Jungwoo Kang Extended Essay MRI

8/20/2019 Jungwoo Kang Extended Essay MRI

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IB Extended Essay - Mathematics

An investigation into the mathematics of the transformation of signals from protons of hydrogen

atoms in magnetic resonance imaging into an image

Jungwoo Kang

Candidate Number: 000717-0036

Supervisor: Mr. Joseph Khan

Anglo-American School of Moscow

September 9, 2014

Word Count: 3998


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Abstract

This investigation assesses the question “How are the magnetically-induced signals from

protons of hydrogen atoms in a magnetic resonance imaging scan converted into an image?”

To investigate this question, the essay examined the nuances of Magnetic ResonanceImaging (MRI) by researching the magnetically induced physical properties of hydrogen atoms

used in MRI. The methods that could induce and change the signal from this physical

phenomenon, Free Induction Decay (FID), to create different image types were investigated.

However, the investigation focused on how FID signals are mathematically converted to animage, and found that the mathematical principles used were specifically of one branch of

mathematics: Fourier analysis, the study of how functions can be approximated with the sum of

simpler trigonometric functions. To examine this, the Fourier series and the complex form of the

Fourier series were derived, which was used to derive the Fourier Transform and Inverse FourierTransform (used to transform information from the spatial domain into the spatial frequency

domain and vice versa). Next, the ideas of the Fourier Transform and Inverse Fourier Transforms

were applied again to MRI technology to reach a conclusion.It was concluded that the Fourier and Inverse Fourier Transforms are used in

transforming Free Induction Decay signals from protons of hydrogen atoms to form an image.

These signals (waves) in the spatial domain are first Fourier Transformed into k-space, the raw

data space, to the frequency domain, and then Inverse Fourier Transformed back into the spatial

domain to sinusoidal waves, which are stored in a computer. Using these sinusoidal waves, andthe corresponding phase and amplitude data, an image is constructed. Finally, deeper

implications of the applications of Fourier analysis and how it can be used in other fields of

medicine and engineering were briefly considered.

Word Count: 293 Words


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Table of Contents

Abstract ................……………………………………………………………………………. 1

Table of Contents ……………………………………………………………………………... 2

Introduction ………………………………………………………………………....………… 3

Overview of the Process of Magnetic Resonance Imaging …………………...……………… 4

Longitudinal Magnetization of Hydrogen Nuclei …………………………………….. 4

Radiofrequency Pulses ………………………………………………………………... 6

Signal Modification ……………………………….…………………………………... 8

Spatial Encoding ……………………………………………………………………… 10

Fourier Transform …………………………………………………………………………….. 13

Fourier Series …………………………………………………………………………. 13

Complex Exponential Form of the Fourier Series ……………………………………. 19

Approximation Example ……………………………………………………………… 21

Deriving the Fourier Transform and Inverse Fourier Transform …………………….. 24

Sample Fourier Transform Calculation ………………………………………………. 26

Application of the Fourier Transform to Magnetic Resonance Imaging ……………………... 28

MR Signal Fourier Transform ………………………………………………………... 28

Fourier Transform of FID Signal Example …………………………………………... 30

Signals in K-Space …………………………………………………………………… 32

Inverse Fourier Transform of K-Space Data ………………………………………… 32

Conclusion ………………………………………………………………………………….... 34

Bibliography …………………………………………………………………………………. 35


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Overview of the Process of Magnetic Resonance Imaging

MRI is a scanning technique based on the principle of Nuclear Magnetic Resonance

(NMR), a physical phenomenon where atomic nuclei in a static magnetic field absorb and reemit

electromagnetic radiation when applied an electromagnetic (usually radiofrequency, RF) pulse.

This phenomenon however, only occurs in atomic nuclei with spin (intrinsic angular momentum),

such as hydrogen nuclei.3 As hydrogen atoms are the most abundant atoms in the human body,

MRI utilizes the NMR properties of hydrogen to image the human body.4

Longitudinal Magnetization of Hydrogen Nuclei

A nucleus of the hydrogen atom consists of a single proton, a positively charged particle

that spins and produces a magnetic field (magnetic moment).5 6 Although the protons are charged,

their random orientation leads to a lack of a magnetic field. Hence, in order to utilize the NMR

properties of the hydrogen nuclei, a magnetic field must be applied.

