the length of the time that it is applied.
•The resulting ‘transverse magnetization’ has magnitude M0 and begins to precess clockwise in the x – y plane. The components along the x- and y-axes have sinusoidal time dependence with a frequency, w0 (Larmor frequency).
• The RF pulse that tips all the original longitudinal or z-magnetization through an angle of 90o into the transverse, or x – y plane, is called a p/2 or 90o pulse.
Signal reception and relaxation
• After the B1 pulse, the tipped net magnetization will recover back to equilibrium of M0 (i.e. low-energy state) under the sole effect of external B0.
• If one situates a current loop in the y – z plane, the change in the magnetic flux, as result of magnetization recovery, will induce a small current of frequency w0 in the conducting loop, which can then be amplified and post-processed.
• The current within the receiving loop decays exponentially and the measured signal is called free induction decay (FID).
Signal detection
• Once the magnetization has a transverse component, it can be detected.
• It is the spin’s own magnetic field lines that are swept along with the
precession which forms the key to observing the net magnetization.
• Therefore, the physical principles of MR signal detection are derived from Faraday’s law of electromagnetic induction.
• An EMF (electromotive force) is created in any coil through which the spin’s magnetic flux sweeps – i.e. current is induced in the conducting coil.
EMF = – d Fm (t) dt
Signal detection
This is analogous to a magnet moving in & out of a coil.
Signal reception and relaxation
•Two processes are present that drive the net magnetization back to equilibrium of M0 under the sole influence of B0:
• Spin-lattice relaxation } • Spin-spin relaxation }
• B0 inhomogeneity
• The following slides detail these two relaxation processes.
process I
(T1 relaxation)
process II
(T2* relaxation)
Spin – lattice relaxation
• Directly after the application of the B1 pulse, the spins are in a high energy state. The excited spins then exchange the ‘extra’ energy with the neighbouring tissue, resulting in exchange of thermal energy.
• As result of this thermal exchange process, the longitudinal magnetization Mz recovers exponentially to the thermal equilibrium state of M0 with a characteristic time constant T1. The energy exchange rate relates to the mobility of spins within the medium and the value of T1 is thus tissue dependent.
• This process is called spin-lattice relaxation (or T1 relaxation) and it brings the longitudinal magnetization back to its original state that existed before the B1 pulse was applied.
Spin – spin relaxation
• Directly after the B1 pulse excitation, all spins should ideally precess around the z- axis (in the x – y plane) at the same Larmor frequency with their magnetic moments in phase, adding up to the maximum induced signal as the magnetization recovers back to its equilibrium state under the effect of B0.
• However, the macro-molecules and other spins in the tissues generate their own magnetic fields that locally perturb the uniform static magnetic field due to the external source.
• As a result, the excited spins will experience a locally non-uniform magnetic field, which causes some spins to precess faster and others slower compared to the ideal Larmor precession (frequency), resulting in loss of phase coherence in the magnetic moments and thus to the decay of transverse magnetization Mxy.
•This decay effect is known as spin-spin relaxation (or T2 relaxation) and is characterized by an exponential time constant T2 that varies between tens and hundreds of milliseconds and is, like T1, tissue dependent.
B0 inhomogeneity
• The Larmor frequency is dependent on the magnetic field strength B0: w0 = gB0
• Therefore, the Larmor frequency of all nuclei (of the same isotope) in a sample will only be the same if the magnetic field is constant.
• In the real world, it is not possible to produce an absolutely homogeneous magnetic field. A consequence of this is the dephasing of magnetic moments with a characteristic exponential time constant T2’ and hence additional loss of transverse magnetization.
• In a typical MRI system, the homogeneity of the magnetic field, across the sample is of the order of 1 part variation in 5 x 105. For protons at 1.5T, where the resonant frequency is 63.75MHz, the range of frequencies observed may be 63.749936 – 63.750064MHz. That is, the bandwidth of the response is 128Hz.
•The combined dephasing effect due to spin-spin relaxation and local B0 inhomogeneity is known as the T2* relaxation. The relationship between the time
constants can be expressed as:
1=1+1
T * T2 T2 ‘ 2
Dephasing effect of T2* relaxation
• The result of T2* effect is that some nuclei will precess around the static field B0 at a higher frequency, w0+d, and some will precess at a lower frequency, w0-d i.e. the spins soon disperse.
• At some point all magnetic moments are out of phase and cancel each other out, leading to no net transverse magnetization.
Laboratory frame Rotating frame
Mxy =M0e-t/T2
* M =M (1-e-t/T ) z 0 1
T1 and T2* relaxation
After a 90o pulse: (a) decay of transverse magnetization Mxy.
(b) recovery of the longitudinal magnetization Mz;
Transverse relaxation in MR images
• Much of the power of MRI lies in the inherent contrast provided by different relaxation characteristics of tissue.
• For example, T2- weighted images are acquired with a sequence which is sensitive to the T2 relaxation time.
• In the MR image below, cerebrospinal fluid (CSF) has a long T2, due to the freedom of the water molecules to tumble. So CSF appears bright.
Appendix
FYI: Diamagnetic levitation
Courtesy of HFL lab Radboud Uni Nijmegen: https://www.ru.nl/hfml/research/levitation/diamagnetic-levitation/
MR Signal excitation
• It is already established that the net magnetization vector M0 points in the direction of the external static field B0 (i.e. along z-axis), where the spins are in a low energy state.
• In this equilibrium configuration, the longitudinal magnetization is Mz = M0, and there are no transverse (Mx or My) magnetization components. With a B1 excitation pulse along the x’-axis, the net magnetization vector can be flipped by 90o from the z- axis to the y’-axis.
• In this non-equilibrium configuration, Mz = 0 while the net magnetization vector is My. In general, the transverse magnetization is written:
Mxy =Mx +jMy where j= -1