Respiration Physiology (1975) 25, 217-234;
North-Holland Publishing
Companp, Amsterdam
THE EFFECT OF VARIATIONS IN AIRFLOW PATTERN ON GAS EXCHANGE. A THEORETICAL STUDY’
A. DAMOKOSH-GIORDANO, G. S. LONGOBARDO*, N. S. CHERNIACK Cardiovascular-Pulmonary
Diuision, Department of Medicine,
J. BAAN3 and
University of Pennsylcania, Philadelphia,
Pa. 19104. V.S.A.
Abstract. blood
It has been shown that gas exchange
is affected
theoretical
study
by the pattern shows
that
of airflow
between
the alveolar
at the mouth
even in the homogeneous
space
and
in the non-homogeneous lung,
the pattern
pulmonary
capillary
lung. The present
of airflow
can
affect
gas
exchange. When tidal volume, inspiratory and expiratory times remain constant, variations in the pattern of airflow result in significantly different values of steady state arterial PO, and Pc,,. This difference in steady state blood gases is exaggerated by low levels of minute ventilation and by long inspiratory times, but is unaffected by changes in the dilTusion coefficient of the alveolar+apillary membrane. Airflow
pattern
Alveolar+apillary
gas exchange
Breathing pattern Homogeneous lung
At any given level of ventilation, there is a tidal volume and frequency at which the chest bellows perform minimum external work (i.e. the work performed on the moving air). In addition to tidal volume and frequency, the pattern of airflow entering and leaving the lung has also been shown to affect the work of breathing. Yamashiro and Grodins (1971) have demonstrated that the inspiratory pattern that requires the least work is the square wave. The fact that the normal b\reathing pattern is not a square wave, suggests that criteria in addition to the work of breathing operate to determine the contour of airflow. In a lung where regional variations in resistance or compliance exist (that is, the Accepted for publication 16 July 1975. ’ Supported in part by USPHS grant 2 Present
address:
10604, U.S.A. 3 Present address: Hospital
International Laboratory
of the University
HL-08805.
Business for Clinical
of Leiden,
Machines Physiology,
Rynsburgerweg
Corp.
SDD
Department
Headquarters, of Pediatrics
IO, Leiden, The Netherlands. 217
White
Plains,
N.Y.
(Kindergeneeskunde),
218
A. DAMOKOSH-GIORDANO,
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERXIACK
of inspired lung is not “homogeneous”), the pattern of airflow affects the distribution gases and consequently, gas exchange between the lung and pulmonary capillary blood (Hedenstierna and Johansson, 1973; Riley et al.: 1951). The present theoretical study shows that even in a lung where there are no inhomogeneities, the pattern of airflow can affect gas exchange. DuBois et al. (1952) showed that fluctuation in alveolar P co2 during a breathing cycle is a function of instantaneous lung volume and airflow. In addition to the time course of alveolar Po, and Pco,, we have shown that different airflow patterns with the same tidal volume, mspiratory and expiratory time (i.e. frequency) can result in different alveolar-arterial pressure gradients and different blood gas tensions in the steady state. Gas exchange may be another factor, along with work of breathing, which affects the way the pattern of airflow is controlled. Methods The model shown in fig. 1 was used to simulate the effect of various airflow shapes on PO2 and Pco, in the body, when breathing air. The model is comprised of three major parts: the lung, arterial blood and venous blood with the rest of the body tissue. The lung is further subdivided into anatomic dead space and alveolar space compartments. Oxygen and COz move between the alveolar and arterial compartments by diffusion across an alveolar-arterial barrier, consisting of all the tissue lying between the air in the alveolus and the red blood cell in the pulmonary capillary. Everywhere else, the movement of gases is by the convection (bulk flow).
DEAD SPACE
Fig. 1. .Model of the lung (for symbols
see table 1).
of the alveolar-arterial barrier have been derived from P,, and Pco2 g radients mean capillary values. The volume of the alveolar compartment is a function of the rate of airflow and varies over a breath from functional residual capacity (FRC) to functional residual capacity plus tidal volume (FRC + VT). Metabolic rate, FRC and cardiac output are constant throughout all simulations; and all compartments are assumed to be well mixed. The
THE EFFECT OF AIRFLOW
There
PATTERN ON GAS EXCHANGE
are two ways by which the effect of changes
in airflow
219
on the dynamics
of oxygen and carbon dioxide can be simulated in this model. The first is to consider air as being forced into the lungs with a pump (as with a positive pressure respirator). The airflow in this kind of simulation does work on the lung and chest wall, and on the air already in the lung. Consequently, because of compression of alveolar air which causes changes in total pressure in the lung, the effect of air and lung compliance must be taken into account in computing the dynamics of lung gases. This method has been used in studying the effect of lung mechanics and airflow patterns on gas distribution in physical and mathematical models of the lung (Lyager: 1968; Jansson and Jonson, 1972; Hedenstierna and Johansson, 1973). The second way to simulate the effect of airflow patterns is to assume that flow rate at the mouth is the result of drawing air into the lungs. That is, the pleural pressure is assumed to vary in such a way that once compliance and inertance effect have been overcome, the resulting flow at the mouth follows a predetermined pattern. This second approach has been used by previous investigators (Yamamoto and Hori, 197 1; Flumcrfelt and Crandall, 1968; Lin and Cumming, 1973) to describe mathematically the changes in oxygen and carbon dioxide occurring in the lungs during the breathing cycle. This is the approach we have adopted since the air entering the lung does no elastic work, and therefore airflow patterns at the mouth remain undistorted as air moves into the alveolar space. The effect of specific airflow patterns on gas exchange obtained with the two different kinds of simulations need not be the same. However, the major objective of the present study was not to find the optimum pattern of airflow for gas exchange, but rather to determine how different contours of airflow might affect levels of gas exchange. Equations of the system The
following
compartments
equations
describe
of the model.
the changes
For explanation
in Po, and
of symbols,
Pco2 in the various
see table
1.
A. Dead space (during inspiration) dPt, 02 dt dPoco 2 dt
= V ( Poo7 - PD&)/Vn
= V ( Pocoz - Pt&&VD
The term V(Po) is the rate at which gas (0, or C02) enters the dead space from the outside. The term V(pU) is the rate at which gas goes from the dead space into the alveolar space. The volumetric flow of air (V) takes on various values depending on the flow pattern being simulated. The volume of dead space (VD) is 0.17 liters and is constant. The inspired air is assumed to have been brought to body temperature and to be fully saturated.
A. DAMOKOSH-GIORDANO,
220
G. S. LONGOBARDO, J. BAAN AND N. S. CHERNIACK
TABLE
1
List of symbols
Surface area of a diffusing
A
membrane
C
- Solubility of gas in blood (liters/liter.atm) -- Concentration (moles/liter)
D
- Diffusion
f
- frequency of breathing (breaths/minute) Hg) - Diffusion coefficient for oxygen (0.0011 moles/minute~mm -. Diffusion coefficient for carbon dioxide (0.011 moles/minute.mm .- Metabolic rate for oxygen and carbon dioxide (246 ml/minute) _ Pressure (mm Hg) - Blood flow rate (5 liters/minute)
Y
K K’ MR 8 Rg S S’
- Gas constant
VT
- Body temperature - Expirdtory
time
Inspiratory
Hg)
Hg)
cycle (minutes)
(37 “C) (IIIinUteS)
time (minutes)
- Volume (liters) -Tidal volume (liters) Volumetric
ii X
(mcter*/min)
(0.082 atm~liters/mole~“C)
Time for one breathing
Tl v
of a membrane
- Slope of the oxygen dissociation curve (moles/liter. mm Hg) - Slope of the carbon dioxide dissociation curve (molcs/hter.mm
T 5 I-E
constant
-
Thickness
flow rate (1iters;‘minutc) of a diffusing
membrane
(meters)
SpeciJicarion.5 A
- Alveolar arterial
a D
- Dead space
tz
-- Expirdtory
I
- Inspiratory
0
-
v ss
Atmospheric Venous
blood and tissue
- Steady state
Abbretiiations FRC - Functional residual capacity (2.4 liters) Negative triangular airflow pattern NT PT - Positive triangular airflow pattern PTS - Positive triangular inspiratory with square SPT -- Square
wave inspiratory with positive .~___ _
wave expiratory
triangular
expiratory
airflow
pattern
airflow
pattern
THE EFFECT OF AIRFLOW PATTERN ON GAS EXCHANGE
221
Dead space (during expiration) @Do, -
dt
_ -
. v(PD,,-p‘402)PD
dPko, _ . ___ - v(p%02 - PACOJ/VD dt The airflow rate (v) during expiration is negative. B. Alveolar space (during inspiration)
dPAcoz __
dt
_ -
[7j ( pDco2- PA,,,) + K’W
@co,
-
PAcoz)l/VA
These equations describe the change in alveolar PO, and Pco, during inspiration as functions of the rate of gas entering the alveolar space from the dead space, the change in alveolar volume and the net amount of gas diffusing across the alveolarcapillary membrane. VA is the instantaneous volume of the alveolar compartment. Alveolar space (during expiration) @A,, -
dt
dP&o, ~
dt
_ - [KRgT@o,, _ -
[K’Rg@co,
- PAo2)]/VA
-
PAco~)l/VA
During expiration, the increase in partial pressure due to the decreasing volume is compensated for by the decrease in pressure due to the outflow of gas. Consequently, the change in alveolar tension is only a function of diffusion across the lung. C. Arterial compartment
d% _ - [Q(Cvo, - Ca,,) + K(Pb2 - %,)l/SaVa dt
-
dP+,,
dt
= [WkOZ
- Ca&
+
K’(PACo2
-
P+o,)]/S’aVa
The change in arterial PO, and Pcoz is a function of the rate of gas entering and leaving the arterial compartment via the blood flow and the net diffusion between the lung and the blood. The slope of the oxygen dissociation curve for arterial blood (Sa) has been approximated by three straight lines at any one pH (i.e. it is piecewise linear).
