* Experimental Toxicology Division, National Health and Environmental Effects Research Laboratory, Office of Research and Development, U.S. Environmental Protection Agency, Research Triangle Park, North Carolina 27711; and
CIIT Centers for Health Research, P.O. Box 12137, 6 Davis Drive, Research Triangle Park, North Carolina 27709
Received March 26, 2001; accepted August 4, 2001
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ABSTRACT |
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Key Words: formaldehyde; upper respiratory tract; respiratory tract; dosimetry modeling; computational fluid dynamics; mass transport; nasal airway.
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INTRODUCTION |
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Since the lowest level (6 ppm) of formaldehyde that induced SCC in rats was less than an order of magnitude greater than some past occupational exposures, there was significant concern about potential carcinogenic risk to humans. Since the early 1980s, various groups have made qualitative or quantitative assessments of formaldehyde carcinogenicity in humans. For example, the International Agency for Research on Cancer (IARC) first published an assessment of formaldehyde in 1982 and subsequently updated their assessment in 1987 and in 1995 (IARC, 1982, 1987
, 1995
). IARC classifies formaldehyde as probably carcinogenic to humans.
Other groups, such as the U.S. Environmental Protection Agency (U.S. EPA) and Health and Welfare Canada (1987) have developed quantitative risk assessments for formaldehyde. Knowledge of the nature of the exposure-dose-response relationship is integral to quantitative risk assessments. Representations of this relationship have evolved over time as additional knowledge has been brought to bear. In the first quantitative assessment of cancer risk by the U.S. EPA (1987), dose was represented by the atmospheric concentration of formaldehyde. The incidence of nasal SCC in rats from the Kerns et al. (1983) study was the response of interest. The dose-response curve was represented by the linearized multistage model (LMS), an empirical model that does not incorporate any mechanistic data. Significant uncertainty in risk assessment estimates obtained in this earlier approach arose both from the lack of data on interspecies differences in dose and the lack of knowledge of a mechanistic basis for the dose-response curve.
In the early 1990s, the U.S. EPA (1991) developed a risk assessment for the carcinogenicity of formaldehyde that used DNAprotein cross-links (DPX) formed by formaldehyde as an indirect measure of dose. DPX data were available in rats and monkeys. Using DPX data helped decrease uncertainties arising from species extrapolation given that the monkey nose is more similar in structure to the human nose than is the rat nose. The unit risks developed by EPA based on rat and monkey DPX data were 6-fold and 50-fold lower, respectively, than the unit risk calculated in the agency's 1987 assessment.
Concerns about the use of DPX based upon acute exposures of rodents to formaldehyde to represent dose when chronic exposures of people are of interest led to additional research on DPX formation with repeated exposures to formaldehyde. In 1994, Casanova and coworkers showed that rats exposed to a single 3-h exposure of 0.7 or 2 ppm formaldehyde had levels of DPX similar to those in rats exposed subchronically (6 h/day, 5 days/week for 59 exposures). At higher exposure levels, the yield of DPX in chronically exposed animals was about half that of naive rats. These results supported the use of DPX data from acute or chronic formaldehyde exposure studies in the development of risk estimates within the range of current human exposures.
In the meantime, the U.S. EPA (1991) updated their risk assessment using DPX data for both rats and monkeys but continued to use the LMS model. Risk estimates in the 1991 EPA assessment were about 6-fold and 50-fold lower than those obtained in 1987 depending on whether the rat or monkey DPX data were used, respectively. EPA used DPX as an internal dosimeter to help decrease the uncertainty associated with interspecies extrapolation. Despite the highly nonlinear relationship between DPX and tumor incidence, concern about the possible mutagenic potential of formaldehyde at low exposure levels was a major factor in the EPA's continued use of the LMS model.
