a model for the distribution and transport of co2 in the body and the ventilatory response to co2
TRANSCRIPT
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
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
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
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).
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
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,,.
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
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
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
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)
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-. $_
IO 0
56 C. M. E. MATTHEWS t?t ai.
:. Using eq. (31)
(324 Vi 2 = .00067u (Pi-(Pi), x &)~exp&$+)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.
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.
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
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.
Tim
e P
I Px
Min
P!)
TA
BL
E
1
Subj
ect
M.C
.
P%
Ps
PA
C8
G
cz
C3
Cd
c,j
cn
H,+
H
;~
H;
H:
H4T
H
s’
Reb
reat
hing
fo
r 2.
67
min
0 47
.5
41.0
51
.1
45.9
48
.3
52.3
,4
95
,519
,5
12
,521
,5
34
.240
8 .2
413
44.6
53
.0
48.2
49
.9
52.6
10
2.5
107.
6
0.5
52.5
48
.0
51.6
46
.2
50.1
53
.3
.516
,5
20
.513
,5
22
,537
.2
409
.241
7 50
.0
53.3
48
.4
51.6
53
.4
102.
5 10
7.8
I 55
.6
51.2
53
.0
46.4
52
.4
55.0
.5
24
,522
,5
13
.525
,5
42
.241
2 .2
431
52.5
54
.6
48.7
53
.7
54.6
10
2.7
108.
6
1.5
58.1
53
.6
54.8
46
.7
54.5
56
.9
,529
,5
25
,513
.5
29
,546
.2
416
.245
3 54
.5
56.1
48
.9
55.5
56
.1
102.
9 10
9.8
2 60
.4
55.9
56
.1
46.9
56
.5
58.8
,5
33
.528
,5
13
,532
.5
49
.242
0 .2
480
56.4
57
.7
49.1
57
.1
57.6
10
3.2
111.
3
2.5
62.6
58
.1
58.7
47
.1
58.5
60
.6
,537
,5
32
,514
.5
35
.551
.2
425
.251
1 58
.2
59.3
49
.3
58.7
59
.2
103.
4 11
3.0
3 20
.0
34.8
59
.5
47.0
56
.3
58.2
.4
63
,534
.5
13
,536
.5
42
.242
9 .2
539
40.5
60
.0
49.2
56
.4
57.7
10
3.7
114.
6
3.5
33.6
35
.6
56.6
46
.2
50.2
53
.2
.468
,5
29
,511
,5
31
,529
.2
426
.253
1 41
.O
57.6
48
.6
50.9
54
.1
103.
5 11
4.2
4 41
.4
41.3
53
.8
46.4
49
.8
52.4
.4
92
,525
,5
11
.527
,5
27
.242
2 .2
511
45.1
55
.1
48.8
50
.8
53.5
10
3.3
113.
0
6 40
.6
41.2
52
.3
46.5
49
.6
52.4
.4
94
.521
,5
12
.524
.5
31
.242
l .2
471
44.8
53
.9
48.8
50
.9
53.1
10
3.2
110.
8
10
40.8
41
.1
51.6
46
.4
48.8
52
.4
,494
,5
20
,513
,5
22
,533
.2
419
.243
4 44
.7
53.3
48
.6
50.3
52
.8
103.
1 10
8.8
CO
Z
inha
latio
n 6.
6”;
for
28
min
0 41
.0
41.0
51
.1
45.9
48
.3
52.3
,4
95
.519
,5
12
,521
,5
34
.240
8 .2
413
44.6
53
.0
48.2
49
.9
52.6
10
2.5
107.
6
0.5
52.8
48
.3
51.5
46
.1
50.0
53
.2
.517
,5
20
.513
.5
22
,537
.2
409
.241
6 50
.3
53.3
48
.4
51.5
53
.3
102.
5 10
7.8
1 52
.4
49.6
52
.8
46.4
52
.2
54.9
.5
20
,522
,5
13
,525
,5
41
.241
2 .2
430
51.4
54
.5
48.6
53
.4
54.5
10
2.7
108.
