Biomedical Engineering Reference

In-Depth Information

decreasing exponentially with distance. The directivity function

in the far field is given by

22

J

1
(

π

a

sin( )

λ

D

S
=

(5.2)

2

π

a

sin(

)

λ

where
J
1
is the first order Bessel function, θ is the angle that the

line joining the point of interest to the center of the transducer

makes with the axis, and λ is the acoustic wavelength in the

propagation medium. For most therapeutic ultrasound appli-

cations involving plane transducers, the tissues being targeted

lie within the near field, and so uniform ultrasound exposure is

most easily achieved by moving the transducer.

FIGURE 5.2
A spherical bowl, multi-element focused ultrasound trans-

ducer. The central aperture allows for the insertion of an imaging device.

5.2.5 Focused Fields

5.2.4 plane transducers

The majority of plane transducers are formed from circular

discs. It is usually assumed that the plane face of these sources

responds linearly to an applied alternating voltage and vibrates

as a whole with uniform amplitude and phase. The field from

such a transducer is commonly described as being composed

of two regions: the near field (Fresnel zone) and the far field

(Fraunhofer zone). Interference effects in the near field result

in complex pressure distributions both on axis and off axis.

Theoretical prediction of the near field is complex. The intensity

on axis can be approximated to

5.2.5.1 Focused Bowl transducers

The simplest way to achieve a focused field is to use a single-

element spherical curved shell of piezoelectric or piezoceramic

material. It is possible to calculate the field from such a trans-

ducer geometry (O'Neill 1949, Penttinen and Luukala 1976).

Figure 5.3 illustrates the pressure distribution within a focused

ultrasound field, both along the sound axis (a) and in a trans-

verse plane through the focal peak (b).

The ultrasound focus lies on the central axis, near the cen-

ter of curvature of the bowl. As
2
π
h/λ
increases (where
h
is the

depth of the bowl, as shown in Figure 5.4), the focal point moves

closer to the geometric center of curvature. The gain factor (
G
,

the ratio of the intensity at the center of curvature to the average

intensity at the transducer surface,
I
C
/I
0
) is given by

I

I

π

λ

z

=

sin(

2

azz

2

+

2

−

)

(5.1)

0

where
z
is the axial distance and
I
z
is the intensity at point
z
.
I
z

oscillates in amplitude between
I
0
and zero. The intensity var-

ies smoothly on axis in the far field, with the intensity on axis

2

I

I

2

π

λ

h

C

=

.

(5.3)

0

(a)

4

Horizontal

(x) position

from focal

peak (mm)

2

0

−2

−4

−25

−20

−15

−10

5

0

5

10

15

20

25

Axial (z) position from focal peak (mm)

(b)

4

2

Vertical (y)

position from

focal peak

(mm)

0

−2

−4

−4

−2

024

Horizontal (x) position from focal peak (mm)

FIGURE 5.3

Pressure distribution close to the focus of a typical focused ultrasound field in the axial (a) and transverse (b) planes.