The steady-state exchange of inert gases across
an in situ canine trachea has recently been shown to be limited equally
by diffusion and perfusion over a wide range (0.01-350) of blood solubilities (
blood;
ml · ml
1 · atm1).
Hence, we hypothesize that the exchange of ethanol
(blood = 1,756 at 37°C) in
the airways depends on the blood flow rate from the bronchial
circulation. To test this hypothesis, the dynamics of the bronchial
circulation were incorporated into an existing model that describes the
simultaneous exchange of heat, water, and a soluble gas in the airways.
A detailed sensitivity analysis of key model parameters was performed
by using the method of Latin hypercube sampling. The model accurately
predicted a previously reported experimental exhalation profile of
ethanol (R2 = 0.991) as well as the end-exhalation airstream temperature (34.6°C). The model predicts that 27, 29, and 44% of exhaled
ethanol in a single exhalation are derived from the tissues of the
mucosa and submucosa, the bronchial circulation, and the tissue
exterior to the submucosa (which would include the pulmonary
circulation), respectively. Although the concentration of ethanol in
the bronchial capillary decreased during inspiration, the three
key model outputs (end-exhaled ethanol concentration, the slope
of phase III, and end-exhaled temperature) were all statistically
insensitive (P > 0.05) to the
parameters describing the bronchial circulation. In contrast, the model
outputs were all sensitive (P < 0.05) to the thickness of tissue separating the core body conditions
from the bronchial smooth muscle. We conclude that both the bronchial circulation and the pulmonary circulation impact soluble gas exchange when the entire conducting airway tree is considered.
mathematical model; Latin hypercube sampling; ethanol; pulmonary
circulation; airways
 |
INTRODUCTION |
GAS EXCHANGE EFFICIENCY is extremely dependent on the
blood solubility (blood; ml
gas · ml
blood1 · atm1)
of the gas. The major effort in respiratory physiology over the past
four decades has been to characterize the exchange of gases with low
(blood <0.1) -to-intermediate
(0.1 < blood < 100) blood
solubility. This effort stemmed from the intermediate solubilities of
the respiratory gases (blood
for O2 = 0.7 and blood for
CO2 = 3). However,
the lungs exchange a wide variety of gases that range from low
solubility, such as sulfurhexafluoride or helium
(blood = 0.01), to high
solubility, such as water vapor (blood = 20,000).
The exchange of low- and intermediate-solubility gases occurs
predominantly in the alveolar regions, with the airways providing a
conduit for movement of gas between the alveoli and the ambient air. In
contrast, the exchange of highly soluble gases (blood >100) occurs primarily
within the conducting airways (1, 6, 30, 31).
The absorption-desorption dynamics of a soluble gas are difficult to
evaluate because of the relative inaccessibility of the airways to
direct experimental measurement. A two-dimensional model of the airways
previously developed in this laboratory and by others (12, 37)
describes the simultaneous exchange of heat, water, and a highly
soluble gas with the pulmonary airways and represents an avenue to
understanding the exchange process. The soluble gas used in the model
simulations is ethyl alcohol because of its high water and blood
solubility (blood = 1,756) and
because of its important applications in the medicolegal arena. The
performance of the model has been compared with axial profiles of air
temperature available in the literature (37) as well as exhalation
ethanol profiles from human subjects (12). In these simulations, the
bronchial capillary bed was assumed to be an infinite source/sink for
ethanol and heat (i.e., no perfusion dependence). Most recently,
experimental and theoretical data suggest that the exchange of gases
spanning a wide range of solubilities (0.01 < blood < 350) demonstrates a
similar perfusion dependence to exchange in the trachea (14, 35). These
results indicate that our previous assumptions related to the bronchial
circulation (infinite sink/source) may not be valid.
In addition, the present model structure lumps the lamina propria and
bronchial epithelium into a single nonperfused layer and does not
include the bronchial smooth muscle as a distinct anatomic layer. Both
the epithelium and the smooth muscle are important anatomic features of
the airways that play critical roles in basic physiology (i.e., mucus
secretion, immune response, airway caliber) and in airway pathology
(i.e., bronchial asthma). Although the epithelium and the smooth muscle
may not be critical to understanding the exchange of inert gases such
as ethanol, they will be important in future airway gas-exchange
simulations involving endogenous gases such as nitric oxide or
pollutant gases such as ozone. Thus the objective of this study is
threefold: 1) to design a more
realistic description of the bronchial circulation for incorporation
into the existing mathematical model;
2) to expand the radial description
of the airway wall to include an epithelial layer, a smooth muscle
layer, and a sink/source that represents the core body; and
3) to perform a detailed sensitivity analysis of the model parameters to determine their relative importance in understanding soluble gas exchange in the airway.
| |
EXPERIMENTAL METHODS |
The experimental methods and data have been previously described and
reported (12). As the focus and goals of this manuscript are
modeling airway gas exchange, only a summary will be presented here.
Six male volunteers without previous history of cardiac or pulmonary
disease and with normal physical examination findings served as
subjects. Each subject ingested enough alcohol in the form of liquor to
achieve a blood alcohol concentration of ~0.09 g/100 ml. After
ingestion of alcohol, the subjects waited ~1 h for absorption to take
place, which was monitored by sequential breath tests.
Ethanol concentration in the exhaled breath was measured with a
commercially available infrared absorption breath-testing instrument
(Intoxilyzer 5000). After passing through the Intoxilyzer 5000, the
exhaled breath entered a wedge spirometer where exhaled volume and flow
rate were measured. Each subject performed a series of
single-exhalation or vital capacity maneuvers where exhalation flow
rate was controlled. In a single-inhalation maneuver, the subject
inhales to total lung capacity then exhales the vital capacity at a
slow constant flow rate to residual volume. The breathing maneuver was
repeated five times, each spaced by ~3 min of quiet nasal tidal
breathing. Blood samples were taken from the antecubital vein at three
points in time after the estimated start of the postabsorptive phase.
