Introduction to MRI Physics

Ian Miller

July 11, 2007

Goal of Today’s Lecture• Relate the concepts of MRI physics to things that engineers

and neurophysiologists already understand• Lay the groundwork for more detailed understanding of

more complicated imaging techniques• Fast spin echo• Echo planar imaging• Volume imaging• 3D time of flight• Diffusion• …

Why this is an Intimidating Topic• There are a myriad of basic physics principles involved,

each of which is individually important and hard to skip over• Need to simultaneously consider two scales

• single particles• aggregates of billions of particles

• There are multiple dimensions involved to keep straight• The math is complicated for non-engineers, and most of it is

vector-based• It’s easy to convince yourself you “get it”

Organization of Discussion• Review of Relevant Basic Physical Principles• Magnetism• Resonance• Image Creation• T1 and T2• Anatomy of an MR Scanner• Future Topics

Current is the Flow of Charged Particles• Abbreviated “I”

dt

dQI

Where• Q is charge• t is time

sodium atom

sodium ions

123456

dt

dQI

6 atoms

3 seconds

Every Current Creates a Magnetic Field

• The current and the magnetic field have to be perpendicular at all points. Therefore• Straight currents magnetic field loops surrounding them• Current loops have straight magnetic fields going through them

I

B

BI

Thermal Energy is Stored as Molecular Motion• At 0 K, all molecular motion ceases.• As heat is added, the molecules move around, absorbing the heat in various

ways:

• The second law of thermodynamics requires that all available mechanisms of thermal energy storage be used

• For simpler molecules, the available options are fewer

Protons are Spinning Charges• Protons have charge and are constantly spinning

• The charge can be thought of as distributed

This is a magnetic moment

Precession• Precession refers to a change in the direction of the axis of

a rotating object.• torque-free• torque-induced

• It occurs when spinning objects experience a moment outside the plane of rotation

Electromagnetic Radiation is Just Light• It’s all made up of photons• It all moves at the same speed• The difference between light we see (visible electromagnetic

radiation) and any other type is the frequency at which the photon oscillates

radio waves

x-rays

visible light

Light Absorption by e- is Quantized• Recall valence shell electron theory from high school

chemistry

• You can get an electron to “jump up” to the next level by supplying energy at the exact wavelength required

Absorption Spectra Show Quantization

visible light

The parts that are missing were absorbed by the

electrons

Emission Spectra Show Quantization

hE

Absorption Experiment

Emission Experiment

The requirement for the precise amount of energy needed is quantization

Exponential Change is Convenient to Study• Exponential change occurs when the rate of change of a

quantity is proportional to the quantity itself

kQdt

dQ

• k can be• Positive (exponential growth)• Negative (exponential decay)

• We should love exponential change because it is relatively easy to study

• If you plot the quantity against time, k can be readily calculated with a few data points, and there is only one degree of freedom

• You often already know two boundary conditions• q at time zero• q at time infinity

• You only need one more!

Recap of Review Material• Current is the Flow of Charged Particles• Every Current Creates a Magnetic Field

• Perpendicular at all points to the current• Thermal Energy is Stored as Molecular Motion• Protons are Spinning Charges

• Protons have a current loop• Protons have a magnetic moment along the axis of rotation

• Precession• Occurs when a spinning object experiences a moment out of the

plane of rotation• Electromagnetic Radiation is Just Light• Light Absorption by particles is Quantized• Absorption Spectra Show Quantization• Emission Spectra Show Quantization• Exponential Change is Convenient to Study

Now we need to scale up to the bulk / macroscopic scale

Magnetic Resonance Imaging• At rest, all protons spin (and translate) because of the presence of thermal

energy. The proton of a single hydrogen atom in the vacuum of space will spin for this reason.

