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and an equation of state (accounting for absorption through to the acoustic wavelength at the highest frequency of interest.
a fractional loss operator or sum of relaxation terms) can Consider the case of transcranial focused ultrasound surgery,
account for the complex wave behavior seen in biological where ultrasound waves are used to destroya small region of
tissue. This includes scattering, refraction, nonlinear wave tissue deep in the brain. The domain of interest encompass-
steepening, and acoustic absorption following a frequency ing the ultrasound transducer and the head is on the order
power law. However, in some cases, additional factors must of 30 cm in each direction. For a center frequency of 650
also be considered, for example, the temperature dependence kHz, this distance is on the order of 130 wavelengths at the
of the tissue material properties, the motion of organs due fundamental frequency and 650 wavelengths at the ﬁfth har-
to breathing or the cardiac cycle, and/or acoustic cavitation monic. Applying the engineering rule of thumb of 20 points
(Maxwell et al., 2012; Gray et al., 2019). Systematically incor- per wavelength (PPW) sometimes used for ﬁnite-element and
porating such extensions into the governing equations in a ﬁnite-difference methods, this corresponds to a computa-
tissue-realistic manner is not straightforward. For modeling tional grid size of 13,000 x 13,000 x 13,000 grid points (more
scenarios that involve bones, the generation, propagation, than 2 trillion degrees of freedom). Simply storing one matrix
and absorption of shear waves must also be considered. In of this size in single-precision floating-point format would
this case, the mass conservation equation and the equation consume 8 terabytes (TB) of memory, and typically several
of state are replaced with a model of viscoelasticity (a gener- such matrices are needed. To put this into context, the current
alization of Hooke’s law). generation MacBook Air comes equipped with 8 gigabytes
(GB) of memory, so 1,000 of them would be required to store
Numerical Methods one matrix at a cost ofmore than one million US dollars! Of
The techniques used to discretize the governing equa- course, supercomputing is not done using consumerlaptops,
tions so that they can be solved by a computer are known but the point remains: problems of this nature can become
as numerical methods. There are many different types of extremely large scale.
numerical methods used in acoustics. These include the
finite-element method, boundary-element method, ﬁ.nite- So why is the engineering rule of thumb to use 20 PPW when
difference method, Green’s function methods, and spectral the Nyquist theorem tells us we should only need two? The
methods (Verweij et al., 2014). The most appropriate choice primary reason is numerical dispersion. This is a numeri-
depends on the problem speciﬁcs, for example, the distri- cal error in which approximations made in the numerical
bution of material properties (e.g., homogeneous, piecewise method cause the modeled ultrasound waves to travel at
constant, or continuously varying), whether the problem is different speeds depending on their frequency (the depen-
linear or nonlinear, whether the source is single frequency dence of sound speed on frequency is known as dispersion).
or broadband, and the scale of the domain of interest. For This dependence means that broadband waves will become
therapeutic ultrasound (which usually involves nonlinear increasingly distorted compared with the true solution as
wave propagation in heterogeneous and absorbing biological they propagate across the computational grid (equivalently,
tissue), the most common approach is to use computationally single frequency waves will travel at the wrong speed). This is
efﬁcient collocation methods, such as the ﬁnite-difference a particular challenge for the large domain sizes encountered
time domain (FDTD) or pseudospectral time domain (PSTD) in therapeutic ultrasound (often hundreds of wavelengths),
methods (Gu and Iing, 2015). These methods can be used to because errors due to numerical dispersion accumulate the
directly solve the governing equations as a system of coupled further the waves travel.
equations, or the equations can be combined and solved as a
generalized wave equation. The former has some advantages For finite-difference methods, provided that the numerical
for numerically imposing radiation conditions at the edge scheme mathematically reduces to the governing equations
of the computational domain (such as a perfectly matched in the limit that the spatial and temporal steps reduce to zero
layer) and for inputs and outputs that depend on the acous- (a condition known as consistency) and that the method is
tic particle velocity (including modeling dipole sources and stable (there are standard mathematical and numerical tools
calculating the vector acoustic intensity). for analyzing stability), the Lax equivalence theorem tells
us that the scheme will be convergent. This means that the
A significant challenge when modeling biomedical ultrasound numerical solution will approach the exact solution of the
is the large distances traveled by the ultrasound waves relative governing equations as the size of the spatial and temporal
Summer 2019 | Anouuuau 'I'b:Iay 1 as
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