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GENERAL ARTICLES

Validation of an Original Mathematical Model of CO₂ Elimination and Dead Space Ventilation

From the University Department of Anesthesia, University Hospital, Nottingham, UK

Address correspondence and reprint requests to Dr. Jonathan G. Hardman, Clinical Senior Lecturer, University Department of Anesthesia, University Hospital, Nottingham, NG7 2UH, UK. Address email to j.hardman{at}nottingham.ac.uk

	Abstract

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We present an original, mathematical model of ventilation and gas-exchange. Our aim was to validate it using data from previous clinical investigations, allowing our use of it in future investigations. The first previous investigation used a low-dead space, double-lumen, tracheal tube (DLT). We matched the model’s PaCO₂ and airway pressures (P_AW) to the patient mean during use of the DLT and a single-lumen tube (SLT). The model’s resulting PaCO₂, PÉCO₂ and P_AW were compared with the patients’ as tidal volume (VT) changed with constant minute volume. The second investigation examined dead space during anesthesia. The model’s VT, respiratory rate, CO₂ production, temperature, and alveolar and anatomical dead spaces were matched to each mechanically ventilated subject. Bias and precision in predictions of PaCO₂ and PÉCO₂ were calculated. The model’s bias in prediction of dead space reduction by the DLT was 6.9%. Bias in prediction of P_AW was 0.1% (peak) and -5.13% (mean), of PaCO₂ was 1.2% (DLT) and 1.5% (SLT) and of PÉCO₂ was 1.7% (DLT) and 1.3% (SLT). Prediction of PaCO₂ and PÉCO₂ in the second investigation (as 95% confidence interval of bias): PaCO₂ -2.6% to 0.8% and PÉCO₂ -4.9% to 1.2%. This validation allows future application of our model in appropriate theoretical investigations.

IMPLICATIONS: We present an original, mathematical model of ventilation and gas exchange. We validate it against previously published clinical data to allow its use in future theoretical investigations where data may be unavailable from patients.

	Introduction

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Estimation of alveolar dead space fraction in the intensive care unit is important and has recently been shown to assist in predicting outcome (1). However, it is seldom performed because of the technical difficulty and time-consuming nature of the technique. A previous investigation has examined the relationship between the arterial to end-tidal CO₂ tension gradient (Pa-E'CO₂) and alveolar dead space fraction (VDalv/VTalv) (2). Significant advances in the complexity and fidelity of physiological modeling have prompted a re-examination of the subject. The major advances in the modeling over that used in our previous investigation are described in Table 1.

	Methods

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The Mathematical Model
The model is an evolution of the Nottingham Physiology Simulator, whose basic components have been described previously (3–5). A detailed description of changes from that described previously is presented in the Appendix.

The validation investigations.
Two validation investigations were performed, each using a previously published, clinical investigation to provide input data. Each validation investigation matched the mathematical model to aspects of the clinical data from previous investigations and examined the model’s ability to reproduce physiological data from the clinical investigation.

Previous, clinical investigation used in validation investigation #1.
The Liebenberg et al. study (6) was used in validation investigation #1. The lungs of 12 healthy anesthetized adults were mechanically ventilated via a tracheal double-lumen tube (DLT). The distal tip of both lumens lay in the trachea. The patients’ lungs were ventilated either via one of the lumens (single-lumen tube, SLT) or in a "reduced dead space" configuration (DLT), with inhalation via one lumen and exhalation via the other.

Ventilation initially comprised tidal volume (VT) 10 mL/kg and respiratory rate (RR) 10 breaths/min. RR was increased sequentially to 13, 21, and 40 breaths/min while VT was reduced to 7.5, 5, and 2.5 mL/kg, ensuring approximately constant expired minute volume. At each VT and RR configuration, peak and mean airway pressures (PAW_peak and PAW_mean) and end-tidal and arterial PCO₂ (PÉCO₂ and PaCO₂) were recorded while patients’ lungs were ventilated in both the "normal" and "reduced dead space" modes with 10-min equilibration for each mode.

