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Longitudinal dispersion in model of central airways during high-frequency ventilation
Respiration Physiology, 84 (1991) 13-29
13
Elsevier RESP 01764
Longitudinal dispersion in model of central airways during high-frequency ventilation. A.M. van der Kooij ~ and S . C . M . Luijendijk 2 'Department of Medical Physiology, State University of Utrecht. Utrecht, The Netherlands and "Department of Puimonology. University Hospital Maastricht, State University of Limburg. Maastricht. The Netherlands (Accepted 27 December 1990) Abstract. We have measured the longitudinal dispersion of boluses of helium, acetylene and sulphur hexafluoride in a plastic model of the human airways -generations zero through six -during high frequency ventilation (HFV). HFV was maintained by a piston pump. Frequency f and tidal volume VT ranged from 2.5 to 25 Hz and from 5 to 20 ml, respectively. Boluses were injected near the entrance of the zeroth generation (trachea), and the dispersion curves were measured by mass spectrometry at the end of the sixth airway generation. The shapes of the bolus dispersion curves could be well described with Gaussian distribution functions. With the exception of the HFV-conditions with VT = 5 ml, the effective dispersion coefficient DDtsr appeared to be independent of the molecular diffusion coefficient. This independency was also found by other investigators in studies with dogs and human subjects. The measured results for DDISPfor different f and VT could be satisfactorily described with the empirical equation DDISP= 0.0617 f°'sV~38 [cm2s- t]. Application of this equation to f and VT values normally applied in man resulted in DDISP values which should be considered to be too small for maintaining eucapnic ventilation in rive. On the basis of this result we believe that during HFV in intubatod subjects gas transport by longitudinal dispersion will be limited to the instrumental dead space - the endotracheal tube inclusive - and a few generations of large bronchi.
Dispersion ofgases in airways; Gas mixing in long airways; High frequency ventilation, and intrapulmonary gas mixing; Inert gases, and mixing in long airways
In the recent past it has been well established that normal blood gases can be maintained during artificial ventilation by applying tidal volumes that are considerably less than the anatomical dead space at increased breathing rates (Bohn et al., 1980; Crawford and Rehder, 1985; Rieke et al., 1983; Rossing et al., 1981). Different mechanisms have been proposed to account for the 02 and CO2 transport in the conductive airways under these ventilatory conditions such as turbulent dispersion, convective dispersion due to asymmetric velocity profiles, Pendelluft, and molecular diffusion (Chang, 1984). These me-
Correspondence to: S.C.M. Luijendijk, Dept. of Pulmonology, University Hospital Maastricht, P.O. Box 5800, 6202 AZ Maastricht, The Netherlands. 0034-5687/91/$03.50 © 1991 Elsevier Science Publishers B.V.
14
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
chanisms may operate in series, i.e. each of them may apply to a portion of the airways (Chang, 1984). In this paper, we will focus on the gas transport in the central airways, and data will be presented which was obtained from measurements performed in a plastic airway model representing the airway generations 0 up to 6 inclusive. It is common practice to present the gas exchange efficacy due to high-frequency ventilation (HFV) by means of an effective diffusion coefficient (Chang, 1984). This assumes that the longitudinal dispersion of gases in the airways during HFV behaves in a way as if it were the result of (augmented) diffusion. The main objective of this study is to verify this assumption, and to the best of our knowledge this has never been performed before. In the central airways, longitudinal mixing of gases during HFV is assumed to be due to turbulent and/or convective dispersion. In the latter case, radial mixing in the airways by molecular diffusion might interfere with longitudinal mixing (Taylor, 1953). We have employed, therefore, tracer gases with different molecular diffusion coefficients, which might reveal the prevailing mechanism of gas mixing in this part of the airways. Further, we will analyse the measured gas exchange efficacy in terms of its dependence on tidal volume (VT) and frequency (f), and the data obtained will be compared with data obtained from measurements by other investigators.
Materials and methods The experiments were performed with a symmetrically branching model of the central airways representing the generations 0 through 6. The different branches of the model were constructed of cylindrical tubes made of clear plastic, and their dimensions corresponded to those of Weibel's model A (table 1)(Weibel, 1963). At each branch point, the angle between the two daughter branches was equal to 700 (Horsfield and Cumming, 1967). After construction, the volume of the airway model was 52 ml. The branches of the sixth generation were extended with flexible tubes (fig. 1). Length and inner diameter of these tubes were 19 cm and 5 ram, respectively. The purpose of these TABLE I Dimensions of airway model. Generation number
Number of tubes per generation
Length of tubes mm
Diameter of tubes mm
0 ! 2 3 4 5 6
! 2 4 8 16 32 64
120.0 47.6 19.0 7,6 12.7 10,7 9.0
18.0 12.2 8.3 5.6 4.5 3.5 2.8
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
15
extension tubes is discussed elsewhere in this section. Further, the zeroth generation was extended with a brass tube (fig. 1). Volume and inner diameter ofthis tube corresponded to the volume of the airway model and the inner diameter of the zeroth generation. An oscillatory flow of room air in the model was maintained by means of a piston which was moved up and down in the brass tube. The piston shaft was connected to a fly wheel which was driven by a DC-motor (fig. 1). A long shaft was used to approximate a sinusoidai displacement of the piston. Different values for VT were adjusted by changing the radial position of the end of the piston shaft on the flywheel. Different values for f were obtained by varying the supply voltage of the DC-motor. The airway model was placed upright with the zeroth generation underneath (fig. 1). The applied tracer gases were injected into the oscillatory flow near the inlet of the airway model. The main parts that formed the injection system were a hollow needle and a trigger mechanism (fig. 1). The hollow needle was placed in two holes in the opposite walls of the brass tube near the connection with the zeroth generation (figs.
(PARTLY DRAWN)
MASS SPECTROMETER/ ADJUSTMENT~~
TRIGGER MECHANISM RELAY
RESERVOIR WITH TRACER GAS PISTON----t!
II
Ii
', OPTICAL " o O.
DEVICE
ADJUSTMENTFOR TIDAL VOLUME
I
FLY- WHEEL Fig. 1. Schematic drawing of experimental set-up.
16
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
1 and 2). This construction made it possible to move the needle perpendicular to the oscillatory flow by means of elastics which were attached to a handle which in turn was attached to one end of the needle (fig. 1). In the stand-by position, this handle was held by the trigger mechanism, and after release the needle was shot by the force of the elastics. A constant flow of tracer gas was supplied to the injection needle and escaped through a small hole in the wall of the needle (fig. 2). Before and after bolus injection, the tracer gas entered the exhaust tubes B and A (fig. 2), respectively. These tubes were continuously flushed with air with the help of a vacuum pump (fig. 2). During bolus injection the outlet of the needle moved from tube B to tube A. Thereby it passed the inside of the brass tube, which took less than 1 msec. In all experimental conditions the peak flow in the airway model was less than 1 L . s e c - ~, therefore, the injected bolus of tracer gas mixed with less than 1 ml of air during the period of injection. The bolus dispersion curves were measured at the other end of the model. As a consequence, the contribution of the initial volume of the bolus (< 1 ml) to the shape of the measured dispersion curves may be neglected. The frequency f was determined from the output signal of an optical device that was used to detect the mid-position of the piston (fig. 1). Frequency was measured by a counter (Philips PM 6620). The optical device was further used to determine the moment of injection: After pressing the start button, the optical device initialized the trigger mechanism to release the injection needle at the next mid-position of the piston. After injection, the tracer gas dispersed both into the airway model and into the brass tube. The dispersion curves were measured by use of a mass spectrometer (Balzers QMG 511). To that end the tip ofthe sample capillary ofthe mass spectrometer was placed in a well-fitting hole in one out of the 64 flexible tubes close to the corresponding branch of the sixth generation of the airway model (fig. 1). The output signal of the mass spectrometer was digitized on-line by means of a microcomputer (Zilog MCZI/05) and stored on floppy disk for off-line data analysis. The applied sample rate ranged from 200 up to 1000 samples per second dependent on the rate of change of the concentration of the tracer gas in the steep part of the dispersion curve.
In view of the oscillatory nature of the flow, the flexible extension tubes made it
BULKFLOW ~
~
INJECTIONNEEDLE
T~ACEROeSSUPPLY
I11 TOVACUUMPUMP
Fig. 2. Schematic drawing of bolus injection system. The dashed ring represents the cross-section of the brass tube which is connected to the zeroth generation of the airway model. The outlet ofthe injection needle is shown in the position it takes prior to bolus injection. The tubes B and A serve to remove the outflowing tracer gas before and after bolus injection, respectively.
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
17
possible to measure the overall dispersion of the whole airway model. From a certain time after injection onwards, however, tracer gas will be lost from the model as soon as it reaches the outlets of these extension tubes. For this reason, we restricted the data analysis to the first part of the measured dispersion curve that is in between the beginning and the maximum of the curve. Le., the total volume of the extension tubes ( = 239 ml) was chosen sufficiently large so that the aforementioned loss of tracer gas could not possibly affect the shape of the first part of the measured dispersion curves. All data files were corrected for the transit-time ofthe gas through the sample capillary of the mass spectrometer which amounted to 90 ms. In all our experiments the difference of time between the maximum and the onset of the dispersion curve exceeded 2 sec. Therefore, no corrections were made for the time constant of the mass spectrometer which was less than 20 ms for the tracer gases applied in this study. The experiments were carried out for three tracer gases (He, C2H2 and SF6), four tidal volumes (5, 10, 15 and 20 ml), and six frequencies (2.5, 5, 10, 15, 20, and 25 Hz). These tracer gases cover a wide range of molecular diffusion coefficients (D): DsF6 = 0.101 cm 2. sec- ~and DH,~ = 0.725 cm 2" sec- ~in air at one atmosphere and at 37 °C (Worth and Piiper, 1978). D r : . : is not known. The molecular weight of C,H2, however, is close to that of CO. Therefore, Dr:a, will be close to Dco which is equal to 0.233 cm2" sec- ~ in air at one atmosphere and at 37 °C (Worth and Piiper, 1978). For statistical reasons, each experiment was performed 5 times.
