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

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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.

It 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 1include 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 MM ...... 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 MR 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 displacement ...... 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 List of Abbreviations

us ultrasound ...... 1 MW magnetic resonance imaging ...... 2 NMR nuclear magnetic resonance ...... 3 RF radio frequency ...... 3 PC Mm phase contrast MRI ...... 6 PCA phase contrast angiography ...... 8 PGSE pulsed gradient spin echo ...... 8 kHz kilohertz ...... 15 MHz megahert.z ...... 15 SNR 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 at 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 MNto 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:

WL = -7Bo (1.2) where the constant of proportionality between tlie magnetic field and frequency is y, the gyromagnetic 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 Tlrelaxation, 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 (Mz)returns to its equilibrium value. This relaxation process is approximated by an exponential growth with the tirne constant Tl. T2relaxation 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 Tldecay also contributes to T2 relaxation, T2 5 Tl. Both Tiand

T2are dependent upon local tissue properties, and hence are the major source of contrat in clinical

MR images. Table 1.1 lists the Tland 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 constant 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 MxYthrough 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 Imm.

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:

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

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 at 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 to 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 acquiring 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 MNis 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 PCA, 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 demonstrate 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 at 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 effective 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 MM 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] hademonstrated 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 an 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 heterogeneous 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 amplitudes 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 to 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, it 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 the 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 currently 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 calibrated 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

\ultrasound4 pulse ultr&ound ultrasound digital transducer absorber oscilloscope

Figure 1.4: Experimental apparatus used for measuring ultrasonic fields with a calibrated hydrophone.

A calibrated hydrophone is a very common device for acquiring pressure distributions because 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 absorber I 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. Knowing the amplitude absorption coefficient (O)and the heat capacity (C), the local intensity 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 at 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 at 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 indicated 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 MM 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 10-~ 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 at 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 calculations 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 ultrasound 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 at 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 10-~nanometers for a diagnostic field (200rn~s/0.0001~n-~)to - 100 nanometers for a therapeutic ultrasound field (800~m-~/0.5m~z).In order to assess the feasibility of applying phase contrat 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 MR 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 TE 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 108rad - 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 ultrasouncj 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 produced in order to successfully image a particular ultrasonic field. For an ultrasound field with

an intensity of 1 W and a frequency of 500 kHz, the particle oscillation amplitude will be <, <, = 36.8nm. To achieve an SNRCbof 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 to 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 the Biot-Savart law, the field produced by a coil composed of N loops of wire dong its axis, can 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 at 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 at 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. It is this radial dependence on L, within a conductor that produces the skin effect.

The impedance of an elemental wire at radial position r, with a self inductance L,(r) and resistance (R) per unit length is given by:

(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). Impedance 4 Z(f,r) Inductance t Ls(r)

-r -a O a r radial position radial position

Figure 2.3: A) Radial dependence of self inductance within a conductor, showing a maximum inductance 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, at 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 haan 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 mR/m Litz Wire 125 mR/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 resonance 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.

Driver Circuit O-5V 515 kHz

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

2.3.2 Ultrasound Transducer

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 at 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 ha. 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 resultant 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 ultrasound 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 measurements acquired using MR phase contrast imaging. The ultrasound intensity at 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 particle 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 apparatus 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 to 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 measurement 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 corresponds 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 Distribution 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 MR technique for measuring t his gradient distribution.

As stated in Section 1.2, the precessional frequency of spins and thus their phase is proportional 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 information 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 an 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, Aeed= c,

needle oscilloscope I I I transducer pulse hydrophone

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 measurements 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 therefore 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 waveform. In order to implement this technique, MR motion sensitive imaging is performed for ultrasonic 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 ais (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 amplitude 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 at 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 ultrasound 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 corresponding to the spatial frequency of the ultrasound wave. By bandpass filtering the image at the ultrasound frequency, the signal is unaffected while the noise at 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 O12345678910 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 demonstrates 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 7 819 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) O12345678910

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 glas 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 quantitative 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 of 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 acoustic 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 MR measured acoustic intensity. Figure 2.19 compares

the ultrasound intensity measured at a fixed point to the corresponding ultrasound transducer excitation 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 - io-iz-i4- Transducer Driving Power (W)

Figure 2.19: MR 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 MR 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 that will be produced for a given magnitude SNR and can be used as an estimate for the overall 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 displacement, 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 frequencies, 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 ultrasonic 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, MNtiming 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 structures, 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 ultrasonic 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 MM 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 MM 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 propagation [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 frequency 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 at 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 at low ultrasound frequencies.

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 distribution 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 scattering 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 application 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 properties, 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 entirely 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 S669-2075Bayview Avenue, Toronto, Ontario, Canada M4N 3M5 Phone (416) 480-5709 Fax (416) 480-5714 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 1WZ), 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 (el0 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 200-1000 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, + t0sin@ 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 ris the duration of the gradient waveform (2W).Thus the spatial distribution

of phase reports the local ultrasound-induceddisplacement 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 IOcm 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-maximumof 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. 1B. 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 I120 nm are observed at the

focus. This translates into a spatial peak temporal average intensity of 11.7 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 I30 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>/"/ NzG, 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 -1000 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. 113.

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, DL; 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: 1854-1 857, 1995.

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): 1038-1042, 1994.

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 Oscillating transducer Absorbing Chamber

Ultrasound RF coi1 field

time @$- @$- 4.2 4.4 4.6 4.8 5 Axial Position (cm) Axial Position (cm) O12345678910

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 ' echo , n/2 7T LL P: 9 k 4.2 4.4 4.6 4.8 5 Axial Position (cm) Axial Position (cm) O12345678910

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

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