Department of Chemical and Biochemical Engineering and Materials
Science, University of California at Irvine, Irvine, California
92697-2575
The relatively
recent detection of nitric oxide (NO) in the exhaled breath has
prompted a great deal of experimentation in an effort to understand the
pulmonary exchange dynamics. There has been very little progress in
theoretical studies to assist in the interpretation of the experimental
results. We have developed a two-compartment model of the lungs in an
effort to explain several fundamental experimental observations. The
model consists of a nonexpansile compartment representing the
conducting airways and an expansile compartment representing the
alveolar region of the lungs. Each compartment is surrounded by a layer
of tissue that is capable of producing and consuming NO. Beyond the
tissue barrier in each compartment is a layer of blood representing the
bronchial circulation or the pulmonary circulation, which are both
considered an infinite sink for NO. All parameters were estimated from
data in the literature, including the production rates of NO in the tissue layers, which were estimated from experimental plots of the
elimination rate of NO at end exhalation (ENO) vs. the
exhalation flow rate (
E). The model
is able to simulate the shape of the NO exhalation profile and to
successfully simulate the following experimental features of endogenous
NO exchange: 1) an inverse relationship between exhaled NO
concentration and E, 2) the dynamic relationship between the phase III slope and
E, and 3) the positive
relationship between ENO and
E. The model predicts that these
relationships can be explained by significant contributions of NO in
the exhaled breath from the nonexpansile airways and the expansile
alveoli. In addition, the model predicts that the relationship between
ENO and E can be used as an
index of the relative contributions of the airways and the alveoli to
exhaled NO.
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INTRODUCTION |
THE IMPORTANCE OF NITRIC OXIDE (NO) in the physiology
of lung function has steadily increased in the past decade.
Experimental work has dominated this influx of information and has
provided critical information related to the cellular source of NO in
the lungs, the specific physiological functions, and the detection of
endogenous NO in the exhaled breath. On the basis of the ability of NO
to relax smooth muscle, there has been interest in using exogenous NO
delivered by inhalation as a therapy in such diseases as pulmonary
hypertension, acute respiratory distress syndrome, and bronchial
asthma. In addition, there has been interest in using endogenously
exhaled NO levels as a noninvasive index of pulmonary inflammation
(13). This interest has generated a need to understand the exchange
dynamics of NO within the lungs.
The exchange dynamics of NO in the lungs are different from those
of other previously studied endogenous gases such as O2 and
CO2 because of specific differences in physical and
biochemical properties. 1) NO, with 11 valence electrons, is
a free radical and is reactive with a number of different endogenous
substrates, including O2, superoxide, thiols, and
metalloproteins. The presence of this array of substrates results in a
half-life (t1/2) of free NO in vivo that is
~0.5-15 s, depending on the specific physiological state (5,
16). 2) NO is actively produced basally and in response to
various stimuli, such as cytokines, by the cells that comprise the
alveolar and airway regions of the lungs. 3) NO binds avidly
to hemoglobin in the blood, the kinetics of which are much different
from those of CO2 and O2, but may play an
important role in the local control of oxygenation (12). Theoretical
modeling has historically played an important role in our understanding
of pulmonary gas exchange dynamics; however, because of the unique
biochemical features of NO, new models must be developed to complete
our understanding of NO exchange. The lone attempt to model NO exchange
in the lungs lumped the entire lung into a single compartment that
effectively represented the alveolar region (11). This model was a
starting point for modeling NO exchange dynamics and provided some
initial insight into NO exchange. However, much further work is needed,
inasmuch as a single-compartment model is not capable of describing the
growing experimental evidence that exhaled endogenous NO is derived
from an alveolar and an airway source. The alveoli and airways are two
distinct regions of the lungs with many different features. From a
gas-exchange perspective, perhaps the most important is the fact that
the alveolar volume is expansile and the airway volume is relatively
nonexpansile. The goal of this study is to develop a relatively simple
two-compartment model of NO exchange dynamics that can 1)
describe the basic features of NO exchange dynamics previously observed
experimentally by us (26) and by others (20, 24), 2) provide
initial insight into the exchange properties of exogenous NO, and
3) provide direction for future experimentation.
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MODEL DEVELOPMENT |
General Structure
A model of the human lungs was developed (Fig.
