Comparing Solutions to the Linear and Nonlinear CFK Equations for Predicting COHb Formation M. V. SMITH* Department of Biostatistics, School of Public Health, University of North Carolina, Chapel Hill, North Carolina 27599 Received 14 July 1988; revised 3 January 1990
ABSTRACT The Coburn, Forster, Kane model of COHb formation as a result of exposure to CO exists in two forms: the linear CFK equation that assumes a constant level of oxyhemoglobin; and the nonlinear CFK equation that allows the oxyhemoglobin level to vary with the carboxyhemoglobin level. Although both equations are currently used, no rigorous analysis exists to show under what conditions the two models determine substantially different solutions. This paper provides such an analysis and shows that the linear model may be used as a reasonable approximation over a much wider range of carboxyhemoglobin levels than had been supposed on physiological grounds.
1.
INTRODUCTION
Inhaled carbon monoxide (CO) competes with oxygen to combine with hemoglobin molecules in the blood forming carboxyhemoglobin (COHb). The toxic effect of carbon monoxide occurs when sufficiently large amounts of COHb are formed and not enough oxygen in the form of oxyhemoglobin (0,Hb) is transported to the body tissues. The processes by which inhaled gases move between the lungs and the blood stream are complex. The rate of diffusion of a gas such as CO across the alveolocapillary membrane depends on the solubility of the gas in the membrane, the diffusion coefficient for the gas in the membrane, the difference in partial pressures across the membrane, and the thickness of the membrane [l]. Many of these quantities do not have uniform values over the entire membrane so estimates represent average values. The overall diffusion rate is referred to as the diffusing capacity or coefficient of the lung for a particular gas (DLCO for carbon monoxide).
*Current
address:
MATHEM4TICAL.
2501 Anne BIOSCIENCES
OElsevier Science Publishing 655 Avenue of the Americas,
Carol Court,
Raleigh,
99:251-263
(1990)
Co., Inc., 1990 New York, NY 10010
NC 27603.
251
252
M. V. SMITH
Both CO and oxygen (0,) in the blood bind with hemoglobin molecules in the red blood cells. The diffusion from the plasma into the red blood cells seems to occur very rapidly [2] and is ignored in the models discussed in this paper. Each hemoglobin molecule has four binding sites. The reaction rates of CO and 0, with hemoglobin differ. Furthermore the reaction rates depend on how many binding sites are occupied and whether these sites are occupied by 0, or CO. (The models we discuss in this paper assume that the different binding rates are all fast enough so that chemical equilibrium can be assumed.) Equations could be written for each process described above, but the resulting complexity would prohibit their utility in applications; therefore simplification is necessary. In 1965 Coburn, Forster, and Kane [3] published the most enduring and most widely used model (CFK model) for the prediction of carboxyhemoglobin (COHb) in humans. The model predicts both the formation of COHb during CO inhalation and the reduction in COHb during CO elimination. The equation used in the original computations assumed that the amount of 0,Hb remained constant at the maximum level, regardless of changes in COHb levels. The resulting equation is linear in the solution [COHbHt). In the original paper the authors suggested that the model would be improved by substituting [O,Hb]-[COHb](t) for the constant 0,Hb parameter. Thejresulting nonlinear equation is integrable, but the solution can not be given explicitly [4]. The original paper [3] validated the linear CFK model against observations made on subjects exposed to very low CO levels over long periods of time. Later investigators tested the nonlinear CFK model against observations made on subjects exposed to a variety of CO profiles including brief exposures to very high CO levels (see e.g., [5, 61). The consensus is that the nonlinear CKF model works quite well even at high levels of CO. However, very little attention has been given to determining the exact conditions under which the linear model becomes inadequate and the nonlinear model is needed. In 1980 Marcus suggested that such a point is reached when the COHb level exceeds the level of free hemoglobin (about 3.6% of the total available hemoglobin) [7]. The suggestion that the linear CFK model could only be used at very low levels of COHb was echoed by Tikuisis in 1987 [S]. In spite of the availability and the proven validity of the nonlinear model, the linear model continues to be used for high levels of COHb when explicit results of specific CO exposure patterns are wanted [8, 91. In this paper we resolve the problem mathematically by determining the range of COHb values for which the solutions of the two models are within a given distance of each other. We begin by resealing the problem so that the only parameters are the initial conditions and a single expression that combines the physiological parameters used in the original model. We
LINEAR
AND NONLINEAR
253
CFK EQUATIONS
restate a comparison theorem of differential equations to which we will frequently refer and present several preliminary lemmas. In the third section we give the principal results of the paper. We then look at a particular case numerically, which will enable us to show some of the results graphically. We close with a brief discussion. 2.