222
A. DAMOKOSH-GIORDANOI
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERNIACK
D. Venous compartment dPv,l -_-_= dt
[O(Cao,
dpvcoz _ -._ dt
- Cvo7) -
[@Caco,
MRI/VVS~~
- Cvco,) + MR]/VvS’v
The change u-r venous P,, and Pco2 is a function of the rate of gas entering and leaving via the blood flow, and the metabolic rate. The Cohr and Haldane effects were taken into account in computing Sa, S’a. Sv and S’v.
(b)
(a)
‘VT
time
(d)
(f)
(e) Fig. 2. The simulated
patterns
(c) square
wave pattern;
triangular
expiration
of airflow:
(d) positive
(a) sinusoidal
triangular
(SF’T); and (f) positive
pattern triangular
pattern:
(b) negative
(PT); (e) square inspiration
triangular
wave inspiration
with square
pattern
(NT);
with positive
wave expiration
(PTS).
The patterns of airflow simulated are shown in fig. 2. These patterns included: sinusnidul, a rough approximation of the normal pattern; negatively sloped triangular pattern (NT), in which maximum flows occur early in inspiration and late in expiration; positicely sloped triangular pattern (PT), in which maximum flows occur late in inspiration and early in expiration; square wave pattern, which results in minimum inspiratory work; and combinations sf‘ the latter two (PTS and SPT patterns), which allow the effects of expiratory and inspiratory flow patterns on gas exchange to be compared. Simulations were done on an IBM 370-168 digital computer, using the 4th order Runga-Kutta integration procedure. Results All simulations of the effect of airflow patterns on O2 and CO2 exchange started at identical values of PO, and Pcoz in the lung and arterial blood. These were chosen
THE EETECT OF AIRFLOW
86
PAlTERN
223
ON GAS EXCHANGE
0 t
I
I
1
4
8
12
16
20
NUMBER
I
24
28
32
36
I 40
28
32
36
40
OF BREATHS
(b)
4
8
I2
I6
20
NUMBER
Fig. 3. The time course (b) over forty breaths,
of end-expiratory
using different patterns
values
24
OF BREATHS
for arterial
Po> (a) and end-expiratory
of airflow. VT = 0.46 liters, f= IO breaths/min,
arterial
P,,,
TI = 0.05 min.
so as to approximate normal physiological values (alveolar Po, = 100 mm Hg; arterial P,, = 90 mm Hg, alveolar Pcoz = 39 mm Hg and arterial Pcoz = 40 mm Hg). Figure 3a shows the simulated breath by breath changes in arterial PO2 at end-expiration, from the initial value to steady state; similar data are shown for the changes in arterial Pcoz in fig. 3b. It can be seen that with a tidal volume of 0.46 liters and a frequency of 10 breaths per minute, the steady state arterial blood gas tensions differ significantly. During the steady state, arterial P,, is highest when the SPT pattern is used and lowest when the sinusoidal pattern is used; the reverse is true for arterial PcoI. Figures 4a and 4b show the changes in arterial and alveolar PO, and Pcol over a steady state breath for each flow pattern. The areas between the alveolar and arterial curves are the same for all flow patterns. That is, in the steady state, all airflow
224
A. DAMOKOSH-GIORDANO,
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERNIACK
(0)
Fig. 4. Changes state breath,
(b)
TIME (Min.) in alveolar and arterial using various
airflow
TlME
Po, (a) and in alveolar patterns.
V~=0.46
and arterial
P,,
(Min.) f (b), during
liters, f= 10 breaths/mm,
T1=0.05
a steady min.
E ; 98ARTERIAL
i?
ALVEOLAR
100
t
ARTERIAL
06
06
ARTERIAL
.I00
.02
.04
.06
.06
.IO
TIME (Min.) Fig. 5. Changes
in alveolar and arterial patterns. V~=0.46
Po, during a non-steady liters. f= 10 breathsimin,
state breath using T1=0.05 min.
various
airflow
THE
EFFECT
OF AIRFLOW
PATTERN
ON GAS
EXCHANGE
225
patterns transferred the same amount of oxygen and carbon dioxide (an amount equal to the metabolic rate). This is expressed mathematically by the following (see table 1 for symbols). MR=fK
-T [PA(t)-Pa(t)]dt
I
“0
In contrast, during the unsteady state, the amounts of oxygen and carbon dioxide transferred by each flow pattern differed. This is illustrated in fig. 5 which shows the changes in alveolar and arterial Po, for each flow pattern during the 10th breath after the beginning of the simulation (a non-steady breath for all patterns). In the non-steady state the areas between curves are not equal for each flow pattern. It can be seen in fig. 6 that the amount of oxygen and carbon dioxide transferred into and out of the blood with the SPT pattern in the 10th breath exceeds
AIRFLOW PATTERNS
02
TRANSFER RATE (ml/mini
Fb I47
state (i.e. during
TRANSFER RATE (ml/mln)
250.9
260.9
250.6
246.4
k
247.9
24 1.9
b
247.0
246.4
244.0
197.1
244.0
179.2
Fig. 6. The effect of variations
steady
CO2
in airflow pattern on the average 0, and CO, transfer the 10th breath). Vr=0.46 liters, f= 10 brcaths;‘min. T1=0.05
rate in the nonmin., MR=246
ml:min.
the metabolic consumption of oxygen and the metabolic production of carbon dioxide. The opposite is true for the sinusoidal pattern. Thus, during the non-steady state, oxygen stores are increased by the SPT pattern, but are diminished by the sinusoidal pattern; while carbon dioxide stores are decreased by the SPT pattern and increased when the pattern is sinusoidal. This transient difference in storage resulting from various flow patterns accounts for the difference in steady state levels of arterial PoI and Pcoz. If after having reached the steady state with one pattern, the pattern of air flow is suddenly changed, the blood gases proceed to a new steady state. This is illustrated in lig. 7, which shows the increase in arterial P,, when the pattern is changed from sinusoidal to the PT pattern. Both the inspiratory and expiratory flow patterns affect gas transfer. This is
226
A. DAMOKOSH-GIORDANO,
76
I 8
G. S. LONGOBARDO,
I
I
I
16
24
32
+
NUMBER
Fig. 7. The effect of suddenly arterial
Po, values. The arrow
changing indicates
the pattern
J. BAAN AND N. S. CHERNIACK
I
I
I
I
I
40
46
56
64
72
OF
BREATHS
of airflow
where the pattern
I
60
on the time course
was switched
from sinusoidal
of end-expiratory to PT. VT=O.4
liters. f= 10 breathsimin.
P,O2
(mmHg)
pa co2
96.0
97.0
40.7
41.3
pa 02
97.5
95.8
95.0
pa co2
40.5
40.5
42.0
pa 02
92.0
96.0
78.6
pa 02
43.4
44.0
45.2
Fig. X. The eNect of changing the steady
99.0 40.4
state arterial
either the inspiratory
(columns)
Pcz and I-‘,,~. Vr=O.46
or the expiratory
liters, f= IO breathslmin,
(rows)
flow pattern
T1=0.05
on
min.
shown in fig. 8 which gives the steady state values of arterial Po, and Pcol at end-expiration, comparing patterns that differ only during the expiratory or inspiratory portion of a breath. It is apparent that the effect of the expiratory flow pattern depends on the inspiratory flow pattern. For example, the effect of changing the expiratory flow pattern from sinusoidal to triangular has a larger effect on steady state gas tensions when the inspiratory pattern is sinusoidal, than it does when the inspiratory pattern is triangular. The effect of altering ventilation, by increasing either tidal volume or frequency. on the steady state values for arterial Po-, and Pcoz is shown in figs. 9 and 10. Figure 9a illustrates the steady state value for arterial Paz achieved by using a variety of patterns, at live different minute ventilations. The higher the minute ventilation, the less the difference produced by the various airflow patterns. At a minute
THE EFFECT
OF AIRFLOW
PATTERt’i
q Omin’6.25
(a)
227
ON GAS EXCHANGE
L/MIN
x
5.0
t
4.6 4.0
3.6
PATTERNS
Fig. 9. (a) The effect of varying minute
ventilations.
airflow
patterns
OF
FLOW
on steady
state
values
for arterial
Po, at different
(b) Differences between the highest and lowest steady state values for arterial produced by varying minute ventilation at each airflow pattern.
