The Suppression of Selected Acoustic
Noise Frequencies in MRI
by
XINGXIAN SHOU
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Dissertation Adviser: Robert W. Brown, Ph.D.
Department of Physics
CASE WESTERN RESERVE UNIVERSITY
January, 2011
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the thesis/dissertation of
______XINGXIAN SHOU candidate for the ______degreePh.D. *.
(signed)______Robert W. Brown (chair of the committee)
______Jeffrey Duerk
______David Farrell
______Harsh Mathur
______Shmaryu Shvartsman
______
(date) ______July 26, 2010
*We also certify that written approval has been obtained for any proprietary material contained therein.
To my parents Jun Shou, Huiqin Bian and my fiancée Shuofen Li
ii
Table of Contents
Chapter 1 An Overview of Magnetic Resonance Imaging ...... 1
1.1 What Is MRI? ...... 1
1.2 Principles of Magnetic Resonance Imaging ...... 2
1.2.1 Spin Precession and Larmor Frequency ...... 2
1.2.2 Rotating Reference Frame and the Bloch Equation ...... 7
1.2.3 Signal Detection ...... 14
1.2.4 Phase Encoding and Fourier Transform ...... 19
1.3 MRI Systems ...... 25
1.3.1 Main Magnet...... 26
1.3.2 Gradient Coil System...... 28
1.3.3 Radiofrequency Coil ...... 30
Chapter 2 Acoustic Noise and Its Reduction in Magnetic Resonance Imaging ...... 35
2.1 Acoustic Noise Characterization ...... 36
2.1.1 Noise Source ...... 37
2.1.2 Noise Pathways ...... 38
2.2 Reduction of Acoustic Noise in MRI ...... 39
2.2.1 Reduction of Acoustic Noise via its Source ...... 39
2.2.2 Reduction of Acoustic Noise via Transmission ...... 44
2.2.3 Reduction of Acoustic Noise at Human Ear ...... 46
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2.2.4 Reduction of Acoustic Noise Using New Sequences ...... 47
Chapter 3 String Model of the Acoustic Noise Vibration ...... 56
3.1 Background ...... 56
3.2 String Model: Equations and Solutions ...... 59
3.3 Solution for Gradients ...... 61
3.3.1 Boxcars ...... 61
3.3.2 Trapezoidal Gradient Solution ...... 67
3.3.3 Follow-up Pulses for Additional Frequency Cancellation ...... 71
3.3.4 General Rules for General Pulses ...... 76
3.3.5 Repeated Pulse: Pulse Trains ...... 78
Chapter 4 String Simulation...... 83
4.1 String Simulation...... 83
4.1.1 Boxcars ...... 83
4.1.2 Single and Double Trapezoids ...... 87
4.1.3 Simulation with Damping Effect ...... 90
4.1.4 Longitudinal and Transverse Gradient Pulse ...... 93
4.1.5 Enhancement of Multiple Pulses ...... 95
Chapter 5 Experimental Design and Results ...... 98
5.1 Experimental Design ...... 98
5.1.1 Experiment Setup ...... 98
5.2 Experimental Results...... 100
iv
5.2.1 Cancellation as A Function of Pulse Timings ...... 100
5.2.2 Cancellations in the Frequency Spectrum ...... 108
5.3 Frequency Response Function ...... 123
5.4 Experiment with a Vacuum System ...... 129
Chapter 6 Discussion and Conclusion ...... 134
6.1 Summary of Results ...... 134
6.2 Discussion ...... 136
6.3 Conclusion ...... 142
v
List of Tables
Table 1.1 List of selected nuclear species with their spins (in units of where the proton has
spin ½), their associated magnetic moments in units of a nuclear magnetonmn , gyromagnetic ratios g (in units of MHz/T), and their relative body abundances...... 4
Table 3.1 Convolution results of boxcars ...... 78
vi
List of Figures
Figure 1.1 Clockwise precession of a proton’s spin along an external magnetic field through
a negative differential df ...... 5
p Figure 1.2 An on-resonance spin flip, as viewed in the (a) laboratory frame and (b) 2
rotating frame, ww= 0 and ww10= 0.05 . In MR applications, the frequency w1 would be
much smaller in relation to the RF frequency, but the spiraling would then be much too dense
to illustrate...... 10
Figure 1.3 An overview of MRI system. The main magnet, the gradient system, and the RF
coil system all have their own shielding, which is not indicated in the figure...... 26
Figure 2.1 (A) The sketch shows the Lorentz forces applied on a loop segments carrying
current at an external magnetic field B along the z-direction, which is perpendicular to the
plane of the loop. (B) The sketch shows an arc-loop setup for x-gradient with balanced
Lorentz force over the whole loop. Given the radii of a and c of the arcs, the separation 2b is adjusted to obtain the desired gradient field strength and linearity...... 41
Figure 3.1 A boxcar gradient pulse with duration t1 ...... 62
Figure 3.2 (a) The top plot shows a 2-ms duration boxcar gradient killing the string vibration with 500 Hz for t > t = 2 ms . (b) The bottom plot shows a 1 ms top boxcar gradient 1
maximally enhancing the 500 Hz string vibration for t > t = 1 ms . In both plots, the thick 1
vii
curve is the vibration induced by the positive Q impulse, the thin curve is the vibration
induced by the negative Q impulse, while the dashed curve is the superposition of the two.66
Figure 3.3 A trapezoidal gradient pulse with ramp time (up and down) tr and flat-top time
ttop ...... 68
Figure 3.4 A double-trapezoid gradient pulse with ramp time tr , flat-top time ttop and separation time D . The discussion in the test explains how the three time scales shown are
connected to three different zeros in the frequency spectrum...... 72
1 Figure 3.5 A four-trapezoid gradient pulse is designed to kill 4 frequencies peaks at , tr
1 1 1 , , and , and all of their harmonics...... 75 ttrtop+ t1 tt12+
Figure 3.6 (a) A symmetric trapezoid pulse as a convolution of two boxcars with identical
heights. (b) A “quadratic” pulse as a convolution of a third boxcar with the symmetric
trapezoid pulse. The timings shown in the two figures are connected to the frequencies that
can be suppressed as discussed in the text and in Table 3.1...... 76
Figure 3.7 The overall response factor introduced in the text is shown as a function of wn for
Tms= 10 , N = 100 . Its maxima are the frequencies supported for the corresponding 100
Hz repetition rate...... 81
Figure 4.1 Following the text and using a boxcar with tmstop = 2 , we expect and confirm in
this figure that the 500 Hz vibration of the string vanishes for tms³ 2 ...... 85
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Figure 4.2 Following the text and using a boxcar with tmstop = 2 , we expect and here that
the 1000 Hz (lowest harmonic of 500 Hz) vibration of the string vanishes for tms³ 2 . ... 85
Figure 4.3 Simple boxcar cancellation (solid curve) with a 2 ms flat-top time for the
cancellation of a 500 Hz vibration. The dashed curve shows the 1 ms flat-top maximum enhancement and the dotted line shows a 1.5 ms flat-top intermediate enhancement...... 86
Figure 4.4 A comparison of the numerical simulation result and the analytical calculation result. The solid curve stands for numerical simulation, which shows an almost invisible numerical error after tms³ 2 . The dashed curve is the result from an analytical calculation, which shows a perfect zero after tms³ 2 . The numerical calculation agrees with the
analytical result within the computational error...... 87
Figure 4.5 The numerical simulation of the frequency components of the amplitude at the
center of a vibrating string as a function of time for a trapezoidal pulse with ramp-up time
tsr = 500 m and flat-top time ttop = 300 ms . According to this prediction, the solid curve
and the dashed curve evolve to a complete cancellation for 2000 Hz and 1250 Hz,
respectively, at the instant where the trapezoidal gradient pulse has been turned off...... 88
Figure 4.6 As in Figure 4.3, but now for a double-trapezoid gradient pulse with ramp-up time
tsr = 500 m , flat-top time tstop = 300 m and D=2000 ms . The solid curve shows the
simulated amplitude of vibration component at 2000 Hz, the dashed curve at 1250 Hz, and
the dotted curve at 500 Hz. The complete cancellation of each component after one circle
(one period) agrees with the theoretical expectations...... 89
ix
Figure 4.7 As in Figure 4.6, but now for a four-trapezoid gradient pulse with ramp-up time
tsr = 500 m , flat-top time tstop = 300 m , tm1 = 2000 s and tm2 = 3000 s . The thick solid
curve shows the simulated amplitude of vibration component at 2000 Hz, the dashed curve at
1250 Hz, the dotted curve at 500 Hz, and the fine solid curve at 200 Hz. The complete cancellation of each component after one circle (one period) agrees with the theoretical expectations...... 90
Figure 4.8 Solid curve shows, for a boxcar with tmstop = 2 , the 500 Hz vibration of the
string vanishes for tms³ 2 . The dashed curve shows result considering the damping factor
k = 50 Hz . The comparison shows that the predicted frequency cancellation is effective
even though we consider damping and in fact can be made equally effective via an
adjustment of the boxcar shape – see the discussion in the text and Figure 4.9...... 91
Figure 4.9 (a) The solid curve represents the cancellation for a standard “flat” boxcar with
damping included, and the dashed curve is for a slightly exponentially modified boxcar (via
-kt the factor e ) for comparison. (b) The string vibration is cancelled more completely with
the “damped” boxcar profile than for the standard boxcar pulse...... 93
Figure 4.10 (a) The current distributions of the longitudinal (solid) and the transverse
(dashed) gradient pulse. (b) Amplitudes of the vibration at 500 Hz driven by the above two
gradients, and subsequently suppressed with both boxcar gradient durations 2 ms...... 96
Figure 4.11 For a double-trapezoidal gradient pulse with ramp-up time tsr = 500 m , a flat-
top time tstop = 300 m and the interval D=2000 ms between trapezoids. The vibration at
x
500 Hz is cancelled by the time scale between the positive and negative trapezoids as shown
with the dashed curve. However, a vibration present at 250 Hz , as shown with the solid
curve, gets enhanced...... 97
Figure 5.1 In (a), the trapezoidal gradient pulse with flat-top time 300 ms and ramp-up time
110 ms enhances the vibration at 1300 Hz, while in (b), the trapezoidal gradient with flat-top
time 650 ms and ramp-up time 110 ms suppresses the same frequency. These results confirm the picture obtained via string cancellation theory...... 103
Figure 5.2 In (a), the trapezoidal y-axis gradient pulse with a flat-top time 1320 ms and ramp time 110 ms suppresses the vibration at 700 Hz, while in (b), the trapezoidal gradient with
flat-top time 610 ms and ramp-up time 110 ms suppresses 1400 Hz peak. The enhancement
of 700 Hz in (b) is more pronounced than the enhancement of 350 Hz in (a), because of the
100 Hz frequency resolution. All results confirm the picture obtained via string cancellation
theory...... 104
Figure 5.3 Comparison of peak acoustic FFT amplitudes at 1300 Hz from experiment and simulations. The experimental result is measured in the frequency power spectrum of
acoustic noise, while the simulation result is generated with MathematicaTM string simulations and MATLAB FFT calculations. The damping factor is chosen as discussed in the text...... 107
Figure 5.4 The above two plots show the resulting amplitudes (arbitrary units) of two
frequency spectra for two different gradient pulse sequences. The pulse profiles are also
shown. The upper one in red (a) corresponds to ramp-up time 110 ms and flat-top time
xi
1200 ms . The lower one in blue (b) corresponds to ramp-up time 530 ms and flat-top time
250 ms and has been designed to suppress the principal noise contributions. The peaks at 60
Hz and 120 Hz in each plot arise from the AC background (as discussed in the text)...... 110
Figure 5.5 The above two plots show the resulting relative dB values of two frequency spectra for two different gradient pulse sequences, with the same parameters as Figure 5.4.
With the log scale, we qualitative observe that the bottom plot corresponds to a relative lower value in dB than the top one, reflecting a noticeable reduction in the acoustic noise. We refer in the text to representative sound files for the comparison...... 111
Figure 5.6 (a) A “quadratic” pulse result for the suppression of 500 Hz, 1200 Hz, and 1900
Hz and all of their harmonic peaks. (b) a wide trapezoidal pulse with 2 ms ramp and 3 ms flat-top time, and a strong “sinc” effect. Both spectra are very low in amplitude and the main contribution is from the AC 60Hz and its harmonics...... 115
Figure 5.7 Comparison of the simulated frequency spectrum of single and double trapezoidal gradient pulses. The solid line stands for the double trapezoidal gradient pulse with
tsr = 590 m , tstop = 250 m and D=2000 ms , while the dashed line stands for the single
trapezoidal pulse with the same tr and ttop ...... 117
Figure 5.8 Comparison in arbitrary units of experimental results for the two spectra corresponding to Figure 5.6. Plot (b) stands for the double trapezoidal gradient pulse with
tsr = 590 m , tstop = 250 m and D=2000 ms , while plot (a) stands for the single
trapezoidal pulse with the same tr and ttop ...... 118
xii
Figure 5.9 An illustration of both zeroing and low-pass filtering of the frequency spectra of
noises with two different trapezoidal gradient pulses. (a) tstsrtop==530 mm, 250 (b)
tstsrtop==110 mm, 650 ...... 120
Figure 5.10 A comparison of simulation results of trapezoidal gradients in Figure 5.9. Dashed curve stands for regular trapezoidal pulse and the solid curve for the other one...... 121
Figure 5.11 A four-trapezoid gradient pulse is designed to kill 500 Hz, 1250 Hz, 1900 Hz,
9100 Hz and their harmonic frequency peaks with parameter tsr = 110 m , tstop = 420 m ,
tm1 = 800 s and tm2 = 1200 s . It is clear that 500 Hz and 1900 Hz peaks are cancelled,
while the 1200Hz and 1300 Hz are not cancelled because they are the odd multiples of the
half frequency enhancement, which are also enhanced...... 123
Figure 5.12 The frequency response spectra of x, y, z-axes as measured with the frequency sweep method. The x- and y-axes spectra are similar since the x-, y-axes gradient coils differ
only by azimuthal rotation. The x- and y-axes spectra are higher value in dB than the z-axis spectrum – see the parenthetical remark in the text...... 127
Figure 5.13 The frequency response spectra obtained using an excitation due to a very short
trapezoidal impulse having tsr = 110 m and tstop = 10 m ...... 128
Figure 5.14 A comparison of frequency spectra of noises with two different trapezoidal
gradient pulses for a vacuum MRI scanner. In (a), the red spectrum shows the cancellation at
1400 Hz and its harmonics with a trapezoidal gradient pulse with tsr = 100 m and
tstop = 600 m . In (b), the blue spectrum shows the cancellation of 1400 Hz and 2200 Hz and xiii
their harmonics with tsr = 450 m and tstop = 250 m . A 3.3 dB noise reduction is achieved using the pulse in (b) compared to (a)...... 130
xiv
Acknowledgement
I would like to express my deepest gratitude to my adviser, Professor Robert W.
Brown, for valuable guidance and enormous support during all these years of my graduate studies in the Physics Department at Case Western Reserve University. Dr. Brown has guided me to the exciting MRI physics world and his passion and profound knowledge of
MRI physics have convinced me to choose MRI physics as my career.
I would like to thank my Ph.D. committee, Dr. Jeffrey Duerk, Dr. Shmaryu
Shvartsman, Dr. Harsh Mathur and Dr. David Farrell for their helpful discussions and advice.
I would like to thank my collaborators Dr. Xin Chen for experiments and data analysis, and Dr. Jamal Derakhshan for his efforts on sequence preparation.
I would like to thank my past colleagues in our research group, including Dr. Victor
Taracila, Dr. Timothy Eagan, Dr. Tanvir Baig and Dr. Xin Chen for many helpful discussions. I would like to thank my current colleagues in our research group Yong Wu,
Minhua Zhu and Zhen Yao for their help, support and friendship.
I would like to thank Dr. Timothy Eagan and Michael Thompson for teaching the course on Magnetic Resonance Imaging in Fall 2006.
I would like to thank Professor Kathleen Kash and Professor Charles Rosenblatt for their help before and during my graduate studies.
xv
I would like to thank our secretaries Mary MacGowan, Patricia Bacevice, Sue Rischar,
Betty Gaffney and Lori Morton for their help during my graduate years.
I would like to express my appreciation to the State of Ohio Third Frontier Innovation
Incentive Fellowship program, Toshiba Medical Research Institute USA Inc. (TMRU), GE
healthcare coils (GEHCC) and ViewRay Inc. (VRI) for financial support during my graduate
study.
I would like to thank Wayne Dannels, Dr. Michael Steckner and Dr. Andrew
Wheaton, for their efforts in creating an internship and help when I worked there. They
broadened my MRI physics background, including pulse sequence simulation and design.
I would like to thank Dr. Victor Taracila, Dr. Fraser Robb, and Dr.Vijay Alagappan for their guidance and help in an internship project on dual-tuned RF coil simulation, design and building.
I would like to thank Dr. Mark Xueming Zou and Dr. Larry Kasuboski for giving me the opportunity to work as an MR imaging scientist at AllTech Medical Systems America,
Inc.
Finally, I would like to thank my family, especially my parents Jun Shou and Huiqin
Bian, and my fiancée Shuofen Li for their education, support and love.
xvi
The Suppression of Selected Acoustic
Noise Frequencies in MRI
Abstract
by
XINGXIAN SHOU
Problems due to Magnetic Resonance Imaging (MRI) acoustic noise have long been an
important concern, both in research and clinical applications. A study is made of certain
dominant frequencies in the acoustic noise spectrum of the MRI system. Motivated by both
spring and string ideas, we investigate whether the contributions to the sound from certain frequencies can be cancelled by the appropriate the gradient pulse sequence design. Ideas for cancelling these frequencies are investigated by carrying out theoretical string calculations.
The MRI gradient assembly is modeled as a string and the gradient pulse sequences as a
driving force for that string. Analytical results are obtained with different input gradient
pulses including boxcars, trapezoids, and multiple trapezoids, along with special “quadratic”
pulses. Pulse trains composed of repetitions of these pulse structures are studied. For
comparison and to test these ideas, both simulations and experiments are carried out to verify
our analytical results for the cancellations of selected frequency peaks. The idea that
vibrations resulting from an impulsive force associated with a ramping up of a gradient pulse
xvii
are shown to be cancelled immediately upon the application of another impulsive force coming from the subsequent appropriately timed ramping down of that pulse is verified both by simulations and experiments. A general approach to suppression of multiple-frequency contributions involving a series of gradient pulses with variable timings is given for the cancellations between pairs of impulse forces. The various examples investigated with string analytics and simulations and the associated MRI experiments are a physical embodiments of general time-invariant linear response theory. The present study also provides a foundation to explain results in previous papers on this subject. The method suggests that a variety of pulse profiles and timing combinations can be used to attenuate important contributions to the acoustic output spectrum.
xviii Chapter 1
An Overview of Magnetic Resonance
Imaging
1.1 What Is MRI?
Magnetic Resonance Imaging (MRI) has been a well-known diagnosis method that is widely used both in clinical application and scientific research. MRI is a noninvasive, versatile and flexible imaging method, which has revolutionary impact on modern diagnostic radiology [1]. With the help of three kinds of magnetic fields, data are collected outside of human body (or other imaging targets). Then the image of the structure, metabolism or function of organs and tissues can be acquired. Also the high speed, high resolution and 3D image and arbitrary direction 2D image all make MRI a great imaging method. MRI is based on the physical phenomenon of nuclear magnetic resonance, which is discovered by Felix Bloch and Edward Purcell independently in 1946
[2, 3]. During the 1970s, the back-projection method was discovered by Paul Lauterbur, the Fourier transform was introduced by Edward Ernst, and the echo-planar imaging was invented by Peter Mansfield.
1
Beginning in the early 1980s, many MRI systems have been built, both for
research and clinical use [4]. The main magnetic field strength has been increased, too.
Now 3 Tesla systems are very popular in clinical use. For scientific research, ultra-high systems are targeted to get higher SNR and better image quality. Besides the main field increase, novel imaging methods have been invented, too. Parallel imaging and functional
MRI have been a principal focus in the past couple of decades. Most recently, a new
“compressed sensing” method has been introduced to MRI since it can significantly reduce imaging time and make MRI even faster.
1.2 Principles of Magnetic Resonance Imaging
1.2.1 Spin Precession and Larmor Frequency
The classical spin precession of a nucleus is a useful picture for understanding the
MRI signal. Spin is an intrinsic property such as mass or charge. Electrons, protons and
neutrons have spins (intrinsic angular momenta), each possessing a spin of ½ in units of
and they can be added as a vector sum. Hence a zero net spin can result when such
particles are paired. Atoms with an even number of protons and neutrons have a net spin
of zero. Therefore, elements, such as 12C and 16O , give zero signal for MRI experiments. Applying a nonzero torque N on a system, its angular momentum changes according to
2
dJ = N (1.1) dt
The magnetic momentum m of a nucleus is proportional to its intrinsic angular momentum J with proportionality given by the gyromagnetic ratio:
m = gJ (1.2)
In Equation (1.2), the gyromagnetic ratio depends on the specific kind of nucleus or particle. See Table 1.1 for different nuclear examples [1]. Since the proton has the highest abundance in the human body, 1H is the most popular for MRI. Although 23Na , 31P , 17O
and 19F are low in abundance and have lower precession frequencies, research shows they
can be “hyperpolarized,” resulting in increase in sensitivity of more than 10,000 folds [5].
