Quantitative Magnetic Resonance Imaging of Ultrasound Acoustic Fields

Conrad Leigh Walker

A thesis submitted in conforrnity with the requirements for the degree of Masters of Science

Graduate department of Medical Biophysics University of Toronto

@ Copyright Conrad Leigh Walker 1997

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Quantitative Magnetic Resonance Imaging of Ultrasound

Acoustic Fields

Conrad Leigh Walker

Masters of Science, 1997

Department of Medical Biophysics

University of Toronto

Abstract

This thesis presents the first demonstration of the use of phase-contrast magnetic resonance

imaging to detect the nanometer scale motions resulting from medical ultrasound. It provides a

new, non-invasive means to visualize and measure ultrasound fields in real tissue. The technique

requires a large oscillating magnetic field gradient of - .5 T/m which oscillates at - 1 MHz. The

fundamental issues in MR visualization of ultrasound fields are outlined and the accuracy and

sensitivity of the technique is demonstrated experirnentally. Based on this apparatus, we show an

ultrasound noise equivalent sensitivity of 3.9 nanometers displacement, 19 kilo Pascals pressure

and 12 milliwatts intensity for a 515 kHz US field. On the basis of potential accuracy, we believe

that it may become a new standard for ultrasound exposimetry and could provide an entirely new

platform for the study of complex interactions of medical ultrasound with tissue.

Acknowledgement s

A thesis dedication is usually written by its author to thank those who played a roïe in the work

the thesis describes. Tragically, the author of this t hesis, Conrad Leigh Walker, is no longer with us

and it has fallen to me, his father, to write his dedication. At the risk of offending, 1 will therefore

use this opportunity to also acknowledge my much beloved son.

Conrad Leigh was a remarkably positive and optimistic young man who embraced life to the

fullest extent while always having time for his fellows, from many of whom tributes have poured

in since his passing with their usual comment being of his cheery love of life. In his brief sojourn

among us, Conrad seemed able to touch many lives and the world, slways in dire need of optimism.

We will be the poorer for his passing.

I t is not for me to judge Conrad's academic achievements and indeed there are very few who

make significant contributions before the age of 23, however by al1 accounts, his efforts have added

a few drops to the ocean of knowledge. The great tragedy is the loss of future contributions. In his

pre-teens, he had already announced his intentions to find a cure for cancer and to win the Nobel

prize. Who knows what great things such unbounded self-confidence and obvious talent could have

achieved? On one of his last nights, deep in the African bush beside a fire beneath a jeweled sky,

we sat and talked 'til almost dawn of the wonderful world of science and of what mysteries might

yet be uncovered.

And now to the usual purpose of a dedication. In his academic life, there is a long list that

1 include with little knowledge of their contributions but with gratefulness for t heir support.

There were his friends in his volleyball gang, in particular Meg Iisuka and Andrea Reid

with whom he spent much time doing assignments. There was much help and discussion with

experimental details in the area of ultrasound physics with David Hope-Simpson and Cash Chin.

Kasia Harasiewicz and Arthur Worthington were helpfuI in making his ultrasound transducer work.

Al1 the students and staff in the MR group, including Mike Leitch, Scott Hinks, Jeff Stainsby, Bruno

Madore, Warren Foltz, Rajiv Chopra, Simon Graham, Atila Ersahin, Chris Macgowan, MarshaIl

Sussman and Rob Peters, were invaluable in understanding the physics of NMR as well as giving

instruction about the aches and pains of the Signa MR xesearch imager. In particular, he would

have recognized Dr. Steven Urchuk, whose help with his very first experiment was invaluable,

and Douglas Henderson for his careful machining. Among the faculty, the particular help of Drs.

Wright, Wood, Henkelman, Yaffe and Burns must be mentioned. His student committee members,

Drs. Hunt and Foster were helpful in guiding his research. Without able support staff, nothing

happens and Merle Casci, Anne Wong-Kerr and Christine Sudeyko must be thanked for their help

with summer students, stipends and university &airs. And of course, there is Dr. Don Plewes, his

supervisor, for whom he had the highest praise, and who 1 personally owe a debt of gratitude for

his wonderful support.

Contrary to popular opinion, Conrad did have a life outside Sunnybrook and his circle

of friends was wide and deep. There were many enduring contacts from Edmonton, SMU and

Kingston, which include the entire Fulton family; the Olsen family; Messrs. Gardiner, Johnson,

Tongue, and Jackson; Jason Reynolds; Andrew Leung; Justin Chant; Erica Watson; Caroline

Doidge; Jamie Partington; Jeffrey Metcalf; Matthew Gibson; and Chris Michelle. Special mention

must go to the McPherson family; to his friend and confidante, Sarah Henschell; to his sister,

Tamara, who was always there for him; and to his dearly beloved, Jean, whose totally dedicated

support gave him the strength to continue through the many crises that student life brings. 1

believe Conrad would also have made mention of his grandparents and his parents, Rose and 1,

who always encouraged him to reach for the stars.

I am sure to have missed others wlio were equally important to Conrad and for that 1

apologize.

For Conrad's loss we are al1 the poorer, but for his life, we are richer and that is what we

should remember. Thank you Conrad for the 23 years of your life; you will live forever in our

hearts.

Conrad Walker, Sr.

Contents

Abst ract ii

Acknowledgements iii

List of Figures viii

List of Tables ix

List of Symbols x

List of Abbreviations xii

Chapter 1 Introduction 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation 1

. . . . . . . . . . . . . . . . . . . . . . . . 1.2 Principles of Magnetic Resonance Imaging 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 General Theory 3

. . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Motion Quantification Using M M 5 . . . . . . . . . . . . . . . . . . . . 1.3 Current Applications of Phase Contrast fmaging 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Diffusion Spectroscopy 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Cochlear Fluid Oscillations 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 MR Elastography 10 . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Principles of Ultrasound Acoustic Fields 11

. . . . . . . . . . . . . . . . . . . . . . . 1.5 Measurement of Ultrasound Acoustic Fields 14 . . . . . . . . . 1.5.1 Acoustic Field Measurement Using a Calibrated Hydrophone 14

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Suspended Sphere Radiometry 16 . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Optical Diffraction Tomography 17

. . . . . . . . . . . . . . 1.5.4 Acoustic Field Measurement Using A Thermocouple 19 . . . . . . . . . . . . . 1.5.5 Summaxy of Ultrasonic Field Measurement Techniques 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Surnmary 21

Chapter 2 Quantitative MRI of US Fields 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 23

. . . . . . . . . . . . . . 2.2 Theoretical Feasibiiity for M R Imaging of Ultrasonic Fields 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Apparatus 28

2.3.1 Construction of the Magnetic Field Gradient Coi1 . . . . . . . . . . . . . . . . 28 2.3.2 Ultrasound Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.3 Supporting Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 ExperimentalMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1 Acquisition of Ultrasound Acoustic Field Maps . . . . . . . . . . . . . . . . . 38 2.4.2 Measurement of the Magnetic Field Gradient . . . . . . . . . . . . . . . . . . 39 2.4.3 Measurement of Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4.4 Verification of Linear Response to Transducer Excitation Power . . . . . . . . 45

2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.1 Acquisition of an Ultrasound Field Map . . . . . . . . . . . . . . . . . . . . . 47 2.5.2 Traveling and Standing Wave Field Distributions . . . . . . . . . . . . . . . . 53 2.5.3 Other Field Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.5.4 Measurement of Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.5.5 Linearity of MR Measurements of Ultrasonic Intensity . . . . . . . . . . . . . 58

2.6 Capabilities and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Chapter 3 Improvements and Future Research 62 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2 Improvements to Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.1 Hardware Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2.2 Improvements to Pulse Sequence Design and Experimental Methodology . . . 63

3.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Appendix A Magnetic Irnaging of Ultrasonic Fields 68

Appendix B CD-ROM 69

List of Figures

1.1 Orientation of bulk magnetization and applied magnetic field . . . . . . . . . . . . . 4 1.2 Timing diagram for a spin-echo MRI pulse sequence . . . . . . . . . . . . . . . . . . 6 1.3 Schematic of ultrasound wave propagation . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Ultrasound field measurement with a hydrophone . . . . . . . . . . . . . . . . . . . . 15 1.5 Ultrasound field measurement by radiation force . . . . . . . . . . . . . . . . . . . . 17 1.6 Ultrasound field measurement by optical diffraction tomography . . . . . . . . . . . 18 1.7 Ultrasound field measurement with a thermocouple . . . . . . . . . . . . . . . . . . . 20

2.1 Gradient requirements for ultrasound detection . . . . . . . . . . . . . . . . . . . . . 27 2.2 Schematic of a simple gradient for ultrasound detection . . . . . . . . . . . . . . . . 29

. . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Schematic of factors dictating skin effect 32 2.4 Circuit diagram for the oscillating gradient power supply . . . . . . . . . . . . . . . . 34 2.5 Schematic of the apparatus for MR detection of ultrasound fields . . . . . . . . . . . 36 2.6 MR pulse sequence for the MR detection of ultrasound fields . . . . . . . . . . . . . 39 2.7 Theoretical and experimental gradient profile for oscillating gradient . . . . . . . . . 41 2.8 Image of gradient distribution within US imaging cavity . . . . . . . . . . . . . . . . 43 2.9 Schematic of sound speed measurement using ultrasound time of flight . . . . . . . . 45 2.10 MR phase image in the absence of any ultrasound field . . . . . . . . . . . . . . . . . 47 2.11 MR phase image and plot of phase and field gradient in the presence of an ultrasound

field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.12 Gradient corrected displacement image and plot of ultrasound pressure and displace-

ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.13 Wavenumber plot of an MR detected ultrmound field . . . . . . . . . . . . . . . . . . 51 2.14 Filtered MR image and plot of ultrasound particle displacement and pressure . . . . 52 2.15 MR image of ultrasound traveling wave . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.16 MR image of ultrasound standing wave . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.17 MR image of an ultrasound field scattering off a 2.5 mm glas cylinder . . . . . . . . 56 2.18 Cornparison of sound speed measured by MR and ultrasound . . . . . . . . . . . . . 57 2.19 MR measurement of ultrasound intensity versus transducer power . . . . . . . . . . 58

3.1 Power spectrum of MR phase data of an ultrasound field . . . . . . . . . . . . . . . . 64

viii

List of Tables

1.1 NMR relaxation times for selected materials at 1.5 Tesla . . . . . . . . . . . . . . . . 5 1.2 Cornparison of techniques for ultrasound fields rneasurement . . . . . . . . . . . . . . 21

2.1 Ultrasound field parameters for diagnostic and therapeutic applications . . . . . . . 24 2.2 Cornparison of MR gradients used for imaging and ultrasound field mapping . . . . . 28 2.3 Cornparison of resistance for solid wire and Litz wire at 500 kHz . . . . . . . . . . . 33

List of Symbols

spin angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 dipole magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 gyromagnetic ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 staticmagneticfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 bulk magnetization vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Larmor frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 transverse magnificat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 spin-lattice relation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 spin-spin relaxation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 longitudinal component of magnet izat ion . . . . . . . . . . . . . . . . . . . . . 4 time constant for loss of spin coherence of a free induction decay . . . . . . . 5

. . . . . . . . . . . . . . . . . . . . . duration of oscillating gradient waveform 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . slice selecting gradient 5

freqency encoding gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 phase encoding gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 time between the RF excitation and the peak of the spin echo . . . . . . . . . 5 spinphase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 magnetic gradient waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 spatial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 transerve plane coordinates in the rotating frame . . . . . . . . . . . . . . . . 6 frequency of ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . position vector 7 time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ........ 7 displacement amplitude of the particle . . . . . . . . . . . . . . . . . . . . . . 7 wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 frequency of gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 phase angle between ultrasound and gradient waveforms . . . . . . . . . . . . 7 amplitude of a sinusoidai oscillating magnetic field gradient . . . . . . . . . . 7 time between application of oscillating gradients surrounding a spin echo in- version pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 magnetic field as measured in Tesla . . . . . . . . . . . . . . . . . . . . . . . . 10 laplacian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 partial differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

acoustic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 acoustic pressure amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 frequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 bulk modulus of compressibiIity . . . . . . . . . . . . . . . . . . . . . . . . . . 12 displacement vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ultrasound intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 acoustic impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 equilibrium density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ultrasound absorption coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 19 heat capacity of a material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 watts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 ultrasound intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

. . . . . . . . . . . . . . . standard deviation of the noise in the phase image 25 signal to noise ratio of the magnitude image . . . . . . . . . . . . . . . . . . . 25 signal to noise ratio of an MRI phase image . . . . . . . . . . . . . . . . . . . 26 acoustic impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 spin phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 number of turns in a coi1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 permeability of free space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 skindepth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 perrneability of copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 selfinductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 magnetic flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 electrical resistance (ohms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 peak voltage across the gradient coi1 . . . . . . . . . . . . . . . . . . . . . . . 33 mot ion-encoding gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 peak voltage across the gradient coi1 . . . . . . . . . . . . . . . . . . . . . . . 40 speed of sound in a phantom material . . . . . . . . . . . . . . . . . . . . . . 45 speed of sound in water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

us MW NMR RF PC Mm PCA PGSE kHz MHz SNR

List of Abbreviations

ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 magnetic resonance imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . nuclear magnetic resonance 3 radio frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 phase contrast MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . phase contrast angiography 8 pulsed gradient spin echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 kilohertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 megahert. z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 signal to noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xii

Chapter 1

Introduction

1.1 Motivation

The applications of ultrasound (US) in the medical field are widespread and growing rapidly. Ultra-

sound provides valuable diagnostic insight into anatomy and disease achieving imaging resolutions

ranging from millimeters for routine diagnostic applications [l] to tens of microns for high frequency

applications [2]. More recently, high power ultrasound has been used in a number of therapeutic

applications, including lithotripsy, physiotherapy and hyperthermia, as well as a means of thermal

ablation for minimally invasive thermal therapy [3,4]. The success of these diverse applications

is largely determined by the ability to craft specific acoustical field patterns within tissue in a

controlled and predictable fashion. For imaging or therapy in homogeneous media, acoustic fields

can be approximately predicted on the basis of classical diffraction theory [5] and verified with

invasive sensors [6,7] implanted within the tissue. In transparent media, direct observation of the

acoustic field can be achieved with Schlieren methods [8] or optical diffraction tomography [9].

These techniques, however, are not applicable to human tissues that are neither transparent nor

homogeneous. The ability to provide an accurate non-invasive means of visualizing the propagation

of longitudinal ultrasonic waves in tissue would fiil a critical need in the development of optimized

ultrasound t herapeutic and imaging strategies.

The key to visualizing longitudinal acoustic propagation is in the development of a means of

mapping the minute particle displacements that accornpany propagating ultrasound waves. Such

displacements are small, typically on the order of tens of nanometers, placing stringent requirements

on the sensitivity of the detection method. Proton rnagnetic resonance imaging (MRI), however,

has been used to detect small motions through phase variations induced in the presence of switched

magnetic field gradients [IO]. This technique has been applied to the imaging of spatial and temporal

flow distributions [II], to the quantification of brain [12] and muscle [13] tissue motion, and to the

measurement of cardiac strain [14]. Denk [15] used oscillating magnetic field gradients to detect

oscillatory flow in rat cochlea a t frequencies up to 4.6 kHz. Muthupillai [16,17] recently imaged

low frequency shear waves in ex-vivo tissue samples using oscillating gradients phase-locked to an

acoustic stimulus. In this case, the oscillation frequency was limited to 1.1 kHz, with shear wave

propagation speeds of a few meters/second in tissue and motion amplitudes ranging frorn 200-

1000 nanometers. For the case of longitudinal wave propagation relevant to medical ult rasound,

wave speeds and frequencies are typically three orders of magnitude higher with corresponding

motion amplitudes on the order of tens of nanometers. It is the purpose of this work to develop

a quantitative and non-invasive method for the visualization of ultrasound acoustic fields using

magnetic resonance imaging.

In order to understand the necessary concepts of motion detection with MRI, Section 1.2 will

review the physics of MRI and then outline the use of time varying magnetic field gradients to detect

rnicroscopic oscillatory motion. Section 1.3 introduces some of the current applications of M N to

the measurement of periodic motion. This discussion provides insight into the enormous capability

of MR to quantify small motions and motivates its extension to the detection of the minute periodic

motions that arise in the presence of ultrasound. Section 1.4 presents an introduction to the

physics of ultrasound acoustic fields while Section 1.5 describes the current methods for measuring

ultrasound fields, including a review of their relative strengths and weaknesses. This chapter closes

with a brief summary of the remaining sections of the thesis.

1.2 Principles of Magnetic Resonance Imaging

The following description of magnetic resonance imaging (MRI) contains only those det ails which

are necessary to understand the concepts of imaging ultrasound acoustic fields. For a more complete

description of the subject, refer to one of the excellent review papeïs or texts listed at the end of

the thesis [18-201.

1.2.1 General Theory

Magnetic resonance imaging (MM) is based upon the phenomenon of nuclear magnetic resonance

(NMR). Nuclei with an odd number of protons or neutrons have a spin angular momentum (S)

and an associated dipole magnetic moment (p ) :

where y is the gyromagnetic ratio and is dependent upon the nucleus being imaged. The most

common nucleus used for medical MRI is the hydrogen proton due to its prevalence within the

body (primarily in water). When a sample containing such protons (called spins) is placed in a

static magnetic field (Bo), a magnetization moment (M) is produced. Classically, this can be viewed

as the net alignment of an ensemble of dipole magnetic moments with Bo, such that M = C p.

