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Physics 5143 – Magnetic Resonance Imaging

Magnetic Resonance: A type of radio frequency spectroscopy, which involves nuclear behaviour (magnetic moments) in magnetic fields. Unlike many other types of spectroscopy, MR, is time based, not merely energy based. Magnetic Resonance Imaging: Extension of magnetic resonance, which permits resolution of nuclear behaviour (density) with space. “Nuclear” has been dropped from most nomenclature because of unfortunate public connotations. MR is presently used and studied in three distinct scientific communities:

Radiology Physics Chemistry Phenomena discovered in 1948, Physics. Fourier transform NMR, 1968, Chemistry. MRI, Radiology post 1973. Radiology: Multi billion dollar industry per year, premiere imaging modality.

From radiology sophisticated, reliable hardware. Physics: Ideas!! Chemistry: Chemical structure, relaxation time behaviour, early instrument

development.

Radiology Physics Chemistry

UNB MRI

Draw on knowledge, abilities all three areas

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In MRI we will frequently encounter the ideas of labeling space by frequency and phase. We will review the mathematics for their description. Complex Numbers

Quantity with a real and imaginary component

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z = a + ib

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z = (a,b)

Where i is formally

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−1 , i2 = −1

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z1 + zz = (a1 + a2,b1 + b2)

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z1 ⋅ zz = (a1a2 − b1b2,a1b2 + a2b1)

Commutative, associative, distributive laws of addition, multiplication all hold.

Argand Diagram

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z = a + ib, often written

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a = Re(z)

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b = Im(z)

Represent complex number geometrically in a plane.

x axis real y axis imaginary

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z = a + ib

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z*complex conjugate

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z = a − ib

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The complex number may be considered a vector in the complex plane. Norm of the vector

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z z = a2 + b2

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z = a2 + b2 Phase angle

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θ

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tanθ = b /a

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θ = arctan(b /a)

The sum of complex numbers is the sum of the vectors in the Argand Diagram. We can represent

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z = a + ib in polar coordinates

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z = z (cosθ + isinθ)

Euler Notation

sin, cos can be expanded into infinite series

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sin x = x − x3

3!+x 5

5!−x 7

7!+ ... (1)

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cos x =1− x2

2!+x 4

4!−x 6

6!+ ... (2)

Expansion of an exponential is likewise

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exp x =1+x1!

+x 2

2!+x 3

3!+ ... (3)

So,

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exp[iz] =1+ iz +(iz)2

2!+(iz)3

3!+ ...

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= 1− z2

2!+z4

4!−z6

6!+ ...

⎛

⎝ ⎜

⎞

⎠ ⎟ +

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i(z − z3

3!+z5

5!+z7

7!+ ...)

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= cosz + isin z So,

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z = z exp(iθ ) The product of two complex numbers

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z1⋅ z2 = z1 exp(iθ1)⋅ z2 exp(iθ2)

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= z1 z2 exp(i(θ1 +θ2)) Multiply moduli, add phase.

So,

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z = z exp(iθ ) could represent a signal of amplitude

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z and phase

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θ . The signal could be time dependent.

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z(t) = z exp(iθ(t))

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= z exp(iω⋅ t)

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Where

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ω is angular velocity.

If

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ω is +, counterclockwise If

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ω is -, clockwise

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ω = 2πν where

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ν is the frequency, revolutions per second.

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z(t) can have a phase offset.

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z(t) = z exp(i(ω⋅ t + φ))

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= z exp(iwt)⋅ exp(iφ)

Fourier Transforms

We often encounter systems where we make a time based measurement and wish to know the frequency.

frequency

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↔ time Domains connected by FT.

More generally, reciprocal space representations of our signal are connected by FT.

Space

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↔ reciprocal space

cm

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↔cm -1

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Analytical Fourier Transform

We can always think about the results of our manipulations in terms of continuous functions.

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f (kx ) = f (x)e−i2πkxxdx−∞

∞

∫

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f (x) = f (kx )e+i2πkxxdkx

−∞

∞

∫

FT decomposes f(x) into its consinusoidal and sinusoidal components at all frequencies kx .