To understand hyperpolarization with a quantum mechanical picture, the spin nuclei
orient themselves in an external magnetic field for two possible states: parallel and anti-
parallel. Each of these states is, to an approximation of 1ppm, equally populated at body or room temperatures, and the MRI signal is proportional to the difference in the populations of the two states. Hyperpolarization is the process of forcing a nuclear system to increase the population difference by four-orders-of-magnitude or more, which will significantly increase the MRI signal. As this hyperpolarization can be utilized for nuclei with low-frequency and abundance, hyperpolarization has become a very important technique in MRI.
3
We consider now the details of the spin precession. The torque applied on a magnetic moment m in a constant magnetic field B is given by:
NB=´m (1.3)
Magnetic Abundance in human Nucleus Spin g (MHz/T) moment body
Hydrogen 1H 1/2 2.793 42.58 88 M
Sodium 23Na 3/2 2.216 11.27 80 mM
Phosphorus 31P 1/2 1.131 17.25 75 mM
Oxygen 17O 5/2 -1.893 -5.77 15 mM
Fluorine 19F 1/2 2.627 40.08 4 mM
Table 1.1 List of selected nuclear species with their spins (in units of where the proton has spin
½), their associated magnetic moments in units of a nuclear magneton mn , gyromagnetic ratios g (in units of MHz/T), and their relative body abundances.
Combining (1.1), (1.2) and (1.3), we find that the magnetic momentum changes due to an external magnetic field B can be expressed by the following:
dm =´gm B (1.4) dt
4
This is a simple version of the more general Bloch equation we will discuss later.
Mathematically, since the time-rate change of a rotating vector is proportional to a cross product involving that vector, which leaves the magnitude mm= unchanged.
Figure 1.1 Clockwise precession of a proton’s spin along an external magnetic field through a negative differential df .
Figure 1.1 shows a clockwise precession of proton’s spin along an external magnetic field B to illustrate (1.4). The external magnetic field points along the z-direction, and q is
5
the angle between B and magnetic momentum m , while df is the angle subtended bydm . The instantaneous change in the magnetic moment dm can be calculated via (1.4), and
the direction of dm is perpendicular to the plane which is defined by m and B . Their magnitudes are m and B , respectively. On the other hand, from the rotation geometry in
Figure 1.1, the magnetic moment change can also be calculated given df and q , which is:
dddmmqfmqf==sin sin (1.5)
From Equation (1.4)
dBdtBdtmgm=´ = gmqsin (1.6)
A comparison with (1.5) and (1.6) gives gfBdt= d , which leads to the well-known
Larmor precession:
df wg==B (1.7) dt
For a proton, the gyromagnetic ratio is:
g =´2.675 108 rad / s / T (1.8)
In calculations relating the cycle frequencies with magnetic field, “gamma-bar” is used
more often for resonance frequency calculation with unit Hz.
6
g g ==42.58MHz / T (1.9) 2p
Here T is the Tesla unit of magnetic field strength.B0 is the external magnetic field along
z-axis. If B0 is constant, the solution (the “phase”) for precession angle change df is:
fwf()tt=-00 + (1.10)
Here wg00= B is the precession frequency, f0 is the initial phase of the spin, while the minus sign shows left-handed precession along B0 . The phase f()t is very important for
MRI signal evolution. The Larmor precession frequency is defined as:
wwgL ºº00B (1.11)
1.2.2 Rotating Reference Frame and the Bloch Equation
At the end of the previous section, the Larmor frequency is defined in terms of the
gyromagnetic ratio and the external magnetic field. Besides the main magnetic field,
there are also gradient fields and radiofrequency fields needed for imaging. In order to explain the principles more clearly, the rotating reference frame is introduced and considered instead of the laboratory frame, which is generally static.
We ultimately want to derive the magnetic momentum rotating formula in the laboratory reference frame. Define the Cartesian coordinates in that frame as (x, y, z) and,
7
by contrast, use primed coordinates (x¢,y¢ ,z¢ ) for a rotating reference frame with angular
frequencyw . xˆ ,yˆ ,zˆ,xˆ¢,yˆ¢ ,zˆ¢ are the respective unit vectors for the above two coordinate systems. In the rotating reference frame, Equation (1.4) can be replaced by: æö ' çdm÷ W ç ÷ =´+gm ()B (1.12) èøç dt ÷ g
In which,W=-gBzˆ is the left-handed reference rotating frequency about z-axis. The
radiofrequency (RF) field, which is perpendicular to the main magnetic field, is used to
tip (i.e., precess around an orthogonal direction) the magnetic momentum down to the
transverse plane. This produces a signal “excitation.” Mathematically, assuming this RF
field along x'-axis is applied with amplitude B1 , in addition to the omnipresent main
magnetic field strength B , we express Equation (1.12) as:
' æödm ç ÷ ¢¢ˆ (1.13) ç ÷ =´gm ((BBzBx01 - ) + ) èøç dt ÷
where B xˆ¢ is a circularly polarized magnetic field (which rotates with angular frequency 1
w in the laboratory frame),
¢ Bx11ˆ =- B(cos xww t y sin) t (1.14)
8
Such fields can be generated by “quadrature” RF coils. If the reference frequency for the
rotating frame is chosen to be the Larmor frequency of the external magnetic fieldB0
(this is the “resonance” of “MRI”), there is effectively no magnetic field left along the z'-
axis. Only B1 remains, which as noted is along the x'-axis. This indicates that the precession about the z'-axis has disappeared in the rotating reference frame. On the other
hand, a new precession along x'-axis can be observed with frequency wg11= B . In this
kind of reference frame, it is easy to trace the magnetic moment’s change upon time,
using Equation (1.13), since only the magnetic field perpendicular to the field in z'-axis
has to be considered. For example, if the initial magnetic moment points in the z-direction
with a value m0 , a continuous application of an RF field B1 is rotates the moment
downward through the y'-axis, and then past the negative z'-axis and back up to the z'- axis and so forth, staying all the while in the plane formed by y'-axis and z'-axes. Given that the RF field is applied for duration t , the exact angular position of rotation can be calculated, which is:
ft()= gB1 t (1.15)
Usually angles like 90 and 180 degrees are selected for imaging. The corresponding RF
pulse duration can be calculated easily from the given RF field B1 . For illustration, we can plot the evolution of magnetic moment both in laboratory and rotating frame. See
Figure (1.2).
9
p Figure 1.2 An on-resonance spin flip, as viewed in the (a) laboratory frame and (b) rotating 2
frame, ww= 0 and ww10= 0.05 . In MR applications, the frequency w1 would be much smaller in relation to the RF frequency, but the spiraling would then be much too dense to illustrate.
So far, we have considered an isolated proton, within a classical magnetic
moment rotating model, in an external magnetic field. In fact, there are so many protons
in human tissue, even in a single voxel, which is the volume element relevant for imaging
resolution. So we need to define a concept to describe proton density, and here we can use the magnetizationM , which is the average proton magnetic moment density. Assume
there are N protons in volume V and that they all experience the same external magnetic
10
field, so they have the same precession frequency. They give rise to the single
magnetization vector for MR imaging:
1 N M = m å i (1.16) V i
This set of spins is called an “isochromat,” where all the spins have the same precession
phase. Summing (1.4) over the volume V with N protons, but without considering the
protons interaction, we get:
dM =´gMB (1.17) dt ext
In general, Bext is along the z-direction with a very spatially uniform and time- independent value, BBzext = 0 ˆ. The cross product in (1.17) has no contribution from the magnetization component along z-axis. So (1.17) can be rewritten as:
dM ^ =´gMB (1.18) dt ^ ext
The above discussion has not taken into account the “spin-lattice relaxation time” T1 and the “spin-spin relaxation time” T2, which are critical in MR imaging. Thus we introduce
the Bloch equation where the T1 and T2 effects are included. With the introduction of the
z-direction value of equilibrium magnetization, M0 , and of the proton interactions with
lattice, the magnetization along z-axis changes at a rate proportional to the difference
MM0 - z , 11
dM 1 z (1.19) =-()MM0 z dt T1
If the external field B is parallel to the z-direction, the solution of (1.19) is
--tT//11 tT Mtzz()=+- M (0) e M0 (1 e ) (1.20)
Equation (1.20) is very important in order to understand the longitudinal relaxation of an isochromat which is initially not at its equilibrium state, in other word, after the application of an RF pulse.
Spin-spin interaction is introduced to account for the fact that neighboring spins experience slightly different local magnetic field. After a 90-degree RF pulse is applied, the external field variance leads to different precession frequencies and results in phase variations between the spins. This will reduce the overall magnetization vector in the transverse plane. The characterization of this transverse magnetization reduction needs
another empirical parameterT2 , which is called the “spin-spin relaxation time.” Adding an extra term to the (1.18) gives us an expression including the “spin-spin” effect:
dM^ 1 =´-gMB^^ext M (1.21) dt T2
It is easier to solve (1.21) in the rotating frame where it simplifies to
dM ^ 1 ()'=- M ^ (1.22) dt T2
12
This shows any initial magnetization in the transverse plane will decay exponentially with a decay factorT2 . Mt^() is dependent on its initial value M ^(0) referring to time
t = 0 :
-tT/ 2 Mt^^()= M (0) e (1.23)
Considering both longitudinal and transverse relaxation times, T1 and T2 , we have a
complete version of Bloch equation given by:
dM 11 (1.24) =´+gMBext() M0 - Mz z ˆ - M^ dt T12 T
Assuming the constant external fieldBz0ˆ, we can solve (1.24) and obtain:
-tT/ 2 Mtxx( )=+ e ( M (0)cosww00 t M y (0)sin t ) (1.25)
-tT/ 2 Mtyy( )=- e ( M (0) cosww00 t M x (0) sin t ) (1.26)
--tT//11 tT Mtzz()=+- M (0) e M0 (1 e ) (1.27)
The equilibrium state can be evaluated with the limit t ¥, where all the exponentials
vanish, leaving the steady-state solution:
MMxy()¥= () ¥= 0, MM z () ¥= 0 (1.28)
13
The magnetization has finally relaxed along the z-direction. The general solution can
also be written down for the overall transverse magnetization, in order to show the phase
evolution. Let us define:
Mt+()=+ Mtxy () iMt () (1.29)
With (1.25) and (1.26), the solution for a static field for the complex transverse
magnetization (1.29) can be expressed as:
--ittTw02/ itf() Mt++^()== e M (0) Mte () (1.30)
The magnitude of transverse magnetization is thus given by:
-tT/ 2 Mt^^()= e M (0) (1.31)
with the phase
fwf()tit=-0 + (0) (1.32)
The evolution of the phase in time plays an extremely important role in MRI.
1.2.3 Signal Detection
Besides understanding these basic principles of magnetic moment evolution in an
external magnetic field, we also need to know that how the signal is detected. First of all, magnetic moment inside sample is always immersed in the large external magnetic field
B0. The precession of the moment in the large field will induce an electromotive force 14
(emf) in any nearby coil. The emf is equal to the negative of the magnetic flux change rate:
df emf =- (1.33) dt
where the magnetic flux can be calculated by integrating over the area that the coil
encloses,
f =⋅BdS (1.34) òcoil area
The magnetization of a sample has an effective current density as (1.35), which generates
a magnetic field in space.
J (r,t) = ´M(r,t) (1.35) M
The curl operation in (1.35) computes the net ‘circulation’ of the magnetization. In this
way, we can calculate the current density of the magnetization, which is used to obtain the vector potential at position r from a current density source.
m 3 Jr()¢ Ar()= 0 d r¢ (1.36) 4p ò rr- ¢
From this, the magnetic field is calculated by:
BA=´ (1.37)
Applying Stokes’ theorem to (1.37), we can rewrite (1.34) as:
15
f =⋅=´⋅=⋅BdS() A dS dlA (1.38) òòarea area ò
We get a very convenient and illuminating form for the flux fM through a coil by substituting A as given by (1.36) and integrating by parts. We obtain:
é ù m ´¢¢Mr() f =⋅dlêú0 d3 r¢ M êú òòêú4p rr- ¢ ëû é ù m 3 ê Mr()¢ 1 ú =⋅´-´0 dr¢¢ dlê ( ) ( ¢ ) Mr ( ¢ )ú 4p òòê rr--¢¢ rr ú ë û (1.39) éù m 3 êú1 =⋅-´0 dr¢¢ dlêú ( ) Mr ( ¢ ) 4p òòêúrr- ¢ ëû éù m 3 êúdl =0 drMr¢¢ ( )⋅´êú ¢ ( ) 4p òòêúr - r¢ ë û
Notice that the vector potential that would be due to the receive coil for unit current has
naturally arisen:
RECEIVE COIL m0 dl A(r¢) = ò (1.40) UNIT CURRENT 4p r -r¢ Now the evaluating (1.36) at position r ¢ for a line integral over current path shows that the curl of (1.40) is actually BRECEIVE COIL , which is defined as the magnetic field per unit UNIT CURRENT current that would be produced by the coil at the point r ¢ :
æ ÷ö RECEIVE COIL ç m dl ÷ B = ´¢ ç 0 ÷ (1.41) UNIT CURRENT ç ò ÷ ç4p r - r¢ ÷ è ø
16
Finally, the flux can be written as:
f (t) = d 3r BRECEIVE COIL (r )⋅ M(r,t) (1.42) M ò UNIT CURRENT sample
where now the time dependence of the magnetization is made explicit. This is the so-
called principle of reciprocity. Thus the flux through the detection coil produced by
magnetization in the sample can be determined by integrating the product of the sample’s
magnetization and the field produced over the sample by the detection coil, carrying unit
current. Thus the emf is:
dd emf=-f () t =- d3 rM ( r , t ) ⋅ Breceive ( r ) (1.43) dtM dt òsample
Equation (1.43) is very useful to understand how MRI signal can be picked up by a
receive coil.
Let us assume that the sample is placed in a static uniform magnetic field, and that
it is tipped by some RF pulse. At time t, there exist three orthogonal magnetic
components, Mrtx (,), Mrty (,) and Mrtz (,). So (1.43) can be rewritten as:
d signalµ- dr3 éùBBBreceive() rM (,) rt + receive () rM (,) rt + receive () rM (,) rt (1.44) dt ò ëûêúxx yy zz
Recall the solutions for longitudinal and transverse magnetization with relaxation from
(1.20) and (1.30), we have:
17
--tT/()11 r tT /() r Mrtzz(,)=+- e Mr (,0) (1 e ) M0 (1.45)
--tT/20 () r iwwf t --+ tT / 200 () r i t i () r Mrt++( , )== e e Mr ( ,0) e e Mr ^ ( ,0) (1.46)
The time derivative in (1.44) can be taken inside the integral, thus we can make some
approximation prior to integration. For the main magnetic field at the Tesla level, the
Larmor frequency w0 is at least four orders-of-magnitude larger than typical values of
1/T1 and 1/T2. Thus the derivative of the relaxation exponent factors can definitely be
neglected compared with the derivative of the precession frequency exponent factor. We
have the dominant derivative as below:
-tT/() r signalµ-+-wwfwf d3 re2 M(,0) réBBreceive ()sin( r t ()) r receive ()cos( r t ()) r ù 00000ò ^ ëê xyûú (1.47)
In other words, we see the dominant signal in the receive coil is induced by the rapid
Larmor oscillations of the transverse magnetization. This also explains why the
transverse magnetization M+ is always referenced as the ‘signal.’ The signal can be
further simplified to:
-tT/ signalµ+-wwqf d3 re2 M(,0) rB ()sin( r t () r ()) r (1.48) 000ò ^^ B
Here, B (r ) and q (r ) are the magnitude and the phase of the complex representation of ^ B
the two transverse components of the field that would be produced by a unit current
flowing through the detector coil. We therefore observe the MRI signal is dependent on 18
the Larmor frequency of the main magnetic field, the sample properties, the so-called
“sensitivity profile” (the field spatial distribution) of the receive coil, the transmit coil profile (which entered through the tipping angle over the sample), and the phase of the magnetization. It is useful to write an expression for an overall uniform system. For a homogeneous sample with volume Vs, assume that all the variables in (1.48) are independent of r, then
-tT/ 2 signalµ+-wwqf000 Vs e M^^B sin( t B ) (1.49)
This enables us to discuss qualitative, and even sometimes quantitative, aspects of the
MRI modeling.
1.2.4 Phase Encoding and Fourier Transform
The rapid oscillations at the frequency w in the above signal expression (1.48) 0 must be removed, in practice, by a step of ‘demodulation.’ This step is the multiplication
of the signal by a sinusoid or cosinusoid with a frequency at or near w0 . The demodulation with a sinusoid generates a real channel signal, while the cosinusoid demodulation generates an imaginary channel signal. The real and imaginary forms of the two channel signals, can be combined into a complex signal as:
st()=+ sre () t is im () t (1.50)
19
In place of the previous signal expression (1.48), the complex modulated signal has spatial dependence in general, which is:
-W-+-tT/ () r i ((wf ) t () r q ()) r st()µ w dre3 200 M ( r ,0)B( re ) B (1.51) 0 ò ^^ where W is the demodulation frequency. This complex signal can be further defined as an equation by incorporating a proportionality constant Lincluding the gain factors from the electronic detection system. With the transmitting and receiving RF coils assumed to
be sufficiently uniform, the initial magnetization phase f0 , the receive directional phase
qB , and the receive field magnitude B^ are all independent of position. With the relaxation neglected, we have:
st()=Lw B drM3((,)) (,0) r eitW+f rt (1.52) 0 ^^ò
Accumulated phase f(r,t) is defined in the counterclockwise directions according to:
t f(r,t) =-ò dt¢w(r,t) (1.53) 0
If there is only a uniform static field, ww= 0 , the precession frequency is independent of position. The purpose of adding a gradient field is to have the accumulated phase change with position.
20
We now will introduce the proton spin density and its relationship with
equilibrium magnetization M0. The resulting expression is involved with temperature T
and static field B0, which is as following:
1 g22 Mr(,0)== Mr ()r () r B (1.54) ^ 004 kT 0
r0()r is the number of proton spins per unit volume in 3D dimensions. With the combination of (1.52) and (1.54), we can rewrite the signal as:
st()= ò dr3((,))r ( re )itW+f rt (1.55)
Here the effective spin density r()r is defined:
1 g22 r(r) º w0LB^M0(r) = w0LB^r0(r) B0 (1.56) 4 kT
Spatial dependence of the receive field, as well as the relaxation factors, can be included
in the above definition. A simple example for the phase factor in (1.55) is to assume it
only depends on one dimension, which can be defined as the z-coordinate. The signal
then is simplified as:
st()= ò dzr () zeit((,))W+f rt (1.57)
Correspondingly, the effective 1D spin density is:
rr()zdxdyrº òò () (1.58)
21
The integration is carried over all regions with nonzero spin density. The eventual
purpose of imaging is to determine the spin density r()z distribution of a sample from the measurement of the signal as a function of time. A very important step is to connect the spin precession to its position, which is achieved by adding a gradient magnetic field linearly proportional to position z.
By adding the linear magnetic field, the Larmor frequency of a spin will also be linearly proportional to the position z. The maximum of the linearly changing magnetic field at any point in patient bore is comparably smaller than the static field. We can express this kind of magnetic field as:
Bztz (,)=+ B0 zGt () (1.59)
The quantity G is the constant gradient in the z-direction as defined below:
GBzzzº¶/ ¶ (1.60)
So the position dependent precession frequency can be written as:
wwg(,)zt=+0 zGt () (1.61)
The use of a gradient to establish a relation, such as (1.61), between the position of spins
along some certain direction and their precession frequency is referred to as a “frequency
encoding” along that direction. The additional phase accumulated by applying the
gradient field for time t is
22
t fg(,)zt=- z dtGt¢¢ ( ) (1.62) G ò 0
It is important to note here that the gradient is assumed to be applied only after the initial
RF excitation at t = 0.
With the modulation frequencyW=w0 , and the 1D assumption, (1.57) can be simplified
to
iztf (,) st()= ò dzr () ze G (1.63)
We can see from the above expression that the phase is only dependent on the gradient
field and time. With the explicit z dependence of the phase in (1.62), we can rewrite
(1.63) as
sk()= ò dzr () ze-ikz2p (1.64)
where the time dependence is included in the spatial frequency
t kt()= g ò dtGt¢¢ ( ) (1.65) 0
The signal s(k) is the Fourier transform of the spin density if the applied gradient field is linearly dependent on the position z. Thus the spin density is also called “Fourier
encoded” along z by the linear gradient. This Fourier transform between the linear field
and the spin density boosts MRI technology. Mathematically, we can easily get the spin
density by taking an inverse Fourier transform, which is
23
r()zdkske= ò ()+ikz2p (1.66)
The signal s(k) and the image r()z are a ‘Fourier transform pair.’ It is important to note
that any direction can be chosen to be a 1D image direction as long as the gradient field is
linear along that direction. The image can be reconstructed by taking the Fourier
transform of collected k-space data.
