# Dosimetric Evaluation in Micro-CT Studies Used in Preclinical Molecular Imaging
## 2 Materials and Methods
### Dosimetric Measurements
#### 2.3.2 Half-Value Layer and Effective Energy  
The X-ray tube's HVL and effective energy (_Eeff_) were measured using a 6 cc ionization chamber (model 20X6-6M, Radcal) connected to an electrometer (2026C, Radcal), which automatically corrects readings of the effects of pressure and temperature. This chamber was calibrated in air kerma, with traceability to LCIE, in France [22]. The chamber's sensitive area was centered at the FOV of the X-ray radiation field at 29 cm below the tube window (at the isocenter). Measurements were performed using the 2D-acquisition modality of the microCT unit for 35 and 45 kV voltages at either 0.2 mA or 0.4 mA.  
Exposure measurements were taken as a function of aluminum thickness, using high purity (3N) aluminum sheets (10 \(\times\) 10 cm\({}^{2}\), 0.1 mm thickness) (Lot Q15684, ESPI Metals). These sheets were placed on the window of the X-ray tube. Air kerma was calculated as: \(K_{air}\)(mGy) = 0.00876 mR [17]. For the 35 kV beam, the air kerma measurements were acquired by adding 0.1 mm aluminum sheets in a range of 0-0.5 mm, while for the 45 kV beam, the measurements were taken in a range of 0-0.6 mm; all measurements were in quadruplicate. Given the polychromatic nature of an X-ray beam, the distribution of the exposure values (X) as a function of the filter thickness (s) does not fit a first-order decreasing exponential function, as in the case of a monoenergetic beam, but rather an exponential function of order n \(>\) 1, i.e., a sum of \(n\) decreasing exponentials [23,24]. Consequently, HVL estimation, measured in mm of aluminum, was calculated using the following equation [25,26]:  
\[\mathrm{HVL}=\frac{s_{b}\ln(2\kappa_{a}/\kappa_{b})-s_{a}\ln(2\kappa_{b}/ \kappa_{b})}{\ln(\kappa_{a}/\kappa_{b})}, \tag{6}\]  
where \(K_{0}\) corresponds to the average air kerma measured without added filtration, \(K_{a}\) is the average kerma, which is slightly greater than half of \(K_{0}\) and is obtained at a filtration thickness of \(s_{a}\), and \(K_{b}\) is the average kerma, slightly less than half of the \(K_{0}\) obtained at an \(s_{b}\) thickness. The uncertainty associated with the HVL was calculated as:  
\[\delta_{\mathrm{HVL}}=\frac{\sqrt{\left(\frac{s_{b}-\mathrm{HVL}}{K_{a}} \delta_{K_{a}}\right)^{2}+\left(\frac{s_{a}-\mathrm{HVL}}{K_{b}}\delta_{K_{b} }\right)^{2}+\left(\frac{s_{b}-s_{a}}{K_{0}}\delta_{K_{b}}\right)^{2}+L(\delta _{s})^{2}}}{\ln(\kappa_{a}/\kappa_{b})} \tag{7}\]  
with \(\delta_{K_{a}}\), \(\delta_{Kb}\), and \(\delta_{K0}\) as the standard deviations for the respective average kerma, \(\delta_{s}\) as the standard deviation associated with the aluminum thickness (reported by the manufacturer as 5% of the nominal thickness value), and \(L\) expressed as:  
\[L=|\ln(2\kappa_{a}/\kappa_{b})|^{2}+|\ln(2\kappa_{a}/\kappa_{b})|^{2} \tag{8}\]  
The aluminum's effective linear attenuation coefficients (\(\mu_{Al}\)) were first estimated from the HVL measurements used to calculate the X-ray beam's effective energy (_Eeff_). In this way, the relationship between both quantities was \(\mu_{Al}=\ln(2)/\mathrm{HVL}\)[2]. These values were interpolated into a standard curve of effective energy as a function of the linear attenuation coefficient of aluminum in the energy range 13.5-18.8 keV, obtained from the values reported by the NIST [14].