Biomedical Engineering Reference
In-Depth Information
increases the amount of computer memory and the computation
time. Furthermore, direct access to the normal particle veloc-
ity is restricted to planar transducer and array geometries. Thus,
the utility of
h
u
(
x ,y,z
) in Equation 6.23 or
H
u
(
k
x
,
k
y
,
z
) in Equation
6.26 for angular spectrum simulations of pressure fields evalu-
ated for thermal therapy is limited, especially if the numerical
errors are considered.
With the proper collection of input parameters, the angu-
lar spectrum approach rapidly and accurately computes pres-
sures in large 3D volumes for simulations of thermal therapy.
One important parameter for these simulations is the distance
between adjacent spatial grid points (i.e., the spatial sampling
rate). For time harmonic pressure simulations, the distance
between two adjacent spatial samples must be no greater than
λ/2, where λ is the wavelength in the medium. If the distance
between adjacent spatial samples is any larger, then the pressure
field will be aliased, and the temperatures computed with the bio-
heat transfer equation will be erroneous. This restriction on the
spatial sampling, which is not specific to the angular spectrum
approach, must be satisfied by any pressure field calculation that
involves subsequent temperature calculations. Other parameters
include the size of the computational grid and the location of the
input pressure plane. For computations in large 3D volumes, the
input pressure plane for angular spectrum calculations should
be located about one wavelength (λ) from the phased array, and
the source pressure plane should be larger than the transducer
or phased array (Zeng and McGough 2009). This combination of
parameters, when applied to the angular spectrum calculations,
typically achieves errors less than or equal to 1% throughout the
computational grid when the input pressure plane is computed
using the fast near-field method. The reduction in the computa-
tion time achieved when the angular spectrum approach is com-
bined with the fast near-field method can reach several orders of
magnitude relative to the Rayleigh-Sommerfeld integral or the
rectangular radiator method, especially for large phased arrays
evaluated on large computational grids. This reduction in com-
putation time is highly desirable for thermal therapy calcula-
tions, which can be very time consuming.
The angular spectrum approach is also amenable to certain
enhancements. For example, if the input pressure plane calcula-
tions for each element in a phased array are stored in advance,
then the total input pressure planes for a wide variety of focal
patterns are quickly obtained via superposition (Vyas and
Christensen 2008). This approach significantly reduces the com-
putation time when evaluating a large number of focal patterns
as in (Jennings and McGough 2010, Zeng et al. 2010). The angu-
lar spectrum approach also models wave propagation in flat, lay-
ered media (Clement and Hynynen 2003, Vecchio et al. 1994).
Angular spectrum models in layered media typically only retain
the transmitted component at planar boundaries, discarding
the reflected component. This approach provides an estimate
of the transmitted power under the assumption that the reflected
power is relatively small and that multiple reflections are negli-
gible. Angular spectrum models of propagation in inhomoge-
neous tissue are also under development. One such model is
the hybrid angular spectrum approach (Vyas and Christensen
2008), which considers different values for the speed of sound,
absorption, and density in each voxel. The hybrid angular spec-
trum approach propagates back and forth between the pressure
evaluated in the spatial domain and the 2D Fourier transform of
the pressure to account for the differences in the material prop-
erties for each voxel. This approach enables propagation through
curved layers and inhomogeneous tissues.
6.2.5 Nonlinear Ultrasound propagation
Nonlinear propagation effects occur in many therapeutic ultra-
sound applications, where these effects are described by several
different numerical models. Some nonlinear models simulate
diffraction with the angular spectrum approach (Christopher
and Parker 1991b, Zemp et al. 2003), and then attenuation and
nonlinear interactions are calculated separately. These nonlin-
ear pressure simulations incorporate a relatively small number
of harmonics (5-10) to describe nonlinear propagation with-
out shock wave formation and a much larger number (at least
30 to 50) when modeling shock waves (Christopher and Parker
1991b). Nonlinear ultrasound propagation is also described by
the KZK equation, which is a popular model for therapeutic
ultrasound that simulates transient and time-harmonic nonlin-
ear propagation for axisymmetric ultrasound sources (Lee and
Hamilton 1995). The KZK equation is solved either in the fre-
quency domain (Aanonsen et al. 1984, Khokhlova et al. 2001)
or the time domain (Cleveland et al. 1996, Lee and Hamilton
1995), where the effects of diffraction, frequency-dependent
attenuation and dispersion, and nonlinearity are computed
separately at each propagation step. The KZK equation has also
been extended to 3D Cartesian geometries (Yang and Cleveland
2005). Nonlinear ultrasound propagation for non-axisymmetric
sources is also simulated with the Westervelt equation (Huijssen
and Verweij 2010), which models full-wave nonlinear propaga-
tion. Nonlinear simulations are very time consuming, especially
when compared to linear simulations, and for this reason, most
simulations of nonlinear ultrasound are evaluated for 2D prob-
lems with cylindrical symmetry. Nonlinear ultrasound simula-
tions are also evaluated in 3D despite the long simulation times.
6.2.6 Software programs
Several software programs are available for simulations of thera-
peutic ultrasound. Some of these are commercial products, and
others were developed by academic researchers. Most simulation
programs were originally developed for other applications and
then adapted to therapeutic ultrasound. One commercial prod-
uct, PZFlex (Wojcik et al. 1993), is a finite element package that
models the electromechanical behavior of ultrasound transduc-
ers, and SPFlex is a separate module that calculates the pressure
fields generated by these transducers (Mould et al. 1999). Many
other commercial finite element packages exist, but these typically
have steep learning curves, and significant user effort is required
when extending them for therapeutic ultrasound simulations.
