List o f s y m b o l s VA = volume of air in lungs F A = C O 2 concentration in lung air /'1 = C O 2 concentration in inspired air C~ = C O 2 concentration in mixed venous blood C. -- C O 2 concentration in arterial blood PA = C 0 2 partial pressure in lung air P1 = C 0 2 partial pressure in inspired a i r Pv = C 0 2 partial pressure in mixed venous blood P. = C 0 2 partial pressure in arterial blood (~ = volume flow rate of blood J = C O 2 current leaving blood in lungs Je~ = C 0 2 current entering blood from tissues l"1 = circulation time "~ z2 mean tissue-to-lung circulation time) "cl ~ 2z2 c~ = d C / d P for C O 2 in blood" y = d F / d P for C O 2 in air (c~/7 taken as 5) = scaled mean lung volum~ = 7VA/~J 12 = scaled mean inspiratory flow rate ( = 7(dVA/dt)i./~t) f = breathing frequency 4> = fraction of breathing cycle occupied by inspiration (taken as 1/3)

known to affect C O 2 transport on the mean value and oscillating component of mixed venous and arterial p C O 2. Analogue rather than digital electronic circuitry has been used throughout. A preliminary demonstration of the analogue has been previously reported (SUMMERS and WARD, 19831. P!

t

PA lungs

vA

T

ID

I t

T1 I

I

1 Introduction

IFMBE: 1985

Medical & Biological Engineering & Computing

.._J

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First received 24th April and in final form 26th September 1984

9

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circulation

I l

THIS paper describes a simplified model of the system for carbon dioxide transport in the body and its representation by an electrical analogue. The analogue aims to reproduce the qualitative and, within the limits set by various inbuilt approximations, the quantitative effects of the variables

1

Fig. 1

July 1985

tissues

The simple model

301

2 Simple model (Note: all gas concentration and partial pressure symbols refer to CO 2, i.e. PA -- PACO2 etc.l Many authors have analysed models of carbon dioxide flow in the lungs and circulation (see, for example, YAMAMOTO,1960; GROD1NSet a~., 1967; CUNNINGHAM, 1975; DICKINSON, 1977; SAUNDERS et al., 1980). The model discussed in this paper is shown in Fig. 1. The lungs are represented by a single alveolar compartment with volume VAand CO2 partial pressure PA vented to an atmosphere with CO 2 partial pressure P r Instantaneous mixing within this compartment is assumed. There is no specific dead space compartment in the model. In real life, at the start of inspiration, alveolar gas remaining in the dead space from the previous expiration enters the alveolar compartment. As this dead space gas is of similar composition to the alveolar gas already there it does not directly alter the alveolar CO 2 tension but affects it by delaying the arrival of fresh air. Hence the effect of a dead space on alveolar and arterial CO 2 partial pressure can be mimicked by lengtheniL, g expiratory time and shortening inspiratory time. In this model the ratio of inspiratory to expiratory time is fixed at 1 : 2. Blood circulates between tissues and lungs at a volume flow rate 0. The tissues are represented by a single compartment which adds a CO 2 current J ~ to the arterial blood. The mixed venous blood emerging from this compartment returns to the lungs after a circulatory delay z z. A CO2 current J leaves the mixed venous blood to enter the alveolar compartment. The lung-to-lung circulation time is r a.

Using these relationships and the additional equations

J = ICy-Ca] 0 (for the CO 2 leaving the blood in the lungs) and

PA=Pa (describing the equilibrium reached between blood and lung air) eqn. 1 reduces to

dP, 7V.~T

dVA

= c~[P~-Pa] Q + 7[P~- P,] ~ - , for inspiration, = c~[P~- Po] Q, for expiration.

These equations can be simplified by defining

VA=~X ' -~ VA to v ; ,dP. ~ =

give

dVA

[P~- P.] Q + [P~- PA] ~ - - , for inspiration,

= [P~- Pa] 0 , for expiration

. (2t

P~ and P, are also linked via the circulation. The relationship

C~[t] = C . [ t - ~,] + @ [ t -

T2]

can be written in terms of partial pressures as

3 Mathematical analysis

Jex

dVA = rate of change of lung volume dt = volume flow of air into lungs (since the net gas flow from blood to lung air is negligiblej. The rate of change of the amount of carbon dioxide in the lungs is thus

d [VAFA] = j + Fj d VA dt dt

9 (3t

(The brackets indicate the appropriate time at which each expression must be evaluated and the overbars on C, and/5, indicate that small fluctuations in CO 2 level associated with the breathing cycle have been lost during the circulation time.) Eqns. 2 and 3 describe this simple model.

