A new drift-flux model for particle transport and deposition in human airways

Particle deposition and transport in human airways isfrequently modeled numerically by the Lagrangian approach. Current formulations of such models always require some ad hoc assumptions, and they are computationally expensive. A new drift-flux model is developed and incorporated into a commercial finite volume code. Because it is Eulerian in nature, the model is able to simulate particle deposition patterns, distribution and transport both spatially and temporally. Brownian diffusion, gravitational settling, and electrostatic force are three major particle deposition mechanisms in human airways. The model is validated against analytical results for three deposition mechanisms in a straight tube prior to applying the method to a single bifurcation G3-G4. Two laminar flows with Reynolds numbers 500 and 2000 are simulated. Particle concentration contour deposition pattern, and enhancement factor are evaluated. To demonstrate how the diffusion and settling influence the deposition and transport along the bifurcation, particle sizes from 1 nm to 10 microm are studied. Different deposition mechanisms can be combined into the mass conversation equation. Combined deposition efficiency for the three mechanisms simultaneously was evaluated and compared with two commonly used empirical expressions.     

Further considerations in a theoretical description of gas transport in lung airways

A discrete one-dimensional model of convection-diffusion in branching alveolar ducts is described and it is shown that, for a suitable choice of effective axial dispersion, the solution closely approximates that for an axially symmetric representation, at least for Peclet numbers Pe<1. Following earlier work a composite model of a uniform lung is formed by matching such a respiratory pathway (now having the more convenient one-dimensional form) onto a trumpet representation of the conducting airways. Enhanced mixing due to heart action, and isotropic volume changes of trumpet (in addition to the pathway) during breathing are additional factors included. Calculations are made of O₂ concentrations during steady-state breathing and of the concentration of inert gas during single breath wash-out of a gas mixture containing it. Predicted alveolar levels in each case agree extremely well with published data, although no alveolar slope is obtained for the inert gas.     

Volume of the conducting airways calculated from morphometric parameters

A method of calculating the volume of a tree distal to a cut at the origin of a branch, using branching, diameter and length ratios, has been developed. The method was applied to bronchial tree casts from human, dog, sheep, hamster, and rat lungs. It was found that the exponenta in the equation weight=k×diameter ^a is approximately equal to 3.0 in sheep lung casts, as found by Hooper (1977), but it is greater than 3.0 in casts from the other four species.     

Secondary velocity fields in the conducting airways of the human lung

An understanding of flow and dispersion in the human respiratory airways is necessary to assess the toxicological impact of inhaled particulate matter as well as to optimize the design of inhalable pharmaceutical aerosols and their delivery systems. Secondary flows affect dispersion in the lung by mixing solute in the lumen cross section. The goal of this study is to measure and interpret these secondary velocity fields using in vitro lung models. Particle image velocimetry experiments were conducted in a three-generational, anatomically accurate model of the conducting region of the lung. Inspiration and expiration flows were examined under steady and oscillatory flow conditions. Results illustrate secondary flow fields as a function of flow direction, Reynolds number, axial location with respect to the bifurcation junction, generation, branch, phase in the oscillatory cycle, and Womersley number. Critical Dean number for the formation of secondary vortices in the airways, as well as the strength and development length of secondary flow, is characterized. The normalized secondary velocity magnitude was similar on inspiration and expiration and did not vary appreciably with generation or branch. Oscillatory flow fields were not significantly different from corresponding steady flow fields up to a Womersley number of 1 and no instabilities related to shear were detected on flow reversal. These observations were qualitatively interpreted with respect to the simple streaming, augmented dispersion, and steady streaming convective dispersion mechanisms.     

Convective dispersion during steady flow in the conducting airways of the human lung

The adverse health effects of inhaled particulate matter from the environment depend on its dispersion, transport, and deposition in the human airways. Similarly, precise targeting of deposition sites by pulmonary drug delivery systems also relies on characterizing the dispersion and transport of therapeutic aerosols in the respiratory tract. A variety of mechanisms may contribute to convective dispersion in the lung; simple axial streaming, augmented dispersion, and steady streaming are investigated in this effort. Flow visualization of a bolus during inhalation and exhalation, and dispersion measurements were conducted during steady flow in a three-generational, anatomically accurate in vitro model of the conducting airways to support this goal. Control variables included Reynolds number, flow direction, generation, and branch. Experiments illustrate transport patterns in the lumen cross section and map their relation to dispersion metrics. These results indicate that simple axial streaming, rather than augmented dispersion, is the dominant steady convective dispersion mechanism in symmetric Weibel generations 7-13 during normal respiration. Experimental evidence supports the branching nature of the airways as a possible contributor to steady streaming in the lung.     

