TY - JOUR
T1 - The four-dimensional kirschner-panetta type cancer model
T2 - How to obtain tumor eradication?
AU - Krishchenko, Alexander P.
AU - Starkov, Konstantin E.
N1 - Publisher Copyright:
© 2018 American Institute of Mathematical Sciences. All rights reserved.
PY - 2018/10
Y1 - 2018/10
N2 - In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters s1,2 under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on s1,2 under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results.
AB - In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters s1,2 under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on s1,2 under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results.
KW - Compact invariant set
KW - Global stability
KW - Kirschner-Panetta tumor model
KW - Tumor eradication
KW - Tumor persistence
KW - Ultimate dynamics
UR - http://www.scopus.com/inward/record.url?scp=85048826897&partnerID=8YFLogxK
U2 - 10.3934/mbe.2018057
DO - 10.3934/mbe.2018057
M3 - Artículo
C2 - 30380309
SN - 1547-1063
VL - 15
SP - 1243
EP - 1254
JO - Mathematical Biosciences and Engineering
JF - Mathematical Biosciences and Engineering
IS - 5
ER -