Department of Human Oncology, University of Wisconsin, Madison, WI, USA.
Phys Med Biol. 2011 Apr 7;56(7):2161-81. doi: 10.1088/0031-9155/56/7/017. Epub 2011 Mar 9.
The underlying requirements for successful implementation of any efficient tumour motion management strategy are regularity and reproducibility of a patient's breathing pattern. The physiological act of breathing is controlled by multiple nonlinear feedback and feed-forward couplings. It would therefore be appropriate to analyse the breathing pattern of lung cancer patients in the light of nonlinear dynamical system theory. The purpose of this paper is to analyse the one-dimensional respiratory time series of lung cancer patients based on nonlinear dynamics and delay coordinate state space embedding. It is very important to select a suitable pair of embedding dimension 'm' and time delay 'τ' when performing a state space reconstruction. Appropriate time delay and embedding dimension were obtained using well-established methods, namely mutual information and the false nearest neighbour method, respectively. Establishing stationarity and determinism in a given scalar time series is a prerequisite to demonstrating that the nonlinear dynamical system that gave rise to the scalar time series exhibits a sensitive dependence on initial conditions, i.e. is chaotic. Hence, once an appropriate state space embedding of the dynamical system has been reconstructed, we show that the time series of the nonlinear dynamical systems under study are both stationary and deterministic in nature. Once both criteria are established, we proceed to calculate the largest Lyapunov exponent (LLE), which is an invariant quantity under time delay embedding. The LLE for all 16 patients is positive, which along with stationarity and determinism establishes the fact that the time series of a lung cancer patient's breathing pattern is not random or irregular, but rather it is deterministic in nature albeit chaotic. These results indicate that chaotic characteristics exist in the respiratory waveform and techniques based on state space dynamics should be employed for tumour motion management.
成功实施任何高效肿瘤运动管理策略的基本要求是患者呼吸模式的规律性和可重复性。呼吸的生理行为受到多个非线性反馈和前馈耦合的控制。因此，根据非线性动力系统理论分析肺癌患者的呼吸模式是合适的。本文的目的是基于非线性动力学和延迟坐标状态空间嵌入来分析肺癌患者的一维呼吸时间序列。在进行状态空间重建时，选择合适的嵌入维数“m”和时间延迟“τ”非常重要。使用成熟的方法，即互信息和虚假最近邻方法，分别获得了适当的时间延迟和嵌入维数。在证明产生标量时间序列的非线性动力系统对初始条件具有敏感依赖性，即混沌之前，建立给定标量时间序列的平稳性和决定性是前提条件。因此，一旦重建了适当的动力系统状态空间嵌入，我们就表明所研究的非线性动力系统的时间序列在本质上是平稳和确定的。一旦确立了这两个标准，我们就可以计算最大李雅普诺夫指数（LLE），它是时间延迟嵌入下的不变量。所有 16 名患者的 LLE 均为正，这与平稳性和决定性一起证明了肺癌患者呼吸模式的时间序列不是随机或不规则的，而是确定性的，尽管是混沌的。这些结果表明，混沌特征存在于呼吸波形中，应采用基于状态空间动力学的技术进行肿瘤运动管理。