An inverting basket model for AE1 transport

We describe an "inverting basket" model for transport in the erythrocyte anion exchanger, AE1. The inverting basket is formed by the side chains of three putative key residues, two positively (Lys 826 and Arg 730) and one negatively (Glu 681) charged residue. We have tentatively chosen seven transmembrane helices, TM1, TM2, TM4, TM8, TM10, TM12 and TM13 to form a conical channel using the well-established Glu 681 of TM8 and candidates Lys 826 and Arg 730 of TM12-13 and TM10, respectively, to form the inverting basket. We assume that these residues bind to an anion and shift from outward facing (C(o)) to inward facing (C(i)) conformation without significant backbone movements to transport an anion across the membrane. The transition of the complex (residues and ion) from outward facing (C(o)) to inward facing (C(i)) constitutes one "basket" inversion. The barrier to inversion is composed of two major components: that of the anhydrous complex, which we refer to as a steric energy barrier and a dehydration effect due to the removal of charges in the complex from water in the channel. The steric barrier is dependent on the side chain charge and configuration and on the ion charge and size. The dehydration effect, for our model, ameliorates the steric barrier, in the case of the empty complex but less so for the monovalent and divalent ions. We conclude, that it is possible for a seven-helix bundle to have a steric barrier to basket inversion, but that hydration effects in thin hydrophobic barrier models may be more complex than usually envisioned.     

An allosteric pore model for sugar transport in human erythrocytes

Glucose transport in human erythrocytes is characterized by a marked asymmetry in the $\text{V}$ and $\text{K}{}_{\mathrm{<mtext>m</mtext>}}$ values for entry and for exit. In addition, they show a high $\text{K}{}_{\mathrm{<mtext>m</mtext>}}$ and a high $\text{V}$ for equilibrium exchange but low $\text{K}{}_{\mathrm{<mtext>m</mtext>}}$ values for infinite cis and for infinite trans exit and entry. An allosteric pore model has been proposed to account for these characteristics. In this model, substrate-induced conformational changes destabilize the interfaces between protein subunits (the pore gates).Pores doubly occupied from inside destabilize the transport gates and result in high $\text{K}{}_{\mathrm{<mtext>m</mtext>}}$ and high $\text{V}$ transport parameters. This effect is less marked when pores are doubly occupied from outside and therefore transport asymmetry results.     

Aerosol transport and deposition in sequentially bifurcating airways

Deposition patterns and efficiencies of a dilute suspension of inhaled particles in three-dimensional double bifurcating airway models for both in-plane and 90 deg out-of-plane configurations have been numerically simulated assuming steady, laminar, constant-property air flow with symmetry about the first bifurcation. Particle diameters of 3, 5, and 7 microns were used in the simulation, while the inlet Stokes and Reynolds numbers varied from 0.037 to 0.23 and 500 to 2000, respectively. Comparisons between these results and experimental data based on the same geometric configuration showed good agreement. The overall trend of the particle deposition efficiency, i.e., an exponential increase with Stokes number, was somewhat similar for all bifurcations. However, the deposition efficiency of the first bifurcation was always larger than that of the second bifurcation, while in general the particle efficiency of the out-of-plane configuration was larger than that of the in-plane configuration. The local deposition patterns consistently showed that the majority of the deposition occurred in the carinal region. The distribution pattern in the first bifurcation for both configurations were symmetric about the carina, which was a direct result of the uniaxial flow at the inlet. The deposition patterns about the second carina showed increased asymmetry due to highly nonuniform flow generated by the first bifurcation and were extremely sensitive to bifurcation orientation. Based on the deposition variations between bifurcation levels and orientations, the use of single bifurcation models was determined to be inadequate to resolve the complex fluid-particle interactions that occur in multigenerational airways.     

