Viscoelasticity and fractal structure in a model of human lungs

Abstract

This paper provides a model of the human respiratory system by taking into account the fractal structure of the airways and the viscoelastic properties of the tissue. The self-similarity of airway distribution is admitted up to the 24th generation. Due to periodic breathing which results in sinusoidal excitation of the respiratory system, an electrical equivalent model is developed. The periodic current in this electrical network, that preserves the geometry of the human respiratory tree, is equivalent to the oscillatory air-flow. The model is expressed by Navier–Stokes equations under cylindrical symmetry, linked with an equation responsible for the motion of viscoelastic tissue of airway walls. By use of both electro-mechanical analogies, the total impedance of the respiratory system is determined and compared to the measured data in the clinical range of 4–48 Hz, as well as in the low-frequency range of 0.1–5 Hz. We propose also a lumped model of fractional orders, which is able to capture frequency-dependent variations in both clinical as well as in the low-frequency ranges. The models proposed in this paper can be further used to determine the effects of disease on the lung morphology.