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A DYNAMIC MODEL OF CARBON DIOXIDE TRANSPORT IN THE BLOOD S.E. Rees 2 , S. Andreassen\ R. Hovorka 2 , E.R. Carson

b

2

" Centre for Measurement and Information in Medicine, City University, London, UK Depanment of Medical Informatics and Image Analysis, Aalborg University, Denmark

Abstract: This paper describes a model of carbon dioxide (C02) transport in the blood. Equations are used to represent the flow of CO 2 between tissues, blood and lung alveoli, and the chemistry associated with CO2 buffering. The dynamic model is used to represent the effects of hypo- and hyper-ventilation on end tidal COz concentration (FE'COz). Time constants describing the response to perturbation and equilibrium concentrations of FEC0 2 are shown to be consistent with those reported in the literature. Keywords: Mathematical, modelling, simulation, dynamic, pH control.

1. INTRODUCTION

representing CO 2 in tissues. venous blood and lung alveoli.

Determining the optimal ventilatory pattern for patients residing in the intensive care unit (ICU) is a complex process whereby the clinician aims to adequately oxygenate the patient without causing lung damage due to high pressures, acidosis. or a toxic reaction to oxygen. To make decisions in the presence of these conflicting goals the clinician requires an understanding of lung function and oxygen/carbon dioxide (C0 2) transport, and the response of these systems to perturbations in ventilation. Mathematical models can improve our understanding of these systems, and act as useful tools when predicting the effects of ventilatory perturbation. This paper addresses these problems further, describing a new physiologically-based model of CO 2 transport, which can be used to predict changes in CO 2 concentrations during hypo- and hyper-ventilation.

2. THE CARBON DIOXIDE MODEL

v co, '

The carbon dioxide (C0 2) model, illustrated in figure 1, is a compartmental model with three compartments

Fp

Fig 1. The physiology covered by the carbon dioxide model.

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total CO 2 (tC0 2) and the normal reference value (ntC02), i.e. (EC02 = tC0 2 - ntC0 2). By using the concept of carbon dioxide excess, mass balance equations can be written to describe the relationship between CO2 in lung capillary, arterial, and tissue capillary blood.

Two types of equation are required to describe CO 2 transport and buffering, dynamic equations describing the flow of CO 2 between compartments and chemical equations describing the buffering of CO 2 in the blood.

I

·I-A

• Ac

=

(1)

(2)

(5) (6) (7)

dldt ECO;Y = (ECO~ - ECO;Y)QN my

(8)

To complete the description of CO 2 dynamics equations are required to describe alveolar ventilation, i.e.

Equation 2 describes the rate of change of mass of COz in the tissue compartment (mtC0 2), where V~o, is the tissue production of CO 2 and V~~, is the transport of COz from the tissues (t) to the tissue capillaries (c). A further equation (eqn 4) describes the relationship between COz pressure (PtC0 2) and the amount of CO 2 (mtC0 2) in the tissues. This IS assumed to be a linear relationship such that

* dldt mtC0 2

•

Equation 5 states that the excess CO 2 in lung capillaries (ECO~ is equal to the excess CO 2 in mixed venous blood (EC:?i') less any CO 2 transported to lung alveoli (Veo,) , where blood flow through lung capillaries is represented by cardiac output (0) corrected for the fraction "shunted" past the lung alveoli (s). Equation 6 describes excess of CO2 in the arterial blood following mixing of lung capillary and "shunted" mixed venous blood. Equation 7 states that the excess CO 2 in tissue capillaries (ECO~) is equal to the excess CO 2 in arterial blood (ECO;) plus any CO2 .transponed to the (·c tissue capillaries from the tissues (Vco,). The change in excess CO2 in the mixed venous compartment can be described using a differential equation by accounting for the excess CO 2 flowing into (ECO~ .0) and out of (ECO~Y. 0) the mixed venous pool, and the volume of mixed venous blood (Vmv).

