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A Mathematical model for carbon dioxide exchange during mechanical ventilation with Tracheal Gas Insufflation (TGI)
MATHEMATICAL COMPUTER MODELLING PERGAMON
Mathematical
and
Computer
Modelling
29 (1999)
45-61
A Mathematical Model for Carbon Dioxide Exchange during Mechanical Ventilation with Tracheal Gas Insufflation (TGI) P. S. CROOKE* AND S. HOTA Department of Mathematics, Vanderbilt University Nashville, TN 37240, U.S.A. J. J. MARINI+ AND J. R. HOTCHKISS Division of Pulmonary and Critical Care Medicine, University of Minnesota and Department of Pulmonary and Critical Care Medicine, Regions Hospital 640 Jackson Street, St. Paul, MN 55101, U.S.A. (Received
and accepted
July
1998)
Abstract-A
mathematical model is presented to study the effect of Tracheal Gas Insufflation (TGI) on the alveolar ventilation during passive, mechanical ventilation. Equations are derive that relate the physiologic dead space and alveolar ventilation to the physiologic parameters of the patient (compliance, inspiratory and expiratory resistances, anatomical dead space, cardiac output, and shunting), the ventilator settings (frequency, inspiratory time fraction, and pressure waveform), and the catheter settings (flow rate and flow duration). It is shown that optimal alveolar ventilation exists for certain frequencies of ventilation and these frequencies are dependent on the catheter flow. @ 1999 Elsevier Science Ltd. All rights reserved. Keywords-Mechanical cheal gas insufflation,
ventilation, Carbon dioxide exchange, Alveolar ventilation, Shunt.
Two-compartment
model,
Tra-
NOMENCLATURE catheter flow
VT
tidal volume = V+ + V$
&
VF
tidal volume of compartment u=a,b
QP
pulmonary blood flow
0s
shunted blood flow
fs
fraction of cardiac output that is shunting the lungs (QS = fsG)
in compartment
w
proportioning parameter for cardiac output to lungs
+ as (cardiac
9
perfusion ratio = @/a*
(T
fraction of time that catheter flows,
T
tidal volume ratio = V+/V$
VA
total alveolar ventilation
ti;
alveolar ventilation V=a,b
$
total perfusion = a, output)
Qy
perfusion to compartment
v = a, b
tc = ate
*Supported in part by a subcontract to Vanderbilt University from the Ramsey Foundation. fsupported by NIH SCOR HL50152-01 and the Ramsey Foundation. 0895-7177/99/$
- see front matter @ 1999 Elsevier Science Ltd. All rights reserved
PII: SO895-7177(99)00054-O
Typeset
by d&-T@
46
P.
V&l
total anatomicaf VS,,
v&h,
dead space =
+ VA,”
anatomical dead space assigned to compartment u = a, b
VDPtIYS physiologic dead space volume of catheter flush V2
of V&,
V,”
vcoz P&O,
fractional concentration of CO2 in alveolar gas of compartment v=a,b fractional concentration expired alveolar gas
of CO2
conversion factor, 863 mm Hg slope of CO2 dissociation
VP = BQ&
volume of catheter v,b = (1 - SfQ&
s. CROOKE et Uf.
Bush of V,“,,
curve
y-intercept CuNe
of CO2 dissociation
rate of CO2 elimination
inspiratory
time fraction,
partial pressure of CO2 in alveolar
tilttot
gas
frequency of ventilation
pLo2
partial pressure of CO2 in afveofar gas in compartment v = o, b
compliance tr=i&,b
P WZOZ
partial pressure of CO2 in arterial blood
inspiratory resistance of compartment v=a,b
pressure of CO2 in arterial p:coz partial blood entering compartment u=a,b
P WOZ
partiai pressure of CO2 in endcapillary region
P&O,
partial pressure of COG in mixed expired gas
c QCO2
concentration blood
cvco,
of comp~m~nt
expiratory resistance ment u = a, b
of compart-
airway resistance
*co2
part&I pressure of CO2 in mixed venous blood
GO2
d =
of CO2 in arterial
applied airway pressure end expiratory mentv=a,b inspiratory expiratory
time time
total time of one cycle, ttot = catheter
concentration of CO2 in arterial blood from eomp~ment v
pressure of compart-
&-I-&
flow time
fraction of catheter V,a,,OIs
flow Rushing
concentration of CO2 in mixed venous blood
1. INTRODUCTION Numerous mathematical models for the mechanical ventilation of patients with acute respiratory failure have appeared in the scientific and medical literature (see, e.g., (l-S]). These models have used to predict clinical relevant outcomes such as work of breathing, auto-PEEP, tidal volume, and mean alveolar pressure as functions of physiologic variables (compliance of the lung, inspiratory and expiratory resistances) and clinician-determined inputs (applied airway pressure, ventilatory frequency, inspiratory time). Such models have focused on mechanical outcomes of ventilation, without attending to gas exchange. Pulmonary gas exchange has also been the object of considerable attention in the modeling literature [?,9-121. Much of this work has focused on the elimination of 602, providing models for physiologic dead space with alveolar ventilation as the primary outcome of interest. fn general, these models treat clinically accessible variables (frequency, tidal volume, inspiratory time fraction) as arbitrary inputs. In ,the current paper, a mathematical model is presented which expresses important clinical outcomes (alveolar ~~~~~ physiologic dead space, and alveolar ventilation) as functions of respiratory system char~teristi~s~ pulmonary vascular perfusion, and clinician selected ventilator settings. The interplay between pulmonary perfusion, lung mechanical characteristics, and the selected ventilatory pattern in determining effective CO2 removal is thus highlighted. While the current model is configured to calculate the average values for alveolar Pco, , it can easily be adapted to compute this outcome in a dynamic fashion-anafogous to the commonly-used technique of capnography.
Carbon Dioxide Exchange
In the clinical alveolar during
setting,
pressures,
there
leading
mechanical
has been increasing
to the use of smaller
ventilation
[13-181.
While
awareness tidal
this
47
of the untoward
volumes
approach
and pressure has improved
effects of elevated limited mortality
strategies in some
studies, it limits the clearance of CO2 with potential deleterious consequences (191. A useful adjunctive technique in the setting of a pressure-limited ventilatory strategy is tracheal gas insufflation (TGI), where a small catheter placed near the carina flushes CO2 from the anatomic dead space during ematical modeling of vascular perfused
ventilation [20-271. Relatively little attention has been directed to the mathof TGI, particularly in multicompa~ment models incorporating the effects
perfusion pulmonary
Accurate
on CO2 clearance.
In this paper,
we model
TGI
in a multicompartment,
system.
determination
of alveolar
ventilation
very important for the clinician. Tracheal adjunctive technique [20-271 for mechanical
in cases of acute respiratory
failure is potentially
gas insufflation (TGI) is a potentially an important ventilation in patients with acute lung injury [13-181.
In this technique, fresh gas flowing through a small catheter near the carina flushes the anatomic dead space during expiration, thus decreasing the end-expiratory CO2 in this portion of the dead space so that a higher concentration of fresh gas is taken into the lungs during the next breathe. Using the mathematical model mentioned above that incorporates the clinical set variables for mechanical ventilation, we derive a model that can demonstrate the effects of TGI on alveolar ventilation.
2. THE
MODEL
During mechanical ventilation pressures are applied across the lungs to drive fresh gas into the system; waste gas is allowed to empty passively. Gas exchange occurs only in the alveoli and terminal airways, not in the conducting airways. The nongas exchanging, conducting airways constitute the u~ato~~c dead space; the gas volume cyclically residing within these regions does not contribute to alveolar ventilation. mismatching of alveolar perfusion and ventilation further decreases effective alveolar ventilation; this effect is termed u~~eo~a~ dead space. The sum of alveolar dead space and anatomic dead space is the ~~~s~o~og~c dead space. The elective alveolar uent~Zut~o~ is the quantity of fresh inspired gas that reaches alveolar units having sufficient perfusion to allow effective CO2 exchange between alveolar gas and blood. Wasted ventilation represents that fraction of total ventilation lost in the physiologic dead space and unable to participate in CO:! clearance. Thus, effective alveolar ventilation is the total ventilation minus the wasted ventilation. Assessment of wasted ventilation requires knowledge of the arterial CO2 tension (P,cOz ) and the alveolar CO2 tension (F’A~~, ). In health, these values are generally nearly equal. However, in individuals with abnormal matching between regional ventilation and regional perfusion, the difference between PA~.,~ and PaC02 widens. Furthermore, with regional heterogeneity of perfusion, there is no longer a unique value for ~~~~~ (it will differ between regions). Ventilation-perfusion mismatching is a significant problem in many common disease processes-acute lung injury, emphysema, pulmonary thromboembolism, sepsis, and shock, to name several. This model allows explicit determination of the effects such ventilation-perfusion mismatching has on gas exchange, and does so in the context of inputs (ventilator settings and TGI parameters of flow and timing) amenable
to clinical
manipulation.
Determination of compartments PACT, becomes more complex in the setting of TGI. Here, fresh gas is delivered to the carina by two pathways: via the proximal airway and through a small, deeply positioned catheter. Catheter flow, which is commonly restricted to the expiratory phase, dilutes the CO2 residing within the proximal anatomic dead space (~~s~~ng the dead space), thereby increasing effective alveolar ventilation. Catheter effectiveness is gauged by the increase in effective alveolar ventilation afforded by a given catheter flow.
48
P. S. CROOKE et al.
The pulmonary system is assumed to consist of two compartments a and b which are ventilated and perfused. In Figure 1, a diagram of pulmonary system is presented. The dotted portion denotes the lungs and airways and the solid lines symbolize the flow of blood to and from the lungs.
Q Figure 1. Two-compartmental
lung system.
Many different quantities are used in the modelling. A list of the notation for the quantities can be found at the end of the paper. In general, superscripts are used to denote compartmental quantities and subscripts are used to indicate physiologic tags. Also, V denote volumes, Q! flows, P partial pressures, C compliances, and R resistances. 2.1.
Model
Assumptions
To calculate the COz partial pressure in each compartment
of the lung, the following assump-
tions are made. 1. The (C” 2. The that
pulmonary system is comprised of two compartments, each with its own compliance or Cb), inspiratory and expiratory resistances (RF,Rf , Rz,R$$. two compartments are connected at a branch point that leads to a common airway is assumed to have the same resistance on inspiration and expiration (Raw).
3. The resistances and compliances are assumed to be constants, and the only compartment interaction is assumed to occur at the branch point. 4. During expiration it will be assumed that the pressure at the airway opening is zero (referenced to atmospheric pressure). 5. It is assumed that the pressure changes due to elastic forces in the compartments are directly proportional to the volume changes, and the pressure changes due to resistive forces are directly proportional to the changes in the flow. 6. At time t = 0, an arbitrary pressure Paw(t) is applied to the opening of the common airway by the ventilator. This pressure is applied to the airway opening over a time interval, 0 5 t 5 ti, which is the inspiratory part of the breathing cycle. At time t = ti,
49
Carbon Dioxide Exchange
the applied airway pressure is released and expiration takes place passively during the time interval, ti 5 t 5 ttot. 7. The anatomic dead spaces are parameters that are characteristic of the patient and hence, are inputs to the model. They can be simply functions of the tidal volumes for each compartment. The total anatomical dead space gas is shared in common by the two compartments so that the portion allotted to compartment a is VE,, and to compartment b, Vi,, . These volumes are assumed to be inputs to the model. The refinement of anatomic dead spaces for each compartment is built into the model; however, determining values for I%,” and Vi,, is very difficult and for most applications of the model, one would choose v&, = VA,, = VDan 12. 8. The flow of fresh gas that washes the anatomical dead space of each compartment are inputs to the model and are denoted by & and Q”, so that the total flow of the catheter is 0, = 0: + @. The catheter flows only during the end of the expiratory part of the breathing cycle and the time of flow is denoted by t,. The volume of fresh gas delivered to each anatomical dead space is then V: = C& = sQ,t, and V,” = &itc = (1 - s)Q,t, where s is a input parameter such that 0 5 s 5 1. The effect of the catheter is to flush the anatomical dead space of each compartment so that the partial pressure of CO2 in the anatomical dead space is proportional to the partial pressures of the alveolar gas from each compartment so that
PDanc02 = (,!$;)CL,
+ (&y&J p;,.,
.