The above 1D image example shows that we need to apply a sufficiently strong
linear field for a long enough time period to cover the sufficient range of k values. In a
2D image, we need two sets of gradient fields to cover the k-space over the image area. In
3D volume image, we need three sets of gradient fields for 3D space. Below, we give the expressions of three sets of gradient fields and their k-space value.
t kt()= g Gtdt (¢¢ ) xxò
t kt()= g Gtdt (¢¢ ) (1.67) yyò
t kt()= g Gtdt (¢¢ ) zzò
With three sets of k values in x, y, z direction, the 3D Fourier transform can be written as
sk()= ò dr32r () re-⋅ikrp (1.68)
or
24
-++ikxkykz2(p ) s(,, k k k )= dxdydzr (,,) x y z e xyz (1.69) xyz òòò
After all the data are collected properly in the time domain, the image can be
reconstructed by taking the inverse Fourier transform as below:
rˆ()rdkske= 32 ()ikrp ⋅ (1.70) ò m
Here, skm ()is the measured discrete data and the rˆ()r is the corresponding reconstructed data, which could be a good estimate of the physical spin density r()r .
1.3 MRI Systems
In the previous sections, we have discussed the general physics principles of MR
imaging. You might notice that several magnetic fields are critical to MR imaging. The three kinds of magnetic fields are generated by corresponding magnet coils, namely, the main magnet coil, the radiofrequency (RF) coil and the gradient coil, the key components
of MRI hardware. Also, the additional software corresponding to the RF and gradient
pulse sequence, the data collection, the data processing, and the final imaging
reconstruction is very important too. Here, let us explain in detail the hardware for the
three coil systems: the main magnet, the gradient system, and the RF system. See Figure
(1.3) for an overall sketch of the MRI system.
25
Figure 1.3 An overview of MRI system. The main magnet, the gradient system, and the RF coil system all have their own shielding, which is not indicated in the figure.
1.3.1 Main Magnet
The main magnet is required to generate an extremely uniform and large magnetic field inside the patient bore. A permanent magnet is usually used for low fields up to about 0.3 T, and a superconducting magnet is normally required for fields of 1.5 T and above [4]. The advantage of a high field is a high signal-to-noise ratio and high resolution. However, the increases in cost, RF penetration, specific absorption rate (SAR)
26
of energy, and acoustic noise are disadvantages at higher field MRI. Depending on the
main use of the MRI system or MRI-hybrid system, optimal field strength can be chosen.
The most important criteria of designing main magnet are its homogeneity, which
is defined by the ratio of maximum variation of magnetic field in a given area of interest.
BB- homogeneityº 0,max 0,min 106 (ppm ) (1.71) B0,mean
An accepted homogeneity level for contemporary whole-body clinical imaging machines
might be given as 5ppm over a 50 cm spherical volume at 1.5 T [4]. This shows that the static field used in MRI must be very homogeneous. The desire to have such a homogeneity field is to avoid the field distortion effects.
Besides the homogeneity requirement, temporal stability, patent access and comfort, cost effectiveness, and the shielding of fields generated outside the imaging environment are important aspects too. Shimming can remove the small inhomogeneities, which are present in the field. Shielding is used to constrain the huge magnetic field to a close proximity to the scanner system. For a superconducting magnet, the liquid helium and nitrogen cryogenics are additional complications.
27
1.3.2 Gradient Coil System
A gradient system is also a crucial part of the MRI system. The magnetic gradient field system normally consists of three orthogonal magnetic fields, which are designed to produce gradient magnetic fields in x-, y-, z-directions. These magnetic fields vary in time t, and also spatially linearly. For example, z-gradient magnetic field can be expressed as:
Grz ()= Gz (1.72)
Here, Gz is proportional to the z-position while independent on the x-, y-positions.
Gradient strength is always in the range milli-Tesla per meter, mT/m. The time rate of change of the gradient magnetic field is called the “gradient slew rate,” which determines the rise time of a trapezoidal gradient pulse through the relation:
Gsrmax =*t (1.73)
Gmax is the maximum gradient strength, and sr is the slew rate, in unit mT/m/ms. The maximum gradient recently can reach up to about 40 mT/m at 250 A, with the slew rate as fast as 150 mT/m/ms. Practical gradient pulses require rise time as short as possible since the rise part of gradient is usually not used for imaging. For fast imaging, the duration of the entire gradient pulse is supposed to be as short as possible, but still managing to cover k-space. Thus, a large magnitude and short rise time are very important to achieve fast imaging.
28
Gradient coils are designed in terms of the following set of criteria. Inductance is
the self-inductance of the wires of an individual gradient coil axis. The inductance
determines the speed of the gradient pulse, especially the turn on/off time. The minimum
rise time of a gradient is determined by the inductance, current and amplifier voltage.
Sensitivity is the ratio of the generated gradient strength divided by the current through
the coil. It is usually expressed in the unit mT/m/A. In general, high sensitivity is needed
since the maximum gradient field must be created by a given fixed amount of current
from amplifier system. Gradient uniformity is typically defined over the diameter of
spherical volume (DSV) where the gradient field must be sufficient linear to allow image
reconstruction with an acceptable level of spatial distortion. The typical DSV of a
standard whole-body gradient coil is between 40 and 50 cm. In contrast to the main
magnetic field, non-uniformity about 5% is still sufficient to meet requirements and yield
an acceptable level of distortion [1]. Additionally, gradient coils have electrical resistance
and dissipate energy during operation. Lower resistance gradients produce lower power
dissipation, but water cooling remains a necessity for most gradient systems. The gradient
coils are conducting wires mounted on cylinder formers, while epoxy is generally used
for the mounting, providing sufficient stiffness for the control of the vibrations generated from the Lorentz forces present. A z-gradient coil is a pair of coils with opposite current
flows. The x-, y-gradient coils are similar to each other, differing by a 90-degree rotation
in the azimuthal direction. The currents flowing through the x-, y-gradient coils are symmetric regarding to the center plane, while they are asymmetric for the z-gradient
29
coil. Hundreds of amps are needed in order to generate gradient fields lying in the range of several tens of mT/m.
Since fast imaging methods are used more frequently, a strong gradient performance is needed for MRI. Subject to other constraints, maximizing the gradient strength and the slew rate are key to superior performance. The phase, and hence the gradient strength needs to increase as the main field increases (which leads to greater main field nonuniformity). The slew rate needs to increase in order to decrease the ramp time and control the overall imaging time. However, an increase in either the maximum gradient strength or the slew rate worsens the acoustic noise problem. Thus the control of acoustic noise in the MRI system is paramount and we will discuss the methods of reducing acoustic noise of MRI in Chapter 2.
1.3.3 Radiofrequency Coil
RF coils are key components in MRI system. They have two main purposes, one as transmitter for generating rotating RF pulses to excite the spins in a sample, and the other one as receiver for the detection of RF signals from that sample. Rather naturally, coils with the first purpose are called transmit coils and those with the second purpose are referred to as receive coils. Of course, one single coil can serve both as a transmit and receive system at the same time, and is sometimes called a transceiver coil.
30
In order to get a uniform magnetization excitation, we need a very uniform
transmitting field, which is almost always produced by a birdcage coil system. See Figure
(1.4) for a general configuration of the birdcage coil. The number of legs is usually 8, 16
or 32, and these legs’ ends at each side are connected with a circle loop, which is called
an end-ring. The desired, and most useful, mode arising from the large number of current
degrees of freedom, is that of a current distribution which is sinusoidal in the azimuthal
angle. The current in the i-th leg thus has the time dependence:
Iti(, )=+- I0 sin(wp t 2 ( i 1)/ N ) (1.74)
N is the leg number of the birdcage. The magnetic field is perpendicular to the legs,
which are parallel to the main magnetic field (the z-direction). This field direction is just
what is wanted in order to tip the spin away from the z-axis. In frequencies of 64 MHz
and below, the uniformity of transverse magnetic can be generated within the aforementioned DSV, the spherical volume of interest. Another advantage of the birdcage is that it can be driven in “quadrature” mode, which means the field generated by birdcage is circularly polarized, and all the field can be used for imaging. For signal detection, a so-called surface coil has its advantage over the birdcage or volume coil.
First of all, since it is closer to the imaging area, the receive field generated by a unit current (recall the statement of reciprocity) is greater than that of the volume coil.
Secondly, due to the smaller size of surface coil, it picks up less noise from the imaging area than does the volume coil.
31
Figure 1.4 A 16-leg high-pass birdcage configuration is sketched. All the capacitors are
distributed along the end-ring and placed between the axial leg connections to the end-ring.
By definition, the SNR is proportional to the signal and inversely proportional to the noise, so the SNR of a surface coil is normally bigger than a volume coil. The main requirements for designing surface coils correspond to maximizing both SNR and the coil gain factor Q. The surface coil can only receive a signal in an imaging area whose size is comparable to its own size. Thus, in order to receive an MR signal from a larger view, such as the spine, a “phased array” is needed. A phased array is a group of surface coils, which can detect signals over a larger imaging area. A combined signal is reconstructed from all the signals and phases associated with each and every single coil. Different methods of imaging reconstruction are discussed in detail in the original phased-array paper [6]. Because the elements in the phased array are near to each other, mutual 32
inductance interferes with their operation and there is noise cross-talk among the coils.
The noise term between two coupled coils can be approximately expressed by iMIw 12 2 .
Here, w is the resonance frequency of the coil, M12 is the mutual inductance between the
two coils, and I2 is the current flow through coil 2 induced by the current in coil 1
through mutual inductance [7]. To eliminate noise cross-talk, there are several methods
that can be used. Adjacent coils may be partially overlapped in order to cancel the
coupling between them (mathematically, the term M12 becomes zero) and the
corresponding noise is minimized by this decoupling method. For next nearest and more
distant neighbor coils, overlap method cannot be applied. However, a low-input
impedance preamplifier may be used to make the induced current I2 zero. The
explanation for this suppression is that the low-input impedance preamplifier with the
matching capacitor and inductor constitutes a resonance circuit at the resonance
frequency at which the coil is operated. This parallel circuit presents a very large
impedance to the outside loop circuit and consequently there is no current through coil 2
induced by coil 1. With this method, the coupling between any two coils can be
eliminated. The challenge is to build, place and tune such preamps for each coil to be
decoupled.
33
Works Cited:
1. E. M. Haacke, R. W. Brown, M. R. Thompson, and R. Venkatesan, Magnetic
Resonance Imaging: Physical Principles and Sequence Design, Wiley &
Sons, 1999.
2. J. P. Hornak, The Basics of MRI, http://www.cis.rit.edu/htbooks/mri/ .
3. J. Jin, Electromagnetic Analysis and Design in Magnetic Resonance Imaging,
CRC Press, 1999.
4. Z. Liang, and P. C. Lauterbur, Principles of Magnetic Resonance Imaging: A
Signal Processing Perspective, IEEE Press, 1999.
5. J. H. Ardenkjaer-Larson, B. Fridlund, A. Gram, G. Hansson, L. Hansson, M.
H. Lerche, R. Servin, M. Thaning, and K. Golman. Increase in signal-to-noise
ratio of > 10,000 times in liquid-state NMR, Proceedings of the National
Academy of Sciences of the United States of America 100:10158–10163,
2003.
6. P. B. Roemer, W. A. Edelstein, C. E. Hayes, S. P. Souza, and O. M. Mueller,
The NMR Phased Array, Magnetic Resonance in Medicine 16:192-225, 1990.
7. Hiroyuki Fujita, New Horizons in MR Technology: RF Coil Designs and
Trends, Magnetic Resonance in Medical Sciences 6(1):29-42, 2007.
34
Chapter 2
Acoustic Noise and Its Reduction in
Magnetic Resonance Imaging
The acoustic noise produced by an MRI system, which is dominated by the
vibration of gradient coil former, is a long-standing problem [1, 2] due to its
unpleasantness and even painfulness, and is exacerbated by both higher field strengths
and faster imaging methods. Experiments have shown that this noise can reach more than
120 dB in high field systems with fast imaging sequences, which is painful not only for
patients, but also for operators and other health service staff. While the MRI acoustic noise problems start with serious patient discomfort (earplugs are generally needed), patients tend to react with sudden motion every time a new gradient sequence is begun, and that motion is detrimental to obtaining quality imaging. For functional MRI of the brain this motion is especially damaging. For studies of the auditory system, acoustic noise generated during fMRI can interfere with assessments of this activation by introducing uncontrolled extraneous sounds [2]. Thus acoustic noise has been a challenging issue for applications of fast sequence methods and high field MRI. In this chapter, we first will talk about the acoustic noise characterization including the noise source, pathway and its measurement. Different methods of reducing acoustic noise and 35
the difficulties encountered in maintaining high-quality performance of MRI with such methods will be discussed.
2.1 Acoustic Noise Characterization
A conventional cylindrical MRI system has three gradient coils and their shielding coils in order to produce three orthogonal linear fields for spatial encoding
inside the patient, without producing significant magnetic fields outside of gradient bore.
During an imaging sequence, the currents flowing through the gradients are turned on and off rapidly to perform the imaging procedure, such as phase encoding and frequency encoding. As we have discussed, gradient coils with large current flows are placed inside the huge static field B0 environment; thus, they experience significantly strong magnetic
forces – the Lorentz forces on currents due to external magnetic fields. As a result of fast
switching of currents flowing in gradient coils, the rapidly changing Lorentz forces result
in significant vibrations on the gradient coil formers, which produces loud acoustic noise.
Especially aggravating is that the large gradient former bore acts like a huge loudspeaker
when it is operated. This is the chief source of gradient acoustic noise produced in MRI
system. In order to understand in depth the MRI acoustic noise, we need to characterize it
in detail.
36
2.1.1 Noise Source
Humans hear sound through the ear drum vibration instigated by an external air
vibrational wave. A normal (youthful) human ear can hear sound with frequencies from
20 Hz to 20 KHz, which is called the audible range. One important property of noise is
defined as loudness, usually with the unit of dB (decibel, a relative sound-pressure level).
To get a quantitative view of the loudness of sound, the sound pressure level is usually measured in mPa in a linear scale. However, dBA (the sound filtered so as to approximate a human ear is called A-weighted, and the corresponding unit is the dBA) is
a frequently used unit defined with a logarithmic scale, which has the following relationship with the linear sound pressure:
SPL= 20 log( P10 / P ) (2.1)
Here P0 is the common reference sound pressure in air with its value 20 mPa and P1 is
the measured value in same unit mPa . Noise is officially an unwanted sound, such as the
sounds from a running engine, percussive piling, and so on. MRI acoustic noise is just
such a definite unwanted, unpleasant sound.
To characterize acoustic noise, we need to figure out the noise sources and
pathways for transmission. As explained above, the gradient acoustic noise is the main
source in the standard MRI system. Besides, the eddy-current induced vibrations of metal
structure such as the body RF coil and the cryostat inner bore are additional and
37
important acoustic noise sources [4]. These components of MR system are necessarily
very stiff, to suppress their motion, so the vibration amplitudes are very small compared
to their sizes, and are reported to be less than 50 microns for the gradient coil former[5].
Despite the small amplitudes, they still carry a large amount of energy to be dissipated
into the air, finally reaching the human ear as a form of noise. Besides these two main
noise sources, the pump and air-handling system can generate noise, too, which exists
even when the MRI system is not being operated [2]. This secondary source of noise is
not the principal concern of the present thesis.
2.1.2 Noise Pathways
In summary, most of the MRI noise originates from the vibrations of solid
segments such as the gradient coil former, the conducting cryostat inner bore, and the RF
coils. Those components vibrate when rapidly changing Lorentz forces are applied on them. After we understand how acoustic noise is generated by the currents carried by the gradient coil former and the eddy-currents induced in the RF coil and cryostat inner bore,
we also need to understand how this noise is transmitted to the surrounding environment.
There are mainly two pathways by which the vibrations of the source can be transmitted
to the patient bore [4]. In standard MRI system, there are holes in the end-caps covering
the magnet-to-patient bore tube, so the acoustical vibration can be transmitted through
these holes to the patient. Thus the vibration of the noise source is heard by the patient or
system operator through the air. The gradient coil former and its shielding are mounted 38
on the cryostat inner bore with stiff aluminum brackets at each end. And cryostat inner
bore is connected to the patient bore via the end-caps. The coil formers and the metallic
cryostat inner bore and RF body coil can also transmit the vibrations from the source to
the patient bore via a solid mechanical pathway.
2.2 Reduction of Acoustic Noise in MRI
We understand the sources and pathways of acoustic noise in MRI system, and that the noise is a challenge for MRI. Many methods have been developed to reduce it.
As MRI system consists of two subsystems, hardware and software, there are mainly two
solutions involving their modifications during the imaging procedure. From the source-
pathways point of view, there are possible ways such as reducing the noise at its sources,
reducing the noise transmission from the source to human ear, and reducing the noise at
human ear using earplugs, for example. We will discuss all of these techniques in detail.
2.2.1 Reduction of Acoustic Noise via its Source
In thinking about the noise source-pathways pattern in an MRI system, it is
natural for us to consider reducing the acoustic noise at its source. There are several ways
to modify the hardware in order to get lower noise at the source. The vibrations of
gradient coil former are induced by the Lorentz forces. One idea is to balance the Lorentz
39
forces using return gradient current paths, without affecting the gradient performance.
Mansfield and his research group have invented a novel gradient set to achieve this
balance [6]. The Lorentz force applied to a conductor with differential length dl carrying
current I in a magnetic field B can be expressed as:
| F | = | Idl ´B | = IdlB sin q (2.2)
where q is the angle between the magnetic field B and the local direction of the current I.
For a conducting loop with currents flowing, the total Lorentz force can be calculated by integrating the above formula over the loop. Since the net force applied to the loop is zero
– and this is the key to the method - and all parts of that loop are mechanically attached to insulating material, the vibration induced by Lorentz force will be minimized significantly. See Figure 2.1(A) for the illustration of the Lorentz force of a simple loop.
A set of x-gradient arc-loops is considered; a y-gradient would correspond to a rotation by
90 degrees around the z-direction. In Figure 2.1(B), the separation 2b is optimized by numerically calculating the gradient field strength and linearity in the area of interest, given the radii a and c of the arcs. Although the overall force applied on the paired arcs is balanced, local Lorentz forces still compress or expand the connecting slab between the arc wires. This local vibration is strongly dependent on the material used to mount the wires. The authors discovered that supporting the set of arcs with stiffer material, such as ceramics and glass, can attenuate more noise than with the more pliable solid polystyrene. In all, experiment has shown that, with this set of gradient design, the
40
acoustic noise is actively attenuated by 13 dB at 0.5 T, compared to that of a standard
gradient system. The attenuation is especially more efficient at lower frequencies.
Figure 2.1 (A) The sketch shows the Lorentz forces applied on a loop segments carrying current at an external magnetic field B along the z-direction, which is perpendicular to the plane of the loop. (B) The sketch shows an arc-loop setup for x-gradient with balanced Lorentz force over the whole loop. Given the radii of a and c of the arcs, the separation 2b is adjusted to obtain the desired gradient field strength and linearity.
The above method of designing new sets of arcs with balanced Lorentz force works well,
but the gradient coil former is necessarily thickened reducing the effective patient bore.
In addition, it is difficult to modify extant MRI systems to incorporate the new arc
designs.
We will return later in this chapter to the use of a vacuum enclosure for the
suppression of noise [4, 7]. For the moment, we will assume an MRI system with vacuum
enclosed gradient coil. For this system, the major source of acoustic noise is the eddy-
current-induced vibrations of conducting cryostat inner bore and RF body coil. These two 41
sources are about 10-15 dB louder than any other noise source, with almost equal contributions from each source. For this kind of system, the reduction of noise at the source (the gradient coil former) is significantly attenuated via the vacuum enclosure, in which the sound has no medium for being transmitted to the outside environment. The
Lorentz force balancing method discussed above is unnecessary for a vacuum system, and the eddy currents can be eliminated in the surround material by utilizing a non- conducting cryostat bore where there is no Lorentz force if there is no current flowing through it. Experiments have been carried out on a standard LX-generation, 1.5 T GE product magnet with an electrically conducting, stainless steel inner cryostat bore. For comparison, a non-conducting, 3.8mm thick fiberglass inner bore is used instead of the
standard conducting inner bore. Additionally, a low-eddy-current RF coil has been
designed to further reduce the acoustic noise. The standard GE RF body coil incorporates
5 cm wide copper strips, which are very close to the inside of gradient assembly. Large
eddy-currents are induced in the wide copper strips, leading to large Lorentz forces and
louder noise. In the new design, the RF body birdcage coil is built with a 6.4 mm OD Cu
circular wire, which significantly reduces the area associating with the pulsed gradient
field. Thus eddy-currents induced in this new coil, and the associated acoustic noise, are
suppressed.
