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Estimation of pulmonary diffusion resistance and shunt in an oxygen status model
ELSES’IER
Computer Methods and Programs in Biomedicine 51 (1996) 95-105
Estimation of pulmonary diffusion resistance and shunt in an oxygen status model S. Andreassen*a,J. Egeberg”, M.P. Schrijtera, P.T. Andersen” aDepartment
of Medical
Informatics hDepartment
and Image Analysis, Aalboq Uniuersi& Fr. Bajersuej of Intensive Care, Aalborg Sygehus. DK-9000 Aalborg,
70. DK-9220 Denmark
Aalborg,
Denmark
Abstract A compartment model of the transport of oxygen from the alveoli to the tissues is described. In patients with both pulmonary shunt and alveolar resistance to diffusion of oxygen, the model is used to simulate their response to variations in the inspired oxygen fraction. These simulation results are compared to the responses from a patient with respiratory malfunction, indicating that the method can identify patients where not only a pulmonary shunt but also a high alveolar resistance to diffusion of oxygen is clinically significant. Estimation of pulmonary shunt and oxygen diffusion resistance can be done in two different implementations of the model. In the first implementation the estimates are generated by numerical solution of the equations of the compartment model. In the second implementation the equations have been used to construct a causal probabilistic net where biological uncertainties and uncertainties in the measurements can be represented. Keywords:
Oxygen diffusion resistance; Oxygen status model; Pulmonary shunt; Respiratory modelling
1. Introduction One of the primary goals of artificial ventilation is to ensure adequate oxygenation of the arterial blood. Incomplete arterial oxygenation may result either from pulmonary shunt or from
an increased resistance to diffusion of oxygen through the alveolar membrane [l]. In patients with ventilatory malfunction it would be attractive to know the quantitative contribution of these -*Corresponding 0169-2607/96/$15.00 PfISO169-2607(96)01765-S
author.
two mechanisms, since therapeutic interventions aimed at correcting either of these problems are different. Unfortunately the methods available for measuring diffusion resistance are time consuming and technically demanding, requiring the use of an indicator gas and gas analysis equipment [2,3]. One practical approach to this problem has been to ascribe the inadequate oxygenation to shunt alone [4], ignoring the possible contribution from the alveolar resistance to diffusion. Instead we have chosen to use the functionality provided by the oxygen status algorithm (OSA) [51 as a starting point and have added diffusion resis-
Q 1996 Elsevier Science Ireland Ltd. All rights reserved.
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tance and a simple description of circulation, oxygen consumption and the ventilator to the model. The resulting model, the oxygen status model (OSM), can be used to simulate the patient’s response to different therapeutic interventions (simulation mode). The model can also be ‘solved”, using the patient’s response to variations of inspired oxygen fraction as input data, thus providing estimates for shunt and diffusion resistance ~diagoostic mode).
2.17. The aygen SMLJSmodel
The OSM is a compartment model with two reservoirs of oxygen. The first reservoir represents the oxygen in the alveolar space and the second reservoir represents the oxygen stored in the mixed venous blood. For each of these reservoirs a differential equation is used to describe the change of oxygen fraction or concentration, respec~ive~. .The differential equation describing the mass balance for the alveolar compartment is given as Eq. P in Fig. 1. d/d@O;)
= (&OS- A - QO,” - ‘),WA
(1)
equation 1 states that the derivative of the fraction of oxygen in the alveoli FOF can be calculated by accounting for the flow of oxygen QOi-* into the alveolar compartment (A> from the environment (e) and for the flow of oxygen QOfmc out of the alveolar compartment into the lung capillaries (c). The oxygen is distributed over the alveolar volume VA. The superscript E denotes Expired to indicate that we assume that the alveolar oxygen fraction FOF can be approximated by the endtidal expired oxygen fraction. This is a reasonable assumption for patients without lung diseases, but is clearly less valid for patients with severe obst~ctive pulmonary disease. The differential equation describing the mass balance for the oxygen concentration COT in the
Fig. 1. The physiology covered by the OSM. The vignettes illustrate the physiological mechanisms transposing the oxygen from the ventilator to the tissues. See text for detailed explanation.
mixed venous (mv> compartment structure: d,‘dt(cO!J”)
has the same
= (CO; - COT) CO/P”’
(2)
The net flow of oxygen into the mixed venous compartment per time unit is the venous (v) oxygen concentration CO; minus the mixed venous oxygen concentration COY, multiplied by the cardiac output CO. The volume of the mixed venous compartment is vmv, To complete the OSM model, we need to provide equations that specify the relations between the variables in Eqs. 1 and 2. These equations can be divided into five blocks describing: (9 alveolar ventilation, (ii) gas exchange over the alveolar membrane, (iii) o~genation of the blood, (iv) shunting of pulmonary capillary by mixed venous blood, and (v> oxygenation of the tissues. Each block is depicted as a vignette in Fig. 1. The alveolar ventilation is described by two equations: vellt ,f( QCI-
~-tidal - Vdead)
* = Vent(FO:
- FO:)
(31 (4)
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Eq. 3 states that the amount of air, Vent, that reaches the alveoli per time unit can be calculated as the difference between the tidal volume vtida’ and the dead-space vdead, multiplied by the respiratory frequency f. Eq. 4 states that the net amount of oxygen QOP-” that reaches the alveoli can be calculated as the product of the amount of air that reaches the alveoli (Vent) and the difference between inspired, FO;, and expired, FO:, oxygen fractions. The gas exchange over the alveolar membrane is described by a diffusion equation that specifies the partial pressure of oxygen in the lung capillaries PO;: PO; = FOE BP - Rdi, QO” - ’
1:5)
This equation converts the expired oxygen fraction, FOF, to partial pressure by multiplying it with BP (barometric pressure). The second term in the equation accounts for the drop in oxygen pressure over the alveolar membrane. This drop is the oxygen diffusion resistance Rdi, multiplied with the oxygen flow from the alveoli into the lung capillaries, QO; - ‘. Oxygenation of the blood is described by three equations: 0,cap = cHb - cMetHb - cCOHb SO; = ODC(p0; :(f,(T’)
x x
f,(pH’)
cf’)
CO; = ~0; O+ap +pOS ff0,
x
(6)
f,(BE’)
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factors. Three variables of major importance for the shape of the ODC curve, capillary pH (pH’1, base excess (BE”) and temperature CT’), have been incorporated explicitly in the model, while remaining variables affecting the ODC have been lumped into the correction factor cf’. It was assumed that the effect of the variables could be taken into account by multiplying the partial oxygen pressure ~0; by correction factors depending on each of the variables. The non-linear correction functions f,, f, and f, as well as the general shape of the ODC were taken from Nunn
HI. Eq. 8 states that the concentration of oxygen in the blood, CO?, is both due to the oxygen bound to haemoglobin and to the physically dissolved oxygen. The contribution from the physically dissolved oxygen is the product of the partial oxygen pressure, PO,, and the oxygen solubility coefficient of oxygen in blood a0,. Finally it should be mentioned that equations identical to Eqs. 7 and 8 can be used to specify the relation between oxygen pressure and saturation everywhere in the blood stream. In particular the equations for the saturation SOS and the oxygen concentration CO; for the arterial (a) blood are of practical interest, since arterial blood samples are routinely taken in patients with respiratory malfunction: SO; = ODC(p0; x f,(T”)
(7)
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x
f,(pHa)
x f,(BE”)
x cf’)
CO; = sOi 0,cap +pO; ‘~0,
(7a) (8al
(8)
Eq. 6 calculates the oxygen capacity 0,cap of the blood as the total haemoglobin concentration cHb minus the inactivated haemoglobin, either in the form of methaemoglobin @MetI-%) or in the form of haemoglobin bound to carbon monoxide (CCOHb). The oxygen saturation of haemoglobin sOi in the lung capillaries is described by the oxygen dissociation curve, ODC (Eq. 7). The ODC specifies the relation between the partial pressure of oxygen and the oxygen saturation of haemoglobin. The ODC is influenced by a number of
The shunting of pulmonary capillary blood by mixed venous blood is described by a shunt equation: CO; = CO; - shunt(cOS - cO!J”)
(9)
Eq. 9 is a shunt equation, which keeps track of the mass balance for the mixing of capillary and mixed venous blood. The oxygen concentration of arterial blood, CO;, is lower than the oxygen concentration of lung capillary blood, CO;, due to shunting (shunt) of mixed venous blood with lower oxygen concentration, COT.