DPX observed at proximal portions of the rhesus monkey lower respiratory tract (Casanova et al., 1991) suggested that, in addition to the upper respiratory tract, the lower respiratory tract may be at risk. In addition, some epidemiologic studies (Blair et al., 1986
, 1990
; Gardner et al., 1993
) reported an increase in lung cancer associated with formaldehyde exposure while others reported no such increases (Collins et al., 1997
; Stayner et al., 1988
). Thus, an assessment of potential risk from formaldehyde exposure based on dosimetry information throughout the respiratory tract is appropriate.
Formaldehyde dosimetry models needed to be developed for humans to underpin a biologically based, 2-stage clonal growth model for formaldehyde carcinogenicity. Researchers at the Chemical Industry Institute of Toxicology (CIIT) Centers for Health Research have developed a 3-dimensional, anatomically accurate CFD model for nasal airflow in the human (Subramaniam et al., 1998). This model has been used to estimate the dose or flux (rate of transport) of formaldehyde into tissue at specific locations in the nasal region (Kimbell et al., 2001a
,b
). In the present work, we describe a 1-dimensional dosimetry model developed for the uptake of formaldehyde in the respiratory tract of humans and present predicted fluxes that can be used (along with the nasal fluxes of Kimbell et al., 2001b
) in a formaldehyde risk assessment.
The dosimetry model presented estimates the flux of formaldehyde to tissue in each airway passage, airway, and airspace of the respiratory tract by the use of 1-dimensional mass transport equations applied to a Weibel-type anatomical model of the lower respiratory tract, augmented by an upper respiratory tract. This 1-dimensional dosimetry model was calibrated so that results for the 1-dimensional nasal airway agreed with the CFD model predictions of Kimbell et al. (2001b) for a 3-dimensional nasal airway model during inspiration. Since humans switch to oronasal breathing when ventilatory demand exceeds about 35 l/min (Niinimaa et al., 1980, 1981
), a portion of the inhaled air is not filtered by the nose, thereby reducing nasal airway flux, but increasing penetration of formaldehyde to the lower respiratory tract. Thus, the 1-dimensional dosimetry model was developed in such a way that oronasal breathing can be taken into account. This capability is critical because a risk assessment for humans needs to account for the various breathing patterns that are associated with human activity levels. For the present work, flux to tissue is determined for 4 daily activity levels.
For a risk assessment, regional flux predictions would consist of the nasal fluxes described by Kimbell et al. (2001b) and the 1-dimensional predictions for the rest of the respiratory tract (RT), presented here. This allows for another iteration of quantitative risk estimation that maximally incorporates toxicological, mechanistic, and dosimetric data on formaldehyde to develop a biologically based dose-response model. Further, cancer risk estimates, based on these regional flux predictions, will incorporate significantly less uncertainty in interspecies extrapolation of response than previous risk assessments.
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METHODS |
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Definition of hypothetical human and activity states.
The exposed human was defined as an adult Caucasian male having a height, age, and weight of 176 cm, 30 years, and 73 kg, respectively (ICRP66, 1994). This definition determined the choice of other characteristics such as ventilation and respiratory tract anatomical dimensions. Depending on a person's occupation or daily routine, time will be spent at different levels of exertion; further, depending on the occupation, the time spent at similar levels of exertion generally will be different. Different activities or levels of exertion can result in substantially different doses of inhaled formaldehyde to respiratory tract tissues. In order to account for variations in dose due to occupation and/or daily routine, simulations were carried out for 4 exertion levels or activity states given by ICRP66 (1994) for the hypothetical adult male. The activities are given in Table 1
, along with minute volume, inspired nasal airflow rate, tidal volumes, breathing frequency, and other information, all of which were needed to perform a simulation for each activity state.
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Computational fluid dynamics nasal passage model.
CFD modeling of the nasal passages is described in detail elsewhere (Kimbell et al., 2001a,b
; Subramaniam et al., 1998
). A brief summary is given here. Local nasal flux of formaldehyde (HCHO) from inhaled air to the air-lining interface was estimated from CFD steady-state simulations of unidirectional inspiratory airflow and HCHO transport equations conducted using a 3-dimensional, anatomically accurate reconstruction of the nasal passages of an adult human male.
