Vol. 87, Issue 1, 36-46, July 1999
Estimation of inspiratory pressure drop in neonatal and
pediatric endotracheal tubes
Pierre-Henri
Jarreau1,2,
Bruno
Louis2,
Gilles
Dassieu3,
Luc
Desfrere1,
Perre W.
Blanchard1,
Guy
Moriette1,
Daniel
Isabey2, and
Alain
Harf2,4
1 Service de Médecine
Néonatale, Centre Hospitalier Universitaire Cochin-Port
Royal, Assistance Publique-Hôpitaux de Paris-Université
Paris V, 75014 Paris; 2 Institut
National de la Santé et de la Recherche Médicale (INSERM) U
492, 94000 Créteil;
3 Service de Réanimation
Néonatale, Centre Hospitalier Intercommunal, 94000 Créteil;
and 4 Service de
Physiologie-Explorations Fonctionnelles, Centre Hospitalier
Universitaire Henri Mondor, Assistance Publique-Hôpitaux de
Paris-Université Paris XII, 94000 Créteil, France
 |
ABSTRACT |
Endotracheal tubes
(ETTs) constitute a resistive extra load for intubated patients. The
ETT pressure drop (
PETT) is
usually described by empirical equations that are specific to one ETT only. Our laboratory previously showed that, in adult ETTs,
PETT is given by the Blasius
formula (F. Lofaso, B. Louis, L. Brochard, A. Harf, and D. Isabey.
Am. Rev. Respir. Dis. 146:
974-979, 1992). Here, we also propose a general
formulation for neonatal and pediatric ETTs on the basis of
adimensional analysis of the pressure-flow relationship. Pressure and
flow were directly measured in seven ETTs (internal diameter:
2.5-7.0 mm). The measured pressure drop was compared with the
predicted drop given by general laws for a curved tube. In neonatal
ETTs (2.5-3.5 mm) the flow regime is laminar. The
PETT can be estimated by the
Ito formula, which replaces Poiseuille's law for curved tubes. For
pediatric ETTs (4.0-7.0 mm),
PETT depends on the following
flow regime: for laminar flow, it must be calculated by the Ito
formula, and for turbulent flow, by the Blasius formula. Both formulas
allow for ETT geometry and gas properties.
fluid mechanics; laminar flow; turbulent flow; Ito formula; Blasius
formula
| |
INTRODUCTION |
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 |
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 = Pke + Pvd, where P is pressure drop and Pke and
Pvd 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,
Pvd, 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,
Pvd is given by the the Blasius
formula, whereas other formulas such as the one proposed by White (27)
describe Pvd 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 |
Experimental Setup
Neonatal- and pediatric-size ETTs, with an ID ranging from 2.5 to 7.0 mm, were tested with their natural curvature. This curvature was
measured by comparing the natural curvature with calibrated circles
drawn on a sheet of paper. ETT ID and length were systematically checked by the acoustic reflection method, already validated for adult
ETTs (19, 25) as well as for neonatal ETTs (13). This method gives the
area of the inner cross section, which was converted into diameter
assuming a pure circular geometry. Table 1
indicates the characteristics of the different ETTs used.
View this table:
[in this window]
[in a new window]
|
Table 1.
Characteristics of the endotracheal tubes studied, maximal flows
studied and their corresponding Reynolds numbers, and values of
critical Reynolds numbers for each ETT
|
|
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
The relationship between pressure drop and flow was obtained by direct
measurement. Once the kinetic energy calculated by Eq. 1 (see APPENDIX) had
been subtracted, we compared the resulting pressure drop to the
predicted pressure drop given by the Blasius formula
(Eq. 4) or the Ito formula
(Eq. 6). Both formulas are given in
the APPENDIX. Flow rate was expressed
as a normalized value, represented by the Reynolds number
(Eq. 3) for turbulent flow and by
the Dean number (which includes the Reynolds number and curvature,
Eq. 5 in the
APPENDIX) for laminar flow. Pressure
drop was normalized by the equivalent pressure drop given by
Poiseuille's law (Eq. 2) for
laminar flow and by the equivalent pressure drop given by the Blasius
formula for turbulent flow. To separate the laminar from the turbulent
flow, the criterion used was the critical Reynolds number, defined in
Eq. 8 (see
APPENDIX).
