## 2 Test Objects and Methods
### Radioscopic Transparency of the Test Object
#### 2.2.1 Basic Formulas
The fundamental difference between the counting and integral recording modes is that the detector output in the first case is already a digital signal (DS), and in the second case, it is an analog signal (AS), which is converted into a DS using an analog-to-digital converter (ADC). The X-ray detector is characterized by the effective atomic number of the material \(Z_{d}\) and the mass thickness \(\rho_{d}h_{d}\). The pre-filter has similar characteristics--\(Z_{f}\) and mass thickness \(\rho h_{f}\). These characteristics are the same for counting and integral registration modes. It was emphasized above that the ADC transforms AS \(I_{E}\) into DS \(D_{E}\). The analog-to-digital converter is characterized by the bit depth \(k_{\rm ADC}\), which determines the range of digital signal from 0 to \(2^{k_{\rm ADC}}-1\). The random nature of the registered AS causes the introduction of the \(C_{\rm ADC}\) parameter, \(C_{\rm ADC}>1\), designed to protect against exceeding the DS level \(2^{k_{\rm ADC}}-1\). Let us introduce the notation  
\[f_{N}^{*}(E,E_{\max},Z_{f},\rho_{f}H_{f},Z_{d},\rho_{d}H_{d})=f_{N}(E,E_{\max} )\exp\Bigl{(}-m(E,Z_{f})\rho_{f}H_{f}\Bigr{)}\epsilon(Z_{d},\rho_{d}h_{d}), \tag{7}\]  
where \(\epsilon(Z_{d},\rho_{d}h_{d})=1-\exp(-m(E,Z_{d})\rho_{d}H_{d})\) is the detection efficiency of the detector. With the introduced parameters and (7) formulas (3), (6) take the form  
\[I_{N}(E_{\max},\rho H,Z)=\left[N_{0}\int\limits_{0}^{E_{\max}}f_{N}^{*}(E,E_{ \max},Z_{f},\rho_{f}H_{f},Z_{d},\rho_{d}H_{d})\exp(-m(E,Z)\rho H)dE\right], \tag{8}\]  
\[T_{N}(E_{\max},\rho H,Z)\approx\frac{\int\limits_{0}^{E_{\max}}f_{N}^{*}(E,E_{ \max},Z_{f},\rho_{f}H_{f},Z_{d},\rho_{d}H_{d})\exp(-m(E,Z)\rho H)dE}{\int \limits_{0}^{E_{\max}}f_{N}^{*}(E,E_{\max},Z_{f},\rho_{f}H_{f},Z_{d},\rho_{d} H_{d})dE}, \tag{9}\]  
Since the digitization of the \(I_{E}\) signal changes its essence, we denote the new signal by the symbol \(D_{E}\). The expression (4) is transformed as follows\[D_{E}(E_{\max},\rho H,Z)=\left[\frac{N_{0}\int\limits_{0}^{E_{\max}}E_{ab}(\mathbf{p_ {d}},E)f_{N}^{*}(E,E_{\max},Z_{f},\rho_{f}H_{f},Z_{d},\rho_{d}H_{d})\exp(-m(E,Z) \rho H)dE}{\Delta_{\mathrm{ADC}}}\right], \tag{10}\]  
where \(\Delta_{\mathrm{ADC}}\) is AS, equivalent to the digital unit; \(\mathbf{p_{d}}\) is a vector of linear dimensions of the sensitive element of the detector; \(E_{ab}(\mathbf{p_{d}},E)\) is the energy transferred to the scintillator by a registered photon with energy \(E\). The energy \(E_{ab}(\mathbf{p_{d}},E)\) depends on the materials and dimensions of the detector's array, but this fact does not play a special role in illustrating metal artifacts, revealing their nature, and developing ways to compensate for them, so \(E_{ab}(\mathbf{p_{d}},E)\approx E\). The maximum signal level \(I_{E}\) is achieved during measurements without TO; more precisely, the air between the radiation source and the detector acts as TO. Air attenuation is usually neglected, so the signal \(\Delta_{\mathrm{ADC}}\) is estimated using the formula  
\[\Delta_{\mathrm{ADC}}=C_{\mathrm{ADC}}\int\limits_{0}^{E_{\max}}E_{ab}( \mathbf{p_{d}},E)f_{N}^{*}(E,E_{\max},Z_{f},\rho_{f}H_{f},Z_{d},\rho_{d}H_{d} )dE/\Big{(}2^{k_{\mathrm{ADC}}}-1\Big{)}. \tag{11}\]  
The physical meaning of the parameter \(\Delta_{\mathrm{ADC}}\) lies in the fact that the signal \(I_{E}\) of a lower level \(\Delta_{\mathrm{ADC}}\) becomes invisible to the ADC, and the signal \(D_{E}=0\). The radioscopic transparency of the TO for the integral photon detection mode is described by the expression  
\[I_{E}(E_{\text{max}},\rho H,Z)=N_{0}\int\limits_{0}^{E_{\text{max}}}Ef_{N}(E,E _{\text{max}})\exp(-m(E,Z)\rho H)dE, \tag{4}\]0  
The set of formulas (1)-(12) is a mathematical model of radioscopic transparency and is the basis of a simulation model designed to study the magnitude of a metal artifact and answer the question posed in the article title.