UNIVERSITY OF SOUTHAMPTON

FACULTY OF ENGINEERING AND APPLIED SCIENCE INSTITUTE OF SOUND AND VIBRATION RESEARCH

NOISE IN MRI SCANNERS

Thomas Peuvrel

A dissertation submitted in partial fulfilment of the requirements for the degree of Master of Science by instructional course.

ABSTRACT

This work investigates the vibration and acoustic behaviour of different MRI scanners devoted to functional Magnetic Resonance Imaging. There is nowadays a need to control the excessive noise produced by this equipment since one of the current medical research objectives is to monitor the response of the human brain to visual or auditory stimuli. Such equipment usually works in high magnetic field (3 Tesla) and as a consequence of its imaging process (EPI), considerable “Lorentz Forces” are produced on the gradient coil system which therefore vibrates and radiates sound intensively. With noise levels reaching a range of 120-130 dB(A), the response of the human brain is therefore disturbed and the safety of the patient could not be guaranteed. The project is concerned with the evaluation of this noise annoyance by measuring sound pressure levels inside the bore of three different scanners working with a 3 Tesla magnetic environment and using Echo-Planar Imaging sequences. This assessment is accompanied by a vibration investigation into the gradient coil systems. An experimental protocol has been developed so as to conduct accurately the measurements. It has been aimed at shedding more insight on the vibration and acoustic behaviour by measuring the response at different positions on the structure. Besides the experimental part of the project, a theoretical overview of the system is presented by means of description of an analytical model of vibration and sound radiation mechanisms. An attempt at comparing the results with the theoretical analysis is made. Conclusions have to be drawn with care since large assumptions have been made so as to facilitate the methodology of the investigation. Despite this limitation the investigation has however revealed important information such as a predicted involvement of particular structural modes in the noise generation. Digital processing of the recorded signals emphasises the frequency characteristics of the dynamic and acoustic responses of the structure to the particular excitation that constitutes the EPI sequence. A strong correlation between the imaging sequence and the induced vibration and noise has been found which suggests the feasibility of appropriate noise control and design enhancement of MRI scanners. This work therefore sets the stage for further studies of this complex system which should then be directed to the achievement of exhaustive FE model or to the development of any function that could describe the coupling between Imaging sequences and structural response of the gradient coil system. This should bring considerable progress to MRI scanner design. Other useful information, coming from further analysis of the recorded signals, such as the Loss Factor, Impulse Response Function, is also given.

ACKNOWLEDGEMENTS

I would like to acknowledge Dr. Matthew Wright for his supervision and guidance during this project and all the people who made this project possible and helped me during the measurements : Mr. John Foster and Prof. Alan Palmer at the Institute of Hearing Research, University of Nottingham ; Dr. Peter Jezzard at the Centre for Functional Magnetic Resonance Imaging of the Brain, University of Oxford ; Dr. Adrian Carpenter at the Wolfson Brain Imaging Center, University of Cambridge. I would also like to thank Dr. Keith Holland, Mr. Rob Stansbridge and Mr. John Fythian at the ISVR for providing with the equipment needed for the investigation. This work is dedicated to my parents who have supported me during all my studies and to my grand parents who have always showed me that life has to be lived simply. A special thanks to Dave and Cedric, my classmates and friends, who have supported me during this year spent in Southampton and who have given a grateful contribution to my project.

TABLE OF CONTENTS

1. INTRODUCTION .......................................................................................................1 1.1 LITERATURE REVIEW............................................................................................2 1.1.1 Magnetic Resonance Imaging: Overview and System Architecture. ..........................3 1.1.1.1 Magnetic Resonance Imaging - Definition. ...............................................3 1.1.1.2 System architecture. ...................................................................................3 1.1.1.3 Characteristics of subsystems involved in the generation of noise. ..........4 1.1.2 Measurement of Noise Levels ....................................................................................8 1.1.2.1 Results found in the literature ....................................................................8 1.1.2.2 Noise generation process ...........................................................................9 1.1.2.3 Noise measurement procedures ...............................................................10 1.1.3 Issues: Noise and Vibration Control, Acoustic Modelling and Design of MRI Scanners ..............................................................................................................................11 2. THEORY .....................................................................................................................14 2.1 STRUCTURAL VIBRATION ..................................................................................14 2.1.1 Free Vibration of Circular Cylindrical Shells ..........................................................14 2.1.2 Natural Frequency and Mode Shape ........................................................................18 2.2 WAVE PROPAGATION ..........................................................................................20 2.2.1 Wave propagation in Thin-Wall Circular Cylindrical Shells ...................................20 2.2.2 Waves in infinite flat surface ....................................................................................21 2.2.3 Flexural Wave Propagation in a Circular Cylindrical Shells ..................................23

2.3 SOUND RADIATION ................................................................................................24 2.3.1 Sound Radiation from Circular Cylindrical Shells....................................................24 2.3.2 Acoustic Modes: Cross-Modes in Circular Cross Section Ducts ..............................28

2.3.3 Coupling between Shell Modes and Acoustic Duct Modes.......................................32

3. EXPERIMENTAL PROCEDURE FOR VIBRATION AND ACOUSTIC MEASUREMENTS ON MRI SCANNERS....................................................................35 3.1 PRESENTATION OF THE SCANNERS AND THEIR IMAGING SETTINGS 35 3.1.1 A few words about functional Magnetic Resonance Imaging Scanners ...................35 3.1.2 The Echo-Planar Imaging sequences .......................................................................36 a. MRC-IHR Nottingham fMRI Scanner EPI sequence .....................................37 b. FMRIB Oxford fMRI Scanner EPI sequence .................................................39 c. WBIC Cambridge fMRI Scanner....................................................................40

3.1.3 Presentation of the Scanners and the design of their relative Gradient coil system. ..41 a. MRC-IHR Nottingham fMRI Scanner system..................................................41 b. FMRIB Oxford fMRI Scanner system..............................................................43 c.WBIC Cambridge fMRI Scanner system...........................................................45 3.2 EXPERIMENTAL PROTOCOLS...........................................................................46 3.2.1 MRC-IHR Nottingham Acoustic Measurements .......................................................46 3.2.2 MRC-IHR Nottingham Vibration Measurements......................................................48 3.2.3 FMRIB Oxford Acoustic and Vibration Measurements............................................51 3.2.4 WBIC Cambridge Acoustic and Vibration Measurements .......................................54 4. RESULTS OF THE MEASUREMENTS ...................................................................56 4.1 RESULTS GIVEN BY THE EXPERIMENTAL ARRANGEMENTS. ................56 4.1.1 Acoustic cap arrangement and Background noise ....................................................56 4.1.2 “Noise floor” arrangement for the vibration measurement ......................................56 4.2 ACOUSTIC MEASUREMENTS ..............................................................................58 4.2.1 MRC-IHR Nottingham scanner..................................................................................58 4.2.2 FMRIB Oxford scanner..............................................................................................60 4.2.3 WBIC Cambridge scanner .........................................................................................62 4.2.4 General comments on the noise measured.................................................................65 4.3 VIBRATION MEASUREMENTS ............................................................................65 4.3.1 Accelerations and displacements of the vibration - Relationship.............................65 4.3.2 Vibration Results.......................................................................................................65 a. MRC-IHR Nottingham scanner ........................................................................66 b. FMRIB Oxford scanner ....................................................................................69 c. WBIC Cambridge scanner ................................................................................76

4.3.3 General comments on the vibration measured. ........................................................82 5. FURTHER INVESTIGATION AND ANALYSIS....................................................84 5.1 CORRELATION BETWEEN IMAGING SEQUENCE, ACOUSTIC AND VIBRATION RESPONSES. ............................................................................................84 5.2 MODAL ANALYSIS. .................................................................................................87 5.2.1 Computation of the structural and acoustic modes frequencies................................87 5.2.2 Discussion on the validity of the model and the results obtained..............................88 5.3 FURTHER INVESTIGATION. ................................................................................89 5.3.1 Damping of the Gradient coil systems. ......................................................................89 5.3.2 Further investigation with Pure Tones and Impulses. ...............................................97 6. CONCLUSION............................................................................................................101 6.1 NOISE AND VIBRATION CONTROL, OVERVIEW AND PROPOSAL FOR FURTHER INVESTIGATION OF THE PROBLEM. ...............................................101 6.2 CONCLUSION..........................................................................................................103 REFERENCES ...............................................................................................................104 NOTE: The appendixes are presented in the Technical Memorandum of the project. The measurement tables are included in this Memo as well as references of measurements made on the scanners, recorded on DAT tapes and edited in .wav files on CDs.

1. INTRODUCTION In medical diagnostics practice, the Magnetic Resonance Imaging scanner plays an increasingly important role. MRI has become a widely used clinical imaging system capable of yielding tomographic images of excellent spatial resolution. The development of MRI hardware and sequence design continues at a fast pace and the last decade has seen the use of MRI to image brain function (functional MRI process). However, one major problem in magnetic resonance imaging equipment is the high level of noise produced during the scanning process (up to 90-140 dB(A)). In the future, it is expected that new, faster scanning techniques, especially those used for functional neuro-imaging (Echo-Planar Imaging, Echo-Volumnar Imaging), will increase the noise problem. Different surveys have shown that the noise seems to be produced by the strong vibration of the so-called gradient coil system. The investigation of the noise in MRI Scanners has therefore focused on two major points ; the strength and nature of vibration of the gradient coil system and the resulting acoustic noise produced inside the bore where the patient’s head should be positioned. The three scanners assessed during this project are all used for functional MRI, but nevertheless differ by their geometry, structure and the design of the imaging pulse sequence. It is therefore difficult to relate all the measurements and to draw generalised conclusions about the structural and acoustic behaviour of scanner coil systems. However, it is possible to compare the "correlation" between the electrical excitation pulse applied to the gradient coils and the induced vibration and/or acoustic responses. The investigation of three gradient coil systems revealed some important characteristics by means of digital processing of recorded signals. Nevertheless, it is also important to set the results in theoretical context. Theoretical studies of the Gradient coil behaviour are therefore presented during the dissertation in both structural and acoustic domains. They therefore set the foundation for any further analysis concerning any noise control possibilities. The dynamic complexity of gradient coil systems has required important assumptions. The system can be assumed to be an axisymetric cylindrical shell. Even though, the theoretical analysis is still subject not to match with the measured behaviour of the system. Modelling then appears to be another alternative, usually providing accurate prediction of noise production of MRI Scanners. However, modelling the gradient coil system involves a large amount of important consideration (such as thickness of layers, geometry, physical properties of each constitutive material) which were unknown most of the time, and therefore made modelling works beyond the scope of this investigation. Nevertheless, the description of the theoretical models aims to provide an overview of the physics behind the sound radiation process and aims to introduce the concept of mode shapes, natural frequencies and cut-on frequencies which appear to have a consequent importance in the noise generation mechanism. The following literature review surveys the noise investigations of MRI scanners made during the last fifteen years and underlines the important research currently made in the domain of vibration control of the gradient coil system. Some works will be more deeply examined in this dissertation as they are in the logical progression of the theory

already reported and also as they represent realistic improvement in noise control of MRI Scanners. These researches usually involves analytical and/or modelling works. They will then constitute an important background for the critical analysis of the three Scanners’ behaviours. The literature review also gives an overview of the Magnetic Resonance Imaging process and all the parameters that constitute important references in the investigation. Finally, during the project, a particular investigation was made at the feasibility of the measurements inside this hostile environment for the equipment and transducers that constitutes the 3 Tesla high magnetic field. An experimental protocol has then been developed for each scanner measurement session in order to set up correctly the measurement procedure but also to avoid any artefactual signal induced by the high magnetic strength. These protocols will also be described in the dissertation as they represented a valuable contribution to the project in itself. 1.1 LITERATURE REVIEW. Different surveys have shown that the noise seems to be produced by the strong vibration of the so-called gradient coil system. More precisely, the MRI process involves current flowing through the gradient coils that are rapidly switched in a complex pattern and, since they do so within a large magnetic field, Lorentz forces deform the gradient coils. These movements translate into intense acoustic noise. The literature on “noise in MRI scanners”, over the last fifteen years, underlines the important issue of assessing and controlling the noise level. The calculation of safe noise dosage was first the main objective of the acoustic surveys as they aimed at estimate the potential “hearing loss” resulting from the MR Imaging. The problem of the influence of noise levels on medical diagnostic came with the development of functional MRI. In fact, functional imaging aims at investigating accurately the response of the brain to sensory stimuli (for instance, auditive stimuli, lights) and image changes occurring during motor tasks. High noise levels created by the MRI system therefore represent a source of annoyance for medical investigations. This problem could be tackled by designing effective noise cancellation system based on adaptive or active sound cancellation or by controlling noise at source with MRI system design efforts. Important assessment of the acoustics of fMRI scanners have been made by the Magnetic Resonance Center (MRC) and the Institute of Hearing Research (IHR) in Nottingham (U.K.). The following literature review will aim to give an introductory review of the previous works on noise problem in MRI scanners in order to set up the background to further acoustic improvements on the fMRI scanners studied during this project. This literature review will provide a brief description of the system architecture of MRI scanners which is essential to understand the parameters influencing the noise level inside the bore of the scanner. Measurements of noise levels made on different type of scanners during the last fifteen years will be reviewed so as to give us a quantitative and qualitative appraisal of the problem. Procedures and techniques of measurement will also be discussed. Finally, the current literature provides different solutions to noise issues in gradient-coil system by developing mathematical model for better design or by elaborating noise cancellation systems which have already been designed and installed on both routine and functional MRI scanners. They involve combination of active and passive techniques, adaptive noise cancellation system or active control of sound power radiated from the cylinder shell. Theses techniques described in the literature will be commented and will give an idea of the issues and the developments to consider upon.

1.1.1 Magnetic Resonance Imaging Overview and System Architecture. 1.1.1.1 Magnetic Resonance Imaging - Definition. [1] MRI stems from the application of nuclear magnetic resonance (NMR) to radiological imaging. The adjective ‘Magnetic’ refers to the use of an assortment of magnetic fields and ‘resonance’ refers to the need to match the radio-frequency of an oscillating magnetic field to the ‘precessional’ frequency of the spin of some nucleus (hence the ‘nuclear’) in a tissue molecule. More precisely, the spin of a hydrogen nucleus (proton) in a magnetic field “precesses” (circular motion of the axis of rotation of a spinning body about another fixed axis ) about that field at the “Lamor frequency” which, in turn, depends linearly on the magnitude of the field itself. The idea of imaging is simple ; If a spatially varying magnetic field is introduced across the object, the Lamor frequencies are also spatially varying. Then, the different frequency components of the signal could be separated to give spatial information about the object. 1.1.1.2 System architecture.[2] Even if MR Imaging is nowadays executed using a large amount of different imaging techniques, we can still sketch a global system architecture ; Firstly, the patient support which brings the patient in the region where the magnetic field is homogeneous. In most case the static field magnet has a cylindrical bore and its center is called “isocenter” (homogeneous field). The magnet is situated in a RF-screened room (radio-frequency) to avoid spurious signals from the surroundings (from foreign transmitters or even from the driving electronics of the system itself). The system must be told the required scan, including geometrical parameters, the imaging method, and the sequence via the user interface (host computer). For new imaging methods the gradient field strength as a function of time for each direction and the RF waveform must be entered into the system and written in the memory of the host computer. All this information are downloaded to the “spectrometer”, which consists of the “front head controller” (controlling the magnet, gradients, RF transmitter and receiver, RF coils switches, etc.) and the data acquisition around the receiver. The Magnetic Resonance Imaging mechanism is described below. There is first an “initialisation phase” where the RF receiving coils are tuned. After this phase, the sequence can start switching on the selection gradient. All other components (except of course the magnet) are switched off. When the selection gradient reaches its required value, the RF power amplifier is activated. The input signal of the power amplifier is a harmonic signal with frequency ω0. This signal is modulated in the waveform generator and fed into the power amplifier. The modulated RF signal drives the transmitted coil so that the desired RF magnetic field, BRF , is emitted. When the RF pulse is ready the output line of the power amplifier is switched to the matched load. Also the selection gradient is now switched off. The result of these actions is that the spins in a single slice are excited. Now the dephasing gradient and the phase-encode gradient are

switched on to the strength required by the spectrometer. When these gradient pulses have their required surface, they are switched off and the focusing pulse can now be applied. This 180° RF pulse is usually slice selective and applies to the same slice as the excitation pulse. After the focusing pulse, the read-out gradient is switched on. During this gradient pulse the receiver coil is in a tuned state and the receiver is activated. The magnetisation is measured during the time the read-out gradient is switched on and the ADC samples the received signal at a certain point in time ts. The samples are sent to the array processor, the circuit that performs the Fast Fourier Transform, and the image results from this transformation, which is sent to the viewing section. Figure 1 shows a simple schematic front view and cross-section of an MRI scanner. A more precise representation of the system architecture of MRI scanner is given by Figure 2.

Figure 1: MRI Scanner From Kuijpers A.H. Acoustic modeling and design of MRI scanners. [16]

1.1.1.3 Characteristics of subsystems involved in the generation of noise.[3] As the gradient coils system is the source of the acoustic noise, it appears important to described the technical characteristics of the gradient system and the MR Imaging parameters leading the RF excitation pulse which therefore drive the intensity of the Lorentzian forces applied on the coils. The design of the Magnet, RF coils (transmit and receive coils), gradient coil system, magnetic field-gradients (Slice select, Phase-encoding and Read-out gradients), pulse sequence (TR, TE, FOV parameters) determine the image quality and clinical utility but also the levels of the unwanted noise. Magnet design. The magnet is the heart of the MRI system. Magnet manufacturers tend not to publish details of the technical properties in the MRI literature. However, the following trends

have been obvious to users from the early 1980s to the present : (1) an increase in the static field strength used, from 0.15 T to 1.0 and 1.5 T in most new European and North American installations ; (2) a trend away from resistive magnets to high-field superconducting systems, and low-field open imagers using innovative permanent magnets and iron-cored electromagnets ; (3) significant improvements in static field homogeneity and temporal stability ; (4) the introduction of self-shielding superconducting designs to reduce the area affected by stray fields, without the need for massive iron shielding. Note for (1) : the interest in higher fields stems from the fact that the signal to noise ratio (SNR for the Imaging process) increases with the field strength. Magnet design innovations have contributed both to improved image quality and to reduced installation costs. Radio-frequency coils. Good homogeneity is desirable for both transmitting and receiving coils. Good RF receive coils should also give high signal to noise ratio (in imaging), while transmit coils should minimize RF power deposition. The principal innovation since the 1980s was the Circularly Polarised RF magnetic field design. Appropriate coil choice is essential for the right balance between Field-of-View (FOV) and Signal to Noise Ratio.

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Figure 2: Schematic diagram of a MRI scanner.

The Gradient coil system. Description. The gradient coils are magnets used to make deliberate variations in the strength of the main magnetic field. There are three gradient magnets, one in each of the X, Y, and Z planes. The arrangement of the magnets means that each individual point within the MRI scanner is subject to a slightly different, but known, magnetic field. The “readout” gradient coil is namely the X gradient coil and covers the X-plane which runs left to right through the patient. The “Phase encoding” gradient coil is namely the Y gradient coil and covers the Y-plane which is the plane at 90-degrees to the ground and running through the patient from the front to the back. The “Slice select” gradient coil is namely the Z gradient coil and covers the Z-plane which is running through the bore and longitudinally through the patient. The gradient coils are firstly required to produce a linear variation in field along one direction. Linear variation in field along the direction of the field (labelled z-axis) is usually produced by a Maxwell coil. This consists of a pair of coils separated by 1.73 times their radius as shown in Figure 3. Current flows in the opposite sense in the two coils, and produces a very linear gradient. To produce a linear gradient in the other two axes requires wires running along the bore of the magnet. This is best done using a saddle-coil, such as the coils shown in Figure 3. This consists of four saddles running along the bore of the magnet which produces a linear variation in Bz along the x or y axis, depending on the axial orientation. This configuration produces a very linear field at the central plane, but this linearity is lost rapidly away from this. In order to improve this, a number of pairs can be used which have different axial separations.

