Abstract
Clark, Justin S., Yuxiang J. Lin, Michael J. Criddle, Antonio G. Cutillo, Adelbert H. Bigler, Fred L. Farr, and Attilio D. Renzetti, Jr. Cardiac output and mixed venous oxygen content measurements by a tracer bolus method: theory. J. Appl. Physiol. 83(3): 884–896, 1997.—We present a bolus method of inertgas delivery to the lungs that facilitates application of multiple inert gases and the multiple inertgasexchange technique (MIGET) model to noninvasive measurements of cardiac output (CO) and central mixed venous oxygen content
 multiple inertgasexchange technique
 ventilationperfusion ratio
a practical, noninvasive method for measuring cardiac output (CO) and mixed venous oxygen content
Stout et al. (15) introduced an opencircuit, multiplebreath (MB) method for determining Q˙eff that does not require patient cooperation or introduce hyperventilation. However, the same recirculation dilemma cited for the RB method also exists for the MB method.
As a solution to the recirculation dilemma, Zwart et al. (21) introduced a steadystate (SS), opencircuit method for determining the overall ventilationperfusion ratio (V˙/Q˙). The method utilizes a timemodulated inertgas concentration profile introduced at the airway. The V˙/Q˙ is determined from comparisons of peaktopeak amplitudes of the steadystate expired and inspired inertgas concentration profiles. Ventilation is measured independently. Reduction of recirculation error is controlled by systemic filtering efficiency, which is maximized by selection of the inert gas. A high filtering efficiency is obtained through the use of the inert gas halothane, which has a high tissuetoblood solubility ratio. Thus, the SS method removes the recirculation error vs. data availability dilemmas of both the RB and MB methods while eliminating the need for subject cooperation.
A second limitation of both the MB and SS models, as reported, is their failure to account for V˙/Q˙distribution inhomogeneities. Whereas the method of Zwart et al. (21) has been reported to be capable of extension to a multigas, multicompartment model system for dealing with inhomogeneity, no such extension has yet been reported. This may be due, in part, to difficulties in finding gases that have sufficient filtering efficiencies.
This paper presents a combined method for measuring CO and
MODEL DESCRIPTION
The tracer bolus is injected into the airway (synchronized with inspiration) for a series of sequential breaths and then ceased for an equal number of breaths, approximating a squarewave input of period T, as shown in Fig. 1.
Calculation of average endcapillary perfusion
( Q ˙ c ¯ )
andV˙a.
With use of the parallel gasexchange model of MIGET (18), the mass balance equation for an inert tracer corresponding to lungcompartment i, approximated by continuous ventilation (seeappendix
for comparison with discontinuous ventilation model), is given by
where P_{i}(t ) is partial pressure of tracer gas in the alveoli of compartment i;Pc_{i}(t ) is partial pressure of tracer gas in the endcapillary blood of compartment i;
For inert gases, attainment of equilibrium between capillary blood and alveolar gas is normally assumed (19), providing the equality Pc_{i} = P_{i}. Equating Pc_{i}(t ) and Pc(t ), ignoring the time dependence of mixed venous pressure
To the extent that time invariance forV˙_{i},Q˙_{i}, and V_{i} can be assumed over a measurement cycle, Eq. 2
approximates a firstorder, linear differential equation where steadystate response to a squarewave input of a tracer gas j (a cycle consisting of consecutive bolus breaths followed by an equal number of nonbolus breaths as indicated in Fig. 1), having a peaktopeak amplitude of inspiratory pressure of gas j(Pi
_{j}), is given by
where time constant (τ) is
and where r_{ij} is defined as
where T is the squarewave period (not to be confused with the duration of a bolus impulse), c_{ij}(t ) is the recirculation contribution from
For N gasexchange compartments, the measurable expired alveolar partial pressure [Pa
_{j}(t )] is the flowweighted average of the P_{ij}(t ) profiles, given by
which, when combined with Eq. 3