(b)
(ai q Omin’6.25
x A . 0
L/MIN
5.0 4.6 4.0 3.6
PATTERNS
Fig. IO. (a) The effect of variations ferent minute ventilations.
in airflow
(b) Differences
P coL produced
PoL
pattern
between
by varying
minute
OF FLOW
on steady
the highest ventilation
state values
and lowest steady at each airflow
for arterial
PC,? at dif-
state values for arterial pattern.
ventilation of 6.25 liters per minute there is almost no difference in the steady state arterial P,, due to pattern variation; while, at a minute ventilation of 3.6 liters per minute there is a difference of more than 20 mm Hg between the PT and the sinusoidal patterns. From fig. 9b, it can be seen that the steady state values for arterial P,, reached by some patterns are affected more than others by a change in minute ventilation. For example, at a minute ventilation of 4.6, the PT pattern yields a lower steady state value for arterial PO2 than does the SPT pattern; but the
228
A. DAMOKOSH-GIORDANO,
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERNIACK
of 3.6. The SPT pattern is PT pattern yields a higher Po, at a minute ventilation therefore more sensitive to changes in ventilation than is the PT pattern. Overall, 0, exchange with the PT pattern is the least affected by changes in ventilation, whereas the sinusoidal pattern appears to be the most affected. Figure 10 shows the effect of changes in minute ventilation on steady state values for arterial PcoI, as a function of various patterns. Unlike oxygen, the effect of airflow patterns on arterial Pcoz is the same at all minute ventilations (fig. lob).
.I% 0%
(O)
@Ia xl%
(b)
014 51-
100 -
z
go-
E
E
00-
0” a”
70-
1
6o0
v \ I
I
I
.02
.04
06
5049
$
inspiratory
using various
airflow
-
4645-
% 8
44
-
4342-
w
I
4’0
.02
TIME
time on the steady patterns.
I
I
.06
INSPIRATORY
Fig. 11. The effect of varying
40 47
:
aa
-
I
.04
I
.06
II
I
.06
(MIN.1
state values for arterial
PO, (a) and Pco, (b)
V~=0.4 liters, f= IO breaths/min.
Figure 11 shows the effect on steady state values of arterial PO2 and Pco2 of changing the ratio between inspiratory and expiratory time for various patterns at a tidal volume of0.4 liters and a frequenc$ of 10 breaths per minute. When inspiratory time is short (0.02 minutes), there is little difference in the steady state values of arterial Po, and Pcoz among patterns. As inspiratory time is increased, the differences increase. Interestingly, the steady state values of arterial Paz that result from patterns with the same expiratory contour are affected similarly by changes in inspiratory time. That is, almost identical values for arterial Po, result from the SPT and PT patterns at each inspiratory time; the same applies to steady state values of arterial Po, for the PTS and square wave patterns. With an inspiratory time of 0.02 minutes, all patterns maintained relatively high steady state values for arterial PO2 and low values for arterial Pcoz. As inspiratory time is increased, steady state values for arterial Po, at first fall and then increase for all patterns except the NT pattern which became unstable at inspiratory times greater than 0.07 minutes. Steady state values for arterial Pco2 increased as inspiratory time was prolonged to about 0.07 minutes and then fell; again, except for the NT
THE EFFECT OF AIRFLOW
PATTERN
229
ON GAS EXCHANGE
(a)
(b)
80
70 .00055
.ooil .0022 DIFFUSION
Fig. 12. The effect of varying the SPT pattern
the diffusion
(a) with the sinusoidal
.Oll .coo55 0011 .0022 COEFFICIENT holes/Min-mmHg) cuefticient
airflow
pattern min.
on steady
state values for arterial
(b). V~=0.46
.Ol I P,,, comparing
liters, f= IO breaths/min,
T1=0.05
pattern. The sinusoidal pattern does not appear in fig. 11, since varying the inspiratory time of the sinusoidal pattern at a constant frequency alters the contour of the pattern. Increasing the diffusion coefficient for oxygen by a factor of 10 raised the steady state value for arterial PO, by about 9 mm Hg for all the patterns, while the steady state alveolar PO, was correspondingly lowered. Decreasing the diffusion coefficient by a factor of 2, lowered the steady state value for arterial P,, by about 9 mm Hg for all patterns; the steady state alveolar Po, was correspondingly raised. Figure 12 shows the effect of altering the diffusion coefficient on the steady state value for arterial Po, comparing the SPT and the sinusoidal patterns. It can be seen that at each value of the diffusion coeffkient, the relative effect of pattern variation on steady state arterial PO2 remains the same. It can also be seen that the effect of increasing the diffusion coefficient on the steady state Paoz approaches a limit (113 mm Hg for the SPT and 98 for the sinusoid). Using patterns other than those shown in fig. 11 also showed that the difference in steady state values for arterial Po, from pattern to pattern is constant at each value of the diffusion coefficient, even when the steady state alveolar-arterial Paz gradient is very small (i.e. when the diffusion coefficient is large). Steady state values for arterial PC., were virtually unaffected by changes in the diffusion coefficient for COz up to a factor of 10, possibly because the steady state alveolar-arterial gradient for CO, in the model was one tenth of that for oxygen. Discussion
The effect of the pattern of airflow on oxygen and carbon dioxide exchange between
230
A. DAMOKOSH-GIORDANO,
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERNIACK
alveolus and blood has been examined using a mathematical model lung. The results showed that even at the same tidal volume and in inspiratory and expiratory airflow pattern can result in different gas tensions. This difference: due to variation in airflow patterns, low minute ventilation diffusion coefficient.
or by a long inspiratory
of a homogeneous frequency, changes steady state blood is exaggerated by a
time; but is unaffected
by changes
in
Factors in the lung-blood system that affect the dynamics of blood gases over time are: the distribution of air in the lung, the blood distribution in the pulmonary capillaries, and the amount of gas that can be transferred per unit time across the alveolo-capillary membrane. In the simplified model used in the present study, the lung was considered to be homogeneous. Therefore, neither variations in the physical properties ofthediffusingmembrane, nor the ratio of ventilation to perfusion can account for the differences in steady state blood gas tension observed among the various flow patterns. Of the factors that contribute to the rate of diffusion in the model, the following are the ones affected by the pattern of airflow: (1) the airflow rate into (or out of) the alveolar compartment from (and to) the outside; (2) the instantaneous alveolar volume; (3) the concentration of oxygen and carbon dioxide in the blood; and (4) the chemical dissociation of oxygen and carbon dioxide in the blood. These four factors, all of which are functions of the airflow pattern, individually and in combination, alter the time course of the instantaneous alveolar-arterial diffusion gradient: and in turn, gas exchange. It has been observed experimentally that the diffusing capacity of the lung, as assessed by CO inhalation, varies according to whether single breath or steady state measurements are used (Ogilvie et al., 1957; Piiper and Sikand, 1966) even though the diffusion coefficient, a function of the physical properties of the diffusing membrane and those of the diffusing gas, probably remains the same (Filley et al., 1954; Jones and Mead, 1961). Kindig and Hazlett (1974) have shown that some of the discrepancy can theoretically be explained by differences in the breathing pattern used in the two techniques. This is understandable from the results of the present model, where it was seen that the rate of gas transfer across the alveolar-arterial barrier, and therefore the transient diffusing capacity, varies with the pattern of airflow even though the diffusion cocflicient is constant ‘. Only in the steady state, where the amount of O2 or CO, transferred into and out of the blood during a breath is equal to the metabolic rate, is the value of the diffusion coefficient equal to the value of the diffusing capacity. To illustrate how different patterns of flow can alter the time course of the pressure or concentrations gradient of a fluid across a diffusion barrier, consider the tank analogy shown in fig. 13. Tank A in each of the two systems corresponds to the alveolar compartment and tank B to the arterial compartment. Tanks A and B arc separated by a narrow tube, the resistance of which corresponds to the alveolar’ Diffusion coefficient K = (aDA),‘x. diffusing capacity gradient per breath. In the steady state, K=K*.
K* = ( PA - Pa)/MR.
where ( FA
-
Pa) is
the average
THE EFFECT OF AIRFLOW
PATTERN
SYSTEM I
Fig. 13. A tank Initial
volume
analogy
SYSTEM
to show the eNect of flow contours
of fluid in compartment
A is 8 liters,
difference
between
Time (min)
VA (liters)
offluid
II
on diffusion
of a fluid across
in B is 2 liters. and
a barrier.
the initial
height
A and B is 2 meters.
