Nonlinear indicial response of complex nonstationary oscillations as pulmonary hypertension responding to step hypoxia (degree of nonlinearityyFourier spectrumyHilbert spectrumynonlinear relationships)

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 (DpO2) in the breathing gas. For a step change of DpO2 beginning at time t1, the pulmonary blood pressure is a nonlinear function of time and DpO2, which can be written as P(t-t1 z DpO2). An effective method does not exist to examine the nonlinear function P(t-t1 z DpO2). 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 subsequent tissue and mechanical properties remodeling are believed to be the results of these stress and strain changes. Therefore, the exact behavior of blood pressure when the oxygen tension changes is of paramount importance. But blood pressure is stochastic (see Fig. 1). To get a meaningful and precise description of the blood pressure history is the first step. We used a catheter that was implanted in the pulmonary arterial trunk of a rat to get the instantaneous reading of blood pressure continuously over many days in a simulated laboratory chamber. The oxygen tension in the breathing gas was controlled as a constant or varied as a step decrease in time or a step increase in time. Typical blood pressure records are shown in Fig. 1. Previous reports (17) have demonstrated various features of pulmonary hypertension caused by hypoxia, but no mathematical analysis of the blood pressure-oxygen relationship was shown. The present paper shows a systematic use of the method presented in refs. 18 and 19 to the study of the nonlinearity of the pulmonary blood pressure-oxygen tension relationship.