Bz

“Maxwell” Z gradient

Y gradient X gradient coil

Figure 3 : Schematic diagram of the Gradient coil system

If a gradient is required in an axis which is not along x, y or z, then this is achievable by sending currents in the appropriate proportions to Gx, Gy and Gz coils. If for example a transverse gradient G at an angle θ to the X-axis is required, then a gradient Gcosθ should be applied in the x direction, and Gsinθ in y. The first layer (inner surface) of the Gradient coil system usually includes the X gradient coil (Gx) and the second layer (outer surface), the Y gradient coil (Gy). These two parts of the system are gathered together usually with epoxy resin so as to give cohesion to the system but also to reduce the vibration occuring when the applied currents are driving the gradient coils. Characteristics and Requirements. Good gradient coils should be highly linear over a large Field-of-View, able to switch gradient fields on and off extremely rapidly and induce virtually no eddy currents in the large metal structures which enclose them. Rapid imaging methods such as Echo-Planar Imaging (EPI) and fast spin-echo methods are more efficiently performed in systems with low gradient switching times. The eddy currents is an other source of annoyance but in the field of imaging. They are set up in the superconducting magnet windings by the changing magnetic flux leaking from the gradient coils during switching. They cause temporal distortions of the desired gradient waveforms. They are particularly bothersome when performing EPI and other rapid imaging methods. Even if this eddy currents are not directly source of noise in the gradient system, it is worth to notice this parallel induced-effect which may or not interfere on Lorentzian force, but nonetheless it made the gradient coils to be actively shielded so that the magnetic flux did not escape from the inner volume of the cylinder former. This resulted in the design of shielded gradient coil reducing therefore the space available in the system and made this hostile environment to leave very little space for extra acoustic shielding. Pulse Sequence design. Pulse Sequence design is a important parameter of noise generation in MRI scanner. Gradient-echo sequence (GRE), Spin-echo sequence (SE), Echo-planar Imaging (EPI) have different settings but they all create very high sound-pressure levels from stray electromechanical radiation. Additionally, in the near future, Echo-volumnar Imaging (EVI) will be pursed to acquire more information per unit time that will exacerbate the noise problem at source. (Note on the meaning of the word “echo”: in order to form an image, a number of Nuclear Magnetic Resonance signals, or echoes, are required.) Echo-planar imaging (EPI) is an extremely rapid method, gathering all the data to form an image after a single excitation pulse (Ultra-fast single-shot methods). Gradient-echo sequence belongs to the «Short Time-Repetition methods» where sequences collect only one gradient echo per RF excitation but achieved rapid scan times by greatly shortening TR. The Spin-echo sequences are using a number of excitations. Each echo of the echo train is allocated to traversing a particular segment and the pulse sequence is repeated until the echo has collected all the lines in its allocated segment. Spin-echo sequences are often used in shielded magnetic field gradient. GRE an SE sequences are usually found in routine MRI scanners (1.5 T) and EPI method is more devoted to functional RMI scanner (3T). Nevertheless these methods are all characterized by three main parameters which

drive the pulse and therefore the induced-noise level: TR = Time Repetition of the pulse; TE = Echo Time of the pulse; FOV = wanted Field-Of-View. These three parameters have to be taken into account during any noise measurement. They describe the pulse sequence and therefore influence the noise level inside the bore of the scanner. 1.1.2 Measurement of Noise Levels. 1.1.2.1 Results found in the literature. Acoustic surveys about noise in MRI scanners have been made during the last ten years. Hurwitz R. et al (1989, [4]) has made in 1989 an acoustic analysis of the gradient-coil noise in MR Imaging system of 0.35-1.5 T. Noise levels were measured at bore isocenter during a variety of imaging sequences in six MR Imaging systems. Measured noise ranged from 82 to 93 dB on the A-weighted scale and from 84 to 103 dB on the linear scale. McJury M.J. et al (1993 [5], 1995 [6]) made measurements in 1993 of the acoustic noise levels generated on two high field MRI system (1.0 T and 1.5 T) and latter, in 1995, has compared the results between a standard an upgrade gradient sytem. Shellock F.G. et al (1994, [7]) assessed acoustic noise levels occurring in MRI scanners six “worst case” pulse sequences. The highest noise levels during MR imaging occurred during the use of a gradient-echo (GRE) pulse sequence and was 102 dB at the entrance and the exit of the magnet bore and 103 dB at the center (A-weighted scale). Shellock F.G. et al.(1998, [8]) has also reported that scanner noise is as high as 115 dB(A) during Echo-planar imaging at 1.5 T. Very high-field scanners (3T and 4T) are becoming more commonly used for functional MRI and there is an evidence that the sound level is even greater than at 1.5 T. Foster J.R. et al. (1999, [9]) found that 3 T MR scanner generates EPI acoustic noise in the region of 122 to 131 SPL (123 to 132 dB(A)) measured at the position that would be occupied by the patient’s head. These results found in the literature have been listed in the following table according to the strength of the static magnetic field and the parameters of the pulse applied to the gradient coil system.

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0.35 Gradient-Echo sequence 60 20 24 5 93 1030.5 Gradient-Echo sequence 60 20 24 5 85 841 Gradient-Echo sequence 60 20 24 5 82 901 FLASH 2D - GRE based 17 5 21 8 97 1031 Spin-Echo sequence 30 10 7.5 3 93 1031 Spin-Echo sequence 30 15 10 2 92 102

1.5 Gradient-Echo sequence 60 20 24 5 91 941.5 Gradient-Echo sequence 60 20 24 5 84 1001.5 FLASH - GRE based 17 5 21 6 102.5 114.51.5 GRE steady state GRASS std 12.6 3.6 16 3 98 99 ( C-Weighted)1.5 FLASH - GRE based 17 5 50 20 91.5 103.51.5 Spin-Echo sequence 60 5 24 3 unknown 951.5 Spin-Echo sequence 30 15 5 2 97 1071.5 SE High resolution 80 20.3 6 2 90 98 ( C-Weighted)1.5 Echo-planar Imaging unknown unknown unknown unknown 115 unknown

These results have also been plotted so as to figure out the probable effect of the static magnetic field on the noise levels measured inside the bore of different MRI scanners. This roughly confirms the tendency of the induced-noise to grow with the strength of the magnetic field.

Noise Levels measured at the isocenter of MRI scanners

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Moreover, the cited acoustic studies revealed trends concerning the behavior of the noise generation during the different pulse sequences. It has been found that noise levels at bore isocenter increased during sequences employing thinner section thickness, shorter repetition time (TR) and echo time (TE) and large field of view (FOV). One obvious point was that Gradient-echo sequences produce the largest acoustic noise. Moreover, the current trend toward faster imaging techniques and thinner sections will likely generate even greater noise levels. 1.1.2.2 Noise generation process. During imaging, magnetic flux gradients are repeatedly switched on and off at intervals of approximately 5-10 msec. From first principles, current passing through a wire placed in a magnetic field will generate a force orthogonal to the direction of both the field and the current. All the references in the literature have proposed the cause of gradient-coil noise to be pulsating “Lorentz” forces. The induced noise would be caused by the impact of the gradient coil against their mountings during the pulse sequences. Figure 4 gives a simplified explanation of gradient-coil “knocking”. The gradient coil is shown as a single winding for illustrative purposes. When the gradient coil is pulsed by current I, a brief force is created in a radial direction.

Figure 4. Gradient coil noise is therefore complex to assess and some of the literature references suggest procedures and techniques for best measurements. 1.1.2.3 Noise measurement procedures. The first priority of Hurwitz R.et al. study (1989, [4]) was to verify the reliability of the sound-pressure detection instruments. Neither high magnetic flux nor RF pulses interfered with the measurement of ambient of MR imaging gradient-coil noise. Neither the orientation of the microphone nor the plane of imaging was found to be relevant: the magnetic flux had no lasting effect on the acoustic device. But there are still some precautions to take in the layout when measuring MRI noise. Hurwitz R. et al. (1989, [4]) have mounted the microphone on a small plastic stand positioned so that the microphone diaphragm was at the isocenter of the magnet bore and parallel to the static magnetic field. The microphone was connected to the sound level meter with a 30 meters extension cable (24 meters for Shellock F.G. et al (1994, [7])). Measurements in magnetic fields need the equipment to be tested at first. Referring to Hurwitz R. et al recommendation, after instrument calibration, tests have to be conducted in order to exclude any effects of either the static magnetic field or radio-frequency pulse on the function of the system consisting of the microphone, extension cable, and preamplifier. Usually, the sound meter and recording equipment are placed outside the shielded MR system room and beyond a 3 Gauss (0.3 mT) fringe field of the static magnetic field for Hurwitz R.et al, 2 Gauss (0.2 mT) for Shellock F.G. et al and 100 Gauss (10mT) for Foster J.R. et al. In the experiment conducted by McJury M.J. et al (1993 [5], 1995 [6]), the meter used was partially sensitive to the main field of the magnet and so was positioned beyond the 20 G (2mT) boundary at the front of the magnet bore. But a particular aspect of the procedure was that a section of non-magnetic tubing was connected to the built-in condenser microphone to channel noise from the magnetic isocenter. Measurements were made to calibrate the meter and the tubing combination with respect to the frequency dependence and attenuation of sound. Errors resulting from the frequency dependence of the tube were found to be minimal and a simple single value of correction over the frequency range was used. Equipment: requirements, type and settings. For the accuracy of the acoustic noise measurements, the equipment must be magnetically transparent and reflect air-borne sound pressure rather than artefactually induced signals.

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Bo

I

In most of cited surveys, the equipment used for sound-level measurement and calibration was manufactured by Bruel & Kjaer (Denmark): Sound level Meter B&K of type 2230 [4], 2204 [6], 2235 [9] ; Condenser Microphone B&K of type 4155 [4], 4165 [9]. Concerning the effect of magnetic field, B&K gives the following guidance: “When measurements are to be carried out in strong magnetic field, the influence of the field should be checked using a pistonphone, calibrator or silica gel cap to acoustically shield the microphone. The meter should then drop by 10 or 15 dB if the effect of due to the magnetic field is to be assumed negligible. If the damping achieved is considerable less than this and moving the microphone results in a variation in the meter reading, the magnetic field will affect the measurement accuracy”. Surveys follow this guidance. Concretely, Foster J.R. et al have noticed that the acoustic cap attenuated the level of the scanner sound by 24 dB or more. Thus, interference artefacts made a negligible contribution to the measures. Concerning the sound level meter, it has to be set with special application in mind. The surveys (Hurwitz R.et al.[4]; McJury M.J. et al [5] [6]; Foster J.R. et al.[9]) report measurement made in both Linear (dB SPL) and A-weighted (dB(A)) scale with the “Fast” root-mean-square (RMS) time constant (125 ms). Sometimes levels were measured using the “peak” characteristic (<100µs) which tends to give an higher measured level related to short-term peaks in the signal. Choosing the weighting network can be also an important issue when the aim of the measurement is to quantify the noise level regarding to the human ear sensitivity. Originally, four different weighting curves (A, B, C and D) were made to reflect the fact that the human hearing has a level-dependent frequency dependence. Most of the surveys have reported levels in dB(lin) and dB(A). Nevertheless, considering the high sound pressure levels reported previously inside the bore of MRI scanner, the use of “C” weighting network appears to be more appropriate to the investigation. Indeed, the C-curve is designed to follow approximately the equal loudness curve of 100 phons (i.e. the noise is highly perceived), and therefore the “C” weighting network is used for measuring SPLs of 85 dB or more (“A” weighting is used in the range of 20 to 55 dB). However, the “A” weighting network is more widespread in the surveys and used for standardisation. dB(A) can still be employed independently of the noise levels and its difference with C-weighted levels can actually provide direct information about the spectral properties of the sound measured. 1.1.3 Issues: Noise and Vibration Control, Acoustic Modelling and Design of MRI Scanners. At first, in MRI scanners, sound attenuation can be achieved by means of energy absorbing materials place on the surface of the radiating structure. For instance, passive noise reduction method have been applied on the fMRI scanner studied by Foster J.R. et al.[9] by installing a lining of acoustic foam around the gradient. More precisely, the foam was positioned between the shim coils and the gradient coils with the front-facing foam placed towards to the gradient coils (i.e. towards the noise source). This reduced the scanner sound level at the centre of the bore by 10 dB (from 122 to 112 dB SPL). No additional attenuation was gained by placing a foam lining between the gradient coil and the RF coil, nor at the head and foot of the scanner bore. The passive technique works quite well for sounds of medium and high frequencies but is very inefficient at low frequencies because the thickness of the absorbing material necessarily increases with decreasing frequency.

Active noise control is an alternative to this passive strategy where destructive interference is used to reduce the sound pressure field. Naghshineh K.et al (1998 [10]) proposed a more advanced form of this technique that could be applied on the MRI bore. This technique, named ASAC (Active Structural Acoustic Control) is based on altering the vibrations of a noisy structure such that it radiates very little sound. This alteration is achieved by introducing discrete force actuators at select points on the cylindrical shell. These actuators are driven via digital controller that receives the noise or vibration signal from a sensor array located on the surface of the structure. This technique is successful in controlling the sound power radiated from the cylindrical shell. In a paper made by Qiu Jinhao et al (1995, [11]), the idea of controlling the vibration of the shell excited by the Lorentz force has been developped. Simulation on the vibration control of the shell were carried out by using piezoelectric actuators which produced a bending moment or an in plane force when pulse voltages were applied synchronously with the pulse current of the coils. The simulations showed that the vibration level was successfully reduced in the frequency range of 400Hz-1200Hz. In 1995 also, numerical and experimental studies of the vibration and acoustic behaviors of MRI gradient tube were conducted by Ling J.X. et al (1995, [12]) where a 3-dimensional finite element model was developed to study the vibration and acoustic characteristics of the tube using ANSYS (Finite Element Analysis code). Linear transient analysis as well as the coupling dynamic and acoustic analysis has been executed. The linear transient analysis, which deals with the dynamic response of the cylindrical shell under periodic loading, results in the displacement and acceleration times histories of the tube. The coupling analysis predicts the air motion (then the sound wave) on the patient side of the tube. All the numerical results were in good agreement with the experimental data and provided valuable information for evolving the gradient design. Consequently, an optimal design of gradient system has been achieved and provided a 12-15 dBA noise level attenuation compared with previous design. Recently, Mechefske C.K. et al. (2000, [13]) have shown that the sound pressure distributions along the centerline of a MRI gradient coil cylinder obtained by both simulation and measurement are in close agreement. This suggests that simulation and measurements can correlate well and then FE models can particularly be developed so as to perform accurate acoustic simulation and analysis of the Gradient coil system. FE based simulation method should therefore lead to gradient design improvement. Finite Element Method (FEM) or Boundary Element Method (BEM) in the design process of complex structures are more and more employed and will increase with the computational power of the modern computer and maturation of the software. However this trend to use FEM and BEM in the design process has been discussed by Kuijpers A. et al (1998, [14]) who affirm that this methods have a major drawback : they do not give the insight as they do not relate directly design changes to acoustic response changes. Kuijpers A. et al (1998, [14]) and Kessels P.H.L.(1999, [15]) propose that the problem can be tackled by the development of problem domain specific tools. In these tools the geometry of the acoustic domain is restricted, but this simplification a much better understanding of the acoustical phenomena in that domain is gained. The benefits of that approach is presented in Kuijpers A. PhD thesis (1999, [15]) with the development of a mathematical model for the acoustic radiation of a MRI scanner. The model for the noise production of MRI scanners can be subdivided in 2 parts : a structural part which deals with the generation of structural vibrations due to Lorentz forces excitations and with the structural-acoustic optimization techniques (Kessels P.H.L. companion doctoral thesis 1999, [15]) ; and an acoustical part describing the

transformation of the structural vibrations into audible noise (Kuijpers PhD thesis 1999, [16]). Kuijpers affirms that an adequate acoustic model for low-noise design of MRI scanner and its gradient coil must incorporate 2 aspects : the specific geometry of the scanner and its complex boundary conditions. As a first approximation, the geometry of the scanner can be modelled as being axisymetric. In his thesis, Kuijpers proposes the model of a finite duct with opens ends mounted with infinite flanges. This way, it is possible to use duct acoustics theory and achieve increased insight. Moreover, a semi-analytical formulation for the acoustic radiation of the model raised by the author (by using a Fourier transformation technique for the “Helmotz” equation and the velocity boundary condition’s at the duct’s wall) leads to the development of accurate and efficient numerical tools. Secondly, the boundary conditions can be approximated by assuming that the gradient coil (assimilated as walled duct) can vibrate and thus radiate acoustic energy. The 3-dimensional Boundary Element Method was not feasible because of the huge computational requirements. Nevertheless, this was accomplished by utilising the so-called Fourier-BEM method, which exploits the axisymmetric properties of the geometry without axisymmetric boundary conditions, and hence reduces the dimensionality of the problem by one. The parameter studies, which is another important point of the thesis, revealed that the radiated sound power and the sound pressure level in the MRI bore respond similarly to design changes, in contrast with the velocity level. This means that the radiated power is an appropriate design objective function, because it is directly related to the noise that is experienced. Additionally, it enables the use of the radiation modes reduction technique which is employed by the author for the modal description of the acoustics. Finally, the physical and computational characteristics of the model, determined by parameter study and by comparison with Fourier Boundary Element Method (BEM) models, led the application of the mathematical model in the design process to be viable.

x

y

z

r a

θ

Figure 5: Cylindrical Co-ordinates System.

u

w v

2. THEORY. A satisfying theoretical model of the gradient coil system structure is difficult to find. In fact, the complexity of the system, its non-homogeneous layout and its usual dimensions do not permit a total comparison to any analytical model that could seem to be suitable and that could have been developed previously in the literature. It is however obvious that the gradient coil system has a form approximating a circular cylindrical shell. Nevertheless, most of the simple analysis of the dynamic and acoustic behaviour of cylindrical shell were done on shells with thin-wall. It is then a dangerous assumption to compare the gradient coil system to these analytical models, but this gives nevertheless meaningful insight of what could generally occur to the system in terms of vibration or sound propagation. Moreover, this theoretical analysis is mainly justified by the fact that it gives an overview of the problem and describes the parameters on which a possible control development can be based. There is not general agreement in the literature on the linear differential equations which describe the deformation of the shell. A number of theories have arisen and are used. The differences among the theories are due to various assumption made about the form of small terms and the order of terms which are retained in the analysis. The Donnell (1938) and Mushtari (1938) shell theories are the simplest of these theories. The Flügge (1934) and Sanders (1959) shell theories are generally felt to be the most accurate. Nevertheless, over broad ranges of parameters of engineering importance, these theories yield similar results. Each of the shell theories mentioned above describes the motion of the shell in terms of an eighth-order differential equation. Only the Flügge and Donnell theories will be developed in the theoretical part. The Flügge shell theory is indeed much described in the litterature, but is also the foundation of a vibration modes control of cylindrical shell developed by Qiu J. and Tani J. (1995, [11]) which will be commented upon. Donnell shell theory will additionally give simple but nonetheless meaningful prediction of natural frequencies of the studied structures assimilated to cylindrical shells. 2.1 STRUCTURAL VIBRATION. 2.1.1 Free Vibration of Circular Cylindrical Shells. [17]

Let the displacement components of the mid-lane of the shell be u, v, and w in the axial, tangential and radial directions. the motion of the shell is described by the following Flügge's shell equations (Flügge 1960) :

( ) 0)1(

241

12

21

121

21

2

22

2

3

3

2

3

32

2

2

2

2

2

22

2

=∂∂ν−ρ

−

∂θ∂∂ν−

+∂∂

−∂∂ν

+∂θ∂

∂ν++

∂θ∂

+

ν−

+∂∂

tu

Ew

zRh

zRh

zR

zv

Ru

Rh

Rz

s

0)1(

24)3(1

41

211

21

2

22

2

3

2

2

2

2

2

2

2

2

2

2

2

=∂∂ν−ρ

−

∂θ∂∂ν−

−∂θ∂

+

∂∂

+

ν−+

∂θ∂

+∂θ∂

∂ν+

tv

Ew

zRh

R

vzR

hRz

uR

s

0)1(61261212

1

124

)3(24

)1(12

2

22

2

2

4

2

4

4

4

2

22

4

2

2

4

42

4

2

2

22

3

2

2

2

3

3

2

3

32

=∂∂ν−ρ

+

θ∂∂

+θ∂∂

+θ∂∂

∂+

∂∂

++

+

θ∂∂

+∂θ∂∂ν−

−+

θ∂∂

∂ν−+

∂∂ν

+∂∂

−

tw

Ew

Rh

Rh

zRh

zh

Rh

R

vRzR

huzR

hzRzR

h

s

where r, θ, and z are cylindrical coordinates, t is the time, and the physical characteristics of the shells are defined by the mean radius R, wall thickness h, density ρs, Young’s modulus E, and Poisson’s ratio ν. A traveling wave solution is sought for the shell:

]/)(2exp[.cos.

]/)(2exp[.sin.

]/)(2exp[.cos.

lztcinww

lztcinvv

lztcinuu

p

p

p

−πθ=

−πθ=

−πθ=

where u is the deformation parallel to cylinder axis, v is the circumferential deformation and w is the radial deformation. u , v , and w are arbitrary constants to be determined from the equations of motion. Substitution of equations (2.2) into equations (2.1) gives three linear algebraic homogeneous equations for u , v , and w :

=

000

333231

232221

131211

wvu

aaaaaaaaa

,

where

(2.1)

(2.1)

(2.2)

(2.1)

,4

12

1a

,)1(2412

aa

,2

1aa

,12

12

1a

222

222

22

32

3113

2112

222

211

Ω−α

δ+

ν−

+=

αν−

δ−α

δ+να−==

α

ν+

−==

Ω+

δ+

ν−

−α−=

n

ni

ni

n

,]2)(1[12

1a

,)3(24

aa

222222

33

22

3223

Ω−−+α+δ

+=

αν−δ

+==

nn

nn

with

lc

andE

RRh

lR ps π

=ω

ν−ρω=Ω=δ

π=α

2,

)1(,,2

2/12

For a nontrivial solution of these simultaneus equations, the determinant of the coefficients of the unknowns should vanish. The resulting characteristic equation is the frequency equation:

| aij | = 0.