The transport of inhaled formaldehyde in the nasal passages was assumed to occur by inspiratory airflow, molecular diffusion, and airway surface absorption. Airflow velocities were estimated from simulations of steady-state inspiratory airflow using the human nasal CFD model. The airflow simulations required the numerical solution of the Navier-Stokes equations of motion for incompressible Newtonian fluid flow.
Airflow and uptake simulations were conducted for nasal inspiratory rates of 15 and 18 l/min, but simulations at 46 and 50 l/min were beyond available computer resources (Kimbell et al., 2001b). Thus, additional CFD simulations were performed at intermediate inspiratory rates and the uptake results were extrapolated to the higher flow rates. Kimbell et al. (2001b) reported predicted nasal HCHO uptake values as 75.9, 71.8, 56.4, and 55.2% at nasal inspiratory rates of 15.0, 18.0, 46, and 50 l/min, respectively (corresponding to minute volumes of 7.5, 9.0, 50, and 25 l/min).
Identical-Path Respiratory Tract Dosimetry Model
Anatomical model.
Ideally, an anatomical model of the respiratory tract should simulate all of the paths (each being unique) from the upper respiratory tract (URT) entrance to the most distal airspaces. Unfortunately, use of such a model for gases is beyond our present calculational resources. Therefore, anatomical characteristics of the respiratory tract are represented by a single-path, symmetric, or an identical-path anatomical model. For the purpose of this investigation, these types of models are equivalent to the assumption that all paths are identical. Thus, for a given generation or model segment, the dimensions of one airway or airspace and the number of airways or airspaces in the generation completely define the characteristics of the given anatomical model generation. As a consequence, only one path needs to be considered, making the respiratory tract dosimetry modeling of HCHO feasible. The use of identical-path type anatomical models is well established for respiratory tract dosimetry modeling (e.g., Miller et al., 1985; Overton et al., 1996
; Paiva, 1973
; Scherer et al., 1972
, 1988
).
URT model segments are identified by negative integers (Table 2) that increase distally from the nose or mouth entrance to the last URT segment (1) that is just proximal to the trachea, defined as generation 0. Beginning with the trachea and proceeding distally, lower respiratory tract generations are identified by increasing integers. Thus, model segments from the beginning of the respiratory tract to the alveolar sacs are identified by a continuous sequence of increasing integers. Although only the lower respiratory tract (LRT) model segments correspond to generations, URT segments will often be referred to as "generations" for convenience.
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The main URT segments correspond to the "proximal" and "distal" URT. The proximal URT (segment 2) represents either the mouth cavity or the (identical-path) nasal airway passage, but not both. The distal URT (segment 1) is defined as the air passage distal to both the nasal airway and the mouth cavity; it begins at the proximal oropharynx and ends at the proximal end of the trachea. For the mouth cavity and the distal URT, the choice of a constant diameter geometry is consistent with Fredberg et al. (1980). The rationale for also using this simple geometry for the nasal airways relates to use of the CFD predictions, and not the identical-path model predictions, to define local fluxes within the nasal airways; thus, the identical-path model only needs to remove HCHO from the inhaled air at the same rates as predicted by the 3-dimensional CFD model simulations; for this, a detailed structure is not necessary.
The geometry and dimensions of the LRT (Table 3) are based on the identical-path anatomical model of Weibel (1963), with the dimensions isotropically scaled to correspond to regional volumes based on ICRP66 (1994), Hart et al. (1963), and Fredberg et al. (1980). The Weibel model is equivalent to a symmetric branching structure such that each airway or duct bifurcates into identical daughters, with each generation having twice as many airways or ducts and alveoli as the previous generation.
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Mass transport model.
