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``````Modeling Therapeutic Ultrasound
Governing Equations negligible in soft tissue at low megahertz frequencies but can
The mathematical expressions that describe the physics of become signiﬁcant as the wavelength decreases or i.n highly
ultrasound wave propagation in tissue are known as govern- scatteri.ng media such as bone. Overall, the acoustic attenua-
ing equations. These are typically based on the conservation tion in soft biological tissue (which includes both absorption
of momentum (which accounts for the tissue having inertia), and scattering) has been experimentally observed to follow a
the conservation of mass (which accounts for the tissue being frequency power law of the form at“ f V, where f is the frequency
compressible), and an equation of state or pressure-density and the power law exponent y is between 1 and 2. This type of
relationship (which encapsulates the thermodynamics of behavior can be captured in the governing equations by includ-
wave propagation). In many branches of acoustics, these i.ng a distribution of relaxation processes or by using fractional
equations can be simplified by assuming linear wave propaga- derivative loss operators (Holm and Nasholm, 2014). A com-
tion i.n a lossless and homogeneous medium, which leads the monly used rule of thumb is that ultrasound in soft biological
widely known wave equation. In biomedical ultrasound, this tissue is attenuated at a rate of 1 dB/MHZ/cm.
equation is appropriate for studying the spatial distribution of
acoustic pressure from therapeutic ultrasound transducers in Third, in biological tissues, medium properties such as the
water at low output levels. However, for modeling therapeutic sound speed, mass density, and acoustic absorption coeﬂicient
ultrasound in the human body, the simplifying assumptions are heterogeneous across multiple scales. At the microscopic
mentioned above generally no longer hold. level (much smaller than an acoustic wavelength), there are

variations in the acoustic properties between individual cells
First, the pressure amplitudes used i.n biomedical ultrasound and between cells and other tissue constituents such as blood
are often suﬂiciently high to give rise to nonlinear effects (this plasma and the extracellular matrix (the structural scaffold-
is true for both therapeutic and diagnostic applications of i.ng that holds many cells i.n place). These differences give rise
ultrasound). At high acoustic pressures, the stiffness of the to diffusive scattering, which is responsible for the speckle
tissue depends on how much it is being compressed (mate- pattern characteristic of diagnostic ultrasound images. How-
rial nonlinearity) and the cyclical motion of the medium, ever, from an ultrasound therapy perspective, this scattering
due to the acoustic wave, affects the wave speed (convective is generally weak and can be accounted for as part of a phe-
nonlinearity). Together, these cause the sound speed in the nomenological attenuation term in the governing equations.
medium to depend dynamically on the local values of the
acoustic pressure and particle velocity. For example, during At a macroscopic level, different structures within an organ,
the compressional phase of the wave where the particle veloc- such as blood vessels or regions of fatty and fibrous tissue,
ity is positive (i.e., the medium is being displaced i.n the same can also give rise to scattering. There are also differences at
direction as the wave is traveling), the effective sound speed the organ level, again due to variations in the underlying
increases and vice versa. This causes a cumulative distortion tissue constituents and their structure. For example, tissues
in the time-domain waveform, which corresponds to the with a higher proportion of lipids (e.g., fat) typically have a
generation of higher frequency harmonics in the frequency lower sound speed compared with water at body temperature,
domain. From amodeling standpoint, this type of nonlinear- whereas tissues with a higher proportion of proteins (e.g.,
ity can be captured by retaining second-order terms in the liver) have a higher sound speed. These macroscopic varia-
governing equations (Hamilton and Blackstock, 2008). tions in the acoustic properties of tissue can have a significant

imp act on the propagation of focused ultrasound ﬁelds,
Second, biological tissue can strongly attenuate ultrasound including changing the shape, position, and amplitude of the
waves, particularlywaves at megahertz frequencies. The exact focal region (see Figure 1, lmttam). In some cases, the aber-
mechanisms for the absorption i.n tissue are complex and occur rations are so strong that the focus is completely destroyed.
at both the cellular level (e.g.,viscous relative motion and ther- Spatially varying material properties can be included in
ma.l conduction between the cells and their surroundings) and the governing equations by starting with the conservation
the molecular level (e.g., molecular and chemical relaxation). equations and retaining the spatial gradients of the material
These processes cause the gradual degradation of acoustic parameters during the linearization process.
energy into random thermal motion and, consequently, the
attenuation of the wave amplitude. In addition to absorption, The combination of the mass and momentum conservation
acoustic energy is also lost due to scattering. This is generally equations (retaining heterogeneous and nonlinear terms)
as | AA:a|.nIiI:l 'I'b:Iay| Summer 2019

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