4 Electrical analogue where Fi = F l

for inspiration,

= FA

for expiration.

=~ V dFA A~-=

J+[Fj-FA]

4.1 Simplifications dVA dt

. (1)

The relationship between C O 2 concentration and partial pressure in blood can be approximated by C = ctP + fi

for mixed venous blood

and C, = ~P, + fl

for arterial blood.

i.e.

where 7 is a constant.

FA = 7P~

for lung air

and F I = 7P~ 3O2

for inspired air.

for inspiration for expiration

JeJ~

and P~[t] = P , [ t - z l ] + ~

Similarly, the relationship between CO 2 concentration and partial pressure in air is assumed to be of the form F = 7P

dPo

~ ~T = [P~-P~176 = [ P o - P,] 0

where c~, fl are constants.

i.e. Ce = ~Pe+fl

The analogue models a slightly different set of equations in which V~ is replaced by its mean value V,, and dV~dt in the inspiration equation is replaced by its mean value over inspiration f/, i.e. it models

It-r2]

.

. (4j 9

. (3), as before.

This change is not very significant in terms of a comparison with the real situation in the body, since eqns. 2 and 3 are in any case only a rough approximation. 4.2 The circuit The basic circuit of the electrical analogue is shown in Fig. 2. The voltages at various points in the circuit correspond to CO 2 partial pressures in various parts of the

Medical & Biological Engineering & Computing

July 1985

model, as indicated. The various impedances are inversely proportional to the volume flow rates which they represent. Electrical currents in the circuit correspond to carbon dioxide currents in the model.

inspiration. Oscillations in this Pa trace have a mean slope during expiration given by . (6J

L dt /o,, ~ a ~

(and hence a slope during inspiration of I-l/q5 -- 1] times the above). The corresponding P~ value is given by

110 1I1

Jex/qQ 1/o

Fig. 2

I

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The electrical analogue Key: z x = low-pass RC network, mean delay z~; also removes cyclic fluctuations in P~ z2 = low-pass R C network, mean delay "f2 + = adder s = voltage-controlled switch; closed during inspiration, open during expiration

The capacitor F which represents the lung air is charged by a current 11 representing the CO 2 flow J into the lungs from the blood. Its periodic discharge by the current 12 represents the dilution of the CO/-rich lung air by fresh air on inspiration. The time delays z~ and T2 are produced by resistor/capacitor networks, shown in the Figure connected to the inputs of the adder. The voltage which represents P~ is derived by the adder from the sum of voltages representing P,, at a time z~ earIier, and Je~/~Q, the extra CO2 load introduced from the tissues, at a time ~2 earlier. Eqn. 3 is thus represented by the function of the adder. Eqn. 4 is represented by the requirement for continuity of current at point A in the Figure. Changes in cardiac output and ventilation rate can be simulated by varying the resistances labelled 1/Q and l/f/, respectively. The two 1/(~ resistors are ganged so that they can be altered simultaneously. Changes in tissue CO 2 production and inspired pCO2 can be simulated by varying the generators for J~x/~ and Px, respectively. Changes in respiratory frequency can be simulated by varying the frequency f of the oscillator controlling the switching of I 2 via the voltage-controlled switch s.

The quasi-steady-state oscillation slope thus depends principally on Jex, and is essentially independent of the other variables Q, f/, PI and J'. SAUNDERS(1980)describes a similar variation of oscillation slope with tissue CO 2 production in his more complicated computer simulation of CO 2 transport. Suitable choices of component values to model the body at rest produce output traces for P, and P~ which have the expected mean values and, in the case of P,, an acceptable oscillation amplitude (see Figs. 3-6t. 5.2 Transients