As in the previous assumption, it would be difficult to assign catheter flows to each compartment without a detailed picture of the anatomical structure. A simpler approach is to assume that 0: = 6): = Q,/2. However, we incorporate this more general setting for the model. 9. In each compartment, the pressure of CO2 in the alveolar gas is in equilibrium with the partial pressure of CO2 in the arterial blood, i.e., Pi,_, = P,“,, for v = a, b. This 2 2 requirement can be and will be relaxed. 10. The arterial concentration of CO2 is a weighted average of the arterial concentrations of CO2 from each compartment, i.e.,
11. There is no CO2 in the ambient air. 12. The CO2 dissociation curve is a straight line and will be assumed to be of the form: Go, = kPcoz + m, where k and m are constants. 13. A portion of the cardiac output is shunted around the two compartments. In particular, the cardiac output is divided according the equation G = 0, + OS, where OS denotes the shunt blood flow. 2.2. Two-Compartment
Ventilation
Model
For the mechanical ventilation part of the model, we use the model developed in [2]. This is a general mathematical model for the dynamic behavior of a two-compartment respiratory system in response to an arbitrarily applied inspiratory pressure. Using clinical set variables such as frequency, inspiratory time fraction, and physiologic parameters for resistance and compliance, outcome variables such as tidal volume, end-expiratory pressure, mean alveolar pressure can be computed for each compartment. Since the explicit formulas for these outcome ventilation variables are quite complex, the reader is referred to the original paper.
P. S. CRO~KE
50
2.3. Carbon
Dioxide
Transport
For the gas exchange part of the model, Initially,
et al.
we use a modi~cation
of a model
proposed
in (lo].
we will assume that the shunt flow is zero, i.e., OS = 0, i.e., &, = 0. In the last part of
this section,
this assumption
will be relaxed.
The starting point for this part of the model is to the conservation of COs, i.e., the amount of carbon dioxide that leaves the lungs is equal to the amount of carbon dioxide that is leaving the blood. Considering this process for each compartment, compartment a into the airway is ~,J’,Y& and the amount - C&,,
blood is &(CV~~,
). For compartment
the amount of CO2 expelled from of CO2 entering compartment a from
a we have the conservation
law
A similar identity is obtain form compartment b. For gas exchange, it is convenient to deal with the partial pressure of a gas rather than its fractional concentration. We assume that the partial pressure of a gas is proportional to fractional concentration. In particular, the partial pressure of alveolar CO:! gas is related its fractional content by the relation: P_&,, = cyFJ&, where rr is the BTPS-STPD conversion factor and is equal to 863 mm Hg when alveolar ventilation is expressed in milliliters per minute. The net change of carbon dioxide in a compartment due to ventilation is the amount of carbon dioxide leaving the compartment minus the amount entering the compartment. For compartment a, the alveolar ventilation is expressed mathematic~y as pz = f(V+ - VE,,). Using this relation in (2.1) for ~ompa~ment a, we find
(2.2) Using our assumption C&oz = “P&0,
about
the shape of the CO2 dissociation
curve Cv,ol
= kA/,oz
+ m and
+ m, (2.2) becomes
w (pvco, - p&J = 6(VTa - Vi,“) P&,,.
(2.3)
Next, we incorporate the effect of the catheter on the portion of the total anatomical dead space for compartment a. Namely, if Vca is the volume of fresh gas delivered by the catheter to compartment a, VT is the tidal volume, V$! is the portion of the tidal volume that is entering compartment a, and V, = act, is the total volume of fresh gas delivered by the catheter to the total anatomical dead space, then effective anatomical dead space of compartment a is given by
If
v, > VDanf then
to a reduction
the excess fresh gas is vented to the atmosphere in the CO2 in the anatomical dead space.
and effectively
does not lead
At this point, we make a remark about possible future directions for the modeling. The dynamics of expiration, at least for passive ventilation, are known from previous modeling [2]. In particuIar, the instantaneous volume and flow from each compartment are known over the complete expiratory part of the breathing cycle. In interaction between the exiting alveolar gas and entering fresh gas could be modeled. The instantaneous partial pressure of carbon dioxide could then be calculated. The direction for modeling has not been pursued in this paper because of the complexity.
Carbon
Dioxide
51
Exchange
The total partial pressure of carbon dioxide in alveolar gas exhaled (PACT,) is assumed to be equal to the weighted mean of carbon dioxide pressures of two compartments with the weights being the fractions of the tidal volume entering or leaving each compartment.
PACO,=
That is,
(g)pko2 +(gpJLo2.
(2.5)
Similarly, partial pressure of carbon dioxide in arterial blood (Paco2 ) is the mean of carbon dioxide pressure of two compartments weighted by fraction of perfusion
(2.6) At this point, we remark that VD~,, is not necessarily independent of the tidal volume VT and we have assumed that partial pressure of CO:! in the anatomic dead space in each compartment is equal to the partial pressure of CO2 in the alveolar gas of that compartment. Using (2.4)-(2.6), (2.3) b ecomes for compartment a
(2.7)
A similar equation can be computed for compartment b
(
f
fV:! kQbPvcoz = T+kcjb-
(v$+ (1 - &ctc)
pb
@vDa”
ACOz a (VT + Q&J2 (2.8)
f (v:! + (1 - $&)
v+vD,,
It is convenient to define some intermediate symbols that reduce the complexity of (2.7) and (2.8). We define
f
fVTa
hr = T+kf+
(v$+S&t,)
V+VDa,, 2
Q!
A2 =
Aa =
v;vD,, 2
cy
f(V,” + dctc)
v$D,, 2
(
h
+ &&
’
v;vD,, 2
(
I
(2.10)
(2.11)
)
f (v; + (1 - &ctc) o’
(2.9)
’
)
VT”+ (1 - &ctc)
fV$
-+k&“-f(
a
A‘j =
(
VT + &&
VT + &tc
)
(2.12)
52
Using the notation of (2.9)-(2.12) in (2.7) and (2.8), we obtain two linear, inhomogeneous equaand Pi,,
tions in the unknowns, P&,
2
2
and P&,,
As - P&,,
in (2.13),(2.14),
A4 =
(2.14)
kQbfic.02.
we have the following expressions for the partial
Solving for P&,
and Pj,,
pressure of carboi
dioxide it the alveolar gas of each compartment:
(2.15)
and (2.16) Substituting (2.15) and (2.16) in (2.7) and (2.8) and using the notation (2.9)-(2.12),
we obtain
expressions for PA,__, and PaGo
PVC,,
(2.17)
and (2.18)
2.4. Physiologic Dead Space The physiologic dead space is defined as the sum of the anatomical dead space and the alveolar dead space. The anatomical dead space is the volume of inspired air that does not reach of the alveoli. The alveolar dead space is the volume of air that reach the alveoli, but does not undergo gas exchange. Hence, the physiologic dead space represents a portion of the tidal volume that is wasted. The total physiologic dead space, VD~,,,_,can be derived using the Enghoff modification of Bohr equation. This equation expresses the total physiologic dead space as a fraction of the tidal volume (2.19) where PE__.,~is the partial pressure of the expired carbon dioxide. can be written as dead space fraction, VD,,/VT, _
pb
vD&s
I=:
Therefore, the physiologic
(2.20)
VT
Let PC*, denote the carbon dioxide minute ventilation. Therefore, I&oz/f is the total carbon dioxide output in one respiratory cycle. This quantity is related to the minute ventilation by the relation: p&J2 = ~EF&~~. Since &,,, = &ECo,, we have the relatively simple relation for PEW,
53
Carbon Dioxide Exchange
P&O, =.
QVCOa VE
(2.21)
+&dc>
VD,, VT Using (2.21)
PA co2
in (2.20), we find the following expression for the physiologic dead space fraction:
(2.22)
Using (2.17) and (2.18) in (2.22), we obtain the a relatively simple expression for the physiologic dead space fraction
where
2.5.
Alveolar
Ventilation
The alveolar ventilation is the quantity of inspired air reaching alveoli perfused by blood and thus taking part in gas exchange. Therefore, it is defined as the difference of minute ventilation, I& = fI+ and physiologic dead space ventilation, V&h,. = fVDphys. Using the expression obtained for physiologic dead space we can obtain an explicit expression for alveolar ventilation
as
(2.24)
This last identity of (2.24) is an explicit expression for the alveolar ventilation in terms of the clinically set and physiologic parameters. 2.6. Shunting In the above model, we have assumed that the total output from the heart is exposed to the lungs. It is quite easy to modify the model to include a shunting component, i.e., having a certain fraction of the venous blood to bypass the lungs and directly mix with the arterial blood. Let 0;
denote the pulmonary blood flow into compartment a and C$, into compartment
b so that
the total pulmonary flow flow is 0, = o,B + @,. Let o!s denote the flow of blood that is shunted by the lungs so that the cardiac output is given by 0 = 0, + OS. To simplify the notation, we
P. S.
54
CMOKE
et 41.
introduce a ~~~~~~~~~~ ~~~~~~ W, 0 5 w < I, so that @ = woP and @E = (1 - LJ&&_ Let E&BCDzdenote the partid pressure of CO2 in the end-capillary zone of compartment a and P&P,,2 for compartment
b. Alveolar, end-capillary, and arterial partial pressure of CO2 with
shunting are given by the following equations: (2.25) f2.26)
where fS is the fraction of the total cardiac output that is shunted, i.e., QS = f.&, 0 < fS 2 1. The steady state equation for CO2 in compartment a with shunting is then given by
= and P&,,, b. Letting P$&, 2 = x’P~,,~ 1 where 0 < x’, y’ < 1 and using (2.6) for PDF,, the follow&g equations relate P.4cn2 in
In a similar equation, we get for compartment
Y~p~co, ’
each compartment: (2.28)
(2.29)
where
(2.30)
(2‘31)
Thus the partial pressures, P&,,
and Pjc,,
, in presence of shunt&
are
(2.34)
(2.35)
Carbon Dioxide Exchange
Substituting
55
in (2.6) and (2.35), we obtain expressions for the end-capillary
(2.35)-(2.6)
and
alveolar CO:! partial pressures
From (2.6), the arterial CO2 tension can be computed
-I”
Having
Ml
4;
- fJPvco2 f
(@qb.
+ @k&z,)
9,
1
+ 4;
(@a,*b.
-
(@%,
f
Q;@b,)
%s@b,)
I.
(2.38)
the physiologic dead space and alveolar ventilation can be computed from (2.19)
J’aco2,
and (2.24).
3. SIMULATIONS Having explicit expressions for the physiologic dead space and alveolar ventilation in terms of the physiologic parameters and the clinical set variables, simulations are possible that show the effect on these two important outcomes in terms of the clinical inputs. Since the physiologic dead space is related directly to the alveolar ventilation by (2.24), we restrict our simulations to the alveolar ventilation outcome. Typical values of the physiologic parameters have been chosen for the simulation. In Figure 2, the alveolar ventilation as a function of frequency is show for different catheter flow rates (GJ and perfusion ratios (q = &a/Q). Th e catheter flow rates are given in liters per minute. As can be seen in the graphs, there is an optimal frequency of ventilation for each catheter flow rate. It appears that this optimal frequency increases with increasing catheter flow.
