INTRODUCTION

Top Abstract Introduction Methods Results Discussion Appendix References

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 (P_E) was normalized by the concentration of ethanol in the alveolar gas (P_A) (P_A = C_blood/_blood, where C_blood 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

Top Abstract Introduction Methods Results Discussion Appendix References

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

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.

View larger version (28K): [in this window] [in a new window]   View larger version (42K): [in this window] [in a new window]   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 2rz, 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).

(2)

(3)

6

(4)

(5)

1

1

(6)

1

1

(7)

(8)

(9)

View larger version (12K): [in this window] [in a new window]   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.

Computer Simulation

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

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 l_b and , they were assigned a high level of uncertainty (±80%), whereas the diffusivity of ethanol in air, D_e,a, and ethanol in tissue, D_e,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 S_III for the 40 simulations) S_III of the exhalation ethanol profile (_III). All three model outputs were normalized such that the maximum value is one. S_III (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

(11)

(12)

	RESULTS

Top Abstract Introduction Methods Results Discussion Appendix References

Experimental Single-Exhalation Maneuver

View larger version (15K): [in this window] [in a new window]   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.

Model Simulation

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).

View larger version (20K): [in this window] [in a new window]   View larger version (17K): [in this window] [in a new window]   View larger version (15K): [in this window] [in a new window]   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.

The inspiratory and expiratory flux profiles for ethanol demonstrate a bimodal distribution with peaks in the trachea and 12th generation. J_e,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 S_III. 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 (J_br), 2) contribution from the body core to the body tissue (J_body), and 3) contribution from the mucosal and submucosal tissues (J_tiss; i.e., mucus, epithelium, connective tissue, smooth muscle, and body tissue). J_br is bimodal, with peaks in the trachea and 10th generation. J_body 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. J_tiss 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 J_br and J_body.

View larger version (23K): [in this window] [in a new window]   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.

Figure 6 plots the axial distribution of the partial pressure of ethanol in the blood exiting the capillary bed (venous blood, P_c,t and P_c,s) normalized by the incoming P_a at end inspiration and at end expiration for the single-exhalation maneuver. Both P_c,t and P_c,s fall below P_a during inspiration. This result indicates that the exchange process of ethanol is limited, at least in part, by the rate of blood flow from the bronchial circulation and is consistent with our hypothesis and with the results of recent investigations (14, 35). During expiration, the partial pressure of ethanol in the capillaries increases as ethanol is desorbed from airstream back to the airway wall (Fig. 4A) but does not entirely recover to P_a at end expiration.

View larger version (17K): [in this window] [in a new window]   Fig. 6.   Axial distribution of normalized (by arterial pressure) partial pressure of venous blood exiting capillary compartments of both tissue layer (lamina propria) and smooth muscle layer at end inspiration and end expiration. Note that partial pressure falls below that of the arterial blood during inspiration and does not completely recover during expiration. Hence, exchange of ethanol within airway space depends on blood flow rate of bronchial circulation. See Glossary for symbol definitions.

Sensitivity analysis. The exhalation profiles (both ethanol concentration and temperature) for the 40 simulations in the LHS analysis are presented in Fig. 6. Figure 7A represents the range of exhalation ethanol profiles, and Fig. 7B represents the range for the exhalation temperature profile. The partial rank correlation coefficients (and their corresponding P values) from the LHS analysis are summarized in Table 2. Coefficients that have a P value <0.05 are boldfaced, whereas those with a P value >0.95 are italicized.

View larger version (31K): [in this window] [in a new window]   Fig. 7.   Model ouput for the 40 simulations of Latin hypercube sampling. A: exhalation ethanol profile. B: exhalation temperature (T) profile.

View this table: [in this window] [in a new window]   Table 2.   Sensitivity coefficients (i, j )

There is only one parameter, l_b, to which all three of the model outputs are statistically sensitive. From this we conclude that the thickness of the effective buffer layer between the smooth muscle and the core body is one of the most significant parameters in predicting heat and mass exchange. We can use this result to either direct future experiments aimed at measuring this parameter more accurately, or, more likely, use this parameter to optimize the fit of the model and determine what a reasonable value may be. This is, in fact, what was done before the sensitivity analysis in an effort to gain an estimate of its central value.

There are three additional parameters to which both P_E,max and the S_III are sensitive: k^e_m,a, _e,m, and _m,a (P < 0.002 for both P_E,max, and S_III). These results are not surprising and serve to validate qualitatively the performance of the model. P_E,max is also sensitive to l_t (P = 0.033),  (P = 0.007), and  (P = 0.004), whereas an additional parameter to which S_III is sensitive is F (P = 0.001). F is a parameter that scales the lengths and diameters of the airways to maintain a constant ratio of the volume of the conducting airways to the vital capacity. Two additional parameters to which exhaled temperature is sensitive are the local heat transfer coefficient between the mucous layer and the air (h_m,a) (P = 0.002) and  (P = 0.001).

