Sources of Error in Noninvasive Pulmonary Blood Flow Measurements by Partial Rebreathing: A Computer Model Study

We evaluated the partial rebreathing technique using a comprehensive mathematical model of the cardiorespiratory system of a healthy, 70-kg adult male. The model simulates tidal breathing through a branched respiratory tree and incorporates the effects on carbon dioxide dynamics of lung tissue mass, vascular transport delays, multiple body compartments, and realistic blood–gas dissociation curves. It is implemented using Matlab and Simulink (Mathworks, Natick, MA). (See  Appendix Bfor further details of the model.) For the purposes of this study, an additional variable dead space that can be switched in and out to simulate the rebreathing process was incorporated into the model. A block diagram of the model of the cardiorespiratory system is shown in figure 1.

The model was run for 6,000 s with parameters shown in table 1and with the rebreathing dead space bypassed to create a set of initial conditions for cardiac outputs 1.5 l/min and 2–14 l/min in steps of 1 l/min (representing true nonshunted PBF from 1.47 to 13.72 l/min). The initial gas in the additional dead space was air. At each cardiac output, a series of data sets were generated by running the model from an appropriate initial condition and switching the additional dead space in for 50 s at intervals of 180 s. The PBF values were calculated from Petco²and airway carbon dioxide flux averaged over single breaths taken immediately before the start and end of the 10th rebreathing cycles, using equation A3.

The partial rebreathing method assumes that mixed venous blood concentrations of carbon dioxide remain constant immediately before and during the rebreathing phase of each measurement cycle. To examine the effects of changes in mixed venous Pco²caused by recirculation, the model was modified by removing all the body compartments. The mixed venous Pco²and partial pressure of oxygen (Po²) inputs to the pulmonary capillaries were kept constant at values appropriate to the cardiac output. The simulations described above in study 1 were repeated.

The partial rebreathing method is based on the mass balance of carbon dioxide at the pulmonary capillary–alveolar interface but uses measurements in the proximal airway. The duration of rebreathing is selected to be long enough for a quasi–steady state in alveolar Pco²(Paco²), Petco², and airway carbon dioxide flux to be reached but short enough to keep recirculation effects small. To examine the effects of using Petco²and proximal airway carbon dioxide flux measurements instead of Paco²and pulmonary capillary carbon dioxide flux measurements, PBF values were calculated using Paco²and pulmonary capillary carbon dioxide flux averaged over a single breath. No recirculation was included in these simulations.

The time constant for alveolar carbon dioxide turnover depends strongly on PBF because of the high solubility of carbon dioxide in blood. Therefore, the time required to achieve a quasi–steady state in Paco², Petco², and airway carbon dioxide flux after a perturbation is increased when PBF is low and decreased when PBF is high. To examine the effects of rebreathing time, three investigations were performed:

1. Pulmonary blood flow values were calculated using an extended rebreathing time of 300 s. No recirculation was included in these simulations. The simulations described above in study 1 were repeated.

2. The percentage of the step change in Petco²completed during 50 s of rebreathing was determined using results from (1) above.

3. The rebreathing times necessary for Petco²to change through 95% of the change achieved after 300 s of rebreathing were determined for each PBF using results from (1) above. Simulated partial rebreathing measurements with recirculation were repeated using these rebreathing times at each value of PBF.

In all studies, additional data sets were generated using simulated rebreathing dead space volumes of 50, 100, 150, and 200 ml.

Typical proximal airway Pco²and breath-by-breath proximal airway and pulmonary capillary carbon dioxide flux, generated using a PBF value of 4.0 l/min with the additional dead space set at 150 ml, are shown in figure 2for the 10th rebreathing period. The rebreathing period was 50 s, and the rebreathing cycle was repeated every 180 s. The 10th rebreathing period starts at 1,650 s and finishes at 1,700 s. Between 1,620 s and 1,650 s, both Pco²levels and carbon dioxide flux reflect the quasi-equilibrium achieved after the ninth rebreathing cycle. Changed Pco²was read at 1,700 s, which is the end of the 10th rebreathing period. Changed carbon dioxide flux was taken from the last completed breath before 1,700 s.