By definition, the z-axis is defined as the direction of Bo. At equilibrium, M and Bo are in the

same direction; but if they are made to point in different directions, M will exhibit a resonance at

a frequency proportional to IBol, called tlie Larmor frequency (wL) . This can be viewed classically

as a precession of M about the static magnetic field vector in the same way a top precesses in a

gravitat ional field:

W L = -7Bo (1.2)

where the constant of proportionality between tlie magnetic field and frequency is y, the gyro-

magnetic ratio defined above. Figure 1.1 illustrates the precession of M about Bo and defines the

relevant axes.

If a radio-frequency (RF) energy pulse is applied at the Larmor frequency, the spins will

Figure 1.1: Orientation of M in relation to Bo and the defined axes.

absorb energy and move away from alignrnent with Bo. These RF pulses are classified by the angle

through which they are tipped. A 90" pulse is one in which the net magnetization (M) is tipped

90 degrees into the transverse plane. Initially, the dipole moments comprising M are coherent and

the transverse magnetization (Mzg) is a maximum. Subsequently, the two relaxation mechanisms

that act on M are called Tl and T2 relaxation.

In Tl relaxation, random fluctuating magnetic fields at the Larmor frequency stimulate

energy exchange between the spins and the surrounding lattice. The energy gained from the RF

pulse is lost to the lattice and the longitudinal component of the magnetization ( M z ) returns to

its equilibrium value. This relaxation process is approximated by an exponential growth with the

tirne constant Tl.

T2 relaxation results in a decrease in the transverse component of tlie magnetizatiori (Mzg).

Magnetic field coupling between neighbouring spins results in a broadening of the spin resonance

frequency spectrum and leads to spin dephasing and a loss of phase coherence. The consequent

decrease in Mzv is also approximated by an exponential with tlie decay time constant T2. Because

the same mechanism producing Tl decay also contributes to T2 relaxation, T2 5 Tl. Both Ti and

T2 are dependent upon local tissue properties, and hence are the major source of contrat in clinical

MR images. Table 1.1 lists the Tl and T2 relaxation times for water, agar and muscle tissue at

Medium Tl T2 water 2000 ms 2000 ms

2% agar 2000 ms 60 ms muscle 850 rns 47 ms

Table 1.1: ReIaxat ion coefficients for selected media

The observed transverse relaxation time is actually much shorter than T2 due to con-

stant rnagnetic field inhomogeneities across the sample. These inhomogeneities arise from non-

uniformities in the static field and from magnetic susceptibility differences between neighbouring

tissues. The effective relaxation, quantified by the tirne constant T2*, results in a rapid loss of

MR signal. The eEect of these constant inhornogeneities can be canceled by creating a spin echo.

In a spin echo sequence, a 90" RF pulse is applied bringing the magnetization into the transverse

plane. It remains in this plane for time T during which Tz* relaxation occurs. A 180" pulse then

Aips MxY through the origin. After another 7, spins dephased by inhomogeneities are rephased,

producing a strong signal, weakened only by pure T2 relaxation. The NMR signal is detected at

time TE = 27, where Mxy is at a local maximum. This process of excitation with 90" and 180" RF

pulses followed by signal detection is repeated every TR seconds until enough data (typically 128

or 256 repetitions) is generated for a complete MR image data set. Figure 1.2 shows the timing

diagram for a standard MR spin echo pulse sequence. The signals G,lice, Greadout, and GplLase

indicate the application of linear magnet ic field gradients necessary for spatial encoding of MT,.

1.2.2 Motion Quantification Using MRI

As stated above, upon entering the transverse plane the spins begin to precess with an angular

frequency w ~ . Consider the spins in a frame of reference (with axes a', y' and z ) rotating at wr;.

The transverse magnetization will be stationary in this frarne and aligned with the -y' axis after

a 90" RF pulse about XI. Now consider applying a magnetic field that changes in its magnitude in

a linear fashion with distance. This linearly changing magnetic field is referred to as a magnetic

field gradient and these can be applied with the direction of change in any arbitrary orientation. In

I m m .

Figure 1.2: Timing diagram for a standard spin echo pulse sequence.

the presence of a gradient, spins will precess at different frequencies depending upon their spatial

positions (see equation 1.2). Their transverse component of magnetization (Mq) will therefore

accumulate or lose phase relative to neighbouring spins. This phase (4,) is dependent upon the

spatial magnetic field gradient ( G ) and can be expressed as a function of position and time:

In the case of static spins, there will be a linear phase variation across the object following the

application of a linear gradient. However, if some spins are moving, their phase accumulation will

be more complex and is dependent upon the nature of the motion as welI as the MR pulse sequence

used. The term phase contrast MM (PC MRI) refers to any technique that exploits the spatial

variation of phase across an object (typically arising from spatially dependent motion) as a form

of contrast.

In the case of an ultrasound acoustic field with frequency fU, which propagates without any

non-linear interactions with the medium, the spins will oscillate in sinusoidal motion as described

by equation 1.11. The phase acquired by these moving spins will be:

$8 (r, T ) = 1' G(t) - e,,, sin(2s fut - k r) dt

where r is the position, t is time,

is the displacement amplitude of the particle, and k is the wave number.

In a time independent magnetic field gradient, the integral over an integer number of cycles will be

zero, as will the net phase accumulation. However, if the gradient oscillates a t frequency f ~ , the

spins will accumulate phase as given by:

where 8 is an arbitrary phase shift in the gradient waveform. Integrating over time T yields:

G~ . cm sinc[~( fu - fG)] cos(k r + 8) (1.6)

A phase image of the transverse magnetization will exhibit a sinusoidal variation across space with

a wavenumber equal to that of the ultrasound wave. The amplitude of this phase wave will be

proportional t o the amplitude of spin motion, suggesting the possibility of using phase contrast

MRI for quantification of ultrasound induced particle displacements. The phase is also dependent

on the frequency difference between the ultrasound and the gradient, which implies a sensitivity

only to motions occurring at a frequency near that of the gradient. This filtering effect can be quite

significant for long integration times.

The above derivation indicates that by applying an oscillating magnetic field gradient and ac-

quiring an MR image of the phase of transverse magnetization, ultrasound-induced particle motions

at the gradient frequency can be detected. Knowing the magnetic field gradient and integration

time, the local particle motion amplitude can be quantified, yielding spatial images of ultrasound

acoustic fields.

The concept of using oscillating magnetic field gradients to spatially encode periodic motion

is not new and the next section will describe some current applications of this technique. These

applications will demonstrate its power and sensitivity, further motivating the extension of phase

contrast MRI to the quantitative imaging of ultrasonic fields.

1.3 Current Applications of Phase Contrast Imaging

Phase contrast M N is a very common technique used for obtaining quantitative information about

particle motion and flow [21]. The most prevalent application of PC MRI is in the field of phase

contrast angiography (PCA) [22]. In PC A, constant motion encoding magnet ic field gradients are

applied providing sensitization to constant velocity blood Aow. PCA is therefore quite different

from the quantification of ultrasound induced moIecular motion that requires oscillating gradients

for detecting periodic particle oscillations. This section will focus specifically on MR phase contrast

applications that yield quantitative information about periodic motion and are thus, relevant to

ultrasonic field measurement. Any technique aimed at the measurement of ultrasonic fields through

displacement imaging requires significant motion sensitivity to allow quantitative determination of

minute ultrasound induced molecular oscillations. The applications discussed below will demon-

strate the high sensitivity of PC MRI to periodic motion and its capability for making quantitative

rneasurements of these motions.

1.3.1 Diffusion Spectroscopy

Diffusion measurements within tissue are of interest for identifying tissue structure [23], pathol-

ogy [24] and quantifying Auid exchange rates across membranes [25]. Diffusion constants are used

to describe the ability of molecules to move within a medium. If a mediutn is free of significant

obstructions and the molecules of interest are small, then they will move rapidly throughout. How-

ever, if there are structures that prevent or restrict the motion of molecules, the diffusional motion

will be limited and the mean diffusion lengtli will be reduced.

The most common means of probing these diffusional motions using MRI is with the use

of a pulsed gradient spin echo (PGSE) technique [26]. In this method, large gradients are applied

on either side of a 180' pulse, separated by a time A. The first gradient pulse dephases the spins

within the medium and the molecules are allowed to diffuse. Stationary spins will be rephased

by the second gradient pulse, but spins which have been displaced due to diffusion will not be

fully rephased. The amount of signal reduction from dephasing is dependent upon the distance the

molecules have traveled. This technique suffers from a significant limitation because of the finite

time required to apply the large diffusion sensitizing gradients. This irnplies that short diffusion

times can not be measured.

Callaghan and Stepinski [27] have proposed using oscillating gradients to probe the frequency

content of molecular diffusion in water. In this method, oscillating gradients are applied a t a range

of frequencies. In their random walk motion through water, molecules coIlide freely with each

ot her. In pure water, the displacement probability function, which describes the probability of a

molecule undergoing a displacement r in a given time, is a zero mean Gaussian. Using oscillating

gradients, only those molecules undergoing collisions at a frequency near that of the gradient will

contribute to MR signal reduction through spin dephasing. This is because these molecules will be

in significantly different positions during each half of a sinusoidal gradient lobe. Molecules colliding

much more slowly will not have a significant net displacement over one half gradient cycle and they

will be rephased by the next half cycle. Measuring the resultant signal loss enables calculation of

the frequency dependence of the diffusion coefficient. Because many cycles of the motion encoding

gradient can be applied within one experiment, sensitivity to small motion is significantly increased,

permitting measurement of diffusion coefficients for diffusion times much shorter than is possible

with a traditional PGSE technique.

Callaghan [27] has demonstrated the use of this technique for the rneasurement of the ef-

fective diffusion coefficient of water moving through a tube filled with closely packed spheres up

to frequencies of 1.67 kHz with motion encoding gradients of 6.78 T/ni. While the frequencies of

interest in the diffusion experiment are significantly lower tlian those for ultrasound irnaging, this

application does demonstrate the excellent frequency selectivity of MR motion sensitive imaging.

Furthermore, it illustrates the ability to quantify small periodic motions, which is fundamental to

the imaging of ultrasonic fields.

1 A.2 Cochlear Fluid Oscillations

Analysis of the motion of endocochlear fluid may provide a useful clinical means of diagnosing

pathologies of the inner ear. Phase contrast M M with oscillating motion encoding gradients has

been used to non-invasively quantify the periodic fluid oscillations within the cochlea of rats [15]. In

this application, a high intensity acoustic stimulus was applied to the ears of rats inducing periodic

oscillation of the endocochlear fluid. Using a motion-encoding gradient with a strength of 0.47T/m,

phase locked to the acoustic stimulus, MR phase images were acquired with motion sensitization in

al1 three directions. The stimulatory frequencies of interest ranged from 2.5 to 4.6 kHz. Using this

technique, functional images of cochlear mechanics could be obtained directly and non-invasively.

These results demonstrate the capability of phase contrast MRI to obtain non-invasive

measurements of periodic motion in complex heterogeneous structures. However, the frequency of

these oscillations is at least one hundred times lower than that required for imaging the motion

associated with ultrasonic fields.

1.3.3 MR Elastography

The distribution of mechanical strain throughout an object is of great interest in many fields of

science and engineering [28]. In medicine, tissue mechanical properties are of fundamental interest

in the detection of tumours. Tumours and other pathologies often exhibit elastic moduli that are

significantly different from healthy tissue and manual palpation is frequently used to probe these

tissue mechanical differences and identify mslignancies. Muthupillai [16,17] h a demonstrated the

use of phase contrast MRI with oscillating gradients as a means of non-iiivasively measuring the

mechanical properties of ex-vivo tissue samples. In this procedure, a shear wave is induced into

the tissue using a n oscillating mechanical stimulus. This wave induces periodic motion of particles

within the tissue in the same way that a propagating longitudinal ultrasound wave does. Standard

clinical imaging gradients (approxirnately 1 G/cm) are phase locked to the mechanical stimulus to

record waves at frequencies up to 1.1 kHz.

This experiment is very similar to that proposed for imaging ultrasonic fields. Their results

demonstrate the validity of using phase contrat MRI for accurate, quantitative and non-invasive

measurements of periodic displacements. Furthermore, by phase shifting the gradients with respect

to the mechanical stimulus, the temporal evolution of shear wave propagation through heteroge-

neous media could be visualized. Once again, the range of frequencies used in these experiments

was significantly lower than that required for ultrasonic field imaging. The corresponding shear

wave propagation speed was on the order of a few meters per second in tissue, with motion ampli-

tudes ranging from 200-1000 nm. Longitudinal ultrasound has wave speeds and frequencies three

orders of magnitude higher with motion amplitudes on the order of tens of nanometers. Successful

imaging of these ultrasound fields will require large, stable rnotion-encoding gradients capable of

oscillating at high frequencies t o achieve a signifiant increase in motion sensitivity over present

techniques.

1.4 Principles of Ultrasound Acoustic Fields

In order to evaluate the current techniques used for detection, measurement and visualization of

ultrasonic fields, i t is useful to understand the nature of these fields. Ultrasound is an acoustic wave

phenornenon, and as such, it involves the propagation of energy through the periodic oscillation

of particles within the medium of interest. It is reasonable to consider this medium as being

composed of small volume elements or voxels. These voxels are srnall enough that field properties

(such as pressure and density) are constant over the volume of the voxel, but large enough to include

millions of molecules (such that t he voxel is continuous with its surroundings). In the presence of a

stimulating ultrasound source, periodic pressure variations are induced within the medium. In the

case of acoustic waves in water and tissue, these pressure variations occur primarily in the direction

of wave propagation (longitudinal mode propagation). Using Newton's second law, an equation

describing the propagation of pressure variations can be derived:

with the solution:

where p is the acoustic wave pressure,

c is the speed of sound,

r is the position,

t is time,

p, is the pressure amplitude, w is the angular frequency (w = 27r f ), and

k is the wave number.

Hence, the pressure disturbance is in the form of a traveling wave that is sinusoidal in both time

and space. The propagation speed of this wave is given by:

where f is the frequency, and X is the wavelength.

If frequency and wavelength are known, then the speed of an ultrasound wave can be calculated.

The traveling pressure wave induces motion of the particles comprising the medium. The pressure

and particle displacement are related by Hookes law for a Auid:

where Bc is the bulk modulus of the medium expressing its compressibility and is the three

dimensional displacement vector. The wave of particle displacement is therefore given by:

Thus the particles within the medium oscillate in response to the pressure wave, but do not undergo

any net displacement over a complete period. Figure 1.3 illustrates the key parameters defining an

ultrasound wave.

While there is no net displacement of particles, there is a flow of energy as a result of the

propagating pressure wave. The ultrasound intensity 1, expressing the average power flow per unit

Figure 1.3: The key parameters defining an ultrasound acoustic wave.

area, can be calculated by considering the kinetic and potential energy of the wave. This intensity

is given by:

where 2, = pc is the characteristic wave impedance of the medium, with p being its equilibrium

densi ty.

The above derivations are based upon the assurnption that the wave is propagating in an

infinite, homogeneous medium. If this is not the case (as in tissue), reflection and scattering of the

wave occurs. These reflected and scat tered waves interfere both constructively and des tructively,

producing a complex acoustic field distribut ion.

In the particular case of a homogeneous medium with a smooth, inflexible back surface, the

ultrasound wave is completely reflected. In this case, the acoustic field is described by forward and

backward traveling waves occurring simultaneously. The combined wave is called a standing wave:

This wave no longer propagates with time. The positions at which there is no change in the pressure

(and no particle displacement) are called nodes. Undergoing maximal displacenient, the antinodes

have a peak amplitude twice that of a corresponding traveling wave. In complex heterogeneous me-

dia with a smooth reflecting back surface, the resultant wave will have both standing and traveling

components because of non-ideal refiection and interference.

1.5 Measurement of Ult rasound Acoustic Fields

The previous section described the fundamentals of ultrasound acoustic fields. Extensive research

into the detection and measurement of these fields has been performed and many methods are cur-

rently used. The measurement of the spatial distribution of acoustic fields is particularly challenging

and four common methods can be identified [29] as listed below:

1. Measurement Using a Calibrated Hydrophone,

2. Suspended Sphere Radiometry,

3. Optical Diffraction Tomography, and

4. Measurement Using a Thermocouple.

Each of these methods is useful in particular circumstances. The following subsections present the

fundamentals of the above measurement techniques, as well as their main strengths and weaknesses.

Examination of these techniques illustrates the need for a new method that is capable of non-

invasively imaging ultrasonic field distributions in heterogeneous media such as tissue.

1.5.1 Acoustic Field Measurernent Using a Calibrated Hydrophone

A calibrated hydrophone is a smaU transducer capable of transforming mechanical oscillations into

an electrical signal. For the purpose of acoustic field measurement, these oscillations are induced

by an incident ultrasound wave and the output voltage is calibrated in terms of absolute pressure.

Figure 1.4 shows a typical experimental setup for acoustic field measurement using a cali-

brated hydrophone [30]. The ultrasound transducer of interest is placed within a large water tank.

A material with high ultrasound absorption properties is placed at the rear of the tank to absorb

the induced traveling waves, preventing reflections and interference. The calibrated liydrophone

is placed within the tank and scanned over the acoustic field in a raster fashion. Voltage (and

therefore pressure) measurements are made at regular intervals, building up a quantitative spatial

map of ultrasonic pressure.

calibrated hydrophone 4

\ultrasound pulse

ultr&ound ultrasound transducer absorber

digital oscilloscope

Figure 1.4: Experimental apparatus used for measuring ultrasonic fields with a calibrated hy- drophone.

