Eugene a Multicompartment Model of Carboxyhemoglobin and Carboxymyoglobin Responses to Inhalation of Carbon Monoxide

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A multicompartment model of carboxyhemoglobin and

carboxymyoglobin responses to inhalation of carbon monoxide

Eugene N. Bruce1 and Margaret C. Bruce2

1Center for Biomedical Engineering, and 2 Department of

Pediatrics, University of Kentucky, Lexington, Kentucky 40506

Submitted 3 March 2003; accepted in final form 5 May 2003

Bruce, Eugene N., and Margaret C. Bruce. A multi-compartment model of carboxyhemoglobin and carboxymyo-globin responses to inhalation of carbon monoxide. J Appl

Physiol 95: 1235–1247, 2003. First published May 16, 2003;10.1152/japplphysiol.00217.2003.—We have developed amodel that predicts the distribution of carbon monoxide (CO)in the body resulting from acute inhalation exposures to CO.The model includes a lung compartment, arterial and venousblood compartments, and muscle and nonmuscle soft tissueswith both vascular and nonvascular subcompartments. Inthe model, CO is allowed to diffuse between the vascular andnonvascular subcompartments of the tissues and to combinewith myoglobin in the nonvascular subcompartment of mus-cle tissue. The oxyhemoglobin dissociation curve is repre-sented by a modified Hill equation whose parameters arefunctions of the carboxyhemoglobin (HbCO) level. Values forskeletal muscle mass and cardiac output are calculated fromprediction formulas based on age, weight, and height of individual subjects. We demonstrate that the model fits datafrom CO rebreathing studies when diffusion of CO into themuscle compartment is considered. The model also fits re-sponses of HbCO to single or multiple exposures to COlasting for a few minutes each. In addition, the model repro-duces reported differences between arterial and venousHbCO levels and replicates predictions from the Coburn-Forster-Kane equation for CO exposures of a 1- to 83-hduration. In contrast to approaches based on the Coburn-Forster-Kane equation, the present model predicts uptakeand distribution of CO in both vascular and tissue compart-ments during inhalation of either constant or variable levelsof CO.

Coburn-Forster-Kane equation; myoglobin; blood volume

ALTHOUGH THE ORIGINAL PURPOSE of the Coburn-Forster-Kane (CFK) equation was to predict the rate of endog-enous carbon monoxide (CO) production (7), this equa-tion has since been used, with varying degrees of suc-cess, to predict the rate of carboxyhemoglobin (HbCO)

formation during inhalation exposure to CO. In one of the first studies to use the CFK equation as a predic-tive model, Peterson and Stewart (26) reported thatHbCO levels in sedentary men exposed to 25–1,000parts/million (ppm) for 0.5–24 h were in reasonableagreement with predicted values. Other investigatorshave also demonstrated that rates of HbCO formationpredicted by this model were in basic agreement with

results obtained in CO inhalation exposure experi-ments (4, 14, 27, 36, 37). As a result of these studies,the CFK equation has often been used under circum-stances quite different from the one for which themodel was initially intended.

Despite the usefulness of the CFK model for predict-ing HbCO levels in response to long exposures to lowCO concentrations as well as brief exposures to highconcentrations, the model has several limitations. Be-cause the CFK equation provides a first-order approx-imation to the slow dynamics of CO transport andstorage, it is unlikely to accurately predict a rapidincrease in HbCO level. In addition, this equation doesnot accurately predict either arterial or venous HbCOlevels but a value between the two (4). Furthermore, adhoc modifications of the CFK equations have often beennecessary to improve the fit of the model to the dataacquired. For example, in agreement with predictionsbased on the CFK equation, Tikuisis et al. (37) foundthat HbCO levels measured shortly after multiple ex-posures were a function of total CO dose rather thanconcentrations of inhaled CO. These observations sug-gested that the CFK equation could be used to predict

percent HbCO even when CO levels varied consider-ably during the exposure period. To improve the fit of their data to the values predicted by the CFK equation,the authors applied values from repeated measures of each subject’s alveolar ventilation to the model. De-spite this modification of the CFK equation, the pre-dicted values for percent HbCO were systematicallygreater than the data at most observation times. Fur-thermore, the responses to such large but brief stimulihave limited usefulness because they are not likely toreflect the full range of properties of the physiologicalsystem.

Another limitation of the CFK model is that it doesnot include extravascular storage sites for CO. Muscle

myoglobin (Mb) contains one heme group per moleculeand is, therefore, a potential binding site for CO. Inhumans, muscle accounts for 41% of total body mass. Assuming a value of 4.7 mg Mb/g wet wt of muscle (23),a 70-kg man would be expected to have 135 g of Mb,suggesting that muscle could be a significant storagesite for CO. Peterson and Stewart’s observation (26)

Address for reprint requests and other correspondence: E. N.Bruce, Center for Biomedical Engineering, University of Kentucky,Lexington, KY 40506-0070 (E-mail: [email protected]).

The costs of publication of this article were defrayed in part by thepayment of page charges. The article must therefore be herebymarked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734solely to indicate this fact.

J Appl Physiol 95: 1235–1247, 2003.First published May 16, 2003; 10.1152/japplphysiol.00217.2003.

8750-7587/03 $5.00 Copyright © 2003 the American Physiological Societyhttp://www.jap.org 1235


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that the half-time for washout of CO was almost 30%greater than predicted by the CFK equation (320 vs.

252 min) is consistent with the concept that CO wasbeing removed from two compartments rather than oneduring the excretion phase.

CO exposure continues to be a significant cause of morbidity and mortality. Although the CFK equationcan be used to estimate the rate of formation of HbCOunder a variety of exposure conditions, it is dif ficult topredict clinical outcome on the basis of the informationprovided by the CFK equation alone. A more compre-hensive model of CO uptake, distribution, and elimi-nation will be necessary to guide treatment strategiesmore effectively. The objective of the present study wasto develop a predictive model of CO exposure that takes

into account the possibility that a significant fraction of inspired CO can be bound to muscle Mb.

