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  • br whereas using different competition intensities

    2020-08-12


    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 MPDL3280A 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.
    AD CELLS
    AI CELLS
    AD and AI cells
    Time (days)
    AD CELLS
    AI CELLS
    AD and AI cells
    Time (days)
    AD CELLS
    AI CELLS
    AD and AI cells
    AD and AI cells
    AD and AI cells
    AD and AI cells 
    AD CELLS
    AI CELLS
    Time (days)
    AD CELLS
    AI CELLS
    Time (days)
    AD CELLS AI CELLS 14
    Time (days)
    AD and AI cells
    AD and AI cells
    AD and AI cells 
    AD CELLS
    AI CELLS
    Time (days)
    AD CELLS
    AI CELLS
    Time (days)
    AD CELLS
    AI CELLS
    Time (days)
    Tanaka et al. [42] have used numerical simulations in their stochastic model to study the dynamics of prostate tumor cells under intermittent androgen suppression therapy (IAS). But in our analysis we have used analytic method to give more details about tumor dynamics (threshold conditions for extinction and persistence) under continuous androgen therapy (CAS). Our model has also demonstrated that the stochastic noise can be responsible for the variability of the response to the treatment from one patient to another.
    Finally, we point out certain limitations in our model and assumptions. First, because of the absence of clinical trials that show the property of stochastic noise, we include only the environmental noises. It will be interesting to consider the noise that influences the mutation from AD cells to AI cells. Another limitation of the assumptions is the constant competition coe cients because there are only qualitative descriptions for the competition between AD cells and AI cells in literature. Therefore, there need more experiments to confirm our theoretical results.
    It will be very interesting to study the stochastic model with IAS therapy, or to combine immunotherapy with ADT [12,35,37] in a SDE model to get new insights into effects on treatments of prostate cancer patients. We leave these as future research.
    Acknowledgments
    We are very grateful to the anonymous referees for careful reading and valuable comments which have led to important improvements of our original manuscript.
    Appendix A. Proof of Theorem MPDL3280A 3.1
    To obtain the equilibria of system (3.1), we solve the following system:
    r a
    r
    X
    m uX
    K
    There exists the trivial equilibrium point E0 = (0, 0), which is unstable. This means that CAS cannot eliminate the tu-mor cells. The boundary equilibrium point E1 = (0, K ) always exists. To find a positive equilibrium point, we consider the following system
    K
    r
    K
    Set
    From the first equation in (A.2) we obtain
    Substituting (A.3) into the second equation in (A.2) we get the quadratic equation
    where
    K
    By (A.3) we see that Q must be positive and (A.4) should be solved in (0, KQ/α) for the existence of a positive equilibrium. Thus, we look for positive equilibria under these conditions. First, if Q > α, (A.5)
    KQ
    Appendix B. Proof of Theorem 3.2
    It is easy to verify the first statement. Let us consider the case where (3.2) holds. The variational matrix at E2 is
    K
    K
    K
    K
    Since