412
J.L. Aroyan et al.
In this equation, Kij,k = (lIh)Cij,k is a nondimensionalized version of the spatially varying acoustic velocity Cij,k; Pij,k is the spatially varying density; and
1= !!.t and h = ax = Lly = Llz are the temporal and spatial grid increments,
respectively. This scheme is fourth order in the spatial derivatives of pressure, second order in the spatial derivatives of density, and second order in
the time derivative of pressure. The first three lines of Eq. 1b constitute the
scheme for the homogeneous wave equation. This homogeneous scheme
was used outside the tissue regions of the grid and greater than two points
away from the grid edges. A second order spatial derivative scheme was
applied at the next-to-Iast grid points. Absorbing boundary conditions were
applied at the extreme grid edges to reduce reflections from grid boundaries to very low levels.
Figure 1O.la illustrates a typical simulation grid layout. The region of the
grid containing the tissue model is indicated. Except for a few trials on halfresolution grids, a 1.5-mm cubic grid corresponding to the finest X-ray CT
data voxel size was used in the forehead simulations. For the inverse simulations, the front face of the grid was used as a flat source to ensonify the
tissue model region with a cosine-windowed "echo" at a fixed frequency
and angle of return. A windowed aperture function was used to reduce the
near-field diffraction of this source. The energy density at the focal maxima
found in the inverse simulations were two to three orders of magnitude
above variations caused by near-field diffraction and uneven ensonification
of the model region.
Several different quantities may be calculated from the simulated pressure field and visualized with graphics packages in order to illustrate the
patterns of sound propagation within the tissue models. For example, one
may visualize the time-averaged magnitude of the total acoustic energy
density Wtotal as the sum of the potential acoustic energy density W potential
and the kinetic acoustic energy density Wkinetic (Pierce 1981; Morse and
Ingard 1968)
1
2
P
2
Weotal =Wpotenrial + Wkinetic =--2 [Re(p)] +-ZIRe(u)1
(Z)
ZpC
where the vector fluid velocity u = Vp/(irop). Alternatively, one may visualize the time-averaged magnitude of the potential energy density Wpotential =
[Re(p)f/(Zpc
2 ), as in Aroyan (1996). In this chapter, we will visualize either
the total or the potential acoustic energy density, although all three forms
of energy density (potential, kinetic, or total) and/or the acoustic intensity
vector yield significant information regarding propagation within tissues.
In order to compute the emitted far-field patterns of various source and
tissue models, the simulated pressure field and its normal derivative over
a rectangular surface immediately surrounding the tissue region of the
grid was used as input to boundary integral extrapolation programs. The
J.L. Aroyan et al.
In this equation, Kij,k = (lIh)Cij,k is a nondimensionalized version of the spatially varying acoustic velocity Cij,k; Pij,k is the spatially varying density; and
1= !!.t and h = ax = Lly = Llz are the temporal and spatial grid increments,
respectively. This scheme is fourth order in the spatial derivatives of pressure, second order in the spatial derivatives of density, and second order in
the time derivative of pressure. The first three lines of Eq. 1b constitute the
scheme for the homogeneous wave equation. This homogeneous scheme
was used outside the tissue regions of the grid and greater than two points
away from the grid edges. A second order spatial derivative scheme was
applied at the next-to-Iast grid points. Absorbing boundary conditions were
applied at the extreme grid edges to reduce reflections from grid boundaries to very low levels.
Figure 1O.la illustrates a typical simulation grid layout. The region of the
grid containing the tissue model is indicated. Except for a few trials on halfresolution grids, a 1.5-mm cubic grid corresponding to the finest X-ray CT
data voxel size was used in the forehead simulations. For the inverse simulations, the front face of the grid was used as a flat source to ensonify the
tissue model region with a cosine-windowed "echo" at a fixed frequency
and angle of return. A windowed aperture function was used to reduce the
near-field diffraction of this source. The energy density at the focal maxima
found in the inverse simulations were two to three orders of magnitude
above variations caused by near-field diffraction and uneven ensonification
of the model region.
Several different quantities may be calculated from the simulated pressure field and visualized with graphics packages in order to illustrate the
patterns of sound propagation within the tissue models. For example, one
may visualize the time-averaged magnitude of the total acoustic energy
density Wtotal as the sum of the potential acoustic energy density W potential
and the kinetic acoustic energy density Wkinetic (Pierce 1981; Morse and
Ingard 1968)
1
2
P
2
Weotal =Wpotenrial + Wkinetic =--2 [Re(p)] +-ZIRe(u)1
(Z)
ZpC
where the vector fluid velocity u = Vp/(irop). Alternatively, one may visualize the time-averaged magnitude of the potential energy density Wpotential =
[Re(p)f/(Zpc
2 ), as in Aroyan (1996). In this chapter, we will visualize either
the total or the potential acoustic energy density, although all three forms
of energy density (potential, kinetic, or total) and/or the acoustic intensity
vector yield significant information regarding propagation within tissues.
In order to compute the emitted far-field patterns of various source and
tissue models, the simulated pressure field and its normal derivative over
a rectangular surface immediately surrounding the tissue region of the
grid was used as input to boundary integral extrapolation programs. The