If one of the parameters defining such a state is changed, P, and P~ will tend asymptotically to a new quasi-steady state, several circulation times being necessary for the new state to become established. This is shown in Fig. 3. In the particular case shown (a change of ventilation rate1 it can be seen that although the size (i.e. slope) of oscillations in the P, trace is almost independent of f/(ile, of the ventilation rate) in the quasi-steady state, this is not true in the transient condition. A sudden decrease of 12 gives a reduction in the slope of oscillations, and vice Versa--this effect disappearing as the new steady state is reached. Fig.4 shows the effect of inserting a single large breath (three times resting tidal volume) in an otherwise regular sequence of breaths under rest conditions. As described above, the slope of oscillations in this transient condition shows a corresponding increase. 5.3 Possible control systems

At first sight, eqn. 6 suggests that a useful control signal to vary the ventilation rate to compensate for changes in tissue

JNNN

5 Behaviour of the analogue 2.5 ~ ' 5.1 S t e a d y states

The voltages representing P, and P~ are conveniently displayed against time on a chart recorder. It can be shown that, in a quasi-steady state, the Pa trace has a mean level given by

0 T,

,60s

, time

p.

P, +

Jegr

. (5)

Fig. 3

where 4) is the fraction of the breathing cycle occupied by

Medical & Biological Engineering & Computing

July 1985

Transition between two quasi-steady states. At time T the ventilation rate is halved, j other parameters remaining unchanged. The final condition corresponds to the body at rest

303

CO2 production could be derived from the oscillation slope. A control equation of the form

L jo,,

6 Comparison w i t h real data

where k is a constant, should give (substituting in eqn. 5)

i.e. approximately constant, as required. However, the transient behaviour renders such a system unworkable in kPo

7.5

Po s-o

~

2.5

~

,.10s , time

Fig. 4

SAUNDERS (1980~ describes the use of a similar control equation for his computer simulation in which such instability appears not to arise.

Disturbance ofrest condition by a Single large breath

this simple form, since breathing irregularities would change the oscillation slope and hence bring about an unwarranted further increase or decrease in ventilation rate, giving rise to a potentially catastrophic positive-feedback loop. (It has been suggested that in the body's control system a phasesensitive response in the monitoring of pCO 2 oscillations by the carotid body could be involved in overcoming this problem and maintaining stability (CUNNINGHAM,1975). The increase in pCO 2 oscillation slope due to a single large breath, for example, could then be compensated for by an associated increase in lung-to-carotid-body circulation time, causing the large-slope signal to arrive at the carotid body at a less sensitive part of its response cycle. Although it seems unlikely that such a system could exactly compensate, all that is required in practice is some form of overcompensation since the unstable positive-feedback loop described above would then become a stable negative-feedback loop3

7,Pol 6 ~P;

"

6.1 Human

In Fig. 5, the two upper traces are P~ and P, from the analogue, showing the transition from rest to exercise (Je~ increased by a factor of three and f / b y three to compensateL The lowest trace shows P, as an arterial pH record (shown inverted), monitored by means of a left brachial artery cannula for a healthy human subject (BANDet al., 1980). It shows a similar transition from rest to exercise (Je~ increased by 2.7 in this case~. 6.2 Cat Fig. 6 shows the simulation of a more unusual e v e n t - alternate breaths with 5 per cent and 0 per cent CO 2 inspired fraction. The two upper traces are P~ and P, from the analogue, showing the transition from rest to a period of seven such alternating breath pairs and part of the subsequent recovery. During the period when the lung air is refreshed on only every second breath the P, oscillation frequency falls to only half that of the breathing frequency, the latter remaining constant throughout the simulation. The lowest trace in Fig. 6 shows the corresponding record from an arterial pH electrode in an anaesthetised cat. Eight alternating breath pairs are followed by a more rapid recovery than displayed by the analogue (whose parameters are chosen to match an uncontrolled human system rather than the controlled cat systemj. A halving of the oscillation frequency and an associated amplitude increase are again observed. The more rapid recovery in the controlled whole animal is almost certainly due to the ventilatory increase during this manoeuvre: in the model, ventilation was held constant.