Blood alcohol concentration was subsequently measured with a gas
chromatograph (Perkin Elmer model 3920) by using headspace analysis
(21).
For the purposes of this paper, a single representative exhalation
profile from a human subject was of interest to test the overall
performance of the model before the sensitivity analysis. Thus the
exhalation profiles from the six subjects (30 profiles together) were
condensed into a single profile as follows. First, a simple smoothing
routine (average of 10 nearest neighbors) was performed on each exhaled
profile. Next, the expired partial pressure of ethanol
(PE) was normalized by the
concentration of ethanol in the alveolar gas
(PA)
(PA = Cblood/blood,
where Cblood is the measured
venous blood concentration of ethanol). Thus the normalized
concentration of ethanol in the air
(
E)
is plotted as function of exhaled volume (V). The five exhaled profiles
for each subject were then truncated to the smallest exhaled volume of
the group and consolidated into a single profile by taking the mean
E at
one-tenth exhaled volume intervals. Finally, the consolidated profiles
from each subject were combined by averaging the
E
across all subjects. As each subject had a different exhaled volume,
the final representative profile (see Fig. 3) has error bars associated
with each axis.
| |
ANALYTICAL METHODS |
Glossary
| Ac |
Surface area for exchange between the connective tissue or the smooth
muscle and the capillaries
(cm2)
|
| Ad |
Surface area of cylinder of diameter d
and length  z
(cm2)
|
| Ac,s |
Surface area between capillaries and smooth muscle tissue in length
z
(cm2)
|
| Ac,t |
Surface area between capillaries and connective tissue in length
z
(cm2)
|
| blood |
Solubility of gas in blood (ml gas · ml
blood1 · atm1)
|
| b |
Solubility of gas in body-tissue layer (ml gas · ml
blood1 · atm1)
|
| e |
Solubility of gas in epithelium (ml gas · ml
blood1 · atm1)
|
| g |
Solubility of gas in air (1 ml gas · ml
blood1 · atm1
at 1 atm pressure)
|
| ij |
Partial rank correlation coefficient
|
| m |
Solubility of gas in mucous layer (ml gas · ml
blood1 · atm1)
|
| s |
Solubility of gas in smooth muscle layer (ml gas · ml
blood1 · atm1)
|
| t |
Solubility of gas in connective tissue layer (ml
gas · ml
blood1 · atm1)
|
| C |
Molar concentration of tissue (assumed to have the properties of water)
(mol/cm3)
|
| Cblood |
Concentration of ethanol in circulating blood (ml ethanol/ml blood)
|
| Ce |
Molar density of ethanol (mass density divided by molecular weight)
(mol/cm3)
|
 |
Molar heat capacity of dry air
(J · mol1 · K1)
|
 |
Molar heat capacity of ethanol vapor
(J · mol1 · K1)
|
 |
Molar heat capacity of liquid ethanol
(J · mol1 · K1)
|
 |
Molar heat capacity of liquid water
(J · mol1 · K1)
|
 |
Molar heat capacity of water vapor
(J · mol1 · K1)
|
 |
 gp,w  gp,da
(J · mol1 · K1)
|
 |
gp,e gp,da
(J · mol1 · K1)
|
| d |
Diameter of the airway (cm)
|
 |
Scaling factor in the range 1.1-2.5, which increases the surface
area of the airway lumen due to invaginations
|
| De,a |
Diffusivity of ethanol in air
(cm2/s)
|
| De,w |
Diffusivity of ethanol in water
(cm2/s)
|
| De,t |
Diffusivity of ethanol in lung tissue
(cm2/s)
|
 c,t |
Ratio of the surface area of the capillaries to that of a cylinder with
the same radius in the connective tissue
|
| c,s |
Ratio of the surface area of the capillaries to that of a cylinder with
the same radius in the smooth muscle
|
 c,t |
Ratio between the volume of the capillaries and the total volume of
tissue
|
| c,s |
Ratio between the volume of the capillaries and the total volume of
smooth muscle
|
| F |
Weighting factor for bronchial blood flow in connective tissue
|
| F |
Scaling factor to maintain a constant ratio of conducting airway space
to vital capacity
|
| g |
Airway generation number
|
 |
Scaling factor that allows the thickness of the body layer for energy
transfer
(*lb) to
be larger than that of mass transfer (lb)
|
| Hv,e |
Latent heat of evaporation for ethanol (J/mol)
|
| Hv,w |
Latent heat of evaporation for water (J/mol)
|
| hm,a |
Local heat transfer coefficient between the lumen wall and the air
(J · s1 · K1 · m2)
|
| hb,b |
Overall heat transfer coefficient between the body layer and the body
(J · s1 · K1 · m2)
|
| hb,s |
Overall heat transfer coefficient between the body layer and the smooth
muscle
(J · s1 · K1 · m2)
|
| he,m |
Overall heat transfer coefficient between the epithelium and mucus
(J · s1 · K1 · m2)
|
| hs,t |
Overall heat transfer coefficient between the perfusive tissue and
smooth muscle
(J · s1 · K1 · m2)
|
| ht,e |
Overall heat transfer coefficient between the perfusive tissue and
epithelium layer
(J · s1 · K1 · m2)
|
| je |
Molar flux of ethanol from the mucous surface in control volume
(mol · s1 · cm2)
|
| jfluid |
Total molar flux of fluid into the control volume
(mol · s1 · cm2)
|
| Jbody |
Total molar flux of ethanol from the body core during both inspiration
and expiration (mol/breath)
|
| Jbr |
Total molar flux of ethanol from the bronchial circulation during both
inspiration and expiration (mol/breath)
|
| Je,exp |
Total molar flux of ethanol from the mucous surface in an airway
generation during expiration (mol/breath)
|
| Je,insp |
Total molar flux of ethanol from the mucous surface in an airway
generation during expiration (mol/breath)
|
| Jh,exp |
Total molar flux of heat from the mucous surface in an airway
generation during expiration (J/breath)
|
| Jh,insp |
Total molar flux of heat from the mucous surface in an airway
generation during inspiration (J/breath)
|
| Jtiss |
Total molar flux of ethanol from the bronchial mucosa and submucosal
tissue layers during both inspiration and expiration of single
exhalation (mol/breath)
|
| Jw,exp |
Total molar flux of water from the mucous surface in an airway
generation during expiration (mol/breath)
|
| Jw,insp |
Total molar flux of water from the mucous surface in an airway