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Together, the shunt equation (Eq. 9) and the three equations describing the oxygenation of the blood (Eqs. 6, 7 and 8) provide the functionality of the oxygen status algorithm [4,51. The oxygenation of the tissues is described by a single equation: CO; = CO; - QO; - ’ /CO
(10)
Equation 10 accounts for the reduction in oxygen concentration in venous blood, CO;, relative to arterial blood, CO;, due to the oxygen demand, i.e., the flow of oxygen QOiet from the tissue capillaries to the tissues, divided by cardiac output, co. 2.2. The steady-state equations The model described by Eqs. 1 through 10 is a dynamic model. This means that it is able to describe the time course of the variables after a change in the conditions. However, after a change, the oxygen reservoirs reach equilibrium after a few minutes, and it may therefore be sufficient to consider the steady-state solutions of the equations. In the steady-state all derivatives are zero. When this condition is inserted into Eqs. 1 and 2 it follows that CO; becomes equal to COT. In addition, all the oxygen flows QOt-‘, QOg-A and QO;- t become equal. In the following, CO: will be used to denote the common value of CO: and COT, and Q02 will be used to denote the common value for the oxygen flows. Below Eqs. 3 through 10, which also apply to the steady-state situation are repeated, except that Eq. 4 has been algebraically manipulated to give Eq. 12, and Eqs. 9 and 10 have been manipulated to give Eq. 17: vent
=f
( T/tidal
T/dead)
(11)
FOF = FO; - QO,/Vent
(12)
~0s = FOG BP - Rdi, QO2
(13)
0,cap = cHb - cMetHb - cCOHb
(14)
SO; = ODUpO;
_
x f,(pH?
x f,(BE’)
x f.,(TC) x cfc>
(15)
CO; = sOi 0,cap +pOS ‘~0~
(16)
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CO: = CO; - QO, /(CO
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(1 - shunt))
CO; = CO; - shunt (CO: - CO; >
(17) (18)
In addition to these equations, Eqs. 7a and 8a still apply for the arterial oxygen pressure and concentration. The steady-state equations, Eqs. 11 through 18, form a system of eight non-linear equations which can be solved by numerical methods. 2.-.‘3 CPN implementation
of the steady-state model
In this section an alternative implementation of the equations will be presented, where the equations will be represented as a causal probabilistic network (CPN) [6-81. First, we shall establish the structure of the CPN. In the CPN terminology we shall call a variable that appears on the left side of the equality sign a child, and the variables that appear on the right side will be called the child’s parents. In a CPN each variable is represented by a node in the network, and each child receives directed links (arrows) from each of its parents. Inspection of the CPN in Fig. 2 reveals that the structure of the CPN matches the structure of the 8 steady-state equations. For example, it can be seen that SO; receives arrows from each of its parents, PO:, pHc, BEC, TC, and cf’. This reflects the structure of Eq. 15. One of the properties of CPNs that make them suited for medical applications is that uncertainty can be expressed and handled in CPNs [9]. We will exploit the ability of CPNs to handle two different types of uncertainty. The first type of uncertainty has to do with uncertainty in the relations between the variables, as specified by the steady-state equations. The second type of uncertainty has to do with the uncertainty in the parameters. For example, the exact volume of the dead-space may not be known, but we would still like to be able to perform some computations, taking into account that the volume of the deadspace is uncertain. Handling of both of these types of uncertainty requires that we consider the variables in the model to be stochastic, rather than deterministic,
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cHb
q
?!!I
co2’I
Fig. 2. The OSM implemented as a CPN. The boxes represent stochastic variables, which are connected with arrows. The arrows represent conditional probabilities that specify the relations between the variables.