TABLE The time course of the transfer
volume
231
ON GAS EXCHANGE
2
from tank A to tank B (fig. l3), and the change over 4 minutes. Va (liters)
Ah (m)
TR (litersimin)
of the A-B gradient
Total amount transferred (liters)
_- - .._ SYSTEM
I
0
8.00
2.00
1.00
0.50
I 2
9.50 10.94
2.50 3.10
I.25 1.53
0.56 0.60
3
12.34
3.66
I.51
0.75
4
13.59
4.42
2.38
1.19
0 1 2
8.00 9.50 10.88
2.00 2.50 3.13
1.00 1.25 2.40
0.50 0.62 1.20
3
11.68
4.33
2.07
1.03
4
12.65
5.36
I .87
0.93
SYSTEM
3.60
II
VA = volume of fluid in A; VB = volume of fluid in B; Ah = height dilference and TR = transfer
4.28 or gradient,
between
A and B;
rate of the fluid from A to B.
arterial diffusion barrier. The initial volume of fluid in tanks A and B in the two systems are the same (corresponding to the A-a gradient). At time equal to 0, the valve between the two tahks is opened, and at the same time more fluid is introduced into tank A at the same constant rate in both systems. The shape of tank A in each system is analogous to two different shapes of airflow. Table 2 shows what happens to the height difference between A and B (the pressure gradient) and the amount of fluid transferred int.o B in systems I and 11 over a fixed period of time (inspiration). As expected, the time course followed by the gradient and the total amount transferred into tank B after a fixed period of time in the two systems is quite different. If the valve between the two compartments is open for a very short period of time (inspiratory time short), there can be very little difference in the
232
A. DAMOKOSH-GIORDANO,
G. S. LONGORARDO,
J. BAAN AND N. S. CHERNIACK
amounts transferred into B in the two systems. This was seen in fig. 11. In fig. 13, tank A (the alveolar compartment) is rigid and its volume is independent of flow. When flow and volume are varying simultaneously, as in the real system, the effect of airflow pattern on the alveolar-arterial Paz and Pco2 gradient can be viewed as an example of parametric pumping (Wilhelm et al., 1968). Parametric pumping is the action of one oscillatory field upon another, such that the phase relationship between the two fields causes dynamic coupling, and process amplification. The concept of parametric pumping has been proposed as a mechanism for active transport, that is “amplified” passive transport, of ions across cellular membranes (Wilhelm, 1966). In the present situation, the airflow pattern can be viewed as governing the phase and amplitude relationship among the oscillating fields of flow, volume and the alveolar--arterial gradient thereby causing gas transfer across the lung to be ‘amplified’ or ‘reduced’ by various degrees. The effect of variation in airflow pattern on steady state values for arterial PO, decreased at minute ventilations greater than 6 liters per minute, a near-normal value for human subjects (fig. 9). This result suggests that during normal ventilation, blood gas tensions will be largely unaffected by airflow patterns. It may also explain why the criterion of minimum work appears to determine patterns of breathing in normal individuals (Otis, 1964). In disease states, however, where alveolar ventilation is low, airflow pattern may be critical in determining blood gas tensions. Although a separate study of airflow patterns in patients with lung disease or idiopathicalveolar hypoventilation has not been’performed, it is possible that specific airflow patterns are used to improve gas transfer, even at the cost of additional work. It was also shown in figs. 9 and 10 that as minute ventilation is decreased, differences between’steady state values for arterial PO2 increase non-linearly from pattern to pattern., This can be attributed to the non-linearity of the oxygen dissociation curve. At high minute ventilation (and corresponding high values for arterial Po2) small fluctuations in the alveolar-arterial gradient, such as those caused by pattern variation. will not appreciably affect blood O2 content because of the flatness of the oxygen dissociation curve in that region. At low minute ventilations, however, the steepness of the dissociation curve can account for the large variations in steady state values for arterial Po, produced by small pertubations to the alveolararterial gradient. In contrast, the linearity of the CO, dissociation curve in the region of simulation can explain why changes in airflow pattern affect equally steady state values for arterial PcoJ at all minute ventilations. The effect of the shape of dissociation curves on gas transfer has been extensively investigated by West (1969/70), who noted that differences in shape had a large effect on gas transfer in the homogeneous lung. In contrast to increasing minute ventilation, increasing the diffusion coefficient has the same relative effect for all patterns on steady state values for arterial PU, (fig. 12). The interaction between airflow pattern and the time course of the alveolararterial gradient is unaffected by changes in the diffusion coefficient. Instead, in-
THE EFFECT OF AIRFLOW
233
PATTERN ON GAS EXCHANGE
creasing the diffusion coefficient causes more gas to be transferred at any one gradient; i.e., changes in the diffusion coefficient cause a reapportionment of gas between the alveolar and arterial compartments, whereas changes in ventilation cause change in both compartments. In the present
theoretical
study,
we have shown
that
variations
in the pattern
of
airflow do affect gas exchange in the homogeneous lung. From the results, it would be premature to predict which airflow pattern is most efhcient for gas exchange, since our model is a simplified version of the real system. Inhomogeneities in the lung and chest due to regional variations in resistance and compliance have not been included. However, one would expect that in disease states the effect of airflow pattern on gas exchange would be magnified. Pulsatilc blood flow and periodicity in capillary blood volume may also be important factors in gas exchange, in both the homogeneous and inhomogeneous lung. If one views the effect of airflow pattern on the alveolar-arterial gradient as an example of parametric pumping, the pulsatile blood flow and varying blood volume can be considered as additional oscillating fields (along with breathing pattern) that may damp out or enhance changes in gas transfer. References DuBois, A. B., A. G. Britt and W. 0. Fenn (1952). Alveolar CO, during the respiratory cycle. J. Appl. Physiol. 4: 535 -548. Filley, G. F., D. J. Maclntosh and G. W. Wright (1954). Carbon monoxide uptake and pulmonary diffusing capacity
in normal
Filley, G. F., D. B. Bigelow,
subjects
at rest and during
exercise. J. C/in. Inuest. 33: 530 539.
D. E. Olson and L. M. Lacquet
(1968). Pulmonary
gas transport.
Am. Rev.
Respir. Dis. 98: 480489. Flumerfelt,
R. W. and E. D. Crandall
(1968).
An analysis
of external
respiration
in man. Marh. Biosci.
3 : 205-230. Hedenstierna,
G. and H. Johansson
(1973).
Different
in a lung model study. Actu Anaesthesiol. Jansson,L.
and B. Jonson
(1972). A theoretical
Dis. 53 : 237-246. Jones, R. S. and J. Mead (1961). A theoretical of pulmonary Kindig,
diffusing
capacity
N. B. and D. R. Hazlett
diffusing
capacity.
flow patterns
study
of flow patterns
and experimental
by the single breath
(1974). The effects of breathing
S. (1968).
of ventilators.
analysis
method.
Sconrl. .I. Respir.
of anomalies
in the estimation
Q. J. Exp. Physiol. 46: 131-143. pattern
in the estimation
of pulmonary
Q. J. Exp. Physiol. 59: 31 l-329.
Lin, K. H. and G. Cumming (1973). A model of time varying a respiratory cycle at rest. Respir. Physiol. 17: 93 112. Lyager,
and their effect on gas distribution
Scanrl. 17: 190-196.
Influence
of flow pattern
on the distribution
gas exchange
in the human
of respiratory
air during
positive-pressure ventilation. Acta AnaesrhesioL Scard. 12: 191-211. Ogilvie, C. M., R. E. Forster, W. S. Blakemore and J. W. Morton (1957). A standardized technique for the clinical J. Clin. Invest. 36: I-17.
measurement
Otis, A. B. (1964). The work of breathing.
of the diffusing In: Handbook
capacity
of Physiology.
intermittent
breath
of the lung for carbon Baltimore,
lung during
Williams
holding
monoxide. and Wilkins
Co., Sec. III. Vol. I. p. 463. Piiper, J. and R. S. Sikand (1966). Determination lung: theory. Respir. Physiol. 1: 75 87. Riley, R. L, A. Cournard oxygen
and carbon
and K. W. Donald dioxide
of D co by the single breath (1951).
Analysis
in gas and blood of lungs:
of factors
methods.
method
affecting
in inhomogeneous partial
pressures
J. Appl. I’hq’siol. 4: 102 120.
of
234
A. DAMOKOSH-GIORDANO, G. S. LONGOBARDO, J. BAAN AND N. S. CHERNIACK
West, J. B. (1969/70). The effect of slope and shape of dissociation curve on pulmonary gas exchange. Respir. Physiol. 8: 66-85.