generation during inspiration (mol/breath)
|
| kem,a |
Local mass transfer coefficient of ethanol from the lung walls to the
airway
(mol · s1 · m2 · mole-fraction1)
|
| kwm,a |
Local mass transfer coefficient of water from the lung wall to the
airway
(mol · s1 · m2 · mole-fraction1)
|
| kb,b |
Overall mass transfer coefficient between the body and the body layer
(cm/s)
|
| kb,s |
Overall mass transfer coefficient between the smooth muscle and the
body layer (cm/s)
|
| ke,m |
Overall mass transfer coefficient between the epithelium layer and the
mucous layer (cm/s)
|
| ks,t |
Overall mass transfer coefficient between the smooth muscle and the
tissue layer (cm/s)
|
| kt,e |
Overall mass transfer coefficient between the epithelium layer and the
tissue layer (cm/s)
|
 w |
Thermal conductivity of water
(J · m1 · K1 · s1)
|
| lb |
Thickness of body layer (cm)
|
| lm |
Thickness of the mucous layer (cm)
|
| le |
Thickness of epithelium (cm)
|
| lt |
Thickness of connective tissue layer (cm)
|
| ls |
Thickness of smooth muscle (cm)
|
 c,t |
Capillary-tissue partition coefficient
|
| s,t |
Smooth muscle-tissue partition coefficient
|
| e,m |
Partition coefficient of ethanol between the epithelium layer and the
mucous layer
|
| m,a |
Partition coefficient of ethanol between mucus and air
|
| Mw |
Molecular weight of water (g/mol)
|
| n |
Compartment number
|
| nc |
Number of capillaries
|
| N |
Molar density of air (mol/l)
|
 |
Molar flow rate of air (mol/s)
|
| Pamb |
Ambient pressure (atm)
|
| Pa |
Arterial partial pressure of ethanol (atm)
|
| PE |
Expired partial pressure of ethanol (atm)
|
| PE,max |
Maximum expired partial pressure of ethanol (atm)
|
| E,max |
Normalized (by arterial partial pressure) maximum expired partial
pressure of ethanol
|
| Pl |
Airway luminal partial pressure of ethanol (atm)
|
| Pe |
Partial pressure of ethanol in the epithelium (atm)
|
| Pm |
Partial pressure of ethanol in the mucus (atm)
|
| Ps |
Partial pressure of ethanol in the smooth muscle (atm)
|
| Pt |
Partial pressure of ethanol in the connective tissue (atm)
|
| Pc,s |
Partial pressure of ethanol in the capillary (venous) of the smooth
muscle (atm)
|
| Pc,t |
Partial pressure of ethanol in the capillary (venous) of the connective
tissue (atm)
|
 |
Normalized blood flow (ml · ml
tissue1 · s1)
|
 br,s |
Bronchial blood flow to a control element of the smooth muscle (ml/s)
|
| br,t |
Bronchial blood flow to a control element of connective tissue (ml/s)
|
 br |
Total bronchial blood flow (1 ml/s)
|
| br,s |
Total bronchial blood flow to the smooth muscle (0.5 ml/s)
|
| br,t |
Total bronchial blood flow to the connective tissue (0.5 ml/s)
|
| r |
Radius of the airway (cm)
|
| rc |
Average radius of a capillary (cm)
|
| R |
Ideal gas constant (8.314 J · K1 · mol1)
|
 w |
Density of water (g/ml)
|
 |
Secretion rate of fluid from the epithelium to mucus
(mol · s1 · cm2)
|
| Tl |
Temperature of air in the control volume of the lumen (K)
|
| Tb |
Average temperature of the body layer (K)
|
| Tbody |
Temperature of the core body and arterial blood (K)
|
| Tc |
Temperature of blood in the capillary (K)
|
| Te |
Average temperature of the epithelium in the control volume (K)
|
| TE |
Expired temperature (K)
|
| TE,max |
Maximum (or end-expired) temperature (K)
|
 E,max |
Normalized (by body temperature) maximum (or end-expired) temperature
|
| Tm |
Average temperature of mucus in the control volume (K)
|
| Ts |
Average temperature of the smooth muscle layer (K)
|
| Tt |
Average temperature of the connective tissue layer (K)
|
| Tc,s |
Temperature of the smooth muscle capillary blood (K)
|
| Tc,t |
Temperature of the connective tissue capillary blood (K)
|
 s |
Residence time of blood in the smooth muscle capillary (s)
|
| t |
Residence time of blood in the connective tissue capillary (s)
|
| V |
Expired volume (ml)
|
| Vee |
Lung volume at end-expiration (ml)
|
| Vei |
Lung volume at end-inspiration (ml)
|
 |
Volumetric flow rate of the airstream (ml/s)
|
| Vc,t |
Volume that capillaries occupy in the control element of the connective
tissue (ml)
|
| Vc,s |
Volume that capillaries occupy in the control element of the smooth
muscle (ml)
|
| Vm |
Volume of control element of mucus (liters)
|
| VT,t |
Total volume of the control element of connective tissue control
element (ml)
|
| Vt |
Volume that tissue mass occupies in the control element (ml)
|
| Xa |
Average mole fraction of ethanol in arterial blood (equal to
Xbody)
|
| Xb |
Average mole fraction of ethanol in the body-tissue layer
|
| Xbody |
Average mole fraction of ethanol in the core body (equal to
Xa)
|
| Xc,t |
Average mole fraction of ethanol in the connective tissue capillary
|
| Xc,s |
Average mole fraction of ethanol in the smooth muscle capillary
|
| Xm |
Average mole fraction of ethanol in mucus
|
| Xe |
Average mole fraction of ethanol in epithelium
|
| Xs |
Average mole fraction of ethanol in the smooth muscle layer
|
| Xt |
Average mole fraction of ethanol in the connective tissue layer
|
| Ye |
Mole fraction of ethanol in air
|
| Ye,wall |
Mole fraction of ethanol at the mucus-air interface
|
| Yw |
Mole fraction of water in the air
|
| Yw,wall |
Mole fraction of water at the mucus-air interface
|
| z |
Length of control element (cm)
|
Lung Model
The mathematical model is described in detail elsewhere (13, 37). The
important new features, including a detailed description of the new
bronchial circulation and additional radial layers, are described in
detail here; the final governing equations are derived, in brief, and
summarized in the APPENDIX. The model
describes the simultaneous exchange of heat, water, and an inert
gas with the airways. The initial detailed description and
sensitivity analysis were described by Tsu et al. (37). Since then, the model has had several modifications, each one adding a new level of
sophistication as our understanding of airway gas exchange has
improved.