Equation 1 states that the change in the fractional concentration of CO 2 in the alveoli (FEC0 2) is calculated by accounting for the flow of carbon dioxide ('i~) into the alveolar compartment (A) i!'L,m lung capillaries (1), and for the flow of CO 2 ( VCo,) out of the alveolar compartment into the environment (e) where VA represents the alveolar volume. This equation uses the assumption that the fractional concentration of CO2 is the same in end tidal and alveolar gases. A further assumption is that end tidal CO 2 is measured relative to dry air. Using this assumption the partial pressure of CO 2 in the alveoli (PAC0 2) can be calculated from FEC0 2, the barometric pressure (BP=101.3 kPa) and the pressure of saturated water vapour (PH20 =6.3 kPa), i.e

dldt PtCO: = K

·I-A

~

Differential equations are used to describe the change in CO2 mass or concentration in the lung alveoli and tissue compartments dldt FEC0 2 = (Vco,- VCo,)NA . tp - t-c dldt mtC0 2 Vco, - Vco,

my

EC0 2 = EC0 2 - Veo, I (Q(1-S» EC022 = S EC0 2my + (1-s) ECOIc2 c: -1< . EC0 2 = EC0 2 + Vco,lQ

2.1. Equations describing CO 2 dynamics

VA=f(VTVD) • Ac • VCO, = VA (FECO z - FrC0 2)

(9) (10)

Equation 9 describes the volume of air (VA) flowing into the alveoli per minute, as a product of the respiratory frequency (f) and the tidal volume (VT) minus the anatomical dead space (VD). .,..., Equation 10 describes carbon dioxide removal (Veo,) in terms of VA and the difference between the fractional concentration of CO 2 in the expired (E) and inspired en gases.

(4)

The constant K is fixed at a value of 0.21 kPalmmol. This value results in a model predicted increase in arterial PC0 2 of 0.4-0.8 kPalmin on complete cessation of ventilation, a rate of increase which is consistent with that reported by Nunn (1987).

2.2 Equations describing the chemistry of CO 2 transport in the blood

Carbon dioxide is transported in blood in physical solution and as bicarbonate (HCOl ). For blood under normal conditions (pH in the plasma =7.4, PCO!= 5.33 kPa, Hb = 9.3 mmoIII) the normal total CO 2 concentration (ntC0 2) can be calculated from the normal concentrations of physically dissolved CO 2 and HCO l in blood. ntC0 2 can then be used as a reference value to define the excess carbon dioxide (EC0 2) in the blood as the difference between the

The majority of CO 2 produced by the tissues is transponed in the blood as bicarbonate. CO: conversion to bicarbonate is one of a complex set of reactions regulating the acid-base chemistry of the blood. A model describing the acid-base chemistry has been built (Rees, et al., 1996) and is illustrated in figure 2.

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A further equation is required to describe the relationship between pH in the plasma and erythrocyte (equation 18). This equation has been experimentally determined (Funder and Weith 1966, Gunn 1989) and accounts for the swelling of the erythrocyte under acidotic conditions

ERYrHROCYI'E

H., + HC03.• - . H., + NBB. - .

H 20. + C02.. HNBB.

pH.,= pHp + log

PLASMA

Fig. 2. Reactions describing the acid-base chemistry of the blood (see text for explanation). This model represents the acid-base chemistry in terms of chemical reactions describing bicarbonate (HC03) and lumped non-bicarbonate (NBB, HNBB) buffers in both the plasma (p) and erythrocyte (e) fractions of the blood. For each of the reaction equations illustrated in figure 2 a HendersonHasselbalch equation can be written to describe the equilibrium position of all acids and bases both before and after a perturbation (Eqs. 11-14).

(18)

Total carbon dioxide (tCOV is similarly defined as the sum of physically dissolved CO 2 and ~icarbonate (HC0 3) in both fractions of blood, i.e.

(11) (12) (13) (14)

Using the calculated values of all buffer acids and bases at normal conditions, values for the normal buffer base (nBB) and the normal total CO: (ntC02) can be calculated. This can only be done if values of fp and f. are known. Siggaard-Andersen (1974) has formulated a relationship between f. and the hemoglobin concentrations in plasma (Hb~) and erythrocyte (Hb.), i.e.