Alternatively, the frequency equation can be written in the functional form:

F(Ω, α, n, ν, δ) = 0

The shell, undergoing free vibration, can be defined in a variety of ways, as shown in Figures 6. The vibration of the shell can consist of a number of waves distributed around the circumference as shown in Figures 6a for n = 0, 1, 2, 3, and 4. n is the number of circumferential waves in the mode shape. In the axial direction, the deformation of the shell consists of a number of waves distributed along the length of a generator, as shown in Figures 6b for 1/2 l, l, 3/2 l, where l is the wave length and in other terms, for m = 1, m = 2 and m = 3, where m is the number of axial half-waves in the mode shape. For a given shell, δ and ν are fixed. The natural frequency Ω depends on α and n only. Two special cases are considered : Circumferential modes. Consider those modes of oscillations that are independent of the axial coordinate z ; i.e. those modes having frequencies corresponding to an infinite phase velocity. The equations for these modes are obtained by setting α = 0 in equation (2.5) ; i.e.

(2.3)

(2.4)

(2.3)

(2.5)

0)1(

12)1(

121

21

121

2222

2222

24

222

2

=−δ

+Ω

−

δ++−Ω

ν−

δ+=Ω

nnnn

n

Symmetric modes. For n = 0, equation (2.5) can be written

.012

)1(12

1)(

02

14

1

23

24

2222

222

2

=

α

δ+να−

+

δ−−Ωα−Ω

=α

ν−

δ+−Ω

n

and

and

(2.6)

n = 0 n = 2

n = 3

n = 1

n = 4

Breathing Beam bending or Translation

Ovalling

Figure 6a: Circumferential Nodal Pattern

Figures 6: Nodal Patterns of Shell with Simply Supported ends without Axial Constraint. From Blevins R.D. Formulas for natural Frequency and mode shape[18].

(2.7)

2.1.2 Natural Frequency and Mode Shape. [18] Recalling the formula for the natural frequency of the cylindrical shell Ω :

mnps f

lc

withE

R ,

2/12

22

,)1(

π=π

=ω

ν−ρω=Ω

The natural frequency of the shell therefore depends on the formula of Ω in each structural mode. The general description of the structural modes of the cylindrical shell by Circumferential and Symmetric modes is somewhat not precise. Refering to Donnell shell theory, these two generalised modes (Circumferential and Symmetric) can be precisely decomposed in a series of particular modes which are nevertheless generally coupled during the vibration of the shell. Each of them provides an individual formulation of the parameter Ω given by the Donnell theory. We will confine our attention on the modes which are likely to occur and be responsible for intensive noise radiation. The natural

m = 1 m = 2 m = 3

(1/2) l l (3/2) l

Figure 6b: Axial Nodal Pattern

Figure 6c: Nodal Arrangement for n = 3 and m = 4

Axial Node

Circumferential Node

(2.8)

frequencies of the concerned modes will therefore be computed. Chapter 5 will present the results obtained as the analytical solutions for the three Gradient coil systems. The natural frequency can be expressed as :

2/1

2,

, )1(2

ν−ρπ

Ω=

s

mnmn

ER

f

The following modes are analytically supposed to occur in a simply supported Cylindrical Shell without Axial Constraint. These formulas are appropriate for long cylinders where the ratio L/(mR) (effective shell length) is higher than 8. Torsion Modes. (n = 0)

LRm

mnπν−

=Ω 2/1

2/1

, 2)1( with n = 0 and m = 1, 2, 3,....

Axial Modes. (n = 0)

LRmmn

2/12, )1( ν−π=Ω with n = 0 and m = 1, 2, 3,....

Radial Modes. (n = 0)

1m,n =Ω with n = 0 and m = 1, 2, 3,.... Bending Modes. (n = 1)

2

2

2/1

2/122

, 2)1(

LRm

mnν−π

=Ω with n = 1 and m = 1, 2, 3,....

Radial-Axial Modes. (n = 2, 3, 4, 5, ...) (Modes dependent on the circumferential angle θ)

22

2/1422

2

242

,

12)1(

nL

Rm

LRmn

Rh

LRm

mn

+

π

π

+

+

π

ν−

=Ω

with n = 2, 3, 4, ... and m = 1, 2, 3,.... Note: For higher circumferential modes, n = 2, 3, 4, ..., the deformation of the shell is dominated by radial motion.

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

2.2 WAVE PROPAGATION. 2.2.1 Wave propagation in Thin-Wall Circular Cylindrical Shells. [19] Flexural waves and “Ring” Frequency. If the ratio of cylinder diameter to wall thickness is large, wave propagation involving distortion of the cross section is of practical importance even at relatively low audio frequencies. If the wall thicknesss is uniform, the allowable spatial form of distortion of across section must be periodic in the length of the circumference. The axial, tangential, and radial displacements of the wall must vary with axial position z and azimutal angle θ as :

( ) ( ) ( )[ ] ),cos(,,,, Φ+θ= nzWzVzUwvu where n is known as the circumferential mode order and 0 ≤ n ≤ ∞. At any frequency, three forms of wave having a given n may propagate along an in vacuo cylindrical shell. Each has different ratios of U, V, W which vary with frequency (Leissa, 1973). It is nearly always necessary to consider the three displacement u,v,w of a cylindrical shell. The principal reason is that a radial displacement of the wall of a cylindrical shell creates tensile or compressive tangential and axial membrane stresses, depending upon whether the displacement is positive or negative, because the length of a circumference is proportional to its radius. For the purposes of studying sound radiation and response behaviour of cylindrical shells that do not contain dense fluids, it is only necessary to consider the vibration waves in which the radial displacement w is dominant, the so-called “flexural waves”. The speed of propagation of these waves in the direction of the longitudinal axis of the cylinder is very dependent upon the circumferential mode order n because the relative contributions to strain energy of membrane strain and wall flexure depend upon this parameter. In addition, a cylindrical shell exhibits waveguide behaviour in that a mode of propagation having a given n cannot propagate freely below its “cutoff frequency”, at which the corresponding axial wavenumber and group velocity are zero. A cutoff frequency corresponds to a natural frequency at which the modal pattern consists of a set of 2n nodal lines lying along equally spaced generators of a uniform shell of infinite length. the cutoff frequency of the n = 0, so-called “breathing mode” of a shell is termed the “ring frequency” and is given by :

,/'1 dcf r π=

where d is the cylinder diameter. At this frequency, radial hooplike resonance occurs. In dynamic shell theory, frequency are usually made non-dimensional by dividing by fr : Ω = f/fr . Then, below Ω = 1, the breathing mode cannot propagate, although axial and tangential n = 0 modes can. One result of the membrane strain on flexural waves propagation speed in cylindrical shells is that the phase velocity of flexural waves in modes of low n can be much greater than the speed of sound in a surrounding fluid at frequencies well below the flat-plate critical frequency, as given the particular combination of shell material, wall thickness and speed of sound in the fluid. The effect is that such modes can radiate rather effectively.

(2.15)

(2.16)

2.2.2 Waves in infinite flat surface. The vibration of most mechanical systems exhibit complex frequency dependent spatial distributions of amplitude and phase. It is therefore useful to compare actual structures with infinite flat surfaces. In parts 2.3.1 & 2.3.3, a comparison of flat-plate and circular cylindrical-shell flexural wavenumber behaviour will be done which will lead to a qualitative discussion of the consequent difference between radiation behaviours. It is therefore helpful to recall the principal characteristics of wave propagation in infinite flat surface and the concept of radiation efficiency, bending or flexural waves and critical frequency. The most important type of vibration of plates is flexural or bending waves. This is governed by a fourth order differential equation, e.g. for one-dimensional waves on a homogeneous plate :

( ) ( )txftwh

xwEh

m ,112 2

2

4

4

2

3

=∂∂

ρ+∂∂

ν−

where h is the thickness , ρm is the density, E is the Young’s modulus, and ν is the Poisson’s ratio. A free wave solution of equation (2.17) is of the following form :

( ) tjjkxeAetxw ω±=, This leads to an expression between wavenumber k = 2π/λ (“spatial frequency”) and (“temporal”) frequency ω :

( ) 0112

242

3

=ωρ−ν−

hkEhm

This equation is called “Characteristic Equation”. The dependence of k = 2π/λ on ω = 2π/T is known as the “Dispersion Relation”:

( ) 2/1

4/1

2

2112ω

ρν−=

Ehk m

For acoustic waves in a fluid (e.g. air) k = ω/co where co is the speed of sound, independent of frequency. These waves are non-dispersive, and all frequency components travel at the same speed. Bending or fluxural waves are dispersive and therefore the wavespeed depends on the frequency. The corresponding wavespeed cb can be expressed as follows :

( )2/1

4/1

2

2

112ω

ρν−

=ω

=mb

bEh

kc

(2.22)

(2.19)

(2.20)

(2.18)

(2.21)

(2.17)

“Dispersion curves”, which plots the flexural wavenumber kb as a function of the frequency ω, can then be drawn and present the following characteristics illustrated by Figure 7. The critical frequency fc of a bending wave is defined as the frequency at which the bending wavenumber equals the wavenumber in air. For a homogeneous material, the critical frequency can be expressed as :

( )Eh

cf mo

c

22 1122

ν−ρπ

=

The critical frequency is therefore dependent of the material characteristics (E, ν,...) Radiation Efficiency. The sound power radiated from a plate can be expressed as follows:

σρ= 2vScw oorad

where S is the total surface area, 2v is the surface-averaged mean square normal

velocity and σ is called the radiation efficiency. When σ = 1, a structure radiates as efficiently as the infinite flat surface. At low frequencies, σ << 1 and at high frequencies σ tends to 1. Radiation from bending waves. At low frequencies, i.e. fc << f , σ << 1. For an infinite plate with a free bending wave, the sound radiation is zero. This is because of the “Acoustic short-circuiting” between the radiation from maxima and minima of the vibration pattern. For a finite plate, the edges and corners results in a net radiation of sound.

(2.23)

Frequency [Hz]

Bending Wavenumber kb Sound

k Bending Wave kb ≅ ω1/2

fc

Figure 7: Dispersion Curve for bending waves.

At high frequencies, i.e. f > fc , σ tends to 1. Sound is radiated at an angle θ to the plate as illustrated in Figure 8. The relationship between acoustic and bending waves can be expressed with the following relations :

θ=θλ=λ

sin,sin

airb

bair

kk

2.2.3 Flexural Wave Propagation in a Circular Cylindrical Shell. [19] Flexural-type waves propagating in a uniform cylindrical shell may be characterised by axial and circumferential wavenumbers kz and ks, as shown in Figure 9. The wavefronts of free propagating waves of wavenumber kcs form a helical pattern like a barber’s pole, the angle between the wavevector and a generator of the cylinder being given by θ = tan-1(ks/kz). Characteristic waveguide modes can be formed by the superposition of helically propagating waves having the same axial wavenumber components kz and oppositely directed circumferential wavenumber components ± ks. The closure of the shell in the circumferential coordinate direction requires that the wave variables be continuous around the circumference and that the characteristic

x

y

z

r

a

θ

Figure 9: Cylindrical Coordinates System

kz

ks kcs

Φ

θ

Radiated Sound

Figure 8: Sound Radiation of Plate

At the critical frequency sin θ = 1 , i.e. sound is radiated parallel to the plate.

circumferential patterns take the form sin(kss) or cos(kss), where s = aφ and ks = n/a (n = 0, 1, 2, 3, ... , ∞), so that an integer number of complete circumferential wavelengths, λs = 2π/ks = 2πa/n, fit around the circumference. The wavenumber relationship is :

( )22222 / ankkkk csscsz −=−= where kcs is the wavenumber of the two propagating helical wave components. The two most important cylindrical-shell parameters are the non-dimensional frequency Ω = ωa/cl’= ω/ωr and the non-dimensional thickness parameter a12/h=β , where h is the wall thickness and a the mean radius of the shell, cl’the longitudinal wavespeed in a plate of the shell material, and ωr = cl’/a (rad.s-1) the ring frequency (2.16). The physical significance of these parameters is as follows: The ring frequency ωr is the lowest frequency at which an n = 0 axisymmetric-mode resonance can occur. The longitudinal wavelength in the shell wall equals the shell circumference when Ω = 1, and an n = 0 breathing, or hoop, resonance occurs at this frequency. Note that ωr is depedent upon the cylinder radius, but not on the wall thickness. An important property is that the ring frequency separates frequency regions in which wall curvature effects are more, and less, important. This property will be commented in more details in the next part. 2.3 SOUND RADIATION. 2.3.1 Sound Radiation from Circular Cylindrical Shells. [19] The influence of shell curvature on sound radiation derives primarily from its effect on the flexural wave dispersion characteristics, particularly at low wavenumbers. Curvature generally increases flexural wave phase velocities through the mechanism of mid-plane strain, with a consequent increase of radiation efficiency. Associated with the increase in wave speed is a reduction in the density of natural frequencies. The surface radial velocity of a wave propagating axially in a cylindrical shell of infinite length may be represented as

)],(exp[cos),,( zktjnvtzv znn −ωΦ=Φ n = 0, 1, 2, 3, ... for the cylindrical coordinate system shown in Figure 8. Sound energy can be radiated as long as the surface axial wavenumber kz is less than the acoustic wavenumber k. The cosinusoidal variation with Φ in equation (2.25) results from the interference between circumferential wavenumber components travelling in opposite directions around the cylinder; that is to say that equation (2.25) represents the interference field of two helical waves of equal and opposite circumferential wavenumber and equal axial wavenumber. Under the conditions:

kz < k, ks = n/a > k, and kz2 + ks

2 > k2,

(2.26)

(2.25)

(2.24)

which are not relevant to the n = 0 mode, adjacent zones of positive and negative volume velocity distributed around the circumference, seen in Figures 6, do not completly cancel. However, their close proximity, in terms of acoustic wavelength, makes their radiation very inefficient. The n = 0 “breathing” mode radiates as a line monopole; the n = 1 “bending” mode radiates as a line dipole; the n = 2 “ovalling” mode radiates as a line quadrupole, and so on, the efficiency of radiation at any frequency decreasing with increase in the order of the equivalent source. The corresponding expressions for power radiated per unit length, for ka << 1 and kz << k, are

.)(321

,)(41

,)(21

22

522

21

321

220

vkacaP

vkacaP

vkacaP

on

on

oon

ρπ=

ρπ=

ρπ=

=

=

=

The value of the circumferential wavenumber ks = n/a (n is the circumferential number and a is the cylindrical radius) for a given n decreases with the increase of cylinder radius. Hence large radius cylinder modes of order n can satisfy the condition

kz2 + ks

2 < k2

at frequencies for which the equivalent modes of smaller cylinders, having the same radial wavenumber, give kz

2 + ks2 > k2. In this case, inter-zone cancellation does not occur and

the cylinder radiates with a radiation resistance close to ρoc per unit area. This shows that it is not sufficient that the frequency at which a cylinder vibrates in the bending modes (n = 1) should exceed the critical frequency based upon the bending waves phase velocity in order to radiate efficiently; it is also necessary that:

(k2 - kz2)1/2 a > 1.

For thin-wall cylinders in which the wall thickness is much smaller than the radius, the n = 1 bending-mode axial wavenumber is:

kz = (2ρω2 / a2E)1/4

The transition from inefficient to efficient radiation occurs rather rapidly, as Figure 10 indicates, so that for most practical purposes sound radiation from transversely vibrating slender bodies can be considered to be negligible if equation (2.29) is not satisfied. Similarly, only when equation (2.28) is satisfied will radiation from the cross-sectional distortion be significant. Note: The theory, which have been described so far, only intends to give an overview of the sound radiation mechanism and its characteristics that could maybe tally with the experimental investigation. Assimilating the MRI gradient coil system as a thin cylindrical

(2.27)

(2.28)

(2.29)

(2.30)

shell somewhat simplifies the vibration and the sound radiation of the studied systems. The assumption of thin walls cannot usually be justified. Nevertheless, the information of a radiation efficiency dependent on the structural mode, the properties of the dispersion relationship, the characteristics of the wavenumber diagram constitute a meaningful knowledge which enables to elaborate correctly the procedure of investigation. The following theoretical parts will see the development of the dispersion relationship for thin cylindrical shell, cut-off frequencies of both structural and acoustic modes. They aim to introduce the coupling between shell modes and acoustic modes which will inevitably occur, but they also set the context in which decoupling or any other control can be worked out.

Dispersion Relationship and Cutoff Frequencies of Flexural modes. There are many thin-shell equation of varying degree of complexity, the differences arising largely from differences in the assumed strain-displacement relationships (Leissa, 1973). In this part, Frequency and Dispersion relationships derived in relatively simple form will be used to emphasise the features of cylindrical shell behaviour that cause the sound transmission characteristics to differ markedly from those of a flat plate. One approximate form of thin shell equation (Helck, 1962) yields the following flexural-wave dispersion relationship between the non-dimensional axial wavenumber kza and the non-dimensional frequency Ω, for given non-dimensional circumference wavenumber ksa = n :

( ) ( )( )( ) ( )( ) ( )[ ]

( )

ν−

ν−−ν−−+β+

+ν−=Ω

12241

22222

2

22

222 nnak

nakak

zz

z

(2.31)

Figure 10: Radiation Efficiencies of Uniformly Vibrating Cylinders. From Fahy F. “Sound and Structural Vibration”

in which ν is Poisson’s ratio. This expression is accurate for thin cylindrical shells (β << 10-1) and for values of n ≥ 2. The first term on the right-hand side of the equation is associated with membrane strain energy and the second, which contains β2, is associated with the strain energy of wall flexure. The cross sectional resonance frequencies, or cutoff frequencies of an infinitely long cylinder, are given by equation (2.31) with kz = 0, which corresponds to an infinite axial wavelength. It is most important to observe that these frequencies are determined by strain energy of wall flexure: they correspond to Rayleigh’s inextensional mode frequencies, which were derived by assuming that the median surface of the shell wall did not strain. The resulting equation for Ω2 can be expressed as follows:

ν+

−ν−

ν−−β=Ω 42

422 241

1211

nnnn

Except for the lowest-order modes, the cutoff frequencies are reasonably well approximated by the formula :

2n nβ≅Ω

Equation (2.33) also indicates that the number of cutoff or cross-sectional resonance frequencies below the “ring frequency” is given approximately by:

2/1−β≅rn Wavenumber Diagram. Wavenumber diagram can be very useful in helping to identify those forms of vibration most effective in radiating sound. In order to illustrate the form of the shell wavenumber diagram, equation (2.32) is simplified by the omission of the less-important flexure term. The approximate form of dispersion equation is [Heckl, 1962] :

( ) ( )[ ] ( ) ( )[ ] 2222/122/1

222/122/1

42/1 )(Ω≅β+β+

β+β

β naknak

akz

z

z

where β2 = h2/12a2, Ω = f/fr , and h is the wall thickness. This produces a wavenumber diagram of the form shown in Figure 11.

(2.35)

(2.32)

(2.33)

(2.34)

The curvature of a cylindrical shell produces coupling between radial, axial, and circumferential motions, and there are consequently three coupled equations of motion and three classes of propagating waves (Leissa, 1973). Although only the radial motion of the shell determines the sound radiation, the form of the radial motion, and the associated dispersion characteristics, are significantly affected by mid-plane strains, especially at frequencies well below the ring frequency a2/'cf 1r π= . These so-called membrane effects considerably raise the phase velocities of waves whose displacement is predominantly radial, so much so in some cases, that these waves have supersonic phase velocities at frequency well below the critical frequency based upon the shell wall considered as a flat plate. Above the ring frequency, curvature effects disappear and the shell vibrates like an equivalent flat plate. The membrane effect on wave speed is seen in the bending of frequency loci toward the origin. A strange consequence of this behaviour is that at one frequency two helical waves of the same axial wavenumber, but different circumferential wavenumber can propagate. Waves of low circumferential wavenumber involve greater membrane strain energy than waves of higher circumferential wavenumber. This leads to a rather unexpected variation of natural frequency with axial and circumferential wavelength. 2.3.2 Acoustic Modes: Cross-Modes in Circular Cross Section Ducts. [20] In the Cross-modes of Circular Cross Section Ducts, the Velocity Potential Φ has to satisfy the following propagation equation:

( ) ( ) 0,,,1,,, 2

2

22 =

∂θΦ∂

−θΦ∇t

tzrc

tzro

where the operator ∇2. can be expressed in the cylindrical coordinates system

(2.36)

Figure 11: Universal constant-frequency loci for flexural waves in thin-walled circular cylindrical shells (n > 1). Fahy F. “Sound and Structural Vibration”

(See Figure 5, p13) as follows:

2

2

2

2

22 ..1.1.

zrrr

rr ∂∂

+θ∂∂

+

∂∂

∂∂

=∇

The relation between the Particle Velocity and the Velocity Potential is given by:

( ) ( )tzrtzru ,,,,,, θΦ∇=θ where the operator ∇. can be expressed in the following cylindrical coordinates:

zrr ∂∂

+θ∂∂

+∂∂

=∇..1..

The equation that relates acoustic pressure and velocity potential is:

( ) ( )t

tzrtzrp o ∂θΦ∂

ρ=θ,,,,,,

When harmonic motion is assumed, equation (2.36) reduces to the Helmotz equation in cylindrical coordinates.

