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  • Spectinomycin br In conclusion the above simulations demonstrate that the optimal

    2022-05-18


    In conclusion, the above simulations demonstrate that the optimal on-treatment period u exists when there is a possi-bility that the elimination of tumor Spectinomycin can be achieved by CAS. Furthermore, the reduction of androgen may give a better result in the case where AD cells may exclude AI cells for the higher u.
    6. Discussions
    In this paper, we propose a stochastic competition model for prostate cancer with androgen deprivation therapy. Firstly, we introduce different competition intensities for AD cells and AI cells because they have different functions and pathways [16,48]. Secondly, since the growth of tumors is sensitive to certain fluctuations such as temperature, radiation and chemical products, oxygen supply and nutrients [27–29,41,42], we incorporate the stochastic noises to find their influences on tumor dynamics. By using Lyapunov functions and the comparison principle for SDEs, the threshold conditions for extinction and persistence in mean for AD and AI cells are established and su cient conditions for the existence of an ergodic stationary distribution of system (2.2) are derived.
    Based on the analytical results of system (2.2), we conclude that there are two possible outcomes of using androgen deprivation therapy. The first one is that under the larger noise σ 2, prostate Spectinomycin cancer could be successfully treated with androgen deprivation therapy as shown in Theorem 4.8 (when u and σ 1 satisfy (4.9), AD cells go to extinction and AI
    AD cells
    AD cells 
    AI cells
    AI cells
    Time (days)
    Time (days)
    cells become extinct also). Secondly, in case that σ 2 is small or (4.9) is not satisfied, the treatment is unable to elimi-nate the tumor where the resistance cells (AI) persist, as shown in Theorems 4.8 and 4.9. Then, the competition inten-sities and the value of androgen suppression u determine the persistence and extinction of AD cells besides the persis-tence of AI cells as Theorem 4.9 indicates. Extinction of AD cells in this case (relapse) is more dangerous than their per-sistence because the tumor becomes resistant (dominance of AI cells) and does not respond to treatment in a short pe-riod of time. If the mutation rate from AI cells to AD cells is ignored in the model of Rutter and Kuang [37] (very small in the on-treatment period), we conclude that using only CAS therapy leads to relapse (special case for our model when
    α = β = 1), whereas using different competition intensities in our model gives the possibility of the existence and the sta-bility of a positive equilibrium point in ODE model and the persistence of both AD cells and AI cells in SDE model. In addition, the results predict that the weaker competitive ability of AI cells gives more possibility of preventing the relapse and reducing the severity of the tumor by using only CAS. Moreover, in [37] CAS could not lead to a cure without im-munotherapy, but in our SDE model we found that under larger noises, CAS could eliminate the tumor even without using immunotherapy.
    The su cient conditions on the existence of a stationary distribution strengthen and reflect the prediction that the small noises may imply the stability in stochastic sense and the large noises may destabilize the system and lead to a cure.
    On the other hand, numerical simulations have shown that a low androgen environment (high rate of androgen suppres-sion) could lead to the disappearance of AD cells and the dominance of resistant cells while a suppression of a medium androgen level could prevent the relapse. This also can be seen from the expression (4.9) which determines the extinction
    of AD cells. In addition, it should be noted that MAB (u = 1) leads to androgen-independent (fatal) prostate cancer in a short time, except in the case of the intensity σ 2 is large.