The majority of non-bicarbonate buffers in plasma are protein and in erythrocyte haemoglobin (Hb) (Siggaard-Andersen, 1974). Assuming that the concentrations of Hb and protein are constant then it follows that: NBBp+ HNBBp= Cp NBB.+ HNBB.= C.

(3.094 - 0.335 pH,,)

Blood under normal conditions has pHp=7.4 and PC02.p=5.33 kPa (C02.p=1.2 mmoVl). Using these values equations 11-18 can be solved uniquely to obtain estimates of all buffer acids and bases in normal conditions (i.e. C02.. = 1.02 mmoVl, PH. = 7.19, HC03•p= 23.93 mmoVl, HC03•• = 12.49 mmoVl, NBBp= 17.75 mmoVl, HNBBp= 6.01 mmoVl. NEB. = 43.55 mmoVl, HNBB. = 295.17 mmoVl). These values define a reference point at which the total buffer base and the total carbon dioxide in the system are said to be normal. Total buffer base is defined as the sum of base (HC~+NBB) in both the plasma (fp) and erythrocyte (f.) fractions of blood. i.e.

Hp + HC0 3•p - . H20p + C02.p Hp + NBBp -r-+ HNBBp

pHp = pKHco).p + loglo (HC0 3•p/C02.p) pH., = pKHcO).e + loglo (HC0 3•• /C02..) pHp = pKNBBp + loglo (NBBpIHNBBp) pH., = pKNBBc + 10glO (NBB.IHNBB.)

10

(15) (16)

(21) Values for the constants Cp= 23 .76 (mmoVl) and C.= 338.72 (mmoVl), and other model parameters (pKNBBc =8.02, pKNBBp = 6.93) have been determined from data describing plasma and whole blood under normal conditions and by assuming a dissociation constant for bicarbonate in plasma and erythrocyte equal to pKHcO).p, pKHco).c: = 6.1 as described by Rees et al. (1996).

In addition f.+fp= 1 and the concentration of Hb in the erythrocyte is assumed to be constant (Hb.=21 mmoVl, (Siggaard-Andersen, 1974). In normal conditions where Hb p=9.3mmoVl. it follows that f. = 0.44, fp=0.56, nBB=48 .04 mmoVl, and ntCO:=19.98 mmoVl. Perturbation of the system can be defined using the normal buffer base (nBB) and the normal total carbon dioxide (ntC02) as reference. Addition or removal of strong acid being described as an excess of base (BE) above normal i.e.

In addition it is assumed that the partial pressures of CO 2 in the plasma and erythrocyte are equal so that

By using the definition of the solubility coefficients in plasma (Clp= 0.225) and erythrocyte (a.. = 0.191) this equation can be re-written as

BE = BB - nBB

(22)

addition or removal of carbon dioxide being similarly defined as an excess of total CO 2 (ECO:) above normal, i.e.

(17)

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Table 1 - Equilibration of blood at a new

(23) In summary, blood has an acid-base status which can be completely described by three variables: the increase in strong acid relative to nonnal conditions, i.e. the base excess (BE); the increase in total CO~ relative to nonnal conditions, i.e. the CO, exces~ (EC0 2); and the haemoglobin concentrati;n (Hb), which describes the fraction of blood in plasma and erythrocyte. Using the values of BE ECO, and Hb with variables describing blood at n~nnal ~eference conditions (pHp=7.4 PCO~5.33 kPa), equations 1123 can be solved uniquely to detennine the concentration of all buffer acids and bases and the pH of both the plasma and erythrocyte fractions of blood.

PCO~

Before Perturbation

Equilibration at PC0 2

Values known

Values known BE Hb

ECO:!

BE

Hb

(nunoVl)

(nunoVl)

(nunoVl)

(mmoVl)

(nunoVl)

(kh)

1.67

0.0

9.3

0.0

9.3

5.30

Model calculated pHp 7.36 PC02 (lcPa) 6.25

PC02

Model calculated 7.40 pHp EC02 (mmoVl) 0.02 1.65 CO2.• (nunoVl)

before and after the perturbation, and the reduction in EC02 necessary to equilibrate the blood at pC0 2 = 5.3 kPa (i.e. C02.J. The same experiment can be perfonned using the Siggaard-Andersen nomograrn. and results in predictions of HC03•p, pHp equivalent to those perfonned by the model (as illustrated by Rees et al. 1996).