PRELIMINARY
RESULTS
The linear CFK model is usually written d[COHb] dr
_ I’CO VB
I -PZCO B 1/B
as follows, [COHb]PCO, [O,Hb]MB I’B ’
(1)
where [COHb](r)
= the concentration of carboxyhemoglobin in the blood in ml CO/ml blood. I/CO = rate of endogenous CO production in ml CO/min c.007) PZCO =partial pressure of inhaled CO in mm Hg B=l/DLCO+PZ./VA DECO =diffusion coefficient of CO in lungs in ml/mm Hg/min (30) PL = barometric pressure of dry gases, in mm Hg (713) I/A = alveolar ventilation in ml/min (4957) M = Haldane affinity coefficient (240) I/B = blood volume in ml (5500) PCO, =average partial oxygen pressure in the blood in mm Hg (100) [O,Hb] =maximum amount of O2 that can bind with hemoglobin in ml 0, /ml blood c.2).
The numbers in the parentheses refer to typical values [5]. COHb levels are commonly written as a percentage of the maximum saturation of hemoglobin possible (e.g., [3]-[6], [S], [9]). Accordingly we rescale the solution to Y(T) = lOO[COHb]/[O,Hb]. The above equation is then written more simply as
where K and R refer to combinations of the physiological change in the time scale (t = rR/lOO)gives, &/dt=E-YI,
~(0)
=y,,dO,E),
parameters.
A
(2)
254
M. V. SMITH
where E = 100 K/R and is the steady state COHb level predicted by the linear equation under constant exposure conditions. (Throughout this discussion the subscripts 1 and IZ will refer to the linear and nonlinear models respectively.) We note that E is a linear function of the inhaled CO level, PZCO, so that the linear steady state may theoretically be larger than 100%. The nonlinear model is formed by substituting [O,Hb]-[COHb] for [O,Hb] in Equation (1). The same groupings of parameters and scale changes applied to the nonlinear CFK model give, dy,/dt=E-100y,/(100-y,),y(O)=y,,~[0,100E/(100+E)].
(3)
We note that the steady state value for the nonlinear model may be found in terms of E, namely lOOE/(lOO+ E), and that the nonlinear steady state is bounded by 100%. If ylo = y0 = yno two possibilities may occur: the uptake of CO during which E, 100 > y,,y,, > yc, and the elimination of CO during which 100 > y, > yl, y,, > E. We state a well-known version of the comparison theorem [lo] to which we will often refer: LEMMit I (Comparison Theorem) Let f and g be solutions of the following differential equations, y’=F(x,y),
z’=G(x,z),
respectively, satisfying the same initial condition f(a) = g(a). Suppose further that F or G satisfies a Lipschitz condition and that F(x, y) < G(x, y) for all x, y in a given domain. Then f(x) < g(x) for x > a.
We present LEMMA
the following lemmas.
2
Let y, and y,, be solutions to Equations
(2) and
(3) respectively and
y,(O) = y, = y,(O). Then y&t) > y,(t) for all t b 0. Proof The comparison theorem is applied to Equations note that 100 > y, b 0, during both uptake and elimination LEMMA
3
Let yi = E, - y, and y$ = E, - y,, y&t)- y,(t) < E, - E, for all t & 0. 3.
(2) and (3). We of CO.
with ~~(0) = ~~(0). Zf E, > E,,
then
MAIN RESULTS
PROPOSITION
1
Let y, and y, be solutions to Equations (2) and (3) respectively, with E and E) as the corresponding steady states. Assume yfo = yno = y.
lOOE/(lOO+
255 2 y,, > y,
(i.e.
CO is entering the blood).
yr(t)-y,(t) y, G=y,, > E, where y, and y, are solutions to Equations (2) and (3) respectively. If yio = y,,, = y. and y,~(-6+4-)/2, theny,-y,,6.
(7)
The quantity a is a real number that can be adjusted to sharpen the bound within given intervals of y,. Proof: We begin with Equation (6) of the previous proposition and approximate the term 1002/(D - y, + 100) by 1002[(l/(a + loo))--((D - y, - a)/(lOO+ a)2)], its Taylor expansion at y, + a. The error term is easily computed to be
1
(8)
D-y/+100
We hence consider
the following differential
x(t)‘=-y,-100+1002
&[
Using Lemma 1 we see that easily integrated to give,
D(t) 2 x(t)
equation, x-y,-a (100+a)2
I’
for all t 2 0. Equation
(9)
(9) is
The result again follows when x(t) is written in terms of yl. To help choose a, 6 may be substituted for D in (8), and the error term bounded by E,
(
6 - y,1+ 100 I( 6-y/+100 (a+lOO)
-1 12 GE.
(10)
The a-value defined by (9) depends on yI, and intervals of yl corresponding to different a-values can easily be calculated for each 6 and E.
258 4.
