Between tumour cells and tumour-infiltrating cytotoxicT-lymphocytes can be modelled as shown
Between tumour cells and tumour-infiltrating cytotoxicT-lymphocytes can be modelled as shown in Figure 1 (see: [13]), where T denotes a tumour cell, E denotes an effector cell (CTL), C denotes the complex formed, T denotes a dead tumour cell and E denotes a dead effector cell. The following assumptions are made: ?the complexes C consist of a tumour cell and a CTL forming at a rate k + . The parameter k + consists of the encounter rate between a tumour cell and a CTL, the probability that the CTL recognizes the tumour cell as a “non-self” entity, and also the probability that the tumour cell forms a complex with the CTLs ?the break-up of complexes can lead to a situation where both the tumour cell and the CTLs are alive with a rate k – ?the break-up of complexes can lead to a situation where either the immune cell or the tumour cell survives the encounter with a rate k ?the probability that a tumour cell is killed is p, and correspondingly the probability that a CTL is killed (i.e. the tumour cells survives) is (1 – p) Using the Law of Mass Action, this leads to the following system of differential equations describing these specific kinetic interactions: t C = k + ET – (k – + k)C t T = -k + ET + k – C + k(1 – p)C t E = -k + ET + k – C + kpC The key idea proposed in this paper which develops the work of [13], is that a proportion of the tumour cells that survive an encounter with a CTL are more resistant (1)Figure 1 Basic local lymphocyte-cancer cell interactions. Schematic diagram of the basic local lymphocyte-cancer cell interactions.Al-Tameemi et al. Biology Direct 2012, 7:31 http://www.biology-direct.com/content/7/1/Page 4 ofto any future attacks by CTLs. Consequently, the phenotypic properties of these new “enhanced” tumour cells will be different from those of the “naive” tumour cells. Specifically, we make the additional assumptions: ?their probability of being killed (previously the parameter p ) is smaller ?their probability of being recognized and also of forming a complex with a CTL (embedded in the parameter k + ) is smaller Moreover, we shall also assume that the proliferation rate of CTLs stimulated by the presence of the complexes is also smaller. We denote the naive tumour cells by T0 (t) and the non-naive tumour cells by Ti , where i stands for the number of previous encounters with the CTLs. We assume that the fitness of tumour cells increases PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/27735993 up to a maximum number of encounters N, implying that we consider in total 1 + N “classes” of tumour cells, T0 , T1 , . . . , TN . The new kinetic relationships of our model are illustrated in Figure 2 and are Vesnarinone chemical information characterized by the following new groups of parameters: ?the rate of formation of complexes [ETi ]: ki+ . We assume that ki+ is constant or decreasing with index i, + with kN 0; ?the probability that a tumour cell of the i -th class is killed: pi . We assume that pi is decreasing with index i, with pN 0; ?the probability of transition Ti Ti+1 to the state i : i . We assume that i is increasing for 0 i N – 1. Since we have assumed N classes of tumour cells, N = 0. As far as the temporal dynamics of the tumour cells, CTLs and complexes is concerned, once again using the Law of Mass Action, the kinetic scheme of Figure 2 can betranslated into the following system of ordinary differential equations:T0 + = -k0 ET0 + k – C0 + k(1 – 0 )(1 – p0 )C0 t Ti = -ki+ ETi + ki-1 (1 – pi-1 )Ci-1 t + (k – + k(1 – i )(1 – pi ))Ci Cl = kl+ ETl – (k – + k)Cl t N N N E + – = -E k.