1.5.1 Generation of the Simulated Arterial Structure

Considering the goal of simulating different arterial structures, we can classify them into three groups: tissue structures, nontissue structures, and artifacts. The spatial distribution of the scatterer number with a given DBC, a (R, &, Z) at point (R^ &, Z), has the following contributions:

where A(R), B(R, 0, Z), and C(R) are the contributions of tissue structures, nontissue structures, and artifacts respectively.

1. Tissue scatterers. These are determined by the contribution of the normal artery structures, corresponding to lumen, intima, media, and adventi-tia. Figure 1.20 shows a fc-layers spatial distribution of the scatterers for a simulated arterial image. These scatterers are simulated as radial Gaussian

distributions [23] centered in the average radius Rk and having standard deviation nk corresponding to each arterial structures. Tissue scatterers are represented by:

where ak is the maximal number of scatterers at R = Rk, k is the kth radial simulated tissue layer, and Rk is the radial layer average position.

2. Nontissue scatterers. These contributions can be made by structures formed by spatial calcium accumulation, which are characterized as having greater DBC density than the rest of the arterial structures. They are simulated by a Gaussian distribution in the radial, angular, and longitudinal arterial positions of the simulated structure:

( 1 ((R - Ri)2 (© - ©m)2 (Z - Zn)2)\ F(R, &, Z) = exp ^ . iJ -^^ + ^-^^)

where (i, m, n) correspond to the radial, angular, and longitudinal axes directions, (io, mo, no) are the structures number in radial, angular, and longitudinal directions, (bl, cm, d„) are the scatterer numbers that have a maximum at R = Ri ,& = ©m, and Z = Zn, (¡i, ym, vn) are the radial, angular, and longitudinal standard deviations, and (Ri, &m, Zn) are the radial, angular, and longitudinal average positions.

3. Artifacts scatterers. In our model we consider the artifact caused by the sheathing transducer:

ao ~ \ 2a0 ' where ao is the scatterers number that has a maximum at R = Ro, ao is the artifact standard deviation, and Ro is the artifact radial average position.

1.5.2 1D Echogram Generation

To obtain a 1D echogram, an ultrasound pulse is generated in accordance with Eq. (1.4) and emitted from the transducer position. The pulse moves

Figure 1.21: The 1D echogramis obtained by fixing the angular position ©0 = © of the ultrasound beam (a). The total signal S(t) is only generated by the scatter-ers N© located at an angular position ©a < ©0 < ©b. The intensity distribution decreases with the depth penetration and the scatterers numbers N© through the beam path (b).

Figure 1.21: The 1D echogramis obtained by fixing the angular position ©0 = © of the ultrasound beam (a). The total signal S(t) is only generated by the scatter-ers N© located at an angular position ©a < ©0 < ©b. The intensity distribution decreases with the depth penetration and the scatterers numbers N© through the beam path (b).

axially through scatterers (Fig. 1.21(a)) and its intensity distribution decreases (Fig. 1.21(b)) with the penetration depth and the scatterers numbers in the ultrasound path given by Eq. (1.8). The echo amplitude is registered by the transducers (Fig. 1.22) as a signal function of time S(t) (Eq. 1.13). The value is transformed to penetration depth replacing t = x/c and normalized to gray scale. The spatial distribution of cross-section scatterers, a, is generated by i

lo 8

lo 8

Normalized time units

Figure 1.22: The corresponding echoes are finally transformed to normalized echo amplitude and then to gray-level scale versus time or penetration depth.

Figure 1.22: The corresponding echoes are finally transformed to normalized echo amplitude and then to gray-level scale versus time or penetration depth.

using Eq. (1.11). Figure 1.21 shows the simulations of N scatterers located in (Ri, &a < &j < ®b):

S(t, eo) = ' (">&> ± & j ) exp ( ) s-nct - «

where &o = (&a + &b)/2, Co defines the transducer constant parameters, and N& is the total scatterers number at the angular position 0a < & < 0b for a radial position Ri. The sum only operates on the scatterers located in the angular position 6a < & < 0b that is the focal transducer zone (Figs. 1.9(b) and 1.13). Therefore, Ne is the total scatterers number in this region. Equation (1.13) canbe written as a function of the penetration depth, replacing t = x/c. Equation (1.13) can be rewritten on gray-level scale as:

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