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 inert-gas delivery to the lungs that facilitates application of multiple inert gases and the multiple inert-gas-exchange technique (MIGET) model to noninvasive measurements of cardiac output (CO) and central mixed venous oxygen content Reduction in recirculation error is made possible by 1) replacement of sinusoidal input functions with impulse inputs and2) replacement of steady-state analyses with transient analyses. Recirculation error reduction increases the inert-gas selection to include common gases without unusually high (and difficult to find) tissue-to-blood partition coefficients for maximizing the systemic filtering efficiency. This paper also presents a practical method for determining the recirculation contributions to inert expired profiles in animals and determining their specific contributions to errors in the calculations of CO and from simulations applied to published ventilation-perfusion ratio (V˙/Q˙) profiles. Recirculation errors from common gases were found to be reducible to the order of 5% or less for both CO and whereas simulation studies indicate that measurement bias contributions from recirculation, V˙/Q˙ mismatch, and the V˙/Q˙ extraction process can be limited to 15% for subjects with severeV˙/Q˙ mismatch and high inspired oxygen fraction levels. These studies demonstrate a decreasing influence of V˙/Q˙ mismatch on parameter extraction bias as the number of inert gases are increased. However, the influence of measurement uncertainty on parameter extraction error limits improvement to six gases.
- multiple inert-gas-exchange technique
- ventilation-perfusion ratio
a practical, noninvasive method for measuring cardiac output (CO) and mixed venous oxygen content could provide an alternative to central venous catheterization, thereby eliminating a serious risk factor associated with monitoring CO and/or Viewed as the sum of effective pulmonary blood flow (Q˙eff ) and effective shunt (Q˙s), CO can be measured noninvasively by using 1) current or improved inert-gas techniques for measuring Q˙eff, and2) a noninvasive technique for measuring Q˙s, which depends on noninvasive acquisition of However, current inert-gas techniques for measuring Q˙eff have found little clinical acceptance. The rebreathing (RB) methods require subject cooperation, which limits their use. Hyperventilation associated with RB modifies the cardiopulmonary status of the patient, including alteration of CO (7). Errors due to recirculation also present a dilemma to this method, since avoidance of recirculation error reduces the amount of data available for analysis.
Stout et al. (15) introduced an open-circuit, multiple-breath (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 steady-state (SS), open-circuit method for determining the overall ventilation-perfusion ratio (V˙/Q˙). The method utilizes a time-modulated inert-gas concentration profile introduced at the airway. The V˙/Q˙ is determined from comparisons of peak-to-peak amplitudes of the steady-state expired and inspired inert-gas 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 tissue-to-blood 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 that depends heavily on its ability to deal with both recirculation and inhomogeneity. By combining transient features of the MB method and steady-state features of the SS method, systemic filtering efficiency is increased sufficiently to extend the choice of inert tracers to common gases. This increase in gas selection makes the multiple inert-gas-exchange technique (MIGET) model (18) a practical means for dealing with V˙/Q˙ inhomogeneity. Both transient and steady-state data are provided by bolus delivery of inert gases (tracers) to the airway by using a bolus-generation device [previously reported for measuring alveolar ventilation (V˙a), oxygen uptake, and carbon dioxide production (1)]. Application of the bolus method further simplifies the measurement of CO by removing the requirement of an airtight seal and the need for a sinusoidal inert-gas concentration generator. This paper also provides 1) an experimental verification of the recirculation error-reduction claims of the bolus method, and2) a computer simulation analysis of the bolus method’s ability to cope effectively with inhomogeneity inV˙/Q˙.
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 square-wave input of period T, as shown in Fig. 1.
Calculation of average end-capillary perfusion andV˙a.
With use of the parallel gas-exchange 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 Equation 1
where Pi(t ) is partial pressure of tracer gas in the alveoli of compartment i;Pci(t ) is partial pressure of tracer gas in the end-capillary blood of compartment i; is partial pressure of tracer gas in the mixed venous blood; Pi(t ) is partial pressure of tracer gas in inspired air; Q˙i is pulmonary perfusion of compartment i;V˙i i,V˙i is inspired and expired ventilation of compartment i, respectively; λb, λti is the blood-to-gas and tissue-to-gas Ostwald partition coefficients, respectively; Vi is the effective gas volume of compartment i, equal to Vgi + λti jVtii; and Vgi, Vtii is alveolar gas and alveolar tissue volumes of compartment i, respectively.