Axial structure. The axial structure
of the model is unchanged and consists of a symmetrical bifurcating
structure through eighteen Weibel generations. The respiratory
bronchioles and alveoli are currently lumped together into a single
respiratory unit that is justified for heat and highly soluble gas
exchange. The dimensions (lengths and diameters) of the airways for the
upper respiratory tract (nasal and oral) are taken from Hanna (15) and
those for the lower respiratory tract are from Weibel (39). Because the volume of the conducting airways increases with increasing lung volume,
the dimensions of the lower airways are scaled by using the parameter
F such that the ratio of the volume of
the conducting airways to the vital capacity is maintained constant.
The vital capacity of the lungs used by Weibel was ~5,075 ml; hence,
F is defined as
|
(1)
|
where
VC is the vital capacity of the lungs being simulated. The lengths and
diameters from Weibel's data are then multiplied by
F thus maintaining a constant ratio of
length to diameter as well as a constant ratio of conducting airway
volume (proportional to length × diameter2) to vital capacity.
The upper and lower respiratory tract and airways are divided into 480 axial control volumes, as depicted in Fig.
1, A
and B.
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|
Fig. 1.
Model control volume. A: previous
model that includes only 4 radial layers and assumes that bronchial
capillary bed is an infinite source or sink for heat and ethanol.
B: new model that now includes 7 radial layers and a dynamic description of bronchial circulation to
both lamina propria and smooth muscle, and a body compartment that
represents an infinite sink or source of ethanol and heat. Blood enters
capillary compartments with a partial pressure
(Pa), ethanol then diffuses
across a series of radial resistance before entering the passing
airstream. Approximate linear partial pressure profiles in each radial
compartment are depicted. Mass transfer by diffusion between radial
layers is described by product of an overall transfer coefficient and
the partial pressure difference between compartments at midpoints.
Shaded region in airway lumen represents aerodynamic resistance due to
mucus-air interface. See Glossary for
symbol definitions.
|
|
Radial structure.
GENERAL. The previous radial structure of the airway
control volume is depicted in Fig.
1A and consisted of four
compartments: 1) the airway lumen;
2) a thin layer of mucus;
3) a nonperfused tissue layer that
represents the respiratory epithelium, basement membrane, and any
connective tissue before reaching 4)
the capillary bed of the bronchial circulation. The capillary bed was
considered an infinite source or sink for heat and the soluble gas;
that is, the temperature and concentration of the soluble gas in the
bronchial circulation were fixed.
In the new model (Fig. 1B), the
airways are now divided into seven radial compartments:
1) the airway lumen,
2) a thin mucous layer,
3) the epithelium,
4) a connective tissue layer (i.e., the lamina propria) perfused by the bronchial circulation,
5) the bronchial smooth muscle layer
perfused by the bronchial circulation, 6) a body-tissue layer that acts as
a buffer to heat and mass transport between the core body conditions
and the smooth muscle layer, and 7)
the core-body layer.
Over the majority of the airway tree, the radius of the airway lumen is
much larger than the thickness of the radial layers; thus the surface
area for exchange between the radial compartments within each control
volume is practically constant and approximately equal to
2
rz,
where r is the radius of the airway
lumen and z is the axial length of
the control volume (with the notable exception of the mucus-air
interface, see below). All radial tissue or liquid-phase layers are
considered a dilute binary mixture of water and a soluble gas
(ethanol). Axial diffusion is neglected, except in the gas phase.
Radial transport between the layers occurs by molecular diffusion
(Fick's first law) and secretion and/or filtration (see
below and APPENDIX for specific
details in each layer). The inert-gas concentration and temperature
gradients within each radial layer are considered linear between the
midpoint and the interface of the adjacent compartment(s) (see
APPENDIX).
AIRWAY LUMEN.
The air is considered a system of dry air, water vapor, and a single
inert gas. The small exchange of respiratory gases with the airways is
considered negligible. At ambient pressure and over the range of
temperatures expected within the lung, air behaves as an ideal gas.
Longitudinal or axial diffusive transport is included in the gas phase.
An effective axial diffusion coefficient (De,a,eff) is
used, which reflects the experimental observations of Scherer et al.
