deletions | additions       diff --git a/M&M - analytical.tex b/M&M - analytical.tex  index 440b930..ac127fc 100644  --- a/M&M - analytical.tex  +++ b/M&M - analytical.tex 
...
\subsubsection{Analytical closed formulation}  From the estimates values  of $\bar{N}_\text{ion}$ and $\bar{N}_\text{GNP}$ (see also the Results section) it follows that the probability to observe more than one ionization per cell is very small. Hence it is possible, for not too high doses, to retain only the first terms of the number of ionization per cell distribution, and to write the fraction of surviving cells, due to the GNP ionizations, as \begin{equation}  S_\text{GNP} = (1-\phi_1-\phi_2) + \phi_1 \exp(-N_\text{leth}(1)) + \phi_2 \exp(-N_\text{leth}(2)) + \cdots 
...
\phi_2 &= \frac{\bar{N}_\text{ion}^2}{2} e^{-\bar{N}_\text{ion}}  \end{align}  where $\bar{N}_\text{ion} = \bar{N}_\text{ion}(D) = \bar{N}^0_\text{ion} \bar{N}_\text{GNP} D$ is the dose dependent expected number of ionizations per cell, which was assumed to follow a Poisson distribution. Using equation \ref{eq_nleth} we estimated $N_\text{leth}(1) = 16$ and $ N_\text{leth}(2) = 64$ for 6 MV, and $N_\text{leth}(1) = 11$ and $N_\text{leth}(2) = 43$ for 15 MV photons. These numbers indicate that, even if it is rare (see Results), a single GNP ionization is very effective in killing the cell. Thus, the survival probability of a cell which has experienced one or more ionizations from a nanoparticle is approximable with zero. The survival of a cell with gold nanoparticles can then be evaluated by  \begin{equation}\label{eq_sapprox}  S_\text{GNP} \backsimeq 1 - \bar{N}^0_\text{ion}\bar{N}_\text{GNP}D \cdot \exp(-\bar{N}^0_\text{ion}\bar{N}_\text{GNP}D) - (\bar{N}^0_\text{ion}\bar{N}_\text{GNP}D)^2 \cdot \exp(-\bar{N}^0_\text{ion}\bar{N}_\text{GNP}D) / 2  \end{equation}  By writing the surviving fraction as $S_\text{GNP} = \exp(-\alpha_\text{GNP}D-\beta_\text{GNP}D^2)$, then performing a Taylor expansion of both left and right sides of Equation \ref{eq_sapprox} and retaining only the linear and quadratic terms in the dose, the following relations are obtained  \begin{align}\label{eq_analytical}  \alpha_\text{GNP} &= \bar{N}^0_\text{ion} \bar{N}_\text{GNP} \\  \beta_\text{GNP} &= 0  \end{align}  Since the quadratic terms is zero, it is possible to obtain the net survival by just multiplying the fractions due to the bare gamma radiation with those due to the GNP ionizations, $S = S_\gamma \times S_\text{GNP}$ (additive approximation), whose LQ parameters in presence of GNPs are expressed as  \begin{align}\label{eq_analytical2}  \alpha &= \alpha_\gamma + \bar{N}^0_\text{ion} \bar{N}_\text{GNP} \\  \beta &= \beta_\gamma  \end{align}  In this approximation the quadratic term does not vary, and the additional contributions of the linear term are easily evaluated by the knowledge of the ionization rate per nanoparticle per Gy and the number of nanoparticles present in the system. Once those are fixed, the LQ parameters can be assessed regardless of the cell type under consideration. That is, the model can be applied to different cell lines, as long as the cells response to radiation without GNPs is known.  In equation \ref{eq_analytical2} all the physical informations needed for the evaluation of GNPs radiobiological effects coalesce in only two parameters. An interesting consequence is that the explicit dependences on the shape of the photon energy spectra and on the GNPs size are not relevant, as these informations are compressed in a single parameter $\bar{N}^0_\text{ion}$, \textit{i.e.}, the average ionization per nanoparticle per Gy. The other physical parameter, $\bar{N}_\text{GNP}$, represents the average number of GNP that potentially impact on single cell damage, and could be associated to the GNPs located inside or nearby the cell nucleus.