# Engineering Acoustics/Biomedical Ultrasound

## Biomedical Ultrasound

This chapter of the Engineering Acoustics Wikibook provides a brief
overview of biomedical ultrasound applications along with some
introductory acoustical analysis for ultrasound beams. As a whole, the
field of Biomedical Ultrasound is one that provides a wealth of topics
for study involving many base disciplines. As such, this limited entry
does not cover all aspects of Biomedical Ultrasound, but instead chooses
to focus on providing readers with an introductory understanding, from
which additional study of the topic is possible. For readers interested
in a more thorough reference on Biomedical Ultrasound the 2007 text by
Cobbold [^1] is suggested.

## Diagnostic Applications

The most well know application of Biomedical Ultrasound is in medical
imaging, also known as ultrasonography. For a list of specific
applications of ultrasonography refer to the corresponding Wikipedia
entry. The following section
provides a qualitative description of the acoustical process used to
generate and capture sound signals used in producing ultrasound images.

An ultrasound transducer emits a short pulse of high frequency sound.
Depending on application the wave frequency ranges between 1 MHz and
15 MHz.[^2] As the emitted sound waves propagate they will be partially
reflected or scattered by any variation in acoustic impedance, *ρc*,
that is encountered. In the context of to biomedical imaging, this
corresponds to anywhere there are density changes in the body: e.g. the
connection of bone to muscle, blood cells in blood plasma, small
structures in organs, etc.[^3]

The behavior of the reflected wave depends largely on the size of the
reflective feature and the wavelength of the emitted sound wave. When
the wavelength is short relative to the reflective structure reflections
will be governed according to the principals of acoustic transmission
and reflection with normal or oblique interfaces.[^4] When the
wavelength is long relative to the structure the principals of acoustic
scattering [^5] are applicable. The latter condition, which occurs for
small reflection sources, sets the requirement for frequencies used in
ultrasound imaging. As discussed by Cobbold,[^6] analysis for a planar
wave incident on a spherical reflection source of effective radius *a*,
shows the acoustic intensity of the scattered wave, *I~s~*, varies
according to:

\
$$I_s \propto  \frac{a^6}{\lambda ^4}$$

This relation shows that when wavelength is long relative to a scatter
sources effective radius, the scattered energy becomes very small, thus
negligible amounts of the incident wave will be reflected back to the
transducer. To reliably capture a feature in an ultrasound image the
emitted wavelength must be smaller than the features of interest. Other
consideration for wavelength are also applicable: due to attenuation of
the propagating wave lower frequencies offer greater imaging depth,
while higher frequencies (with smaller wavelength) offer increase
ability for lateral focusing of the emitted beam (small beam width at
focus, see below).[^7] Table 1 gives the correlation between frequency
and wavelength in water for several frequencies used in ultrasound
imaging (*λ* = *c*/*f*).

\
{\| class=\"wikitable\" style=\"text-align: center;\" align=\"center\"
\|+ Table 1: Medical ultrasound frequencies and corresponding
wavelengths. \|- \| Frequency (MHz) \|\| 1 \|\| 2 \|\| 5 \|\| 8 \|\| 10
\|\| 12 \|\| 15 \|- \| Wavelength (mm) \|\| 1.50 \|\| 0.75 \|\| 0.30
\|\| 0.19 \|\| 0.15 \|\| 0.13 \|\| 0.10 \|}

After the wave burst is transmitted the ultrasound transducer can act as
a receiver, much like a microphone or hydrophone. Waves reflected off
structures and density gradients are returned to the transducer and
recorded. The delay time between the transmitted and received signal is
correlated to the distance of the reflection source, while the intensity
of the received signal is correlated to the reflection sources acoustic
impedance and size.[^8] In instances where doppler ultrasonography is
utilized, the frequency shift between the transmitted and received
signals can be correlated to the velocity of the reflection source.

Modern ultrasonography use arrays of small transducers, each of which
are individually electronically controlled to achieve an effect know as
beamforming. When using this technique,
control of the phase relation between array elements results in control
over the emitted beam\'s direction and focal depth.[^9] To produce a
two-dimensional ultrasound image, the ultrasound beam focal position is
swept through a region, and the recorded reflected waves are correlated
to the particular focal locations. The exact process by which this
general concept is accomplished varies with each ultrasonography
instrument. Figure 1 provides a sample 2D image produced by the sweeping
of the focal location through a 2D plane.