(29) that account for enhanced axial diffusion due to secondardy
convective flows induced by the bifurcations
|
(2)
|
|
(3)
|
where
NPe is the Peclet
number
[ud/De,a,
where u is the mean axial velocity of
the airstream (cm/s), d is the airway
diameter, and
De,a is the
molecular diffusion coefficient of ethanol in air at 37°C (0.128 cm2/s)]. Transport between
the mucous layer and gas phase is described with heat and mass transfer
coefficients. The heat transfer coefficient (h), is taken from an
empirical correlation derived by Ingenito (18), and the corresponding
mass transfer coefficient is calculated from the Chilton-Colburn
analogy (5).
During the inspiratory and expiratory phases of respiration, the lung
expands and contracts to draw and expel air from the alveoli. This
results in a slight stretching and compressing of the airway walls. In
addition, during bronchoconstriction there is conservation of the
airway lumen perimeter. As a result, the luminal surface of the airway
wall contains luminal folds or currugations (19). It is unclear from
the report of Weibel (39) whether luminal folds were considered in the
measurement of airway diameters and, hence, in the calculation of
luminal surface area. Thus, is a scaling parameter that accounts
for enhanced surface area due to mucosal folds such that = 1 when
there are no mucosal folds. To attain a rough estimate of , we
empirically examined cross-sectional histological images of the airways
(9) by measuring the perimeter of the airways with and without
consideration of the mucosal folds. It was observed that the degree to
which the wall is corrugated increases as the generation number
increases until approximately the 12th generation. The value of was
found to be ~2.5 in the small airways; was found to be 1.1 at the trachea and was scaled linearly to the 12th generation to a value of
2.5. The value was then held constant at 2.5 for the
remainder of the airway tree. It should also be noted that
incorporating into the surface-area calculation improved the
model's ability to simulate the phase III slope
(SIII; see
RESULTS, Single
exhalation). Derivation of the energy and mass
balance within the airway lumen as well as the remaining layers can be
found in APPENDIX.
MUCUS.
Because mucus is ~95% water, the physical properties of the mucous
layer (subscript m) are equivalent to these of water. A variable mucous
layer thickness is incorporated into the model to account for local
hydration and dehydration. Fluid is secreted into the mucous layer from
the epithelium if the thickness of the mucus falls below a minimum
value. The minimum value
(lm) is 10 µm
in the trachea (25) and is scaled to smaller values in the lower
generations such that the volume of mucus in each generation is
equivalent to that in the trachea. This assumption is based on the
observations that the mucous layer is thinner in smaller airways (32)
and that the volume of mucus in each generation is constantly swept
caudally toward larger airways, including the trachea.
BRONCHIAL EPITHELIUM.
The physical properties of the epithelium (subscript e) are assumed to
be equal to these of water, with the exception of solubility and
diffusivity. Because blood and tissue have similar water and lipid
contents, the tissue is assumed to have the solubility properties of
blood. The diffusion coefficient of ethanol in the respiratory mucosa
has been experimentally determined to be 5.63 × 106
cm2/s, which is approximately
one-third of the diffusion coefficient of ethanol in water (11). The
epithelium secretes fluid to the mucous layer to maintain a minimum
thickness of the mucous. In order for the epithelium to maintain a
constant volume (or thickness), an equimolar volume of fluid enters the
epithelium from the adjacent perfused connective tissue layer. The
thickness of the epithelium (le), is
determined from data of Gastineau et al. (10) (100 µm in the trachea,
20 µm in the bronchioles).
PERFUSED CONNECTIVE TISSUE AND SMOOTH MUSCLE.
All physical properties of the connective tissue (subscript t) and
smooth muscle (subscript s) are assumed to be equal to those of water,
with the exception of solubility and diffusivity, as described above
for the epithelium. Fluid is secreted from the nonperfused tissue layer
to the epithelium, as described above, and replaced by filtration
from the bronchial circulation within the connective tissue layer. No
fluid is secreted from the smooth muscle layer because of its
anatomical distance from the mucous layer relative to the perfused
connective tissue.
The blood flow to the connective tissue and smooth muscle layers is
modeled as an evenly dispersed network of capillaries that supplies
blood at the condition of the body and exits at a new condition, which
is determined by the dynamics of heat and mass transfer. The bronchial
blood flow to the smooth muscle layer (br,s)
is assumed to be equal to that of the connective tissue (br,t) on a
unit volume of tissue basis, such that the the sum of the two
circulations for the entire airway tree
(br) is equal to 1 ml/s [~1% of the cardiac output (23)]. It has been
previously demonstrated that ~90% of the blood flow to the smooth
muscle originates from the bronchial circulation (38). The average radius of the capillary
(rc) is set
equal to 10 µm (22), and the mean residence time () of the blood
in the control volume is set to 1 s (40). The number of capillaries
(nc) necessary to achieve the above stipulations is then calculated as simply the
ratio of the total volume of the capillary bed and the volume of one
capillary
|
(4)
|
where
br is the bronchial
blood flow to each control volume. Once the number of capillaries is
known, the surface area for exchange between the connective tissue or
the smooth muscle and the capillaries
(Ac) is simply
nc(2rcz).
If Eq. 2 is substituted into the
definition of Ac,
then the bronchial circulation in each layer (smooth muscle or
connective tissue) can be described by four parameters
(Ac, ,
rc, and
br) by the
following relationship
|
(5)
|
Hence,
if any three parameters are chosen, the fourth is fixed. In our
analysis, Ac is
calculated by Eq. 5, and the three remaining parameters are incorporated into the sensitivity analysis (Table 1).
Blood entering the connective tissue and smooth muscle layers has a
partial pressure of gas that is equal to
Pa and body temperature (Tbody, 37°C). Blood within
the capillaries is considered well mixed (no axial gradient within each
control volume); it exchanges heat and mass with the tissue compartment
and exits with a partial pressure
Pc (equivalent to venous) and
temperature of blood in the capillary
(Tc).