Data analysis As mentioned in the Introduction, the main objective of this study was to investigate whether gas transport in the central airways due to HFV can be described as if it were the result of a diffusion process. The data analysis, therefore, started from the following equation: C (t) = M (2 g~(t)) -°'5 exp [ -(Vo - V(t))2/2 ~(t)]
(l)
where C(t) represents the concentration of the tracer gas at the inlet of the sample capillary t seconds after the injection of the tracer gas, M is the total amount of tracer gas, Vo is the volume of the airway model, and ~(t) is the volume variance at time t of the underlying Gaussian distribution function. The measured dispersion curves showed periodic variations. These variations were caused by the oscillatory flow which moved the concentration profile of the tracer gas in the airway model up and down the model. For the same reason the volume between the centre of this concentration profile and the site of gas sampling changed continuously. This is accounted for by the term V(t) in eq. (1), and V(t) should be directly related to the volume displacement of the piston. Therefore, V(t) is given by: V(t) = 0.SVTsin 2 rift
(2)
18
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
In practice, the amplitude of the periodic variations in C(t) was attenuated because the time constant of the mass spectrometer which ranged from 10 to 20 msec for the different tracer gases was too large with regard to the applied values for f in particular at the higher frequencies (f> 10 Hz). For this reason, we have modified eq. (1) by substituting for (Vo - V(t))2 the mean value which is equal to (Vo2 + 0.125VT2). The volume variance ~(t)is related to the dispersion coefficient Dmsp(t) according to: ~ ( t ) - - 2Ao2 ff~ Dmsv(t) dt
(3)
where Ao is the cross-section of the zeroth generation (Subramanian and Gill, 1976). In a previous study we found that at a constant flow in the airway model DDtsp was a function of the flow (Kooij and Luijendijk, 1985). This means that in the present experiments DD~sp(t) was a periodic function of time due to the periodicity of the flow with a time-period T equal to 1/f seconds. I.e., for each time period T the integral in eq. (3) will show the same value, and the long-term mean value of Dmsp(t) can then be obtained from:
DDISP
-"
T-
I
.fT Dmsp(t) dt
(4)
In 80% of the measurements the tracer gases reached the outlet of the airway model after 8T seconds (range 3-88T). Hence, the data analysis was in general related to a time-interval for which t > 8T. For such t-values ~(t) can then be well approximated by:
(s) because ~(t) increases monotonously with increasing t as Dmsp(t)>0 for any t. This approximation of ~(t) will be further discussed in the Appendix by taking a particular example for DDtsp(t). Substitutions of the mean value of IVo - V(t)] 2 and the result for ~(t) into eq. (1) yields: C(t) = M (4gAo2Dmspt) -°,5 exp [ -(V~ + 0.125VT2)/(4AoaDmspt)]
(6)
This expression for C(t) was fitted with the measured dispersion curves by adjusting the parameters M and Dmsp with the help of a non-linear least squares algorithm (Brown and Dennis, 1972). This curve-fitting was applied to each individual dispersion curve, for examples see fig. 3. M depends on the amplitude but not on the shape of the dispersion curve. As a consequence, the results for M are of no concern for this study and will, therefore, not be presented. For statistical analysis we used Student's t-test for paired observations.
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
19
Results Figure 3 shows six measured dispersion curves, two for each tracer gas. In each panel, the smooth curve represents the best least squares fit through the data points for the relationship between C and t given by eq. (6). The noisy structure of the measured dispersion curves is due, at least in part, to the variations in C caused by the rapid, periodic movement of the concentration profile in the airway model. Leaving this aside, there are still other deviations between the measured and fitted dispersion curves which are related to the average behaviour of measured C as a function of t (fig. 3). As an indication of the magnitude of these deviations we have determined for each measurement the maximal deviation between measured and fitted dispersion curves expressed as a percentage of the maximal value of C included in the analysis. These percentages ranged from 1 to 20% with an average value of 5 % and a standard deviation of 4%. Thus, in general, the deviations between the measured and fitted dispersion curves were small. As mentioned in Methods, each experiment was performed five times, and inspection of the graphic representations, as in fig. 3, of each set of five measurements showed that the deviations between measured and fitted dispersion curves were not i
:
~1'0-1 t :15OHz VT: 0.020t
i
/
,
0.5
, f
He 06
3b
1,0- ! :25Hz VT:00051
'
"5~P/P ~
C
,,f"
/~
/"~
0~
i,,,"f
--
~o
,c2.~
C2H2
0~" '
'
;
10 i
~,/ 0
'
o
10- f :25Hz 00051 VT:
0
§
10 f :150Hz
o
O~
He
60
0
sF6 '
:tlsl
-tls)
Fig. 3. Representative plots of individual bolus dispersion curves for different f, VT and tracer gases. The smooth curve in each panel represents the result of the curve-fitting with eq. (6).
20
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
systematic, on the contrary, they behaved differently and randomly for the different measurements. On the basis of all these results we may conclude that the basic shape of the measured dispersion curves can be satisfactorily described with eq. (6). In turn, this means that the measured dispersion curves can be characterized by the value of DD~Sp obtained from the curve fitting. For. each set of five identical measurements we have computed the coefficient of variation (CV = SD/mean) of the five values for DD~Sp obtained. The mean value of CV for all three gases and all ventilator settings together was 6.6% (_+ 3.9%). The mean values for Dr)~sp obtained from the different sets of identical measurements were fitted with the empirical equation: DDISP "- Kf~VT~
(7)
The values of K, a and fl were 617" 10 -4, 0.8 and 1.38, respectively. To evaluate the impact of molecular diffusion on DD~SP we have plotted the measured values of Dn~sP v e r s u s the predicted values of Dn~sp in figs. 4 through 7, where predicted DD~Sp is given by: DDISP (predicted) = 617.10- 4f°'8VTl'3s
(8)
No systematic deviations are seen in fig. 4 between the values of DDISPfor the different tracer gases for DDtsP > 10. This range of DDtsp-values includes nearly all data obtained with VT ~ 10 ml, and, as a consequence, the range of DDlsp-Values between 1 and 10 is mainly related to data obtained with VT - 5 ml (figs. 5-7). For VT -- 5 ml, the values of DDtsp for He are substantially larger than the correspon:.uag values for C2H2 and SF6 (fig. 4). Evidently, only for this small tidal volume are the results for DDISP affected by molecular diffusion. All data points for VT ~ 10 ml are close to the line ofidentity (figs. 5-7). This means that for VT > l0 ml the dependence of DD~sp on f and VT may be well described by eq. (8). The data shown in figs. 4-7 were obtained from measurements at a single outlet. A few additional measurements were performed at other outlets. The results of these measurements did not differ substantially from those obtained at the outlet selected for the bulk of the measurements.
Discussion
To answer the question whether longitudinal dispersion by HFV behaves as if it were the result of diffusion we have fitted our measured dispersion curves with eq. (6). This equation has been derived from the Gaussian distribution function given by eq. (l). It is well known that this function underlies the description of all kinds of random processes such as diffusion. A perfect fit between the measured and fitted dispersion
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
100
j
-~
x
!
,¢"o
! l,-~ /
,
4w
0.4
Fig. 4. Measured
•
/, 2
,
1
I
x :He + :C2H2
/"
I/: -
/~-
, /o°+
| 1-1
0•
21
~ w 4
I
[
I , , i m , , w I 10 20 402 1 100 ]~OlsP(predicted}[cm s- ]
predicted values of the mean dispersion coefficient for the three tracer gases. Full line represents the line of identity. Predicted values of DD=sP were computed with eq. (8). versus
~00 "
/
+/
He
40x" 20-
/,
°
U
±/~.
10-
+,/"
4-
° x7 "+
-
°
-/o
/~.~
,_/ ,~
ae" 2-
/
/
"
VT (ml}
-:20
/
Oz.: 0,4
I
I
+ 15 ,o
:
I
I
I
1
2
4
I
I
I
I
I
I
I
10 20 40 100 Doxsp(predicted)[cm 2 s-1 ]
Fig. 5. Measured v e r s u s predicted values of the mean dispersion coefficient for helium. This diagram shows the results for different VT. For each value of VT, the data points from left to right correspond to increasing f. For further explanation see fig. 4.
22
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK 100
C2H2
4o
4-
x
x . t.
20-
~
10_ +
~,
-
,,
18 2-
/o o
D
V T (roll ~ 20 + 15
o
A o
1-
0.4
I
14
Fig. 6. Measured
1
2
4
I
I
I
I
10 5
I
I
I
10 20 40 100 DozsP ( predicted ) [ cm 2 s-1 ]
predicted values of the mean dispersion coefficient for acetylene. For further explanation see fig. 5.
versus
~oo. SF6 t,020IO-
Q
I,~ 2/
VTlml )
o
a
10
o
5
1o
0.4 0,4
Fig. 7. Measured
versus
I I;
1
i 2
; 4
I
i;
i I I I 10 20 40 100 "DozsP(predicted)[cm2 s-l]
predicted values of the dispersion coefficient for sulphur hexafluoride. For further explanation see fig. 5.