For inert gases, attainment of equilibrium between capillary blood and alveolar gas is normally assumed (19), providing the equality Pci = Pi. Equating Pci(t ) and Pc(t ), ignoring the time dependence of mixed venous pressure (the source of recirculation addressed below), and arranging Eq. 1 into standard form provides Equation 2
To the extent that time invariance forV˙i,Q˙i, and Vi can be assumed over a measurement cycle, Eq. 2 approximates a first-order, linear differential equation where steady-state response to a square-wave 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 peak-to-peak amplitude of inspiratory pressure of gas j(Pi j), is given by Equation 3
where time constant (τ) is Equation 4
and where rij is defined as Equation 5
where T is the square-wave period (not to be confused with the duration of a bolus impulse), cij(t ) is the recirculation contribution from and Ri is defined as the ratioV˙i i/V˙i.
For N gas-exchange compartments, the measurable expired alveolar partial pressure [Pa j(t )] is the flow-weighted average of the Pij(t ) profiles, given by Equation 6
which, when combined with Eq. 3 and divided by Pi j, becomes Equation 7
where V˙a is given by ∑ V˙i , and Cj is the total recirculation contribution to Pa j(t ) divided by Pi j. If the Cij(t ) terms are essentially equal to Cij(0), Cj(t ) is essentially equal to Cj(0), and the Cj term can be eliminated by subtracting Pa j(0)/Pi jfrom both sides of Eq. 7, giving Equation 8
The magnitude of the recirculation problem is determined by the extent to which Cj(T) is approximated by Cj(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 Equation 9
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 Ri] 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 Ri are definable in terms of (V˙/Q˙)i values (as well as mixed venous and inspired values of O2 and CO2), the presence of Ri in Eqs.8 and 9 has essentially no influence on theV˙/Q˙ distribution parameter extraction sensitivity by model parameter-fitting processes applied to both transient and equilibrium data and Eq. sets 8 and9, as described in methods. Volume parameters become by-products of the method.
Once the V˙/Q˙ distribution has been obtained, is obtainable from the sum of the Q˙ielements given by Equation 10
where V˙a is obtained by applying mass balance to the reference (insoluble) tracer, as previously reported (1), giving Equation 11
where is the average rate of bolus delivery and is the average alveolar partial pressure of the reference tracer, ands g is the Ostwald gas phase solubility coefficient. Because is under system control (and therefore is known) and is measurable, V˙a is obtainable from Eq.11. Note that no accounting of series dead space is required when the bolus is synchronized with inspiration to assure complete passage of the tracer bolus into the gas-exchange regions of the lung and that the term of Eq. 11 substitutes for the usual airtight airway seal requirement to achieve V˙a and measurements. However, it should be further noted that the authors of Ref. 1 point out that use of Eq. 11 is subject to considerable error in the presence of pulmonary disease, presumably because of the lack of ventilation concurrence among ventilation units (seediscussion).
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 O2-CO2 diagram. Subroutines published by Olszowka and Farhi (8) provided the means to mathematically model the O2-CO2 diagram and, thereby, calculate the expected values of Po 2and Pco 2 of compartment i(Po 2 i and Pco 2 i, respectively), for a given (V˙/Q˙)i value [given values of and mixed venous CO2 content and blood chemical variables hemoglobin (Hb), base excess, and Po 2 at 50% Hb saturation (P50)]. From mass balance, Ri can be calculated in terms of gas fractions of O2 and CO2 by Equation 12
where Fo 2 i and Fco 2 i are fractions of O2 and CO2 in compartment i,respectively; and Fi O2 is inspired fraction of O2.
Our approach to measuring is an iterative one, involving adjustment of the input values and until calculated predictions Po 2 i and Pco 2 i and the corresponding predictions of ventilation-weighted Pa O2 and Pa CO2(given by Eqs. 13 and 14 below) are consistent with measured expired O2and CO2 profiles. Predictions of Pa O2 and Pa CO2 are given by Equation 13
and Equation 14
It should again be noted that representation of Pa O2 and Pa CO2 in the respective expired time profiles of O2 and CO2 is complicated by the lack of ventilation concurrence associated with pulmonary disease (see discussion and Ref. 1).
The procedure for calculating being a by-product of the procedure) now consists of first assuming (or measuring from a peripheral blood sample) the blood chemical values and performing the following iterative procedure.
1) Assume starting values for and
2) Calculate the Po 2 i, Pco 2 i, for each compartment, starting with a V˙/Q˙ distribution calculated based on uniform Ri values. (For uniform R, the influence of R on the right-hand 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).
4) Adjust and parameters until matches in step 3 are obtained.