METHODS

Model Development

Structure of model. The model comprises fi ve major com-partments: lungs (alveolar), arterial blood, mixed venousblood, muscle tissue, and other soft tissues (Fig. 1). CO entersthe lungs via the alveolar ventilation and diffuses into thepulmonary capillary blood according to the prevailing COdiffusion capacity of the lung (DLCO), establishing an end-capillary CO concentration (CecCO). Both hemoglobin (Hb)-

bound and dissolved CO are taken into account. End-capil-lary PO2 (PecO2) is assumed to equal arterial PO2, which is

specified as a parameter. After a time delay, end-capillaryblood undergoes mixing in the arterial compartment with atime constant determined by the ratio of the volume of thecompartment to the blood inflow rate [i.e., cardiac output(Q̇)]. Thus arterial levels of oxyhemoglobin (HbO2) and HbCOare established. Arterial blood flows into two parallel com-partments, both of which are divided into vascular and ex-travascular (tissue) subcompartments. Inflowing blood un-dergoes mixing in the vascular subcompartments, and COcan diffuse from the vascular to the extravascular tissuesubcompartments. This diffusion is governed by two COdiffusion coef ficients: DmCO for muscle tissue and DotCO fornonmuscle (other) tissue. In nonmuscle tissue, CO exists onlyin dissolved form. In muscle tissue, CO may also combine

with Mb in competition with oxygen (O2). Muscle O2 tension(PmO2) is a parameter of the model. For both Mb and Hb, thecorresponding Haldane equation is satisfied in each relevantcompartment. Venous outflows from the two tissue compart-ments are combined at the entrance to the mixed venouscompartment, where further mixing occurs. After anothertime delay, this blood returns to the lungs.

Mass balance equations. All compartments are repre-sented by equations for conservation of mass for CO. In thealveolar compartment

V LdC A COt

dt PICOt P A COt V ˙ A /PB COfluxLBt (1)

Fig. 1. The structure of the model of carbon mon-oxide (CO) storage and transport. PecCO, end-capillary partial pressure of CO. See Tables 1 and2 for remaining definitions.

1236 MULTICOMPARTMENT MODEL OF CO UPTAKE AND DISTRIBUTION

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where V L is lung volume, C A CO is the alveolar CO concentra-tion, P A CO is the alveolar partial pressure of CO (PCO), PICOis inhaled PCO, t is time, V ˙ A is alveolar ventilation, PB isbarometric pressure, and COfluxLB(t) is the CO flux fromlungs to blood, defined as

COfluxLBt P A CO 1 K v̄PecCOt

K v̄Pv̄COt d v̄ DLCO( 2)

where PecCO is end-capillary PCO, d v̄ is the mean transportdelay in mixed venous blood, Pv̄CO is mixed venous PCO, and

K v̄ is used to apportion the effective pulmonary capillary PCObetween the mixed venous and end-capillary pressures. In all

simulations here, K v̄ 0.5 (see Tables 1 and 2 for definitionsof variables and nominal values of parameters).

Vascular mixing in the pulmonary capillary compartmentis ignored and CecCO is determined by adding COfluxLB tothe mixed venous blood, which enters the lungs after themean transport delay, d v̄, from venous blood compartments.Thus

CecCOtCmvCOt d v̄COfluxLB /Q̇ ( 3)

The four blood volume compartments (arterial, mixed ve-nous, and two vascular subcompartments of the tissues) areeach described by an equation of the form

Table 1. Glossary of variables

C jCO jec,a,v̄,vm,vot, A Concentration of CO (ml/ml, BTPS) in end-capillary (ec), arterial (a), and mixed venous (v̄ ) blood,muscle venous (vm) and other tissue venous (vot) out flow, and lung alveolar ( A ) compartment

HbCO j ja,vm,vot,v̄ Carboxyhemoglobin concentration (ml CO/ml blood, BTPS) in arterial (a), muscle venous (vm), othertissue venous (vot), and mixed venous (v̄) compartments

MbCO Carboxymoglobin (ml CO/ml tissue, BTPS)d j ja,v̄ Time delay (min) between the end-capillary and arterial compartments (a), and between the mixed

venous and pulmonary arterial compartments (v̄)HbO2 j ja,vm,vot,v̄ Oxyhemoglobin concentration (ml CO/ml blood, BTPS) in arterial (a), muscle venous (vm), other

tissue venous (vot), and mixed venous (v̄) compartmentsP jO2 ja,A,m,ot,v̄ Pressure (Torr) of O2 in the arterial (a), alveolar ( A ), muscle (m) and other (ot) tissue, and mixed

venous (v̄) compartmentsP jCO ja,m,ot,v̄ Pressure (Torr) of CO in the arterial (a), alveolar muscle (m), and other (ot) tissue, and mixed

venous (v̄) compartments

Table 2. Parameters and their default values

Parameter Definition Default Value Ref.

DLCO Lung diffusion capacity for CO,normoxia

30 ml min1 Torr1 26

Lung diffusion capacity for CO,hyperoxia

15 ml min1 Torr1 21

DmCO Muscle diffusion capacity for CO See textHb Concentration of Hb in blood Measured for each subjectMb Myoglobin concentration 4.7 mg/g wet wt muscle 23MH Haldane af finity ratio for Hb 218 35MM Af finity ratio for Mb 36 12n Hill exponent for Hb See text 20HbO2 Oxygen capacity of Hb 1.38 ml O2 /g Hb 28PB Barometric pressure 760 TorrPaO2 Partial pressure of O2 (arterial) 100 Torr (air) 35

500 Torr (hyperoxia) 39PmO2 Partial pressure of O2 (muscle) 20 Torr (air) 3

30 Torr (hyperoxia) 39P50Hb Partial pressure of O2 at 50% Hb sat. See text 20P50Mb Partial pressure of O2 at 50% Mb sat. 2.32 Torr 30SO2 Solubility of O2 in plasma 3.14 105 ml Torr 11SCO Solubility of CO in plasma 2.35 105 ml/Torr 11Q̇ Cardiac output Regression on body wt and gender 19Q̇m Blood flow to muscle See text 2

10Q̇ot Blood flow to other tissues See text 210

V j , j a, v̄, vm, vot, A , L,m, ot

Volume (ml, BTPS) of the arterial (a)and mixed venous (v̄) blood, muscle

venous (vm) and other (vot) tissue vascular compartments, and lung alveolar ( A ), lung (L), muscle (m),and other (ot) tissue nonvascularcompartments

Estimated from data; see text

V ˙ O2 Metabolic rate 225 ml/min 10 V ˙ CO Endogenous CO production 7 l/min 7 V ˙ A Alveolar ventilation Measured for each subject (see text)

sat., Saturation.

