Khoo-82.pdf

Factors inducing periodic breathing in humans: a general model

MICHAEL C. K. KHOO, RICHARD E. KRONAUER, KINGMAN P. STROHL, AND ARTHUR S. SLUTSKY Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138; Department of Medicine, Case Western Reserve University, Cleveland, Ohio 44109; and Departments of Medicine, West Roxbury Veterans Administration Hospital and Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115

KHOO,MICHAEL C.K., RICHARD E. KRONAUER,KINGMAN P. STROHL, AND ARTHUR S. SLUTSKY. Factors inducingpe- riodic breathing in humans: a general model. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 53(3): 644-659, 1982.-A general model is developed to account for all kinds of periodic breathing (PB) resulting from instability in respiratory control: in normals during sleep and on acute exposure to high altitude, in sleeping infants, and in patients with cardiovascular or neurologic lesions. It is found that in almost every case the ventilatory oscillation is mediated predominantly by the peripheral controller. System stability is decreased by hypoxia, hypercapnia, increased lung washout times, prolonged lung-chemoreceptor delays, and high controller sensitivity. Stability is enhanced by large lung CO2 and O2 storage volumes but little affected by body tissue stores. Using our own measurements of lung-ear delays, the model predicts that the mean cycle time of PB decreases from about 30 s at sea level to 20 s at 14,000 ft, in excellent agreement with data from other studies. Allometric scaling of the relevant parameters also shows close agreement between model predictions and data obtained on infants.

respiratory control; Cheyne-Stokes; control theory; hypoxia; sleep; altitude

IT HAS BEEN one and a half decades since a number of models (6, 21, 23) were proposed to explain periodic breathing (PB) in terms of instability in the respiratory control system. The underlying approach was to consider the system as one incorporating negative feedback: changes in ventilation (“the disturbance”) from some set point produce corresponding changes in the blood gas tensions, which in turn evoke a “corrective action” from the controller to restore ventilatory drive to its original level. Since there is a delay before the controller senses the signal and responds, the timing and magnitude of the corrective action are critical parameters in determining whether the disturbance would be damped out or would produce a self-sustaining oscillation. Experiments on an- imals have shown that artificially altering these parameters (such as prolonging the lung-to-chemoreceptor circulatory delay or enhancing ventilatory gain by neurological means) can produce Cheyne-Stokes respiration (CSR) (8, 9). These results are consistent with observations that CSR occurs frequently in humans with congestive heart failure (29) and brain stem lesions (3).

However, PB is also known to occur under normal physiological conditions in adults and infants during sleep (27, 32), and in both awake and sleeping normals at high altitude (12, 37, 39). Such phenomena would be impossible to simulate in Milhorn and Guyton’s model (23), since abnormally high controller gains or abnormally long circulatory delays are needed to produce instability. For PB to be induced in Horgan and Lange’s model (16) an unrealistic ventilatory response curve had to be assumed. The model of Longobardo et al. (21) successfully simulated CSR under a variety of clinical and normal conditions. However, they demonstrated this only for cases in which a relatively l-arge initial disturbance (a period of forced hyperventilation followed by apnea) was required to drive the model into an oscillatory mode. By contrast, observations of PB show that sustained episodes often appear and disappear in an appar- ently spontaneous manner. In the present model, we demonstrate that even under normal physiological conditions the overall system gain can be easily enhanced, thus increasing the chances for spontaneous oscillations to occur. We have based the model on accurate physiological data and are able to show that the predicted periodicities of these oscillations agree closely with observed values. Whereas the previous models of PB pre- supposed only one controller, we have included the effects of both central and peripheral chemoreceptors. This undoubtedly addresses the important issue of which controller generally mediates an instability.

