deletions | additions       diff --git a/subsubsection_TCP_NTCP_models_One__.tex b/subsubsection_TCP_NTCP_models_One__.tex  index 7be3727..3f1eead 100644  --- a/subsubsection_TCP_NTCP_models_One__.tex  +++ b/subsubsection_TCP_NTCP_models_One__.tex 
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\noindent where $\gamma$ is a parameter associated with the density of clonogens prior irradiation, $D_{i}$ the RBE-weighted total dose deposed in the \textit{i}-th voxel, $\alpha_\gamma$ and $\beta_\gamma$ the LQ parameters \cite{Bengt_K_Lind_Panayiotis_Mavroidis_1999} accounting for the linear and quadratic response of a cell, and $N_f$ the number of fractions. In this formulation, the clonogens repopulating during treatment are neglected and full repair between fractions is assumed.  A phenomenological method to estimate NTCP values for both uniform and nonuniform irradiation of many critical tissues is provided by the Lyman-Kutcher-Burman (LKB) model \cite{Lyman 1985, Kutcher 1985,Kutcher  and Burman 1989, Mohan 1989,Mohan  et al 1992, Deasy 2000, Li 1992,Deasy 2000,Li  et al. 2003, and 2003,and  Luxton et al. 2008}, which can be described by three equations \begin{align}  \label{ntcp} 
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D_{eff} &= \left( \sum_{i} v_i D_i^{\frac{1}{n}} \right)^n  \end{align}  \noindent where $v_i$ and $D_i$ are the fractional volume and dose of each bin used to tabulate the Dose-Volume Histograms (DVHs), $n$ a parameter describing the magnitude of the volume effect, $D_{eff}$ the Equivalent Uniform Dose (EUD), $TD_{50}$ the uniform dose required for 50\% inactivation of a tumor cell population, and $m$ a measure of the slope of the sigmoidal curve represented by the integral of the normal distribution. The fractionation is implicitly considered through a LQ-based correction of DVHs \cite{Wheldon et al. 1998, Yorke 1998,Yorke  2001}, converting the total dose into an isoeffective dose in 2-Gy fractions \begin{multline}  \label{lqed2}