1Department of Anaesthesia, The University of Sydney, NSW 2060, Australia. 2Drägerwerk AG, Moislinger Allee 5355, D-23542 Lübeck, Germany*Corresponding author
Accepted for publication: April 17, 2000
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Abstract |
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Br J Anaesth 2000; 85: 3718
Keywords: measurement techniques, pulmonary arterial; lung, blood flow; lung, dead space; lung, volume; computers
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Introduction |
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Oxygen (O2) has a very low bloodgas partition coefficient of 0.024 in fully saturated arterial blood. In healthy well oxygenated lungs, variations in alveolar PO2 result in very small variations in pulmonary venous O2 content when high haemoglobin saturation (PO2>100 mm Hg) is maintained. If FIO2 is varied, then the O2 flux from the airway into the alveolar compartment can be considered to consist of two components: a constant unidirectional flux which brings mixed venous blood to arterial saturation; and a varying bi-directional flux which changes the fraction of O2 in the alveolar compartment. Hahn5 showed that if arterial haemoglobin is well saturated and oxygen uptake is constant, the constant unidirectional flux may be ignored and O2 can be used as an approximately insoluble indicator gas for the measurement of lung volumes. This technique has been verified experimentally in healthy adults.6
It would be convenient if N2O and O2 could be used as indicator gases in the same concentrations as they are commonly used during anaesthesia. We analysed the sinusoidal technique theoretically and used detailed computer modelling of dynamic multi-component gas exchange to investigate the systematic errors in estimates of dead space, alveolar volume and PBF that result when the sinusoidal technique is used as published with mean inspired fractions of indicator gases (O2 and N2O) greater than 10% as is commonly the case during general anaesthesia.
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Patients and methods |
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Computer
simulation
Computer simulation was used to investigate the
systematic errors introduced by the following assumptions in the derivation
of Hahns equations: (i) the indicator gases are present at low
concentrations; (ii) the absorption of the soluble indicator gas does not
affect expiratory flow; (iii) each indicator gas behaves independently of
all the other gases in the lung. The computer model (see Appendix)
comprises four simultaneous differential equations describing the mass
balance of four gases (O2, CO2, N2 and
N2O) in a single perfectly mixed constant-volume alveolar
compartment subjected to continuous inspiratory ventilation
V·I with a mixture of O2, N2
and N2O. These equations describe the complete mass balance of
the four gases and do not rely on the above assumptions. N2O is
exchanged with a constant PBF (Q·p) and the mixed
venous partial pressure of N2O is kept constant at the mean
inspired value. Hence, once steady-state sinusoidal equilibrium is
reached by the model, the mean N2O flux becomes zero. The body
compartment is assumed to be large enough to filter out all perturbations
in gas concentrations.7
A proportion (30%) of the total inspiratory gas flow bypasses the
alveolar compartment and mixes continuously with expired gas to represent
dead space ventilation, leaving a net inspiratory alveolar flow of
V·AI. Shunted blood flow, lung tissue absorption of
N2O and the effects of water vapour are not included in the
model as they do not influence the validity of this study. Constant oxygen
consumption is modelled by the removal of oxygen from the alveolar
compartment at a rate of 250 ml min1.
CO2 is added to the alveolar compartment at a rate equal to the
O2 consumption (RQ=1). Both N2
(bloodgas partition coefficient=0.012) and O2
(bloodgas partition coefficient in saturated blood=0.024) are
assumed to be insoluble, and the bloodgas partition coefficient of
N2O is assumed to be 0.47. The sum of gas partial pressures is
equal to atmospheric pressure at all times.
The model was implemented using Matlab and Simulink software (The MathWorks, Natick, MA, USA) and solved with the integration routine ode15 with variable step size and an absolute tolerance parameter of 1x106.
The inspired gas composition was
modulated sinusoidally with a peak-to-peak amplitude of 0.02 and
a period of 120 s which has been suggested to be the optimum period
for this technique.8
Alveolar volume was maintained constant at 2.5 litres by adjusting the
alveolar expiratory flow V·AE at each time step.
The initial values of the gas fractions in the lungs were set to the mean
inspired values and the model run for 1200 s to achieve
steady-state sinusoidal conditions. The relative amplitudes and phases
of the sinusoidal components of the inspired, mixed expired and
end-expired indicator gas fractions were estimated from the last
120 s of the simulation by fitting the expression
Psin (t +
) to
simulated partial pressure values using the GaussNewton method
(Matlab, The MathWorks), and these values were used to recover dead space
using equation (A9). Alveolar volume and PBF were estimated from the
amplitudes of the sinusoids using the following sets of equations (see
Appendix): (i) equations (A9), (A8) and (A6) evaluated sequentially; (ii)
equations (A9), (A8) and (A10) evaluated sequentially; and (iii) evaluation
of equation (A9) followed by simultaneous solution of equations (A10) and
(A13) by an interval bisection technique (Matlab, The MathWorks).