M. V. SMITH NUMERICAL
EXAMPLES
We recall that the parameter E used throughout the paper is the steady state solution of the linear CFK equation and may be greater than 100. In terms of the parameters given with the original statement of the model (Equation l), E = lOOM(B I/CO + PICO)/PCO,. For an individual whose physiological parameters have the typical values given in the second section we can say that the endogenous COHb level (the level attained solely by metabolic processes without inhaled CO) is about .3%. Throughout this section, unless otherwise specified, the linear model will be said to provide a “good” approximation of the nonlinear model if the two solutions (both in percent of available COHb) are no further than one percent apart, i.e., 6 is one. Example 1. We choose a simple case of CO uptake in which the initial percent COHb is the endogenous level (.3%), and the individual is suddenly exposed to a constant concentration of CO. Proposition 1 assures us that the linear equation provides a good approximation for any initial COHb level as long as E is less than about 10.5%. Using the parameter values of the second section the corresponding level of inhaled CO is about 60 ppm. In the event that the combination of parameter values and inhaled CO concentration produces an E value greater than 10.5% Proposition 1 no longer applies. However, as shown by Proposition 3, the linear equation may still provide a sufficient approximation over a considerable range of COHb values. In Figure 1 this range is shown graphically for E values between zero and 100. Note that yI will always lie between y0 and E, so that only the lower triangular region is of interest. The region between the bottom curve and the E-axis is defined by Proposition 3 and represents the range of COHb values predicted by the linear equation that provides a “good” approximation for the nonlinear solution. We see as expected that for E values less than 10.5% the range of COHb values reaches the linear equilibrium, i.e., the linear approximation is good throughout the entire trajectory. For the larger E value of 50% the solution to the linear CFK equation will be adequate as long as the values it predicts remain less than about 20%. For the parameter values of the second section an E value of 50 corresponds to an inhaled CO concentration of about 290 ppm. The area above the middle curve is defined by Proposition 4 and shows values of the linear solution that do not meet the one-percent approximation criterion. Values computed by Equation (2) in this range should always be recomputed by the nonlinear equation. The region between the lower two curves is not covered by the results of this paper. The value of the a-parameter used in computing the middle curve was set at - 14. The error E defined by inequality (10) is less than .0005% as long as the computed y-value is less than 31%. The same curves are shown in Figure 2 over a
LINEAR
AND NONLINEAR
CFK EQUATIONS
solutions
259
are more than
solutions
20
1% apart
are less than
40
1%
60
apart
80 E
FIG. 1. Region of 1% approximation during uptake of CO for ya =.3. The bottom region is defined by inequality (5) in the text. The top region is defined by inequality (7) in the text, using a = - 14.
100
Yl 80 solutions are more than
1% apart
60
40
solutions 20
are less than
1% apart
/
I 1-
I
500
I
1000
1500 E
FIG. 2. Region of 1% approximation during uptake of CO for y, = .3. The bottom region is defined by inequality (5) in the text. The top region is defined by inequality (7) in the text, using a = - 25.
260
M. V. SMITH
100 -
YI
solutions are less than
1X apart
solutions
20
40
are more than
1%
apart
60
80 yo
FIG. 3. Region of 1% approximation during elimination of CO with E =.3. The slender top region is defined by inequality (5) in the text. The triangular region to the right is defined by inequality (7) in the text, using a = - 14.
larger interval for E. The a-value of -25 provides that than .Ol% as long as the computed yl-value is less than reference COHb concentrations of 40% and above are for a resting individual. Concentrations below that may exercise ill].
the error E is less 68%. As a point of usually dangerous be dangerous with
Example 2. To study the elimination of CO we assume a relatively high initial COHb level, then remove the CO source thereby giving a lower value for E (say .3%). Proposition 2 shows that the linear solution provides a “good” approximation to the nonlinear solution as long as ye is less than about 9.5% regardless of the E value. Figure 3 shows the regions defined by Propositions 3 and 4 during elimination of CO. In this case the percentage of COHb decreases from the initial level so that, again, only the lower triangular part of the graph is of interest. The area above the middle curve is defined by Proposition 3 and gives the combinations of y, and y, for which the present paper shows that the linear CFK gives an adequate approximation. The region to the right of the lower curve is defined by Proposition 4 and shows the pairs (ya, yr) for
LINEAR
AND NONLINEAR
261
CFK EQUATIONS
which the present paper shows that the solutions are more than 1% apart. For the a-value of - 14, the error in inequality (10) stays below .005% as long as the computed y-value is less than 55%. Note that Proposition 4 does not rule out a good approximation for very low y-values or for the expected y-values near y,. This is reasonable because the linear and nonlinear solutions will tend towards equilibria that are very close together. The nonlinear equilibrium value corresponding to E = .3% is .299%. 5.