1) that includes enough detail to describe
the following basic features of NO exchange observed experimentally:
1) breath holding before exhalation creates an initial spike
in NO concentration during exhalation of the airway volume,
2) exhaled NO concentration is an inverse function of
exhalation flow rate (E), and
3) elimination rate of NO from the lungs (ENO) is
a positive function of E. These basic
observations are consistent with the alveolar and the airway regions of
the lungs as sources of exhaled NO. Hence, the model (Fig. 1) consists
of two main compartments: 1) a rigid or nonexpansile
compartment representing the conducting zone or airways [trachea
through generation 17 as defined by Weibel (27)] and
2) a flexible or expansile compartment representing the
respiratory bronchioles and alveolar region (generation 18 and beyond). Both compartments are surrounded by a layer of tissue
representing the bronchial mucosa in the airway compartment and the
alveolar membrane in the alveolar compartment. Blood representing the
bronchial circulation and the pulmonary circulation is distal to the
tissue in the airway compartment and alveolar compartment,
respectively. Cells present in the airway mucosa and the alveolar
membrane are capable of producing endogenous NO (8, 23); hence, we
assume that NO is produced at a constant rate per unit volume of tissue by the tissue surrounding the airway and alveolar compartments. The
endogenously produced NO can follow one of three paths: 1) consumption through reaction with substrates within the tissue compartments, 2) diffusion toward the pulmonary or bronchial
circulations, where it reacts instantaneously and irreversibly with the
hemoglobin of the blood, or 3) diffusion toward the
airstream, where it evaporates and enters the alveolar volume or airway
volume. Consequently, there will be a net flux of NO between the tissue
and the airstream in both compartments. The direction of the flux
depends on the relative concentrations of NO in the tissue and gas
phases. Hence, NO concentration in the exhaled breath will depend on
two additive mechanisms: 1) the exchange of NO in the
alveolar compartment and 2) the conditioning of the alveolar
gas as it is convected through the airway compartment. On the basis of
this general structure, a series of mass balances on the tissue and gas
phase of the airway compartment and the alveolar compartment produce
the governing equations for the model.
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Fig. 1.
Schematic of 2-compartment model for nitric oxide (NO) pulmonary
exchange. First compartment represents relatively nonexpansile
conducting airways; second compartment represents expansile alveoli.
Each compartment is adjacent to a layer of tissue that is capable of
producing and consuming NO. Exterior to tissue is a layer of blood that
represents bronchial or pulmonary circulation and serves as an infinite
sink for NO. E and
I, expiratory and inspiratory flow,
respectively; CE and CI, expiratory and
inspiratory concentration, respectively; Cair and
Calv, airway and alveolar concentration, respectively;
Vair and Valv, airway and alveolar volume,
respectively; Jt:g,air and
Jt:g,alv, total flux of NO from tissue to air and
from alveolar tissue, respectively; t, time; V, volume.
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Airway Compartment
Tissue phase.
The airway compartment is modeled as a cylindrical tube, and the
surrounding tissue (Fig. 2) is modeled as a
homogenous compartment of uniform thickness
(Lt,air) with aqueous physical properties. NO is
produced uniformly in position and constant in time by the tissue at a
rate 
t,air
(mol · s
1 · cm3).
NO reacts with several substrates including O2,
superoxide, thiols, and metalloproteins such as guanylate cyclase (5); hence, NO is consumed by chemical reaction at a rate
t,air
(mol · s1 · cm3).
The reaction with O2 is second order in NO concentration,
and this reaction is the rate-limiting step in the reaction with
endogenous thiols such as glutathione (14). At physiological NO
concentrations, only the reactions with superoxide and
metalloproteins are significant (2); both are first order in NO
concentration. Hence, the t,air can be
expressed as
kCt(t, x, z), where
k is an overall first-order rate constant
(s1), Ct(t, x, z)
is the concentration of NO in the airway tissue, x is the
distance from the bronchial blood compartment, z is the axial
position, and t is time (s).
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Fig. 2.
Schematic of tissue layer in airway compartment. NO is produced from
cellular sources at a constant rate in position and time
(t,air). NO can then be transported by
diffusion and enter bronchial circulation or gas phase of airway lumen,
or it can be consumed by chemical reaction that is 1st order in NO
concentration. Solid line, approximate radial concentration profile of
NO in blood (b), tissue (t), and gas (g) phases under ambient inspired
conditions. Radial gradient in NO concentration at tissue-air interface
is proportional to flux of NO into airstream.
Lt,air, thickness of tissue surrounding airway
compartment;  t:g, partition coefficient of NO between
tissue and air; Cw, concentration at airway-tissue
interface; Ct, concentration in tissue.