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V idCiCOt

dt Ci

intCiCOt Q̇i Fluxit ( 4)

where i (i a, v̄, vm, or vot) indicates one of the compart-ments, V is volume, Ci

in is the concentration of CO in bloodentering the compartment, and Fluxi(t) is the rate of diffu-sion of CO out of the vascular volume. For the arterial andmixed venous compartments, this rate is zero. Otherwise

Flux vmt PtcCOt PmCOt DmCO

Flux vott PotcCOt PotCOt DotCO(5)

with PtcCO(t) and PotcCO(t) representing the algebraic meansof the partial pressures of CO in the arterial inflow and

venous outflow for the muscle and other tissue compart-ments, respectively. Little information is available regarding DmCO and DotCO. Because O2 diffusivity is similar in plasmaand muscle tissue (20), the model assumes that these COdiffusion coef ficients are equal. CO concentrations are appor-tioned into dissolved and Hb-bound components so that

CiCOtHbCOit SCO PiCOt (6)

where SCO is the solubility of CO in plasma.CO concentration at the input to the mixed venous com-

partment is determined by a weighted summation of theconcentrations of CO in the venous outflow from the twotissue compartments. Thus

HbCO v̄,intHbCO vott Q̇ot/Q̇HbCO vmt Q̇m/Q̇ (7 )

This input concentration then passes through the mixed venous compartment and returns to the lungs after a delay(d v̄). In this and all vascular compartments, the variablesPO2, PCO, HbCO, and HbO2 are required to satisfy the Hal-dane relationship

MH PCO

HbCO

PO2

HbO2(8)

PCO is determined by first solving for the other variables,assuming that PCO is constant across an integration step,then updating its value via the Haldane equation. Althoughsmall relative to the exposures being considered, endogenousCO production has been included by assuming that all en-dogenous CO is delivered to the mixed venous compartment.

Similar compartmental mass balance equations are writ-ten for the HbO2 concentration. It is assumed that O2 dif-fuses from the vascular to the extravascular subcompart-ments of the tissues at rates just suf ficient to meet metabolicdemands. Total body metabolism is set at 225 ml O2 /min andis divided between the two tissue compartments in the sameratio as that of the blood flows to these compartments.

The various blood flows are assumed to be constant. Q̇ isestimated via a regression approach (see Eq. 12) and thefraction of Q̇ that flows to the muscle tissue is varied to obtainthe best fit of the model to each data set.

Model of O 2 dissociation curves of Hb and Mb. The pres-ence of HbCO not only reduces the maximum amount of O2that Hb can bind but also changes the shape of the O2dissociation curve (29). Except at low PO2 values, a reason-able approximation to this curve for any given level of HbCO(15, 20) is the Hill equation

CO2 CO2max PO2 /P50

n

1 PO2 /P50n (9)

where P50 is the PO2 necessary to half-saturate the Hb notbound to CO. Thus we derived a set of equations to represent

the O2 dissociation relationship in the presence of CO bydetermining the dependence of the Hill parameters on theHbCO level and then applying a correction to the Hill equa-tion at low pressures. The O2 dissociation curves for 0, 20, 40,and 60% HbCO (29) were digitized, and the Hill equation wasfit to each curve individually by using the fmin function of MATLAB. CO2,max was reduced in direct proportion to thepercentage of HbCO. The values of P50 and n were then fit tofunctions of the form

P50 ap11 expap2 %HbCO1

n an1 an2 e xp0.025 %HbCO(10)

by using fmin to estimate the parameters ap1, ap2, an1, andan2. By trial and error, it was found that multiplying the Hillequation by the function

f PO2 1 0.25 5PO2 /P5031 (11)

produced a satisfactory fit at low pressures. Note that f (PO2)is 1 for PO2 20 Torr. Figure 2 shows graphs of P50 and nas functions of percent HbCO and of actual dissociationcurves, and the approximations at four levels of HbCO.

The O2 dissociation curve for Mb is also represented by aHill equation (Fig. 5 of ref. 30), but it is assumed that itsparameters are independent of the carboxymyoglobin(MbCO) level (except for the maximum O2-binding capacity).

At each time step of the simulation, the dissociation curve forMb and the Haldane equation for Mb are satisfied simulta-neously via an implicit solution using an iterative procedure.

Calculation of subject-specific and other parameters. Nom-inal values for parameters are presented in Table 2. Whenavailable, subject-specific values of model parameters wereobtained from the literature or directly from the investiga-tors. Usually, the age, weight, and height of a subject wereavailable, and often one or more additional parameters, such asHb concentration, total blood volume, Q̇, or ventilation, weresupplied by investigators. In some cases, average values for agroup of subjects were used. Predictive formulas were used toestimate Q̇ (when it was not measured) and tissue volume of skeletal muscle (Vm). As a function of body weight (BW), age(A), height (HT), and gender (G; G has a value of 1 for a maleand 0 for a female subject)

Q̇ 54.1 7.9G BW 1400 200 G (12)

in ml/min (19), and

Vm 10000.244 BW 7.80 HT

6.6 G 0.098 A 3.3 /1.04(13)

in ml (17). The volume of nonmuscle tissue is assumed to beproportional to Vm

Vot 9.6/29.1 Vm (14)

where 9.6 and 29.1 liters are the nominal normal values fornonmuscle and muscle tissue volumes for a 70-kg man (10).For obese subjects (body mass index 30), Vm is calculatedby using the alternative parameter values for the preceding equation as given by Lee et al. (17).

