313
Ultrasound Imaging
sides of the tissue. In reflection tomography, the detected signal is collected in the wake
of the input pulse and is collected by the same transducer as the one that emitted
the ultrasound pulse. The delay between emitted and detected pulsed train is often
referred to as “time of flight.” As discussed later, one can measure end-to-end delays and
use tomographic techniques to calculate the delays in each point of the tissue structure. In other words, once the delay for each tissue is computed, a very informative
tomographic imaging technique can be applied, which can reveal invaluable details
about the physical properties of the tissues under study.
Another useful concept in ultrasound is the acoustic impedance. In acoustic wave
propagation, the factor that limits the speed of the sound is often expressed using
a variable called acoustic impedance of a medium, Z A . The characteristic acoustic
impedance depends on the density of the medium and the ease with which motion
can be transferred from a point to a neighboring point. This quantity can be related
to the speed of the wave as follows:
Z A = rV
(16.4)
The characteristic impedance is often used to describe the mathematical formulation
of the wave propagation, as discussed later in this chapter.
16.4.2 WAVE EQUATION
The ultrasound transducer produces a pressure wave that is both a function of location and time, P(x, y, z, t). Acoustic wave propagation in a medium such as a biological tissue is governed by the following wave equation:
2
2
2
2
2
∂ P ∂ P ∂ P 1 ∂ P
∇ P =
+
+
=
(16.5)
2
2
2
2
2
∂x
∂y
∂z
v ∂t
Often the initial assumption in forming the previous equation is that the wave propagates in the direction of one of the coordinates such as x-direction, while the beam
is not confined to one dimension only. The solution to Equation 16.5 can have many
different forms, but one typical solution of this equation can be described as follows:
⎛ 2pt
⎞
0
k x k y k z
z
=
− y
) (16.6)
P z t
( , ) = P sin
− x − y −
P 0 sin(wt − k x k y k z
x
− z
⎝ ⎜ l
⎠ ⎟
where
ω = 2πf is the angular frequency
w
k = is the wave number
V
Theoretically, the wave number k in the three Cartesian directions can have different values. This is due to the fact that the speed of the sound wave can be different
in different directions due to the type of tissues encountered in these directions.
Ultrasound Imaging
sides of the tissue. In reflection tomography, the detected signal is collected in the wake
of the input pulse and is collected by the same transducer as the one that emitted
the ultrasound pulse. The delay between emitted and detected pulsed train is often
referred to as “time of flight.” As discussed later, one can measure end-to-end delays and
use tomographic techniques to calculate the delays in each point of the tissue structure. In other words, once the delay for each tissue is computed, a very informative
tomographic imaging technique can be applied, which can reveal invaluable details
about the physical properties of the tissues under study.
Another useful concept in ultrasound is the acoustic impedance. In acoustic wave
propagation, the factor that limits the speed of the sound is often expressed using
a variable called acoustic impedance of a medium, Z A . The characteristic acoustic
impedance depends on the density of the medium and the ease with which motion
can be transferred from a point to a neighboring point. This quantity can be related
to the speed of the wave as follows:
Z A = rV
(16.4)
The characteristic impedance is often used to describe the mathematical formulation
of the wave propagation, as discussed later in this chapter.
16.4.2 WAVE EQUATION
The ultrasound transducer produces a pressure wave that is both a function of location and time, P(x, y, z, t). Acoustic wave propagation in a medium such as a biological tissue is governed by the following wave equation:
2
2
2
2
2
∂ P ∂ P ∂ P 1 ∂ P
∇ P =
+
+
=
(16.5)
2
2
2
2
2
∂x
∂y
∂z
v ∂t
Often the initial assumption in forming the previous equation is that the wave propagates in the direction of one of the coordinates such as x-direction, while the beam
is not confined to one dimension only. The solution to Equation 16.5 can have many
different forms, but one typical solution of this equation can be described as follows:
⎛ 2pt
⎞
0
k x k y k z
z
=
− y
) (16.6)
P z t
( , ) = P sin
− x − y −
P 0 sin(wt − k x k y k z
x
− z
⎝ ⎜ l
⎠ ⎟
where
ω = 2πf is the angular frequency
w
k = is the wave number
V
Theoretically, the wave number k in the three Cartesian directions can have different values. This is due to the fact that the speed of the sound wave can be different
in different directions due to the type of tissues encountered in these directions.