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f (kx ) = f (x)cos(2πkxx)dx − i−∞

∞

∫ f (x)sin(2πkxx)dx−∞

∞

∫

In this form integral can be evaluated analytically.

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Poularikas, Alexander D., ed. The Transforms and Applications Handbook.

2nd ed. United States: CRC Press, 2000

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Question 1 Show analytically that the FT of the boxcar function yields a Sinc function. What happens to the Sinc as the width of our original boxcar function increases? Note: There are important symmetry properties to the FT!!

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Discrete versus Continuous FT Experimentally we deal with discrete functions, not continuous functions. Conceptually, working with continuous functions is sometimes more convenient.

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f (ν) = f (τ )e− i2π

ντN

τ =0

N −1

∑

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f (τ) =1N

f (ν )ei2π

ντN

ν =0

N −1

∑

We usually do not concern ourselves with proper normalization of any of the FTs discrete FT or Fast FT. Computationally the discrete FT may be performed very quickly using the Cooley, Tukey algorithm, 1965. Discrete transform with data set a power of 2

128 27

256 28

512 29 1024 210

2056 211 4096 212

Called, typically, the fast Fourier Transformation FFT.

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Callaghan, Paul T. Principles of Nuclear Magnetic Resonance Microscopy.

Oxford: Clarendon Press, 1991.

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Callaghan, Paul T. Principles of Nuclear Magnetic Resonance Microscopy.

Oxford: Clarendon Press, 1991.

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Transform Pairs

Nishimura, Dwight G. Introduction to Magnetic Resonance Imaging. Stanford

University, 1994. There are many tables of transform pairs: -analytical -graphical

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Karu, Zoher Z. Signals and Systems Made Ridiculously Simple. Cambridge, MA:

Zizi Press, 1995.

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Fourier Transform Properties

Karu, Zoher Z. Signals and Systems Made Ridiculously Simple. Cambridge, MA: Zizi Press, 1995.

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Question 2 Calculate the analytical Fourier Transform of the function

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f = Ae−a t The result is a Lorentzian, which can be characterized by a full width half maximum–often called the “linewidth”. Relate the linewidth to the rate contrast a. Note a time constant is the reciprocal of a rate. The result of your calculation is much simpler when you consider the function above for

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−∞ < t < ∞ . Why is the result more complicated when the function is restricted to

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t > 0, and

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f = 0 for

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t < 0 . A short written answer will suffice. Both of these cases occur frequently in magnetic resonance.

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Special Functions:

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δ Dirac delta function

Special sieving properties

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δ(x − b)⋅ f (x)dx = f (b)−∞

∞

∫

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δ(x)dx =1−∞

∞

∫

Unit area, exists only at a point.

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δ(x) = 0 if

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x ≠ 0

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δ(x − b) = 0 if

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x ≠ b

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Poularikas, Alexander D., ed. The Transforms and Applications Handbook. 2nd ed. United States: CRC Press, 2000

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Shah function

Picket fence (sampling function). A periodic function

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III(x) = δ(x − n)n∑ n an integer

Special Properties of the Fourier transform:

One of the most important is convolution. Multiplication in one domain is equivalent to convolution in the reciprocal domains.

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y(t) = x(τ)h(t −τ)dτ−∞

∞

∫

Faltung, folding integral.

Convolution theorem:

The Fourier transformation of the product of two functions is a convolution of the two transformed functions in conjugate space.

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y(t) = h(t)∗ x(t)

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Y ( f ) = H( f )X( f ) convolution in the multiplication in the time domain frequency domain

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y(t) = h(t)x(t)

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Y ( f ) = H( f )∗X( f ) Multiplication in convolution in time domain frequency domain

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Graphical Flip/Shift Method of Convolution

Karu, Zoher Z. Signals and Systems Made Ridiculously Simple. Cambridge, MA:

Zizi Press, 1995.

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Continuous-Time Convolution

Karu, Zoher Z. Signals and Systems Made Ridiculously Simple. Cambridge, MA:

Zizi Press, 1995.