5
2 52
.1
50.2
55
.3
46.8
54
.3
56.9
,5
18
,526
,5
13
.529
,5
44
.241
9 .2
469
52.1
56
.5
49.0
55
.2
56.2
10
3.1
110.
7
5 52
.2
51.5
58
.5
47.8
57
.6
58.7
,5
19
.533
.5
13
,536
.5
42
.244
1 .2
565
53.4
59
.0
50.1
57
.8
58.2
10
4.3
1 16.
0
10
52.5
52
.6
60.6
49
.4
60.3
59
.4
.519
,5
36
.513
.5
41
.539
.2
478
.264
7 54
.5
60.8
51
.7
59.9
59
.3
106.
4 12
0.6
15
52.6
53
.0
61.6
50
.9
61.4
59
.7
.518
,5
38
,515
,5
42
,538
.2
511
.268
0 55
.0
61.6
53
.1
60.9
59
.7
108.
2 12
2.5
20
52.1
53
.3
62.1
52
.2
61.9
59
.9
,518
,5
38
.518
.5
43
,537
.2
541
.269
4 55
.3
62.0
54
.3
61.4
59
.9
109.
9 12
3.3
HQ
-
28
52.7
53
.5
62.5
54
.2
62.3
60
.0
,518
.5
38
.521
.5
43
.536
.2
583
.27&
t 55
.5
62.5
55
.8
61.7
60
.2
112.
2 12
3.8
28.5
26
.1
34.9
61
.4
53.3
56
.5
55.4
.4
58
.536
.5
21
,539
,5
25
.258
2 .2
691
41.0
61
.6
55.0
56
.3
56.7
11
2.2
123.
1
29
31.0
39
.5
58.4
52
.9
53.3
53
.1
,478
.5
32
,521
.5
34
.519
.2
575
.265
6 44
.5
59.1
54
.6
53.7
54
.9
111.
8 12
1.1
30
39.0
41
.6
55.8
52
.8
52.8
52
.9
.490
,5
21
.522
.5
30
522
.256
6 .2
598
45.6
56
.9
54.4
53
.6
54.4
11
1.3
117.
9
38
40.5
41
.2
52.5
50
.8
49.0
52
.5
.492
.5
20
.525
,5
20
.531
.2
522
.244
1 45
.0
54.2
52
.1
50.7
53
.1
108.
8 10
9.1
Hyp
erve
ntila
tion*
at
23.
5 lin
es
min
tot
al
vent
ilatio
n (R
TPS)
fo
r 14
.4 m
in
0 41
.0
41.0
51
.1
45.9
48
.3
52.3
0.5
24.7
30
.8
51.2
45
.2
45.6
50
.7
1 24
.3
29.1
50
.3
44.3
42
.7
48.5
2 23
.5
27.4
47
.8
42.8
39
.5
45.5
5 22
.1
24.2
43
.3
38.3
34
.2
40.1
10
20.4
20
.8
38.0
31
.1
29.0
34
.6
14
19.3
18
.6
34.5
26
.3
26.1
31
.2
14.5
24
.1
18.7
34
.1
25.8
25
.8
30.8
15
36.2
30
.0
32.5
26
.8
28.3
31
.6
16
31.2
33
.1
33.7
27
.8
33.7
35
.1
18
42.5
38
.1
38.7
29
.2
39.0
41
.9
20
48.1
43
.5
43.9
30
.6
44.3
48
.7
30
41.8
40
.7
48.8
35
.6
41.7
52
.1
- --
_
P =
Pc
o2 i
n m
m
Hg
C
=
bica
rbon
ate
conc
entr
atio
n (I
I)
H+
=
hydr
ogen
io
n co
ncen
trat
ion
(nan
omol
es
I)
See
fig.
i f
or n
umbe
ring
of
poo
ls.