In each model segment, HCHO transport and uptake are approximated by a 1-dimensional convection-dispersion equation that accounts for inhalation and exhalation, expansion and contraction, convection, dispersion, molecular diffusion, and absorption at the air-liquid lining surface. Inhalation and exhalation times are assumed equal; airflow rates are constant and equal to twice the minute volume during both inhalation and exhalation; i.e., the airflow rate is in the form of a square wave. The rationale for this latter assumption is to maintain consistency with the CFD simulations, which used constant airflow rates. A mathematical description of the model is given in the Appendix.
In order to account for oronasal breathing in the identical-path model, two simulations were used. In 1 simulation, the nasal airway model was used for the proximal URT; in the other, the mouth cavity model was used for this region (see Table 2). In these simulations, the airflow rate that would occur in the mouth cavity or in the nasal airways during oronasal breathing was taken into account. The airflow rate in all segments distal to the proximal URT corresponded to that expected for the full minute volume of 50 l/min. The same airflow rate split was assumed for both inhalation and exhalation. For each corresponding segment distal to the proximal URT, the fluxes of HCHO from both simulations were added to obtain the estimated dose for oronasal breathing.
The rationale for the approach to oronasal breathing dosimetry is presented in the Appendix. Also in the Appendix, issues concerning the accuracy of the numerical procedure used to solve the equation of mass transport, solution convergence, and mass conservation are discussed.
Calibration of identical-path nasal passage uptake to CFD nasal passage uptake.
For the identical-path simulation results to be meaningful, the percent uptake predictions for the identical-path nasal airway were required to agree with the CFD predictions. To obtain this, overall mass transfer coefficients were estimated for the identical-path nasal airway (segment 2, Table 2), so that percent uptakes were in agreement with the CFD results.
The identical-path dosimetry model was used to simulate the CFD results, but the model had the following modifications: (1) Except for the nasal airway model, uptake in all other model segments was required to be 0. (2) Parameters were set to obtain the correct inhaled airflow rates and to obtain steady state by the end of the inhalation phase (which was made long as necessary to obtain steady state). Simulations were carried out for different values of the identical-path nasal airways overall mass transport coefficients until a value was found for which the percent uptake at steady state matched the appropriate CFD simulation value to within 0.2%. The resulting overall identical-path nasal airway mass transfer coefficients, corresponding to minute volumes of 7.5, 9.0, 25, and 50 l/min (nasal steady-state inspired rates of 15, 18, 50, and 46 l/min), were 1.68, 1.78, 2.98, and 2.83 cm/s, respectively.
As a check on the agreement between the 1-dimensional dosimetry calibration and CFD simulation results, the predicted average nasal airway surface fluxes were compared. Table 4 presents the CFD model and calibration results for comparison in columns 3 and 4, respectively; these results indicate that the CFD predictions and identical-path average nasal passage surface fluxes differ by less than 0.7% (remaining table entries are discussed later).
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RESULTS |
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For the identical-path model, calibration results and nasal uptake during the inhalation phase of a complete breath are presented in Table 4 for comparison. The calibration surface fluxes (column 4) are approximately twice the fluxes during inhalation for which the averaging time is the breath time (Table 4
, column 5). However, since the mass was absorbed during inhalation, which is assumed to be one-half the breath time, the average surface flux based on the inhalation time is twice that in column 5. Based on this time averaging, the calibration results (column 4) and the surface flux during inhalation (twice the values in column 5) differ by less than 3% for nasal breathing and about 7% for oronasal breathing. That they differ is expected because the calibration simulations and the simulations that resulted in column 5 were carried out with different conditions (i.e., no absorption distal to the nasal airways occurred for the calibration simulations, whereas for the other identical-path simulations, the distal airways did absorb).