A calibrated hydrophone is a very common device for acquiring pressure distributions be-

cause it is inexpensive, relatively easy to implement and it provides quantitative acoustic pressure

measurements. These measurements, however, are invasive since the hydrophone must be placed

within the acoustic field for rneasurement. In general, it is desirable to have a hydrophone diarneter

significantly smaller than the ultrasound wavelength [31] so its presence will have a negligible effect

on the acoustic field spatial distribution. Furthermore, acoustic fields exhibit pressure variations on

the order of a wavelength and a hydrophone will rneasure an average pressure value over its active

area [32]. However, a hydrophone makes meaurements of an average pressure over a srnall ares. In

most real applications this is not possible. For medical ultrasound frequencies, wavelengths ranging

from 3 mm at 500 kHz to 0.15 mm at 10 MHz are produced. Cornmonly used hydrophones range in

size from 0.5 to 1 mm [33], which is a significant fraction of the ultrasound wavelength, particularly

at higher frequencies.

The need to scan the hydrophone throughout an acoustic field to acquire a quantitative

spatial map necessitates measurernent in a fluid medium. In non-fluid media (such as tissue),

acoustic field mapping is restricted to hydrophone measurements made in a few discrete points. As

rnentioned in Section 1.4, the acoustic field distributions within heterogeneous structures such as

tissue are complex. In order to adequately quantify these complex fields, it is necessary to visualize

their entire extent. The inability to acquire high resolution quantitative maps of ultrasonic fields

in non-fluid media is a significant limitation of calibrated hydrophones.

1.5.2 Suspended Sphere Radiometry

Suspended sphere radiometry falls into the general category of radiation force balance methods [34].

It is particularly useful for measuring the spatial distribution of the time averaged intensity [35] of

an acoustic field. In this method, the average force exerted on a srnall sphere by an acoustic field

is measured. Theoretical analysis of radiation force and its relation to acoustic field parameters is

complex, but the underlying principle is easily understood.

The radiation force acting on an object in a radiation field will be the surface integral of

infinitesimal forces exerted on its surface. By measuring the displacement of a small object as a

result of the radiation force, that force and the corresponding local acoustic field intensity can be

calculated [36]. Figure 1.5 shows the experimentd setup used for radiation force measurernent. The

ultrasound transducer of interest is placed in a large water tank with appropriate absorbing material

covering the back and sides. A small sphere (approximately 3 mm in diameter) is suspended

within the tank from a thin wire such that it hangs vertically. A signal is then applied to the

transducer, inducing an acoustic field and displacing the sphere. By measuring the displacement of

the supporting wire required to return the sphere to its original position, it is possible to calculate

the local radiation force and intensity. A spatial intensity map is acquired by moving the sphere

throughout the acoustic field in a raster scanning pattern.

The radiation force technique, when used to acquire acoustic field spatial distributions,

suffers from the same disadvantages as the calibrated hydrophone met hod described above. In

addition to the difficulty of converting the measured force into an acoustic field parameter, the

I I

transducer I ultrasound I

absorber I I

ultrasound 3mm diameter wave sphere

water tank

Figure 1.5: Experimental apparatus used for measuring ultrasonic fields with a suspended sphere radiometer.

detection sphere must be of adequate size to undergo a measurable deflection. This results in a

tradeoff between sensitivity and spatial averaging over the sphere cross-section. Furthermore, the

sphere must be free to move in under the influence of a radiation force, and thus this technique

is only applicable in a fluid medium, preventing its application to the memurement of ultrasound

spatial distributions in tissue.

1.5.3 Optical Diffraction Tomography

Optical diffraction tomography is a technique uscd for non-invasively obtaining quantitative spa-

tial pressure distributions within an acoustic field. It falls into the general category of Schlieren

techniques [37], but is modified to provide both amplitude and phase information of the irnaged

ultrasound wave. In the presence of an acoustic field, the optical refractive index of a medium is

changed [38,39]. Portions of the wave experiencing compression and rarefaction exhibit different

optical refractive indices. These differences lead to a pressure dependent diffraction. The integrated

optical effect is a complex quantity which describes the magnitude and phase of an optical beam

that has passed through an acoustic field and can be used to determine the acoustic displacement

distribution of the field [40]. Figure 1.6 shows the experimental apparatus used to determine an

acoustic field distribution wit h optical diffraction tomography. A light source and a detector are

placed on opposite sides of an acoustic field of interest. The photo detector records the integrated

optical effect produced by the acoustic field by rotating the light source and detector through 360'

and recording these projections for many angIes for which a spatial map of the integrated optical

effect can be obtained. A back projection of this data, followed by an inverse Fourier transform,

produces the pressure distribution of the acoustic field [41,42].

ultrasound transducer

Figure 1.6: Experimental apparatus used for optical diffraction tomography.

This technique is particularly powerful as it provides quantitative images of acoustic field

distributions in non-fluid media. However, it suffers from the significant limitation that it requires

an optically transparent medium, and therefore can not be applied to the imaging of acoustic field

distributions in tissue.

1.5.4 Acoustic Field Measurement Using A Thermocouple

Ultrasound will produce heat within an absorbing medium, resulting in a local temperature el-

evat ion.

intensity

Knowing the amplitude absorption coefficient (O) and the heat capacity (C), the local

can be determined from the measured rate of temperature elevation [43]:

Figure 1.7 shows the experimental setup used for thermocouple measurements of local intensity

within an acoustic field. The ultrasound transducer is placed in a water tank with absorbing

sides. A thermocouple is placed in the centre of a small chamber containing a highly absorbing

material (such as castor oil). This chamber is separated from the medium of interest by a thin mylar

membrane. The transducer is pulsed and the rate of temperature elevation within the thermocouple

chamber is recorded. This heating slope is used to calculate the local ultrasound intensity. A

spatial map of ultrasound intensity is acquired by scanning this thermocouple chamber throughout

the acoustic field and repeating the experiment a t many discrete points.

Equation 1.14 is based upon the assumption that al1 heat remains where it is produced and

does not diffuse. The rate of thermal diffusion is dependent upon local temperature gradients.

In order to ensure that this diffusion does not occur, it is riecessary to make the therrnocouple

chamber large enough that significant temperature gradients are not present near the thermocouple.

This, combined with short heating times, limits the error due to heat transport away from the

thermocouple, but also reduces the technique's spatial resolution. Another error arises due to

the presence of the thermocouple in an acoustic field. Viscosity a t the interface between the

thermocouple and the surrounding medium results in an increased initial temperature rise [44].

For this reason, the rate of temperature elevation is measured between approximately 0.3 and 1

1 water I I

ultras ound ultrasound transducer absorber -

Figure 1.7: Experimental apparatus used for ultrasonic field measurement with a t hermocouple.

s, where both thermal diffusion and viscosity induced heating at the thermocouple interface are

negligible [45].

1.5.5 Summary of Ultrasonic Field Measurement Techniques

Table 1.2 [46] lists several methods for measuring acoustic field spatial distributions. Also indi-

cated are the acoustic field parameters that they measure, their sensitivity, resolution and the

requirements t hey impose on the ultrasound propagation media.

From this table, it can be observed that there is no current method capable of non-invasively

acquiring acoustic field spatial distributions in opaque, heterogeneous materials such as tissue.

There is a distinct need for such a technique. As will be shown in the remainder of this thesis,

phase contrast magnetic resonance imaging with intense oscillating gradients is capable of non-

invasively quantifying the extremely small particle oscillations induced within an ultrasonic field.

Using this technique, we will show that M M can be be used to acquire a spatial map of ultrasonic

particle displacement amplitudes, pressure amplitudes and intensity based on fundamental NMR

constants, together with a knowledge of imaging timing parameters and the strength and spatial

distribution of the oscillating gradient.

Technique Measured Sensitivity Resolution Media Quantity ( W cm-') (mm) Requirements

Calibrated P 10-Io 0.5 fluid Hydrophone

Suspended Sphere 1 1 0 - ~ 2 fluid Radiometer

Optical Diffraction P IO-^ O. 1 optically Tomography transparent

Thermoprobe 1 10-1 O. 1 fluid

Table 1.2: Cornparison of current techniques for measuring the spatial distribution of ultrasound acous t ic fields.

1.6 Summary

This thesis continues with two chapters. Chapter 2 will outline in detail the use of phase contrast

MR methods to detect and visualize an ultrasound field form, a 515 kHz focused transducer. This

chapter is composed of the following five sections:

A theoretical study of the technical feasibility of detecting ultrasound field with MR,

The design considerations of the oscillating gradient and its electronic driver,

The design of the US field mapping apparatus and a discussion of the experiments to be

performed with this apparatus,

The results of the experiments under varying conditions of ultrasoiind amplitude and media

of different speeds of sound, and

A brief section on the fundamental limitations of the current apparatus in terms of minimal

ultrasound displacement amplitude, power density and pressure that can be detected.

Chapter 3 concfudes the thesis by suggesting techniques for the further irnprovement of this concept

based on changes to the experimental apparatus, alternative approaches to MR pulse sequence

design and improved data processing. This chapter closes with a brief discussion of future directions

and applications of MR mapping of US fields. Finally, Appendix A is the current preprint of a

papa summarizing the major points of Chapter 2, which hm been accepted for publication by

Ultrasaund in Medicine and Biology. Appendix B is a CD-ROM to play Quicktime animations of

propagating ultrasound fields.

Chapter 2

Quantitative Magnet ic Resonance

Imaging of Ultrasound Acoust ic Fields

2.1 Introduction

A non-invasive met hod for quant itatively imaging ultrasound acoustic fields in heterogeneous tissues

has many potential applications as indicated in Section 1.1. As described in Section 1.5, current

methods for rneasuring ultrasonic fields are either invasive or require transparent media. The

ability to provide an accurate and non-invasive means of visualizing the propagation of longitudinal

ultrasonic waves in tissue would fil1 a critical need in the development of optimized ultrasound

therapeutic and imaging strategies and as a basic tool in the study of ultrasound biophysics.

The capability of phase contrast MR to quantify small motions [47] suggests its use as an

imaging tool for probing the molecular oscillations induced in the presence of an ultrasonic field.

However, these motions are a t least 10 times smaller and 1000 fold higher frequency than those

currently measured [16] in MR elastography, so that their detection will be difficult. Section 2.2

provides a theoretical analysis aimed at establishing the feasibility of this technique. These calcu-

lations suggest that acoustic field mapping using phase contrast MRI is indeed possible, although

specialized hardware capable of producing large magnetic field gradients that oscillate at ultrasound

frequencies will be necessary. Section 2.3 discusses the design and implementation of coils capable

of yielding these gradients and the related hardware necessary for producing and imaging ultrasonic

fields. The experimental techniques and image processing necessary to obt ain quantitative images

of uItrasonic field distributions and improve their visual quality is described in Section 2.4. This

section also discusses the experimental techniques used for external validation of t hese quantit a-

tive MR measurements. The results of ultrasonic field imaging experiments demonstrating the wide

range of phenornena that can be observed are presented in Section 2.5. Quantitative speed of sound

measurements obtained from the MR data compare well with those made using an ultrasound time

of flight technique. F'urthermore, the expected linear relationship between MR measured ultra-

sound intensity and the transducer excitation power is demonstrated. These results illustrate that

this technique can be used for making quantitative measurements of ultrasound acoustic fields and

suggest that MRI can provide a new and non-invasive method for ultrasound exposimetry and the

basic study of ultrasound biophysics in tissue.

Theoretical Feasibility for MR Imaging of Ultrasonic Fields

Medical ultrasound can be divided into two main regions based upon its intended applications.

Table 2.1 indicates the two regions and their associated acoustic field parameters. Diagnostic

ultrasound applications use low intensities and relatively high frequencies [48] to achieve high

spatial and temporal resolution while producing negligible thermal effects on insonated tissues.

Therapeutic ultrasound makes use of much higher intensity ultrasonic fields a t lower frequencies 1491

to deposit significant amounts of energy within tissues and induce localized thermal effects.

Region Frequency Intensity Displacement Amplitude (nm) diagnostic 1 - 200 MHz 0.0001 - 1 W cm-' 9 x w3 - 18

therapeutic 0.5 - 2 MHz 1 - 800 W cm-2 9 - 102

Table 2.1: Comparison of ultrasonic field parameters for diagnostic and therapeutic applications.

In the absence of any non-linear acoustic interactions, the oscillation displacement amplitude

for particles (c ) in an acoustic field depends upon the ultrasound frequency ( w ) and intensity (1)

or pressure (P), as:

where p and c are the tissue density and ultrasound speed of sound respectively. The corresponding

displacement amplitudes are indicated in Table 2.1 for a tissue with a speed of sound of 1550

m/sec that is typical of muscle. These values were calculated for the range of parameters that

would span the largest and smallest amplitudes for the diagnostic and therapeutic range within

the indicated frequency and intensity values. This shows t hat the displacement amplitudes can

be very small ranging with the minimum displacement amplitude of N 1 0 - ~ nanometers for a

diagnostic field ( 2 0 0 r n ~ s / 0 . 0 0 0 1 ~ n - ~ ) to - 100 nanometers for a therapeutic ultrasound field

( 8 0 0 ~ m - ~ / 0 . 5 m ~ z ) . In order to assess the feasibility of applying phase cont ra t MR to the

irnaging of these ultrasound fields, it is necessary to estimate the minimum oscillating field gradient

amplitude needed to detect these motion amplitudes for a given MR imaging sequence.

The fundamental motion sensitivity limit for a given magnetic field gradient is determined

by the amplitude of the random phase noise present in an MR image. Motion that produces phase

accumulations of less than this noise cannot be detected. This phase noise (+,) is related to the

signal to noise ratio (SNRM) [50] in a magnitude image and well approximated

The SNRM is dependent upon two major factors: 1) the sensitivity of the RF coil receiving the M R

signal, and 2) the MR signal amplitude. The RF coil sensitivity is determined by its siae, geometry,

and tuning. The MR signal amplitude in a spin echo pulse sequence is dependent upon pulse

sequence timing parameters T E and TR and the tissue Tl and 2'2 as described in Section 1.2. For

example, using a motion sensitive spin echo pulse sequence with TE = 120 ms and TR = 2000 ms,

a small RF coil located near the imaged sample can achieve a magnitude SNRhf of approximately

50, yielding a phase noise of 4, = 0.014 radiuns. Subtraction of two phase images with opposite

motion encoding is necessary to remove any phase accumulations unrelated to the periodic motion.

This subtraction increases the phase noise by a factor of fi, resulting in a final phase noise of

4, = 0.20 radiuns. This represents the smallest phase accumulation that can be detected and the

phase caused by ultrasound-induced motion must at least equal or preferably exceed this value.

This can be expressed in terms of a phase image (SNR+ ). For the purposes of this feasibility study,

it is reasonable to expect an SNR+ of at least 10. Equation 1.6 describes the phase signal (4,)

amplitude produced by periodic motion with ampIitude cm occurring in an oscillating magnetic

field gradient of amplitude (Go). Rearranging this equation gives:

where y = 2.675 x 108 rad - T-' . s-' is the proton gyromagnetic ratio and T is the duration of

the motion encoding gradient. Expressing Equation 2.4 in terms of intensity (1) and frequency ( f )

(frorn Equation 1.12) yields the minimum required motion encoding gradient amplitude:

where 2, is the acoustic impedance (150 kg cmd2 s-' in water). Equation 2.5 suggests that the size

of the gradient can be reduced by increasing the duration of motion encoding (7). Unfortunately,

this parameter is constrained by T2 relaxation. In general, TE is chosen to provide adequate time

to accumulate phase without suffering excessive attenuation from T2 relaxation. This balance is

achieved when TE T2. Thus, the duration of the oscillating gradients is approximately limited

to T2. Using a spin echo experiment, it is possible to place the motion-encoding gradients on

both sides of the spin echo 180" pulse, which in Our case results in a maximum gradient duration of

approximately r = 100 rns. Figure 2.1 shows the gradient required to produce a phase accumulation

(4,) ten times that of the phase noise (SNR+ = 10). The horizontal axis is expressed in terms of

The amplitude of standard clinical imaging gradients is indicated in Figure 2.1. Evidently,

the ability to perform ultrasonic field imaging using phase contrast MR will be highly dependent

upon the production of large motion-encoding gradients. Also indicated in Figure 2.1 are the

Figure 2.1: Dependence of required motion-encoding gradient strength on ultrasounc j intensity anc -

frequency. The dashed line represents the amplitude of current clinical imaging gradients. Also indicated on the plot are the operating regions for diagnostic and therapeutic ultrasound.

diagnostic and therapeutic ultrasound regions previously discussed. The higher intensities and

lower frequencies used in ultrasound therapy are particularly amenable to MR ultrasonic field

imaging as they are less demanding on gradient strength.

From Figure 2.1, it is possible to estimate the motion-encoding gradient that must be pro-

duced in order to successfully image a particular ultrasonic field. For an ultrasound field with

an intensity of 1 W and a frequency of 500 k H z , the particle oscillation amplitude will be

<, = 36.8nm. To achieve an SNRCb of 10, the required motion-encoding gradient must have an

amplitude of Go = 0 . 4 ~ m-1 and oscillate at 500kHz. Table 2.2 compares the properties of the

proposed gradient t o typical imaging gradients in current clinical prüctice.

While the required gradient is indeed large and must resonate at a very Iiigh frequency, its

design and construction are not impossible. In particular, the volume over which the gradient must

Parameter Clinical MRI US Field Mapping Frequency (kHz) 2 500 Gradient (T/m) 0.01 0.40

Volume whole body US field ( 4 ( 207000) ( 17000)

Table 2.2: Comparison of current clinical MR imaging gradients to the motion-encoding gradient required to produce an SNR@ of 10 for a 1 W 500kHz ultrasonic field.

exist is significantly reduced for field mapping. This reduced volume can be traded for larger, faster

gradients. The following section discusses the engineering challenges associated with building the

required motion-encoding gradient and related apparatus for ultrasonic field imaging.