VA
VA
8
Q
6
-0
4
_...... 5
2
.--_ 20
40
60
f
10
20
40
60
f
Figure 2. Alveolar ventilation (L/min) versus frequency (l/ min) for different perfusion ratios and catheter flow rates. Parameters: R,” = 25cm HzO/L/s, Rt = iOcm HzO/L/s, RE = 40cm HzO/L/s, Rt = 20cm HzO/L,fs, R,, = 6cm HzO/ L/s, Ca = O.O2L/cm H20, Cb = O.O3L/cm H20, Pset = 20cm H20, VD,, = 0.25 L, a = 4.5 L/min, H20.
a8 = 0 L/min,
cy = 1208 cm H20, and k = 0.0032 ml/cm
In Figure 3, the alveolar ventilation is plotted as a function of the inspiratory time fraction for different catheter flow rates and perfusion ratios. As can be seen, there appears to be a optimal inspiratory time fraction. The maximal alveolar ventilation that is achieved using the TGI is affected by the perfusion ratio (q). This is expected because of the asymmetry of the compliances and resistances.
P. S. CROOKE et at.
56
VA
VA
6
..-...
s
____ f()
4
4
-..._ 5
2
2
____ fO
0.25
0.5
i
d
0.75
0.25
Figure 3. Alveolar ventilation (L/min) versus inspiratory ent perfusion ratios and catheter flow rates. Parameters:
time fraction RF =
25cm
0.5
0.75
d
for differHzO/L/s,
Rt = 1Ocm I&O/L/s, Rz = 40cm HzO/L/s, Rt = 20cm HsO/L/s, R,, =I. 6cm HzO/L/s, Ca = O.O2L/cm HsO, Cb = O.O3L/cm HeO, Pset = 20cm HeO, V& = 0.25 L, 4 = 4.5 L/min, H20.
ss
= 0 L/min,
CY= 1208 cm HsO, and k = 0.0032 ml/cm
The alveolar ventilation as functions of frequency and inspiratory time fraction as functions of different flow times of the catheter (750/o, 50%, and 25% of the expiratory part of the breathing cycle) have been computed. For the physiologic parameters chosen above, there is very change in the alveolar ventilation. A measure of the ventilation-perfusion unevenness is the arterial-Alveolar CO2 dijference (a - WC02 The relative change is %(a - A)DC02
= &e,
-
(3.1)
pAcol.
is defined as
%(a - A)DC02
=
P
ace;
- pAcoz
.
WC+
(3.2)
In Figures 4 and 5, the absolute and relative change in the arterial-Alveolar difference is plotted as a functions of frequency and inspiratory time fraction. (a-A)DCOz
(a-A)DCOz
Qc
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 20
40
60
f ’
0.25
0.5
0.75
d
Figure 4. (a - A) Pco,, (cm H20) versus f (I/min) and d for different catheter flow rates. Parameters: Rr = 25, Rib = IOcm H20/L/s, Rz = 40cm HzO/L/s, Co = 0.02 L/cm HsO, C6 = 0.03 L/cm R: = 20cm HsO/L/s, &, = 6 cm HzO/L/s, e. = 0 L/min, a = 1208cm H20,Pset = 20 cm Hs0, VB,, = 0.25L, 4 = 4.5L/min, H20, and k = O.~32ml/cm H20. %(a-A)DCOz
/
6
I
/
I
%(a-?DC02
,
I
/
I
I
I
iq=o.51
/
1
4 2 20
40
60
f
0.25
0.5
Figure 5. Percentage change in (a - A) Pcoa versus f (l/ min) and D for different catheter flow rates. Parameters: Rio = 25cm HzO/L/s, Ri = 1Ocm H2O/L/s, Rz = 40cm HaO/L/s, h!: = 20cm HsO/L/s, Raw = 6cm HsO/L/s, Ca = O.O2L/cm HsO, Cb = O.O3L/cm HsO, Pset = 20cm HsO, VD,, = 0.25L, 0 = 4.5L/min, OS = OL/min,
a! = 1208cm HsO, and k = O.O032ml/cm
HsO.
0.75
Carbon Dioxide Exchange Other
modes
same physiologic
of ventilation p~ameters
of ventilation,
the outcome
time fraction
are presented. %(a-A)DCOz
can be used with the model. as in the previous variable
simulation,
%(a - A)DCOz
57 For example,
in Figure
but using a accelerating
as functions
of frequency
6, using the pressure
mode
and inspiratory
%(a-A)DCOz
Figure 6. Percentage change in (a - A) PCO, versus f (I/ min) and D for different catheter flow rates. Parameters: RT = 25cm HzO/L/s, R$ = 1Ocm H20/L/s, Rz = 40 cm H20/Ljs, R: = 20 cm H20/L/s, Raw = 6 cm H20/L/s, C” = 0.02 L/cm H20, Cb = 0.03 L/cm H20, P,,(L) = l.S(t/ti) + 5cm H20, ~~~~ = 0.25L, 4 = 4.5 L/min, Gs = 0 L/min, o = 1208 cm J&O, and k = 0.003 ml/cm H20.
I
VA
60 Figure 7. Alveolar ventilation (L/min) as a function of f (l/min) and d fraction. Parameters: R4 = 25cm HzO/L/s, R,” = 1Ocm HzO/Lfs, Rz = 40cm HzO/L/s, Rk = 20cm H20/L/s, Raw = 6cm HzO/L/s, C” = O.O2L/cm H20, Cb = 0.03 L/cm H20, Pset = 20cm H20, V,,, =0.25L,Cj=4.5, &=O, &==5, q= 0.5, D = 0.75, a = 1208 cm H20, and k = 0.0032 ml/cm H20.
Other simulations alveolar ventilation
One can vary the shunting and observe the effect on the are possible. for different catheter flows. It was observed from these simulations that
relatively small shunting (O-10%) did not produce significant changes in the alveolar ventilation as functions of the frequency or inspiratory time fraction. Finally, the alveolar ventilation is a function of several clinical set variables and physiologic parameters. The behavior of this key outcome variable is can be viewed as surfaces in higher
58
P. S. CROOKE et al.
dimensional spaces.
In Figure 7, the alveolar ventilation as a function of both frequency and
inspiratory time fraction for Qc = 5 is plotted.
4. DISCUSSION The model developed in this paper predicts clinically important outcomes as functions of respiratory system mechanical and perfusion characteristics and clinically accessible inputs (ventilator and TGI settings). This model is a composite of a general ~~comp~tment chanical ventilation and a tw~compartment
model of passive me-
model of pulmonary CO2 exchange which includes
the possibilities of nonhomogeneous perfusion and ventilation. As such, it represents a significant advance over previous models of TGI. The model suggests several patterns of behavior which are of clinical importance, and allows quantization of their significance. The maximal differences in alveolar ventilation are of modest magnitude (O-l L/min).
However, changes in alveolar ventila-
tion of this magnitude may well be of clinical significance in the gravely ill, hypercapnic patient who is being ventilated with a low tidal volume, pressure limited strategy-the population in whom TGI is most often employed.
4.1.Critique
of the Model
Several features of the model are subject to criticism. Foremost among such concerns is that the model has not been validated in vivo. We have previously validated two components of the model (two compa~ment models of mechanical ventilation and the model for anatomic dead space flushing) in a mechanical lung analog [3,28]. In contrast, the diseased lung is composed of many interdependent compartments, any of which may depart from linear behavior, which are arranged in a geometric complex structure. Rigorous in vivo validation of the model would require the fine scale measurement of regional alveolar pressures and small airway resistances, as well as local vascular perfusion characteristics. Acquisition of such information would be problematic. The independent two-compartment airspace configuration we modeled is clearly a gross oversimplification of the multiple interactive compartments comprising the diseased lung. However, there is data that such a model may have clinical utility [29-321. Similarly, while the perfusion model (two perfused compartments plus shunt) is an oversimplification, it represents the least complex perfusion analysis applicable to a two compartment model. The clarity of a model including two perfused compartments plus shunt is the reason that it is commonly used for educational purposes. Because it allows for heterogeneity of airspace and vascular factors, the current model represents a signi~c~t
advance over previous models of TGI.
We have assumed linear behavior in our mechanics model, and have not incorporated the effects of nonlinearity of resistance or compliance, or the effects of compartmental interaction. While subject to criticism, this approximation is likely not unreasonable for two reasons. First, the work of Bake, Rossi, and Shardonofsky and their respective colleagues suggests that constancy of impedance parameters may be an acceptable initial approximation for respiratory system behavior [29,33,34]. Furthermore, our previous work in this area indicates that inclusion of plausible nonlinear behaviors does not affect the qualitative behavior of ventilatory distribution or pressure distribution-the major mechanics outcomes of consequence in the current model [3]. Similarly, we have not considered the potential viscoelastic behavior of the pulmonary parenchyma. Viscoelastic behavior, in which the pulmonary parenchyma displays timedependent, non-Newtonian behavior (such as stress relaxation), has been recently and elegantly addressed 135,361. Such behavior is clearly important from the standpoint of mechanics, and likely will affect ventilatory distribution. However, the inclusion of viscoelastic behavior in the current two compartment model would require knowledge of regional variation in viscoelastic time constants. The effects of regional viscoelastic behavior may either augment or attenuate the distributional effects of Newtonian mechanics, depending on the magnitude and alignment of viscoelastic time
Carbon Dioxide Exchange constant
heterogeneity.
Detailed
ingiy, we have not addressed 4.2.
Clinical
information
this complex
is not available issue in the current
59
regarding
these questions;
aceord-
model.
Implications
The model predicts that there is an optimal ventilatory frequency, which maximizes alveolar ventilation, for a given set of respiratory system mechanical and perfusion characteristics. As noted, the optimal frequency rises as TGI flow is increased, in accord with our previous findings in a unicompartmental
model of TGI
the slope of I& ws frequency optimization optimal
may be of clinical
[28]. This optimal
significance.
value is likely due to the balance
is flushed each minute
value occurs within
(f) is quite steep in this neighborhood,
and the decreasing
As detailed between efficiency
in our previous
the number
the clinical
suggesting
that
cycfe as frequency
and
frequency-
work, the presence
of times the anatomic
of each flushing
range,
of an
dead space increases.
Similarly, the model predicts the existence of an optimal inspiratory time fraction for a given set of associated inputs. Again, the optimal value arises within the clinical range, and may be associated with clinically significant increases in alveolar ventilation. The presence of an optimal inspiratory time fraction arises from the interplay between allowing adequate expiratory time for complete flushing of the anatomic dead space and minimizing end expiratory residual flow from the alveolar compartments into the anatomic dead space. The presence of residual endexpiratory flow (and its detrimental effect on TGI efficacy) become more important as expiratory resistance increases (data not shown). Thus, the model would suggest that TGI would not be as effective in diseases characterized by high expiratory resistances, such as emphysema or asthma. An identical effect can be seen as compartmental mechanical heterogeneity (expiratory time constants) increases (data not shown). The inclusion of pulmonary perfusion and a shunt pathway allowed consideration of the consequences of nonhomogeneous pulmonary blood flow and shunt as regards alveolar ventilation. Shunt had little effect, whereas asymmetry of perfusion significantly reduced the effectiveness of TGI. This is of considerable importance, as many disease processes (shock, sepsis, emphysema, pulmonary embol~m) may lead to derange distribution of pulmonary perfusion, decreasing the potential benefits of TGI. Large-scale maldistribution of pulmonary perfusion, as reflected in the model, theoretically may be improved by changing the position of the patient (i.e., prone or lateral positioning). Small scale (regional) maidistribution of perfusion may require pharmacologic intervention. In summary, we present a two-compartment model for TGI which includes heterogeneity of pulmonary ventilation and perfusion. While obviously an oversimplification of clinical reality, the model suggests that clinically relevant changes in ventilator settings may lead to clinically significant increases in TGI effectiveness. Certain patient characteristics (high expiratory resistance, mismatching of ventilation and perfusion, or compartmental mechanical heterogeneity) may render TGI less efficacious. Furthermore, the model provides a theoretical framework relating patient characteristics, ventilator settings, and clinically important outcomes. Such a framework may offer the opportunity to develop rational optimization algorithms for the use of TGI. Reconfiguration of the model to interpret dynamic information, pa~icularly regarding the end-expiratory acquire detailed information regarding expired PC-J-,, , may allow the clinician to noninvasively ventilation-perfusion matching from the capnogram.