The axial distribution of blood flow to the smooth muscle is uniform;
that is, each control volume has the identical blood flow on a unit
volume of tissue basis (0.0407 ml blood · ml
tissue1 · s1).
The axial distribution of the remaining bronchial circulation to the
connective tissue layer is described by an exponential dependence on
axial position recently described by Bernard et al. (2). The blood flow
rate to control volume x is defined by
the following relationship
|
(6)
|
where
VT,t(x)
is the total volume of tissue in the control volume
(cm3),
F(x) is a weighting factor given by
Bernard et al. (2), and
br,t is the mean
control volume blood flow on a unit volume of tissue basis (0.0407 ml
blood · ml
tissue1 · s1).
These parameters are defined by the following relationships
|
(7)
|
|
(8)
|
|
(9)
|
where
d(x)
is the airway diameter (mm). Figure 2 plots
F as a function of airway generation. Note that the blood flow per unit
volume of tissue in the upper airways is approximately an order of
magnitude smaller than in the bronchioles and that the mean normalized
blood flow (br,t) occurs at approximately
generation 7 (F 1). Details
of the energy and mass balances within the connective tissue and smooth
muscle leading to the governing equations can be found in
APPENDIX.
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Fig. 2.
Blood flow to connective tissue normalized by tissue volume is
inversely related to airway diameter, based on work by Bernard et al.
(2). F represents tissue-volume-normalized blood flow to each
compartment normalized by mean value of tissue-volume-normalized flow
for entire airway tree. See Glossary
for symbol definitions.
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BODY-TISSUE.
A layer of "body-tissue" (subscript b) was added into the model
as a buffer between the main body compartment and the radial compartments. The body-tissue does not represent a distinct anatomical layer that is present in vivo; rather, the extra layer is a necessary model construct, since the radial distance from the smooth muscle at
which the conditions of the body (i.e., 37°C and
Pa) exist is unknown. The
thickness of the body layer,
lb, represents
the distance from the smooth muscle interface, where the partial
pressure of ethanol and the temperature are constant at
Pa and
Tbody, respectively. Because
thermal diffusivity is approximately two orders of magnitude larger
than the mass diffusivity of ethanol, the effective thickness of the
body layer will be larger for heat transfer. This effect is accounted
for by a scaling factor (), such that the thickness of the body
layer for heat transfer is
lb.
CORE BODY.
The core body layer represents an infinite sink or source of heat and
mass in the model. The partial pressure of ethanol and the temperature
are considered constant at Pa and
37°C, respectively.
Boundary conditions. To calculate
concentrations of the species at the airway wall, local vapor-liquid
equilibrium is assumed at the air-mucus interface. Raoult's law is
applied to water, and w is used
for the soluble gas. The ratio of solubilities between blood and
connective tissue
(blood/t)
and between blood and smooth muscle
(blood/s)
is set equal to unity, and between the epithelium and mucus
(e/m)
is set equal to the
blood/w ratio. At the interface between each of the compartments, it is assumed
that there is no net accumulation of energy or mass such that flux of
energy and mass between adjacent compartments is continuous. During
inspiration, the concentration of the soluble gas in the ambient air is
considered to be zero. The temperature and water content of the
inspired air are user-controlled variables but are held constant during
a simulation. During expiration, the air that leaves the alveoli has
the following properties: 1) it is
fully saturated with water, 2) its
temperature is equal to body temperature, and
3) the partial pressure of the
soluble gas is in equilibrium with the blood as described by
blood.
Computer Simulation
The mass and energy balances produce 16 dependent variables and 2 independent variables (time and position). The
APPENDIX summarizes the sixteen
coupled partial differential equations. The system of partial
differential equations is solved numerically by using a UNIX-based
computer. The spatial derivatives are handled with upstream finite
differencing, whereas the time derivatives are solved using LSODE, a
time-integration software package developed by Hindmarsh (17).
Before simulating a single exhalation maneuver, the model must simulate
30 tidal breaths to reach steady-state conditions for temperature and
concentration profiles in the airway lumen, mucus, and tissue regions
(36). A respiratory rate of 12 breaths/min, a sinusoidal flow waveform,
and a tidal volume approximated as 10% of the subject's vital
capacity (16) were used. For all simulations, inspired air temperature
was 23°C and relative humidity 50%. Inspired volume, expired
volume, inhalation time, and exhalation time must be specified to
simulate a single exhalation maneuver. Inspired volume was determined
based on the assumption that each subject inhaled to total lung
capacity; inspired volume can then be approximated as 65% (16) of the
subject's mean vital capacity (0.65 × 5,400 ml = 3,510 ml).
Expired volume was equal to the mean minimum value for all six subjects
(4,160 ml), as described in the EXPERIMENTAL
METHODS and RESULTS.
Inhalation time was equal to that during tidal breathing (2.5 s), and
inhalation flow rate was assumed constant at a value equal to inspired
volume divided by inspiration time. The exhalation time (20.8 s) for
each condensed single-exhalation maneuver can be determined by dividing
the expired volume by the mean flow rate (200 ml/s) of the experimental exhalation
maneuvers.
Sensitivity Analysis
The method of Latin hypercube sampling (LHS) (27), was chosen to
perform the sensitivity and uncertainty analysis. The advantage of
using LHS rather than Monte Carlo for sensitivity analysis is that it
substantially reduces the number of simulations needed for an adequate
analysis of numerical or computer models (27). LHS has been used
successfully in the field of atmospheric chemistry to analyze the
sensitivity and uncertainty in complex atmospheric models (7).
Table 1 summarizes the characteristics (central values and uncertainty
ranges) of 20 parameters in the model judged to have the greatest
uncertainty or impact on the model output. Broadly, the 20 parameters
include those that describe the bronchial circulation and physical
features of the airways, such as solubility, diffusivity, surface area,
and diffusing distance. The choice of uncertainty ranges is subjective
and based on the method used to obtain the central value. For example,
since there is no information on the parameters
lb and , they
were assigned a high level of uncertainty (±80%), whereas the
diffusivity of ethanol in air,
De,a, and ethanol in tissue, De,t,
which were based on careful experimental measurements, were assigned an
uncertainty range of only ±10%.