For each model generation, Table 5 lists the average surface fluxes predicted by the identical-path dosimetry model as well as the results of the two simulations that were combined to estimate the oronasal breathing fluxes. In this table and in Figure 1
the URT segments designated in Table 2
as numerical segments (3) have been omitted (their purpose is calculational and they do not absorb HCHO). In Table 5
, column 1 (columns are indicated in top row of table) specifies the model generation number. Columns 2, 3, 4, and 5 are the fluxes predicted for the 4 ventilation states. Column 5 is also the sum of the values in columns 7 and 9, as indicated by the equal and plus signs in columns 6 and 8, respectively. Columns 7 and 9 are the fluxes of the nasal and oral path simulations that were combined to estimate the oronasal fluxes given in column 5. "N/a" is placed in column 5, generation 2, because there are 2 proximal URT fluxes for oronasal breathing, the nasal airway and the mouth cavity fluxes; these fluxes are listed for generation 2 in columns 7 and 9, respectively.
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From Figure 1 and Table 4
we observe the following: Except for oronasal breathing, the highest surface fluxes are predicted for the nasal airways and mouth cavity. As minute volume increases, HCHO penetrates further into the respiratory tract. For nasal breathing, the average nasal airway flux is at least 2.5 to 3 times greater than predicted for any other model segment or generation. During oronasal breathing (50 l/min minute volume), 46% of the inhaled air flows through the nasal airways (ICRP66, 1994
), which corresponds to a rate that is less than for the 25 l/min nasal breathing state. This results in nasal flux during oronasal breathing being less than that for the 25 l/min minute volume nasal breathing state. The peak fluxes that occur at generation 3 are due to 2 competing factors. From generation 1 distally, the overall mass transfer coefficient increases, which tends to increase flux distally. On the other hand, absorption by proximal generations reduces the mass available for absorption. Hence, at some point, the fluxes must decrease, which occurs in generation 3 in the current simulations. The distal decrease in flux is such that the fluxes to pulmonary region surfaces are predicted to be several orders of magnitude smaller than the maximum lower respiratory tract fluxes.
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DISCUSSION |
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Cyclic breathing simulations conducted using the identical-path model supported the use of CFD steady-state inspiratory dosimetry simulation results as a basis to estimate nasal uptake of formaldehyde during cyclic breathing. This conclusion is based on the identical-path model predictions that during exhalation nasal airway fluxes increase by less than 3% over fluxes during inspiration. Thus, the assumption that the use of the CFD predictions contributes insignificantly to error in estimating nasal flux for a complete breath is reasonable.
The calibration of the identical-path model to the CFD predictions resulted in identical-path model predictions of nasal airway % uptake that were within 0.2% of the CFD model predictions and in average surface fluxes that were within 0.7% of CFD model predictions. Thus, any contribution to uncertainty from the assumption that these two models were appropriately matched is considered low.
Quantification of anatomical sources of uncertainty involves determining the extent to which normal variation in these model inputs affects flux. Based on Overton et al. (1996), we infer that for ± 2 standard deviations in dead space volume the pulmonary region fluxes could be 60% less to 180% more than the flux predicted for an average dead space volume. These investigators also showed that the sensitivity of the fractional uptake due to change in dead space volume decreases as the TB region mass transfer coefficient increases, but the sensitivity of the pulmonary region uptake increases. Since the HCHO mass transfer coefficients are at least 4 times as large as those used by Overton et al. (1996) for ozone, we infer that uncertainty in the dead space volume has a negligible effect on fractional uptake, but can affect the pulmonary region distribution of flux.
Another anatomical issue is the assumption of identical paths. As this assumption does not strictly hold in reality, there would be in each generation a distribution of fluxes due to the different paths and different airways of the human LRT. An estimated upper limit to which flux may vary in a given generation may be inferred from Overton and Graham (1995) who simulated ozone uptake in the asymmetric branching airways of rats. They predicted that the ratio of the highest to lowest flux in the same TB generation could be as high as 7, the average ratio being 3. Since the human LRT is considered much more symmetric than that of the rat (Phalen and Oldham, 1983), flux variation among airways in a human TB generation should be less than for the rat.