The volumetric carrying capacity of Hb for O2 or CO iscalculated as the product of Hb concentration and the nom-inal maximum O2 content, 1.38 ml O2 /g Hb. For Mb, thecarrying capacity is proportional to that of Hb multiplied bythe ratio of the molecular weights of Hb and Mb (i.e., 64,500and 17,000, respectively) and divided by four. The concentra-tion of Mb varies with the type of muscle, and an average

value of 0.0047 g/g wet weight was used (23).




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Arterial and muscle PO2 are parameters of the model. Arterial PO2 was set at 100 Torr during normoxia and 500Torr in hyperoxia. In normoxic resting conditions, muscle PO2has been estimated to be 20 Torr (3). In hyperoxia, musclePO2 increases much less than arterial PO2 (39), and a value of 30 Torr has been used.

Estimation of other cardiovascular parameters. The distri-bution of the total blood volume among the four vascular com-partments and the distribution of Q̇ between the two tissue

compartments are unknown and were estimated for eachsubject. Initial values were determined by assuming thatthe muscle tissue corresponds to the slow vascular com-partment of Smith et al. (33) and that the nonmuscle tissuecorresponds to the fast compartment of that model. The

values for blood flow used by Smith et al. (33) roughlycorrespond to those used by El-Hefnawy et al. (10) formuscle and nonmuscle tissue compartments, and the val-ues for blood volumes roughly correspond to the relativesizes of the tissue volumes. Then, for each data set, themodel was run repeatedly to determine the best visual fitto the data, attempting to utilize parameters that werewithin 30% of the initial values. Although this process didnot optimize parameter estimates in a formal sense, it diddemonstrate that comparable values for these parameters

can be found that reasonably fit the data from severalsubjects representing three greatly different types of ex-perimental situations.

Blood leaving the pulmonary vein returns to the pulmo-nary artery after an elapsed time that depends on the tissuethat it perfuses. This time lapse is likely to be unimportant indetermining HbCO levels after tens of minutes or hours of CO exposure, but one can expect it to have a significant rolein responses to rebreathing of CO [e.g., Burge and Skinner.(6)]. The average total time to traverse the circulation (Tc)equals the total blood volume divided by Q̇. Through washin/ washout, each blood compartment imposes an effective timedelay that is a function of its time constant, the compartmen-

tal blood volume divided by its blood flow. By considering thehalf-time of the step response as the effective delay, the delaytimes for the arterial, mixed venous, and parallel tissuecompartments were calculated. The difference between Tcand the sum of these three delay times is assumed to repre-sent an additional transport delay and is apportioned in aratio of 1:2 between the arterial and mixed venous compart-ments (10, 22). When the total time delay is not matched toTc, the predicted HbCO response to CO rebreathing is no-

ticeably more sensitive to the relative distribution of blood volume among the compartments than it is in the finalmodel.

Implementation. The model was implemented with the Advanced Continuous Simulation Language (ACSL) and the ACSLMath simulation environment. A fourth-order Runge-Kutta integration algorithm was used with a fixed step size of 0.02 min. Smaller step sizes were tested to ensure thataccuracy was achieved with the one chosen.

Testing the Model

Data sources. Predictions from the model were comparedwith published observations on human subjects. CO re-breathing in hyperoxia was used by Burge and Skinner (6) to

estimate total blood volume, and data from three of theirsubjects who rebreathed for 40 min were supplied to us by theauthors. Benignus et al. (4) exposed human volunteers to6,683 ppm CO for 4–6 min and compared HbCO levelsobtained from simultaneous arterial and venous samples atseveral times during and after the exposure. These data alsohave been supplied to us by the authors. HbCO responses tolong-term (e.g., 1–83 h) CO exposures can be described ade-quately by the CFK equation, and model predictions havebeen compared with solutions of this equation presented inFig. 2 of Peterson and Stewart (27) by digitizing the curvesfrom this figure. Finally, Tikuisis et al. (37) compared theresponses to 5-min exposures of 1,500 ppm CO with those to

Fig. 2. A: actual (solid line) and predicted (dashed line) oxygen (O2) dissociation curves of Hb at various levels (0,20, 40, and 60%) of HbCO. See text for prediction equations. Nonlinear regression relationships (see text) for theparameters of Hill’s equation [PO2 necessary to reach 50% Hb saturation (P50; B) and n (C)] as a function of HbCOlevel.




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1-min exposures of 7,500 ppm, and model predictions werecompared with their findings.

Validation protocol. The model was tested first by using the CO rebreathing data (6). After available subject-specificparameters for each of the three subjects were assigned, a setof cardiovascular volumes was found that provided accept-able fits, which allowed for small intersubject variationsin the fraction of Q̇ assigned to the muscle compartment(Q̇mfrac). The value of the diffusion coef ficient of CO from

vascular to extravascular tissue subcompartments, DmCO,was also estimated. These parameters were then utilized tomodel the arterial and venous HbCO data from Benignus etal. (4). Because subjects in this latter study exhibited a rangeof temporal responses, it was necessary to adjust the cardio-

vascular volumes to fit the variety of responses. Blood volumeand flow parameters determined from fitting the rebreathing data also were used to predict the venous HbCO responses to25, 200, and 1,000 ppm CO for up to 5,000 min. Thesepredictions were compared with the corresponding CFK so-lutions (27). Parameters not given by Peterson and Stewart(27) were assigned typical values (see Table 2). In all cases,we compared model predictions obtained when using theestimated value of DmCO with those that resulted whenDmCO was set equal to zero.

RESULTS

Validation of Model

CO rebreathing. The subjects of Burge and Skinner(6) rebreathed from an external circuit containing 100% O2 into which a fixed bolus of CO was injected attime zero. For three of these subjects who rebreathedCO for 40 min, the following specific information wasavailable: age, gender, height, weight, dose of CO ad-ministered, barometric pressure, Hb concentration,and total blood volume. DLCO is known to decreaseconsiderably in hyperoxia (21), and a value of 15 was

assumed. Q˙

at rest also decreases in hyperoxia by 15%(18). To simulate rebreathing, the lung volume of themodel (1,500 ml) was augmented by the volume of theexternal circuit (3,500 ml) and alveolar ventilation wasset to zero. The fraction of the total blood volume ineach of the four vascular compartments and the Q̇mfracare unknown parameters. Resting Q̇mfrac was set ini-tially to 0.20 (2). The total blood volume was dividedinitially among the four compartments by associating the muscle compartment with the slow vascular com-partment of Smith et al. (33) and the nonmuscle tissuecompartment with their fast compartment. These ini-tial fractional distributions of total vascular volumewere adjusted until a set of satisfactory parameters,

which were similar for the three subjects, was found.Q̇mfrac was allowed to change during these trials aslong as the time constant of the muscle vascular com-partment remained slower than that of the nonmuscle vascular compartment.