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
23
curves would, therefore, mean that in our experiments the result of longitudinal gas mixing by HFV is equivalent to that by (augmented) diffusion. As a function of time the bolus dispersion curves show a steep, initial part up to the maximum of C(t), Cmax, and thereafter C(t) decays very gradually. In our data analysis we have used only this steep part of the dispersion curves as, in particular, the shape ofthis part is suitable to study deviations from eq. (6). In addition, the shape ofthe final part of the bolus dispersion curves is undoubtedly affected by loss of tracer gas from the airway model at the open ends of the flexible extension tubes of the sixth generation airways and by back-dispersion from the cylinder of the piston pump (fig. 1). For all measurements the data analysis included concentrations between 0 and 80 to 100 percent of Cmax. Thus, most of the steep, initial part of the bolus dispersion curves was included in the data analysis. In quite a few cases a nearly perfect agreement was obtained between the measured dispersion curve and the best fitting curve through the data points for eq. (6). Two representative examples are shown in fig. 3 one for C2H2 and one for SF6 both related to f = 15 Hz and VT = 20 ml. A more detailed analysis showed that- leaving aside the rapid variations in C - the maximal difference between measured and fitted data was 5~/o of Cma x (mean value, see Results). Considerably larger differences (> 12%)were found only rarely. As these percentages refer to the maximal differences obtained for the different measurements, it will be clear, that in general the mean difference between measured and fitted data for individual measurements was still considerably less than 5 % C,l,,x. As already mentioned in Results, no systematic deviations were seen between the measured and fitted dispersion curves when the results of a single set of five identical measurements were compared. Summarizing it may be said that the measured dispersion curves could be satisfactorily described by eq. (6). That is, the net effect of HFV on longitudinal gas mixing in the central airways is approximately identical to that of augmented diffusion. We have studied the dependence of DD~sp on D by using tracer gases with different molecular weights. We preferred this to using a single tracer gas in combination with carrier gases with different densities, e.g. SF6 versus He (Palosk! et ai., 1987), because changes in carrier gas might also involve density-related changes in the velocity profiles of the carrier gas in the airways. For VT - 5 ml, (measured) DD~sP for He is significantly larger than DD~sp for C2H2 or SF6 (P < 0.05; fig. 4). Evidently, under these conditions molecular diffusion in the axial direction of the airways contributes substantially to the longitudinal mixing of the bolus in the airway model and this should especially apply to He. This is a plausible explanation as the D-value for He ( - 0.725 cm 2. sec - i) is comparable in magnitude to the smallest DD~sp-values for He ( ~ 2.5 cm 2. sec - I ) shown in fig. 4. In a previous experiment with the same airway model we observed that at a constant flow in the model the longitudinal dispersion for He was smaller than that for SF6 (Kooij and Luijendijk, 1985). This inverse relationship between DD~sp and D was ascribed to the impact of radial diffusion on the mixing process as radial diffusion interferes with longitudinal dispersion (Taylor laminar dispersion). Such an inverse relationship, however, is not
24
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
shown by the results in fig. 4. That is, in the present experiments Taylor laminar dispersion was of no importance. For DD~SP > 10 no significant differences (P > 0.10) were found between the data of the different tracer gases. Evidently, for DDIsv > 10 any contribution of diffusive mixing to DDISP is completely overruled by that of convective mixing. The above described behaviour of DDIsP for the different tracer gases is consistent with the experimental data published by Scherer and Haselton (1982). These authors studied the movements of buoyant tracer-beads in an oscillating flow in Plexiglas models ofthe bronchial airways. The flow consisted of a glycerine-water solution whose composition was adjusted to different values of the kinematic viscosity ranging from 0.00. up to 7 cm 2. s-~. The measurements were performed by photographing the movements of the beads. Next, the photographs were used to determine the distance L that the tip of the particle tongue was convected forward into the model per cycle averaged over the first five flow cycles after injection of the beads. A curvi-linear relationship was found between the dimensionless quantity L(A/VT) and Reynolds number Re, where A is cross-sectional area of tube. In our airway model, Re ~ 117 in the zeroth generation for f = 2.5 Hz and VT = 5 ml. For this value of Re, Scherer and Haselton (1982) found L(A/VT) to be equal to about 0.17. For A = (Ao) = 2.54 cm 2 and VT = 5 ml this results in L = 0.33 cm per cycle. The average displacement of particles by diffusion LD in one direction in time l/f is given by (Scherer and Haselton, 1982): LD = ~/~ D/f
(9)
For f = 2.5 Hz, this results in LD = 0.76 cm and LD = 0.28 cm for He and SF6, respectively. Thus, the above discussed experimental findings predict that for the HFV conditions that correspond to the lower range of our measured DDtsp-values the contributions of diffusion and convection to longitudinal dispersion are comparable in magnitude. Accordingly, for these HFV conditions DDtsp for He should be substantially larger than Dotsp for SF6. It can be further derived from the data given by Scherer and Haselton (1982) that for f -- 25 Hz and VT = 20 ml L/Lo will amount to about 5.5 and 15 for He and SF6, respectively. Hence, in the upper range of our measured Do~sp" values DDtsP should be virtually independent of D. Thus, we may conclude that our findings concerning the dependence of DD~sP on D are consistent with the results obtained by Scherer and Haselton (1982) in a fairly different experiment. The findings for DotsP < 10 were obtained with the smallest tidal volume ( - 5 ml) used in our measurements. Tidal volumes normally applied in HFV are larger than 20 mi. The rest of the discussion will, therefore, be exclusively related to our findings for DD~Sp > 10. Knopp etal. (1983) showed in intubated dogs that there were no consistent differences in the time courses of washout between He and SF6 or between He and Ar using oscillation frequencies between 10 and 30 Hz and tidal volumes ranging from about 40 to 100 ml. It is clear that these findings are in full agreement with our findings namely that DDtsP is independent of D for D < 0.725 cm2. sec- ~. The dependence of the dispersion coefficient on frequency and tidal volume has been described by several authors (Berdine et aL, 1983; Jaeger et aL, 1983, 1984; Kamm
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
25
et a/., 1984; Mitzner eta/., 1984; Paloski eta/., 1987; Reisfeld and Ultman, 1988; Slutsky et ai., 1980, 1981). The empirical eq. (7) was adopted from their publications, and we have shown that also our data for DDISP Can be satisfactorily described with this equation for DrnsP > 10 cm 2- sec - ~when the proper values for the parameters K, and,8 are used (eq. (8), fig. 4). This finding now facilitates the comparison ofour results with those reported in the literature. Most publications show that 0.7 _< 0c_< 1.0 and 1.0 _<,8 _< 2.0 (table 2). Our results for 0cand//are thus compatible with these general findings. Further, most investigators and we too find that a
DD, sP = K*f~ (VT/A)~
(10)
Evid¢~aly, VT/A and not VT is a major determinant of the longitudinal mixing process in HFV. This is understandable as VT/A represents the mean, axial displacement of the
gas molecules during one oscillation. Thus, we have used K* instead of K to compare our results to those reported by other investigators. The results for K* (table 2) were thus calculated with the help of eq. (10). In most papers on intubated dogs, however, some data are missing so that K* cannot be computed. Our result for K* (-0.223) is in the middle of the wide range (0.03-3.06) of K* values shown in table 2. It may be concluded from table 2 that data for K*, a and p obtained from the different investigations show considerable, mutual differences, and this especially applies to the data for K*. These differences may, at least in part, be due to the diversity of applied models - dog lungs, straight tubes and networks of branching tubes. In addition, in a number of the studies cited in table 2 the values for a and/~ were assumed rather than calculated (Jaeger et al., 1983, 1984; Mitzner et al., 1984; Slutsky et al., 1980), and this also affects the results for K*. The consequence of the things mentior, e~J is that the most reliable results for K*, ~ and fl reported so far refer to measurements in networks of branching tubes. In this category we have used the most realistic model of the central airways as the other investigators (Kamm et al., 1984; Paloski et al., 1987) used models with much less branchings and constructed of tubes of equal length and diameter for all airway generations. One should realize that eq. (1) refers to models that describe dispersion of gases by
26
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK TABLE 2 Survey of the 0c and fl values reported in the literature.
K*
a
,//
M aterial
References
3.06 0.030 0.076 0.275 0.340 0.223
0.17 1.00 0.8 1.0 1.08 1.0 1.0 0.71 0.94 0.8
1.74 1.00 1.5 2.0 1.49 2.0 2.0 1.93 1.66 1.38
intubated dogs intubated dogs intubated dogs intubated dogs intubated dogs straight tubes straight tubes network of branching tubes network of branching tubes network of branching tubes
Reisfeld and Ultman (1988) Slutsky et al. (1980) Berdine et al. (1983) Jaeger et al. (1983) Slutsky et al. (1981) Jaeger et al. (1984) Mitzner et al. (1984) Paloski et al. (1987) Kamm et al. (1984) present study
and//are the exponents of f and VT in the empirical equation for DDISP, see eq. (7). The values for K* were computed with the help of eq. (10).