variables. To handle the first type of uncertainty, we replace the ‘hard’ functional relations, specified by the equations, by ‘soft’ specifications of conditional probability distributions. For example, Eq. 15 is replaced by a specification of the conditional probability distribution for sOi given the parents PO;, pH’, BE’, TC and cfC: p(sO; 1 PO;, pH’, BE’, TC, cf’)
(l9)
In the limiting case, where the conditional probability distribution is very narrow, the conditional probability distribution specified by Eq. 19 is identical to the deterministic relation specified by Eq. 15. The uncertainty in SOS reflects both measurements errors and the influence of factors that in principle could have been taken into account in the model, but were omitted. In the case of ~0; such factors could for example be the fraction of foetal haemoglobin and the concentration of 2,3diphosphoglycerate. Both of these factors were omitted from the model in the interest of simplicity, and now appear as a contribution to the uncertainty in the conditional probability distribution for SO& The conditional probability dis-
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tribution specified in the CPN was chosen to reflect the experimental observation that for high oxygen pressures, the saturation is close to 100% and has a standard deviation that is less than 1%. For lower oxygen pressures the standard deviation of sOi is about 3% [lo]. The second type of uncertainty, the uncertainty of the parameters, is taken into account by giving each parameter an a priori probability distribution. Calculations in the CPN will take into account both the uncertainty stemming from the parameters and the uncertainty from the ‘soft’ relations between the variables. When measurements are inserted into the CPN, this results in a Bayesian updating of the probability distributions of all variables, including those that we usually refer to as parameters. In general, insertion of measurements into the CPN replaces the wide a priori probability distributions with more narrow probability distributions that are compatible with the measurements. The CPN was implemented in the Hugin shell [11,12]. CPNs can only handle continuous stochastic variables without incurring unacceptable computational costs, if the variables are linearly related. Since most relations between variables in the OSM model are non-linear, the continuous variables must be converted to discrete variables. This is done by sampling the variables, and using the sampling points as states in the corresponding discrete variables [13,141. The sampling involves some compromises [14]. If the sampling is too sparse, the computational accuracy suffers, and if the sampling is too dense, the number of states in the discrete variables becomes too large. This causes the requirements for computation time to increase exponentially. :3. Results 3.1. Comparison of simulation results and patient data
The OSM can to variations in section we shall the response of
be used to simulate the response the model’s parameters. In this restrict ourselves to simulating the model to different inspired
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oxygen concentrations for a range of values for shunt and oxygen diffusion resistance, Rditi. At the end of the section the simulation results will be compared to data from a patient with respira.. tory problems. In all simulations it will be assumed, unless otherwise noted, that the parameters of the model have the values given in Table 1. The simulation results are essentially identical. for the two implementations of the model, and only results from the implementation based on the equations will be shown. Figure 3 shows how the arterial oxygen saturation SO; depends on the fraction of inspired oxygen FO:, when the oxygen diffusion resistance &, is zero. For the curve labelled 0% in Fig. 3 the shunt is also zero, and the curve is therefore identical to the oxygen dissociation curve. The remaining curves in Fig. 3 show how increasing pulmonary shunting decreases the arterial oxygen saturation. In Fig. 4A is shown the situation where it is diffusion resistance, rather than shunt, that is the reason for the incomplete oxygenation of the arterial blood. It is assumed that the shunt is zero, and again the curve for R,i, = 0 corresponds to oxygen dissociation curve. The remaining curves show how increasing Rdi, reduces the
Table 1 Standard parameter values for the OSM = 14 mini’
f v
tidal
@%I
g2 cHb cMetHb cCOHb CrO? pH” pHC BE” BE’ T’l TC cf8 cfC BP
=
0.5
-0.15
= = = = = = = = = = = = =
I I
5 I/mm 11.5 mmol/min 9.0 mmol/l 0 mmol/l 0 mmol/l 0.0102 mmol/(l 7.4 7.4 0 mmol/l 0 mmol/l 37°C 37°C 1 I = 101.3 kPa
kPa)
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Fig. 3. Arterial oxygen saturation (SO;) as a function of inspired oxygen fraction FO:. Rdift = 0 and shunt ranging from 0% to 40% in steps of 5%. The curve at the top corresponds to a shunt of 0% and the curve at the bottom to a shunt of 40%. Other parameters as given in Table 1.
arterial oxygenation for Rditi up to 330 kPa/(l/min). With such a high Rdi, even 100% inspired oxygen is not sufficient to ensure a complete oxygenation of the arterial blood. The curves in Fig. 4A are considerably steeper than those in Fig. 3. This seems reasonable: when shunting is the main problem, then the capillary blood that is not shunted is already fully oxygenated. Increasing the inspired oxygen fraction only increases the capillary oxygen concentration a little because a small amount of oxygen will be dissolved physically. When diffusion resistance is the main problem, then an increase in inspired oxygen fraction will markedly increase the diffusion of oxygen across the alveolar membrane. The model can also be used to simulate situations where a patient has a combination of shunt and an increased diffusion resistance. An example is shown in Fig. 4B, where simulations were performed for a shunt of 15% and a range of diffusion resistances. In Fig. 5 data from a patient with pulmonary dysfunction are plotted. The data were obtained during adjustment of the inspired oxygen fraction and measurements of blood gases were made from five arterial punctures, taken at approxi-
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i
98
96
94
92
92
90
88 20
30
40
50
@I
70
80
100
FO$
(S)
Fig. 5. Arterial oxygen saturation (SO;) inspired oxygen fraction FO:. As in Fig. terS from a patient as given in Table 2. from the patient are plotted (filled circles). mately going through the two lowest data tion performed with shunt = 18% and RJitt
Table 2 Data for a patient pericardiac cavity Time 20
30
40
50
60
FOT’
70
80
90
after
t,
thoracotomy
as a function of 3. but with parameThe measurements The curve approxipoints is a simula= 64 kPa/(l/min).
with
Time
exploration
of the
t2
1w
(7oh)
Fig. 4. Arterial oxygen saturation (SO:) as a function of inspired oxygen fraction FO:. A: Shunt = 0% and Rdirr range from 0 to 330 kPa/(l/min) in steps of 30 kPa/(l/min). The curve at the top corresponds to an Rdi, of 0 kPa/(l/min) and the curve at the bottom corresponds to an Rdi, of 330 kPa/(l/min). B: Shunt = 15% and R,,, range is from 0 to 330 kPa/(l/min) in steps of 30 kPa/(l/min). Other parameters as given in Table 1.
mately 15-min intervals, to allow the oxygen to equilibrate. The data are plotted on top of simulations where shunt alone causes the reduced oxygenation. The curves are a little displaced relative to the curves in Fig. 3. This is because the curves in Fig. 3 were plotted for a patient with standard hemodynamic parameters as given in Table 1, while the curves in Fig. 5 were calculated for the parameters for the patient as given in Table 2. It is apparent that the sharp drop in
FO: SO; cHb pH3 BEa T” CO /,ttdal QO2
% % mmol/l 7.38 mmol/l “C I/min I mmol/min
25 90.9 6.6 7.37 5.0 37.3 5.2 0.34 8.14
35 95.1 6.6 4.8 37.4 5.4 0.34 8.38
Arterial blood samples were taken twice (t, and t,) with a 15-min interval. Data not stated are identical to values given in Table 2, except that it is assumed that pHC = pH”. BEC = EIEa and TC = T”. The patient had a Swan-Ganz catheter, enabling estimation of cardiac output, CO. and the oxygen demand, QO,.
oxygenation for inspired oxygen fractions less than 40% in the data from the patient can not be explained by shunt alone. The curve that approximately goes through the two lowest of the patient’s data points are from a simulation with shunt =
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18% and &, = 64 kPa/(l/min). This means that the response of this patient to variations in inspired oxygen fraction is better explained by assuming that the patient has both a significant pulmonary shunt and a significant diffusion problem, rather than by assuming that the patient only has an increased shunt.
