Mathematical Model of the Bronchial Tree with Alveoli. Deposition of Anthropogenic Particles in the Lungs

The mathematical model of the bronchial tree developed by the authors includes alveolar sacs. Alveolar sacs begin to appear in the human bronchial tree starting from the 15th generation of bronchi. Their number increases geometrically as they move down the bronchial tree, reaching a maximum in the terminal bronchi. The total number of sacs is more than 500 million. Human respiratory physiology imposes a number of conditions on the mathematical model of alveolar respiration. These conditions are uniform ventilation of the alveoli and minimal work associated with breathing. Therefore, the model must satisfy the condition of equal pressure in all 500 million alveolar sacs. Based on these conditions, a model of human alveolar inhalation was constructed. The alveolar pressure during inhalation, as determined in the model, is lower than atmospheric pressure, which is consistent with physiological data.
Based on this model, calculations were made of the deposition of anthropogenic particles (cement dust and soot) in the lungs, which allows the risks to the respiratory system to be assessed. It has been shown that a decrease in air flow velocity (a decrease in inhaled air flow) contributes to an increase in particle deposition in the upper parts of the lungs. An increase in inhaled air flow reduces particle deposition, but at the same time, particles are ``driven'' into the lower parts of the lungs. Particle density has a negligible effect on their deposition compared to particle size and air flow velocity.

Keywords

initial-boundary value problems; connected graphs; eigenvalues and eigenfunctions of operators; discrete and semi-bounded operators; Galerkin method; regularized trace method.

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