Validation investigation #1.
We matched the model to the mean of the results of this investigation because insufficient data were provided from each subject to allow matching to individuals. Assumption of some physiological values was required where they were not provided in the study. These are included in the methodological description below.

1. The model was configured as a 1.75-m, 75-kg subject. Oxygen consumption was set at 250 mL/min and CO₂ production at 200 mL/min.
   
2. Intrinsic compartmental compliances and resistances were equal throughout the 500 compartments and gravitational effects on alveolar units determined the distribution of ventilation. Cardiac index was fixed at 2.73 L/min/m^-2 and was distributed through the 500 "alveolar" compartments such that the ventilation-perfusion (VQ) distribution closely resembled that of the healthy supine subject (7): mean VQ ratio was 1.38 with a standard deviation of 0.16.
   
3. The model was matched to the mean of the patient data at VT 10 mL/kg and RR 10 breaths/min. This matching included adjusting compartmental compliances and resistances to match the model’s PAW_peak and PAW_mean during mechanical ventilation via the SLT (standard dead space). In addition, PaCO₂ was matched to the mean values observed in the clinical investigation (in both SLT and DLT configurations) by adjusting the volume of the model’s VDanat laminae. Model VDanat (calculated using Fowler’s technique on its PÉCO₂ versus time trace) was subsequently recorded for comparison with the clinical study’s measured reduction in dead space caused by modification of the DLT.
   
4. After this matching, VT and RR were adjusted as in the clinical study. At each combination of RR and VT, in SLT and DLT configurations, the model’s values for PaCO₂ and PÉCO₂ were recorded. PAW_peak and PAW_mean were recorded in the SLT configuration.
   

Calculation using current, standard methods (validation investigation #1).
Predictions of PaCO₂ after the adjustments in RR and VT were also calculated using the equation new PaCO₂ = old PaCO₂ x old alveolar minute volume/new alveolar minute volume (Equation 1) in combination with Nunn and Hill’s conclusions on the variation of VDanat with VT in the anesthetized patient (8).

Previous, clinical investigation used in validation investigation #2.
The Nunn and Hill study (8) was used in validation investigation #2. Temperature, CO₂ production, PaCO₂, PÉCO₂, VT, and RR were measured using standard techniques in 12 healthy, anesthetized subjects during elective, general anesthesia. VDalv and VDanat were measured using the Bohr-Enghoff equation (9) and Fowler’s technique (10). The aim of this investigation had been to investigate the variation in dead spaces during anesthesia and the variation of dead space volumes during alteration in VT.

Validation investigation #2.
Provision of sufficient physiological detail allowed matching of the model to individual subjects whose lungs were mechanically ventilated. The mathematical model does not currently include algorithms for gas flow and VQ variations during spontaneous ventilation, which differ markedly from those in patients whose lungs are mechanically ventilated. Eleven subjects had a complete data set and were incorporated as input data (8). Assumption of some physiological values was required where they were not provided in the study. These are included in the methodological description below.

1. The model was configured as a 1.75-m, 75-kg subject and cardiac index was fixed at 2.73 L/min/m^-2.
   
2. Model VT, RR, CO₂ production, and body temperature were matched to the subject’s values.
   
3. The model’s VDanat laminae were adjusted until the total VDanat (calculated using Fowler’s technique on its PÉCO₂ versus time trace) matched the measured VDanat of the subject.
   
4. Model VDalv was adjusted by varying compartmental inlet resistances and vascular resistances in opposite directions until the value obtained by a combination of Fowler’s technique and the Bohr-Enghoff equation matched that observed in the subject. Inlet and vascular resistances were distributed through the alveolar units in a lognormal fashion, producing a realistically physiological VQ distribution in each case.
   
5. Once equilibrium was attained (defined as total body flux of O₂ and CO₂ <0.1 mL/min), PaCO₂ and PÉCO₂ were recorded.
   

Calculation using current, standard methods (validation investigation #1).
Additionally, prediction of PaCO₂ was made using a conventional formula (Equation 2):

Bias was calculated as the mean of the model’s error in predicting the variable under consideration, and 95% limits of agreement were calculated as the 95% confidence interval (CI_95%) of the bias. CI_95% was calculated as mean ± 1.95 x SD.