2.3 Apparat us

Results of the feasibility analysis of Section 2.2 suggest that phase contrast MR imaging of ultrasonic

fields is possible; however, the most significant challenge will be the construction of the required

gradient coil. Fortunately, the demand for large, fast gradients is offset by the reduced imaging

volume over which these gradients must extend. This section discusses the engineering challenges

associated with the construction of the required gradient coils and the related apparatus necessary

to produce and image ultrasonic fields.

2.3.1 Construction of the Magnetic Field Gradient Coi1

A simple gradient is composed of two loops of current carrying wire with radii a, separated by a

distance L as shown in Figure 2.2A. In this configuration the gradient will be produced dong the

z-axis which is parallel to the (Bo) field direction. The current I in eacli coil flows in opposite

directions, producing equal but opposing magnetic fields. Using

produced by a coil composed of N loops of wire dong its axis, can

the Biot-Savart law, the field

be calculated:

where po = 47r x IO-' ~ m - ' is the permeability of free space. The total magnetic field produced by

the two coils is the superposition of the fields from each coil as shown in Figure 2.2B. The magnetic

field reaches a minimum and maximum at the center of the left and right coil respectively. At

a point equi-distant from each coil, the opposing fields exactly cancel each other and the total

magnetic field is zero. The spatial derivative of this field distribution with respect to the z-ais

yields the magnetic field gradient (G) . The greater the number of loops comprising each coil, the

larger the gradient will be.

N Loops

Relative Axial Position

Figure 2.2: A) The orientation of two current carrying coils used to produce a magnetic field gradient. The current flows in opposite directions in eacli coil. B) An axial profile of the magnetic field (solid line) and gradient (dashed line) produced by the coils in (A). The opposing fields cancel at a point equi-distant from each coil. This is also the position of maximum gradient.

While it is necessary to have a gradient that is large enough to provide adequate motion

sensitivity, it is also important that this gradient exist over an acceptable volume. For this reason, a

gradient cornposed of only two windings as shown in Figure 2.2 is inappropriate. Rather, a gradient

is wound with many turns distributed over the length of the gradient coil. By varying the spacing

between neighbouring loops of wire in each coil such that loops near the center of the gradient

coi1 are more widely spaced than those further out, the gradient volume can be maximized. A

computer software package [51] was used to optimize the placement of the wire loops to rnaxirnize

the gradient amplitude and volume. Using this design package, a gradient coil layout was designed

using 48 loops of wire. The coil diameter and length were 4.5 and 10 cm respectively. The full-

width-half-maximum of the gradient, which describes the length over which the gradient is at least

half of its maximum value, was 5.6 cm.

The amplitude of the gradient is also proportions1 to the current flowing within the coils.

Maximizing the gradient produced requires maximizing the current. This can be accomplished in

two ways:

1. Minimizing the coil resistance, and

2. Maximizing the driving voltage.

The optimization of these two components will be discussed in the following sub-sections.

Minimizing the Coi1 Resistance

High frequency current within a conductor is constrained to its surface through the skin effect [52].

This reduces the effective wire cross-section A, and substantially increases the ac resistance, which

is inversely proportional to A. The skin depth d describes the depth from the surface of a conductor

a t which the electric field has been reduced to l /e of its surface value:

where a is the electrical conductivity (5.7 x 107 S m-' in copper), and p is the magnetic permeability

(e for copper). 63% of the ac current is carried within one skin depth of the surface. At a

frequency of 500kHq the skin depth in copper is reduced to 93pm, significantly reducing the

effective wire cross-section and resulting in high ac resistance.

Litz wire [53] is a high frequency wire designed to minimize the skin effect, resulting in

substantially lower ac resistances. Use of Litz wire for the gradient coi1 windings enables the

production of larger gradients than otherwise possible. In order to understand how Litz wire

exhibits a low ac resistance, it is useful to understand why the skin effect arises.

The skin effect is a consequence of self inductance within a wire. Current flowing in a

conductor establishes a magnetic field with magnetic flux lines that form concentric rings coaxial

with the wire. Consider a wire being composed of many individual wire elements, each carrying

current. The self inductance (L,) of a current element within the wire is defined as the total flux

(a) linking that elernent, divided by its current (1):

Only flux lines outside a particular current element will enclose it and contribute to its inductance.

Thus, the total flux surrounding a wire element is obtained by integrating the flux from a radial

point infinitely distant up to the point in question. The linked flux is therefore greatest a t the

center of the wire and decreases with increasing radial distance. The self-inductance, which is

proportional to this linked flux (Equation 2.8), is shown in Figure 2.3A as a function of radial

position. L, is a maximum at the center of the conductor and decreases with radial position. I t is

this radial dependence on L, within a conductor that produces the skin effect.

The impedance of an elemental wire at radial position r,

resistance (R) per unit length is given by:

with a self inductance L,(r) and

(2.9)

The inductive term introduces a frequency dependence to the impedance. At dc, al1 of current

elements within a wire will have equal impedance and the current will be uniformly distributed

throughout the conductor. As the frequency is increased, the impedance of the central wire elements

that have a greater inductance, will rise more quickly than tliose near the surface. Current now

flows near the surface where the resistance is lower. This is the source of the skin effect. The radial

dependence of conductor impedance is shown in Figure 2.3B for bot11 dc current (dashed line) and

ac current (solid line).

Inductance t Ls(r)

radial position

Impedance 4 Z(f,r)

-r -a O a r radial position

Figure 2.3: A) Radial dependence of self inductance within a conductor, showing a maximum induc- tance at the center of the wire. B) Impedance within a wire at dc (dashed line) and ac frequencies (solid line). At dc, there is no radial dependence of impedance. However, a t ac frequencies, the impedance is maximized at the center of the wire. This radial impedance variation is the source of the skin effect.

In order to minimize the skin effect, it is necessary to equalize the elemental impedance

throughout a conductor. Litz wire achieves this goal. It is composed of many strands of individually

insulated wire and is fabricated such that each individual strand h a an equal probability per unit

distance of being located at any radial position. This winding pattern is periodic over a distance

of now less than a wavelength of the voltage distributions over the wire for a given frequency. The

linked flux, and thus the inductance per unit length, is now equal for al1 strands and the current

remains uniformly distributed across the wire. The end result is a significantly lower ac resistance

than traditional single or multistrand wires. The Litz wire used for the gradient coil was composed

of 60 strands of individually insulated 0.13 mm diameter wire. Table 2.3 indicates the resistance

reduction realized by using Litz wire instead of standard single strand copper wire at 5OOkHz.

Table 2.3 demonstrates that the use of Litz wire in the gradient coil windings creates a

significant reduction in the winding resistance at 500 kHz enabling more efficient transmission of

current through the gradient coils.

Wire Type Resistance at 500 kHz - -

Single Strand Copper 270 m R / m Litz Wire 125 m R / m

Table 2.3: Cornparison of ac wire resistance for Litz wire and equivalent single strand copper wire.

Maximizing the Driving Voltage

The coil current required to produce a given magnetic fieId gradient cm be estimated using the

Biot-Savart law. In the coil used, a current of 14A, is required to achieve a gradient of O.4OT cm-'

with the gradient coil described in Section 2.3.1. This coil consists of 48 turns of Litz wire with an

inductance of approximately 39pW. If this coil is driven at a frequency of 500 kHz, the peak power

required to produce the magnetic field will be:

For the given coil configuration, a peak power of Pm = 25.4kW is required. This is clearly an

unreasonable power requirement. This magnetic field can be created much more efficiently using

a resonant circuit composed of an inductor and capacitor whose values are chosen to produce an

oscillation at the resonant frequency. The energy within the circuit oscillates between the capacitor

and the inductor. It is only necessary to provide enough power to replace losses due to resistance

within the circuit. The resistance of the above coil is 880 mR, implying that the power required to

drive the resonant circuit will be 180 W, which can be easily provided.

Given the coil inductance ( L ) , the appropriate capacitance (C) is chosen to provide a reso-

nance at the desired frequency:

which results in a required capacitance of 2.46nF for a resonance frequency of 515kHz. The

maximum voltage (V,) produced within the circuit can be calculated giveti the current I =

fi&,, cos(2n f t ) flowing within the coil, using the following equation:

The maximum voltage produced within the circuit is V, = 4851V. This significant voltage requires

special high voltage capacitors capable of withstanding large electric potentials without dielectric

breakdown. Dielectric breakdown wit hin the capacitors is currently the limiting factor preventing

the production of larger gradients.

The small amount of power required to maintain a constant current within the resonant

circuit can be provided from a dc power supply (30Vdc) and a transistor switching circuit driven

at the resonance frequency. The transistor circuit provides current pulses to the coil every cycle

to replace resistive losses. A schematic of the transistor circuit is shown in Figure 2.4. The two

transistors operate asynchronously, switching the dc power supply in and out of the gradient coil

circuit every cycle. The driver circuit, which ensures that the two transistors are not on at the

same time, eliminates shorts across the power supply terminals. The output current waveform is a

square wave, but the resonant gradient coil behaves as a filter ensuring that the circulating current

is a sinusoid at the resonant frequency.

Figure 2.4: Circuit schematic of gradient coil assembly and associated transistor driving circuit.

Driver Circuit

2.3.2 Ultrasound Transducer

O - 5 V 515 kHz

The other fundamental component in the ultrasonic field mapping apparatus is the ultrasound

transducer that is responsible for generating the ultrasound field. As described above in Section 1.2,

- -

the ultrasound transducer and magnetic field gradients must oscillate a t the same frequency. The

transducer is 5 cm in diameter with a focal length of 10 cm, resonant at a frequency of 515 kHz

with a bandwidth of 52 kHz and h a . an input impedance of 76 G? on resonance. Power is provided

to the transducer using an ENI2000 RF amplifier, capable of providing approximately 400 W of

continuous power.

2.3.3 Supporting Apparatus

Figure 2.5 shows the apparatus used for ultrasound acoustic field imaging. The central region is

composed of three concentric cylinders. The inner cylinder supports the RF coil that is used for

transmission of the RF waveform responsible for tipping the spins into the transverse plane and

receiving the resultant MR signal after motion encoding (see Section 1.2). The middle cylinder

contains the gradient coil. Placement of the RF coil inside the gradient coil improves the SNRM,

allowing greater sensitivity to the minute motions induced in the presence of ultrasound acoustic

fields. Cooling oil placed within this cavity cools the gradient coil during operation. The ultrasound

transducer is located at one end of the cylinder and positioned so that the ultrasound focus is near

the center of the gradient coil where the magnetic field gradient is a maximum. The diameter of the

inner cylinder is chosen such that rays drawn from the outer edges of the transducer to the focal

point would not interfere with the cylinder edges. This is an approximation of the beam diameter

at that point. The inner cylinder is filled with agar to support ultrasound wave propagation. The

dashed box in Figure 2.5 indicates the imaging field of view.

In order to study different wave propagation characteristics within the imaging region, the

apparatus was designed to support two different reflectors near the exit surface of the inner cylinder

as shown in Figure 2.5. By the choice of reflector geometry, it is possible to produce either standing

or traveling waves. Standing waves are generated when the wave passes tlirough the imaging cavity

and then is reflected back into the same imaging cavity. As the incident and reflected waves have

approximately equal amplitude, these two waves will interfere producing a complex wave pattern

of spatial nodes and anti-nodes. This wave pattern is characterized by a modulation of the wave

Oscillating Oil Ultrasound transducer

field

Figure 2.5: Schematic of apparatus used for ultrasound acoustic field imaging. The dashed box indicates the imaging field of view.

amplitude with time that does not appear to propagate in space. To obtain this reflection, a smooth

Lucite surface was used to cover the exit surface of the irnaging cavity. Conversely traveling waves

are produced when there is very little reflection. In this case, the reflector was removed to allow the

incident wave to propagate through the cavity where it was absorbed in a diagonally oriented, rough

gel-air surface. As the reflector is now absent, the wave appears to propagate in the imaging cavity.

Finally, more complex ultrasound fields can be seen by inserting scattering objects in the field to

provide images of wave propagation that exhibit reflection, scat tering and diffraction throughout

the imaging cavity.

Using the apparatus described above, MR phase images of ultrasound acoustic fields were

obtained. The next section describes the experimental techniques used to acquire these motion-

sensitive phase images, and the post processing necessary to convert them into quantitative maps

of particle motion and pressure amplitude within an ultrasonic field.

Experimental Methods

The apparatus described in Section 2.3 was used to obtain MR phase images of ultrasound acoustic

fields. This section describes the experimental techniques used to acquire and process these phase

images to produce spatial displacement, pressure and intensity maps of ultrasonic fields. The

experimental techniques associated with ultrasonic field imaging can be divided into 3 main areas:

1. Acquisition of raw phase images of ultrasonic fields,

2. Processing of these phase images to yield quantitative displacement, pressure and intensity

values describing the ultrasonic fields, and

3. Validation of the quantitative MR measurements of ultrasonic field parameters.

These areas are the subject of the next 4 sections. Section 2.4.1 describes the MR pulse sequence

used to obtain raw motion encoded phase images and to visualize their temporal evolution. The

image subtraction necessary to remove phase accumulations unrelated to motion is also discussed.

Knowledge of the spatial variation in the motion-encoding gradient is required to convert the re-

sultant phase images to quantitative ultrasound field parameters such as displacement, pressure

or intensity. A technique for determining the spatial variation of the motion-encoding gradients

is presented in Section 2.4.2. Using this information, the raw images can be translated into ul-

trasound field maps expressed in terms of particle displacement, local pressure and ultrasound

intensity. Once these field maps are obtained, it is important to validate their accuracy. One of the

quantitative ultrasound field measurements that can be made from the MR data is the ultrasound

wave propagation speed. The speed of sound can also be easily measured using an ultrasound time

of flight experiment. Section 2.4.3 describes both of these methods for quantifying the speed of

sound. Comparison of the results from the two difFerent methods cari be used to validate the MR

data. Section 2.4.4 presents a technique for verifying the linearity of ultrasound intensity measure-

ments acquired using MR phase contrast imaging. The ultrasound intensity a t a fixed point should

scale linearly with transducer excitation power. Validation of this fact provides further evidence

suggesting that MR acoustic field measurements are quantitatively accurate.

2.4.1 Acquisition of Ultrasound Acoustic Field Maps

Section 1.2 describes a technique for using MR phase imaging to detect the small, periodic par-

ticle motions induced by an ultrasound wave. MR motion sensitive ultrasonic field imaging was

implernented using a Signa 1.5T (General Electric Medical Systems) MR imager and the appara-

tus described in Section 2.3. The MR pulse sequence consisted of a standard spin echo sequence

with additional externally applied waveforms driving the resonant gradient coi1 and ultrasound

transducer. Figure 2.6 shows the MR pulse sequence used to implement this technique. The lines

labeled: RF, Gslice, Gphase and Greadout, represent the RF and the slice select, phase encode and

readout gradients in a standard spin echo pulse sequence. As stated in Section 1.2, the phase

signal produced due to periodic motion is dependent upon T (the duration of motion encoding).

The motion encoding gradient (Gmotion) and ultrasound (US) waveforms are placed within the

pulse sequence to completely fil1 the empty space between the RF excitation and the signal readout

without overlapping any of the other imaging waveforms. A spin echo pulse sequence with a TE

of 120 ms allows for 100 ms of motion induced phase accumulation. In order to maintain phase

accumulation after the n RF pulse that flips the spins 180' through the origin, it is necessary to

also phase shift the ultrasound waveforrn by 180' relative to the motion encoding gradient. A dual

channel arbitrary function generator (AWG2020, Tektronix) triggered from the MR imager every

TR = 2000 ms was used to provide the ultrasound and motion encoding gradient waveforms.

The temporal evolution of the ultrasound wave can be resolved by shifting the phase (8) of

the ultrasound excitation waveform with respect to the gradient waveform. By creating a movie

loop from a set of images obtained with different phase offsets, the propagation of an ultrasound

wave through space can be visualized.

Using the pulse sequence shown in Figure 2.6, a raw MR image is obtained. Particle motion

amplitude is encoded as phase in this image, but phase accumulations from sources other than

periodic motion are also present (see Section 1.2). Accumulations unrelated t o the ultrasonic field

can be removed by acquiring a second phase image with no ultrasound excitation. Pixel by pixel

subtraction of the two images removes phase that is constant from image to image. Only the

Figure 2.6: The timing of the application of 100 ms of 515 kHz ultrasound and gradient waveforms shown in relation to the spin-echo MR pulse sequence used for imaging. Adjustments of the parameter 9 were used to visualize the temporal evolution of ultrasonic fields.

phase signal resulting from ultrasound-induced motion remains. In order to translate this phase

information into acoustic field parameters such as particle displacement, local pressure or intensity,

it is necessary to know the motion-encoding gradient strength throughout the imaging field of view.

Equation 2.4 can then be used to calculate the ultrasound displacement amplitude. Section 2.4.2

describes a technique for obtaining the spatial distribution of gradient strength that can be used

to convert the raw phase images into ultrasound field maps.

2.4.2 Measurement of the Magnetic Field Gradient

The imaging technique described in Section 2.4.1 results in an MR phase image where phase is

proportional to particle motion occurring at the ultrasound frequency. In order to translate this

phase to a quantitative ultrasonic field map, it is necessary to know the local magnetic field gradient

strength throughout the imaging field of view. The measurement of this field distribution requires

two steps:

1. Absolute measurement of the magnetic field gradient along the central coil axis, and

2. Relative measurement of the spatial distribution of the magnetic field gradient throughout

the imaging field of view that can be scaled by the axial absolute measurement indicated

above.

The following two subsections will describe the experimental details associated with each of these

steps.

