deletions | additions       diff --git a/M&M - LEM.tex b/M&M - LEM.tex  index 4bd57fe..838fae1 100644  --- a/M&M - LEM.tex  +++ b/M&M - LEM.tex 
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Assuming a Poisson statistics, for an uniform dose D, the damage which occurs within the cell can be described as $S(D) = \exp(-N_\text{leth}(D))$, where $S$ is the cell survival probability and $N_\text{leth}$ the average yield of lethal lesions. In the framework of a Linear Quadratic (LQ) parametrization, the number of lethal lesions caused by a uniform dose is $N_\text{leth}(D) = -\ln(S) = \alpha D + \beta D^2$, where $\alpha$ and $\beta$ are the cell type and radiation specific LQ parameters. The LEM evaluates the total number of lethal lesions $N_\text{leth}$ induced by the inhomogeneous energy deposition by introducing a LQ parametrization also for the local effect, which are integrated over the cell nucleus volume  \begin{equation}  \label{eq_lem} %\label{eq_lem}  N_\text{leth}  = %=  \int_{V_N} n_\text{leth}(D_r)\ \text{d}r = %=  -\int_{V_N} (\alpha_\gamma D_r + \beta_\gamma D_r^2)\ \frac{\text{d}r}{V_N} \end{equation}  where $n_\text{leth}$ is the density of lesions induced by a local dose $D_r$ delivered in a (supposedly infinitesimal) sub-volume located at a point $r$, and $V_N$ is the volume of the cell nucleus. Thus, in case of a uniform dose, the equation gives back the standard LQ parametrization describing the response to photon irradiation of the cell as a whole, but, when inhomogeneous doses such as those given by GNPs are introduced, the quadratic term causes greater level of damage, explaining why locally heterogeneous doses induce greater biological effects. Following these premises and assuming a uniform radiosensitivity in the nucleus volume, the parameters $\alpha_\gamma$ and $\beta_\gamma$ in Equation \ref{eq_lem} can be identified with the experimentally observed LQ parameters for the irradiation of cells without the GNPs. In order to evaluate the surviving fractions in presence of GNPs, experimental cell control survival parameters from MDA-MB-231 breast cancer cells for 160 kVp, 6 MV and 15 MV irradiations were used \cite{Jain_2011} (see Table \ref{tab_LQ}). The effective radius of each nucleus was set to $R_N=3.5\text{ }\mu\text{m}$ \cite{Fu_2012}.