( ) ( ) 0,,,,,, 22 =θψ+θψ∇ tzrktzr

where Φ(r, θ, z, t) = ψ (r, θ, z) e jωt and k = ω/co . Writing ψ (r, θ, z) in the form:

( ) ( ) ( ) ( )zHGrFzr θ=θψ ,, Substituing this expression into the equation (2.41) and separating the variables yields to three ordinary differential equations of the forms:

( ) ( ) 022

2

=+ zHkdz

zHdz

( ) ( ) 022

2

=θ+θθ Gm

dGd

( ) ( ) ( ) 01

2

222 =

−−+

∂∂

∂∂ rF

rmkk

rrFr

rr z

Since no Boundary Conditions are assumed for the z direction, the general solution to (2.43) is used:

( ) zjkz

zjkz

zz eBeAzH += − No definite Boundary Conditions are specified for the θ direction. However, there is a periodicity requirement such that:

(2.42)

(2.37)

(2.43)

(2.38)

(2.44)

(2.39)

(2.45)

(2.40)

(2.46)

(2.41)

( ) ( )π=θ==θ 20 GG

This results in a solution for equation (2.44) of the form:

( ) ( ) )sin(.cos. θ+θ=θ θθ mBmAG Rearranging equation (2.45) yields:

( ) ( ) ( ) ( )rFmrdr

rdFrdr

rFdr 2222

22 −η++

where η2 = k2 - kz2 .

This equation is Bessel’s equation of order m. Its solution is given by:

( ) ( ) ( )η+η= rYBrJArF mrmr Jm(rη) is Bessel function of the first order m. Ym(rη) is Bessel function of the second order m, or so-called “Neumann” function. One of the restrictions on the solution to equation (2.49) for the problem at hand is the function F(r) must be bounded at r = 0. Since, Ym(rη) is unbounded at r = 0, this term is deleted and the solution of equation (2.49) becomes

( ) ( )η= rJArF mr A rigid duct wall is located at r = a. Thus the Particle Velocity in the r direction at r = a must be equal zero.

( ) ( ) ( ) ( )( )r

ezHGrFr

tzruorutj

rarr ∂θ∂

=∂θΦ∂

==ω

=,,,0|

( ) ( )( )[ ]

0=η

=⇒dr

aJAddr

adF ma

This Boundary Condition is satisfied when:

( )

0=β

drdJ mnm

Where βmn = aηmn

The following table gives values of βmn for which the equation ( )

0=β

drdJ mnm is satisfied.

βmn n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

m = 0 0 3.8317 7.01558 10.17346 13.32369 16.47063 19.61585m = 1 1.84118 5.33144 8.53632 11.70600 14.86359 18.01553 21.16437m = 2 3.05424 6.70613 9.96947 13.17037 16.34752 19.51291 22.67158m = 3 4.20119 8.01524 11.34592 14.58585 17.78875 20.97248 24.14490m = 4 5.31755 9.28240 12.68191 15.96411 19.19063 22.40103 25.58976m = 5 6.41562 10.51986 13.98719 17.31284 20.57551 23.80358 27.01031

(2.52)

(2.53)

(2.47)

(2.48)

(2.49)

(2.50)

(2.51)

m = 6 7.50127 11.73494 15.26818 18.63744 21.93172 25.18393 28.40987 It is noted that ηmn = βmn / a. F(r) for the m,n mode can be written:

( )

β=

arJArF mnmmnmn

The total solution for the Cross-Modes in a Circular Duct that are travelling in the positive z direction is:

( ) ( )( )

( )∑∑∞

=

−ω∞

=

β

θθ

=θΦ0 0 sin

cos,,,

m

zktj

nmnmmn

zearJ

mm

Atzr

Velocity Potential

The Mode Function for the m,n Cross-Modes is given by:

( ) ( )( )

β

θθ

=θψarJ

mm

r mnmmn sincos

,

The above equation implies that either the sine or cosine term can be used for the mode function. Cutoff Frequency. The Cutoff Frequency for a m,n Cross-Mode is obtained from the expression:

η2 = k2 - kz2 = (βmn / a)2 or

22

β−=

akk mn

z

kz must be a real value number. Thus a

k mnβ≥ and

oo cf

ck π

=ω

=2 .

The Cutoff Frequency can therefore be expressed as follows:

( )ac

f omncmn π

β=

2

This frequency only depends on the radius a and therefore on the considered m,n mode. Recalling equation (2.40) and equation (2.55) :

( ) ( )t

tzrtzrp o ∂θΦ∂

ρ=θ,,,,,, and ( ) ( )

( )( )∑∑

∞

=

−ω∞

=

β

θθ

=θΦ0 0 sin

cos,,,

m

zktj

nmnmmn

zearJ

mm

Atzr

(2.57)

(2.58)

(2.54)

(2.55)

(2.56)

It is possible to express the cross-mode proper to a rigid-walled cylindrical shell in terms of its acoustic pressure:

( ) ( )( )

( )zktjmnmmnmn

zearJ

mm

Ptzrp −ω

β

θθ

=θsincos

,,,

The term (βmn r/a) of the above equation (2.59) is equal to ηmnr with ηmn

2 = k2 - kz2 .

We can therefore rearrange equation (2.59):

( ) ( )( ) ( ) ( )zktj

mnmmnmnzerJ

mm

Ptzrp −ωηθθ

=θsincos

,,,

where ηmn is defined as the radial wavenumber and determined by the zero normal-particle wall boundary condition as solution of the equation (2.53) when r = a. The characteristic solutions are multi-valued for a given m. m indicates the number of diametral pressure nodes and n the number of concentric circular pressure nodes. The patterns are illustrated in Figure 12.

2.3.3 Coupling between Shell Modes and Acoustic Duct Modes. The acoustic coupling between the fluid contained in a cylinder and the shell is very much dependent upon the relative axial phase speeds of the waveguide modes in the two media. The radial and axial wavenumbers satisfy the acoustic wave equation:

222mnzkk η+=

The cutoff frequencies below which a particular mode cannot propagate freely and carry energy in a infinitely long duct are given by:

0222 =η−= mnz kk , or mnmnaka β=η=

(2.59)

(2.60)

(2.61)

(2.62)

n = 1, m = 0 n = 2, m = 0 n = 3, m = 0

n = 1, m =1 n = 2, m=1 n = 3, m = 1

n = 1, m = 2 n = 2, m = 2 n = 3, m = 2

Figure 12: Fourier-Bessel mode shapes for the pressure inside the duct.

Values of βmn were given in the above table (p29). In terms of the ring frequency of the shell (ωr = cl’/a (rad.s-1)), the equation (2.62) can be written in the term of the non-dimensional frequency :

β=

η==

λπ

=ω

=ωω

=Ω '''''

2

l

imn

l

imn

l

i

l

i

lrmn c

ccc

ac

kacc

acca

where ci is the speed of sound in the contained fluid and cl’ the longitudinal wavespeed in a plate of the shell material. For instance, equation (2.63) and the value of βmn for the m = 1, n = 0 give the lowest cutoff frequency :

=Ω '0,1 84.1

l

i

cc

The dispersion relationship (2.61) can be represented graphically by the non-dimensional axial wavenumber kz against the non-dimensional frequency Ωmn , i.e.:

kz2 a2 = Ωmn

2 (cl’/ cl)2 - βmn2

but the appropriate values of βmn and cl’/cl must be employed. There is unfortunately no universal form of combined structural and acoustic wavenumber diagram. Nevertheless, the acoustic duct modes and the shell wall modes do not exist independently in a fluid-filled duct. The dispersion diagrams for the shell waves and the fluid waves may be superposed as shown in Figure 13. Only waves of equal circumferential order m may couple. Figure 13 shows that equality of fluid and shell axial wavenumbers can occur : This is a “coincidence” condition.

(2.63)

(2.64)

In most practical cases, coincidence between the lower-order shell modes and the acoustic modes of low radial order n occurs at frequencies close to the acoustic mode cutoff frequencies because menbrane effects keep the slope of the structural wavenumber curve low , whereas the acoustic wavenumber curves rise rapidely with Ω. The lowest possible coincidence frequency, corresponding to coincidence between the beam-bending mode and the (1,0) acoustic mode, corresponds closely to the cutoff frequency of this mode, which is given by equation (2.64). Between this frequency and the ring frequency, a number of further such coincidences can occur; any one shell mode can be coincident with all the acoustic modes of equal circumferential order m and increasing radial order n. The relationship between the internal acoustic and shell wavenumbers is of cruicial importance in determining the coupling of the media.

shell and fluid share same m value of n irrelevant

β = h /√12.a

Figure 13: Illustration of coincidence between shell and fluid modes.Fahy F. “Sound and Structural Vibration”

3. EXPERIMENTAL PROCEDURE FOR VIBRATION AND ACOUSTIC MEASUREMENTS ON MRI SCANNERS. Noise and Vibration measurements inside any MRI scanners require particular attention to the working environment. The bore and surroundings of the scanner are hostile to any measurements made with metallic component devices such as transducers, microphones, cables, recorders. The 3 Tesla high magnetic field is a source of interference when the aim of the investigation is to obtain reliable measurements. Therefore a specific protocol has been developed so as to ensure enough accuracy for these measurements. It is however important, while establishing the experimental procedure, to be aware of the expected signals to record. This procedure should contribute to a correct choice of transducers or other equipment to use for a successful measurement session. A strong correlation between the acoustic or vibration signals and the impulse sequence of the imaging process was already expected before making the measurements. The imaging process and its characteristics have therefore been primarly studied. The following part will present an overview of each scanner design, the gradient coil system in particular, and “Echo-Planar Imaging” sequences used in the “functional Magnetic Resonance Imaging” process. This will be helpful to qualitatively understand the results presented in Chapter 4 and should also give insights of what could be achieved in the design of the imaging sequence so as to control the Lorentz forces and therefore the induced vibration and noise. 3.1 PRESENTATION OF THE SCANNERS AND THEIR IMAGING SETTINGS. 3.1.1 A few words about functional Magnetic Resonance Imaging Scanners. All the assessed scanners were devoted to functional Magnetic Resonance Imaging. FMRI is a technique for determining which parts of the brain are activated by different types of physical sensation or activity, such as sight, sound or the movement of a subject's fingers. The “brain mapping” is achieved by setting up an advanced MRI Scanner in a special way so that the increased blood flow to the activated areas of the brain shows up on functional MRI scans. The subject in a typical experiment will lie in the magnet and a particular form of stimulation will be set up. For example, the subject may wear headphones so that sounds can be played during the experiment. Then, MRI images of the subject's brain are taken. Firstly, a high resolution single scan is taken. This is used later as a background for highlighting the brain areas which were activated by the stimulus. Next, a series of low resolution scans are taken over time, for example, 150 scans, one every 5 seconds. For some of these scans, the stimulus will be performed, and for some of the scans, the stimulus will be absent. The low resolution brain images in the two cases can be compared, to see which parts of the brain were activated by the stimulus. After the experiment has finished, the set of images is analysed. Firstly, the raw input images from the MRI scanner require mathematical transformation (Fourier transformation, a kind of spatial "inversion") to reconstruct the images into "real space", so that the images look like brains. The rest of the analysis is done using a series of tools which correct for distortions in the images, remove the effect of the subject moving their head during the experiment, and compare the low resolution images taken when the stimulus was off with those taken when it was on. The final statistical image shows up bright in those parts of the brain which were activated by this experiment. The functional imaging procedure therefore involves special settings of the sequence and therefore characteristically drives the gradient coil system responsible of the high noise generation.

3.1.2 The Echo-Planar Imaging sequences. Echo-planar imaging (EPI) is an extremely rapid method, gathering all the data to form an image from a single excitation pulse (Ultra-fast single-shot methods). EPI is fundamentally just a method of spatial encoding. Tomographic image formation requires spatial encoding in three dimensions. In most cases, one dimension is determined by slice selective excitation (refer to Figure 14 for axis labels). Briefly, a radio frequency excitation pulse with a narrow frequency range is transmitted to the subject in the presence of a spatial magnetic field gradient. Because the magnetic resonance phenomenon depends on an exact match between the radio frequency excitation pulse frequency and the proton spin frequency, which depends in turn, on the local magnetic field, this pulse will excite the Magnetic Resonance signal over a correspondingly narrow range of locations: an imaging slice. The differences between EPI and conventional imaging occur in the remaining “in-plane” spatial encoding.

One dimension of spatial encoding is achieved by slice selective excitation (the “Slice Selection” axis). The other two are encoded by phase and frequency. The Slice Selection axis is also referred as the “Z” axis. The Readout axis is variously labeled the “Frequency” or “X” axis; the Phase Encoding axis is sometimes labeled the “Y” axis. When a magnetic field gradient is applied across this excited slice, it will cause the spin frequency to be a function of position. The spatial resolution (pixel size) of an MR image depends on the product (actually the integral) of the imaging gradient amplitudes and their “on” duration (for example, a gradient of 0.5 gauss/cm, left “on” for 10 msec yields a spatial resolution of 0.47 mm). This reflects spatial encoding along one in-plane dimension only: the “Readout” direction. In Fourier transform imaging, the encoding for the second in-plane dimension is created by applying a brief gradient pulse (along a second gradient axis) before each readout line. For 128 lines of resolution in this axis, 128 separate lines must be acquired, each for 10 msec. The total readout duration is therefore 128 x 10 msec, or 1.28 seconds. In EPI, much larger gradient amplitudes are used. A gradient of about 2.5 gauss/cm is typical. With five times the gradient amplitude, the encoding duration can be reduced by five and fold to 10/5 = 2 msec/line, so that the total spatial encoding time for our reference image is reduced from 1.28 seconds to 256 msec. In practice, however, this is not a practical configuration. Most significantly, the gradients cannot instantly reach such large magnitudes, and the rise time therefore becomes a significant fraction of the readout duration. Secondly, the decay of the MR signal during

Slice Selection

Figure 14: The three axes used for spatial encoding of MR images.

Phase

Readout

Matrix

readout introduces blurring into the images. Because of these tradeoffs, most EPI studies are performed at somewhat lower resolution. In-plane voxel (elementary volume of the matrix or thicknesss of the slice, i.e. spatial resolution) sizes between 1.5 and 3 mm are typical. Characteristics of the Echo-Planar Imaging Sequence used during the acoustic and vibration investigation of the Scanners. a. MRC-IHR Nottingham fMRI Scanner EPI sequence. The EPI pulse sequence used was a Modulus Blipped Echo-planar Single-pulse Technique (MBEST) sequence. It enables a rapid switching of a strong gradient to form a series of gradient echoes (GE), each with a different degree of phase encoding, which can be reconstructed to form an image. Its characteristics in terms of waveform components and duration are described as follows. The “Slice select” is a single square wave immediately before the “Broadening” and “Switch”. The “Broadening” or “Phase encode” to refer as above, is a small triangular waveform at each zero-crossing of the “Switch”. The “Switch” or “Readout” to refer as above, is a 1.9 kHz sinusoid for 65 whole waves plus some build-up waves (67 waves, approximately 35 ms).

0.01 0.02 0.03 0.04 0.05 0.06 0.07-1

-0.5

0

0.5

1Figure 15 : EPI Signal Components

Time [seconds]

Arb

itrar

y un

its"Switch" Electrical Signal

0.01 0.02 0.03 0.04 0.05 0.06 0.07-1

-0.5

0

0.5

1

Time [seconds]

Arb

itrar

y un

its

"Slice" Select Signal

0.01 0.02 0.03 0.04 0.05 0.06 0.07-0.2

-0.1

0

0.1

0.2

Time [seconds]

Arb

itrar

y un

its

"Broadening" Electrical Signal

The total duration of the sequence is determined as we have seen by the spatial resolution and the size of the matrix. The dimension of the matrix is 128x128x16 slices/volume, giving a spatial resolution of 4x4x4 mm (voxel size). These 16 slices constitute a total signal of approximately 1.1s length of time which can be decomposed in 16 slices of 35 ms each plus 16x32 ms of gap between slices. The Echo time (TE) and Repetition time (TR) have to be mentioned as they constitute an important reference but also as they influence the noise perceived by the patient during the scans. The Echo time is determined by the 1.923 Hz “Switch” frequency and is 36 ms. The Repetition time which is the total start-to-start time for volumes is 5 s. The three main waveforms of the Nottingham EPI sequence have been recorded during the measurements as they are “driving” the noise production and moreover constitute a useful reference for the analysis of other signals. Figure 15 shows the recorded waveforms constituting the EPI sequence in the time domain. Additionally their spectrum in the frequency domain are presented in Figure 16. The EPI sequence is referred as the “Continuous” signal in the Measurement Tables given in Appendix.

0 1923 3846 5769 7692 9615 11538 13461 1538410

-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Figure 16 : PSD of EPI Signal Components

Frequency [Hertz]

Mag

nitu

de [

Arb

itrar

y U

nits

]Switch signal Slice select signalBroadening signal

Frequency Resolution: 16 Hz

b. FMRIB Oxford fMRI Scanner EPI sequence. The EPI sequence used for MR imaging with the Oxford scanner provides a resolution of 64x64 (2 Dimensions of the Matrix) and enables 3 mm of thickness for each slice (Voxel). The Echo time (TE) was 50 ms and the Repetition time (TR) was 500 ms. Another information was the bandwidth of the sequence which was 100 kHz for the EPI sequence referred as EPI 1 in the measurements table in Appendix ???, and 200 kHz for the EPI 2. The Field-of-View parameter (FOV) was 192x192 mm. These two sequences are the most employed and also noisiest sequences used by the FMRIB. The “Readout” signal, previously called “Switch” signal is a 32 “square-wave” signal of 647.5 Hz which lasts approximately 50 ms. The “Broadening” signal is named “Phase encode” here. Figure 17 and 18 illustrate the signals composing the EPI and their relative spectrum in the frequency domain.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1

-0.5

0

0.5

1Figure 17 : EPI1 Signal Components

Time [seconds]

Arb

itrar

y un

its

"Readout" Electrical Signal

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1

-0.5

0

0.5

Time [seconds]

Arb

itrar

y un

its

"Slice" Select Signal

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1

-0.5

0

0.5

1

Time [seconds]

Arb

itrar

y un

its

"Phase encode" Signal

0 647 1294 1941 2588 3235 3882 4529 5176 5823 6470 7117 7764 8411 9058 970510352109991164612293129401358714234148811552810

-15

10-10

10-5

100

Figure 18: PSD of EPI1 Signal Components

Frequency [Hertz]

Mag

nitu

de [

Arb

itrar

y U

nits

]

Readout signal Slice select signalPhase encode signal

Frequency Resolution : 21 Hz

c. WBIC Cambridge fMRI Scanner. Two EPI sequences were used in Cambridge. The EPI1 provides a resolution of 128x128 (2 Dimensions) and enables 5 mm of thickness for each slice. 23 Slices were executed during the Imaging Process. The Repetition Time is 160 ms and the Field-Of-View was set to be 25 cm. The bandwidth of the sequence was 200 kHz. The other sequence used is referred as EPI 2 in the Appendixes and is characterised by a Matrix of 128x64 dimension with a thickness of 5 mm, a repetition time TR of 4s and an echo time TE of 30 ms. The Field-Of-View was 25 cm with a spectral bandwidth of 142857 Hz. 21 Slices were executed during each imaging sequence. Unfortunately, the components of these EPI sequences were not recorded on the DAT for technical reason.

3.1.3 Presentation of the Scanners and the design of their relative Gradient coil system. The assessed scanners were all designed specifically for ultra-fast imaging for brain functional magnetic resonance imaging studies. The point of interest, while investigating the vibration and acoustical behaviour of MRI scanner, is the Gradient coil system. a. MRC-IHR Nottingham fMRI Scanner system. The scanner is a 3 T Whole body imaging and spectroscopy system. The 3 Tesla magnet is a superconducting magnet cooled by liquid helium and nitrogene and provides a strong highly homogeneous magnetic field.

Gradient coil System. Figure 19 describes the Head Gradient coil. Its length, radius and thickness are important parameters for the analytical investigation of its vibration and acoustical behaviours. The RF coil, which broadcasts the RF signal to the patient and/or receives the return signal, is inserted inside the bore of Gradient coil system, where the patient’s head is positioned during the imaging process. The wires made of copper, the epoxy resin and the cooling system confer its anisotropic constitution.

During the experiments, the settings of the EPI sequence lead the gradient coils to be driven differently in terms of nature and amplitude of currents. Indeed, the imaging process decides the excitation of the gradient coils. Three major class of images are usually obtained in MR Imaging. They are named “Coronal”, “Transverse” and “Sagittal”. Figure 20 illustrates these three different type of brain images.

395 mm

445 mm

25 mm 25 mm

20 mm

155 mm

175 mm

700 mm

Wires

350 mm

Wires

Wires Wires

Inner

Outer Core

Figure 19 : Dimensions of the Gradient Head coil system

•Isocenter

Each of these type of images involves the gradient coils to be particularly driven by the 3 main components of the EPI sequence. Table 1 indicates the relative settings of the Nottingham Gradient coil system for each type of images.

X-gradient coil (Gx)

Y-gradient coil (Gy)

Z-gradient coil (Gz)

Transverse Readout signal (Switch)

Phase encode signal (Broadening)

Slice select signal

Coronal

Slice select signal Phase encode signal

Readout signal

Sagittal

Phase encode signal Slice select signal Readout signal

Table 1 : Nottingham Scanner Imaging settings for EPI sequence.