Statistics
The pressure drop obtained by direct measurements and the predicted
pressure drop given by Eq. 4 or
6 in the
APPENDIX were compared by plotting the
difference between the two values against the mean of the two values,
as recommended by Bland and Altman (2).
| |
RESULTS |
Laminar Flow
We first analyzed the data obtained under laminar flow conditions,
i.e., for the Reynolds number below the critical Reynolds number, which
represent all the flows encountered in clinical practice for 2.5- to
3.5-mm ETTs and the lower flows for 4.0- to 7.0-mm ETTs (Table 1).
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
The conditions of turbulent flow, i.e., for the Reynolds number
exceeding the critical number, were only present for the higher flows studied for 4.0- to 7.0-mm ETTs.
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 |
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/(PPoiseuille) is far above one, except for flows close to zero, where
PPoiseuille 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.
Given the flows usually used in clinical practice, and the flow values
for the critical Reynolds number for curved tubes (Table 1), the
laminar flow in a curved pipe predicted by the Ito formula (Eq. 6) applies to neonatal ETTs,
the diameters of which range from 2.5 to 3.5 mm. For pediatric ETTs
(4.0- to 7.0-mm ID), the Ito formula can be applied as long as the
Reynolds number is below the critical Reynolds number, i.e., as long as
the flow does not exceed the flow corresponding to this critical
number. We therefore noted the value of this flow for a given ETT
geometry (Table 1).
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.
In contrast to laminar flow, we found that, for turbulent flows, the
calculated effect of the curvature was negligible. In the range of
geometric curvatures and flows tested, the
Pvd value predicted by White
(27) for turbulent flows in curved tubes was indeed extremely close to
the value given by the Blasius formula (see
APPENDIX). It can therefore be
concluded that the effect of the ETT curvature on the pressure drop is
slight and can be disregarded. In adult ETTs the pressure drop along
the tube can be estimated by the Blasius formula. In this study, we
extended the results for turbulent flows to ETTs with an ID of
4.0-7.0 mm.
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 (Pent) 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 Pent 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
Pent 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.
The flow becomes turbulent for a Reynolds number exceeding the critical
number. Taking into account the gas properties of air and assuming a
curvature diameter of 14 cm, this number is reached when
This
means that, under the simplified conditions already defined above, the
flow for a standard ETT becomes turbulent when it exceeds the following
flows
Note that these flows are indicative. They correspond to the
manufacturer's values of ETT ID instead of the values given in Table
1, which correspond to the measurement of ETT ID by the acoustic
reflection method.
When a Reynolds number is above the critical Reynolds number, i.e., for
turbulent flows, the Blasius formula must be used. Taking into account
the gas properties of air, the pressure drop can be estimated
(cmH2O) with the following
dimensional form
To evaluate the pressure drop due to the ETT connector plus the ETT,
the pressure drop due to kinetic energy must be added to both the Ito
and Blasius formulas. This drop can be written in a simplified form,
assuming a standard ETT connector with a diameter of 1.15 cm at the top
Prediction of the ETT Pressure Drop, ETT Resistance, and WOB Due to
ETT (ETT WOB) in Clinical Practice
The formulas proposed in the present study (see
APPENDIX) allow calculation of the
pressure value required to overcome ETT resistance for a given flow and
calculation of the pressure at the distal end of the ETT, under any
conditions where ETT geometry (ID, curvature diameter, and length) and
gas properties are known. This means that calculation of the pressure
drop due to the ETT gives the minimal level of inspiratory pressure
required to overcome ETT resistance, as shown below.
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.
Several groups (22, 23, 30) have provided empirical equations such as
the Rohrer equation (P = K1 · · K2 · 2)
in which the coefficients
K1 and
K2 are both
computed from the fitting of the data for a single ETT. Such empirical
equations lack physical significance. They describe the pressure
drop-flow relationship for a particular curvature of one specific ETT
but do not account for variations in ETT ID, length, or curvature or
for changes in gas composition. The use of such dimensional pressure-flow relationships in the present study would have led to
seven distinct curves to describe the ETTs tested (Fig. 2), thus
introducing seven distinct pairs of Rohrer coefficients. It is true
that the report by Sly et al. (22) takes into account the
diameters and lengths of the ETTs, but the results cover a full page of
the study, and the large number of multiple coefficient pairs given
cannot easily be used in clinical practice. By contrast, the formulas
proposed in the present work are applicable to all neonatal and
pediatric ETTs. For example, we compared the results given by Sly et
al. to the values obtained with the Ito formula, to which we added the
pressure drop due to kinetic energy, and found that differences between
the pressure drops were <15%. Note that, because entry effects and
other nonlinear processes were disregarded, the formulas we propose do
not provide the exact pressure drop, but they do give an estimation
that is very close to the measured values.
