A mathematical model for the first derivative wave analysis of the volumetric capnogram from the perspective of erythrocyte motion profiles

Abstract

Current trends in monitoring system are leading to the adoption of volumetric capnogram (Vcap). The first derivative wave analysis (FDWA) of Vcap represented the cardiogenic oscillations (CarO) as a propagated wave and the slope of phase III (S_III) as a constant. Until today the genesis of CarO and S_III is however under debate. In this study, we defined motion profiles of erythrocytes in the pulmonary parenchyma as pulsated-run and random-walk, on the basis of which we obtained a new mathematical expression describing FDWA of Vcap. The mathematical model of Vcap provided theoretical explanation concerned with motion profiles of erythrocytes about the genesis of CarO and S_III. As the results, the mathematical model predicted the close relationship between S_III and the transfer factor of carbon monoxide, which will be used for estimating validity of this mathematical model. In addition, the velocity of propagated wave in the phase III was suggested as a new physiological variable to estimate elastic properties of pulmonary arterioles, and a new measuring method of V_D was proposed based on the theoretical reason, as well. Clinical investigations of the new V_D to test its efficacy of monitoring are needed.

Keywords: Cardiogenic oscillation; Dead space; First derivative wave analysis; Mathematical biosciences; Phase III slope; Physiology; Pulsated-run; Random-walk; Volumetric capnogram.

Fig. 1

First Derivative Wave Analysis (FDWA) for a single breath test (SBT) for nitrogen (N₂) and for carbon dioxide (CO₂). A: A sample of FDWA-N₂ from Wada et al. (2015); Both usual drawing graph of SBT-N₂ (the dot line) and corresponding FDWA-N₂ graph (solid line) of 62-year old healthy man are shown. Note that cardiogenic oscillations (CarO) were seen in phases III and IV by FDWA-N₂. B: Both a SBT-CO₂ curve (dot line) of a 54-year-old healthy male and its corresponding FDWA-CO₂ (solid line) was displayed in Fig. 2B; note that FDWA is composed of the oscillated part and the constant one in the phase III, and the wave length of cardiogenic oscillation is measured as λ from CarO of the latter part of phase III.

Fig. 2

Capillary Network and Percolation. A: A large number of erythrocytes come in the capillary beds by pulsated-runs through the precapillary arterioles, and are distributed in capillary beds by random walks. Two types of trajectory of each erythrocyte by random-walk are presented; the first (blue) trajectory is a path from an arteriole to a venule, another (red) is a path of sustained stay in the capillary bed. B: A sample of Monte Carlo simulation by Sedgewick and Wayne (2016), who showed the site vacancy probability p versus the percolation probability for 100-by-100 random grid with a Java programming environment. This graph indicates that pulmonary capillary perfusion would appear by all-or-none style, which was reported by Presson et al. (1997).

Fig. 3

Dead Space. Carbon dioxide from erythrocytes is excreted into the alveoli constantly in the lung parenchyma during a breath. Since the flow rate of carbon dioxide ($E_{AC{O}_2}/T$) is proportional to the slope of phase III, $E_{AC{O}_2}$ during a breath is evaluable as the area of trapezoid 1-2-3-4. The area under Vcap (AUVcap) is the total volume of exhaled CO₂ from the pulmonary parenchyma during a breath. The dead space normalized by tidal volume (V_T) is expressed by the ratio $\left(1-AU{V}_{cap}/E_{AC{O}_2}\right)$.

Fig. 4

Difference among Bohr, Enghoff, and our approaches to Dead space. The line discriminating the dead space from the tidal volume is created so that the area A equals to the area B. In the case of Enghoff, the line is created so that the area C equals to the area D. The dead space (V_D) is different among approaches, and V_D of our model seems the smallest.