INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY METHODS RESULTS DISCUSSION REFERENCES APPENDIX

PRECISE KNOWLEDGE of the resistance to flow in endotracheal tubes (ETTs) is important for a rational approach to the clinical management of critically ill patients under mechanical ventilation, because of the pressure drop inside the ETT; this drop means that the pressure measurement given by the ventilator does not indicate the pressure at the distal end of the ETT. ETT resistance increases total airway resistance and must be taken into account when the patient's respiratory system resistance is being evaluated (16, 18, 23). Furthermore, this additional resistance increases the patient's work of breathing (WOB) (3, 6). This is particularly important for spontaneous ventilatory cycles, which may constitute most of the respiratory cycles during the intermittent mandatory ventilation generally used for neonates.

For adult ETTs, a mechanical approach has been used, to allow precise characterization of the relationship among ETT pressure drop, flow, inner geometry, and the physical properties of gas (17). It was found that the flow regime can be considered as turbulent and described with the Blasius equation, in which ETT pressure drop is proportional to ^1.75, with representing flow. To our knowledge, such an approach is lacking in the neonatal and pediatric fields. Although resistance measurements are available from the literature (6, 15, 22, 26), they have so far been interpreted with empirical equations, generally Rohrer equations, the coefficients of which must be computed for each ETT and that are altered by any change in ETT length, diameter, or other parameters (22). We propose to establish a general law, based on the conventional adimensional fluid mechanical approach to analysis of the pressure-flow relationship (20).

We found that flows encountered in clinical practice can be considered as laminar for neonates intubated with ETTs that have an internal diameter (ID) of 2.5, 3.0, or 3.5 mm. Under these conditions, the ETT pressure drop cannot be estimated by the Poiseuille formula, because it does not take the curvature of the tube into account, but with the Ito formula, the dimensional form of which defines pressure drop as a polynomial function of ^0.5, ^0.5, and ^1.5. For larger ETTs, 4.0- to 7.0-mm ID, the clinical range of flows includes both the laminar and turbulent flows: the Ito formula applies to laminar flows, and the Blasius formula to turbulent flows.

These general laws relate the ETT pressure drop to flow, inner geometry, and gas physical properties and can be applied to any ETT without changing the coefficients of the equation. Evaluating the pressure drop in neonatal and pediatric ETTs makes it possible to predict the part played by the ETT in total airway resistance, as well as the additional WOB imposed on the patient. Furthermore, this evaluation could be used to compensate for the ETT pressure drop with new types of assisted ventilatory modes.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY METHODS RESULTS DISCUSSION REFERENCES APPENDIX

In the fluid dynamic theory, it is usual to divide the pressure drop in a duct into two components. The first is reversible and is caused by the changes in kinetic energy between two distinct cross sections of a tube during a steady incompressible flow. The second component is the friction pressure drop or viscous dissipation term generated by the friction between the gas and the wall. Accordingly, P = P_ke + P_vd, where P is pressure drop and P_ke and P_vd are pressure drop due to changes in kinetic energy and to viscous dissipation, respectively.

The pressure drop in a tube caused by the viscous dissipation, P_vd, has been extensively studied (4, 9, 11, 12, 18, 20, 21, 27). It depends on whether the flow is laminar or turbulent, on the geometry of the tube, i.e., its ID, curvature diameter and length, and on the thermodynamic characteristics of the gas. In all cases, the transition between laminar and turbulent flow depends on the value of the Reynolds number, an adimensional number representing fluid velocity. Below a critical Reynolds number value, the flow is laminar, and above it the flow is turbulent. For curved tubes, the critical value of the Reynolds number is higher than for straight tubes and depends on the ratio of the curvature diameter to the ETT ID.

Both adult and neonatal ETTs are curved, to permit adaptation to the normal morphology of the upper airways. Under conditions of laminar flow in a straight tube, the pressure drop is given by the well-known Poiseuille's law, whereas in a curved tube it is given by the Ito formula, which indicates a larger pressure drop. Under conditions of turbulent flow in a straight tube, P_vd is given by the the Blasius formula, whereas other formulas such as the one proposed by White (27) describe P_vd for curved tubes. However, as demonstrated below (see APPENDIX), we found that, in the clinical range of flows and ETT geometry, the pressure drop computed with the White formula is virtually the same as that computed with the Blasius formula.

