a model for the distribution and transport of co2 in the body and the ventilatory response to co2

43
Re.@wtion Physiology (1968/l 969) 6, 45-87 ; Nourh-Holland Publishing Company, Amsterdam A MODEL FOR THE DISTRIBUTION AND TRANSPORT OF CO, IN THE BODY AND THE VENTILATORY RESPONSE TO CO, C. M. E. MATTHEWS, G. LASZLO, E. J. M. CAMPBELL AND D. J. C. READ M. R.C. Cyc.lofron Unit and Ro~wl Postgraduate Medical School, Hammersmith Hospital, Dwane Road, London, W. 12, England Abstract. A mathematical model has been constructed which simulates changes in ventilation, Pu)~, bicarbonate and hydrogen ion concentrations in different body compartments during re- breathing, COZ inhalation and hyperventilation. The model is based on equations for the balance of ions. Henderson’s equation, and Van Slyke’s equation for buffering power applied to Con, bicarbon- ate, and haemoglobin buffer in a system of body compartments. The system of compartments is based on the results obtained in experiments with “C labelled Con. described in the previous paper. The equations are simulated on a digital computer. The computer values match a number of experimental results and the model clarifies the relations between the various parameters involved. Body compartments CO2 stores Model of CO2 distribution Studies of the CO, concentration in the body fluids, of the kinetics of CO, distribution, and of the control of ventilation have led to the formulation of various models. Our studies of the kinetics of “CO, distribution (FOWLE, MATTHEWS and CAMPBELL, 1964: MATTHEWS et ul., 1968) of the apparent CO, dissociation curve of the body (FOWLL-: and CAMPBELL (1964); LASZLO, CLARK and CAMPBELL, to be published), and of the ventilatory response to CO, (READ, 1967) have made us increasingly aware of the incompleteness and incompatibility of these existing models and we have therefore sought to develop an improved one. Ovbiously such a model is likely to be more complicated than most earlier ones but this handicap is lessened by the availability of a computer. The purposes of this paper are first to describe the model and secondly to show how far the interpretation of data in the different fields are consistent with a single overall formulation. Difficulty in achieving such compatibility might point the why for future experimental studies. The model The present model combines many of the features of earlier systems, and in addition Accepiedfbr publication 8 July 1968. 45

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Page 1: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

Re.@wtion Physiology (1968/l 969) 6, 45-87 ; Nourh-Holland Publishing Company, Amsterdam

A MODEL FOR THE DISTRIBUTION AND TRANSPORT OF CO,

IN THE BODY AND THE VENTILATORY RESPONSE TO CO,

C. M. E. MATTHEWS, G. LASZLO, E. J. M. CAMPBELL AND D. J. C. READ

M. R.C. Cyc.lofron Unit and Ro~wl Postgraduate Medical School, Hammersmith Hospital, Dwane Road, London, W. 12, England

Abstract. A mathematical model has been constructed which simulates changes in ventilation, Pu)~, bicarbonate and hydrogen ion concentrations in different body compartments during re- breathing, COZ inhalation and hyperventilation. The model is based on equations for the balance of ions. Henderson’s equation, and Van Slyke’s equation for buffering power applied to Con, bicarbon- ate, and haemoglobin buffer in a system of body compartments. The system of compartments is based on the results obtained in experiments with “C labelled Con. described in the previous paper. The equations are simulated on a digital computer. The computer values match a number of experimental results and the model clarifies the relations between the various parameters involved.

Body compartments CO2 stores

Model of CO2 distribution

Studies of the CO, concentration in the body fluids, of the kinetics of CO, distribution,

and of the control of ventilation have led to the formulation of various models. Our

studies of the kinetics of “CO, distribution (FOWLE, MATTHEWS and CAMPBELL, 1964:

MATTHEWS et ul., 1968) of the apparent CO, dissociation curve of the body (FOWLL-:

and CAMPBELL (1964); LASZLO, CLARK and CAMPBELL, to be published), and of the

ventilatory response to CO, (READ, 1967) have made us increasingly aware of the

incompleteness and incompatibility of these existing models and we have therefore

sought to develop an improved one. Ovbiously such a model is likely to be more

complicated than most earlier ones but this handicap is lessened by the availability of

a computer.

The purposes of this paper are first to describe the model and secondly to show

how far the interpretation of data in the different fields are consistent with a single

overall formulation. Difficulty in achieving such compatibility might point the why

for future experimental studies.

The model

The present model combines many of the features of earlier systems, and in addition

Accepiedfbr publication 8 July 1968.

45

Page 2: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

46 C. M. E. MATTHEWS et d

introduces some new concepts. Body CO, is divided into a number of compartments

as shown in fig. 1. Anatomically these consist of lungs, arterial blood, venous blood,

brain, well perfused tissues other than brain, and poorly perfused tissues (i.e. muscle).

Each of these compartments? except lungs and brain, is further subdivided. In the

venous and arterial blood, dissolved CO, and bicarbonate form separate pools.

The tissue compartments are divided into two pools, one containing extracellular

dissolved CO, and extracellular bicarbonate plus intracellular dissolved CO,, and

the other containing intracellular bicarbonate. Thus the intracellular and extracellular

P car are assumed to be the same.

The slow exchange between the pools is assumed to be due to the slow rate of hy-

dration of CO, in the absence of carbonic anhydrase ( ROUGHTON, 1964), since it is

known that muscle cells contain little carbonic anhydrase. This is discussed in detail

later and also in the previous paper (MATTHEWS et cd., 1968).

Another feature of the present model is the simulation of an effect observed by

JONES et al. (1967). They found that during rebreathing from a small bag the arterial

P co2 is 2-4 mm Hg lower than the alveolar PcoL, and suggested that this was due to

incomplete buffering in pulmonary capillary blood. Due to the short transit time in

the pulmonary capillaries there may not be time for complete exchange with the

bicarbonate and haemoglobin buffer anions in the red cells. This effect was simulated

simply by reducing the buffering capacity of p~~imonary capillary blood compared

with that of arterial blood (see Discussion).

The exact anatomical significance of the pool from which ventilation is controlled

is not specified in this model. Thus the “brain” pool may be regarded as an operational

pool; i.e. it is the simplest system required to match the experimental results, and the

real system will certainly be more complex. All other pools represent CO, in definite

anatomical sites so that the amount of CO, in each of them is known at least ap-

proximately. Hence the number of variables in the model is not as great as might

appear at first sight. Receptors sensitive to hypoxia are not included in the model, and

hypoxic conditions are not simulated.

The behaviour of the model has been compared with data from four different

kinds of experiment: rebreathing from a small bag; inhalation of CO,; hyperventi-

lation; and tracer experiments with “CO, (MATTHEWS rt d., 1968). Experimental

data include changes of alveolar PCo2, ventilation, and bicarbonate and hydrogen ion

concentrations in arterial blood. Some of these data came from experiments reported

here and some from published work. Since there is considerable variation in results

in different subjects we have tried as far as possible to compare several experiments in

the same individual.