The difference between the simulated true PBF and the PBF calculated from rebreathing measurements (table 2) is shown in figure 3(curve a). The results show errors that vary systematically with the true PBF. At low simulated PBF (< 4.5 l/min), the partial rebreathing technique tends to overestimate PBF, while at high cardiac outputs (> 4.5 l/min), the technique tends to underestimate PBF. The results indicate that the measurements are most accurate with systematic error less than 10%, when the simulated PBF is between 3 and 6 l/min.

The effects of removing recirculation are shown in curve b of figure 3(additional dead space set to 150 ml). Removal of recirculation clearly reduces the systematic error in the measurements. At low simulated PBF (< 4.5 l/min), study 2 produced systematic errors approximately 75% less than those of study 1, while at high cardiac outputs (> 4.5 l/min), study 2 produced systematic errors approximately 60% better than those of study 1. They indicate, however, that even in the absence of recirculation, the partial rebreathing technique tends to overestimate PBF at low cardiac outputs and underestimate PBF at high cardiac outputs.

Results obtained using Paco²as an estimate of arterial Pco²and true pulmonary capillary carbon dioxide flux (without recirculation) in the differential Fick equation, shown in curve c of figure 3, are clearly more accurate than the results from either study 1 or study 2. PBF is, however, still underestimated at high cardiac outputs and overestimated at low cardiac outputs.

Results obtained using arterial Pco²and pulmonary capillary carbon dioxide flux (without recirculation) in the direct Fick equation, shown in curve d of figure 3, produced excellent results across the entire range examined.

1. Figure 4compares results obtained with a rebreathing time of 50 s (using alveolar Pco²and carbon dioxide flux measurements) with equivalent results obtained with a 300-s rebreathing time (curves a and b, respectively). The systematic errors after 300 s of rebreathing are very small for low PBF but are very similar to the 50-s rebreathing results for PBF greater than approximately 6 l/min. PBF is underestimated over the entire range after 300 s of rebreathing.

2. The percentage of the step change in Petco²achieved after 50 s of rebreathing is shown in figure 5. The result demonstrates that for low PBF, 50 s is not long enough for the Pco²level to reach a quasi-equilibrium (95% of complete change). For PBF greater than approximately 3.5 l/min, 50 s is long enough to achieve an adequate quasi-equilibrium.

3. The rebreathing time (quantized into whole breaths) required for Pco²to change through at least 95% of the complete step is shown in the bar chart in figure 6. The result demonstrates an approximate inverse relation between the PBF and the required rebreathing time. For PBF less than approximately 3 l/min, the required rebreathing period is longer than 50 s, and for PBF greater than approximately 3 l/min, the required rebreathing period is less than 50 s.

Pulmonary blood flow measurements made from Petco²and airway carbon dioxide flux using the variable rebreathing period shown in the chart with recirculation are also shown in figure 6(curve c). For the entire PBF range, the results fall between curve a, depicting standard end-tidal measurements, and curve b, depicting no-recirculation end-tidal measurements. This result demonstrates that the variable rebreathing method improves the standard method, and in some cases, the improvement is as good as the no-recirculation case.

In all the studies, no PBF measurement varied by more than 5% when the volume of the additional dead space was varied between 50 and 200 ml. All further discussion relates only to results obtained with additional dead space of 150 ml.

This study found that PBF determined by the partial rebreathing technique is overestimated at low cardiac outputs and underestimated at high cardiac outputs, which was observed experimentally by Tachibana et al. , 6Nilsson, 7and van Heerden et al.  8This technique yields PBF measurements that are biased toward normal values, which is a characteristic that is not desirable in the clinical environment. Under normal conditions, measured PBF is within 10% of the true value in the simulated adult when the true PBF is in the approximate range of 3–5.5 l/min. This limitation in the measurement range is not predictable from the differential indirect carbon dioxide Fick principle.

