10. Discussion 987 11. Conclusions 991 References 993A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the 'empirical mode decomposition' method with which any complicated data set can be decomposed into a finite and often small number of 'intrinsic mode functions' that admit well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes. With the Hilbert transform, the 'instrinic mode functions' yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert spectrum.In this method, the main conceptual innovations are the introduction of 'intrinsic mode functions' based on local properties of the signal, which makes the instantaneous frequency meaningful; and the introduction of the instantaneous frequencies for complicated data sets, which eliminate the need for spurious harmonics to represent nonlinear and non-stationary signals. Examples from the numerical results of the classical nonlinear equation systems and data representing natural phenomena are given to demonstrate the power of this new method. Classical nonlinear system data are especially interesting, for they serve to illustrate the roles played by the nonlinear and non-stationary effects in the energy-frequency-time distribution.

We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.

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Almost all variables in biology are nonstationarily stochastic. For these variables, the conventional tools leave us a feeling that some valuable information is thrown away and that a complex phenomenon is presented imprecisely. Here, we apply recent advances initially made in the study of ocean waves to study the blood pressure waves in the lung. We note first that, in a long wave train, the handling of the local mean is of predominant importance. It is shown that a signal can be described by a sum of a series of intrinsic mode functions, each of which has zero local mean at all times. The process of deriving this series is called the ''empirical mode decomposition method.'' Conventionally, Fourier analysis represents the data by sine and cosine functions, but no instantaneous frequency can be defined. In the new way, the data are represented by intrinsic mode functions, to which Hilbert transform can be used. Titchmarsh [Titchmarsh, E. C. (1948) Introduction to the Theory of Fourier Integrals (Oxford Univ. Press, Oxford)] has shown that a signal and i times its Hilbert transform together define a complex variable. From that complex variable, the instantaneous frequency, instantaneous amplitude, Hilbert spectrum, and marginal Hilbert spectrum have been defined. In addition, the Gumbel extremevalue statistics are applied. We present all of these features of the blood pressure records here for the reader to see how they look. In the future, we have to learn how these features change with disease or interventions.We recorded the blood pressure in the pulmonary arterial trunk (between the pulmonic valve and the bifurcation point of the right and left pulmonary arteries). The recording was done continuously with an implanted catheter as a part of a research plan to study the remodeling of the three layers of vascular tissues of the arterial wall in response to changes of stresses in the tissues (1-6). Pulmonary arteries were chosen as an object of tissue engineering research because blood pressure in pulmonary arteries can be changed quickly and noninvasively by changing the oxygen concentration in the gas that the animal breathes. Blood pressure is a major parameter related to the stress distribution in the blood vessel wall. The present article is focused on the analysis of the blood pressure records of a normal rat breathing normal atmosphere at sea level. Fig. 1A shows a record over a 24-h period. Fig. 1 B and C show segments recorded at an expanded time scale. It is seen that the amplitude and frequency are variable. The changes are nonstationary, and definitions are needed to know what the heart rate, the mean blood pressure, and the amplitude of pressure oscillations are. Our objective is to see how these quantities can be characterized mathematically. MATERIALS AND EXPERIMENTAL METHODSFor the purpose of long term recording of blood pressure, a catheter must be implanted into an artery. The Riva-Rocci cuff inflation method of blood pressure measurement based on Korotkoff sounds cannot provide the desi...

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 (⌬pO 2 ) in the breathing gas. For a step change of ⌬pO 2 beginning at time t 1 , the pulmonary blood pressure is a nonlinear function of time and ⌬pO 2 , which can be written as P(t-t 1 ͦ ⌬pO 2 ). An effective method does not exist to examine the nonlinear function P(t-t 1 ͦ ⌬pO 2 ). 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.In biomedical science, we often have to deal with variables that are stochastic, oscillatory, and nonstationary and the relationship of these variables to other chemical, mechanical, physical, and pharmacological variables. In the cardiovascular system blood pressure is such a variable. This paper illustrates the mathematical approach to deal with the question of linearity or nonlinearity of the dependence of blood pressure on other variables. As a specific illustration, we consider the changes that occur in the lung when a sea-level dwelling animal is flown to a ski resort at a higher altitude where the partial pressure of oxygen in the gas that the animal breathes is lower. What happens is that the pulmonary arterial blood pressure becomes higher (1-3), the arterial blood vessel wall becomes thicker (3-5), the different layers of the arteries thicken with different rates and different courses of time (2-6), the mechanical properties of the blood vessel wall change with specific historical courses (7-9), cells in the wall modify, grow, proliferate, or move (5, 6, 10-13), intercellular matrix and interstitial space change (14, 15), the stress and strain distribution in the vessel wall change with time in a specific way (16), and because of cellular and extracellular changes the zero-stress state of the blood vessel wall changes with time (7-9). The crucial fact is the blood pressure change, because the blood pressure imposes load on the blood vessel wall, causing stress and strain, and the sub...