Computational model of trachea-alveoli gas movement during spontaneous breathing
Introduction
The recent appearance of the SARS COV-2 pandemic has shown the importance of studying all topics around the transport and exchange of gases in the respiratory system. Therefore, the proper understanding and description of gas transport is essential both for the adequate diagnosis of respiratory diseases and for the correct operation of mechanical ventilation equipment.
The measurement of O2 and CO2 at the alveolar level has been the subject of multiple investigations, focused in many cases on the single-breath nitrogen wash curve (single-breath N2 test) and not on the continuous determination for its use, as done in patients under mechanical ventilation. In 1909, Bohr required to know the alveolar tensions of O2 to make it possible to predict the profile of oxygen changes along the pulmonary capillary (Bohr, 1909). In 1911, Krogh and Lindhart experimentally calculated concentrations of these gases at the alveolar level (Krogh and Lindhard, 1914). In 1946, a theoretical study of the concentration of alveolar gases in height was published (Fenn et al., 1946). In 1949, quantitative analysis of pulmonary gas exchange was made possible, applicable to both normal and pathological conditions (Riley and Cournand, 1949). In 1952, the alveolar concentration of CO2 was experimentally determined (DuBOIS and Britt, 1952) and in 1954, a mathematical analysis of O2 during respiration was made (Chilton et al., 1954). In 1964, Bouhuys published in the Physiology Handbook a chapter on the distribution of inspired gas (Bouhuys, 1964) where the phenomenon of stratification in the lung was analyzed. In 1966, the diffusion equation (using O2 and N2) in various models (including conical models) of the airway was analyzed, but the convective component of oxygen movement was not involved in their calculations (Cumming et al., 1966). In 1970, Nye showed to what extent pulmonary gas exchange can be influenced by the way the pattern of air flow was disposed with respect to time during each breath (Nye, 1970). In 1973, the direct measurements in the alveoli in dogs was carried out (Engel et al., 1973) and Paiva (Paiva, 1973) published an equation for the mass transport in the lungs considering convective and diffusive motion of gases, but made significant simplifications.
Later, in 1975, the inclusion of the apparent diffusivity coefficient (ADC) in the complete mass transport equation was proposed (Yu, 1975). In 1987, a review of the theoretical studies of gas mixture and distribution of ventilation in the lung was published (Paiva and Engel, 1987). In 2005, a mathematical framework to link between the models that had previously been developed separately at different scales and at different degrees of complexity was established (Ben-Tal, 2006). To date, the theoretical basis used for mouth-alveolus gas transport is based on the published work of Butler and Tsuda (2011). They proposed the Eq. (1) in one dimension:where: is the average mass concentration, is the total speed of the mixture, A is the cross-sectional area and α represents the rate of mass loss. The first two terms on the right side of the equation (R.H.S.) are precisely those that appear in the original and complete convection-diffusion equation. As can be seen, effective diffusion ) is assumed and not molecular diffusion, implying that regardless of the interaction between the velocity field and the gas concentration gradients within the lung, the net diffusive flow behaves like a simple one-dimensional current. Therefore, implicit considerations related to are included like airway diameters and velocity (Taylor, 1953). The third term emerges explicitly due to the variation in area and the with the axial distance and the last term is required whenever there is a series exchange between gas and tissue or blood. West & Prisk (West and Prisk, 2018) in developed a new non-invasive method for the evaluation of gas exchange using pulse oximetry and expired gas from the patient. Noel proposed a diffusive convective gas exchange model that took into account the alveolar-capillary exchange to evaluate the interplay between optimal ventilation and gas transport in the lung (Noël and Mauroy, 2019). Fitz (Fitz-Clarke, 2020) presented a well-structured gas exchange model to analyze the effect of tidal volume during rescue ventilation.
In the present work, the last two terms of the Eq. (1) that represent the radial diffusion has been neglected and a value for the diffusivity (D) of CO2 and O2 in the mixture of alveolar gases has been found, considering the water vapor. Using Matlab® script, a model of the mass transport of O2 and CO2 has been simulated, which uses the symmetric dichotomy model of the airway proposed by Weibel (1963). Some other considerations were made assuming uniformity in the radial coordinate in each branch of each generation and a variable speed function has been used that allows to simulate, in a more adequate way, a healthy patient under spontaneous breathing. The model includes the entire route of the airway, containing the alveoli at the last generation (23) and disregarding gas exchange in any other generation.
Outlining a simplified but effective and manageable model of pulmonary gas transport and its interaction with respiratory rate and diaphragmatic movement, the gap in the state of knowledge was addressed. This model allowed a realistic simulation of the flow of the different gases. By incorporating these computational simulations, it is shown how the simulation of the human pulmonary system can be detailed and computationally manageable. Mechanical ventilation of the lung would acquire a great tool with the development of dynamic and accurate measurement of alveolar gases. Thus, the main contribution of this work lies in the fact that it was possible to develop a phenomenological model that simulates the transport of gases from alveoli to trachea. This allows estimating the CO2 concentration in the alveoli from easily measurable variables such as gas flow, concentrations in the mouth and frequencies, which can become a potential diagnostic tool.
Section snippets
Glossary
C Concentration of gas
Average mass concentration
Total speed of the mixture
G Mass generation
A Cross-sectional area
Effective diffusion
BPM Breaths per minute
yi Mole fraction of the gas
Di,j Diffusivity value of each pair of gases
Di,m Diffusivity value of gas present in the mixture of respiratory gases
FiO2 Fraction of inspired oxygen
Flow velocity in the branch x–y
Cross-sectional area of the branch x–y
t Time
This section describes the geometric model used, the way the
Results
The results were grouped in seven figures (Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10). For the results shown in Fig. 4, Fig. 5, Fig. 6, Fig. 7 the following convention was used:
Test point A: Solid black line. A point immediately after the inlet and outlet of gases at the alveolar level.
Test point B: Blue line. Inlet of generation 22.
Test point D: Fuchsia line. Inlet of generation 17.
Test point F: Red line. Inlet of generation 3.
Test point G: Thick dashed line. Inlet of generation
Discussion
Given the requirement to make a series of considerations, as is necessary in any mathematical analysis of a real problem, in this paper several were considered in which their validity and implications of all of them have been discussed by various authors. They are summarized in:
1. Constant CO2 and O2 concentrations at the alveolar and tracheal levels.
2. The Weibel model was used (Weibel, 1963).
3. The diameters in the airways are corrected according to Wiggs (Wiggs et al., 1990; La Force and
Conclusions
The current computational simulation tools allow to olve complex models of respiratory physiology, allowing the development of new hypotheses that improve the knowledge of this important organ. The results shown here are consistent with the information available today about this phenomenon of CO2 and O2 transport between the inlet of gases at the trachea and the alveoli.
Models could be developed under this methodology that consider radial diffusion and differential modeling between the more
Acknowledgement
This projet was supported by Fondo de Investigación en Salud (FIS), convocatoria 807-2018 Colciencias, Colombia (project 133380764110, contract 801-2018).
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