Instituto Superior de Engenharia de Lisboa - ISEL, ADM and CEAUL, Rua Conselheiro Emidio Navarro, 1, 1959-007 Lisboa, Portugal.
Math Biosci Eng. 2013 Apr;10(2):379-98. doi: 10.3934/mbe.2013.10.379.
In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta • (p,q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p=2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta• (2,q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.
在这项工作中，提出了一种新的概率和动力学方法来扩展 Gompertz 定律。研究了一类广义的概率密度函数，记为 Beta•(p,q)，它与 Tsoularis-Wallace 模型的右手边成正比。特别地，对于 p=2，研究扩展到了 Weibull 和 Frechet 类型的极值模型。这些由微分方程描述的模型与超 Gompertz 生长模型成正比。证明了 Beta•(2,q)密度是贝塔混合的幂，其动力学由概率的非线性耦合决定。使用符号动力学技术进行动力学分析，并使用拓扑熵测量系统复杂性。一般来说，恶性肿瘤的自然史通过分岔图反映出来，其中可以识别出回归、稳定、分岔、混沌和终点的区域。