Axial Profile of the Magnetic Field Gradient

Absolute determination of the magnetic field gradient along the central coil axis requires the mea-

surement of the magnetic field distribution along that axis. This can be done using a small Faraday

coil. The spatial derivative of this rnagnetic field yields the gradient.

Faradays law (Equation 2.13) describes the voltage (V) generated across a coil experiencing

a changing magnetic flux (Q):

where N is the number of wire loops in the coil,

A is the cross sectional area of the coil, and

B is the magnetic field.

In the case of a magnetic field that is oscillating sinusoidally with a frequency ( f ) and amplitude

(BI), the peak voltage induced in the coil will be V, = -27rf . NAB'. Measuring this voltage

induced in a small coiI, it is possible to calcuhte the local magnetic field. By moving a Faraday

coil along the central axis of the gradient windings in srnall increments, the axial profile of the

magnetic field is obtained. The spatial derivative of this profile with respect to axial position yields

the magnetic field gradient. In order to minimize the noise introduced through differentiation, the

magnetic field profile is fitted to a 6th order poiynomial. Symbolic differentiation of the resultant

polynomial produces a very low noise estimate of the axial rnagnetic field gradient profile. The

solid line in Figure 2.7 shows the axial gradient profile obtained from Faraday coil magnetic field

rneasurements. The dashed line indicates a theoretical gradient profile calculated using the Biot-

Savart law as described in Section 2.3.1. As expected, the two curves are in reasonable agreement.

The differences between these curves are due to imperfect winding and current distributions in the

experimental gradient coil.

Distance (cm)

Figure 2.7: Profile of magnetic field gradient along central axis of gradient coil. Solid line cor- responds to experimental measurement made using a Faraday coil. Dashed line is a theoretical calculation for the same gradient coil. Differences between the two curves are due to non-ideal current distributions in the gradient coil

Figure 2.7 shows the absolute measurement of magnetic gradient along the central axis of

the experimental gradient coil. It is clear that this gradient will provide adequate motion sensitivity

to detect therapeutic ultrasound fields as indicated in Figure 2.1. In order to translate MR phase

measurements into ultrasonic field data, it is necessary to know the spatial distribution of this

gradient throughout the imaging field of view. Section 2.4.2 describes a technique for measuring

the relative amplitude of the gradient throughout space. The absolute measurement described

above is then used to scale this relative distribution, yielding the spatial gradient distribution

required for conversion of phase images to ultrasonic field data.

Spatial Distr ibut ion of the Magnetic Field Gradient

The technique described above provides an absolute measurement of the magnetic field gradient

along the central axis of the motion encoding gradient coil. In order to produce quantitative rnaps

of motion induced in the presence of an ultrasonic fieId, it is necessary to know the rnagnetic field

gradient throughout the irnaging field of view. This section will describe an M R technique for

measuring t his gradient distribution.

As stated in Section 1.2, the precessional frequency of spins and thus their phase is pro-

portional to the local magnetic field strength. After an initial 90" RF pulse, al1 spins are aligned.

Following a short pulse of the motion encoding gradient, the magnetization phase in each volume

element will be proportional to the local magnetic field strength. In order to obtain spatial in-

formation about the gradient, an MR phase image is formed using a 256ps, 1 ampere gradient

pulse within each TR of a standard spin echo MR pulse sequence. Subtraction of a second phase

image where there is no gradient pulsing removes phase variations due to time invariant sources,

leaving a resultant spatial map where phase is proportional to local magnetic field strength. Each

axial profile from this map is fitted to a 6th order polynomial and differentiated, as described in

Section 2.4.2, to obtain a relative value of the motion encoding gradient throughout the irnaging

field of view. This relative gradient map is then scaled by the absolute field profile (Figure 2.7)

obtained at the center of the gradient coil. Figure 2.8 shows a spatial map of the motion encoding

magnetic field gradient obtained using t his technique.

Using the absolute measurement of magnetic field gradient along a central profile of the

gradient coil (obtained with a Faraday coil), combined with the relative spatial gradient distribution

(obtained from a n MR phase experiment), the local gradient throughout the imaging field of view

is determined. This enables translation of MR phase measurernents into quantitative ultrasonic

field data including displacement amplitude, acoustic pressure and local ultrasound intensity.

Motion Encoding Gradient (Tlm)

-0.50 -0.30 -0.1 O O 0.10 0.30 0.50

Figure 2.8: Spatial distribution of motion encoding magnetic field gradient obtained from an MR phase imaging experiment . Pixel intensity corresponds to gradient amplitude.

2.4.3 Measurement of Speed of Sound

The speed of an ultrasound wave is determined by its frequency as well as the medium in which it

is propagating. This speed of sound can be calculated if the ultrasound frequency and wavelength

is known (Equation 1.9). MR measured ultrasonic field data shows the spatial distribution of

an ultrasound wave and hence, enables direct measurement of the wavelength. The frequency is

also known and it is therefore possible to calculate the speed of sound. This speed can also be

measured in an ultrasound time of flight experiment, providing an excellent way of validating the

MR measurements. Using phantoms composed of 2% agar combined with O - 50% glycerol, it is

possible to produce materials with ultrasound speeds ranging from 1498 m/s (for pure agar) to

approximately 1780 m/s (50% glycerol by weight). In this way, a large range of speeds could be

measured using both MR and ultrasound techniques described in the next two sections, permitting

validation of MR measurements.

MR Measurement of Speed of Sound

As stated in Section 1.4, in order to determine the ultrasound speed, it is necessary to know

accurately the wavelength and frequency of the ultrasound wave. The frequency is determined

by the resonant frequency of the gradient coi1 used for motion encoding and is therefore known

precisely. MR measurement of ultrasound speed becomes a matter of accurately measuring the

ultrasound wavelength. The MR data provides an image of the spatial distribution of an ultrasonic

field at a single frequency. An axial profile through this data is sinusoidally varying. The wavelength

of its variation is the ultrasound wavelength of interest and can be mexured directly from an image.

A more accurate method of obtaining the speed of sound involves obtaining the Fourier

transform of an axial ultrasound wave profile. The Fourier spectrum of the MR data contains a

sharp peak at the wavenumber (k) of the ultrasound field. Measuring the location of this peak yields

the wavelength (ikl = 27r/X) and thus the speed of sound. The precision of this result is dependent

upon the linewidth of the spectral peak. This linewidth is determined by two factors: the bandwidth

of the ultrasound wave and the frequency domain resolution. The ultrasound wave bandwidth is

very narrow due to the sharp filtering effect of the motion encoding gradient (see Section 1.2). The

primary contribution to the spectral linewidth is due to poor frequency resolution, which is limited

by the imaging fieId of view.

Using this technique, quantitative measurements of ultrasound speed can be made from the

MR data. The next section will describe an ultrasound method for measuring the speed of sound

that can be used to validate the MR measurement.

Ultrasound Measurement of Speed of Sound

The speed of sound can also be measured very accurately using ultrasound with a time of Aight

method. Figure 2.9 shows an experimental setup to measure the speed of soiind in a block of agar

phantom material. The ultrasound transducer and a needle hydrophone are placed in a water tank.

A short pulse of ultrasound is transmitted from the transducer. The hydrophone, located near

the ultrasound beam focus, detects the pulse that is then recorded on a digital oscilloscope. A

block of phantom material with a higher speed of sound and a known thickness, placed between the

transducer and the hydrophone, advances the arriva1 time of ultrasound pulses. Cross-correlation

of hydrophone waveforms obtained with and without the phantom block yields the tirne sliift (At )

that arises due to the different sound speeds in water and in the phantom material. At is related

to the speed of sound in the phantom (cp) , the speed of sound in water (h) and the thickness of

the phantom block (L) by the following equation:

Using this formula, the speed of sound in the phantom material can be calculated. This result is

compared to the speed of sound measurement made using MR, providing a means of validating the

MR data.

phantom water material speed = c,

A e e d = c,

needle transducer pulse hydrophone oscilloscope I I I

Figure 2.9: Experimental apparatus used for measuring the speed of sound using ultrasound time of flight.

2.4.4 Verification of Linear Response to Transducer Excitation Power

The quantitative accuracy of acoustic field parameters obt ained frorn MR phase contrast imaging

is difficult to determine. Other methods for measuring these parameters suffer from significant

limitations as described in Section 1.5. In particular, there is no non-invasive technique capable of

measuring the ultrasonic fields within the MR motion sensitive imaging apparatus. Point measure-

ments at one location can be made using a calibrated hydrophone, but their invasive nature may

significantly alter the ultrasonic field distribution. In order to obtain accurate results, it is there-

fore imperative to make both the MR and hydrophone measurements of acoustic pressure with the

hydrophone embedded in the imaging apparatus. In this way, the boundary conditions and thus

the acoust ic field, is identical for each experiment. Unfortunately, MR compatible hydrophones are

not available, preventing their use in the strong static and switched magnetic fields required for

MR imaging.

An alternative to verifying the absolute accuracy of MR ultrasonic field measurements is to

demonstrate the linearity of these measurements witli respect to the transducer excitation wave-

form. In order to implement this technique, MR motion sensitive imaging is performed for ultra-

sonic fields produced by a range of excitation voltages. Measuring the excitation voltage across the

transducer permits calculation of the corresponding driving power, given the transducer impedance

(76il). Following conversion of the acquired phase images into ultrasound displacement maps (as

described in Section 2.4.1), three cycles of ultrasound waveform selected from a region of uniform

displacement amplitude within the image were fitted to a sinusoid. The resultant amplitude of this

fitted curve was used to calculate the local ultrasound intensity from Equation 1.12. The error bars

on the ultrasound intensity, obtained from MR, are determined by the quality of the sinusoidal

fits. The true ultrasound intensity is linearly related to ultrasound transducer excitation power

and verification of this relationship for the MR measured intensity can provide further evidence

dernonstrating the validity of MR measurements of acoustic field parameters.

Results and Discussion

In order to demonstrate the use of MR for imaging ultrasound acoustic fields, experiments were

carried out using the apparatus described in Section 2.3. The results of these experiments can be

divided into 5 main areas which will be presented in the following sections. Section 2.5.1 describes

the irnaging of a traveiing ultrasound wave. Each of the processing steps required to convert the

raw MR phase images into quantitative images showing acoustic pressure are described, and their

effect on the final image is illustrated. These processing steps are also performed on al1 subsequent

images. Section 2.5.2 compares the ultrasonic fields obtained using both the traveling and standing

wave boundary conditions described in Section 2.3. The distributions are significantly different,

indicating the wide range of ultrasonic fields which can be visualized using MR. Section 2.5.3 shows

the field pattern obtained when a glass rod is placed near the ultrasound focus. This pattern is very

complex and illustrates the capability of MR phase contrast imaging to visualize scattering and

reflection of ultrasonic fields in heterogeneous media. In Section 2.5.4, the capability of MR to yield

accurate measurements of ultrasound speed is demonstrated. Section 2.5.5 shows the relationship

between ultrasonic intensity measured using MR and transducer excitation power. The linearity

of this relationship provides further evidence to the validity of making quantitative ultrasonic field

measurements using MR. These results illustrate the capability of MR phase contrast irnaging for

non-invasively visualizing and quantifying the propagation of ultrasonic fields in heterogeneous

material.

2.5.1 Acquisition of an Ultrasound Field Map

The imaging apparatus was setup as described in Section 2.3. The back surface of the agar was

oriented at an oblique angle and roughened to prevent coherent reflection of ultrasound waves into

the imaging volume. Under these conditions, the ultrasonic field within the cavity behaves as a

propagating wave. This section describes the processing steps required to obtain an ultrasonic field

map from raw traveling wave phase images and demonstrates the effect of each processing step on

the final image. Figure 2.10 shows a raw phase image acquired in the absence of an ultrasonic field.

The pixel intensity scaling was chosen to emphasize the random fluctuations associated with phase

noise in the raw MR image.

Figure 2.10: Raw phase image acquired in the absence of an ultrasonic field. Imaging field of view is as indicated by the dashed box in Figure 2.5.

When 40 W of electrical power was applied to the ultrasound transducer, the image shown

in Figure 2.11A was obtained. It is now possible to visualize the phase accumulations acquired due

to particle oscillations in the presence of the motion encoding gradient. The pixel intensity scaling

is identical to that in Figure 2.10 to illustrate the relative amplitude of the signal and noise. The

solid line in Figure 2.11B shows a profile through the center of the ultrasound field (dashed line in

A). Also shown is a profile of the motion encoding gradient (dashed line) at the same location. The

negligible gradient at axial positions of approximateiy 1.5 and 8.5 cm reduces motion sensitivity in

these regions and results in decreased ultrasound induced phase accumulations.

Axial Position (cm)

Figure 2.11: A) Raw phase image acquired in the presence of a traveling ultrasound wave. B) A plot of phase accumulation along a central a i s (solid line), and gradient profile at the same location (dashed line). Regions of decreased phase accumulation at 1.5 and 8.5 cm are a consequence of reduced motion encoding gradient strength.

The motion encoding gradient strength at each location, acquired as described in Sec-

tion 2.4.2, was used in Equation 1.6 to correct for gradient non-lineaïity and convert the phase

accumulation at each pixel to the corresponding displacement amplitude. The resultant image is

shown in Figure 2.12A. Division of the phase signal by the small motion encoding gradient am-

plitude near 1.5 and 8.5 cm resulted in large noise spikes in these regions. While these spikes did

not interfere with adjacent regions, they did interfere with the gray-scale display of these images

which attempts to apply an eight bit display range over the entire range of image data. This would

in turn reduce the contrast of the ultrasound field in regions outside the noise spikes. In order to

elirninate this problem, these noise spikes were removed from the final image data by setting the

image data to zero wherever the gradient amplitude fell below a minimum value. The exact choice

of this minimum value is not critical; however, it was found empirically that setting this threshold

a t 20% of the maximum gradient produced good images without any perturbation of the gray-scale

contrast. These regions can be seen as gray bands of constant intensity indicated by the arrows at

locations around 1.5 and 8.5 cm. Figure 2.12B shows a profile through the central axis of the ultra-

sound field (dashed line in A). Using equations Equation 2.1 and Equation 2.2, we can express this

data in terms of displacement amplitude and pressure. Gradient non-linearity compensation has

been applied, producing valid displacement and pressure amplitudes at al1 locations except where

the gradient was below 20% of the maximum and the phase signal was set to zero. This image

quantitatively describes the spatial distribution of the ultrasonic field. However, the presence of

noise within the image is distracting. Knowing the specific nature of the signal enables extraction

of the signal from the noise, improving the signal to noise ratio (SNR). The next section describes

a technique for using image filtering to increase the SNR.

Image Filtering

As described above, it is possible to use a-priori knowledge of the expected signal to preferentially

select the signal present within noisy images. In the case of ultrasonic field images, it is known that

the signal of interest is composed of a single sinusoidal frequency. This implies that in the Fourier

domain, the signal is contained within a narrow band of spatial frequencies. The noise however,

is uniformly distributed t hroughout the ent ire spatial frequency spectrum. Figure 2.13 shows the

Pressure Fpa)

-

Displacment (nm)

Fourier transform (solid line) of an axial profile through the ultrasonic field in Figure 2.12. As

expected, the majority of the image power is located within a very narrow spectral band corre-

sponding to the spatial frequency of the ultrasound wave. By bandpass filtering the image a t the

ultrasound frequency, the signal is unaffected while the noise a t higher and lower frequencies is

reduced. The dashed line in Figure 2.13 shows a hamming windowed bandpass filter which, when

applied to this data, reduces noise outside the filter bandwidth. The bandwidth is broad to avoid

altering the original frequency content of the signal. The filter center frequency was selected to be

at the peak of the ultrasound spectrum.

\ lkl = 2 1.2 cm-'

\ E

\ FWHM = 0.8 cm-'

Wavenumber (cm")

Figure 2.13: Spatial frequency domain content of acoustic field map.

Figure 2.14A shows the result of using this Hamming windowed bandpass filter on the

ultrasonic field image of Figure 2.12A. The high frequency noise, which appears as speckle within

that image, has been reduced. In particular, in regions where the signal is small the filtered image

enables visualisation of the wavefront where it was Iost in the noise before. Figure 2.1413 shows a

pressure and displacement profile through the central axis of the ultrasound field (dashed line in

A). The focus is clearly seen as a peak in the pressure amplitude at a position of 3.5 cm. It can

also be seen in Figure 2.14A as a change in the concavity of the ultrasound wavefront from concave

before 3.5 cm, to convex beyond.

I I I I I I I I I I

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

Axial Position (cm)

Figure 2.14: A) Ultrasonic field image filtered using a Hamming windowed bandpass filter with a center frequency corresponding to the peak in the signal frequency spectrum. The image shows increased SNR due to elimination of high and low frequency noise. B) An axial profile through the filtered ultrasound field (dashed line in A) showing displacement and pressure amplitude. The focus can be clearly seen as a peak in the pressure amplitude at a position of 3.5 cm.

The sequence of processing steps described in Section 2.5.1 were used to process al1 acquired

uItrasonic field images. The following two sections will demonstrate the wide range of field config-

urations that can be imaged. Section 2.5.2 will show the effect of changing the reflectivity of the

back surface of the cavity. Section 2.5.3 will show the ultrasound field pattern produced when a

cylindrical scattering object is placed within the field near the focus. These images will illustrate

the ability of MR motion sensitive imaging to visualize a wide range of acoustic field distributions.

2.5.2 Traveling and Standing Wave Field Distributions

Section 2.5.1 demonstrated the processing steps performed to acquire an image of an ultrasonic

field. The steps were used to process images of ultrasound fields obtained under different cavity

conditions. This section will illustrate the capability of MR motion sensitive imaging to visualize

the wide range of field distributions that arise under different boundary conditions.

