* The K. S. Crump Group, ICF Consulting, Research Triangle Park, North Carolina 27709;
University of Delaware, 126 Spencer Lab, Newark, Delaware 19716
Received March 30, 1999; accepted November 1, 1999
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ABSTRACT |
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Key Words: particulate air pollution; dosimetry model; dispersion; deposition; control volume (CV); human airways.
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INTRODUCTION |
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Deposition models can be broadly classified as trumpet models (Taulbee and Yu, 1975) or compartmental models (Anjilvel and Asgharian, 1995
). Trumpet models are single-path models where the lung is approximated by a one-dimensional variable cross-section channel (which resembles a trumpet) with the cross-sectional area being a function of the generation. Trumpet models track particles confined within a small control volume (CV) and simulate the breathing process as the movement of this CV into and out of the channel. These models use a convection-diffusion type differential equation, with appropriate boundary conditions, to compute the transport and deposition of aerosol particles from the CV onto the respiratory tract.
Compartment models are models where each airway unit is represented by a discrete and well-mixed compartment. The lung as a whole is represented by a series of such compartments in a treelike dichotomous structure. Analytical expressions are derived to account for particle loss due to the combined action of impaction, diffusion, and sedimentation in these compartments. The overall particle deposition in the respiratory tract is obtained as the cumulative sum of deposition in each airway unit as the inhaled volume moves sequentially through these compartments. Using a compartmental approach, it is easy to represent an anatomically accurate airway network, with random airway dimensions and inhomogenous ventilation. Such a detailed representation cannot be handled using trumpet models, as individual airway units within a given generation are not differentiated in these models.
Few deposition models take dispersion into account. Edwards (1995) provides a review of the literature. In that work, dispersion in airways was likened to that in packed beds. In dispersion, an aerosol bolus introduced into the lung during inhalation shows a broadening in the concentration profile during exhalation (Brand et al., 1997; Heyder et al., 1988
). Dispersion arises due to a convective mixing process between the inhaled bolus and the residual air in the lung (Heyder et al., 1988
). Aerosol deposition and dispersion are processes that occur concurrently in the airways. As will be shown, dispersion has a strong influence on the regional deposition pattern of the aerosols in the lung.
Many dispersion mechanisms have been proposed (Sarangapani and Wexler, 1999) but few agree with the available experimental data without tuning adjustable parameters. Sarangapani and Wexler (1999) recently proposed a mechanism for dispersion based only on simple fluid mechanical principles. Briefly, dispersion in the human airways is caused by an irreversibility in the particle velocity profile between inhalation and exhalation. This primarily is an outcome of the nature of the secondary fluid flow in a bifurcating geometry. Weak secondary motion during inhalation results in axial streaming of the inhaled particles into the daughter branches, causing inhaled particles to be transported to deeper regions of the lung than would be expected by the mean fluid front (Scherer and Haselton, 1982
). The relative contribution to dispersion of mixing processes and streaming processes during inhalation crucially determines how far a bolus penetrates into the airways and consequently where in the pulmonary airways the particles deposit (Edwards, 1994
; Sarangapani and Wexler, 1999
). Studies on aerosol bolus transport in hollow airway casts by Briant and Lippmann (1992) support this hypothesis. Using 0.5-µm aerosol particles as nondiffusible tracers of convective flow, they have shown that inhaled aerosols penetrate along the axial core as a jet, much deeper than the volumetric depth of inhalation alone would predict.
Compartment models assume that the inhaled particles are confined within the mean fluid front and move with a velocity equal to the mean flow velocity. Such an approximation neglects axial streaming during inhalation and thus does not account for one of the factors leading to dispersion. Similarly, trumpet models also assume that the CV moves with the mean fluid velocity, once again neglecting the effect of axial streaming. However, trumpet models introduce an apparent diffusivity term to account for aerosol dispersion (Taulbee and Yu, 1975). This effective diffusivity is obtained from nitrogen bolus experiments under fully developed laminar flow conditions in a straight tube (Ultman, 1985
). For a gas bolus, augmented diffusion arises due to the coupling between axial convection and radial diffusion. Diffusion will not cause aerosol dispersion because the diffusivity of particles is orders of magnitude lower than that of gases (Sarangapani and Wexler, 1999
). Hence, an effective diffusivity derived from gas bolus experiments cannot be applied directly to model particle dispersion in the human airways. In this paper we develop a deposition model that incorporates dispersion and then study its influence on regional particle dosimetry in the lung.