The simulation study was conducted in two parts using inspired gas mixtures comprising N2O, O2 and N2 under the following conditions.
(i) N2O was oscillated in anti-phase with N2 such that FIN2O + FIN2 was constant. FIO2 was not modulated. The mean FIN2 was kept constant at 0.01 while the mean FIN2O was first set to 0.01 and then varied from 0.1 to 0.7 in steps of 0.1. FIO2 was selected to make up the balance. This condition represents the case in which an inert gas (in this case N2) is used as an insoluble indicator gas at low concentration, and N2O as a soluble indicator gas at concentrations ranging between those typical of an indicator gas and those typical of an anaesthetic agent. These conditions were used to evaluate the performance of equations (A9), (A8), (A6) and (A10) as a function of mean FIN2O when the mean inspired fraction of the insoluble gas was low.
(ii) N2O was oscillated in anti-phase with
O2 such that
FIN2O +
FIO2 was constant. The mean
FIO2 took values of 0.2, 0.25 and
0.3, while mean FIN2O
was first set to 0.01 and then varied from 0.1 to 0.7 in steps of 0.1.
FIN2 was selected to make up the
balance and was not modulated. This condition represents the case in which
O2 is used as an insoluble indicator gas at physiological
concentrations and N2O as a soluble indicator gas at
concentrations ranging between that of a typical indicator gas and that of
an anaesthetic agent. The per unit sensitivities of PBF to errors in
alveolar volume and gas partial pressure measurements, and of alveolar
volume to errors in partial pressure measurements were calculated
numerically using a forward difference numerical technique as follows. If
y0 = y(x0)
and
y1 = y(x0 + x)
where
x is small compared with x0, then
the per unit sensitivity of y to errors in x is given
by
Each condition was simulated with PBF at values of 1, 5 and 10 litre min1 and the recovered PBF compared with the true value. Only systematic errors (referred to in this report simply as errors) related to approximations in the simplified equations were assessed.
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Results |
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(i) When N2 at a mean inspired fraction of 0.01 was used as the insoluble indicator gas, the absolute values of the systematic errors in estimates of dead space (equation (A9)) and alveolar volume (equation (A8)) were less than 0.5% for 0.01 < mean FIN2O < 0.70. PBF calculated from simulation results by the approximate equation (equation (A6)) and plotted in Fig. 2 exhibited systematic errors which were negative and increased in magnitude in approximate proportion to increasing mean FIN2O. The systematic errors in the corrected equation (equation (A10)) were substantially smaller (less than 3.5%).
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Discussion |
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The
difference between the performance of equations (A6) and (A10) in Fig.
2 can be explained as
follows. When the alveolar N2O concentration exceeds its mean
value, N2O flows from the alveolar gas into the blood, causing
expiratory flow to decrease. Similarly, a decrease in alveolar
N2O concentration causes expiratory flow to increase. Hence,
there is a sinusoidal component of alveolar expiratory flow superimposed on
the mean flow. In a subject in whom FRC is determined by the balance of
elastic and muscular forces, we would expect to observe an equivalent
behaviour superimposed on tidal gas movements. Expiratory flow carries
N2O out of the alveoli at a rate proportional to the absolute
alveolar N2O concentration, hence the sinusoidal component of
the expiratory flow causes a sinusoidal flux of N2O to be
superimposed on the mean expiratory flux, disturbing the mass balance of
N2O in the lung in proportion to the mean N2O
fraction. In equation (A10) this effect appears as the modifying term
(1 )1.
When the inspired fraction of N2O approaches 1
(
= 1), PBF
causes no change in the alveolar partial pressure of N2O and
hence a 100% error results. The simplified mass balance equation
(equation (A4)) is strictly true only when the indicator gas is present at
negligibly low concentrations and modulated at very low amplitudes.
Although the corrected equation apparently performs well under some
conditions in the model study (curves (ii) in Fig. 2), the sensitivity to errors in measurements
of the soluble gas concentration increases in inverse proportion to
(1
), hence this
approach is impractical at high
.
Equation (A13) is a complex (magnitude and phase) solution to the steady-state sinusoidal mass balance in the alveolar compartment when a soluble gas is modulated in anti-phase to an insoluble gas in the presence of at least one additional insoluble gas. If PBF is known, then equation (A13) can be solved for alveolar volume, or equations (A13) and (A10) can be solved simultaneously for alveolar volume and PBF when a third gas is present in the lungs.