• Entropy dictates that the spins within a group of protons are not organized

• Going from a single proton to a group of protons will yield all possible orientations (which sum to zero)

Net Magnetic Field (M)

Net Magnetic Field (M)

MRI: Application of a Magnetic Field• Now let’s return to what happens with a proton when you

apply an external magnetic field

• There are two effects here• Alignment of spins

with the external magnetic field

• Precession, because the moment experienced by the proton is out of the plane of rotation

They are related, but different

B0

MRI: Determinants of Spin Rate• The speed of precession of a spinning body in a field is

called the Larmor frequency, and we know a few things about it• Zero when B0 = 0• Increases as magnetic field increases• We could do an experiment and plot the relationship

between B0 and precessional frequency

• For protons wL is approximately 42 MHz/Tesla

0BL • Larmor freqency, and is dictated by

wL

B0

Slope = gyromagnetic ratio

B0

MRI: Scaling up to PopulationsPopulation of ProtonsSingle Proton

No

Ext

erna

l Fie

ldE

xter

nal F

ield

= B

0

M M

MM

Simplification• We can’t stop a proton from spinning, so let’s simplify our diagram

will now be

will now be

MRI: Scaling up to Populations

• In a big population of protons, more line up with the field than against, but there is a distribution of both

• Thermodynamics will tell us what the ratio is

M

Ext

erna

l Fie

ld =

B0

B0

Difference in Energy Levels

100,000

100,000

B0

1 T, 37C

100,000

100,006

kT

E

N

N

e k = Boltzmann Constant

100,000

100,0020

100,000

100,050

Net = 0

This is 500 ppm (small!)

MRI works because we have Avagadro’s numbers of protons

We’ve Seen This Before

hE

• The excitation of proton spins is a quantized system• So what is the frequency (v) needed to cause this excitation?

MRI: Combining Precession and Quanta

• Gyromagnetic ratio = gamma = 42.58 MHz/T• The excitation frequency (for an individual proton) is going to

depend on the magnetic field (the individual proton) experiences

• This is the Resonance frequency• MRI

wL

B0

Slope = gyromagnetic ratio

Understanding Resonance by Analogy

• The proton is like a tetherball• If you hit the tetherball at random intervals, it’s net vector is

random• If you hit the tetherball at exactly the right interval (equal to

it’s period of rotation around the pole), each hit is additive and makes the ball go higher and higher

• Think about the different scales: single tetherball vs. billions of tetherballs

MRI: Identifying the EMF of Interest

Simplification• We’ve used B0 enough that we know what it is: a homogeneous external magnetic field

Ext

erna

l Fie

ld =

B0

B0

will now be

Recipe for an NMR Experiment1) Put sample in big magnetic field2) Transmit radio waves into sample (saturate the protons)3) Turn off radio wave transmitter4) Receive radio waves re-transmitted by subject (“relaxation”)

• Emission experiment• Who cares? (what are the applications of spectra?)• Immediately recognized that it could tell us about the local

magnetic environment of hydrogen protons

B0

You get a spectrum because each hydrogen atom has a different local environment

Optimizing the NMR Experiment• Parallelizing the process: shooting the whole spectrum at

once• We would like to

• Take a complex EMF wave• (Group of photons with different Frequencies)• Break it up into component frequencies• Use components to predict identity

• We don’t have a prism for radio waves• Enter… the Fourier Transform

• Put in amplitude-time data• Get out amplitude-frequency data

Made possible by state-of-the-art computer

processors

Optimizing the NMR Experiment

B0

A prism is an example of a fourier transformSo is the cochlea

Lauterbur’s Insight• Conventional NMR used spectra to make

inferences about local magnetic perturbations in a uniform magnetic field

• If the magnetic field was instead made to vary with position, then the resonant frequency spectrum would instead tell you about location in a uniform population of protons

• Great idea…

MRI: Slice Selection• The Larmor frequency is dependent on B0, the external

magnetic field

0BL • By varying B0 over the subject, we can choose a frequency

that will only excite a particular part of the subject

• Different values of B0 will tune the photons to require different energies (w)

• If we only give one frequency of EMF, only one slice will be excited”Z

MRI: Slice Selection• Now we have an excited plane of protons• We reset B0 so that the field is uniform• We wait for the energy to be re-transmitted as a radio signal

• The results are not very exciting

• The whole signal is at the frequency we put in

• It starts loud, and exponentially decays

Z

X

Y

SX

SY

Questions?