In the dosimetry model, gas phase mass transfer coefficients and dispersion coefficients were based on formulas derived by investigators using conditions often different than those in the present work. For example, the URT gas phase mass transfer coefficients given by Nuckols (1981) are not defined locally, as used as in the present investigation; they are representative coefficients for the regions from the nostrils or lips to the trachea. Further the values of the air speed and diameter (Appendix; Equations 11 and 12
) used to calculate the transfer coefficient are defined by Nuckols (1981) to be those of the trachea. We used local values for these parameters, which is reasonable, since the mass transfer coefficient is positively correlated with air speed and diameter. A method for converting formulas and their predictions obtained under one set of conditions to different conditions would be useful.
Respiratory tract air was approximated as being dry with no contaminants other than HCHO. In reality, respiratory tract air contains water, carbon dioxide, and inhaled contaminants that may affect the transport of HCHO. Under normal conditions, gaseous contaminants are generally at low enough partial pressures to have little effect on the mass transfer of HCHO. Conceivably, aerosols could be a source of HCHO. In addition, moving throughout the airways, aerosols have the potential to absorb and desorb HCHO, altering the distribution of HCHO uptake. The absorption or desorption of water vapor at the air-liquid lining interface may enhance or reduce, respectively, the transfer of HCHO to this interface. Although the inclusion of these processes are beyond the scope of this investigation, we consider the assumption of dry air with no contaminants as being a reasonable approximation.
Overall, the use of local respiratory tract dosimetry modeling in calculating human cancer risk estimates should represent a significant reduction in uncertainty over previous risk assessments. The 1987 and 1991 formaldehyde cancer risk assessments (U.S. EPA, 1987U.S. EPA, 1991) were based on the linearized multistage model using as measures of dose the inhaled concentration of formaldehyde and DPX, respectively. Major uncertainties in these risk assessments arose from the assumptions that uneven distribution of inhaled formaldehyde over the nasal surface and differences in these distributions between rats and humans were not consequential. Site-specific nasal lesions observed in rats and primates (Monticello et al., 1989, 1991
) suggest that there is significant local species-specific variation in formaldehyde. The same can be inferred about the LRT.
In summary, HCHO, which is used widely and extensively in various manufacturing processes, has been shown to be a nasal carcinogen in rats and mice. Further, studies in monkeys suggest that the LRT may be at risk and some epidemiological studies have reported an increase in lung cancer associated with HCHO, whereas other studies have not. Thus, an assessment of possible human risk to HCHO exposure throughout the respiratory tract is desirable, requiring dosimetry information in the RT. To this end, two types of dosimetry models were used to provide predictions of local HCHO surface fluxes (dose). The first type of dosimetry model is based on a 1-dimensional equation of mass transport applied to each generation airway and airway passage of a symmetric, bifurcating RT anatomical model. This model is the subject of the present investigation. For the second type, CFD techniques were used to estimate total uptake and local surface fluxes in a 3-dimensional model of the nasal region (Kimbell et al., 2001a,b
). The 1-dimensional model was calibrated so that the predicted uptake in its nasal airways model agreed with the human nasal region uptake results of the CFD simulations of Kimbell et al. (2001b). Simulations for both model types were carried out for an adult human male and 4 activity states (4 different sets of ventilatory parameters). The two types of modeling approaches were made consistent by requiring that the 1-dimensional version of the nasal passages have the same uptake during inspiration as the CFD results for each human activity level. Thus, the estimated surface fluxes for the human are defined as the CFD predictions for the nasal region plus the 1-dimensional model predictions of local flux in the rest of the RT.
Results obtained include the following: (1) For each activity state, more than 95% of the inhaled HCHO is predicted to be retained by the RT. (2) The CFD predictions for inspiration, modified to account for the difference in inspiration and complete breath times, are predicted by the 1-dimensional to be a good approximation to uptake by the nasal airways during a single breath. (3) In the lower respiratory tract, flux is predicted to increase for several generations and then decrease rapidly. (4) Compared to first pulmonary region generation fluxes, the first few tracheobronchial generations fluxes are over 1000 times larger. Further, there essentially is no flux in the alveolar sacs. (5) The predicted fluxes, based on the 1-dimensional model that are presented here, can be used in a biologically based dose-response model for human carcinogenesis and should reduce uncertainty in a risk assessment.