To perform the LHS analysis, the model simulates the exhalation profile
two times the number of free or uncertain parameters. Thus, for 20 parameters, the model simulates the exhalation profile under 40 different conditions or sets of parameter values. The values for each
parameter during each of the 40 simulations are chosen by using the
following algorithm. Each variable is assigned a series of random
numbers between 1 and 40 without replacement (each number is used only
once). The random number is then converted into a multiplying factor (a
factor that multiplies the central value), which is based on the
uncertainty range defined for the variable (see Table 1). For example,
if a random number of 1 appears under a variable that has an
uncertainty of 20% for a specific run, then the multiplying factor for
the variable used in that run would be 0.80. During each run, the
choice for the specific multiplier of the central value is chosen
completely randomly but without replacement.
The last step in the sensitivity analysis is to determine a
quantitative sensitivity index for each of the 20 parameters and establish a threshold to identify those parameters to which the model
output is sensitive and those to which the model output is insensitive.
In LHS, the sensitivity index for each parameter is the respective
partial-rank correlation coefficient,
i, j, as defined by the
following relationship
|
(10)
|
where
Y is the value of the model output
variable, 
is a constant, X is the
value for the model input variables, the superscript k refers to the simulation number
(i.e., 1-40), and the subscript i
refers to the specific model output. Linear least squares regression is
implemented to determine the values for
i,j, then a simple statistical
test (t-statistic) is employed to
determine whether each i,j is
statistically different (P < 0.05) from zero. If i,j is different
from zero, then we can conclude that it has a significant impact on
model output i. Three model outputs
were chosen that best reflect heat and mass transfer dynamics and that
can also be easily measured experimentally: 1) normalized (by body temperature)
end-exhaled airstream temperature (E,max);
2) normalized (by alveolar partial
pressure) end-exhaled airstream partial pressure of ethanol
(Emax);
and 3) the normalized (by the
maximum value of
SIII for the 40 simulations) SIII
of the exhalation ethanol profile
(
III). All
three model outputs were normalized such that the maximum value is one.
SIII (liters1) is calculated
from a linear least squares fit of the model output over the last
one-half of the exhaled volume.
Several of the parameters in the model are calculated based on other
parameters. For example, the thickness of the connective tissue layer
is calculated as a fraction of the thickness of the epithelium layer.
To obtain a sensitivity coefficient that is representative of only that
parameter, each parameter is changed independently of the others. For
example, the tissue thickness is calculated from the base value of the
epithelium thickness and is not dependent on how the epithelium
thickness is varied for the 40 simulations.
Airway Ethanol Flux
As a subject inhales, the airstream has the potential to absorb ethanol
from the airways, more specifically, from the mucous layer that lines
the airways. Over the course of an entire inspiration, each airway
generation will contribute to the overall flux of ethanol from the
mucus to the air. Over the course of an expiration, a portion of the
ethanol absorbed on inspiration is desorbed back to the airways.
The total flux of ethanol (mol/breath) from airway compartment
n during either inspiration,
Je,insp(n),
or expiration, Je,exp(n),
is simply the number of airway branches in the generation multiplied by
the sum of the flux from each individual control volume
|
(11)
|
|
(12)
|
where
je is the flux of
ethanol from the mucous surface in each control volume as defined by
the local mass transfer coefficient (see
APPENDIX, Eq. A3); is the volumetric flow rate
of air (cm3/s) during inspiration
and expiration, respectively; Vee
is the lung volume at end expiration,
Vei is the lung volume at end
inspiration, and g is the generation
number. The differential time element dt has been transformed to a
differential volume by dV = dt. Similar
expressions can be easily derived to define the total molar flux of
water (Jw,insp
and Jw,exp) and
heat (Jh,insp and Jh,exp) from
each airway generation during a breath, as well the flux of ethanol
between other compartments, most notably, the flux of ethanol from the
bronchial capillaries
(Jbr) and the
flux of ethanol from the body core
(Jbody).
| |
RESULTS |
Experimental Single-Exhalation Maneuver
Detailed results of the experimental protocol have been previously
published (12); hence, only the salient results will be presented here.
The mean age, weight, vital capacity, and minimum exhaled volume for
the single-exhalation maneuver for the six subjects were 30 ± 10 (SD) yr, 78 ± 14 kg, 5,400 ± 740 ml, and 4,160 ± 810 ml, respectively. The range for the mean
exhaled flow rates for the six subjects was 140-320 ml/s, and the
mean exhaled flow rate for all six subjects was 200 ± 70 ml/s. The average exhalation profile (as described in
EXPERIMENTAL METHODS) is depicted in
Fig. 3. Note that the concentration of
ethanol in the exhaled breath increases immediately after the start of
exhalation, demonstrating exchange in the airway space, and that phase
III has a positive slope similar to that of other gases such as
CO2 and
N2. The mechanism underlying the
positive SIII of
ethanol differs from relatively insoluble gases such as
CO2 and
N2 and is related to a temporal heterogeneity in the partial pressure of ethanol in the airway tissue
during exhalation. This mechanism is described in much greater detail
in a previous paper (12).
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Fig. 3.
Experimental and model-predicted exhaled ethanol profiles for a
representative human subject. Experimental data are given by ( ).
Model fit to data (solid line) was optimized by using thickness of
body-tissue layer,
lb, as no
experimental estimate of this parameter exists. This method is
validated by sensitivity analysis.
E,
normalized expired partial pressure of ethanol;
E,max,
normalized maximal expired partial pressure of ethanol.