The basis for local stability analysis. From a rigorous viewpoint, the development of an accurate model of the respiratory system necessitates the use of a digital or analog computer since it is extremely difficult, if not impossible, to solve the nonlinear problem analytically. However, this approach often leaves one with very little physical insight into the mechanics of the problem. For instance, computer simulations may demonstrate a dependence of cycle time on circulatory delay, but unless the calculations are made over a wide range of circulation times, it may not be clear what form the dependence takes. On the other hand, an analytic solution would show explicitly the relationship between the two vari- ables.

Thus, in contrast to the computer-based approach of 0161-7567/82/0000-0000$1.25 Copyright 0 1982 the American Physiological Society

A MODEL OF PERIODIC BREATHING IN HUMANS 645

previous studies, we have adopted a method of solution that proceeds along the following lines. By restricting the analysis to a sufficiently small regime of arterial CO2 and Oa partial pressures (Pace, and Pao,) it is possible to linearize the equations describing the nonlinear system. The stability properties of the linearized system can then be described in terms of analytic expressions and exam- ined by means of a frequency response method. Let us assume that a small disturbance displaces the system away from its original equilibrium state to some neigh- boring state that lies within the same Pace, and Pao., regime. Then we say that the system is locally stable if in the course of time it returns to its original equilibrium state. If it moves away from the equilibrium point, it is locally unstable. A classic theorem of Liapunov in stability theory (20, 25) states that the information obtained from the linearized equations of a nonlinear system is sufficient to give a correct answer to the question of its stability in the immediate vicinity of any equilibrium point. It follows that the stability properties of the system over a wider regime of Pace, and Pao, can be deduced by applying this quasi-linearization technique to several different equilibrium points.

In this paper, we are solely interested in the factors that affect system stability and increase the chances for the initiation of oscillations at selected equilibrium states. Therefore, the application of a local stability analysis is completely valid and adequate for this purpose. Events that occur subsequent to the onset of instability (e.g., the development of limit cycles) lie beyond the scope of the present study. Thus it should be emphasized that all statements about stability in the following sections nec- essarily refer to local stability unless specified otherwise. Furthermore, quantitative predictions of cycle time and system gain should be interpreted as estimates of these parameters immediately following the initiation of oscillatory behavior. However, at the same time it is important to note that the process of quasi-linearization produces a maximum error of only 25% in these estimates, if the peak-to-peak fluctuations in Pace, and Pao, are kept within 10 Torr and if the oscillation is nonapneic.

In an experimental study on unacclimatized healthy subjects at high altitude (37), we found a systematic but small dependence of cycle time on strength of the oscillation. The development of a low-level nonapneic oscillation into a strong oscillation in which the apneic dura- tion accounted for half the cycle time produced a maximum increase in periodicity of about 5 s. This means that the application of our analysis to situations in which strong nonlinearities prevail would result in errors of only 25%, consistent with our earlier error estimate. The evi- dence is further strengthened by the close agre.ement between our model predictions and published data (see RESULTS), demonstrating clearly the robustness of our method of local stability analysis.

Instability criteria for negative- ft:edback systems. The model used in our analysis is represented schematically in Fig. 1. However, for purposes of conceptual clarity, it is best to consider a simplified version first (Fig. 2A). This still retains all the essential features of a negative- feedback system: a “disturbance” (u), which increases alveolar venti@tion by ADA from some steady-state equilibrium level (VA), will lower Pace, by APaco, and thus evoke a negative “corrective action” (e) from the controller to suppress the disturbance.

The stability of the system is tested by subjecting it to small sinusoidal disturbances [i.e., u = u sin(27;rt/T)] of different cycle times (T). The controller responds with a corrective action e = e sin(2rt/T + +), which occurs only after a total phase lag (+) or equivalently a time delay of T+/271. The ratio of the magnitude of e to that of u (i.e., e/u) is termed the system loop gain (LG). Case I of Fig. 2B represents an inherently stable system where LG is less than unity and + is less than 180”: the effect of a disturbance is eventually suppressed by attenuation and cancellative interference. Random disturbances to such a system may produce the effect of “bursts” of oscillations, a phenomenon which has been noted in a number of experimental studies (4, 15, 36). In case II, + is 180” and LG is unity- this allows the disturbance to persist indefinitely, giving rise to a self-sustained oscillation. When + is 180” and LG is greater than unity (case III)

- c

Central pho, P Brain Tissue ace, Lung- to- Brain

7 (Medullary) 4 Compartment + Transport 4 Controller Delay

FIG. 1. Schematic representation of respiratory control system, which as- sumes that total ventilatory response is simply the sum of peripheral and central components. See APPENDIX I for symbols.