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Kdiff(kPa/l/min)
3.2. Estimation of shunt and Rdi,
We now want to consider the possibility of calculating estimates for shunt and Rdiff from data obtained during variations in inspired oxygen fraction. Data from the patient given in Fig. 5 will be used as an example. Table 2 gives the measured values corresponding to the two data points in Fig. 5 with the lowest inspired oxygen fraction, 25% and 35%, respectively. We assume that pH, base excess and temperature in the blood in the lung capillaries (pH’, BE’, and T’) are identical to pH, base excess and temperature in the arterial blood (pH”, BE”, and T”). Inspection of the eight steady-state equations, Eqs. 31 through 18, reveals that we have eight equations with nine unknowns, seven variables (Vent, FOF, PO:, O,cap, sOi, CO; and cOV,>and 2 parameters (shunt and Rditi). This gives a single infinity of solutions to the set of equations. From Fig. 6A we can see that if Rdi, is zero, then the measured oxygen saturation sOi = 90.9% can be explained by assuming that shunt = 31%. Alternatively, if shunt = 0, then R,i, = 78 kPa/(l/min). The measured oxygen saturation can equally well be explained by any combination of shunt and Rdi, that falls on the line in Fig. 6A connecting those two points. We now consider the measurements taken 15 min later (t2 column in Table 2). When the 8 steady-state equations are applied to those measurements, we get another single infinity of solutions for shunt and Rdiff, as indicated by the line in Fig. 6B. The only solution to the complete set of 16 equations generated by applying the steady-state equations to the two sets of measurements in Table 2 can be found as the intersection of the two lines in Figs. 6A and 6B, as shown in Fig. 6C. The intersection corresponds to shunt = 18% and Rdi,, = 64 kPa/(l/min). This demon-
shunt
(%I
Fig 6. The full lines give solutions for shunt and R,,, when the steady-state equations are applied to the measurements in Table 2. The dashed lines indicate the solutions for SO: 1% higher or lower than actually measured. A: Solutions based on the first set of measurements in Table 2. B: Solutions based on the second set of measurements. C: Superposition of the solutions. The intersection of the full lines gives the solution to shunt and Rdi, that fits with both sets of measurements.
strates that estimates of shunt and Rdi, can be obtained from measurements that are routinely available. The line that approximately fits the patient data in Fig. 5 is plotted for these two values of shunt and Rdi,.
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3.3. Sensitivity analysis
Unfortunately the calculations do not reveal how reliable the estimates of shunt and R,,, are. To answer this question a multidimensional sensitivity analysis should be carried out, where all possible variations of all the measured and estimated parameters are considered. We will not attempt to do this, but we will consider the effect of the uncertainty in the measured oxygen saturation. We will use the assumption that due to the combined effect of measurement error and biological variability of haemoglobin’s oxygen dissociation curve we have an uncertainty of 1% on the measured oxygen saturation sOi. The dashed lines in Fig. 6A and 6B indicate how the solutions for shunt and Rdi, would have been affected if the measured ~0; had been 1% higher or lower. This gives a considerable uncertainty in the parameter estimates as can be seen from Fig. 6C.
50
0 0
5
IO
0
5
IO
0
5
IO
20
25
30
35
40
20
25
30
35
‘lo
20
2.5
30
35
150
50
0 15
100
xl
0
3.4. Estimation by the CPN
Estimates of shunt and R,,, can also be provided by the CPN implementation of the steadystate equations. Insertion of the first set of measurements from Table 2 into the CPN (Fig. 2) gives a contour plot of the joint probability distribution (Fig. 7A). The contour lines are not smooth in the plot. This is due to the previously mentioned compromise between the number of states in the discrete stochastic variables and the computational accuracy. Comparison of Fig. 6A and Fig. 7A shows that the set of equations and the CPN arrive at approximately the same solution. The contour plot provides the same information as provided by the sensitivity analysis, except that the CPN in principle takes into account the uncertainty from all parameters simultaneously. Insertion of the second set of measurements in Table 2 gives Fig. 7B. The CPN can also be used to solve the 16 equations that are generated when the eight steady-state equations are applied to both sets of measurements. To do this, two identical copies of the CPN are constructed, where the two copies share shunt and R,i, as common parameter nodes. Figure 7C shows the joint probability dis-
shunt
C‘S)
Fig. 7. Contour plots of the joint probability of shunt and A: Joint probability after insertion of the first set of measurements. B: Joint probability after insertion of the second set of measurements. C: Joint probability after insertion of both sets of measurements. R J,ti.
tribution for shunt and Rdi, when both sets of measurements are inserted in the double CPN. 4. Discussion
A model has been developed that can be used to simulate how the oxygen status depends on a number of physiological parameters. The model contains the pulmonary shunt equation (Eq. 9) and three equations for the oxygenation of the blood (Eqs. 6,7 and 8). Together, these equations allow determination of the same parameters as the oxygen status algorithm [4,5]. In addition the model contains a diffusion equation that describes the diffusion of oxygen over the alveolar membrane and a simple description of circulation, oxygen consumption and the ventilator. The model shows that the response to variations in inspired oxygen fraction depends on
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whether the dominating respiratory problem is pulmonary shunting or increased resistance to diffusion of oxygen. It is proposed that this difference in the response to variations in inspired oxygen fraction can be used to estimate pulmonary shunt and diffusion resistance, simply by taking an arterial blood sample at two different inspired oxygen fractions. Data from the patient shown in Fig. 5 indicate that the patient’s response to variations in inspired oxygen fraction is much better accounted for if it is assumed that both pulmonary shunt and diffusion resistance contribute to the respiratory problems, rather than pulmonary shunt alone. Distinguishing between pulmonary shunt and di~sion resistance as the primary cause of insufficient oxygenation of the arterial blood may be of clinical importance, since therapeutic interventions such as positive end-expiratory pressure, inhalation of nitric oxide, surfactant therapy, adjustment of inspired oxygen fraction, etc. then can be selected to address the most important cause. The model has the implicit assumption that the lung is uniformly affected. This assumption is shared with the oxygen status algo~thm, but it must be expected that both shunt and diffusion resistance are different in different regions of the lung. How important this is for the model is not clear, although it is encouraging that an estimation of the parameters based on two out of the five measurements in Fig. 5 can reasonably predict the oxygen samrations for the remaining three measurements. It would be tempting to use all five measurement, covering a wide range of inspired oxygen fractions, to determine shunt and &,. This would create some technical problems, because the set of equations would become overdetermined, with more equations than unknowns. If the equations are solved, using the CPN representation, this will not be a problem. The CPN will then find the solution for shunt and R,,, that best fits all the measurements. As previously mentioned this approach also has the advantage that it provides an assessment of the reliability of the estimates. However, until further experiments have given a better understanding of the physiological interpretation of the diffusion resistance, it is cautiously noted that at least the model can provide
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a prediction of the patient’s response to variations in inspired oxygen fraction around a couple of measurements in the range of !30-100% oxygen saturation. Until it has been experimentally verified in a number of patients how well the model describes the patients’ responses to large vacation in inspired oxygen fraction the physiological interpretation of shunt and R,i, should be done cautiously. References
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Hugh Expert, HUGIN Runtime, Users Guide, version 2.0. (Hugin Expert A/S, Aalborg, Denmark, 1991). [ 131 K G. Olesen and S. Andreassen, Specification of models in large expert systems based on causal probabilistic nttworks. Artificial Intelligence Med. 5 (1993) 269-281.
