# Dosimetric Evaluation in Micro-CT Studies Used in Preclinical Molecular Imaging
## 2 Materials and Methods
### Theoretical Calculations  
#### 2.2.1 X-ray Spectra  
The non-filtered X-ray spectra from the Apogee Series 5000 tube with W anode have been calculated previously [13]. Here, the spectrum was parameterized using [PERSON]'s third-order polynomial expressions [8; 9; 10]. The interpolation coefficients reported by [PERSON] et al. [1; 13] were conveniently used to calculate the Albira's X-ray spectra, generated at 35 and 45 kV. The output (\(\zeta(E)\)) of each calculation represented a discrete distribution of the photon fluence (\(\Phi(E)\)) (for each energy) that made up the spectrum, normalized by 1 mAs (i.e., \(\zeta(E)=\Phi(E)\) (mAs)\({}^{-1}\)) at 1 m from the tube window. Every \(\zeta(E)\) of the spectrum was analytically filtered by 0.5 mm of aluminum, using its linear coefficients of attenuation, \(\mu_{\rm{AI}}(E)\), reported by the NIST [14].  
_Total air kerma and absorbed dose in air_  
By using the calculated X-ray spectral distributions at 35 and 45 kV, the total air kerma (\(K_{air}\)) at 1 m was calculated, and the total air kerma per mAs (\(\Gamma(E)\)) was estimated using:  
\[\Gamma(E)=\frac{K_{air}(E)}{mAs}=\kappa\int_{E_{min}}^{E_{max}}\xi(E)\left( \frac{\mu_{tr}(E)}{\rho}\right)_{air}EdE\, \tag{1}\]  
where \((\mu_{tr}(E)/\rho)_{air}\) is the mass energy-transfer coefficient for air; \(E_{min}\) and \(E_{max}\) integration limits correspond to the X-ray spectrum's minimum and maximum energy values, respectively, and \(\kappa=1.602\times 10^{-11}\) Gy kg/eV is a unit-conversion factor. For a known X-ray spectrum, the \(\Gamma(E)\) expression means that the total air kerma can be calculated by integrating its spectral distribution. Since the simulated spectra are discrete distributions, an integration method based on the composite Simpson's rule [15] was used. For typical diagnostic X-ray energies and material media of low atomic number Z, \((\mu_{en}(E)/\rho)=(\mu_{tr}(E)/\rho)\)[8; 16; 17]; then \((\mu_{en}(E)/\rho)_{air}\) in the 2-50 keV range, reported by NIST [14], were plotted in 5 to 6 subintervals to obtain the curve of best fit for the distribution. The criterion used to determine the best fit was to consider those values with the correlation coefficient R\({}^{2}\) closest to 1.0.  
Assuming charged particle equilibrium (CPE) conditions: \(K_{air}=\) the total dose in the air (\(D_{air}\)), then Equation (1) was rewritten to calculate total air dose per mAs (\(\Gamma^{D}(E)\)):  
\[\Gamma^{D}(E)=\frac{D_{air}(E)}{mAs}=\kappa\int_{E_{min}}^{E_{max}}\xi(E) \left(\frac{\mu_{en}(E)}{\rho}\right)_{air}EdE \tag{2}\]  
where \((\mu_{en}(E)/\rho)_{air}\) is the mass energy-absorption coefficient. To estimate the \(D_{air}(E)\) (mAs)\({}^{-1}\), the \((\mu_{en}(E)/\rho)_{air}\) was obtained from the NIST, combined with those reported by [PERSON] et al. [16], and then plotted and adjusted in the same way as mentioned above. The results were used to interpolate the \((\mu_{tr}(E)/\rho)_{air}\), and \((\mu_{en}(E)/\rho)_{air}\) values for those energy values that make up the X-ray spectra.  
_Total absorbed dose in the air at isocenter_  
The following expression was used to calculate \(D_{air}\) at the isocenter of the microCT unit:  
\[D_{air}(SID,\text{mAs})=F_{SID}F_{\text{mAs}}\frac{D_{air}}{\text{mAs}}, \tag{3}\]  
This expression considers the source-isocenter distance (\(SID=290\) mm), as well as the effects of the current on the cathode and the time during which it irradiates the object of study (mAs), where \(F_{SID}=(1/SID)^{2}\) indicates the correction factor by the beam decreasing intensity with the square of the distance and \(F_{mAs}=\) I t, with I, the current filament (in units of mA) and t, the irradiation time (in seconds) during the tomographic study.