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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5814_Библиотеки_им_академика_М_И_Перельмана
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37Chapter two: Fundamentals of acoustic propagation
vf
c
2(cos)
Doppler eect
(a) (b)
Observer
' '
S
p
Figure 2.23 A stationary observer perceives a change in frequency of a wave
emitted by a moving source toward the observer resulting from a change in
wavelength from λ to λ’. (a) The source is stationary. (b) The source is moving at
a velocity v.
S
p
For the situation where the velocity is making an angle of θ relative
to the direction of sound propagation, as shown in Figure 2.23(b), v in
Equation (2.43) should be replaced by v(cosθ):
f
=
d
θ
(2.44)
The Doppler effect is used in ultrasonic Doppler devices for the measurement and imaging of blood ow transcutaneously, i.e., without penetrating the skin in any manner. In these devices ultrasonic waves are launched
into a blood vessel by an ultrasonic transducer, and the scattered radiation
from the moving red cells is detected by either the same transducer or a
separate transducer. Appropriate instrumentation is incorporated to extract
the Doppler frequency, which is proportional to the red cell velocity.
References and Further Reading Materials
American Institute of Ultrasound in Medicine. Safety considerations for diagnostic
ultrasound. Laurel, MD: AIUM, 1984.
Dunn F and Goss SA. Denitions of terms and measurements of acoustical quanti-
ties. In Greenleaf JF (ed.), Tissue characterization with ultrasound. Boca Raton,
FL: CRC Press, 1986, pp. 1–14.
38 Diagnostic ultrasound: imaging and blood ow measurements
Fei DY and Shung KK. Ultrasonic backscatter from mammalian tissues. J Acoust
Soc Am 1985; 78: 871–877.
Fields S and Dunn F. Correlation of echographic visualizability of tissue with
biological composition and physiological state. J Acoust Soc Am 1973; 54:
809–811.
Geleskie JV and Shung KK. Further studies on the acoustic impedance of major
bovine blood vessel walls. J Acoust Soc Am 1982; 71: 467–470.
Goss SA and O’Brien WD Jr. Direct ultrasonic velocity measurements of mamma-
lian collagen threads. J Acoust Soc Am 1979; 65: 507–511.
Greenleaf JA. Tissue characterization with ultrasound. Boca Raton, FL: CRC Press, 1986.
Hamilton MF and Blackstock DT. Nonlinear acoustics. San Diego: Academic Press, 1998.
Hete B and Shung KK. Scattering of ultrasound from skeletal muscle tissue. IEEE
Trans Ultrasonics Ferroelect Freq Cont 1993; 40: 354–365.
Kossoff G. Radiation force. In Reid JM and Sikov MR (eds.), Interaction of ultra-
sound and biological tissues. Rockville, MD: FDA, 1972, pp. 159–161.
Malecki I. Physical foundations of technical acoustics. New York: Pergman Press, 1969.
Morse PM and Ingard KU. Theoretical acoustics. New York: McGraw Hill, 1968.
Nightingale K. Acoustic radiation force impulse (ARFI) imaging: A review. Curr
Med Imaging Rev 2011; 7: 328–339.
Pierce A. Acoustics: An introduction to physical principles and applications. New York:
McGraw Hill, 1986.
Shung KK and Thieme GA. Ultrasonic scattering by biological tissues. Boca Raton,
FL: CRC Press, 1993.
Tranquart F, Grenier N, Eder V, and Pourcelot L. Clinical use of ultrasound tissue
harmonic imaging. Ultrasonics Med Biol 1999; 25: 889–894.
Twersky V. Acoustic bulk parameters in distributions of pair-correlated scatterers.
J Acoust Soc Am 1978; 64: 1710–1719.
Westervelt PJ. The theory of steady forces caused by sound waves. J Acoust Soc Am
1951; 23: 312–315.
Wickline SA, Perez JE, and Miller JG. Cardiovascular tissue characterization in
vivo. In Shung KK and Thieme GA (eds.), Ultrasonic scattering in biological
tissues. Boca Raton, FL: CRC Press, 1993, pp. 313–345.
Yuan YW and Shung KK. Ultrasonic backscatter from owing whole blood. I.