Figure 2.15A shows the ultrasonic field pattern obtained with a roughened, oblique back

surface. Under these conditions, there is very little ultrasound reflection and the ultrasonic field

propagates as a traveling wave. The focal position (arrow) can be clearly visualized as a change in

the curvature of the wavefront, from concave before the focus, to convex beyond. By shifting the

relative phase of the gradient and ultrasound, the propagation of this wave in time can be visualized.

Figure 2.15B shows the propagation of several cycles of ultrasound wave (dashed line in A) over

time. The diagonal line is indicative of a traveling wave propagating in the forward direction. The

speed of sound estimated from the dope of this advancing wavefront is approximately 1530rn/s in

this case.

When the ultrasound cavity was modified to generate reflections from the exit surface, the

field pattern changed significantly (Figure 2.16A). In particular, the curvature of the wavefronts

disappeared due to the superposition of forward and refiected waves with approximately equal

amplitude. Plotting the wave throughout an ultrasound period, from the indicated region of this

image, resulted in a completely different propagation pattern (Figure 2.16B). In this case, the

cavity configuration resulted in two wavefronts moving in opposite directions that generated nodes

periodically in time and space. This is consistent with a standing wave.

The different field patterns produced in the presence of traveling and standing waves demon-

strates the immense power of MR motion sensitive phase imaging to visualize ultrasonic fields. In

the next section, a field resulting from the introduction of a scattering cylinder to the cavity will

be demonstrated. Again, the field patterns are changed significantly.

Axial Position (cm) O 4 5 6 7 8 1 9 10

4.2 4.4 4.6 4.8 5 Axial Position (cm)

Figure 2.15: A) A gradient corrected, filtered MR phase image showing an ultrasound field. The focal region is indicated by the large arrow. The small arrows indicate regions where ]GI < 20% of the maximum gradient and the phase signal was set to zero. B) Temporal evolution of 3 cycles of ultrasound wave (from solid white line in A) obtained by varying 8. The sloping lines indicate a traveling wave moving away from the transducer.

Axial Position (cm) O 1 2 3 4 5 6 7 8 9 1 0

6.6 6.8 7.0 7.2 7.4 Axial Position (cm)

Figure 2.16: A) A gradient corrected, MR ultrasound field image with the cavity adjusted to produce strongly reflected waves. The superposition of forward and reflected waves of approximately equal amplitude flatten the curvature of the ultrasound wavefronts. B) Temporal evolution of ultrasound wave (fmm solid line in A). The checkerboard pattern indicates the presence of two waves moving in opposite directions, generating periodic pressure iiodes and anti-nodes that are characteristic of standing waves.

2.5.3 Other Field Distributions

One of the most intriguing features of this technique is its ability to visualize ultrasound scattering

in complex structures such as tissue. Figure 2.17 illustrates this feature, showing the acoustic field

pattern resulting from the introduction of a 2.5 mm cylindrical g las rod into the ultrasound cavity,

near the acoustic field focus. This image shows a region of intense reflection (arrow-a) from the

glass cylinder and the presence of clear diffraction nodes beyond the cylinder (arrow-b).

transducer glass rod

Figure 2.17: Gradient-corrected MR image of an US field scattering from a 2.5 mm diameter glas rod in the center of the imaging cavity. Imaging studies of the wave evolution show an intense region of backscatter from the rod evident as a standing wave (arrow-a). Diffraction nodes are evident in the near field region (arrow b).

The above images illustrate the abiIity to visualize complex wave scattering in heterogeneous

media that is provided by MR motion sensitive imaging. This technique is non-invasive, and thus

enables measurement of ultrasound acoustic fields without disturbing the nature of the field. The

other important aspect of ultrasonic field imaging using motion sensitive MR is the quantitative

nature of the measurements that can be achieved. The next section will illustrate the ability to

make accurate, quantitative and non-invasive measurements of ultrasound wave speeds.

2.5.4 Measurement of Speed of Sound

In order to validate the quantitative aspect of this technique, it is useful to compare them with quan-

titative measurements made using ot her well established methods. As discussed in Section 2.4.3,

the speed of sound can be determined from the MR ultrasonic field map and verified using an

ultrasound time of flight experiment. Figure 2.18 shows a plot of the speed of sound obtained

from the MR data versus that measured using an ultrasound time of flight method. The error

bars in the MR data are determined by the full-width-half-maximum of the wavenumber spectral

peak (Figure 2.13) and represent sampling limitations in the data. The solid line corresponds to a

line of unity, corresponding to perfect agreement between the MR and ultrasound measurements.

The data compares well, demonstrating the validity of making quantitative measurements of the

ultrasound wave speed using MR.

1450 1500 1550 1600 1650 1700 1750 1800 1850 Ultrasound speed o f sound (m/s)

Figure 2.18: The speed of sound determined by MR phase irnaging versus a direct hydrophone time of flight measurement. Error bars in the MR data are determined by the full-width-half-maximum of the wavenumber spectral peak and represent sampling limitations in the data. The solid line corresponds to a dope of unity.

2.5.5 Linearity of MR Measurements of Ultrasonic Intensity

MR motion sensitive imaging is a technique capable of making quantitative measurements of acous-

tic field parameters including speed of sound, particle displacement, acoustic pressure amplitude

and ultrasound intensity. While absolute accuracy of the MR phase contrast methods has been

demonstrated for flow and elastography [l6], validating quantitative levels of ultrasound power by

MR against direct measurements is difficult due to the lack of non-invasive sensors capable of non-

invasively measuring acoustic fields in magnetic fields. However, it is possible to demonstrate the

linear relationship between input power and M R measured acoustic intensity. Figure 2.19 compares

the ultrasound intensity measured at a fixed point to the corresponding ultrasound transducer ex-

citation power, obtained as described in Section 2.4.4. A linear fit to this data (solid line) yielded

a correlation coefficient of r = 0.9978. Thus, MR rneasurements of ultrasonic intensity are linearly

related to the input power, as expected.

- 5 . 4 5 - f - i o - i z - i 4 - Transducer Driving Power (W)

Figure 2.19: M R determinat ion of ultrasonic field intensi ty versus transducer excitation power. Error bars reflect the uncertainty of sinusoidal fits to MR displacernent data. The solid line is a linear fit to the data, with r = 0.9978.

2.6 Capabilities and Limitations

The motions induced in the presence of an ultrasound field are extremely small, on the order of

tens of nanometers. The sensitivity of phase contrast MR to the detection of these motions is of

fundamental importance for determining potential applications of this technique. This sensitivity is

limited the motion-encoding gradient strength and duration and by the signal to noise ratio (SNR)

achieved in the corresponding MR magnitude images.

Random noise fluctuations within the real and imaginary components of the M R signal lead

to noise in the calculated magnitude and phase MR images. The phase noise 4, is related to the

SNR in the magnitude image by the following equation [50]:

where 4, is the phase noise in radians. Equation 2.15 describes the phase noise present in a single

phase image. During the formation of an ultrasonic field map, two phase images are subtracted.

The noise from each image is additive, resulting in a increase in the final phase noise. The

total phase noise present in an ultrasonic field map acquired using phase contrast MR is therefore

described as follows:

This relationship enables calculation of the phase noise

tude SNR and can be used as an estimate for the overall

that will be produced for a given magni-

sensitivity of phase contrast MRI to small

motions. In the current apparatus, the magnitude SNR was typically on the order of 50. This

implies that phase noise will be approximately 0.02 radians, producing a noise equivalent displace-

ment, pressure and intensity of 3.9 nm, 19 kPa or 12 rn W respectively. This does not limit

the imaging of therapeutic ultrasound (1 - 800 W cm-2) [48], nor does it prevent the detection of

power densities typical of diagnostic imaging (0.1 - 1000 mW cms2) [49], but it is hampered by the

short pulse duration typical of these applications. The next section wilI describe some potential

improvements to the imaging apparatus and pulse sequence that may reduce these detection limits.

There is a definite need for a technique that is capable of non-invasively imaging ultrasonic field

patterns in heterogeneous tissue. Section 2.2 evaluated the theoretical feasibility of using phase

contrast MR for the detection of the minute particle motions induced in the presence of ultrasonic

fields. These calculations indicate that very large gradients, capable of oscillating at high frequen-

cies, are required. However, the design and construction of a gradient coi1 capable of producing

t hese gradients is not impossible. Section 2.3 addressed the engineering considerations associated

with the construction of such gradients and described the related apparatus necessary for US field

imaging. Using the experimental techniques described in Section 2.4, the first MR images of ul-

trasonic field distributions were acquired. The ability of MR phase contrast imaging to visualize a

wide range field distributions, including traveling, standing and scattered waves, was demonstrated

in Section 2.5.

These data and images illustrate the feasibility of using magnetic resonance methods to

visualize non-invasively the spatial distribution of particle displacement, pressure and intensity

within an ultrasound field. One of the strengths of phase contrast MR is its ability to accurately

determine particle motion. This allows one to quantify US field parameters based solely on the

proton gyrornagnetic ratio, M N timing and gradient parameters. When combined with a knowledge

of the tissue density, this method provides accurate measures of the power and acoustic pressure in

tissue without the need for calibrated hydrophone measurements. Unlike optical techniques that

are only capable of visualizing the US field in optically transparent media, this method is directly

applicable to the visualization of US field propagation within heterogeneous tissues. Furthermore,

by phase shifting the ultrasound and gradient waveforms, the temporal evolution of an ultrasonic

field can be resolved.

The primary challenge associated wi t h t his technique is the development of stable oscillating

gradients having adequate strength to detect the small molecuhr motions induced by ultrasound at

clinically relevant power levels. The amplitude resolution is dictated by the gradient strength and

the phase noise within the MR images. Chapter 3 begins with an analysis of the limitations of MR

phase contrast imaging as applied to ultrasonic field imaging. Potential techniques for increasing

motion sensitivity will be addressed in Section 3.2.

The results and data already acquired are restricted by the small irnaging volume provided

by the current RF and motion encoding gradient coils. Ultrasound field distributions are highly

dependent on the boundary conditions of the cavities in which they are contained. In order to

adequately visualize the interaction of ultrasound fields with complex heterogeneous tissue struc-

tures, it is necessary to have a large irnaging volume that does not interfere with the ultrasonic

field. Section 3.2.1 discusses the design considerations associated with building a new apparatus

that provides a larger imaging volume, therefore enhancing the visualization of ultrasonic field

distributions.

The ability to image non-invasively the spatial distribution and temporal evolution of ultra-

sonic fields in heterogeneous materials has many applications. Section 3.3 discusses the potential

for using this technique for the imaging of non-linear ultrasound interactions with tissue, thus

indicating one of its many future directions.

Chapter 3

Improvement s and Future Research

3.1 Overview

There is a definite need for a technique that enables the direct and non-invasive visualization of

ultrasound acoustic fields in heterogeneous media such as tissue. Phase contrast MRI provides

a means of achieving this goal by quantifying the minute motions induced in the presence of an

ultrasound field. The motions are extremely small and large magnetic field gradients, oscillating at

high frequencies, are necessary to achieve the required sensitivity. An apparatus producing a peak

gradient of 0.4 T/m was designed and constructed. Imaging using this apparatus demonstrated

the ability of phase contrast M M to quantify ultrasound-induced motion and illustrates the wide

range of ultrasonic fields which can be visualized.

These results suggest the application of phase contrast MRI to the quantification of US field

parameters and for visualizing the temporal evolution of ukrasound fields. This chapter discusses

the limitations, potential improvements and future directions of this technique. Section 3.2 describes

improvements to the imaging apparatus and experimental methods that will extend the application

of this technique to the imaging of new ultrasound field configurations. Potential future applications

of phase contrast M M will be described in Section 3.3.

3.2 Improvements to Apparatus

The imaging apparatus described in Section 2.3 demonstrated the feasibility of using phase contrast

MR for the visualization of ultrasonic fields. However, in order to successfully apply this technique

to the imaging of ultrasound interactions with tissue, several improvements must be made. These

improvements can be divided into two main areas: 1) hardware improvements, and 2) improvements

in pulse sequence design and experimental methodology.

3.2.1 Hardware Improvements

Phase contrast imaging is only sensitive to motion occuring in the direction of the motion-encoding

magnetic field gradient (Section 1.2). In the present apparatus, motion encoding only occurs in the

axial direction. Ultrasound interactions within tissue will produce longitudinal waves propagating

in al1 directions and may also induce mode conversion from longitudinal to transverse wave propaga-

tion [54]. Complete characterization of tissue-ultrasound interactions requires motion sensitization

in al1 three dimensions. This implies the need for motion-encoding gradients in three orthogonal

directions. Furthermore, the current motion-encoding gradient provides motion sensitivity over

an imaging volume of 70 cm3 and limits adequate visualization of ultrasound propagation in large

tissue samples.

We are currently in the process of designing a new imaging apparatus that will provide three

orthogonal motion encoding gradients of 0.4 T/m over a volume which is 10 cm in diameter and 20

cm in length. In order to do this, approximately 3000 W of dc power will produce 60 amperes within

the gradient coils and generate coi1 voltages of approximately 45 kV. This will enable unimpeded

visualization of ultrasonic field interactions with tissue.

3.2.2 Improvements to Pulse Sequence Design and Experimental Methodology

The sinusoidal phase variations induced by the ultrasound field occur at a specific spatial frequency

and are dependent upon the ultrasound wavelength. This spatial frequency corresponds to a specific

region of MR data k-space. Figure 3.1 shows a vertical profile through the center of k-space at

kz = O . This profile demonstrates the localization of the MR signal at specific regions of k-space.

One means of improving the efficiency of ultrasonic field irnaging is to perform non-uniform sampling

of k-space. By acquiring more data samples in regions where the signal is located, the SNR at that

spatial frequency can be improved. Bandpass filtering around the frequency of interest will then

reduce high and low frequency noise, significantly improving the final image quality.

ultrasound peak

Wave Number (cm-')

Figure 3.1: Central profile through k-space at k, = O, showing peaks at dc and at the ultrasound wave number. Non-uniform k-space sampling at the ultrasound peaks can improve SNR at these spatial frequencies.

Ultrasonic field irnaging using phase contrast MR presently requires relatively thick slices (5

mm). This is limited by the current imaging gradient strengths and SNR issues. These thick slices

mean that the acquired ultrasound field is averaged over the slice, implying that fine field structure

is attenuated. Three dimensional imaging sequences provide the capability to produce significantly

thinner slices while still maintaining SNR. Using an additional phase encoding gradient in the sIice

selection direction, square imaging voxels can be localized in three dimensions with sizes on the

order of 0.3 mm per edge.

A new apparatus with a larger imaging volume and ultrasound cavity, combined with non-

uniform k-space sampling and a three dimensional acquisition, will enable the unimpeded imaging

of ultrasound-tissue interactions with high resolution in al1 three directions. Tliese improvements

will also increase the sensitivity of this technique for the detection of small motions induced by

ultrasound at higher frequencies.

3.3 Future Directions

One of the most unique properties of phase contrast MRI for ultrasonic field imaging is its frequency

selectivity. This technique is only sensitive to motion occurring at the same frequency as the

gradient oscillations. This provides a novel and exciting means of quantifying non-linear ultrasound

behaviour .

The wave equation describing ultrasound wave propagation, as described in Section 1.4,

is based upon the assumption that there is a linear relationship between pressure and density

within the medium. In real media, with waves of a finite amplitude, this is not the case. The

relationship can be expressed as a Taylor series and the higher order terms produce the non-linear

behaviour. Because of this non-linear behaviour, the speed of sound is pressure dependent. The

wave velocity is higher in regions of compression than it is in rarefied positions. This behaviour

distorts the ultrasound waveform, transferring energy from the fundamental frequency to higher

order harmonics. As the wave advances, regions of compression approach regions of rarefaction and

a shock wave is established [55]. The shocking depth is defined as the depth at which the ratio

of power in the fundamental and second harmonics is maximized. As the wave penetrates further

into the medium, the higher order liarmonics are attenuated more rapidly because of their higher

frequency, and it becomes sinusoidal again.

Non-linear ultrasound waves and tlieir interaction within tissue is of great interest in medicine [55],

particularly in therapeutic applications where the intensities and thus pressure amplitudes are high.

Phase contrast MRI of ultrasonic fields may provide great insight into the nature of these non-linear

ultrasound-tissue interactions. As described in Section 1 -2, t his imaging technique is highly fre-

quency selective, achieving bandwidths on the order of 5 Hz. The filtering effect is highly ammenable

to spectroscopic imaging. The propagation of second and higher ultrasound harmonics can be easily

visualized by oscillating the motion-encoding gradient a t these higher frequencies. This technique

can be used to selectively image the temporal and spatial evolution of the non-linear cornponents

within an ultrasound wave.

The major drawback of this technique is the reduced power and displacement amplitude

present within the harmonics. In order to detect the resultant extremely small ultrasound-induced

motions, very large gradients capable of oscillating at high frequencies are required. However,

with the improvements in the gradient coils and experimental techniques described in Section 3.2,

imaging of non-linear ultrasound-tissue interactions is feasible, particularly a t low ultrasound fre-

quencies.

3.4 Conclusion

A method for the direct and non-invasive imaging of ultrasonic fields based on phase contrast MRI

has been demonstrated. This imaging technique uses an oscillating gradient to quantify the minute

motions induced in the presence of an ultrasound wave. Using this method, the spatial distri-

bution of particle displacement, ultrasound pressure and intensity were rneasured. Furthermore,

the ability to visualize the temporal evolution of these fields in complex heterogeneous structures

was demonstrated. The noise equivalent displacement, pressure and intensity of this technique are

3.9 nm, 19 kPa and 12 W cm-2 respectively, which describes the smallest ultrasound fields that can

be detected. This sensitivity is currently limited by the motion-encoding gradient strength and the

imaging coi1 sensitivity, but improvements to the apparatus, pulse sequence and signal processing

methods could result in increased motion sensitization. The present apparatus is only capable of

imaging periodic motion occuring in the axial direction at the fundamental ultrasound frequency. A

new apparatus wit h a larger imaging volume and t hree orthogonal motion-encodiiig gradients will

enable unimpeded visualization of linear and non-linear ultrasound interactions with heterogeneous

tissue.