2.3 Using the acid-base model as pan of the dynamic CO2 model The acid-base model (equations 11-23) is used as an integral part of the dynamic model of CO 2 transport. CO2 is transported into blood at the tissues capillaries and removed from blood at the lung capillaries in processes involving equilibration of PC0 2 between tissues and tissue capillaries and between luner alveoli '" blood and lung capillaries. In the arteries and veins with different excess of CO 2 (but the same BE and Hb) is mixed and a new equilibrium is reached for the acid-base chemistry. Examples of the use of the acid base model (eqns 11-23) to simulate both these situations are give below.

Using the acid-base model to describe the mixing of shunted mixed venous and blood from the lung capillaries. The majority of blood entering the arteries is from lung capillaries. The remainder is blood which has been "shunted" past the lung directly from the mixed venous pool. End capillary and shunted venous blood are mixed in the arteries and a new acid-base status is achieved. Since CO 2 is the only acid or base added or removed in the lung capillaries then it follows that lung capillary and mixed venous blood will have the same base excess (BE) and haemoglobin concentration (Hb), varying only in their concentration of EC0 2. Mixing these pools will therefore result in arterial values of BE and Hb which are unchanged, and a value of ECO:! which can be calculated from the sum of ECO, enteriM the arterial pool from the mixed venous (s -* ECO~v). and lun!! I capillary I-s)* ECO:!) blood, i.e equation 6, where s represents the fraction of blood shunted past the lungs.

Using the acid-base model to describe diffusion of CO 2 at lung and tissue capillaries.

Blood entering lung capillaries has an acid-base status which can be completely described by the total base excess (BE). CO 2 excess (ECO:!) and haemoglobin concentration (Hb). As blood passes through the capillaries a PCO, gradient between luner capillaries and alveoli causes ~n- amount of CO 2 (V~~ in figure 1) to diffuse into lung alveoli. resultin!! in an equilibrium position for end capillary blood -where PC0 2 in alveoli and lung capillary blood are equ~valent. an~I~COf has been reduced by an amount eqUivalent to Vco,l(Q(l-s» (see equation 5).

«

Table 2 illustrates a model simulated experiment where 20% mixed venous blood at BE=O mmoln EC0 2 = 1.67 mmoln and Hb=9.3 mmolll is mixed with 80% end lung capillary blood at BE: 0 mrnoln ECO;f= 0.02 mmolll and Hb= 9.3 mmolll.

The acid-base model (eqns 11 -23) can be used to simulate the titration of mixed venous blood at a known BE, EC0 2 and Hb to a new pCO:!. Table 1 illustrates a model simulated experiment where venous blood at BE= 0.0 mmolll, ECO,=1.67 mmolll and Hb=9.3 mmolll is equilibrated -with alveolar pC0 2= 5.30 kPa.

Equation 6 is used to calculate the excess CO, in the arterial blood (ECO~, the acid-base model is then used to calculate the concentrations of all buffer acids and bases in the mixed venous, end lung capillary and arterial pools.

Using the model it is possible to calculate the concentrations of all buffer acids and bases both

60

Table 2 - Mixing two pools of blood with different ECO z

38

~[;d

36

Venous blood

Lung blood

Capillary

Arterial blood

505

34

ZOo

:9'32

4.19 4.52

no

0.5

1

30

Values known ECO~v 1.67 BE 0.0 Hb 9.3

Values known ECO~ 0.02 BE 0.0 Hb 9.3

Values known BE 0.0 Hb 9.3

828 u

w "-26

~~

24 22

1

605

Model calculated pHp PC0 2

Model calculated 7.40 pHp 7.39 5.30 PC02 5.48 ECO; 0.35 (All values reported in nunolll except PC0 2 (in kPa) and pH)

18

399"-

3

3n~

... ;45 3.19 2.'12

2.56

20

Model calculated pHp 7.36 PC0 2 6.25

4.25

0

20

40

60

80

100

1.0 2.39

lv.E(mon)

Fig 3. Model predicted end tidal CO2 pressure (PEC0 2) and concentration (FEC0 2) following hyper- (0-60 minutes) and hypo-ventilation (60-120 minutes). Vignettes illustrates the change in PEC02 following the first minute of each experiment and hence the "fast" first exponential of the response.