The Primary Magnetic Field (B0) of an MRI, usually of strength 1, 1.5 or 3 Tesla (SI unit

of magnetic strength), causes protons to align with B0.7 Most protons are parallel to B0 and are in

a low-energy state; however, some protons, due to increased regional energy (possibly from

increased heat) line up antiparallel to B0 and are in a high-energy state. Protons aligned in B0 do

not simply point parallel or antiparallel to B0, rather, they precess (spin at an angle) around the z-

3 Scientific American. 1999. What exactly is the 'spin' of subatomic particles such as electrons and protons? Does ithave any physical significance, analogous to the spin of a planet? October 21. Accessed July 11, 2014.

http://www.scientificamerican.com/article/what-exactly-is-the-spin/.4 Earl Frieden. 1972. "The Chemical Elements of Life." Scientific Amerian 52-60.5 National Health Services. n.d. NHS. Accessed July 8, 2014. http://www.nhs.uk/Conditions/MRI-scan/Pages/How-

does-it-work.aspx.6 R Nave. n.d. Nuclear Magnetic Resonance. Accessed July 15, 2014. http://hyperphysics.phy-

astr.gsu.edu/hbase/nuclear/nmr.html.7 “Introduction to MRI Physics,” YouTube video, 8:39, posted by "Lightbox Radiology Education," September 24,

2011, http://www.youtube.com/watch?v=Ok9ILIYzmaY


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axis at a rate, Larmor Frequency (LF), of 42.58MHz per 1 Tesla.8 Nonetheless, this proton

alignment creates a Net Magnetic Vector (M) in the direction of B0. The creation of M in the

long axis (z-axis) is called longitudinal magnetization.

9

10

8 R Nave. n.d. Nuclear Magnetic Resonance. Accessed July 15, 2014. http://hyperphysics.phy-

astr.gsu.edu/hbase/nuclear/nmr.html.9 “Introduction to MRI Physics,” YouTube video, 8:39, posted by "Lightbox Radiology Education," September 24,

2011, http://www.youtube.com/watch?v=Ok9ILIYzmaY10 Ibid.

Diagram 1: Normal proton orientation (left) versus parallel/antiparallel proton orientation

with B0 (right)

Diagram 2: Proton precessing around z-axis


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Radiofrequency Pulses

After protons align with B0, transverse magnetization is created to achieve resonance.

Thus, a continuous 90o RF pulse equal to the LF of the protons is transmitted from the RF coil to

the protons. As protons absorb energy from the 90o RF pulse, 50% of them move into the high

energy state, and longitudinal magnetization disappears. Then, protons begin to precess in-phase

rather than out-of-phase, creating transverse magnetization and a weak, but measurable current.

Subsequently, the 90o RF pulse is removed and protons return to thermodynamic equilibrium in a

process called relaxation. There are two types of relaxation that occurs: Spin-Spin (transverse)

and Spin-Lattice (longitudinal).11

Spin-Spin relaxation occurs first as protons, with same positive charge, repel each other

and precess out-of-phase, leading to transverse magnetization decay which is modelled by an

exponential decay curve. The time required to reach strength (≈37%) of transversemagnetization with 90o RF pulse is time constant T2 (a tissue-specific value unaffected by field

strength).12

Then, Spin-Lattice relaxation occurs as some protons in the high-energy state revert to

their original low-energy state, releasing RF energy as heat to the surrounding lattice and leading

to longitudinal magnetization recovery which is modelled by an exponential curve. The time

required to reach 1 strength (≈63%) of the original longitudinal magnetization is timeconstant T1 (a tissue-specific value that increases with stronger magnetic fields).13 Overall,

11 Dennis Hoa. n.d. MRI step-by-step, interactive course on magnetic resonance imaging. Accessed August 27, 2014.

http://www.imaios.com/en/e-Courses/e-MRI.12 Ibid.13 Ibid.


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relaxation produces a signal called Free Induction Decay (FID) that can be measured by the RF

coil.

