The previous reports studied only the apoptosis process in
The previous reports studied only the apoptosis process in a single cell. In our case we generalized these models inspired by the results of Albeck et al. (2008) and Raychaudhuri et al. (2008), to study the apoptosis cell death after photodynamic therapy in those areas of the tumor that are not exposed to high doses of light and/or photosensitizer. The analysis of apoptosis in a tumor region is obtained by studying the apoptosis pathways in a connected group of Pimasertib that interact which each other. We include the cell to cell variability, the connection between the two pathways, the influence of the surrounding environment and the specific conditions for apoptosis in PDT. Experimental results show that in PDT, the most effective photosensitizing agents tend to localize in mitochondria, the endoplasmic reticulum or lysosomes where apoptosis may be induced through an intrinsic pathway (shaded in Fig. 1) (Castano, Demidova, Hamblin, 2005, Yang, Palasuberniam, Kraus, Chen, 2015). The activation of the intrinsic pathway starts with the permeabilization of the mitochondria and the release of cytochrome C (CytC) into the cytoplasm. The permeabilization of the membrane is regulated by the Bcl-2 protein family that promotes or inhibits apoptosis by their direct action on the outer mitochondrial membrane channels. In particular, Bax forms the pore, while Bcl-2 inhibits its formation. The subsequent release of Cytochrome C inside the citoplasm of the cells induces the formation of a multiprotein complex, called apoptosome, that activates the caspase cascade through caspase-9 (C9) leading to Apoptosis. Moreover, after the permeabilization of the membrane, SMAC protein (Second Mitochondria-derived Activator of Caspases in the cytoplasma) is released into the cytoplasm and they bind to the inhibitors of apoptosis in the cell (XIAPs), preventing the XIAPs from arresting the process, and allowing apoptosis to proceed (Castano, Demidova, Hamblin, 2005, Mroz, Yaroslavsky, Kharkwal, Hamblin, 2011). Usually the activation of the intrinsic pathway of apoptosis is modeled starting by directly releasing cytochrome C into the cytoplasm. Instead we want to highlight the role of the PDT in the process. We consider that in a healthy cell there is an equilibrium between the concentrations of the molecules Bid, Bax, Bcl-2, and the complexes Bcl2:tBid and Bcl2:Bax. This equilibrium guarantees that the amounts of tBid and Bax2 present in the cell are small enough to inhibit the apoptosis. The role of PDT in this case is to increase the oxidative stress in the cell breaking this equilibrium. Experimental evidence shows that after PDT, the ratio of Bax/Bcl-2 increases (Mroz et al., 2011) activating the production of tBid. Therefore, in our model the intrinsic pathway is activated with any initial concentration of tBid different from zero and this will be assumed proportional to the photodynamic damage. On the other hand, the so called extrinsic pathway (not shaded in Fig. 1) is another pathway that leads to apoptosis. Experimental results (Agostinis, Berg, Cengel, Foster, Girotti, Gollnick, Hahn, Hamblin, Juzeniene, Kessel, Korbelik, Moan, Mroz, Nowis, Piette, Wilson, Golab, 2011, Almeida, Manadas, Carvalho, Duarte, 2004, Oleinick, Morris, Belichenko, 2002) suggest that when death ligands bind to the death cell receptors in the membrane, induce the formation of Death Inducing Signaling Complexes (DISC) that activates caspase-8 (C8). C8 activates the effector caspases C3 that starts the degradation cascade that leads again to cell death by apoptosis. Moreover Caspase-6 (C6) cleaves C8 in a positive feedback loop that amplify this activation signal. The appearance of the death ligands is associated with the presence of death neighbouring cells and this is taken into account in our model. Our tumor consists in a bi-dimensional network of interacting cells. We tried our simulations with systems of (20 × 20 and 100 × 100) and did not find important differences in the results (See Appendix B). Each cell in the lattice has it own internal (bio-chemical) dynamics defined by the discussion above and represented in Table 1 and by the molecules listed in Table 2. The initial concentration of each molecule in each cell was generated using a random number that satisfied a Gaussian distribution with the mean listed in Table 3 for each molecule and standard deviation equal to the ten percent of the mean.