diffusion, and, as a con:+equence, this equation has little to do with the actual gas mixing processes which take place during HFV within our airway model. The final equation that was used for curve-fitting (eq. 6) may be considered to describe the diffusion of a bolus of a hypothetical gas with D = Dt)~sp in a tube with cross-section Ao, where C(t) represents the concentration at time t at a distance ! = (V 2 + 0.125VT2)°'5. Ao ~ from the site of bolus injection. That is, in the data analysis our airway model is considered to be a kind of black box with an input - the injected bolus - and an output - the measured dispersion curve. Our results show that this output can be satisfactorily described with eq. (6). This means that whatever the exact nature of the mixing processes in our airway model during H FV may be its net effect for gas transport across the model is identical to that of a diffusion process. For this reason, the results of the curve-fitting may be used, among other things, to compute the conductance of our airway model for gas transport by HFV. The (total) cross-sections of the generations zero through six show only slight mutual differences (Weibel, 1963). Their mean value is approximately equal to that of the cross-section of the zeroth generation. This generation also forms the larger part of our airway model, therefore, we have used the value of the cross-section of the zeroth generation (Ao --- 2.54 c m 2) in the computation Of D D l s p [eq. (6)]. The conductance of the airway model for gas transport by H FV can then be computed from: Cg,,~ = 6.95 10-
DDlspAol
2--
-- |
(11)
where Cg,,~ is expressed in mi (STPD)mmHg- ~. min- ~. Substitution of i, Ao and Vo in eq. (11) then results in: Cg,,~ = 8.66 10- 3DDisp" [ 1 + 0.125 (VT/Vo)2] -0"5
(12)
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
27
Studies in intubated human subjects have shown that normocapnic conditions can be maintained by HFV by using a frequency of 12 Hz and a tidal volume of 1.2 ml per kg body weight (Crawford and Rehder, 1985; Rehder and Didier, 1984). In the experiments of Rehder and Didier (1984) subjects had a mean body weight of 82 kg, therefore, the applied tidal volumes in these subjects will have been close to 100 ml. Substitution of f = 12 and VT = 100 into eq. (8) yields a predicted value for DD~sP equal to 259.2 cm 2. s - ', and substitution of this value into eq. (11) results in a predicted value for Cg..,sequal to 1.86 ml (STPD)mmHg- !. min- !. This result for Cgas corresponds to a CO2 transport across the airway model of 74 ml (STPD)min-' for a partial pressure difference of 40 mmHg. CO2 production in anesthetized subjects weighing 82 kg will be at least twice as much. Thus, our data are incompatible with the in-vivo observations. The predicted value of DD,Sp for f = 12 Hz and VT = 100 ml is out of the range of measured values for DD,SP, therefore, DD~Sp might have been underestimated. For this reason, we have also computed DDiSp from:
DDISP = 50.5 (f/25)2(VT/20) 2
(13)
Using this equation, the extrapolation starts from the largest, measured value of DDlsP ( = 50.5 cm2" sec- ' for f = 25 Hz and VT = 20 mi), and further the largest values of g(= 1) and [3(= 2) shown in table 2 are used. Application of eq. (13) results in DD,sp = 606 cm 2. s e c - ' for f = 12 Hz and VT = 100 ml, which corresponds to a CO2 transport of 173 ml (STPD)min- i for a partial pressure difference of 40 mmHg across the model. It is obvious, that even this larger value of DDISP is still incompatible with the in-vivo observations as the generations 0 through 6 constitute only about one-third of the volume of the conductive airways. We think, therefore, that in normocapnic ventilated subjects longitudinal dispersion cannot possibly be the sole or even the main gas transport mechanism in HFV. Lehr etaL (1985) showed by cinematographic tochniques in excised dog lungs that there were gross phase differences between the tidal movements of the different lung regions in HFV. Further, they found that the sum of regional tidal volumes exceeded the applied oscillatory tidal volume at the trachea thus demonstrating that HFV is accompanied by gas movements among lung regions high-frequency Pendelluft. This is not the only evidence of high-frequency Pendelluft reported in the literature. Other reports dealing with this phenomenon which were published prior to the publication of Lehr et ai. (1985) have been extensively reviewed by Chang (1984). Kaethner et al. (1984)observed that most of the total concentration drop of He and SF 6 during washout in dogs occurred along the endotracheal tube and a few generations of large bronchi. This observation is compatible with our findings of small values for Cg.,.~in the central airways at normal values for f and VT for human subjects, and we agree with Kaethner et al. (1984) that in intubated subjects the instrumental dead space - tube etc. - together with the larger airways will provide the main gas transport resistance during HFV. In summary, we think that during HFV gas tcansport by longitudinal dispersion will be restricted to the upper airways, the trachea - o r the endotracheal tube in intubatcd subjects - and a few generations oflarge bronchi,
28
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
whereas in the remaining conducting airways gas transport will mainly take place by a different mechanism. This mechanism may be Pendelluft as was already suggested by Chang (1984). In particular the combination of turbulent dispersion and PendeUuft is compatible with further observations on HFV, e.g., the independence of HFV of molecular diffusion.
Conclusion In this study we have validated that gas transport resulting from longitudinal dispersion of gases in the airways during high-frequency ventilation may be described in a way as if it were the result of augmented, longitudinal diffusion. In addition, we have shown that for frequencies and tidal volumes normally used in high-frequency ventilation the effc~.ti~,, di~,permon coefficient is independent of molecular diffusion. The quantitative data further suggest that the importance of longitudinal dispersion for gas transport in high-frequency ventilation is limited to a few generations of large airways.
Appendix in a previous investigation with the same airway model we found that at a constant flow in the model DDtsp was proportional to LV[ with a slight dependence on the direction of the flow, i.e. inspiratory versus expiratory. If we assume that this also applies to HFV then Dr, tsp(t) can be approximated by: m
DD,s,(t) = D=,,s,,(n/2)Icos 2~ftl
(14)
where Dl~lsp is again the mean value of DD,m,(t). Substitution of eq. (14) into eq. (3) yields:
~(t) = 2A~,~,,,m, f',
(7t/2) Lcos 2 n f f l d t
(15)
Further, f r,4 (n/2)Icos 2nftldt - T/4
(16)
)
anu every period of T/4 seconds the value of the integral in eq. (15) increases with this same amount. Thus, to a first approximation ~(t) increases linearly with time so that ~ ( t ) can be written as:
~v2(t) =
2-2AoDl~lsp
[t + A(t)]
(17)
where A(t) represents the deviation. A(t) is a periodic function, and it can easily be shown that IA(t)/tl < 0.053/(ft) - 0.053. T/t. Thus the inaccuracy ofthe linear approximation o f f ( t ) [eq. (5)] decreases rapidly with time, and after two cycles (t > 2T) this inaccuracy has already decreased to about 2%.
References Berdine, G. B., J. L. Lehr and J. M. Dr.~zen (1983). Ethane washout by high frequency (2-40 Hz) low volume (20-100 ml) ventilation (HFV) in excised dog lungs. Fed. Proc. 42: 763.
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
29
Bohn, D.J., K. Miyasaka, B. E. Marchak, W. K. Thompson, A. B. Froese and A.C. Bryan (i 980). Ventilation by high-frequency oscillation. J. Appl. Physiol. 48: 710-716. Brown, K.M. and J.E. Dennis (1972). Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximations. Numer. Math. 18: 289-297. Chang, H. K. (1984). Mechanisms of gas transport during ventilation by high-frequency oscillation. J. Appl. Physiol. 56: 553-563. Crawford, M. and K. Rehder (1985). High-frequency small-volume ventilation in anesthetized humans. Anesthesiology 62: 298-304. Horsfield, K. and G. Cumming (1967). Angles of branching and diameters of branches in the human bronchial tree. Bull. Math. Biophys. 29: 245-259. Jaeger, M.J., M. Banner and J. Gallagher (1983). Alveolar ventilation in high frequency studies. Fed. Prec. 42: 1351. Jaeger, M.J., U. H. Kurzweg and M.J. Banner (1984). Transport of gases in high-frequency ventilation. Crit. Care Med. 12: 708-710. Kaethner, T., J. Kohl and P. Scheid (1984). Gas concentration profiles along airways of dog lungs during high-frequency ventilation. J. Appl. Physiol. 56: 1491-1499. Kamm, R. D., J. Collins, J. Whang, A. S. Slutsky and M. Greiner (1984). Gas transport during oscillatory flow in a network of branching tubes. J. Bimnech. Engin. 106: 315-320. Knopp, T.J., T. Kaethner, M. Meyer, K. Rehder and P. Scheid (1983). Gas mixing in the airways of dog lungs during high-frequency ventilation. J. Appl. Physiol. 55: ! 141-1146. Kooij, A. M. van der and S.C.M. Luijendijk (1985). Longitudinal dispersion of gases measured in a model of the bronchial airways. J. Appl. Physiol. 59: 1343-1349. Lehr, J. L., J.P. Butler, P.A. Westerman, S.L. Satz and J.M. Drazen (1985). Photographic measurement of pleurai surface motion during lung oscillation. J. Appl. Physiol. 59: 623-633. Menon, A. S., M. E. Weber and H. K. Chang (1984). Model study of flow dynamics in human central airways. Part !!I: Oscillatory velocity profiles. Respir. Physiol. 55: 255-275. Mitzner, W., S. Permutt and G. Weinmann (1984). Gas transport during high-frequency ventilation: theoretical model and experimental validation. Ann. Biomed. Eng. 12: 407-419. Paloski, W. H., R.B. Slosberg and R.D. Kamm (1987). Effects of gas properties and waveform asymmetry on gas transport in a branching tube network. J. Appl. Physiol. 62: 892-901. Rehder, K. and E.P. Didier (1984). Gas transport and pulmonary perfusion during high-frequency ventilation in humans. J. Appl. Physiol. 57: 1231-1237. Reisfeld, B. and J.S. Ultman (1988). Longitudinal mixing in dog lungs during high-frequency forced flow oscillation. Respir. Physiol. 71: 269-286. Rieke, H., C. Hook and M. Meyer (1983). Pulmonary gas exchange during high.frequency ventilation in dogs. Respir. Physiol. 54: 1-17. Rossing, T. H., A. S. Slutsky, J. L. Lehr, P. A. Drinker, R. Kamm and J. M. Drazen (1981). Tidal volume and frequency dependence of carbon dioxide elimination by high-frequency ventilation. N. Engl. J. bled. 305: 1375-1379. Scherer, P.W. and F.R. Haselton (1982). Convective exchange in oscillatory flow through bronchial-tree models. J. Appl. Physiol. 53: 1023-1033. Slutsky, A. S., J. M. Drazen, R.H. Ingrain Jr, R.D. Kamm, A.H. Shapiro, J.J. Fredberg, S.H. Loring and J. Lehr (1980). Effective pulmonary ventilation with small-volume oscillations at high frequency. Science 209:609-61 !. Slutsky, A. S., R.D. Kamm, T.H. Rossing, S.H. Loring, J. Lehr, A.H. Shapiro, R.H. Ingram Jr and J. M. Drazen (1981). Effects of frequency, tidal volume and lung volume on CO2 elimination in dogs by high frequency (2-30 Hz), low tidal volume ventilation. J. Clin. Invest. 68: 1475-1484. Subramanian, R. S. and W.N. Gill (1976). Unsteady convective diffusion in non-Newtonian flows. Can. J. Chem. Eng. 54: 121-125. Taylor, G. I. (1953). Dispersion of soluble matter in a solvent flowing slowly through a tube. Prec. R. Soc. London. A. Math. Phys. Sci. 219: 186-203. Weibel, E.R. (1963). Morphometry of the human lung. New York: Academic Press. Worth, H. and J. Piiper (1978). Diffusion of helium, carbon monoxide and sulphur hexafluoride in gas mixtures similar to alveolar gas. Respir. Physiol. 32: 155-166.