In clinical practice, flow in the ETT is oscillatory. However, under certain conditions, oscillatory flow can be considered to be quasi-steady, i.e., it behaves instantaneously like steady flow. Criteria for quasi-steady-flow conditions in airways have already been extensively reported (10, 20) and are described in detail in the APPENDIX. For the purposes of this study, we can consider breathing flow in the ETT to be quasi-steady.

All equations and details of the formulas are given in the APPENDIX.

	METHODS

TOP ABSTRACT INTRODUCTION THEORY METHODS RESULTS DISCUSSION REFERENCES APPENDIX

Experimental Setup

The pressure-flow relationship was studied for the ranges of flow rates used in clinical practice, as well as for higher flow rates (Table 1), so as not to exclude any clinical situation. The range of flow studied therefore included and extended beyond the entire clinical range for each ETT. Pressure drop and flow were measured by using setups A and B (Fig. 1).

View larger version (22K): [in this window] [in a new window]   Fig. 1.   Schematic representation of setups used to measure pressure drop-flow relationship. A: setup A allowed pressure-flow measurement under conditions close to clinical reality. Ventilator circuit was connected to an endotracheal tube (ETT), the tip of which was open to the atmosphere, through a standard ETT connector. Compressed air was used to deliver a predetermined steady flow (), which was measured with a pneumotachograph placed on inspiratory line, to inspiratory line of standard ventilator circuit. Expiratory line was obstructed. Pressure was measured at 2 sites with 2 differential pressure transducers connected to holes 1 mm in diameter drilled in inspiratory line upstream of Y piece and in ETT connector. B: setup B allowed pressure drop measurement through a standard ETT connector and ETT, the tip of which was open to the atmosphere. Compressed air was used to deliver a predetermined steady measured with a pneumotachograph placed just upstream of ETT connector. Pressure was measured with a differential pressure transducer connected to a hole 1 mm in diameter drilled in standard ETT connector.

Setup A allowed pressure and flow measurements under conditions close to clinical conditions. A ventilator circuit was connected via a standard ETT connector to the ETT studied, the tip of which was open to the atmosphere. Compressed air was used to deliver a predetermined steady flow to the inspiratory line of standard ventilator circuits. These comprised a neonatal circuit for 2.5- to 3.5-mm ETTs (Electro Medical Equipment, Brighton, UK), a pediatric circuit (Siemens, Erlangen, Germany) for 4.0-mm ETTs, and an adult circuit for 5.0- to 7.0-mm ETTs (DAR SPA, Mirandola, Italy). The predetermined flows were measured with a screen pneumotachograph (PT 180/10, Jaeger, Würzburg, Germany), which was linear up to flows of 500 ml/s. For larger flows, we used a Fleisch no. 1 pneumotachograph (SensorMedics, Marne-La-Vallée, France), linear up to flows of 1,350 ml/s. Both pneumotachographs were placed on the inspiratory line and connected to a differential pressure transducer (Validyne DP-45, ±2 cmH₂O, Validyne, Northridge, CA). The expiratory line was obstructed. Pressure was measured at two sites with two differential pressure transducers (Validyne DP-45, ±22.5 cmH₂O, Validyne) connected to a hole 1 mm in diameter drilled, first, in the inspiratory line 2 cm upstream of the Y piece for the neonatal and pediatric circuits and 3 cm for the adult circuit, and, second, in the ETT connector.

To assess the effect of the Y piece, we used setup B, in which the pneumotachograph was directly linked to the ETT connector. It allowed pressure drop measurement through a standard ETT connector and the studied ETT, the tip of which was open to the atmosphere. Compressed air was used to deliver a predetermined steady flow measured with the same pneumotachographs as above, which were placed upstream of the ETT connector. Pressure was measured with the same differential pressure transducer as above, connected to a hole 1 mm in diameter drilled in the standard ETT connector.