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Convolution by Inspection

Aside from convolving with impulses or doublets, there are other times when it is actually quite easy to just write down the result of a convolution. This method relies on a mental picture of flipping and shifting and works particularly well for signals with “box-like” shapes. Start by visualizing flipping and sliding one of the signals (choose the simpler looking one) through the other signal. Stop at appropriate breakpoints (e.g. step transitions in the signal), multiply the signals, and then compute the area of the product. Note, this is not the same thing as looking for the area of the overlap. Plot the result as a single point on the axes for the answer. Repeat this procedure for all appropriate transition points and simply connect the dots.

Karu, Zoher Z. Signals and Systems Made Ridiculously Simple. Cambridge, MA:

Zizi Press, 1995.

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Transform of the product is equal to convolution of the individual transforms. Fukushima, Eiichi, and Stephen B.W. Roeder. Experimental Pulse NMR: A Nuts

and Bolts Approach. Reading, MA: Addison-Wesley Publishing Company, 1981.

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↓FT

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↓FT

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↓FT

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A ‘noisy’ spectrum of 2,5 anhydromannitol (b) and its Fourier partner, a ‘noisy’ free induction decay (a).

Homans, S.W. A Dictionary of Concepts in NMR. Oxford: Clarendon Press, 1992.

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Multiplication of the noisy free induction decay by a decaying exponential function (a) gives rise to a free induction decay (b) with fewer noise components. The corresponding spectrum (c) has decreased noise but broader lines.

Homans, S.W. A Dictionary of Concepts in NMR. Oxford: Clarendon Press, 1992.

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Definite Integral The definite integral of a function from

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−∞ to

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+∞ is equal to the value of its transform at the origin

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f (x)dx = F(0)−∞

∞

∫

Where F(0) is transformed function evaluated at zero.

Proof

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f (x)dx = f (x)e−i2πxkx dx−∞

∞

∫−∞

∞

∫

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kx = 0

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= F(0)

This is of importance to us Time domain signal, exponential decay

Initial amplitude is equal to area of the peak in the spectrum.

Issues related to Signal Processing and FT Pablo Prado once acquired a 19F profile of gas phase Freon 22. The profile (1D image) looked like

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There arose the question. What is the origin of the spike in the center of the profile? How to get rid of it? What are legitimate steps to remove the spike? You should conclusively identify any unwanted image feature as artifacts before you take steps to remove them.

How to remove it (no knowledge of origin)

-‘airbrush’ in Photoshop or other program -scale image to hide, both are difficult to do with a 1D image.

-Blur in image domain by convolution with a Lorentzain or Gaussian. Spread the intensity around!

On the way to thinking about the origin-

What will the Gaussian or Lorentzian look like or do in reciprocal space?

Lorentzain

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FT← → ⎯ exponential Gaussian

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FT← → ⎯ Gaussian

Consider the Lorentzain

-In reciprocal space multiply by exponential. -Signal at large

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kx →0

A hint: Extremes of k-space

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→0 remove spike at origin. Confirm spike is at the origin, and is not image centered, by translating object. What is the FT of a DC offset?

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exp − kx

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f (x)

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f (kx ) ________________________________________________

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ei2πk0x

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FT← → ⎯

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δ(kx − k0)

1

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FT← → ⎯

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δ(kx ) ________________________________________________

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δ(x − x0)

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FT← → ⎯

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e−i2πkxx0

A spike at

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x0 Let’s set

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x0 = 0

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δ(x)

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FT← → ⎯

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e−i2πkx 0

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e0 = 1

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δ(x)

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FT← → ⎯ 1 Can we think about it another way?

As we increase the period of our function in kx space, eventually the frequency is so low it is a constant in our observed kx space. The image displays a spike at

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x = 0.

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How to get rid of the spike? -convolution with a reason -baseline connect prior to FT

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Data should be zero at extremes of kx. If not the reason is an instrumental artifact. Calculate offset from data points at extreme of data record. Subtract uniformly the mean. Instrument phase cycle (more later).

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DC offset DC offset