13
Elsevier RESP 01764
Longitudinal dispersion in model of central airways during high-frequency ventilation. A.M. van der Kooij ~ and S . C . M . Luijendijk 2 'Department of Medical Physiology, State University of Utrecht. Utrecht, The Netherlands and "Department of Puimonology. University Hospital Maastricht, State University of Limburg. Maastricht. The Netherlands (Accepted 27 December 1990) Abstract. We have measured the longitudinal dispersion of boluses of helium, acetylene and sulphur hexafluoride in a plastic model of the human airways -generations zero through six -during high frequency ventilation (HFV). HFV was maintained by a piston pump. Frequency f and tidal volume VT ranged from 2.5 to 25 Hz and from 5 to 20 ml, respectively. Boluses were injected near the entrance of the zeroth generation (trachea), and the dispersion curves were measured by mass spectrometry at the end of the sixth airway generation. The shapes of the bolus dispersion curves could be well described with Gaussian distribution functions. With the exception of the HFV-conditions with VT = 5 ml, the effective dispersion coefficient DDtsr appeared to be independent of the molecular diffusion coefficient. This independency was also found by other investigators in studies with dogs and human subjects. The measured results for DDISPfor different f and VT could be satisfactorily described with the empirical equation DDISP= 0.0617 f°'sV~38 [cm2s- t]. Application of this equation to f and VT values normally applied in man resulted in DDISP values which should be considered to be too small for maintaining eucapnic ventilation in rive. On the basis of this result we believe that during HFV in intubatod subjects gas transport by longitudinal dispersion will be limited to the instrumental dead space - the endotracheal tube inclusive - and a few generations of large bronchi.
Dispersion ofgases in airways; Gas mixing in long airways; High frequency ventilation, and intrapulmonary gas mixing; Inert gases, and mixing in long airways
In the recent past it has been well established that normal blood gases can be maintained during artificial ventilation by applying tidal volumes that are considerably less than the anatomical dead space at increased breathing rates (Bohn et al., 1980; Crawford and Rehder, 1985; Rieke et al., 1983; Rossing et al., 1981). Different mechanisms have been proposed to account for the 02 and CO2 transport in the conductive airways under these ventilatory conditions such as turbulent dispersion, convective dispersion due to asymmetric velocity profiles, Pendelluft, and molecular diffusion (Chang, 1984). These me-
Correspondence to: S.C.M. Luijendijk, Dept. of Pulmonology, University Hospital Maastricht, P.O. Box 5800, 6202 AZ Maastricht, The Netherlands. 0034-5687/91/$03.50 © 1991 Elsevier Science Publishers B.V.
14
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
chanisms may operate in series, i.e. each of them may apply to a portion of the airways (Chang, 1984). In this paper, we will focus on the gas transport in the central airways, and data will be presented which was obtained from measurements performed in a plastic airway model representing the airway generations 0 up to 6 inclusive. It is common practice to present the gas exchange efficacy due to high-frequency ventilation (HFV) by means of an effective diffusion coefficient (Chang, 1984). This assumes that the longitudinal dispersion of gases in the airways during HFV behaves in a way as if it were the result of (augmented) diffusion. The main objective of this study is to verify this assumption, and to the best of our knowledge this has never been performed before. In the central airways, longitudinal mixing of gases during HFV is assumed to be due to turbulent and/or convective dispersion. In the latter case, radial mixing in the airways by molecular diffusion might interfere with longitudinal mixing (Taylor, 1953). We have employed, therefore, tracer gases with different molecular diffusion coefficients, which might reveal the prevailing mechanism of gas mixing in this part of the airways. Further, we will analyse the measured gas exchange efficacy in terms of its dependence on tidal volume (VT) and frequency (f), and the data obtained will be compared with data obtained from measurements by other investigators.
Materials and methods The experiments were performed with a symmetrically branching model of the central airways representing the generations 0 through 6. The different branches of the model were constructed of cylindrical tubes made of clear plastic, and their dimensions corresponded to those of Weibel's model A (table 1)(Weibel, 1963). At each branch point, the angle between the two daughter branches was equal to 700 (Horsfield and Cumming, 1967). After construction, the volume of the airway model was 52 ml. The branches of the sixth generation were extended with flexible tubes (fig. 1). Length and inner diameter of these tubes were 19 cm and 5 ram, respectively. The purpose of these TABLE I Dimensions of airway model. Generation number
Number of tubes per generation
Length of tubes mm
Diameter of tubes mm
0 ! 2 3 4 5 6
! 2 4 8 16 32 64
120.0 47.6 19.0 7,6 12.7 10,7 9.0
18.0 12.2 8.3 5.6 4.5 3.5 2.8
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
15
extension tubes is discussed elsewhere in this section. Further, the zeroth generation was extended with a brass tube (fig. 1). Volume and inner diameter ofthis tube corresponded to the volume of the airway model and the inner diameter of the zeroth generation. An oscillatory flow of room air in the model was maintained by means of a piston which was moved up and down in the brass tube. The piston shaft was connected to a fly wheel which was driven by a DC-motor (fig. 1). A long shaft was used to approximate a sinusoidai displacement of the piston. Different values for VT were adjusted by changing the radial position of the end of the piston shaft on the flywheel. Different values for f were obtained by varying the supply voltage of the DC-motor. The airway model was placed upright with the zeroth generation underneath (fig. 1). The applied tracer gases were injected into the oscillatory flow near the inlet of the airway model. The main parts that formed the injection system were a hollow needle and a trigger mechanism (fig. 1). The hollow needle was placed in two holes in the opposite walls of the brass tube near the connection with the zeroth generation (figs.
(PARTLY DRAWN)
MASS SPECTROMETER/ ADJUSTMENT~~
TRIGGER MECHANISM RELAY
RESERVOIR WITH TRACER GAS PISTON----t!
II
Ii
', OPTICAL " o O.
DEVICE
ADJUSTMENTFOR TIDAL VOLUME
I
FLY- WHEEL Fig. 1. Schematic drawing of experimental set-up.
16
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
1 and 2). This construction made it possible to move the needle perpendicular to the oscillatory flow by means of elastics which were attached to a handle which in turn was attached to one end of the needle (fig. 1). In the stand-by position, this handle was held by the trigger mechanism, and after release the needle was shot by the force of the elastics. A constant flow of tracer gas was supplied to the injection needle and escaped through a small hole in the wall of the needle (fig. 2). Before and after bolus injection, the tracer gas entered the exhaust tubes B and A (fig. 2), respectively. These tubes were continuously flushed with air with the help of a vacuum pump (fig. 2). During bolus injection the outlet of the needle moved from tube B to tube A. Thereby it passed the inside of the brass tube, which took less than 1 msec. In all experimental conditions the peak flow in the airway model was less than 1 L . s e c - ~, therefore, the injected bolus of tracer gas mixed with less than 1 ml of air during the period of injection. The bolus dispersion curves were measured at the other end of the model. As a consequence, the contribution of the initial volume of the bolus (< 1 ml) to the shape of the measured dispersion curves may be neglected. The frequency f was determined from the output signal of an optical device that was used to detect the mid-position of the piston (fig. 1). Frequency was measured by a counter (Philips PM 6620). The optical device was further used to determine the moment of injection: After pressing the start button, the optical device initialized the trigger mechanism to release the injection needle at the next mid-position of the piston. After injection, the tracer gas dispersed both into the airway model and into the brass tube. The dispersion curves were measured by use of a mass spectrometer (Balzers QMG 511). To that end the tip ofthe sample capillary ofthe mass spectrometer was placed in a well-fitting hole in one out of the 64 flexible tubes close to the corresponding branch of the sixth generation of the airway model (fig. 1). The output signal of the mass spectrometer was digitized on-line by means of a microcomputer (Zilog MCZI/05) and stored on floppy disk for off-line data analysis. The applied sample rate ranged from 200 up to 1000 samples per second dependent on the rate of change of the concentration of the tracer gas in the steep part of the dispersion curve.
In view of the oscillatory nature of the flow, the flexible extension tubes made it
BULKFLOW ~
~
INJECTIONNEEDLE
T~ACEROeSSUPPLY
I11 TOVACUUMPUMP
Fig. 2. Schematic drawing of bolus injection system. The dashed ring represents the cross-section of the brass tube which is connected to the zeroth generation of the airway model. The outlet ofthe injection needle is shown in the position it takes prior to bolus injection. The tubes B and A serve to remove the outflowing tracer gas before and after bolus injection, respectively.
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
17
possible to measure the overall dispersion of the whole airway model. From a certain time after injection onwards, however, tracer gas will be lost from the model as soon as it reaches the outlets of these extension tubes. For this reason, we restricted the data analysis to the first part of the measured dispersion curve that is in between the beginning and the maximum of the curve. Le., the total volume of the extension tubes ( = 239 ml) was chosen sufficiently large so that the aforementioned loss of tracer gas could not possibly affect the shape of the first part of the measured dispersion curves. All data files were corrected for the transit-time ofthe gas through the sample capillary of the mass spectrometer which amounted to 90 ms. In all our experiments the difference of time between the maximum and the onset of the dispersion curve exceeded 2 sec. Therefore, no corrections were made for the time constant of the mass spectrometer which was less than 20 ms for the tracer gases applied in this study. The experiments were carried out for three tracer gases (He, C2H2 and SF6), four tidal volumes (5, 10, 15 and 20 ml), and six frequencies (2.5, 5, 10, 15, 20, and 25 Hz). These tracer gases cover a wide range of molecular diffusion coefficients (D): DsF6 = 0.101 cm 2. sec- ~and DH,~ = 0.725 cm 2" sec- ~in air at one atmosphere and at 37 °C (Worth and Piiper, 1978). D r : . : is not known. The molecular weight of C,H2, however, is close to that of CO. Therefore, Dr:a, will be close to Dco which is equal to 0.233 cm2" sec- ~ in air at one atmosphere and at 37 °C (Worth and Piiper, 1978). For statistical reasons, each experiment was performed 5 times.