5. CONCLUSIONS In this paper, we have derived one of the first mathematical models for tracheal gas insufflation that permits the input of physiologic parameters (resistances, compliances, shunt, perfusion ratio, etc.) and clinical set variables (frequency, inspiratory time fraction, mode of ventilation) to produce key ventilation outcomes such as alveolar ventilation. The model is a synthesis of a general, two-compartment, mechanical ventilation model with a sophisticated two-alveoli CO2
P. S. CROOKE et al.
60
exchange
modeI.
The resulting
shows that there are optimal fraction,
to produce
TGI
model
demonstrates
the effects of the catheter
settings for the clinical variables,
the best alveolar ventilation
frequency
and/or
flow and
inspiratory
time
for the patient.
REFERENCES 1. P.S. Crooke and J.J. Marini, A nonlinear mathematical model of pressure preset ventilation: Description and limiting values for key outcome variables, Math. Mod. Meth. Appt. Ski. 3, 839-859, (1993). 2. P.S. Crooke, J.D. Head and J.J. Marini, A general tw~comp~tment model for mecbanicd ventilation, lathe. Cornput. ~odell~~g 24 (7), I-18, (1996). 3. J.R. Hot&kiss et al., Implications of a bipbaic two-comp~ment model of constant flow ventilation for the clinical setting, J. Grit. Care 9, 114-123, (1994). 4. V.K. Jain et al., Models of the lung, Brit. J. Anaesth. 43, 816-817, (1971). 5. J.J. Marini, P.S. Crooke and J.D. Truwit, Determinants and limits of pressure-preset ventilation: A mathematical model of pressure control, J. Appl. f%ySiOk 67, 1081-1092, (1989). 6. U. Morgenstern and S. Kaiser, Mathematical modelling of ventilation mechanics, Int. J. Clin. Monit. Corn@. 12, 105-112, (1995). 7. M.H. Reid anti H.B. Hechtman, A multicompartment analysis of the lung, Med. Biol. Eng. 12, 405-414, (1974). 8. A.A. Wald, T.W. Murphy and V.D. Mazzia, A theoretical study of controlled ventilation, IEEE 74ans. Biomed. Eng. 15, 237-248, (1968). 9. M. Sharan, S. Selvakumar and M.P. Singh, Mathematics model for the computation of alveolar partial pressure of carbon monoxide, Int. J. Biomed. Camp. 26, 135-147, (1990). 10. K. Suwa and H.H. Benedixen, A mathematical analysis of physiological dead-space in a lung model, J. Appi. Physiol. 24, 549-555, (1968). 11. K. Suwa and H.H. Benedixen, Change in PacO, with mechanical dead space during artificial ventilation, J. Appl. Physiol. 24, 556-561, (1968). 12. J.B. West, Ventilation-perfusion inequality and overall gas exchange in computer models of the lung, Reap, Physiol. 7, 88-110, (1969). 13. J.J. Marini, Pressure-targeted, lung-protective ventilatory support in acute lung injury, Chest 105, 109S115S, (1994). 14. J.C. Parker et al., Lung edema caused by high peak inspiratory pressure in dogs, Am. Rev. Respir. Dis. X42, 321-328, (1990). 15. K.J. Peevy, L.A. Hernandez, A.A. Moise and J.C. Parker, Barotrauma and microvascular injury in lungs of nonadult rabbits: Effect of ventilation pattern, Crit. Care Med. 18, 634-637, (1990). 16. A.E. Taylor, K. Rehder, R.E. Hyatt and J.C. Parker, Clinical Respiratory Physiology, W. B. Saunders Company, (1989). 17. F.E. Udwadia, Diagnosis and Management of Acute Respiratory Failure, Oxford University Press, Oxford, (1979). 18. J.B. West, Respiratory Physiology, Williams and Wilkins Company, Baltimore, MD, (1985). 19. M.B.P. Amato et al., Beneficial effects of the open lung approach with low distending pressures in ARDS: A prospective randomized study on mechanical ventilation, Am. J. Respir. Grit. Cure Med. 152, 1835-1846, (1995). 20. W.C. Burke et al., Modes of tracheal gas insufflation: Comparison of continuous and phase-specific gas injection in normal dogs, Am. Rev. Respir. Dis. 148, 562-568, (1993). 21. N. Gaviely, D. Larsson and J.B. Grotberg, Gas exchange by tracheal insufflation in a ventilatory failure dog model, J. Clin. Invest. 90, 2376-2383, (1992). 22. A. Hurewitz, E. Bergofsky and E. Vomero, Airway insufflation: Increasing flow rates progressively reduce dead spsce in respiration failure, Am. Rev. Respir. Dis. 144, 1229-1233, (1991). 23. A. Larsen, Elimination of apparatus dead space-a simple method for improving CO2 removal with increasing airway pressure, Acta Anaesthesiol. &and 36, 796-799, (1992). 24. A. Nahum et al., Lung mechanisms and gas exchange during pressure-control ventilation in dogs: Augmentation of CO1 elimination by an intratracheal catheter, Am. Rev. Respir. Dis. 146, 965-973, (1992). 25. A. Nahum et al., Effect of catheter flow direction on COz removal during tracheal gas insufflation in dogs, J. Appl. Physiol. 75, 1238-1246, (1993). 26. G. Nakos et al., Respiratory effects of tracheal gas insulation in spontaneousIy breathing COPD patients, Intensive Care Med. 21, 904-912, (1995). 27. S.A. Ravenscraft et al., Tracheal gas insufflation: Catheter effectiveness determined by expiratory flush volume, Am. J. Respir. Grit. Care Med. 153, 1817-1824, (1996). 28. J.R. Hotchkiss, P.S. Crooke and J.J. Marini, Theoretical interactions between ventilator settings and proximal dead space ventilation during tracheal gas insufflation, Intensive Care Med. 22, 1112-1119, (1996). 29. D. Bake, L. Wood, B. Murphy et al., Effect of inspiratory flow rate on regional distribution of inspired gas, J. Appl. Physiol. 37, 8-17, (1974). 30. J.H. Batea, M. Decramer, D. Chartrand et al., Volume-time profile during relaxed expiration in the normal dog, J. Appl. Physiot. 59, 732-737, (1985).
Carbon Dioxide Exchange
61
31. G.L. Chelucci, F. Brunet, J. Dall’Ava-Santucci et al., A single compartment model cannot describe expiration in intubated, paralyzed humans, Eur. Resp. J. 4, 458-464, (1991). 32. T. Conolly, B. Bake, L. Wood et a$., Regional distribution of a 133Xe labelled gas volume inspired at constant flow rates, &and. .I. Respir. Dis. 56, 150-159, (1975). 33. A. Rossi, S.B. Gottfried, B.D. Higgs et al., Respiratory mechanics in mechanically ventilated patients with respiratory failure, J. Appl. Physiol. 58, 1849-1858, (1985). 34. F.R. Shardonofsky, J. Sato and J.H. Bates, Quasi-static pressure-volume hysteresis in the canine respiratory system in vivo, J. Appl. Physiol. 68, 2230-2236, (1990). 35. B. Jonson, L. Beydon, K. Brauer et al., Mechanics of the respiratory system in healthy anesthetized humans with emphasis on viscoelastic properties, I. Appl. Physiol. 75, 132-140, (1993). 36. L. Beydon, C. Svantesson, K. Brauer et al., Respiratory mechanics in patients ventilated for critical lung disease, Eur. Resp. J. 9, 262-273, (1996). 37. P.F. Crossman, L.S. Bushnell and J. Hedley-White, Dead space during artificial ventilation: Gas compression and mechanical dead space, J. Appl. Physiol. 28, 94-97, (1970). 38. S. Damanto and G. Cumming, Tidal volume change and gas mixing in the lung, Respiration 56, 173-181, (1989). 39. J.W. Evans, D.G. Cantor and J.R. Norman, The dead space in a compartmental lung model, Sull. Math. i3i0phy. 29, 711-718, (1967). 40. J.W. Evans, P.D. Wagner and J.B. West, Conditions for reduction of pulmonary gas transfer by ventilationperfusion inequality, .I. App~. P~ys~ol. 36, 533-537, (1974). 41. R. Fletcher, B. Johnson, G. Cummings and J. Brewa, The concept of dead space with special reference to the single breath test for carbon dioxide, B&&h J. Anaesthesia 53, 77-88, (1981). 42. H.V. Forster, L.G. Pan, G.E. Bisgard, C. Flynn and R.E. Hoffer, Effect of reducing anatomical dead space on arterial Pcoz during CO2 inhalation, I. Appl. Physiol. 61, 728-733, (1986). 43. G. Hedenstierna, The anatomical and alveolar dead space during respirator treatment. Influence of respiratory frequency, minute volume and tracheal pressure, British J. Anaesthesia 47, 993-999, (1975). 44. G.F. Filley, D.B. Bigelow, D.E. Olson and L.M. Lacquet, Pulmonary gas transport, Am. Rev. Resp. Dis. 98, 480-489, (1969). 45. K. Holk, S.G. Einarsson, K.L. Stevensson, J.P. Bengtson and 0. Stenqvist, Continuous non-invasive monitoring of energy expenditure, oxygen consumption and alveolar ventilation during controlled ventilation: Validation in an oxygen consuming lung model, Acta Anaesthes. &and. 40 (5), 530-537, (1996). 46. V.K. Jain and SK. Guha, A study of intermittent positive pressure ventilation, Med. Biol. Eng. 8, 575-583, (1970). 47. P.M. Kibby, A mathematical model for CO2 exchange during the initial stages of rebreathing, Resp. Physiol. 3, 243-255, (1967). 48. G. Nakos et aI., Tracheal gas insufflation reduces the tidal volume while P,,_o, is maintained constant, Intensive Care Med. 20, 407-413, (1994). 49. M.I. Noble, F. Langley, M. Buckman, P. Vernon, A. Seed, R. Jewkes, A. Bisdee and G. Cumming, Comparison of ventilation/perfusion lung-imaging and dead-space measurements in airway disease, C&n. Sci. 60, 17-23, (1981). 50. E.B. Olson, Physiological dead space increases during initial hours of chronic hypoxemia with or without hypocapnia, .I. Appl. Physiol. 77, 1526-1531, (1994). 51. V. Taskar et al., Dynamics of carbon dioxide elimination following ventilator resetting, Chest. 108, 196-202, (1995). 52. G.G. Weinmann, W. Mitsner and S. Permutt, Physiological dead space during high-frequency ventilation in dogs, J. Appl. Physiol. 57 (3), 881-887, (1984). 53. R.L. Wilkins and J.R. Dexter, Respiratory Disease. Principles of Patient Care, F.A. Davis Company, Philadelphia, PA, (1993).