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The transport of NO from the site of production to the bronchial blood
or the airstream occurs by molecular diffusion described by Fick's
first law. The axial and angular gradients of NO concentration in the
tissue are considered negligible. Hence, the transport of NO in the
tissue can be sufficiently described by a one-dimensional diffusion
equation. The small tissue thickness in comparison with the airway
radius allows the use of simple Cartesian coordinates. Because of the
fast reaction of the NO with hemoglobin present in abundance in the
blood, the concentration of free NO at the interface between the blood
and the tissue will be very close to zero. For the interface between
the airway lumen and the tissue, we assume instant thermodynamic
equilibrium governed by Henry's law. Under these assumptions, a
differential mass balance of NO in the tissue produces the following
second-order partial differential equation
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(1)
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with the following boundary conditions:
Ct(t, 0) = 0, Ct(t,
Lt,air) = Cw(t, z). Dt is the
molecular diffusion coefficient of NO in tissue
(3.3 × 105 cm2/s) (16), and
Cw is the concentration of NO at the interface between the
airway and tissue. Cw(t, z) will depend on
the concentration in the gas phase; as a result, Eq. 1 must be
solved simultaneously with the NO mass balance in the gas phase of the
airway compartment (see below). The solution to Eq. 1 can be
simplified by assuming that Ct is in the steady state. This
approximation is valid if the times of inspiration and expiration are
much greater than ~0.6 s (the time for the concentration profile in
the tissue to reach 90% of its steady-state value). This is due to the
relatively small thickness of tissue relative to the rate of diffusion,
and a proof is provided in APPENDIX A. With the assumption of steady state in the tissue, Eq. 1 reduces to
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(2)
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The solution of Eq. 2 is
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(3a)
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where
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(3b)
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(3c)
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Once the concentration profile in the tissue is known, using
Fick's first law of diffusion, one can easily determine that the flux
of NO from the tissue to the air ( Jt:g,air) is a
linear function of Cw
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(4a)
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where
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(4b)
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(4c)
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Because of the very low solubility of NO in water and
tissue, the radial transport of NO is not limited by diffusion in the
gas phase (APPENDIX B) (6). Hence, Eq. 4 becomes
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(5)
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where t:g is the partition coefficient of NO
between the tissue and the air at 37°C and Cair is the
bulk gas concentration of NO in the airway lumen. From Eq. 5,
the flux of NO per unit airway surface area and per unit time between
the airway tissue and the airway lumen is a linear function of the bulk
gas concentration. As the concentration in the bulk gas of the airway
lumen increases, the amount of NO that is consumed by the pulmonary
blood or through reactions with substrates in the airway tissue
increases; thus Jt:g,air decreases. For
Cair greater than a/(bt:g),
the consumption overcomes the endogenous production rates, resulting in
net flux of NO from the airway to the tissue.
Gas phase.
The airway compartment is modeled as a cylinder of constant total
volume Vair and total surface area
As,air. As described above, the absorption of NO
by the airstream per unit surface area at any given point of the airway
is described by Eq. 5. If we assume plug flow (i.e., no radial
gradient in the velocity or concentration), then a differential mass
balance of NO over a unit volume will give
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(6a)
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which can be transformed to a more convenient form by
substituting dV = dzAc
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(6b)
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where is volumetric flow rate of air
(I for inspiration and
E for expiration), As
is the wall surface area per unit axial distance,
Ac is airway volume per unit axial distance, and V
is axial position in units of cumulative volume. The ratio
As/Ac (surface area per
unit airway volume) increases in the human lung as the radius of each
airway branch (r) decreases with generation number
(As/Ac = 2/r).
Hence, a more realistic description of the airway compartment is not as
a perfect cylinder with constant As/Ac but, rather, as a
cylinder whereby As/Ac
increases with increasing axial position (or decreasing radius). To
preserve an analytic solution to the model equations,
As/Ac is approximated by
a linear function of V (R2 = 0.94) with use of
the airway dimension data of Weibel (28): As/Ac = aI + bIV for inspiration and
As/Ac = aE + bEV for expiration (aE = aI + bIVair,
bE = 
bI). Hence, the total
surface area of the airway compartment (As,air) is
described by
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(7)
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Equation 6 is subject to the following arbitrary
initial and boundary conditions: Cair(0, V) = f (V) and Cair(t, 0) = g(t), where f (V) is the
concentration profile of NO in the upper compartment before the
inspiration (V = 0 at the mouth) or expiration (V = 0 at the
entrance of the alveolar compartment) and g(t) is the
inspired concentration or the alveolar concentration for inspiration
(I) or expiration (E), respectively. The solution to Eq. 6 for
constant is
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(8a)
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for case I (t < V /)
and
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(8b)
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for case II (t > V /),
where
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(8c)
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(8d)
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Case I describes the convective emptying of the
airways at the beginning of inspiration or expiration; case II
describes the convection of the alveolar gas or inspired gas through
the airway compartment during expiration or inspiration, respectively. For the linearly increasing and decreasing flow rate maneuvers, E is not constant. However, an analytic
solution is still attainable and is presented in APPENDIX
C.
Alveolar Compartment
Gas phase.
In modeling the alveolar region (Fig. 1), we assume a well-mixed
compartment of variable volume [Valv(t)]. The NO
concentration in the alveolar gas [Calv(t)] is
uniform in position but is time dependent. NO can enter or leave the
compartment through convective flow during inspiration or expiration,
respectively, and can exchange with the alveolar tissue by diffusion.