.495
.5
19
,512
.5
21
.534
.2
408
.241
3
.457
.5
18
.512
.5
19
.530
24
06
.240
8
,451
.5
15
.513
,5
16
,525
.2
397
.238
7
447
.510
.5
13
SO
9 .5
19
.236
9 .2
331
.440
,4
99
,510
.4
96
.511
.2
269
.217
5
.431
.4
86
.494
.4
84
.505
.2
087
.I99
9
.425
.4
76
.478
.4
79
.501
.1
948
.I90
2
.427
.4
75
.476
.4
78
.500
.I
932
.189
1
,478
.4
76
.475
.4
79
.504
.I
923
.188
7
.489
.4
8 1
.473
.4
88
.518
.I
932
.192
6 .5
02
,494
.4
72
.502
.5
38
.I96
4 a2
046
.514
,5
07
.473
.5
15
.551
.2
001
.217
9
.500
.5
18
.482
.5
25
.541
.2
146
.239
1 _
.___
.
44.6
53
.0
48.2
49
.9
52.6
10
2.5
107.
6
36.2
53
.2
47.5
47
.2
51.4
10
2.4
107.
3
34.7
52
.5
46.5
44
.5
49.7
10
1.8
106.
2
32.9
50
.4
44.9
41
.7
47.1
10
0.3
103.
0 p
29.6
46
.6
40.3
37
.0
42.2
94
.7
94.4
2
25.9
42
.0
33.9
32
.2
36.8
84
.5
84.5
8
23.5
38
.9
29.6
29
.3
33.5
76
.8
79.1
;
23.6
38
.6
29.1
29
.0
33.1
75
.9
78.5
3
33.4
36
.8
30.3
31
.8
33.8
75
.4
78.3
‘j
36.3
37
.8
31.8
37
.1
36.5
75
.9
80.4
40.8
42
.1
33.3
41
.7
41.9
77
.7
87.1
$ z
45.5
46
.6
34.8
46
.3
41.4
79
.8
94.6
43.8
50
.7
39.7
48
.8
51.7
87
.8
106.
4
* 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
hyp
erve
ntila
tion
wer
e se
t eq
ual
to e
xper
imen
tal
valu
es.
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.
MODEL OF CO, DISTRIUUTION
i
I 0 .- - x
I
‘h.-__- I .._-.. 0 0 VI 9
a .-
I
1”
I 0
64 C. M. E. MATTHEWS et d.
x
x
i
x ”
MODEL OF CO, DISTRtBUTION 65
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
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).
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.).
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.
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).
MODEL OF CO, DISTRIBUTION
4 0 0
72 C. M. E. MATTHEWS et d.
t--
s 8
- _.
: cl .o
”
u I;
.c_ N
8
I__ ‘_L
8 8
MODEL OF CO, DISTRIBUTION
C. M. E. MATTHEWS et Cd.
MODEL OF CO, DISTRIBUTION
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
MODEL OF CO, DISTRIBUTION
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
$-
a
r I 20
1
-A
IO
30
40
Min
utes
Fig.
14
. (b
) V
aria
tion
of co
mpu
ted
vent
ilatio
n an
d PCO,
duri
ng
CO
2 in
hala
tion
for
diff
eren
t ca
rdia
c ou
tput
s of
a)
1.
5 x
norm
al,
b)
norm
al
and
C)
half
no
rmal
.
MODEL OF CO, DISTRIBUTION
-3o m 9 P ru a P 04 _ - - - 0 m
-0
TA
BLE
3
Dia
gram
of
m
odel
Aut
hors
Mod
el
Poo
ls
Exc
hang
e ra
tes
depe
nd
on :
Lung
s T
issu
es
i___
-__
_-_t
I-
I
GR
oDIh
s e-
I al
.
biG
OB
AR
D0
Cf 01
.
DE
FA
RE
S c
f al
.
MIL
HO
RX
e/
al.
FA
RH
I an
d R
AW
~
LosG
osnR
uo
er a
l.
