Scatter response function

The scatter point spread function (SPSF) was calculated for a series of uniform phan- toms in a thickness range (0.1 to 10cm) and glandularities of 0% and 100% using a narrow pencil beam geometry as shown in Figure 6.3(a), which illustrates the position for three different projection angles. As described in the literature, MC simulation represents a flexible option to determine SPSFs [24, 46, 82, 122], hence the GEANT4 toolkit was used in this work to calculate the scatter response functions.

As described by Sechopoulos et al. [82], the SPSF is distributed symmetrically for a 0◦ projection angle. However, it becomes more asymmetric as the projection angle φ (perpendicular to the detector surface) increases. This phenomenon is illustrated in Figures 6.3(b) and 6.3(c), where the thick black curved lines represent regions with the same energy. For this reason, the spatial distributions of the SP SF s were recorded using polar coordinates (r, θ) for different radii r and angles θ in steps of 0.05mm (∆r) and 3◦(∆θ) respectively. The intermediate values were calculated by simple linear interpolation due to the low spatial frequency composition of the scatter distribution in mammography images [46].

6.1.1.1 Validation of SPSFs with literature

A set of SPSFs was validated against SPSF results calculated by Sechopoulos et al. [82] and Boone and Cooper [19]. As described by these authors, the SPSFs were calculated using an X-ray pencil beam experiment (see Section 3.3.2). In this setup, a narrow beam hits a geometry which is made up of a circular shape phantom (radius 116mm) of different compositions, an air gap (between the bottom of the circular phantom and the image receptor) and an ideal image receptor for photon scoring. Five MC simulations of 109 photons each was used.

In order to calculate the SPSFs for this work, all the energy within the image receptor was recorded in annuli in steps of 1mm radii (see Figure 6.3(b)). For each annulus, the

124 Chapter 6. Convolution-based scatter prediction for DBT

(a) Pencil beams diagram

(b) SPSF φ = 0◦ (c) SPSF φ > 0◦

Figure 6.3: (a) represents the side-views of three pencil beam experiments for a uniform slab phantom of thickness T and glandularity G. Top-view diagrams of scatter response for φ = 0◦and φ > 0◦are shown in (b) and (c) respectively. Each curve represents the distribution of a given energy. Note the isotropic scatter response for φ = 0◦ and the loss of circular symmetry for φ > 0◦ as the X-ray source tilts towards the left edge of the page.

total energy deposited by the scattered X-ray photons was normalized by the area of the ring (A). For example, the area of the nth ring (An) would correspond to

An= (πrn2) − (πr2n−1), (6.1)

where rn and rn−1 are the radii of an the n and n − 1 rings respectively.

Then, the results from the above area normalisation was divided by the total energy deposited by primary X-ray photons recorded within the image receptor. Thus, the SPSF has units of mm−2.

6.1. Idealised DBT geometry 125

The setup of the different validations described by [19, 82] used in this work is shown in Table 6.1.

Table 6.1: Configuration of the experiments used to validate the SPSF with Sechopoulos et al. [82] and Boone and Cooper [19].

Exp Energy Spectrum(kVp) Breast Thickness (cm) Glandularity (%) Air gap (mm)

A 26 Mo/Mo 5 0, 50, 100 & H2O 10

B 26 Mo/Mo, 32 Rh/Rh 5 50 10

C 26 Mo/Mo 4 50 0, 10, 20 & 30

D 26 Mo/Mo 2, 4, 6 & 8 50 10

Sample comparisons of SPSF with the literature are illustrated in Figure 6.4. These results correspond to experiment D described in Table 6.1 using breast thicknesses of 2 and 8cm. The error bars for each of the points calculated in this work were also illustrated. However, they are too small as the y-axis is presented in a logarithmic scale. Therefore, the SEM of the SPSF is illustrated as a function of distance in the top right corner of each of the graphs. Note how the errors increase with distance as less number of photons reach the detector due to the larger absorption.

A quantitative comparison between the SPSF results from this work (Diaz) and Se- chopoulos et al. [82] is presented in Table 6.2, where the SPR values were calculated after integrating the area under each SPSF curve using a circular field of view of radius 100mm. The average SEMs (in %) associated to the results for this work are shown in brackets. Sechopoulos et al. described their SEM errors as 0.4% and 1.6% for the 5cm and 8cm phantoms respectively (50% glandularity and energy beam of 27 kVp).

Table 6.2: Comparison of SPR values for Sechopoulos and results from

this work using a circular field of view of radius 100mm. SEMs (in %) of this work are shown in brackets. Maximum and minimum differences are highlighted in red.

Exp. label Area under the curve (SPR) Difference (%) Sechopoulos Diaz (SEM,%)

A G=0 % 5.33×10−1 5.23×10−1(0.2) 1.9 G=50 % 5.57×10−1 5.40×10−1(0.4) 3.0 G=100 % 5.92×10−1 _5.68×10−1_{(0.9) 4.0} H2O 5.91×10−1 5.89×10−1(0.6) 0.3 B 26MoMo 5.57×10 −1 _5.40×10−1_{(0.4) 3.0} 32RhRh 5.70×10−1 5.55×10−1(0.2) 2.6 C AG=0mm 4.57×10−1 _4.43×10−1_{(0.8) 3.0} AG=10mm 4.55×10−1 4.41×10−1(0.3) 3.0 AG=20mm 4.48×10−1 4.36×10−1(0.2) 2.7 AG=30mm 4.37×10−1 _4.26×10−1_{(0.2) 2.5} D T=2cm 2.44×10−1 2.39×10−1(0.2) 2.2 T=4cm 4.55×10−1 _4.41×10−1_{(0.4) 3.0} T=6cm 6.63×10−1 6.40×10−1(0.5) 3.5 T=8cm 8.72×10−1 8.43×10−1(0.8) 3.3

It was observed that the largest discrepancy found was 4.0% for 100% glandular breast tissue and, on the other hand, the minimum difference observed was 0.3% for a breast phantom filled in with water. In this work, the breast composition from Hammer- stein [54] was employed. However, Sechopoulos et al. did not mentioned the adipose and glandular compositions they used. It is suspected that the discrepancies come

126 Chapter 6. Convolution-based scatter prediction for DBT

(a)

(b)

Figure 6.4: (a) and (b) illustrate the SPSFs using a D configuration (see Table 6.1) using a breast thickness of 2 and 8cm respectively. In each one, the SEM associated to the MC simulations of this work as a function of dis- tance is shown in the top right corner. Results from this work, Sechopoulos et al. [82] and Boone and Cooper [19] are plotted as Diaz, Sechopoulos and Boone respectively.

from differences in the breast tissue composition as good agreement was found when simulating water.

The close agreement of the results with the literature suggested that the SPSF generated in this work can be used with confidence in the aforementioned convolution-based scatter calculations.

6.1. Idealised DBT geometry 127