Wilhelm, R. H. (1966). Parametric pumping a model for active transport. In: Intracellular Transport. New York, Academic Press, p. 199. Wilhelm, R. H., A. W. Rice, R. W. Rolke and N. H. Sweed (1968). Parametric pumping, a dynamic principle for separating fluid mixtures. Ind. Eng. Chem. Fundam. 7: 337-349. Yamamoto, W. and T. Hori (1971). Phasic air movement of respiratory regulation of carbon dioxide balance. Comput. Biomed. Res. 3: 699-717. Yamashiro, S. M. and F. S. Grodins (1971). Optimal regulation of respiratory airflow. J. Appl. Physiol. 30: 597-602.
North-Holland Publishing
Companp, Amsterdam
THE EFFECT OF VARIATIONS IN AIRFLOW PATTERN ON GAS EXCHANGE. A THEORETICAL STUDY’
A. DAMOKOSH-GIORDANO, G. S. LONGOBARDO*, N. S. CHERNIACK Cardiovascular-Pulmonary
Diuision, Department of Medicine,
J. BAAN3 and
University of Pennsylcania, Philadelphia,
Pa. 19104. V.S.A.
Abstract. blood
It has been shown that gas exchange
is affected
theoretical
study
by the pattern shows
that
of airflow
between
the alveolar
at the mouth
even in the homogeneous
space
and
in the non-homogeneous lung,
the pattern
pulmonary
capillary
lung. The present
of airflow
can
affect
gas
exchange. When tidal volume, inspiratory and expiratory times remain constant, variations in the pattern of airflow result in significantly different values of steady state arterial PO, and Pc,,. This difference in steady state blood gases is exaggerated by low levels of minute ventilation and by long inspiratory times, but is unaffected by changes in the dilTusion coefficient of the alveolar+apillary membrane. Airflow
pattern
Alveolar+apillary
gas exchange
Breathing pattern Homogeneous lung
At any given level of ventilation, there is a tidal volume and frequency at which the chest bellows perform minimum external work (i.e. the work performed on the moving air). In addition to tidal volume and frequency, the pattern of airflow entering and leaving the lung has also been shown to affect the work of breathing. Yamashiro and Grodins (1971) have demonstrated that the inspiratory pattern that requires the least work is the square wave. The fact that the normal b\reathing pattern is not a square wave, suggests that criteria in addition to the work of breathing operate to determine the contour of airflow. In a lung where regional variations in resistance or compliance exist (that is, the Accepted for publication 16 July 1975. ’ Supported in part by USPHS grant 2 Present
address:
10604, U.S.A. 3 Present address: Hospital
International Laboratory
of the University
HL-08805.
Business for Clinical
of Leiden,
Machines Physiology,
Rynsburgerweg
Corp.
SDD
Department
Headquarters, of Pediatrics
IO, Leiden, The Netherlands. 217
White
Plains,
N.Y.
(Kindergeneeskunde),
218
A. DAMOKOSH-GIORDANO,
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERXIACK
of inspired lung is not “homogeneous”), the pattern of airflow affects the distribution gases and consequently, gas exchange between the lung and pulmonary capillary blood (Hedenstierna and Johansson, 1973; Riley et al.: 1951). The present theoretical study shows that even in a lung where there are no inhomogeneities, the pattern of airflow can affect gas exchange. DuBois et al. (1952) showed that fluctuation in alveolar P co2 during a breathing cycle is a function of instantaneous lung volume and airflow. In addition to the time course of alveolar Po, and Pco,, we have shown that different airflow patterns with the same tidal volume, mspiratory and expiratory time (i.e. frequency) can result in different alveolar-arterial pressure gradients and different blood gas tensions in the steady state. Gas exchange may be another factor, along with work of breathing, which affects the way the pattern of airflow is controlled. Methods The model shown in fig. 1 was used to simulate the effect of various airflow shapes on PO2 and Pco, in the body, when breathing air. The model is comprised of three major parts: the lung, arterial blood and venous blood with the rest of the body tissue. The lung is further subdivided into anatomic dead space and alveolar space compartments. Oxygen and COz move between the alveolar and arterial compartments by diffusion across an alveolar-arterial barrier, consisting of all the tissue lying between the air in the alveolus and the red blood cell in the pulmonary capillary. Everywhere else, the movement of gases is by the convection (bulk flow).
DEAD SPACE
Fig. 1. .Model of the lung (for symbols
see table 1).
of the alveolar-arterial barrier have been derived from P,, and Pco2 g radients mean capillary values. The volume of the alveolar compartment is a function of the rate of airflow and varies over a breath from functional residual capacity (FRC) to functional residual capacity plus tidal volume (FRC + VT). Metabolic rate, FRC and cardiac output are constant throughout all simulations; and all compartments are assumed to be well mixed. The
THE EFFECT OF AIRFLOW
There
PATTERN ON GAS EXCHANGE
are two ways by which the effect of changes
in airflow
219
on the dynamics
of oxygen and carbon dioxide can be simulated in this model. The first is to consider air as being forced into the lungs with a pump (as with a positive pressure respirator). The airflow in this kind of simulation does work on the lung and chest wall, and on the air already in the lung. Consequently, because of compression of alveolar air which causes changes in total pressure in the lung, the effect of air and lung compliance must be taken into account in computing the dynamics of lung gases. This method has been used in studying the effect of lung mechanics and airflow patterns on gas distribution in physical and mathematical models of the lung (Lyager: 1968; Jansson and Jonson, 1972; Hedenstierna and Johansson, 1973). The second way to simulate the effect of airflow patterns is to assume that flow rate at the mouth is the result of drawing air into the lungs. That is, the pleural pressure is assumed to vary in such a way that once compliance and inertance effect have been overcome, the resulting flow at the mouth follows a predetermined pattern. This second approach has been used by previous investigators (Yamamoto and Hori, 197 1; Flumcrfelt and Crandall, 1968; Lin and Cumming, 1973) to describe mathematically the changes in oxygen and carbon dioxide occurring in the lungs during the breathing cycle. This is the approach we have adopted since the air entering the lung does no elastic work, and therefore airflow patterns at the mouth remain undistorted as air moves into the alveolar space. The effect of specific airflow patterns on gas exchange obtained with the two different kinds of simulations need not be the same. However, the major objective of the present study was not to find the optimum pattern of airflow for gas exchange, but rather to determine how different contours of airflow might affect levels of gas exchange. Equations of the system The
following
compartments
equations
describe
of the model.
the changes
For explanation
in Po, and
of symbols,
Pco2 in the various
see table
1.
A. Dead space (during inspiration) dPt, 02 dt dPoco 2 dt
= V ( Poo7 - PD&)/Vn
= V ( Pocoz - Pt&&VD
The term V(Po) is the rate at which gas (0, or C02) enters the dead space from the outside. The term V(pU) is the rate at which gas goes from the dead space into the alveolar space. The volumetric flow of air (V) takes on various values depending on the flow pattern being simulated. The volume of dead space (VD) is 0.17 liters and is constant. The inspired air is assumed to have been brought to body temperature and to be fully saturated.
A. DAMOKOSH-GIORDANO,
220
G. S. LONGOBARDO, J. BAAN AND N. S. CHERNIACK
TABLE
1
List of symbols
Surface area of a diffusing
A
membrane
C
- Solubility of gas in blood (liters/liter.atm) -- Concentration (moles/liter)
D
- Diffusion
f
- frequency of breathing (breaths/minute) Hg) - Diffusion coefficient for oxygen (0.0011 moles/minute~mm -. Diffusion coefficient for carbon dioxide (0.011 moles/minute.mm .- Metabolic rate for oxygen and carbon dioxide (246 ml/minute) _ Pressure (mm Hg) - Blood flow rate (5 liters/minute)
Y
K K’ MR 8 Rg S S’
- Gas constant
VT
- Body temperature - Expirdtory
time
Inspiratory
Hg)
Hg)
cycle (minutes)
(37 “C) (IIIinUteS)
time (minutes)
- Volume (liters) -Tidal volume (liters) Volumetric
ii X
(mcter*/min)
(0.082 atm~liters/mole~“C)
Time for one breathing
Tl v
of a membrane
- Slope of the oxygen dissociation curve (moles/liter. mm Hg) - Slope of the carbon dioxide dissociation curve (molcs/hter.mm
T 5 I-E
constant
-
Thickness
flow rate (1iters;‘minutc) of a diffusing
membrane
(meters)
SpeciJicarion.5 A
- Alveolar arterial
a D
- Dead space
tz
-- Expirdtory
I
- Inspiratory
0
-
v ss
Atmospheric Venous
blood and tissue
- Steady state
Abbretiiations FRC - Functional residual capacity (2.4 liters) Negative triangular airflow pattern NT PT - Positive triangular airflow pattern PTS - Positive triangular inspiratory with square SPT -- Square
wave inspiratory with positive .~___ _
wave expiratory
triangular
expiratory
airflow
pattern
airflow
pattern
THE EFFECT OF AIRFLOW PATTERN ON GAS EXCHANGE
221
Dead space (during expiration) @Do, -
dt
_ -
. v(PD,,-p‘402)PD
dPko, _ . ___ - v(p%02 - PACOJ/VD dt The airflow rate (v) during expiration is negative. B. Alveolar space (during inspiration)
dPAcoz __
dt
_ -
[7j ( pDco2- PA,,,) + K’W
@co,
-
PAcoz)l/VA
These equations describe the change in alveolar PO, and Pco, during inspiration as functions of the rate of gas entering the alveolar space from the dead space, the change in alveolar volume and the net amount of gas diffusing across the alveolarcapillary membrane. VA is the instantaneous volume of the alveolar compartment. Alveolar space (during expiration) @A,, -
dt
dP&o, ~
dt
_ - [KRgT@o,, _ -
[K’Rg@co,
- PAo2)]/VA
-
PAco~)l/VA
During expiration, the increase in partial pressure due to the decreasing volume is compensated for by the decrease in pressure due to the outflow of gas. Consequently, the change in alveolar tension is only a function of diffusion across the lung. C. Arterial compartment
d% _ - [Q(Cvo, - Ca,,) + K(Pb2 - %,)l/SaVa dt
-
dP+,,
dt
= [WkOZ
- Ca&
+
K’(PACo2
-
P+o,)]/S’aVa
The change in arterial PO, and Pcoz is a function of the rate of gas entering and leaving the arterial compartment via the blood flow and the net diffusion between the lung and the blood. The slope of the oxygen dissociation curve for arterial blood (Sa) has been approximated by three straight lines at any one pH (i.e. it is piecewise linear).