and divided by Pi
_{j}, becomes
where V˙a is given by ∑ V˙_{i }, and C_{j} is the total recirculation contribution to Pa
_{j}(t ) divided by Pi
_{j}. If the C_{ij}(t ) terms are essentially equal to C_{ij}(0), C_{j}(t ) is essentially equal to C_{j}(0), and the C_{j} term can be eliminated by subtracting Pa
_{j}(0)/Pi
_{j}from both sides of Eq. 7, giving
The magnitude of the recirculation problem is determined by the extent to which C_{j}(T) is approximated by C_{j}(0), which for a given tracer is dependent on the magnitude of T. Note that if recirculation placed no restriction on the size of T, allowing T to become large relative to τ, for t = T/2 Eq. 8
approaches
which, for n soluble tracers, constitutes a set ofn independent equations [analogous to the excretion equation set of MIGET (18) except for the addition of the factor R_{i}] from which a distribution of ventilationV˙/V˙a with respect to theV˙/Q˙ is theoretically obtainable without consideration of the transient data. (However, it is the use of transient data that allows T to be small enough to reduce the uncertainty from recirculation error to within acceptable limits.) Because the values of R_{i} are definable in terms of (V˙/Q˙)_{i} values (as well as mixed venous and inspired values of O_{2} and CO_{2}), the presence of R_{i} in Eqs.8 and 9 has essentially no influence on theV˙/Q˙ distribution parameter extraction sensitivity by model parameterfitting processes applied to both transient and equilibrium data and Eq. sets 8 and9, as described in methods. Volume parameters become byproducts of the method.
Once the V˙/Q˙ distribution has been obtained,
where V˙a is obtained by applying mass balance to the reference (insoluble) tracer, as previously reported (1), giving
where
Determination of
C v ¯ O 2 .
About four decades ago, Rahn and Fenn (10) established empirical relationships for both alveolar Po
_{2} and Pco
_{2}(Pa
_{O2} and Pa
_{CO2}, respectively) as functions ofV˙/Q˙ for given input variables associated with mixed venous blood. These relationships formed what became known as the O_{2}CO_{2} diagram. Subroutines published by Olszowka and Farhi (8) provided the means to mathematically model the O_{2}CO_{2} diagram and, thereby, calculate the expected values of Po
_{2}and Pco
_{2} of compartment i(Po
_{2 i} and Pco
_{2 i}, respectively), for a given (V˙/Q˙)_{i} value [given values of
where Fo _{2 i} and Fco _{2 i} are fractions of O_{2} and CO_{2} in compartment i,respectively; and Fi _{O2} is inspired fraction of O_{2}.
Our approach to measuring
and
It should again be noted that representation of Pa _{O2} and Pa _{CO2} in the respective expired time profiles of O_{2} and CO_{2} is complicated by the lack of ventilation concurrence associated with pulmonary disease (see discussion and Ref. 1).
The procedure for calculating
1) Assume starting values for
2) Calculate the Po _{2 i}, Pco _{2 i}, for each compartment, starting with a V˙/Q˙ distribution calculated based on uniform R_{i} values. (For uniform R, the influence of R on the righthand side of Eqs. 8 and 9 is canceled by the influence of R on Pi _{j}, as shown in Eqs. 19 and 20 of methods).
3) Compare measured Pa _{O2}and Pa _{CO2} values against predicted values according to Eqs. 13 and 14.
4) Adjust
5) Recalculate the V˙/Q˙distribution based on R_{i} values calculated fromEq. 12 by using Po _{2 i} and Pco _{2 i} values obtained in step 2.
6) Repeat steps 2–4 by using theV˙/Q˙ and R distributions obtained instep 5. (Although this highly convergent process can be repeated to achieve higher accuracy, in practice, repetition is not required).
Calculation of CO.
Calculation of CO involves determination of the physiological shuntQ˙s and adding it to
where Ca_{O2} is arterial O_{2} content, and by mass balance, the average endcapillary content
where O_{2} content in compartment i(Co
_{2 i}) values are calculated from
and where O_{2} saturation in compartment i(So _{2 i}) values are obtained from1) Po _{2 i} (determined above) and 2) knowledge of the O_{2} disassociation function’s relationship to Pco _{2 i}and blood chemical parameters base excess, P_{50}, and Hb (13).