\
!Figure 1: 2D Obstetric ultrasound
image.{width="420"}

## Clinical and Therapeutic Applications

A number of important clinical and therapeutic applications make use of
high intensity, focused ultrasound beams. In many of these applications,
the therapeutic effect is achieved due the heat generation associated
with dissipation of the high intensity acoustic beam. In some
applications, such as
lithotripsy the
therapeutic effect is obtained from acoustic non-linearity, causing
wave deformation
and shock wave
formation.
This effect is discussed in more detail in a section to
follow.

Provided below is a partial list of Therapeutic applications of
ultrasound:

-   Ultrasound is sometimes used to clean teeth in dental
    hygenist.
-   Focused ultrasound may be used to generate highly localized heating
    to treat cysts and tumors (benign or malignant). This is known as
    Focused Ultrasound Surgery (FUS) or High Intensity Focused
    Ultrasound (HIFU). These procedures generally
    use lower frequencies than medical diagnostic ultrasound (from
    250 kHz to 2000 kHz), but significantly higher energies.
-   Focused ultrasound may be used to break up kidney stones by
    lithotripsy.
-   Ultrasound may be used for cataract treatment by
    Phacoemulsification.
-   Additional physiological effects of low-intensity
    ultrasound have
    recently been discovered, such as the ability to stimulate
    bone-growth and the potential to disrupt the blood-brain
    barrier for drug delivery.

## Acoustic Properties of Ultrasound Beams

As a first approximation, an ultrasound beam can be considered as
resulting from flat circular piston oscillating on an infinite baffle.
In practice, such a system would lead to relatively high diffusion of
the sound beam, severe side lobes, and an inability to choose a focal
length of the acoustic energy. In current biomedical applications the
use of phased arrays is a common approach stemming from the more general
field know as beamforming. Despite the limitations of planar
transducers, their relatively simple analysis serves well to illustrate
the basic properties of any formed beam and the challenges of designing
more advanced systems.

The analytical approach utilized for the simple cylindrical transducer
appears in many acoustics reference Texts, such as those by Pierce,[^10]
Kinsler et al.[^11] and Cheeke.[^12] The sound field solution is
obtained by first considering the sound emitted by the harmonic motion
of a single point source (small sphere) vibrating in free space. The
resulting sound pressure field from this point source is:

\
$$\boldsymbol{P} = i\left(\frac{\rho_o c U_o k a^2}{r}  \right )e^{-ikr}$$

$$I(r) = \frac{1}{2}\rho_o c_o U_o^2 \left(\frac{a}{r} \right )^2 \left(ka \right )^2$$

Where ***P***(*r*) is the harmonic pressure amplitude at the radial
distance *r*, *ρ~o~* is the fluid density, *c~o~* is the fluid sound
speed, *U~o~* is the maximum velocity of the spherical source, *a* is
the sphere radius, and *k* = 2*πf*/*c~o~* is the wave number. In the
preceding equations, *i* = -1^1/2^, which incorporates both amplitude an
phase into the harmonic pressure variable.

To apply this result to the ultrasound transducer as a cylindrical
radiator, each differential section of the cylinder surface can be
considered as a separate spherical source. The resulting sound field
from this approximation is the integral sum of each spherical source. In
general the resulting equation cannot be analytically integrated;
however, when considering regions of the field for r \>\> a, where a is
now the cylinder radius, a simple result is found. Forgoing the full
derivation (for reference see Kinsler [^13] or Cheeke [^14]), the
equations for the produced sound field and acoustic intensity are:

\
$$\boldsymbol{P}(r, \theta) = i \left( \tfrac{1}{2}\rho_o c_o U_o \frac{a}{r}ka \right ) H(\theta)e^{-ikr},$$

$$H(\theta) = \frac{2J_1(ka\sin\theta)}{ka \sin \theta},$$

$$I(r,\theta) = \frac{|\boldsymbol{P}(r)|^2}{2\rho_oc_o},$$

where *H*(*θ*) is the directivity function, *J~1~* is the Bessel
Function of the first
kind, and *I*(*r*) is the
acoustic intensity in W/m^2^. Physically the directivity function
represents the pressure amplitude for beam angles not parallel to the
cylinder axis. It is worthwhile to note that roots of the Bessel
function produce certain beam angles with zero amplitude; the regions
between these angles are known as side lobes, with the on axis component
know as the main lobe. Physically, lobes result from the phase
interaction of waves originating from different parts of the cylindrical
transducer, and are in some ways analogous to pressure nodes in simple
harmonic waves.