After an analysis that is identical to that presented for the airway
tissue compartment, it can be shown that the flux of NO between the
alveolar gas and the tissue phase ( Jt:g,alv) is
a linear function of the bulk gas phase concentration (see Eq. 5). Then the net total flux (mol/s) of NO from the alveolar
tissue
( Jt:g,alvAs,alv,
where As,alv is the surface area of the alveolar
region) can be described by a combination of two terms
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(9)
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where app,alv (mol/s) is the
apparent production rate of NO is the tissue (defined as the flux of NO
into the alveolar compartment if the concentration of NO was zero in
the compartment) and DLNO is the diffusing
capacity
(mol · s1 · mol1 · cm3)
of NO in the alveolar region. The method for determining the alveolar
flux contrasts with that used in the airway compartment only in the
manner of determining the coefficients of the linear function (i.e.,
app,alv and DLNO
in Eq. 9). In the alveolar compartment the coefficients can
be determined experimentally rather than calculated from the geometric
characteristics of the alveolar tissue and the endogenous production
rate, as was done in the airway compartment.
DLNO has been experimentally measured (3, 7) and is equal to ~2,100
mol · s1 · mol1 · cm3,
and app,alv can be determined from the
steady-state alveolar concentration (see Eq. 12).
A differential mass balance for NO in the alveolar compartment
(valid for inspiration and expiration) is then
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(10)
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subject to the following initial condition:
Calv(0) = Calv,0. Valv(t)
is the volume of the alveolar compartment, and Cair,end is
the concentration of NO in the air that exits the airway compartment. The volume change due to exchange of the respiratory gases has been
neglected. Equation 10 is valid for inspiration and
expiration (I = 0) as well as for
nonconstant flow rates.
The analytic solution of Eq. 10 for constant flow rates and for
constant Cair,end is as
follows
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(11a)
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for case I (inspiration)
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(11b)
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for case II (expiration), where 
= DLNO/j
( j = I or E), and
![+ <FR><NU><A><AC>S</AC><AC>˙</AC></A><SUB>app,alv</SUB></NU><DE>D<SC>l</SC><SUB>NO</SUB></DE></FR> [1 − <IT>e</IT><SUP>−[D<SC>l</SC><SUB>NO</SUB>/V<SUB>alv</SUB>(<IT>t</IT>)]<IT>t</IT></SUP>]](/content/vol85/issue2/fulltext/653/img025.gif)
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(11c)
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for case III (breath hold). Solution of Eq. 10 for linearly time-dependent expiratory flow rate is derived in
APPENDIX D.
Parameter Estimation
Table 1 summarizes the values for key
parameters used in the model simulations. The t:g is
calculated from Henry's law constant for NO between air and water
(25.8 × 106 mmHg NO · mol
NO1 · mol H2O) (18a). The
reaction constant k is calculated from the
t1/2 of NO in tissue. For first-order reactions,
k = ln (2)/t1/2. The values of
t1/2 that have been reported in the literature
range from 0.5 to 15 s, depending on the tissue and local environment
(15, 16, 28). For our simulation, we chose an intermediate value of 4 s. The volume and the surface area of the upper compartment are
calculated from Weibel's anatomic data (28) between the trachea and
generation 17. The coefficients of the linear regression of the
ratio As /Ac were chosen to
satisfy Eq. 7. app,alv and
t,air are more challenging to
estimate. app,alv can be estimated
from the diffusing capacity of NO and the expired steady-state alveolar
concentration (Calv,ss, Eq. 10) if
Calv,ss is known
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(12)
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Hyde et al. (11) used the end-exhaled concentration of NO
after a 15-s breath hold as an estimate of Calv,ss.
However, this approximation is valid only if the contribution from the airways is neglected [recall that the model of Hyde et al. (11) had
only an expansile alveolar compartment]. An alternative technique, which accounts for the contribution from the airways, is to use ENO from the lungs, as described in the companion article
(27)
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(13a)
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(13b)
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where
is the
average (over axial position) Jt:g,air and
E,ee is E
at end exhalation. Hence, an estimate of Calv,ss
(Calv,ee) is attained from the slope of a plot of
ENO vs. E,ee. We
used the elimination rate data from a representative subject
(subject 6) in the companion article (26) for the model simulations. For subject 6, Calv,ss was equal to
8.11 parts/billion (ppb), and thus
app,alv was estimated as 6.7 × 1010 mol/s.
t,air can also be estimated from Eq.
13, with the intercept of the ENO vs.
E,ee plot as an estimate of
is then chosen using an iterative scheme to satisfy Eqs. 4 and 13 simultaneously, for which
t,air is the only unknown parameter. In
the case of subject 6, 
was equal to 2.33 × 1011 mol/s, which produced a value
for t,air of 5.5 × 1013
mol · s1 · cm3.
The model equations can be solved for different breathing patterns of
constant E as well as
E that change linearly in time (see
APPENDICES C AND D). The initial alveolar volume, inspired and expired volumes, and inspired and expired flow
rates are specified by the user. Unless otherwise noted, inspiratory
conditions for all simulations were from functional residual capacity
to total lung capacity (alveolar volume 2,300 and 5,800 ml,
respectively). Functional residual capacity and total lung capacity
were estimated from the vital capacity (VC) of subject 6 (9).