Theory

THEORY OF BUFFER INTERACTIONS

The differential equations of the system were derived from a general equation relating

concentrations of bicarbonate, buffer anion, and dissolved CO,. This general equation

Page 3: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO2 DISTRIBUTION 47

was obtained by combining the Henderson equation, Van Slyke’s equation for buffer- ing power, and equations expressing the balance of COz, bicarbonate and hydrogen ions as follows. This approach is rather similar to that of DANTZIG et al. (1961), but free-energy functions are not used,

It is assumed that concentrations of H,CO, are very small so that the reactions

CO, f H,O = H&O, * H * + HCO, -

may be written

CO,+H,O~H++HCO,-

Also

H++B-;rlHB

where B- represents buffer anion. Let x = [CO, f =con~entration of dissolved CO, in M/l,

Y = [HCO,- ]--concentration of bicarbonate in M/I, Z= [H+] =~onc~ntration of hydrogen ions in M/l, B= [B-l =concentration of buffer anions in M/l, X,, Y,, Z,, B, = instantaneous increase in X, Y, Z, or B due to respective addition

of CO,, bicarbonate, hydrogen ions, or buffer anions to the system.

Subscript 5 denotes initial values. We have used hydrogen ion concentration throughout rather than hydrogen ion

activity. The latter would probably be more realistic for comparison with experimental measurements of pH but the assumption of a constant activity coefficient of 1 is, on present evidence, a reasonable approximation. We have also considered whole blood without separating red cells and plasma. This introduces no important error into the computations because of two fortunate physicochemical facts about blood. Firstly, the CO, dissociation curve of whole blood is virtually parallel to that of the true plasma. Secondly, the relationship of change in bicarbonate concentration for a given change in pH is approximately the same in red cells as in plasma, and therefore changes in the acid base ratio of the blood buffers are mirrored with reasonable accur- acy by changes in plasma pli. Over a reasonably small range of pH change this holds also for hydrogen ion concentration. Hence when comparing calculated and experi- mental values for bicarbonate concentrations we have corrected measured plasma values to mean whole blood values by multiplying by 5.85, a factor derived from the Singer-Hastings nomogram. Plasma hydrogen ion concentrations were also corrected to whole blood values by dividing by 5.85.

The system is completely closed except for the additions of X,, Y,, Z,, and B,, It is assumed that no appreciable quantity of H’ or OH- ions or strong acid or base is added to the system.

ff we designate initial values by the subscript 5, internal exchanges by the subscript

Page 4: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

48

i, and external exchanges by

x-x,=x,+xi

C. M. E. MATTHEWS et Cd.

the subscript e, as defined above, then

with similar equations for Y, Z and B.

Then

xi= -Yi

and

Zi=Y,+Bi.

Therefore. combining these equations, the equations for balance of ions are:

(1) x-X,$-Y-Y,=X,+Y,

(2) Z-Z,,+X-X,-(B-B,,)==Z,+X,-B,

e.g. if the Pco2 is suddenly raised so that an instantaneous increase of the dissolved

CO, concentration of 5 mMj1 occurs, and if 90”,, of this is converted to bicarbonate

with a change in hydrogen ion concentratioil of 1.5x IO-’ mM/l, then the various

changes are as fohows:

X,=5 mM/l Y,=O z,=o B,=O

Xi= -4.50 mM/I= -Yi Zi= 1.5x IO-” mM/I

:. Biz. -Yi=4.50 mM/I

-(X-X,)= -f-(Y-Y,,)=050 mM/l.

The Henderson equation states that:

(3) Z= K,(X/Y)

where K, =apparent dissociation constant for hydration of C02.

Van Slyke’s equation for buffering power /I, gives:

-zg- = 0.434 p

where /j=(buffering power in Slykes) x 10e3

(4) :. B- B, = - .434 /I In(Z/Z,).

Change in buffer anion col~cel~tration replaces added base in the equation as stated by

Van Slyke. Since we have assumed that there will be no addition of large quantities

of non-carbonic acid or hydroxyl ions, and since at the pHs we are concerned with,

concentrations of H+ and OH- ions are much less than those of buffer anions, the

change in buffer anion concentration is equivalent to the amount of acid or base

added. For extracellular CO, the principal buffer is the haemoglobjn in the blood, for

which at a given degree of oxygenation the buffering capacity is constant over the

range of pH to be considered. Hence /I is constant.

Combiningeqs. (l)-(4) and neglecting Z compared with X and Y gives the general

approximate equation :

Y-Y,-Y,-B,= -(X-X,-X,)-B,= -(B-B&=.434/i in x. y (X0 Yo).

Page 5: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION

Let C=concentration of bicarbonate in 1 of CO,/1 water =22.4 Y

and

49

X p = PC02 in mm Hg = --

.03x10 -3’

Then

(5) C -C, -C, - 22.4 B, = - .00067( P - P, - P,) - 22.4 B,

(6) = - 22.4(B - B,)

If changes in C and P are not too large, eq. (7) may be simplified:

Then

(9) C-C,-C,-22.4B, = ?- PC, -c - p 0

.

The equations will first be derived without this approximation. Differentiating eqs.

(% (6) and (7):

.00067 2 + 00067 3 dt ’ dt

(11) 224dB _ 224 dBe dC ; dCC ’ dt __-_ ’ dt dt dt

(12) 9.7 /I dP 9.7 fl dC dC dC, 22.4 dB, Fdt = _-_

C dt + dt - dt dt’

These equations may now be applied to the pools of the model.

Model. The compartments of the model are shown in fig. 1. Some anatomical

compartments are divided into separate pools for dissolved CO, and bicarbonate.

The pools are numbered as follows:

1.

2.

3.

4.

5.

6.

7.

Lungs.

Poorly perfused tissues (muscle), extracellular bicarbonate and dissolved CO, plus

intracellular dissolved COz.

Well perfused tissues (other than brain), extracellular bicarbonate and dissolved

CO, plus intracellular dissolved CO,.

Brain, total COz.

Poorly perfused tissues, intracellular bicarbonate.

Well perfused tissues, intracellular bicarbonate.

Not used in this study. If required brain CO, can be divided into intracellular

Page 6: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

50 r. M. E. MATTHEWS et NI.

bicarbonate (pool 7) and extracellular total CO, plus intracellular dissolved CO,

(pool 4).

8. Arterial blood.

9. Venous blood.

Poorly perfused

Fig. I. Model system. I.C.B. ~~ intracellular bicarbonate. E.C. total extracellular CO2 plus intracellular dissolved Con. Dotted arrou indicates control of ventilation from F’u)~ in brain pool.

Subscripts are used to denote pools. Subscripts a and v denote pulmonary end-

capillary and systemic venous blood respectively.

c= concentration of bicarbonate (I of COJ water)

v= Volume (I)

P= P co2 (mm Hg) Q= blood flow (Ijmin)

F= fractional concentration of CO, in gas phase (SI-PD).

The first closed system considered is a small section of blood flowing through the

lung capillaries from the pulmonary arterial to the pulmonary venous end. No

bicarbonate is added to this blood from outside, but due to the Haldane effect, there

is an effective change of buffer anion which is represented by B,,.

Page 7: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION

Therefore for this system,

Ca-Cv+22.4 B,=9.7 jj~.m (;. g)

where BL = buffering capacity for the pulmonary capillary blood,

since C, = Cv =concentration of bicarbonate in mixed venous blood

and C=Ca=concentration of bicarbonate in pulmonary venous blood.

B, must be subtracted from B, in this case because it represents a change in buffer

anion concentration which is not due to a change in hydrogen ion concentration.