The gradients used in this experiment result in a peak aB/Bt of approximately 10"s-',

which is much larger than the recommended limit (960 T s- ' ) [56] for human imaging applications

at these frequencies. Nevert heless, quantitative visualizat ion of ultrasound propagation and scat-

tering in heterogeneous tissue appears feasible and represents a novel approach for improving our

understanding of basic ultrasound interactions with ex-vivo tissue samples. Furthermore, the ap-

plication of spectroscopic imaging methods by applying gradients at harmonies of the ultrasound

frequency may provide new insight into the det ails of non-linear ultrasonic interactions wit h tissue.

These studies have demonstrated the first direct, non-invasive visualizat ion of the nanometer

displacements associated with medical ultrasound. The ability to quantify ultrasound field proper-

ties, based solely on fundamental constants and well known MR imaging parameters, is attractive.

Furthermore, the resolution of subtle aspects of the spatial and temporal evolution of ultrasonic

scattering within complex media is a novel aspect of this technique. Phase sensitive MR imaging

t herefore presents an ent irely new and non-invasive approach to ult rasound exposimetr y.

Appendix A

Magnetic Imaging of Ultrasonic Fields

C.L. Walker, F.S. Foster, D.B. Plewes, Magnetic Resonance Imaging of Ultrasonic Fields, Ultra-

sound in Medicine and Biology. In press, 1997.

MAGNETIC RESONANCE IMAGING OF ULTRASONIC FIELDS

CL Walker, FS Foster, DB Plewes Department of Medical Biophysics

University of Toronto Sunnybrook Health Science Centre

2075 Bayview Avenue Toronto, Ontario, Canada, M4N 3M5

Ultrasound in Medicine and Biology Manuscript #97-88 Submitted - June 1997, Accepted - July 1997,

Revised - August 1997.

Address Correspondence to: DB Plewes

Sunnybrook Health Science Centre S669-2075 Bayview Avenue, Toronto, Ontario, Canada

M4N 3M5 Phone (41 6) 480-5709

Fax (41 6) 480-571 4 E-mail: dbpO srcl.sunnybrook.utoronto.ca

A nuclear magnetic resonance imaging (MRI) method is described that allows non-

invasive, quantitative mapping of medical ultrasound (US) fields in tissue. Application

of a resonant rnagnetic field gradient operating at the US frequency permits detection of

nanometer motions associated with ultrasound, and allows direct measurement of

absolute pressure, intensity, and speed-of-sound. By altering gradient timing, the

propagation of US fields in time and space can be observed, which enables tracking of

US scattering phenornena in media. An experimental apparatus was constructed that

combined a 515 kHz focused US transducer configured with its focus in the centre of a

small bore oscillating gradient. This provided an oscillating gradient with a peak gradient

strength of 0.40 T/m over a useable imaging volume of 61 cm3. When used in

conjunction with 1.5 T clinical MR imaging system, this apparatus allowed the clear

visualization of the focused US field within this volume and its propagation with time.

Current limits of sensitivity indicate a noise equivalent sensitivity of 3.8 nm in

displacement amplitude, 19 kPa in pressure amplitude and 12 mW/cm2 in acoustic

intensity. These studies indicate that MRI can provide a new, non-invasive method for

US exposimetry and.the basic study of ultrasound biophysics in tissue.

KEYWORDS: Phase-contrast MRI, ultrasound exposimetry, motion detection,

oscillating magnetic field gradients.

Page 1

-.-------- - - - - -

The applications of ultrasound (US) in the medical field are widespread and

growing rapidly. The success of these diverse applications is largely determined by the

ability to craft specific acoustical field patterns within tissue in a controlled and

predictable fashion. The ability to observe a field pattern in acoustically heterogeneous

tissues is important to understand the effect of phase aberrations on spatial resolution

in imaging applications and on the deposition of thermal energy in high intensity

focused ultrasound (HIFU) applications. Validation of US field patterns in homogeneous

media can be predicted on the basis of classical diffraction theory (Hunt et al. 1983)

and verified with invasive sensors(Schafer and Lewin 1988) (Fry and Fry 1954)

implanted within the tissue. In transparent media, direct observation of the acoustic

field can be achieved with optical methods (Raman and Nath 1935, Breazeale and

Heideman 1959) or optical diff faction tomography (Pitts 1994). However, these

techniques collectively suffer from being either local, invasive, or not applicable to

human tissues which are neither transparent nor homogeneous. The ability to provide

an accurate non-invasive means of visualizing the propagation of longitudinal ultrasonic

waves in tissue and to quantify its intensity distribution would fiIl an important need in

the development of optirnized ultrasound therapeutic and imaging strategies. In this

paper, we report and demonstrate the use of motion-sensitive magnetic resonance

imaging (MRI) to provide the first direct visualization of longitudinal ultrasound

propagation in tissue equivalent rnaterials.

Page 2

The key to visualizing longitudinal acoustic wave propagation is the mapping of

the minute particle displacements that accompany propagating ultrasound waves.

Such displacements are small, typically on the order of tens of nanometers, and place

stringent requirements on motion detection sensitivity. Nuclear magnetic resonance

signals can be influenced by motion through the application of a magnetic field gradient

superimposed on the static, uniform magnetic field which is used for spin polarization

(Hahn 1960). This concept has been used with proton MRI to4mage fluid flow (Frayne

et al. 1995), brain (Enzmann and Pelc 1 WZ), and muscle (Pelc et al. 1992) tissue

motion, as well as the measurement of cardiac tissue strain (Wedeen 1992). Several

authors have used oscillating magnetic field gradients to detect oscillatory fluid flow or

viscoelastic tissue motion. Specifically, acoustic oscillatory fluid flow in the rat cochlea

(Denk et al. 1993) has been demonstrated at frequencies up to 4.6 kHz. Similarly, the

viscoelastic properties of tissue have been measured by monitoring the wave velocity of

slow (el 0 mlseconds), low frequency shear waves (cl .1 kHz) (Lewa and de Certaines

1996; Muthupillai et al. 1995; Muthupillai et al. 1996) generated by mechanical

stimulation. In these cases, the oscillation frequency was limited to a few kilohertz with

motion amplitudes ranging from 200-1 000 nanometers. However, for the case of

longitudinal wave propagation relevant to medical ultrasound, the wave speed and

frequency are typically three orders of magnitude higher, with corresponding smaller

mdion amplitudes on the order of tens of nanometers.

Page 3

frequency is proportional to the local magnetic field. By using a magnetic field gradient

in addition to a uniform polarizing field, the resultant distribution of frequencies encodes

the spatial distribution of spin density. Similarly, when spins are moving in the presence

of this gradient, the phase of the spins indicates the history of spin location in their

movement through a gradient. More specifically, the phase of transverse

magnetization, 0, in the presence of a magnetic field gradient is determined by:

where y is the gyromagnetic ratio for protons, r(t) describes the temporal dependence

of the spin position, and G(r,f) is the wavefom of the applied gradient at r. If we

consider the spins at equilibriurn position r, to be undergoing hamonic motion of

amplitude 6 arising from a plane ultrasound wave with angular frequency o and

wavenumber kr, the displacement can be written as:

c(r,t) = r, + t0 sin@ t + k, *r).

By applying many cycles (N) of an oscillating gradient G'= Gosin(wt + 4, and

substituting into eqn (l), we generate a phase distribution throughout the MR image

given by:

Page 4

where r is the duration of the gradient waveform ( 2 W ) . Thus the spatial distribution

of phase reports the local ultrasound-induced displacement amplitude. The applied

phase 8, between the ultrasound and gradient waveforms, controls the time and the

location where the phase accumulation from the US motion and the applied gradient

rnaximize. Thus, by generating multiple images while sweeping this parameter over an

US cycle, the temporal evolution of the ultrasonic field can be observed. The measured

displacement amplitude (G) can be used to estimate the spatial distribution of acoustic

pressure amplitude p, and acoustic intensity 1 according to:

p(r, 0) = pc G(r, 0) (5)

where c is the speed of sound in a medium with density p. The ultrasound wave speed

can be inferred from previou&neasurernentç or it can be determined directly from the

MR image using the measured wavenumber (kJ and the known ultrasound frequency.

Note that the detemination of the absolute US displacement, pressure, and intensity

depends only on a fundamental constant y, knowledge of the applied rnagnetic field

gradient G and its duration T. Thus, the accuracy of the approach is guaranteed without

Page 5

EXPERIMENTAL PROCEDURES

To demonstrate this concept, we constructed an MR irnaging apparatus

containing a 5 cm diameter 0.51 5 MHz focused ultrasound transducer (Fig. 1A). The

US wave was focused at a depth of I O cm within a 3 cm diameter cylindrical chamber

containing a tissue-equivalent agar mixture. The back surface of the agar was oriented

at an oblique angle and roughened to scatter and absorb ultrasound waves after one

passage through the cavity, thus eliminating wave reflections. An oil-cooled, resonant

gradient coi1 tuned to operate at the ultrasound frequency was designed to fit around

the agar chamber. The gradient field was oriented parallel to the direction of ultrasound

propagation to encode motions arising from the longitudinal ultrasound waves. A

pulsed RF amplifier was designed to deliver up to 20 amps of current at the resonant

frequency which produced maximum gradient strengths of 0.40 Tesldmeter at the US

focus, with a gradient full-width-half-maximum of 5.6 cm. MR measurements of the

gradient distribution were used to correct spatial variations in gradient strength and

. provide a uniforrn displacernent sensitivity over the imaging volume. An RF coil used

for spin excitation and detection was p[aced between the gradient coil and the

ultrasound cavity. A spin-echo sequence with a repetition time of 2000 milliseconds

and an echo time of 120 milliseconds was used for imaging with in-plane resolution of

0.31 mm and a section thickness of 5 mm. The ultrasound and phase-locked gradient

Page 6

synchronously with the imaging sequence as shown in Fig. 1 B. These waveforms were

applied before and after the spin echo refocusing pulse, with a phase reversal of the

ultrasound waveform following the refocusing pulse. A second phase image was

acquired without ultrasound. Subtraction of these two images yielded a final phase

image which suppressed al1 phase variations over the imaging volume unrelated to the

ultrasound field. The resultant phase image was corrected for variations in the gradient

magnitude throughout the imaging volume. ln regions where the gradient approached

zero, the corrected image noise became large due to gradient compensation. In order

to indicate these regions, the final phase was set to zero wherever 101 < 20% of the

maximum gradient.

An ultrasound field image for the 515 kHz transducer is shown in Fig. 2A. The

applied electrical power delivered to the transducer was 40 watts with the gradient

applied for 100 ms. The results demonstrate a complex field pattern throughout the

imaging volume. The ultrasound focus can be seen as a reversal of wavefront

cutvature on opposite sides of the focal region (arrow). A profile of pressure and

displacement amplitude along the central axis of the imaging cavity is shown in Fig. 2B.

Maximum pressures of 600 kPa and displacements of I 120 nm are observed at the

focus. This translates into a spatial peak temporal average intensity of 1 1.7 watts/cm2.

The temporal evolution of the ultrasound wave can be resolved by shifting the phase (8)

of the ultrasound with respect to the gradient waveform. This is shown in Fig. 2C which

Page 7

---..-..--.---- -- -Q - - - - - - Y a

calculated from the slope of this advancing wavefront is 1530 m/s in this case.

When the ultrasound cavity is modified to generate reflections from the exit

surface, the field pattern changes significantly (Fig. 3A). In paiticular, the cuwature of

the wavefronts disappear due to the superposition of fotward and reflected waves with

approximately equal amplitude. Plotting the wave throughout an ultrasound period,

from the indicated region of this image, results in a completely different propagation

pattern (Fig. 38). In this case, the cavity configuration results in two wavefronts moving

in opposite directions, which generates nodes periodically in time and space. This is

consistent with a standing wave.

An alternative method of measuring the speed of sound involves performing a

Fourier transform of an axial pressure profile. A Fourier spectrum of the waveforrn in

Fig. 2A is given in Fig. 4. A sharp peak at a wavenumber of 21.2 t 0.4 cm" with a

corresponding wave speed, dlkl of 1530 I 30 meterdsecond, is demonstrated. Speed

of sound measurements from MR data were obtained for a number of tissue equivalent

materials (agar containing 0-50% glycerol) with velocities ranging from 1475 to 1780

mis. These results compared well with direct measurements'of ultrasound velocity

obtained from an ultrasound pulsed time-of-flight experiment (Ye et al. 1995) (Fig. 5).

In order to test the linearity of MR acoustic pressure measurements, we compared the

US intensity at a fixed point, determined from eqn (5) using a sinusoidal fit of the MR

displacement data, to the corresponding ultrasound transducer driving power (Fig. 6).

Page 8

indicating a linear relationship as expected.

One of the most intriguing features of this technique is its ability to visualize

ultrasound scattering in complex geometries. Figure 6 illustrates this feature, showing

the acoustic field pattern resulting from the introduction of a 2.5 mm cylindrical glass

rod into the ultrasound cavity near the acoustic field focus. This image shows a region

of intense reflection (arrow-a) from the glass cylinder and the presence of clear

diffraction nodes beyond the cylinder (arrow-b).

DISCUSSION

These data and images illustrate the feasibility of using magnetic resonance

methods for the non-invasive visuaIization of the spatial distribution of particle

displacement, pressure, and intensity of an ultrasound field. By applying these

techniques to rnulti-section MR imaging, three dimensional US field visualization is

feasible. This technique allows one to quantify US particle displacement solely on the

proton gyromagnetic ratio and MRI timing and gradient parameters. When combined

with a knowledge of the tissue density, this method provides accurate measures of the

power and acoustic pressure in tissue without the need for calibrated hydrophone

rneasurements. Unlike optical techniques which are only capable of detecting the US

field in optically transparent media, this technique is directly applicable to the

visualization of US field propagation within heterogeneous tissues. Furthemore, by

Page 9

ultrasonic field can be resolved.

The primary challenge associated with this technique is the development of a

stable oscillating gradient having adequate strength to detect the small molecular

motions induced by ultrasound at clinically relevant power levels. This gradient strength,

together with the phase noise of the MR images, dictates the minimum detectable

power level. Specifically, the variance of the phase noise is well approxirnated by 1/(2

SNR2) (Scott et al. 1992) where SNR is the signal-to-noise ratio of the magnitude MR

image. From eqn (3), the corresponding standard deviation of particle displacement &,,

is then given by a>/"/ N z G , SNR . Using eqns (5) and (6), yields the variance of the

pressure and power density. In these experiments, a typical magnitude image SNR of

50 was achieved which corresponds to a noise equivalent displacement amplitude,

pressure amplitude, and intensity of 3.9 nanometers, 19 kPa, and 12 mW/cm2

respectively. This is well within the range of application of therapeutic ultrasound (1-

800 W/cm2) (Vykhodtseva et al. 1994) for thermal therapy. In addition, this allows

detection of continuous applications of US at power densities typical of diagnostic

irnaging (0.1 -1 000 mW/cm2) (Hykes et al. 1992). lmprovernents in data averaging, and

increased gradient strength will further reduce this detection limit.

The magnetic field gradient used in this experiment resulted in a peak aB/at of

-1 O4 Tesldsec, which is much larger than the recommended limit (960 Teslalsec)

(International Electrotechnical Commission) for human imaging applications at these

Page 10

scattering in heterogeneous tissue seems to be feasible and would represent a novel

approach to improving Our understanding of basic ultrasound interactions with ex-vivo

tissue samples in a tissue-equivalent medium.

These studies have demonstrated the first direct non-invasive visualization of the

nanometer displacements associated with medical ultrasound in a tissue equivalent

medium. The ability to quantify ultrasound field properties based solely on fundamental

constants and well-known MR imaging parameters is attractive. Furthemore, the ability

to resolve subtle aspects of the spatial and temporal evolution of ultrasonic scattering

within complex media is a novel aspect of this technique. Phase sensitive MR irnaging

therefore presents an entirely new and non-invasive approach to ultrasound

exposimetry.

ACKNOWLEDGMENTS

The authors gratefully acknowledge helpful discussions with John Hunt, Graham

Wright, and Mike Bronskill, as well as the mechanical craftsmanship of Doug

Henderson. This research was supported by grants from the Medical Research Council

of Canada and the Terry Fox Foundation of the National Cancer Institute of Canada.

FS Foster is a Terry Fox scientist of the NCIC. This paper is dedicated to the memory

of Conrad Leigh Walker. He will be remembered as a gifted student, an excellent

scientist and a wonderful human being.

Page 11

Breazeale, M.A.; Heideman, €.A. Optical methods for measurement of sound pressure

in liquids. J. Acoust. Soc. Am. 31 :24-33, 1959.

Denk, W.; KeoIian, R.M.; Ogawa, S.; Jelinski, L.W. Oscillatory flow in the cochlea

visualized by a magnetic resonance imaginglechnique. Proc. Natl. Acad. Sci.

Enzmann, DR.; Pelc, N.J. Brain motion: measurement with phase-contrast MR

imaging. Radiology, 185(3):653-660, 1992.