The control inspiratory conditions also included the following
parameter values: I = 2,000 cm3/s, Calv,0 = 8.3 ppb, and CI = 15 ppb, where Calv,0 is initial alveolar concentration and
CI is inspiratory concentration. Calv,0 can be
estimated by first setting its value to zero and then simulating several rebreathing maneuvers (tidal volume = 500 ml, respiratory rate = 12 breaths/min) by setting the initial concentration to the
end-expiratory concentration of the previous maneuver. A steady-state value is quickly achieved, which can serve as Calv,0 for
the remaining simulations.
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RESULTS |
Simulation of the Single Exhalation
Figure 3 depicts four different
experimental single-exhalation maneuvers of the representative subject
(26) and the model prediction of these maneuvers with the same set of
parameters (Table 1) in each case. Recall that no particular
optimization procedure has been performed to simulate the experimental
profiles. All parameters were estimated from the literature or from the elimination rate data from this particular subject. In Figure 3,
A and B, the subject exhales at a constant low flow
rate (control) and a constant high flow rate, respectively. In Fig.
3C, the control maneuver is performed after a 15-s breath hold,
and in Fig. 3D E is decreasing
linearly with time on the basis of the flow signal from this particular
profile.
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Fig. 3.
Model simulation (solid lines) of 4 different experimental exhalation
NO profiles in which E is independent
variable. A: control maneuver, which represents a constant
E of ~200 cm3/s. B:
high flow rate maneuver in which E is
still constant but is increased to ~450 cm3/s.
C: preexpiratory 15-s breath hold followed by a constant
E of ~200 cm3/s.
D: linear decrease of E in time
during course of exhalation with a slope of approximately 40
cm3 · s1 · s
and an average E of ~300
cm3/s. CE, expiratory concentration; ppb,
parts/billion.
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At a constant E (Fig. 3, A and
B) the model replicates the initial rise in exhaled NO
concentration (beginning at CI) as the airway compartment
is exhaled (first 200 ml or ~1 s). This rise represents an axial NO
gradient due to the absorption of NO by the airstream from the airway
tissue. After this initial peak, there is a step change as air from the
alveolar compartment reaches the exhalate. This value (10 ppb) is
larger than the steady-state alveolar concentration (8.11 ppb) and the
initial or preexpiratory concentration (8.3 ppb) due to absorption of
NO from the airway tissue as the airstream is convected through the
airways. The model-predicted and experimental expiratory concentration
(CE) then steadily declines over the remaining portion of
the exhalation. After a 15-s breath hold (Fig. 3C) the model
predicts a large initial peak due to accumulation of NO in the airway
compartment followed by a similar decline, although less pronounced, in
CE over the remaining exhalation. When
E decreases linearly in time (Fig.
3D) the model is able to simulate the initial fall in
CE during phase III followed by a steadily increasing rise.
Flow Rate Dependence of the Average Phase III
Concentration
In Fig. 4, average phase III concentrations
with respect to time
![[<OVL>C</OVL><SUB>E</SUB>(<IT>t</IT>)]](/content/vol85/issue2/fulltext/653/img032.gif)
or
volume ![[<OVL>C</OVL><SUB>E</SUB>(V)]](/content/vol85/issue2/fulltext/653/img033.gif)
are plotted
as a function of E over a wide
range of constant E. 
is estimated
over a specified time interval (exhalation time of 2-8 s) and
over a specified
volume interval (exhalation volume of 20-60% of
VC), as described in the companion article (26). For very low or
very high E, where phase III does not include the above intervals, the maximum allowable interval is used. The experimental data from the companion article are
also reported for reference. There is a good agreement between our experimental data and our simulations for intermediate flow
rates. Over the entire flow rate range there is an
exponential decrease for
and
with
E. This result is consistent with the
experimental data (end-exhaled NO concentration) presented by Silkoff
et al. (24) over the range 4.2-1,500 cm3/s. For
low E (<500 cm3/s),
has a slightly
larger value. In the high-E region
(>500 cm3/s),
is slightly higher
and has a small positive slope. In the intermediate region the slope of
vs. flow (0.004
ppb · ml1 · s1)
is steeper than for
(0.002
ppb · ml1 · s1),
which is in agreement with our experimental data (see Fig. 4 in Ref.
26).
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Fig. 4.
Dependence of average NO concentration in phase III
 with respect to
time (t) and volume (V) over a wide range of
E.