222
A. DAMOKOSH-GIORDANOI
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERNIACK
D. Venous compartment dPv,l -_-_= dt
[O(Cao,
dpvcoz _ -._ dt
- Cvo7) -
[@Caco,
MRI/VVS~~
- Cvco,) + MR]/VvS’v
The change u-r venous P,, and Pco2 is a function of the rate of gas entering and leaving via the blood flow, and the metabolic rate. The Cohr and Haldane effects were taken into account in computing Sa, S’a. Sv and S’v.
(b)
(a)
‘VT
time
(d)
(f)
(e) Fig. 2. The simulated
patterns
(c) square
wave pattern;
triangular
expiration
of airflow:
(d) positive
(a) sinusoidal
triangular
(SF’T); and (f) positive
pattern triangular
pattern:
(b) negative
(PT); (e) square inspiration
triangular
wave inspiration
with square
pattern
(NT);
with positive
wave expiration
(PTS).
The patterns of airflow simulated are shown in fig. 2. These patterns included: sinusnidul, a rough approximation of the normal pattern; negatively sloped triangular pattern (NT), in which maximum flows occur early in inspiration and late in expiration; positicely sloped triangular pattern (PT), in which maximum flows occur late in inspiration and early in expiration; square wave pattern, which results in minimum inspiratory work; and combinations sf‘ the latter two (PTS and SPT patterns), which allow the effects of expiratory and inspiratory flow patterns on gas exchange to be compared. Simulations were done on an IBM 370-168 digital computer, using the 4th order Runga-Kutta integration procedure. Results All simulations of the effect of airflow patterns on O2 and CO2 exchange started at identical values of PO, and Pcoz in the lung and arterial blood. These were chosen
THE EETECT OF AIRFLOW
86
PAlTERN
223
ON GAS EXCHANGE
0 t
I
I
1
4
8
12
16
20
NUMBER
I
24
28
32
36
I 40
28
32
36
40
OF BREATHS
(b)
4
8
I2
I6
20
NUMBER
Fig. 3. The time course (b) over forty breaths,
of end-expiratory
using different patterns
values
24
OF BREATHS
for arterial
Po> (a) and end-expiratory
of airflow. VT = 0.46 liters, f= IO breaths/min,
arterial
P,,,
TI = 0.05 min.
so as to approximate normal physiological values (alveolar Po, = 100 mm Hg; arterial P,, = 90 mm Hg, alveolar Pcoz = 39 mm Hg and arterial Pcoz = 40 mm Hg). Figure 3a shows the simulated breath by breath changes in arterial PO2 at end-expiration, from the initial value to steady state; similar data are shown for the changes in arterial Pcoz in fig. 3b. It can be seen that with a tidal volume of 0.46 liters and a frequency of 10 breaths per minute, the steady state arterial blood gas tensions differ significantly. During the steady state, arterial P,, is highest when the SPT pattern is used and lowest when the sinusoidal pattern is used; the reverse is true for arterial PcoI. Figures 4a and 4b show the changes in arterial and alveolar PO, and Pcol over a steady state breath for each flow pattern. The areas between the alveolar and arterial curves are the same for all flow patterns. That is, in the steady state, all airflow
224
A. DAMOKOSH-GIORDANO,
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERNIACK
(0)
Fig. 4. Changes state breath,
(b)
TIME (Min.) in alveolar and arterial using various
airflow
TlME
Po, (a) and in alveolar patterns.
V~=0.46
and arterial
P,,
(Min.) f (b), during
liters, f= 10 breaths/mm,
T1=0.05
a steady min.
E ; 98ARTERIAL
i?
ALVEOLAR
100
t
ARTERIAL
06
06
ARTERIAL
.I00
.02
.04
.06
.06
.IO
TIME (Min.) Fig. 5. Changes
in alveolar and arterial patterns. V~=0.46
Po, during a non-steady liters. f= 10 breathsimin,
state breath using T1=0.05 min.
various
airflow
THE
EFFECT
OF AIRFLOW
PATTERN
ON GAS
EXCHANGE
225
patterns transferred the same amount of oxygen and carbon dioxide (an amount equal to the metabolic rate). This is expressed mathematically by the following (see table 1 for symbols). MR=fK
-T [PA(t)-Pa(t)]dt
I
“0
In contrast, during the unsteady state, the amounts of oxygen and carbon dioxide transferred by each flow pattern differed. This is illustrated in fig. 5 which shows the changes in alveolar and arterial Po, for each flow pattern during the 10th breath after the beginning of the simulation (a non-steady breath for all patterns). In the non-steady state the areas between curves are not equal for each flow pattern. It can be seen in fig. 6 that the amount of oxygen and carbon dioxide transferred into and out of the blood with the SPT pattern in the 10th breath exceeds
AIRFLOW PATTERNS
02
TRANSFER RATE (ml/mini
Fb I47
state (i.e. during
TRANSFER RATE (ml/mln)
250.9
260.9
250.6
246.4
k
247.9
24 1.9
b
247.0
246.4
244.0
197.1
244.0
179.2
Fig. 6. The effect of variations
steady
CO2
in airflow pattern on the average 0, and CO, transfer the 10th breath). Vr=0.46 liters, f= 10 brcaths;‘min. T1=0.05
rate in the nonmin., MR=246
ml:min.
the metabolic consumption of oxygen and the metabolic production of carbon dioxide. The opposite is true for the sinusoidal pattern. Thus, during the non-steady state, oxygen stores are increased by the SPT pattern, but are diminished by the sinusoidal pattern; while carbon dioxide stores are decreased by the SPT pattern and increased when the pattern is sinusoidal. This transient difference in storage resulting from various flow patterns accounts for the difference in steady state levels of arterial PoI and Pcoz. If after having reached the steady state with one pattern, the pattern of air flow is suddenly changed, the blood gases proceed to a new steady state. This is illustrated in lig. 7, which shows the increase in arterial P,, when the pattern is changed from sinusoidal to the PT pattern. Both the inspiratory and expiratory flow patterns affect gas transfer. This is
226
A. DAMOKOSH-GIORDANO,
76
I 8
G. S. LONGOBARDO,
I
I
I
16
24
32
+
NUMBER
Fig. 7. The effect of suddenly arterial
Po, values. The arrow
changing indicates
the pattern
J. BAAN AND N. S. CHERNIACK
I
I
I
I
I
40
46
56
64
72
OF
BREATHS
of airflow
where the pattern
I
60
on the time course
was switched
from sinusoidal
of end-expiratory to PT. VT=O.4
liters. f= 10 breathsimin.