14

However, as MRI utilizes magnetic gradients (later discussed in the Spatial Encoding

section), the resulting inhomogeneity in the magnetic field causes FID signals to actually decay

faster than T2 predicts because of the destructive interference it creates.15

Thus, a modified T2* time constant takes into consideration both tissue-specific times of

normal T2 decay and accelerated spin dephasing due to inhomogeneities. Yet, an 180o RF pulse

(antiparallel to B0) can also reverse the effect of static magnetic field inhomogeneities by

rephasing spins.16

During Spin-Spin relaxation, field inhomogeneities cause some protons to spin faster than

others and dephase at time TE/2 after the 90o RF pulse. To counter this, applying an 180o RF

14 FDA. 2014. A Primer on Medical Device Interactions with Magnetic Resonance Imaging Systems. May 8.

Accessed July 16, 2014.

http://www.fda.gov/medicaldevices/deviceregulationandguidance/guidancedocuments/ucm107721.htm.15 Dennis Hoa. n.d. MRI step-by-step, interactive course on magnetic resonance imaging. Accessed August 27, 2014.

http://www.imaios.com/en/e-Courses/e-MRI.16 Ibid

Diagram 3: Graph of FID


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pulse causes protons to flip vertically, causing M to shift from Z+ to Z-. Hence, protons moving

faster lag behind protons moving slower, and at time TE after the 90o RF pulse, the spins are

back in phase, but shortly after begin to dephase. However, at TE (Echo Time), the signal

sampled is not as strong as the initial intensity as the relaxation is entirely due to Spin-Spin

relaxation.17

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Signal Modification

The process of repetitively applying the 90o and 180o RF pulses is called a Spin Echo

sequence and contains 2 parameters: TR (repetition time: the time between two 90o RF pulses)

and TE (echo time: the time between 90o RF pulse and MR signal sampling). Through

manipulation of the two parameters, tissue signals can be modified.

T1-weighting emphasizes differences in T1 by shortening TR and TE, limiting complete

Spin-Lattice relaxation. As different tissues have varied T1, tissues with shorter T1 recover

longitudinal magnetization more than tissues with longer T1 and thus has greater transverse


http://www.imaios.com/en/e-Courses/e-MRI.18 Ibid.

Diagram 4: Graph of FID with T2 and T2* decay


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magnetization amplitude after a subsequent excitation. Therefore, tissue contrast depends largely

on T1 differences. 19

Contrastingly, T2-weighting lengthens TR and TE to highlight differences in T2. Longer

TR allows tissues to achieve full Spin-Lattice relaxation, thus accentuating variances in Spin-

Spin relaxation. Then, the longer TE is able to pick up dissimilar Spin-Spin relaxations between

tissues. Therefore, T2 differences affects tissue contrast.20

Finally, proton density-weighting (PD-weighting) lengthens TR but shortens TE,

delineating proton density differences. As higher proton density leads to faster transverse

magnetization decay (by lengthening TR and allowing full Spin-Lattice relaxation) the short TE

recognizes differences in proton density. Thus, PD variance influences tissue contrast.21

Nonetheless, MRI images do not completely rely on one weighting type; rather, the

produced images are a combination of effects of T1, T2 and PD. For instance, a tissue in a

predominantly T1-weighted image, a tissue with long T1 and T2 (water) is dark while a tissue



Diagram 5: Graphs of MR Signal with different weightings


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with short T1 and long T2 (fat) is bright. However, in a predominantly T2-weighted image, the

former is bright and the latter is grey.22

Spatial Encoding

In order to create an image from MR signals, the precise location of FID must be isolated

through spatial encoding. This process consists of 3 steps: slice selection, phase encoding, and

frequency encoding.

First, a slice-selection gradient (SSG) is applied orthogonal to the slice plane, causing

protons to precess in a frequency relative to SSG in each slice. Then, a RF wave (selective pulse)

with frequency equal to the precession frequency of the desired slice is applied. The selective

pulse only excites the protons in the desired slice, shifting magnetization and thus, isolating the

slice.23 However, in the case of a selective pulse less than 180o, the dispersion of the resonance

frequency causes protons begin to dephase. To counteract this, an antiparallel gradient in the

same axis that is half the surface (amplitude x time) of the original gradient is applied.24

Secondly, a phase encoding gradient (PEG) is applied to the slice for a short amount of

time, which encodes by utilizing different rates of change of phase for the various signal

measurements. The PEG changes spin resonance frequencies and causes dephasing, resulting in

protons antiparallel to PEG to spin out-of-phase. This is important as this can be used to find the

rate of change of phase (equal to frequency), thus pseudo-frequency encoding the image slice

and utilizing a FEG twice would not allow the derivation of the image, since multiple voxels




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would have the same frequency).25

26

Finally, a frequency encoding gradient (FEG) is applied to the slice for a limited period.