When a non-respiring lung is ventilated with a modulated binary mixture of a soluble and insoluble indicator gas at constant barometric pressure, the modulation amplitudes of the indicators cannot be different. Hence, only one independent measurement can be made from which only one unknown can be estimated. In a respiring lung, the presence of CO2 transforms the alveolar gas into a ternary mixture, and therefore in principle both volume and blood flow estimates are possible. However, the large sensitivity to measurement errors (Fig. 6) makes this approach impractical.
Figure 3 shows that when the insoluble indicator gas (in this case O2) has a mean concentration greater than is typically used for indicator gases and is modulated at low amplitude in anti-phase with a soluble gas, then substantial but perhaps clinically acceptable (less than 10%) systematic errors in lung volume estimates result, even when the soluble gas is at a low mean concentration. These errors are approximately proportional to PBF and mean FIO2 and are caused by the sinusoidal component of the expiratory flow. These observations are related to the last term in equation (A12) which is neglected in equation (A8). When the interdependence between the indicator gases is not neglected (equation (A13)) the systematic errors are less than 1%, although the magnitude of the sensitivity to errors in O2 measurements is greater than unity (Fig. 7). The sensitivity to errors in soluble gas measurements exceeds unity only when FIN2O >0.5.
When the insoluble indicator gas is used at physiological inspired fractions (in this case O2 at 0.200.30), the errors in PBF estimated by equation (A6) (Fig. 4) are similar in pattern but slightly larger in magnitude than those obtained with N2 as insoluble indicator at an inspired fraction of 0.01 (Fig. 2). The negative bias of the corrected equation (A10) (curves (ii) in Fig. 4) results from errors in alveolar volume estimates (Fig. 3). Figure 5 shows that estimates obtained with equation (A10) are extremely sensitive to errors in alveolar volume under the conditions of this study, particularly at high FIN2O and low cardiac output. When alveolar volume and PBF are estimated simultaneously from noise-free measurements of soluble and insoluble gases modulated in anti-phase, then the systematic errors in PBF are small. However, the sensitivity analysis (Fig. 6) suggests that this technique is extremely sensitive to measurement error, particularly at low values of PBF.
This study has identified an upper limit on the performance of the sinusoidal forcing technique under idealized conditions, but has not examined all possible causes of error. Effects that were ignored, including non-equilibrium of the soluble indicator gas throughout the body, the solubility of the indicators in lung tissue, pulmonary shunt and variations in O2 consumption and in RQ, are likely to increase the uncertainty in estimates of cardiopulmonary variables. Pulmonary disease leading to maldistribution of ventilation and V·/Q· mismatch, and variations in FRC during anaesthesia are also likely to cause additional systematic and random errors. Tidal breathing, the limited dynamic response of gas analysers and measurement noise limit the accuracy with which mixed expired and alveolar gas composition can be measured, and hence also adversely affect the uncertainty in the estimates, particularly when the alveolar plateau slopes steeply.
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Conclusions |
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Acknowledgements |
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Appendix |
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Equations for recovering lung volumes and pulmonary blood flow
With reference to the
model in Fig. 1, the
mass balance of a single indicator gas with bloodgas partition
coefficient in a perfectly mixed single-compartment lung is
given by:
where
PI(t), PA(t) and
P(t) are the
time varying inspired, alveolar and mixed venous partial pressures of the
indicator gas. Alveolar volume VA and continuous
inspiratory alveolar ventilation (V·AI) are assumed
to be constant. V·AE(t) is expiratory
ventilation. Storage of indicator gas in lung tissue and pulmonary
capillary blood is ignored.
In the absence of metabolic gas exchange, or if the respiratory quotient is unity, the alveolar expiratory flow can be written as:
where
Q·p(PA(t) P
(t))
is the volumetric rate of uptake of the soluble indicator gas (in this case
N2O) by the pulmonary blood.
Under steady-state
sinusoidal conditions when the indicator gas is equilibrated throughout the
instrumentation and all the tissues of the subject, then
PI(t) and PA(t)
can be written as
PI(t) = +
PI(t),
PA(t) =
+
PA(t),
respectively, where
is the mean
partial pressure which is equal in all compartments, and
P(t) indicates the time-varying perturbation
about the mean in each compartment. If the frequency of the sinusoid is
high, the perturbations are strongly attenuated in the systemic
circulation, and mixed venous content of the indicator gas is constant and
P
(t)
0.