FO
NE
, M
AT
TW
E$\
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
&
Oth
er
Tvv
o:
Fas
t (m
aini
)
eXtr
a~el
fUla
r)
You
im
ainl
y
I~lT~C~lllll~~)
Eig
ht:
Lung
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
.
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
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
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).
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).
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
ADLER, S., A. ROY and A. S. RELMAN (1965). Intracellular acid-base regulation. I. The response of
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/.
J. Med. 212: 6-12.
BROOKS, C. McC., F. F. KAO and 9. B. LLOYD (1965). Cerebrospinal fluid and the regulation of
ventilation. Oxford, Blackwell.
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.
CONWAY, E. J. and P. J. FEARON (1944). The acid labile CO2 in mammalian muscle and the pH of the
muscle fibre. J. Physiol. (London) 103: 274-289.
DANTZIG, G. B., J. C. DEHAVEN, I. COOPER, S. M. JOHNSON, E. C. DEAND and H. E. KANTER (1961).
A mathematical model of the human external respiratory system. Perspecf. Biol. Med. 4: 324-376.
DEFARES, J. G., H. E. DERKSEN and J. W. DUYFF (1960). Cerebral blood flow in the regulation of
respiration. (Studies in the regulation of respiration, I). Acta. Plrysiol. Pharmucol. New/. 9:
321-360.
86 (‘. M. E. MATTHEWS et d.
DRIPPS, R. D. and J. H. COMROE (1947). The respiratory and circulatory response of normal man to
inhalation of 7.6 and 10.4 per cent CO2 with a comparison of the maximal ventilation produced
by severe muscular exercise, inhalation of CO2 and maximal voluntary hyperventilation. ADZ.
J. Phymol. 149: 43-51.
FARHI, L. E. and H. RAHN (1960). Dynamics of changes in carbon dioxide stores. Anrsthesio/o,yy 21 : 604-6 14.
FINI<. B. R. (1961). Influence of cerebral activity in wakefulness on regulation of breathing. J. Appl.
Physiol. I 6 : I 5-20.
FOLVL~. A. S. E. and E. J. M. CAMPBEI.L (1964). The immediate carbon dioxide storage capacity of
man. Clitr. Sc,i. 27: 41 -49.
Fow~t, A. S. E., C. M. E. MATTHEWS and E. J. M. CAMPBELL (1964). The rapid distribution of
:‘HtO and “COa in the body in relation to the immediate carbon dioxide storage capacity.
C/in. Sri. 27: 5 I-65.
GRODINS, F. S., J. S. GRAY, K. R. SCHROEDER, A. L. NORINS and R. W. JONES (1954). Respiratory
responses to CO2 inhalation. A theoretical study of a nonlinear biological regulator. J. A&.
P/I)~sio/. 7: 283 308.
GRODINS, F. S.. J. BUELL and A. J. BART (1967). Mathematical analysis and digital simulation of the
respiratory control system. J. Appl. Phy.sio/. 22: 260-276.
GURTNER, G. H., S. H. SONY; and L. E. FARHI (1967). Alveolar-to-mixed venous Pro1 difference
during rebreathing. P/rysio/qyi.vt 10: 190.
Handbook of Respiration (1958). edited by P. L. Altman, J. F. Gibson and C. C. Wang. Philadelphia
and London, W. B. Saunders Co., p. 64.
JONES, N. L., G. J. R. MCHARC~Y. A. NAIMARK and E. J. M. CAMP~FLL (1966). Physiological dead
space and alveolar-arterial gas pressure differences during exercise. C/in. Sci. 31 : 19-29.
JONES, N. L., E. J. M. CAMPBELL. G. J. R. M~HARDY, B. E. Hicc;s and M. CLODE (1967). The es-
timation of carbon dioxide pressure of mixed venous blood during exercise. C/i/r. Sri. 32:
31 l-327.
KFTY. S. S. and C. F. SCHMIDT (1946). The effects of active and passive hyperventilation on cerebral
blood flow, cerebral oxygen consumption, cardiac output and blood pressure of normal young
men. J. C/i/l. fwwst. 25: 107-l 19.