The FEG (orthogonal to both PEG and SSG) acts like the PEG, causing dephasing by changing

spin resonance frequencies, making protons antiparallel to the FEG spin out-of-phase. Hence, in

a slice 1 proton thick, each proton precesses at a different speeds and thus, FID can be isolated in

the image. These signals are transferred into K-Space (the representation of spatial frequency

information in using amplitude, frequency and phase information), and then from K-Space into

an image.27


http://www.imaios.com/en/e-Courses/e-MRI.26 D M Higgins. 2014. The How K-Space Works Tutorial. Accessed September 10, 2014.

http://www.revisemri.com/tutorials/how_k_space_works/.27 Dennis Hoa. n.d. MRI step-by-step, interactive course on magnetic resonance imaging. Accessed August 27, 2014.

http://www.imaios.com/en/e-Courses/e-MRI.

Diagram 6: Demonstrates why the frequency encoding cannot be used for both dimensions.


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These signals are transformed through the Fourier Transform and Inverse Fourier

Transforms. The former translates the spatial information of the signals from the protons of

hydrogen nuclei into spatial frequency, while the latter does the opposite. Therefore, before any

further description takes place about the processes of MRI, it is essential to discuss the Fourier

transform and its related components.

Diagram 7: K-Space is the transformed values of the MR signal, with the x-axis representing

the FEG and the y-axis representing the PEG


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Fourier Transform

Fourier Series

In mathematics, series of simpler functions can represent a more complex function. For

example, Taylor series, utilize powers of x to represent other functions and thus are called power

series. Contrastingly, Fourier series utilizes trigonometric functions (cosine and sine) to represent

other functions and thus are called trigonometric series. Simply put, the Fourier series is a

method to represent a wavelike function through the decomposition of a periodic function (or

signal) to an infinite set of sine and cosine waves (and thus, complex exponentials).

The principle of linear superposition states that the total output equals the linear

combination of the corresponding outputs of individual inputs. Thus, an infinite series of

sine/cosine functions expressing some periodic function f with period T , constants a, b and

integer n (shown below) was first described by Joseph Fourier.

x a a cos

a cos

⋯ b sin

b sin

…

The above equation can further be simplified by substituting the value of with ω0, the symbol

for angular frequency.

x a a cos a cos2 a cos3 ⋯ b sin b sin2 b sin3 ⋯

And finally, the above equation can be expressed as a summation.

∴ x a a= cos b sin 1


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The use of nω0 leads to another essential aspect of the Fourier series: harmonics. Harmonics are

the integer multiple of fundamental frequency , where the nth harmonic is the nth multiple ofthe fundamental frequency. The more harmonics there are, the better the approximation becomes.

0,±1,±2,±3,… One aspect to consider is the properties of the summation even and odd functions. The

sum of odd functions yields an odd function and the sum of even functions yields an even

function. However, the sum of both odd and even functions yields a function that neither odd nor

even. Thus, if f ( x) is odd, its Fourier series only includes sine terms; if f ( x) is even, its Fourier

series only includes cosine terms; and if f ( x) is neither odd nor even, its Fourier series includes

both sine and cosine terms.

Next, evaluating coefficient values of a0, an, and bn, the Fourier series can be used to

approximate a function. The value of a0 can be determined by integrating Equation 1 over a

period, − to .

∫ − ∫ a− ∫ ∑ a= cos b sin

− a ∑ a= ∫ cos− ∑ b= ∫ sin− a ∑ a= sin sin ∑ b= cos cos

a ∑ a= sinsin ∑ b= coscos a , sin 0 ∴ a 1

− 2


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To determine an for n≥1, Equation 1 is multiplied by cos ω0mx (for m≥1) and

integrated from− to for period T.

∫ − ∫ a− ∫ ∑ a= cos b sin− ∫ cos− a ∫ cos

− ∑ a= ∫ coscos−

∑ = ∫ sincos− 3 From Equation 3, an is isolated by solving each term.