In fact B, will depend on the change in oxygen saturation, and therefore if there is

a steady state for oxygen,

22.4 B - = H- Q

where b=a constant, and A=rate of production of CO2 due to metabolism in l/min.

(13) :.Ca-Cv=9.7 PLln (g. g) - 7 = -22.4(Ba-Bv) .

Considering the alveolar air plus capillary blood (of negligible volume) as a closed

system and using eq. (IO):

v dF, ~ =Q(Cv-Ca)-‘?(F,-F)+22.4QX,-Qx.O0067(Pa-Pv)-F,$$, ’ dt

where \i = alveolar ventilation in I/min,

F=fractional CO, concentration in inspired air, and

X, is the extra concentration of dissolved CO2 produced by the effect of

oxygenation on the carbamino COz. (Carbamino CO, can be neglected

compared with the total concentration of CO2 as bicarbonate, but not

compared with the change in bicarbonate concentration from mixed venous

to arterial blood). Like B,, X, will depend on the change in oxygen satu-

ration so that

CA xc=-. 22.4 Q

(14) dF, ’

:. dt = g(Cv-Ca-.00067(Pa-Pv))-\;(F,-F)+ F - 2%. 1 I

Also Pa=P, =F,(PB-47)

where PU = barometric pressure.

(ii) Tissues (excluding extracellular bicarbonate)

a) For a closed system consisting of a section of blood flowing through the capillaries

Page 8: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

52 C. M. E. MATTHEWS et cri.

of an organ, C, is not zero since bicarbonate will exchange with the extracellular

bicarbonate. Therefore for this system, in the i’h pool.

(15) bAi

-22.4(Bi-B,)- 7 Qi

=9.7p,m ($.2).

Here the value of B, is again subtracted from the change in B as explained on page 5 I,

but in this case B,, is negative.

b) A different closed system may also be considered consisting of the extracellular

fluid and the capillary blood in CO2 equilibrium with it. For this system there is a

continuous inflow and outflow of buffer and bicarbonate in the arterial and venous

blood. Also since intracellular dissolved CO, is assumed to exchange rapidly with

extracellular COZ, there is an inflow of dissolved CO, produced by metabolism. There

is also inflow or outflow of CO, exchanging with the intracellular dissolved CO, and

bicarbonate pools. Unlike (a) above where the oxygenation of the closed system

changes with time, there is no change in buffer anion due to the Haldane effect in this

closed system, since a steady state is assumed for oxygen. However, there is a removal

of dissolved CO, due to the uptake of CO* by carbamino.

Using eq. (12) for the closed system we are considering in the ilh pool, (i=2,3,4),

(16) 9.7 /Ii dP, 9.7 /Ii dCi -P-dt = -s;-dl + z - +(C,-(:i)- -VT 22.4 Oi (BB _ Bi) .

, I I

Also using eq. (IO) and allowing for CO, taken up by carbamino,

(17)

+ .00067 Qi ( P, - P. vi ’

dCi+j .-p -.00067(1+ +)% dt

Ai( I -c) )+ -77.

Intracellular PcoZ is assumed to be the same as extracellular Pco2.

Combining eqs. (IS), (16) and (I 7),

(18) In (,?. 3, -bAi

.00067i)i(P,- Pi)+A,(I -c)-V~+~ dCi+,

-dt

‘::i”i +.00067(1+ G;)(l+ +)]).

(For i =4, the brain pool, Vi + 3 = 0.)

(iii) Arteriul blood and U~IZOUS blood

The venous blood entering the lung capillaries is assumed to be in equilibrium with

Page 9: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION 53

pool 9 so that subscripts V and 9 are equivalent.

Using eq. (IO):

(19) Q $$ = .@a-C,)-.00067 $ Q

8 + .00067 T ( Pa - PS) .

8

-I- ~(~2P,+~~P,-tQ4P,-QP,). 9

Also using eq. (12)

w 9.7 & dP8 9.7 /I8 dC8 (I ___ - = - --...-

P, dt C8 dt + L$ - +ca-C,)-

8

Z+Ra-B,j. 8

(22) 9.7 &, dP, 9.7 /I9 dCg + dC,

__- - = ___ _.___ ___ - P, dt Cg dt dt

- ~(Q,B,+Q,B,+Q,B,-QB,) 9

From eqs. (IO), (11)

(23) 22.4% . = 22.4 $ (Ba - BR) + ‘00067 Tf - .00067 $ (Pa - P,f . 8 8

(24) 22.4 % = 7 (Q2B2+(53B3+Q4B4-QB9)~.00067 qf 9

- ~(Q*P~+QZP3+Q4P4-QP9) * 9

Hence by putting dB/dt=O in the steady state,

(25) 22.4((Ba), - (B&j = .00067((Pa), - (P,)J = (C&, - (Ca),

(26j 22.4 Qz@,k, -Q&W;+ QJB& -0%) i

= .00067 x i

&(P*)o f Q3(P& + WP4)0 i2

-(P&J

Page 10: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

54 C. M. E. MATTHEWS et 01.

The following equations, similar to eq. (13) may also be obtained using eqs. (6) and

(9):

(28) -22.4(&-(B,),)=9.7 j191n ( $??. $fF . 1

From eq. (I 3)

-22.4(Ba-Bv)=9.7 /kin (fi, 2) - $- =Ckl-CV ,

From eq. (15)

-22.4(B,-B,)=9.7 p, in ’ Pi c,

i

bAi ,%. c, t T.

Combining these equations:

t.00067 @Pa- P8))-@(C8)0-(Cv)~)

+ Q ‘(

,9.7 /jq 111

li i v 8

x .00067(42PZ+~~P,+Q4P4-QP9)

Page 11: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION 55

(iv) Intracellular bicarbonate pools

For the intracellular bicarbonate pool similar equations are applied, except that the

Henderson equation is used only for initial values of hydrogen ion concentration and

not in the transient state. In eq. (7) /3 now represents the buffering power of intra-

cellular buffers for COz, and this buffering power is assumed to be constant. It is

also assumed that C,=O and B, =O, i.e. that the flow of bicarbonate ions and buffer

anions across the cell membrane is negligibly small.

Therefore

(31) C-&=9.7 ~ln(Z/Zo)=9.7 fill1 (f . 2) . 0

Since it is assumed that exchange with the bicarbonate pool is limited by rate of

hydration of CO,, then in the i”’ pool (i=5,6),

dY, vi dt =ux,-VW, )

where W is the molar concentration of H,CO, and u and v are the velocity constants

for the first reaction in which H,CO, is formed:

H,O+CO, L H&O, + H+ +HCO; .

It is assumed that the change in H,CO, concentration is negligibly small compared

with the change in bicarbonate concentration (MAREN, 1967).

If it is assumed that the second reaction is infinitely rapid with equilibrium constant

K,, then

The overall equilibrium constant for the hydration of CO2

Also the Henderson equation can be used for initial values (although not for transient

values), so that

.‘.Vi 2 = .OOO67u pi _ ;!%T-. $_

IO 0

Page 12: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

56 C. M. E. MATTHEWS t?t ai.

:. Using eq. (31)

(324 Vi 2 = .00067u (Pi-(Pi), x &)~exp&$+)j.