Parameters to which the model output is statistically insensitive (P > 0.95) also provide useful information. Each model output is insensitive to at least one parameter. P_E,max is insensitive to the heat transfer coefficient (P = 0.97) and the thermal conductivity of the tissue (P = 0.97). This result is expected and can be used to qualitatively validate the performance of the model. T_E,max is insensitive to  (P = 0.99), whereas S_III is insensitive to the thickness of the mucous layer (P = 0.99).

It is important to note that, although many of the parameters listed in Table 2 are not statistically significant, this does not mean that they are unimportant and have no impact on the model. What it does mean is that within their uncertainty range there was no particularly strong correlation between the input variable and the selected model output.

	DISCUSSION

Top Abstract Introduction Methods Results Discussion Appendix References

Blood Flow Rate and Perfusion Limitation

Blood and/or Water Solubility

Diffusion Coefficient

6

Thickness of Diffusion Barrier

The tremendous sensitivity of all model outputs on the thickness of the body-tissue layer indicates that the core body conditions are the greatest contributors to the exchange of ethanol and heat. In particular, the pulmonary circulation is in close proximity to the airways, beginning at approximately the 4th generation, and, due to its large volumetric flow rate (two orders of magnitude larger than the bronchial circulation), represents an enormous source of ethanol and heat. In addition, the pulmonary circulation has been previously shown experimentally to have a large impact on intra-airway heat exchange (33). Hence, the pulmonary circulation dictates the thickness of the body layer in the model and, thus, has a larger impact on ethanol exchange than does the bronchial circulation.

As l_b is not a true anatomic barrier but, rather, a model construct, it is difficult to comment on the absolute value attained by the model. However, the value of 1.68 mm for the transfer of ethanol in the trachea certainly seems reasonable. In addition, the value of 1.34 cm for heat transfer also seems reasonable based on the data of Solway et al. (34), who reported cooling of the esophageal lumen during cold air hyperpnea. One simplification that may be a source of error is the use of a single value for l_b and  in the entire airway tree. The fact that the exchange of both ethanol and heat is heterogeneous in the axial direction (see Fig. 4) indicates that the distance from the smooth muscle where the core body conditions exist (i.e., _b ) is also heterogeneous. For example, the exchange of heat is essentially complete by the 12th generation; hence, l_b may be effectively zero (and not 0.39 cm) in the bronchioles and larger in the oropharynx and trachea, where the exchange of heat is substantial. The impact of introducing an axial dependence to l_b will be addressed in future simulations.

Local Mass Transfer Coefficient

The correlation predicts that the mass transfer coefficient decreases monotonically as airstream velocity decreases. However, as airstream velocity decreases and flow transitions from turbulent to laminar (transition occurs at approximately Re <2,100), the heat and mass transfer coefficients reach an asymptote and become independent of further decreases in velocity (3, 14). Thus, in generations 7-18, where airflow is very slow (Re <5 during inspiration to total lung capacity in 2.5 s), it is very likely that the correlation predicts a mass transfer coefficient that is too small. In addition, the impact of the mass transfer coefficient on the resistance to airway gas exchange is substantial for a gas that has blood solubility as high as that of ethanol (14), as solubility in the gas phase is independent of solubility in water and tissue. These two concepts, combined with the exceptionally small P values for k (0.001 and 0.002 for P_E,max, and S_III, respectively) demonstrate the extreme importance of k^e_m,a in accurately describing airway gas-exchange dynamics. The probable underestimation in k^e_m,a might also explain the observed improved ability of the model to simulate the S_III when the surface area of the smaller airways is enhanced by . The flux of ethanol from the airway wall is proportional to the product k^e_m,a A_d; hence, an improved simulation of the S_III could also have been attained with a larger value for k^e_m,a in the smaller airways. Additional studies to more accurately describe the factors that influence k^e_m,a are planned for the future.

Surface Area

F is a scaling factor that increases the size of the airway tree based on the subject's vital capacity, such that the volume of the airways is always the same percentage of the vital capacity. Both the lengths and the diameters of the airways are increased as the vital capacity is increased. Interestingly, an increase in F (which increases the total surface area of the airway lumen) does not have a statistical impact on P_E,max but it does have a very significant impact on S_III. This is the opposite response to . The difference can be attributed to the effect of increasing the volume (as F does) of the airways and not just the surface area (as  does). Recall that by increasing F, the volume of the airways is increased by increasing both the length and diameter of the airway. A larger d corresponds to an increase in the Reynolds number (proportional to d) but a decrease in the linear velocity of the flow (inversely proportional to d²). Hence, the net result is a decrease in the Reynolds number, which results in a smaller mass transfer coefficient (proportional to the Re^0.688). Hence, the larger surface area for recovery of ethanol is partially offset by a reduced efficiency of transfer across the gas-phase film resistance. The rate of ethanol recovery remains enhanced, such that the increase in F results in a smaller S_III.

Conclusions

The results of this study suggest a more prominent role of the pulmonary circulation in the exchange of highly soluble gases than previously thought when the arterial blood is the source of the gas. Additional experimental information, in particular on the heat and mass transfer coefficients in the lower airways, is needed for future simulations.