Attributing 40% of the blood volume to the muscle vascular compartment and 25% to the nonmuscle vas-cular compartment produced predictions that closelymatched the data for all three subjects. Figure 3 dem-onstrates that the predicted HbCO level in the muscle venous blood (%HbCOvm) closely matched all datapoints for the two female subjects and all points except

those at 2.5 and 5 min for the male subject. Q̇mfracranged from 0.25 to 0.30. The overshoot in the datafrom the male subject can be modeled by increasing Q̇mfrac to a level that causes the muscle vascular com-partment to become the “fast” compartment, as shownby the dashed curve in Fig. 3 A.

For each subject, the most appropriate value forDmCO was determined by comparing the predicted

Fig. 3. Muscle venous %HbCO vs. time during CO rebreathing.Model predictions (solid lines) and measured levels from antecubital

vein samples (F) for 3 subjects from Burge and Skinner (6). In allcases, 40% of total blood volume is assigned to the muscle compartmentand 25% to the nonmuscle tissue compartment. Fraction of cardiacoutput assigne to muscle compartment (Q̇mfrac) 0.25( A),0.30 ( B), and0.25 (C). Dashed line in A is the response of arterial %HbCO.




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responses for a range of values. An example from onesubject is presented in Fig. 4. For both female subjects,a value of 0.8 ml min1 cmH2O

1 reasonably ap-proximated the slope of %HbCOvm from 8 to 40 min.For the male subject, estimated DmCO 1.75. Thepredicted responses of several variables for subject 2are shown in Fig. 5. After 10 min of rebreathing, thepredicted percentage of MbCO (%MbCO) is 2.61, from

which one may calculate that 0.86 ml of CO has boundto Mb since the start of rebreathing.

Brief inhalation of a high level of CO. Benignus et al.(4) exposed subjects to 6,683 ppm of CO in room air for4–6 min and acquired multiple simultaneous samplesof arterial (radial artery) and venous (antecubital vein)blood during and after the exposure. The subjects ex-hibited a wide range of temporal responses of arterialand venous HbCO; however, in all subjects, the venousHbCO level lagged the arterial HbCO during CO expo-sure, and the two measurements converged at variedtimes after the CO exposure terminated. The following parameters were available for each subject: age, gen-der (all male), weight, height, Hb concentration, total

blood volume, Q̇, V ˙ A , and DLCO. The physiological param-eters, however, had been measured 1 day before COexposure. For validation of the model, predicted arte-rial and muscle venous %HbCO levels were comparedwith data from two subjects (subjects 114 and 120)representative of those having small or moderate dif-ferences between arterial and venous %HbCO re-sponses (Fig. 6).

The distribution of blood volumes determined fromthe CO-rebreathing studies provided the starting val-ues for these simulations, and the best visual fits to thedata were obtained with similar final values. It wasnecessary to assume that Q̇mfrac was larger than the

corresponding value used for the three subjects of

Burge and Skinner (6) (50% larger for subject 114;100% larger for subject 120). Also, the observed de-crease in %HbCO after termination of the CO exposurewas faster than the model predicted using values of DmCO that were estimated from the CO-rebreathing studies. Because it is possible that these subjects mayhave hyperventilated during this period, no furtherestimation of DmCOwas warranted. Finally, as was the

case for the CO-rebreathing data, it was noted that thelevel of %HbCO after the CO exposure was terminatedwas very sensitive to the total amount of blood in thefour vascular compartments. For subject 120, the vas-cular compartments comprised 92.5% of the measuredtotal blood volume of the subject, whereas for subject114 this value was 97.5%. [For the subjects of Burgeand Skinner (6), the corresponding value was 85% forall 3 subjects.]

Comparison to CFK solutions. A graph presented byPeterson and Stewart (Ref. 27 and Fig. 2 therein)shows %HbCO predicted by the CFK equation vs. timefor various levels of inspired CO for exposures lasting 5,000 min. We chose the two extremes (25 and 1,000

ppm) and an intermediate value (200 ppm) for compar-ison with our model. Some parameter values are givenin Fig. 2 of the Stewart and Peterson paper (27); theremainder were taken from the simulations of COrebreathing described above, except that values forDLCO, PmO

2, and Q̇ in normoxia were used. The subject

was assumed to be a 30-yr-old (middle of their agerange) man, with a height of 1.8 m and blood volume of 74 ml/kg. The two models produce very similar resultsfor these long exposure times, although the predictionsof the present model are somewhat lower over much of the transient part of the response at all three inspired

Fig. 4. Model predictions of muscle venous %HbCO during CO re-breathing for subject 2 of Fig. 3 at various values of CO diffusioncapacity from blood to muscle tissue (DmCO). F, Data points fromFig. 3 B.

Fig. 5. Model predictions of HbCO in arterial (%HbCOa), mixed venous (%HbCOv̄), and muscle venous (%HbCOvm) compartmentsand of %MbCO vs. time during CO rebreathing for subject 2 of Burgeand Skinner (6).




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CO levels (Fig. 7). Not surprisingly, when DmCO 0(as the CFK equation assumes), the two predictions areessentially identical in the steady state. When DmCO1.75, there is a small decrease in the predicted steady-state level of %HbCO, but the difference between thetwo cases is probably too small to be detectable from

actual data. Nonetheless, there is a significant effect of allowing for uptake of CO by muscle even at this slowrate. For the three CO exposure levels (25, 200, and

1,000 ppm), the predicted %MbCO levels after 5,000min are 2.37, 14.5, and 42.1%, respectively.