With an unbiased value of arterial O_{2} saturation, obtained noninvasively by pulse oximetry (or by arterial sample), an unbiased calculation of Ca_{O2} becomes available through Eq. 17
(with “a” substituted for i where arterial Po
_{2} is iteratively calculated from Ca_{O2} by its functional relationship to the O_{2} dissociation curve). Thus, with all contents of Eq.15
determined, CO is calculated by adding
METHODS
V˙/Q˙ distribution extraction procedures.
Extraction of the V˙/Q˙ distribution from Eq. 8
first requires determination of the input Pi
_{j} values that are not directly measurable by the bolus method. The process for obtaining the input values begins with application of Eq. 7
to the insoluble tracer (where j = 1, λ_{b1} = 0, and C_{1} = 0). Solving Eq. 7
for Pa
_{1}(T/2)/Pi
_{1} and Pa
_{1}(0)/Pi
_{1} and combining gives
Solving for Pi
_{1} then gives
where Pa
_{1}(0) and Pa
_{1}(T/2) are the measurable minimum and maximum points on the Pa
_{1}(t ) profile. The Pi
_{j} values for the soluble tracers are then calculated by
where G_{j} represents the measurable tracer supply gas fraction ratios of the soluble tracers to the insoluble tracer, measurable by the multiplegas analyzer (with sufficient sample dilution to satisfy the dynamic range limitation of the analyzer).
Note that when Eqs. 20 and 21 are combined with theV˙/Q˙ distribution defining Eqs.8 and 9, the R_{i} factors cancel as the R_{i} distribution approaches uniformity. This fact, plus the fact that the influence of a nonuniform R_{i}distribution on (V˙/Q˙) extraction is small, is the basis for the iterative procedure for including R_{i} in theV˙/Q˙ distribution extraction method described below.
With the Pi _{j} values determined, two methods for extracting the V˙/Q˙distribution become available. The simplest method mathematically involves use of Eq. set 9 with an estimate of Pa _{j}(∞) substituted for Pa _{j}(T/2). The Pa _{j}(∞) values for the soluble inert tracers are estimated by fitting the multiexponential expiration profile of each tracer to a single exponential and by accepting the 3τ extrapolation point as 95% of each respective Pa _{j}(∞) value; whereas the Pa _{j}(∞) for the insoluble tracer is equal to Pi _{1} (see Eq. 7 ), which is calculated directly by Eq. 20. (This is quite fortunate, since the insoluble tracer is always the farthest from equilibrium.) TheV˙/Q˙ distribution is then calculated by use of Eq. set 9 (2) in an analogous manner to that of MIGET. The reliability of this extrapolation equilibration method for V˙/Q˙ distribution extraction is dependent on the accuracy of the Pa(∞) extrapolation, which decreases as 1) T is reduced to meet recirculationbased uncertainty specifications, and 2) lungs become progressively less uniform in terms of V˙, Q˙, and V.
With the V˙/Q˙ distribution determined, and
V˙a and
For situations in which the extrapolations may be inadequate, the accuracy dependencies on T and the degree of nonuniformity can largely be overcome by adding volume parameters directly to the parameterfitting process applied to transient data and Eq. set8, which we refer to as “the transient method.” An outline of this method is as follows.
1) Total gas volume (∑ Vg_{i}) is obtained from insolublegas data only. Advantage is taken of the fact that the reference Pa(∞) value is available, being equal to Pi _{1} (as described above).
2) Input values for the Downhill Simplex Method (9) are determined by randomly selecting 50 sets of initial values and comparing with the data, with ∑ Vg_{i}constrained by the value of step 1 and ∑ V˙_{i} constrained by Eq.11. The set of values corresponding to the lowest error is selected for step 3.
3) The results of step 2 are applied to the Downhill Simplex Method. The resulting output values are then applied as inputs to the Downhill Simplex Method.
4) Step 3 is repeated two times, using the previous output values as inputs.