If ci-(ci)o is small then 9.7 pi

(32b) v, ds N ooo67 u

‘dt ’ .

Intracellular hydrogen ion concentration,

z, = .00067 K; (33) , c.

1

(v) Cerebral bloodflow and muscle blood.flobj’

The variation of cerebral blood flow with arterial Pco2 was included in the model.

The following equation was used. which matches the combined data of KETY and

SCHMIDT (1946, 1948), SHAPIRO, WASSERMAN and PATTERSON (1966), and WASSERMAN

and PATTERSON (I 967).

(34) Q4=(QJ0.{.41 e(‘105(p4-(p4)o)_t.59} .

The variation of muscle blood flow and flow to other organs was also simulated

using the following equations which match the curves given by TENNEY and LAMB

(1964)

(34a) Q,=(Q,), x e-“~396(p-p0) _

(34b) Q,=tQJo+<Q2)o-Q2

(34c) Q =Qz+Q3+Q4.

(vi) Controller equation

Ventilation is assumed to be proportional to Pcoz in the brain pool minus a threshold

P 022’

Hence

I

\i = k,(P, - PT) if P, > P,

(35) ii =o if P,<P,

where V = alveolar ventilation in l/min

k, = controller sensitivity

P, = threshold brain Pco2.

Page 13: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO2 DISTRIBUTION 57

Model system in the unsteady state

Eqs. (13), (14), (17)-(20), (29), (30), (32)-(35), give the behaviour of the system in the

unsteady state.

Model system in the steady state

The steady state equations are obtained by putting the differentials equal to zero, and

are as follows:

(36) v(C, - F) = A.

(37) pv= pa cv exp Cv-Ca-bA/Q

’ Ca . ! TpL . 1

(38) Cv = Ca + .00067( Pa - Pv) + A(1 -c)

Q .

For i = 2,3,4.

(39) Pi=P,. $exp Ci -C, - bA,/Q,

8 9.7 p, .

(40) Ci = Ca + .00067(Pa - Pi) + A,(1 -c)

Qi

(41) c, = Ca + .00067(Pa - P,).

(42) C,= K, ,V,CZ

K j/l 52 5

(43) .00067uP, = K .00067uP, = K

GV, 2s

CY3 36 .

It was assumed that in the normal steady state P, = Pa.

The effective value of u was derived from eq. (43).

COMPUTER PROGRAMME

A digital computer programme in Algol was written to solve the equations, for the

steady state and then for the unsteady state. The approximate eqs. (9) and (32b) were

used, and it was found that this approximation had very little effect on the results.

This programme is available on request. It was run both on an Elliott 4100 and on

Atlas. During the development of the model an analogue computer was also used.

The digital programme prints out values for Pco2, bicarbonate concentration and

hydrogen ion concentration in each pool every half minute. Cerebral blood flow is

also printed.

Page 14: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

58 C. M. E. MATTHEWS et 01.

Use qf ’ ‘CO, data and simulation of experiments

The results obtained from the “CO, experiments in the previous paper (MATTHEWS

et al., 1968) were used to fix the pool sizes and exchange rates when simulating the

other experiments for each subject. In the “CO, model it was not necessary to dis-

tinguish between bicarbonate, dissolved CO, in the extracellular pool, and dissolved

CO, in the intracellular pool since these all exchange rapidly. In the present model

this distinction is required, but the total CO, was adjusted to the value found to

match the “CO2 data. Another difference between the present model and that in

the previous paper is the addition of a separate venous blood pool. The total CO, in

pool 3 was reduced in the present model so that the sum of the total CO, in pool 3

plus the venous blood pool was the same as that in pool 3 in the “CO, model. It was

found that this rearrangement of the pools made little difference to computed results

for the “CO, experiment. The values of K,, and K,j were adjusted to keep the flow of

CO, between pools 3 and 6 and pools 2 and 5 the same as for the “CO, experiments.

Apart from these adjustments, only parameters such as controller sensitivity, which

do not affect the “CO, experiments were varied to match the results for the other

experiments. The CO, production rate and the initial alveolar CO, concentration

were set at the measured values for each experiment. Total ventilation at IITPS was

calculated from alveolar ventilation at S1PD, using an equation which fits the data of

JONES et a/. (1966) to correct for dead space. Allowance was made for instrumental

dead space.

Variation of parumeters

Each parameter was varied in turn to investigate its effect on the results. Ln some cases

it was necessary to vary another parameter at the same time. For example if steady

state cerebral blood flow is changed there is a change in cerebral Pcor. If threshold

brain Pco2 for the controller equation is kept constant, this will lead to ridiculously

large changes in ventilation and alveolar P,, *; hence it is necessary to change thres-

hold Pcoz also. This means that if it is possible to have different values of steady state

cerebral blood flow with reasonable values of alveolar ProI and ventilation, then

threshold Pro1 must vary also.

E.yperimental

The subjects of all experiments were ourselves and professional colleagues. 6 out of

the 8 in the previous paper carried out all experiments, except hyperventilation which

was performed by only four. All studies were performed with the subject sitting, afrer

half hour’s rest in the postabsorptive state. Records of resting ventilation and metab-

olism were obtained. The subject then rebreathed a mixture of 7”,, CO, in 0, for

3 min from a 6 litre bag in a bottle arranged to record expired ventilation on a Tissot

spirometer. CO, inhalation experiments were then performed for 30 min on a con-

ventional circuit using 7”,, CO2 in 0, mixtures. Hyperventilation experiments were

performed on a separate occasion. Ventilation and end-tidal PcoZ were followed for

IO min after return to normal air breathing. Respired gas volumes were measured with

Page 15: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO2 DISTRIBUTION 59

a Tissot spirometer. End-tidal and expired CO, concentrations were measured with

a continuous rapid infra-red analyser (URAS, Godart-CPl) and expired 0, concen-

tration with a paramagnetic analyser (Servomex, DCL), both being calibrated with

0,; CO,; N, mixtures analysed on a Lloyd Haldane apparatus. Conventional

circuits were used for experiments on CO, inhalation and hyperventilation. Ven-

tilation and gas exchange were calculated in the standard manner.

Results

co, DISSOCIATION CURVE

A theoretical CO, dissociation curve is obtained from eq. (5) by putting B,=O and

C,=O since there is no addition of buffer or bicarbonate to the system.

Then :

(44) c-c,=9.,,1*,(~.~).

This equation is not used as such in the model, eqs. (5))(7) being used instead. How-

ever, it may be used as a test of the validity of the equations.

Figure 2 shows a plot of concentration, (C), against Pco,, (P), compared with

experimental data compiled by R. W. ROOT, (Handbook of Respiration, 1958);

there is quite good agreement. In the calculation the buffering capacity was assumed

to be 22.5 Slykes. Initial values of C and P are normalised at C= .605 l/l and P = 45 mm

Hg.

.7 r

.6 I

.!5

Tot01 COn .4 I Concentration

1/l. ,3 1

, 1 / , I I

0 10 20 30 40 50 60 70 80

P co2 mmHg

Fig. 2. Theoretical COz dissociation curve (continuous line) compared with experimental points ( Y ). Total COZ concentration in I:1 of blood is plotted against blood Pco2.

Page 16: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

Tim

e P

I Px

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P!)