The “Coronal” Image settings has only been used during the measurements. These settings differs from scanners. The vibration and acoustical responses are then expected to be different depending of the wanted multiplanar reconstruction of the brain. b. FMRIB Oxford fMRI Scanner system. The FMRIB scanner is an advanced 3T Varian/Siemens MR imaging system. It is equipped with both a body gradient coil and a fast head gradient coil insert.

Figure 20: Illustration of the different type of Brain images obtained by MRI.

From AOS Magazine, Review of Medical Imaging Techniques.

Gradient coil System. The FMRIB Gradient coil system is different from the MRC-IHR system by its non-uniform layout. This system does not have a symmetrical disposition of the gradient coils as Figure 21 shows. The wires are settled differently inside the Gradient coil system compared to the other head coils. It reinforced the dissymmetric loading of the Lorentz forces on the system. One specificity of this system is that the Imaging isocenter (and therefore the zone of perceived sound) is 180 mm away from the foot end and therefore let a wide unoccupied space behind the head in the Gradient system bore. 180 mm

•Isocenter

350 mm

595 mm

802 mm

1030 mm

540 mm 379 mm

Wiring

The EPI sequences and the Imaging settings interfere as well on the vibration and noise production. Three different EPI settings (i.e: three different type of images) has been used during the measurements in Oxford. Another EPI sequence (ref: EPI2, only on RF coil) and others type of signals (such as an impulse and few pure tones) have also been used. The following Table 2 gives a description of the characteristic electrical loading on each gradient coil referring to its corresponding class of images. The acoustical and vibration behaviour of the scanner to these three set of gradient settings have been investigated during the measurements in Oxford. The results and details of the measurements are displayed in the Tables in Appendixes.

X-gradient coil (Gx) Y-gradient coil (Gy) Z-gradient coil (Gz)

Transverse Phase encode signal

Readout signal

Slice select signal

Coronal

Readout signal Phase encode signal

Slice select signal

Sagittal

Slice select signal Phase encode signal Readout signal

Table 2 : FMRIB Oxford Scanner Imaging settings for EPI sequence.

c. WBIC Cambridge fMRI Scanner system. The WBIC scanner is a BRUCKER 3 Tesla MRI System. The imaging settings and their requirement on the driving of the gradient coils are presented in the following Table.

Figure 21: Dimensions of the Gradient Head coil system.

X-gradient coil (Gx) Y-gradient coil (Gy) Z-gradient coil (Gz)

Axial Phase encode signal

Readout signal

Slice select signal

Coronal

Readout signal Slice select signal Phase encode signal

Sagittal

Slice select signal Phase encode signal Readout signal

3.2. EXPERIMENTAL PROTOCOLS. The experimental protocol was developed for the first set of measurements which took place in the University of Nottingham at the Magnetic Resonance Centre. The protocol describes the experimental procedure which was used to investigate the scanner in terms of its vibration and acoustical behaviour. The other sets of measurements made on Oxford FMRIB scanner and Cambridge WBIC scanner followed the same procedure. However, some of the equipment were changed so as to qualitatively and quantitatively improve the measurements. The changing executed for each other set of measurements will also be described.

3.2.1 MRC-IHR Nottingham Acoustic Measurements. Equipment and Procedure. The equipment has been selected in order to avoid any artefactual signals that could have been induced by the high magnetic environment (See Table 3). Moreover, the procedure made sure that the electrical equipment which contained ferro-magnetic components was progressively brought into the scanner room so as to control their potential attraction by the 3T static field or the damage of the high-field strength.

Half-inch Microphone - Free-field response Bruel and Kjaer Type 4155 2 Long Screened connecting cables Bruel and Kjaer 20 dB Attenuator Bruel and Kjaer Type ZF 0023 Sound Level Meter with digital display, Weighting network and time constants set for Type 1 measurements

Bruel and Kjaer Type 2230 S/N # 1033330

Sound Level Calibrator SPL 94 dB at 1000 Hz Bruel and Kjaer Type 4230 Microphone Support: Wood Rod (9x9 mm square section) + Strings + Blocks of wood

2 Channel Digital Audio Tape Recorder (DAT) AIWA HHB 1 PRO Stereo Headphones Pro-Luxe PX 921 Acoustic Cap : constructed with a calibrator insert and a plastic dehumidifier cap. (or silica-gel cap)

Bruel and Kjaer Type DB 0311

Table 3 : Equipment used for the acoustical investigation of Nottingham Scanner.

To make valid measurements in that hostile environment, the procedure takes into account the factors that depend upon the accurate representation of the acoustic situation. The calibration of the entire arrangement (Microphone, Cables and Sound Level Meter) was executed in the control room away from the high magnetic field. The Sound Level Meter and the DAT recorder were positioned approximately 5 meters away from the scanner and linked to the microphone by 2 screened cables. Only the microphone and a certain length of cable were in the bore of the scanner. To verify and avoid any contamination of the recorded signal, two careful operations were conducted. First, the influence of the field has been checked by using a calibrator or silica-gel cap to acoustically shield the microphone. The meter reading has then to drop by 10 to 15 dB if the effect due to the magnetic field is assumed negligible. Secondly, the cable and microphone were attached to a wooden rod suspended by 2 strings (fixed at the top-ends of the scanner), which was adjusted to position the microphone and cable along the axis of the RF coil. The cable was fixed straight onto the rod so as to avoid the scanner field-induced voltage. This wooden rod was also used for microphone placement inside the RF & gradient coils system and constituted a useful position ruler. The layout of the scanner room is shown in the following Figure 22.

The background noise was recorded inside the bore in “Fast” RMS time constant mode in order to ensure that it was sufficiently below the measurements that were taken. Then, the measurements of noise at three positions were conducted. The long axis of the microphone was aligned with the axis of the coil and pointed towards approximately the point where the top of the patient’s head should be. Sound pressure levels and equipment settings (DAT recording level, weighting and times constants) were noted on the Acoustic Measurement Table given in Appendixes. It was intended to take the sound pressure levels using both linear and A-weighted scales, and with “Fast” root-mean-square (RMS) time constant (effective averaging time 125ms), Peak (<100 µs) and Impulse time constant modes. The latter is designed to represent human perception and response to impulsive noise. For further temporal and frequency spectrum analysis, the left channel of the DAT tape recorded the microphone acoustic signal while the right channel recorded the electrical signals driving the 3 gradients coils picked off at current monitoring points. Figure 23 indicates and assigns the position points of the microphone along the axis of the RF & gradient coils system.

1 m

> 5 m

1.5 m

Moving trolley bed Gradient coils (Head position)

Scanner bore

Wall

2 m

Control Room

SLM & DAT

Wooden Ruler Screened Cables

Figure 22: Disposition in the Scanning room [MRC Nottingham].

3.2.2 MRC-IHR Nottingham Vibration Measurements. Equipment and Procedure. As well as the acoustics measurements, the equipment has been chosen in order to minimise the artefactual signals and damages that could have occurred during the measurements. The following Table 4 gives the reference of equipment chosen with particular application in mind to perform thorough and meaningful vibration measurements:

Piezoelectric Accelerometer Bruel and Kjaer Type 4375 SN # 1585785

Charges Amplifier Bruel and Kjaer Type 2635 SN # 814950

2 Accelerometer Leads

“Green” 318 pF

Melted Wax for accelerometer attachment 2 Channels Digital Audio Tape recorder (DAT) AIWA HHB1 PRO 2 Channels Oscilloscope HAMEG 20 MHz HM 203/5

SN # 22994

Table 4 : Equipment used for the vibration investigation of Nottingham Scanner. The important magnetically-transparent requirement for the vibration transducer is difficult to meet. Most of the accelerometers usually used for vibration measurements are piezoelectric devices made of metallic components. Nevertheless, according to the

123

0.40 m

0.45 m

Wooden ruler

Patient support side Patients’head side

Figure 23: Microphone positions along the axis.

String

Gradient coils RF coil

Microphone Position

3 T scanner bore

manufacturer, B&K accelerometers are very insensitive to magnetic fields. The sensitivity lies between 0.5 and 30 ms-2 per Tesla (0.005 and 0.3 g per Kgauss). However, there will be some induced motion of the seismic masses because it will have some magnetic permeability. Moreover, it is expected that currents will be induced in the accelerometer cable. It is therefore a good policy to check the “background noise” level of the vibration measurement system. This is done by mounting the accelerometer on a non-vibrating object and measuring the “apparent” vibration level of this arrangement. By using this procedure, the influence of the magnetic field on the accelerometer and cables can also be investigated. The idea was then to put an accelerometer isolated from vibration by a layer of plastic foam inside the RF & Gradient coils’ bore. Figure 24 describes this arrangement. If during the switching pulse sequence, the “apparent” vibration level of the installation is less than one third of the measured vibration (i.e. its “noise floor” is at least 10 dB below

RF Coil inner surface or Gradient coil system inner surface

Figure 25: Position of accelerometer on the inner surface of the RF coil.

Accelerometer

1.2

1.1

1.3

2.1

Patient’s head support

0.40 m

1.4

45° 1.1 1.2

1.3

1.4

3.1

3 Tesla Static Magnetic field + Switching fields Environment

Foam

Figure 24: “Noise Floor” Arrangement.

Accelerometer

the vibration levels to measure), the actual vibrations measurements can be obtained with reasonably good accuracy. The installation has to be rigidly attached inside the bore in order to avoid its attraction on the inner surface of the coils (due to electromagnetic induced motion). Following this stage, the accelerometer was mounted on different places on the inner surface of the coils. The mounting of the accelerometer on the surface was done with melted wax which allowed quick test vibration measurements and provided good dynamic response. The location of the mounting should provide a short and rigid vibration transmission path to the vibrating source avoiding any compliance and damping elements present in the structure. Figures 25 & 26 define the positions which have been chosen for the accelerometer mountings on the inner surface of the RF coil and the gradient coils system and also on the inner surface of the main bore, where vibrating behaviour appeared to engender important consequences in the noise generation. The inner surface of the RF coil is made of “Perspex”, and the inner surface of the Gradient coils system is made of “Epoxy”, both flat and robust. The inner surface of the main bore is made of aluminium sticky-backed tape. Therefore, the nature of the test surfaces are all suitable for wax mountings and should provide accurate vibration measurements. The temperature of surfaces should not cause any decrease in the coupling stiffness or detachment of the transducer. The random important vibrations should occur in the frequency range 0-5 kHz. Therefore the accelerometer B&K Type 4375 has been chosen for its matched frequency response and its sensitivity to record the expected acceleration. The charge amplifier B&K Type 2635 enables the integration of the acceleration signal measured by the accelerometer using its integration network and will convert the measurements into acceleration, velocity or displacement. The acceleration data, relevant for further analysis, was then recorded on the left channel of the DAT tape; the right channel recorded the electrical signal. The positions of the accelerometers on the inner surface of the RF coil and on the inner surface of the gradient coils system have been chosen in accordance with the expected main excitation of vibration modes. At one end of the coil, measurements of the vibration were taken at four different positions separated by a angle of 45° [Positions : 1.1, 1.2, 1.3, 1.4]. The Charge Amplifier, set in acceleration, and an oscilloscope connected to its output monitored the vibrations at these

0.9 m (inner ∅)

0.27 m (inner ∅ )

RF+ Grd coils

Main Bore

Support

Square section Al. runners

Main scanner bore

Gradient Coils

RF Head Coil

Accelerometer

Figure 26: Position of accelerometer on the inner surface of the main bore.

4

Air cavities

positions and revealed the position of the most important acceleration. The corresponding accelerations were recorded on the DAT tape. On the middle of the RF and Gradient coils’surfaces, acceleration at the longitudinally equivalent positions [2.1, 2.2, 2.3, 2.4] was also recorded. Finally, the acceleration at the other end was measured and referred as position [3.1]. The settings of the charge amplifier (Measurement mode, Low and High frequency limits, Amplifier Sensitivity) of each positions were written on the Vibration Measurements Table given in Appendixes. 3.2.3 FMRIB Oxford Acoustic and Vibration Measurements. The FMRIB scanner was driven by many other signals than the typical EPI sequences. Impulse signal and pure tones were used so as to give more insight of the structural vibration behaviour of the gradient coil system. A 8 channel DAT recorder was then used in order to gain some time and to make the measurements easier. The microphone, via the Sound Level Meter, was delivering the acoustic signal in Channel 1; four accelerometers, via four respective charge amplifiers, enabled us to record the acceleration of the vibration at 4 positions simultaneously [Ch.2, Ch3., Ch4., Ch5.]. Finally, the EPI sequence electrical components or other signals were recorded on Channel 6. Details are provided in the Measurements tables given in Appendix. Figure 27 illustrates the disposition of the equipment in the scanner and control rooms. In fact, the equipment was far away from the scanner bore (more than 10 meters) as it was possible for the cables and leads connecting the accelerometers and Microphone to go through a duct in the wall separating the control room from the scanner room. The SLM, charge amplifiers, DAT recorder, Oscilloscope were then positioned inside the control room which is shielded against the high magnetic field. Figure 28 described the positions of the accelerometers an microphone in accordance to the references written in the measurements tables. Tables 5 & 6 give the technical references of the equipment used when assessing Oxford FMRIB scanner.

SLM Ch. AmpsDAT

Moving trolley bed

Scanner bore

Gradient coils (Head position)

Wooden Ruler

10 m

Screened Cables & Acc. Leads

Half-inch Microphone - Free-field response Bruel and Kjaer Type 4155 3 Long Screened connecting cables (3x4m) Bruel and Kjaer Type AO 0128 Sound Level Meter with digital display, Weighting network and time constants set for Type 1 measurements

Bruel and Kjaer Type 2230 S/N # 1033330

Sound Level Calibrator SPL 94 dB at 1000 Hz Bruel and Kjaer Type 4230 Microphone Support: Wood Rod (9x9 mm square section) + Blocks of wood

8 Channel DAT DATA recorder TEAC RD-135T ISVR S/N # 0091

Stereo Headphones Pro-Luxe PX 921 Acoustic Cap : constructed with a calibrator insert and a plastic dehumidifier cap. (or silica-gel cap)

Bruel and Kjaer Type DB 0311

Table 5 : Equipment used for the acoustical investigation of FMRIB Oxford scanner.

4 Piezoelectric Accelerometers 1: B&K Type 4375 SN # 1585787

2: B&K Type 4375 SN # 1239034 3: B&K Type 4375 SN # 972930

Figure 28: Position of accelerometers and the microphone inside FMRIB Oxford.

180 mm

1.2

1.1

1.3

2.1

1.4

3.1

2.2

2.3

2.4

4.1

Position 1 Position 2 Position 3

MicrophoneAccelerometer

RF Coil inner surface or

Gradient coil system

Patient’s head support

180 mm 180mm

4: B&K Type 4375 SN # unknown 4 Charges Amplifiers Bruel and Kjaer Type 2635

Accelerometer Leads.

B&K

Melted Wax for Accelerometer Attachment 8 Channel DAT DATA Recorder TEAC RD-135T

ISVR S/N # 0091 Digital Storage Oscilloscope Tektronix 2212 60 Mhz

ISVR SN # 0096

Table 6 : Equipment used for the vibration investigation of FMRIB Oxford scanner.

Disposition of the Equipment in the FMRIB Control Room.

3.2.4 WBIC Cambridge Acoustic and Vibration Measurements. The measurements at Cambridge WBIC followed the same experimental procedure as in Oxford and used the same equipment. Naturally, the layout of the scanning room was different and the gradient coil system as well. Figure 29 illustrates the disposition in the scanning room. The gradient coil system has its isocenter 162 mm away from the "foot" end of the scanner. Therefore, the position [1.1], [2.1], [3.1], [4.1] are 162 mm distant from each others. Unfortunately, the exact dimensions of the gradient coil system are unknown. The last picture shows a close look at the disposition of the Acoustic an Vibration equipment inside the bore of the scanner.

Console

SLM

Ch. Amps

DAT

Figure 29 : Disposition in the Scanning room (WBIC Cambridge).

Moving trolley bed

Scanner bore

Gradient coils (Head position)

Wooden Ruler

5 m

Screened Cables & Acc. Leads

1.5 m

Operating Room

Examination Room

Disposition of the equipment inside the scanner bore for the investigation of its vibration and acoustic behaviours (WBIC Cambridge Scanner).

4. RESULTS OF THE MEASUREMENTS. The result of the measurements made on the different scanners will be given in this chapter. The first session of measurements took place at the Magnetic Resonance Centre, University of Nottingham, the 1st July 2000. A second session was organised at the Oxford Centre for Functional Magnetic Resonance Imaging of the Brain, University of Oxford, the 20th September 2000. Finally, the last measurements of the project were made at the Wolfson Brain Imaging Centre, University of Cambridge, the 16th October 2000. Sound pressure levels inside the bore and acceleration of the vibration on the inner surface of the gradient coil system and/or the RF coil were measured and recorded following the guidance elaborated in the experimental protocol. Working in this high magnetic environment requires special awareness of the possible inaccuracy of the signals delivered by the transducers. First, the results of the arrangements, which have been used in order to avoid the errors, will be presented and will confirm the accuracy of the experimental procedure. Acoustic and Vibration results will then be given and commented upon in the context of their relative MR imaging settings, their relative position in the system and of course their strength. 4.1 RESULTS GIVEN BY THE EXPERIMENTAL ARRANGEMENTS.

4.1.1 Acoustic cap arrangement and Background noise. The influence of the 3T magnetic field was checked by using a silica-gel cap to acoustically shield the microphone. The meter reading had dropped by more than 30 dB (linear) when the microphone, fitted with its acoustic cap, was positioned inside the MRC Nottingham scanner (See Appendix 1). Therefore the effect due to the magnetic field on the acoustic measurement was assumed negligible. It is also important in any acoustic investigation to check the “background noise level” of the environment. The background noise levels were at least 30 dB (linear) lower than any of the noise levels recorded in all the operating scanners. This allows correct noise measurements. 4.1.2 “Noise floor” arrangement for the vibration measurement. The “background vibration” level of the vibration measurement system was checked by using an accelerometer isolated from vibration by a layer of plastic foam inside the RF & Gradient coils’ bore. While the scanners were operating, the vibration level delivered by this accelerometer was recorded. A comparison between this level and the vibration level usually recorded in Nottingham on the inner surface of the RF coil and the gradient coil system are presented in Figure 30 & 31. Over the frequency range [0 - 16000 Hz], the noise floor level was found to be 108.3 dB lower than any usual vibration level of the RF coil in average. Compared to any usual vibration level of the gradient coil system, it was found to be 78.4 dB lower in average. This confirmed that the use of the B&K accelerometer type 4375 was appropriate to measure the acceleration of the vibration in the MRI scanner system.

0 2000 4000 6000 8000 10000 12000 14000 160000

50

100

150

Frequency [Hertz]

Mag

nitu

de [

dB r

e N

oise

Flo

or]

Figure 30 : Strength of the RF Vibration Signal compared to the noise floor recorded by the accelerometer

0 2000 4000 6000 8000 10000 12000 14000 1600020

40

60

80

100

120

140Figure 31: Strength of the Gradient Coil Vib. Signal compared to the Noise Floor recorded by the accelerometer

Frequency [Hertz]

Mag

nitu

de [

dB r

e N

oise

Flo

or]

4.2 ACOUSTIC MEASUREMENTS. 4.2.1 MRC-IHR Nottingham scanner. In Nottingham, only one pulse sequence was selected for testing the noise level inside the scanner bore. This sequence is referred as “Continuous” signal in Appendix 1. There is nevertheless a peculiarity to the noise assessment in Nottingham. Indeed, two different measurements sessions were conducted as a layer of acoustic foam has already been installed by the Institute of Hearing Research on the inner surface of the main bore in order to reduce the high noise level. Therefore, noise measurements were made with and without the foam so as to quantify the noise improvement of this passive control, but also to measure effectively the noise induced by the vibration of the gradient coil system. The results of the measurements are displayed in Table 7.

Pulse Sequence Position Weighting Acoustic Noise Level - dB SPL Fast Impulse Peak

Measurements without the acoustic foam layer on the main bore Continuous 1 Lin. 127.8 128.8 134.5

A-Weighted 129.2 130.0 135.4 2 Lin. 134.2 135.6 141.4 A-Weighted 135.4 136.8 142.6 3 Lin. 127.3 128.3 136.8

A-Weighted 128.5 129.6 137.5 Measurements with the acoustic foam layer on the main bore

Continuous 2 Lin. 126.8 NA NA A-Weighted 128.1 NA NA

Table 7

Appendix 1 gives further details concerning the measurements. The acoustic foam layer installed on the inner surface of the 3T main bore provides a good attenuation of the noise sensed inside the head coil, in which the patients’ head is usually positioned, with measured attenuation of 7.3 dB(A). A weighted scale should be used in this context so as to mimic the response of the auditory system. In a structural context, it is better to measure the sound pressure level in dB linear as any frequency component of the noise contributes at the same scale to this measure. Figure 33 displays the acoustic spectrum of each position of the microphone inside the bore after the acoustic foam was removed. This was obtained by digital signal processing of the signals recorded onto the DAT recorder (Figure 32). The Power Spectral Densities of these three signals have shown a dominant frequency of 1923 Hz plus harmonics. The frequency corresponds to the Switching frequency of the EPI signal.