Secretion and ion transport in airways during inflammation

Water and secretions interact in airways to produce the sol and gel layers that allow for entrapment of foreign materials and subsequent clearance by ciliary movement and by cough. Active Cl ion transport produces fluid, and this process is activated by products of mast cells (leukotrienes), eosinophils (major basic protein), and by other inflammatory mediators (prostaglandins, bradykinin). Gland secretions produce the bulk of the volume of secretions. Airway irritation stimulates gland secretion reflexly via vagal muscarinic pathways. Recently, the sensory nerves have been discovered to release substance P and other neuropeptides when the airways are irritated. The stimulatory effects of neuropeptides on gland secretion (and on other inflammatory sites) are modulated by enkephalinase a membrane-bound enzyme that cleaves neuropeptides and thereby inactivates them. Up- or down-regulation of enkephalinase is predicted to change the degree of inflammatory response to neuropeptides. Finally, the cell surface of airway epithelial cells have been discovered to secrete large molecular weight glycoconjugates; these secreted products are increased markedly by a series of proteinases produced by inflammatory cells (neutrophils, mast cells) and by bacteria. Their exact physiologic roles are still unknown but they may contribute to the bulk and viscoelastic properties of airway secretions, and they may serve an important role in bacterial, viral and inflammatory cell adhesion.     

An axisymmetric boundary integral model for incompressible linear viscoelasticity: application to the micropipette aspiration contact problem

The micropipette aspiration test has been used extensively in recent years as a means of quantifying cellular mechanics and molecular interactions at the microscopic scale. However, previous studies have generally modeled the cell as an infinite half-space in order to develop an analytical solution for a viscoelastic solid cell. In this study, an axisymmetric boundary integral formulation of the governing equations of incompressible linear viscoelasticity is presented and used to simulate the micropipette aspiration contact problem. The cell is idealized as a homogeneous and isotropic continuum with constitutive equation given by three-parameter (E, tau 1, tau 2) standard linear viscoelasticity. The formulation is used to develop a computational model via a "correspondence principle" in which the solution is written as the sum of a homogeneous (elastic) part and a nonhomogeneous part, which depends only on past values of the solution. Via a time-marching scheme, the solution of the viscoelastic problem is obtained by employing an elastic boundary element method with modified boundary conditions. The accuracy and convergence of the time-marching scheme are verified using an analytical solution. An incremental reformulation of the scheme is presented to facilitate the simulation of micropipette aspiration, a nonlinear contact problem. In contrast to the halfspace model (Sato et al., 1990), this computational model accounts for nonlinearities in the cell response that result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell-micropipette contact region, and curvature of the edges of the micropipette (inner radius a, edge curvature radius epsilon). Using 60 quadratic boundary elements, a micropipette aspiration creep test with ramp time t* = 0.1 s and ramp pressure p*/E = 0.8 is simulated for the cases a/R = 0.3, 0.4, 0.5 using mean parameter values for primary chondrocytes. Comparisons to the half-space model indicate that the computational model predicts an aspiration length that is less stiff during the initial ramp response (t = 0-1 s) but more stiff at equilibrium (t = 200 s). Overall, the ramp and equilibrium predictions of aspiration length by the computational model are fairly insensitive to aspect ratio a/R but can differ from the half-space model by up to 20 percent. This computational approach may be readily extended to account for more complex geometries or inhomogeneities in cellular properties.     

A biomechanical model of the upper airways for simulating laryngoscopy

This paper describes a three-dimensional finite element model of the human upper airways during rigid laryngoscopy. In this procedure, an anaesthetist uses a rigid blade to displace and compress the tongue of the patient, and then inserts a tube into the larynx to allow controlled ventilation of the lungs during an operation. A realistic model of the main biomechanical aspects involved would help anaesthetists in training and in predicting difficult cases in advance. For this purpose, the finite element method was used to model structures such as the tongue, ligaments, larynx, vocal cords, bony landmarks, laryngoscope blade, and their inter-relationships, based on data extracted from X-ray, MRI, and photographic records. The model has been used to investigate how the tongue tissue behaves in response to the insertion of the laryngoscope blade, when it is subjected to a variety of loading conditions. In particular, the mechanical behaviour of the soft tissue of the tongue was simulated, from simple linear elastic material to complex non-linear viscoelastic material. The results show that, within a specific set of tongue material parameters, the simulated outcome can be successfully related to the view of the vocal cords achieved during real laryngoscopies on normal subjects, and on artificially induced difficult laryngoscopy, created by extending the upper incisors teeth experimentally.     