TA

BL

E

1

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ect

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Page 17: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

28

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* T

hese

re

sults

ar

e fo

r th

e sa

me

CO

Z p

rodu

ctio

n ra

te

as

for

the

othe

r re

sults

in

th

e ta

ble.

T

hey

are

not

the

sam

e as

th

ose

show

n in

fi

g.

5, w

here

oI

C

O2

prod

uctio

n ra

te

and

leve

l of

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erve

ntila

tion

wer

e se

t eq

ual

to e

xper

imen

tal

valu

es.

Page 18: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

62 C. M. E. MATTHEWS ef d.

SIMULATION OF REBREATHING AND co, INHALATION

The results for one subject are shown in figs. 3-5 and tables 1-2. In the figures ex-

perimental data for ventilation and Pco2 are compared with computed values obtained

using the “COZ data as described above. It can be seen that the agreement is quite

good. A few systematic discrepancies between experimental and computed data were

noted. In most subjects the computed ventilation rise during CO, inhalation was too

rapid; either the beginning of the ventilation curve matched and the experimental

points continued to rise after the computed values had become constant, or the final

values matched while the computed curve increased too rapidly at the beginning. This

effect is not seen for the subject shown in figs. 3-5. There was no systematic difference

between experimental curves and computed points for the Pco2 curves during CO,

inhalation, and they were usually matched quite well, although the computed under-

shoot was sometimes too large or too small. In the rebreathing experiments the Pco,

undershoot tended to be too large, as shown in fig. 3 and in two subjects the intercept

of the rising part of the Pcoa curve was too low. In the hyperventilation experiments

the computed values of ventilation were zero for several minutes after the end of the

period of hyperventilation, whereas the experimental ventilation was always greater

than zero. The discrepancies seen in the Pco2 curves follow from this discrepancy in

ventilation. In all these experiments the computed difference between alveolar and

arterial Pco, depended on the value chosen for the buffering capacity for blood in the

TABLE 2

Values of parameters for subject M.C.

Vl

2.5

A2

0.028

vz

3.46

Aa

0.173

V3 Vi VS Vii VU VI, b C

6.67 1.4 26.5 8.5 1.0 3.0’ 0.105 0.245

Al ci;? h:l QJ Kas K;,2 Kst; KG:+

0.044 I.0 4.2 0.7 1.44 0.40 0. I33 0.225

pt. 82 /A3 iit p5 ii,; /ix @!I 10.3 3.08 3.22 23.5 18.4 19.3 18.5* 23.0

V -1 volume in I Cj blood flow in I:I

A -= COz production rate in I/min

K ---- fractional exchange rate

p buffering capacity, given here in slykes

b :: constant for Haldane effect

c -- constant for carbamino effect

See fig. 1 for numbering of pools.

Total ventilation was given by %‘E (BTPS) = 1.23 (9A (STPU) 1 3.5)

* This value corresponds to the ha~in~~~lobin level of the subject. Values of& can be varied consider-

ably with little effect on results.

+ These do not include capillary blood volume.

Page 19: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIUUTION

i

I 0 .- - x

I

‘h.-__- I .._-.. 0 0 VI 9

a .-

I

1”

I 0

Page 20: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

64 C. M. E. MATTHEWS et d.

x

x

i

x ”

Page 21: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRtBUTION 65

Page 22: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

66 C. M. E. MATTHEWS et ~1.

lung. When this was adjusted to match the Pcoz difference observed during re-

breathing, it was found that the computed Pcoz rise also matched the experimental

rise.

On the whole the agreement between experimental and computed values was fairly

good, the biggest discrepancy being the rate of rise of ventilation during CO, in-

halation.

In fig. 6 the computed arterial Pco2 is plotted against arterial hydrogen ion and bi-

carbonate concentrations and compared with the data of BRACKETT, COHEN and

SCHWARTZ (1965), of MANFREDI (1967), and of LASZLO et al. (to be published).

Similar data have been reported using arterialised blood by MICHEL, LLOYD and

CUNNINGHAM (1966). Some of these data were given as plasma or extracellular

concentrations and have been corrected to whole blood concentrations for comparison

with our computed values. The agreement is within the range of experimental varia-

tion.

In fig. 7 computed bicarbonate concentrations are plotted against pH for extra and

intracellular fluid of muscle. These are compared with the data of CLANCY and

BROWN (1966), and MANFREDI (1967). Considering the variation in experimental

results, the computed values fit as well as can be expected.

Figure 8 shows the increase in ventilation for different concentrations of CO2

inhaled when a new steady state is reached. Computed points for an average subject

fit the mean experimental curve given by DRIPPS and C~MROE (1947).

Computed values for brain venous blood Pcoz and total CO, content plotted

against arterial Pcoz and CO, content are shown in fig. 9 and compared with ex-

perimental results of LAMBERTSEN et al. (1953) for internal jugular and arterial blood

for subjects inhaling different concentrations of CO,. The agreement is satisfactory.

Each parameter was varied in turn to determine its effect on the computed ven-

tilation and Pco2 curves and on other results. Many of these had a negligible effect

when varied within the probable range of values which may occur. For some param-

eters there was an appreciable effect and some of these effects are shown in figs. 10-14.

In the rebreathing experiment the effects of varying the parameters on the plot of

ventilation against alveolar or arterial PcOr was similar to the effects found by READ

and LEIGH (1967).

Other effects of variations of the parameters which are not shown in figs. IO-14 are

as follows. Ventilation for both rebreathing and CO, inhalation was decreased with

little effect on PcOa when cerebral blood flow was decreased, or when muscle blood

flow was increased, or when CO, production in brain was increased with total CO2

production constant. Changes in the opposite direction produced the opposite effects,

except for a decrease in muscle blood flow which produced very little effect.

The relations between [HCO; 1, [H + ] and Pco2 for arterial blood and muscle

(figs. 6 and 7) were only very slightly affected by changes in the parameters. When

there was a change in the initial conditions, the new curves were approximately

parallel to those for the standard conditions. However, a decrease in muscle blood

flow did make the muscle extracellular [HCO,]/pH curve in fig. 7 a little flatter

Page 23: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DlSTRl~UTION 67

-7 -

-6 -

_S - +/~~s=~??---~

Arterial bicorb. .4 concentration ,’

I/ 1. @M

,3 -

-2 -

,I -

i_, i / i I

0 IO x) 30 40 50 60 70 80

Arterial Pco, mm Hg

Arterial [i-r+] nM/t

80

70

60

50

40

30

20

IO

Arterial PC+,* mm Hg

Fig. 6. (a) Top: bicarbonate concentration in whole blood plotted against PCO, for arterial blood. (bf Bottom: Hydrogen ion concentration in whole blood plotted against PCO, for arterial blood.

The continuous curves are the computer results, using the initial conditions and the new “steady state” computed values after 2X min of COa inhalation and after 30 min of hyperventilation. In fact a complete steady state is not quite reached in this time. The dotted curves are published experimental data of LASZLO ef al. (to be published) (L), MANFREDI (1967) (M) and BRACKETT et al. (1965) (3).

Page 24: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

68 C. M. E. MATTHEWS et ~1.

*5

Bicarbonate

concentration -4

t. I 2.

