Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois 62026-1653, USA.
Math Biosci Eng. 2013 Jun;10(3):787-802. doi: 10.3934/mbe.2013.10.787.
In this paper, a mathematical model for chemotherapy that takes tumor immune-system interactions into account is considered for a strongly targeted agent. We use a classical model originally formulated by Stepanova, but replace exponential tumor growth with a generalised logistic growth model function depending on a parameter v. This growth function interpolates between a Gompertzian model (in the limit v → 0) and an exponential model (in the limit v → ∞). The dynamics is multi-stable and equilibria and their stability will be investigated depending on the parameter v. Except for small values of v, the system has both an asymptotically stable microscopic (benign) equilibrium point and an asymptotically stable macroscopic (malignant) equilibrium point. The corresponding regions of attraction are separated by the stable manifold of a saddle. The optimal control problem of moving an initial condition that lies in the malignant region into the benign region is formulated and the structure of optimal singular controls is determined.
本文考虑了一种针对强靶向药物的考虑肿瘤免疫系统相互作用的化疗数学模型。我们使用 Stepanova 最初提出的经典模型，但用依赖于参数 v 的广义 logistic 生长模型函数代替指数型肿瘤生长。这个生长函数在 Gompertz 模型（v→0 时的极限）和指数模型（v→∞时的极限）之间进行插值。动力学是多稳定的，我们将根据参数 v 来研究平衡点及其稳定性。除了 v 的小值外，系统既有渐近稳定的微观（良性）平衡点，也有渐近稳定的宏观（恶性）平衡点。对应的吸引区域由鞍点的稳定流形隔开。我们制定了将初始条件从恶性区域移动到良性区域的最优控制问题，并确定了最优奇异控制的结构。