To assess the effect of curvature diameter on pressure drop and to validate our general laws, we studied the pressure-flow relationship with setup B for various curvature diameters of a 2.5-mm ETT, ranging from 6 to 20 cm. To determine the diameter, the ETT was gently curved and fixed on a circle of known diameter drawn on a sheet of paper. The curvature was constant over the entire length of the ETT.

Analysis

Statistics

	RESULTS

TOP ABSTRACT INTRODUCTION THEORY METHODS RESULTS DISCUSSION REFERENCES APPENDIX

Laminar Flow

With setup A for laminar flow, no significant difference was observed between the pressures measured upstream of the Y piece and those measured in the ETT connector. The mean difference between the two measurements was 0.02 cmH₂O with an SD of 0.06 cmH₂O, which corresponds to a mean of 3.1% of the pressure measurement at the ETT connector.

Figure 2 gives a dimensional representation of the pressure-flow relationship.

View larger version (12K): [in this window] [in a new window]   Fig. 2.   Dimensional representation of pressure drop-flow relationship. Pressure drop was measured by using setup A. Unlike adimensional representation, this conventional representation led to construction of 7 curves with 7 equations, each corresponding to 1 ETT in 1 given geometry comprising inner diameter (ID), curvature diameter, and length, and for physical properties of 1 gas. #, ID (mm).

The pressures we measured were extremely close to those predicted by the Ito formula (Fig. 3). Figure 3A shows the measured and predicted data expressed as normalized values (see METHODS). The actual pressure measurement curve was close to the curve corresponding to the pressure drop values predicted by the Ito formula. The difference between the measured and predicted values vs. the mean of both gave a mean of 0.04 cmH₂O with an SD of 0.15 cmH₂O (Fig. 3B). The coefficient of correlation (r) between the two methods was 0.997. 

View larger version (22K): [in this window] [in a new window]   Fig. 3.   A: plot of adimensional pressure-flow relationship by using setup A for 7 ETTs studied in range of laminar flow. Criterion used to separate laminar from turbulent flow was critical Reynolds number (Eq. 8 in APPENDIX). Neonatal ETTs were in laminar flow throughout entire flow range. Once kinetic energy had been subtracted, pressure drop (P) was normalized by equivalent pressure drop in Poiseuille flow (PPoiseuille; Eq. 2) and compared with predicted pressure drop given for a curved tube by Ito equation (solid line). Flow rate was normalized in terms of the Dean number [Re · (D/Dc)0.5], where Re is Reynolds no. and D and Dc are diameter of tube and curvature, respectively. Note that ratio of actual and Ito-predicted P to PPoiseuille was much higher than 1, indicating that PPoiseuille greatly underestimates ETT pressure drop. All formulas and equations are in APPENDIX. B: comparison of pressure drop values minus pressure drop due to kinetic energy changes obtained from direct pressure measurement, using setup A, with predicted pressure drop given for a curved tube by Ito formula. As recommended by Bland and Altman (2), differences between 2 values were plotted against mean of 2 values. Mean of differences between 2 methods was 0.04 cmH2O, with SD of 0.15 cmH2O.

In setup B, the measured values were also not different from the theoretical values, with mean difference of 0.04 cmH₂O, SD of 0.08 cmH₂O, and r = 0.996 (figure not shown).

As shown in Fig. 4A, the measured pressure drops for different curvature diameters were very close to those predicted by the Ito formula. Figure 4B shows the variation of pressure drop curves for different curvature diameters in a dimensional representation of the pressure-flow relationship and emphasizes the influence of curvature diameter on this relationship.

View larger version (20K): [in this window] [in a new window]   Fig. 4.   Effect of Dc on pressure-flow relationship in a 2.5-mm-ID ETT. A: adimensionnal plot. Measured pressure drops for different curvature diameters were very close to those predicted by Ito formula. B: dimensional representation of pressure-flow relationship. Pressure drop increase with decreasing value of curvature.