DISCUSSION
As can be seen from Figures l-3 the linear model provides a good approximation over a larger range of COHb values during the uptake of CO than during its elimination. The reason is clear when we rewrite the nonlinear model (Equation 3) as y,: = E -Y,
- ~,2/(100-
(11)
Y,>
During the uptake of CO, y, tends to be small initially, especially with respect to the value of E. Hence the final term in Equation (11) will be small and the solution similar to that of Equation (2). During the elimination of CO, the value of E is small with respect to initial y, values. Thus the final term in Equation (11) will be large, and the solutions of the two models diverge quickly. Proposition 3 defines the value y? such that during CO uptake all linear model predictions below y[* are guaranteed to be within a set distance of the nonlinear model prediction. In Figures 1 and 2, yr is plotted as a
TABLE Values
for y:’ During
1 Uptake
E (with corresponding
6 = 0.5 y,=.3
Yo = 5 y, = 10
6 = 1.0 6=1.5 s = 2.0 S = 2.5 6 = 3.0 6 = 1.0 6 = 1.0
‘CO concentrations given in Section 2.
1.9 10.0 10.0 10.0 10.0 10.0 10.0 10.0 are computed
500
100
(290)
(582)
16.9 20.5 22.9 24.6 26.1 27.3 20.6 21.0
21.8 26.1 29.9 32.3 34.3 36.0 26.7 21.0
of CO inhaled
CO in ppmt)
500 (2920)
1000 (5842)
2000 (11684)
37.0 45.0 50.2 54.1 57.3 59.9 45.0 45.1
45.4 54.8 60.7 65.1 68.6 71.4
55.1
54.8 54.8
using the values of the physiological
65.5 71.9 76.4 79.8 82.6 65.5 65.5 parameters
262
M. V. SMITH TABLE Values for yf
During
2 Elimination
of CO
Yo
E =.3
E=5 E=lO
10
20
30
40
50
60
6 = 0.5 6 = 1.0
3.4 .3
6=1.5 6 = 2.0 S = 2.5 s = 3.0 6=1.0 6=1.0
.3 .3 .3 .3
16.7 15.4 12.4 8.2
28.8 27.6 26.2 24.8 23.2 21.5 28.0 28.4
39.2 38.5 37.7 36.8 36.0 35.1 38.7 38.8
49.5 49.0 48.5 47.9 47.4 46.8 49.1 49.2
59.7 59.3 59.0 58.6 58.3 57.9
5 10
.3 .3 16.8 18.0
59.4 59.4
function of E for fixed values of 6 ( = 1) and y, (= .3). In general, yr increases with E, although the rate of increase decreases with 6. The effect of different y, values on yi+ seems to be slight during CO uptake. (See Table 1.) During elimination of CO the solutions to both models begin at a relatively high level, then decrease. Proposition 3 guarantees that any linear model prediction above y;” is within the set distance from the nonlinear model prediction. In Figure 3, y,* is plotted as a function of y0 for fixed values of 6 (= 1) and E (= .3). During uptake of CO increasing the tolerated error 6 necessarily increases the region over which the linear model provides a sufficient approximation. Increasing E has more effect during the elimination of CO than increasing y, did during uptake, especially for the more moderate initial values. (See Table 2.)
The author wishes to acknowledge constructive remarks made by Dr. Keith Muller (Biostatistics Department, UNC-CH) and by the editor of this journal.
REFERENCES J. A. Jacquez, Respiratory R. E. Forster, Diffusion
Physiology, Hemisphere-McGraw-Hill, New York, 1979. of gases across the alveolar membrane in Handbook of Physiology, Respiration Vol. IV, L. E. Farhi and S. M. Tenney, eds., Am. Physiol. Sot., Bethesda, Md. 1987. R. F. Coburn, R. E. Forster, and P. B. Kane, Considerations of the physiological variables that determine the blood carboxyhemoglobin concentration in man, Journal
of Clinical Investigation. 44:1899-1910 (1965). K. E. Muller and C. N. Barton, A nonlinear version of the Coburn, Forster model of blood carboxyhemoglobin, Atmos. Environ. 21:1-5 (1987).
and Kane
LINEAR 5
6 7 8 9 10 11
AND NONLINEAR
P. Tikuisis,
H. D. Madill,
CFK EQUATIONS B. J. Gill, W. F. Lewis, and D. M. Kane, A critical
263 analysis
of the use of the CFK equation in predicting COHb formation, Am. Ind. Hyg. Assoc. J. 48208-213 (1987). J. E. Peterson and R. D. Stewart, Predicting the carboxyhemoglobin levels resulting from carbon monoxide exposures, Journal of Applied Physiology, 39:633-638 (1975). A. H. Marcus, Mathematical models for carboxyhemoglobin, Atmos. Environ. 14:841-844 (1980). T. E. Bernard and J. Duker, Modeling carbon monoxide uptake during work, Am. Ind. Hyg. Assoc. J. 42:361-364 (1981). B. E. Saltzman and S. H. Fox, Biological significance of fluctuating concentrations of carbon monoxide, Environ. Sci. Technol. 20:916-923 (1986). G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn, New York, 1962, p. 22. S. J. W. Roughton, Transport of oxygen and carbon monoxide in Handbook of Physiology, Respiration Vol. I, W. 0. Fenn and H. Rahn, ed., Am. Physiol. Sot., Washington D.C., 1964, p. 780.
