Let us consider an ultrasound pulse Po that is emitted at time to with speed c from a point with coordinates (ro, 0o, zo) (Fig. 1.6), and that interacts with the scatterer located at position (R, ©, Z) with the spatial distribution of the differential backscattering cross-section, a(R, ©, Z). The reflected pulse Pi for the ith scatterer is an exact replica  of the transmitted sound pulse Po that will return to the point (ro ,0o, zo) at time (ti — to) and will be out of phase temporarily with respect to the pulse Po by time difference 8 = ti — to between the emitted pulse at ti and the received pulse at to. The time delay 8 is given by
We choose a coordinate system (X, Y, Z) with respect to the emitter/receiver position:
1 = (x - Xo)i, 1 = (y - yo)j, 1 = (z - Zo)k and the corresponding cylindrical coordinates are given by
| R| = VX2 + Y2 + Z2, © = arctan(Y/ X) where X = |, Y = |"f |, and Z = |.
Assuming the Born approximation [11,12], the ultrasound reflected signal S(t, t) for a finite set of N reflecting scatterers with coordinates (R, &, Z) and spatial distribution of the differential backscattering cross-section a (R, &, Z) is given by:
S(R, &,Z, t, t) = j2 ai(R,&, Z)Zi(t, t) (1.2)
where N is the number of scatterers, ai(R, &, Z) is the spatial distribution of the differential backscattering cross-section (DBC) of the ith scatterer located at position (R, &, Z), ^(t, t) is the transducer impulse function, and t 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.
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]. According to the coordinates system illustrated in the far field circular transducer, pressure P(r, 0, t) (Fig. 1.7) can be written as:
where t is time, po is the medium propagation density, c is the sound speed for biological tissue (typically c = 1540 m/sec), vo is the radial speed at a point on the transducer surface, a is the transducer radius, 1 is the propagation vector, defined as k = I11 = 2n/X, where X is the ultrasound wavelength defined as X = c/fo, where fo is ultrasound frequency, m = 2nfo, and Ji(x) is the first class Bessel function. Figure 1.8 shows a graphics of the pressure as a function of v, where v = ka sin(0). In some applications, particularly when discussing biological effects of ultrasound, it is useful to specify the acoustic intensity . The intensity at a location in an ultrasound beam, I, is proportional to the square of the pressure amplitude P. The actual relationship is:
Again, p is the density of the medium and c is the speed of sound. The impulse function Z (t, 8) is generally approximated  by a Gaussian (Fig. 1.9(a)), which envelopes the intensity distribution, and is given by:
where a is the pulse standard deviation. We consider that the beam is colli-mated by 0 = 0a. In our model only the corresponding interval d0 « 0.1° is used that corresponds to the transducer lateral resolution zone (Fig. 1.9(b)). Hence
Figure 1.8: Transducer pressure distribution.
Eq. (1.2) in the transducer coordinate system is based on a discrete representation of the tissue of individual scatterer elements with given position and DBC with respect to the transducer coordinates given by:
^ oAR, &, Z) S(R,&, Z, t, S) = C0J2 , „ , Z (t, S)
where S is given by Eq. (1.1), and Z (t, 8) is the impulse function given by Eq. (1.4). If we consider only the axial intensity contributions, Co can be written as :
where A is the transducer area.
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