Biomedical Engineering Reference
In-Depth Information
Assuming the Born approximation [11, 12], the ultrasound reflected signal
S ( t ) for a finite set of N reflecting scatterers with coordinates ( R ,, Z ) and
spatial distribution of the differential backscattering cross-section σ ( R ,, Z )
is given by:
i = 1 σ i ( R ,, Z ) ζ i ( t )
N
S ( R ,, Z , t ) =
(1.2)
where N is the number of scatterers, σ i ( R ,, Z ) is the spatial distribution of
the differential backscattering cross-section (DBC) of the i th scatterer located
at position ( R ,, Z ), ζ i ( t ) is the transducer impulse function, and τ is the
delay time which leads to constructive and destructive contributions to the
received signal. The Born approximation implies that the scattered echoes are
weak compared to the incident signal and it is possible to use the principle of
superposition to represent the wave scattered by a collection of particles by
adding their respective contribution.
1.3.1 The Ultrasound Pulse
We consider a planar transducer that is mounted inside an infinite baffle, so that
the ultrasound is only radiated in the forward direction. We assumed that the
transducer is excited with uniform particle velocity across its face [13, 14]. Ac-
cording to the coordinates system illustrated in the far field circular transducer,
pressure P ( r ,θ, t ) (Fig. 1.7) can be written as:
2 J 1 ( ka sin( θ ))
ka sin( θ )
exp( j ( w t kr ))
P ( r ,θ, t ) = j ρ o cka 2
v o
2 r
where t is time, ρ o is the medium propagation density, c is the sound speed
for biological tissue (typically c = 1540 m / sec), v o is the radial speed at a point
on the transducer surface, a is the transducer radius, k is the propagation
vector, defined as k =| k |= 2 π/λ , where λ is the ultrasound wavelength defined
as λ = c / f o , where f o is ultrasound frequency, ω = 2 π f o , and J 1 ( x ) is the first
class Bessel function. Figure 1.8 shows a graphics of the pressure as a function
of ν , where ν = ka sin( θ ). In some applications, particularly when discussing
biological effects of ultrasound, it is useful to specify the acoustic intensity [16].
The intensity at a location in an ultrasound beam, I , is proportional to the square
of the pressure amplitude P . The actual relationship is:
P ( r ,θ, t ) 2
2 ρ c
I ( r ,θ, t ) =
(1.3)
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