	Results

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Validation investigation #1 (6).
The model predicted the dead space reduction caused by the modification of the tracheal tube as 49.4 mL, compared with 46.2 mL in the clinical investigation. Bias in predicting PaCO₂, PÉCO₂, PAW_peak, and PAW_mean are shown in Table 2 and Figure 1. The average error (model prediction minus study mean), expressed as a percentage of the SD observed clinically at each measurement point, was 15.4% of SD for PaCO₂, 8.4% of SD for PÉCO₂, and 3.3% of SD for P_AW. No model prediction differed from the clinical study mean by more than 30% of the SD or by 10% of the mean.

View larger version (15K): [in this window] [in a new window]   Figure 1. The model’s predictions of the effect of changing respiratory rate (RR) and tidal volume (VT) on (panel A) PaCO2 and (panel B) PÉCO2 and (panel C) peak and mean airway pressures. Solid lines show data from the clinical investigation. The dashed lines show predictions made by the model. Error bars represent standard deviation within the clinical study group. In panels A and B, the upper lines show values observed using the tracheal tube in single-lumen tube (SLT) configuration and the lower lines show values observed in double-lumen tube (DLT) configuration. In panel C, the upper lines show peak airway pressure and the lower lines show mean airway pressure. Note: model and clinical study data points coincide at 10 breaths/min because this was the point of matching. As RR increases and VT decreases (maintaining constant minute volume) the predictive capacity of the model is demonstrated.

Validation investigation #2.
Data describing the accuracy of the model’s predictions of the clinical subjects’ PaCO₂ and PÉCO₂ are given in Table 3 and Figures 2 and 3. The correlation coefficient was not statistically significant between the measured values of PaCO₂ and PÉCO₂ and the error in their predicted values.

View larger version (11K): [in this window] [in a new window]   Figure 2. Model error in prediction of PaCO2 during mechanical ventilation. Dashed lines represent bias (i.e., mean error) and 95% limits of agreement between measured and predicted values.

View larger version (11K): [in this window] [in a new window]   Figure 3. Model error in prediction of PÉCO2 during mechanical ventilation. Dashed lines represent bias (i.e., mean error) and 95% limits of agreement between measured and predicted values.

	Discussion

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The first investigation providing input data (6) examined the use of a tracheal tube that had been modified to reduce VDanat, the non-gas-exchanging part of the respiratory tract conducting gas between gas-exchanging areas and the atmosphere (11). Arterial PCO₂ and inspiratory pressures were measured during approximately constant minute volume with reducing VT and increasing RR. The clinical investigation provided input data to validate the CO₂ elimination, VDanat and VDalv models, and static and dynamic compliance aspects of the respiratory model. The errors in predicting inspiratory pressures, PÉCO₂ and PaCO₂ were each small (<16% of clinically observed SD). We feel that the model accurately reproduced the data recorded in the clinical study. It also accurately predicted the reduction in VDanat by the of DLT mode versus SLT.

Even a simple model of respiratory physiology would be expected to predict the general behavior observed in the clinical investigation. However, application of Equation 1 in combination with Nunn and Hill’s conclusions on the variation of VDanat with VT in the anesthetized patient (8) produced predictions of PaCO₂ whose errors were much larger than those produced by our modeling. It is clear that this modeling produces a significant advance on the application of current theory.