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APPENDIX |
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Identical-Path Model of Mass Transport
The identical-path anatomical model is divided into sequential segments in which anatomical dimensions are, in general, a function of time throughout each breath. In this work, model segments correspond to airways or airspaces; also, there is a one-to-one correspondence between a LRT model segment and each airway or airspace in a specific LRT generation. Further, each airway or airspace in the same generation or model segment is assumed to have the same transport characteristics, such as airflow rate, effective dispersion coefficient, and the overall mass transfer coefficient.
Equation of mass transport for one model segment or airway.
For an expanding or contracting airway, air passage, or airspace, a coordinate system is defined such that the x-axis is along the direction of the mean air flow and the origin is at the entrance of the airway (for simplicity, the term "airway" is also used for air passage and airspace). The cross-sectional area of the airway, A(t), is perpendicular to the x-axis and is independent of x for the given airway, but is a function of time, t, due to expansion and contraction. The airway length is L(t).
For the mass balance equation, a small compartment in the airway is considered. The left and right boundaries of this compartment, which are perpendicular to the x-axis, are located at x = z(t) and z + l(t), respectively; where
![]() | ((1a)) |
![]() | ((1b)) |
![]() | ((2)) |
Expanding Jz+l in a Taylor series about z in terms of l and using Equation 1b, Equation 2
can be written as
![]() | ((3)) |
![]() | ((4)) |
![]() | ((5)) |
Replacing J in Equation 4 with the right-hand side of Equation 5
and rearranging, we obtain the spatially 1-dimensional convection-dispersion equation that describes the transport of HCHO in each model segment or airway of the RT,
![]() | ((6)) |
The functional dependence of u on z can be determined by considering C in Equation 6, not as HCHO, but as the concentration of an incompressible gas; i.e., air. In this case,
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![]() | ((7)) |
![]() | ((8)) |
The dispersion coefficient is defined as
![]() | ((9)) |
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The overall mass transfer coefficient, Ko (see Equations 2 and 6
), depends on the gas phase mass transfer coefficient (kg) and the mass transfer coefficient of the mucus-coated tissue (km). For the identical-path dosimetry modeling, Ko can be written as (Overton et al., 1987
):
![]() | ((10)) |
The gas phase mass transfer coefficient (kg) is dependent on air speed, the segment's hydraulic diameter (d = 4 x volume/[surface area]), molecular diffusion coefficient, and the inhalation or exhalation state:
![]() | ((11)) |
Nuckols (1981) did not conduct oronasal breathing experiments. We assume for the nasal airway and mouth cavity that the appropriate kg is that for nasal and mouth breathing, respectively, regardless of breathing mode. For the distal URT, which receives airflow from both the nose and mouth during oronasal breathing, the "Nuckols" term in Equation 11 is approximated by the geometric mean of the term for oral breathing and the term for nasal breathing:
![]() | ((12)) |
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At the beginning of the first breath of a simulation (t = 0) the air phase concentration is 0 everywhere except at the entrance, where it is equal to the exposure concentration. Simulations are carried out for several breaths until the solution is periodic. Results from periodic solutions are reported.
Numerical formulation.