5) Recalculate the V˙/Q˙distribution based on Ri 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 as obtained fromEq. 10 above. By mass balance applied to O2 (Fick principle), Q˙s is given by Equation 15
where CaO2 is arterial O2 content, and by mass balance, the average end-capillary content is given by Equation 16
where O2 content in compartment i(Co 2 i) values are calculated from Equation 17
and where O2 saturation in compartment i(So 2 i) values are obtained from1) Po 2 i (determined above) and 2) knowledge of the O2 disassociation function’s relationship to Pco 2 iand blood chemical parameters base excess, P50, and Hb (13).
With an unbiased value of arterial O2 saturation, obtained noninvasively by pulse oximetry (or by arterial sample), an unbiased calculation of CaO2 becomes available through Eq. 17 (with “a” substituted for i where arterial Po 2 is iteratively calculated from CaO2 by its functional relationship to the O2 dissociation curve). Thus, with all contents of Eq.15 determined, CO is calculated by adding to both sides ofEq. 15 and solving explicitly for CO to obtain Equation 18
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, λb1 = 0, and C1 = 0). Solving Eq. 7 for Pa 1(T/2)/Pi 1 and Pa 1(0)/Pi 1 and combining gives Equation 19
Solving for Pi 1 then gives Equation 20
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 Equation 21
where Gj represents the measurable tracer supply gas fraction ratios of the soluble tracers to the insoluble tracer, measurable by the multiple-gas 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 Ri factors cancel as the Ri distribution approaches uniformity. This fact, plus the fact that the influence of a nonuniform Ridistribution on (V˙/Q˙) extraction is small, is the basis for the iterative procedure for including Ri 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 recirculation-based uncertainty specifications, and 2) lungs become progressively less uniform in terms of V˙, Q˙, and V.
With the V˙/Q˙ distribution determined, and determined by Equation 22
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 parameter-fitting 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 (∑ Vgi) is obtained from insoluble-gas 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 ∑ Vgiconstrained 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 measurement can be attained in subjects with severeV˙/Q˙ mismatch appears to be four. Of the four compartments, one represents a shunt (V˙ = 0), whereas the remaining three are ventilation compartments without constraints placed on their respectiveV˙/Q˙ values. The appropriate number of inert gases for extracting theV˙/Q˙ distribution of this four-compartment model is six. This inert-gas number is reduced to five when a dead-space compartment is created from one of the ventilation compartments by restricting its Q˙ to zero. Four gases are appropriate when the number of compartments is reduced to three, whereas three gases are appropriate when one ventilation compartment has its Q˙ restricted to zero. The least number of gases that can be applied to the bolus method is two, which corresponds to a two-compartment model.
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 double-lumen Kottmeier endobronchial tube, providing separate ventilation by a double-cylinder Harvard animal respirator (model 608), as shown in Fig.2. The tidal volumes were adjusted to minimize the expired CO2 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 inert-gas 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 (C2H2)] 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% C2H2, 10% methylvinylether (MVE), 20% dimethylether (DME), 10% sulfurhexafluoride (SF6) with the balance nitrogen. The λb 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 7-Fr Swan-Ganz 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 laboratory comparisons, for arterial blood gas and pH laboratory measurements, and for delivering the thiopental sodium.
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 square-wave bolus input profiles. Tracer loss was determined by comparing the extrapolated equilibrium values of the tissue soluble tracers (C2H2, MVE, and DME) to the tissue-insoluble tracer SF6, with input values normalized to unity.
CO and extraction error simulation study.
The purpose of this study was to separately identify the magnitudes of CO and error contributions due to 1) V˙/Q˙mismatch, 2) method of V˙/Q˙extraction, and 3) recirculation.
Error contribution specific to V˙/Q˙mismatch is the result of the inability of a limited amount of noiseless inert-gas 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 four-compartment models having tracer gases ranging in number from two to six as follows:1) the two-gas set (minimum required for the bolus method) having λb values of 0 and 0.95; 2) the three-gas set (with dead-space compartment) having λb values of 0, 0.95, and 11.1; 3) the four-gas set having λbvalues 0, 0.95, 2.7, and 11.1; the five-gas set (with dead-space compartment) having λb values of 0, 0.47, 0.95, 2.7, and 11.1; and 4) the six-gas set having λb 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, Vgi values were arbitrarily chosen to be proportional to an equally weighted linear combination of Q˙i andV˙i, given by Equation 23
The tissue (Vtii) values were arbitrarily set equal to 0.3Q˙i.
Note, however, that no assumptions regarding the values of Vgi and Vtii are used in the CO and extraction processes.
For the purpose of testing our methods, these published distributions (with added values for Vgi and Vtii) are assumed to be true representations of a normal subject and of three COPD “subjects” having “known” CO and values. By generating Pa(t ) profiles for O2and CO2 as well as for inert tracers in response to bolus delivery, simulation data become available for testing the accuracy of the CO and extraction techniques against known parameters of the simulation studies.
