# Silicon 3D Microdosimeters for Advanced Quality Assurance in Particle Therapy
## 2 Principles of Microdosimetry and Its Application for RBE Prediction  
The microdosimetry concept is related to stochastic deposition of ionizing energy by ions moving in matter. The microdosimetric quantity of interest is lineal energy. Lineal energy describes the stochastic energy deposition by a single event in a micron-size SV along a particle's track, given by:  
\[\text{y}=\frac{E}{\langle l\rangle} \tag{1}\]  
where \(E\) is the energy deposited in an SV from a single event with a mean chord length \(<\)\(l\)\(>\) and its distribution _f_(_y_) for all primary and secondary charged particles. The microdosimetric kinetic model (MKM) developed by [PERSON] [14; 15] can predict RBE in proton and heavy ion radiation fields. A modified MKM proposed by [PERSON] et al. introduced saturation correction to account for the reduction of RBE from overkill effects for events with a very high lineal energy \(y\). The modified MKM was shown to predict the survival fraction of human salivary gland (HSG) cells using the microdosimetric quantities measured with a TEPC [16] for different heavy ion beams from protons to silicon ions. Following the successful results from the modified MKM, several more detailed studies were carried out to estimate the biological effects of carbon ion irradiation using the MKM [17; 18]. RBE calculations based on the MKM are advantageous over other empirical models in that experimentally measured microdosimetric distributions of lineal energies, _f(y)_, are used. If a microdosimetric measurement is performed in a non-tissue equivalent material, such as an SOI or diamond microdosimeter, the recorded spectra must be converted so that they represent a tissue equivalent (TE) material (water or tissue). A number of methods exist to convert from a non-TE to a TE material; however a simple and commonly used method is to apply a single conversion factor that scales the energy deposition. An example conversion factor is 0.58, which converts the microdosimetric measurements made in a proton beam using silicon to striated muscle [19]. In the modified MKM, the survival fraction, S, of certain cells is calculated with the biological model parameters (\(\alpha_{o},\ \beta,\ r_{d}\) and \(y_{0}\)) and is given by:  
\[S=exp\ \left[-\alpha D-\beta D^{2}\right] \tag{2}\]  
\[\alpha=\alpha_{o}+\frac{\beta}{\rho\pi r_{d}^{2}}\ y^{*} \tag{3}\]  
where \(\alpha_{0}\) is a constant that represents the initial slope of the survival fraction curve in the limit of zero LET, \(\beta\) is a constant independent of LET, \(\rho\) is the density of tissue, \(r_{d}\) is the radius of a sub-cellular domain in the MK model and \(y^{*}\) is the restricted dose-mean lineal energy that accounts for cell overkill for \(y>150\) keV/\(\upmu\)m and is calculated from Equation (4) where \(y_{o}=150\) keV/\(\upmu\)m. These MKM model parameters are cell line dependent. For example, the parameters for Human Salivary Gland (HSG) cells are: \(\alpha_{0}=0.13\) Gy\({}^{-1}\), \(\beta=0.05\) Gy\({}^{-2}\), \(\rho=1\) g/cm\({}^{3}\), \(r_{d}=0.42\)\(\upmu\)m.  
\[y^{*}=\frac{y_{0}^{2}\int_{0}^{\infty}\bigl{(}1-exp\bigl{(}-y^{2}/y_{o}^{2} \bigr{)}\bigr{)}f(y)dy}{\int_{0}^{\infty}yf(y)dy} \tag{4}\]  
The RBE of the HSG cells is then calculated from:  
\[RBE_{10}=\frac{D_{10,R}}{D_{10}}=\frac{2\beta D_{10,R}}{\sqrt{\alpha^{2}-4 \beta ln(0.1)-\alpha}} \tag{5}\]  
where \(D_{10}\) is the 10% survival dose of in-vitro cells for the radiation of interest, and \(D_{10,R}\) is the 10% survival dose obtained from the empirical survival curve of HSG cells irradiated by 200 kVp X-rays, which is the reference radiation (\(D_{10,R}=5.0\) Gy for HSG cells). Due to cells having a greater dependence on the dose delivered from protons compared to heavier ions, a dose-dependent form is used for RBE estimations calculated with the MKM.