Equation 6 is solved using finite difference techniques. For the interior points of a segment, spatial derivatives are defined in terms of central differences; thus, Equation 6 becomes
![]() | ((13)) |
At the boundary points of a segment/generation the following equations are used for the left- and right-hand boundaries, respectively:
![]() | ((14)) |
![]() | ((15)) |
The set of first order differential equations obtained as described above were solved numerically using Euler integration. To decrease the effect of the artificial diffusion that results from using first order time integration, the approach of Owen (1984) was used to redefine the dispersion coefficient for numerical integration by replacing D in Equations 13, 14
, and 15
with Dc:
![]() | ((16)) |
Rationale for Using the Results of Combining 2 Simulations to Approximate Dosimetry for Oronasal Breathing
The rationale behind the method of simulating uptake during oronasal breathing is based on 3 assumptions: (1) Assumption 1 is based on the principle of superposition, which (for our purpose) states that if concentrations C1(x,t) and C2(x,t) are each a solution to Equation 6 with the same set of parameters, then (C1(x,t) + C2(x,t)) is also a solution. We assume that this principal applies to the HCHO molecules breathed in through the nose and to those entering by the mouth. That is, in the RT regions distal to the two proximal URTs, nose and mouth molecules can be treated as being independent of each other, their concentrations can be determined independently, and the total concentration at any position and time can be found by adding the two concentrations. For the two sets of molecules, the instantaneous flux to surfaces is given by J1 = K0 x C1(x,t) and J2 = K0 x C2(x,t), and the total flux due to both sets of molecules is J = J1 + J2 = K0 x (C1(x,t) + C2(x,t)). Thus, the total flux at any position and time can be found by adding the two independent fluxes. (2) In the 2-simulation approach, HCHO inhaled through the nose is delivered directly to the proximal URT and the HCHO inhaled through the mouth is delivered directly to the proximal URT and no exchange of HCHO between the nasal airway and the mouth cavity is possible. In actual oronasal breathing, an exchange of HCHO between the mouth cavity and the nasal airway most probably occurs. Thus, we assume that during inhalation, any exchange of HCHO between the nasal airway and the mouth cavity is negligible compared to the transfer from each of the proximal URTs to the distal URT. (3) During exhalation in the 2-simulation approach, all the HCHO mass at the proximal URT and distal URT interface flows into the proximal model segment, which is either the nasal passage or the oral cavity, but not both. In reality, the mass would split along the two different proximal URT paths. Thus, for the 2-simulation approach to be reasonable, proximal URT uptake must be negligible during exhalation. This is supported by the simulation results.
Test of Numerical Procedure and Convergence of Solutions for Identical-Path Dosimetry Model
The computer program that solves the equation describing mass transfer and uptake has been tested by comparing results to known analytical solutions (Overton and Graham, 1995). For these tests, a rigid (nonexpanding/contracting) RT "anatomical" model was used for which there are known solutions to the equation of mass transport. Compared to the known solution, the numerical solution was accurate and showed evidence of converging (absolute difference between analytical and calculated solution decreased) to the analytical solution as the spatial step size and the time step decreased. Convergence is an indication that the numerical formulation of the partial differential equation of mass transfer (Equation 6
) is correct.
For testing expansion/contraction, a simulation was performed with no uptake (overall mass transfer coefficients set to 0) for 20 breaths. Results showed that all concentrations within the RT approached the exposure concentration as the number of breaths increased. At the end of 20 breaths, concentrations were essentially constant everywhere in the RT at all times. Although this is a simple and limited test, the results are consistent with expectations.
The present simulations were tested for independence of spatial and time step size and for mass balance. There were 9 different conditions that required simulations, 4 calibration simulations, 4 nasal-path simulations, and 1 oral-path simulation. We chose to perform simulations for spatial step sizes of 1/2, 1/4, and 1/8 of the original size for each of the original 9 simulations. Corresponding to the spatial step size reduction, the time steps were reduced to insure stable solutions by factors of 1/4, 1/16, and 1/64, respectively. Results indicated that further reduction in step sizes would not significantly alter the results.
For a complete breath, the mass balance error is defined as (mass inhaled mass exhaled the mass left in the RT) divided by the mass inhaled. In all 36 of the simulations (the 9 different simulation conditions multiplied by 4, the number of different spatial step sizes), the absolute value of the mass balance error was less than 12 parts in 1 million.
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NOTES |
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This manuscript has been reviewed in accordance with the policy of the National Health and Environmental Effects Research Laboratory, U.S. Environmental Protection Agency, and approved for publication. Approval does not signify that the contents necessarily reflect the views and policies of the Agency, nor does mention of trade names or commercial products constitute endorsement or recommendation for use.
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