This study has identified four sources of systematic error in the partial rebreathing technique:

1. Recirculation of mixed venous blood with increased carbon dioxide content

2. Carbon dioxide partial pressure gradients and flux differences between the proximal airway and the alveolar compartment

3. Pco²and Po²gradients between the alveolar compartments and pulmonary capillary blood

4. Rebreathing times at low PBFs are inadequate to achieve quasi-equilibrium in the alveolar compartment, and rebreathing times at high PBFs are too long, causing errors due to recirculation.

The partial rebreathing technique assumes that mixed venous Pco²remains constant during a measurement. During rebreathing, carbon dioxide excretion decreases, while carbon dioxide production is maintained; therefore, carbon dioxide is accumulated in the body. The first evidence of this accumulation is an increase in mixed venous Pco²(fig. 2). The increases begin earlier and hence have a greater effect on Paco²when the cardiac output is high. Our simulated measurements, in which we used constant mixed venous Pco²and Po²(effectively simulating a body of infinite capacity for storing carbon dioxide), suggest that increases in mixed venous Pco²during rebreathing cause approximately 60% of the total systematic error in measured PBF at high cardiac outputs (fig. 3, curve b).

End-tidal airway carbon dioxide partial pressure differs from mean Paco², even in an ideal, single-compartment homogenous lung because of the time delay and mixing caused by the gas movement in the airways during tidal expiration. Using simulated alveolar values of Pco²and carbon dioxide flux for PBF calculations further approximately halved the systematic errors in the airway measurements when mixed venous Pco²was fixed (fig. 3, curve c). We found that in our model, the breath-by-breath difference between carbon dioxide flux change in the proximal airway and the carbon dioxide flux change in the pulmonary capillaries during rebreathing was small. The small difference indicates that the component of the systematic error related to proximal airway–alveolar compartment differences is due mainly to Pco²differences.

The residual systematic error can only be due to the alveolar–capillary Pco²gradient. Our cardiorespiratory model simulates an alveolar–capillary diffusion barrier using published diffusion coefficients. 9The alveolar–capillary Pco²gradient increases with cardiac output due to decreased capillary residence times, hence creating a systematic error in PBF that increases with PBF.

The time constant for carbon dioxide in an alveolar compartment is approximately inversely proportional to the product of the solubility of carbon dioxide in blood and the PBF. Hence, we expect that the rebreathing times required to achieve quasi-equilibrium of Pco²in the alveolar compartment should be shorter at high PBF and longer at low PBF, when mixed venous conditions are constant. Figure 4(curve b), in which PBF measurements were obtained using a 300-s rebreathing time and simulated alveolar values with no recirculation, suggests that the systematic errors in measured PBF at low cardiac outputs are almost entirely due to incomplete rebreathing. Figure 5suggests that the rebreathing time could be optimized by making it a function of PBF. Figure 6(curve c), in which rebreathing times follow the values indicated to achieve 95% of equilibrium changes (quantized into integer multiples of complete breaths), shows that shorter rebreathing times at high PBFs substantially reduce the systematic errors caused by recirculation. While the selection of the rebreathing time requires an a priori  knowledge of PBF, a practical approach might be to select the rebreathing time based on past measurements of PBF.

The main limitation of this study is that it was performed using a mathematical model and not live subjects. The cardiorespiratory model used in this study is based on well-documented physiologic and anatomical data, but it does have limitations. The model airway is a lumped approximation to the human respiratory tree, which might affect the magnitude of end-tidal to alveolar Pco²gradients. The model has a fixed cardiac output distribution, which might affect the calculated systematic errors, particularly at low PBF, when cardiac output might be preferentially distributed to organs with short recirculation times. We expect, however, that modeling limitations affect only the magnitude of the error calculations, not their form.