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MATERIALS AND METHODS |
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A Lung Model
The human lung has large variability in airway dimensions, inhomogenous ventilation, asymmetric branching, and numerous other complications. Particle deposition in the lung may depend on all these morphometric and flow parameters. In this paper we develop a compartment model of the lung to keep track of the particle deposition rate in the individual airway units as the inhaled air traverses the airway network. In a compartmental description, each airway unit is viewed as a discrete compartment connected to others in a treelike structure (Anjilvel and Asgharian, 1995). Using this approach we provide a statistical description of the airway network that captures the variability in dimensions and flow rates as in a real lung. A detailed description of the lung model is provided elsewhere (Sarangapani and Wexler, 1999
). Only a brief model description is provided here.
The most widely used morphometric measure for the human airways has been Weibel's symmetric lung model (Weibel, 1963). The Weibel model represents the lung by a symmetric tree structure with regular dichotomous branching, with all branches of one generation having the same morphometric properties. This is a highly idealized representation based on a relatively small amount of morphometric data. Kaye and Phillips (1997) have shown that using Weibel's data leads to erroneous estimates for the aerosol deposition efficiency in the lung. Geometric parameters of airways are subject to both large intra- and intersubject variations. Yeh and Schum (1980) developed a five-lobed lung model based on the comprehensive morphometric data on the human bronchial tree collected by Raabe et al. (1976).
We employ a four-lobe lung model, a slight variation on Yeh and Schum's five-lobe model. In our model the four airway branches at the end of the second generation open into four lobes: the right upper, right lower, left lower, and the left upper. In order to bring the typical morphometric measures in our four-lobe model closer to the real lung, irregular features are superimposed on the typical airway by randomly picking the dimensions for individual airway units from a normal distribution. This normal distribution for the airway length and diameter has a mean value as listed in Sarangapani and Wexler (1999) and a standard deviation set equal to 10% of this mean value (Koblinger and Hofmann, 1985).
The human lung can be broadly divided into a conducting region and an intra-acinar or pulmonary region (Weibel, 1963). Our lung model comprises 23 generations after the trachea, the first 15 generations forming the conducting airways and the rest the intra-acinar region. In the intra-acinar region, about 25% of the volume is contained in the lumen and 75% in the alveoli (Haefeli-Bleuer and Weibel, 1988
). As the volume inhaled into the lung is completely accommodated by the expansion of the alveolar sacs, the percent expansion of the alveolar sacs can be given as the ratio of the inhaled volume to the total alveolar sac volume. As the inhaled air traverses the intra-acinar region, a small fraction of the inhaled volume is displaced from the lumen into the expanding alveolar sacs and is unavailable for penetration into the latter generations. This loss to the inhaled volume is modeled as a fraction of the percent alveolar expansion, called the fractional displacement (FD). Our model assumes a FD of 20%. Penetration of the inhaled air in the lung depends on both the flow rate and the flow profile in the airways. Although under physiologic breathing conditions the flow in the lung is pulsatile, we assume a steady flow rate in all compartments. For a given flow rate at the trachea, the mean flow rate in each subsequent airway segment is obtained by weighing it to the respective lobar volume into which it opens. The actual flow rate is then derived from a random normal distribution using the above mean values and a prescribed standard deviation. The model ensures mass conservation both during inhalation, when the flow is partitioned between two daughter compartments at a bifurcation, and during exhalation, when flow from the two daughter branches merge at the bifurcation.
Various flow regimes exist in the human airways. Turbulent and transitional flows are prevalent in the upper airways, whereas in the pulmonary region we encounter creeping flow. Moreover, entrance effects as the fluid flows from the parent into the daughter airway will influence the initial flow profile in each generation. The entrance length is the length required for the flow to develop from the entrance profile to a parabolic velocity profile. All the above conditions will affect the centerline to mean particle velocity during inhalation and thereby determine the penetration of the inhaled air. Although the flow may not be fully developed in the conducting airways, we neglect the entrance effects and assume a parabolic velocity profile in generation 014, which gives a ratio of centerline velocity to mean fluid velocity of 2. The presence of alveolar sacs surrounding the lumen leads to a centerline to mean fluid velocity of 1.36 in the pulmonary region (Sarangapani and Wexler, 1999).
Modeling Deposition
Using this multipath compartmental lung model, we have developed a computer algorithm to keep track of particle deposition in each airway unit as the particle-laden inhaled air traverses the airway network. Particle loss in each compartment of the tracheobronchial and pulmonary region is primarily due to impaction, diffusion, and sedimentation. In this section we describe the mathematical formulation to compute particle deposition in each airway compartment during inhalation and exhalation due to the combined action of all the deposition mechanisms. The model assumes that particles are introduced at the trachea and does not account for particle loss in the upper respiratory tract (i.e., nasal cavity, nasopharynx, pharynx, and larynx). Because impaction loss is predominant at airway bifurcations, the model assumes that particle loss due to impaction occurs only at the end of each compartment (i.e., at a bifurcation point), whereas that due to sedimentation and diffusion occur simultaneously over the whole length of the compartment.