1) a ∫ cos− ∫ cos− sin sin sin sin

0

2) ∑ a= ∫ coscos−

a. For n ≠ m, the second term equals:

∫ coscos− ∫ cos( )− ∫ cos(

−) (−)− (+)+ −− −−−


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++ +−+ (−)− (+)+ 0 ≠ , ,

b. For n = m, cos is substituted for cos, and the third term equals:∫ coscos − ∫ 1cos2

−

− ,sin2 0

c. ∴ ∫ coscos − 0 ≠

3) ∑ = ∫ sincos−

∫ sincos − ∫ sin( ) − ∫ sin( )

− −(+)+

−(−)−

−+ + −+− + −−

− −−−

− (+)+ (+)+ (−)− (−)− 0

Thus, using these values, Equation 3 is simplified to the below equation (only the value for n =

m is used, as it yields the only nonzero value).


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∴ cos − 2

If n is then substituted for m, the coefficient an is found.

∴ 2 cos

− 4 Finally, to determine bn for n≥1, Equation 1 is multiplied by sin ω0mx (for m≥1) and integrated

from− to .

∫ − ∫ a− ∫ ∑ a= cos b sin− ∫ sin− a ∫ sin

− ∑ a= ∫ cossin−

∑ = ∫ sinsin− 5 Once again, bn of Equation 5 is isolated by solving the equation term-by-term.

1) a ∫ sin− ∫ sin − cos cos c o snπ cos

0

2) ∑ = ∫ cossin −


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∫ cossin − ∫ sin( ) − ∫ sin( )

− −(+)+

−(−)−

−++ −+−+ −−− −−−− (+)+ (+)+ (−)− (−)− 0

3)

∑ b= ∫ sinsin

−

a. For n ≠ m, the third term equals:

∫ sinsin− ∫ cos( )− ∫ cos(

−)

(−

)−

(+

)+

−− −−− ++ +−+ (−)− (+)+ 0 ≠ , ,

b.

For n = m,

sin is substituted for

sin and the second term equals:

∫ sinsin− ∫ 1cos2−


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−

c. ∴ ∫ sinsin − 0 ≠

Thus, using these values, Equation 5 is simplified to the below equation. Like simplifying

Equation 3, only the values for n = m are used, as it yields the only nonzero value.

∴ sin

−

2 If n is then substituted for m, the coefficient an is found.

∴ 2 sin

− 6 Complex Exponential Form of the Fourier Series

To simplify the Fourier series, the coefficients an and bn are combined by expressing the

series in a complex exponential form. Using Euler’s identity, cos s in , and replacingθ with ω0nx, one can derive the following.

e c o s s i n 7

e−

cossin c o s s i n 8 ∴c o s −2 9


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∴s in −2 10 These values can be directly substituted into Equation 1 to convert the Fourier series into a

complex form.

x a ∑ a + b − = a ∑ − − + = 11

Let cn and its conjugate, c-n, equal:

c a b2 12 c− a b2 13

If these values are substituted into Equation 11, f (x) is expressed as a summation of ce,and therefore, as the complex exponential form of Fourier series.

x a ∑ [c c−−]= ∑ c=− as a ≡ c because for n 0, c 14 Then, cn is computed by substituting the values of an and bn into Equation 12.

c −

∫ − ∫ ∫ cos s i n −


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Finally, the values of cos and sin from Equations 9 and 10 are substituted to furthersimplify the equation.

c ∫ cos s i n − ∫ + − − ∫ + − −

∫ (

−

)

− 15

Approximation Example

Using the coefficients, the Fourier series can approximate a periodic function or a non-

periodic function (over a period). For instance, take the square wave shown in Diagram 8.

The function is defined as:

x 1 0 < < 0 0, ±1 < < 0 , x x2π

Diagram 8: Square wave with amplitude of 1 unit and wavelength/period of 2π units


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First, taking into consideration that the function is odd, the Fourier series expansion of this

square wave consists of solely sine terms. Thus, these values of f (x), are plugged into the

formulas for bn.

a ∫ − ∫ − 0 x b ∫ sin−

∫ sin−

2 sin 2 ∫ 1sin cos0 − coscos0 − 1 1

0

Likewise, the same is done with the complex form of Fourier series.

c ∫ (−)− ∫ − ∫ −− ∫ − ∫ −−


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(− 1 1 ) ( − 2)

0 −

Thus, f (x) is expressed as:

x ∑ = − sin21 ∑ −− −=− Taking the 1, 3 and 10 harmonics, the Fourier series approximation expresses the square wave

more accurately as number of harmonics increases.