In addition, it must be emphasized that the general formulas we propose
apply to gases other than air. For instance, using pure oxygen instead
of air slightly raises the resistance, whereas the use of a low-density
gas mixture like heliox is known to reduce it (26, 28). In view of
this, we use the Ito formula to calculate ETT resistance with air and
heliox for an ETT with a diameter of 2.5 mm, a length of 14.8 cm, and a
curvature diameter of 33 cm. For air, we predicted a resistance of 89 cmH2O · l1 · s
and for heliox a resistance of 73 cmH2O · l1 · s.
These values are very close to the 90 and 68 cmH2O · l1 · s,
respectively, measured by Wall (26) for the effect of gas density on
ETT resistance at a flow of 100 ml/s.
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
The importance of the ETT WOB has already been studied (3, 6). It was
clearly demonstrated that reducing the ETT diameter increased the WOB.
On the basis of a flow pattern recorded in an infant, we simulated the
inspiratory pressure drop and calculated the theoretical ETT WOB for
physiological flow patterns in neonatal and pediatric ETTs. Flow was
simulated, assuming a tidal volume for boys with mean French
weight-for-age values and a normal respiratory rate (7), with
intubation by using an ETT of standard size (1). Pressures were
estimated from the physiological flow by using the laws that we have
described in THEORY. WOB was computed by numerical integration. Figure 6 shows
the ETT WOB simulation for ETTs, assuming a tidal volume of 7 ml/kg,
whereas Fig. 7 shows the ETT WOB simulation
for a tidal volume of 12 ml/kg and a 25% increase in the respiratory
rate. In the first case (Fig. 6), the pressure drop is due to kinetic
energy and laminar flow. For all ETTs, the ETT WOB normalized for tidal
volume is relatively constant (between 0.09 and 0.17 mJ/ml), and the
increase in work is mainly due to the pressure drop due to kinetic
energy (from 17% of the total ETT WOB for the 2.5-mm ETT to
28% for the 7.0-mm ETT). In the second case (Fig. 7), the
turbulence occurs in the pediatric ETTs (4.0- to 7.0-mm ID). In these
ETTs, the WOB is mainly due to the turbulent flow (between 43 and 56%
of the total ETT WOB).
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.
|
|
To evaluate the physiological significance of ETT WOB, we also
calculated it from flow recordings obtained in two infants intubated
with an ETT of 2.5-mm ID and who were breathing spontaneously in the
continuous positive airway pressure mode. The data were taken from a
previous study in which we assessed total WOB by using the esophageal
catheter technique (14). In these two patients, the ETT WOB, calculated
from the Ito formula plus the pressure drop due to kinetic energy
value, was found to amount to 13 and 20% of total WOB, respectively
(Table 2). The differences observed between
these two infants were due to the differences in the flow spontaneously
generated. The greater the weight and flow rate, the higher the ETT
WOB. In view of these results, it would have been wise to use a larger
ETT for the infant with a body weight of 1.9 kg if the patient's
cricoid cartilage was large enough to accept it.
View this table:
[in this window]
[in a new window]
|
Table 2.
Work of breathing due to the ETT in 2 patients spontaneously breathing
in the continuous positive airway pressure mode
|
|
ETT WOB cannot be disregarded, particularly for small premature infants
ventilated under intermittent mandatory ventilation, and perhaps under
synchronized intermittent mandatory ventilation, at a slow frequency.
In these ventilatory modes, the portion due to spontaneous ventilatory
cycles may exceed the portion due to mechanical ventilatory cycles. In
these infants, we indeed demonstrated that the WOB in the continuous
positive airway pressure mode was not different from the WOB necessary
in the intermittent mandatory mode at slow frequency (14). In any case,
the additional ETT WOB, better defined as unjustified, could have been
compensated for by an adequate mode of ventilation with a low level of
inspiratory pressure. In adults, it has been shown that the pressure
support level compensating for the increase in ETT WOB varies between 3.4 and 12.4 cmH2O (3). This
additional WOB may be important, as it can lead to underestimation of
an infant's ability to withstand extubation, and it is therefore
important to be able to calculate it. This additional WOB may at least
be taken into account in the process of weaning.
