| Estimates from the parametric data analysis | Text, it follows that the 8* cell dynamics reduce to a subcritical branching process, |
| Model | Mathematically, these dynamics are summarized as a two-type branching process , see also Fig. |
| Model | Since the 8 cells in (1) undergo a critical branching process , their progeny will eventually go extinct (see also the discussion of 8*< cells below). |
| Model | From this it is easy to see that the dynamics of the 8* cell population in the basal layer are governed by the continuous-time critical branching process |
| Space and the impact of clustering | The subcritical branching process model above was derived under the assumption of a well-mixed basal layer where infected cells are surrounded primarily by susceptible cells. |
| Space and the impact of clustering | Immune capacity in the branching process model. |
| V | First, time to clearance is generally longer in the branching process model: the three dotted horizontal lines correspond to the three quartiles for the (M/fi_ = 1)distribution in Fig. |
| V | Only for the (,u/fi_ = 8) -distri-bution, which corresponds to an 8-fold increase in immune capacity, are all three quartiles of the spatial model below the corresponding quartiles of the branching process model. |
| V | This is due to the fact that, in contrast to the branching process model, elimination of an infected cell can trigger division of an 8* cell (with probability 195* > 0), therefore compensating for the loss of the infected cell and delaying clearance. |
| Abstract | To study the joint effect of drug heterogeneity, growth rate, and evolution of resistance, we analyze a multi-type stochastic branching process describing growth of cancer cells in multiple compartments with different drug concentrations and limited migration between compartments. |
| Generating function approach | The backward equations for this branching process are (see 81 Text for how to derive them) |
| Results | Nevertheless, using the multiplicative properties of branching processes , we can calculate the probability of escape and the (conditional) average time to resistance for any given initial conditions of tumor size or metastases (see derivation details in 81 Text). |
| Results | In this way, the two competing pathways can be seen as three-type branching processes , respectively, with different fitness |
| Results | Since we are considering a branching process , larger population size is more likely to generate resistant mutation. |