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Phase III Slope
Figure 5 presents data from the simulation
of linearly changing flow rate profiles. The normalized phase III slope
(slope in phase III normalized by the average concentration in
phase III) in time
![[<OVL><IT>S</IT></OVL><SUB>III,NO</SUB>(<IT>t</IT>)],](/content/vol85/issue2/fulltext/653/img037.gif)
is plotted as a function of the normalized slope of
E
see Fig. 6 of Ref. 26]. To generate the data, the model simulated the
exhalation maneuvers of subject 6, matching the experimental flow rate profile of each maneuver. The simulations are performed for
the breath-hold and the non-breath-hold maneuvers. The experimental data for subject 6 are also shown for comparison. The model is able to simulate the inverse relationship, including the negative intercept at a constant E. The inverse
relationship is due to the nonnegligible contribution from the airways
to exhaled NO and, hence, the flow rate dependence. As
E decreases, CE increases. Thus, if E decreases in time,
CE will increase in time and affect the slope of
phase III. The slight negative slope at constant E is generated from continuous gas
exchange in the alveolar region. In addition, in a fashion similar to
the experimental data, the intercept increases after a 15-s breath
hold. This phenomenon is due to a decrease in the alveolar
concentration toward the steady-state value during the 15-s breath hold
(see Inspiratory Conditions).
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Fig. 5.
Experimental and model-predicted relationship between normalized phase
III slope of exhaled NO concentration in time
![[<OVL><IT>S</IT></OVL><SUB>III,NO</SUB>(<IT>t</IT>)]](/content/vol85/issue2/fulltext/653/img039.gif)
and exhalation flow rate
![[<OVL><IT>S</IT></OVL><SUB>III,<A><AC>V</AC><AC>˙</AC></A><SUB>E</SUB></SUB>(<IT>t</IT>)]](/content/vol85/issue2/fulltext/653/img040.gif) with
and without a 15-s breath hold.
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Elimination Rate
In Fig. 6 we present ENO at the
end of expiration, estimated as the product of end-expiratory
exhaled concentration and E. The data
are derived from model simulations of constant
E (range 50-1,000 cm3/s)
exhalation maneuvers with or without a 15-s breath hold. For E > 500 ml/s only data from the
breath-hold experiments were used to ensure an elapsed time of
8 s from end inspiration (see Inspiratory
Conditions). Least squares linear regression over the resulting
data reveals a highly linear (R2 = 0.99)
dependence of ENO with E.
The slope and intercept of the linear regression line are 8.12 ppb and
0.589 × 106 ml/s, respectively, which are in close
agreement with the experimental values from subject 6 (8.11 ppb
and 0.592 × 106 ml/s). The nonzero slope and
intercept are due to significant contributions of exhaled NO from the
expansile alveoli and the nonexpansile airways (Eq. 13).
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Fig. 6.
Model-predicted relationship between elimination rate of NO at end
exhalation (ENO) and E.
ENO is defined as product of end-exhaled concentration and
E. Positive slope is consistent with a
nonzero alveolar concentration of NO; nonzero intercept is consistent
with a nonnegligible flux of NO from airway tissue.
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Inspiratory Conditions
In Fig. 7 the model-predicted effect of the
inspired conditions on the exhalation profile of NO is examined.
Although this represents an extrapolation of the model, this type of
prediction can be used to guide future experimentation. Figure
7A presents the model's prediction for the NO
single-exhalation profile. Three different values for the inhaled
concentration are shown: 0, 15, and 30 ppb. For CI = 0 the
model predicts an initial rise from 0 ppb as the conducting airway
space is emptied, then a more gradual rise to the same plateau value
seen previously for CI = 15 ppb. The result is a positive
slope for phase III. In addition, the model predicts obliteration of
the peak in phase I after inspiration of NO-free air. For
CI = 30 ppb, there is, again, an initial rise in
CE as the airway compartment is emptied, which is then
followed by a steeper decline to the end-exhalation value compared with CI = 15 ppb. Hence, the model predicts that ambient
concentration can significantly affect the peak and the slope of phase
III during exhalation.
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Fig. 7.
Effect of inspired conditions (flow rate, volume, and concentration) on
shape of exhalation profile and alveolar concentration. A:
exhalation profile at 3 different CI. B:
exhalation profile at 3 different
I. C: exhalation
profile at 3 different VI.
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Figure 7B presents the effect of
I on the exhalation profile when
VI, CI, and E
are held constant at the control values (CI = 15 ppb,
E = 250 cm3/s,
VI = 3,500 ml). The model predicts that the slope of
phase III becomes progressively smaller as
I decreases from 2,000 to 100 cm3/s, yet the end-exhalation concentration remains
unchanged. In addition, I affects the
peak in phase I; the maneuver of the slowest
I produces the highest peak.
Figure 7C presents the effect of VI on the
exhalation profile. The model predicts that only the slope of phase III
becomes progressively larger as VI decreases from 3,500 to
500 cm3, while the peak in phase I remains unaffected.
The above results can be understood by examining the alveolar
concentration of NO under these conditions. Figure
8 demonstrates the alveolar concentration
of NO at end inspiration (Calv,ei) under different
conditions of CI and I.