P,O2
(mmHg)
pa co2
96.0
97.0
40.7
41.3
pa 02
97.5
95.8
95.0
pa co2
40.5
40.5
42.0
pa 02
92.0
96.0
78.6
pa 02
43.4
44.0
45.2
Fig. X. The eNect of changing the steady
99.0 40.4
state arterial
either the inspiratory
(columns)
Pcz and I-‘,,~. Vr=O.46
or the expiratory
liters, f= IO breathslmin,
(rows)
flow pattern
T1=0.05
on
min.
shown in fig. 8 which gives the steady state values of arterial Po, and Pcol at end-expiration, comparing patterns that differ only during the expiratory or inspiratory portion of a breath. It is apparent that the effect of the expiratory flow pattern depends on the inspiratory flow pattern. For example, the effect of changing the expiratory flow pattern from sinusoidal to triangular has a larger effect on steady state gas tensions when the inspiratory pattern is sinusoidal, than it does when the inspiratory pattern is triangular. The effect of altering ventilation, by increasing either tidal volume or frequency. on the steady state values for arterial Po-, and Pcoz is shown in figs. 9 and 10. Figure 9a illustrates the steady state value for arterial Paz achieved by using a variety of patterns, at live different minute ventilations. The higher the minute ventilation, the less the difference produced by the various airflow patterns. At a minute
THE EFFECT
OF AIRFLOW
PATTERt’i
q Omin’6.25
(a)
227
ON GAS EXCHANGE
L/MIN
x
5.0
t
4.6 4.0
3.6
PATTERNS
Fig. 9. (a) The effect of varying minute
ventilations.
airflow
patterns
OF
FLOW
on steady
state
values
for arterial
Po, at different
(b) Differences between the highest and lowest steady state values for arterial produced by varying minute ventilation at each airflow pattern.
(b)
(ai q Omin’6.25
x A . 0
L/MIN
5.0 4.6 4.0 3.6
PATTERNS
Fig. IO. (a) The effect of variations ferent minute ventilations.
in airflow
(b) Differences
P coL produced
PoL
pattern
between
by varying
minute
OF FLOW
on steady
the highest ventilation
state values
and lowest steady at each airflow
for arterial
PC,? at dif-
state values for arterial pattern.
ventilation of 6.25 liters per minute there is almost no difference in the steady state arterial P,, due to pattern variation; while, at a minute ventilation of 3.6 liters per minute there is a difference of more than 20 mm Hg between the PT and the sinusoidal patterns. From fig. 9b, it can be seen that the steady state values for arterial P,, reached by some patterns are affected more than others by a change in minute ventilation. For example, at a minute ventilation of 4.6, the PT pattern yields a lower steady state value for arterial PO2 than does the SPT pattern; but the
228
A. DAMOKOSH-GIORDANO,
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERNIACK
of 3.6. The SPT pattern is PT pattern yields a higher Po, at a minute ventilation therefore more sensitive to changes in ventilation than is the PT pattern. Overall, 0, exchange with the PT pattern is the least affected by changes in ventilation, whereas the sinusoidal pattern appears to be the most affected. Figure 10 shows the effect of changes in minute ventilation on steady state values for arterial PcoI, as a function of various patterns. Unlike oxygen, the effect of airflow patterns on arterial Pcoz is the same at all minute ventilations (fig. lob).
.I% 0%
(O)
@Ia xl%
(b)
014 51-
100 -
z
go-
E
E
00-
0” a”
70-
1
6o0
v \ I
I
I
.02
.04
06
5049
$
inspiratory
using various
airflow
-
4645-
% 8
44
-
4342-
w
I
4’0
.02
TIME
time on the steady patterns.
I
I
.06
INSPIRATORY
Fig. 11. The effect of varying
40 47
:
aa
-
I
.04
I
.06
II
I
.06
(MIN.1
state values for arterial
PO, (a) and Pco, (b)
V~=0.4 liters, f= IO breaths/min.
Figure 11 shows the effect on steady state values of arterial PO2 and Pco2 of changing the ratio between inspiratory and expiratory time for various patterns at a tidal volume of0.4 liters and a frequenc$ of 10 breaths per minute. When inspiratory time is short (0.02 minutes), there is little difference in the steady state values of arterial Po, and Pcoz among patterns. As inspiratory time is increased, the differences increase. Interestingly, the steady state values of arterial Paz that result from patterns with the same expiratory contour are affected similarly by changes in inspiratory time. That is, almost identical values for arterial Po, result from the SPT and PT patterns at each inspiratory time; the same applies to steady state values of arterial Po, for the PTS and square wave patterns. With an inspiratory time of 0.02 minutes, all patterns maintained relatively high steady state values for arterial PO2 and low values for arterial Pcoz. As inspiratory time is increased, steady state values for arterial Po, at first fall and then increase for all patterns except the NT pattern which became unstable at inspiratory times greater than 0.07 minutes. Steady state values for arterial Pco2 increased as inspiratory time was prolonged to about 0.07 minutes and then fell; again, except for the NT
THE EFFECT OF AIRFLOW
PATTERN
229
ON GAS EXCHANGE
(a)
(b)
80
70 .00055
.ooil .0022 DIFFUSION
Fig. 12. The effect of varying the SPT pattern
the diffusion
(a) with the sinusoidal
.Oll .coo55 0011 .0022 COEFFICIENT holes/Min-mmHg) cuefticient
airflow
pattern min.
on steady
state values for arterial
(b). V~=0.46
.Ol I P,,, comparing
liters, f= IO breaths/min,
T1=0.05
pattern. The sinusoidal pattern does not appear in fig. 11, since varying the inspiratory time of the sinusoidal pattern at a constant frequency alters the contour of the pattern. Increasing the diffusion coefficient for oxygen by a factor of 10 raised the steady state value for arterial PO, by about 9 mm Hg for all the patterns, while the steady state alveolar PO, was correspondingly lowered. Decreasing the diffusion coefficient by a factor of 2, lowered the steady state value for arterial P,, by about 9 mm Hg for all patterns; the steady state alveolar Po, was correspondingly raised. Figure 12 shows the effect of altering the diffusion coefficient on the steady state value for arterial Po, comparing the SPT and the sinusoidal patterns. It can be seen that at each value of the diffusion coeffkient, the relative effect of pattern variation on steady state arterial PO2 remains the same. It can also be seen that the effect of increasing the diffusion coefficient on the steady state Paoz approaches a limit (113 mm Hg for the SPT and 98 for the sinusoid). Using patterns other than those shown in fig. 11 also showed that the difference in steady state values for arterial Po, from pattern to pattern is constant at each value of the diffusion coefficient, even when the steady state alveolar-arterial Paz gradient is very small (i.e. when the diffusion coefficient is large). Steady state values for arterial PC., were virtually unaffected by changes in the diffusion coefficient for COz up to a factor of 10, possibly because the steady state alveolar-arterial gradient for CO, in the model was one tenth of that for oxygen. Discussion
The effect of the pattern of airflow on oxygen and carbon dioxide exchange between
230
A. DAMOKOSH-GIORDANO,
G. S. LONGOBARDO,
J. BAAN AND N. S. CHERNIACK
alveolus and blood has been examined using a mathematical model lung. The results showed that even at the same tidal volume and in inspiratory and expiratory airflow pattern can result in different gas tensions. This difference: due to variation in airflow patterns, low minute ventilation diffusion coefficient.
or by a long inspiratory
of a homogeneous frequency, changes steady state blood is exaggerated by a
time; but is unaffected
by changes
in
Factors in the lung-blood system that affect the dynamics of blood gases over time are: the distribution of air in the lung, the blood distribution in the pulmonary capillaries, and the amount of gas that can be transferred per unit time across the alveolo-capillary membrane. In the simplified model used in the present study, the lung was considered to be homogeneous. Therefore, neither variations in the physical properties ofthediffusingmembrane, nor the ratio of ventilation to perfusion can account for the differences in steady state blood gas tension observed among the various flow patterns. Of the factors that contribute to the rate of diffusion in the model, the following are the ones affected by the pattern of airflow: (1) the airflow rate into (or out of) the alveolar compartment from (and to) the outside; (2) the instantaneous alveolar volume; (3) the concentration of oxygen and carbon dioxide in the blood; and (4) the chemical dissociation of oxygen and carbon dioxide in the blood. These four factors, all of which are functions of the airflow pattern, individually and in combination, alter the time course of the instantaneous alveolar-arterial diffusion gradient: and in turn, gas exchange. It has been observed experimentally that the diffusing capacity of the lung, as assessed by CO inhalation, varies according to whether single breath or steady state measurements are used (Ogilvie et al., 1957; Piiper and Sikand, 1966) even though the diffusion coefficient, a function of the physical properties of the diffusing membrane and those of the diffusing gas, probably remains the same (Filley et al., 1954; Jones and Mead, 1961). Kindig and Hazlett (1974) have shown that some of the discrepancy can theoretically be explained by differences in the breathing pattern used in the two techniques. This is understandable from the results of the present model, where it was seen that the rate of gas transfer across the alveolar-arterial barrier, and therefore the transient diffusing capacity, varies with the pattern of airflow even though the diffusion cocflicient is constant ‘. Only in the steady state, where the amount of O2 or CO, transferred into and out of the blood during a breath is equal to the metabolic rate, is the value of the diffusion coefficient equal to the value of the diffusing capacity. To illustrate how different patterns of flow can alter the time course of the pressure or concentrations gradient of a fluid across a diffusion barrier, consider the tank analogy shown in fig. 13. Tank A in each of the two systems corresponds to the alveolar compartment and tank B to the arterial compartment. Tanks A and B arc separated by a narrow tube, the resistance of which corresponds to the alveolar’ Diffusion coefficient K = (aDA),‘x. diffusing capacity gradient per breath. In the steady state, K=K*.