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[14] S. Andreassen, Knowledge representation by extended linear models, in Deep models for medical knowledge engineering, ed. E. Keravnou, pp. 129-145 (Elsevier, Amsterdam, 1992).
Computer Methods and Programs in Biomedicine 51 (1996) 95-105
Estimation of pulmonary diffusion resistance and shunt in an oxygen status model S. Andreassen*a,J. Egeberg”, M.P. Schrijtera, P.T. Andersen” aDepartment
of Medical
Informatics hDepartment
and Image Analysis, Aalboq Uniuersi& Fr. Bajersuej of Intensive Care, Aalborg Sygehus. DK-9000 Aalborg,
70. DK-9220 Denmark
Aalborg,
Denmark
Abstract A compartment model of the transport of oxygen from the alveoli to the tissues is described. In patients with both pulmonary shunt and alveolar resistance to diffusion of oxygen, the model is used to simulate their response to variations in the inspired oxygen fraction. These simulation results are compared to the responses from a patient with respiratory malfunction, indicating that the method can identify patients where not only a pulmonary shunt but also a high alveolar resistance to diffusion of oxygen is clinically significant. Estimation of pulmonary shunt and oxygen diffusion resistance can be done in two different implementations of the model. In the first implementation the estimates are generated by numerical solution of the equations of the compartment model. In the second implementation the equations have been used to construct a causal probabilistic net where biological uncertainties and uncertainties in the measurements can be represented. Keywords:
Oxygen diffusion resistance; Oxygen status model; Pulmonary shunt; Respiratory modelling
1. Introduction One of the primary goals of artificial ventilation is to ensure adequate oxygenation of the arterial blood. Incomplete arterial oxygenation may result either from pulmonary shunt or from
an increased resistance to diffusion of oxygen through the alveolar membrane [l]. In patients with ventilatory malfunction it would be attractive to know the quantitative contribution of these -*Corresponding 0169-2607/96/$15.00 PfISO169-2607(96)01765-S
author.
two mechanisms, since therapeutic interventions aimed at correcting either of these problems are different. Unfortunately the methods available for measuring diffusion resistance are time consuming and technically demanding, requiring the use of an indicator gas and gas analysis equipment [2,3]. One practical approach to this problem has been to ascribe the inadequate oxygenation to shunt alone [4], ignoring the possible contribution from the alveolar resistance to diffusion. Instead we have chosen to use the functionality provided by the oxygen status algorithm (OSA) [51 as a starting point and have added diffusion resis-
Q 1996 Elsevier Science Ireland Ltd. All rights reserved.
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tance and a simple description of circulation, oxygen consumption and the ventilator to the model. The resulting model, the oxygen status model (OSM), can be used to simulate the patient’s response to different therapeutic interventions (simulation mode). The model can also be ‘solved”, using the patient’s response to variations of inspired oxygen fraction as input data, thus providing estimates for shunt and diffusion resistance ~diagoostic mode).
2.17. The aygen SMLJSmodel
The OSM is a compartment model with two reservoirs of oxygen. The first reservoir represents the oxygen in the alveolar space and the second reservoir represents the oxygen stored in the mixed venous blood. For each of these reservoirs a differential equation is used to describe the change of oxygen fraction or concentration, respec~ive~. .The differential equation describing the mass balance for the alveolar compartment is given as Eq. P in Fig. 1. d/d@O;)
= (&OS- A - QO,” - ‘),WA
(1)
equation 1 states that the derivative of the fraction of oxygen in the alveoli FOF can be calculated by accounting for the flow of oxygen QOi-* into the alveolar compartment (A> from the environment (e) and for the flow of oxygen QOfmc out of the alveolar compartment into the lung capillaries (c). The oxygen is distributed over the alveolar volume VA. The superscript E denotes Expired to indicate that we assume that the alveolar oxygen fraction FOF can be approximated by the endtidal expired oxygen fraction. This is a reasonable assumption for patients without lung diseases, but is clearly less valid for patients with severe obst~ctive pulmonary disease. The differential equation describing the mass balance for the oxygen concentration COT in the
Fig. 1. The physiology covered by the OSM. The vignettes illustrate the physiological mechanisms transposing the oxygen from the ventilator to the tissues. See text for detailed explanation.
mixed venous (mv> compartment structure: d,‘dt(cO!J”)
has the same
= (CO; - COT) CO/P”’
(2)
The net flow of oxygen into the mixed venous compartment per time unit is the venous (v) oxygen concentration CO; minus the mixed venous oxygen concentration COY, multiplied by the cardiac output CO. The volume of the mixed venous compartment is vmv, To complete the OSM model, we need to provide equations that specify the relations between the variables in Eqs. 1 and 2. These equations can be divided into five blocks describing: (9 alveolar ventilation, (ii) gas exchange over the alveolar membrane, (iii) o~genation of the blood, (iv) shunting of pulmonary capillary by mixed venous blood, and (v> oxygenation of the tissues. Each block is depicted as a vignette in Fig. 1. The alveolar ventilation is described by two equations: vellt ,f( QCI-
~-tidal - Vdead)
* = Vent(FO:
- FO:)
(31 (4)
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Eq. 3 states that the amount of air, Vent, that reaches the alveoli per time unit can be calculated as the difference between the tidal volume vtida’ and the dead-space vdead, multiplied by the respiratory frequency f. Eq. 4 states that the net amount of oxygen QOP-” that reaches the alveoli can be calculated as the product of the amount of air that reaches the alveoli (Vent) and the difference between inspired, FO;, and expired, FO:, oxygen fractions. The gas exchange over the alveolar membrane is described by a diffusion equation that specifies the partial pressure of oxygen in the lung capillaries PO;: PO; = FOE BP - Rdi, QO” - ’
1:5)
This equation converts the expired oxygen fraction, FOF, to partial pressure by multiplying it with BP (barometric pressure). The second term in the equation accounts for the drop in oxygen pressure over the alveolar membrane. This drop is the oxygen diffusion resistance Rdi, multiplied with the oxygen flow from the alveoli into the lung capillaries, QO; - ‘. Oxygenation of the blood is described by three equations: 0,cap = cHb - cMetHb - cCOHb SO; = ODC(p0; :(f,(T’)
x x
f,(pH’)
cf’)
CO; = ~0; O+ap +pOS ff0,
x
(6)
f,(BE’)
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factors. Three variables of major importance for the shape of the ODC curve, capillary pH (pH’1, base excess (BE”) and temperature CT’), have been incorporated explicitly in the model, while remaining variables affecting the ODC have been lumped into the correction factor cf’. It was assumed that the effect of the variables could be taken into account by multiplying the partial oxygen pressure ~0; by correction factors depending on each of the variables. The non-linear correction functions f,, f, and f, as well as the general shape of the ODC were taken from Nunn
HI. Eq. 8 states that the concentration of oxygen in the blood, CO?, is both due to the oxygen bound to haemoglobin and to the physically dissolved oxygen. The contribution from the physically dissolved oxygen is the product of the partial oxygen pressure, PO,, and the oxygen solubility coefficient of oxygen in blood a0,. Finally it should be mentioned that equations identical to Eqs. 7 and 8 can be used to specify the relation between oxygen pressure and saturation everywhere in the blood stream. In particular the equations for the saturation SOS and the oxygen concentration CO; for the arterial (a) blood are of practical interest, since arterial blood samples are routinely taken in patients with respiratory malfunction: SO; = ODC(p0; x f,(T”)
(7)
51 (1996)
x
f,(pHa)
x f,(BE”)
x cf’)
CO; = sOi 0,cap +pO; ‘~0,
(7a) (8al
(8)
Eq. 6 calculates the oxygen capacity 0,cap of the blood as the total haemoglobin concentration cHb minus the inactivated haemoglobin, either in the form of methaemoglobin @MetI-%) or in the form of haemoglobin bound to carbon monoxide (CCOHb). The oxygen saturation of haemoglobin sOi in the lung capillaries is described by the oxygen dissociation curve, ODC (Eq. 7). The ODC specifies the relation between the partial pressure of oxygen and the oxygen saturation of haemoglobin. The ODC is influenced by a number of
The shunting of pulmonary capillary blood by mixed venous blood is described by a shunt equation: CO; = CO; - shunt(cOS - cO!J”)
(9)
Eq. 9 is a shunt equation, which keeps track of the mass balance for the mixing of capillary and mixed venous blood. The oxygen concentration of arterial blood, CO;, is lower than the oxygen concentration of lung capillary blood, CO;, due to shunting (shunt) of mixed venous blood with lower oxygen concentration, COT.