3. USING THE DYNAMIC CO 2 MODEL TO

REPRESENT HYPO- AND HYPERVENTILAnON

36 mmHg), values which are consistent with those reported by Ivanov and Nunn (1968) i.c. FEC02 =20.3 ± 3.6 mmHg and 30.7 ± sd 4.1 mmHg respectively. In both the hyper- and hypo-ventilation experiments the model predicted change in FECO! followed a double exponential. The time constants describing FEC0 2 following hypoventilation are 6 seconds for the first exponential and 10 minutes for the second. The model determined time constant describing the first exponential has a value (6 seconds) somewhat smaller than the mean reported by Ivanov and Nunn (29.4 ± 12.0 seconds). It is however within the range of individual values reported by these authors who found values for this time constant as low as 2 seconds. The model determined time constant describing the second exponential has a value (10 minutes) consistent with the mean reported by Ivanov and Nunn (1968) of 10.6 ± 4.3 minutes.

Ivanov and Nunn (1968) have investigated the response of end tidal CO 2 concentration (FEC00 and pressure (PEC0 2) to step changes in ventilation. In their study patients were hyperventilated (minute volume = 10 Vmin), for periods of varying duration, followed by periods of hypo ventilation (minute volume = 5 Vmin). Hyperventilation resulted in a mean value and standard deviation of PEC0 2 = 20.3 ± 3.6 mmHg. Hypoventilation was characterised by an increased PEC0 2 , the dynamics of which could be describe by a two exponential function. The first exponential was found to have a time constant of 29.4 ± 12.0 seconds, whilst the second had a time constant of 10.6 ± 4.3 minutes which was related to the duration of the previous hyperventilation. The hyper- and hypoventilation experiments described by Ivanov and Nunn (1968) have been simulated using our model of CO 2 transport. The model was first equilibrated at a minute volume = 5 Vmin so that venous pH=7.36, venous PC0 2= 6.25 kPa, arterial pH =7.4, arterial PC0 2= 5.1 kPa and FE'C02 = 5.1 % (PEC0 2 = 36 nunHg). This equilibrium position was consistent with model estimated parameters of shunt = 5.1 'la, VD = 800 ml and a carbon dioxide production rate of "CD! = 0.22 Vmin. Model parameters were then fixed at these values and the model used to simulate the effects of 60 minutes hyper- and hypoventilation at minute volumes of 10 Vmin and 5 Vmin respectively.

4. DISCUSSION AND CONCLUSIONS This paper has described a new dynamic model of CO 2 transport in the blood. The model includes two types of equations representing both CO 2 dynamics and the acid-base chemistry associated with CO2 buffering. The model of acid-base chemistry described here enables any pool of blood to be . completely characterised by its base excess (BE), haemoglobin concentration (Hb), and the excess CO 2 (EC0 2) relative to normal conditions (pHp=7.4, PC0 2 = 5.33 kPa). Using this representation experiments have been performed simulating titration of blood at a new PC0 2, as occurs in lung and tissue capillaries, and the mixing of two pools of blood with different excess of CO 2, as occurs in the arteries and veins. These experiments are used as an integral part of the dynamic model which along with equations keeping

Figure 3 illustrates the model predicted changes in FEC0 2 and PEC0 2 following hyper- and hypoventilation. After hyperventilation the model reaches equilibrium at FEC02= 2.5 % (PE'C02 =19 mmHg), and after hypoventilation at FEC0 2= 5.1 % (PECO! =

61

ECO'2

track of BE, EC02 and Hb are sufficient to describe the transport of CO 2 in the blood.

