Validation and clinical application of a first order step response equation for nitrogen clearance during FRC measurement

Abstract

Objective: To derive a difference equation based on mass conservation and on alveolar tidal volumes for the calculation of Functional Residual Capacity. Derive an equation for the FRC from the difference equation. Furthermore, to derive and validate a step response equation as a solution of the difference equation within the framework of digital signal processing where the FRC is known a priori.

Methods: A difference equation for the calculation of Functional Residual Capacity is derived and solved as step response of a first order system. The step response equation calculates endtidal fractions of nitrogen during multiple breath nitrogen clearance. The step response equation contains the eigenvalue defined as the ratio of FRC to the sum of FRC and alveolar tidal ventilation. Agreement of calculated nitrogen fractions with measured fractions is demonstrated with data from a metabolic lung model, measurements from patients in positive pressure ventilation and volunteers breathing spontaneously. Examples of eigenvalue are given and compared between diseased and healthy lungs and between ventilatory settings.

Results: Comparison of calculated and measured fractions of endtidal nitrogen demonstrates a high degree of agreement in terms of regression and bias and limits of agreement (precision) in Bland & Altman analysis. Examples illustrate the use of the eigenvalue as a possible discriminator between disease states.

Conclusion: The first order step response equation reliably calculates endtidal fractions of nitrogen during washout based on a Functional Residual Capacity. The eigenvalue may be a clinically valuable index alone or in conjunction with other indices in the analysis of respiratory states and may aid in the setting of the ventilator.

Fig. 1

Sampling at two frequencies, the waveform sampling rate and two samples per breath sampling rate. Lower frame: the continuous sampling of FO₂ (left y-axis) and FCO₂ (right y-axis) during a N₂ washout procedure. From these tracings (enlarged in upper frame) two samples of FO₂ and FCO₂ are sampled once during inspiration and as endtidal values for entering into Equations 6 and 7.

Fig. 2

Calculated and measured F_ETN₂ during N₂ washout in metabolic lung model. ΔFN₂ 0.3, FRC 1800 mL, VA_T 270 mL, eigenvalue 0.86, 187 mL/min. The regression equation has a slope of 1.02 and a R² of 0.999.

Fig. 3

Bland & Altman plot of agreement between pairs of measured and calculated F_ETN₂ according to step response derivation in metabolic lung model. The FRC of the lung model was 1800 mL. Bias 0, upper and lower limits of agreement of 0.25 and −0.5.

Fig. 4

Bland & Altman plot of agreement between pairs of measured and calculated F_ETN₂ according to step response derivation in 35 N₂ washout procedures in five patients. Bias of 0.3, upper and lower limits of agreement of 2 and −2.6. The regression equation showed a slope of 1.07, intercept −3 and R² 0.98 with a p < 0.001.

Fig. 5

λ as a function of PEEP in five patients.

Fig. 6

Bland & Altman plot of agreement between pairs of measured and calculated F_ETN₂ according to step response derivation in 16 N₂ washout procedures in three healthy volunteers. Bland & Altman analysis showed bias of −0.16, lower and upper limits of agreement at 3.2 and −3.5. The regression equation showed a slope of 0.97, intercept at 1.9 and R² 0.98 with a p < 0.001.