Many MRI systems have been equipped with conducting cryostat warm bores,
since they are cheaper and easier to make compared to non-conducting warm bores. With
42
this much metallic surrounding material, another mechanism for the suppression of external eddy currents is the introduction of a passive metal shield, in addition to the standard active shielding. This passive shield can be mounted on the outside of the vibration-isolated, vacuum enclosed shielded gradient set [4]. The method leads to a dramatic decrease in MRI acoustic noise. The main assumption is that the mechanical vibration power is proportional to the ohmic power dissipation, which is proportional to the square of currents flowing through the stainless steel bore. For different configurations of copper shielding with a z-gradient coil, numerical calculations have been carried out in estimation of the eddy-currents in, and power dissipation by, the steel bore. Variable thickness of copper passive shielding is investigated in order to get the optimized decrease of dissipation energy. Also two configurations are considered. One is where a copper layer is applied only just outside the gradient assembly and the other is where end caps are attached to, and extend radially inward from, this outer copper layer.
The reduction of power dissipation in the cryostat warm bore is substantial for both configurations. The outer shielding layer with end cap has better performance than without it. In general, thicker copper shielding layers lead to greater reduction of power dissipation than thinner layers. With a copper-shielding layer applied, the eddy-current induced vibration acoustic noise in cryostat warm bore is reduced. Note that eddy- currents generated in this additional copper shielding are not themselves a significant problem, if a) the shielding is enclosed in a vacuum enclosure and/or b) it is firmly mounted on the gradient assembly to suppress vibrations.
43
2.2.2 Reduction of Acoustic Noise via Transmission
An MRI system with a gradient coil assembly contained inside a vacuum chamber
still requires that the gradient assembly be mounted directly and firmly to the floor. The
patient bore and the cryostat warm bore tube can be used as the inner and outer walls of
the vacuum chamber, with a vacuum pump working to maintain low air pressure inside
the chamber, which can significantly reduce the air propagation of acoustic noise. The air
propagation of acoustic noise is the main transmission from the acoustic noise source to
the patient ear. However, the acoustic noise reduction achieved by the vacuum method is
limited to only 6 dB if the gradient assembly is directly mounted on the magnet [7].
Although the air propagation is the main factor, the solid transmission of the acoustic
noise becomes dominant when the air transmission is heavily restricted. Thus we need to
reduce the acoustic noise propagation through a solid pathway when vacuum method is
used. The A-weighted sound level of the acoustic noise has been reduced up to 24 dB
when the gradient assembly is directly mounted onto the floor, for which the sound
propagation through solid material to the human ear is significantly decreased.
Besides the application of a vacuum-enclosed gradient assembly, there are also some novel material solutions that can be used for mechanical damping. Constrained- layer damping (CLD) consists of a thin sheet of flexible, solid material attached by a
lossy elastomer, which is installed on the outside of the magnet cryostat and the inside of
the patient tube. The elastomer can absorb energy from the vibrations of the acoustic
44
resonances. The application of CLD enables mechanical damping and therefore reduces
the amplitudes of resonance peaks in the acoustic noise spectrum. The method is reported to further reduce acoustic noise by 1-2 dBA.
Sound barrier materials are also applied to the outside of magnet cryostat bore, which produces approximately 1-2 dB noise reduction in the patient bore. Furthermore, a viscoelastic damping material can be attached to the outer surface of gradient coil using a topology optimization technique [8]. To simulate, an FE model was built with two layers of solid elements: the gradient coil assembly and the viscoelastic damping material layer.
It was observed that the optimized placement of the viscoelastic damping material suppresses the intrinsic response frequencies of the gradient coil assembly. The maximization of damping effects is to be further studied by these authors.
Another sound barrier material, a micro-perforated panel absorber was used to reduce MRI scanner acoustic noise [9]. In their simulations, it was observed that changing the diameter of holes, the thickness of the plate, and the air gap has significant impact on the reduction of the gradient acoustic noise. These simulation results need to be verified by experiments.
45
2.2.3 Reduction of Acoustic Noise at Human Ear
Besides the source and transmission blocking methods, an obvious method is to
use mufflers and earplugs as worn by the patients to alleviate noise. The mufflers and
earplugs act as a low-pass filter, which is especially efficient for attenuating the high- frequency acoustic noise component. This is the simplest and most economical approach for reducing acoustic noise at the human ear. However, the sound can still be transmitted through the bones, which generates some noise. The material used for the hearing protection devices, and also the placement of the devices, determine the overall effect of reducing the acoustic noise. Thus custom-made hearing protection devices have better performance than the standard devices. This is also useful for the MR workers who must work daily in the scanner room.
Besides such passive methods applied directly to the human ear, there is also an active noise cancelling method called anti-noise [10]. The general idea of this method is to add same length of controlled sound 180 degree out of phase to cancel out the original noise, as applied to the dominant frequencies. A feed-forward system is refined to address MRI scanner noise to verify this noise cancellation algorithm [11, 12]. And dramatic noise reduction was measured for a 4T system with an EPI sequence for the main resonance peaks. One disadvantage of this method is the potential for doubling the noise at human ear for any components that arrive in phase.
46
2.2.4 Reduction of Acoustic Noise Using New Sequences
Above we have discussed various methods for the reduction of acoustic noise in
MRI system, both through the attenuation of the source, the blocking of the transmission,
and the use of hearing protection devices for the human ear. In addition, the reduction of
the acoustic noise can also be achieved by optimizing the MRI sequence. Based on linear
response theory, the gradient sequence pulse parameters and the generated gradient
acoustic noise are correlated [13, 14]. The acoustic response of an MRI gradient system to current pulses is assumed to be linear, which can be measured as a “frequency response function” (FRF) by the method described by Hedeen and Edelstein [1]. Thus, the acoustic noise generated by a particular gradient pulse sequence can be predicted by calculating the product of the Fourier transform of the gradient waveform and the FRF.
The authors’ strategy is to design a gradient pulse, with its FT contains no frequencies for which the amplitude of FRF is high.
Based on the linear response theory discussed above, “soft” pulses with sinusoidal ramps are used, which produces lower noise [15, 16, 17]. This silent sequence was run with spin-echo, gradient-echo and rapid acquisition with relaxation enhancement
(RARE). The acoustic FRF of the whole-body gradient system measured by Hedeen is relatively low, in the frequency below 200 Hz. So soft pulses, for which the FT peaks at low frequencies, but suppressed at high frequency, can lead to overall low noise since the high frequencies are not enhanced by the pulse sequence. In fact, these soft pulses act as
47
low-pass filters, which significantly suppress the high frequency noise components. A
simple method of designing this kind of limited-band, sinusoidal ramp soft pulse is
described. With this design, noise levels can be controlled at about 40 dB for relative
long TE for both spin echo and gradient echo sequences at the field strength of 3 T [15].
This soft pulse duration can reach up to 4 milliseconds. The application of soft gradient
pulse cannot be used for fast imaging sequences such as fast low-angle shot (FLASH)
and echo-planar imaging (EPI). First, the pulse duration is too long, which could be less
than the inter-echo spacing time. Second, the fundamental frequency 1/TR is above 200-
500Hz range, which could lead to an enhancement of acoustic noise.
Although such individual gradient pulses with sinusoidal ramp cannot be applied
to reduce the acoustic noise associated with fast EPI sequences, a novel optimized
gradient pulse was developed for use with EPI involving active acoustic noise control
[16, 18, 19]. The basic idea of active acoustic control in gradient coils is to add
appropriate additional windings in the coil. This setting acts to significantly suppress
certain vibrational modes in solid structure; thus, less acoustic noise is generated without
affecting the efficiency of the gradient coils. This is because the wire spacing of the
additional controlled windings is small enough that the magnetic field generated by the
primary windings is unaffected to any important extent. Experimental results show that
the overall output acoustic noise is substantial reduced if the control windings are properly driven. The principal frequency chosen to be suppressed is naturally the
48
operating frequency of the EPI sequence, in order to maximize the overall reduction of the gradient acoustic noise. Combining the application of the active noise control and a simple sinusoidally modulated gradient pulse, the authors report a reduction of acoustic noise by 37 dB in a 3T system using an EPI sequence with an operating frequency of
3289 Hz. Furthermore, a total of 50 dB acoustic noise reduction can be achieved when the tailored optimized pulse is combined with the active acoustic control method. With this extra 13 dB reduction, there is a trade-off where the ramp of the gradient envelope is lengthened, and the acquisition time becomes longer. However, partial k-space techniques can be applied to reduce this time. This technique reduces acoustic noise far more than conventional methods, although it is restricted to the operating frequency of the active acoustic control.
Without active noise control, there is still another possible way to reduce the acoustic noise from fast sequences like EPI. Experiments have been carried out on a 4 T system with intrinsic mechanical resonance frequencies at 720 Hz and 1220 Hz. A small adjustment of the read-out gradient for an EPI sequence is made to shift the fundamental frequency to avoid the mechanical resonance. A 12 dB reduction was measured by changing the fundamental read-out frequency from 720 Hz to 920 Hz [20]. Schmitter et al. developed a quiet EPI sequence given the measurement of FRF for the whole-body system [21, 22]. A sinusoidal frequency sweep method was applied to measure the FRF from 0 Hz to 5000 Hz. It was found that the acoustic response of all 3 gradient coils is
49
significantly reduced below 1500 Hz. Thus the readout of a sinusoidal gradient EPI sequence should be within this range. With timings comparable to that of standard EPI imaging, the acoustic noise is reduced by more than 20 dB. Furthermore, the main frequency components are located at lower frequencies where the human ear is less sensitive.
A new approach embracing a mechanically rotating DC gradient can also reduce
MRI acoustic noise [23]. While the x- and y- gradients are the main source of acoustic noise, the z-gradient coil is kept static when the other two gradient coils are rotated along z-axis. The rotating of read-out and phase encoding gradient coils requires a new image reconstruction algorithm, but this is quite doable. Experiments were carried out with a 2-
Tesla whole-body MRI system, and indicated that the gradient acoustic noise was effectively suppressed using this method. One limitation, however, is that the slice selection direction is always in the z-direction since only the z-gradient is kept mechanically static.
There are various methods we have discussed above to suppress the acoustic noise in MRI system. However, there is not a single method which can be adapted easily with existing system. Some of them need modification of hardware, while some of them need software change. We try to understand the acoustic noise with a simple model and
50
approach to reduce it without changing the pulse sequence a lot. Our optimization of the gradient pulse sequence is complimented existing hardware and software.
51
Works Cited:
1. R. A. Hedeen and W. A. Edelstein, Characterization and Prediction of
Gradient Acoustic Noise in MR Imagers, Magnetic Resonance in Medicine 37:
7-10, 1997.
2. M. E. Ravicz, J. R. Melcher, and N. Y. Kiang, Acoustic Noise During
Functional Magnetic Resonance Imaging, Journal of the Acoustical Society of
America 108(4):1683-1694, 2000.
3. W. A. Edelstein, T. K. Kidane, V. Taracila, T. N. Baig, T. P. Eagan, Y. N.
Cheng, R. W. Brown, and J. A. Mallick, Active-passive gradient shielding for
MRI acoustic noise reduction, Magnetic Resonance in Medicine 53:1013-1017,
2005.
4. W. A. Edelstein, R. A. Hedeen, R. P. Mallozzi, S. A. El-Hamamsy, R. A.
Ackermann, and T. J. Havens, Making MRI Quieter, Magnetic Resonance
Imaging 20:155-163, 2002.
5. D. Tomasi and T. Ernst, A Simple Theory for Vibration of MRI Gradient Coils,
Brazilian Journal of Physics 36(1A):34-39, 2006.
6. P. Mansfield, B. L. Chapman, R. Bowtell, P. Glover, R. Coxon, P. R. Harvey.
Active acoustic screening: reduction of noise in gradient coils by Lorentz force
balancing. Magnetic Resonance in Medicine 33:276-281, 1995.
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7. A. Katsunuma, H. Takamori, Y. Sakakura, Y. Hamamura, Y. Ogo, R.
Katayama, Quiet MRI with novel acoustic noise reduction, Magnetic Resonance
Materials in Physics, Biology and Medicine 13:139-144, 2002.
8. S. Y. Kim, I. Y. Kim, C. K. Mechefske, and D. H. Lee, Control of Gradient
Coil in Natrual Frequency using a Topology Optimization Technique,
Proceedings International Society for Magnetic Resonance in Medicine 16,
p.1159, 2008.
9. M. Li, C. K. Mechefske, Reduction of MRI Scanner Acoustic Noise using a
Micro-perforated Panel Absorber, Proceedings International Society for
Magnetic Resonance in Medicine 17, p. 1157, 2008.
10. A. M. Goldman, W. E. Gossman, P. C. Friedlander, Reduction of Sound Levels
with Antinoise in MRI Imaging, Radiology 173:549-550, 1989.
11. J-H. Lee, B. W. Rudd, M. Li, J. Osterhage, and T. C. Lim, Experimental Study
of Active Acoustic Noise Control with MRI Compatible Headphones and
Microphones in a 4T MRI Scanner, Proceedings International Society for
Magnetic Resonance in Medicine 16, p.2964, 2008.
12. B. Rudd, M. Lee, T.C. Lim, and J-H. Lee, Experimental Study of Active
Acoustic Noise Control in a 4T MRI Scanner In-Situ, Proceedings International
Society for Magnetic Resonance in Medicine 17, p.4781, 2009.
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13. M. Segbers, C. V. Rizzo S., H. Duifhuis, and H. Hoogduin, Optimized MRI
gradient waveforms for Acoustic Noise Reduction, Proceedings International
Society for Magnetic Resonance in Medicine 16, p.1349, 2008.
14. C.V. Rizzo S., M. J. Versluis, J. M. Hoogduin, H. Duifhuis, Acoustic fMRI
Noise: Linear Time-Invariant System Model, IEEE transactions on Biomedical
Engineering 55:2115-2123, 2008.
15. F. Hennel, F. Girard, and T. Loenneker, “Silent” MRI with soft gradient pulses,
Magnetic Resonance in Medicine 42, 6-10, 1999.
16. B. L. W. Chapman, B. Haywood, P. Mansfield, Optimized Gradient Pulse for
Use With EPI Employing Active Acoustic Control, Magnetic Resonance in
Medicine 50:931-935, 2003.
17. F. Hennel, Fast Spin Echo and Fast Gradient Echo MRI with Low Acoustic
Noise, Journal of Magnetic Resonance Imaging 13:960-966, 2001.
18. P. Mansfield, B. Haywood, R. Coxon, Active Acoustic Control in Gradient
Coils for MRI, Magnetic Resonance in Medicine 46:807-818, 2001.
19. P. Mansfield, B. Haywood, Principles of active acoustic control in gradient
coils design, Magnetic Resonance Materials in Physics, Biology and Medicine
10:147-151, 2000.
20. D. G. Tomasi, T. Ernst, Echo Plannar Imaging at 4 Tesla With Minimum
Acoustic Noise, Journal of Magnetic Resonance Imaging 18:128-130, 2003.
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21. S. Schmitter, E. Diesch, M. Amann, A. Kroll, M. Moayer, L. R. Schad, Silent
echo-planar imaging for auditory FMRI, Magnetic Resonance Materials in
Physics Biology and Medicine 21:317-325, 2008.
22. S. Schmitter, E. Diesch, M. Amann, A. Kroll, M. Moayer, L. R. Schad, Silent
echoplanar imaging for auditory fMRI, Proceedings International Society for
Magnetic Resonance in Medicine 17, p., 2009.
23. Z. H. Cho, S. T. Chung, J. Y. Chung, S. H. Park, J. S. Kim, C. H. Moon, I. K.
Hong, A New Silent Magnetic Resonance Imaging Using a Rotating DC
Gradient, Magnetic Resonance in Medicine 39:317-321, 1998.
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Chapter 3
String Model of the Acoustic Noise Vibration
3.1 Background
In the previous chapter, we have discussed many kinds of passive and active noise cancelling methods. However, there is as yet no practical comprehensive solution to the
MRI acoustic noise problem that addresses all noise sources and is compatible with quality imaging. Although some quiet imaging scanners have been developed and demonstrated, these are not only difficult to build and also require significant adaptations of existing MRI scanners. The purpose of this thesis is to discuss a new method for active noise suppression of selected acoustic noise frequencies. We consider modifications of the gradient pulse sequence itself to provide a noise cancellation mechanism. The idea is to use the same source that produced the vibrations to also cancel them. We do not add external or additional forces or sources of the original noise. The point of our method is to modify the physical source of the noise, where the source is the abrupt changes in the
Lorentz force due to the interaction of the pulsed gradient currents with the main MRI magnetic field. Thus this is a change in the input function in the acoustic noise generation and it does not correspond to an additional filter. Instead of a filter, we modify the
56
gradient pulse input and thus the relevant comparisons are the noise for different pulse
sequence timings, and not of prior/post filter noise signals.
We originally conceived of the gradient method and its self-cancellation
mechanism by considering the “plucking” of springs and strings, followed up by identical
plucks carefully timed to cancel the original oscillation or standing wave. In the example of a spring model the vibrations resulting from an impulsive force could be cancelled upon the application of a counter-impulsive force. This cancellation is most effective at
multiples of half of the spring period, even in the presence of damping, provided the
strength of the follow-up impulse is appropriated adjusted. The spring vibrations from a
series of impulses can be cancelled by follow-up impulses and associated timings. A
more careful mathematical description of the impulsive forces is described below.
A more appropriate model for the normal modes of gradient standing waves is
one where both the spatial and temporal oscillations are considered, like a vibrating string. In order to consider more closely the standing wave spatial aspects of the vibrating
MRI conductors (coils and surrounding metallic material), we investigate a forced and
damped vibrating string model. In fact, a string approximation has been put forth as a
useful model for the dominant frequencies of both longitudinal and transverse gradient
coils [1]. Currents through the MRI gradient coils are on the cylindrical surface, with
both azimuthal and longitudinal (z-axis) components. With a main magnetic field along
57
z-axis, the resultant Lorentz force on the gradient coil former is in the radial direction.
The gradient coil former (cylinder) is suppressed or expanded radially depending on the
current distribution. For a longitudinal coil, the current is anti-symmetrically distributed
along the z-axis, so the Lorentz force compresses half of the cylinder while it expands the other half. On the other hand, the current of a transverse coil is symmetrically distributed along z-axis, and the Lorentz force actually bends the cylinder with its maximum displacement in the middle. Both ends of the cylinder are fixed either to the magnet or directly to the floor with very stiff material, so the displacements at both ends are zero.
Standing waves are thus induced by the Lorentz forces, with nodes at both ends for all of the vibration frequencies. A string is a reasonable physical picture to describe the vibration of gradient coil former with a single vibration fundamental frequency. So a
combination of strings with separate intrinsic frequencies is a simple but effective model
for the vibrations in the complicated gradient coil former. The gradient vibrational modes
yield a rich spectrum of frequencies where the frequency response ranges over several
kHz. The present discussion provides a framework to understand how to suppress any given frequency peak in that spectrum, and to explain the results of previous experiments on peak cancellations. While we use string analytical solutions for a given frequency as a very useful guide in timing pulse sequences to suppress specific vibrational modes, many conclusions are more general and apply to any linear time-invariant system. The general solution of the string model is a robust embodiment of a linear response system such as the MRI gradient coil and we consider this solution next.
58
3.2 String Model: Equations and Solutions
The vibrating gradient cylinder assembly together with its boundary conditions has a complicated frequency spectrum. Vibrations arising from the magnetic force on the eddy currents induced in the surrounding conducting material in an MRI scanner also have a rich frequency spectrum. Fortunately, a simple damped and driven string model serves as quite a robust guide to the evolution of, and cancellation mechanisms for, a given vibration frequency in the gradient system. The system equation of motion for its transverse displacement y at a given position x and time t is approximately expressed by a driven force damped string equation [1, 2]:
¶¶¶22yxt(,) 2k yxt (,) 1 yxt (,) -- =-fxt(,) (3.1) ¶¶xt22uu¶t 22 where fxt(,) is the external force per unit length applied over the whole string as a function of time. Here, ul= T / is the string wave velocity for tension T and linear mass density l . The damping constant is k . The vibration spectrum for the string is its intrinsic harmonic spectrum, which depends on the boundary conditions (usually imposed on the ends of the string). To connect the string model with the gradient system, Tomasi and Ernst replace the ratio T / l by the ratio E / s , where s is the surface mass density of the coil former (supporting the gradient wire pattern) and E is the Young’s modulus per unit area of that former. The resistive force per unit area, opposing the former’s 59
motion, is interpreted in terms of a viscosity h , which models the energy lost by the
“string”, such that 2=khs.