Data analysis As mentioned in the Introduction, the main objective of this study was to investigate whether gas transport in the central airways due to HFV can be described as if it were the result of a diffusion process. The data analysis, therefore, started from the following equation: C (t) = M (2 g~(t)) -°'5 exp [ -(Vo - V(t))2/2 ~(t)]
(l)
where C(t) represents the concentration of the tracer gas at the inlet of the sample capillary t seconds after the injection of the tracer gas, M is the total amount of tracer gas, Vo is the volume of the airway model, and ~(t) is the volume variance at time t of the underlying Gaussian distribution function. The measured dispersion curves showed periodic variations. These variations were caused by the oscillatory flow which moved the concentration profile of the tracer gas in the airway model up and down the model. For the same reason the volume between the centre of this concentration profile and the site of gas sampling changed continuously. This is accounted for by the term V(t) in eq. (1), and V(t) should be directly related to the volume displacement of the piston. Therefore, V(t) is given by: V(t) = 0.SVTsin 2 rift
(2)
18
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
In practice, the amplitude of the periodic variations in C(t) was attenuated because the time constant of the mass spectrometer which ranged from 10 to 20 msec for the different tracer gases was too large with regard to the applied values for f in particular at the higher frequencies (f> 10 Hz). For this reason, we have modified eq. (1) by substituting for (Vo - V(t))2 the mean value which is equal to (Vo2 + 0.125VT2). The volume variance ~(t)is related to the dispersion coefficient Dmsp(t) according to: ~ ( t ) - - 2Ao2 ff~ Dmsv(t) dt
(3)
where Ao is the cross-section of the zeroth generation (Subramanian and Gill, 1976). In a previous study we found that at a constant flow in the airway model DDtsp was a function of the flow (Kooij and Luijendijk, 1985). This means that in the present experiments DD~sp(t) was a periodic function of time due to the periodicity of the flow with a time-period T equal to 1/f seconds. I.e., for each time period T the integral in eq. (3) will show the same value, and the long-term mean value of Dmsp(t) can then be obtained from:
DDISP
-"
T-
I
.fT Dmsp(t) dt
(4)
In 80% of the measurements the tracer gases reached the outlet of the airway model after 8T seconds (range 3-88T). Hence, the data analysis was in general related to a time-interval for which t > 8T. For such t-values ~(t) can then be well approximated by:
(s) because ~(t) increases monotonously with increasing t as Dmsp(t)>0 for any t. This approximation of ~(t) will be further discussed in the Appendix by taking a particular example for DDtsp(t). Substitutions of the mean value of IVo - V(t)] 2 and the result for ~(t) into eq. (1) yields: C(t) = M (4gAo2Dmspt) -°,5 exp [ -(V~ + 0.125VT2)/(4AoaDmspt)]
(6)
This expression for C(t) was fitted with the measured dispersion curves by adjusting the parameters M and Dmsp with the help of a non-linear least squares algorithm (Brown and Dennis, 1972). This curve-fitting was applied to each individual dispersion curve, for examples see fig. 3. M depends on the amplitude but not on the shape of the dispersion curve. As a consequence, the results for M are of no concern for this study and will, therefore, not be presented. For statistical analysis we used Student's t-test for paired observations.
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
19
Results Figure 3 shows six measured dispersion curves, two for each tracer gas. In each panel, the smooth curve represents the best least squares fit through the data points for the relationship between C and t given by eq. (6). The noisy structure of the measured dispersion curves is due, at least in part, to the variations in C caused by the rapid, periodic movement of the concentration profile in the airway model. Leaving this aside, there are still other deviations between the measured and fitted dispersion curves which are related to the average behaviour of measured C as a function of t (fig. 3). As an indication of the magnitude of these deviations we have determined for each measurement the maximal deviation between measured and fitted dispersion curves expressed as a percentage of the maximal value of C included in the analysis. These percentages ranged from 1 to 20% with an average value of 5 % and a standard deviation of 4%. Thus, in general, the deviations between the measured and fitted dispersion curves were small. As mentioned in Methods, each experiment was performed five times, and inspection of the graphic representations, as in fig. 3, of each set of five measurements showed that the deviations between measured and fitted dispersion curves were not i
:
~1'0-1 t :15OHz VT: 0.020t
i
/
,
0.5
, f
He 06
3b
1,0- ! :25Hz VT:00051
'
"5~P/P ~
C
,,f"
/~
/"~
0~
i,,,"f
--
~o
,c2.~
C2H2
0~" '
'
;
10 i
~,/ 0
'
o
10- f :25Hz 00051 VT:
0
§
10 f :150Hz
o
O~
He
60
0
sF6 '
:tlsl
-tls)
Fig. 3. Representative plots of individual bolus dispersion curves for different f, VT and tracer gases. The smooth curve in each panel represents the result of the curve-fitting with eq. (6).
20
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
systematic, on the contrary, they behaved differently and randomly for the different measurements. On the basis of all these results we may conclude that the basic shape of the measured dispersion curves can be satisfactorily described with eq. (6). In turn, this means that the measured dispersion curves can be characterized by the value of DD~Sp obtained from the curve fitting. For. each set of five identical measurements we have computed the coefficient of variation (CV = SD/mean) of the five values for DD~Sp obtained. The mean value of CV for all three gases and all ventilator settings together was 6.6% (_+ 3.9%). The mean values for Dr)~sp obtained from the different sets of identical measurements were fitted with the empirical equation: DDISP "- Kf~VT~
(7)
The values of K, a and fl were 617" 10 -4, 0.8 and 1.38, respectively. To evaluate the impact of molecular diffusion on DD~SP we have plotted the measured values of Dn~sP v e r s u s the predicted values of Dn~sp in figs. 4 through 7, where predicted DD~Sp is given by: DDISP (predicted) = 617.10- 4f°'8VTl'3s
(8)
No systematic deviations are seen in fig. 4 between the values of DDISPfor the different tracer gases for DDtsP > 10. This range of DDtsp-values includes nearly all data obtained with VT ~ 10 ml, and, as a consequence, the range of DDlsp-Values between 1 and 10 is mainly related to data obtained with VT - 5 ml (figs. 5-7). For VT -- 5 ml, the values of DDtsp for He are substantially larger than the correspon:.uag values for C2H2 and SF6 (fig. 4). Evidently, only for this small tidal volume are the results for DDISP affected by molecular diffusion. All data points for VT ~ 10 ml are close to the line ofidentity (figs. 5-7). This means that for VT > l0 ml the dependence of DD~sp on f and VT may be well described by eq. (8). The data shown in figs. 4-7 were obtained from measurements at a single outlet. A few additional measurements were performed at other outlets. The results of these measurements did not differ substantially from those obtained at the outlet selected for the bulk of the measurements.
Discussion
To answer the question whether longitudinal dispersion by HFV behaves as if it were the result of diffusion we have fitted our measured dispersion curves with eq. (6). This equation has been derived from the Gaussian distribution function given by eq. (l). It is well known that this function underlies the description of all kinds of random processes such as diffusion. A perfect fit between the measured and fitted dispersion
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
100
j
-~
x
!
,¢"o
! l,-~ /
,
4w
0.4
Fig. 4. Measured
•
/, 2
,
1
I
x :He + :C2H2
/"
I/: -
/~-
, /o°+
| 1-1
0•
21
~ w 4
I
[
I , , i m , , w I 10 20 402 1 100 ]~OlsP(predicted}[cm s- ]
predicted values of the mean dispersion coefficient for the three tracer gases. Full line represents the line of identity. Predicted values of DD=sP were computed with eq. (8). versus
~00 "
/
+/
He
40x" 20-
/,
°
U
±/~.
10-
+,/"
4-
° x7 "+
-
°
-/o
/~.~
,_/ ,~
ae" 2-
/
/
"
VT (ml}
-:20
/
Oz.: 0,4
I
I
+ 15 ,o
:
I
I
I
1
2
4
I
I
I
I
I
I
I
10 20 40 100 Doxsp(predicted)[cm 2 s-1 ]
Fig. 5. Measured v e r s u s predicted values of the mean dispersion coefficient for helium. This diagram shows the results for different VT. For each value of VT, the data points from left to right correspond to increasing f. For further explanation see fig. 4.
22
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK 100
C2H2
4o
4-
x
x . t.
20-
~
10_ +
~,
-
,,
18 2-
/o o
D
V T (roll ~ 20 + 15
o
A o
1-
0.4
I
14
Fig. 6. Measured
1
2
4
I
I
I
I
10 5
I
I
I
10 20 40 100 DozsP ( predicted ) [ cm 2 s-1 ]
predicted values of the mean dispersion coefficient for acetylene. For further explanation see fig. 5.
versus
~oo. SF6 t,020IO-
Q
I,~ 2/
VTlml )
o
a
10
o
5
1o
0.4 0,4
Fig. 7. Measured
versus
I I;
1
i 2
; 4
I
i;
i I I I 10 20 40 100 "DozsP(predicted)[cm2 s-l]
predicted values of the dispersion coefficient for sulphur hexafluoride. For further explanation see fig. 5.
