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目的:通过研究由R.P.Garay和R.Lefever所建立的描述免疫系统中效应细胞对肿瘤细胞反应的动力学模型,获得肿瘤细胞个数变化的数学描述。方法:先说明模型解的正性和有界性,用线性系统理论分析平衡点的局部稳定性、构造可以取到无穷大的Liapunov函数,使其沿模型的全导数恒负,判定边界平衡点全局稳定。利用Hopf分支理论说明周期解,即肿瘤细胞数量周期性变化是存在的。结果:假定每单位体积效应细胞与结合体的总数量是常数,而且b>1,即效应细胞绑定癌细胞的比速率与单位体积效应细胞与结合体的总数量之积大于癌细胞增殖的比速率,我们给出了在一定条件下,例如效应细胞与癌细胞的结合体分离的比速率大于其结合的比速率(b≥1),边界平衡点全局稳定,即癌细胞被消灭,恢复健康;其他相应的条件成立时正平衡点稳定或局部周期解存在,即肿瘤细胞数量可以由免疫系统控制。结论:本研究显示建立合理的数学模型,用来解释病理现象是可行的。如果能够被试验数据所证实,这个模型将能够提供一个有效的癌症治疗生物方法。
OBJECTIVE: To obtain a mathematical description of the change of the number of tumor cells by studying the kinetic model of effector cells response to tumor cells established by R. P. Garay and R. Lefever. Methods: Firstly, the positive and the boundedness of the model solution are described. The local stability of the equilibrium point is analyzed by using the linear system theory. The Liapunov function, which can take infinity, is constructed so that it follows the all-derivative constant and negative of the model. stable. Hopf bifurcation theory is used to illustrate the periodic solution, that is, there are periodic changes in the number of tumor cells. Results: Assuming that the total number of effector cells and bound units per unit volume is constant and b> 1, the ratio of the effector cell bound cancer cells to the total number of effector cells bound to the conjugate is greater than the proliferation of cancer cells For specific rates, we show that under certain conditions, for example, the specific rate of binding of effector cells to cancer cells is greater than the specific rate of binding (b ≧ 1), the boundary equilibrium is globally stable, ie the cancer cells are eliminated and recovered Health; the other corresponding conditions are established positive equilibrium point or the existence of local periodic solution, that the number of tumor cells can be controlled by the immune system. Conclusion: This study shows that it is feasible to establish a reasonable mathematical model to explain the pathological phenomena. If confirmed by experimental data, this model will provide an effective biological treatment of cancer.