For the moment, consider a string fixed at its ends, but without any symmetry
imposed on the external force f. The spatial boundary conditions appropriate for
longitudinal (axially anti-symmetric) and transverse (axially symmetric) gradient coils
will not change the conclusions. If we know the complete history of the forces driving the
string, the general solution for a string tied down at the end is given by
lt yxt(,)=- dx dtt fx ( , ) Gxx (, , t t ) (3.2) ò0 000ò-¥
where the Green’s function is
2v2 ¥ sin(ppnx / l )sin( nx / l ) Gxx(, ,) s=Q0 e-ks sin(2pn s ) () s 0 å n (3.3) T n=1 2pnn
In this equation nn is the n-th harmonic eigenfrequency
nu nkpu=-1(/ln )2 (3.4) n 2l
Notice that the Heaviside function Q()s , which is defined as the step function (
Q<(0)0,ss = Q>(0)1 =) imposes a causality condition leading to the upper limit of
t in the t integration of (3.2). The Green’s function vanishes at the fixed ends x = 0 and
xl= to satisfy the imposed zero boundary conditions. (3.2) is the analytical formula to
calculate the string displacement at position x and time t. 60
3.3 Solution for Gradients
We use the string solutions to understand how to control a given frequency of
vibration using the standard pulses that drive it in MRI operation. In this subsection, we
present analytical results for different pulses, assuming the forces applied on the string
are considered constant over the length of the string. While the force on an MRI gradient
will have a spatial dependence owing to the current patterns, the cancellations discussed
later will depend only on the time dependence and we can for convenience consider f constant in space. It is the time dependence of the gradient pulses of which we take advantage. We begin with boxcars, the convolution of which yields trapezoid pulses. The trapezoidal pulse profile is quite standard and important for MR imaging, since a pulse with a flat-top is the work-horse for readout and other gradient pulse sequences in most clinical applications. Still, we have also considered triangular pulses and smoothed out pulses (through convolution of boxcars). The “quadratic pulse” described below is one such smoothed out pulse.
3.3.1 Boxcars
Consider exciting a string with a simplified “boxcar” pulse uniform over the
string’s length and with time dependence illustrated in Figure 3.1. This will allow us to
show most simply the role of the impulsive forces associated with the turning on and off 61
of pulses. With the applied force f(t) independent of x0 (taking on a constant value f0 along the entire string during the time interval 0 to t1, and zero for other times), we can
calculate the integration from (3.2). In detail,
2 lt 2v 1 yxt(,)= f dx dt 00òò00 T (3.5) ¥ sin(ppnx / l )sin( nx / l ) 0 ett--kt()t sin(2pn (-Q- t )) ( t ) å n n=1 2pnn
Figure 3.1 A boxcar gradient pulse with duration t1 .
After switching the order of the summation of different frequency modes and the integral
of the time and the position, and using angular frequencywpnnn= 2 we have:
2sin(/)vnxl2 ¥ p yxt(,)= f 0 å T n=1 wn (3.6) lt dx sin(ptwtt nx / l )1 d e--kt()t sin( ( t-Q- )) ( t ) òò0000 n
After taking the integral of the position along string from 0 to l, we have:
2 ¥ t 2sin(/)2vnxllp 1 yxt(,)=-Q- f detwtt--kt()t sin( ( t ))( t )(3.7) 0 å ò0 n TnnODD=1( ) wpn
62
Finally we need to calculate the integral involved with the time, which is calculated in
two different ways: tt<>11 and tt.
For tt< 1 (during the boxcar), we have
t 1 detwtt--kt()t sin( ( t-Q- )) ( t ) ò0 n t = dse-ks sin(w s ) ò0 n 11()itwk--+ 1 () it wk 1 1 (3.8) =+--()(eenn ) 2iiwknn-+ i wk i wkwk nn -+ i 1 =--+[(ett-kt wwkww cos() sin())] 22 nn n n wkn +
For tt> 1 (after the boxcar applied), we have
t 1 detwtt--kt()t sin( ( t-Q- )) ( t ) ò0 n t = dse-ks sin(w s ) ò n tt- 1 t 1 ()iswk--+ () is wk =-ds() enn e òtt- 2i 1 11()itwk--+ 1 () it wk =+()( eenn (3.9) 2iiwknn-+ i wk 11()()ittwk-- - ()() itt wk +- --eenn11 ) iiwknn-+ wk 1 =--{(et-kt wwk cos() sin(w t )) 22 nn n wkn + --k()tt1 +etttt [wwnn cos( (-+11 )) kw sin( n ( - ))]}
Combining (3.8) and (3.9), we can rewrite the string displacement as an overall solution
63
2sin(/)21vnxll2 ¥ p yxt(,)=´ f 0 Tnå wp22 nODD=1( _) n wkn + -kt {etttt (--wwkwwnn cos( ) sin( nn ))+ Q- (1 ) (3.10)
--k()tt1 +-+-Q-etttttt [wwnn cos( (111 )) kw sin( n ( ))] ( )}
The parameters of the boxcar can now be adjusted to cancel out the very oscillations that were initiated by it. The cancellation is between impulsive forces due to the beginning and end of the pulse. To see this most easily, we consider the simplification where damping is neglected. We have obtained similar cancellations where damping is included, which will be presented in the next chapter. With damping neglected (k = 0), the solution (3.10) becomes
221vl2 ¥ yxtf(0,,)kp== sin(/) nxl ´ 0 Tnå p 3 nODD=1( ) wn
[-+Q-+-Q-wwwnnn cos()()ttt111 ww nn cos(())()] tttt(3.11)
Solution (3.11) can be simplified to
221vl2 ¥ yxtf(0,,)kp== sin(/) nxl ´ 0 Tnå p 2 (3.12) nODD=1( ) wn 2 [2sin (wwwnnnttt / 2)Q-+ (11 ) (cos( ( tt - )) - cos( ttt )) Q- ( 1 )]
where now wpn = nv/ l. At the end of the applied boxcar gradient pulse ()tt= 1 , (3.12)
becomes
221vl2 ¥ yxtf(0,,)kpw== sin(/) nxl ´ 2sin(/2)2 t (3.13) 0 Tnå p 2 n nODD=1( ) wn
64
If we choose the boxcar pulse time duration to be tlv1 = 2/ , then wpntn/2= , leading
2 to sin (wnt / 2)= 0 so that all the terms in the series (3.13) vanish. The vibration caused by the first “semi-impulse” (the rise or leading edge of the boxcar, at t = 0 ) is cancelled
out by immediately following it with a semi-impulse in the opposite direction (the trailing
edge of the boxcar) at one fundamental period 2/pw1 later. In terms of Figure 3.1, the
oscillations would have disappeared for t > t = 2p / w . Figure 3.2a exhibits this 1 1
cancellation for the example tms1 = 2 (i.e., a frequency of 500 Hz). In general, we see
that this will happen for any t1 such that wpntm= 2 , and any integer m; i.e., for any
multiple of the fundamental period 2/pw1 later. Analogously, there is an enhancement
pattern we can observe if we shift the boxcar time duration by pw/ 1 . Figure 3.2b shows that, instead of a cancellation, we can enhance the vibration with a decrease in boxcar width by this amount (a reduction to t = 1 ms enhances the 500 Hz vibration). In 1
general, a boxcar pulse with duration t1 can eliminate the vibration at frequencies 1/t1
along with its harmonics mt/ 1 .
65
Figure 3.2 (a) The top plot shows a 2-ms duration boxcar gradient killing the string vibration with 500 Hz for t > t = 2 ms . (b) The bottom plot shows a 1 ms top boxcar gradient maximally 1 enhancing the 500 Hz string vibration for t > t = 1 ms . In both plots, the thick curve is the 1 vibration induced by the positive Q impulse, the thin curve is the vibration induced by the negative Q impulse, while the dashed curve is the superposition of the two. There are two important physical effects evident in these results. First, by “semi- impulse” we are describing a force that was suddenly applied and sustained, as opposed to a delta function force, which is applied only instantaneously. That is, this is a force is described by a theta or Heaviside function. Instead of “semi-impulse” or some such
66
language, we could call this a Q impulse to describe the Q function force, just as a d
impulse is used as standard nomenclature to describe a d function force. The second
physical effect to which we would like to draw attention is that, after a Q pulse, a spring
(or a point on the string or on the gradient assembly cylinder) is vibrating around a new
equilibrium. Following a second Q impulse that is equal in magnitude and opposite in sign, the system is vibrating now around the original quiescent equilibrium (zero gradient current). The detailed superposition of the two standing waves produced by the two Q impulses can be understood quantitatively by paying attention to these two different equilibrium positions. It is worth saying this again: the net result of the interference is a superposition of the two sinusoids, each of the same frequency but differing in equilibrium position and phase. The net result is itself a sinusoidal function of the time
over the interval TR - see later portions of this chapter.
3.3.2 Trapezoidal Gradient Solution
A gradient pulse for practical use has a trapezoidal shape rather than that of the ideal boxcar. This is due to the gradient limitation that the current flowing through the gradient coil is not able to increase from zero to some certain amount instantaneously.
While this is a bit more complex, it offers us the advantage of another time scale, the ramp-up time (and usually by symmetry this is the same as the ramp-down time). The gradient pulse amplitude and its slew rate determine the ramp time. The additional scale offers the flexibility of cancelling two different frequencies, as we shall see. We can 67
continue to ignore the string damping in explaining the mechanism here. Damping is easy
to incorporate and does not change the conclusions; only small changes in the trapezoidal
shape and timings are needed when modest damping effects are included.
Figure 3.3 A trapezoidal gradient pulse with ramp time (up and down) tr and flat-top time ttop .
In the trapezoid of Figure 3.3, tr is the ramp (up or down) time and ttop is the
plateau or flat-top time. Defining ttrtop+= t1 , we will show below that a trapezoidal
gradient pulse will kill the same frequency wt1,1= 2/p 1 , and its harmonics
wp1,n = 2/nt 1 , that a boxcar of the same duration has suppressed. In addition, we will
see that the vibrations with higher frequency wp2,1 = 2/tr , and its harmonics w2,n , are
likewise killed. In Equation (3.2), the time-dependent force function f is no longer a boxcar and now follows the trapezoidal time dependence under consideration. Since we are concerned about the vibrations of the string after the whole trapezoidal is applied, the
t integration to be performed covers the pulse duration from t = 0 to t =+ttr 1 , where we again assume zero damping (0)k = . As before, the damping effect can be
68
easily modeled and changes none of the conclusions. Starting with (3.7) and a trapezoidal gradient pulse, we obtain
2 ¥ tt+ 2sin(/)2vnxllp rtop fxt(,)=- dtt f ()sin( w ( t t )) å n Tnwpò0 nODD=1( ) n (3.14) 2sin(/)2vnxll2 ¥ p =fBnt ( , ) 0 å TnnODD=1( ) wpn where
tt r t 1 Bnt(,)=-+- sin((wtt t )) d sin(( wtt t )) d òò0 nnt t r r (3.15) tt1 + r tt+-t +-1 r sin(wtt (td )) òt n 1 tr
If wpnt1 = 2 , all amplitudes with frequencies wn are suppressed. The sum of the first and third terms (we are essentially adding two identical triangular areas) can be rewritten as:
ttt+ rrt 1 tt+-t sin(wtt (td-- )) +1 r sin( wtt ( td )) òò0 nnt ttrr1 t r t tt+-()t + t =(sin((wttttd-+ ))11r sin(( w -- tt ))) ò0 nn1 ttrr t1 +-sin(wtt (td )) (3.16) ò n tr tt =-+-r sin(wtt (td ))1 sin( wtt ( td )) òònn 0 tr t =-1 sin(wtt (td )) ò0 n which is exactly the same as that for a boxcar gradient pulse integration. We have verified that a trapezoidal gradient has the same effect as a boxcar as long as the sum of
ramp time and flat-top time is equal to t1 .
69
To address the effect of the additional time scale tr , we return to general
evaluations of the integral in (3.14). For tt>+r t1 , (3.15) can be integrated to give
tt r t 1 Bnt(,)=-+- sin((wtt t )) d sin(( wtt t )) d òò0 nnt tr r
tt1 + r tt+-t +-1 r sin(wtt (td )) òt n 1 tr ttt+ 11rrcos(wt (tt-- ))1 cos( wt ( )) =-nnddtt + (3.17) òò0 t ttrnww r1 n 1 =---- [sin(www (tt )) sin( t ) sin( ( ttt- )) 2 nr n n1 r trnw
+- sin(wn (tt1 ))]
For a trapezoidal gradient pulse, the result of the integration for the string motion after
the pulse is applied ()tt>+1 tr is given by:
2sin(/)2vnxll2 ¥ p yxt(,)= f Bnt (,) 0 å TnnODD=1( ) wpn 2sin(/)2vnxll2 ¥ p =fttt [sin(ww (-- )) sin( ) (3.18) 0 Ttå 3 np nr n r nODD=1( ) wn
---+- sin(wwnr (tt11 t )) sin( n ( tt ))]
When wp1,nrtm= 2 or wp2,ntm 1 = 2 , the above integral is zero and the string ceases to
vibrate. Thus a trapezoidal gradient pulse always kills two vibrational frequencies and all
2pn their harmonics related to the ramp time and flat-top time according to w1,n = and tr
2pm w2,m = , for integers n and m. t1 70
To understand the two cancellation mechanisms, recall that a boxcar effects a
cancellation of the vibration caused by the first Q impulse by the follow-up negative Q
sin(x ) impulse. In frequency domain, this manifests itself as zeros in a sinc function: . A x trapezoidal pulse is the convolution of two boxcars, which helps us to understand how the
two frequencies are killed. One of the boxcars in the convolution has a duration of tr and
the other has t1 ; the picture described earlier in terms of the cancelling theta impulses
within each boxcar continues to hold; the net result is that both boxcars lead to
subsequently suppressing the string vibration at the two prescribed frequencies, respectively. In the frequency domain, we see this through the product of two sinc functions.
3.3.3 Follow-up Pulses for Additional Frequency Cancellation
We now describe how to cancel an additional frequency and its harmonics through the application of a follow-up pulse. The physical picture of Q and anti -Q impulses continues to be helpful. The possibility of a cancellation between the rise and fall of a boxcar immediately generalizes to the possibility of a Q -cancellation between
two different boxcars. Based on this idea, a general approach is immediately apparent.
We can use pairs of pulses, pairs of pairs of pulses, and so forth, to completely kill
additional frequencies of vibration. Later, we’ll discuss the complications this generates
corresponding to possible enhancements of other frequencies. 71
As examples, a sequence with one trapezoid followed by an identical one has
three timings, which can be chosen to suppress three different frequencies. See Figure
3.4. Each trapezoid kills (the same) two frequencies; a third frequency is cancelled out
between the two trapezoids. In short, we add additional time scales that can lead to more
frequency suppression. Consider the double gradient pulse shown in Figure 3.4, where
the second lobe is simply the negative image of the first lobe. The second lobe is applied
D later after the first one is applied.
Figure 3.4 A double-trapezoid gradient pulse with ramp time tr , flat-top time ttop and separation time D . The discussion in the test explains how the three time scales shown are connected to three different zeros in the frequency spectrum.
We are interested in the string vibration after the double pulse is completely
applied, so the integral in our general solution (3.2) of string vibration can be evaluated
for ttt>D+2 rtop + . The resulting amplitude is given by
72
2 ¥ D+2t +t 2v sin(pnx /l) 2l r top y(x,t) = dtf(t)sin(w (t - t)) å n T w np ò0 n=1(ODD) n (3.19) 2v 2 ¥ sin(pnx /l) 2l =f C(n,t) 0 å T w np n=1(ODD) n
in which the integral Cnt(,)for non-constant force f()t is:
Cnt(,)=+ I12 I (3.20)
Since the integral is the superposition of the vibrations of the two gradient pulses, we
recall the result for a single trapezoidal gradient pulse (3.17),
1 Itttttttt=-----+-[sin(wwww ( )) sin( ) sin( ( )) sin( ( ))](3.21) 1 2 nr n n11 r n trnw
in which, ttt1 =+rtop. I 2 can be calculated by integrating the non-constant force from
time D to D+2ttrtop + as below:
D+tt D+ r t -D 1 Itdtd=-sin(wtt ( - )) - sin( wtt ( - )) 2 òòDD+nnt tr r
D+tt1 + r D+tt + -t --1 r sin(wtt (td )) òD+t n 1 tr D+tt D+ D cos(wt (tt-- ))rrt cos( wt ( )) =-nn + ttww rnDD rn D+t1 (3.22) D+t 1 r cos(wt (tt-- )) cos( wt ( )) -+nndt òD trnww n D+tr D+tt + D+ tt + D+tt +cos(wt ( t - ))11rrt cos( wt ( t - )) +-1 rn n ttrnww rn D+tt11 D+ D+tt + 1 1 r cos(wt (t - )) + n dt òD+t trn1 w
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In the above expression, the sum of the first, second, fourth, fifth, and sixth terms leads to
their cancellation, and we then have:
D+ttt D+ + 11rrcos(wt (tt-- ))1 cos( wt ( )) Idd=-nntt + 2 òòDD+t ttrnrww1 n 1 = ( - sin(ww (tt -D- )) + sin( ( t -D )) 2 nr n (3.23) wnrt
+-D----D- sin(wwnr (ttt11 )) sin( n ( tt )))
We observe that the integral result Cnt(,)vanishes, if any ONE of the following relations holds:
ww==nin w,1 for i = 1, 2, 3 where
222ppp ww1,1==, 2,1, w 3,1 = (3.24) tttrrtop+D
For either of the first two, II12==0 , separately. For the third, I1 cancels I 2 . In all cases, we can track the cancellations of the Q and anti -Qimpulses.
Thus the string with fundamental frequency w stops vibrating under any
conditions shown in (3.24). This follows from the vanishing of all of the harmonic
contributions to the solution (3.2) under (3.24). The result is that a double gradient pulse
222ppp can kill the three different frequencies given by , , . tttrrtop+D
74
1 Figure 3.5 A four-trapezoid gradient pulse is designed to kill 4 frequencies peaks at , tr 1 1 1 , , and , and all of their harmonics. ttrtop+ t1 tt12+
Furthermore, a four-trapezoid pulse can kill four different frequencies,
1111 , , , , where the fourth frequency is cancelled out between the tttrrtop++ttt112 two pairs of trapezoids, and so on. See Figure 3.5. An analytical calculation can be carried out similar to the method we use for a double-trapezoid pulse, which is the result of superposition of the amplitudes of four single trapezoidal pulses. Once the pulse becomes longer we also need to consider the on-off ratio of the gradient pulse (during the
TR cycle). In considering multiple pulses, we have the challenge where nullifying a particular frequency f will enhance the frequency f/2. The reinforcement of given frequency, however, may not increase acoustic noise, if the system frequency response function does not support that frequency. 75
3.3.4 General Rules for General Pulses
A convenient building block for the discussion of more general pulses is a simple
boxcar pulse. Again neglecting the vibration decay time (i.e., the damping effect), we
expect that, if the boxcar time duration matches the period of the fundamental string
frequency, the string vibration caused by the front end (step-up) of the boxcar is completely suppressed immediately following the back end (step-down) of the boxcar.
All the harmonics are killed along with the fundamental frequency. If the boxcar time is a multiple of this period, the same conclusion is reached, but again only after the boxcar is turned off.
Figure 3.6 (a) A symmetric trapezoid pulse as a convolution of two boxcars with identical heights. (b) A “quadratic” pulse as a convolution of a third boxcar with the symmetric trapezoid pulse. The timings shown in the two figures are connected to the frequencies that can be suppressed as discussed in the text and in Table 3.1.
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The boxcar is a building block in that we can construct a standard symmetrical
trapezoidal pulse through a convolution of two boxcars. See Figure 3.6.a. With this
construction, we can understand that two frequencies can be suppressed; the two
frequencies correspond to the inverse of the two boxcar time-widths, respectively. After convolution, the ramp time (up or down) is equal to the shorter width of the two boxcars and the sum of one ramp time plus the flat-top time is equal to the longer width of the two boxcars. In the limit of equal widths, we convolve to a triangular pulse. In summary,
a general trapezoid with ramp time tr and flat-top time ttop leads to spectral zeros at the
1 1 frequencies and and their harmonics. tr ttrtop+
Consider pulses made up of successive convolutions of boxcars, noting that a
general symmetric trapezoid is the convolution of two boxcars. We can find the pattern
by building on the frequency killed by each boxcar pulse. The general rule, for the
frequencies that are nullified by a pulse consisting of convolutions of n individually
timed boxcars, is as follows. All frequencies given by the series
11 1 1 , , , , will be killed, where ti is the time tttttt11212+++ 3 tt 12 +++ tn
duration of the i-th boxcar. But keep in mind that standard imaging utilizing a pulse made
up of a large number of convolutions is limited, since the total time duration of the
resultant pulse, which is tt12++ + tn , will grow too large. In addition, if data must
77
be taken over the plateau (the flat-top) of the read-out gradient pulse, the time interval
n given by Max{0, 2 Max ( t )- t } decreases with the number of convolutions. iiå 1
Notwithstanding these difficulties, the hierarchy of timings and the plateau time duration
for the n-convolution pulse, is also shown in Table 3.1.
Width of boxcars Duration of Plateau of resultant convolved Frequencies killed resultant pulse pulse t1 1/t1 t1 t1 t1, t2 1/t1, 1/t2 t1-t2 t1 + t2 t1, t2, t3 1/t1, 1/t2, 1/t3 Max(t1-t2-t3, 0) t1+ t2+ t3 t1, t2, t3, t4 1/t1, 1/t2, 1/t3, 1/t4 Max(t1-t2-t3-t4, 0) t1+ t2+ t3+ t4 1/t1, 1/t2, 1/t3, 1/t4, Max(t1-t2-t3-t4-t5, t1, t2, t3, t4, t5 t1+ t2+ t3+ t4+ t5 1/t5 0) Assume t1 is the maximum width of all the boxcars.