646 KHOO, KRONAUER, STROHL, AND SLUTSKY

A ‘Disturbonce’

u (t)

Controlled System:

Lungs and Tissues

.

- Pa

co2 - APO,,

2 9 2

‘Corrective Action’ -e(t)

L

?A- e(t)

1

Lung-To- , Cont fol ler 1 Controller

/

B CASE I: M180°,LG<l CASE II +=180°, LG =I CASE III += 180°, LG>I

I

n 2-3

rl \ \ /’ t b’ ‘2

LAP oco 2

2-3

oco2 34’

-\ /-, v

/ / \ 1’

\

I

t

I

FIG. 2. A: simplified version of respiratory control model, with only one feedback loop. Note that ADA = u(t) - e(t). B: conditions for

corresponding changes in Pace:, [seen at (2)], which are delayed before

stability and instability in model of A. Changes in VA at (I) produce being sensed by controller [at (3)].

A MODEL OF PERIODIC BREATHING IN HUMANiS

a self-sustained oscillation would continually grow in amplitude until it is bounded by saturating nonlinearities in the chemoreceptors, the gas exchange process, and the mechanical properties of the system.

The necessary and sufficient conditions for the system to become unstable (i.e., for any oscillatory behavior to be self-sustaining) can be stated as follows. 1) Loop gain must be at least unity. 2) The total phase lag must be 180".

The derivation of these instability criteria can be found in most textbooks on control theory.

Under a given set of physiological conditions and within a given operating regime (defined by Pace,, and Pao,), LG and + for the system are dependent on the cycle time (T) of the disturbance. This information can be represented pictorially in terms of a vector of length (e/u) oriented at an angle (measured clockwise) to the horizontal. The locus traced by the tip of this vector for values of T ranging from zero to infinity represents the response of the system to disturbances of all possible frequencies. Such a polar diagram is known as the NV-

d

@= 270°

647

quist Plot. Two examples are shown in Fig. 3. The first is designated Sl and refers to a particular set of physiological conditions in sleep (see Table I). When the disturbance cycle time is 50 s, + is 125” and LG is 1.1. But when T is 29 s, + and LG become 180" and 0.7, respectively. Thus, if the system is subjected to a ventilatory disturbance (such as a sigh), it would oscillate transiently with a fundamental cycle time of 29 s. However, this oscillation would not be self-sustaining and would eventually decay, since both the above instability criteria have not been satisfied together. On the other hand, if we simulate a condition (C) of congestive heart failure by halving cardiac output and consequently doubling the lung-to-chemoreceptor transport lags, LG becomes greater than unity at + = 180”. The system is now unstable, with a self-sustained and growing oscillation of fundamental cycle time T = 55 s. The point at which the Nyquist Plot intersects the negative horizontal axis (i.e., LG at + = 180”) is therefore critically important in determining whether the system is unstable and at what fundamental periodicity it would oscillate.

\ \

\ \ FIG. 3. Representation of system sta-

\ bility in terms of the Nyquist Plot. Two sets of conditions are shown: simulated

I condition of congestive heart failure (C)

@ = 360° and sleep state at sea level (52). Each

I !P=OO number corresponding to a cZosed circZe

I on the plot represents cycle time of sinusoidal perturbation. Point where LG = 1 at + = 180” is designated as x. Broken circZe represents locus of all points where LG = 1.


