Sensitivity and optimality analysis of breathing scenarios for 1D or 0D models of gas diffusion in the lung

In the present work we propose a new nonlinear coupled 1D model to describe lung ventilation and the transport and diffusion of both oxygen and carbon dioxide in the bronchial tree through the blood. It takes into account the so-called Bohr-Haldane effect, which induces a strong coupling of oxygen and carbon dioxide, and is driven by the applied pleural pressure. The ability of this model to reproduce standard acknowledged values in healthy situations and normal breathing scenario is provided. One key aspect is that, contrary to its 0D counterpart, it naturally takes into account mixing of gases along the tree and a time delay as the gases have to be transported before reaching the alveoli. We investigate the sensitivity of the 1D model to model parameters, its behavior at exercise and recover standard washout carbon dioxide curves. We also investigate the sensitivity of both the 1D model and its 0D counterpart with respect to the breathing pattern by considering two types of pleural applied pressure: a piecewise constant one and a piecewise exponential one for various values of the breathing period, inspiratory ratio and pressure amplitude. We finally explore which cost functions the observed stereotypical breathing scenario for normal breathing in healthy situations may optimize, emphasizing the fact that it should be a combination of several criteria: low effort and small lung distension while maintaining average carbon dioxide arterial partial pressure at a given level. The paper concludes with a discussion on the proposed model, its limitations and further works.