Turbulent Flow

In setup A, a small insignificant difference was observed between the pressure recorded upstream of the Y piece and the pressure recorded in the ETT connector (mean difference between the 2 measurements = 0.32 cmH₂O, SD = 0.24 cmH₂O). This difference constituted a mean of 3.4% of the pressure measured in the ETT connector.

Our results are in agreement with the Blasius formula, as we found that, throughout the entire flow range studied, the difference between the actual measurements and those predicted by this formula vs. the mean of the two methods gave a mean of 0.02 cmH₂O and an SD of 0.50 cmH₂O (Fig. 5). The r between the two methods was 0.99. 

View larger version (18K): [in this window] [in a new window]   Fig. 5.   A: plots of adimensional pressure-flow relationship by using setup A for ETTs with 4.0- to 7.0-mm ID in range of turbulent flow. Pressure drop and flow were obtained by direct measurement. Once pressure drop due to kinetic energy change had been subtracted, it was compared with predicted pressure drop given by Blasius formula (Eq. 4 in APPENDIX). Pressure drop was normalized by equivalent pressure drop given by Blasius formula (PBlasius). Criterion used to separate laminar from turbulent flow was critical Reynolds number (Eq. 8). Solid line, values predicted by Blasius formula. B: comparison of values for pressure drop minus pressure drop due to kinetic energy changes obtained from direct pressure measurements by using setup A and predicted pressure drop given by Blasius formula. As recommended by Bland and Altman (2), differences between 2 values were plotted against mean of 2 values. Mean of differences between 2 methods was 0.02 cmH2O, with SD of 0.50 cmH2O.

In setup B, we found that the Blasius formula slightly overestimates our experimental results. Throughout the entire range of clinical values for flow, the mean difference between the measured and predicted values was 0.20 cmH₂O, with an SD of 0.18 cmH₂O and r = 0.99.

	DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY METHODS RESULTS DISCUSSION REFERENCES APPENDIX

Our laboratory previously demonstrated that, in adult ETTs, the inspiratory pressure drop can be satisfactorily estimated by the Blasius formula, which, for turbulent flows, relates ETT geometry and gas properties to the pressure drop (17). In the present study, we used an identical approach for neonatal and pediatric ETTs. Our main observation was that, for neonatal ETTs 2.5-3.5 mm in diameter, 1) the flow is laminar; and 2) because of the curvature of the tube, the pressure drop is higher than predicted by the Poiseuille flow. We found that, in neonatal ETTs, the pressure drop can be estimated by the Ito formula, in which the ETT pressure drop is a polynomial function of ^0.5, ^0.5, , and ^1.5. In pediatric ETTs (4.0- to 7.0-mm ID), we observed that physiological flows are successively laminar and turbulent during inspiration, and consequently the ETT pressure drop can be estimated by the Ito formula for laminar flows and by the Blasius formula for turbulent flows.

Flow Regimes

Laminar flow. By calculating the critical Reynolds number (Eq. 8 in the APPENDIX), we found that the flow regime in neonatal ETTs was laminar for a curved tube. The excellent agreement (Fig. 3) observed in the physiological flow range between our pressure drop measurement and the values predicted by the Ito formula (Eq. 6 in the APPENDIX) indicates that the flow in neonatal ETTs was laminar for a curved tube. At first sight, this result seems to disagree with the findings of several groups (22, 23, 30) who used empirical equations such as the Rohrer equation (P = K₁ ·  + K₂ · ²) to describe the resistive properties of neonatal ETTs. The Rohrer constants K₁ and K₂ are indeed generally assumed to refer to the factors determining laminar and nonlaminar flow, respectively (24). However, this interpretation of the significance of the Rohrer equation coefficients might indicate that the neonatal flow is not only laminar, which would be erroneous. The laminar flow in a curved tube is a good example of the fact that laminar flow does not mean that there is a linear relationship between pressure drop and flow. Furthermore, our results clearly show that laminar flow is not synonymous with Poiseuille flow, and the use of the conventional Poiseuille equation (Eq. 2) does not account for the pressure drop due to the ETT curvature. Figure 4 clearly illustrates the influence of curvature on pressure drop, and, obviously, in Fig. 3, the ratio P/(P_Poiseuille) is far above one, except for flows close to zero, where P_Poiseuille is the Poiseuille pressure drop. The use of the Poiseuille flow (Eq. 2) would have underestimated the pressure drop by a factor that can reach three for the larger flows (Fig. 3). Moreover, compared with the Rohrer equation, the Ito formula has the advantage of taking into account the circular inner geometry of the ETT (diameter and length), its curvature, and the gas properties. The ETTs we studied had in vitro curvatures. In clinical situations, this curvature varies with the position of the patient's head. Furthermore, this curvature is probably not constant along the tube. Because all these variations along the tube cannot be integrated, we propose to use a mean curvature diameter. We tried to evaluate this curvature diameter on 20 magnetic resonance imaging scans of children, from premature newborn infants to 10-yr-old children. In a first approximation, the mean range of curvature diameter encountered in clinical situations can be estimated as 8-18 cm. The difference in pressure drop between these two extremes (variation of up to 100%) is <16%. This difference can be disregarded in clinical practice, especially when it is compared with the error induced by Poiseuille's law (see Fig. 3A). This slight difference induced by the curvature diameter variation, therefore, allows the use of a mean curvature diameter for calculation of the pressure drop.