Table 3.1 Convolution results of boxcars
3.3.5 Repeated Pulse: Pulse Trains
In a typical MRI experiment, a pulse sequence consists of a given group of pulses
repeated hundreds of times in order to cover the entire Fourier k-space for image reconstruction. To compare to the experiments, we need the response of a string to a
pulse train of N trapezoids with the repetition time TTR = . The vibration amplitude
after all of the pulses are applied can be evaluated by superposing the effects from each
single trapezoidal pulse. This response is given by:
2sin(/)2vnxll2 ¥ p yxt(,)= f Dnt (,) 0 å (3.25) TnnODD=1( ) wpn
with
78
N -1 1 Dnt( , )=---- [sin(ww ( t t iT ) sin( ( t iT ) å 2 nr n (3.26) i=0 wnrt
----+-- sin(wwnr (tt11 t iT ) sin( n ( tt iT )]
for tN>-(1) Ttt ++1 r
Starting from the first term in (3.26), we have
N -1 sin(w (tt-- iT )) å nr i=0 N -1 1 ittiTww()-- - ittiT () -- =-å()eenr nr 2i i=0 -iTwwnnNN iT 11()ittww()-----ee itt () 1() =-[]eenr nr (3.27) -iTww iT 2i 11--eenn 1 =-+---{sin(wwnr (tt )) sin( nr ( tt ( N 1) T )) 2(1- cos(wnT ))
-----+sin(wwnr (tt NT )) sin( nr ( tt T ))}
With resulting expressions that are similar to (3.27), the other terms in (3.26) can be
calculated by substituting tt- r with tt, -- t11 tr , t - t, so we have:
NTw sin(n ) 1 NT T Dnt(,)=--+2 {sin((w t t )) T w w2t nr22 sin(n ) nr 2 NT T NT T --+----+ sin(ww (tttt )) sin( ( )) (3.28) nnr221 22 NT T ---+ sin(w (tt ))} n 1 22
79
NTw sin(n ) 2pm Consider the effect of the overall factor 2 in (3.28). When w , the Tw n T sin(n ) 2
NTw sin(n ) limit of the factor 2 is -N , when m is odd and N when m is even, which Tw sin(n ) 2
1 shows the expected peaks in any steady-state vibrating system with frequency and its T
harmonics. Notice also that this factor vanishes whenever wnT is an odd multiple of p with N even. In Figure 3.7, it is indeed observed for Tms= 10 that the amplitude is
maximum at 100 Hz and its harmonics, which is the repetition-rate effect. The expected
zeros from this factor are mixed in with other zeros in the response function. (The
1 1 1 vibration at frequency is enhanced while is cancelled; the odd harmonics of T 2T 2T
1 are killed and the harmonics of are enhanced.) These are not the frequency-killing T
mechanisms described earlier; they are present for any long pulse trains with fixed
repetition time.
80
Figure 3.7 The overall response factor introduced in the text is shown as a function of wn for Tms= 10 , N = 100 . Its maxima are the frequencies supported for the corresponding 100 Hz repetition rate.
81
Works Cited:
1. D. Tomasi, T. Ernst, A simple theory for vibration of MRI gradient coils.
Brazilian Journal of Phhysics 36(1A):34-39, 2006.
2. P. Morse, H. Feshbach, Methods of theoretical physics, McGraw-Hill Book
Company, Inc., p.1339, 1953.
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Chapter 4
String Simulation
4.1 String Simulation
The next step in the study of the various cancellation mechanisms described earlier is their examination and verification via numerical computations. We later test the method with MRI experiments, which is discussed in the next chapter. The wave equation for the string was solved with MathematicaTM programming routines. We saw in the preceding chapter that the gradient pulse parameters are essential to determine what frequencies are to be cancelled. Boxcars, trapezoids, multiple pulses, and pulse trains are simulated and discussed in detail in this chapter. All the results are in good agreement with the theory and analytical results that we carried out in the previous chapter.
4.1.1 Boxcars
For boxcar gradient pulse, we investigate the dependence of the cancellation on
the flat-top time duration ttop . In Equation (3.2), we are free to employ a unit system with following parameters: k = 0 (no damping), f()()1 force== l length with the ends
83
of the string fixed at xl= 0, , and with time units (frequency units) in milliseconds (Hz)
unless otherwise stated. The string velocity is then v = 2n1 according to Equation (3.4).
The string equation simplifies to
¶¶22yxt(,) 1 yxt (,) -=-1 (4.1) ¶¶xt222u
We choose tt1 ==top 2 ms. In Figure 4.1, we can see that the amplitude of oscillation at center of the string vanishes for tms³ 2 , to within the errors of the numerical method of calculation (less than 1%). This is maximum destructive interference between the two standing waves - recall Figure 3.2 (a). As our theory shows in Chapter 3, the timing of this boxcar can also kill all the harmonics. Figure 4.2 shows an example of the cancellation at 1000 Hz, which is the next harmonic for 500 Hz. We can compare the
cancellation at tmstop = 2 to the absence of cancellation at other flat-top times
tmstop ¹ 2 . Figure 4.3 shows this comparison where the two other flat-top times are 1)
tmstop = 1 , a time where the vibration due to the second Q impulse enhances maximally (rather than cancelling) the first Q impulse vibration and 2) the value
tmstop = 1.5 where there is a constructive interference that is less than maximum.
84
Figure 4.1 Following the text and using a boxcar with tmstop = 2 , we expect and confirm in this figure that the 500 Hz vibration of the string vanishes for tms³ 2 .
Figure 4.2 Following the text and using a boxcar with tmstop = 2 , we expect and here that the 1000 Hz (lowest harmonic of 500 Hz) vibration of the string vanishes for tms³ 2 .
85
The result we showed in the previous page is from the numerically solved string
equation by MathematicaTM, without damping. (We include dampling in a subsequent
discussion.) To compare, we can calculate (3.11) with summing a sufficient number of
harmonic terms associated with wn . To save calculation time, but still remaining quite
accurate, we choose the upper limit for n as 100. The result is in excellent agreement
(better than 0.2%) with the numerical solution for yxt(,) during the time interval where
the pulse is applied. See Figure 4.4. While the summing method gives a perfect
cancellation with a zero subsequent amplitude, the numerical solution generates a tiny but nonzero ripple amplitude (less than 1% of the maximum amplitude) remaining after the boxcar gradient pulse is turned off.
Figure 4.3 Simple boxcar cancellation (solid curve) with a 2 ms flat-top time for the cancellation of a 500 Hz vibration. The dashed curve shows the 1 ms flat-top maximum enhancement and the dotted line shows a 1.5 ms flat-top intermediate enhancement.
86
Figure 4.4 A comparison of the numerical simulation result and the analytical calculation result. The solid curve stands for numerical simulation, which shows an almost invisible numerical error after tms³ 2 . The dashed curve is the result from an analytical calculation, which shows a perfect zero after tms³ 2 . The numerical calculation agrees with the analytical result within the computational error.
4.1.2 Single and Double Trapezoids
In the comparison of the analytical and numerical results for trapezoidal pulses, we consider both a single trapezoid and a two-trapezoid set. A solution for a single
trapezoidal pulse, with ramp time tr and flat-top time ttop leads to a cancellation of both
1 1 of the frequency peaks at and . In Figure 4.5, the results showing the tr ttrtop+ cancellation of the two different frequencies superimposed on the same graph. Inside the 87
simpler boxcar amplitude, the net amplitude within the trapezoid time duration was
approximately sinusoidal in behavior. Here, however, the amplitudes - see Figure 4.5 -
especially for the higher frequency where its peak amplitude and the trapezoid are flat at
the same time, have individual contributions associated with the different times for the four cusps of the trapezoid.
Figure 4.5 The numerical simulation of the frequency components of the amplitude at the center
of a vibrating string as a function of time for a trapezoidal pulse with ramp-up time tsr = 500 m and flat-top time ttop = 300 ms . According to this prediction, the solid curve and the dashed curve evolve to a complete cancellation for 2000 Hz and 1250 Hz, respectively, at the instant where the trapezoidal gradient pulse has been turned off.
Recall from Chapter 3 that we could cancel additional frequencies by adding
negative copies of the original pulse. If we add a follow-up negative identical trapezoidal
88
gradient pulse, with additional time interval D , then an extra frequency 1 is D
cancelled. A simulation showing all three cancellations of target frequencies is illustrated
in Figure 4.6.
Figure 4.6 As in Figure 4.3, but now for a double-trapezoid gradient pulse with ramp-up time
tsr = 500 m , flat-top time tstop = 300 m and D=2000 ms . The solid curve shows the simulated amplitude of vibration component at 2000 Hz, the dashed curve at 1250 Hz, and the dotted curve at 500 Hz. The complete cancellation of each component after one circle (one period) agrees with the theoretical expectations.
If we add a follow-up negative identical double-trapezoid gradient pulse, with
1 additional time t2 , then an extra frequency is cancelled. A simulation ()tt12+
showing all three cancellations of target frequencies is illustrated in Figure 4.7.
89
Figure 4.7 As in Figure 4.6, but now for a four-trapezoid gradient pulse with ramp-up time
tsr = 500 m , flat-top time tstop = 300 m , tm1 = 2000 s and tm2 = 3000 s . The thick solid curve shows the simulated amplitude of vibration component at 2000 Hz, the dashed curve at 1250 Hz, the dotted curve at 500 Hz, and the fine solid curve at 200 Hz. The complete cancellation of each component after one circle (one period) agrees with the theoretical expectations.
4.1.3 Simulation with Damping Effect
The method described in this thesis can be applied equally well for damped vibrations. Various examples exhibiting the cancellation mechanisms discussed above, but also including the damping effects and the different spatial excitations for longitudinal and transverse gradients, have been verified by numerical solutions of the
90
string equation. Here we will show the simulation of a boxcar gradient with a reasonable damping effect considered.
Figure 4.8 Solid curve shows, for a boxcar with tmstop = 2 , the 500 Hz vibration of the string vanishes for tms³ 2 . The dashed curve shows result considering the damping factor k = 50 Hz . The comparison shows that the predicted frequency cancellation is effective even though we consider damping and in fact can be made equally effective via an adjustment of the boxcar shape – see the discussion in the text and Figure 4.9.
Figure 4.8 illustrates the comparison of the simulations of a boxcar gradient with and without a damping effect. The string vibration frequency is still 500 Hz with boxcar duration 2 ms, and the damping factor k = 50 Hz . The solid curve shows the complete cancellation of vibration, and the dashed curve shows a substantial cancellation of the same frequency vibration. Specifically, we find that for damping factors as small as 1/10 of the vibration frequency, cancellations down to the level of a few percent can still be
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effected. Furthermore, according to (3.2) and (3.3), small simultaneous adjustments of the boxcar shape can be made such that we obtain complete cancellation. The procedure is to multiply the boxcar gradient with exponential decay factor e-kt , which will then
eliminate the time-dependence of t in the integral evolved with Green’s function. This leads to cancellation at the same level as that for an undamped string. This agrees perfectly with the numerical simulations, as shown in Figure 4.9. In Figure 4.9 (a), the solid curve stands for a standard boxcar pulse with duration 2 ms, while the dashed curve
-kt is for the boxcar adjusted by multiplication with the factor e , for comparison. For the
same damping factor 50 Hz, the standard boxcar cancellation corresponding to the solid
curve in Figure 4.9 (b) is visually not as complete as that for the adjusted boxcar. In
general, damping does not change our conclusions about general frequency rules if the
pulse profiles can be tweaked in a manner resembling the exponential correction just
mentioned. We have carried out experiments to verify this as well. See Chapter 5.
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Figure 4.9 (a) The solid curve represents the cancellation for a standard “flat” boxcar with damping included, and the dashed curve is for a slightly exponentially modified boxcar (via the -kt factor e ) for comparison. (b) The string vibration is cancelled more completely with the “damped” boxcar profile than for the standard boxcar pulse.
4.1.4 Longitudinal and Transverse Gradient Pulse
In all of above simulations and discussions, our assumption is that the driving force on the string is spatially constant; i.e., it is independent of position z along the 93
string. In fact, the current distribution patterns are different for the gradient coils. For the
longitudinal gradient, the current has an anti-symmetric and approximately sinusoidally
distributed along the z-axis, while the current patterns are symmetric for the transverse
gradient (and approximately co-sinusoidal in z). In mathematical form, and for z = 0 at one end, these approximations are f(z) = sin(2pz) and f(z) = sin(pz), respectively, for
unit string length. See Figure 4.10 (a) for an illustration of the two different current
distributions. It is evident that, for the longitudinal gradient, the current has zeros at both ends and the center of the string, while for the transverse gradient, while the current has zeros at the both ends and a maximum value at center. For the longitudinal gradient coil,
therefore, the Lorentz forces applied on it are compressing one half of the gradient
assembly while expanding the other half. For the transverse gradient, the Lorentz force is
bending the center of the coil former in and out. Considering this spatially dependent
current distribution, a simulation was carried out to calculate the amplitude at z = 0.25.
The reason we choose this point instead of the previous choice of the center point is that the amplitude is always zero for the longitudinal current distribution. Figure 4.10 (b) shows the 500 Hz vibration amplitude comparison of the longitudinal (the solid curve) and the transverse (the dashed curve) before and after the boxcar gradient pulse (with time duration 2 ms) is applied. It is clear that the vibrations from both gradients are cancelled to within 10% of their original amplitudes. This confirms that our cancellation algorithm is still valid for a practical non-constant position-dependent force. Furthermore, for the same gradient current amplitude, the vibration from transverse gradient is larger
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than that from longitudinal gradient during the pulse is applied. This simulation agrees
with the experiment result that the transverse gradient acoustic noise is louder, which we
will discuss later in next chapter.
4.1.5 Enhancement of Multiple Pulses
We explained in Chapter 3 how double-trapezoidal gradient pulses can be timed
to kill three different vibration frequencies corresponding to the three time scales.
Although these frequencies are suppressed, other frequencies can get enhanced. Every time a vibration frequency at f is cancelled, there is the possibility that any vibration
frequency at f /2is enhanced. To get enhancement, however, a given frequency has to
have been in the intrinsic frequency response function, in order to be excited by the
gradient pulse.
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(a)
(b)
Figure 4.10 (a) The current distributions of the longitudinal (solid) and the transverse (dashed) gradient pulse. (b) Amplitudes of the vibration at 500 Hz driven by the above two gradients, and subsequently suppressed with both boxcar gradient durations 2 ms.
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We use the double-trapezoidal gradient pulse with the same parameters as in
Figure 4.6. The vibration at 500 Hz is cancelled as predicted, as shown by the dashed
curve in Figure 4.11. The solid curve, on the other hand, refers to a vibration at 250 Hz,
which is half the cancellation frequency, and thus becomes enhanced. This phenomenon
is similar to what we have in Figure 3.2, where 1 ms duration boxcar enhances the
vibration at 500 Hz. This is an unavoidable situation, if the original frequency spectrum
supports a given peak with half the frequency we are trying to suppress.
Figure 4.11 For a double-trapezoidal gradient pulse with ramp-up time tsr = 500 m , a flat-top
time tstop = 300 m and the interval D=2000 ms between trapezoids. The vibration at 500 Hz is cancelled by the time scale between the positive and negative trapezoids as shown with the dashed curve. However, a vibration present at 250 Hz , as shown with the solid curve, gets enhanced.
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Chapter 5
Experimental Design and Results
Because the amplitude of sound in the air is coupled to the MRI gradient coil
vibrations, the mechanism described in the previous chapter for the suppression of a
given vibrational frequency using particular pulse timings is expected to kill that frequency contribution to the acoustical noise produced by gradient coils assembly. This conclusion is borne out by the experiments we have performed and the description of these results is contained in this section. We have been able to confirm the predicted cancellation of frequency peaks for all of the above pulse sequence configurations. The use of a string model is shown to be an excellent guide to all results and for the general interference found between the semi-impulses.
5.1 Experimental Design
5.1.1 Experiment Setup
The measurements were performed on a 1.5T system (Espree, Siemens Medical
Solutions, Erlangen, Germany), located in the Department of Radiology, University
Hospitals, Cleveland, OH, United States. There are no RF pulses required. Gradient strengths are 18 mT/m for most of the measurements, and the gradient pulse trains were
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run typically with repetition time TmsR = 10 . The corresponding 100-Hz repetition rate
is a common choice for imaging, but we have also carried out tests for a range of
¢ practical values of TsR . (Note also that the repetition rate will limit the frequency
spectrum measure – see below.) The total acquisition time is usually taken to be one
¢ hundred TsR in length; thus, an averaging procedure is both appropriate and accurate,
with little error introduced by the averaging or from any fraction of a period left over in
the difference between the total time and the multiple of TR . Our assumptions have been verified in examples where we reduced the total time (for example, 0.3 second in
comparison with 0.6 second), keeping the overall time duration large compared to TR in
order to cover k-space.
Various measurements were carried out regarding x, y, z gradients. The acoustic noise from these gradient pulse trains have been measured with a microphone placed at the wall of the scanner room. While the level of the frequency response would depend on the microphone position, the suppression of selected frequencies can be measured at arbitrary locations. The signal was acquired using an audio software program (Sound
Forge 9.0, Sony, Japan) with a sampling frequency of 192 KHz and analyzed with an
TM FFT (fast Fourier Transform) using MATLAB . With variable ramp-up times tr (the ramp-down time is always symmetrically equal to the ramp-down time) and a variable
flat-top time ttop , single trapezoid pulses have led to the expected oscillations and zeros 99
1 1 associated with the two frequencies, and , and their harmonics. Both a tr ttrtop+
standard trapezoidal gradient pulse and a damping-factor-adjusted trapezoidal gradient
have been studied to test our theory. Additionally, two-trapezoid pulses, four-trapezoid
pulses, “quadratic” pulse and pulse trains with different TRs have all been investigated.
One significant electrical background is the omnipresent 60-cycle Helium pump whose noise is picked up and observed in all of the spectra, which does not, however, interfere with cancellation mechanisms of interest to us. It is an additive (linear) effect at low
frequency.
5.2 Experimental Results
5.2.1 Cancellation as A Function of Pulse Timings
As a representative test, we analyze, and compare to experiment, the 1300 Hz
acoustical spectrum peak for a longitudinal-gradient single trapezoidal pulse with
variable flat-top time ttop . As we already discussed in Chapters 3 and 4, a trapezoidal
gradient pulse with ramp-up time tr and flat-top time ttop can kill two frequency peaks at
1 1 1 1 and . In the meantime, the half frequencies and peaks should tr ttrtop+ 2tr 2(ttrtop+ )
become enhanced. For a 1300 Hz peak, as the experimental result shows in Figure 5.1
(a), the trapezoidal gradient pulse with flat-top time 300 ms and ramp-up time 110 ms 100
enhances the vibration at 1300 Hz, while in Figure 5.1 (b), the trapezoidal gradient with
flat-top time 650 ms and ramp-up time 110 ms suppresses the same frequency. We
notice that the amplitude at 1300 Hz in Figure 5.1 (b) is very much suppressed,
quantitatively which is 3.5% compared to the amplitude at the same frequency in Figure
5.1 (a). There is a small residual amplitude left, which is understandable if for no other
reason than the fact that the flat-top time is not perfectly timed. If the flat-top time in our experiment is 1000 / 1.3-= 110 659 ms , the resultant 1300 Hz vibration amplitude
would be zero, within the resolution of our plots. Our theory about string cancellation and
enhancement is experimentally verified.
In a presentation of experimental results, it is important to discuss errors. In our
measurements, there are possible factors that might affect the experimental results. These
factors include the MRI scanner, the gradient pulses, position of microphone, the volume
of the recording software, and the possible error in data analysis. As we carried out the
experiments in several different times in a 2-years range, we are very careful dealing with
data from different measurement dates. In all of our results discussed, we only compare
the results that were measured at the same date. These data have been collected in the exactly same measurement environment. As far as the frequency spectrum is concerned,
we have observed that experiments carried out months apart; the spectrum can have shifts
in amplitude and position for the prominent peaks, even for exactly the same parameters.
Such discrepancies can occur after scanner hardware and software upgrades have been
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made and with even small differences in the positioning of the microphone [1]. The
Helium pump, which is a primary source of a 60 Hz and its harmonics, is regularly
serviced along with the gradient coils and other components. Nevertheless, our
experimental goal is to verify whether we can suppress the selected extant frequency peaks for a given gradient pulse sequence according to our string modeling. We found
that our results are completely reproducible and no discrepancy with the expectations has
been found at any time.
The experiments we carried out include measurements of the noise from x, y, z- gradients. However, the results we discussed here focus on the z-gradient. Our string
cancellation theory is verified with all gradients, and we have obtained similar data from different gradient axes. Below, we show one example for a transverse gradient experiment. Figure 5.2 (a) shows the noise frequency spectrum from a y-axis trapezoidal
gradient pulse train with tsr = 110 m and tstop = 1320 m . The frequency peak at 700 Hz
is clearly suppressed according to our frequency cancellation formula
1 (110+= 1320)msHz 704 . The harmonics of 700 Hz at 1400 Hz, 2100 Hz, 2800 Hz and 3500 Hz are also cancelled. Figure 5.2 (b) shows the noise spectrum with a different
tstop = 610 m from the choice in (a), where now the frequency peak at 1400 Hz is cancelled. Note that the half frequency at 700 Hz is noticeably enhanced, in agreement
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with our predictions. The 350 Hz enhancement in (a) is less noticeable, owing to the 100
Hz frequency steps – see the TR discussion in the previous chapter and in Chapter 6.
