Biomedical Engineering Reference
In-Depth Information
c
is the sound velocity. If a wave of intensity
I
is incident on an
inhomogeneity in tissue, the total power scattered or absorbed
is proportional to the intensity. The constant of proportional-
ity is known as the scattering or absorption cross section (σ
s
or
σ
a
). From these definitions we see that, for a beam of power
W
,
and cross-sectional area
A
, the power scattered by the
i
ith inho-
mogeneity is
σ
si
A
, and that absorbed by the
j
th is
σ
aj
A
. If there
are
n
i
scattering and
n
j
absorbing inhomogeneities per unit vol-
ume, then the power scattered per unit volume is
i
is one that gives an overall acoustic attenuation in the path of
approximately 10 dB.
5.3.1 reflection and refraction
When an ultrasound beam meets an interface between two
regions of different acoustic impedance at normal incidence, the
fraction of energy reflected is described by the reflection coef-
ficient
R
given by
nW
A
σ
and that
i i
nW
A
σ
. In the absence of multiple
scattering, the absorption cross section per unit volume, μ
a
, is
given by
µ
absorbed per unit volume is
Σ
j
j j
2
ZZ
ZZ
−
+
R
=
2
1
(5.13)
, and for scattering the cross section per unit
=
Σ
n
σ
a
j
j
j
2
1
volume μ
s
is
i
n
σ
.
For a slice of tissue Δ
x
thick, the total power scattered, Δ
W
s
,
is given by
i
i
where
Z
1
is the impedance of the first medium and
Z
2
is that of
the second medium. The proportion of transmitted energy,
T
, is
then described by
=µ
and the total absorbed power, Δ
W
a
,
is
µ
a
Wx
. The total loss of power, Δ
W
, can then be written as:
WWx
s
s
(
)
W
=µ+µ
W
x
.
(5.7)
TR
ZZ
ZZ
4
s
a
=− =
1
12
.
(5.14)
(
)
2
+
Integrating Equation 5.2 for a target of finite thickness, we get
2
1
When the beam hits an interface at an angle that is not normal
to the surface, it changes propagation direction in the second
medium. If the beam angle to the normal in medium 1 is θ
1
, and
that in medium 2 is θ
2
, then these angles are related by Snell's law:
(
)
−+
µµ
x
WWe
=
(5.8)
as
0
where
W
0
is the power at position
x
= 0.
This may be rewritten as
sinsin
θ
θ
WWe
=
−
µ
x
(5.9)
1
=
2
(5.15)
0
c
c
1
2
where μ is the attenuation coefficient (also referred to as the
intensity coefficient) and
µµ µ
where
c
1
and
c
2
are the sound velocities in the two media. A criti-
cal angle θ
c
is reached for which θ
2
is 90
o
, that is, the beam does
not enter medium 2, but travels parallel to the interface. When
θ
1
≥ θ
c
, no sound energy enters the second medium. θ
c
is given by
=
a s
. An amplitude coefficient,
α, can also be defined where μ
= 2α
and α is the sum of scat-
tering and absorption coefficients (
ααα
=
a s
). In soft tissues,
absorption represents between 60% and 80% of the attenuation.
From Equation 5.9:
c
c
θ=
−
1
sin.
1
(5.16)
c
1
W
W
2
µ=−
ln
(5.10)
x
0
and so the units of μ are cm
−1
, or nepers cm
−1
. In these units μ is
numerically twice α. If
W
0
is expressed in decibels,
5.3.2 Non-Linear propagation
When particle oscillations are small, as, for example, might be
expected in the very low intensity beams used in bone fracture
healing, the acoustic propagation is linear. However, for most
medical applications (both diagnostic and therapeutic) propaga-
tion is nonlinear. Since the speed of sound depends on density,
compressions travel faster than rarefactions, and thus a wave
that is initially sinusoidal becomes distorted as each compres-
sion “catches up” with the preceding rarefaction. In the limit,
a pressure discontinuity, or shock is formed. The linear wave
equation can be expressed as:
10
W
W
µ=−
log
(5.11)
10
x
0
20
P
P
α=−
log
(5.12)
10
x
0
and the units become dB cm
−1
. Here μ and α are numerically
equal. The conversion between the two units is μ dB cm
−1
=
4.343μ neper cm
−1
, α dB cm
−1
= 8.686α neper cm
−1
.
The choice of exposure frequency to obtain a specific inten-
sity level in a volume in which tissue heating is sought is a com-
promise between the desire to keep the frequency low in order
to minimize attenuation in the tissues overlying the target and
the wish to maximize energy absorption within this region. Hill
(1994) showed that for HIFU exposures, the frequency of choice
−=
−
ρρ
ρ
PP A
0
(5.17)
0
0
∂
∂
p
=
where
A
=
ρ
ρ
c
2
, and ρ, ρ
0
are the instantaneous and
0
ρ
0
0
equilibrium densities.