To maximize computational efficiency and minimize potential convergence problems, the number of compartments is set to provide a total number of V˙_{i} andQ˙_{i} unknowns equal to the number of independent equations of Eq. sets 8 and 9.However, improved efficiency and convergence trade off against decreased accuracy as the number of compartments decreases. Based on the results of simulation studies described below, the largest number of compartments for which a significant improvement in accuracy of CO and
Experimental procedure for measuring recirculation.
A lobe isolation preparation was devised, allowing the recirculation component of bolus injections to be viewed separately from the injected tracer profiles. This was accomplished by isolating the left lobes of the lungs of dogs from the right lobes by a doublelumen Kottmeier endobronchial tube, providing separate ventilation by a doublecylinder Harvard animal respirator (model 608), as shown in Fig.2. The tidal volumes were adjusted to minimize the expired CO_{2} difference between left and right lobes. Separation was checked by introducing helium in the right tube and measuring the helium concentration in the left. Zero concentration in the left tube indicated adequate separation. Separation was routinely checked for throughout each experiment.
With this animal preparation, bolus measurements were made on each of the lungs separately; the advantage of this preparation for studying recirculation error was that both lungs, being perfused with blood having common mixed venous tracer values, provided a direct measurement of the tracer venous return signal from the noninjected lobes. For example, if the right and left lobes were matched in terms ofV˙/Q˙, the recirculation profile of the injected set was provided by the expired inertgas profile of the noninjected set. The general procedure for determining the recirculation contribution to the expired profile is described inappendix .
Measurements of recirculation profiles of soluble tracers were obtained in five dogs. The injector was adjusted to inject a bolus volume of 5 ml/breath in 300 ms, delayed 10 ms from the initiation of inspiration. Such bolus injections were provided for 16 consecutive breaths (with the respirator set at 11 breaths/min), followed by no injections for an equal number of breaths. This provided a total period T of ∼3 min, which was approximately equal to 6τ [for acetylene (C_{2}H_{2})] for the five dogs of the study. Experiments in which T was decreased to 1.5 min were performed to establish the systemic filtering efficiency as a function of T. The bolus consisted of 10% C_{2}H_{2}, 10% methylvinylether (MVE), 20% dimethylether (DME), 10% sulfurhexafluoride (SF_{6}) with the balance nitrogen. The λ_{b} values for these gases were determined by mass balance (using the inverse of the manometric Van Slyke method).
Mongrel dogs of either sex were anesthetized with pentobarbital sodium (30 mg/kg body wt delivered in divided doses). Further doses of thiopental sodium were given during the experiment to maintain an absent corneal reflex. A 7Fr SwanGanz catheter was introduced into the pulmonary artery (via the femoral vein), and an arterial catheter was inserted into the aorta via the femoral artery for obtaining central venous blood samples for
Procedure for testing against tracer loss to tissue of the conducting airways.
To ensure against irreversible tracer loss to tissue of the conducting airways, flow studies were performed at the conclusion of the recirculation studies by continuing to ventilate the dogs for a few minutes after their hearts were stopped (by KCl injection) and by measuring the expired tracer profiles in response to the same squarewave bolus input profiles. Tracer loss was determined by comparing the extrapolated equilibrium values of the tissue soluble tracers (C_{2}H_{2}, MVE, and DME) to the tissueinsoluble tracer SF_{6}, with input values normalized to unity.
CO and
C v ¯ O 2
extraction error simulation study.
The purpose of this study was to separately identify the magnitudes of CO and
Error contribution specific to V˙/Q˙mismatch is the result of the inability of a limited amount of noiseless inertgas data to uniquely define realV˙/Q˙ distributions. However, as pointed out by Evans and Wagner (3), the resolution ofV˙/Q˙ distribution extractions should not be generalized from hypothetical distributions but can only be determined by using real data. Therefore, in this study, error contributions specific to V˙/Q˙mismatch were obtained by applying Eq. set 9(equilibrium measurements) to four publishedV˙/Q˙ distributions representing one normal subject (17), and three subjects with chronic obstructive pulmonary disease (COPD) (16) ranging from mild to severe with Fi _{O2} values ranging from 0.21 to 0.70.