7 Conclusion This analogue produces an output which demonstrates many of the features of CO 2 transport in the body including the breath-by-breath p,CO 2 oscillations. It has been suggested (YAMAMOTO,1960) that these oscillations, varying as they do with tissue CO 2 production, could form a control signal in exercise. The analogue demonstrates the way in which the oscillations are affected both immediately and in the subsequent steady state not only by alterations in CO2 production but also by other variables such as ventilation and cardiac output. A particular feature is the instantaneous

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time Fig. 5

304

Onset oJ exercise. Comparison of simulation (upper tracest and human data (lowest trace). Note that the pH trace is shown inverted, i.e. a downward displacement of the trace corresponds to increasin9 alkalinity. ( The pH trace, derived from Nature, 283 (5742), p. 85 9 1980, Macmillan Journals Ltd,, is used with permission)

time

Fig. 6

Alternate breaths o J 5 per cent and 0 per c e n t C O 2. Comparison of simulation (upper traces) and eat data (lowest trace~. Note that the pH trace is shown inverted, i.e. a downward displacement oJthe trace corresponds to increasing alkalinity

Medical & Biological Engineering & Computing

July 1985

display of the output to oscilloscope or pen recorder. Previously, models demonstrating the behaviour of respiratory oscillations in CO2 (e.g. SAUNDERS, 1980~ have required sophisticated and expensive computer backup. The use of a circuit composed of relatively few elements also facilitates understanding of the complete simulation and the interaction of its component parts. Despite the simplifications already detailed, the analogue produces results in good agreement with real experimental data. A limitation of the analogue stems from the use of lowpass R C networks to realise the time delays. This results in fixed delays rather than delays which change with blood flow 0 , as happens in real life. This means that the time scale for paCO 2 oscillations is not always the same as that for longterm trends in p,CO z and p~CO 2. This behaviour, together with the absence of a 'dead time' in the RC-simulated delays, means that care must be taken in interpreting the timing in some simulated events. However, compared with conventional 'bucket brigade' delay lines, the R C networks do have the advantage of simulating a range of delays, as occurs in real life as blood returns from tissues with different circulation times. We believe that our simulation is sufficiently realistic to prove useful in the formulation and testing of hypotheses regarding the regulation of CO 2 levels. It is also useful for teaching purposes to demonstrate the effects of changes in variables known to affect CO 2 transport on the mean level and oscillatory component of p C O 2 at points around the body. The possibility exists for the incorporation into the analogue of additional circuitry to model possible control systems or of linking the analogue to a computer to act as the controller.

GROD1NS, F. S., BUELL, J. and BART, A.J. (1967} Mathematical analysis and digital simulation of the respiratory control system. J. Appl. Physiol., 22, 260-276. SAUNDERS, K. B. (1980) Oscillations of arterial C O 2 tension in a respiratory model: some implications for the control of breathing in exercise. J. Theor. Biol., 84, 163-179. SAUNDERS,K. B., BALl,H. N. and CARSON,E. R. (1980) A breathing model of the respiratory system: the controlled system. Ibid., 84, 135 161. SUMMERS,I. R. and WARD,J. (1983) A simple electrical analogue of CO 2 transport in the body. J. Physiol., 334, 2P. YAMAMOTO,W. S. (1960~ Mathematical analysis of the time course of alveolar CO 2. J. Appl. Physiol., 15, 215-219.

Authors" biographies Ian Summers was born in Warwickshire, England in 1952. He received an M.A. from the University of Oxford and a Ph.D. from the University of London with a thesis on combustion noise. After a year at the Royal National Institute for the Deaf, London he moved in 1978 to become a Lecturer in Physics at the University of Exeter. His other research interests include the electrical simulation of deafness and the development of aids for the deaf. He is also interested in musical instruments and performing styles from around the world.

References BAND, D.M., WOLFF, C. B., WARD, J., COCHRANE, G. M. and PRIOR, J. (1980) Respiratory oscillations in arterial carbon dioxide tension as a control signal in exercise. Nature, London, 283, 84-85. CUNNINGHAM,D. J. C. (19751 A model illustrating the importance of timing in the regulation of breathing. Ibid., 253, 440-442. DICKINSON, C.J. (1977) A computer model of human respiration. MTP Press.

Medical & Biological Engineering & Computing

July 1985

Jane Ward was born in Surrey, England in 1952. She received a B.Sc. in 1974 and M.B. Ch.B. in 1977 from the University of Manchester. Having completed her medical training in London, she joined Guy's Hospital Medical School where she is currently lecturing in physiology. Her research interests include cardiovascular reflexes and the control of breathing.

305