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Andrea Attili added M&M - TCP-NTCP.tex
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\subsubsection{Clinical response models}
The models used to quantify the TCP/NTCP are based on Poisson distribution assumptions (Munro and Gilbert 1961), which take into account the tumor characteristics and organ architecture (Kallman et al 1992). Poisson-based models were used in the present analysis since they are able to evaluate individual response to radiation and directly use the cell survival probability computed at the voxel level
\begin{align}
\text{TCP}^{(j)} &= \exp\left( -\text{d}v_i \sum_{i \in V(j)} n^{(j)}_i \right) \\
\text{NTCP}^{(j)}
\end{align}
TCP(j)=exp(-dvi i ∈ V(j) n(j)i) eq_TCP
NTCP(j) =(1 -∏ (1-exp(-n(j)i)s(j))dvi )1/s(j) eq_NTCP
where dv_i is the volume relative to each voxel, V^(j) the total volume taken under consideration, j the index pointing to the specific organ/tissue and s^(j) the relative seriality parameter of the organ in consideration. The expected number of surviving clonogens n_i^(j) was calculated as
n(j)i=exp(-exp(e(j)-(j)Di-(j)Di2Nf))
where ɣ^(j) is a parameter associated with the density of clonogens prior irradiation, D_i the total dose deposed in the i-th voxel, α and β the LQ parameters [Lin99] 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 has also been assumed.
Poisson-based models were used in the present analysis since they are able to evaluate individual response to radiation and directly use the cell survival probability computed at the voxel level.