Included in the study were two, three, and fourcompartment models having tracer gases ranging in number from two to six as follows:1) the twogas set (minimum required for the bolus method) having λ_{b} values of 0 and 0.95; 2) the threegas set (with deadspace compartment) having λ_{b} values of 0, 0.95, and 11.1; 3) the fourgas set having λ_{b}values 0, 0.95, 2.7, and 11.1; the fivegas set (with deadspace compartment) having λ_{b} values of 0, 0.47, 0.95, 2.7, and 11.1; and 4) the sixgas set having λ_{b} values of 0, 0.2, 0.47, 0.95, 2.7, and 11.1.
Comparisons of the extrapolation equilibrium and transientV˙/Q˙ extraction methods were simulated by applying these methods separately to the four publishedV˙/Q˙ distributions described above. Without literature guidance, Vg_{i} values were arbitrarily chosen to be proportional to an equally weighted linear combination of Q˙_{i} andV˙_{i}, given by
The tissue (Vti_{i}) values were arbitrarily set equal to 0.3Q˙_{i}.
Note, however, that no assumptions regarding the values of Vg_{i} and Vti_{i} are used in the CO and
For the purpose of testing our methods, these published distributions (with added values for Vg_{i} and Vti_{i}) are assumed to be true representations of a normal subject and of three COPD “subjects” having “known” CO and
Errors specific to V˙/Q˙ mismatch were obtained by application of Eq. 9
(which ignores recirculation and is not affected by volumes). The two methods ofV˙/Q˙ extraction were compared for three values of T. Simulations were performed using the fivegas, fourcompartment model and the threegas, threecompartment model described above. Error contributions from recirculation were then quantitated by adding tracer recirculation profiles (based on the dual lung measurements) to their respective simulated expired profiles generated for the V˙/Q˙ extraction method comparison studies. The recirculation profiles for each tracer were generated from the fivedog study averages. The recirculation CO and
Noise simulation procedure.
As pointed out by Jaliwala et al. (4), measurement error becomes a limiting factor in the construction of meaningfulV˙/Q˙ distributions from inertgas data as the number of gases increases. However, the insensitivity ofV˙/Q˙ distribution distortion (from both nonuniqueness and noise) in the prediction of arterial blood gases, as described by these authors, should also apply to
Two major measurement error sources for MIGET are 1) the measurement of inert blood gas pressures that are influenced by both the precision of gas and blood volume proportioning and measurement accuracy of gas chromatography; and 2) the uncertainty of the blood solubility values, for which there is significant intersubject variability. For the bolus method, the measurement of gas samples is less limiting, since the step of converting dissolved gases to the gas state does not apply. However, the influence of inertgas blood solubility uncertainty is the same for the bolus method as for MIGET. Therefore, controlling uncertainty of the blood solubility values was the mechanism chosen for adding systematic noise to quantify the effect of noise on CO and
Simulation studies having two levels of noise (SDs of 2 and 4%) were performed on the normal subject and on three COPD distributions. Total CO and
RESULTS
Recirculation error study.
Recirculation components of the Pa(t ) profiles corresponding to C_{2}H_{2}, MVE, and DME are presented separately in Fig. 3, with their corresponding Pa(t ) profiles. The Pa(t ) curves have been normalized to unity, and the relative magnitudes of the recirculation components have been multiplied by a factor of five for better visualization. In this range of T, the reduction in relative recirculation amplitude for each gas was found to equal 0.45 per reduction in T by a factor of two. The ratio of the recirculation amplitude to the Pa(T/2) − Pa(0) difference (T = 180 s) for each dog, is given in Table 1. The impact of recirculation on the measurements of CO and
Tracer loss to tissue study.
The influence of stopped blood flow on inert tracer Paprofiles for a typical dog experiment is shown in Fig.4
A; their reference (normal blood flow) profiles are shown in Fig. 4
B. The time constants displayed in Fig. 4
A are in the order of the λ_{bj} values. Figure 4
A demonstrates that the P_{j}(t )/Pi
_{j}values for all the tracers approach unity (including DME, even though equilibrium is not quite obtained in T/2 because of its large tissue gas volume), indicating no significant irreversible gas loss to the tissue. This conclusion is confirmed by the calculation of
CO and
C v ¯ O 2
extraction error resulting fromV˙/Q˙ mismatch.