1.25 1.3 1.35 1.4

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8 P os it ion [2]

1.3 1.35 1.4 1.45

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8 P os it ion [1 ]

1.3 1.35 1.4

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

F ig u re 3 2 : Aco u stic s ig n a ls in s id e th e b o re - 1 S lice -

Tim e [s ec onds ]

Arb

itrar

y un

its

P os it ion [3 ]

0 1923 3846 5769 7692 9615 11538 13461 15384

-240

-220

-200

-180

-160

-140

-120

-100

-80

-60

-40Figure 33 : Power Spectral Density of Acoustic signals at the three different positions along the coil axis

Frequency [Hertz]

Mag

nitu

de [

dB]

Arb

itrar

yMiddle (1)End (2) End (3)

4.2.2 FMRIB Oxford scanner. In Oxford, two different EPI pulse sequences were selected for the acoustic investigation of the scanner. They are referred as EPI1 in Appendix 4 and EPI2 in Appendix 5. These two sequences were described in Chapter 3. The results of the measurements are displayed in Table 8, and further details are given in Appendix 4 & 5. A feature of these measurements is that different imaging settings of the scanner have been acoustically measured, so each pulse sequence presents 3 different imaging settings: transverse, oblique and sagittal. All of them make the gradient coil system vibrate differently and therefore the noise production differs.

Pulse Sequence Position Weighting Acoustic Noise Level - FAST dB SPL

Transverse Orientation

Oblique Orientation

Sagittal Orientation

EPI 1 1 Lin. 117.8 111.4 120.0 A-Weighted 118.7 112.5 121.1 2 Lin. 117.2 116.8 116.9

A-Weighted 117.2 116.7 117.8 3 Lin. 118.8 112.3 117.4 A-Weighted 119.0 113.0 118.4

EPI 2 2 Lin. 118.2 114.2 116.3 A-Weighted 118.4 115.2 117.4 3 Lin. 122.1 113.1 114.7 A-Weighted 123.2 114.3 115.5

Table 8

Figures 34, 35 & 36 display Power Spectral Densities of the acoustic signals measured and recorded while the pulse sequence EPI1 was operating. Each of these figures corresponds to a special imaging orientation. The Transversal and Oblique imaging processes result in the generation of noise with a dominant frequency of 1941 Hz which corresponds in particular to the second harmonic of the readout signal frequency (647 Hz), wherever the microphone is positioned. However, the sagittal imaging process gives the noise a different configuration. At the isocenter, or further away inside the bore (Position 3), the main frequency of the noise is shifted to 3235 Hz which corresponds to the fourth harmonic of 647 Hz, whereas the frequency of noise at the end of bore (Position 1) keeps its dominant frequency at 1941 Hz.

0 647 1294 1941 2588 3235 3882 4529 5176 5823 6470 7117 7764 8411 9058 9705-260

-240

-220

-200

-180

-160

-140

-120

-100

-80Figure 34 : PSD of Acoustic Signal - Pos.[1]&[2]&[3] in the Scanner Bore - Transversal Imaging

Frequency [Hertz]

Mag

nitu

de [

dB]

- A

rbitr

ary

Pos ition [1] - End Position [2] - IsocenterPosition [3]

0 647 1294 1941 2588 3235 3882 4529 5176 5823 6470 7117 7764 8411 9058 9705-260

-240

-220

-200

-180

-160

-140

-120

-100

-80Figure 35 : PSD of Acoustic Signal - Pos.[1]&[2]&[3] in the Scanner Bore - Oblique Imaging

Frequency [Hertz]

Mag

nitu

de [

dB]

- A

rbitr

ary

Pos ition [1] - End Position [2] - IsocenterPosition [3]

0 781 1562 2343 3124 3905 4686 5467 6248 7029 7810 8591 9372-260

-240

-220

-200

-180

-160

-140

-120

-100

-80Figure 36 : PSD of Acoustic Signal - Pos.[1]&[2]&[3] in the Scanner Bore - Axial Imaging

Frequency [Hertz]

Mag

nitu

de [

dB]

- A

rbitr

ary

Position [1] - End Position [2] - IsocenterPosition [3]

4.2.3 WBIC Cambridge scanner. In Cambridge, two different EPI pulse sequences were selected for the acoustic investigation of the scanner. They are referred as EPI1 in Appendix 22 and EPI2 in Appendix 23. These two sequences were described in Chapter 3. The results of the measurements are displayed in Table 9, and further details are given in Appendix 22 & 23. Three different imaging settings, Axial, Coronal and Sagittal have been “acoustically” investigated. This also resulted in various noises in terms of frequency components and levels. Figures 37, 38 & 39 display the computed Power Spectral Densities of these signals. It appears that the main peaks are correlated with the harmonics of the fundamental frequency 781 Hz. The distribution of the harmonics in terms of noise constitution, depends on the position inside the bore and the imaging settings. Generally, three main peaks contribute mostly to the noise concentrating the power at 781, 2343 and 3905 Hz, regardless of the position of the microphone inside the bore. Nevertheless, with an Axial or Coronal orientation of the image, the noise at the end has a main 781 Hz component whereas noises at isocenter and further inside the bore, keep a 2343 Hz frequency main component. Sagittal settings give the isocenter a 781 Hz component, while the end and further position inside the bore keep the 2343 Hz main component.

Pulse Sequence

Position Weighting Acoustic Noise Level - FAST dB SPL

Axial Orientation

Coronal Orientation

Sagittal Orientation

EPI 1 1 Lin. 116.4 106.5 115.9 A-Weighted 116.1 106.9 116.9 2 Lin. 117.5 112.8 116.3 A-Weighted 117.8 113.7 116.4 3 Lin. 116.2 111.0 110.4 A-Weighted 117.0 112.0 111.0 EPI 2 1 Lin. 111.7 104.1 110.8 A-Weighted 110.8 104.8 111.7 2 Lin. 113.0 112.0 117.4 A-Weighted 112.4 113.1 118.5 3 Lin. 117.2 111.2 115.9 A-Weighted 116.8 111.4 116.2

Table 9

0 781 1562 2343 3124 3905 4686 5467 6248 7029 7810 8591 9372-260

-240

-220

-200

-180

-160

-140

-120

-100

-80Figure 37 : PSD of Acoustic Signal - Pos.[1]&[2]&[3] in the Scanner Bore - Axial Imaging

Frequency [Hertz]

Mag

nitu

de [

dB]

- A

rbitr

ary

Position [1] - End Position [2] - IsocenterPosition [3]

0 781 1562 2343 3124 3905 4686 5467 6248 7029 7810 8591 9372-280

-260

-240

-220

-200

-180

-160

-140

-120

-100

-80Figure 38 : PSD of Acoustic Signals - Pos.[1]&[2]&[3] in the Scanner Bore - Coronal Imaging

Frequency [Hertz]

Mag

nitu

de [

dB]

- A

rbitr

ary

Pos ition [1] - End Position [2] - IsocenterPosition [3]

0 781 1562 2343 3124 3905 4686 5467 6248 7029 7810 8591 9372-280

-260

-240

-220

-200

-180

-160

-140

-120

-100

-80Figure 39 : PSD of Acoustic Signals - Pos.[1]&[2]&[3] in the Scanner Bore - Sagittal Imaging

Frequency [Hertz]

Mag

nitu

de [

dB]

- A

rbitr

ary

Position [1] - End Position [2] - IsocenterPosition [3]

4.2.4 General comments on the noise measured. The noise sensed inside the head coil can be considered to be very high. In general, most of the Echo-planar imaging sequences lead to a generation of noise higher than 110 dB(A). The highest sound pressure level was recorded in Nottingham to be 134.4 dB(A) (Fast SLM setting) and moreover with some peaks which can attain 142.6 dB(A). Most of the EPI sequences are very noisy, and there is a tendency for the sound pressure to be higher at the isocenter position inside the bore. Nevertheless, as the gradient coil system differs from scanner to scanner, it is difficult to accept this observation as a general assumption. The isocenter is not always the zone of highest sound pressure. The end of the bore and the inside can also represent a region of intens noise, depending on the scanner setting. It is however possible to predict when high noise will occur according to the imaging settings. It appeared that Sagittal orientation for Oxford FMRIB scanner and Axial orientation for Cambridge WBIC scanner induce higher sound pressure levels. This can be tallied with the fact that each imaging setting has it own configuration of gradient coils excitation. The Readout gradient which is known to cause the change of magnetic flux, and as a consequence to generate the “Lorentz forces”, is applied on either X, Y or Z gradient coil depending on the MR imaging. The Gradient coil system is a complex system where each gradient coil has a defined position and layout. Therefore they are expected to contribute differently to the vibration of the whole structure and then to the induced noise in terms of strength and frequency constitution. The studies of vibration levels, which is coming next, should give more insight.

4.3 VIBRATION MEASUREMENTS. 4.3.1 Accelerations and displacements of the vibration - Relationship. For the convenience of the experiment, the vibration was measured in acceleration by using the usual combination of an accelerometer and a charge amplifier. The acceleration of each vibration was recorded and its amplitude was displayed on a oscilloscope. Setting the charge amplifier to measure the acceleration allowed the conversion of the acceleration into velocity or displacement afterwards. To find the displacement from the acceleration, we just have to divide the acceleration level by the absolute value of the squared natural frequency. The natural frequency of the system will be taken as the main frequency component of the vibration found by computing the Power Spectral Density of the acceleration signal. 4.3.2 Vibration Results. The studies focused on the vibration of inner surfaces of both Gradient coil system and RF coil. It is actually important to know the range of vibration levels of the RF coil for patients’safety reason, even if this is not really relevant from a structural dynamic point of view. In fact, there is not a direct path for the vibration to propagate from the inner surface of the gradient coil system to the inner surface of the RF coil. There is sometimes an air gap between the two surfaces (outer and inner) of the RF coil if surfaces exist. The RF coils are usually designed with non homogeneous thickness. Nevertheless the ends of the RF coil are usually in direct contact with the gradient coil system. The wires are fixed with perspex material which gives homogeneity to the RF coil. Some RF shielding layer can be inserted between the RF coil and the Gradient coil system and therefore this could damp the original vibration from the gradient coils. Measurements of the RF coils’ vibration were made in Nottingham and Oxford, but the RF of Cambridge scanner system do not provide any surfaces of contact so its vibration was not measured. The same measurement procedure was used in both RF coil and Gradient coil investigations. Briefly, accelerometers were positioned according to a particular pattern (defined in Chapter 3) which was supposed to give more information on the structural mode excitation of the system. a. MRC-IHR Nottingham scanner. These measurements were made only with a audio DAT recorder and one accelerometer. Then, the levels of the measured acceleration had to be taken visually during the sequence from the signal displayed on a oscilloscope. The difficulty was to take the amplitude of the signal in such a short time. They may not correspond to the correct value of the acceleration but they still constitute a good estimation of the range of vibration levels of the system. The procedure was improved during the measurements sessions in Oxford and Cambridge. Table 10 displays the results obtained for the RF coil vibrations. More information are provided in Appendix 3. Figure 40 shows the Power Spectrum Densities of the recorded signals.

Position Acceleration Velocity Displacement

m.s-2 RMS mm.s-1 RMS µm RMS RF Coil Vibration Levels - Positions [1] - End of the RF coil - Foot end

1.1 212 17.5 1.45 1.2 424 35.1 2.90 1.3 460 38.0 3.14 1.4 212 17.5 1.45

RF Coil Vibration Levels - Positions [2] - Middle of the RF coil - Isocenter 2.1 424 35.1 2.90 2.2 495 40.9 3.39 2.3 619 51.2 4.23 2.4 424 35.1 2.91

RF Coil Vibration Levels - Position [3] - End of the RF coil - Cooling system End 3.1 354 29.3 2.42

Table 10

The vibration level of the Gradient coil system was much higher but kept the same frequency components. Table 11 displays the results of the vibration measurements made on the inner surface of the Gradient coil system. Figures 41, 42 and 43 are representing the PSDs obtained from signals recorded at different points of the system. Detailed information are given in Appendix 3.

Position Acceleration m.s-2 RMS

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Gradient Coil System Vibration Levels - Positions [1] - End of the coil - Foot end 1.1 2828 234.1 19.4 1.2 2121 175.5 14.5 1.3 2475 204.8 16.9 1.4 2828 234.1 19.4

Gradient Coil System Vibration Levels - Positions [2] - Middle of the coil - Isocenter 2.1 1414 117.2 9.68 2.2 1061 87.7 7.26 2.3 1061 87.7 7.26 2.4 1768 146.3 12.1

Gradient Coil System Vibration Levels - Position [3] - Cooling system End 3.1 1061 87.7 7.26

Table 11

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Figure 40 : PSDs of the vibration signals taken at different position on the inner surface of the RF coil.

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Like the acoustic signal, the vibration has a dominant frequency of 1923 Hz which corresponds to the switching frequency component of the tested EPI sequence. Harmonics of this fundamental frequency can also be noticed. The nature of the gradient coil vibration is preserved in the RF coil. They all have the same characteristics in terms of frequency but a discrepancy can be noticed by comparing the levels of vibration of both systems. Lower levels of vibration were expected on the inner surface of the RF coil but it seems that the vibration does not follow the same pattern. Whereas the recorded displacements of the RF coil middle circumference are higher than their corresponding values positions at the ends, the vibration of the Gradient coil system seems to increase at the "foot" end of the system. This could be due to the imaging setting and therefore to a particular excitation of one coil inside the Gradient system by the Switching electrical signal. Quantitatively, the vibration can be considered to be quite high. Levels of acceleration from 210 to 620 m.s-2 RMS (1.45 µm to 4.25 µm RMS in displacement) were measured on the inside surface of the RF Head Coil. The Gradient coil system gave much

higher levels : 1060 to 2830 m.s-2 RMS (7.25 µm to 19.4 µm RMS in displacement) on the inside surface. The introduction of the acoustic foam layer leads the vibration of the RF coil to fall down noticeably from a displacement of 1.45 µm to 1.09 µm at the same "foot end top" position (1.1). In addition, the vibration of the inner surface of the main bore, where the acoustic foam layer was originally placed, was estimated to be 0.3 µm in displacement with the same frequency characteristics. This may reveal the existence of induced-vibration of the whole MRI system caused by the vibration of the gradient coils. Furthermore, as the acoustic investigation revealed a noise improvement by placing this acoustic foam at this particular region of the system, one could think of an eventual involvement of sound wave reflections in the noise generation. b. FMRIB Oxford scanner. The measurement procedure was different in Oxford. In order to improve and multiply the measurements, an 8 channel DAT recorder and a storage oscilloscope were used. This equipment enabled to take simultaneously the vibration levels delivered by four accelerometers at four different positions and then to easily monitor the signals on the oscilloscope display thereafter. Two different type of EPI sequence were then assessed. Moreover, an investigation of the vibration engendered by different type of images settings could then be undertaken. Indeed, it was possible to run the scanner so as to give Transversal, Oblique and Sagittal MR images. There is then a possibility to find the difference that could occur in the dynamic response of the system when the latter is excited by the readout signal in different arrangements, i.e. different excited gradient coils (Gx, Gy, Gz). Appendixes 6, 7 and 8 give details on the measurements concerning the vibration of the Gradient coil system while the EPI 1 was operating for different type of images. The RF coil vibration behaviour has only been assessed on the "foot end" positions while the sequence EPI 1 was operating (See Appendix 15 & 19). The different following Tables will display the results of the measured vibrations and the Figures, their relative frequency spectrum. Transversal Imaging. Table 12 shows the results obtained when assessing the Gradient coil system excited under the EPI1 sequence. More information is given in Appendix 16. Figure 44 gives an idea of the possible correlation that exists between one acceleration signal (Vibration at the [1.2] position), and the Readout and Phase encode signals from the Imaging sequence. The figures also display the Power Spectral Densities of the different acceleration signals recorded on the inner surface of the system.

Position Acceleration m.s-2 RMS

Velocity mm.s-1 RMS

Displacement µm RMS

Gradient Coil System Vibration Levels - Positions [1] - End of the coil - Foot end 1.1 23.8 1.96 0.160 1.2 28.9 2.37 0.194 1.3 17.2 1.40 0.115 1.4 24.2 1.98 0.163

Gradient Coil System Vibration Levels - Positions [2] - Isocenter of the coils system 2.1 28.3 2.32 0.190 2.2 28.9 2.37 0.194 2.3 29.3 2.40 0.197 2.4 28.8 2.36 0.194

Gradient Coil System Vibration Levels - Position [3] - Longitudinal Axis 1.1 15.9 1.30 0.107 2.1 138.2 11.3 0.929 3.1 73.7 6.04 0.496 4.1 67.3 5.51 0.452

Table 12 - Transversal Imaging.

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Figures 44 : PSD of the vibration signals taken at different position on the inner surface of the Oxford Gradient coil system while processing Transversal Imaging. EPI1.

During the Transversal Imaging Process, the vibration response of the Gradient coil system seems be homogeneous. Indeed, accelerations and corresponding displacement levels do not show an obvious discrepancy between them. They are however much lower than the ones measured in Nottingham. The levels range between 17 and 74 m.s-2 for the acceleration which usually correspond to an estimated displacement in the order of 0.2 µm. One could distinguish a resonance state at 1941 Hz, which seems to convey most of the vibration power. This frequency corresponds to the second harmonic of the readout signal frequency (647 Hz). The distribution of the vibration power can sometimes be shifted to 3235 Hz depending of the position on the inner surface of the system.

Oblique Imaging. (Appendix 17)

Position Acceleration m.s-2 RMS

Velocity mm.s-1 RMS

Displacement µm RMS

Gradient Coil System Vibration Levels - Positions [1] - End of the coil - Foot end 1.1 28.4 2.32 0.190 1.2 23.7 1.94 0.159 1.3 18.3 1.49 0.123 1.4 16.3 1.34 0.109

Gradient Coil System Vibration Levels - Positions [2] - Isocenter of the coils system 2.1 28.6 2.34 0.192 2.2 20.6 1.69 0.138 2.3 28.6 2.34 0.192 2.4 25.4 2.08 0.170

Gradient Coil System Vibration Levels - Position [3] - Longitudinal Axis 1.1 18.5 1.51 0.124 2.1 47.2 3.87 0.317 3.1 22.7 1.86 0.152 4.1 42.4 3.47 0.285

Table 13 - Oblique Imaging.

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Figures 45: PSD of the vibration signals taken at different position on the inner surface of the Oxford Gradient coil system while processing Oblique Imaging. EPI1.

The Oblique Imaging process seems to cause the same range of vibration levels as the Transversal Imaging process. With acceleration from 16 to 48 m.s-2 and displacements of the order of 0.2 µm, the system behaves similarly and follows the same frequency distribution with the dominant frequencies 1941Hz and 3235 Hz (respectively, second and fourth harmonics of the fundamental frequency 647 Hz). Therefore, one could expect the existence of some effective radiation modes in this frequency region. Nonetheless, some vibration appear to be widespread in the frequency domain, for instance, 5823 Hz, 7117 Hz, 8411 Hz, especially for those position around the isocenter of the scanner bore.

Sagittal Imaging. (Appendix 18)

Position Acceleration m.s-2 RMS

Velocity mm.s-1 RMS

Displacement µm RMS

Gradient Coil System Vibration Levels - Positions [1] - End of the coil - Foot end 1.1 29.4 1.03 0.0363 1.2 30.4 1.49 0.0737 1.3 30.5 1.07 0.0376 1.4 32.3 1.13 0.0398

Gradient Coil System Vibration Levels - Positions [2] - Isocenter of the coils system 2.1 27.7 1.36 0.0671 2.2 30.9 1.08 0.0381 2.3 32.0 1.13 0.0395 2.4 31.4 1.10 0.0387

Gradient Coil System Vibration Levels - Position [3] - Longitudinal Axis 1.1 47.2 2.32 0.114 2.1 45.9 2.26 0.111 3.1 47.4 3.88 0.318 4.1 34.9 1.22 0.043

Table 14 - Sagittal Imaging.

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Figures 46: PSD of the vibration signals taken at different position on the inner surface of the Oxford Gradient coil system while processing Sagittal Imaging. EPI1.

Unlike the Transversal and Oblique Imaging Processes, Sagittal Imaging seems to cause a different response of the system. In fact, the vibration of the coils system are covering a wider frequency range. Excitation of higher harmonics can be distinguished regardless of the location on the inner surface. Although the acceleration are still varying in the same range of levels, the displacements are quantitatively lower than the others caused by different Imaging Processes. Nevertheless, this statement is only valid if the assumption of a harmonic excitation is verified which should be difficult to meet knowing the complexity of the system.

c.WBIC Cambridge scanner. An equivalent procedure was used in Cambridge. Different imaging settings were therefore similarly assessed as the previous ones in Oxford. The investigation focused on the structural behaviour of the Gradient coil system exited by two different EPI sequences referred as EPI1 and EPI2 in the concerned Appendixes. The vibration caused by EPI1 will be discussed in the following part in accordance to the particular imaging setting employed. Axial Imaging. (Appendix 28) Table 15 resumes the vibration levels recorded on the inner surface of the Gradient coil system while "EPI1 - Axial" Magnetic Resonance process was running the scanner. The PSDs of the signals are presented in Figures 47 which give moreover an example of one of the vibration records in the time domain and a close look at its nature.