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Together, the shunt equation (Eq. 9) and the three equations describing the oxygenation of the blood (Eqs. 6, 7 and 8) provide the functionality of the oxygen status algorithm [4,51. The oxygenation of the tissues is described by a single equation: CO; = CO; - QO; - ’ /CO
(10)
Equation 10 accounts for the reduction in oxygen concentration in venous blood, CO;, relative to arterial blood, CO;, due to the oxygen demand, i.e., the flow of oxygen QOiet from the tissue capillaries to the tissues, divided by cardiac output, co. 2.2. The steady-state equations The model described by Eqs. 1 through 10 is a dynamic model. This means that it is able to describe the time course of the variables after a change in the conditions. However, after a change, the oxygen reservoirs reach equilibrium after a few minutes, and it may therefore be sufficient to consider the steady-state solutions of the equations. In the steady-state all derivatives are zero. When this condition is inserted into Eqs. 1 and 2 it follows that CO; becomes equal to COT. In addition, all the oxygen flows QOt-‘, QOg-A and QO;- t become equal. In the following, CO: will be used to denote the common value of CO: and COT, and Q02 will be used to denote the common value for the oxygen flows. Below Eqs. 3 through 10, which also apply to the steady-state situation are repeated, except that Eq. 4 has been algebraically manipulated to give Eq. 12, and Eqs. 9 and 10 have been manipulated to give Eq. 17: vent
=f
( T/tidal
T/dead)
(11)
FOF = FO; - QO,/Vent
(12)
~0s = FOG BP - Rdi, QO2
(13)
0,cap = cHb - cMetHb - cCOHb
(14)
SO; = ODUpO;
_
x f,(pH?
x f,(BE’)
x f.,(TC) x cfc>
(15)
CO; = sOi 0,cap +pOS ‘~0~
(16)
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CO: = CO; - QO, /(CO
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(1 - shunt))
CO; = CO; - shunt (CO: - CO; >
(17) (18)
In addition to these equations, Eqs. 7a and 8a still apply for the arterial oxygen pressure and concentration. The steady-state equations, Eqs. 11 through 18, form a system of eight non-linear equations which can be solved by numerical methods. 2.-.‘3 CPN implementation
of the steady-state model
In this section an alternative implementation of the equations will be presented, where the equations will be represented as a causal probabilistic network (CPN) [6-81. First, we shall establish the structure of the CPN. In the CPN terminology we shall call a variable that appears on the left side of the equality sign a child, and the variables that appear on the right side will be called the child’s parents. In a CPN each variable is represented by a node in the network, and each child receives directed links (arrows) from each of its parents. Inspection of the CPN in Fig. 2 reveals that the structure of the CPN matches the structure of the 8 steady-state equations. For example, it can be seen that SO; receives arrows from each of its parents, PO:, pHc, BEC, TC, and cf’. This reflects the structure of Eq. 15. One of the properties of CPNs that make them suited for medical applications is that uncertainty can be expressed and handled in CPNs [9]. We will exploit the ability of CPNs to handle two different types of uncertainty. The first type of uncertainty has to do with uncertainty in the relations between the variables, as specified by the steady-state equations. The second type of uncertainty has to do with the uncertainty in the parameters. For example, the exact volume of the dead-space may not be known, but we would still like to be able to perform some computations, taking into account that the volume of the deadspace is uncertain. Handling of both of these types of uncertainty requires that we consider the variables in the model to be stochastic, rather than deterministic,
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cHb
q
?!!I
co2’I
Fig. 2. The OSM implemented as a CPN. The boxes represent stochastic variables, which are connected with arrows. The arrows represent conditional probabilities that specify the relations between the variables.
