A NONLINEAR MATHEMATICAL MODEL OF PRESSURE PRESET VENTILATION: DESCRIPTION AND LIMITING VALUES FOR KEY OUTCOME VARIABLES

Abstract

In recent years, several new forms of pressure preset mechanical ventilation (PPV) have been introduced to clinical practice. Although these modes are widely employed in patient care, clinical decision making remains a largely empirical “trial and error” process. Existing predictive equations for ventilation are questionably accurate, in part because most attempts to model ventilation have assumed constant values for inspiratory and expiratory resistance, even though the pressure-flow relationship is clearly nonlinear in biological systems. In this paper, we present and analyze a nonlinear mathematical model of PPV which accounts for the interactive behavior of inspiratory and expiratory half cycles. It comprises a set of nonlinear differential equations which incorporate a variably nonlinear relationship between the resistive component of the applied pressure and flow rate. This model is compared to our previously described biphasic (linear) exponential model of PPV (J. Appl. Physiol.67 (1989) 1081–0192) which serves to link the clinical “input” variables of pressure level, frequency, inspiratory time fraction, and impedance with the key “outcome” variables of clinical interest: tidal volume, minute ventilation, power, airway pressure, mean alveolar pressure, and expiratory alveolar pressure. Predictive differences arise between linear and nonlinear formulations. Although general closed-form solutions for several of the outcome variables could not be obtained, implicit expressions are derived. Explicit derivations are also presented for selected flow exponents of interest. Furthermore, we found that the limiting values for each outcome variable as a function of cycling frequency could be derived explicitly, once the exponent for flow is uniquely specified.