Mathematical
and
Computer
Modelling
29 (1999)
45-61
A Mathematical Model for Carbon Dioxide Exchange during Mechanical Ventilation with Tracheal Gas Insufflation (TGI) P. S. CROOKE* AND S. HOTA Department of Mathematics, Vanderbilt University Nashville, TN 37240, U.S.A. J. J. MARINI+ AND J. R. HOTCHKISS Division of Pulmonary and Critical Care Medicine, University of Minnesota and Department of Pulmonary and Critical Care Medicine, Regions Hospital 640 Jackson Street, St. Paul, MN 55101, U.S.A. (Received
and accepted
July
1998)
Abstract-A
mathematical model is presented to study the effect of Tracheal Gas Insufflation (TGI) on the alveolar ventilation during passive, mechanical ventilation. Equations are derive that relate the physiologic dead space and alveolar ventilation to the physiologic parameters of the patient (compliance, inspiratory and expiratory resistances, anatomical dead space, cardiac output, and shunting), the ventilator settings (frequency, inspiratory time fraction, and pressure waveform), and the catheter settings (flow rate and flow duration). It is shown that optimal alveolar ventilation exists for certain frequencies of ventilation and these frequencies are dependent on the catheter flow. @ 1999 Elsevier Science Ltd. All rights reserved. Keywords-Mechanical cheal gas insufflation,
ventilation, Carbon dioxide exchange, Alveolar ventilation, Shunt.
Two-compartment
model,
Tra-
NOMENCLATURE catheter flow
VT
tidal volume = V+ + V$
&
VF
tidal volume of compartment u=a,b
QP
pulmonary blood flow
0s
shunted blood flow
fs
fraction of cardiac output that is shunting the lungs (QS = fsG)
in compartment
w
proportioning parameter for cardiac output to lungs
+ as (cardiac
9
perfusion ratio = @/a*
(T
fraction of time that catheter flows,
T
tidal volume ratio = V+/V$
VA
total alveolar ventilation
ti;
alveolar ventilation V=a,b
$
total perfusion = a, output)
Qy
perfusion to compartment
v = a, b
tc = ate
*Supported in part by a subcontract to Vanderbilt University from the Ramsey Foundation. fsupported by NIH SCOR HL50152-01 and the Ramsey Foundation. 0895-7177/99/$
- see front matter @ 1999 Elsevier Science Ltd. All rights reserved
PII: SO895-7177(99)00054-O
Typeset
by d&-T@
46
P.
V&l
total anatomicaf VS,,
v&h,
dead space =
+ VA,”
anatomical dead space assigned to compartment u = a, b
VDPtIYS physiologic dead space volume of catheter flush V2
of V&,
V,”
vcoz P&O,
fractional concentration of CO2 in alveolar gas of compartment v=a,b fractional concentration expired alveolar gas
of CO2
conversion factor, 863 mm Hg slope of CO2 dissociation
VP = BQ&
volume of catheter v,b = (1 - SfQ&
s. CROOKE et Uf.
Bush of V,“,,
curve
y-intercept CuNe
of CO2 dissociation
rate of CO2 elimination
inspiratory
time fraction,
partial pressure of CO2 in alveolar
tilttot
gas
frequency of ventilation
pLo2
partial pressure of CO2 in afveofar gas in compartment v = o, b
compliance tr=i&,b
P WZOZ
partial pressure of CO2 in arterial blood
inspiratory resistance of compartment v=a,b
pressure of CO2 in arterial p:coz partial blood entering compartment u=a,b
P WOZ
partiai pressure of CO2 in endcapillary region
P&O,
partial pressure of COG in mixed expired gas
c QCO2
concentration blood
cvco,
of comp~m~nt
expiratory resistance ment u = a, b
of compart-
airway resistance
*co2
part&I pressure of CO2 in mixed venous blood
GO2
d =
of CO2 in arterial
applied airway pressure end expiratory mentv=a,b inspiratory expiratory
time time
total time of one cycle, ttot = catheter
concentration of CO2 in arterial blood from eomp~ment v
pressure of compart-
&-I-&
flow time
fraction of catheter V,a,,OIs
flow Rushing
concentration of CO2 in mixed venous blood
1. INTRODUCTION Numerous mathematical models for the mechanical ventilation of patients with acute respiratory failure have appeared in the scientific and medical literature (see, e.g., (l-S]). These models have used to predict clinical relevant outcomes such as work of breathing, auto-PEEP, tidal volume, and mean alveolar pressure as functions of physiologic variables (compliance of the lung, inspiratory and expiratory resistances) and clinician-determined inputs (applied airway pressure, ventilatory frequency, inspiratory time). Such models have focused on mechanical outcomes of ventilation, without attending to gas exchange. Pulmonary gas exchange has also been the object of considerable attention in the modeling literature [?,9-121. Much of this work has focused on the elimination of 602, providing models for physiologic dead space with alveolar ventilation as the primary outcome of interest. fn general, these models treat clinically accessible variables (frequency, tidal volume, inspiratory time fraction) as arbitrary inputs. In ,the current paper, a mathematical model is presented which expresses important clinical outcomes (alveolar ~~~~~ physiologic dead space, and alveolar ventilation) as functions of respiratory system char~teristi~s~ pulmonary vascular perfusion, and clinician selected ventilator settings. The interplay between pulmonary perfusion, lung mechanical characteristics, and the selected ventilatory pattern in determining effective CO2 removal is thus highlighted. While the current model is configured to calculate the average values for alveolar Pco, , it can easily be adapted to compute this outcome in a dynamic fashion-anafogous to the commonly-used technique of capnography.
Carbon Dioxide Exchange
In the clinical alveolar during
setting,
pressures,
there
leading
mechanical
has been increasing
to the use of smaller
ventilation
[13-181.
While
awareness tidal
this
47
of the untoward
volumes
approach
and pressure has improved
effects of elevated limited mortality
strategies in some
studies, it limits the clearance of CO2 with potential deleterious consequences (191. A useful adjunctive technique in the setting of a pressure-limited ventilatory strategy is tracheal gas insufflation (TGI), where a small catheter placed near the carina flushes CO2 from the anatomic dead space during ematical modeling of vascular perfused
ventilation [20-271. Relatively little attention has been directed to the mathof TGI, particularly in multicompa~ment models incorporating the effects
perfusion pulmonary
Accurate
on CO2 clearance.
In this paper,
we model
TGI
in a multicompartment,
system.
determination
of alveolar
ventilation
very important for the clinician. Tracheal adjunctive technique [20-271 for mechanical
in cases of acute respiratory
failure is potentially
gas insufflation (TGI) is a potentially an important ventilation in patients with acute lung injury [13-181.
In this technique, fresh gas flowing through a small catheter near the carina flushes the anatomic dead space during expiration, thus decreasing the end-expiratory CO2 in this portion of the dead space so that a higher concentration of fresh gas is taken into the lungs during the next breathe. Using the mathematical model mentioned above that incorporates the clinical set variables for mechanical ventilation, we derive a model that can demonstrate the effects of TGI on alveolar ventilation.
2. THE
MODEL
During mechanical ventilation pressures are applied across the lungs to drive fresh gas into the system; waste gas is allowed to empty passively. Gas exchange occurs only in the alveoli and terminal airways, not in the conducting airways. The nongas exchanging, conducting airways constitute the u~ato~~c dead space; the gas volume cyclically residing within these regions does not contribute to alveolar ventilation. mismatching of alveolar perfusion and ventilation further decreases effective alveolar ventilation; this effect is termed u~~eo~a~ dead space. The sum of alveolar dead space and anatomic dead space is the ~~~s~o~og~c dead space. The elective alveolar uent~Zut~o~ is the quantity of fresh inspired gas that reaches alveolar units having sufficient perfusion to allow effective CO2 exchange between alveolar gas and blood. Wasted ventilation represents that fraction of total ventilation lost in the physiologic dead space and unable to participate in CO:! clearance. Thus, effective alveolar ventilation is the total ventilation minus the wasted ventilation. Assessment of wasted ventilation requires knowledge of the arterial CO2 tension (P,cOz ) and the alveolar CO2 tension (F’A~~, ). In health, these values are generally nearly equal. However, in individuals with abnormal matching between regional ventilation and regional perfusion, the difference between PA~.,~ and PaC02 widens. Furthermore, with regional heterogeneity of perfusion, there is no longer a unique value for ~~~~~ (it will differ between regions). Ventilation-perfusion mismatching is a significant problem in many common disease processes-acute lung injury, emphysema, pulmonary thromboembolism, sepsis, and shock, to name several. This model allows explicit determination of the effects such ventilation-perfusion mismatching has on gas exchange, and does so in the context of inputs (ventilator settings and TGI parameters of flow and timing) amenable
to clinical
manipulation.
Determination of compartments PACT, becomes more complex in the setting of TGI. Here, fresh gas is delivered to the carina by two pathways: via the proximal airway and through a small, deeply positioned catheter. Catheter flow, which is commonly restricted to the expiratory phase, dilutes the CO2 residing within the proximal anatomic dead space (~~s~~ng the dead space), thereby increasing effective alveolar ventilation. Catheter effectiveness is gauged by the increase in effective alveolar ventilation afforded by a given catheter flow.
48
P. S. CROOKE et al.
The pulmonary system is assumed to consist of two compartments a and b which are ventilated and perfused. In Figure 1, a diagram of pulmonary system is presented. The dotted portion denotes the lungs and airways and the solid lines symbolize the flow of blood to and from the lungs.
Q Figure 1. Two-compartmental
lung system.
Many different quantities are used in the modelling. A list of the notation for the quantities can be found at the end of the paper. In general, superscripts are used to denote compartmental quantities and subscripts are used to indicate physiologic tags. Also, V denote volumes, Q! flows, P partial pressures, C compliances, and R resistances. 2.1.
Model
Assumptions
To calculate the COz partial pressure in each compartment
of the lung, the following assump-
tions are made. 1. The (C” 2. The that
pulmonary system is comprised of two compartments, each with its own compliance or Cb), inspiratory and expiratory resistances (RF,Rf , Rz,R$$. two compartments are connected at a branch point that leads to a common airway is assumed to have the same resistance on inspiration and expiration (Raw).
3. The resistances and compliances are assumed to be constants, and the only compartment interaction is assumed to occur at the branch point. 4. During expiration it will be assumed that the pressure at the airway opening is zero (referenced to atmospheric pressure). 5. It is assumed that the pressure changes due to elastic forces in the compartments are directly proportional to the volume changes, and the pressure changes due to resistive forces are directly proportional to the changes in the flow. 6. At time t = 0, an arbitrary pressure Paw(t) is applied to the opening of the common airway by the ventilator. This pressure is applied to the airway opening over a time interval, 0 5 t 5 ti, which is the inspiratory part of the breathing cycle. At time t = ti,
49
Carbon Dioxide Exchange
the applied airway pressure is released and expiration takes place passively during the time interval, ti 5 t 5 ttot. 7. The anatomic dead spaces are parameters that are characteristic of the patient and hence, are inputs to the model. They can be simply functions of the tidal volumes for each compartment. The total anatomical dead space gas is shared in common by the two compartments so that the portion allotted to compartment a is VE,, and to compartment b, Vi,, . These volumes are assumed to be inputs to the model. The refinement of anatomic dead spaces for each compartment is built into the model; however, determining values for I%,” and Vi,, is very difficult and for most applications of the model, one would choose v&, = VA,, = VDan 12. 8. The flow of fresh gas that washes the anatomical dead space of each compartment are inputs to the model and are denoted by & and Q”, so that the total flow of the catheter is 0, = 0: + @. The catheter flows only during the end of the expiratory part of the breathing cycle and the time of flow is denoted by t,. The volume of fresh gas delivered to each anatomical dead space is then V: = C& = sQ,t, and V,” = &itc = (1 - s)Q,t, where s is a input parameter such that 0 5 s 5 1. The effect of the catheter is to flush the anatomical dead space of each compartment so that the partial pressure of CO2 in the anatomical dead space is proportional to the partial pressures of the alveolar gas from each compartment so that
PDanc02 = (,!$;)CL,
+ (&y&J p;,.,
.
