Numerical methods for nonlinear wave propagation in ultrasound
Abstract
The intensities associated with the propagation of diagnostic and therapeutic ultrasound
pulses are large enough to require a nonlinear description. As a nonlinear wave propagates
it distorts, creating harmonics and eventually acoustic shocks. Harmonics can be used
to generate images with improved spatial resolution and less clutter. The energy from
nonlinear waves is deposited in a different way than in the linear case which modifies
predictions for in situ acoustic exposure. Tissue heating and radiation force depend
on this intensity. High intensity shock waves are essential for stone communition
with lithotripsy because it depends on the shear gradients caused by the pressure
differentials and on the peak negative pressures for cavitation. The work presented
in this dissertation investigates numerical simulations that solve nonlinear ultrasonic
wave propagation in both the strongly nonlinear regime, where shocks develop, and
the weakly nonlinear regime, where the acoustic attenuation prevents the formation
of pressure discontinuities.
The Rankine-Hugoniot relation for shock wave propagation describes the shock speed
of a nonlinear wave. This dissertation investigates time domain numerical methods
that solve the nonlinear parabolic wave equation, or the Khokhlov-Zabolotskaya-Kuznetsov
(KZK) equation, and the conditions they require to satisfy the Rankine-Hugoniot relation.
Two numerical methods commonly used in hyperbolic conservation laws are adapted to
solve the KZK equation: Godunov's method and the monotonic upwind scheme for conservation
laws (MUSCL). It is shown that they satisfy the Rankine-Hugoniot relation regardless
of attenuation. These two methods are compared with the current implicit solution
based method. When the attenuation is small, such as in water, the current method
requires a degree of grid refinement that is computationally impractical. All three
numerical methods are compared in simulations for lithotripters and high intensity
focused ultrasound (HIFU) where the attenuation is small compared to the nonlinearity
because much of the propagation occurs in water. The simulations are performed on
grid sizes that are consistent with present-day computational resources but are not
sufficiently refined for the current method to satisfy the Rankine-Hugoniot condition.
It is shown that satisfying the Rankine-Hugoniot conditions has a significant impact
on metrics relevant to lithotripsy (such as peak pressures), and HIFU (intensity).
Because the Godunov and MUSCL schemes satisfy the Rankine-Hugoniot conditions on coarse
grids they are particularly advantageous for three dimensional simulations.
The propagation of focused and intense ultrasound beams is determined by nonlinearity,
diffraction, and absorption. Most descriptions of nonlinear wave propagation in ultrasound,
such as the KZK equation, rely on quadratic nonlinearity. At diagnostic and some therapeutic
amplitudes the quadratic, or B/A, term dominates the nonlinear term. However, when
the amplitudes are sufficiently large, such as in shock wave lithotripsy, the cubic,
or C/A, term becomes significant. Conventionally the parabolic wave equation has only
included the quadratic terms. This dissertation establishes a time domain numerical
method that solves the parabolic wave equation with cubic nonlinearity in an attenuating
medium. The differences between solutions of the quadratic and cubic equations for
a focused lithotripter in a water bath are investigated.
A study of numerical solutions to the linear full-wave equation and the KZK or parabolic
wave equation is presented. Finite difference time domain methods are used to calculate
the acoustic field emitted from a diagnostic ultrasound transducer. Results are compared
to Field II, a simulation package that has been used extensively to linearly model
transducers in ultrasound. The simulation of the parabolic equation can accurately
predict the lateral beamplot for large F-numbers but exhibits errors for small F-numbers.
It also overestimates the depth at which the focus occurs. It is shown that the finite
difference solution of the full-wave equation is accurate for small and large F-numbers.
The lateral beamplots and axial intensities are in excellent agreement with the Field
II simulations. For these reasons the KZK equation is abandoned in favor of the full-wave
equation to describe nonlinear propagation for ultrasound imaging.