ABSTRACT The Coburn, Forster, Kane model of COHb formation as a result of exposure to CO exists in two forms: the linear CFK equation that assumes a constant level of oxyhemoglobin; and the nonlinear CFK equation that allows the oxyhemoglobin level to vary with the carboxyhemoglobin level. Although both equations are currently used, no rigorous analysis exists to show under what conditions the two models determine substantially different solutions. This paper provides such an analysis and shows that the linear model may be used as a reasonable approximation over a much wider range of carboxyhemoglobin levels than had been supposed on physiological grounds.
1.
INTRODUCTION
Inhaled carbon monoxide (CO) competes with oxygen to combine with hemoglobin molecules in the blood forming carboxyhemoglobin (COHb). The toxic effect of carbon monoxide occurs when sufficiently large amounts of COHb are formed and not enough oxygen in the form of oxyhemoglobin (0,Hb) is transported to the body tissues. The processes by which inhaled gases move between the lungs and the blood stream are complex. The rate of diffusion of a gas such as CO across the alveolocapillary membrane depends on the solubility of the gas in the membrane, the diffusion coefficient for the gas in the membrane, the difference in partial pressures across the membrane, and the thickness of the membrane [l]. Many of these quantities do not have uniform values over the entire membrane so estimates represent average values. The overall diffusion rate is referred to as the diffusing capacity or coefficient of the lung for a particular gas (DLCO for carbon monoxide).
*Current
address:
MATHEM4TICAL.
2501 Anne BIOSCIENCES
OElsevier Science Publishing 655 Avenue of the Americas,
Carol Court,
Raleigh,
99:251-263
(1990)
Co., Inc., 1990 New York, NY 10010
NC 27603.
251
252
M. V. SMITH
Both CO and oxygen (0,) in the blood bind with hemoglobin molecules in the red blood cells. The diffusion from the plasma into the red blood cells seems to occur very rapidly [2] and is ignored in the models discussed in this paper. Each hemoglobin molecule has four binding sites. The reaction rates of CO and 0, with hemoglobin differ. Furthermore the reaction rates depend on how many binding sites are occupied and whether these sites are occupied by 0, or CO. (The models we discuss in this paper assume that the different binding rates are all fast enough so that chemical equilibrium can be assumed.) Equations could be written for each process described above, but the resulting complexity would prohibit their utility in applications; therefore simplification is necessary. In 1965 Coburn, Forster, and Kane [3] published the most enduring and most widely used model (CFK model) for the prediction of carboxyhemoglobin (COHb) in humans. The model predicts both the formation of COHb during CO inhalation and the reduction in COHb during CO elimination. The equation used in the original computations assumed that the amount of 0,Hb remained constant at the maximum level, regardless of changes in COHb levels. The resulting equation is linear in the solution [COHbHt). In the original paper the authors suggested that the model would be improved by substituting [O,Hb]-[COHb](t) for the constant 0,Hb parameter. Thejresulting nonlinear equation is integrable, but the solution can not be given explicitly [4]. The original paper [3] validated the linear CFK model against observations made on subjects exposed to very low CO levels over long periods of time. Later investigators tested the nonlinear CFK model against observations made on subjects exposed to a variety of CO profiles including brief exposures to very high CO levels (see e.g., [5, 61). The consensus is that the nonlinear CKF model works quite well even at high levels of CO. However, very little attention has been given to determining the exact conditions under which the linear model becomes inadequate and the nonlinear model is needed. In 1980 Marcus suggested that such a point is reached when the COHb level exceeds the level of free hemoglobin (about 3.6% of the total available hemoglobin) [7]. The suggestion that the linear CFK model could only be used at very low levels of COHb was echoed by Tikuisis in 1987 [S]. In spite of the availability and the proven validity of the nonlinear model, the linear model continues to be used for high levels of COHb when explicit results of specific CO exposure patterns are wanted [8, 91. In this paper we resolve the problem mathematically by determining the range of COHb values for which the solutions of the two models are within a given distance of each other. We begin by resealing the problem so that the only parameters are the initial conditions and a single expression that combines the physiological parameters used in the original model. We
LINEAR
AND NONLINEAR
253
CFK EQUATIONS
restate a comparison theorem of differential equations to which we will frequently refer and present several preliminary lemmas. In the third section we give the principal results of the paper. We then look at a particular case numerically, which will enable us to show some of the results graphically. We close with a brief discussion. 2.