Figure 5.1 In (a), the trapezoidal gradient pulse with flat-top time 300 ms and ramp-up time 110 ms enhances the vibration at 1300 Hz, while in (b), the trapezoidal gradient with flat-top time 650 ms and ramp-up time 110 ms suppresses the same frequency. These results confirm the picture obtained via string cancellation theory.
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Figure 5.2 In (a), the trapezoidal y-axis gradient pulse with a flat-top time 1320 ms and ramp time 110 ms suppresses the vibration at 700 Hz, while in (b), the trapezoidal gradient with flat- top time 610 ms and ramp-up time 110 ms suppresses 1400 Hz peak. The enhancement of 700 Hz in (b) is more pronounced than the enhancement of 350 Hz in (a), because of the 100 Hz frequency resolution. All results confirm the picture obtained via string cancellation theory.
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The simple analysis described in previous chapters for an undamped oscillator
with a single trapezoidal pulse predicts that the interference mechanism between the Q
impulses is sinusoidal. (This corresponds to TR =¥.) That is, the net magnitude of the
vibration peak (the amplitude in the Fast Fourier Transform) will be a sinusoidal function
of ttop , as we move from maxima to minima and back again. The 1300 Hz peak
magnitude is of the form sin(2p (ttTtop+ r ) /0 ) , with the theoretically predicted value
ramp time tsr = 110 m and THzs0 ==2 / 1300 1538 m , which is verified by a
numerical solution for the motion of a single point on a string. For a repeated series of
gradient pulses, however, along with the sinusoidal behavior there is a steady decrease in
the magnitude of the peak as a function of time, even for an undamped oscillator. We
focus on this case (i.e. finite TR ), which is relevant for practical imaging, in our
simulations and measurements.
The string simulation for our example is analyzed via an FFT for discrete
numerical data xt()over the time interval TmsR = 10 with a sampling frequency the
same as in the experiment, 192 kHz. All string simulation results refer to the motion of
the center of the string, although any other point would serve the purpose. The flat-top
time of the gradient trapezoidal pulse is varied from 50 ms to 3500 ms with 50 ms
intervals, which is the same as the experiment. The ramp time is fixed at tsr = 110 m .
Neglecting damping for the moment, the string oscillates at 1300 Hz from the moment 105
the trapezoid is ramped up, and continues after it is ramped down, but with a change in its
maximum displacement depending on the degree of interference between the ramps. Now
the corresponding amplitude in the FFT is an average over TR and, as a function of ttop , there is a monotonic decrease in the peak amplitude (and correspondingly a monotonic increase in the minimum) even for zero decay in the intrinsic vibrating coil structure!
This is because the vibration set into motion by the ramp-up time Q impulse of the
gradient is then interfered with by the Q impulse due to the ramp-down, leading to either
a greater or lesser net amplitude depending on the relative phase (the flat-top time
between the two ramps.) Since the FFT amplitude represents an average over the
repetition time TR , the average peak decreases as we increase the flat-top time ttop : for
larger ttop , the gradient vibrates for a larger fraction of the time at the lower amplitude.
Similarly, the minimum grows as we increase ttop , since the gradient vibrates for a larger
fraction of time at the larger minimum then. These conclusions from simulation analysis
are in agreement with experiment.
In Figure 5.3, we plot both the experimental and simulation results, where we
have included the (small) effect of a string damping factor k = 100 Hz . This damping
factor is consistent with the range of values found in [2], and has been chosen to improve
the fit at the 5% level. We see that there is excellent agreement between the measurement
and the calculation. Referring to the previous discussion about the amplitude dependence
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of sin(2p (ttTtop+ r ) /0 ) , we constructed nonlinear fits to both simulation and
experimental plots. We obtain tsr = 109 m and Ts0 = 1537 m by fitting the simulation
plot and tsr = 110 m , Ts0 = 1546 m by fitting the experimental plot. Both the results are
in good agreement with the theoretical prediction tsr = 110 m and Ts0 = 1538 m . It is emphasized that such fits to the data can be used to characterize sets of k , for a given
gradient system. For the amplitude dependence of sin(2p (ttTtop+ r ) /0 ) , we also need
to point out that this dependence on the variable flat-top time ttop agrees with the Fourier
transform of a trapezoidal excitation, which is reported in [3, 4].
Figure 5.3 Comparison of peak acoustic FFT amplitudes at 1300 Hz from experiment and simulations. The experimental result is measured in the frequency power spectrum of acoustic noise, while the simulation result is generated with MathematicaTM string simulations and MATLAB FFT calculations. The damping factor is chosen as discussed in the text.
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5.2.2 Cancellations in the Frequency Spectrum
In carrying out various gradient pulse sequence excitations for the Siemens
scanner, we found (by visual inspection) the dominant scanner resonance frequencies for
longitudinal operation to be 1300 Hz, its harmonic at 2600 Hz, 1900 Hz and 3500 Hz.
The tactic is, therefore, to design pulse timings that cancel these frequencies. Recall that a
11 symmetric trapezoidal pulse will kill the two frequencies , and it is therefore tttrrtop+
desired that the timings be chosen to nullify the dominant mechanical vibrational
frequencies of the gradient system considered. Figure 5.4 allows the comparison of frequency spectra for two different gradient pulse sequences. While the standard plots are
in dB (see Figure 5.5), we find linear plots useful for examining zeros in the frequency
spectrum that have arisen from our cancellation mechanism. It is immediately noticed in
the figure that the frequency spectrum is restricted to multiples of 100 Hz, owing to the
repetition time TR for the steady-state pulse sequence. The red plot stands for the
trapezoid gradient pulse with ramp time tsr = 110 m and the flat-top time
tstop = 1200 m , which gives a fairly typical (repeated) pulse profile, which is shown in
the figure. The principal noise components in the vicinity of 1300 Hz, 1900 Hz, 2600 Hz
and 3500 Hz are not suppressed. According to our previous analysis, this typical pulse
must suppress something, and in this instance, it suppresses the frequencies 800 Hz and
the rather high 9100 Hz (763 Hz and 9090 Hz before rounding off to the nearest 100 Hz
according to the repetition rate spectral resolution) and their harmonics. Note that the 108
harmonics have to be calculated carefully before comparing to the nearest 100 Hz to avoid a significant round-off error. It is observed in Figure 5.4a that the neighborhoods of
800 Hz and 1600 Hz are suppressed as expected. (The 9100 Hz suppression is not in the figure’s frequency range.) Also note that the cancellations take place over neighborhoods of the targeted frequencies, and not just single spectrum lines. These two neighborhoods,
and, a fortiori, the 9100 Hz neighborhood, are not important contributions to the acoustic
noise. The blue plot, on the other hand, does address the main contributors to the noise.
The blue plot in Figure 5.4b corresponds to tsr = 530 m and tstop = 250 m . The
associated repeated pulse leads to the suppression of the three important acoustic peaks or
clusters, 1300 Hz, 1900 Hz and 2600 Hz (1282 Hz, 1887 Hz and 2564 Hz before the
round-off to the nearest 100 Hz.) This is clearly evident in the comparison of the two plots in Figure 5.4. Again, it is not just a given spectral line but rather the neighborhood around the targeted frequencies that is smothered.
In a comparison between the two experiments illustrated in Figure 5.4, there is a marked reduction in acoustic noise in going from the “typical” pulse to the specially designed pulse. Whether we refer to reduction in terms of dB (typically about 30-40 dB per peak), or to the verdict of a listener, the general conclusion is that a marked reduction in sound can be achieved when at least three of the most dominant frequency peaks have been suppressed.
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Figure 5.4 The above two plots show the resulting amplitudes (arbitrary units) of two frequency spectra for two different gradient pulse sequences. The pulse profiles are also shown. The upper one in red (a) corresponds to ramp-up time 110 ms and flat-top time 1200 ms . The lower one in blue (b) corresponds to ramp-up time 530 ms and flat-top time 250 ms and has been designed to suppress the principal noise contributions. The peaks at 60 Hz and 120 Hz in each plot arise from the AC background (as discussed in the text).
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Figure 5.5 The above two plots show the resulting relative dB values of two frequency spectra for two different gradient pulse sequences, with the same parameters as Figure 5.4. With the log scale, we qualitative observe that the bottom plot corresponds to a relative lower value in dB than the top one, reflecting a noticeable reduction in the acoustic noise. We refer in the text to representative sound files for the comparison.
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To determine the overall noise reduction, we used a sound level meter (Model 40,
ANSI specification S1.4, Type 2, Pulsar Instruments) to measure directly the noise from the sound files for the three-peak suppression experiment with A-type frequency weighting. The difference was expected to be the relevant measurement and relatively independent of volume. We found that the average difference was (13.2 2.9) dB for four measurements. The measurements differed by changing the overall volume, and volume change was such that the louder (quieter) sound range from 74 to 90 dB (59-81 dB). This is consistent with the 10.3 dB difference from the power integration method carried out in MATLAB (including the 60 Hz peak.) This is also consistent with an estimated reduction of 8.5 dB based on an estimated formula for the overall SPL (sound pressure level) over a series of peaks given in [5]. The overall sound pressure level of several peaks can be calculated by the following equation:
N 1 SPL /10 SPL = 10 log [ 10i ] 10 å (5.1) N i=1
in which SPLi is the value for each frequency peak i in the spectrum, and N represents the total number of frequency peaks that are averaged together.
A question immediately comes to mind in the above comparison. While it is clear that the individual frequencies have been suppressed in agreement with the theoretical ideas, the noise in an MRI experiment comes from, in frequency space, a series of frequency clusters in the product of the input gradient pulse sequence transform and the 112
system frequency response. The suppression of noise found in the comparison illustrated
in Figure 5.4 (b) is not just due to the suppression of two frequency clusters and their harmonics. It is also due to the overall narrowing, in frequency space, of the trapezoidal pulse profile for larger ramp time. This necessarily occurs when we need a time scale large enough to kill a lower frequency. We refer to this as a “sinc” effect, arising from additional sinc factors, which suppress more and more of the higher frequency spectrum, completely independently of any given peak cancellation. This is connected to the well- known fact that smoothing out a pulse will certainly make less noise [6]. In our experiments, we carried out two measurements that show the sinc effect. We convolved 3 boxcars to a “quadratic” pulse, which can suppress the frequency peaks at 500 Hz,
1200Hz and 1900 Hz and their harmonics. The length of this pulse is 3.24 ms and the flat-top is only 0.64 ms, which becomes impractical for real imaging. However, it does still show the frequency cancellations expected through its equivalence to a convolution of boxcars. We plot the frequency spectrum of this pulse in Figure 5.6 (a), and observe that most of the frequency peaks are suppressed. The high peak in the low frequency is the AC 60 Hz peaks and its harmonics, which, as we mentioned, results from the helium pump. There are a few, small peaks remaining at frequency 400 Hz, 700 Hz, 800 Hz, and
1300 Hz; the majority of the original noise peaks are cancelled. We also measured the noise from a trapezoidal gradient pulse with 2 ms ramp time and 3 ms top time. This pulse is designed to kill frequencies at 200 Hz and 500 Hz and their harmonic frequencies. Likewise, all the dominant peaks are cancelled except the 60 Hz and its
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harmonics. While there are some small peaks left at 700 Hz, 900 Hz, 1100 Hz and 1300
Hz, the noise with this wide trapezoidal gradient is already close to the background noise.
Quantitatively, we can calculate the sum of the FFT amplitude over the entire frequency range. Using the background noise as a reference, we find the sum over all the amplitudes of the remaining peaks in the “quadratic” spectrum exceeds the corresponding sum over the background noise spectrum by only 6.4%. The sum over all the amplitudes of the peaks for the aforementioned wide pulse with 2 ms ramp time and 3 ms top time yields almost exactly the same answer as that for the quadratic pulse. Again, we emphasize the need to distinguish between sinc effects, coming from broader, smoother pulses, and individual frequency cancellations.
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Figure 5.6 (a) A “quadratic” pulse result for the suppression of 500 Hz, 1200 Hz, and 1900 Hz and all of their harmonic peaks. (b) a wide trapezoidal pulse with 2 ms ramp and 3 ms flat-top time, and a strong “sinc” effect. Both spectra are very low in amplitude and the main contribution is from the AC 60Hz and its harmonics.
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Another question is whether the suppression of additional frequencies (clusters)
improves matters. We saw that convolving more boxcars to make a richer pulse with
more zeros is not only impractical to produce electronically but is also useless for
imaging, since it is so broad. Instead, we ask if a double-trapezoid pulse sequence of the
kind discussed previously could make the scanner quieter in comparison with a single
pulse. To answer this, we do both simulations and experiments. We perform a numerical
Fourier transform of both a single trapezoidal gradient pulse and a double-trapezoid
gradient pulse and plot the results in Figure 5.7. Each trapezoid has the same timings, and
these timings are similar to those in Figure 5.4 (b). The solid line represents the transform
of the double-trapezoid gradient pulse with tsr = 590 m , tstop = 250 m and
D=2000 ms , while the dashed line stands for the single trapezoid with the same tr and
ttop . For comparison, the experimental sound spectra have been measured and plotted in
Figure 5.8. A qualitative judgment is that the overall noise level of the two spectra is
similar, despite the fact that the double trapezoidal gradient pulse kills an additional frequency and its harmonics. Quantitatively, the overall FFT amplitude for the double
trapezoidal gradient pulse is 5.8% larger than that for the single gradient pulse. Although
the double trapezoidal gradient pulse kills an additional frequency at 500 Hz, it leads to enhancements that are counterproductive. For instance, it enhances the 750 Hz peak.
The net result of the more complicated enhancements along with the additional
cancellations makes the overall FFT amplitude a bit larger. The conclusion is that it is
not an easy matter to use multiple trapezoids to nullify a larger set of frequencies. There 116
are then a larger set of potential enhancements that have to compared carefully to the
frequency response spectrum of the scanner.
Figure 5.7 Comparison of the simulated frequency spectrum of single and double trapezoidal gradient pulses. The solid line stands for the double trapezoidal gradient pulse with tsr = 590 m
, tstop = 250 m and D=2000 ms , while the dashed line stands for the single trapezoidal pulse
with the same tr and ttop .
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Figure 5.8 Comparison in arbitrary units of experimental results for the two spectra corresponding to Figure 5.6. Plot (b) stands for the double trapezoidal gradient pulse with tsr = 590 m ,
tstop = 250 m and D=2000 ms , while plot (a) stands for the single trapezoidal pulse with the
same tr and ttop .
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We return to single trapezoidal pulses, with more attention paid to imaging
constraints. We compare two trapezoidal gradient pulses with different ramp and flat-top times, but with the same height and area. Although it requires some data being taken during the ramp times for readout gradients, the idea is to maintain the same k-space coverage and maximum gradient strength. With the same input level, we will compare
the outputs depending on the gradient pulse parameters: tr and ttop . In detail, one
trapezoidal pulse is designed to kill frequency peaks at 1300 Hz and 1900 Hz and their
harmonics, while the other one is just a standard pulse with short ramp-up and ramp-
down time. Figure 5.9 shows the experimental comparison of their corresponding noise
frequency spectra. The red spectrum is the result of zeroing the frequencies at 1300 Hz
and 1900 Hz and their harmonics, as well as the suppression of frequencies above 2000
Hz. The blue spectrum with its shorter ramps shows the zeroing of 1300 Hz and its
harmonics, but much less higher-frequency suppression. There is a marked noise
reduction between the regular gradients and specially designed gradients. The noise
difference was measured with a dB meter and is found to be relatively independent of the
overall sound volume. The average difference was (9.2 1.5) dB for four
measurements. The measurements differed by changing the overall sound volume, and
the volume change was such that the louder (quieter) sound ranged from 65 to 87 dB (57
to 78 dB). This is consistent with an estimated reduction of 7.5 dB based on the formula
(5.1). Figure 5.10 compares the dB levels found in simulation for the above two
trapezoidal gradients (the overall scale is arbitrary).
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Figure 5.9 An illustration of both zeroing and low-pass filtering of the frequency spectra of noises with two different trapezoidal gradient pulses. (a) tstsrtop==530 mm, 250 (b)
tstsrtop==110 mm, 650
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Figure 5.10 A comparison of simulation results of trapezoidal gradients in Figure 5.9. Dashed curve stands for regular trapezoidal pulse and the solid curve for the other one.
In the previous chapter, we have mentioned that multiple gradient pulses can be
combined to cancel more frequency peaks. Here, we have an experimental result from a
four-trapezoid gradient pulse. The frequency spectrum of the noise from this pulse is
shown in Figure 5.11. The parameters of this four-trapezoid gradient pulse are
tsr = 110 m , tstop = 420 m , tm1 = 800 s and tm2 = 1200 s , which are designed to kill
500 Hz, 1250 Hz, 1900 Hz, 9100 Hz and their harmonic frequency peaks. It is clear that
500 Hz and 1900 Hz peaks are cancelled as expected, while the 1200Hz and 1300 Hz are not cancelled owing to the concomitant half-frequency enhancement. Here, when a 500
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Hz frequency peak is cancelled according to the timing ()2tt12+= ms , its half
frequency 250 Hz and its odd harmonics are enhanced due to the half-frequency
enhancement rule we have previously discussed. (Recall that the enhancement of a half- frequency carries with it the enhancement of the odd-multiple harmonics – we refer the reader back to Chapter 4) That is the reason why we can see peaks at 700 Hz, 800 Hz, and 1200 Hz, 1300 Hz, as these peaks are near the enhancement frequency 750 Hz and
1250 Hz. A natural question might be asked why there are still obvious frequency peaks
at 1200 Hz and 1300 Hz, given that tm1 = 800 s is designed to cancel this frequency. In
fact, when there are cancellations and enhancements going on at the same frequency, here
for 1250 Hz, these vibration amplitudes are added, not multiplied. Thus, at 1250 Hz, the
contribution from the half frequency enhancement due to 500 Hz cancellation is shown in
the spectrum. In all, multiple-pulse frequency cancellation is more complicated compared
to single pulse cancellation, since there are more frequencies involved both with
cancellations and enhancements from the larger number of different timings. It is
therefore a challenge to apply multiple pulses idea to get cancellations of the dominant
peaks in the noise spectrum, and at the same time avoiding the enhancement of other
peaks in the frequency response spectrum – see a discussion later in this chapter.
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Figure 5.11 A four-trapezoid gradient pulse is designed to kill 500 Hz, 1250 Hz, 1900 Hz, 9100
Hz and their harmonic frequency peaks with parameter tsr = 110 m , tstop = 420 m ,
tm1 = 800 s and tm2 = 1200 s . It is clear that 500 Hz and 1900 Hz peaks are cancelled, while the 1200Hz and 1300 Hz are not cancelled because they are the odd multiples of the half frequency enhancement, which are also enhanced.
5.3 Frequency Response Function
We have focused on the dominant frequencies of an MR scanner’s vibrations.
MRI acoustic noise has more complicated frequency spectrum, which must be ultimately
considered in order to reduce acoustic noise to a really quiet level. Getting back to the
basic problem, an MRI system is a linear time-invariant system with external magnetic
forces applied to it. The MR scanner responds to the gradient pulse trains as an input, and
the output will depend upon the hardware-determined acoustic noise frequency response
123
function of the MR system. The frequency response function is intrinsic and is essential
to understanding the acoustic noise and its reduction. To understand the frequency
spectrum, note that the output response y(t) of a linear time-invariant system to an
arbitrary input excitation function x(t) is the convolution of x(t) with the system’s impulse
response function h(t): yt()=Ä xt () ht () . The power spectrum is therefore
Sf()= XfHf () () in terms of the respective Fourier transforms, where Hf()is called frequency response function. This shows a direct relation between the output acoustic noise power spectrum and the input gradient pulse spectrum, and is useful for understanding and predicting the related experimental results. Since there are three gradient coils working when the scanner is running, the total acoustic noise generated from the gradients can be written as:
Sf()= XfHf () () (5.2) totalå i i ixyz= ,,
The Fourier transform of a Delta function d()t is a constant. Theoretically, the frequency
response function of a gradient can be determined by measuring the acoustic noise output
excited by a Delta impulse. This method is actually impractical because an ideal delta
impulse is difficult to generate with a gradient coil system. So, although we cannot get a
perfect Delta pulse, we can apply a gradient pulse with very short flat-top time 10 ms [7].
Alternatively, we can use a method called the frequency sweep method [8, 9]. A
sinusoidal gradient pulse sequence is used while increasing its frequency from 0 to 5000
Hz, and the amplitude of the pulse is fixed at 1 mT/m. The time rate of change of the
124
gradient amplitude increases as the frequency increases, so we need to keep the gradient
amplitude at a lower level in order not to exceed the gradient slew rate limit. The
mathematical expression for this kind of pulse is Gatsin(p 2 ) with the varying frequency
at /2. All three gradient axes are measured.