Frayne, R.; Steinman, D.A.; Ethier, C.R.; Rutt, B.K. Accuracy of MR phase contrast

velocity measurements for unsteady flow. J. Magn. Reson. Imaging, 5(4):428-431,

Fry, W.S.; Fry, R.B. Determination of absolute sound levels and acoustic absorption by

thermocouple probes. J. Acoust. Soc. Am., 26:294-317, 1954.

Hahn, E.L. Detection of c-water motion by nuclear precession. J. Geophys. Res.

Hill, C.R. Physical Principles of Medical Ultrasonics. Chichester: Ellis Horwood; 1986,

pp. 11 3.

Hunt, J.W.; Arditi, M.; Foster, F.S. Ultrasound transducers for pulse-echo rnedical

imaging. IEEE Trans. Biomed. Eng., BME-30(8):453-481, 1983.

Hykes, D L ; Hedrick, W.R.; Starchman, D.E. Ultrasound Physics and Instrumentation,

St. Louis: Moseby-Year Book Inc., 2" edition, 1992, pp. 195.

Page 12

Lewa, C.J.; de Certaines. J.D. Viscoelastic property detection by elastic displacement

NMR measurements. J. Magn. Reson. lmaging, 6(4):652-656, 1996.

Muthupillai, R.; Lomas, D.J.; Rossrnan, P.J.; Greenleaf, J.F.; Manduca, A.; Ehman, R.L.

Magnetic resonance elastography by direct visualization of propagating acoustic strain

waves. Science, 269: 1 854-1 857, 1 995.

Muthupillai, R.; Rossman, P.J.; Lomas, D.J.; Greenleaf, J.F.; Riederer, S.J.; Ehman,

R.L. Magnetic resonance imaging of transverse acoustic strain waves. Magn. Reson.

Med. 36(2):266-274, 1996.

Pelc, L.R.; Sayre, J.; Yun, K.; Castro, L.J.; Herfkens, R.J.; Miller, D.C.; Pelc, N.J.

Evaluation of myocardial motion tracking with cine-phase contrast magnetic resonance

imaging. Invest. Radiol., 29(12): 1038-1 042, 1 994.

Pitts T, Greenleaf J, Lu JY, Kinnick R., Tornographic Schlieren lrnaging for

Measurement of Beam Pressure and Intensity., IEEE Ultrasonics Symposium pg, 1665,

1994.

Raman CV, Nath NS, The diffraction of light by high frequency ultrasonic waves. Proc.

Indian Acad. Sci 11, 406, 1935.

Schafer, M.E.; Lewin, P.A. Computerized system for measuring the acoustic output

from diagnostic ultrasound equipment. IEEE Trans. Ultrason. Ferroelectr. Freq.

Control, 35(2):102-109, 1988.

Scott, G.C.; Joy, M.L.; Armstrong, R.I.; Henkelman, R.M. Sensitivity to magnetic

Page 13

Vykhodtseva, N.I.; Hynynen, K.; Damianou, C. Pulse duration and peak intensity during

focused ultrasound surgery: theoretical and experirnental effects in rabbit brain in vivo.

Ultrasound Med. & Biol., 20(9):987-1000, 1994.

Wedeen, V.J. Magnetic resonance imaging of myocardial kinematics. Technique to

detect, localize, and quantify the strain rates of the active human myocardium. Magn.

Reson. Med., 27(1):52-67, 1992.

Ye, S.G.; Harasiewicz, K.A.; Pavlin, C.J.; Foster, F.S. Ultrasound characterization of

normal ocular tissue in the f requency range f rom 50 MHz'to 100 MHz. IEEE Trans.

Ultrason. Ferroelectr. Freq. Control, 42:8-14, 1995.

Page 14

Figure 1. (A) Schernatic of the M WUS apparatus. The US transducer and imaging

cavity were 5 cm and 3 cm in diameter respectively. The dashed box

indicates the imaging field of view. (B) The timing of the application of N =

50,000 cycles of 515 kHz ultrasound and gradient waveforms is shown in

relation to the spin-echo MR sequence used for imaging. Adjustments of

the parameter Bwere used to visualize the temporal evolution of US

fields.

Figure 2. (A) A gradient-corrected MR phase image showing an ultrasound field.

The focal region is indicated by the large arrow. The small arrows indicate

regions where IGI < 20% of the maximum gradient and the phase signal

was set to zero. (B) A plot of pressure and displacement along a central

axis. (C) Temporal evolution of 3 cycles of ultrasound wave (frorn solid

white line in A) obtained by varying 0. Sloping lines indicate a traveling

wave moving away from the transducer (located at -5 cm).

Figure 3. (A) A gradient-corrected, MR ultrasound field image with the cavity

adjusted to produce strongly reflected waves. (B) Temporal evolution of

ultrasound wave (from solid white line in A). Checkerboard pattern

Page 15

generating periodic pressure nodes that are characteristic of standing

waves.

Figure 4. The wavenumber spectrum (Fourier transform) of the data from Fig. 28

indicates a peak at 21.2 cm-' with a full-width-half-maximum of 0.8 cm".

This corresponds to an ultrasound speed of 1530 i 30 m/s.

Figure 5. The speed of sound determined by MR phase imaging versus a direct

hydrophone time-of-flight measurement. Error bars in the MR data are

determined by the full-width-half-maximum of the wavenumber spectral

peak and represent sampling limitations in the data. The solid line

corresponds to a slope of unity.

Figure 6. MR detemination of ultrasonic field intensity versus transducer excitation

power. Error bars reflect the uncertainty of sinusoidal fits to MR

displacement data. The solid line is a lin& fit to the data, with r =

0.9978.

Figure 7. Gradient-corrected MR image of an US field scattering from a 2.5 mm

diameter glass rod in the center of the imaging cavity. lmaging studies of

Page 16

evident as a standing wave (arrow a). Diffraction nodes are evident in the

near field region (arrow b).

Page 17

Ultrasound transducer

Oscillating Absorbing Chamber

Ultrasound RF coi1 field

time @$-

4.2 4.4 4.6 4.8 5 Axial Position (cm)

Axial Position (cm) O 1 2 3 4 5 6 7 8 9 1 0

6.6 6.8 7.0 7.2 7.4 Axial Position (cm)

FWHM = 0.8 cm" -+

Wavenumber (llcm)

1450 1500 1550 1600 1650 1700 1750 1800 1850 Ultrasound speed of sound (mls)

Transducer Driving Power (W)

Ultrasound Oscillating

field

, n/2 7T ' echo

LL P: 9

k

4.2 4.4 4.6 4.8 5 Axial Position (cm)

Axial Position (cm) O 1 2 3 4 5 6 7 8 9 1 0

6.6 6.8 7.0 7.2 7.4 Axial Position (cm)

FWHM = 0.8 cm" +

Wavenumber (1 /cm)

1450 1500 1550 1600 1650 1700 1750 1800 1850 Ultrasound speed of sound (mis)

Transducer Driving Power (W)

Appendix B

CD-ROM

The CD-ROM attached to this thesis contains Quicktime files to allow inspection of ultrasound

field propagation in the form of a movie loop. Playing these files with the Quicktime viewer on

a PC will provide clearer interpretation of the field propagation studies outlined in Chapter 2. A

PC with a CD-ROM drive and standard SVGA graphics adapter is needed to read and play the

files. Alternatively, a standard Macintosh with a CD-ROM drive can be used. DOS and Macintosh

versions of the Quicktime player are included. The three files are:

Wavel.mov This contains a movie loop that corresponds to Figure 2.15 and

shows the propagation of a 515 kHz ultrasound field as it propagates

through the MR imaging cavity in the form of a traveling wave. Each

movie frame represents a time shift of 242 rnicroseconds.

Wave2.mov This contains a view loop that corresponds to Figure 2.16 and shows

the propagation of a 515 kHz ultrasound field in the imaging cavity

with a reflecting surface placed at the exit port of the cavity. This

generates a standing wave.

Wave3.mov This contains a movie Ioop t hat corresponds to Figure 2.17 and shows

the scattering of the ultrasound field of Figure 2.15 with the addition

of a 2.5 mm glass cylinder.

Bibliography

[l] C. Hill. Phgsical Principles of Medical Ultrasonics, (Ellis Horwood, Chichester, 1986), p. 113.

[2] S. Ye, K. Karasiewicz, C. Pavlin, and F. Foster. "Ultrasound characterization of normal ocular tissue is the frequency range from 50 MHz to 100 MHz", IEEE Transactions on Ultrasonics, Ferroelectronics, and Frequency Control 42, 8-14 (1995).

[3] N. Sanghvi, K. Hynynenn, and F. Lizzi. "New development in therapeutic ultrasound", IEEE Engineering in Medicine and Biology , 83-92 (1996).

[4] F. Duck and K. Martin. "Exposure values for medical devices". In M. Zisken and P. Lewin, editors, Ultrasound Exposirnetry, (CRC Press, 1993), chapter 11, p. 338.

[5] F. Duck and K. Martin. "Exposure values for medical devices". In M. Zisken and P. Lewin, editors, Ultrasound Exposimetry, (CRC Press, 1993), chapter 11, p. 323.

[6] M. Schafer and P. Lewin. "Computerized system for measuring the acoustic output from diagnostic ultrasound equipment", IEEE Transactions on Ultrasonics, Ferroelectronics, and Frequency Control 35, 102-109 (1988).

[7] B. Herman and G. Harris. "Calibration of miniature ultrasonic receivers using planar scanning technique", Journal of the Acoustical Society of America 72, 1357 (1982).

[8] M. Breazeale and E. Hiedeman. L'Opticd methods for measureinent of sound pressure in liquids", Journai of the Acoustical Society of America 31, 24-33 (1959).

[9] T. Pitts, J. Greenleaf, J. Lu, and R. Kinnick. "Tomographic Schlieren imaging for measurement of bean pressure and intensity". In Proceedings of the 1994 IEEE Ultrasonics Symposium, volume 3, pp. 1665-1668, New York, 1994. IEEE, IEEE.

[IO] E. Hahn. "Detection of sea-water motion by nuclear precession", Journal of Geophyiscal Research 65, 776-777 (1960).

[Il] R. Frayne, D. Steinman, C. Ethier, and B. Rutt. "Accuracy of MR phase contrast", Journal of Magnetic Resonance Imaging 5, 428-431 (1995).

[12] D. Enzmann and N. Pelc. "Brain motion: Measurement with phase-contrast M R imaging", Radiology 185, 653-660 (1992).

[13] L. Pelc, J. Sayre, K. Yun, L. Castro, R. Herfkens, D. Miller, and N. Pelc. "Evaluation of my- ocardial motion tracking wit h cine-phase contrast magnetic resonance imaging" , Investigative Radiology 29, 1038-1042 (1994).

[14] V. Wedeen. "Magnetic resonance imaging of myocardial kinematics: Technique to detect, localize, and quantify the strain rates of the active human myocardium", Magnetic Resonance in Medicine 27, 52-67 (1992).

[15] W. Denk, R. Keoliam, S. Ogawa, and L. Jelinski. L'Oscillatory flow in the cochlea visualized by a magnetic resonance imaging technique". In Proceedings of the National Academg of Science, volume 90, pp. 1595-1598. National Academy of Science, 1993.

[16] R. Muthupillai, D. Lomas, P. Rossman, J. Greenleaf, A. Manduca, and R. Ehman. "Magnetic resonance elastography by direct visualization of propagating acuostic strain waves", Science 269, 1854-1857 (1995).

[17] R. Muthupillai, P. Rossman, D. Lomas, J. Greenleaf, S. Riederer, and R. Ehman. lLMagnetic resonance imaging of transverse acoustic strain waves", Magnetic Resonance in Medicine 36, 266-274 (1996).

El81 D. Stark and W. Bradley, editors. Magnetic Resonance Imaging, (Mosby, 1992).

[19] C. Chen, D. Hoult, and A. Hilger. Biomedical Magnetic Resonance Technologg, (Bristol, 1989).

[20] P. Mansfield and P. Morris. NMR imaging in biomedicine, (Academic Press, 1982).

[21] P. Moran. "A flow velocity zeugmatographic interlace for NMR imaging in humans", Magnetic Resonance Imaging 1, 197-203 (1982).

[22] C. Dumoulin and H. Hart. "Magnetic resonance angiography" , Radiology 161, 717-720 (1986).

[23] T. Chenevert, J. Brunberg, and J. Pipe. "Anisotropic diffusion within hurnan white matter: demonstration with NMR technique in-vivo", Radiology 177, 401-405 (1990).

[24] M. Moseley, J. Kucharczyk, J. Mintorovitch, Y. Cohen, J. Kurhanewicz, N. Derugin, H. Asgari, and D. Norman. "Diffusion-weighted M R imaging of acute stroke: correction with T2-weighted and magnetic susceptibility-enhanced MR imaging in cats", American Journal of NEuroradi- ology 11, 423-429 (1990).

[25] K. Brownstein and C. Tarr. "Importance of classical diffusion in NMR studies in biological cells", Physics Review 19A, 2446 (1979).

[26] E. Stejskal. "Use of spin-echoes in a pulsed magnetic field gradient to study anisotropic, restricted diffusion and fiow", Journal of Chernical Physics 43, 3597 (1965).

[27] P. Callaghan and J. Stepinski. "F'requency-domain analysis of spin motion using modulated gradient NMR", Journal of Magnetic Resonance, Series A 117, 118-122 (1995).

[28] Y. Fung. Biomechanics: mechanical properties of living tissues, (Springer-Verlag, 1992).

[29] M. Zisken and P. Lewin, editors. Ultrasound Exposimetry, (CRC Press, 1993).

[30] G. Harris. Wydrophone measurement in diagnostic ultrasound field", IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 35, 87 (1988).

[31] K. Beissner. "Maximum hyrophone size in ultrasonic field measurement" , Acoustica 59, 61 (1985).

[32] G. Harris. "A discussion of procedure for ultrasonic intensity and power calculations from miniature hydrophone rneasurements", Ultrasound in Medicine and Biology 11, 803 (1983).

[33] D. Shombert and G. Harris. Wse of miniature hydrophones to determine peak intensities typical of medical ultrasound devices", IEEE Tkansactions on Ultrasonics, Ferroelectrics, and F'requency Control33, 287 (1986).

[34] L. Brillouin. "Sur les tensions de radiation", Annals of Physics (Paris) 4, 528 (1925).

[35] K. Beissner. "On the time-average cacoustic pressure", Acoustica 57, 1 (1985).

[36] P. Cctrson and R. Banjavic. Radiation force balance system for precise acuostic power mea- surement in diagnostic ultrasound, (American Institute of Physics, 1980).

[37] M. Born and E. Wolf. Principles of Optics, Electrornagnetic Theory of Propagation, Interfer- ence and Difraction of Light, (Pergamon Press, 1970), 4th edition.

[38] R. Lucas and P. Biquard. "Properietes iques des milieux solides et liquides soumis aux vibra- tions elastiques ultrasonores", Journal of Phys. Radium 3, 464 (1932).

[39] C. Raman and N. Nath. "The diffraction of liglit by sound waves of high frequency". In Proceeding of the Indian Academg of Science, volume 2, p. 406. Indian Academy of Science, 1935.

[40] R. Reibold and W. Molkenstruck. "Laser interferrometric rneasurements and cornputerized evaluation of ultrasonic displacements", Acoustica 49, 205 (1981).

[41] R. Reibold and W. Molkenstruck. "Light diffraction tornography applied to the investigation of ultrasoic fields, 1. continuous waves" , Acoustica 56, 180 (1984).

[42] R. Reibold. "Light diffraction tomography applied to the investigation of ultrasonic fields, II. standing waves" , Acoustica 63, 283 (1987).

[43] K. Parker. LLUltrasonic attenuation and absorption in liver tissue", Ultrasound in Medicine and Biology 9, 363-369 (1983).

[44] W. Fry and R. F'ry. "Determination of absolute sound levels and acoustic absorption by thermocouple probes. Theory", Journal of the Acoustical Society of America 26, 294-310 (1954).

[45] W. Fry and R. Fry. "Determination of absolute sound levels and acoustic absorption by thermocouple probes. Experiment", Journal of the Acoustical Society of America 26, 311-317 (1954).

[46] K. Brendel and G. Ludwig. "Hydrophone measurements". In M. Zisken and P. Lewin, editors, Ultrasound Exposimetry, (CRC Press, 1993), chapter 5A, p. 117.

[47] D. Plewes, 1. Betty, S. Urchuk, and 1. Soutar. "Visualizing tissue compliance with MR", Journal of Magnetic Resonance Imaging 5, 733-738 (1995).

[48] F. Foster. "Medical irnaging with ultrasound" , Physics in Canada , 182-189 (1995).

[49] N. Vykhodtseva, K. Hynynen, and C. Damianou. "Pulse durationa and peak intensity during focused ultrasound surgery: theoretical and experimental effects in rabbit brain in vivo", Ultrasound in Medicine and Biology 20, 987-1000 (1994).

[50] G . Scott, M. Joy, R. Armstrong, and R. Henkelman. "Sensitivity to magnetic resonance current-density imaging", Journal of Magnetic Resonance Imaging 97, 235-254 (1992).

[51] "TESOR gradient coi1 design". Magnus Software Corporation.

[52] W. Smythe. Static and dynamic electricity, (McGraw-Hill Book Company, l968), p. 371.

[53] Howard W. Sams and Company, Inc. Reference data for radio engineers, Section 13-17, 1985.

[54] A. Redford and D. Drumheller. Introduction to elastic wave propagation, (John Wiley and Sons, Ltd., 1994), p. 96.

[55] P. Wells. "Physics of ultrasound". In M. Zisken and P. Lewin, editors, Ultrasound Exposimetry, (CRC Press, 1993), chapter 2, pp. 33-36.

[56] International Electrotechnical Commission. IEC 601-2-33, 1st edition.

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