If infinitely many harmonics are used to approximate, the series will converge to f(x), given that

f(x) is integrable. Convergence theory states: if f(x) is piecewise smooth over period [-T, T ], then

the Fourier series converges to average value++− at the point of discontinuity, x0, and

to the periodic extension through 2T of f (x) if f (x) = f (x + 2T ).

Deriving the Fourier Transform and Inverse Fourier Transform

One problem that mathematicians faced was that, while the Fourier series provided a way

to approximate periodic functions, it is limited as it cannot approximate non-periodic functions

Diagram 9: Fourier approximations of diagram 8. Note how the approximation improves as

the number of harmonics increases from 1 (red) to 3 (purple) to 10 (blue)


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beyond a period. Thus, mathematicians derived the Fourier Transform from the complex

exponential form of the Fourier Series, which can approximate non-periodic functions from -∞ to

∞.

Taking the complex exponential Fourier series from Equation 14 and substituting the value

of cn (Equation 15), the result is:

∑ c=− ∑ ∫ (−)− =−

Then, is replaced with (definition of angular frequency). ∑ ∫ (−)− =− Now, take into consideration the definition of the nth harmonic.

2 0,±1,±2,±3,… From this, the separation of these frequencies can be calculated.

1 2 As T approaches ∞, the frequency separation (∆ω) decreases, ∆ω nears 0 and all the frequency

harmonics are represented, corresponding to a Riemann sum. Therefore, the function can be

simplified into a double integral representation as ω0 is replaced with ∆ω and the summation of

nω0 (discrete frequencies) is replaced by an integral over all frequencies. Now, nω0 is replaced

with a general variable of frequency, ω.

12 (−)−

− 16


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The inner integral of Equation 16 can be expressed as , representing a function dependenton frequency, ω (when integral is evaluated for x = ±∞, x disappears and only ω remains).

(−)− 17Equation 17 is the Fourier Transform (FT) that shifts information in the time dimension, x, into a

information the frequency dimension, ω. When Equation 17 is plugged back into Equation 16,

the result is the Inverse Fourier Transform (IFT) which transforms information in the frequency

domain, ω, to information in the time domain, x.

∴ 12 − 18 When information is in the frequency domain, the y-value (amplitude) of the matching x-

value (frequency) denotes the amplitude of the frequency forming the part of the approximation.

28

Sample Fourier Transform Calculation



Diagram 10: The FT transforms the sinusoidal wave (blue) in the time domain to the

frequency domain (red), while the IFT does the opposite. Note that amplitude does not change

and that both graphs are equivalent but are in different domains.


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Take the function (graphed above) in the time domain, x, {− ≥ 0 < 0 .Plugging this into the FT, it yields:

∫ −(−) ∫ (−)− ∫ −+ ∫ −− −+ → ∞0 − 0 → ∞ 0 −+ − 0 + −

++−+− + Hence, for the values of frequencies (ω) that are plugged in, the value of F( ω ) corresponds to the

amplitudes of the frequencies, yielding the graph below.


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Application of the Fourier Transform to Magnetic Resonance Imaging

As the Fourier Transform has been derived, the original problem of the conversion of MR

signals into, and out of k-space can be investigated. The reason that the FT is used is because it

can turn MR signals information, which is in the space domain, and transform it into the

frequency domain, which is temporarily stored into k-space. Then, using the IFT, information in

the frequency domain is transformed back into the spatial domain to reconstruct the image in a

computer. Furthermore, another benefit of this method is that it provides a shorter and more

efficient way of transformation. Rather than superimposing all frequencies, which would require

65536 calculations for a 256*256 pixel image, the Fourier transforms provide a shortcut to

construct the image.

MR Signal Fourier Transform

Diagram 11: Graph of the frequency (x-axis) of f(x) calculated above, against the amplitude(y-axis).


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First, in order for the Fourier Transform to be applied to MR Imaging, the time domain is

replaced with a space domain (time variable becomes x-coordinate) and the frequency becomes

spatial frequency. As frequency is the inverse of the time it takes for a sinusoidal wave to repeat

(period), the spatial frequency is the inverse of the space required for the intensity of an image to

change.

29

Through this transformation of MR Signals, its 3 essential points of information can be

found: amplitude (determines signal strength); frequency (determines relative position in slice);

and phase (determines relative position in slice). First, the FID signal is Fourier transformed in

the frequency encoding direction (X-axis) to derive the frequency. Subsequently, the signal is

Fourier transformed in the phase encoding direction(Y-axis), to derive the location of the signal

in the PEG.