To illustrate the phenomena of side lobes in ultrasound beams, the
resulting directivity function and acoustic intensity is calculated for
a 1 MHz beam transmitted into water using a 1 cm radius transducer.
Figure 2 plots the Directivity function, while Figure 3 plots the
acoustic intensity relative to the intensity at the transducer surface.

\
! Figure 2: Beam function for a 1 cm radius cylinder radiating 1 MHz
ultrasound into
water.{width="520"}

! Figure 3: Normalized acoustic intensity field for a 1 cm radius
cylinder radiating 1 MHz ultrasound into
water.{width="520"}

For the purposes of diagnostic and therapeutic ultrasound the presence
of side lobes is an undesirable effect. In diagnostic imaging wave
reflection originating from the side lobes can be misinterpreted as
reflections from the main beam, and act to reduce the resulting image
quality. In therapeutic applications, side lobes represent energy
dissipation on regions not intended to be effected. To reduce the
effects of side lobes, ultrasound devices use transducer designs based
on beamforming theory,
making the analysis substantially more complicated than the simple
cylindrical transducer discussed. One technique to reduce side lobes is
the use of a phased array to focus the main at a particular depth, thus
reducing the relative magnitude of side lobes. Another technique known
as acoustic shadowing reduces side lobes by emitting lower amplitude
waves near the edge of the transducer. As will be discussed in a
proceeding section, an emerging technique to enhance focusing and reduce
side lobes is the purposeful consideration of nonlinear acoustic effects
in ultrasound beams.[^15][^16]

## Nonlinear Acoustics in Biomedical Ultrasound

In many fields related to application of acoustic theory, the assumption
of linear wave propagation is sufficient. In Biomedical ultrasound
however, the propagation of sound waves is often accompanied by
progressive wave distortion resulting from nonlinear, finite amplitude,
effects. The nonlinear effect of most interest in many diagnostic
applications is the generations of harmonics in the ultrasound beam. As
a primer to this section, a review of the acoustic parameter of
nonlinearity,
and harmonic
generation is
suggested.

Nonlinearities relevant to Biomedical Ultrasound are relatively weak,
making their effects on the propagating acoustic wave cumulative with
distance. For appreciable harmonic generation to occur four conditions
should be met:

-   Sufficient pressure and velocity amplitude. The waves emitted for
    almost all applications of biomedical ultrasound meet this
    requirement.[^17]
-   Sufficient propagation distance with near planar wave conditions.
    For directional beams, such as those used in ultrasonography, this
    conditions is approximately met within the Rayleigh Distance, *x* =
    1/2 *ka^2^*, on the main lobe.[^18] Furthermore, harmonic generation
    is proportional to the number of wavelengths propagated and not the
    absolute distance. The ultrasonic frequencies utilized have very
    short wavelengths, for example, a 10 MHz wave must propagate over
    500 wavelengths for a focal depth of 10 cm.
-   Sufficient value of the parameter of
    nonlinearity,
    B/A. For the same acoustic intensity, a material with a higher value
    of B/A will produce harmonics more quickly. The value of B/A in
    water is ten times that in air, and B/A for some biological tissues
    can be double that of water.
-   Low acoustic absorption. In many tissues with high values for B/A
    there are also high values for acoustic absorption. As the extent of
    wave dissipation increases with frequency, generated harmonics are
    absorbed more readily than the fundamental frequency. This effect
    reduces the influence of B/A in biological tissues relative to that
    in low-loss fluids.[^19]

Reviewing these conditions it can be seen that in many circumstances,
harmonic generation will be appreciable in biomedical ultrasound. Two
developing applications that make use of this harmonic generation are:

-   Use of harmonic content in recorded ultrasonography signals. As
    acoustic intensity and propagation distance is highest on the main
    lobe, harmonic generation occurs most significantly on the main
    lobe, and is smaller on side lobes. As a result, the beam pattern
    produced by 2nd harmonics is more directional than the beam produced
    by the fundamental frequency. This allows for potential improvement
    in the resulting image.[^20]
-   The analysis of harmonic profiles for tissue characterization using
    the B/A parameter. Referring to The Acoustic Parameter of
    Nonlinearity,
    values of B/A vary for tissues that have otherwise similar acoustic
    impedance. As a result, harmonic content in ultrasound waves has the
    potential to produce images correlated to the B/A parameter of
    tissues. Practical realization of this concept is an area in
    development, as current imaging methods are unable to utilize this
    potential.[^21]

## External links

-   Medical Ultrasonic
    Transducers

## References


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