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MRI. Vector Review. z. x. y. Vector Review (2). (a scalar). The Dot Product. The Cross Product. (a vector). (a scalar). MR: Classical Description: Magnetic Moments. Intuitively current, but nuclear spin operator in quantum mechanics. Spin angular momentum =.

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Vector Review




Vector Review (2)

(a scalar)

The Dot Product

The Cross Product

(a vector)

(a scalar)

MR: Classical Description: Magnetic Moments

Intuitively current, but nuclear spin

operator in quantum mechanics

Spin angular momentum =

Planck’s constant / 2

NMR is exhibited in atoms with odd # of protons or neutrons.

Spin angular momentum creates a

dipole magnetic moment

= gyromagnetic ratio : the ratio of the dipole moment to angular momentum

Which atoms have this phenomenon?

1H - abundant, largest signal



Model proton as a ring of current.

MR: Classical Description: Magnetic Fields

How do we create and detect these moments?

Magnetic Fields used in MR:

1) Static main fieldBo

2) Radio frequency (RF) field B1

3) Gradient fieldsGx, Gy, Gz

MR: Classical Description: Magnetic Fields: Bo

1) Static main field Bo

without Bo, spins are randomly oriented.


net magnetization

with Bo,

a) spins align w/ Bo (polarization)

b) spins exhibit precessional behavior

- a resonance phenomena

Reference Frame




MR: Energy of Magnetic Moment



z: longitudinal

x,y: transverse

Alignment Convention:



At equilibrium,

Energy of Magnetic Moment in is equal to the dot product

quantum mechanics - quantized states

MR: Energy states of 1H

Energy of Magnetic Moment in

Hydrogen has two quantized currents,

Bo field creates 2 energy states for Hydrogen where

energy separation

resonance frequency fo

MR: Nuclei spin states

There are two populations of nuclei:

n+ - called parallel

n- - called anti parallel

higher energy



lower energy

Which state will nuclei tend to go to? For B= 1.0T

Boltzman distribution:

Slightly more will end up in the lower energy state. We call the

net difference “aligned spins”.

Only a net of 7 in 2*106 protons are aligned for H+ at 1.0 Tesla.

(consider 1 million +3 in parallel and 1 million -3 anti-parallel. But...

There is a lot of a water!!!

  • 18 g of water is approximately 18 ml and has approximately 2 moles of hydrogen protons

  • Consider the protons in 1mm x 1 mm x 1 mm cube.

  • 2*6.62*1023*1/1000*1/18 = 7.73 x1019 protons/mm3

  • If we have 7 excesses protons per 2 million protons, we get .25 million billion protons per cubic millimeter!!!!

Magnetic Resonance: Spins

We refer to these nuclei as spins.

At equilibrium,

- more interesting -

What if was not parallel to Bo?

We return to classical physics...

- view each spin as a magnetic dipole (a tiny bar magnet)

MR: Intro: Classical Physics: Top analogy

Spins in a magnetic field are analogous to a spinning top in a gravitational field.

(gravity - similar to Bo)

Top precesses about

MR:Classical Physics


View each spin as a magnetic dipole (a tiny bar magnet). Assume we can get dipoles away from B0 .Classical physics describes the

torque of a dipole in a B field as

Torque is defined as

Multiply both sides by

Now sum over all

Partial Bloch Equation: Describes interaction of M and B

  • Above: Portion of the Bloch Equation

  • Explains how to change the direction of the magnetization vector M with applied magnetic fields, B.

Bloch Equations – Homogenous Material

  • Let us solve the Bloch equation for some interesting cases. In the first case, let’s use an arbitrary M vector, a homogenous material, and consider only the static magnetic field.

  • Ignoring T1 and T2 relaxation, consider the following case.

It’s important to visualize the components of the vector M

at different times in the sequence.


The Solved Bloch Equations


The Solved Bloch Equations

A solution to the series of differential equations is:

where M0 refers to the initial conditions. (M0 refers to the

equilibrium magnetization when no RF has been applied for

some time. Some time would be several T1 relaxation



Try two initial conditions

1.First mimics spins in equilibrium position along z

2.First mimics spins right after a 90 degree RF excitation

Solution to 1)

1.First mimics spins in equilibrium position along z


Conclusion: Since M and B start in the same direction,

there cross product is zero. Nothing will change

Solution to 2

1.First mimics spins right after a 90 degree RF excitation

Conclusion: This solution describes a circular path for the

transverse magnetization. M and B are constantly perpendicular. This drives a circular motion.

Sample Torso Coil




MR: Classical Physics: Precession

Solution to differential equation:

rotates (precesses) about

Precessional frequency:

is known as the Larmor frequency.


for 1H

Usually, Bo = .1 to 3 Tesla

So, at 1 Tesla,

fo = 42.57 MHz for 1H

1 Tesla = 104 Gauss

MR: RF Magnetic field

The RF Magnetic Field, also known as the B1 field

To excite nuclei ,

apply rotating field at o in x-y plane. (transverse plane)

B1 radiofrequency field tuned to Larmor frequency and applied in transverse (xy) plane induces nutation (at Larmor frequency) of magnetization vector as it tips away from the z-axis.