variables. To handle the first type of uncertainty, we replace the ‘hard’ functional relations, specified by the equations, by ‘soft’ specifications of conditional probability distributions. For example, Eq. 15 is replaced by a specification of the conditional probability distribution for sOi given the parents PO;, pH’, BE’, TC and cfC: p(sO; 1 PO;, pH’, BE’, TC, cf’)
(l9)
In the limiting case, where the conditional probability distribution is very narrow, the conditional probability distribution specified by Eq. 19 is identical to the deterministic relation specified by Eq. 15. The uncertainty in SOS reflects both measurements errors and the influence of factors that in principle could have been taken into account in the model, but were omitted. In the case of ~0; such factors could for example be the fraction of foetal haemoglobin and the concentration of 2,3diphosphoglycerate. Both of these factors were omitted from the model in the interest of simplicity, and now appear as a contribution to the uncertainty in the conditional probability distribution for SO& The conditional probability dis-
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tribution specified in the CPN was chosen to reflect the experimental observation that for high oxygen pressures, the saturation is close to 100% and has a standard deviation that is less than 1%. For lower oxygen pressures the standard deviation of sOi is about 3% [lo]. The second type of uncertainty, the uncertainty of the parameters, is taken into account by giving each parameter an a priori probability distribution. Calculations in the CPN will take into account both the uncertainty stemming from the parameters and the uncertainty from the ‘soft’ relations between the variables. When measurements are inserted into the CPN, this results in a Bayesian updating of the probability distributions of all variables, including those that we usually refer to as parameters. In general, insertion of measurements into the CPN replaces the wide a priori probability distributions with more narrow probability distributions that are compatible with the measurements. The CPN was implemented in the Hugin shell [11,12]. CPNs can only handle continuous stochastic variables without incurring unacceptable computational costs, if the variables are linearly related. Since most relations between variables in the OSM model are non-linear, the continuous variables must be converted to discrete variables. This is done by sampling the variables, and using the sampling points as states in the corresponding discrete variables [13,141. The sampling involves some compromises [14]. If the sampling is too sparse, the computational accuracy suffers, and if the sampling is too dense, the number of states in the discrete variables becomes too large. This causes the requirements for computation time to increase exponentially. :3. Results 3.1. Comparison of simulation results and patient data
The OSM can to variations in section we shall the response of
be used to simulate the response the model’s parameters. In this restrict ourselves to simulating the model to different inspired
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oxygen concentrations for a range of values for shunt and oxygen diffusion resistance, Rditi. At the end of the section the simulation results will be compared to data from a patient with respira.. tory problems. In all simulations it will be assumed, unless otherwise noted, that the parameters of the model have the values given in Table 1. The simulation results are essentially identical. for the two implementations of the model, and only results from the implementation based on the equations will be shown. Figure 3 shows how the arterial oxygen saturation SO; depends on the fraction of inspired oxygen FO:, when the oxygen diffusion resistance &, is zero. For the curve labelled 0% in Fig. 3 the shunt is also zero, and the curve is therefore identical to the oxygen dissociation curve. The remaining curves in Fig. 3 show how increasing pulmonary shunting decreases the arterial oxygen saturation. In Fig. 4A is shown the situation where it is diffusion resistance, rather than shunt, that is the reason for the incomplete oxygenation of the arterial blood. It is assumed that the shunt is zero, and again the curve for R,i, = 0 corresponds to oxygen dissociation curve. The remaining curves show how increasing Rdi, reduces the
Table 1 Standard parameter values for the OSM = 14 mini’
f v
tidal
@%I
g2 cHb cMetHb cCOHb CrO? pH” pHC BE” BE’ T’l TC cf8 cfC BP
=
0.5
-0.15
= = = = = = = = = = = = =
I I
5 I/mm 11.5 mmol/min 9.0 mmol/l 0 mmol/l 0 mmol/l 0.0102 mmol/(l 7.4 7.4 0 mmol/l 0 mmol/l 37°C 37°C 1 I = 101.3 kPa
kPa)
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Fig. 3. Arterial oxygen saturation (SO;) as a function of inspired oxygen fraction FO:. Rdift = 0 and shunt ranging from 0% to 40% in steps of 5%. The curve at the top corresponds to a shunt of 0% and the curve at the bottom to a shunt of 40%. Other parameters as given in Table 1.
arterial oxygenation for Rditi up to 330 kPa/(l/min). With such a high Rdi, even 100% inspired oxygen is not sufficient to ensure a complete oxygenation of the arterial blood. The curves in Fig. 4A are considerably steeper than those in Fig. 3. This seems reasonable: when shunting is the main problem, then the capillary blood that is not shunted is already fully oxygenated. Increasing the inspired oxygen fraction only increases the capillary oxygen concentration a little because a small amount of oxygen will be dissolved physically. When diffusion resistance is the main problem, then an increase in inspired oxygen fraction will markedly increase the diffusion of oxygen across the alveolar membrane. The model can also be used to simulate situations where a patient has a combination of shunt and an increased diffusion resistance. An example is shown in Fig. 4B, where simulations were performed for a shunt of 15% and a range of diffusion resistances. In Fig. 5 data from a patient with pulmonary dysfunction are plotted. The data were obtained during adjustment of the inspired oxygen fraction and measurements of blood gases were made from five arterial punctures, taken at approxi-
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101
i
98
96
94
92
92
90
88 20
30
40
50
@I
70
80
100
FO$
(S)
Fig. 5. Arterial oxygen saturation (SO;) inspired oxygen fraction FO:. As in Fig. terS from a patient as given in Table 2. from the patient are plotted (filled circles). mately going through the two lowest data tion performed with shunt = 18% and RJitt
Table 2 Data for a patient pericardiac cavity Time 20
30
40
50
60
FOT’
70
80
90
after
t,
thoracotomy
as a function of 3. but with parameThe measurements The curve approxipoints is a simula= 64 kPa/(l/min).
with
Time
exploration
of the
t2
1w
(7oh)
Fig. 4. Arterial oxygen saturation (SO:) as a function of inspired oxygen fraction FO:. A: Shunt = 0% and Rdirr range from 0 to 330 kPa/(l/min) in steps of 30 kPa/(l/min). The curve at the top corresponds to an Rdi, of 0 kPa/(l/min) and the curve at the bottom corresponds to an Rdi, of 330 kPa/(l/min). B: Shunt = 15% and R,,, range is from 0 to 330 kPa/(l/min) in steps of 30 kPa/(l/min). Other parameters as given in Table 1.
mately 15-min intervals, to allow the oxygen to equilibrate. The data are plotted on top of simulations where shunt alone causes the reduced oxygenation. The curves are a little displaced relative to the curves in Fig. 3. This is because the curves in Fig. 3 were plotted for a patient with standard hemodynamic parameters as given in Table 1, while the curves in Fig. 5 were calculated for the parameters for the patient as given in Table 2. It is apparent that the sharp drop in
FO: SO; cHb pH3 BEa T” CO /,ttdal QO2
% % mmol/l 7.38 mmol/l “C I/min I mmol/min
25 90.9 6.6 7.37 5.0 37.3 5.2 0.34 8.14
35 95.1 6.6 4.8 37.4 5.4 0.34 8.38
Arterial blood samples were taken twice (t, and t,) with a 15-min interval. Data not stated are identical to values given in Table 2, except that it is assumed that pHC = pH”. BEC = EIEa and TC = T”. The patient had a Swan-Ganz catheter, enabling estimation of cardiac output, CO. and the oxygen demand, QO,.
oxygenation for inspired oxygen fractions less than 40% in the data from the patient can not be explained by shunt alone. The curve that approximately goes through the two lowest of the patient’s data points are from a simulation with shunt =
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18% and &, = 64 kPa/(l/min). This means that the response of this patient to variations in inspired oxygen fraction is better explained by assuming that the patient has both a significant pulmonary shunt and a significant diffusion problem, rather than by assuming that the patient only has an increased shunt.