In summary, the pressure drop due to the ETT may be evaluated by
general formulas that take account of flow, ETT geometry, and gas
properties. When the ETT pressure drop is known, ETT resistance can be
separated from total resistance, and the ETT WOB can easily be
calculated from the measurement of the inspiratory flow. All these
formulas can be easily used with a pocket calculator, thus allowing
fast calculation of these different parameters in clinical practice.
| |
APPENDIX |
ETT Pressure Drop
The pressure drop in the region between two distinct cross sections of
a tube during a steady incompressible flow can be estimated from the
Cotton-Fortier formula (also called the generalized Bernouilli formula)
(4). When the hydrostatic pressure gradient is disregarded, which is
usual when the fluid is a gas, this formula is used to calculate the
pressure drop between the two cross sections
S1 and
S2 and
becomes
|
(1)
|
where is the flow rate,
Dp is the rate of
energy dissipation in the region between the cross sections
S1 and
S2, is the fluid density, 
is the
average velocity of section i defined
by = /Ai, where
Ai designates the cross-sectional area,
i is the coefficient of the kinetic energy associated with section
i, and
pi is the pressure in section
i.
P, the pressure drop in the region between the two cross sections,
is divided into two components: the first one is
Pvd, representing viscous
dissipation, which is always negative, and the second is
Pke, representing the pressure
drop due to changes in kinetic energy.
The kinetic energy term, Pke,
is given by the second term of Eq. 1
above, in which
and
u is the fluid velocity. The
coefficient i depends on the
velocity profile and varies between one (flat profile) and two
(parabolic profile). In view of the entry length generally accepted in fluid mechanics, ~10-15 diameters for
laminar flow and the respective length of the ETT, for calculation of
kinetic energy we considered that the flow was flat in the ETT, as the entry length was larger than the ETT length, except for the 2.5-mm-ID ETT for which the ETT length was close to the entry length. We therefore considered ETT to be
equal to 1, except for the 2.5-mm-ID ETT, for which
ETT was considered to be equal
to 1.5. This analysis, which can also be interpreted as a larger entry
effect in the 2.5-mm-ID ETT connector and is confirmed by the fact that
the ETT pressure drop, corrected for kinetic energy measured for
various diameters of the curvature
(Dc), i.e.,
when PETT varies but not Pke, in 2.5- and 3.0-mm-ID ETTs
perfectly agrees with the pressure drop predicted by the Ito formula.
The viscous dissipation term,
Pvd, depends on the flow regime.
In a long straight tube in laminar flow (Poiseuille flow),
Pvd is given by the well-known
formula (4, 21)
|
(2)
|
with
|
(3)
|
as
the Reynolds number. D is tube
diameter, L is tube length, and 
is
the kinematic viscosity ( = µ/, where µ is the dynamic viscosity). This formula is valid as long as the Reynolds number is
<2,300.
For larger Reynolds numbers (>2,300) the flow becomes turbulent, and
Eq. 2 is no longer valid. For a
hydraulically smooth tube, Eq. 2
should be replaced by the Blasius formula (4, 21)
|
(4)
|
In a curved tube, the presence of centrifugal forces makes
Pvd larger than in a straight
tube. The characteristic dimensionless variable that determines the
effect of the curvature is the Dean number (De)
|
(5)
|
When flow is laminar and
Dc is the
diameter of the curvature, the Ito formula (12) gives
Pvd
as
|
(6)
|
In its dimensional form, the Ito formula can be
written
|
(7)
|
and is a polynomial function of
1.5,
1,
0.5,
0, and
0.5.
The critical Reynolds number, i.e., the number for which the flow
becomes turbulent, is larger in a curved tube than in a straight tube.
It depends on the ratio
D/Dc.
Ito proposed the following formula
|
(8)
|
where
Rec is the critical Reynolds number.
For turbulent flow, Ito showed that the dimensionless variable is
|
(9)
|
For
this flow, different empirical formulas were proposed to compute
Pvd. White proposed (27)
|
(10)
|
For low
Re · (D/Dc)2
values, the pressure drop given by the White formula is approximated by
the Blasius formula (Eq. 4), which
is simpler to use. In the clinical range,
Re · (D/Dc)2
does not exceed 10, and discrepancy between the two formulas does not
exceed 12% at the maximal flow rate.