A full-wave equation that describes nonlinear propagation in a heterogeneous attenuating
medium is solved numerically with finite differences in the time domain (FDTD). Three
dimensional solutions of the equation are verified with water tank measurements of
a commercial diagnostic ultrasound transducer and are shown to be in excellent agreement
in terms of the fundamental and harmonic acoustic fields, and the power spectrum at
the focus. The linear and nonlinear components of the algorithm are also verified
independently. In the linear non-attenuating regime solutions match simulations from
Field II to within 0.3 dB. Nonlinear plane wave propagation is shown to closely match
results from the Galerkin method up to four times the fundamental frequency. In addition
to thermoviscous attenuation we present a numerical solution of the relaxation attenuation
laws that allows modeling of arbitrary frequency dependent attenuation, such as that
observed in tissue. A perfectly matched layer (PML) is implemented at the boundaries
with a novel numerical implementation that allows the PML to be used with high order
discretizations. A -78 dB reduction in the reflected amplitude is demonstrated. The
numerical algorithm is used to simulate a diagnostic ultrasound pulse propagating
through a histologically measured representation of human abdominal wall with spatial
variation in the speed of sound, attenuation, nonlinearity, and density. An ultrasound
image is created in silico using the same physical and algorithmic process used in
an ultrasound scanner: a series of pulses are transmitted through heterogeneous scattering
tissue and the received echoes are used in a delay-and-sum beamforming algorithm to
generate images. The resulting harmonic image exhibits characteristic improvement
in lesion boundary definition and contrast when compared to the fundamental image.
We demonstrate a mechanism of harmonic image quality improvement by showing that the
harmonic point spread function is less sensitive to reverberation clutter.
Numerical solutions of the nonlinear full-wave equation in a heterogeneous attenuating
medium are used to simulate the propagation of diagnostic ultrasound pulses through
a measured representation of the human abdomen with heterogeneities in speed of sound,
attenuation, density, and nonlinearity. Conventional delay-and-sum beamforming is
used to generate point spread functions (PSF) from a point target located at the focus.
These PSFs reveal that, for the particular imaging system considered, the primary
source of degradation in fundamental imaging is due to reverberation from near-field
structures. Compared to the harmonic PSF the mean magnitude of the reverberation clutter
in the fundamental PSF is 26 dB higher. An artificial medium with uniform velocity
but unchanged impedance characteristics is used to show that for the fundamental PSF
the primary source of degradation is phase aberration. Ultrasound images are created
in silico and these beamformed images are compared to images obtained from convolution
of the PSF with a scatterer field to demonstrate that a very large portion of the
PSF must be used to accurately represent the clutter observed in conventional imaging.
Conventional delay-and-sum beamforming is used to generate images of an anechoic lesion
located beneath the abdominal layer for various transducer configurations. Point spread
functions (PSF) and estimates of the contrast to noise ratio (CNR) are used to quantify
and determine the sources of improvement between harmonic and fundamental imaging.
Simulations indicate that reducing the pressure amplitude at the transducer surface
has no discernible effect on image quality. It is shown that when the aperture is
reduced there is an increase in the image degradation due to reverberation clutter
in the fundamental and an increase in the effects of reverberation and phase aberration
in the harmonic. A doubling of the transmit frequency shows that the harmonic lesion
CNR becomes worse than the fundamental CNR due to increases in pulse lengthening and
phase aberration.
Acoustic Radiation Force Impulse (ARFI) imaging uses brief, high intensity, focused
ultrasound pulses to generate a radiation force that displaces tissue. Nonlinear propagation
of acoustic pulses transfers energy to higher frequencies where it is preferentially
absorbed by tissue. The radiation force is proportional to the absorbed energy. This
dissertation examines the effects of nonlinearity on the displacements induced by
radiation force with various ultrasound transducer configurations. A three dimensional
numerical method that simulates nonlinear acoustic propagation is used to calculate
the intensity and absorption losses for typical ARFI pulses. It is demonstrated that
nonlinearity has a relatively small effect on the intensity but increases estimates
of the loss by up to a factor of 20. The intensity fields obtained from the acoustic
simulations are used as an input to a finite element method (FEM) model of the mechanical
tissue response to a radiation force excitation. These simulations show that including
nonlinearity in the acoustic intensity significantly reduces predictions of the displacement
without having a significant impact on the lateral and elevation resolution.
Type
DissertationPermalink
https://hdl.handle.net/10161/457Citation
Pinton, Gianmarco (2007). Numerical methods for nonlinear wave propagation in ultrasound. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/457.Collections
More Info
Show full item record
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.
Rights for Collection: Duke Dissertations
Works are deposited here by their authors, and represent their research and opinions, not that of Duke University. Some materials and descriptions may include offensive content. More info