Dependence on shear rate and hematocrit. J Acoust Soc Am 1988; 84: 52–58.
chapter three
Ultrasonic transducers and arrays
All ultrasonic imaging systems require a device called an ultrasonic transducer to convert electrical energy into ultrasonic or acoustic energy and
vice versa. Ultrasonic transducers come in a variety of forms and sizes,
ranging from single-element transducers for mechanical scanning to linear arrays to multidimensional arrays for electronic scanning. Although
the performance of an ultrasonic scanner is critically dependent upon
transducers/arrays, array/transducer performance has been one of the
bottlenecks that has prevented current ultrasonic imagers from reaching their theoretical resolution limit. The primary reason is that medical ultrasonic array/transducer design and fabrication are processes of a
broad interdisciplinary nature. They require knowledge from a variety of
disciplines, such as acoustics and vibration, electrical engineering, material sciences and engineering, medical imaging, and anatomy and physiology. It is no wonder that even to date the design of transducers is still
mostly empirical, generally involving a trial-and-error approach. Many
manufacturers regard their expertise in designing and manufacturing
transducers as a trade secret of the highest order.
The most critical component of an ultrasonic transducer is a piezo-
electric element.
3.1 Piezoelectric effect
The phenomenon that a material upon the application of an electrical eld
changes its physical dimensions and vice versa is known as the piezoelectric effect (pressure-electric effect) (Cady, 1964; Kino, 1987). The piezoelectric effect was discovered by French physicists Pierre and Jacques
Curie in 1880. The direct and reverse piezoelectric effects are illustrated in
Figure3.1(a) and (b), respectively. The direct effect refers to the phenomenon in which the application of a stress causes a net charge to appear
across the electrodes, whereas the inverse effect concerns the production
of a strain upon the application of a potential difference across the electrodes. Certain naturally occurring crystals, such as quartz and tourmaline, are piezoelectric. In these single crystals where a lattice structure is
repeated at a regular manner, forming a 3D pattern throughout the material, the molecular structure does not exhibit a separation of charges; i.e.,
the center of the negative charge coincides with that of the positive charge.
39
40 Diagnostic ultrasound: imaging and blood ow measurements
s
(a)
+ + + + + +
Strai
Strai
Potential
es
(b)
– +
+
– +
–
– – – – – –
Stress
Stress
Charge
Charge
+ + + + + +
+
–
– – – – ––
Direct Piezoelectric Effect
n
n
Reverse Piezoelectric Effect
Electrode
Electrod
Figure 3.1 (a) Direct piezoelectric effect where a stress induces a charge separation. (b) Reverse piezoelectric effect where a potential difference across the electrodes induces a strain.
Thus, in the normal state it is neutral. Upon the application of a strain,
the lattice structure is disturbed, causing a charge separation. Similarly,
an applied external electric eld causes the centers of the positive and
negative charges to separate, resulting in a displacement, as shown in
Figure 3.2. Naturally occurring piezoelectric crystals are seldom used
today as transducer materials in diagnostic ultrasonic imaging because of
their weak piezoelectric properties.
In a class of materials called ferroelectric materials, which are polycrystalline, the periodicity of the crystal lattices is disrupted at the socalled grain boundaries (Safari and Akdogan, 2008). Figure3.3 shows the
electron micrograph of lead zirconate titanate, Pb(Zr, Ti)O3 or PZT, consisting of many grains, each of which is composed of many domains, as
41Chapter three: Ultrasonic transducers and arrays
(a) (b)
Upon strain
Grains
Fine grain Coarse grain
Neutral state
–
+
–
–
–
–
–
+
+
–
Figure 3.2 Atomic structure of a natural piezoelectric crystal (a) is distorted upon
the application of external strain producing a charge separation (b).
illustrated in Figure3.4. Ferroelectric materials have spontaneous polarization or electric dipoles as opposed to paraelectric materials, which have
no spontaneous dipoles. The most popular ferroelectric material is PZT,
which possesses very strong piezoelectric properties following a preparation step called poling. The physical reason that the piezoelectric phenomenon occurs in a ferroelectric material can be idealistically explained
by considering that a ferroelectric material consists of many domains,
Figure 3.3 Electron micrograms of PZT materials with ne grain and coarse
grain size.
42 Diagnostic ultrasound: imaging and blood ow measurements
omains
+
–
Figure 3.4 Each grain of a ferroelectric piezoelectric material consists of many
domains.
D
+
–
each possessing innumerable electric dipoles, resulting in a net electric
dipole. In the virgin state of the material, it is electrically neutral. A net
dipole or polarization can be produced by poling or applying a strong
electrical eld in a certain direction to align the dipoles to that direction.