MRI: What We Have So Far• We have selectively charged a slab of brain with

radiofrequency EMF• We turned off the magnet, and got signals back from all over

the slab• What we need is to know where each signal came from

Z

• Right now, all we can “see” is the net magnetization vector, but we can see it in several directions: x, y, z

• Therefore we can measure signal averaged over the whole slice

• That’s not a very interesting picture

A Trick Necessary to Continue

hE

• The energy absorbed by an excited particle is determined by the field it is acting against in order to become excited• Electrons: attraction with nucleus• Spinning top: gravity• Protons in NMR: magnetic field

• Note that the last one is very easily manipulated

Changing the Rules in Midstream

hE

• By changing the strength of the magnetic field (re-writing the rules of attraction in mid-stream), the protons can be manipulated “on the fly”• Increase the field to increase precession speed• Increase the field to increase their resonance frequency

• Consider increasing gravity on a spinning top

MRI: Getting Coordinates in Plane (X)• We need to revisit Lauterbur’s idea with the trick we just

learned in order to use frequency to map location • Once we get the spins saturated, we can vary B0 over x

XZ

• B0 still points in the same direction, but make it stronger on one side of the patient than the other

• This is changing the rules in midstream: the protons are already saturated / excited, and now we’re altering the field on them

• We know from our simple experiments that the protons exposed to the weaker field will precess less quickly

net magnetization

vector, M

MRI: Getting Coordinates in Plane (X)

X

Z

• Protons will spontaneously revert to the lower energy state

• Protons at x=0 will be relaxing in a strong field, and give off high-frequency EMF

• Protons at x=1 will be relaxing in a weak field, and give off low-frequency EMF

Z

• Then we can use the FT to separate the frequencies and identify the signal strength at each x-coordinate

net magnetization

vector, M

MRI: Getting Coordinates in Plane (X)• All frequencies in the output signal come at once, and is a plot of signal strength per

unit time

X

Z

• Now, the frequency of the emission tells us the aggregate signal for each x-column

MRI: Getting Coordinates in Plane (Y)• Tying the x coordinate to the frequency is called frequency encoding• It would be nice if we could just do the same thing with the Y direction, but we can’t

X

Y

• Since magnetic field vectors add, putting a second gradient in the y direction is indistinguishable from doing a single gradient at an angle

• No new information is captured, and the Fourier Transform can’t distinguish them unless the frequencies are unique

A Transient Gradient Changes the Phase• If we change the field strength on a magnet that is already

precessing, we can make its precession change speed• If we increase it again, it speeds back up

B0 B0

• This is what we mean when we say that two protons are out of phase

MRI: Getting Coordinates in Plane (Y)• Instead, we will introduce a gradient in the y direction temporarily• This slows changes the speed for a moment, but then the frequency returns to what it had been• Holding back some spins in this way creates a phase shift in the spins, which we can exploit later

X

Y

X

Y

MRI: Getting Coordinates in Plane (Y)• This technique gives us an extra degree of freedom• The Fourier transform does not know how to process phase information, but it does preserve it• We then do a second Fourier transform in order to obtain the information we want• Let’s use an example to understand the 2D Fourier Transform

The 2-D Fourier Transform

Plo

t A

-t

1D FTby row

FT

by column

The 2-D Fourier Transform

FT

by column

Plo

t A

-t

1D FTby row

MRI: What we End Up With• We now have a plot of signal as a function of time at each

individual voxel• We know that the signal will decay unequally in different

tissues, so we get signal-vs-position plots at multiple time points, calculate the rate of decay, and give that pixel a shade near white if the decay constant is large, and a shade near black if the decay constant is short