The predicted influences of V˙/Q˙abnormality and Fi
_{O2} on CO, and
CO and
C v ¯ O 2
extraction error minimization in the presence of recirculation.
In addition to the V˙/Q˙ limitations represented in Table 2, there are the limitations imposed by the reduction of T for reducing the influence of recirculation on the CO and
Comparison of Table 3 results with those of Table 2 (note that Table 2results are represented in the T = ∞ columns of Table 3 except for COPD_{H} with 20% shunt added) shows no significant increase in error with reduced T associated with the transient method. However, the extrapolated equilibrium method appears to be unreliable for applications in which more than three tracer gases are involved and is particularly influenced by the length of T.
Even though the recirculation contribution to the Pa(t ) profiles decreases proportionally with decreasing T, the results of Table 4 fail to show much advantage of T values <180 s for reducing the cumulative errors from CO and
Other conclusions to be drawn are 1) three tracer gases are probably adequate for monitoring CO and
Noise study.
Table 5, top, presents cumulative CO and
Note that the noise contribution to measurement uncertainty is greatest for the sixgas COPD_{HL} distribution. It equals the bias level at a noise level of 4% when T is set at 180 s. The same is true for the threegas normal distribution, for which the errors are essentially negligible. For comparison, the error results for the same noise contributions are presented in Table 5, bottom, for T = 90 s. The modest improvements in bias error from reduced venous return appear to be generally eroded by the increased sensitivity to noise.
DISCUSSION
The recirculation study demonstrates that the bolus method, in which common gases are used, has the capability to contain the recirculation contribution of CO and
As the number of gases is increased to reduce measurement bias in diseased lungs, noise plays a more significant role for increasing measurement uncertainty. The continuous measurement characteristic of the method provides a tradeoff between filtering efficiency of random noise and monitoring frequency response. However, filtering cannot reduce errors from the systematic noise sources of analyzer calibration error or uncertainty in values of blood solubility coefficients. Fortunately, the noise contributions of these sources are under experimental control and, in principle, are reducible to the needs of the study. In this study, 2% was chosen as the smallest systematic noise level presented, as blood solubility measurements are measurable within this uncertainty level with the use of semiautomatic instrumentation (unpublished observations). Calibration accuracies of 1% and less are practical for a massspectrometer analyzer. Therefore, the error estimates corresponding to 2% noise levels (Table 5,top and bottom) should be achievable when solubility coefficients are measured for individual subjects. Otherwise, the 4% values are likely more applicable.
Whereas multiple inert gases can also be applied to steadystate methods, the availability of transient data and the opportunity it provides for reducing T provides the bolus method with significant advantages. For illustration, we compare the bolus method to that described by Zwart et al. (21) in which a 3min period was cited as being optimal. In this example, the term containing lung volume information,
The bolus method is not without problems, however. Cited limitations of the MIGET model are transferred to the bolus method. For example, the MIGET model assumes that gasexchange units are 100% concurrent. Whereas the MIGET method’s sensitivity to lack of concurrence (ventilation asynchrony) has been reported to be small (5), the bolus method accentuates this sensitivity because of the additional influence of bolus timing on the distribution of bolus delivery. Fortunately, the sensitivity to bolus timing also provides the opportunity of addressing the problem quantitatively for the purpose of correcting the influence of asynchrony on CO and
Other problems cited for the MIGET model include 1) not accounting for uptake of highly soluble inert gases in conducting airways (12), and 2) neglecting diffusionlimited gas mixing. The problem of soluble gas uptake by airway tissue has been previously shown to be negligible for diethylether (12), a result confirmed by this study for DME and the bolus tracers of lower tissue solubility.