Oscillatory Flow
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 discussed (10, 20).
Steady or unsteady conditions depend on the balance between
1) the viscous force added to the
convective acceleration, which alone causes the fluid to respond
instantaneously; and 2) the inertial
forces, which cause the fluid to lag. For fully developed laminar
oscillating flow in a straight tube (29), the balance between viscous
and initial forces is given by the Womersley parameter = D/
,
where D is the diameter, 
is the
pulsation, and is the kinetic viscosity. It is only when is
much greater than one that the unsteadiness of flow cannot be
neglected. It has been shown (10, 29) that an increase in resistance
over the entire flow occurs only for 
4. Such a criterion applied
to the ETT introduces a higher bound of frequency (fmax)
to ensure the quasi-steady condition. In air, this higher bound is
roughly
The presence of curvature in the tube or singular losses (for example
entry phenomena), as in our study, increase convective accelerations
and viscous loss in contrast to inertial forces. They therefore tend to
increase the higher bound of frequency. In summary, for the purposes of
our study, flow in the ETT can be considered to be
quasi-steady.
| |
ACKNOWLEDGEMENTS |
This work was performed while P. W. Blanchard was on sabbatical
from the Université de Sherbrooke, Québec, Canada.
| |
FOOTNOTES |
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: P.-H. Jarreau,
Service de Médecine Néonatale, Hôpital
Cochin-Port-Royal, 123, Bd de Port-Royal, 75679 Paris Cedex 14, France
(E-mail: pierre-henri.jarreau{at}cch.ap-hop-paris.fr).
Received 23 September 1998; accepted in final form 18 February
1999.
| |
REFERENCES |
1.
American Heart Association Emergency Cardiac Care Committee and Subcommittees.
Guidelines for cardiopulmonary resuscitation and emergency cardiac care.
JAMA
268:
2262-2275,
1992[Abstract/Free Full Text].
2.
Bland, J.,
and
D. Altman.
Statistical method for assessing agreement between two methods of clinical measurement.
Lancet
1:
307-310,
1986[Medline].
3.
Brochard, L.,
F. Rua,
H. Lorino,
F. Lemaire,
and
A. Harf.
Inspiratory pressure support compensates for the additional work of breathing caused by the endotracheal tube.
Anesthesiology
75:
739-745,
1991[Medline].
4.
Comolet, R.
Mécanique Expérimentale des Fluides, Dynamique des Fluides Réels, Turbomachine. Paris: Masson, 1982.
5.
Danan, C.,
G. Dassieu,
J.-C. Janaud,
and
L. Brochard.
Efficacy of dead-space washout in mechanically ventilated premature newborns.
Am. J. Respir. Crit. Care Med.
153:
1571-1576,
1996[Abstract].
6.
Farstad, T.,
and
D. Bratlid.
Effects of endotracheal tube size and ventilator settings on the mechanics of a test system during intermittent flow ventilation.
Pediatr. Pulmonol.
11:
15-21,
1991[Medline].
7.
Gregory, G. A.
Pediatric Anesthesia. New York: Churchill Livingstone, 1983, p. 391-398.
8.
Guttmann, J.,
L. Eberhard,
B. Fabry,
W. Bertschmann,
and
G. Wolff.
Continuous calculation of intratracheal pressure in tracheally intubated patients.
Anesthesiology
79:
503-513,
1993[Medline].
9.
Idel'cik, I. E.
Mémento des Pertes de Charge. Coefficient de Pertes de Charge Singulières et de Pertes de Charge par Frottement. Paris: Eyrolles, 1986.
10.
Isabey, D.,
and
H. K. Chang.
Steady and unsteady pressure-flow relationships in central airways.
J. Appl. Physiol.
51:
1338-1348,
1981[Abstract/Free Full Text].
11.
Ito, H.
Friction factor for turbulent flow in curved pipes.
J. Basic. Eng.
81:
123-134,
1959.
12.
Ito, H.
Laminar flow in curved pipes.
Z. Angev. Math. Mech.
49:
653-663,
1969.
13.
Jarreau, P.-H.,
B. Louis,
L. Desfrere,
D. Isabey,
A. Harf,
and
G. Moriette.
Acoustic method to monitor the longitudinal area profile of neonatal endotracheal tubes (Abstract).
Pediatr. Res.
41:
35A,
1997.
14.