Exact matching of the model’s predictions for PaCO₂, PAW_peak, and PAW_mean at the various RR and VT combinations from clinical investigation #1 would be surprising and coincidental because the model was matched to the mean of a heterogeneous group. Reasons for inexact matching, other than the expected population variation, include the assumption of incorrect physiological data for subjects such as the VQ distribution and oxygen consumption. The difficulty of prediction is further increased by the induction of general anesthesia in the patient population, which introduces further physiological variation. Error in the assumption of incorrect population physiological data is probably unavoidable. We considered that the accuracy of our blinded, prospective matching of PaCO₂, PÉCO₂, and P_AW (all of which were within 10% of the original, clinical observations in all cases) was acceptably accurate and that this accurate matching represented acceptable validation of pertinent parts of this lung model. In particular, matching of the model to the clinical finding of an increasing PaCO₂ during reducing VT and increasing RR (with constant minute volume) implies that use of a "constant-volume, non-mixing, poly-laminar" model of VDanat is acceptable in this context.

The second investigation used to provide input data (8) was the only investigation that we could find where sufficient detail was provided of each subject to allow us to match the model to individual subjects. Thus, very little blinded assumption of physiological data was required. The model’s accurate prospective predictions of PaCO₂ (95% limits of agreement: -2.6% to 0.8%) and PÉCO₂ (95% limits of agreement: -4.9% to 1.2%) during mechanical ventilation represent very convincing validation of pertinent aspects of the model. As above, attempts at prediction of the PaCO₂ using conventional means (Equation 2) produced a result whose error was much larger than that produced using our mathematical model.

A limitation of this model validation is that our matching and predictions have been in healthy subjects, whereas subsequent investigations will deal with deranged physiology and disease. First, both sets of clinical data were obtained from subjects under general anesthesia, a situation in which respiratory function is significantly disturbed. Second, this validation in healthy, anesthetized subjects allows us to claim adequate validation of the processes of the model. As long as the model is well matched to the physiological factors in patients with respiratory diseases then estimations using the model will have similar accuracy to those presented here. Finally, there is a paucity of data of sufficient detail to perform this type of validation at all, and it is not feasible to validate the model in the presence of a variety of diseases.

This validation assumes that PÉCO₂ is determined absolutely by physiological circumstances. In practice, however, PÉCO₂ may be measured at different sites within a breathing system, giving different values (12). Incomplete exhalation may yield a mixture of serial dead space gas and alveolar gas, causing a misleading reduction in PÉCO₂. Measuring PÉCO₂ at a standardized site within the breathing system and encouraging complete exhalation whenever possible by the use of an adequate expiratory time will help assure a more reliable PÉCO₂. Despite the difficulties in interpreting single PÉCO₂ values, trends within subjects are likely to be accurate.

In conclusion, we have validated important parts of an original, multicompartmental mathematical lung model by accurately reproducing the following in healthy, anesthetized subjects:

• The complex variation of PaCO₂, PÉCO₂, and P_AW in patients during changing VT and RR, by blinded matching the model to the mean of the patient group and reproducing the clinical study protocol.
  
• PaCO₂ and PÉCO₂ during steady-state conditions under mechanical ventilation after detailed matching of the model to individual subjects’ temperature, CO₂ production, VT, RR, Vdanat, and VDalv.
  

This validation allows us to recommend the use of this model of pulmonary physiology for further theoretical investigations into methods of mechanical ventilation, CO₂ clearance and alveolar pressures. The investigation of new methods of estimating VDalv is particularly important in monitoring patients on intensive care units with acute respiratory distress syndrome (1).

	Appendix

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The model uses a mass-conserving, iterative, analytical method. Each iteration represents 1 ms of real, physiological time. During this period, gas-flow into and out of each of the 500 "alveolar" compartments is calculated from airway pressure, compartment "bronchiolar" resistance, compartment compliance, and compartment volume. Bronchiolar flow is laminar or turbulent as dictated by the Reynolds number. Each of the 500 bronchioles communicates directly with the lowest lamina of the VDanat. Communication between alveolar compartments takes place via the VDanat. The VDanat comprises 250 non-mixing laminae arranged in a sequential pattern from the airway (e.g., mouth or nose) to the opening into the 500 "alveolar" compartments. Mixing between the VDanat laminae was assumed not to occur during this investigation. Entry of gas into each "alveolar" compartment results in an immediate and complete mixing with the compartmental contents. This "constant-volume, non-mixing, poly-laminar" model of VDanat ventilation is original, simple and computationally efficient.