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 five-gas, four-compartment model and the three-gas, three-compartment 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 five-dog study averages. The recirculation CO and results, extracted by using both the transient and extrapolated equilibrium methods, were compared with the without-recirculation results of the previous study.
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 inert-gas 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 and CO. There, a similar simulation study was designed to test the influence of noise-induced distribution shape changes on calculated CO and values.
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 inert-gas 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 extraction error.
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 errors for three-, five-, and six-inert-gas sets, having Fi O2 of 0.21, 0.40, and 0.70, respectively, for periods T of 180 and 90 s were calculated for the transient method. The noise contributions to extraction error were compared with previous noise studies pertaining to the extrapolated equilibrium method (2).
Recirculation error study.
Recirculation components of the Pa(t ) profiles corresponding to C2H2, 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 is described below.
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 λbj values. Figure 4 A demonstrates that the Pj(t )/Pi jvalues 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 which in this instance produces a value not significantly different from zero.
CO and extraction error resulting fromV˙/Q˙ mismatch.
The predicted influences of V˙/Q˙abnormality and Fi O2 on CO, and calculations are presented in Table 2. The calculations are based on Eq. set 9, which eliminates the influence of T. Q˙s/Q˙ calculations are presented to illustrate the influence of the shunt compartment on the CO and results. Additionally, a true shunt (such as that represented by atelectasis) of 20% was added to the published highV˙/Q˙ region distribution in COPD (COPDH) (16) to further illustrate the influence of the shunt compartment at high Fi O2levels. [Room-air high-low (HL) pattern data are not presented in Table 2 because subjects having such distributions require elevated Fi O2 to maintain physiologically compatible values.] Note, however, that the benefit of the shunt compartment is reduced as the Fi O2 is increased to 0.40 and virtually eliminated for an Fi O2 of 0.70 (except when true shunts are present). Because physiological shunting is reduced with increased Fi O2 (the rationale for increasing the level of inspired O2), more tracer gases are required to compensate for the reduced sensitivity of O2 data for exposing lowV˙/Q˙ units. This is well illustrated in Table 2, which demonstrates an advantage to using up to six inert tracers for the COPDHL as well as the COPDLdistribution patterns at an Fi O2 of 0.70.
CO and 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 measurements. Because the limitations associated with reducing T are intrinsically tied to both the extraction method and the patterns and magnitudes ofV˙/Q˙ nonuniformity, comparisons of transient and extrapolated equilibrium methods’ extraction errors (for four values of T) for theV˙/Q˙ distributions represented in Table 2 are presented in Table3, without recirculation. (The added influences of recirculation, provided by the recirculation experimental data described above, are presented in Table4.)
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 COPDH 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 extraction by the transient method. The extrapolated equilibrium method definitely favors the 180-s value of T for minimizing extraction error. Furthermore, its practical use appears to be limited to a maximum of two gas-exchanging compartments.
Other conclusions to be drawn are 1) three tracer gases are probably adequate for monitoring CO and for most patients breathing room air, whereas 2) use of up to six tracer gases should be of advantage for monitoring patients with severe pulmonary disease who are on high Fi O2.
Table 5, top, presents cumulative CO and measurement errors for 2 and 4% noise added toV˙/Q˙ mismatch, presented for the transient method for T = 180 s. Included are three-gas, five-gas, and six-gas results for respective Fi O2values of 0.21, 0.40, and 1.70, encompassing the full range of expected extraction error for the two noise levels. As seen from comparisons with Tables 4 and 2, the noise contributes insignificantly to the error bias. Table 5, top, demonstrates a modest increase in measurement uncertainty of the six-gas data compared with the five-gas, with the largest differences corresponding to the most severeV˙/Q˙ mismatch (COPDHL). [These results are not significantly different from the error vs. noise determinations corresponding to the extrapolated equilibrium method (2)].
Note that the noise contribution to measurement uncertainty is greatest for the six-gas COPDHL 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 three-gas 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.
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 measurement errors to within 5% when transient data are fully utilized. The simulation results also indicate that measurement bias contributed by the combination of recirculation, V˙/Q˙mismatch, and the V˙/Q˙extraction process can be limited to ∼10%. For normal lungs, bias in CO and calculations should be negligible when limiting the distribution representations to four compartments (definable by 3 gases). However, the predicted results for diseased lungs may be optimistic without inclusion of asynchronous ventilation parameters to the model. Also, experimental studies should be extended to patients receiving high inspired levels of O2 before the full clinical utility of the bolus method can be determined.