An analysis of the adequacy of the asymmetric carrier model for sugar transport

In 1972, Lieb, W. R.; Stein, W. D. (Biochim. Biophys. Acta 265, 187–207) in their review of sugar transport in human erythrocytes concluded that the conventional two-state carrier model was inconsistent with the experimental data available at that time. Since then, other papers have appeared which question the validity of the model. In this paper, we give a brief derivation of the equations describing the two-state carrier model, and analyze the predictions of the model in the classical experiments, i.e. zero-trans, infinite-cis, and equilibrium exchange. We show that the estimate of the half saturatiion constant of 2.8 mM for glucose at the inner face of the human red cell membrane for the infinite-cis procedure reported by Hankin, B. L., Lieb, W. R. and Stein, W. D ((1972) Biochim. Biophys. Acta 288, 114–126) is unreliable. We note that all of the other experimental findings are consistent with the asymmetric carrier model.     

Particle transport modeling in pulmonary airways with high-order elements

Inhaled particles can be either harmful (e.g., smoke, exhaust, viruses) or beneficial (e.g., a therapeutic drug). The accurate and computationally efficient simulation of particle transport and deposition remains a challenge because it requires the simultaneous solution of the Navier-Stokes equations and multiple advection-diffusion mass transport equations when the particles are modeled as multiple mono-dispersed populations. The solution of these equations requires that multiple length scales be resolved since the ratio of advection to diffusion varies among the different equations. Here, the spectral element method is examined because the high-order approximation provides greater flexibility in resolving multiple length scales. The problem geometry is based on the Weibel model A of the human airway for convergence tests and the first three generations of a typical rat airway for experimental validation. Particles in the size range 1 to 100 nm are simulated for deposition results. The particle concentration and flux were determined using meshes of varying coarseness to represent the geometry along with basis polynomials of order 5 to 11. The higher-order elements accurately propagate the short wavelengths contained in the advection-diffusion solution without sacrificing efficiency for the more computationally expensive Navier-Stokes solution. As the diffusion coefficient in the advection-diffusion equation decreases (i.e., particle size increases) the advantages of the spectral elements become apparent for the coupled system.     

A permeation-diffusion-reaction model of gas transport in cellular tissue of plant materials

Gas transport in fruit tissue is governed by both diffusion and permeation. The latter phenomenon is caused by overall pressure gradients which may develop due to the large difference in O(2) and CO(2) diffusivity during controlled atmosphere storage of the fruit. A measurement set-up for tissue permeation based on unsteady-state gas exchange was developed. The gas permeability of pear tissue was determined based on an analytical gas transport model. The overall gas transport in pear tissue samples was validated using a finite element model describing simultaneous O(2), CO(2), and N(2) gas transport, taking into account O(2) consumption and CO(2) production due to respiration. The results showed that the model described the experimentally determined permeability of N(2) very well. The average experimentally determined values for permeation of skin, cortex samples, and the vascular bundle samples were (2.17+/-1.71)x10(-19) m(2), (2.35+/-1.96)x10(-19) m(2), and (4.51+/-3.12)x10(-17) m(2), respectively. The permeation-diffusion-reaction model can be applied to study gas transport in intact pears in relation to product quality.     

An analysis of the adequacy of the asymmetric carrier model for sugar transport.

In 1972, Lieb, W. R. and Stein, W. D. (Biochim, Biophys. Acta 265, 187-207) in their review of sugar transport in human erythrocytes concluded that the conventional two-state carrier model was inconsistent with the experimental data available at that time. Since then, other papers have appeared which question the validity of the model. In this paper, we give a brief derivation of the equations describing the two-state carrier model, and analyze the predictions of the model in the classical experiments, i.e. zero-trans, infinite-cis, and equilibrium exchange. We show that the estimate of the half saturation constant of 2.8 mM for glucose at the inner face of the human red cell membrane for the infinite-cis procedure reported by Hankin, B.L., Liev, W.R. and Stein, W.D ((1972) Biochim. Biophys. Acta 288, 114-126) is unreliable. We note that all of the other experimental findings are consistent with the asymmetric carrier model.