The detail for the sinusoidal gradient pulse is as follows. For a gradient pulse with expression,
Gt()= G sin(p at2 ) (5.3)
its frequency is dependent on time t, and is given by at /2. The gradient amplitude
change rate is the time derivative of G(t):
dG() t sr() t== 2pp atG cos( at 2 ) (5.4) dt
The possible maximum slew rate occurs at tmax and the value is 2patmax , where tmax is
the length of the gradient pulse train. Additionally, there is a relation between the gradient pulse train length and the maximum frequency of the sequence waveform:
fatmax= max /2 (5.5)
So we can find an equation for the maximum value of G in terms of the maximum
frequency covered in the spectrum. That is:
sr G = max (5.6) max 4pf max 125
in terms of the system parameter srmax and the maximum frequency in the frequency
response spectrum. The gradient amplitude is determined by Equation (5.6).
As we have discussed for the previous experimental setup, we have measured the
frequency response function corresponding to all the gradient axes: x, y and z. The
original data were analyzed by MATLAB to evaluate the frequency response function.
Figure 5.12 shows that the frequency response spectra for the x-, y-, z-gradients measured
with the frequency sweep method. The x- and y-axes spectra are similar to each other since the x-, y-axes gradient coils are for the most part interchanged via ninety-degree
rotations in the azimuthal direction. The x- and y-axes spectra are higher value in dB than
the z-axis spectrum (there are different gradient currents and, also, note the longitudinal
gradient must always have a node in the center where the transverse gradient can have its maximum amplitude). In general, the peak distributions are different for the transverse (x,
y-axes) and longitudinal (z-axis) frequency response spectra. The dominant peaks for the
transverse gradient spectrum is around 1400 Hz, 2100 Hz, and 2800 Hz, while there are
1300 Hz, 1900 Hz, 2600 Hz and 3500 Hz for longitudinal gradient spectrum. One
drawback of this frequency sweep method is that it can only cover a certain frequency
range up to some maximum, which can be estimated by (5.6). Compared to this method,
Figure 5.13 shows the frequency response spectra obtained via a very short trapezoidal
impulse with tsr = 110 m and tstop = 10 m . This impulse can excite a broad frequency
range up to the Nyquist frequency decided by sampling frequency.
126
Figure 5.12 The frequency response spectra of x, y, z-axes as measured with the frequency sweep method. The x- and y-axes spectra are similar since the x-, y-axes gradient coils differ only by azimuthal rotation. The x- and y-axes spectra are higher value in dB than the z-axis spectrum – see the parenthetical remark in the text.
127
Figure 5.13 The frequency response spectra obtained using an excitation due to a very short trapezoidal impulse having tsr = 110 m and tstop = 10 m .
128
5.4 Experiment with a Vacuum System
Besides the experiments carried out with a conventional 1.5 T Siemens system,
we have also performed experiments with a 3.0 T Toshiba research system that includes a
vacuum embedded gradient assembly. The measurement procedures for the vacuum
system are similar to what we did for the conventional system. Trapezoidal gradients with varied ramp and flat-top time are used for the z-axis gradient in our experiment, and the gradient strength is set at 15 mT/m. As an example to show the effectiveness of our selected frequency cancellation method, we compare two trapezoidal gradients with different ramp and flat-top times, while with the same height and area. In detail, one
trapezoidal pulse is designed with tsr = 450 m and tstop = 260 m to kill frequency peaks
at 1400 Hz and 2200 Hz and their harmonics, while the other is a typical trapezoidal
pulse with short ramp-up and ramp-down time tsr = 100 m and flat-top time
tstop = 260 m . Figure 5.14 shows the comparison of the noise frequency spectra resulting
from trains with these pulses. The spectrum in (a) shows the zeroing of 1400 Hz and its
harmonics, while the blue spectrum in (b) is the result of the suppression of frequencies at
1400 Hz and 2200 Hz and their harmonics. There is a noticeable noise reduction between
the regular and specially designed gradients. An average of 3.3 dB is measured with a
sound level meter with A-weighting, which is consistent with a reduction of 3.8 dB based
on the Matlab power integration method.
129
Figure 5.14 A comparison of frequency spectra of noises with two different trapezoidal gradient pulses for a vacuum MRI scanner. In (a), the red spectrum shows the cancellation at 1400 Hz and its harmonics with a trapezoidal gradient pulse with tsr = 100 m and tstop = 600 m . In (b), the blue spectrum shows the cancellation of 1400 Hz and 2200 Hz and their harmonics with
tsr = 450 m and tstop = 250 m . A 3.3 dB noise reduction is achieved using the pulse in (b) compared to (a).
130
Is has been generally assumed that the main source of MRI acoustic noise is the
vibrating gradient assembly [10]. However, the eddy-current-induced vibrations of metal
structures outside or inside of the gradient are additional important sources of noise.
These become dominant for a sealed vacuum gradient system and yield noise levels that
are still too loud. Patient comfort remains an issue, for example. In our vacuum system,
the RF body coil and shield is inside of the vacuum bore so the eddy currents induced in
it cause noticeable acoustic noise in the proximity of the scanner. Eddy currents can also
be induced in the outside metallic structure such as a cryostat inner bore, which contributes to the acoustic noise too. From our experiments with the scanner
incorporating a vacuum gradient system, we have verified that the pulse sequence
cancellation mechanism discussed in this paper is effective for the sources of the Lorentz- force-induced noise both on and away from the gradient coil.
131
Works Cited:
1. R. A. Hedeen, W. A. Edelstein, Characterization and prediction of gradient
acoustic noise in MR imagers, Magnetic Resonance in Medicine 37:7-10, 1997.
2. D. Tomasi and T. Ernst, A Simple Theory for Vibration of MRI Gradient Coils,
Brazilian Journal of Physics 36(1A):34-39, 2006.
3. Y. Wu, B. A. Chronik, C. Bowen, C. K. Mechefske, and B. K. Rutt, Gradient-
Induced Acoustic and Magnetic Field Fluctuations in a 4T Whole-Body MR
Imager, Magnetic Resonance in Medicine 44:532-536, 2000.
4. A. Barnett, Comments on “Gradient-Induced Acoustic and Magnetic Field
Fluctuations in a 4T Whole-Body MR Imager”, Magnetic Resonance in Medicine
44:207, 2000.
5. W. Li, C. K. Mechefske, C. Gazdzinski, B. K. Rutt, Acoustic noise analysis and
prediction in a 4-T whole-body MRI imager, Concepts in Magnetic Resonance
Part B 21B(1):19-25, 2004.
6. F. Hennel, F. Girard, T. Loenneker, “Silent” MRI with soft gradient pulses,
Magnetic Resonance in Medicine 42:6-10, 1999.
7. M. Segbers, C. V. Rizzo S., H. Duifhuis, and H. Hoogduin, Optimized MRI
gradient waveforms for Acoustic Noise Reduction, Proceedings International
Society for Magnetic Resonance in Medicine 16, p. 1349, 2008.
132
8. S. Schmitter, M. Mueller, W. Semmler, M. Bock, Maximum sound pressure levels
at 7 Tesla-What’s all this fuss about? Proceedings International Society for
Magnetic Resonance in Medicine 17, p. 3029, 2009.
9. S. Schmitter, E. Diesch, M. Amann, A. Kroll, M. Moayer, L. R. Schad, Silent
echo-planar imaging for auditory FMRI, Magnetic Resonance Materials in
Physics, Biology and Medicine 21:317-325, 2008.
10. W. A. Edelstein, R. A. Hedeen, R. P. Mallozzi, S. A. El-Hamamsy, R. A.
Ackermann, and T. J. Havens, Making MRI Quieter, Magnetic Resonance
Imaging 20:155-163, 2002.
133
Chapter 6
Discussion and Conclusion
6.1 Summary of Results
Using the forced damped string model as a guide, successive convolutions of
boxcars are seen to lead to a series of zeros in the frequency spectrum. The series
corresponds to the interference between the respective Q impulses associated with
turning the gradient on and off. With the appropriate timings, a boxcar can kill one
frequency (and its harmonics), a trapezoidal pulse can kill two different frequencies and
their harmonics, and the convolution of each additional boxcar with the original pulse can
be used to cancel out one more frequency and its harmonics. A different route that is
verified by the modeling is to add a follow-up trapezoidal gradient lobe composed of
ramp-up and ramp-down force impulses that cancel the respective vibrations caused by
the first trapezoidal gradient pulse. A pair of pulses can be added to this first pair to kill
another frequency, and two more pairs can be added to these two pairs for yet another
frequency zero, and so forth. For our trapezoidal gradient experiments, most of our results show 30 – 40 dB reduction for a single frequency peak, with an overall noise
suppression about 10 dB.
134
In a previous paper [1], it was observed that the dominant peak heights in the acoustic noise spectrum “fluctuated” periodically with respect to the trapezoidal gradient
impulse flat-top width, “strong suggesting that mechanical resonance of the gradient coil
structure are the source of this acoustic energy.” The present study confirms this
connection and, moreover, shows the fluctuations to be damped sinusoids. Through our string simulation and experiment data analysis, we also understand that the decay of
peaks in the sinusoidal plot is due to the increase of the flat-top time, not because of the
damping factor in gradient coil vibration. In all, the experimental results and their Fourier
analysis are in excellent accord with the string model simulation.
We have alluded to the fact that the output response yt() of a linear time-invariant system to an arbitrary input excitation function xt()is the convolution of xt() with the system’s impulse response function ht(): yt()=Ä xt () ht ()[1]. The power spectrum is
2 therefore Pf()= XfHf () () in terms of the respective Fourier transforms. This shows a
direct relation between the output acoustic noise power spectrum and the gradient pulse
spectrum, and is useful for understanding and predicting the related experimental results.
When a boxcar gradient input excitation is used, the noise power spectrum is proportional
to the square of a sinc function, which shows a general suppression of noise, especially
for high frequency components. As the width of the boxcar pulse increases, the suppression becomes more effective due to a narrowing of the sinc profile. A trapezoidal gradient (the convolution of two boxcars) with fairly long ramp times reduces the noise significantly due to a “double” sinc effect in the power spectrum. A “quadratic” pulse 135
(the convolution of three boxcars) has such severe high frequency suppression (three sinc factors in Xf()) that its sound is spectacularly suppressed, but at the high cost of 3-4 ms pulse widths (coming from long ramp times).
6.2 Discussion
We have chosen standard pulse sequence profiles and timings in order to be compatible with practical imaging. For example, in an EPI sequence (echo-planar imaging where speed is of the essence), a double trapezoidal readout pulse with
tsr = 200 m and tmstop = 1 with TmsR = 2.8 could be used for practical imaging.
Our range of timings is compatible with such examples. The TR freedom has been
utilized by other groups to reduce acoustic noise. In [3], TR was carefully selected to avoid certain dominant broad resonances. For our system, these dominant peaks are at
480 60 Hz and 800 100 Hz , and they are evident in the frequency response function data of Figure 5.13. The acoustic noise can be reduced as much as 11 dB by
changing TR to an appropriate value when the fundamental frequency and its harmonics avoid these peaks. In our work, we have a fundamental frequency usually at 100 Hz, where only high harmonics match the two peaks. In [4], a narrow-band readout gradient
pulse was used for EPI imaging, with selected TR to match the readout fundamental frequency to the trough of the frequency response spectrum. In a recently published
136
MRM paper [5] on acoustic noise cancellation, our previous abstracts introducing the
concept of the cancellation of selected acoustic noise frequencies by the pairs of semi-
impulses and with spring and string physical motivation have been cited. These authors
have carried out an EPI-like sequence with a double trapezoid pulse. By adjusting the
sum of flat-top time and ramp time, one dominant frequency peak in the frequency
response spectrum was suppressed. At the same time, by adjusting the separation between
the two pulses, the fundamental frequency determined by TR is placed at the minimum of
the frequency response spectrum. A 12 dB noise reduction is achieved by using this EPI-
like pulse. This confirms the potential application of double trapezoid pulse to EPI
sequence. We have considered and understood these TR effects through our theoretical
calculations and numerical simulations. Recall that the fundamental frequency at 1/TR
and its harmonics are enhanced by a factor N in our amplitude formulas.
Besides the relevance of the double trapezoid pulse to an EPI sequence gradient
read-out, this pulse also is the building block for the phase encoding gradient in balanced
SSFP (steady state free precession) [6, 7]. The net gradient area on any axis is zero under
this building block. (The repetition time TR in this thesis is actually equal to the inter
echo spacing in the balanced SSFP sequence.) That is, the positive and negative lobes of
this double trapezoid pulse have the same gradient area magnitude, although the common
magnitude can be varied to collect different echoes. The separation time between these
two lobes is usually equal to the flat-top time of the readout gradient, which maintains a 137
constant value independent of echo number. By adjusting the phase encoding gradient pulse ramp time and flat-time, with the adjustable separation time between the two lobes, along with the decided inter echo spacing (TR), there is possibility that the gradient acoustic noise in balanced SSFP could be reduced by our double trapezoid gradient pulse noise cancellation mechanism. Additionally, spiral k-space acquisition is always used for fast imaging sequence. Quiet imaging method with interleaved spiral read-out has been achieved by Oesterle [8]. Starting with a single shot spiral read-out gradient with maximum slew rate at the start of sequence that produces noise pressure levels over 110 dB, a longer ramp was used with decreased slew rate and interleaved spiral trajectories to reduce acoustic noise. T2*-decay in this spiral acquisition sets an upper limit for the acquisition time so the upper limit of the ramp time needs to be considered. 72 dB(A) for the spiral sequence has been achieved compared to a standard gradient echo sequence with a maximum SPL at 94 dB(A). The Fourier transform of this long smooth ramp corresponds to lower fundamental frequencies when we consider this as input of acoustic noise. With a ramp as long as 5 ms, its fundamental frequency is as low as 200 Hz, which is located away from the peaks of the system’s frequency response spectrum. This is the key to achieving excellent acoustic noise reduction by using a long ramp time.
It is important to discuss the uncertainties involved with our experimental results.
One systematic error arises because data are sometimes taken months apart, giving rise to changes in the frequency peak distribution and amplitude. Upgrades in the scanner
138
components and software as well as “instabilities” in the scanner operation due to
temperature changes can affect the experimental results. There are also changes in laptop
performance; two different laptops were used for recording the sound. However, the
purpose of our experiments is to verify that selected frequency peaks can be cancelled
with designed pulses, including trapezoidal pulses, multiple pulses, and “quadratic”
pulse. Despite day-to-day and month-to-month changes in the scanner and detector
operations, all results still show the expected cancellations to within a few percent
(typically, 2-3%) corresponding to the timings we described in detail in the previous
chapters. This is quite robust, when one takes into account the discrete resolution in
determining the frequency. Note that we took the measurements of these variable flat-top
trapezoidal gradient pulses with fixed 110 ms ramp time, and we changed the flat-top
time in steps of 50 ms . Therefore it is generally not possible to find timings where the
sum of ramp and flat-top time match exactly the frequency peaks in our spectrum. Recall
also that all measurements are multiples of 100 Hz due to the repetition time
TmsR = 10 . Another factor arises from taking a Fourier transform of numerical
calculated cancellation and enhancement data. The FFT peak amplitude of the
cancellation data is 2.6% compared to that of the enhancement data. This tells us that the
FFT peak amplitude cannot directly refer to the amplitude of the vibration after the gradient pulse is applied. Instead, this FFT amplitude stands for an overall effect of the vibration, including the vibration before the follow-up cancelling gradient semi-impulse
139
is applied. Although our experiments show the FFT amplitude to be 2-3 % or more of the
peak to be nullified, it actually confirms that the vibration is cancelled to within 1%.
To understand most of the discussion in this paper, we really only need “springs”
instead of “strings.” The modeling “time-invariant” linear model exemplified by an undamped spring perfectly well explains how specific frequencies are nullified. In the paper by Tomasi and Ernst [9], the dominant vibration frequencies have been modeled usefully in a string picture, including the differences between the (axially anti-symmetric) longitudinal gradient currents and the (axially symmetric) transverse gradient currents.
The symmetries of these different gradients require a spatial description, which we expand upon in the next paragraph. We propose that the general method of using combinations of gradient pulses in all directions for suppressing acoustic noise is more robust than what we have already demonstrated, and a string picture is very useful in exploring new possibilities.
It is recalled that we have used a uniform force profile in the sample calculations.
However, referring back at (3.2), only the n = 1 term survives for the purely transverse
string mode (the symmetric mode) and only the n = 2 term remains for the longitudinal
mode (the anti-symmetric mode). This brings up the interesting point that we expect that
the spatial sinusoidal symmetries of the gradient pulses do not excite the harmonics as
much as a constant force does. This emphasizes the usefulness of the string model in
140
addressing the spatial, as well as the temporal aspects, of the gradient. Although the
frequency cancellations themselves need only the results of linear time-invariant analysis
(for example, the formula in [10] is sufficient for the basic trapezoidal analysis), the
string model informs the understanding of force impulse cancellations even for arbitrary
current profiles over the gradient, and is fertile ground for looking at more ways in which
the gradient vibrations can be damped by gradient pulse interference. While the single mode, n = 1 for transverse or n = 2 for longitudinal, can only be obtained if there are
perfect sinusoidal current distributions, the observation of more frequency peaks at higher frequencies could suggest these are fundamental excitations. A detailed understanding of the mechanical vibrations of a gradient coil and surrounding metallic structures would help in deciding what dominant peaks should be cancelled out.
The utilization of multiple pulses for the cancellation of three or more frequencies requires not only longer sequences in time, but also generates new time scales, which can enhance other frequencies in the output response spectrum. Nevertheless, a rich variety of
possibilities in pulse sequence design, including the TR freedom, may lead to an
optimized solution for the significant suppression of MRI acoustic noise within the constraint of imaging sequences. A key point is that we can use some of the degrees of freedom for noise suppression and some for k-space coverage at the same time. Another key point is that further noise reduction may be achieved by combining our method with active noise control or passive noise shielding such as a vacuum chamber system.
141
6.3 Conclusion
The interfering standing-wave picture, produced by an incident force semi-
impulse (Q impulse) over the string/gradient coil assembly and subsequently damped by
a second force semi-impulse, has motivated this research. It has led to successful
comparisons between theory - analytical and numerical simulations with springs and
strings - and experiment, where specific gradient normal modes have been killed off with
appropriate gradient pulse timings. This includes successful nullification of the vibration
modes excited by eddy currents in the surrounding MRI metal structures.
By measuring the MR system’s frequency response spectrum, we can simulate the
acoustic noise with a known excitation input gradient pulse. This is carried out by
2 applying the formula Pf()= XfHf () () to calculate the power in the frequency
domain for one gradient axis. In reality, if the frequency response spectra of all three
gradient axes can be measured accurately, we can calculate the overall gradient acoustic
noise for the entire gradient system, which is
2 Pf()=++ XfH ()xyz () f YfH () () f ZfH () () f . Thus the powers of different input
pulses can be simulated and they can be compared in dB scale. This algorithm can be
written in Matlab and it can be useful for assessing the estimated noise level for any new
pulse sequence without actually running the scanner. With this modest step, pulse sequence designers can save scanner time and money.
142
Works Cited:
1. Y. Wu, B. A. Chronik, C. Bowen, C. K. Mechefske, B. K. Rutt, Gradient-Induced
Acoustic and Magnetic Field Fluctuations in a 4T Whole-Body MR imager,
Magnetic Resonance in Medicine 44:532-536, 2000.
2. R. A. Hedeen and W. A. Edelstein, Characterization and Prediction of Gradient
Acoustic Noise in MR Imagers, Magnetic Resonance in Medicine 37:7-10, 1997.
3. J. Smink, G-J. Plattel, P. R. Harvey, and P. Limpens, General method for acoustic
noise reduction by avoiding resonance peaks, Proceedings in International Society for
Magnetic Resonance in Medicine 15, p.1088, 2007.
4. S. Schmitter, E. Diesch, M. Amann, A. Kroll, M. Moayer, L. R. Schad, Silent
echo-planar imaging for auditory FMRI, Magnetic Resonance Materials in Physics,
Biology and Medicine 21:317-325, 2008.
5. M. Segbers, C. V. Rizzo S., H. Duifhuis and J. M. Hoogduin, Shaping and Timing
Gradient Pulses to Reduce MRI Acoustic Noise, Magnetic Resonance in Medicine 64:
546-553, 2010.
6. E. M. Haacke, R. W. Brown, M. R. Thompson, and R. Venkatesan, Magnetic
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143
8. C. Oesterle, F. Hennel, and J. Hennig, Quiet imaging with interleaved spiral read-
out, Magnetic Resonance Imaging 19: 1333-1337, 2001.
9. D. Tomasi, T. Ernst, A simple theory for vibration of MRI gradient coils.
Brazilian Journal of Phhysics 36(1A):34-39, 2006.
10. Alan Barnett, Comments on “Gradient-Induced Acoustic and Magnetic Field
Fluctuations in a 4T Whole-Body MR Imager”, Magnetic Resonance in Medicine
46:207, 2001.
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