AMAC model calibration using NIST data - Simulation: Validating and calibrating the model

5.4 Simulation: Validating and calibrating the model

5.4.1 AMAC model calibration using NIST data

I was required to estimate the photoelectric absorption coefficient,K1, Compton scat-

tering coefficient, K2, energy exponent constant, m, and atomic number exponent

constant,n, for the AMAC model (5.1) using the selected reference materials and the two simulated spectra. K1 and K2 constants estimate the contribution of PE and CS

effects to the model. The numerical fits to NIST data formandn, subsequently, lies in the range [3, 3.5] and [3, 4] [Cho et al., 1975].

The attenuation coefficients (µ) of the reference materials (ω) were obtained from the NIST database in the energy range [1, 120] keV for the simulations. I fit (in a least square sense) the AMAC model (5.1) with the NIST data to calibrate the model for our reference materials as shown in Eqn. 5.9.

minimise {K1,K2,m,n}_ε,ω,

∑

_i Wε(E)kµ ω NIST(s,E)− pω_(s_,_E₎ Em −c ω_(s_,_E₎_f KN(E)k2. (5.9)

Initially titanium dominated the fits therefore, to prevent excessive influence of materials with higher atomic number, the error calculation was modified to minimise the relative error i.e., min(δµ

µ)

2 _{over all pixels,} _i_{. The estimated parameters are}

K1 = 13.96 (keV

3_cm2

gr ), K2 = 0.30 (cm

gr ), m = 3.00 and n = 3.20. Figure 5.7 shows a plot of attenuation coefficients according to NIST data (red line) and the model (dashed blue line) fits applying the estimated K1, K2, m and n constants for glass,

acrylic, titanium and marble to be discussed in this section.

To improve the calibration I decided to apply the spectra influence, so I weighted Eqn. 5.9 with the absolute difference of the spectra [W(E)] as shown in Fig. 5.6 (dashed cyan line). This gives lower weight to the overlapping energy region between lower and higher energy spectra than the effective PE and CS regions (Fig. 5.6). Adding the spectra weight also means the maximum K-edge in our reference materials (Ti; approximately 5 keV) is avoided through applying spectra weight. For these simulations shown in Fig. 5.6, that model our experimental setup, the spectra start at about 10 keV so the K-edges do not impact the results when I apply the spectra

weight to AMAC model calibration. I used different spectral weights, e.g., absolute difference, squared difference and square root of the absolute difference of the dual spectra and to assess the accuracy of results compare to no weight. The average relative error ofρandZestimation showed the absolute difference spectra is lowest with around 0.6% lower error compared to no spectral weight.

I note here that the AMAC model (5.1) does not account for increased absorption above the K-edges of materials. Within our range of reference materials, Ti has the highest K-edge at 4.99 keV and marble the next highest K-edge at 2.48 keV (see Fig. 5.7c and d). Figure 5.7b shows a plot of the fitting results for glass, which was one of the reference materials. Glass and aluminium have K-edges at 1.84 and 1.56 keV, respectively, which are well below the transmission energies in our setup. The average relative error between the AMAC model estimated attenuation coefficients and the NIST attenuation coefficients in the energy range [1, 120] keV is 16.50%, however the relative error in energy range [10, 120] keV is 1.96% as shown in table 5.1. As noted, the inaccuracy is mostly because the model does not account for K-edge absorption of titanium and marble, therefore, our selected spectra (Fig. 5.6) reduced calibration error, due to K-edges, by approximately 14%.

I also note that using the attenuation coefficient values of AMAC model for calibration will result in totally compatible µ values with that of the model, however using NIST attenuation coefficients for calibration (even after avoiding K-edges; energy range of [10, 120] keV) shows 1.96% average relative error between NIST and the model attenuation coefficients. In this case the inaccuracy is mostly due to inade- quacy of the AMAC model. A noticeable error of 5% in energy range of [10, 120] keV shown in table 5.1 for acrylic is because it does not match the Alvarez and Macovski model. Acrylic doesn’t have a K-edge in [10, 120] keV energy range. In the absence of any spectral information corresponding to the mismatch between the acrylic attenuation with the AMAC model, we can mention that acrylic has a very low attenuation that possibly cause this error. The estimated atomic number is a mathematical model to best reflect the observed attenuation measurements, rather than an actual phys- ical reality (e.g., there is no fractional atomic number elements). Figure 5.8 is the flowchart of the procedure of the AMAC model calibration explained in this section.

(a) (b)

Figure 5.7: Plot of ln(µ)versus (E = [1, 120]keV) of NIST data (red line) and AMAC model (blue dashed line) fits applying the the estimatedK1,K2,mandnconstants in

section 5.4.1 for a) glass, b) acrylic, c) titanium and d) marble.

Material names

Estimated model-data

relative error

in [1, 120] keV

Estimated model-data

relative error

in [10, 120] keV

Al

18.75 %

1.29 %

C

2.98 %

2.79 %

Marble

25.71 %

0.85 %

Acr

5.52 %

5.00 %

Teflon

1.81 %

1.23 %

Glass

5.80 %

1.26 %

Ti

54.98 %

1.29 %

Average

16.50 %

1.96 %

Table 5.1: Relative error between NIST attenuation coefficient and AMAC model for reference materials using AMAC model (Section 5.4.1).

Data Acquisition; from NIST Attenuation coefficient [µ

(

)

] of 7 reference material (ω) Fit µ

(

)

on AMAC model; Eqn. 5.9 Weight of Dual- energy spectra;

?

from Fig. 5.1 Theoretical ρ and Zvalues of material ω Parameter estimation; (K1,K2,m,n)

to Fig. 5.10 & 5.12

Calibrate the AMAC model

Calibrated AMAC model

Figure 5.8: Flowchart of the AMAC model calibration (Section 5.4.1).

In document Development of energy selective techniques in x ray computed tomography (Page 128-131)