Calv,ss is 8.11 ppb. For CI in excess of ~5
ppb, there is an increase in Calv,ei above Calv,ss. Hence, during exhalation there is a net negative
flux of NO (e.g., toward the pulmonary blood) in the alveolar
compartment as DLNOCalv > app,alv (Eq. 9). The result is
a progressively decreasing Calv, and hence
CE, during exhalation until Calv reaches Calv,ss. At this point, CE reaches a plateau
value of 10 ppb (for the control E). The
opposite scenario occurs for CI less than ~5 ppb. Under
these conditions, Calv,ei is below Calv,ss,
and DLNOCalv < app,alv. The result is an exponential
rise in Calv and CE during exhalation. The time
constant for this exponential decline or rise is equal to
Valv(t)/DLNO or ~1-3
s over the course of exhalation. Thus the model predicts that an
exhalation time of ~8 s will achieve an alveolar concentration that
differs by <1% of the steady-state concentration. In addition, the
difference between Calv,ei and Calv,ss for any
given inspired concentration is increased with
I. As a result, phase III slope is also
affected by I (Fig. 7B).
Concentration Profile in the Airway Tissue
The steady-state concentration profile in the airway tissue, as
predicted by the model, for different inhaled NO concentrations is
shown in Fig. 9. The radial distance
(x) is considered zero at the blood-tissue interface. For
CI less than ~418 ppb (derived from zero net flux or the
ratio a/b from Eq. 4), which is usually the
case for inspiration of ambient air, there is a net positive flux of NO
or absorption of NO by the airstream. For CI greater than
~418 ppb (i.e., when exogenous NO is inspired for clinical application), the net flux of NO is negative, or there is desorption of
NO from the airstream to the airway tissue. At these higher concentrations the tissue concentration profile is approximately linear
when t1/2 is 4 s. Under these circumstances, the
consumption rate of NO within the tissue is small compared with the
flux of NO from the airstream.
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Fig. 9.
Model-predicted steady-state radial concentration profile in airway
tissue as a function of CI. A: low concentrations
[0 < CI < 1 part/million (ppm)]. B: high
concentrations (CI = 50 or 100 ppm).
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Exogenous NO
In Fig. 10A we present the model
predictions for the fate of exogenous NO at concentrations that are
representative of clinical situations [CI = 10
parts/million (ppm)]. A single rebreathing maneuver of the tidal
volume (VI = 500 ml) is simulated, while I and E are
held constant (I and
E = 250 ml/s). The model predicts that
55.1% of inspired NO (90% of the NO that actually reaches the
alveoli) is absorbed by the pulmonary blood in the alveoli. In
contrast, only a very small portion (0.4%) of the inspired NO is
absorbed by the airway. The remaining 44.5% of inspired NO is exhaled.
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Fig. 10.
A: distribution of absorbed inhaled exogenous NO
(CI = 10 ppm) between alveoli and airway compartments for
tidal breathing. B: effect of flow rate and inspiratory
concentration on the percent NO absorbed by airway tissue
(I%,air). C: effect of inspiratory volume and
breath-hold time on I%,air.
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In Fig. 10, B and C, the percentage of NO absorbed by
the airway tissue is calculated under different conditions. In Fig.
10B, CI and flow rate change from 10 to 100 ppm and
from 50 to 1,000 ml/s, respectively. In Fig. 10C, simulations
are performed twice, with and without a 15-s breath hold, with
different inspired volumes (VI = 250-1,000 ml). The
percentage of inspired NO absorbed by the airway tissue
(I%,air) is inversely related to
E, VI, and
I and is a positive function of
breath-hold time and CI. Maximum absorption (8% of total
NO inspired) in the airway tissue occurs when residence time in the
airway compartment is maximized and contact with the alveolar
compartment is minimized: VI of 250 ml and 15-s breath
hold.
Reaction Rate Constant in Tissue
To investigate the sensitivity of the exhalation profile on the
reaction kinetics in the tissue, we varied t1/2
between 0.5 and 15 s, which corresponds to a range in k from
1.38 to 0.046 s1. Figure
11 demonstrates that
t1/2 values <4 s affect significantly the tissue
concentration profile. Decreasing t1/2 corresponds to a higher k or more rapid consumption of NO. The result is a reduced average concentration of NO in the tissue and an increased radial gradient at the airway wall. For low (CI 
418
ppb) inspired concentrations, the flux from the tissue to the airway is
reduced. For high (CI 
418 ppb) concentrations, the
flux to the tissue from the airstream increases.
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Fig. 11.
Effect of reaction rate [or half-life (t1/2)] on
radial tissue concentration profile for 2 different gas phase
concentrations: 100 ppb (A) and 50 ppm (B).