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Model Simulation
Single exhalation. Estimates for the
central values of all model parameters were available (see Table 1)
from data in the literature, except for the parameters associated with
the thickness of the body layer
(lb and ).
Hence, lb and were utilized to perform an initial optimization of the fit of the
model prediction to the experimental exhaled ethanol profile and the
end-exhaled temperature, respectively, by minimizing the sum of squares
of the error. Initially, the model prediction of the exhalation profile overestimated the initial concentration of ethanol in phase III and,
hence, produced a smaller
SIII. This
systematic error in the model could be overcome by enhancing the
desorption of ethanol to airway wall in the smaller airways by
incorporating the parameter in the calculation of the airway wall
surface area Ad,
as described above. The optimal fit of the model is presented in Fig.
3. The model predicts well the shape of the single exhalation with a coefficient of determination
(R2) of 0.991. The optimal value for
lb was 14 times
the thickness of the epithelium
le. Then, in the
trachea, where le = 100 µm, lb = 1.4 mm, and in the bronchioles (generations
11-18), where le = 20 µm,
lb = 0.28 mm. To
simultaneously match the reported value of 34.6°C for the mean
end-exhaled temperature of the breath (20), the optimal value for was 9; in other words, the thickness of the body layer for heat
transfer (product
lb ) is
126 times the value of the epithelium. In the trachea and bronchioles,
this would correspond to ~1.26 and 0.25 cm, respectively.
Axial flux distribution. Model
predictions for the axial flux distribution from the mucus to the
airstream for the single-exhalation maneuver (vital capacity of 5,400 ml, exhaled volume of 4,160 ml, and a flow rate of 200 ml/s) for
ethanol, water, and heat are presented in Fig.
4,
A-C,
respectively. A positive flux denotes transfer of ethanol, water, or
heat from the mucus to the airstream (absorption), a negative flux
denotes flux from the airstream to the mucus (desorption).
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Fig. 4.
Axial flux (J) distribution for
ethanol during inspiration and expiration for ethanol
(A;
Je), water
(B;
Jw), and heat
(C;
Jh). Note
bimodal distribution in A-C and that
all 3 components reach a local equilibrium with body conditions before
reaching alveolar region. Hence, model predicts that the exchange of
ethanol, water, and heat occurs entirely within airway space.
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The inspiratory and expiratory flux profiles for ethanol demonstrate a
bimodal distribution with peaks in the trachea and 12th generation.
Je,insp becomes
progressively smaller after the 9th generation and is nearly zero in
17th generation. Thus the model predicts that the incoming airstream
reaches a local equilibrium with the capillary blood before reaching
the alveoli; thus the exchange of ethanol in the lungs occurs entirely
within the conducting airway space. During expiration, a portion (33%)
of the ethanol absorbed by the airstream during inspiration is
desorbed. The rate of desorption during exhalation decreases,
thus accounting for the positive
SIII. The total
amount of ethanol eliminated from the lungs during the
single-exhalation maneuver is 36.1 µmol/breath. This pattern of
absorption-desorption is similar to prior predictions made by the model
(12).
The inspiratory and expiratory flux profiles for water and heat are
similar to these of ethanol, with the notable exception that they
predict a local equilibrium with the core body conditions proximal to
those of ethanol. For water and heat, the peak flux occurs in the
trachea and generation 6, and the flux
is nearly zero by generation 13.
Hence, the model predicts that inspired air is fully warmed and
humidified by approximately the 13th generation.
Figure 5 plots the net (inspiration plus
expiration) axial flux distribution of ethanol into the airstream
subdivided into three potential sources:
1) contribution from the bronchial
circulation (Jbr),
2) contribution from the body core
to the body tissue
(Jbody), and
3) contribution from the mucosal and
submucosal tissues
(Jtiss; i.e.,
mucus, epithelium, connective tissue, smooth muscle, and body tissue).
Jbr is bimodal,
with peaks in the trachea and 10th generation.
Jbody does not
have a significant contribution in the extrathoracic airways but,
rather, a rapid rise beginning in the 5th generation to a peak in the
11th generation. The sum of the flux of ethanol from the bronchial
circulation and from the body core for the entire airway tree is 10.3 and 16.0 µmol/breath, respectively (29 and 44% of total
ethanol eliminated). The remaining ethanol eliminated in the
exhaled airstream (9.8 µmol/breath or 27%) of the transient single
exhalation arises from the tissues of the mucosa and submucosa.
Jtiss is positive
in the mouth, oropharynx, and generations
9-16 but is negative in the trachea and
generations 1-8. Although not
shown, the flux is positive during both inspiration and expiration for
both Jbr and
Jbody.
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Fig. 5.
Net axial flux distribution of ethanol into airstream during both
inspiration and expiration. For each generation, net accumulation in
airstream is a combination of contribution from body core
(Jbody),
bronchial circulation
(Jbr), and
surrounding mucosal and submucosal tissues
(Jtiss, mucus,
epithelium, connective tissue, smooth muscle, and body layer). Hence,
net accumulation presented here is equal to net flux presented in Fig.
4A from bronchial circulation
(Jbr), the core
body (Jbody),
and the mucosal and submucosal tissues during single exhalation
maneuver. Positive flux denotes flux from blood to the adjacent tissues
or from body core to body tissue layer. In the case of
Jtiss, a positive
value denotes a net increase in moles of ethanol present in mucosal and
submucosal tissue (mol/breaths). Note that
Jbody is
~2-fold larger than
Jbr in the airway
tree but is ~0 in upper respiratory tract. This is consistent with
the presence of pulmonary circulation surrounding intrathoracic
airways.
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|
Figure 6 plots the axial distribution of the partial
pressure of ethanol in the blood exiting the capillary bed (venous
blood, Pc,t and
Pc,s) normalized by the incoming
Pa at end inspiration and at end
expiration for the single-exhalation maneuver. Bot
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Abstract
Introduction
