Modeling the hallmarks of avascular tumors

The paper models each cell’s event sequence as Bernoulli trials with success probability q_i = b_i/(b_i + d_i) and forces q_i = 0 after η_i proliferations, then derives Ri = q_i(1 − q_i^{η_i})/(1 − q_i) (its Eq. 10) and Rpop = (1/Ncells)∑_i (b_i/d_i)(1 − q_i^{η_i}) (its Eq. 11). These are stated explicitly and match the candidate’s formulas and modeling assumptions, including the same “final-step” handling for the replicative limit . The candidate arrives at the identical Ri via the tail-sum formula rather than the paper’s truncated-expectation sum, but both yield the same closed form. A minor implicit assumption in the paper—and used by the candidate—is that b_i and d_i (hence q_i) are stationary across the Bernoulli sequence until termination; the paper’s notation treats q_i as fixed during that computation but does not dwell on time-variation under changing microenvironmental conditions (cf. the conditions for proliferation/apoptosis) .