This net polarization disappears when the temperature is raised above a
material-specic level called Curie temperature, because above this temperature the directions of the net dipoles of all domains become random,
thus producing no net dipole. After the material is poled, producing a
net dipole for each domain, an electrical potential difference applied
across a slab of piezoelectric material realigns the dipoles in the material
to a preferential direction, resulting in a deformation or a change in the
thickness of the slab. Conversely, a stress that causes a deformation of
the material and reorientation of the dipoles induces a net charge across
the electrodes.
Poling or polarization of a ferroelectric material is carried out by heating it to a temperature just above the Curie temperature of the material,
and then allowing it to cool down slowly in the presence of a strong electric eld, typically in the order of 20 kV/cm, applied in the direction in
which the piezoelectric effect is required. The electrical eld is usually
applied to the material by means of two electrodes. This process aligns
the dipoles along the direction of the electrical eld. Poling is typically
done in oil to prevent electric arcing. There are a great variety of ferroelectric materials. Barium titanate (BaTiO3) was among the rst that were
developed. Single-crystal ferroelectric materials such as lead metaniobate
(PbNb2O6) and lithium niobate (LiNbO3) have also been found to possess
strong piezoelectric properties.
Certain piezoelectric properties of PZT can be enhanced by doping.
As a result, many types of PZT are commercially available.
The relationship between the applied electric eld and the resultant
polarization is material specic and, in general, has a characteristic in the
form of a hysteresis loop, shown in Figure3.5. Remanent polarization and
coercive eld are dened, respectively, as the value of polarization when
30
Voltage (V)
ercive eld
43Chapter three: Ultrasonic transducers and arrays
20
10
)
2
0
P (µC/cm
–10
–20
–30
–400 –200 0200 400
Figure 3.5 Polarization (P) versus electrical eld or voltage (V) loop of a ferroelec-
tric piezoelectric material.
Remanent
polarization
Co
the electric eld = 0 or voltage = 0 after poling and the value of the electric
eld when there is no polarization. Remanent polarization is a measure
of the piezoelectric strength of a material, whereas coercive eld is a measure of the propensity of a material to depole. For both, the larger, the
better.
Many ferroelectric materials have different ferroelectric states depending on the constituent composition. For instance, PZT has three different
states: cubic, rhombohedral, and tetragonal, depending on the PbTiO3
and PbZrO3 composition, as shown in Figure3.6. Its piezoelectric properties are the best at the morphotropic phase boundary (MPB) between the
rhombohedral and tetragonal states, where the volume concentrations of
PbTiO3 and PbZrO3 are 48 and 52%, respectively. The state of a ferroelectric material may shift from one to another at a certain temperature called
phase transition temperature. Curie temperature is one of the transition
temperatures. For PZT it is the transition temperature between the cubic
and rhombohedral states when the PbTiO3 concentration is lower than
48%. Above this temperature, the state becomes cubic and there is no net
dipole. For example the Curie temperature for BaTiO3 is 120°C, but there is
a transition temperature at 10°C.
In order to dene and better understand the physical meaning of the
piezoelectric properties of a material, the constitutive equations that govern the piezoelectric effect must be examined.
44 Diagnostic ultrasound: imaging and blood ow measurements
Temperature [°C]
3
500
450
400
350
300
250
200
150
100
50
PbZrO
Rhombohedral
010203040506070
3
Cubic
Pb(Ti
MPB
0.48Zr0.52)O3
Morphotropic phase boundary
Tetragonal
80 90 100
PbTiO
Figure 3.6 PZT has three ferroelectric states.
3.2 Piezoelectric constitutive equation
Since the piezoelectric effect involves the interaction between electric
elds and the mechanical deformation, the constitutive equations relate
the electric properties to mechanical properties of the material.
