A mathematical model to detect inspiratory flow limitation during sleep

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Abstract
The physiological significance of inspiratory flow limitation (IFL) has recently been recognized, but methods of detecting IFL can be subjective. We sought to develop a mathematical model of the upper airway pressure-flow relationship that would objectively detect flow limitation. We present a theoretical discussion that predicts that a polynomial function [F(P) = AP3 + BP2 + CP + D, where F(P) is flow and P is supraglottic pressure] best characterizes the pressure-flow relationship and allows for the objective detection of IFL. In protocol 1, step 1, we performed curve-fitting of the pressure-flow relationship of 20 breaths to 5 mathematical functions and found that highest correlation coefficients ( R 2) for quadratic (0.88 ± 0.10) and polynomial (0.91 ± 0.05; P < 0.05 for both compared with the other functions) functions. In step 2, we performed error-fit calculations on 50 breaths by comparing the quadratic and polynomial functions and found that the error fit was lowest for the polynomial function (3.3 ± 0.06 vs. 21.1 ± 19.0%; P < 0.001). In protocol 2, we performed sensitivity/specificity analysis on two sets of breaths (50 and 544 breaths) by comparing the mathematical determination of IFL to manual determination. Mathematical determination of IFL had high sensitivity and specificity and a positive predictive value (>99% for each). We conclude that a polynomial function can be used to predict the relationship between pressure and flow in the upper airway and objectively determine the presence of IFL.