PRELIMINARY
RESULTS
The linear CFK model is usually written d[COHb] dr
_ I’CO VB
I -PZCO B 1/B
as follows, [COHb]PCO, [O,Hb]MB I’B ’
(1)
where [COHb](r)
= the concentration of carboxyhemoglobin in the blood in ml CO/ml blood. I/CO = rate of endogenous CO production in ml CO/min c.007) PZCO =partial pressure of inhaled CO in mm Hg B=l/DLCO+PZ./VA DECO =diffusion coefficient of CO in lungs in ml/mm Hg/min (30) PL = barometric pressure of dry gases, in mm Hg (713) I/A = alveolar ventilation in ml/min (4957) M = Haldane affinity coefficient (240) I/B = blood volume in ml (5500) PCO, =average partial oxygen pressure in the blood in mm Hg (100) [O,Hb] =maximum amount of O2 that can bind with hemoglobin in ml 0, /ml blood c.2).
The numbers in the parentheses refer to typical values [5]. COHb levels are commonly written as a percentage of the maximum saturation of hemoglobin possible (e.g., [3]-[6], [S], [9]). Accordingly we rescale the solution to Y(T) = lOO[COHb]/[O,Hb]. The above equation is then written more simply as
where K and R refer to combinations of the physiological change in the time scale (t = rR/lOO)gives, &/dt=E-YI,
~(0)
=y,,dO,E),
parameters.
A
(2)
254
M. V. SMITH
where E = 100 K/R and is the steady state COHb level predicted by the linear equation under constant exposure conditions. (Throughout this discussion the subscripts 1 and IZ will refer to the linear and nonlinear models respectively.) We note that E is a linear function of the inhaled CO level, PZCO, so that the linear steady state may theoretically be larger than 100%. The nonlinear model is formed by substituting [O,Hb]-[COHb] for [O,Hb] in Equation (1). The same groupings of parameters and scale changes applied to the nonlinear CFK model give, dy,/dt=E-100y,/(100-y,),y(O)=y,,~[0,100E/(100+E)].
(3)
We note that the steady state value for the nonlinear model may be found in terms of E, namely lOOE/(lOO+ E), and that the nonlinear steady state is bounded by 100%. If ylo = y0 = yno two possibilities may occur: the uptake of CO during which E, 100 > y,,y,, > yc, and the elimination of CO during which 100 > y, > yl, y,, > E. We state a well-known version of the comparison theorem [lo] to which we will often refer: LEMMit I (Comparison Theorem) Let f and g be solutions of the following differential equations, y’=F(x,y),
z’=G(x,z),
respectively, satisfying the same initial condition f(a) = g(a). Suppose further that F or G satisfies a Lipschitz condition and that F(x, y) < G(x, y) for all x, y in a given domain. Then f(x) < g(x) for x > a.
We present LEMMA
the following lemmas.
2
Let y, and y,, be solutions to Equations
(2) and
(3) respectively and
y,(O) = y, = y,(O). Then y&t) > y,(t) for all t b 0. Proof The comparison theorem is applied to Equations note that 100 > y, b 0, during both uptake and elimination LEMMA
3
Let yi = E, - y, and y$ = E, - y,, y&t)- y,(t) < E, - E, for all t & 0. 3.
(2) and (3). We of CO.
with ~~(0) = ~~(0). Zf E, > E,,
then
MAIN RESULTS
PROPOSITION
1
Let y, and y, be solutions to Equations (2) and (3) respectively, with E and E) as the corresponding steady states. Assume yfo = yno = y.
lOOE/(lOO+
255 2 y,, > y,
(i.e.
CO is entering the blood).
yr(t)-y,(t) y, G=y,, > E, where y, and y, are solutions to Equations (2) and (3) respectively. If yio = y,,, = y. and y,~(-6+4-)/2, theny,-y,,6.
(7)
The quantity a is a real number that can be adjusted to sharpen the bound within given intervals of y,. Proof: We begin with Equation (6) of the previous proposition and approximate the term 1002/(D - y, + 100) by 1002[(l/(a + loo))--((D - y, - a)/(lOO+ a)2)], its Taylor expansion at y, + a. The error term is easily computed to be
1
(8)
D-y/+100
We hence consider
the following differential
x(t)‘=-y,-100+1002
&[
Using Lemma 1 we see that easily integrated to give,
D(t) 2 x(t)
equation, x-y,-a (100+a)2
I’
for all t 2 0. Equation
(9)
(9) is
The result again follows when x(t) is written in terms of yl. To help choose a, 6 may be substituted for D in (8), and the error term bounded by E,
(
6 - y,1+ 100 I( 6-y/+100 (a+lOO)
-1 12 GE.
(10)
The a-value defined by (9) depends on yI, and intervals of yl corresponding to different a-values can easily be calculated for each 6 and E.
258 4.