Position Acceleration m.s-2 RMS

Velocity mm.s-1 RMS

Displacement µm RMS

Gradient Coil System Vibration Levels - Positions [1] - End of the coil - Foot end 1.1 83.9 17.1 3.480 1.2 68.5 2.79 0.113 1.3 36.5 2.48 0.168 1.4 50.4 2.05 0.084

Gradient Coil System Vibration Levels - Positions [2] - Isocenter of the coils system 2.1 97.5 19.8 4.030 2.2 82.6 5.61 0.381 2.3 33.8 2.29 0.156 2.4 28.3 5.77 1.170

Gradient Coil System Vibration Levels - Position [3] - Longitudinal Axis 1.1 85.2 17.4 3.540 2.1 110.1 22.4 4.570 3.1 77.1 5.23 0.355 4.1 84.6 3.45 0.140

Table 15 - Axial Imaging.

The spectra show that the vibrations are usually important at harmonics of the 781 Hz fundamental frequency. Indeed, it is obvious that every vibration dominates at 781 Hz, 2343 Hz (Third harmonic), 3905 Hz (Fifth harmonic) and 5467 Hz (Seventh harmonic). However, there is a large discrepancy in the levels. Acceleration were found to range from 15 to 97 m.s-2 and the corresponding displacements, from 0.08 to 4 µm. This large discrepancy could reveal some excitation of characteristic structural modes which should be in accordance with the data. This will be discussed latter. Nevertheless, these data already confirms that the isocenter is still the location of the highest vibration of the system.

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Figures 47 : PSDs of the vibration signals taken at different position on the inner surface

of the Cambridge Gradient coil system while processing Axial Imaging. EPI1. Coronal Imaging. (Appendix 29) Table 16 presents the results obtained when the scanner was set up to give a Coronal MR image. The relative PSDs are given in the Figures 48.

Position Acceleration m.s-2 RMS

Velocity mm.s-1 RMS

Displacement µm RMS

Gradient Coil System Vibration Levels - Positions [1] - End of the coil - Foot end 1.1 17.4 0.708 0.028 1.2 43.8 8.93 1.820 1.3 41.3 1.68 0.068 1.4 38.1 1.55 0.063

Gradient Coil System Vibration Levels - Positions [2] - Isocenter of the coils system 2.1 51.9 3.53 0.239 2.2 51.4 10.5 2.130 2.3 38.7 2.63 0.178 2.4 45.6 1.86 0.076

Gradient Coil System Vibration Levels - Position [3] - Longitudinal Axis 1.1 15.3 0.625 0.025 2.1 46.9 3.18 0.216 3.1 14.9 1.02 0.069 4.1 18.6 0.756 0.031

Table 16 - Coronal Imaging. The levels of vibration are lower than the ones produced by the axial imaging. Accelerations from 15 to 50 m.s-2 were measured which correspond to a range of displacement from 0.025 to 2 µm. The spectrum reveal the same frequency distribution as previously in which the total power of each signal is shared by the harmonics of the fundamental frequency 781 Hz. In general, these frequencies are the odd harmonics of the fundamental, but it seems that the other harmonics are also involved in the vibration process while progressing on the longitudinal axis, further away from the end and isocenter of the system.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.4

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0.0635 0.064 0.0645 0.065 0.0655 0.066 0.0665 0.067-0.3

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0 7 8 1 1 5 6 2 2 3 4 3 3 1 2 4 3 9 0 5 4 6 8 6 5 4 6 7 6 2 4 8 7 0 2 9 7 8 1 0 8 5 9 1 9 3 7 2- 2 2 0

- 2 0 0

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F r e q u e n c y [ H e r t z ]

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dB]

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P o s i t i o n [ 1 . 1 ]P o s i t i o n [ 1 . 2 ]P o s i t i o n [ 1 . 3 ]P o s i t i o n [ 1 . 4 ]

0 7 8 1 1 5 6 2 2 3 4 3 3 1 2 4 3 9 0 5 4 6 8 6 5 4 6 7 6 2 4 8 7 0 2 9 7 8 1 0 8 5 9 1 9 3 7 2- 2 4 0

- 2 2 0

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F r e q u e n c y [ H e r t z ]

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dB]

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P o s i t i o n [ 2 . 1 ]P o s i t i o n [ 2 . 2 ]P o s i t i o n [ 2 . 3 ]P o s i t i o n [ 2 . 4 ]

0 7 8 1 1 5 6 2 2 3 4 3 3 1 2 4 3 9 0 5 4 6 8 6 5 4 6 7 6 2 4 8 7 0 2 9 7 8 1 0 8 5 9 1 9 3 7 2- 2 6 0

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F r e q u e n c y [ H e r t z ]

Mag

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dB]

- A

rbitr

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P o s i t i o n [ 1 . 1 ]P o s i t i o n [ 2 . 1 ]P o s i t i o n [ 3 . 1 ]P o s i t i o n [ 4 . 1 ]

Figures 48 : PSD of the vibration signals taken at different position on the inner surface of the Cambridge Gradient coil system while processing Coronal Imaging. EPI1.

Sagittal Imaging. (Appendix 30) Table 17 summarises the levels recorded when the sagittal Imaging was operating. Figures 49 display the PSDs of the acceleration measured on the inner surface.

Position Acceleration m.s-2 RMS

Velocity mm.s-1 RMS

Displacement µm RMS

Gradient Coil System Vibration Levels - Positions [1] - End of the coil - Foot end 1.1 32.5 2.21 0.149 1.2 37.1 2.52 0.171 1.3 43.9 8.93 1.822 1.4 23.7 1.61 0.109

Gradient Coil System Vibration Levels - Positions [2] - Isocenter of the coils system 2.1 61.1 12.4 2.536 2.2 60.2 4.08 0.277 2.3 59.9 2.44 0.099 2.4 76.9 3.14 0.127

Gradient Coil System Vibration Levels - Position [3] - Longitudinal Axis 1.1 34.9 2.37 0.160 2.1 56.9 11.6 2.362 3.1 89.1 6.05 0.411 4.1 50.7 3.44 0.233

Table 17 - Sagittal Imaging.

The vibration levels are similar to the ones measured with the Axial Imaging process. Levels range between 23 and 89 m.s-2 for the acceleration and between 0.1 and 2.5 µm for the corresponding displacement. The dominant frequencies are still harmonics of 781 Hz frequency, but the power seems to be concentrated under three main peaks at 781 Hz, 2343 Hz and 3905 Hz. The distribution is then not so homogeneous as the previous cases in terms of frequency. We can even notice an isolated peak at 8581 Hz which appears to give a significant contribution to the vibration of the system.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.8

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0.0595 0.06 0.0605 0.061 0.0615 0.062 0.0625 0.063 0.0635 0.064

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0 7 8 1 1 5 6 2 2 3 4 3 3 1 2 4 3 9 0 5 4 6 8 6 5 4 6 7 6 2 4 8 7 0 2 9 7 8 1 0 8 5 9 1 9 3 7 2-2 4 0

-2 2 0

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-8 0

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dB]

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P o s it io n [ 1 . 1 ]P o s it io n [ 1 . 2 ]P o s it io n [ 1 . 3 ]P o s it io n [ 1 . 4 ]

0 7 8 1 1 5 6 2 2 3 4 3 3 1 2 4 3 9 0 5 4 6 8 6 5 4 6 7 6 2 4 8 7 0 2 9 7 8 1 0 8 5 9 1 9 3 7 2-2 4 0

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dB]

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P o s it io n [ 2 .1 ]P o s it io n [ 2 .2 ]P o s it io n [ 2 .3 ]P o s it io n [ 2 .4 ]

0 7 8 1 1 5 6 2 2 3 4 3 3 1 2 4 3 9 0 5 4 6 8 6 5 4 6 7 6 2 4 8 7 0 2 9 7 8 1 0 8 5 9 1 9 3 7 2-2 4 0

-2 2 0

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F re q u e n c y [ H e rt z ]

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dB]

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P o s i t io n [ 1 . 1 ]P o s i t io n [ 2 . 1 ]P o s i t io n [ 3 . 1 ]P o s i t io n [ 4 . 1 ]

Figures 49 : PSD of the vibration signals taken at different position on the inner surface of

the Cambridge Gradient coil system while processing Sagittal Imaging. EPI1.

4.3.3 General comments on the vibration measured. According to these results, general observations can be drawn about the vibration behaviour of the gradient coil system. The obvious point is that the system responds to the excitation pulse (i.e. Readout, Switching signals) by producing vibration in which the frequency distribution is in an important correlation with the pulse. The harmonics of the

Readout, Switching frequency are dominant. Depending on the Gradient coil system, certain harmonics appeared to contribute particularly to the frequency nature of the vibration. In Nottingham, the fundamental frequency was the main frequency component and the power was then distributed proportionally to the harmonics. Oxford and Cambridge scanners, however have shifted this frequency dominance to higher harmonics. The Oxford scanner vibration convey its power to the even harmonics (second, fourth, ...) which can vary depending on the Imaging. The Cambridge scanner distributes its vibration power to the odd harmonics of the fundamental (third, fifth, seventh, ...). However, there is somewhat a similarity in all these vibration differences. They are all dominant in the frequency range from 600 Hz to 5000 Hz. This might help to figure the structural modes which are expected to be mostly excited. The computation of the natural frequency of the gradient coil system in Chapter 5 should give more accuracy in finding out the modes involved in the vibration, even if this computation is subjected to important assumptions. Nevertheless, the vibration amplitude can already inform on the nature of the Circumferential and Axial modes. Referring to the particular location of the accelerometers and their respective levels of amplitude, the pattern of the structure distortion can be then revealed. One could distinguish either the n = 2 Circumferential mode, ovalling pattern (in Oxford and Cambridge) or the n = 3 Circumferential mode (in Nottingham). The existence of these modes cannot be confirmed with a total confidence as possible experimental errors could have occurred during the measurements. However, the existence of a m = 1 Axial mode can be somewhat confirmed at least in Nottingham, by the highest amplitude recorded at the isocenter position of the system which can be assimilated to the antinode of the pattern. The excitation of higher Axial modes such as m = 2 or m = 3 can also be assumed as the Oxford and Cambridge Gradient coil systems have isocenters that are not the exact symmetrical centres of the systems.

5. FURTHER INVESTIGATION AND ANALYSIS. During the measurements, it was sometimes possible to investigate the scanner by applying different "non-imaging" sequences which could have shed more insight on the structural behaviour but also on the linearity of the mechanical system. In Oxford, pure tones and sort of impulse sequences were used in order to reveal the characteristics of the system. However, this attempt was not successful every time, but a large amount of important parameters have then been calculated and seemed to be relevant. Consequently, a simple overview of the system behaviour has then been achieved. The accuracy of the analysis can be discussed given the important assumptions taken in order to simplify the analytical procedure. At least, this results can be taken as a foundation for directing further studies. The Computation of Coherence functions should tell about the correlation between the Vibration or Acoustic signals and the EPI component signals. In a dynamic context, damping of the systems has been evaluated via the calculation of the Loss Factor. An attempt to know more about the phase difference between the recorded signals has been tried but unfortunately led to irrelevant results. All these results will be described and discussed in the followings parts. Beside this, the computation of the natural frequencies of the gradient coil structure and acoustic cut-off frequencies of the duct have been executed according to the models and formulas given in Chapter 2. This aims to strengthen the assumption in which the involvement of special modes in the noise and vibration generation in MRI scanner is presumed. 5.1 CORRELATION BETWEEN IMAGING SEQUENCE, ACOUSTIC AND VIBRATION RESPONSES. Correlation between the Imaging Sequences, Acoustics and Dynamics of Nottingham and Oxford Scanners. The correlation between the input (EPI sequence) and the output (Sound Pressure Level or Acceleration) of the measuring system has been estimated for Nottingham and Oxford MRI scanner. Indeed, the EPI component signals were recorded on the channels of the DAT recorder and it was therefore possible to compare all these signals thereafter in the frequency domain. Figure 50 and Figure 51 represent respectively the comparison between one of the acoustic signal and one of the vibration signal with the electrical components of the Nottingham EPI sequence. Figure 52 and Figure 53 plot the same comparison obtained with the Oxford scanner. The most obvious correlation appears to be between the acceleration or sound pressure signals and the EPI Readout (Switching) signal in both cases. Nonetheless, the Phase encode (or Broadening) signal appreciably contribute to the frequency nature of the Acoustic and Vibration Signals. In order to quantity this correlation properties between the signals, the Coherence functions has been computed in part 5.1.2 and should reveal at which frequency, signals are in total accordance.

0 1923 3846 5769 7692 9615 11538 13461 1538410

-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Figure 50 : PSD of Nottingham Acoustic signal and EPI Electrical components signals

Frequency [Hertz ]

Arb

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itsAcoustic s ignal Switch elec. s ignal S lice elec . s ignal B roadening elec. s ignal

0 1923 3846 5769 7692 9615 11538 13461 1538410

-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Figure 51 : PSD of Vibration signal [Gradient Coil] and EPI Component Electrical signals

Frequency [Hertz]

Mag

nitu

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Arb

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nits

]Vibration signal Switch elec. signal Slice elec. signal Broadening elec. signal

0 647 1294 1941 2588 3235 3882 4529 5176 5823 6470 7117 7764 8411 9058 970510

-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

Figure 52 : PSD of Acoustic Signal and EPI1 Components Electrical Signals

Frequency [Hertz]

Arb

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itsAcou. S ignal Pos[2]Readout Signal Phase Encode SignalSlice Select S ignal

0 647 1294 1941 2588 3235 3882 4529 5176 5823 6470 7117 7764 8411 9058 970510

-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Figure 53 : PSD of Vibration Signal [Gradient Coil] and EPI1 Components Electrical Signals

Frequency [Hertz]

Arb

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itsV ib. S ignal Pos[1.2]Readout Signal Phase Encode Signal Slice Select Signal

5.2 MODAL ANALYSIS. 5.2.1 Computation of the structural and acoustic modes frequencies. The computation of the natural frequencies is subject to large assumptions. Indeed, the analytical estimation of these frequencies was based on formulas raised by the "Donnell Shell Theory" which compares the scanner to a Simply Supported Cylindrical Shell without Axial Constraint. Another important assumption is made on the constitution of the gradient coil system. In order to simplify the calculation of the natural frequencies, a model of the system as a cylindrical shell with a single layer of epoxy has been assumed. These large assumptions will be discussed in part 5.2.2 . Table 18 & 20 displays the structural natural frequencies obtained respectively for the Nottingham and Oxford scanner and in accordance with the formula (2.14) given in Chapter 2 (p.18) describing the Radial-Axial Modes. For information, the acoustic cut-off frequencies of the bore has also been computed in order to estimate the possible coupling that might occur between the acoustics and the dynamics of the system. The results obtained from the formula (2.58) given in Chapter 2 (p.29) are showed in Table 19 & 21. The “Matlab” program used for this computation is given in Appendix 31. a. MRC Nottingham scanner.

Natural Frequencies (Hz)

m = 1 m = 2 m = 3 m = 4 m = 5

n = 2 1499 3413 6539 10934 16601

n = 3 2434 4355 7509 11917 17589 n = 4 3812 5721 8882 13298 18974 n = 5 5597 7497 10659 15078 20756 n = 6 7781 9677 12837 17257 22937 n = 7 10362 12257 15415 19835 25515 n = 8 13341 15235 18392 22811 28492 n = 9 16717 18610 21767 26186 31866

Table 18

where n is the number of circumferential waves and m, is the number of axial half waves in the mode shape.

Acoustic Cut-off Frequencies (Hz)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

m = 0 0 1195 2188 3173 4156 5137 6119 m = 1 574 1663 2662 3651 4636 5619 6602 m = 2 952 2091 3109 4108 5099 6086 7072 m = 3 1310 2500 3539 4550 5549 6542 7531 m = 4 1658 2895 3956 4979 5986 6987 7982 m = 5 2001 3281 4363 5400 6418 7425 8425 m = 6 2340 3660 4762 5813 6841 7856 8862

Table 19

where n indicates the number of diametrical pressure nodes and m the number of concentric circular pressure nodes in Fourier-Bessel mode shapes for the pressure inside the duct. b. FMRIB Oxford scanner.

Natural Frequencies (Hz)

m = 1 m = 2 m = 3 m = 4 m = 5

n = 2 690 1001 1464 2050 2778 n = 3 1423 1681 2113 2706 3454 n = 4 2463 2712 3130 3714 4461 n = 5 3802 4049 4462 5041 5785 n = 6 5439 5685 6097 6673 7414 n = 7 7373 7619 8030 8605 9346 n = 8 9605 9851 10261 10836 11576 n = 9 12134 12380 12791 13366 14105

Table 20

Acoustic Cut-off

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

Frequencies (Hz) m = 0 0 1103 2021 2930 3838 4744 5650 m = 1 530 1535 2459 3372 4281 5189 6096 m = 2 879 1931 2872 3794 4709 5621 6531 m = 3 1210 2309 3268 4201 5124 6041 6955 m = 4 1531 2674 3653 4598 5528 6453 7371 m = 5 1848 3030 4029 4987 5927 6857 7781 m = 6 2160 3380 4398 5369 6318 7254 8184

Table 21

5.2.2 Discussion on the validity of the model and the results obtained. The "Donnell" Formula for Cylindrical shells involves the Young's modulus (Modulus of elasticity) and the Poisson's ratio which are both characteristics of the material. By assuming the homogeneity of composition of the gradient coil system, the accuracy of the estimation is somewhat compromised. An appreciable difference in the natural frequencies appears by changing the nature of the material from epoxy to copper. Indeed, the gradient coil system is usually designed with these two materials, but it is rather difficult to know exactly the proportion of each material in the whole complex system and even more to calculate the resulting properties of the combination. Moreover the geometrical disposition inside the system (thickness of the layers, wiring of each gradient coils) might influence the density and other constitutive properties. The Young's modulus or Poisson's ratio is also difficult to establish in this context. The prediction of the natural frequencies of the Radial-Axial Modes cannot then be accurate. Nevertheless, previous studies has showed that attributing the composition of a simple epoxy layer to a modelled system has almost given the same results as the experimental measurements. On the other hand, when changing the material from epoxy to copper, the formula tends to give higher frequencies for the same mode shape. By reaching a compromise, we can at least find the frequency range in which certain mode shapes appear. Concerning the scanner studied in this project, the measurement has already indicated the possible importance of the "Ovalling" mode shape (n = 2, m =1,2,3,...) and the n = 3, m = 1,2,3,... Circumferential mode shape. By looking at the frequency ranges in which these modes can be found, we can assure more the involvement of these modes. For instance, the MRC Nottingham scanner showed stronger vibration responses in the 1900Hz - 4500Hz in which the Radial-Axial modes of the orders n = 2,3,4 ; m = 1,2 can be found (Table 18). With the FMRIB Oxford scanner, the vibration responses were estimated to be maximum in the frequency range 1900Hz - 6000Hz which contains the involved harmonics given earlier. However, in this case the suspected modes are appearing at higher frequencies (Table 20) and therefore contradict the supposition in that case. Nevertheless, if we include in the model a part of copper, this should increase the values and then the particular modes may be met. Overall, referring to the measurements and the analytical analysis of the system, one could somewhat confirm an important involvement of these modes in the noise and vibration generation. 5.3 FURTHER INVESTIGATION.

5.3.1 Damping of the Gradient coil systems. Calculation of the Loss Factor. Theory and Assumptions. For investigating the damping of a mechanical system, it is usual to calculate its Loss Factor. The damping can be calculated from the slope of the envelope of the decaying vibration signal. The following part will present briefly the theory from which the LF is deduced and therefore recall the assumption taken in order to simplify the analysis of the Gradient coil system. The simplest damped system is the Single-Degree-Of-Freedom system. It can be represented by the following Figure 58, with a mass m, a spring of stiffness k and a viscous damper of damping constant c. If we consider the free vibration of this system, the equation of motion is:

0=++ kxxcxm &&&

The underdamped displacement of the mass is given by

( )φ+ω= ζω− tsinXex dtn

where ζ is the damping ratio and is equals to :

nm2cω

=ζ ( 0 < ζ < 1 )

ωn is the undamped natural frequency and is equals to:

mk

n =ω

ωd is the damped natural frequency and is equals to:

2nd 1 ζ−ω=ω

and φ is the phase angle.

m

k c

Figure 58

The free vibration of a SDOF system is decaying with the time and its envelope can be described by this two equations which respectively represent its positive and its negative parts.

( )x t x eotn= −ζω ; ( )x t x eo

tn= − −ζω Consider the positive part of the envelope: ( )x t x eo

tn= −ζω

The next maximum will occur one period Td

=

2πω

later and the value of x will be then:

( ) ( )x t T x eot Tn+ = − +ζω

Therefore, ( )

( )x t

x t Te enT

+= ≈ζω πζ2 ( Note that : ζω ζω

πω

πζn nd

T = ≈2

2 )

And, ( )

( )logex t

x t T+

= 2πζ . The damping can then be expressed in terms of a

logarithmic decrement : ( )

( )ζπ

=+

12

logex t

x t T

Consider an arbitrary reference for x : xref .

( )

log logeref

eo

refn

x tx

xx

t

=

− ζω

Now,

( )( )( )

( )log

loglog

log.e x

xe

x= =10

10

100 4343

Therefore, ( )

log log .10 10 0 4343x tx

xx

tref

o

refn

=

− ζω

or ( )

20 20 8 685910 10log log .x tx

xx

tref

o

refn

=

− ζω

Therefore, by plotting the decaying signal in the log10 scale and in accordance with the latter formula, one could deduce the damping of the system by calculating the slope of the graph. Two other parameters have to be introduced. Indeed, the slope of the graph can be expressed as follows:

σ ζω= =8 685960

60. n T

where, T60 is called the Reverberation Time which is the time for the vibration to decay by 60 dB.

