6.003 Signal Processing

6.003 Signal Processing

Week 12, Lecture B:

Magnetic Resonance Imaging

6.003 Fall 2020

MRI is a Fourier Transform

Each pixel in a conventional camera reports the amount of light at a particular position in space. The collection of pixels represents a spatial mapping of light intensity and produces an image of space.

MRI is different.

An MRI scanner collects data that represent samples in Fourier space. The collection of measurements provides the Fourier transform of an image.

Today’s goal is to motivate how MRI works and why MRI images are different,

Magnetic Resonance Imaging

MRI is formerly know as Nuclear Magnetic Resonance Imaging, derives from (nuclear) spin angular momentum and associated magnetic dipole moment, m Proper treatment of spins: Quantum Mechanics

Here: Classical picture of “charged, spinning sphere”

gives rise to current loop that creates a magnetic dipole moment, m Signal Source in MRI: Hydrogen 1_H

MRI in medicine is imaging of water

Spins in a strong magnetic field, B

₀

Spins in earth’s magnetic field:

Randomly oriented, no net magnetization

Spins in a strong, external field B₀

Net alignment of spins in the presence of B₀ yields signal source in MRI.

Spins after RF excitation, B

₁

𝜔₀ = 𝛾𝐵₀

Key relationship is the Larmor equation:

𝑚

𝛾

2𝜋 = 42.58 MHz/T

Signal Detection

Faraday’s and Lenz’ laws:

Φ(𝑡) ∝ Mcos(𝜔𝑡)

𝑉 = − 𝑑𝑁Φ 𝑑𝑡

𝑠 𝑡 ∝ 𝑀𝜔 ∙ sin(𝜔𝑡)

Image Encoding

Imaging: We want to estimate 𝑀 as a function of x, y, z …

Gradient Fields for Frequency Encoding of Spatial Information: Intentional spatial variation in the precession frequency of spins

Frequency and spatial location map 1-to-1 in the presence of a constant gradient field

Image Encoding

𝑥-gradient coil produces a linearly varying z-directed field that is characterized by its slope,𝐺_𝑥, as a function of 𝑥

Uniform Magnet Field from 𝑥 -gradient coil

Total field, sum of 𝐵₀

and 𝐺_𝑥 ∙ 𝑥 directed along Ƹ𝑧

Image Encoding

𝑥-gradient coil produces a linearly varying z-directed field that is characterized by its slope,𝐺_𝑥, as a function of 𝑥

Uniform Magnet Field from 𝑥 -gradient coil

Total field, sum of 𝐵₀

and 𝐺_𝑥 ∙ 𝑥 directed along Ƹ𝑧

Mapping Space to Frequency via Gradient Field

i.e.: ω = 𝜔₀ + 𝛾𝐺_𝑥𝑥

After demodulation

ω = 𝛾𝐺_𝑥𝑥

Total field, sum of 𝐵₀

and 𝐺_𝑥 ∙ 𝑥 directed along Ƹ𝑧

Mapping Space to Frequency via Gradient Field

i.e.: ω = 𝜔₀ + 𝛾𝐺_𝑥𝑥

After demodulation

ω = 𝛾𝐺_𝑥𝑥

Total field, sum of 𝐵₀

and 𝐺_𝑥 ∙ 𝑥 directed along Ƹ𝑧

G: 10-50mT/m switched in ~100 sec

𝑚(𝑥, 𝑦)𝑒−𝑗𝛾𝐺𝑥𝑥𝑡𝑒−𝑗𝛾𝐺𝑦𝑦𝑡

Mapping Space to Frequency via Gradient Field

𝑠(𝑡) ∝ ∬ 𝑚(𝑥, 𝑦)𝑒−𝑗(𝛾𝐺𝑥𝑥+𝛾𝐺𝑦𝑦)𝑡𝑑𝑥𝑑𝑦

𝑠(𝑘_𝑥, 𝑘_𝑦) ∝ ∬ 𝑚(𝑥, 𝑦)𝑒−𝑗2𝜋(𝑘𝑥𝑥+𝑘𝑦𝑦)𝑑𝑥𝑑𝑦

𝑘_𝑥 𝑡 = 𝛾 2𝜋 න₀

𝑡

𝐺_𝑥 𝜏 𝑑𝜏 , 𝑘𝑦(𝑡) = 𝛾 2𝜋 න₀

𝑡

Mapping Space to Frequency via Gradient Field

𝑠(𝑡) ∝ ∬ 𝑚(𝑥, 𝑦)𝑒−𝑗(𝛾𝐺𝑥𝑥+𝛾𝐺𝑦𝑦)𝑡𝑑𝑥𝑑𝑦

𝑠(𝑘_𝑥, 𝑘_𝑦) ∝ ∬ 𝑚(𝑥, 𝑦)𝑒−𝑗2𝜋(𝑘𝑥𝑥+𝑘𝑦𝑦)𝑑𝑥𝑑𝑦

To learn more:

https://www.youtube.com/watch?v=TQegSF 4ZiIQ

Example Image

By collecting data as 𝐺_𝑥 and 𝐺_𝑦 are varied, we can assemble a 256×256 array of k-space data 𝑀[𝑘_𝑥, 𝑘_𝑦] of the following form.

These direct measurements do NOT represent the image. They represent the Fourier transform of the image.