K* = ( PA - Pa)/MR.
where ( FA
-
Pa) is
the average
THE EFFECT OF AIRFLOW
PATTERN
SYSTEM I
Fig. 13. A tank Initial
volume
analogy
SYSTEM
to show the eNect of flow contours
of fluid in compartment
A is 8 liters,
difference
between
Time (min)
VA (liters)
offluid
II
on diffusion
of a fluid across
in B is 2 liters. and
a barrier.
the initial
height
A and B is 2 meters.
TABLE The time course of the transfer
volume
231
ON GAS EXCHANGE
2
from tank A to tank B (fig. l3), and the change over 4 minutes. Va (liters)
Ah (m)
TR (litersimin)
of the A-B gradient
Total amount transferred (liters)
_- - .._ SYSTEM
I
0
8.00
2.00
1.00
0.50
I 2
9.50 10.94
2.50 3.10
I.25 1.53
0.56 0.60
3
12.34
3.66
I.51
0.75
4
13.59
4.42
2.38
1.19
0 1 2
8.00 9.50 10.88
2.00 2.50 3.13
1.00 1.25 2.40
0.50 0.62 1.20
3
11.68
4.33
2.07
1.03
4
12.65
5.36
I .87
0.93
SYSTEM
3.60
II
VA = volume of fluid in A; VB = volume of fluid in B; Ah = height dilference and TR = transfer
4.28 or gradient,
between
A and B;
rate of the fluid from A to B.
arterial diffusion barrier. The initial volume of fluid in tanks A and B in the two systems are the same (corresponding to the A-a gradient). At time equal to 0, the valve between the two tahks is opened, and at the same time more fluid is introduced into tank A at the same constant rate in both systems. The shape of tank A in each system is analogous to two different shapes of airflow. Table 2 shows what happens to the height difference between A and B (the pressure gradient) and the amount of fluid transferred int.o B in systems I and 11 over a fixed period of time (inspiration). As expected, the time course followed by the gradient and the total amount transferred into tank B after a fixed period of time in the two systems is quite different. If the valve between the two compartments is open for a very short period of time (inspiratory time short), there can be very little difference in the
232
A. DAMOKOSH-GIORDANO,
G. S. LONGORARDO,
J. BAAN AND N. S. CHERNIACK
amounts transferred into B in the two systems. This was seen in fig. 11. In fig. 13, tank A (the alveolar compartment) is rigid and its volume is independent of flow. When flow and volume are varying simultaneously, as in the real system, the effect of airflow pattern on the alveolar-arterial Paz and Pco2 gradient can be viewed as an example of parametric pumping (Wilhelm et al., 1968). Parametric pumping is the action of one oscillatory field upon another, such that the phase relationship between the two fields causes dynamic coupling, and process amplification. The concept of parametric pumping has been proposed as a mechanism for active transport, that is “amplified” passive transport, of ions across cellular membranes (Wilhelm, 1966). In the present situation, the airflow pattern can be viewed as governing the phase and amplitude relationship among the oscillating fields of flow, volume and the alveolar--arterial gradient thereby causing gas transfer across the lung to be ‘amplified’ or ‘reduced’ by various degrees. The effect of variation in airflow pattern on steady state values for arterial PO, decreased at minute ventilations greater than 6 liters per minute, a near-normal value for human subjects (fig. 9). This result suggests that during normal ventilation, blood gas tensions will be largely unaffected by airflow patterns. It may also explain why the criterion of minimum work appears to determine patterns of breathing in normal individuals (Otis, 1964). In disease states, however, where alveolar ventilation is low, airflow pattern may be critical in determining blood gas tensions. Although a separate study of airflow patterns in patients with lung disease or idiopathicalveolar hypoventilation has not been’performed, it is possible that specific airflow patterns are used to improve gas transfer, even at the cost of additional work. It was also shown in figs. 9 and 10 that as minute ventilation is decreased, differences between’steady state values for arterial PO2 increase non-linearly from pattern to pattern., This can be attributed to the non-linearity of the oxygen dissociation curve. At high minute ventilation (and corresponding high values for arterial Po2) small fluctuations in the alveolar-arterial gradient, such as those caused by pattern variation. will not appreciably affect blood O2 content because of the flatness of the oxygen dissociation curve in that region. At low minute ventilations, however, the steepness of the dissociation curve can account for the large variations in steady state values for arterial Po, produced by small pertubations to the alveolararterial gradient. In contrast, the linearity of the CO, dissociation curve in the region of simulation can explain why changes in airflow pattern affect equally steady state values for arterial PcoJ at all minute ventilations. The effect of the shape of dissociation curves on gas transfer has been extensively investigated by West (1969/70), who noted that differences in shape had a large effect on gas transfer in the homogeneous lung. In contrast to increasing minute ventilation, increasing the diffusion coefficient has the same relative effect for all patterns on steady state values for arterial PU, (fig. 12). The interaction between airflow pattern and the time course of the alveolararterial gradient is unaffected by changes in the diffusion coefficient. Instead, in-
THE EFFECT OF AIRFLOW
233
PATTERN ON GAS EXCHANGE
creasing the diffusion coefficient causes more gas to be transferred at any one gradient; i.e., changes in the diffusion coefficient cause a reapportionment of gas between the alveolar and arterial compartments, whereas changes in ventilation cause change in both compartments. In the present
theoretical
study,
we have shown
that
variations
in the pattern
of
airflow do affect gas exchange in the homogeneous lung. From the results, it would be premature to predict which airflow pattern is most efhcient for gas exchange, since our model is a simplified version of the real system. Inhomogeneities in the lung and chest due to regional variations in resistance and compliance have not been included. However, one would expect that in disease states the effect of airflow pattern on gas exchange would be magnified. Pulsatilc blood flow and periodicity in capillary blood volume may also be important factors in gas exchange, in both the homogeneous and inhomogeneous lung. If one views the effect of airflow pattern on the alveolar-arterial gradient as an example of parametric pumping, the pulsatile blood flow and varying blood volume can be considered as additional oscillating fields (along with breathing pattern) that may damp out or enhance changes in gas transfer. References DuBois, A. B., A. G. Britt and W. 0. Fenn (1952). Alveolar CO, during the respiratory cycle. J. Appl. Physiol. 4: 535 -548. Filley, G. F., D. J. Maclntosh and G. W. Wright (1954). Carbon monoxide uptake and pulmonary diffusing capacity
in normal
Filley, G. F., D. B. Bigelow,
subjects
at rest and during
exercise. J. C/in. Inuest. 33: 530 539.
D. E. Olson and L. M. Lacquet
(1968). Pulmonary
gas transport.
Am. Rev.
Respir. Dis. 98: 480489. Flumerfelt,
R. W. and E. D. Crandall
(1968).
An analysis
of external
respiration
in man. Marh. Biosci.
3 : 205-230. Hedenstierna,
G. and H. Johansson
(1973).
Different
in a lung model study. Actu Anaesthesiol. Jansson,L.
and B. Jonson
(1972). A theoretical
Dis. 53 : 237-246. Jones, R. S. and J. Mead (1961). A theoretical of pulmonary Kindig,
diffusing
capacity
N. B. and D. R. Hazlett
diffusing
capacity.
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study
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by the single breath
(1974). The effects of breathing
S. (1968).
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Sconrl. .I. Respir.
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Lin, K. H. and G. Cumming (1973). A model of time varying a respiratory cycle at rest. Respir. Physiol. 17: 93 112. Lyager,
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Scanrl. 17: 190-196.
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Otis, A. B. (1964). The work of breathing.
of the diffusing In: Handbook
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holding
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Co., Sec. III. Vol. I. p. 463. Piiper, J. and R. S. Sikand (1966). Determination lung: theory. Respir. Physiol. 1: 75 87. Riley, R. L, A. Cournard oxygen
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West, J. B. (1969/70). The effect of slope and shape of dissociation curve on pulmonary gas exchange. Respir. Physiol. 8: 66-85.
Wilhelm, R. H. (1966). Parametric pumping a model for active transport. In: Intracellular Transport. New York, Academic Press, p. 199. Wilhelm, R. H., A. W. Rice, R. W. Rolke and N. H. Sweed (1968). Parametric pumping, a dynamic principle for separating fluid mixtures. Ind. Eng. Chem. Fundam. 7: 337-349. Yamamoto, W. and T. Hori (1971). Phasic air movement of respiratory regulation of carbon dioxide balance. Comput. Biomed. Res. 3: 699-717. Yamashiro, S. M. and F. S. Grodins (1971). Optimal regulation of respiratory airflow. J. Appl. Physiol. 30: 597-602.
