Simulation of nanoparticle transport in airways using Petrov–Galerkin finite element methods

Abstract

The transport and deposition properties of nanoparticles with a range of aerodynamic diameters ( 1 nm ≤ d ≤ 150 nm) were studied for the human airways. A finite element code was developed that solved both the Navier-Stokes and advection-diffusion equations monolithically. When modeling nanoparticle transport in the airways, the finite element method becomes unstable, and, in order resolve this issue, various stabilization methods were considered in terms of accuracy and computational cost. The stabilization methods considered here include the streamline upwind, streamline upwind Petrov-Galerkin, and Galerkin least squares approaches. In order to compare the various stabilization approaches, the approximate solution from each stabilization approach was compared to the analytical Graetz solution, which is a model for monodispersed, dilute particle transport in a straight cylinder. The optimal stabilization method, especially with regard to accuracy, was found to be the Galerkin least squares approach for the Graetz problem when the Péclet number was larger than 10(4). In the human airways geometry, the Galerkin least squares stabilization approach once more provided the most accurate approximate solution for particles with an aerodynamic diameter of 10 nm or larger, but mesh size had a much greater effect on accuracy than the choice of stabilization method. The choice of stabilization method had a greater impact than mesh size for particles with an aerodynamic diameter 10 nm or smaller, but the most accurate stabilization method was streamline upwind Petrov-Galerkin in these cases.

A rapid simulation of nano‐particle transport in a two‐dimensional human airway using POD/Galerkin reduced‐order models

“…Thus, there still remains an unknown amount of particles within the airway in the form of deposition. The amount of the deposition can be computed from the difference in the mass flux between the inlets and outlets as follows Here, DF denotes the deposition fraction, is the average concentration in the element face i = 1,..., n b c , where n b c is the total number of element faces on the boundary, and Q i is the flow rate that exists on the boundary. This arrangement will be studied in more detail in the next section.…”

“…However, with the complexity of the human tracheobronchial tree, the computational efficiency is still a challenge even with simplified geometries. For example, in‐house code for simulation of the human airway (G0‐G3) based on Weibel's model was studied in ; the model used approximately 100,000 elements, and a single simulation was completed in 72–120 h. The study in employed a user‐enhanced commercial finite volume‐based program, that is, CFX4.4, and performed simulations using the oral airway model. This model had 420,000 cells in the oral airway (including the oral cavity, pharynx, larynx, and trachea) and 670,000 cells in the four‐generation airway (the triple bifurcation that represents Generations G0 (trachea) to G3).…”

“…By introducing τ into the governing equations, the FEM solutions become stable against spurious oscillations in the velocity field. This method is used by Rajaraman and Heys for simulating particle transport and deposition in the human airway. However, their approach could lack generality and accuracy because they assumed a constant τ and attempted to find an optimal value in the range of (0,1).…”

Particle transport in the human respiratory tract: formulation of a nodal inverse distance weighted Eulerian–Lagrangian transport and implementation of the Wind–Kessel algorithm for an oral delivery

“…Currently, the deposition in the human airway tract can be estimated using ‘quasi’ experiments , empirical approaches , computational approaches , or a mix of the previously mentioned approaches . The shortcomings of the previous approaches are as follows: The quasi experiments are performed on respiratory tract casts/models.…”

“…Currently, the deposition in the human airway tract can be estimated using 'quasi' experiments [5,6], empirical approaches [7][8][9], computational approaches [10][11][12][13][14][15], or a mix of the previously mentioned approaches [16][17][18][19]. The shortcomings of the previous approaches are as follows:…”