As in the previous assumption, it would be difficult to assign catheter flows to each compartment without a detailed picture of the anatomical structure. A simpler approach is to assume that 0: = 6): = Q,/2. However, we incorporate this more general setting for the model. 9. In each compartment, the pressure of CO2 in the alveolar gas is in equilibrium with the partial pressure of CO2 in the arterial blood, i.e., Pi,_, = P,“,, for v = a, b. This 2 2 requirement can be and will be relaxed. 10. The arterial concentration of CO2 is a weighted average of the arterial concentrations of CO2 from each compartment, i.e.,
11. There is no CO2 in the ambient air. 12. The CO2 dissociation curve is a straight line and will be assumed to be of the form: Go, = kPcoz + m, where k and m are constants. 13. A portion of the cardiac output is shunted around the two compartments. In particular, the cardiac output is divided according the equation G = 0, + OS, where OS denotes the shunt blood flow. 2.2. Two-Compartment
Ventilation
Model
For the mechanical ventilation part of the model, we use the model developed in [2]. This is a general mathematical model for the dynamic behavior of a two-compartment respiratory system in response to an arbitrarily applied inspiratory pressure. Using clinical set variables such as frequency, inspiratory time fraction, and physiologic parameters for resistance and compliance, outcome variables such as tidal volume, end-expiratory pressure, mean alveolar pressure can be computed for each compartment. Since the explicit formulas for these outcome ventilation variables are quite complex, the reader is referred to the original paper.
P. S. CRO~KE
50
2.3. Carbon
Dioxide
Transport
For the gas exchange part of the model, Initially,
et al.
we use a modi~cation
of a model
proposed
in (lo].
we will assume that the shunt flow is zero, i.e., OS = 0, i.e., &, = 0. In the last part of
this section,
this assumption
will be relaxed.
The starting point for this part of the model is to the conservation of COs, i.e., the amount of carbon dioxide that leaves the lungs is equal to the amount of carbon dioxide that is leaving the blood. Considering this process for each compartment, compartment a into the airway is ~,J’,Y& and the amount - C&,,
blood is &(CV~~,
). For compartment
the amount of CO2 expelled from of CO2 entering compartment a from
a we have the conservation
law
A similar identity is obtain form compartment b. For gas exchange, it is convenient to deal with the partial pressure of a gas rather than its fractional concentration. We assume that the partial pressure of a gas is proportional to fractional concentration. In particular, the partial pressure of alveolar CO:! gas is related its fractional content by the relation: P_&,, = cyFJ&, where rr is the BTPS-STPD conversion factor and is equal to 863 mm Hg when alveolar ventilation is expressed in milliliters per minute. The net change of carbon dioxide in a compartment due to ventilation is the amount of carbon dioxide leaving the compartment minus the amount entering the compartment. For compartment a, the alveolar ventilation is expressed mathematic~y as pz = f(V+ - VE,,). Using this relation in (2.1) for ~ompa~ment a, we find
(2.2) Using our assumption C&oz = “P&0,
about
the shape of the CO2 dissociation
curve Cv,ol
= kA/,oz
+ m and
+ m, (2.2) becomes
w (pvco, - p&J = 6(VTa - Vi,“) P&,,.
(2.3)
Next, we incorporate the effect of the catheter on the portion of the total anatomical dead space for compartment a. Namely, if Vca is the volume of fresh gas delivered by the catheter to compartment a, VT is the tidal volume, V$! is the portion of the tidal volume that is entering compartment a, and V, = act, is the total volume of fresh gas delivered by the catheter to the total anatomical dead space, then effective anatomical dead space of compartment a is given by
If
v, > VDanf then
to a reduction
the excess fresh gas is vented to the atmosphere in the CO2 in the anatomical dead space.
and effectively
does not lead
At this point, we make a remark about possible future directions for the modeling. The dynamics of expiration, at least for passive ventilation, are known from previous modeling [2]. In particuIar, the instantaneous volume and flow from each compartment are known over the complete expiratory part of the breathing cycle. In interaction between the exiting alveolar gas and entering fresh gas could be modeled. The instantaneous partial pressure of carbon dioxide could then be calculated. The direction for modeling has not been pursued in this paper because of the complexity.
Carbon
Dioxide
51
Exchange
The total partial pressure of carbon dioxide in alveolar gas exhaled (PACT,) is assumed to be equal to the weighted mean of carbon dioxide pressures of two compartments with the weights being the fractions of the tidal volume entering or leaving each compartment.
PACO,=
That is,
(g)pko2 +(gpJLo2.
(2.5)
Similarly, partial pressure of carbon dioxide in arterial blood (Paco2 ) is the mean of carbon dioxide pressure of two compartments weighted by fraction of perfusion
(2.6) At this point, we remark that VD~,, is not necessarily independent of the tidal volume VT and we have assumed that partial pressure of CO:! in the anatomic dead space in each compartment is equal to the partial pressure of CO2 in the alveolar gas of that compartment. Using (2.4)-(2.6), (2.3) b ecomes for compartment a
(2.7)
A similar equation can be computed for compartment b
(
f
fV:! kQbPvcoz = T+kcjb-
(v$+ (1 - &ctc)
pb
@vDa”
ACOz a (VT + Q&J2 (2.8)
f (v:! + (1 - $&)
v+vD,,
It is convenient to define some intermediate symbols that reduce the complexity of (2.7) and (2.8). We define
f
fVTa
hr = T+kf+
(v$+S&t,)
V+VDa,, 2
Q!
A2 =
Aa =
v;vD,, 2
cy
f(V,” + dctc)
v$D,, 2
(
h
+ &&
’
v;vD,, 2
(
I
(2.10)
(2.11)
)
f (v; + (1 - &ctc) o’
(2.9)
’
)
VT”+ (1 - &ctc)
fV$
-+k&“-f(
a
A‘j =
(
VT + &&
VT + &tc
)
(2.12)
52
Using the notation of (2.9)-(2.12) in (2.7) and (2.8), we obtain two linear, inhomogeneous equaand Pi,,
tions in the unknowns, P&,
2
2
and P&,,
As - P&,,
in (2.13),(2.14),
A4 =
(2.14)
kQbfic.02.
we have the following expressions for the partial
Solving for P&,
and Pj,,
pressure of carboi
dioxide it the alveolar gas of each compartment:
(2.15)
and (2.16) Substituting (2.15) and (2.16) in (2.7) and (2.8) and using the notation (2.9)-(2.12),
we obtain
expressions for PA,__, and PaGo
PVC,,
(2.17)
and (2.18)
2.4. Physiologic Dead Space The physiologic dead space is defined as the sum of the anatomical dead space and the alveolar dead space. The anatomical dead space is the volume of inspired air that does not reach of the alveoli. The alveolar dead space is the volume of air that reach the alveoli, but does not undergo gas exchange. Hence, the physiologic dead space represents a portion of the tidal volume that is wasted. The total physiologic dead space, VD~,,,_,can be derived using the Enghoff modification of Bohr equation. This equation expresses the total physiologic dead space as a fraction of the tidal volume (2.19) where PE__.,~is the partial pressure of the expired carbon dioxide. can be written as dead space fraction, VD,,/VT, _
pb
vD&s
I=:
Therefore, the physiologic
(2.20)
VT
Let PC*, denote the carbon dioxide minute ventilation. Therefore, I&oz/f is the total carbon dioxide output in one respiratory cycle. This quantity is related to the minute ventilation by the relation: p&J2 = ~EF&~~. Since &,,, = &ECo,, we have the relatively simple relation for PEW,
53
Carbon Dioxide Exchange
P&O, =.
QVCOa VE
(2.21)
+&dc>
VD,, VT Using (2.21)
PA co2
in (2.20), we find the following expression for the physiologic dead space fraction:
(2.22)
Using (2.17) and (2.18) in (2.22), we obtain the a relatively simple expression for the physiologic dead space fraction
where
2.5.
Alveolar
Ventilation
The alveolar ventilation is the quantity of inspired air reaching alveoli perfused by blood and thus taking part in gas exchange. Therefore, it is defined as the difference of minute ventilation, I& = fI+ and physiologic dead space ventilation, V&h,. = fVDphys. Using the expression obtained for physiologic dead space we can obtain an explicit expression for alveolar ventilation
as
(2.24)
This last identity of (2.24) is an explicit expression for the alveolar ventilation in terms of the clinically set and physiologic parameters. 2.6. Shunting In the above model, we have assumed that the total output from the heart is exposed to the lungs. It is quite easy to modify the model to include a shunting component, i.e., having a certain fraction of the venous blood to bypass the lungs and directly mix with the arterial blood. Let 0;
denote the pulmonary blood flow into compartment a and C$, into compartment
b so that
the total pulmonary flow flow is 0, = o,B + @,. Let o!s denote the flow of blood that is shunted by the lungs so that the cardiac output is given by 0 = 0, + OS. To simplify the notation, we
P. S.
54
CMOKE
et 41.
introduce a ~~~~~~~~~~ ~~~~~~ W, 0 5 w < I, so that @ = woP and @E = (1 - LJ&&_ Let E&BCDzdenote the partid pressure of CO2 in the end-capillary zone of compartment a and P&P,,2 for compartment
b. Alveolar, end-capillary, and arterial partial pressure of CO2 with
shunting are given by the following equations: (2.25) f2.26)
where fS is the fraction of the total cardiac output that is shunted, i.e., QS = f.&, 0 < fS 2 1. The steady state equation for CO2 in compartment a with shunting is then given by
= and P&,,, b. Letting P$&, 2 = x’P~,,~ 1 where 0 < x’, y’ < 1 and using (2.6) for PDF,, the follow&g equations relate P.4cn2 in
In a similar equation, we get for compartment
Y~p~co, ’
each compartment: (2.28)
(2.29)
where
(2.30)
(2‘31)
Thus the partial pressures, P&,,
and Pjc,,
, in presence of shunt&
are
(2.34)
(2.35)
Carbon Dioxide Exchange
Substituting
55
in (2.6) and (2.35), we obtain expressions for the end-capillary
(2.35)-(2.6)
and
alveolar CO:! partial pressures
From (2.6), the arterial CO2 tension can be computed
-I”
Having
Ml
4;
- fJPvco2 f
(@qb.
+ @k&z,)
9,
1
+ 4;
(@a,*b.
-
(@%,
f
Q;@b,)
%s@b,)
I.
(2.38)
the physiologic dead space and alveolar ventilation can be computed from (2.19)
J’aco2,
and (2.24).
3. SIMULATIONS Having explicit expressions for the physiologic dead space and alveolar ventilation in terms of the physiologic parameters and the clinical set variables, simulations are possible that show the effect on these two important outcomes in terms of the clinical inputs. Since the physiologic dead space is related directly to the alveolar ventilation by (2.24), we restrict our simulations to the alveolar ventilation outcome. Typical values of the physiologic parameters have been chosen for the simulation. In Figure 2, the alveolar ventilation as a function of frequency is show for different catheter flow rates (GJ and perfusion ratios (q = &a/Q). Th e catheter flow rates are given in liters per minute. As can be seen in the graphs, there is an optimal frequency of ventilation for each catheter flow rate. It appears that this optimal frequency increases with increasing catheter flow.