Analysis of sensitivities to parameters. The sensitiv-ity of the predicted transient response of %HbCOvm to variations in the circulatory volumes and other param-eters was examined for the CO-rebreathing procedure.

[The sensitivities observed for the Benignus et al. (4)protocol were qualitatively similar to those reportedhere for the rebreathing protocol.] Varying the Hal-dane coef ficients over typical ranges reported in theliterature (MH: 218–248, MM: 25 –36) changed the slowrate of decay of %HbCO (Fig. 8 A), but only slightly. Onthe other hand, changing Q̇mfrac while other parameterswere held constant had a substantial effect on the riseof %HbCO early during CO rebreathing. As muscleblood flow increases, the muscle venous HbCO levelrises more quickly since turnover in the muscle vascu-lar compartment is faster (Fig. 8 B). Changing PmO

2

has a noticeable but small effect on the slow decay of

%HbCO (Fig. 8C) because CO competes more success-fully for Mb at lower O2 tensions. Reducing PmO2 can

offset the effect on the decay rate of a simultaneousreduction of DmCO, but the fit to the earlier data pointsbecomes more problematic if DmCO and PmO

2 are

changed from the values used in the above simulations(not shown). Changing DLCO alters the initial rise of the response (Fig. 8 D), as expected, but the “plateau”level and later decay are not affected. That is, after apseudoequilibrium is reached, the rate of transfer of CO from the lungs to the pulmonary circulation is verysmall, and the dominant factor is a slow leak of CO out

Fig. 6. Arterial (radial artery; E) and venous (antecubi-tal vein; F) %HbCO vs. time for 2 subjects from thestudy of Benignus et al. (4). Model predictions of arte-rial (dashed lines) and venous (solid lines) %HbCO. A:estimated fraction of blood volume in muscle 40%; innonmuscle 32.5%; Q̇mfrac 0.20. B: estimated frac-tion of blood volume in muscle 35%; in nonmuscle 32.5%; Q̇mfrac 0.40.

Fig. 7. Calculated solutions of Coburn-Forster-Kane equation (F) for3 levels of inspired CO, taken from Fig. 2 of Peterson and Stewart(26), and predictions of %HbCOvm from the model (dashed lines:DmCO 0; solid lines: DmCO 1.75) using parameters given in thatfigure.




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of the circulatory system (which is modeled here as aflux into the muscle compartment).

The plateau level of the response during CO re-breathing is strongly affected by the distribution of blood volume among the circulatory compartments. Asseen in Fig. 8 F , the distribution of blood volume be-tween the muscle and other tissue compartments has a

significant effect on the plateau level. Consequently,the data provide a strong criterion for selecting theseparameters. The ratio of mixed venous to arterial vol-umes also affects the plateau level, but less strongly(Fig. 8 E). It should be noted that altering one volumewithout making a compensatory change elsewhere tokeep total blood volume constant can have an indepen-

Fig. 8. Sensitivity of the model solutions for subject 2 of Fig. 3 to changes in values of the follow-ing parameters: MH and MM ( A); Q̇mfrac ( B) ; PmO2(C); DLCO ( D); ratio of mixed venous to arterialblood volume ( E); and ratio of muscle to non-muscle tissue blood volume ( F ). Note the ex-panded vertical scales in A and C.




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dent effect on the response. In addition, because chang-ing these volumes also alters the compartmental timeconstants, the effective delays can change. In thepresent model, the total delay is kept constant by theaddition of a pure time delay in the mixed venouscirculation, as described previously. Without this addi-tional compensation, the effects of changing the circu-latory volumes are more profound.

Comparison of two different exposure protocols.Tikuisis et al. (37) compared the HbCO responses of human subjects to two different pulses of CO for whichthe areas of the pulses were equal. Thus one pulse hada CO level fi ve times the other, but a duration of 1 mincompared with 5 min for the latter. Each pulse of COwas administered fi ve times at 8-min intervals. A sim-ulation of this experiment, using nominal parameter values, is presented in Fig. 9. The simulation repro-duces their experimental finding that both pulses pro-duce the same %HbCOvm level by the end of each8-min interval. The simulation also predicts that the%MbCO will differ in the two situations (%MbCO 1.83 and 1.76 at 40 min for the larger and smaller CO

pulses, respectively). It should be noted that it is alsopossible to devise two different exposures that wouldyield different %HbCO levels but the same %MbCOlevel. Thus %MbCO cannot be predicted on the basis of %HbCO alone.

DISCUSSION

Structure and Validation of Model

The transport and storage of CO in the body isdominated by the af finity of Hb for CO, and priorquantitative models, starting with the CFK equation,have emphasized this mechanism. Although the pa-rameters of the CFK equation can be adjusted to fit

measurements of venous %HbCO for CO exposures of tens of minutes or longer (14, 27), this equation doesnot accurately predict either venous or arterial%HbCO data for shorter exposures even when all ormost of the needed parameters are determined exper-imentally (4). Neither is the CFK equation able topredict the difference between venous and arterialHbCO levels (4). Smith et al. (33) recognized the need

to account for multiple compartments in the circula-tory system to model arteriovenous differences. Thepresent model adopts a similar framework but demon-strates that four circulatory compartments (instead of nine) are suf ficient to reproduce available data ontransient CO exposures. The reduced number of com-partments simplifies the estimation of parameters atthe loss of some physiological isomorphism.