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Kdiff(kPa/l/min)
3.2. Estimation of shunt and Rdi,
We now want to consider the possibility of calculating estimates for shunt and Rdiff from data obtained during variations in inspired oxygen fraction. Data from the patient given in Fig. 5 will be used as an example. Table 2 gives the measured values corresponding to the two data points in Fig. 5 with the lowest inspired oxygen fraction, 25% and 35%, respectively. We assume that pH, base excess and temperature in the blood in the lung capillaries (pH’, BE’, and T’) are identical to pH, base excess and temperature in the arterial blood (pH”, BE”, and T”). Inspection of the eight steady-state equations, Eqs. 31 through 18, reveals that we have eight equations with nine unknowns, seven variables (Vent, FOF, PO:, O,cap, sOi, CO; and cOV,>and 2 parameters (shunt and Rditi). This gives a single infinity of solutions to the set of equations. From Fig. 6A we can see that if Rdi, is zero, then the measured oxygen saturation sOi = 90.9% can be explained by assuming that shunt = 31%. Alternatively, if shunt = 0, then R,i, = 78 kPa/(l/min). The measured oxygen saturation can equally well be explained by any combination of shunt and Rdi, that falls on the line in Fig. 6A connecting those two points. We now consider the measurements taken 15 min later (t2 column in Table 2). When the 8 steady-state equations are applied to those measurements, we get another single infinity of solutions for shunt and Rdiff, as indicated by the line in Fig. 6B. The only solution to the complete set of 16 equations generated by applying the steady-state equations to the two sets of measurements in Table 2 can be found as the intersection of the two lines in Figs. 6A and 6B, as shown in Fig. 6C. The intersection corresponds to shunt = 18% and Rdi,, = 64 kPa/(l/min). This demon-
shunt
(%I
Fig 6. The full lines give solutions for shunt and R,,, when the steady-state equations are applied to the measurements in Table 2. The dashed lines indicate the solutions for SO: 1% higher or lower than actually measured. A: Solutions based on the first set of measurements in Table 2. B: Solutions based on the second set of measurements. C: Superposition of the solutions. The intersection of the full lines gives the solution to shunt and Rdi, that fits with both sets of measurements.
strates that estimates of shunt and Rdi, can be obtained from measurements that are routinely available. The line that approximately fits the patient data in Fig. 5 is plotted for these two values of shunt and Rdi,.
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3.3. Sensitivity analysis
Unfortunately the calculations do not reveal how reliable the estimates of shunt and R,,, are. To answer this question a multidimensional sensitivity analysis should be carried out, where all possible variations of all the measured and estimated parameters are considered. We will not attempt to do this, but we will consider the effect of the uncertainty in the measured oxygen saturation. We will use the assumption that due to the combined effect of measurement error and biological variability of haemoglobin’s oxygen dissociation curve we have an uncertainty of 1% on the measured oxygen saturation sOi. The dashed lines in Fig. 6A and 6B indicate how the solutions for shunt and Rdi, would have been affected if the measured ~0; had been 1% higher or lower. This gives a considerable uncertainty in the parameter estimates as can be seen from Fig. 6C.
50
0 0
5
IO
0
5
IO
0
5
IO
20
25
30
35
40
20
25
30
35
‘lo
20
2.5
30
35
150
50
0 15
100
xl
0
3.4. Estimation by the CPN
Estimates of shunt and R,,, can also be provided by the CPN implementation of the steadystate equations. Insertion of the first set of measurements from Table 2 into the CPN (Fig. 2) gives a contour plot of the joint probability distribution (Fig. 7A). The contour lines are not smooth in the plot. This is due to the previously mentioned compromise between the number of states in the discrete stochastic variables and the computational accuracy. Comparison of Fig. 6A and Fig. 7A shows that the set of equations and the CPN arrive at approximately the same solution. The contour plot provides the same information as provided by the sensitivity analysis, except that the CPN in principle takes into account the uncertainty from all parameters simultaneously. Insertion of the second set of measurements in Table 2 gives Fig. 7B. The CPN can also be used to solve the 16 equations that are generated when the eight steady-state equations are applied to both sets of measurements. To do this, two identical copies of the CPN are constructed, where the two copies share shunt and R,i, as common parameter nodes. Figure 7C shows the joint probability dis-
shunt
C‘S)
Fig. 7. Contour plots of the joint probability of shunt and A: Joint probability after insertion of the first set of measurements. B: Joint probability after insertion of the second set of measurements. C: Joint probability after insertion of both sets of measurements. R J,ti.
tribution for shunt and Rdi, when both sets of measurements are inserted in the double CPN. 4. Discussion
A model has been developed that can be used to simulate how the oxygen status depends on a number of physiological parameters. The model contains the pulmonary shunt equation (Eq. 9) and three equations for the oxygenation of the blood (Eqs. 6,7 and 8). Together, these equations allow determination of the same parameters as the oxygen status algorithm [4,5]. In addition the model contains a diffusion equation that describes the diffusion of oxygen over the alveolar membrane and a simple description of circulation, oxygen consumption and the ventilator. The model shows that the response to variations in inspired oxygen fraction depends on
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whether the dominating respiratory problem is pulmonary shunting or increased resistance to diffusion of oxygen. It is proposed that this difference in the response to variations in inspired oxygen fraction can be used to estimate pulmonary shunt and diffusion resistance, simply by taking an arterial blood sample at two different inspired oxygen fractions. Data from the patient shown in Fig. 5 indicate that the patient’s response to variations in inspired oxygen fraction is much better accounted for if it is assumed that both pulmonary shunt and diffusion resistance contribute to the respiratory problems, rather than pulmonary shunt alone. Distinguishing between pulmonary shunt and di~sion resistance as the primary cause of insufficient oxygenation of the arterial blood may be of clinical importance, since therapeutic interventions such as positive end-expiratory pressure, inhalation of nitric oxide, surfactant therapy, adjustment of inspired oxygen fraction, etc. then can be selected to address the most important cause. The model has the implicit assumption that the lung is uniformly affected. This assumption is shared with the oxygen status algo~thm, but it must be expected that both shunt and diffusion resistance are different in different regions of the lung. How important this is for the model is not clear, although it is encouraging that an estimation of the parameters based on two out of the five measurements in Fig. 5 can reasonably predict the oxygen samrations for the remaining three measurements. It would be tempting to use all five measurement, covering a wide range of inspired oxygen fractions, to determine shunt and &,. This would create some technical problems, because the set of equations would become overdetermined, with more equations than unknowns. If the equations are solved, using the CPN representation, this will not be a problem. The CPN will then find the solution for shunt and R,,, that best fits all the measurements. As previously mentioned this approach also has the advantage that it provides an assessment of the reliability of the estimates. However, until further experiments have given a better understanding of the physiological interpretation of the diffusion resistance, it is cautiously noted that at least the model can provide
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a prediction of the patient’s response to variations in inspired oxygen fraction around a couple of measurements in the range of !30-100% oxygen saturation. Until it has been experimentally verified in a number of patients how well the model describes the patients’ responses to large vacation in inspired oxygen fraction the physiological interpretation of shunt and R,i, should be done cautiously. References
[ll J.F. Nunn, Applied respiratory physiology, 3rd Edn. (Butterworth,
1987).
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