Example Image

The inverse transform of 𝑀[𝑘_𝑥, 𝑘_𝑦] reveals the underlying image 𝑚[𝑥, 𝑦]. The reconstructed image has both real and imaginary parts because of phase delays in the RF signal path (not

considered here).

The magnitude of the

resulting image is a better measure of 𝑚[𝑥, 𝑦].

Resolution and Field of View (FOV)

∆𝑥 ∝ 1

𝑘_{𝑥,𝑚𝑎𝑥} ,

The spatial resolution is limited by maximum frequency:

∆𝑦 ∝ 1 𝑘_{𝑦,𝑚𝑎𝑥} ∆𝑘_𝑦∝ 1

𝐹𝑂𝑉_𝑦

The spatial FOV is the inverse of step size in k:

∆𝑘_𝑥∝ 1 𝐹𝑂𝑉_𝑥 ,

Scanning Time

How long does it take to obtain an image like the one on the previous slide? Typically, one can measure an entire row after a single RF excitation.

If RF excitation occurs once every 2 seconds, then the total acquisition time would be 256 × 2 seconds, which is approximately 8.5 minutes.

This is a long time even for a healthy young adult. What about a child? Or a patient with uncontrolled tremors?

Accelerating Imaging

An important area of current research is in decreasing the time required to capture an image.

One idea for accelerating imaging is to intentionally under-sample the frequency representation.

Accelerating Imaging

Let 𝐹[𝑘_𝑟, 𝑘𝑐] represent the original k-space data and 𝐺[𝑘_𝑟, 𝑘𝑐] represent the k-space data with odd numbered columns set to zero.

𝐺 𝑘_𝑟, 𝑘𝑐 = 𝐹[𝑘_𝑟, 𝑘𝑐] ∙ 1+(−1)𝑘𝑐

2

𝑔 𝑟, 𝑐 = ෍

𝑘_𝑟,𝑘𝑐

1

2𝐹[𝑘𝑟, 𝑘𝑐] ∙ 1 + 𝑒

𝑗𝜋𝑘_{𝑐 ∙ 𝑒}𝑗2𝜋𝑘_{𝑅 𝑟}𝑟 _𝑒𝑗2𝜋𝑘_{𝐶 𝑐}𝑐

= 1

2𝐹[𝑘𝑟, 𝑘𝑐] ∙ 1 + 𝑒

𝑗𝜋𝑘_𝑐

= 1

2𝑓 𝑟, 𝑐 + ෍

𝑘_𝑟,𝑘𝑐

1

2𝐹[𝑘𝑟, 𝑘𝑐] ∙ 𝑒

𝑗𝜋𝑘_{𝑐 ∙ 𝑒}𝑗2𝜋𝑘_{𝑅 𝑟}𝑟 _𝑒𝑗2𝜋𝑘_{𝐶 𝑐}𝑐

= 1

2𝑓 𝑟, 𝑐 + ෍

𝑘_𝑟,𝑘𝑐

1

2𝐹[𝑘𝑟, 𝑘𝑐] ∙ 𝑒

𝑗2𝜋𝑘_{𝑅 𝑟}𝑟

𝑒𝑗2𝜋𝑘𝐶 (𝑐+𝑐 𝐶 2)

= 1

2𝑓 𝑟, 𝑐 + 1

2𝑓[𝑟, 𝑐 + 𝐶

2 𝑚𝑜𝑑 𝐶]

Multi-Coil Imaging

Multiple readout coils provide additional data without increasing imaging time.

The readout of s(t) is fast compared to generating the B-fields and letting the spins relax. Thus adding readout coils has little effect on imaging time.

Consider two coils, one on each side of the head. The left coil will be more sensitive to the left portions of the brain, and vice versa.

In 1D, our coil sensitivities could have the following form:

Images From Coils 1 and 2

Since coil 1 is only sensitive to the left part of the brain, the image produced with data from coil 1 shows just the left half of the brain.

If we only measure 𝐹₁[𝑘_𝑟, 𝑘𝑐] and 𝐹₂[𝑘_𝑟, 𝑘𝑐] at even-numbered 𝑘_𝑐, then the image from coil 1 will be added to a circularly shifted version of itself, as will the image from coil 2.

Images From Coils 1 and 2

g₁ and g₂ are after omitting odd numbered columns.

Sampling in frequency causes aliasing in space: but the aliased copy no longer overlaps the

Images From Coils 1 and 2

Since we know the sensitivity of the two coils, the aliased part copy can be recognized easily.

For further reading: Anagha Deshmane, Vikas Gulani, Mark A. Griswold and Nicole Seiberlich,

“Parallel MR Imaging”, JOURNAL OF MAGNETIC RESONANCE IMAGING 36:55–72 (2012)

Advantage:

|g₁| was acquired in half the time required for a full-frame full-resolution image. Similar with |g₂|. And |g₁| and |g₂| data were acquired simultaneously!

Summary

6.003 Fall 2020

Magnetic Resonance Imaging is a powerful tool– revealing deep tissue structure while being completely non-invasive.

Improving the imaging speed is an active area of research.

Magnetic Resonance Images are acquired by sampling the Fourier representation of the proton density function.

Naive methods (such as under-sampling) that work in convention spatial imaging modalities are not applicable in MRI.

Magnetic Resonance Imaging can be made faster using multiple readout coils, which enables parallel acquisition of under-sampled k-space data. Modern MRI systems can use as many as 32 coils.