With no electric eld, the stress and strain relationship has been
described in Chapter 2, given by Equations (2.3) to (2.5). Since most materials are anisotropic, this relationship written in tensor form is given by
[κ] = [C][ε] (3.1)
where [κ], [C], and [ε] are, respectively, the stress, the elastic constant, and
the strain tensor. Upon the application of an electric eld, Equation (3.1)
needs to be modied to include the effect of the electric eld. Taking the
electric eld [E] and the strain [ε] as independent variables and the electric
displacement [D] and stress [κ] as dependent variables, the piezoelectric
constitutive equations are given by
[κ] = [CE][ε] – [e][E] (3.2)
[D] = [Kε][E] + [e] [ε] (3.3)
45Chapter three: Ultrasonic transducers and arrays
where [e], [CE], and [Kε] are the piezoelectric stress constant tensor, elastic
constant tensor when the electric eld [E] = 0, and dielectric constant ten-
sor when strain [ε] = 0 or clamped dielectric constant tensor. The physical
meaning of [CE] and [κε] can be readily understood by setting [E] and [ε] = 0,
respectively, in Equations (3.2) and (3.3). Letting [ε] = 0 in Equation (3.2),
[e] = – [κ]/[E] (3.4)
where [e], the piezoelectric stress constant, is the resultant stress change
per unit change in electric eld without strain or while being clamped. It
has the unit of newtons/v-m or coulombs/m2.
The constitutive equation can be written in another form when [κ] and
[E] are treated as independent variables.
[ε] = [d][E] + [γE][κ] (3.5)
[D] = [Kκ][E] + [d][κ] (3.6)
where [Kκ] is the free dielectric constant, which is the dielectric constant
when there is no stress, and [d] = [ε]/[E] is the transmission or piezoelec-
tric strain constant, representing the resultant change in strain per unit
change in electric eld with a unit of coulombs/newton when there is no
stress. [γE] = [ε]/[κ] from Equation (3.5) is the compliance of the material for
[E] = 0 and [γE] = 1/[CE]. The relationship between [e] and [d] can be found
from Equation (3.2) by setting [κ] = 0,
[CE] [ε] – [e] [E] = 0
Therefore,
[e] = [CE] [ε]/[E] = [CE] [d]
If [D] and [κ] are treated as independent variables, the constitutive
equations are given by
[E] = [ακ][D] – [g][κ] (3.7)
[ε] = [g][D] + [γD][κ] (3.8)
where [g] = –[E]/[κ] is the receiving constant with a unit of volt-m/new-
ton, representing the change in electric eld per unit change in applied
stress when [D] = 0; i.e., there is no current or under open circuit conditions. [γD] is the compliance when [D] = 0, and [ακ] = 1/[Kκ] when [κ] = 0.
46 Diagnostic ultrasound: imaging and blood ow measurements
Stored Mechanical Energy
The dielectric constant [K] of a piezoelectric material depends on the
extent of freedom of the material. Two values are often quoted in the literature. If the material is clamped so that it cannot move in response to
an applied eld or the strain is zero, the dielectric constant measured is
designated as the clamped dielectric constant [Kε]. If the material is free to
move without restriction, the dielectric constant measured is denoted as
[Kκ], the free dielectric constant. The transmitting constant and the receiving constant are related by the following relationship (Kino, 1987):
[d] = [g] [Kκ] (3.9)
As was mentioned, these equations are in tensor form because most
piezoelectric materials or crystals are anisotropic. To completely describe
the piezoelectric properties of a material, 18 piezoelectric stress constants
and 18 receiving constants are required. Fortunately, since these materials
usually are symmetric, a smaller number of constants are actually needed.
For instance, there are only ve constants for quartz. For single crystals,
the principal axes are dened by the crystalline axes (Cady, 1964). A plate
cut with its surface perpendicular to the x-axis is called x-cut, and so forth.
The x-, y-, z-directions are indicated by numbers 1, 2, 3. For polarized ferroelectric ceramics, the 3 direction is usually reserved for the polarization
direction. A piezoelectric strain constant, d33, represents the strain produced
in the 3 direction by applying an electric eld in the 3 direction, and d13 is the
strain in the 1 direction produced by an electric eld in the 3 direction when
there is no external stress. Here it is important to note that the piezoelectric
properties of a material depend upon boundary conditions, and therefore
upon the shape of the material. For example, the piezoelectric constant of a
material in the plate form is different from that in the rod form.
The capability of a piezoelectric material to convert one form of energy
into another is determined by its electromechanical coupling coefcient
(ECC), dened as
ECC
Stored Mechancial Energy
=
TotalStored Energy
The total stored energy includes both mechanical and electrical energy.
Therefore,
2
=
ECC
TotalStored Eenergy
It should be noted that this quantity is not the efciency of the
transducer. If the transducer is lossless, its efciency is 100%, but the
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