Turbulent flow. We previously found that, in adult ETTs, turbulent flow is observed during most of the inspiration and that the pressure drop along the ETT is proportional to ^1.75, as predicted by the Blasius formula (17, 25). In the present study, computation of the critical Reynolds numbers showed that turbulent flow is only present in ETTs with an ID above 4.0 mm under the flow conditions chosen for study.

Entry effects. The formulas used in the present study (Ito, Poiseuille, White, and Blasius) assume a fully developed flow. This means that the velocity profile remains the same from the entry to the exit of the ETT. When this profile is not verified, a pressure drop (P_ent) must be added to the kinetic energy term and to the viscous dissipation term to account for the entry flow effect. Many formulas have been proposed to describe this additional pressure drop, which is due to fluid contraction (9, 18). This drop can be rendered by the equation where  is a coefficient that depends on the flow regime, on the ratio of the area before contraction to the area after contraction, and, above all, on the shape of the passage from the upper to the lower section of the ETT connector (rough, smooth, and so on),  is the fluid density, and is the average velocity. If we apply the formula proposed by Idel'cik (9) for a conical curvilinear connector with an angle close to 60° to the ETT connector, the ratio of P_ent to the pressure drop due to the ETT is between 3 and 14%. Our results, which are in agreement with this approximate analysis, suggest that P_ent is negligible. We also observed in setup A that the pressures measured upstream and downstream of the Y piece were identical, suggesting the Y piece is designed to reduce the pressure drop to a minimum. This last point is clinically important, because it suggests that the pressure measurement in the Y piece of the neonatal circuit is well adapted to estimating the real pressure at the entry of the ETT connector.

Kinetic energy. The use of the Ito or Blasius formula alone implies that the pressure drop due to kinetic energy has been subtracted from the measured drop in the ETT plus the connector. Therefore, direct comparison of the measured pressure drop to the predicted value makes it necessary to add the drop due to kinetic energy to the drop calculated with the Ito or Blasius formula. The drop due to kinetic energy is due to the fact that the section at the top of the ETT connector is larger than the section at the bottom. Whereas the diameter at the top of the connector is always the same, the diameter of the lower section, i.e., of the ETT, varies. Here, we observed that the pressure drop due to kinetic energy was between 13 and 37% of the mean total pressure drop, which includes the ETT plus the connector.

Exit effect. Because of the area difference between the end of the ETT and the trachea, an exit effect associated with a variation of kinetic energy is observed. The resulting pressure drop is always positive (18) and close to It is difficult to use this formula without a precise knowledge of the tracheal area, A₂. Use of the Ito (or Blasius) formula may therefore slightly overestimate the ETT pressure drop with the ETT in situ. In this study, we considered that the pressure drop due to the abrupt expansion between the ETT and the trachea is not a specific effect of the ETT but is included in the pressure drop due to the patient's airways.