-3

4LoO 300 200 100 60 50 40 30 20

Fig. 7. Bicarbonate concentration plotted against pH for the muscle pool. The top right pair of lines are for the extracellular pool and the others for the intracellular pool. Continuous lines are computer results using the initial conditions (*) and the new “steady state” computed values after 28 min of COz inhalation (A) and after 30 min of hyperventilation (t). In fact a complete steady state is not quite reached in this time. The dotted lines are from published data of MANFREDI (1967) (M) and

CLANCY and BROWN (1966) (C and B).

1100 -

1000 -

900 -

BOO

Ventilation 700

% normal 600

500

400

300-

200 --

I00

% COz inhaled

Fig. 8. Variation of new steady state ventilation during CO2 inhalation as 7; of control value with concentration of COZ inhaled. The dotted line is from DRIPPS and COMROE (1947) fitted to experi-

mental data, and the continuous line is computed (subject M.C.).

Page 25: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION 69

mm Hg 60

PCCQ 5

brain

c .56-

5 .- -,o On

I-

40L I I

50 60 mm Hg

PC02 arterial

-551 I I I I I I -50 .51 .52 53 .54 .55 .5b

Total CO, content arterial

II.1 L.

Fig. 9. Top: PCO, in brain pool plotted against arterial PCO, from new “steady state” computed values (continuous line) for subject M.C., compared with experimental internal jugular venous PCO, plotted against arterial PCO, (0) from results of LAMBERTSEN et al. (1953). Bottom: Total CO2 content of brain venous blood plotted against that of arterial blood from new “steady state” computed values (continuous line) for subject M.C. compared with experimental total CO2 content of internal jugular venous blood plotted against that of arterial blood, (O), from

results of LAMBERTSEN et al. (1953).

because the fall in extracellular bicarbonate during hyperventilation was reduced.

Changes in buffering capacity in the various pools produced very little effect. In the

hyperventilation experiment when the proportion of the cardiac output flowing to

pool 3 in the steady state is reduced, blood flow to pool 3 tends to fall to a very small

value and eventually to become negative. Clearly the assumptions made about

variations in blood flows (section (v) of theory) are not correct for hyperventilation in

these circumstances. In fact cardiac output increases during hyperventilation (TENNEY

and LAMB, 1965), but the magnitude of the increase in relation to such factors as the

degree of hyperventilation and the changes in Pco2 are not sufficiently documented

for our purpose.

Page 26: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

60

P C

O>

mm

%

50

100,

I j-r

l.p3t

h’y

.--1

90,

901.

I

60

P c02

m

mH

9 50

sot

Pco

, mm

Hg

L_.-

I

____

m__

___.

! I

I 15

0

5 IO

I5

0

5 IO

I5

hIIn

ures

M

inut

es

Fig.

10

. V

aria

tion

of c

ompu

ted

Pco,

du

ring

re

brea

thin

g fo

r (l

eft)

di

ffer

ent

initi

al

conc

entr

atio

ns

CO

Z in

the

bag

; (m

iddl

e)

diff

eren

t va

lues

of

Pco

2 in

the

lu

ng a

nd

(rig

ht)

diff

eren

t ba

g vo

lum

es.

The

num

bers

ne

xt

to e

ach

curv

e sh

ow

the

initi

al

valu

es

of

“/g C

Oa

in b

ag (

left

),

alve

olar

Pc

o,

(mid

dle)

, an

d lu

ng

plus

ba

g vo

lum

es

(rig

ht).

Page 27: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION

4 0 0

Page 28: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

72 C. M. E. MATTHEWS et d.

t--

s 8

- _.

: cl .o

u I;

.c_ N

8

I__ ‘_L

8 8

Page 29: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION

Page 30: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

C. M. E. MATTHEWS et Cd.

Page 31: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION

Page 32: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

76 C. M. E. MATTHEWS et a/.

I

0 aJ

-2

-2

-0

I 8 O 0 0 0 0

In P N -

I” E E

Page 33: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION

Page 34: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

z in

hala

tion

CO

!

100

Ven

tilat

ion

I U

rnin

. j

0’

-..L

._--

--

IO

90

80

60

40

P,o

, m

m+

30

IO

20

30

40

c

Min

utes

CO

? in

ho~

ot~

on

---I

$-

a

r I 20

1

-A

IO

30

40

Min

utes

Fig.

14

. (b

) V

aria

tion

of co

mpu

ted

vent

ilatio

n an

d PCO,

duri

ng

CO

2 in

hala

tion

for

diff

eren

t ca

rdia

c ou

tput

s of

a)

1.

5 x

norm

al,

b)

norm

al

and

C)

half

no

rmal

.

Page 35: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION

-3o m 9 P ru a P 04 _ - - - 0 m

-0

Page 36: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

TA

BLE

3

Dia

gram

of

m

odel

Aut

hors

Mod

el

Poo

ls

Exc

hang

e ra

tes

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nd

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s T

issu

es

i___

-__

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I

GR

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s e-

I al

.

biG

OB

AR

D0

Cf 01

.

DE

FA

RE

S c

f al

.

MIL

HO

RX

e/

al.

FA

RH

I an

d R

AW

~

LosG

osnR

uo

er a

l.

FO

NE

, M

AT

TW

E$\

S

CA

MP

BE

LL

Tw

o:

Llln

gs

Tis

sues

Thr

ee:

Bra

in

LUng

S

Tis

sues

FIX

:

Lung

s

Bra

in

Hea

rt

MU

&

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er

Tvv

o:

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t (m

aini

)

eXtr

a~el

fUla

r)

You

im

ainl

y

I~lT~C~lllll~~)

Eig

ht:

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5

Art

eria

l bl

ood

Ven

ous

bloo

d

Poo

riy

perf

used

Intr

acel

lula

r

bica

rh.

Res

t

Wel

l pe

rfus

ed:

intr

acel

lula

r

bica

rh.

Res

t

Bra

in.

Blo

od

flow

and

volu

mes

Blo

od

flons

and

volu

mes

.

Cer

ebra

l bl

ood

flou

vari

es

Hith

I’< o

1

Blo

od

flow

s

and

~oI

umes

.