M. V. SMITH NUMERICAL
EXAMPLES
We recall that the parameter E used throughout the paper is the steady state solution of the linear CFK equation and may be greater than 100. In terms of the parameters given with the original statement of the model (Equation l), E = lOOM(B I/CO + PICO)/PCO,. For an individual whose physiological parameters have the typical values given in the second section we can say that the endogenous COHb level (the level attained solely by metabolic processes without inhaled CO) is about .3%. Throughout this section, unless otherwise specified, the linear model will be said to provide a “good” approximation of the nonlinear model if the two solutions (both in percent of available COHb) are no further than one percent apart, i.e., 6 is one. Example 1. We choose a simple case of CO uptake in which the initial percent COHb is the endogenous level (.3%), and the individual is suddenly exposed to a constant concentration of CO. Proposition 1 assures us that the linear equation provides a good approximation for any initial COHb level as long as E is less than about 10.5%. Using the parameter values of the second section the corresponding level of inhaled CO is about 60 ppm. In the event that the combination of parameter values and inhaled CO concentration produces an E value greater than 10.5% Proposition 1 no longer applies. However, as shown by Proposition 3, the linear equation may still provide a sufficient approximation over a considerable range of COHb values. In Figure 1 this range is shown graphically for E values between zero and 100. Note that yI will always lie between y0 and E, so that only the lower triangular region is of interest. The region between the bottom curve and the E-axis is defined by Proposition 3 and represents the range of COHb values predicted by the linear equation that provides a “good” approximation for the nonlinear solution. We see as expected that for E values less than 10.5% the range of COHb values reaches the linear equilibrium, i.e., the linear approximation is good throughout the entire trajectory. For the larger E value of 50% the solution to the linear CFK equation will be adequate as long as the values it predicts remain less than about 20%. For the parameter values of the second section an E value of 50 corresponds to an inhaled CO concentration of about 290 ppm. The area above the middle curve is defined by Proposition 4 and shows values of the linear solution that do not meet the one-percent approximation criterion. Values computed by Equation (2) in this range should always be recomputed by the nonlinear equation. The region between the lower two curves is not covered by the results of this paper. The value of the a-parameter used in computing the middle curve was set at - 14. The error E defined by inequality (10) is less than .0005% as long as the computed y-value is less than 31%. The same curves are shown in Figure 2 over a
LINEAR
AND NONLINEAR
CFK EQUATIONS
solutions
259
are more than
solutions
20
1% apart
are less than
40
1%
60
apart
80 E
FIG. 1. Region of 1% approximation during uptake of CO for ya =.3. The bottom region is defined by inequality (5) in the text. The top region is defined by inequality (7) in the text, using a = - 14.
100
Yl 80 solutions are more than
1% apart
60
40
solutions 20
are less than
1% apart
/
I 1-
I
500
I
1000
1500 E
FIG. 2. Region of 1% approximation during uptake of CO for y, = .3. The bottom region is defined by inequality (5) in the text. The top region is defined by inequality (7) in the text, using a = - 25.
260
M. V. SMITH
100 -
YI
solutions are less than
1X apart
solutions
20
40
are more than
1%
apart
60
80 yo
FIG. 3. Region of 1% approximation during elimination of CO with E =.3. The slender top region is defined by inequality (5) in the text. The triangular region to the right is defined by inequality (7) in the text, using a = - 14.
larger interval for E. The a-value of -25 provides that than .Ol% as long as the computed yl-value is less than reference COHb concentrations of 40% and above are for a resting individual. Concentrations below that may exercise ill].
the error E is less 68%. As a point of usually dangerous be dangerous with
Example 2. To study the elimination of CO we assume a relatively high initial COHb level, then remove the CO source thereby giving a lower value for E (say .3%). Proposition 2 shows that the linear solution provides a “good” approximation to the nonlinear solution as long as ye is less than about 9.5% regardless of the E value. Figure 3 shows the regions defined by Propositions 3 and 4 during elimination of CO. In this case the percentage of COHb decreases from the initial level so that, again, only the lower triangular part of the graph is of interest. The area above the middle curve is defined by Proposition 3 and gives the combinations of y, and y, for which the present paper shows that the linear CFK gives an adequate approximation. The region to the right of the lower curve is defined by Proposition 4 and shows the pairs (ya, yr) for
LINEAR
AND NONLINEAR
261
CFK EQUATIONS
which the present paper shows that the solutions are more than 1% apart. For the a-value of - 14, the error in inequality (10) stays below .005% as long as the computed y-value is less than 55%. Note that Proposition 4 does not rule out a good approximation for very low y-values or for the expected y-values near y,. This is reasonable because the linear and nonlinear solutions will tend towards equilibria that are very close together. The nonlinear equilibrium value corresponding to E = .3% is .299%. 5.