Partial pressures and volumes of nitrogen, oxygen, carbon dioxide and water vapor are included in the model. Incremental volumes of each gas move from the model’s pulmonary capillary to the alveolar compartment or vice versa until the partial pressure in each differs by <1%. The process is repeated for every gas in every pulmonary compartment. The process of equilibration is complicated by the nonlinear solubilities of gases and by alterations in each compartment’s volume during gas equilibration necessitating repeated recalculation of compartmental pressure.

Mass is conserved at every mathematical step. The ideal gas laws (i.e., the constancy of pressure x volume/temperature) are applied within the model to compensate for the effects of changes in respiratory tract pressures and gas temperatures.

The effect of gravity on the interstitial pressures in the lung, and thus the "resting volume" of individual alveolar units, is included in the model (7). The compartmental interstitial pressure is modeled as increasing by 0.3 cm H₂O for every centimeter down the lung from the highest point. The resulting resting distension of the apical alveoli compared to those at the bases produces a realistic distribution of volume and ventilation.

Several studies suggest that differences between static and dynamic compliance are determined by viscoelastic behavior of pulmonary tissues in addition to the intrapulmonary gas redistribution seen in diseased lungs (13,14). A simple model of tissue plasticity is included. An "elastance multiplier" is used to scale the intracompartmental pressure. When compartmental volume is greater than "resting volume" (that volume assumed when distending pressure equals zero), the elastance multiplier decreases, during each 1 ms time-slice, by 0.00002 x Sampling interval x current volume/basic volume. Thus when compartmental volume is double the "resting volume" the elastance multiplier decreases by 0.00004 per millisecond. The elastance multiplier is allowed to decrease as far as 0.8 (14). To counteract this "relaxing" effect of alveolar compartmental distension, the elastance multiplier constantly increases towards its basic value of unity with a half-time of 2 s (14,15). Consequently, the elastance of ventilating compartments is continually changing during ventilation closely mimicking the behavior described by D’Angelo et al. (13) and by Milic-Emili et al. (14).

Compartmental compliance was calculated as follows: If CurrentVol > BasicVol then Pressure = 10 x (IntP + Pull + Elastance x ((MaxVol - BasicVol)/(MaxVol - CurrentVol) - 1))). Otherwise Pressure = 10 x (IntP + Pull + 1.5 x Elastance x (1 - BasicVol/CurrentVol)), where the resulting pressure is cm H₂O greater than atmospheric pressure and volume is mL. CurrentVol represents the current alveolar unit volume; MaxVol represents the maximum volume of the alveolar unit; BasicVol represents the volume of the unit at a distending pressure of zero; lntP is the interstitial pressure of each unit; Pull is the gravity-induced negative pressure for each unit’s interstitium; Elastance is the unit’s pressure change for a change in volume at the current position on the unit’s elastance curve.

Oxygen content was calculated as: n = PO₂ x 10^{(0.48*(pH - 7.4) - 0.024 * (temp - 37) - 0.0013 * BE)}; SO₂ = 1/(1 + 55.4667/(N x (N² + 2.6667))); CO₂ = 1.34 x SO₂ x Hb + 0.2 x PO₂ (16); where PO₂ denotes partial pressure of oxygen (kPa), SO₂ denotes oxygen saturation of hemoglobin (0%–100%), CO₂ denotes oxygen content of blood (mL/L), temp denotes blood temperature (°C), BE denotes base excess (mmol/L) and Hb denotes hemoglobin concentration in blood (g/L). This algorithm provided the best compromise between accuracy and computational efficiency (17).

	References

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a colleague Similar articles in this journal Similar articles in Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Request Permissions Citing Articles Citing Articles via HighWire Citing Articles via Web of Science (4) Citing Articles via Google Scholar Google Scholar Articles by Hardman, J. G. Articles by Aitkenhead, A. R. Search for Related Content PubMed PubMed Citation Articles by Hardman, J. G. Articles by Aitkenhead, A. R. Related Collections Monitoring (Non-cardiac)