Substituting
for V·AI, PI,
PA and
P in equation (A1) and
introducing the alveolar time constant
= VA/V·AI
yields:
If
the perturbation amplitude is small and
<< 1, then
equation (A3) reduces to:
Equation (A4) is a linear
first-order differential equation. Steady-state sinusoidal
modulation of PI(t) at a frequency
rad s1 results in sinusoidal perturbation of
PA with amplitude given by:
Solving for Q·P yields:
If the inspired indicator is
insoluble ( = 0) and the concentration of the
soluble gas is negligible, then equation (A5) reduces
to:
Solving equation (A7) for alveolar volume yields:
Alveolar ventilation V·A is estimated by subtracting dead space ventilation V·D from the total ventilation V·E. Dead space ventilation is given by:2
where A and
are the phase differences
between the inspiratory and alveolar, and inspiratory and mixed expiratory
sinusoids, respectively, and
P
is the amplitude of
the sinusoidally modulated mixed expired partial pressure of the indicator
gas. Equation (A9) is a form of the standard Bohr dead space
equation.
Therefore, if the inspiratory concentration of an insoluble indicator gas is modulated sinusoidally about an arbitrary mean value and measurements of the amplitude and phase of the alveolar and mixed expiratory concentrations are made, dead space ventilation can be recovered using equation (A9) and alveolar ventilation estimated. Alveolar volume is estimated with equation (A8) and, if a soluble gas is oscillated simultaneously at low mean concentration, pulmonary blood flow can be estimated using equation (A6).
If
is not small then equation (A6)
becomes:
Most of the Oxford work2 68 subsequent to Bartons papers3 4 is based on equations (A4)(A9).
Simultaneous solution
Equation
(A10) allows the estimation of pulmonary blood flow from measurements of
inspired and alveolar partial pressures of a sinusoidally modulated soluble
gas at any mean partial pressure less than one atmosphere, if is
known. If we wish to estimate VA simultaneously from
measurements of an insoluble indicator gas modulated in anti-phase
with the soluble gas, then the mass balances of the two gases must be
considered simultaneously. Using the subscripts 1 and 2 to denote the
soluble and insoluble gases, respectively, and neglecting second-order
small terms, equation (A3) can be written as:
for the soluble gas and
for
the insoluble gas. Equations (A11) and (A12) must be satisfied
simultaneously. Transforming to the Laplace domain under steady-state
sinusoidal conditions and noting that
PI1(t) =
PI2(t)
when the soluble and insoluble gases are modulated in anti-phase,
equations (A10) and (A11) can be solved simultaneously to
yield:
where s is the Laplace
variable and the superscript * indicates a transformed
variable. Under steady-state sinusoidal conditions
s = i, where
The model
The model is based on four simultaneous equations describing the complete mass balance of O2, CO2, N2O and N2 in a constant-volume, single-compartment well mixed lung at constant temperature and pressure. No approximations are made other than to assume that all the gases obey the perfect gas law.
N2 is assumed to be
insoluble. PN2O is assumed
to be constant and equal to the mean inspired partial pressure of
N2. The total pressure in the alveolar compartment is always
equal to atmospheric pressure:
All gases are assumed to be ideal. For the jth gas:
Inspiratory alveolar ventilation is given by:
Expiratory alveolar ventilation (V·AE) is adjusted in each time step to maintain constant alveolar volume.
Total expiratory ventilation is given by:
In the mixed expired gas the partial pressure of the jth component is given by:
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References |
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2 Williams EM, Aspel JB, Burrough SML, et al. Assessment of cardiorespiratory function using oscillating inert gas forcing signals. J Appl Physiol 1994; 76: 21309
3 Barton SA, Black AMS, Hahn CEW. Dynamic models and solutions for evaluating ventilation, perfusion, and mass transfer in the lung Part I: the models. IEEE Trans Biomed Eng 1988; 35: 4507[ISI][Medline]
4 Barton SA, Black AMS, Hahn CEW. Dynamic models and solutions for evaluating ventilation, perfusion, and mass transfer in the lung Part II: analog solutions. IEEE Trans Biomed Eng 1988; 35: 45865[ISI][Medline]
5 Hahn CEW. Oxygen respiratory gas analysis by sine-wave measurement: a theoretical model. J Appl Physiol 1996; 81: 98597
6 Williams EM, Hammilton R, Sutton L, Viale JP, Hahn CEW. Alveolar and dead space volume measured by oscillations of inspired oxygen in awake adults. Am J Respir Crit Care Med 1997; 156: 18349
7 Hahn CEW, Black AMS, Barton SA, Scott I. Gas exchange in a three-compartment lung model analyzed by forcing sinusoids of N2O. J Appl Physiol 1993; 75: 186376[Abstract]
8 Williams EM, Hahn CEW. Measurements of cardiorespiratory function using single frequency gas concentration forcing signals. Adv Exp Med Biol 1994; 361: 18795[Medline]