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DISCUSSION |
Model Validation
The model is able to simulate the different single-exhalation maneuvers
(Fig. 3) extremely well. This finding becomes even more significant if
one considers that only rough approximations of critical
parameters were used (DLNO,
app,alv, and
t,air) that were acquired by independent
means. No adjustment of model parameters was made to optimize the model
prediction of the experimental exhalation profile. The data are
presented for only a representative subject, inasmuch as our goal was
only to simulate the fundamental features of the exhalation profile
under dynamically changing conditions. The small differences between
the experimental profiles and the simulations can be attributed to
model simplifications and parameter estimation.
The model assumes that the lung consists of two well-defined, separate
regions: a rigid airway compartment and a well-mixed, expansile
alveolar compartment. Such a division represents an idealization of the
actual lungs. For example, the volume of the airways has been shown to
change during breathing, there is a transitional region (respiratory
bronchioles) between the airways and the alveolar region, and there may
be axial concentration gradients within the alveolar region (22).
Although our model does not consider these features, the error
introduced is minimal in relation to the model's ability to describe
the basic features of NO exchange.
The model appears to fail to predict the exact value of the initial
peak (phase I) in an inconsistent fashion. The peak is underestimated
in Fig. 3, A, B, and D, and overestimated in
the breath-hold maneuver (Fig. 3C). Because our model does
not include axial diffusion, one would expect the predicted peaks to be
overestimated. Axial dispersion and mixing, especially at the interface
between the alveolar and the airway regions, would create a broader
peak width and hence a smaller peak height. The underestimation of the
peak in Fig. 3, A, B, and D, is an artifact
that arises from the fact that the model simulates a step change for
the flow, from inspiration to expiration, while in reality a small
period of decreasing I or zero flow might
have occurred. In fact, a small breath-hold time (~0.5 s) between
inspiration and expiration may represent more accurately a
single-exhalation maneuver and will increase the height of the peak in
phase I.
The model's simulation of phase III under the wide range of flow
conditions is quite reasonable. Although we measured the average
concentration in phase III experimentally over only a small range of
flow rates, there is good agreement (Fig. 4). In addition, over a wider
range of flow rates, the model is able to predict the same functional
dependence reported by Silkoff et al. (24).
The phase III slope under constant-flow conditions and under
dynamically changing flow conditions is similar to the experimental data (Fig. 5), but several important differences exist. First, the
intercept (phase III slope at constant E)
is larger for the experimental data and importantly the model predicts
a zero intercept after a 15-s breath hold. Second, the slope of
vs.
is steeper for the experimental data with and without a 15-s breath hold. Both of these discrepancies are likely due to simplifications in
the model structure or parameter estimation.
The model predicts a zero intercept after a breath hold simply because
a period of 15 s is long enough for Calv to reach its steady-state value (as discussed previously). The structure of the
model only allows for two mechanisms to create a phase III slope:
1) continuing gas exchange and 2) nonconstant
absorption from the airway tissue. Continuing gas exchange is
eliminated if Calv reaches Calv,ss, and the
model predicts that, under ambient or physiological conditions
(CI less than ~50 ppb), the absorption of NO from the
airway tissue is essentially constant. Hence, the presence of a nonzero
intercept experimentally after a 15-s breath hold is evidence that
there are other mechanisms affecting gas exchange that cannot be
completely neglected. For example, inhomogeneity in alveolar
concentrations due to heterogeneous distributions of
ventilation-to-volume ratios has been shown to contribute to the
positive phase III slope of gases with similar physical characteristics (18, 19). In addition, axial or serial heterogeneity in the concentration of the gas in the respiratory region (respiratory bronchioles through the terminal alveoli) has been shown to contribute to the phase III slope of CO2, SF6, and He
(22). The current model, of course, assumes a well-mixed alveolar
compartment of uniform concentration that empties through a single
path. The model also neglects axial diffusion, which may serve to
increase Calv above its steady-state value during a
breath-hold maneuver.
Regarding the slope of
vs.
a number of possible reasons can justify the small difference with the experimental data. DLNO has been demonstrated
experimentally to be a positive function of lung volume (3). This
contrasts with CO and is due to the relatively rapid kinetics of NO
with hemoglobin. The rapid kinetics create a greater dependence of
DLNO with surface area available for diffusion.
The model assumes a constant DLNO throughout
the exhalation. In addition, DLNO varies among
subjects. The value of DLNO for subject
6 may be different from the mean value from Borland et al. (3) used
in the simulations. The value of DLNO can
affect the phase III slope by affecting the rate at which
Calv approaches Calv,ss [time constant is
equal to Valv(t)/DLNO].
Inspiratory conditions affect the phase III slope (Fig. 7),
Top
Abstract
Introduction
![C<SUB>alv</SUB>(<IT>t</IT>) = C<SUB>alv</SUB>(0)<IT>e</IT><SUP>−[D<SC>l</SC><SUB>NO</SUB>/V<SUB>alv</SUB>(<IT>t</IT>)]<IT>t</IT></SUP>](/content/vol85/issue2/fulltext/653/img024.gif)
