deletions | additions       diff --git a/M&M - LEM.tex b/M&M - LEM.tex  index b35f13d..b1b6fa7 100644  --- a/M&M - LEM.tex  +++ b/M&M - LEM.tex 
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The Local Effect Model (LEM) \cite{Scholz_1996}, originally devised to predict the biological response of a cell irradiated with high LET (Linear Energy Transfer) charged particles, accounts for dose heterogeneities along ionizing particle tracks, and has been applied to the high-dose spots located near GNPs to determine the additional cell killing induced by the nanoparticles presence \cite{McMahon_2011,Lechtman_2013}. The LEM assumes that the cell response to inhomogeneous radiation dose on the micro-scale is similar to the response of the cell population as a whole to sparsely ionizing radiation.  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} is 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 $N_\text{leth}(D) = -\ln(S) = \alpha_\gamma \alpha  D + \beta_gamma \beta  D^2$, where $\alpha_\gamma$ $\alpha$  and $\beta_\gamma$ $\beta$  are the cell type and radiation specific LQ parameters. The LEM evaluates the total number of lethal lesions $N_\text{leth}$ induced by inhomogeneous by integrating the effects given by a local dose over the whole cell nucleus volume \begin{equation}  N_\text{leth} = \int_{V_N} n_\text{leth}(D_r)\frac{\text{d}r}{V_N} = -\int_{V_N} (\alpha_\gamma D_r + \beta_\gamma D_r^2)\frac{\text{d}r}{V_N}  \end{equation}