The present model augments that of Smith et al. (33)in several regards. First, we have explicitly modeledthe dynamics of CO storage in the lung and its depen-dence on ventilation and on the PCO of mixed venousblood. Second, equations for the O2-dissociation curveand its dependence on the level of CO have been de-

veloped in a form that permits us to relax the assump-tion that Hb is saturated; thus it applies to both arte-rial and venous blood and in hyperoxia as well asnormoxia. The Haldane relationship is also taken intoaccount. Consequently, the model calculates HbO2 lev-els as well as HbCO levels. Third, the model includes amuscle tissue compartment containing Mb and allowsfor a small flux of CO into this compartment. TheHaldane equation for Mb is included in the model.Fourth, the model accounts for dissolved CO in bloodand tissues. Fifth, when necessary, prediction formu-las are used to estimate parameters on the basis of ageand body dimensions. Sixth, total mean transport de-

lay in the circulation is assessed and maintained con-stant as circulatory volumes are adjusted.We validated the model by using data from transient

CO exposures, then demonstrated that the model couldalso predict HbCO levels in response to inhalationexposures of up to 5,000 min. Thus this model providesa more complete representation of the dynamics of COtransport and storage than previous models. Except forthe early overshoot in the arterial HbCO data from themale subject in the Burge and Skinner study (6), theparameter sets that were found to provide visuallygood fits to the data from CO rebreathing were verysimilar, suggesting that the model structure capturesthe dynamics adequately. The overshoot reported for

the male subject was predicted well by the %HbCO inthe venous blood from the nonmuscle tissue, suggest-ing the possibility that these data represent arterial-ized venous blood. Alternatively, it would be possible toreproduce the overshoot by increasing Q̇mfrac, but itwas decided to maintain the muscle compartment asthe “slow” vascular compartment. Furthermore, theparameters determined for CO rebreathing were alsosimilar to those that provided good fits to the data frominhalation of a brief, high-intensity pulse of CO andwere adequate for predicting the CFK solutions underthree levels of long-term CO inhalation.

Fig. 9. Simulation of the experimental study of Tikuisis et al. (37)comparing %HbCO vm vs. time for 2 separate exposure protocols.Each protocol comprises 5 periods of CO inhalation indicated by thelong (5-min duration) and short (1-min duration) horizontal dashesabove the x-axis. CO level was 1,431 ppm during the longer expo-sures and 7,216 ppm during the shorter ones. At 8, 16, 24, 32, and 40min, %HbCO is the same for both protocols.




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Some of the differences between our model predic-tions and the experimental data may be related to the values for total blood volume used in the model. For allsubjects, the sum of the volumes of the four circulatorycompartments was 100% of the total blood volumereported for each subject. The experimental methodsfor determining total blood volume, however, may havebeen inconsistent with the CO exposure that was

tested. For example, Benignus et al. (4) measuredblood volume using a dilution method wherein thelabeled red blood cells were allowed a much longer timeto circulate than the duration of the CO exposure.Therefore, the measured blood volume might havebeen larger than the accessible blood volume during the CO exposure. The difference between the modeledand measured blood volumes was no more than 10% forthe two subjects from this study who were examined indetail; this difference might well have been due to thepresence of a very slowly perfused compartment thatdid not significantly affect the response to 4–6 min of CO inhalation.

An overestimate of total blood volume also couldexplain the deviation of the model predictions from theCFK equation during the initial part of the response(Fig. 7). Because the CFK equation does not allow forflux of CO to muscle, it is possible that the blood volume found to produce a good fit to data using thatequation systematically overestimates a subject’s ac-tual blood volume. If it did not, then the predicted%HbCO would be too high (just as it is higher than thepredictions of the present model). The blood volumes of our model also were less than the reported blood vol-umes for the subjects from the rebreathing study (6).By not accounting for loss of CO to muscle, that studymight have overestimated blood volume by a few per-

cent. Also, the calculated volume in that study dependssignificantly on an assumption about the venous-to-body hematocrit correction factor. Thus the blood vol-umes used in our model are not unreasonable. In fact,a formal statistical estimation of the parameters of themodel could be utilized as an alternative estimate of blood volume.

Our present model does not include the effect of Hbsaturation on the Haldane coef ficient (9, 13, 25). How-ever, based on Fig. 4 B of Di Cera et al. (9), the changein the Haldane coef ficient is 20% over the range of conditions we considered. Furthermore, Mb is effec-tively saturated with O2 or CO for the muscle PO2

values actually used. Given the small sensitivity of model responses to the Haldane coef ficient values (Fig.8 A), these effects can be ignored in the situationsmodeled in this study.

PCO2 and pH also affect both M (1, 16) and theexponent n of the Hill equation (24). We have assumedthat pH and PCO2 are constant, but the small effects onM should be included in a future model that would beapplied to other conditions where these variablesmight change significantly. The effect of changes in pHon n for Hb is not negligible, but the pH effect on n forMb is quite small (30).

Interpretation of DmCO

In the model, DmCO represents all the mechanismsthrough which CO could “leak” out of the circulatorysystem. We assume that the primary mechanism of leakage is diffusion into tissues. Because of the pres-ence of Mb, virtually all of this CO is found as MbCO.Two lines of evidence support the conclusion that

DmCO is not zero (as the CFK equation assumes). First,there is a small but steady decline of HbCO in the 10-to 40-min time period during CO rebreathing despiteendogenous CO production. The rate of decline wasgreater in the male subject whose calculated musclemass was 29.15 kg than in the two female subjectswhose muscle masses were 21.83 and 20.52 kg. Themodel could not reproduce this decline when DmCOwasset equal to zero. Second, the ratios of MbCO to HbCOpredicted by the model after long-term CO exposuresare compatible with findings in muscle samples ob-tained from anesthetized dogs and rats similarly ex-posed to CO (8, 34). Although these nonhuman data arehighly variable, MbCO levels were far from negligible,

ranging from 38 to 153% of HbCO levels. In the model,similar levels of MbCO were realized with DmCO val-ues that were much lower than typical DLCO values.

DmCO should depend on multiple physiological fac-tors, among which are the following: diffusion of COacross the capillary wall, diffusion of CO within tissue,rate of combination of CO with Mb, and the totalsurface area for diffusion. Although the total surfacearea of capillaries in muscle is unknown, the fact thatDmCO was estimated to be an order of magnitudesmaller than DLCO suggests that diffusion rates of COfrom capillaries into tissues are indeed small. Further-more, because Hb binds CO so effectively, PaCO istypically much smaller than P A CO, and therefore thedriving pressure for diffusion is much smaller in tissuecapillaries than in the lung. Nonetheless, as notedabove, the model predicts that exposures lasting morethan a few minutes will be associated with measurableincreases in MbCO levels. For the CO-rebreathing studies, however, the predicted increase in %MbCO at10 min was only 0.71, 0.61, and 0.57% for the male andtwo female subjects, respectively, which representedthe binding of an additional 1.33, 0.86, and 0.76 ml,respectively, of CO to Mb. These amounts are smallenough compared with the 60 or 70 ml of CO originallyinjected into the rebreathing circuit that the loss of COfrom the circulation has only a slight effect on the

estimation of total blood volume. Note, however, thatafter 40 min of CO rebreathing, the increase of %MbCOwas approximately one-third of that of %HbCO.