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
23
curves would, therefore, mean that in our experiments the result of longitudinal gas mixing by HFV is equivalent to that by (augmented) diffusion. As a function of time the bolus dispersion curves show a steep, initial part up to the maximum of C(t), Cmax, and thereafter C(t) decays very gradually. In our data analysis we have used only this steep part of the dispersion curves as, in particular, the shape ofthis part is suitable to study deviations from eq. (6). In addition, the shape ofthe final part of the bolus dispersion curves is undoubtedly affected by loss of tracer gas from the airway model at the open ends of the flexible extension tubes of the sixth generation airways and by back-dispersion from the cylinder of the piston pump (fig. 1). For all measurements the data analysis included concentrations between 0 and 80 to 100 percent of Cmax. Thus, most of the steep, initial part of the bolus dispersion curves was included in the data analysis. In quite a few cases a nearly perfect agreement was obtained between the measured dispersion curve and the best fitting curve through the data points for eq. (6). Two representative examples are shown in fig. 3 one for C2H2 and one for SF6 both related to f = 15 Hz and VT = 20 ml. A more detailed analysis showed that- leaving aside the rapid variations in C - the maximal difference between measured and fitted data was 5~/o of Cma x (mean value, see Results). Considerably larger differences (> 12%)were found only rarely. As these percentages refer to the maximal differences obtained for the different measurements, it will be clear, that in general the mean difference between measured and fitted data for individual measurements was still considerably less than 5 % C,l,,x. As already mentioned in Results, no systematic deviations were seen between the measured and fitted dispersion curves when the results of a single set of five identical measurements were compared. Summarizing it may be said that the measured dispersion curves could be satisfactorily described by eq. (6). That is, the net effect of HFV on longitudinal gas mixing in the central airways is approximately identical to that of augmented diffusion. We have studied the dependence of DD~sp on D by using tracer gases with different molecular weights. We preferred this to using a single tracer gas in combination with carrier gases with different densities, e.g. SF6 versus He (Palosk! et ai., 1987), because changes in carrier gas might also involve density-related changes in the velocity profiles of the carrier gas in the airways. For VT - 5 ml, (measured) DD~sP for He is significantly larger than DD~sp for C2H2 or SF6 (P < 0.05; fig. 4). Evidently, under these conditions molecular diffusion in the axial direction of the airways contributes substantially to the longitudinal mixing of the bolus in the airway model and this should especially apply to He. This is a plausible explanation as the D-value for He ( - 0.725 cm 2. sec - i) is comparable in magnitude to the smallest DD~sp-values for He ( ~ 2.5 cm 2. sec - I ) shown in fig. 4. In a previous experiment with the same airway model we observed that at a constant flow in the model the longitudinal dispersion for He was smaller than that for SF6 (Kooij and Luijendijk, 1985). This inverse relationship between DD~sp and D was ascribed to the impact of radial diffusion on the mixing process as radial diffusion interferes with longitudinal dispersion (Taylor laminar dispersion). Such an inverse relationship, however, is not
24
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
shown by the results in fig. 4. That is, in the present experiments Taylor laminar dispersion was of no importance. For DD~SP > 10 no significant differences (P > 0.10) were found between the data of the different tracer gases. Evidently, for DDIsv > 10 any contribution of diffusive mixing to DDISP is completely overruled by that of convective mixing. The above described behaviour of DDIsP for the different tracer gases is consistent with the experimental data published by Scherer and Haselton (1982). These authors studied the movements of buoyant tracer-beads in an oscillating flow in Plexiglas models ofthe bronchial airways. The flow consisted of a glycerine-water solution whose composition was adjusted to different values of the kinematic viscosity ranging from 0.00. up to 7 cm 2. s-~. The measurements were performed by photographing the movements of the beads. Next, the photographs were used to determine the distance L that the tip of the particle tongue was convected forward into the model per cycle averaged over the first five flow cycles after injection of the beads. A curvi-linear relationship was found between the dimensionless quantity L(A/VT) and Reynolds number Re, where A is cross-sectional area of tube. In our airway model, Re ~ 117 in the zeroth generation for f = 2.5 Hz and VT = 5 ml. For this value of Re, Scherer and Haselton (1982) found L(A/VT) to be equal to about 0.17. For A = (Ao) = 2.54 cm 2 and VT = 5 ml this results in L = 0.33 cm per cycle. The average displacement of particles by diffusion LD in one direction in time l/f is given by (Scherer and Haselton, 1982): LD = ~/~ D/f
(9)
For f = 2.5 Hz, this results in LD = 0.76 cm and LD = 0.28 cm for He and SF6, respectively. Thus, the above discussed experimental findings predict that for the HFV conditions that correspond to the lower range of our measured DDtsp-values the contributions of diffusion and convection to longitudinal dispersion are comparable in magnitude. Accordingly, for these HFV conditions DDtsp for He should be substantially larger than Dotsp for SF6. It can be further derived from the data given by Scherer and Haselton (1982) that for f -- 25 Hz and VT = 20 ml L/Lo will amount to about 5.5 and 15 for He and SF6, respectively. Hence, in the upper range of our measured Do~sp" values DDtsP should be virtually independent of D. Thus, we may conclude that our findings concerning the dependence of DD~sP on D are consistent with the results obtained by Scherer and Haselton (1982) in a fairly different experiment. The findings for DotsP < 10 were obtained with the smallest tidal volume ( - 5 ml) used in our measurements. Tidal volumes normally applied in HFV are larger than 20 mi. The rest of the discussion will, therefore, be exclusively related to our findings for DD~Sp > 10. Knopp etal. (1983) showed in intubated dogs that there were no consistent differences in the time courses of washout between He and SF6 or between He and Ar using oscillation frequencies between 10 and 30 Hz and tidal volumes ranging from about 40 to 100 ml. It is clear that these findings are in full agreement with our findings namely that DDtsP is independent of D for D < 0.725 cm2. sec- ~. The dependence of the dispersion coefficient on frequency and tidal volume has been described by several authors (Berdine et aL, 1983; Jaeger et aL, 1983, 1984; Kamm
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
25
et a/., 1984; Mitzner eta/., 1984; Paloski eta/., 1987; Reisfeld and Ultman, 1988; Slutsky et ai., 1980, 1981). The empirical eq. (7) was adopted from their publications, and we have shown that also our data for DDISP Can be satisfactorily described with this equation for DrnsP > 10 cm 2- sec - ~when the proper values for the parameters K, and,8 are used (eq. (8), fig. 4). This finding now facilitates the comparison ofour results with those reported in the literature. Most publications show that 0.7 _< 0c_< 1.0 and 1.0 _<,8 _< 2.0 (table 2). Our results for 0cand//are thus compatible with these general findings. Further, most investigators and we too find that a
DD, sP = K*f~ (VT/A)~
(10)
Evid¢~aly, VT/A and not VT is a major determinant of the longitudinal mixing process in HFV. This is understandable as VT/A represents the mean, axial displacement of the
gas molecules during one oscillation. Thus, we have used K* instead of K to compare our results to those reported by other investigators. The results for K* (table 2) were thus calculated with the help of eq. (10). In most papers on intubated dogs, however, some data are missing so that K* cannot be computed. Our result for K* (-0.223) is in the middle of the wide range (0.03-3.06) of K* values shown in table 2. It may be concluded from table 2 that data for K*, a and p obtained from the different investigations show considerable, mutual differences, and this especially applies to the data for K*. These differences may, at least in part, be due to the diversity of applied models - dog lungs, straight tubes and networks of branching tubes. In addition, in a number of the studies cited in table 2 the values for a and/~ were assumed rather than calculated (Jaeger et al., 1983, 1984; Mitzner et al., 1984; Slutsky et al., 1980), and this also affects the results for K*. The consequence of the things mentior, e~J is that the most reliable results for K*, ~ and fl reported so far refer to measurements in networks of branching tubes. In this category we have used the most realistic model of the central airways as the other investigators (Kamm et al., 1984; Paloski et al., 1987) used models with much less branchings and constructed of tubes of equal length and diameter for all airway generations. One should realize that eq. (1) refers to models that describe dispersion of gases by
26
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK TABLE 2 Survey of the 0c and fl values reported in the literature.
K*
a
,//
M aterial
References
3.06 0.030 0.076 0.275 0.340 0.223
0.17 1.00 0.8 1.0 1.08 1.0 1.0 0.71 0.94 0.8
1.74 1.00 1.5 2.0 1.49 2.0 2.0 1.93 1.66 1.38
intubated dogs intubated dogs intubated dogs intubated dogs intubated dogs straight tubes straight tubes network of branching tubes network of branching tubes network of branching tubes
Reisfeld and Ultman (1988) Slutsky et al. (1980) Berdine et al. (1983) Jaeger et al. (1983) Slutsky et al. (1981) Jaeger et al. (1984) Mitzner et al. (1984) Paloski et al. (1987) Kamm et al. (1984) present study
and//are the exponents of f and VT in the empirical equation for DDISP, see eq. (7). The values for K* were computed with the help of eq. (10).