During inhalation, particle deposition due to sedimentation and diffusion are computed first, using the flow volume and particle concentration at the compartment entrance. The particle loss due to impaction is computed next, using these quantities at the compartment exit after accounting for the fractional displacement and the decrease in particle concentration by the combined action of the former two mechanisms. Impaction is assumed to be independent of the other two deposition mechanisms. The model uses empirical relations for impaction efficiency during inhalation derived as a function of particle Stokes number. These were obtained using flow experiments in a bifurcating geometry by Kim and Iglesias (1989). Diffusion and sedimentation are assumed to occur concurrently and analytical expressions giving the combined deposition efficiency are used in the model (Appendix).
As stated earlier, the number of particles depositing in a given compartment is a product of the total number of particles entering the compartment and the particle deposition efficiency in the compartment. If Qi is the flow rate entering the ith compartment, Ci is the uniform particle concentration at the compartment entrance, and ti is the time taken by the tip of the fluid front to reach the entrance to the ith compartment, then the total number of particles crossing this compartment during an inhalation time T is . The number of particles depositing due to the combined action of diffusion and sedimentation is
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where imp is the impaction efficiency. The uniform particle concentration entering the next compartment is
.
During exhalation, loss due to impaction at the bifurcation is accounted for prior to the loss due to the combined action of diffusion and sedimentation. If and
are the flow rates in the two daughter compartments and
and
the exit particle concentrations, then the flow averaged particle concentration at the entrance to the parent compartment is
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RESULTS AND DISCUSSION |
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Autopsy studies on morphologically normal nonsmoking adults from the general population have shown high concentration of exogenous fine-mode mineral particles in the apical segmental bronchus of the lung (Churg et al., 1990). Analysis of insoluble particles in bronchoalveolar lavage fluid from non-occupationally exposed individuals (Dumortier et al., 1994
; Falchi et al., 1996
) and a more rigorous count of the retained particles in the apical segments of the lung parenchyma in 10 never-smoking adults using analytical electron microscopy (Churg and Brauer, 1997
) have also shown a predominance on fine-mode particles in the lung. It is important to understand how fine-mode particles are transported and deposited in the apical regions of the pulmonary airways. The effect of dispersion on deposition may help address some of these questions.
The model simulations shown in Figure 2 were conducted with and without accounting for the added penetration of the fluid front due to axial streaming. Incorporating axial streaming increases the overall particle deposition for the coarse mode and the ultrafines and provides a better fit to the experimental measurement for these two size ranges, but the improvement is only marginal for the fine-mode particles (Fig. 2
). We had stated previously that dispersion is a result of both axial streaming during inhalation and radial mixing during exhalation. The flow patterns that give rise to this phenomenon are inherent to a bifurcating geometry. Dispersion causes mixing and transports particles from the inhaled air to the residual volume in the lung. Under normal breathing conditions the inhaled volume is equal to the exhaled volume. For resting-phase breathing (minute ventilation ~7.5 l/min) the inhaled volume over a period of 2 s is about 500 ml; roughly the same amount is also exhaled. Dispersion causes this 500 ml of particle-laden inhaled air to penetrate to a depth of 650 ml at the end of inhalation. As only 500 ml of this particle-laden air is exhaled from the lung at the end of the breathing cycle, a residual volume of 150 ml of particle-laden air is entrained in the lung (Fig. 3
). During the next breathing cycle the effective inhaled volume of particle laden air is 650 ml. This in turn will penetrate axially to a depth of 850 ml, resulting in an particle-laden entrained volume of 350 ml at the end of the second breathing cycle. Qualitatively speaking, over a series of breathing cycles these trapped particles will move distally and finally flood the lung with particles (Scherer and Haselton, 1982
). The total deposited fraction in the lung is then a sum of the particles that deposit on the airway surface by the combined action of the three deposition mechanisms during a ventilation cycle and the particles that are entrained at the end of the cycle and eventually deposit in the lung.
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APPENDIX |
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The empirical relations used in the model for impaction deposition efficiency in terms of particle stokes number are obtained from Kim et al. (1989):
Inhalation:
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Exhalation:
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ACKNOWLEDGMENTS |
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NOTES |
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REFERENCES |
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