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 trade-off 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 mass-spectrometer 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 steady-state 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 3-min period was cited as being optimal. In this example, the term containing lung volume information, (where ω is angular frequency, and τL is lung time constant), and the term containing V˙/Q˙information, (1 + λbQ˙/V˙)2, are equally represented in the data. (Separation of the two terms to evaluate the second requires additional data acquired from superposition of a higher frequency sine wave.) However, when the bolus method is applied to this same example, the (1 + λbQ˙/V˙) term would be represented by 95% of the data, with only 5% having a lung volume contribution. By reducing the period to 1.5 min, the ratio is still 78–22% in favor of the (1 + λbQ˙/V˙) term; whereas such a reduction increases the (lung volume) contribution of the method by Zwart et al. by a factor of four. The ability of the bolus method to adequately measure CO with the period reduced to 1.5 min allows the recirculation error for any given gas to be less than the method of Zwart et al. at a 3-min period by the factor 0.29 [0.29 comes from the product of the reduction in amplitude (1/2.3), due to improved filtering efficiency of the fundamental and ratio of the amplitude of the fundamental to that of the square wave (0.64)]. This is lower than the minimum reduction fraction of 40% needed to meet Zwart et al.’s selection criteria given by λT/λb ≥ 2.5.
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 gas-exchange 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 calculations. Measurement of expired profiles as a function of bolus timing can provide an extra dimension of information for extracting phasing parameters of an extended MIGET model that incorporates asynchronous ventilation. Although the animal studies (unpublished observations) exhibited no significant sensitivity to variations in bolus timing (which has prevented testing of models incorporating asynchronous ventilation), we expect that incorporation of asynchronous ventilation into the model may be beneficial for application of the bolus method to patients having significant pulmonary disease.
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 diffusion-limited 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 diffusion-limited 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 SF6 (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 SF6 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 Although this influence is usually small (<5% in the animal studies; unpublished observation), the problem can be entirely dismissed by application of a discontinuous ventilation extension to the MIGET model (described in appendix ), made possible by the transient characteristics of bolus-input scheme. However, a more significant benefit of this discontinuous ventilation extension is the opportunity it provides for incorporating breath-to-breath variations in model parameters such as ventilation. Furthermore, parameter calculations can be updated as often as each one-half cycle. This is in contrast to requiring a few cycles of system steady state for proper applications of the steady-state analytical technique. The ability of the bolus method to follow rapid changes in CO has been observed (unpublished observations). With the additional simplicity of the subject interface, potential applications of the bolus method range from critical care to ambulatory settings.
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 catalytic-based 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.
We gratefully acknowledge the support of the National Heart, Lung, and Blood Institute and the Deseret Foundation.
Address for reprint requests: J. S. Clark, Department of Medical Informatics, 825 N. 300 W., Suite 420, Salt Lake City, UT 84103-1414.
- Copyright © 1997 the American Physiological Society
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 Equation A1
where ΔVik is the volume change of the ith compartment during the kth breath cycle, andh Lij is the ith parallel gas-exchange element for gas j, having the generalized form Equation A2
where k is the breath index and periodt k is defined by the summation Equation A3
where Tk is the period of thekth breath, rik is the gas dilution ratio of the ith compartment and kth breath, and Equation A4
where Vik is the volume of theith compartment, and Π(rik) is a product function, given by Equation A5
τ′ijk is the time constant for uptake of gas j by blood flow Qik of theith compartment and kth breath, and Equation A6
where λj is the blood-gas partition coefficient of the jth gas.
For the special case of uniform breathing (ΔVik = ΔV˙i,Q˙ik = Q˙i),Eq. EA1 simplifies to Equation A7
To better illustrate comparisons to MIGET, Eq. EA7 is further simplified by the approximation of continuous gas exchange by letting ΔVi (and Tk) approach zero, giving Equation A8
where V˙i (defined as ΔVik/Tk) is the ventilation of the ith compartment, and τij 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 Equation A9
which is Eq. 8 of the text.
Note also that for the discontinuous-ventilation but uniform-breathing model (Eq. EA7 ), ΔVi replacesV˙i as the set of ventilation unknowns in the continuous-ventilation 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.
The recirculation profile of the injected lung is determined first by calculating the mixed venous tracer pressure profile by the deconvolution equation Equation B1
where is the measured expired tracer profile of the noninjected lobes in response to andh′(t ) is the unit impulse response given by Equation B2
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 is then given by the convolution equation Equation B3
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 gives Equation B4
The parameters of h′(t ) andh"(t ) are obtained by alternately providing square-wave 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.]