Huang W, Shen Z, Huang N E, Fung Y C
Department of Bioengineering, University of California at San Diego, La Jolla, CA 92093-0412, USA.
Proc Natl Acad Sci U S A. 1999 Mar 2;96(5):1834-9. doi: 10.1073/pnas.96.5.1834.
This paper is devoted to the quantization of the degree of nonlinearity of the relationship between two biological variables when one of the variables is a complex nonstationary oscillatory signal. An example of the situation is the indicial responses of pulmonary blood pressure (P) to step changes of oxygen tension (DeltapO2) in the breathing gas. For a step change of DeltapO2 beginning at time t1, the pulmonary blood pressure is a nonlinear function of time and DeltapO2, which can be written as P(t-t1 | DeltapO2). An effective method does not exist to examine the nonlinear function P(t-t1 | DeltapO2). A systematic approach is proposed here. The definitions of mean trends and oscillations about the means are the keys. With these keys a practical method of calculation is devised. We fit the mean trends of blood pressure with analytic functions of time, whose nonlinearity with respect to the oxygen level is clarified here. The associated oscillations about the mean can be transformed into Hilbert spectrum. An integration of the square of the Hilbert spectrum over frequency yields a measure of oscillatory energy, which is also a function of time, whose mean trends can be expressed by analytic functions. The degree of nonlinearity of the oscillatory energy with respect to the oxygen level also is clarified here. Theoretical extension of the experimental nonlinear indicial functions to arbitrary history of hypoxia is proposed. Application of the results to tissue remodeling and tissue engineering of blood vessels is discussed.
本文致力于研究当两个生物变量之一为复杂非平稳振荡信号时，二者关系的非线性程度的量化问题。这种情况的一个例子是呼吸气体中氧分压（ΔpO₂）阶跃变化时肺血压（P）的指数响应。对于在时间t₁开始的ΔpO₂阶跃变化，肺血压是时间和ΔpO₂的非线性函数，可写为P(t - t₁ | ΔpO₂)。目前不存在检验非线性函数P(t - t₁ | ΔpO₂)的有效方法。本文提出一种系统方法。均值趋势和围绕均值的振荡的定义是关键。利用这些关键要素设计出一种实用的计算方法。我们用时间的解析函数拟合血压的均值趋势，这里阐明了其相对于氧水平的非线性。围绕均值的相关振荡可转化为希尔伯特谱。希尔伯特谱平方在频率上的积分产生振荡能量的度量，它也是时间的函数，其均值趋势可用解析函数表示。这里也阐明了振荡能量相对于氧水平的非线性程度。提出了将实验非线性指数函数理论扩展到任意缺氧历史情况。讨论了该结果在血管组织重塑和组织工程中的应用。