Diagram 12: On the left, the image requires a long distance to change in intensity (low

spatial frequency). On the right, the image requires a short distance to change in intensity

(high spatial frequency).


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30

Fourier Transform of FID Signal Example

31

Take this voxel located in (3,3). The raw data of the signals for this specific voxel is shown

below: notice how the frequency is the same for all the signals.

30 D M Higgins. 2014. The How K-Space Works Tutorial. Accessed September 10, 2014.

http://www.revisemri.com/tutorials/how_k_space_works/.31 Hornak, Joseph P. 2014. The Basics of MRI. Henrietta, NY: Interactive Learning Software.

Diagram 13: The Fourier Transform process of a wave visualized. Notice in the middle how

the frequencies are identical, yet, they are different as the phase information is different and

hence must be treated separately.


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32

Fourier transforming in the frequency encoding direction, the result is the following: note how

the x-value of the frequency of the peaks are in line with the x-value of the voxel.

33

Taking into account the oscillation of the amplitudes of the frequencies, they can be represented

as a wave to make it more visible.

32 Hornak, Joseph P. 2014. The Basics of MRI. Henrietta, NY: Interactive Learning Software.33 Ibid.


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34

Now, Fourier transforming this wave in the phase encoding direction results in a single peak. Its

location is equal to the location of the voxel.

35

Signals in K-Space

34 Hornak, Joseph P. 2014. The Basics of MRI. Henrietta, NY: Interactive Learning Software.35 Ibid.


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K-Space stores the FT data, recording frequency, phase, and amplitude information. By

definition of k-space, the data near the middle of k-space contains the majority of information.

This is because the lowest spatial frequency data tends to have the largest amplitudes, as they

give the greatest changes in contrast. On the other hand, periphery data has higher frequency but

lower amplitudes since they code for finer details that do not require vast changes in contrast.

These amplitudes are portrayed in k-space by the grayscale color that is given to them, with

white as higher amplitudes and black as low/zero amplitudes.

Inverse Fourier Transform of K-Space Data

The final step in the MRI Image Formation process is performing an IFT on k-space data.

Due to the single-dimensional nature of IFT, it must be done line by line in one direction and

then repeated in another direction. This leads to information returning to the spatial domain and

these sinusoids are used to construct an image using frequency, phase and amplitude information.

The process, while it may sound almost impossible, is visualized below.

36

36 “Every picture is made of waves - Sixty Symbols,” YouTube video, 9:42, posted by "Sixty Symbols," June 4,

2014, http://www.youtube.com/watch?v=mEN7DTdHbAU

Diagram 14: The progression of an image that has been constructed with 9, 36, 64 and 1024

spatial frequencies.


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Conclusion

Hence, it is evident that the magnetically-induced FID signals from protons of hydrogen

atoms are converted into an image through the use of two mathematical transforms: FT and IFT.

The isolated FID signals of the slice being imaged is are transformed into k-space with the FT

and information in the spatial domain is converted into the spatial frequency domain (while other

variables such as amplitude and phase remain constant). Then, the data is transformed out of k-

space into a computer through the IFT, converting information from the spatial frequency

domain into the spatial domain. Finally, the data (consisting of sinusoidal waves and its

amplitude and phase) is used to reconstruct the image.

By investigating this issue, the elegant and “unseen” mathematical elements of medical

imaging were uncovered. Yet, while this investigation did explore the mathematics of the

conversion of the signals of protons to images, it did not uncover the more intricate mathematical

concepts such as the Fast Fourier Transform, and different types of MRI such as MRI

angiography, which also utilizes complex integrals.37 These applications of mathematics only

serve to demonstrate the prominence of mathematics in other scientific fields such as engineering

and medicine. Yet, this notion becomes even more evident since the Fourier Series and Fourier

Transform can be used in a variety of fields. For instance, it can be used in seismology to

determine the seismic activity of a volcano, be used to measure cosmic background radiation and

differentiate it from the random signals that are in the universe or be used to model the process of

how the human ears hear sound (consequently digital music encoding). These examples suggest

the elegance and importance of mathematics, as it is the basis of almost all fields of research.

37 Nave, R. n.d. Nuclear Magnetic Resonance. Accessed September 20, 2014. http://hyperphysics.phy-

astr.gsu.edu/hbase/math/fft.html


35/35

34

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