Therefore,

ζω π

= = =60

8 685960

2 8 685911

60 60 60n n nT f T f T. ..

and the Loss Factor, η, is given by

η ζ= =22 2

60

.f Tn

There are therefore two parameters to know for calculation of the Loss Factor. The T60 which will be obtained from the graph, and the natural frequency of the system. The latter cause some difficulties to find out. In fact, we cannot consider the gradient coil system to act like a Single-Degree-Of-Freedom system. The existence of multiple degrees of freedom and therefore multiple natural frequencies require to chose one of them for the computation of the Loss Factor. The choice is based on the relevance of one of the natural frequencies in the Power Spectrum Density of the signal which is then supposed to be mostly responsible for the vibration of the whole system. The influence of the other modes with their different natural frequencies can be largely observed when plotting the decay of the vibration in the log scale. The slope does not totally correspond to a straight line. A appreciable discrepancy in the results confirms the involvement of other modes. The following part will present the results for the Loss Factor and will display the decaying of the recorded vibration signals. Estimation of the Loss Factor of the different systems. a. MRC Nottingham Gradient coil system.

0 0.005 0.01 0.015 0.02-6

-4

-2

0

2

4

6x 10

-9 Figure 58 : D ecay of the vibration signal w ith time - L inear scale.

Tim e [seconds ]

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G r. Coil Displacem ent[1.1]

0 0.005 0.01 0.015 0.02 0.025-260

-250

-240

-230

-220

-210

-200

-190

-180

-170

-160

Loss Fac tor : 0.0551

Figure 59 : Damping of Gradient coil V ibration [1.1] - Log Scale

Time [seconds]

Dis

plac

emen

t in

[dB

] A

rbitr

ary

units

20*Log10(Displacement)S lope

Figure 58 illustrates one example of the decay of the gradient coil vibration signals. When plotting this decay on the log scale, the slope is easily calculable (Figure 59).

Nevertheless, in some case, it is somewhat difficult to exactly measure the slope. The decaying usually start right after the EPI sequence has stopped to “switch”. The induced vibration therefore decays. Table 22 resumes the Loss Factor found for this gradient coil system.

Position of the accelerometer on the inner surface

Natural Frequency fn (Hz) (Assumed)

Loss Factor (Deduced)

1.1 1923 Hz 0.0551 1.2 1923 Hz 0.0544 1.3 1923 Hz 0.0594 1.4 1923 Hz 0.0538 2.1 1923 Hz 0.0542 2.2 1923 Hz 0.0640 2.3 1923 Hz 0.0549 2.4 1923 Hz 0.0565

Table 22

b. FMRIB Oxford Gradient coil system. Figures 60 and 61 present the decay of the vibration signal in the time domain measured in Oxford. The slope was more difficult to calculate as some modulation in the signal has appeared due to the expected involvement of different natural frequencies of excited modes. Nevertheless, the results obtained seem to give enough accuracy to consider them as meaningful information. Table 23 displays the results.

0.08 0.085 0.09 0.095 0.1

-5

-4

-3

-2

-1

0

1

2

3

4

5

x 10-9 F ig u re 60 : D ecay o f Grad ien t co il V ib ra tio n w ith th e tim e - L in ear S ca le

Tim e [s ec onds ]

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D is plac em ent [1.1]Readout S ignal

0.08 0.082 0.084 0.086 0.088 0.09 0.092 0.094-250

-240

-230

-220

-210

-200

-190

-180

-170

-160

Loss Factor : 0.0545

Figure 61 : Damping of Gradient coil Vibration - Log Scale

Time [seconds]

Dis

plac

emen

t in

[dB

] A

rbitr

ary

units

20*Log10(Displacement)Readout s ignal S lope

Position of the accelerometer on the inner surface

Natural Frequency fn (Hz) (Assumed)

Loss Factor (Deduced)

1.1 1941 0.0545 - 0.0660 1.2 1941 0.0401 1.3 1941 0.0577 1.4 1941 0.0418 2.1 1941 0.0465 2.2 1941 0.0762 2.3 1941 0.0512 2.4 1941 0.0456 3.1 1941 0.0511 4.1 1941 0.0483

Table 23

c. WBIC Cambridge Gradient coil system. Figures 62 and 63 represent the damping of the gradient coil system investigated in Cambridge. Table 24 displays the results obtained for the calculation of the Loss Factor. In this case, the involvement of several frequencies made the slope of the decay somewhat difficult to calculate. However, like the calculation of the Oxford system damping, the

results are considered to be accurate enough so as to give relevant information on the damping of this system. Table 24 displays them.

0.104 0.106 0.108 0.11 0.112 0.114 0.116 0.118 0.12 0.122 0.124

-3

-2

-1

0

1

2

3

x 10-8 Figure 62 : Decay of the Vibration signal with time - Linear Scale

Time [seconds]

Arb

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Displacement [1.1]

0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14-240

-230

-220

-210

-200

-190

-180

-170

-160

-150

-140

Loss Fac tor : 0.0677

Figure 63 : Damping of V ibration [1.1] - Log Scale

Time [seconds]

Dis

plac

emen

t in

[dB

] A

rbitr

ary

units

20*Log10(Displacement)S lope

Position of the accelerometer on the inner surface and

Imaging Settings

Natural Frequency fn (Hz)

(Assumed)

Loss Factor

(Deduced) 1.1

Axial

Coronal Sagittal

781 3985 2343

0.0677 0.0142 0.0248

1.2

Axial Coronal Sagittal

3985 781 2343

0.0168 0.0827 0.0324

1.3

Axial Coronal Sagittal

2343 3905 781

0.0364 0.0198 0.0745

1.4

Axial Coronal Sagittal

3905 3905 2343

0.0135 0.0235 0.0264

2.1

Axial Coronal Sagittal

781 2343 781

0.0895 0.0315 0.0659

3.1

Axial Coronal Sagittal

2343 2343 2343

0.0257 0.0280 0.0408

4.1

Axial Coronal Sagittal

3905 3905 2343

0.0184 0.0209 0.0339

Table 24

General comments on the results obtained for the Loss Factor. The Damping of the studied Gradient coil systems can be estimated to be around 0.05 (5 %) at least for Nottingham and Oxford scanners. These results could have been expected and represent usual Loss Factor for this kind of mechanical system. The estimation of Loss Factor for the Cambridge gradient coil system was more difficult to estimate as the technique for finding the damping involves important assumptions and the use of a single natural frequency. It was showed previously that the power in the vibration signal is distributed over several defined frequencies which are usually sharing almost equivalent energy. Therefore, it was somewhat difficult to agree with this technique and decide which frequency should be considered as the natural frequency of the assumed system. Nevertheless, the results still give an appreciable estimation of the Loss Factor which fluctuates in this case from 0.015 to 0.09 (1.5 % to 9%). 5.3.2 Further investigation with Pure Tones and Impulses. This kind of investigation was made in Oxford where it was possible to drive the Gradient coils with pure tones and a build impulse. This was a attempt to know more about the linearity of the system and the Frequency Response Function of the system which should have been obtained from the Impulse Response Function. In fact, the results did not really lead to be so relevant but still constitute a reasonable source of information. Figure 64 illustrate input of pure tones with frequencies 500Hz, 1000Hz, 2000Hz, 5000Hz and the resulting output signals recorded by accelerometers. Figure 65 illustrate the impulse created to drive the gradient coil and its impulse response recorded by the accelerometers. Finally Figure 66 displays the best spectrum obtained from the Impulse Response Function which should represent the FRF of the system.

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Figures 64 : Pure Tones of 500 Hz, 1000 Hz, 2000 Hz and 5000 Hz as electrical input signals of the Oxford Gradient coil system and respective Response signals recorded by the accelerometers.

0 0.005 0.01 0.015 0.02 0.025-0.1

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Impulse Response Signal - End Position [1.]

Figures 65 : Built Impulse Signal applied on the Y Gradient coil of the Oxford Scanner and resulting Response recorded by an accelerometer placed at the end Pos[1.] of the coil system.

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Figure 66 : Frequency Response Function of Oxford Gradient coil system estimated from the IRF of the Y Gradient coil.

General comments on the "Pure tones and Impulse" Investigation of the Oxford scanner. The "Pure tones" investigation revealed that the linearity of the system is more disturbed at low frequency than higher ones. Indeed, some distortions of the sine wave signal of discrete frequency (500 Hz, 1000 Hz) can be noticed and may tell about the involvement of different modes around these two frequencies. One could deduced that these two discrete sine wave excite some structural modes in the frequency surroundings according to the appreciable distortion and its periodicity. At higher frequency (2000 Hz, 5000 Hz), the linearity seems to be recovered. The impulse investigation can help in this problem of modal density. In fact, the Frequency Response Function can be obtained from the Impulse response and then reveals the natural modes of the system. From Figure 66, one could deduce the modal density which appears to be 22 modes in the frequency range 0 - 10000 Hz. However, precaution in this estimation should be taken as a large number of Impulse Responses measured during the experiment appeared to be unsatisfactory. Figures 65 and 66 are one of the few recorded good signals. Therefore using this technique is somewhat inaccurate and mostly because the Impulse used cannot be a real one for technical and practical reasons.

6. CONCLUSION. 6.1 NOISE AND VIBRATION CONTROL, OVERVIEW AND PROPOSAL FOR FURTHER INVESTIGATION OF THE PROBLEM. The noise and vibration control of the Gradient coil system therefore appears to be quite complex and resolving this problem of high noise and vibration occurring in the bore of MRI scanner, might involve a combination of several control techniques if appreciable improvements are desired. The investigation of the noise and vibration behaviour of the scanners studied in this project has showed important characteristics on which the development of control system should be based. Attempts to control both vibration or noise in MRI scanners have already been made in the past and they showed promising views on what could be achieved in the future. The following part will describe the proposed techniques and their feasibility and possible employment on the studied scanners will be commented. An overview of the problem configuration is presented in order to point out the possible ways of tackling this problem of noise generation. Moreover, connections to the obtained results and characteristics revealed during the investigation of the scanners will be made and should justify the use of most of these control techniques. Figure 67 displays a simplified system which illustrates the noise generating process causing high noise and vibration levels in MRI Scanners.

Figure 67 : Block Diagram of the “Noise Generating Process”. Control systems can be build upon these elements which all cause the scanner to be noisy. Some controlling can be done particularly only on one of the elements or can combine work on several. Passive Method. The easiest technique is usually passive. However, one of the main difficulties in using passive method is the difficulty to control low frequencies which has appeared to be dominant is some cases. Moreover, MRI scanners do not usually provide so much space to install regular acoustic foam layers. Nevertheless, a good sound absorption was achieved on the MRC Nottingham scanner by simply placing a layer of acoustic foam on the inner surface of the superconducting magnet. The investigation of the frequency characteristics revealed a direct correlation between the vibration recorded on the inner surface of the main bore and the one recorded on the Gradient coil system. This has confirmed the direct propagation of sound and the possible contribution of reflective waves in the noise levels. Therefore, any simple passive method such as acoustic layers should be tried if the scanner provides some convenient free space in its structure. The other potentially successful passive method that comes to mind after the investigation is the use of detuning and decoupling techniques. If a passive device can be designed so as to detuned or

Lorentz Forces

Gradient Coils Structure

Vibrations of The Whole System

Air Sound

decoupled the system from the known excitation pulse, this could represent one of the most enhancement in terms of vibration reduction according to the strong correlation observed between the frequency of the readout component of the Imaging sequence and the response of the gradient coil system. Stiffening and damping of the Gradient coil system could also contribute to the vibration attenuation but should maybe alter the design of the structure which is usually devoted to Magnetic Resonance Imaging. Active Method. Active control of noise and vibration could be an ambitious alternative to the passive methods. Experiments showed that passive insulating and absorbing methods are effective in noise reduction of MRI in the frequency range above 1000 Hz. Since low-frequency noise inside the bore is usually directly coupled with the vibration of the Gradient coil system, an alternative way of noise suppression is to actively control the structural vibration of the Gradient coil system and consequently reduce the sound radiation. Some previous work has been achieved by Qui J. et al (1994, [11]) who has carried out some simulation on the vibration control of the shell (Gradient coil system) by using distributed piezoelectric actuators. Multiple distributed bending in-plane piezoelectric actuators were used to suppress a number of modes in a given frequency range simultaneously. Indeed, the investigation has shown the possible involvement of particular modes which occurs in a certain frequency range. Although frequency ranges differs from scanner to scanner depending on the geometrical and structural nature of the Gradient coil system, the modes can be considered to occur at frequencies up to 8000 Hz, with a dominance at 2000 Hz - 4000 Hz. Therefore developing an active control system could give successful noise attenuation even if they need to be adapted to the nature of the structure and the pulse sequence. In fact, analytical modelling are required for the control system development and need to be based on a special equation of motion of the shell, the knowledge of the primary forces applied on the coils, the control forces of the actuators and all the design of plant and algorithm needed to implement successfully an active control system. For instance, Qui J. et al have based their control development on the Flugge’s equation of shell motion described in Chapter 2 and on some in-plane and bending piezoelectric actuators which were respectively coupled with axisymmetrical and asymetrical modes. The undertaken simulation showed that successful reduction of the amplitudes of circumferentially low modes and that a considered number of modes can be completely controlled when enough actuators are used and their sizes, positions and force amplitudes are optimised. As the investigation has put the responsibility of certain circumferential mode in the noise generating process, this simulation work brings large hope in the implementation of a future active vibration control in the Gradient coil system. Design Issues. Referring to the strong correlation observed and pointed out during the investigation, the development of a design protocol of Gradient coil system can be conceived. This constitutes perhaps the most efficient development in reducing the noise and vibration of MRI scanner at source. It is therefore worth to develop a design function which may lead to improve the dynamical and sound radiation response of the Gradient coil system. The obvious development that comes first to mind is to define a system transfer function between the input (i.e. gradient coil current, especially the readout) and output (i.e. acoustic noise or vibration) which will then be incorporated into the coil design protocol. Mechefske C. et al (2000, [13]) propose the Gradient coil design to be based on achieving

specific magnetic gradients at specific locations within the coil while maintaining an overall balance of the Lorentz forces generated. This should be achieved by using a transfer function that relates the coil conductor current to the Lorentz forces generated at all points along the conductor. The design issues can be solved in developing appropriate mathematical tools. Kuijpers et al (1999, [16]) has developed a semi-analytical model to investigate the typical acoustics characteristics of MRI Gradient coil system. Using the Sound Power as a design objective function, an optimum acoustic MRI design has been observed which occurs when there is a mismatch between the vibration level peaks and the acoustic near-cut resonances. According to the author, this mismatch can lead to a noise reduction of 5 dB(A) up to 15 dB(A) depending on whether the excitation is typical, single circumferential harmonic or multiple harmonics. In the project, the importance of the excitation pulse has been shown and can be considered as the obvious noise parameter. However, the mismatch between vibration level peaks and the cut-on frequencies could not be characterised during the investigation. Indeed, the computation of the acoustic cut-on frequencies relies too much on an analytical formula for the acoustics duct which cannot totally model the acoustic behaviour of the MRI Gradient coil system. Nevertheless, if further investigation are more concerned with the bore geometry and the vibration distribution, it will be possible and meaningful to characterise this mismatch as they are the relevant design factors and the parameters affecting the acoustic radiation characteristics. Finally, to complete the proposition of design factors, the casing and the mounting of the Gradient coils inside the MRI whole system are still appearing as important possible design factors which should highly influence the MRI noise. 6.2 CONCLUSION. The investigation of the acoustic and vibration levels has revealed some important information which have given insight into the mechanism of the noise and vibration generation. The involvement of structural modes and particular parameters, such as natural frequencies, acoustics cut-on frequencies, Magnetic Resonance Parameters (TE, TR, Imaging) has been underlined and seemed to be coupled to the noise process. But the most important coupling could be thought to be the one between the acoustics or vibration and the EPI sequence component, more precisely the electrical readout signal which gives, through the generation of the Lorentz forces, the characteristic of the gradient coil vibration response. Harmonics of the readout signal frequency have been found to be excited in most of the cases and their relative existence in a certain frequency range in which natural frequencies of certain modes can be found as well, has led to consider these modes to be responsible of the highest sound radiation. However, the “inexact” prediction of the natural frequencies and the possible errors that could have occurred during the measurements sessions lead to temper this affirmation. The experimental part of this project has nevertheless given important measurements of the noise inside scanners devoted to functional MRI. Resonance imaging using a 3 Tesla magnetic static field and Echo-Planar Imaging sequences can be considered to be highly noisy with Sound Pressure Level which can attain 135 dB(A) inside the head coil and with their corresponding high levels of vibration. These levels are really interfering the diagnostics of the human brain function carried out by functional MRI at medical centres such as the MRC-IHR Nottingham, FMRIB Oxford and WBIC Cambridge. As well as being a source of disturbance for the patient, this high noise levels interfere with the stimuli used in the

medical investigation of brain functions which calls as well for an important need of noise improvement.

REFERENCES Chapter 1. Magnetic Resonance Imaging – Overview and system architecture. [1] • Haacke EM, Brown RW, Thompson MR, Venkatesan R. Magnetic Resonance Imaging – Physical principles and sequence design. Wiley-Liss 1999; Chapter I. [2] • Vlaardingerbroek MT, A. den Boer J. Magnetic Resonance Imaging - Theory and practice, “system architecture”. Springer, 1998; Chapter I, 13-19. [3] • Redpath TW. MRI developments in perspective. British Journal of Radiology, 1997:S70-S80. Measurement of Noise levels. [4] • R Hurwitz, SR Lane, RA Bell, and MN Brant-Zawadzki. Acoustic analysis of gradient-coil noise in MR imaging. Radiology, 1989, 173, pp. 545-548. [5] • M. McJury, A. Blug, C. Joerger, B. Condon, and D. Wyper. Acoustic noise levels during magnetic resonance imaging scanning at 1.5 T. British Journal of Radiology, Vol. 67, Nr. 796, p. 413-415, (1994). [6] • Mc Jury MJ. Acoustic noise levels generated during high field MR Imaging. Clinical Radiology 1995; 50; 331-334. [7] • FG Shellock, SM Morisoli, and M Ziarati. Measurement of acoustic noise during MR imaging: evaluation of six "worst-case" pulse sequences. Radiology 1994 191: 91-93. [8] • FG Shellock, M Ziarati, D Atkinson, D-Y Chen . Determination of gradient magnetic field-induced acoustic noise associated with the use of echo planar and three-dimensional, fast spin techniques. Journal of Magnetic Resonance Imaging 1998; 8 ;1154-1157. [9] • Foster JR, Hall DA, Summerfield AQ, Palmer AR, Bowtell RW. Sound-Level measurements and calculations of safe noise dosage during fMRI at 3T. Journal of Magnetic Resonance Imaging – Special Issue on MR Safety. Sub28/10/99.

Issues: Noise and Vibration Control, Acoustic Modeling and Design of MRI Scanners. [10] • Naghshineh K., Chen W., and Koopmann G.H. Use of acoustic basis functions for active control of sound power radiated from a cylinder shell. Journal of the Acoustical Society of America, 1998, 103(4), pp. 1897-1903. [11] • Qiu Jinhao, Tani Junji. Vibration control of a cylindrical shell used in MRI equipment. Smart Materials and Structures, v 4, 1A, (Mar 1995), p A75-A81 ISSN: 0964-1726. [12] • Ling J. X., Amor W, DeMeester G. Numerical and experimental studies of the vibration and acoustic behaviors of MRI gradient tube. American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, v 84, n 3 Pt A/1(1995), p 311-317. Conference: Proceedings of the 1995 ASME Design Engineering Technical Conference (code 46152). [13] • Mechefske C.K., Wu Y., Rutt B.K. MRI Gradient coil cylinder sound field simulation and measurement. Structural Dynamics recent advances. Proceedings of the 7th International Conference, 2000, Volume II, p 995. [14] • Kuijpers A.H.W.M., Rienstra S.W., Verbeek G.,and Verheij J.W. The acoustic radiation of baffled finite ducts with vibrating walls. Journal of Sound and Vibration, 216(3), pp. 461 – 493. [15] • Kessels P.H.L. Engineering toolbox for structural-acoustic design _ applied to MRI Scanners. Ph.D. Thesis, Eindhoven University of Technology, 1999, the Netherlands. [16] • Kuijpers A.H.W.M. Acoustic modeling and design of MRI scanners. PhD. Thesis, Technische Universiteit Eindoven, 1999. ISBN 90-386-0648-6. Chapter 2. [17] • Shoei-Sheng Chen. Flow-Induced Vibration of Circular Cylindrical Structures. 1987, Chapter 4 “Circular Cylindrical Shells containing fluid”. [18] • Blevins R.D. Formulas for natural frequency and mode shape. 1979.

[19] • Fahy Franck. Sound and Structural Vibration. 1987 Academic Press. [20] • Reynolds D.D. Engineering Principles of Acoustics. Part 9.11, “Cross-Modes in Circular Cross-Section Ducts”