DmCO in the male subject of Burge and Skinner (6)was found to be higher than DmCO for the two femalesubjects. Because of the surface area effect discussedabove, one expects that DmCO will be larger in menbecause of their larger muscle mass. Also, the capillarydensity in muscle may be further increased in malesubjects who exercise frequently (4).

The distribution of inhaled CO into muscle tissue aswell as into multiple circulatory compartments compli-




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cates the prediction of washout profiles after cessationof the exposure. If alveolar ventilation is constant, thewashout profile will not be monoexponential. Further-more, because of the slow transfer of CO between Mband Hb, washout of the muscle tissue compartmentwill be much slower than washout of the circulation.Consequently, the washout profile is expected to ex-hibit a long tail, and the time required for full washout

may be very much longer than twice the half-time of the washout (26, 32).

Comparison to Previous Models

The CFK equation is a first-order approximation tothe dynamics of CO storage in blood under the assump-tions that ventilation, PICO, and PaO

2 are constant and

that no CO leaks out of the blood. It provides a goodapproximation to the HbCO response to a steady levelof inhaled CO (27). Furthermore, when measurementsare made periodically and the equation is solved againfor each time period, one often obtains a reasonablematch to the data (5, 14, 27), although significant

errors may accumulate as this procedure is repeatedduring successive time intervals. Although these ap-proaches are useful approximations, they are limitedbecause 1) the equation must be solved every time ventilation, inspired PCO, or PaO

2 changes; 2) the arte-

riovenous differences cannot be reproduced unless cir-culatory compartments are introduced; and 3) flux of CO into tissue is ignored. Thus these approaches areinef ficient for situations involving frequent changes inPICO or ventilation and are inapplicable for CO re-breathing in which ventilation is effectively zero andPICO changes rapidly and continuously. Finally, theseapproaches would be unable to predict the levels of COin compartments other than the blood, e.g., musclesand brain.

The present model utilizes a standard mass balanceapproach to describe CO storage dynamics. Although itassumes that ventilation and PO2 are constant, extend-ing the model to allow these variables to be time varying would be mathematically trivial. Thus, forexample, the model could be enhanced by including O2transport and stores in the lung compartment and O2flux into the pulmonary capillary blood (22). It is pos-sible to develop more detailed descriptions of lung-capillary exchange (31) and capillary-tissue exchange(20) than the simple structures used here; however, theparameters of these detailed descriptions would not be

well specified by the type of data available from wholebody CO exposures in humans.The present results reinforce the need for precise

measurement of the flux of CO from the circulation intomuscle tissue; however, the predicted slow rate of fluxwill be dif ficult to measure. The amount of Mb inmuscle is another important factor that influenceswhole body distribution of CO. Reported values for Mbconcentrations differ both among species and among muscles in a given species. Model predictions would beimproved by having more precise measurements of average Mb concentrations in human muscle tissues.

In addition, although PmO2 only has a small effect on

HbCO levels in hyperoxia and normoxia, one can an-ticipate that it will be more important in hypoxia.Consequently, better measurements of this variableare necessary.

Our primary purpose was to establish the feasibilityand potential importance of including MbCO formationin any model of CO uptake rather than to provide a

subject-specific clinical or experimental tool. Nonethe-less, one can offer some general guidelines. One ex-pects that the accuracy of the model will be enhancedwhen more of its parameters are measured for eachsubject. The steady-state %HbCO level is most depen-dent on inspired CO level, Hb concentration, ventila-tion, and total blood volume. The latter two variablesmay be unavailable in a clinical setting but should bemeasured in experimental studies. The time course of changes in %HbCO depend also on the distribution of blood flows and blood volumes among the cardiovascu-lar compartments. Aside from Q̇, these flows and vol-umes represent “effective” or average values for muscleand nonmuscle tissue compartments and are not di-

rectly measurable; however, with the use of statisticalestimation methods, average values for these parame-ters (and DmCO) could be determined in a representa-tive population. The regression formulas used in thisstudy to estimate muscle and nonmuscle tissue vol-umes (and Q̇, if necessary) are reasonably accurate forclinical purposes. Finally, as shown in Fig. 8 A, thespecific values chosen for the Haldane coef ficients forblood and muscle and the muscle PO2 have small effectson the responses (in normoxia and hyperoxia), andtherefore it is less critical that precise values for thesecoef ficients be determined.

In conclusion, although changes in HbCO levels dur-

ing CO inhalation can be approximated by the CFK equation, a more complete description of transport andstorage mechanisms of CO is needed to account forHbCO levels when inspired CO levels change rapidlyor when one needs to account for differences betweenarterial and venous HbCO levels. The model presentedhere reproduces whole body responses to both tran-sient and long-term inhalations of CO and appears tobe a minimal structure that can reproduce reportedarteriovenous differences in %HbCO during transientexposures. The present results further suggest thatmeasurable amounts of CO diffuse out of the circula-tory system and into tissue compartments in as littleas 10 min. To ascertain the distribution of CO to tissue

such as the brain (and its effects on O2 delivery), it willbe important to measure more precisely the flux of COfrom the circulation into tissues. It is also likely thatthe CFK equation is inadequate to closely predict thewashout of CO during treatment of CO poisoning,whereas the present model should provide improvedpredictions of the time course of CO washout from thebody. Finally, in contrast to approaches based on theCFK equation, the present model predicts uptake anddistribution of CO in both vascular and tissue compart-ments during inhalation of constant or variable levelsof CO.




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The authors thank Drs. Caroline Burge and Vernon Benignus forproviding both data published in their papers (4, 6) and unpublishedmeasurements of parameter values from their subjects.

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Documents

Eugene a Multicompartment Model of Carboxyhemoglobin and Carboxymyoglobin Responses to Inhalation of Carbon Monoxide