Jarreau, P.-H.,
G. Moriette,
P. Mussat,
C. Mariette,
A. Mohanna,
A. Harf,
and
H. Lorino.
Patient-triggered ventilation decreases the work of breathing in neonates.
Am. J. Respir. Crit. Care Med.
153:
1176-1181,
1996[Abstract].
15.
LeSouef, P. N.,
S. J. England,
and
A. C. Bryan.
Passive respiratory mechanics in newborns and children.
Am. Rev. Respir. Dis.
129:
552-556,
1984[Medline].
16.
LeSouef, P. N.,
S. J. England,
and
A. C. Bryan.
Total resistance of the respiratory system in preterm infants with and without endotracheal tube.
J. Pediatr.
104:
108-111,
1984[Medline].
17.
Lofaso, F.,
B. Louis,
L. Brochard,
A. Harf,
and
D. Isabey.
Use of the Blasius resistance formula to estimate the effective diameter of endotracheal tubes.
Am. Rev. Respir. Dis.
146:
974-979,
1992[Medline].
18.
Loring, S. H.,
E. A. Elliott,
and
J. M. Drazen.
Kinetic energy loss and convective acceleration in respiratory resistance measurements.
Lung
156:
33-42,
1979[Medline].
19.
Louis, B.,
G. M. Glass,
and
J. J. Fredberg.
Pulmonary airway area by the two-microphone acoustic reflection method.
J. Appl. Physiol.
76:
2234-2240,
1994[Abstract/Free Full Text].
20.
Pedley, T. J.,
and
J. M. Drazen.
Aerodynamic theory.
In: Handbook of Physiology. The Respiratory System. Mechanics of Breathing. Bethesda, MD: Am. Physiol. Soc., 1986, sect. 3, vol. III, pt. 1, chapt. 4, p. 41-55.
21.
Schlichting, H.
Boundary-Layer Theory. New York: McGraw-Hill, 1979. (McGraw-Hill Ser. Mech. Eng.)
22.
Sly, P. D.,
K. A. Brown,
J. H. T. Bates,
S. Spier,
and
J. Milic-Emili.
Noninvasive determination of respiratory mechanics during mechanical ventilation of neonates: a review of current and future techniques.
Pediatr. Pulmonol.
4:
39-47,
1988[Medline].
23.
Sullivan, M.,
J. Paliotta,
and
M. Saklad.
Endotracheal tube as a factor in measurement of respiratory mechanics.
J. Appl. Physiol.
41:
590-592,
1976[Abstract/Free Full Text].
24.
Tobin, M. J.
Principles and Practice of Mechanical Ventilation. New York: McGraw-Hill, 1994, p. 967-1003.
25.
Van Surell, C.,
B. Louis,
F. Lofaso,
L. Beydon,
L. Brochard,
A. Harf,
J. J. Fredberg,
and
D. Isabey.
Acoustic method to estimate the longitudinal area profile of endotracheal tubes.
Am. J. Respir. Crit. Care Med.
149:
28-33,
1994[Abstract].
26.
Wall, M. A.
Infant endotracheal tube resistance: effects of changing length, diameter, and gas density.
Crit. Care Med.
8:
38-40,
1980[Medline].
27.
White, C. M.
Fluid friction and its relation to heat transfer.
Trans. Inst. Chem. Eng.
10:
66-85,
1932.
28.
Wolfson, M. R.,
V. K. Bhutani,
T. H. Shaffer,
and
F. Bowen, Jr.
Mechanics and energetics of breathing helium in infants with bronchopulmonary dysplasia.
J. Pediatr.
104:
752-757,
1984[Medline].
29.
Wormesley, J. R.
Method for the calculation of velocity rate of flow and viscous drag in arteries when the pressure gradient is new.
J. Physiol (Lond.)
127:
332-363,
1955.
30.
Wright, P. E.,
J. J. Marini,
and
G. R. Bernard.
In vitro versus in vivo comparison of endotracheal tube airflow resistance.
Am. Rev. Respir. Dis.
140:
10-16,
1989[Medline].
J APPL PHYSIOL 87(1):36-46
8570-7587/99 $5.00
Copyright © 1999 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
![+ 7.782 ⋅ De<SUP>−1</SUP> + 9.097 ⋅ De<SUP>−1.5</SUP> + 5.608 ⋅ De<SUP>−2</SUP> + ⋯)]](/content/vol87/issue1/fulltext/36/img020.gif)