TABLE 1. Values of system parameters used in model calculations

Sea Level (Awake) Sea Level (Sleep) High Altitude (Awake)

Parameter 8,000 ft 14,000 ft Al A2 A3 Sl (C) s2 s3 s4 s5

Ml M2 HI H2

GP 0.504 0.504 0.640 0.504 0.504 0.504 0.504 0.640 0.504 0.504 0.504 0.504 Ip, Torr 35.5 35.5 35.5 35.5 35.5 35.5 35.5 35.5 35.5 35.5 35.5 35.5 GC 0.024 0.024 0.030 0.015 0.015 0.015 0.015 0.019 0.024 0.024 0.024 0.024 Ic, Torr 35.5 35.5 35.5 44.5 44.5 44.5 44.5 44.5 35.5 35.5 35.5 35.5 w, s 6.1 6.1 7.0 6.1 7.0 7.0 7.0 7.0 5.4 5.4 4.9 4.9

(12.2)

rc, s 7.1 7.1 8.2 7.1 8.2 8.2 8.2 8.2 6.4 6.4 5.9 5.9 (14.4)

& 7 1/ S 0.10 0.10 0.085 0.1 0.085 0.085 0.085 0.085 0.15 0.15 0.20 0.20 (0.05)

Vcoz, liters 3.2 3.2 2.3 3.2 3.2 2.3 2.3 2.3 3.2 4.0 3.2 4.0 Voz, liters 2.5 2.5 1.8 2.5 2.5 1.8 1.8 1.8 2.5 3.3 2.5 3.3 PIO~, Torr 150 150 150 150 150 150 150 150 110 110 93 93

pacoz, Torr 40 39 40 46 46 46 46 46 39 39 38 38

Pao,, Torr 100 60 100 85 85 85 80 80 60 60 40 40

VD, l/s 0.038 0.038 0.038 0.030 0.030 0.030 0.030 0.030 0.050 0.050 0.050 0.050 ~______ In all the above cases, Tb = 80 s (Ref. 24); PICO, = 0, 71 = 1 s, 72 = 2 s (Ref. 19); Vko, = 15 liters,&, = 6 liters (Ref. 21). C represents

simulated condition of congestive heart failure. See APPENDIX I for Glossary of Symbols.

Cases Al and A2 in Fig. 6 show that the frequency response and stability characteristics of the system can also be changed drastically with changes in operating regime (Pace, and PaoJ without any alteration in the other parameters.

THE MODEL

The lung compartment. We assume a single homoge- neous alveolar compartment perfused by the total blood flow (Q,.

The effective CO2 storage capacity of this compartment (Vco,) is larger than the gaseous volume (Vo,) by the amount of CO2 dissolved in lung tissue, pulmonary capillary blood, and extravascular lung water. We also assume that the alveolar gas tensions are equilibrated with the arterial gas tensions (symbols defined in APPEN-

DIX I)

dpaco,, 863Q - -

- - -

dt vco2 wco, - Cac0,) +

mo, - Pace., l

vco2 - VA (1)

dPao, 863Q PI0 - Pa0 =- dt vo2

PO, - Ca0,) + to ’ VA (2) 2

where

VA = TjE - TjD (3)

The body tissue compartment. Oxygenated blood leaving the lung compartment perfuses the brain (b) and body tissues (t) compartments (Fig. 1). The PCO~ and PO, of mixed venous blood (PiJr co, and PVo.,) returning to the lungs are assumed to be equilibrated with the corresponding gas tensions in the body tissues compartment.