- lab frame of reference

Image & caption: Nishimura, Fig. 3.2

MR: RF Magnetic field (Laboratory and Rotating Frame)

B1 induces rotation of magnetization towards the transverse plane. Strength and duration of B1 can be set for a 90 degree rotation, leaving M entirely in the xy plane.

See Proton Procession under RF excitation on webpage animation

a) Laboratory frame behavior of M

b) Rotating frame behavior of M

Images & caption: Nishimura, Fig. 3.3

MR: RF General Excitation (Rotating Frame)


By design ,

In the rotating frame, the frame rotates about z axis at o radians/sec

1) B1 applies torque on M

2) M rotates away from z.

(screwdriver analogy)

3) Strength and duration of B1 determines flip angle



This process is referred to as RF excitation.

What determines flip angle?

Typical B1 Strength: B1 ~ .1 G.

A field of .1 G for 1 ms (T above) will produce approximately a 90 degree pulse

Viewing RF Pulse as a Rotation Matrix about X axis

What happens for

What happens for

After a flip of angle a

  • Initial conditions:

  • After flip of angle a:




Precession of induces EMF in the RF coil. (Faraday’s Law)

MR: Detection


Switch RF coil to receive mode.




EMF time signal - Lab frame



for 90 degree excitation

(free induction decay)

Complex m

m is complex.

m =Mx+iMy

Re{m} =MxIm{m}=My

This notation is convenient:

It allows us to represent a two element vector as a scalar.






Transverse Magnetization Component

The transverse magnetization relaxes in the Bloch equation according to

Solution to this equation is :

This is a decaying sinusoid.


Transverse magnetization gives rise to the signal we “readout”.

will precess, but decays.

MR: Detected signal and Relaxation.



Rotating frame

Transverse Component

with time constant T2

After 90º,

MR Relaxation: Transverse time constant T2

T2 values: < 1 ms to 250 ms

What is T2 relaxation?

- z component of field from neighboring dipoles affects the

resonant frequencies.

- spread in resonant frequency (dephasing) happens on the

microscopic level.

- low frequency fluctuations create frequency broadening.

Image Contrast:

Longer T2’s are brighter in T2-weighted imaging, darker in T1-weighted imaging

- spin-spin relaxation

MR: Relaxation: Some sample tissue time constants: T2

T2 of some normal tissue types

Table: Nishimura, Table 4.2

Bloch Equation Solution: Longitudinal Magnetization Relaxation Component

The greater the difference from equilibrium,

the faster the change


Return to Equilibrium

Initial Mz

Doesn’t have to be 0!

Solution: Longitudinal Magnetization Component




Example: What happens with a 180° RF flip?

Effect of T1 on relaxation

- 180° flip angle




T1 Relaxation

MR Relaxation: Longitudinal time constant T1

Relaxation is complicated.

T1 is known as the spin-lattice, or longitudinal time constant.

T1 values: 100 to 2000 ms


- fluctuating fields with neighbors (dipole interaction)

- stimulates energy exchange

n- n+

- energy exchange at resonant frequency.

Image Contrast:

- Long T1’s are dark in T1-weighted images

- Shorter T1’s are brighter

Is |M| constant?

MR Relaxation: More about T2 and T1

T2 is largely independent of Bo


- immobile spins

- low frequency interactions

- rapid T2 decay: T2 < 1 ms

Distilled water

- mobile spins

- slow T2 decay: ~3 s

- ice : T2~10 s

T1 processes contribute to T2, but not vice versa.

T1 processes need to be on the order of a period of the resonant frequency.

Approximate T1 values as a function of Bo

MR: Relaxation: Some sample tissue time constants - T1

gray matter


white matter

Image, caption: Nishimura, Fig. 4.2




Components of M after Excitation

Laboratory Frame

will precess, but decays.

returns to equilibrium

MR: Detected signal and relaxation after 90 degree RF puls.



Rotating frame

Transverse Component

with time constant T2

After 90º,

Longitudinal Component

Mz returns to Mo with time constant T1

After 90º,

MR Contrast Mechanisms

T2-Weighted Coronal Brain

T1-Weighted Coronal Brain

Putting it all together: The Bloch equation

Sums of the phenomena


RF excitation





Changes the direction

of , but not the length.

These change the length of

only, not the direction.

includes Bo, B1, and

Now we will talk about affect of

MR: Polarization and Excitation

What we can do so far:

1) Excite spins using RF field at o

2) Record FID time signal

3) Mxy decays, Mz grows

4) Repeat.

Now, we will work to understand spatial encoding of the signal

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