DISCUSSION
As can be seen from Figures l-3 the linear model provides a good approximation over a larger range of COHb values during the uptake of CO than during its elimination. The reason is clear when we rewrite the nonlinear model (Equation 3) as y,: = E -Y,
- ~,2/(100-
(11)
Y,>
During the uptake of CO, y, tends to be small initially, especially with respect to the value of E. Hence the final term in Equation (11) will be small and the solution similar to that of Equation (2). During the elimination of CO, the value of E is small with respect to initial y, values. Thus the final term in Equation (11) will be large, and the solutions of the two models diverge quickly. Proposition 3 defines the value y? such that during CO uptake all linear model predictions below y[* are guaranteed to be within a set distance of the nonlinear model prediction. In Figures 1 and 2, yr is plotted as a
TABLE Values
for y:’ During
1 Uptake
E (with corresponding
6 = 0.5 y,=.3
Yo = 5 y, = 10
6 = 1.0 6=1.5 s = 2.0 S = 2.5 6 = 3.0 6 = 1.0 6 = 1.0
‘CO concentrations given in Section 2.
1.9 10.0 10.0 10.0 10.0 10.0 10.0 10.0 are computed
500
100
(290)
(582)
16.9 20.5 22.9 24.6 26.1 27.3 20.6 21.0
21.8 26.1 29.9 32.3 34.3 36.0 26.7 21.0
of CO inhaled
CO in ppmt)
500 (2920)
1000 (5842)
2000 (11684)
37.0 45.0 50.2 54.1 57.3 59.9 45.0 45.1
45.4 54.8 60.7 65.1 68.6 71.4
55.1
54.8 54.8
using the values of the physiological
65.5 71.9 76.4 79.8 82.6 65.5 65.5 parameters
262
M. V. SMITH TABLE Values for yf
During
2 Elimination
of CO
Yo
E =.3
E=5 E=lO
10
20
30
40
50
60
6 = 0.5 6 = 1.0
3.4 .3
6=1.5 6 = 2.0 S = 2.5 s = 3.0 6=1.0 6=1.0
.3 .3 .3 .3
16.7 15.4 12.4 8.2
28.8 27.6 26.2 24.8 23.2 21.5 28.0 28.4
39.2 38.5 37.7 36.8 36.0 35.1 38.7 38.8
49.5 49.0 48.5 47.9 47.4 46.8 49.1 49.2
59.7 59.3 59.0 58.6 58.3 57.9
5 10
.3 .3 16.8 18.0
59.4 59.4
function of E for fixed values of 6 ( = 1) and y, (= .3). In general, yr increases with E, although the rate of increase decreases with 6. The effect of different y, values on yi+ seems to be slight during CO uptake. (See Table 1.) During elimination of CO the solutions to both models begin at a relatively high level, then decrease. Proposition 3 guarantees that any linear model prediction above y;” is within the set distance from the nonlinear model prediction. In Figure 3, y,* is plotted as a function of y0 for fixed values of 6 (= 1) and E (= .3). During uptake of CO increasing the tolerated error 6 necessarily increases the region over which the linear model provides a sufficient approximation. Increasing E has more effect during the elimination of CO than increasing y, did during uptake, especially for the more moderate initial values. (See Table 2.)
The author wishes to acknowledge constructive remarks made by Dr. Keith Muller (Biostatistics Department, UNC-CH) and by the editor of this journal.
REFERENCES J. A. Jacquez, Respiratory R. E. Forster, Diffusion
Physiology, Hemisphere-McGraw-Hill, New York, 1979. of gases across the alveolar membrane in Handbook of Physiology, Respiration Vol. IV, L. E. Farhi and S. M. Tenney, eds., Am. Physiol. Sot., Bethesda, Md. 1987. R. F. Coburn, R. E. Forster, and P. B. Kane, Considerations of the physiological variables that determine the blood carboxyhemoglobin concentration in man, Journal
of Clinical Investigation. 44:1899-1910 (1965). K. E. Muller and C. N. Barton, A nonlinear version of the Coburn, Forster model of blood carboxyhemoglobin, Atmos. Environ. 21:1-5 (1987).
and Kane
LINEAR 5
6 7 8 9 10 11
AND NONLINEAR
P. Tikuisis,
H. D. Madill,
CFK EQUATIONS B. J. Gill, W. F. Lewis, and D. M. Kane, A critical
263 analysis
of the use of the CFK equation in predicting COHb formation, Am. Ind. Hyg. Assoc. J. 48208-213 (1987). J. E. Peterson and R. D. Stewart, Predicting the carboxyhemoglobin levels resulting from carbon monoxide exposures, Journal of Applied Physiology, 39:633-638 (1975). A. H. Marcus, Mathematical models for carboxyhemoglobin, Atmos. Environ. 14:841-844 (1980). T. E. Bernard and J. Duker, Modeling carbon monoxide uptake during work, Am. Ind. Hyg. Assoc. J. 42:361-364 (1981). B. E. Saltzman and S. H. Fox, Biological significance of fluctuating concentrations of carbon monoxide, Environ. Sci. Technol. 20:916-923 (1986). G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn, New York, 1962, p. 22. S. J. W. Roughton, Transport of oxygen and carbon monoxide in Handbook of Physiology, Respiration Vol. I, W. 0. Fenn and H. Rahn, ed., Am. Physiol. Sot., Washington D.C., 1964, p. 780.