VA
VA
8
Q
6
-0
4
_...... 5
2
.--_ 20
40
60
f
10
20
40
60
f
Figure 2. Alveolar ventilation (L/min) versus frequency (l/ min) for different perfusion ratios and catheter flow rates. Parameters: R,” = 25cm HzO/L/s, Rt = iOcm HzO/L/s, RE = 40cm HzO/L/s, Rt = 20cm HzO/L,fs, R,, = 6cm HzO/ L/s, Ca = O.O2L/cm H20, Cb = O.O3L/cm H20, Pset = 20cm H20, VD,, = 0.25 L, a = 4.5 L/min, H20.
a8 = 0 L/min,
cy = 1208 cm H20, and k = 0.0032 ml/cm
In Figure 3, the alveolar ventilation is plotted as a function of the inspiratory time fraction for different catheter flow rates and perfusion ratios. As can be seen, there appears to be a optimal inspiratory time fraction. The maximal alveolar ventilation that is achieved using the TGI is affected by the perfusion ratio (q). This is expected because of the asymmetry of the compliances and resistances.
P. S. CROOKE et at.
56
VA
VA
6
..-...
s
____ f()
4
4
-..._ 5
2
2
____ fO
0.25
0.5
i
d
0.75
0.25
Figure 3. Alveolar ventilation (L/min) versus inspiratory ent perfusion ratios and catheter flow rates. Parameters:
time fraction RF =
25cm
0.5
0.75
d
for differHzO/L/s,
Rt = 1Ocm I&O/L/s, Rz = 40cm HzO/L/s, Rt = 20cm HsO/L/s, R,, =I. 6cm HzO/L/s, Ca = O.O2L/cm HsO, Cb = O.O3L/cm HeO, Pset = 20cm HeO, V& = 0.25 L, 4 = 4.5 L/min, H20.
ss
= 0 L/min,
CY= 1208 cm HsO, and k = 0.0032 ml/cm
The alveolar ventilation as functions of frequency and inspiratory time fraction as functions of different flow times of the catheter (750/o, 50%, and 25% of the expiratory part of the breathing cycle) have been computed. For the physiologic parameters chosen above, there is very change in the alveolar ventilation. A measure of the ventilation-perfusion unevenness is the arterial-Alveolar CO2 dijference (a - WC02 The relative change is %(a - A)DC02
= &e,
-
(3.1)
pAcol.
is defined as
%(a - A)DC02
=
P
ace;
- pAcoz
.
WC+
(3.2)
In Figures 4 and 5, the absolute and relative change in the arterial-Alveolar difference is plotted as a functions of frequency and inspiratory time fraction. (a-A)DCOz
(a-A)DCOz
Qc
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 20
40
60
f ’
0.25
0.5
0.75
d
Figure 4. (a - A) Pco,, (cm H20) versus f (I/min) and d for different catheter flow rates. Parameters: Rr = 25, Rib = IOcm H20/L/s, Rz = 40cm HzO/L/s, Co = 0.02 L/cm HsO, C6 = 0.03 L/cm R: = 20cm HsO/L/s, &, = 6 cm HzO/L/s, e. = 0 L/min, a = 1208cm H20,Pset = 20 cm Hs0, VB,, = 0.25L, 4 = 4.5L/min, H20, and k = O.~32ml/cm H20. %(a-A)DCOz
/
6
I
/
I
%(a-?DC02
,
I
/
I
I
I
iq=o.51
/
1
4 2 20
40
60
f
0.25
0.5
Figure 5. Percentage change in (a - A) Pcoa versus f (l/ min) and D for different catheter flow rates. Parameters: Rio = 25cm HzO/L/s, Ri = 1Ocm H2O/L/s, Rz = 40cm HaO/L/s, h!: = 20cm HsO/L/s, Raw = 6cm HsO/L/s, Ca = O.O2L/cm HsO, Cb = O.O3L/cm HsO, Pset = 20cm HsO, VD,, = 0.25L, 0 = 4.5L/min, OS = OL/min,
a! = 1208cm HsO, and k = O.O032ml/cm
HsO.
0.75
Carbon Dioxide Exchange Other
modes
same physiologic
of ventilation p~ameters
of ventilation,
the outcome
time fraction
are presented. %(a-A)DCOz
can be used with the model. as in the previous variable
simulation,
%(a - A)DCOz
57 For example,
in Figure
but using a accelerating
as functions
of frequency
6, using the pressure
mode
and inspiratory
%(a-A)DCOz
Figure 6. Percentage change in (a - A) PCO, versus f (I/ min) and D for different catheter flow rates. Parameters: RT = 25cm HzO/L/s, R$ = 1Ocm H20/L/s, Rz = 40 cm H20/Ljs, R: = 20 cm H20/L/s, Raw = 6 cm H20/L/s, C” = 0.02 L/cm H20, Cb = 0.03 L/cm H20, P,,(L) = l.S(t/ti) + 5cm H20, ~~~~ = 0.25L, 4 = 4.5 L/min, Gs = 0 L/min, o = 1208 cm J&O, and k = 0.003 ml/cm H20.
I
VA
60 Figure 7. Alveolar ventilation (L/min) as a function of f (l/min) and d fraction. Parameters: R4 = 25cm HzO/L/s, R,” = 1Ocm HzO/Lfs, Rz = 40cm HzO/L/s, Rk = 20cm H20/L/s, Raw = 6cm HzO/L/s, C” = O.O2L/cm H20, Cb = 0.03 L/cm H20, Pset = 20cm H20, V,,, =0.25L,Cj=4.5, &=O, &==5, q= 0.5, D = 0.75, a = 1208 cm H20, and k = 0.0032 ml/cm H20.
Other simulations alveolar ventilation
One can vary the shunting and observe the effect on the are possible. for different catheter flows. It was observed from these simulations that
relatively small shunting (O-10%) did not produce significant changes in the alveolar ventilation as functions of the frequency or inspiratory time fraction. Finally, the alveolar ventilation is a function of several clinical set variables and physiologic parameters. The behavior of this key outcome variable is can be viewed as surfaces in higher
58
P. S. CROOKE et al.
dimensional spaces.
In Figure 7, the alveolar ventilation as a function of both frequency and
inspiratory time fraction for Qc = 5 is plotted.
4. DISCUSSION The model developed in this paper predicts clinically important outcomes as functions of respiratory system mechanical and perfusion characteristics and clinically accessible inputs (ventilator and TGI settings). This model is a composite of a general ~~comp~tment chanical ventilation and a tw~compartment
model of passive me-
model of pulmonary CO2 exchange which includes
the possibilities of nonhomogeneous perfusion and ventilation. As such, it represents a significant advance over previous models of TGI. The model suggests several patterns of behavior which are of clinical importance, and allows quantization of their significance. The maximal differences in alveolar ventilation are of modest magnitude (O-l L/min).
However, changes in alveolar ventila-
tion of this magnitude may well be of clinical significance in the gravely ill, hypercapnic patient who is being ventilated with a low tidal volume, pressure limited strategy-the population in whom TGI is most often employed.
4.1.Critique
of the Model
Several features of the model are subject to criticism. Foremost among such concerns is that the model has not been validated in vivo. We have previously validated two components of the model (two compa~ment models of mechanical ventilation and the model for anatomic dead space flushing) in a mechanical lung analog [3,28]. In contrast, the diseased lung is composed of many interdependent compartments, any of which may depart from linear behavior, which are arranged in a geometric complex structure. Rigorous in vivo validation of the model would require the fine scale measurement of regional alveolar pressures and small airway resistances, as well as local vascular perfusion characteristics. Acquisition of such information would be problematic. The independent two-compartment airspace configuration we modeled is clearly a gross oversimplification of the multiple interactive compartments comprising the diseased lung. However, there is data that such a model may have clinical utility [29-321. Similarly, while the perfusion model (two perfused compartments plus shunt) is an oversimplification, it represents the least complex perfusion analysis applicable to a two compartment model. The clarity of a model including two perfused compartments plus shunt is the reason that it is commonly used for educational purposes. Because it allows for heterogeneity of airspace and vascular factors, the current model represents a signi~c~t
advance over previous models of TGI.
We have assumed linear behavior in our mechanics model, and have not incorporated the effects of nonlinearity of resistance or compliance, or the effects of compartmental interaction. While subject to criticism, this approximation is likely not unreasonable for two reasons. First, the work of Bake, Rossi, and Shardonofsky and their respective colleagues suggests that constancy of impedance parameters may be an acceptable initial approximation for respiratory system behavior [29,33,34]. Furthermore, our previous work in this area indicates that inclusion of plausible nonlinear behaviors does not affect the qualitative behavior of ventilatory distribution or pressure distribution-the major mechanics outcomes of consequence in the current model [3]. Similarly, we have not considered the potential viscoelastic behavior of the pulmonary parenchyma. Viscoelastic behavior, in which the pulmonary parenchyma displays timedependent, non-Newtonian behavior (such as stress relaxation), has been recently and elegantly addressed 135,361. Such behavior is clearly important from the standpoint of mechanics, and likely will affect ventilatory distribution. However, the inclusion of viscoelastic behavior in the current two compartment model would require knowledge of regional variation in viscoelastic time constants. The effects of regional viscoelastic behavior may either augment or attenuate the distributional effects of Newtonian mechanics, depending on the magnitude and alignment of viscoelastic time
Carbon Dioxide Exchange constant
heterogeneity.
Detailed
ingiy, we have not addressed 4.2.
Clinical
information
this complex
is not available issue in the current
59
regarding
these questions;
aceord-
model.
Implications
The model predicts that there is an optimal ventilatory frequency, which maximizes alveolar ventilation, for a given set of respiratory system mechanical and perfusion characteristics. As noted, the optimal frequency rises as TGI flow is increased, in accord with our previous findings in a unicompartmental
model of TGI
the slope of I& ws frequency optimization optimal
may be of clinical
[28]. This optimal
significance.
value is likely due to the balance
is flushed each minute
value occurs within
(f) is quite steep in this neighborhood,
and the decreasing
As detailed between efficiency
in our previous
the number
the clinical
suggesting
that
cycfe as frequency
and
frequency-
work, the presence
of times the anatomic
of each flushing
range,
of an
dead space increases.
Similarly, the model predicts the existence of an optimal inspiratory time fraction for a given set of associated inputs. Again, the optimal value arises within the clinical range, and may be associated with clinically significant increases in alveolar ventilation. The presence of an optimal inspiratory time fraction arises from the interplay between allowing adequate expiratory time for complete flushing of the anatomic dead space and minimizing end expiratory residual flow from the alveolar compartments into the anatomic dead space. The presence of residual endexpiratory flow (and its detrimental effect on TGI efficacy) become more important as expiratory resistance increases (data not shown). Thus, the model would suggest that TGI would not be as effective in diseases characterized by high expiratory resistances, such as emphysema or asthma. An identical effect can be seen as compartmental mechanical heterogeneity (expiratory time constants) increases (data not shown). The inclusion of pulmonary perfusion and a shunt pathway allowed consideration of the consequences of nonhomogeneous pulmonary blood flow and shunt as regards alveolar ventilation. Shunt had little effect, whereas asymmetry of perfusion significantly reduced the effectiveness of TGI. This is of considerable importance, as many disease processes (shock, sepsis, emphysema, pulmonary embol~m) may lead to derange distribution of pulmonary perfusion, decreasing the potential benefits of TGI. Large-scale maldistribution of pulmonary perfusion, as reflected in the model, theoretically may be improved by changing the position of the patient (i.e., prone or lateral positioning). Small scale (regional) maidistribution of perfusion may require pharmacologic intervention. In summary, we present a two-compartment model for TGI which includes heterogeneity of pulmonary ventilation and perfusion. While obviously an oversimplification of clinical reality, the model suggests that clinically relevant changes in ventilator settings may lead to clinically significant increases in TGI effectiveness. Certain patient characteristics (high expiratory resistance, mismatching of ventilation and perfusion, or compartmental mechanical heterogeneity) may render TGI less efficacious. Furthermore, the model provides a theoretical framework relating patient characteristics, ventilator settings, and clinically important outcomes. Such a framework may offer the opportunity to develop rational optimization algorithms for the use of TGI. Reconfiguration of the model to interpret dynamic information, pa~icularly regarding the end-expiratory acquire detailed information regarding expired PC-J-,, , may allow the clinician to noninvasively ventilation-perfusion matching from the capnogram.