Dimensional forms of the equations for practical estimation of the pressure drop in a clinical situation. The use of either the adimensional or the dimensional form of the Ito formula (see APPENDIX) to compute the pressure drop may represent a hard challenge for the clinician outfitted with only a pocket calculator. It is possible to propose a simpler formula. Taking into account the gas properties of air with 100% relative humidity at 37°C, and with a barometric pressure of 76 mmHg, and assuming a curvature diameter of 14 cm, a simplified and dimensional form of the Ito formula can be written to estimate the pressure drop (cmH₂O) where is the flow (ml/s), D is the ID of the ETT (mm), and L is its length (cm). Note that this polynomial function of and ^1.5 is derived from Eq. 7 in the APPENDIX by neglecting the terms in ^0.5, ⁰, and ^0.5. Because the introduction of oxygen instead of air does not substantially modify the physical properties of gas, this formula can be used to estimate the pressure drop when oxygen is added to the respiratory gas. However, this formula must not be used when the respiratory gas has very different physical properties, like heliox.

Prediction of the ETT Pressure Drop, ETT Resistance, and WOB Due to ETT (ETT WOB) in Clinical Practice

ETT pressure drop and resistance estimation. Our results indicate that, in any clinical situation, knowledge of ETT geometry and of the flow regime allows accurate evaluation of the pressure drop specifically due to the ETT. In the neonatal range, there is no easy way of measuring this pressure drop. The catheter technique for measuring tracheal pressure, used for adult ETTs (8, 17), is hardly applicable routinely during the neonatal period, even though its use has been attempted (5). If the catheter is too large, it will dramatically reduce the permeability of the ETT and may render ventilation difficult or hazardous and may also modify the flow regime, thus affecting the measurement. If the catheter is very thin, there is a high risk of its being partially or totally obstructed, which also impedes correct measurement.

1

1

1

ETT WOB. The pressure drop calculated according to the Ito formula (or the Blasius formula when appropriate) augmented by the pressure drop due to kinetic energy changes allows estimation of the ETT pressure drop and therefore the ETT WOB. WOB can be calculated from pressure drop and flow as where V is the volume and P is the pressure drop

View larger version (21K): [in this window] [in a new window]   Fig. 6.   Simulated flow (top), theoretical total work of breathing (WOB) due to ETT (ETT WOB; middle), and that normalized by tidal volume (VT; bottom). Pressure drop was computed from flow rate by using Ito or Blasius formula, depending on Reynolds number, plus pressure drop due to kinetic energy changes (see APPENDIX). ETT WOB was calculated by assuming a VT of 7 ml/kg, a physiological respiratory rate (RR) with an inspiratory-to-expiratory ratio of 1/2.75. ETT #2.5: VT = 7 ml, RR = 60 breaths/min; ETT #3.0: VT = 14 ml, RR = 50 breaths/min; ETT #3.5: VT = 35 ml, RR = 35 breaths/min; ETT #4.0: VT = 77 ml, RR = 24 breaths/min; ETT #5.0: VT = 126 ml, RR = 20 breaths/min; ETT #6.0: VT = 172 ml, RR = 18 breaths/min; ETT #7.0: VT = 333 ml, RR = 16 breaths/min.

View larger version (23K): [in this window] [in a new window]   Fig. 7.   ETT WOB was calculated from same formulas as in Fig. 6, assuming a VT of 12 ml/kg and an increase in RR of 25%. ETT #2.5: VT = 12 ml, RR = 75 breaths/min; ETT #3.0: VT = 24 ml, RR = 63 breaths/min; ETT #3.5: VT = 60 ml, RR= 44 breaths/min; ETT #4.0: VT = 131 ml, RR = 30 breaths/min; ETT #5.0: VT = 216 ml, RR = 25 breaths/min ; ETT #6.0: VT = 295 ml, RR = 23 breaths/min; ETT #7.0: VT = 570 ml, RR = 20 breaths/min. Note that scale in middle panel is different from that in middle panel in Fig. 6. Total ETT WOB is much larger than that computed for conditions specified in Fig. 6. Note also appearance of turbulent flow for 4.0- to 7.0-mm ETTs and large amount of related WOB.