• In actuality, the phase-encoding step is done first, because the magnetic gradient it requires is transient. The frequency-encoding step is done last, because it needs to be active when the protons relax (…the readout gradient)

Energy Accounting 101• It is possible to exhaustively inventory where all of the

energy of the RF pulse goes (1st law of thermodynamics)

heat

Spin alignment (work)

Spin alignment (work)

the universe(as seen by the physicist)

protons(spins)

everything else(“the lattice”)

MRI: Details of the Excitation• We can choose how much to excite the protons in the plane

• A big EMF pulse can knock all the spins into the x-y plane

• An even bigger EMF pulse can knock all the spins onto the –z axis

Z

initial net magnetization

vector, M0

90° Pulse

180° Pulse

Or anything in between

Z

Z

Measuring the Net Magnetization Vector• The net magnetization vector can be measured directly by

using orthogonal radio antennas.

Z

X

Y

SX

SY

• This will allow the vector within each voxel (which we’ve just learned how to identify) to be measured in x and y

MRI: Details of the Decay• If we start with a 180° pulse, the decay is exponential and goes from -1 to 1 (two data points are

known)

180° PulseZ

net magnetization

(M=Mz)

Z

Mz

t

Z

MRI: Details of the Decay• If we perform any pulse except 180° pulse, then all protons will get knocked into x-y plane, and precess there• Initially, they will all be in-phase, because they are all knocked away from Z-axis at the same time in the same direction• With time, they will de-phase due to two factors

• Interactions with neighboring protons (random effects)• Imperfect homogeneity of B0 (nonrandom effects)

• This is spin-spin relaxation

90° Pulse

net magnetization

M=Mz

Mxy

t

MRI: Details of the Decay• De-phasing is when the signal is lost because it averages itself out and becomes noise• Here is a visual example of dephasing

Z

in phase out of phaseout of phase

• There are ways to reverse this process, and any sequence which does so will be called an “echo” sequence

• Once the signal is completely dephased, we have randomness

• Even so, a non-zero net vector in the z-direction may still exist

Revisiting Energy Accounting

heat

Spin alignment (work)

Spin alignment (work)

protons(spins)

everything else(“the lattice”)

Radiofrequency in

Radiofrequency out

Spin-Spin relaxation

ZZ

T1 relaxationT2 relaxation

MRI: Details of the Decay

Mz

t

Mxy

t

• We have seen two types of recovery toward equilibrium• Mz recovery starts out low and recovers exponentially back toward

one (because z equilbrium is to line back up with B0)• It happens with all pulse strengths• It reflects energy loss to surrounding molecules

• Random interactions (changes moment to moment, “noise”)• Nonrandom interactions (is static for a given molecule, “bias”)

• Mxy recovery starts out maximal and exponentially decays toward zero

• It happens with all pulse strengths except 180°• It reflects energy loss back to “the universe”

the rate constant for this process is called T1

the rate constant for this process is called T2

Energy Accounting 501

heat

Spin alignment (work)

random effects

Radiofrequency in

Radiofrequency out

staticeffects

T1 relaxation

T2 relaxation

T2*

T2 Effects

random effects

staticeffects

T2 relaxation

T2*Z

T2 relaxation

Z

Echo

180° Pulse

180° Pulse

• Because the static interactions are static, they can be reversed by the 180 degree pulsation

Another Analogy

Anatomy of a ScannerFour main hardware components• Main magnet• RF system• Magnetic field gradient system• Computer system.

What We’ve Covered• Reviewed the most fundamental rules that govern MR

phenomena• Identified how to excite selected photons using

supermagnets and radio waves• Identified how to manipulate the excited photons in order to

“encode” positional information• Frequency encoding• Phase encoding

• Become familiar with 2D Fourier transforms

Future Topics to Explore• Mechanisms of contrast

• Proton Density• T1, T2, T2* in more detail• Anisotropy

• Flow• Diffusion• Tensor mapping

• IV Contrast• Pulse Sequence Diagram Interpretation• Sequence selection, costs and benefits