ECO~

ECO~ BE

The dynamic model of CO2 transport also includes a description of CO 2 production and storage by the tissues, and CO2 removal via ventilation. For a known • tp rate of CO 2 production by the tissues (VCo,) , and known values of other model parameters (shunt, VD) , the model can be used to describe the dynamic response of the system to perturbation in ventilation. In particular, the model has been used to simulate published experiments where subjects were hypo- and hyper-ventilated. Model predictions were shown to be consistent with those reported in the literature both in terms of their dynamic response and the equilibrium concentration of expired CO 2 •

s

Q Vmv

VA f

VT VD NBB HNBB Hb pK

In future the carbon dioxide model could be used as a tool to aid the intensive care clinician in predicting the likely effects of ventilatory perturbation on the acidity of the blood. The CO2 model would be particularly applicable when used in combination with a model predicting the effects of ventilatory perturbation on oxygenation (Andreassen et a1. 1996). Combining these models would enable the clinician to detennine the appropriate compromise between adequate oxygenation and normalisation of pH when deciding upon an appropriate ventilatory strategy.

REFERENCES Andreassen S., J. Egeberg, M.P. Schroter and PT. Andersen (1996). Estimation of pulmonary diffusion resistance and shunt in an oxygen status model. Computer Methods and Programs in Biomedicine 51, 95-105. Funder J. and J.O. Weith (1966). Chloride and hydrogen ion distribution between human red cells and plasma. Acta Physiol. Scand. 68 , 234-235.

GLOSSARY

• Ae

Vc 0, • tp

Vco, · t< Vco,

VA PtCO! PAC0 2 BP PH20 mtC02 HCO) ntC0 2

tC0 2

ECO~ ECO;'

Excess CO 2 in arterial blood Excess CO 2 in lung capillary blood Excess CO 2 in tissue capillary blood Base excess concentration Fraction of total cardiac output "shunted" past lung capillaries Cardiac Output Volume of mixed venous blood Alveolar ventilation Respiratory frequency Tidal volume Anatomical dead space Non':bicarbonate base concentration. Non-bicarbonate acid concentration. Haemoglobin concentration Negative logarithm of dissociation constant Plasma fraction of blood Erythrocyte fraction of blood

Subscripts Plasma p Erythrocyte e Added a

Support for this work was provided by the Engineering and Physical Sciences Research Council.

FlC02 FE'C0 2 PE'CQ, • ~A Vco,

.

Inspired CO2 concentration End tidal CO 2 concentration End tidal CO 2 pressure CO 2 flow from lung capillaries (I) to alveoli (A). CO 2 flow from alveoli (A) to the environment (e). CO 2 tissue production CO 2 flow from tissues (t) to tissue capillaries (c) Alveolar volume Pressure CO 2 in the tissue pool (t) Pressure CO 2 in the alveoli (A) Barometric pressure Pressure of saturated water vapour Mass CO2 in the tissue pool Bicarbonate concentration Total CO 2 concentration at normal conditions (pHp = 7.4, PC0 2= 5.33, Hb=9.3) Total CO 2 concentration in blood Excess CO 2 in lung capillaries Excess CO 2 in mixed yenous blood

Gunn R.B. (1989). Buffer equilibria in red blood cells. In: The regulation of acid-base balance (D.W. Seldin, and G. Giebisch. (Ed.» . 57-67 . Raven Press Ltd. New York. Ivanov S.D., and J.F. Nunn (1968). Influence of duration of hyperventilation on rise time of PCOl after a step reduction in ventilation. Respiration Physiology 5, 243-249. Nunn J.F (1987). Applied Respiratory Physiology, Butterworth & Co., U.K. Rees S.E., S. Andreassen, R. Hovorka, R. Summers and E.R. Carson (1996). Acid-base chemistry of the blood - a general model. Computer Methods and Programs in Biomedicine 51 , 107-119. Siaaaard-Andersen 0 (1974). The Acid·Base Status I:IC of the Blood, Munksgaard, Copenhagen.

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