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) = AP(3) + BP(2) + 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.

We have shown that a polynomial equation, FP = AP3 + BP2 + CP + D, where F is flow and P is pressure, can accurately determine the presence of inspiratory flow limitation (IFL). This equation requires the invasive measurement of supraglottic pressure. We hypothesized that a modification of the equation that substitutes time for pressure would be accurate for the detection of IFL and allow for the noninvasive measurement of upper airway resistance. The modified equation is Ft = At3 + Bt2 + Ct + D, where F is flow and t is time from the onset of inspiration. To test our hypotheses, data analysis was performed as follows on 440 randomly chosen breaths from 18 subjects. First, we performed linear regression and determined that there is a linear relationship between pressure and time in the upper airway (R2 0.96 +/- 0.05, slope 0.96 +/- 0.06), indicating that time can be a surrogate for pressure. Second, we performed curve fitting and found that polynomial equation accurately predicts the relationship between flow and time in the upper airway (R2 0.93 +/- 0.12, error fit 0.02 +/- 0.08). Third, we performed a sensitivity-specificity analysis comparing the mathematical determination of IFL to manual determination using a pressure-flow loop. Mathematical determination had both high sensitivity (96%) and specificity (99%). Fourth, we calculated the upper airway resistance using the polynomial equation and compared the measurement to the manually determined upper airway resistance (also from a pressure-flow loop) using Bland-Altman analysis. Mean difference between calculated and measured upper airway resistance was 0.0 cmH2O x l(-1) x s(-1) (95% confidence interval -0.2, 0.2) with upper and lower limits of agreement of 2.8 cmH2O x l(-1) x s(-1) and -2.8 cmH2O x l(-1) x s(-1). We conclude that a polynomial equation can be used to model the flow-time relationship, allowing for the objective and accurate determination of upper airway resistance and the presence of IFL.

We have previously shown that the pressure-flow relationship of the upper airway during nonrapid eye movement sleep can be characterized by a polynomial equation: F(P) = AP(3) + BP(2) + CP + D. On the basis of fluid mechanic principles, we hypothesized that we could objectively calculate upper airway resistance (R(UA)) using the polynomial equation. We manually measured RUA (mR(UA)) from the first linear portion of a pressure-flow loop in 544 breaths from 20 subjects and compared the mRUA to the R(UA) calculated from the polynomial equation (cRUA). Bland-Altman analysis showed that the mean difference between mR(UA) and cRUA was 0.0 cm H2O/L/s (95% CI, 0.1 to 0.1 cm H2O/L/s) with an upper limit of agreement of 2.0 cm H (2)O/L/s (95% CI, 1.9 to 2.1 cm H2O/L/s) and a lower limit of agreement -2.0 cm H2O/L/s (95% CI, -2.1 to -1.9 cm H2O/L/s). Additional Bland-Altman analyses showed that the agreement between the two measures was excellent for both inspiratory flow-limited and non-flow-limited breaths. We conclude that R(UA) can be measured in a simple, objective, and reproducible fashion from a polynomial function that characterizes the upper airway pressure-flow relationship.