5. CONCLUSIONS In this paper, we have derived one of the first mathematical models for tracheal gas insufflation that permits the input of physiologic parameters (resistances, compliances, shunt, perfusion ratio, etc.) and clinical set variables (frequency, inspiratory time fraction, mode of ventilation) to produce key ventilation outcomes such as alveolar ventilation. The model is a synthesis of a general, two-compartment, mechanical ventilation model with a sophisticated two-alveoli CO2
P. S. CROOKE et al.
60
exchange
modeI.
The resulting
shows that there are optimal fraction,
to produce
TGI
model
demonstrates
the effects of the catheter
settings for the clinical variables,
the best alveolar ventilation
frequency
and/or
flow and
inspiratory
time
for the patient.
REFERENCES 1. P.S. Crooke and J.J. Marini, A nonlinear mathematical model of pressure preset ventilation: Description and limiting values for key outcome variables, Math. Mod. Meth. Appt. Ski. 3, 839-859, (1993). 2. P.S. Crooke, J.D. Head and J.J. Marini, A general tw~comp~tment model for mecbanicd ventilation, lathe. Cornput. ~odell~~g 24 (7), I-18, (1996). 3. J.R. Hot&kiss et al., Implications of a bipbaic two-comp~ment model of constant flow ventilation for the clinical setting, J. Grit. Care 9, 114-123, (1994). 4. V.K. Jain et al., Models of the lung, Brit. J. Anaesth. 43, 816-817, (1971). 5. J.J. Marini, P.S. Crooke and J.D. Truwit, Determinants and limits of pressure-preset ventilation: A mathematical model of pressure control, J. Appl. f%ySiOk 67, 1081-1092, (1989). 6. U. Morgenstern and S. Kaiser, Mathematical modelling of ventilation mechanics, Int. J. Clin. Monit. Corn@. 12, 105-112, (1995). 7. M.H. Reid anti H.B. Hechtman, A multicompartment analysis of the lung, Med. Biol. Eng. 12, 405-414, (1974). 8. A.A. Wald, T.W. Murphy and V.D. Mazzia, A theoretical study of controlled ventilation, IEEE 74ans. Biomed. Eng. 15, 237-248, (1968). 9. M. Sharan, S. Selvakumar and M.P. Singh, Mathematics model for the computation of alveolar partial pressure of carbon monoxide, Int. J. Biomed. Camp. 26, 135-147, (1990). 10. K. Suwa and H.H. Benedixen, A mathematical analysis of physiological dead-space in a lung model, J. Appi. Physiol. 24, 549-555, (1968). 11. K. Suwa and H.H. Benedixen, Change in PacO, with mechanical dead space during artificial ventilation, J. Appl. Physiol. 24, 556-561, (1968). 12. J.B. West, Ventilation-perfusion inequality and overall gas exchange in computer models of the lung, Reap, Physiol. 7, 88-110, (1969). 13. J.J. Marini, Pressure-targeted, lung-protective ventilatory support in acute lung injury, Chest 105, 109S115S, (1994). 14. J.C. Parker et al., Lung edema caused by high peak inspiratory pressure in dogs, Am. Rev. Respir. Dis. X42, 321-328, (1990). 15. K.J. Peevy, L.A. Hernandez, A.A. Moise and J.C. Parker, Barotrauma and microvascular injury in lungs of nonadult rabbits: Effect of ventilation pattern, Crit. Care Med. 18, 634-637, (1990). 16. A.E. Taylor, K. Rehder, R.E. Hyatt and J.C. Parker, Clinical Respiratory Physiology, W. B. Saunders Company, (1989). 17. F.E. Udwadia, Diagnosis and Management of Acute Respiratory Failure, Oxford University Press, Oxford, (1979). 18. J.B. West, Respiratory Physiology, Williams and Wilkins Company, Baltimore, MD, (1985). 19. M.B.P. Amato et al., Beneficial effects of the open lung approach with low distending pressures in ARDS: A prospective randomized study on mechanical ventilation, Am. J. Respir. Grit. Cure Med. 152, 1835-1846, (1995). 20. W.C. Burke et al., Modes of tracheal gas insufflation: Comparison of continuous and phase-specific gas injection in normal dogs, Am. Rev. Respir. Dis. 148, 562-568, (1993). 21. N. Gaviely, D. Larsson and J.B. Grotberg, Gas exchange by tracheal insufflation in a ventilatory failure dog model, J. Clin. Invest. 90, 2376-2383, (1992). 22. A. Hurewitz, E. Bergofsky and E. Vomero, Airway insufflation: Increasing flow rates progressively reduce dead spsce in respiration failure, Am. Rev. Respir. Dis. 144, 1229-1233, (1991). 23. A. Larsen, Elimination of apparatus dead space-a simple method for improving CO2 removal with increasing airway pressure, Acta Anaesthesiol. &and 36, 796-799, (1992). 24. A. Nahum et al., Lung mechanisms and gas exchange during pressure-control ventilation in dogs: Augmentation of CO1 elimination by an intratracheal catheter, Am. Rev. Respir. Dis. 146, 965-973, (1992). 25. A. Nahum et al., Effect of catheter flow direction on COz removal during tracheal gas insufflation in dogs, J. Appl. Physiol. 75, 1238-1246, (1993). 26. G. Nakos et al., Respiratory effects of tracheal gas insulation in spontaneousIy breathing COPD patients, Intensive Care Med. 21, 904-912, (1995). 27. S.A. Ravenscraft et al., Tracheal gas insufflation: Catheter effectiveness determined by expiratory flush volume, Am. J. Respir. Grit. Care Med. 153, 1817-1824, (1996). 28. J.R. Hotchkiss, P.S. Crooke and J.J. Marini, Theoretical interactions between ventilator settings and proximal dead space ventilation during tracheal gas insufflation, Intensive Care Med. 22, 1112-1119, (1996). 29. D. Bake, L. Wood, B. Murphy et al., Effect of inspiratory flow rate on regional distribution of inspired gas, J. Appl. Physiol. 37, 8-17, (1974). 30. J.H. Batea, M. Decramer, D. Chartrand et al., Volume-time profile during relaxed expiration in the normal dog, J. Appl. Physiot. 59, 732-737, (1985).
Carbon Dioxide Exchange
61
31. G.L. Chelucci, F. Brunet, J. Dall’Ava-Santucci et al., A single compartment model cannot describe expiration in intubated, paralyzed humans, Eur. Resp. J. 4, 458-464, (1991). 32. T. Conolly, B. Bake, L. Wood et a$., Regional distribution of a 133Xe labelled gas volume inspired at constant flow rates, &and. .I. Respir. Dis. 56, 150-159, (1975). 33. A. Rossi, S.B. Gottfried, B.D. Higgs et al., Respiratory mechanics in mechanically ventilated patients with respiratory failure, J. Appl. Physiol. 58, 1849-1858, (1985). 34. F.R. Shardonofsky, J. Sato and J.H. Bates, Quasi-static pressure-volume hysteresis in the canine respiratory system in vivo, J. Appl. Physiol. 68, 2230-2236, (1990). 35. B. Jonson, L. Beydon, K. Brauer et al., Mechanics of the respiratory system in healthy anesthetized humans with emphasis on viscoelastic properties, I. Appl. Physiol. 75, 132-140, (1993). 36. L. Beydon, C. Svantesson, K. Brauer et al., Respiratory mechanics in patients ventilated for critical lung disease, Eur. Resp. J. 9, 262-273, (1996). 37. P.F. Crossman, L.S. Bushnell and J. Hedley-White, Dead space during artificial ventilation: Gas compression and mechanical dead space, J. Appl. Physiol. 28, 94-97, (1970). 38. S. Damanto and G. Cumming, Tidal volume change and gas mixing in the lung, Respiration 56, 173-181, (1989). 39. J.W. Evans, D.G. Cantor and J.R. Norman, The dead space in a compartmental lung model, Sull. Math. i3i0phy. 29, 711-718, (1967). 40. J.W. Evans, P.D. Wagner and J.B. West, Conditions for reduction of pulmonary gas transfer by ventilationperfusion inequality, .I. App~. P~ys~ol. 36, 533-537, (1974). 41. R. Fletcher, B. Johnson, G. Cummings and J. Brewa, The concept of dead space with special reference to the single breath test for carbon dioxide, B&&h J. Anaesthesia 53, 77-88, (1981). 42. H.V. Forster, L.G. Pan, G.E. Bisgard, C. Flynn and R.E. Hoffer, Effect of reducing anatomical dead space on arterial Pcoz during CO2 inhalation, I. Appl. Physiol. 61, 728-733, (1986). 43. G. Hedenstierna, The anatomical and alveolar dead space during respirator treatment. Influence of respiratory frequency, minute volume and tracheal pressure, British J. Anaesthesia 47, 993-999, (1975). 44. G.F. Filley, D.B. Bigelow, D.E. Olson and L.M. Lacquet, Pulmonary gas transport, Am. Rev. Resp. Dis. 98, 480-489, (1969). 45. K. Holk, S.G. Einarsson, K.L. Stevensson, J.P. Bengtson and 0. Stenqvist, Continuous non-invasive monitoring of energy expenditure, oxygen consumption and alveolar ventilation during controlled ventilation: Validation in an oxygen consuming lung model, Acta Anaesthes. &and. 40 (5), 530-537, (1996). 46. V.K. Jain and SK. Guha, A study of intermittent positive pressure ventilation, Med. Biol. Eng. 8, 575-583, (1970). 47. P.M. Kibby, A mathematical model for CO2 exchange during the initial stages of rebreathing, Resp. Physiol. 3, 243-255, (1967). 48. G. Nakos et aI., Tracheal gas insufflation reduces the tidal volume while P,,_o, is maintained constant, Intensive Care Med. 20, 407-413, (1994). 49. M.I. Noble, F. Langley, M. Buckman, P. Vernon, A. Seed, R. Jewkes, A. Bisdee and G. Cumming, Comparison of ventilation/perfusion lung-imaging and dead-space measurements in airway disease, C&n. Sci. 60, 17-23, (1981). 50. E.B. Olson, Physiological dead space increases during initial hours of chronic hypoxemia with or without hypocapnia, .I. Appl. Physiol. 77, 1526-1531, (1994). 51. V. Taskar et al., Dynamics of carbon dioxide elimination following ventilator resetting, Chest. 108, 196-202, (1995). 52. G.G. Weinmann, W. Mitsner and S. Permutt, Physiological dead space during high-frequency ventilation in dogs, J. Appl. Physiol. 57 (3), 881-887, (1984). 53. R.L. Wilkins and J.R. Dexter, Respiratory Disease. Principles of Patient Care, F.A. Davis Company, Philadelphia, PA, (1993).