Con

stan

t ce

rebr

al

bloo

d R

ow

Blo

od

flous

an

d

~~~l

umes

an

d ra

te

of

exch

ange

ui

th

i.c.

bica

rbon

ate.

Sim

ulat

es

CO

Z

inha

latio

n co

rrec

tly;

but

m

rebr

eath

mg,

P

co2

rise

is

too

sl

ow.

Too

m

uch

tota

l C

O*.

Doe

s no

t iit

hy

perv

entil

atio

n.

Doe

s no

t fir

K

Oa

data

,

Too

si

mpl

ified

.

Will

on

ly

fit

Pco

2 ri

se

in

rebr

eath

ing

if

slop

es

of

tissu

e di

ssoc

iatio

n cu

rves

equa

l to

th

at

for

e.c.

f.

or

if m

uscl

e

bloo

d R

ow

abno

rmal

ly

low

. T

otal

C

O2

corr

ect,

w

ater

vo

lum

es

inco

rrec

t. D

oes

nor

fit

“CO

n da

ta.

Fits

P

(02

rise

in

rc

brea

rhin

g.

Tot

al

CO

2 an

d ua

ter

corr

ect.

F

its

11C

02

data

. V

entil

ator

y co

ntro

l no

t si

mul

ated

.

Too

~i

mpl

ltied

.

Fits

da

ta

from

re

brea

thin

g,

CO

*

inha

lario

n,

h)pe

rven

tllat

ion,

an

d “C

Oz

expe

rimen

t. T

otal

C

O*

and

wat

er

corr

ecr.

F

its

publ

ishe

d da

ta

for

hydr

o-

gen

ion

and

bica

rbon

atec

once

nlra

rions

.

Page 37: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO2 DISTRIBUTION 81

Discussion

COMPARISON WITH OTHER MODELS

Various models have been used to simulate body CO, stores and the ventilation

produced by an increase in Pco2. These models have recently been reviewed by

YAMAMOTO and RAUB (1967). The first and simplest model was introduced by GRODINS

et al. (1954), with the body CO, divided into two compartments, lungs and tissues,

with ventilation proportional to tissue CO2 concentration. DEFARES, DERKSEN and

DUYFF (1960), added a separate brain pool and simulated variation of cerebral blood

flow with Pco2, and the system was further extended by GRODINS, BUELL and BART

(1967). MILHORN et al. (1965) studied a similar but more complex model including

the effects of changes in oxygen tension. FARHI and RAHN (1960), divided up the

tissues into different pools according to the blood flow per unit volume. FOWLE

et ul. (1964) divided tissue CO2 into two pools on a different basis, one being mainly

extracellular and the other mainly intracellular. Their experiments using ’ 'C labelled

CO, have now been extended and the results are reported in the accompanying paper

(MATTHEWS et ul., 1968). LONGOHARDO, CHERNIACK and STAW (1967) used both a

single tissue pool model and a five tissue pool one similar to that of Farhi and Rahn,

and also concluded that the Pco2 rise during rebreathing could only be matched by

assuming a low slope for the CO, dissociation curve for the tissues with which the

CO, exchanged. The only models which are consistent with the rapid rate of rise of

P co2 during rebreathing from a small bag are those of FOWLE et ul., and of LONGO-

BARDO et al. FOWLE et al. (1964) suggested that a rate limiting process separates

intracellular and extracellular pools of CO,. Of the possible mechanisms, they favoured

the slow hydration of CO, in the absence of carbonic anhydrase in intracellular

fluid. This would also explain the changes in Pro2 during rebreathing observed by

FOWLE and CAMPBELL (1964) and is also put forward by LONGOBARDO et al. (1967) to

explain their findings. In essence, therefore, the distinction is not between intra and

extracellular COz, but between extracellular dissolved CO, and bicarbonate and intra-

cellular dissolved CO, on the one hand and intracellular bicarbonate on the other.

Another model by DANTZIG et al. (1961) considers concentrations of bicarbonate ions,

dissolved CO,, buffers and other substances in the body, but applies only to the

steady state. An early stage of the development of the present model has already been

described briefly (CAMPBELL, MATTHEWS and READ, 1966).

A summary of the main features of some of these models is given in table 3. The

prediction of these models for various experimental conditions may be compared as

follows.

Body composition

With regard to body composition, the models of Grodins et ul. and of Defares appear

to contain rather too much total CO,. The reason for this is that a constant total

CO, concentration throughout the whole body water is assumed, whereas in fact

total CO, concentration is considerably lower in the intracellular pool than in

Page 38: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

82 C. M. E. MATTHEWS ef Cd.

extracellular fluid. On the other hand, FARHI and RAHN have taken a reasonable total

COz, but their total volume of tissue is too small. It is clear that this difficulty will

always arise if total extracellular and intracellular CO, are lumped together in each

tissue. The tissue or water volume might be regarded as a virtual volume which

does not correspond to the amount of fluid actually present, but in this case it is

difficult to define the rate constant for removal of CO, from the tissues due to blood

flow, since this will depend on flow as a fraction of actual fluid volume. The model of

Fowle et al. and the present model avoid these dithculties by having separate intra-

ceIIuIar total CO,, or separate jf7trace~lu~ar bicarbonate pools.

’ ’ CO, experiments

The predictions for the tracer experiment using “CO, have been considered in the

previous paper (MATTHEWS et al., 1968). The models of Grodins and Defares will not

fit this experiment, nor will that of Farhi and Rahn with the pool masses which they

suggest.

Body CO, distribution

The Pco, rise during rebreathing from a small bag is independent of assumptions

regarding ventilatory control and so may be considered under the heading of body

CO2 distribution only. The models of Grodins and Defares et ul. would predict too

slow a rise in P,02. Farhi and Rahn’s model only fitted experimental data for rebreath-

ing if muscle blood flow was made abnormally low. A simple calculation using total

CO2 produced, volume in which it is distributed, and ratio of dissolved CO2 to

bicarbonate, indicates that these models can never match the observed rise in Pco,.

The model of Longobardo ef al. was also unable to match the observed rise in Pfoz

unless the slope of the dissociation curve of intracellular fluid was assumed to be very

Aat. The model of Fowle et rd. and the present model correctly predict this Pcoz rise

using independent data from the “CO2 experiment.

In this context, FARM and RAHN have introduced a major approximation into their

model by assuming that the total CO2 content, partial pressure and slope of the CO,

dissociation curve of whole blood are exactly analogous to the charge, potential and

capacity of a condenser. This would only be true if the CO, dissociation curve were

a straight line passing through the origin; that is, the intercept of this line on the

concentration axis has been ignored. We have not explored the efTects of this ap-

proximation on the predictions but they are probably important.

The results of the present study indicate that the rapid rise of PLO2 is due to several

factors and that the slow exchange with intracellular bicarbonate is only one of these.

Another important factor is the difference between alveolar and arterial PC,,, during

rebreathlng. The at-terra1 Pco2 is below the alveolar Pco,. The explanation of this is

at present uncertain. It may be due, as suggested by JONES et ul. (1967) to incomplete

equilibration between all forms of CO, within the transit time of blood in the pulmon-

ary capillary, or as suggested by GURTNER, SONG and FARHI (1967) to a charged state

at the alveolar capillary membrane which maintains a higher [Pcol] on the alveolar

Page 39: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO, DISTRIBUTION 83

side. Whatever the mechanism, the result is that the Pcoz in blood leaving the pulmon-

ary capillaries is lower than it would be otherwise, so that less CO, is removed from

the alveolar air by the blood. Hence alveolar and arterial Pcoz rise more rapidly. In

our model the fraction of the cardiac output which flows to the different tissues is

also a factor which affects rate of rise of Pco,, although it is not as important as

assumed in the models in which exchange is limited only by blood flow. If more of the

cardiac output is directed to the muscle, there is a larger uptake by the slowly ex-

changing muscle bicarbonate pool, so that during a short period this CO, is virtually

lost from the rapidly exchanging pools and rate of rise of alveolar Pco2 is reduced.

Increasing the total cardiac output without changing the proportion flowing to the

different tissues also reduces the rate of rise of alveolar and arterial Pcor, since more