Because of the nature of this study, the effects of measurement errors are not included. The magnitude of the changes in Petco²and carbon dioxide flux depends on the size of the additional dead space used to cause partial rebreathing. Hence, measurement errors would affect PBF measurements obtained with small additional dead space more than those obtained with larger dead spaces. However, our model does not simulate measurement errors, and thus, the effects of varying the size of the additional dead space in this study are small.

There are a large number of variables that might affect the accuracy of the partial rebreathing method. This study has addressed variables closely associated with the fundamental assumptions that underpin this measurement technique. Further investigation is required to assess the relative effects of other variables such as tidal volume, 6body size, hemoglobin concentrations, and others.

This study found that the partial rebreathing technique for the measurement of PBF is strongly biased toward normal values; it tends to overestimate at low cardiac outputs and underestimate at high cardiac outputs. Three important mechanisms contributing to systematic errors in PBF measurements have been identified: recirculation; alveolar–proximal airway Pco²and carbon dioxide flux differences; and rebreathing times, which are inadequate at low PBF and excessive at high PBF. We have shown that the systematic errors at low PBF can be reduced if rebreathing times are increased and the effects of recirculation can be reduced by limiting the rebreathing time when PBF is high. We suggest that the partial rebreathing technique might be improved either by making the rebreathing time a function of predicted PBF or by correcting calculated cardiac output. Further investigations should explore these possibilities.

In a single-compartment lung under steady state conditions and neglecting fluctuations related to tidal breathing, the Fick mass balance for carbon dioxide can be written: where V̇co²is the airway carbon dioxide flux, Q̇^pis pulmonary blood flow, Cv̄co²is mixed venous carbon dioxide concentration, and Caco²is arterial carbon dioxide concentration.

During partial rebreathing, an additional dead space is added to the patient airway, causing Petco²to rise and airway carbon dioxide excretion to decrease. If Q̇^pand mixed venous carbon dioxide content remain constant during the rebreathing period, a new quasi-equilibrium is established. Equation A1can be applied to the new conditions as follows: where the prime indicates the new quasi-equilibrium conditions. Subtracting equation A2from equation A1and simplifying yields: which can be rearranged as:

where ΔV̇co²=V̇co²−V̇′co²and ΔCaco²=ΔC′aco²− Caco². The change in arterial carbon dioxide concentration was calculated from the change in Petco²using the blood–gas dissociation curves of Olszowka and Fahri. 10 

The model (fig. 1) incorporates a 14-compartment approximation to Weibel's lung model “A,”11simulating diffusive and convective transport and storage of gases in the lungs. The airways terminate in one unperfused and three perfused alveolar compartments. Lung tissue carbon dioxide storage is simulated using two lung tissue compartments. 12Intrapulmonary and anatomical shunt are included. Ventilation and V̇/Q̇ heterogeneity can be simulated by varying the inspired gas distribution and by varying the fraction of the cardiac output perfusing the three perfused alveolar compartments. The model simulates alveolar–capillary diffusion, 9intraventricular and intravascular mixing, 13variable transport delays, 14intravascular storage, 15and carbon dioxide and oxygen storage, production, and consumption in eight anatomically appropriate body compartments. 16Nonlinear blood–gas dissociation curves 10include the Haldane and Bohr effects. The tissue dissociation curves are after Farhi and Rahn 17and after Cherniack and Longobardo. 18The respiratory flow waveform was selected to match a mechanically ventilated subject: constant inspiratory flow followed by exponential expiratory flow. The model has been verified against published human data. 19,20One of the comparisons is shown in figure 7. It shows the Paco²measured in an anesthetized patient after the ventilation was reduced from 21.6 l/min to 4.8 l/min, 20compared with the cardiorespiratory model Paco²after the same ventilation change.