KETY, S. S. and C. F. SCHMIIIT ( 1948). The effects of altered arterial tensions of carbon dioxide and
oxygen on cerebral blood flow and cerebral oxygen consumption of normal young men. J.
C/in. Itwr.vt. 27: 484-492.
LAMBFRTSEN, C. J.. R. H. K~UGH, D. Y. COOPCR, G. L. EMMEL, H. H. LOES(.HCK~ and C. F. SCHMIDT
(1953). Comparison of relationship of respiratory minute volume to Pcc,, and pH of arterial
and internal jugular blood in normal man during hyperventilation produced by low concen-
trations of CO2 at 1 atmosphere and by 0~ at 3.0 atmospheres. J. Appl. Ph.L:sio/. 5: W-813.
LASZLO. G., T. J. H. CLARK and E. J. M. CAMPBELL (To be published)
LONGOBARIX~, G. S., N. S. CHERNIACK and I. STAW (1967). Transients in carbon dioxide stores.
IEEE Truns. Bio-med. Eng. BME-14, 182~191.
MANFR~DI, F. (1967). Effects of hypocapnia and hypercapnia on intracellular acid-base equilibrium
in man. J. Lob. Clin. Med. 69: 304-3 12.
MAREI\, T. H. (1967). Carbonic anhydrase: chemistry, physiology and inhibition. Ph~~~iol. Rev. 47:
595-78 I.
MATTHEWS, C. M. E.. G. LASZLO. E. .I. M. CAMPBELL, P. KIRBY and S. FR~~DMAN (1968). Exchange
of “COZ in arterial blood with body COa pools. Re.spir. Ph~‘.vio/. 6: 29-44.
MICHEL, C. C., B. B. LLOYO and D. J. C. CUNNINGHAM ( 1966). The in ciw carbon dioxide dissociation
curve of true plasma. Respir. Ph.vsio/. I : 121-137.
MILHORN. H. T., R. RENTON. R. Ross and A. C. Guv.roN ( 1965). A mathematical model of the human
respiratory control system. Biop/r~:v. J. 5: 27-46.
MODEL OF CO, DISTRIBUTION 87
PLUM, F., H. W. BROWN and E. SNOEP (1962). The neurological significance of posthyperventilation
apnea. J. A/n. Med. Ass. 181: 1050-1055.
READ, D. J. C. (1967). A clinical method for assessing the ventilatory response to carbon dioxide.
Australas. Ann. Med. 16: 20-32.
READ, D. J. C. and J. LEIGH (1967). Blood-brain tissue PCO, relationships and ventilation during
rebreathing. J. Appl. Physiol. 23: 53-70.
ROUGHTON, F. J. W. (1964). Transport of oxygen and carbon dioxide. In: Handbook of Physiology.
Section 3. Respiration. Vol. 1, edited by W. 0. Fenn and H. Rahn. Washington, D.C., American
Physiological Society, pp. 767-825.
SHAPIRO, W., A. J. WASSERMAN and J. L. PATTERSON (1966). Mechanism and pattern of human
cerebrovascular regulation after rapid changes in blood COn tension. J. C/in. Incest. 45: 913-922.
TENNEY, S. M. and T. W. LAMB (1965). Physiological consequences of hypoventilation and hyper-
ventilation. In: Handbook of Physiology. Section 3. Respiration. Vol. II, edited by W. 0. Fenn
and H. Rahn. Washington, D.C., American Physiological Society, pp. 979-1010.
WASSERMAN, A. J. and J. L. PATTERSON (1967). The cerebral vascular response to reduction in arterial
carbon dioxide tension. J. C/in. Invest. 40: 1297-1303.
YAMAMOTO, W. S. and W. F. RAUB (1967). Models of the regulation of external respiration in mam-
mals. Problems and promises. Computers & Riomed. Res. I : 65-104.