diffusion, and, as a con:+equence, this equation has little to do with the actual gas mixing processes which take place during HFV within our airway model. The final equation that was used for curve-fitting (eq. 6) may be considered to describe the diffusion of a bolus of a hypothetical gas with D = Dt)~sp in a tube with cross-section Ao, where C(t) represents the concentration at time t at a distance ! = (V 2 + 0.125VT2)°'5. Ao ~ from the site of bolus injection. That is, in the data analysis our airway model is considered to be a kind of black box with an input - the injected bolus - and an output - the measured dispersion curve. Our results show that this output can be satisfactorily described with eq. (6). This means that whatever the exact nature of the mixing processes in our airway model during H FV may be its net effect for gas transport across the model is identical to that of a diffusion process. For this reason, the results of the curve-fitting may be used, among other things, to compute the conductance of our airway model for gas transport by HFV. The (total) cross-sections of the generations zero through six show only slight mutual differences (Weibel, 1963). Their mean value is approximately equal to that of the cross-section of the zeroth generation. This generation also forms the larger part of our airway model, therefore, we have used the value of the cross-section of the zeroth generation (Ao --- 2.54 c m 2) in the computation Of D D l s p [eq. (6)]. The conductance of the airway model for gas transport by H FV can then be computed from: Cg,,~ = 6.95 10-
DDlspAol
2--
-- |
(11)
where Cg,,~ is expressed in mi (STPD)mmHg- ~. min- ~. Substitution of i, Ao and Vo in eq. (11) then results in: Cg,,~ = 8.66 10- 3DDisp" [ 1 + 0.125 (VT/Vo)2] -0"5
(12)
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
27
Studies in intubated human subjects have shown that normocapnic conditions can be maintained by HFV by using a frequency of 12 Hz and a tidal volume of 1.2 ml per kg body weight (Crawford and Rehder, 1985; Rehder and Didier, 1984). In the experiments of Rehder and Didier (1984) subjects had a mean body weight of 82 kg, therefore, the applied tidal volumes in these subjects will have been close to 100 ml. Substitution of f = 12 and VT = 100 into eq. (8) yields a predicted value for DD~sP equal to 259.2 cm 2. s - ', and substitution of this value into eq. (11) results in a predicted value for Cg..,sequal to 1.86 ml (STPD)mmHg- !. min- !. This result for Cgas corresponds to a CO2 transport across the airway model of 74 ml (STPD)min-' for a partial pressure difference of 40 mmHg. CO2 production in anesthetized subjects weighing 82 kg will be at least twice as much. Thus, our data are incompatible with the in-vivo observations. The predicted value of DD,Sp for f = 12 Hz and VT = 100 ml is out of the range of measured values for DD,SP, therefore, DD~Sp might have been underestimated. For this reason, we have also computed DDiSp from:
DDISP = 50.5 (f/25)2(VT/20) 2
(13)
Using this equation, the extrapolation starts from the largest, measured value of DDlsP ( = 50.5 cm2" sec- ' for f = 25 Hz and VT = 20 mi), and further the largest values of g(= 1) and [3(= 2) shown in table 2 are used. Application of eq. (13) results in DD,sp = 606 cm 2. s e c - ' for f = 12 Hz and VT = 100 ml, which corresponds to a CO2 transport of 173 ml (STPD)min- i for a partial pressure difference of 40 mmHg across the model. It is obvious, that even this larger value of DDISP is still incompatible with the in-vivo observations as the generations 0 through 6 constitute only about one-third of the volume of the conductive airways. We think, therefore, that in normocapnic ventilated subjects longitudinal dispersion cannot possibly be the sole or even the main gas transport mechanism in HFV. Lehr etaL (1985) showed by cinematographic tochniques in excised dog lungs that there were gross phase differences between the tidal movements of the different lung regions in HFV. Further, they found that the sum of regional tidal volumes exceeded the applied oscillatory tidal volume at the trachea thus demonstrating that HFV is accompanied by gas movements among lung regions high-frequency Pendelluft. This is not the only evidence of high-frequency Pendelluft reported in the literature. Other reports dealing with this phenomenon which were published prior to the publication of Lehr et ai. (1985) have been extensively reviewed by Chang (1984). Kaethner et al. (1984)observed that most of the total concentration drop of He and SF 6 during washout in dogs occurred along the endotracheal tube and a few generations of large bronchi. This observation is compatible with our findings of small values for Cg.,.~in the central airways at normal values for f and VT for human subjects, and we agree with Kaethner et al. (1984) that in intubated subjects the instrumental dead space - tube etc. - together with the larger airways will provide the main gas transport resistance during HFV. In summary, we think that during HFV gas tcansport by longitudinal dispersion will be restricted to the upper airways, the trachea - o r the endotracheal tube in intubatcd subjects - and a few generations oflarge bronchi,
28
A.M. VAN DER KOOIJ AND S.C.M. LUIJENDIJK
whereas in the remaining conducting airways gas transport will mainly take place by a different mechanism. This mechanism may be Pendelluft as was already suggested by Chang (1984). In particular the combination of turbulent dispersion and PendeUuft is compatible with further observations on HFV, e.g., the independence of HFV of molecular diffusion.
Conclusion In this study we have validated that gas transport resulting from longitudinal dispersion of gases in the airways during high-frequency ventilation may be described in a way as if it were the result of augmented, longitudinal diffusion. In addition, we have shown that for frequencies and tidal volumes normally used in high-frequency ventilation the effc~.ti~,, di~,permon coefficient is independent of molecular diffusion. The quantitative data further suggest that the importance of longitudinal dispersion for gas transport in high-frequency ventilation is limited to a few generations of large airways.
Appendix in a previous investigation with the same airway model we found that at a constant flow in the model DDtsp was proportional to LV[ with a slight dependence on the direction of the flow, i.e. inspiratory versus expiratory. If we assume that this also applies to HFV then Dr, tsp(t) can be approximated by: m
DD,s,(t) = D=,,s,,(n/2)Icos 2~ftl
(14)
where Dl~lsp is again the mean value of DD,m,(t). Substitution of eq. (14) into eq. (3) yields:
~(t) = 2A~,~,,,m, f',
(7t/2) Lcos 2 n f f l d t
(15)
Further, f r,4 (n/2)Icos 2nftldt - T/4
(16)
)
anu every period of T/4 seconds the value of the integral in eq. (15) increases with this same amount. Thus, to a first approximation ~(t) increases linearly with time so that ~ ( t ) can be written as:
~v2(t) =
2-2AoDl~lsp
[t + A(t)]
(17)
where A(t) represents the deviation. A(t) is a periodic function, and it can easily be shown that IA(t)/tl < 0.053/(ft) - 0.053. T/t. Thus the inaccuracy ofthe linear approximation o f f ( t ) [eq. (5)] decreases rapidly with time, and after two cycles (t > 2T) this inaccuracy has already decreased to about 2%.
References Berdine, G. B., J. L. Lehr and J. M. Dr.~zen (1983). Ethane washout by high frequency (2-40 Hz) low volume (20-100 ml) ventilation (HFV) in excised dog lungs. Fed. Proc. 42: 763.
DISPERSION OF GASES IN HIGH FREQUENCY VENTILATION
29
Bohn, D.J., K. Miyasaka, B. E. Marchak, W. K. Thompson, A. B. Froese and A.C. Bryan (i 980). Ventilation by high-frequency oscillation. J. Appl. Physiol. 48: 710-716. Brown, K.M. and J.E. Dennis (1972). Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximations. Numer. Math. 18: 289-297. Chang, H. K. (1984). Mechanisms of gas transport during ventilation by high-frequency oscillation. J. Appl. Physiol. 56: 553-563. Crawford, M. and K. Rehder (1985). High-frequency small-volume ventilation in anesthetized humans. Anesthesiology 62: 298-304. Horsfield, K. and G. Cumming (1967). Angles of branching and diameters of branches in the human bronchial tree. Bull. Math. Biophys. 29: 245-259. Jaeger, M.J., M. Banner and J. Gallagher (1983). Alveolar ventilation in high frequency studies. Fed. Prec. 42: 1351. Jaeger, M.J., U. H. Kurzweg and M.J. Banner (1984). Transport of gases in high-frequency ventilation. Crit. Care Med. 12: 708-710. Kaethner, T., J. Kohl and P. Scheid (1984). Gas concentration profiles along airways of dog lungs during high-frequency ventilation. J. Appl. Physiol. 56: 1491-1499. Kamm, R. D., J. Collins, J. Whang, A. S. Slutsky and M. Greiner (1984). Gas transport during oscillatory flow in a network of branching tubes. J. Bimnech. Engin. 106: 315-320. Knopp, T.J., T. Kaethner, M. Meyer, K. Rehder and P. Scheid (1983). Gas mixing in the airways of dog lungs during high-frequency ventilation. J. Appl. Physiol. 55: ! 141-1146. Kooij, A. M. van der and S.C.M. Luijendijk (1985). Longitudinal dispersion of gases measured in a model of the bronchial airways. J. Appl. Physiol. 59: 1343-1349. Lehr, J. L., J.P. Butler, P.A. Westerman, S.L. Satz and J.M. Drazen (1985). Photographic measurement of pleurai surface motion during lung oscillation. J. Appl. Physiol. 59: 623-633. Menon, A. S., M. E. Weber and H. K. Chang (1984). Model study of flow dynamics in human central airways. Part !!I: Oscillatory velocity profiles. Respir. Physiol. 55: 255-275. Mitzner, W., S. Permutt and G. Weinmann (1984). Gas transport during high-frequency ventilation: theoretical model and experimental validation. Ann. Biomed. Eng. 12: 407-419. Paloski, W. H., R.B. Slosberg and R.D. Kamm (1987). Effects of gas properties and waveform asymmetry on gas transport in a branching tube network. J. Appl. Physiol. 62: 892-901. Rehder, K. and E.P. Didier (1984). Gas transport and pulmonary perfusion during high-frequency ventilation in humans. J. Appl. Physiol. 57: 1231-1237. Reisfeld, B. and J.S. Ultman (1988). Longitudinal mixing in dog lungs during high-frequency forced flow oscillation. Respir. Physiol. 71: 269-286. Rieke, H., C. Hook and M. Meyer (1983). Pulmonary gas exchange during high.frequency ventilation in dogs. Respir. Physiol. 54: 1-17. Rossing, T. H., A. S. Slutsky, J. L. Lehr, P. A. Drinker, R. Kamm and J. M. Drazen (1981). Tidal volume and frequency dependence of carbon dioxide elimination by high-frequency ventilation. N. Engl. J. bled. 305: 1375-1379. Scherer, P.W. and F.R. Haselton (1982). Convective exchange in oscillatory flow through bronchial-tree models. J. Appl. Physiol. 53: 1023-1033. Slutsky, A. S., J. M. Drazen, R.H. Ingrain Jr, R.D. Kamm, A.H. Shapiro, J.J. Fredberg, S.H. Loring and J. Lehr (1980). Effective pulmonary ventilation with small-volume oscillations at high frequency. Science 209:609-61 !. Slutsky, A. S., R.D. Kamm, T.H. Rossing, S.H. Loring, J. Lehr, A.H. Shapiro, R.H. Ingram Jr and J. M. Drazen (1981). Effects of frequency, tidal volume and lung volume on CO2 elimination in dogs by high frequency (2-30 Hz), low tidal volume ventilation. J. Clin. Invest. 68: 1475-1484. Subramanian, R. S. and W.N. Gill (1976). Unsteady convective diffusion in non-Newtonian flows. Can. J. Chem. Eng. 54: 121-125. Taylor, G. I. (1953). Dispersion of soluble matter in a solvent flowing slowly through a tube. Prec. R. Soc. London. A. Math. Phys. Sci. 219: 186-203. Weibel, E.R. (1963). Morphometry of the human lung. New York: Academic Press. Worth, H. and J. Piiper (1978). Diffusion of helium, carbon monoxide and sulphur hexafluoride in gas mixtures similar to alveolar gas. Respir. Physiol. 32: 155-166.