Our approach to minimizing the problem of diffusionlimited gas mixing has been to use tracers of similar molecular weight when possible and to make corrections of the excretion data based on molecular weight differences (11) when matching tracers according to molecular weight is not convenient. An example of the latter has been the use of SF_{6} (mol wt = 128) for the insoluble gas in place of argon to achieve improved measurement sensitivity. Accordingly, 2.5% corrections to the Pa(t ) profiles were made on all SF_{6} data. (Whereas this correction influences the measurements of V˙ and theV˙/Q˙ distribution, it has no effect on the measurement of CO.)
The assumption of continuous ventilation has some influence on the calculation of CO and
The potential for widespread clinical application of the bolus method also depends on availability of a practical gas analyzer. Whereas recent developments in mass spectrometry have produced gas analyzers having the portability and ruggedness necessary to meet the physical requirements of hospital applications (14), less expensive gas analysis could greatly expand clinical applications. In this category are catalyticbased microsensors (6, 20), which, hopefully, will be available in the near future. However, the bolus method, supported by any practical gas analyzer, has the potential to provide a noninvasive alternative to central venous catheterization for obtaining CO and central venous blood gas data.
Acknowledgments
We gratefully acknowledge the support of the National Heart, Lung, and Blood Institute and the Deseret Foundation.
Footnotes

Address for reprint requests: J. S. Clark, Department of Medical Informatics, 825 N. 300 W., Suite 420, Salt Lake City, UT 841031414.
 Copyright © 1997 the American Physiological Society
Appendix
Generalized Response to a Bolus Input
When mass balance is applied to an inert tracer gas of index j, transfer function response to h
_{Lj} is given by
where ΔV_{ik} is the volume change of the ith compartment during the kth breath cycle, andh
_{Lij} is the ith parallel gasexchange element for gas j, having the generalized form
where k is the breath index and periodt
_{k} is defined by the summation
where T_{k} is the period of thekth breath, r_{ik} is the gas dilution ratio of the ith compartment and kth breath, and
where V_{ik} is the volume of theith compartment, and Π(r_{ik}) is a product function, given by
τ′_{ijk} is the time constant for uptake of gas j by blood flow Q_{ik} of theith compartment and kth breath, and
where λ_{j} is the bloodgas partition coefficient of the jth gas.
For the special case of uniform breathing (ΔV_{ik} = ΔV˙_{i},Q˙_{ik} = Q˙_{i}),Eq. EA1
simplifies to
To better illustrate comparisons to MIGET, Eq. EA7
is further simplified by the approximation of continuous gas exchange by letting ΔV_{i} (and T_{k}) approach zero, giving
where V˙_{i} (defined as ΔV_{ik}/T_{k}) is the ventilation of the ith compartment, and τ_{ij} is the total rate constant, defined in the text by Eq. 4.
Note, for comparison with MIGET, that integrating Eq. EA8
over the period T gives
which is Eq. 8 of the text.
Note also that for the discontinuousventilation but uniformbreathing model (Eq. EA7 ), ΔV_{i} replacesV˙_{i} as the set of ventilation unknowns in the continuousventilation model (Eq. 8 ), which gives both models the same number of variables for extraction. However, in practice, calculation of CO by use of Eqs. 8 or A7usually produces results within 5%. Probably of more significance is the application of Eq. EA1 for future applications involving nonuniform breathing.
Appendix
The recirculation profile of the injected lung is determined first by calculating the mixed venous tracer pressure profile
where
where τ′ is given by Eq. 4
with ′ designating noninjected lobe parameters. The venous return component of the expired tracer concentration profile of the injected lobes
where h"(t ) is the unit impulse response of the injected lobes represented by Eq. EB2
(where the symbol " designates injected lobe parameters). Substituting Eq.EB1
into Eq. EB3
to eliminate
The parameters of h′(t ) andh"(t ) are obtained by alternately providing squarewave inputs to both sets of lobes and application of Eq.EB2. [Note that if the two sets of lobes are matched in terms ofV˙/Q˙ distribution, i.e.,h′(t ) = h"(t ), the measured noninjected expired tracer profile equals the recirculation component of the injected expired profile.]