CO2 is now removed from the lung by the blood flow. The alveolar-arterial Pcoz

difference also depends on cardiac output, and is much increased when cardiac

output is decreased.

Hyperventilation

The computed fall in alveolar Pcoz during hyperventilation cannot be compared so

well with experimental results since accurate estimates of mean alveolar Pcoz from

end tidal Pco, are difficult to obtain and because CO, production rate tends to

increase. However, it can be said that quite good agreement is obtained with the pre-

sent model, whereas the other models, with the possible exception of that of Farhi

and Rahn, would not fit at all.

The fact that breathing does not stop after the period of hyperventilation, as found

by others, is probably due to the operation of neural mechanisms independent of

those affected by chemical factors, which this model does not consider (PLUM,

BROWN and SNOEP, 1962; FINK, 1961).

Vefltil~tory control

The simulation of control of ventilation may be tested by considering the results for

changes in ventilation during rebreathing and CO, inhalation and the changes in

P during CO, inhalation. The models of Grodins and Defares fit the changes

dGy;ng CO, inhalation; changes during rebreathing were not examined but they could

not have been correctly simulated since the Pco, rise would not fit. Ventilatory control

was not included in the models of Farhi and Rahn and Fowle et al. The present

model gives quite good agreement with experiment, except for the difference in rate

of rise of ventilation during CO2 inhalation. This discrepancy may reflect the sim-

plicity of the model controller pool compared with what is known about the actual

controller site (BROOKS, KAO and LLOYD, 1965). It could also be due to a gradual

increase in CO, production rate during the experiment but this would have to be

approximately doubled to account for the observed effect. In an attempt to estimate

CO, production we have measured oxygen consumption during sustained hyper-

capnia and it appears that it may increase by about a factor of two but the measure-

ments are technically difficult (LASZLO, in preparation).

Page 40: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

84 c‘. M. E. MATTHEWS et cl!.

Using a simpler model of changes in receptor Pro, in which an empirical function

representing changes in arterial Pco, is fed in, READ and LEIGH (1967) reach similar

conclusions.

VARIATION OF PARAMETERS

The curves for ventilation and P co2 plotted against time and also the curves shown

in figs. 6 and 7 are only slightly affected by variation of the parameters within the

range of expected values. except for the conditions mentioned on p. 66. Thus blood

flows and CO2 production rates and the initial conditions seem to be the most im-

portant factors in determining the shapes of these curves.

Thus it appears that the slow exchange with the intracellular bicarbonate pool is

not essential for simulation of these experiments, although it is required to simulate

the “CO, experiments. However, the effect of varying several parameters together

has not been investigated, and it is possible for example that when blood flows are

much increased, that variations in jntracellul~~r exchange rate might then become

more important. A discussion as to whether the present model is “blood flow limited”

or not has little meaning. In some circumstances changes in blood flow have an im-

portant effect, whereas in other circumstances changes in blood flow have little effect

and intraceIlular exchange rate becomes more important.

The interactions with buffers and variation in hydrogen ion concentrations have not

previously been simulated in the transient state. The present model indicates that these

changes can be explained quite satisfactorily on the basis of classical concepts of total

ion balance and buffering capacity. Other mechanisms may exist such as active trans-

port of hydrogen ions out of the cell which maintains the ratio of intracellular to

extracellular hydrogen ions and bicarbonate ion out of Donnan equilibrium. The

factors which determine the intracellular [Hf ] and [HCO,] are poorly understood

(CLANCY and BROWN, 1966). In spite of early (CONWAY and FEARON, 1944) and recent

(CARTER ef al., 1967) work suggesting that intracellular pH may be approximately 6.0,

the bulk of the evidence at present suggests that [Hi ] or [HCO, ] are rnai]ltaiIled out

of Donnan equilibrium and that intracellular pH is about 7.0 in the resting condition.

This may require active transport of H + or HCO$ across the cell membrane. We

believe that the ’ ‘CO2 experiments suggest that HCOY is not transported actively since

this would lead to a rapid exchange of the label with the intracellular pool which we do

not find experimei~tally. However, we cannot contribute any evidence regarding H+.

Nevertheless, we have been able to simulate observed changes in extracellular acid base

state in response to a rise of body Pcoz without postulating any bulk trans-cellular

movement of hydrogen ions as distinct from bicarbonate (fig. 6). While the evidence

for human subjects over the range of Pco, from 40 to 80 mm Hg is conflicting, a

review of the literature suggests that there is little exchange of extracellular Hf for

intracellular Na’ or K+ during acute hypercapnia (ARMs~-R(~N~ ef ctl., 1966; LASZLO

el a/., in preparation).

Page 41: A model for the distribution and transport of CO2 in the body and the ventilatory response to CO2

MODEL OF CO2 DISTRIBUTION 85

We have not studied experimentally the changes in intracellular [H+] and [HCO; ]

during hypercapnia. However, the computed results are compared with the experi-

mental data of other authors (figs. 6 and 7), and considering the variation in the

experimental data they agree as well as can be expected. The studies of ADLER, ROY

and RELMAN (1965) on isolated muscle suggest that over the range of Pco, of 40-80

mm Hg there is little change in [Hf ] implying an increase in [HCO,] which main-

tains the Pco,: [HCO,] ratio unchanged. This is not consistent with the results of

other authors quoted and has not been included in the figures.

In conclusion therefore our model appears to fit the experimental data reasonably

well and to clarify the relationship between parameters observed in a number of

different types of experiment. However, more experimental work is needed on the

changes in intracellular CO, during hypercapnia.

Acknowledgements

We are grateful to our colleagues both for their help and for their co-operation as

subjects, and especially to Dr. M. CLODE for carrying out the hyperventilation ex-

periments; to Mr. M. LONGY for help with the initial stages of the computer program-

me, to Mr. B. KELLY for modifying and running the programme on Atlas, and to the

Computer Centre of the Royal Postgraduate Medical School for facilities. We also

thank Miss A. HART, Miss H. POPE, Messrs. G. FOR%, J. BATRA, S. GUNASEKRA and

A. RANICAR for technical assistance, and Mr. D. D. VONBERG for his encouragement.

References

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muscle cells to changes in CO2 tension or extracellular bicarbonate concentration. .I. Clirz.

Ir1lYst. 44: S-30.

ARMSTRONG, B. W., J. G. MOHLER, R. C. JUNG and J. REMMERS (1966). The in riro carbon-dioxide

titration curve. Lancet (i), 759-761.

BRACKFTT, N. C., J. J. COHEN and W. B. SCHWARTZ (1965). Carbon dioxide titration curve of normal

man: effect of increasing degrees of acute hypercapnia on acid-base equilibrium. Nrbr En,<r/.

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BROOKS, C. McC., F. F. KAO and 9. B. LLOYD (1965). Cerebrospinal fluid and the regulation of

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CAMPBELL, E. J. M., C. M. E. MATTHEWS and D. READ (1966). Analogue computer studies of COZ

stores and control of ventilation. J. P~~~~sio/. (London) 184: 55-56 P.

CARTER, N. W., F. C. RECTOR, D. S. CAMPION and D. W. SELDIN (1967). Measurement of intra-

cellular pH of skeletal muscle with pH-sensitive microelectrodes. J. C/in. Incvst. 46: 920-933.

CLANCY, R. L. and E. B. BROWN (1966). In Go COZ buffer curves of skeletal and cardiac muscle. Anl. J. Physiol. 21 I: 1309-1312.

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DRIPPS, R. D. and J. H. COMROE (1947). The respiratory and circulatory response of normal man to

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FOLVL~. A. S. E. and E. J. M. CAMPBEI.L (1964). The immediate carbon dioxide storage capacity of

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