The metabolic production rate for CO2 (MRcoJ and consumption rate of 02 (MRoJ are constant under resting conditions

d&o & 2 M&o., -=3- dt vtco.,

(Cac0, - CVCO,) + - vtco,

(4)

dcvo, & MRo, =- dt vto,

Go, - CTO,) - 7 (5) 02

The dissociation curves. The dissociation curves for CO2 and 02 in blood with normal hemoglobin content can be represented approximately by the following linear (or piece-wise linear) functions (21)

Cw0, = Kc02*Paco,~ + 0.244 (6)

Ca 0, = 0.00025 9 Pao, + 0.1728 (70 < Pa0,)

= 0.00067 l Pao, + 0.1434 (55 < Pa0, < 70) (7)

= 0.00211 l Pao, + 0.0662 (35 < Pao, < 55)

where the gas tensions (Pace,, Pao,) are given in Torr, and the blood gas concentrations (Caco,, Cao.,) in ml/ml whole blood. The same relationships are assumed to hold in mixed venous blood, body tissues, and brain tissue. Kco2 in Ey. 6 represents the slope of the physiological CO2 dissociation curve and therefore implicitly accounts for the Haldane effect. The value for Kc02 under nor- moxie conditions is 0.0065. At an equivalent altitude of 14,000 ft, Kc02 can increase by 20% to a value of 0.0084. Our calculations, however, show that this produces at most a 2% decrease in predicted cycle times. The effect of changes in Kc02 with hypoxia may thus be neglected.

Circulatory mixing and delay. The Pace, and Pao,, waveforms of the arterial blood leaving the lung corn: partment will be time-delayed and distorted before de- tection by the chemoreceptors. The pure time delay (up from lung to carotid body, or 7c from lung to brain) can be attributed to the convective transport process, whereas the distortion effect is due to mixing in the heart and in the arterial vasculature. Lange et al. (19) have proposed that the lung-to-chemoreceptor transfer function can be modeled in terms of two mixing chambers (with time constants 71 for the heart and 72 for the


arteries) in series with each other and with a pure time delay (up or 7~). They obtained the values of 1 and 2 s for 71 and 72, respectively, from dye-dilution experiments on catheterized humans. The transfer function is defined <by Eq. A3 in APPENDIX II. If a bolus of dye is injected into the pulmonary venules, up (or 7~) would represent the absolute delay before the first appearance of the dye at the peripheral (or central) controller.

The peripheral and central controllers. Rebuck et al. (30) and Severinghaus and Crawford (34) have shown that minute ventilation (VE) is a linear function of Pco2 under isoxic conditions and a linear function of arterial 02 saturation (SaoJ under isocapnic conditions. Using Severinghaus’ formula for the relationship between Saoe a and Pao.,

Sa 0‘2 = lO()(l - 2~4~~0~05pa%) (8)

it_can be shown that the steady-state ventilatory response (VE) to Pace, and Pao, takes the following form

f?E = Gpe-o*05Pao~( Pace, - IP) + Gc(Paco, - Ic) (9)

It has been found (30) that a wide range of values for Gp and Gc exists across individuals. The parameter values used in the present analysis provide a reasonable com- promise between the results obtained in different studies (10, 30, 34): Gp = 0.504, Gc = 0.024 and Ip = 1~ = 35.5, if TE is measured in liters/second.

The representation in Eq. 9 separates the hypoxia- dependent term from the hypoxia-independent term. We will assume that the former can be attributed to the peripheral controller, while the latter reflects the steady state response of the central controller. Miller et al. (24) have reviewed the validity of such an assumption and have conciuded that it is probably justifiable in man.

It is generally accepted that the carotid chemoreceptors respond directly and very rapidly to changes in Pco2 (or equivalently, pH) and Po2 in the blood that perfuses them (28). Thus we will represent the peripheral ventilatory response by

VP = Gpe-o-05Pa&( paEo, - Ip) (10)

where Pa&, and Pa& are the arterial blood gas tensions at the chemoreceptor site.

From Eq. 9, we know that the steady-state central reponse to a step in Pace, is given by

7

V c = Gc( Pa:,, - 1~) = G&%%, - Ic) (11)

We will assume that the central controller responds only to the tissue Pco2 (Pbco,) of the brain compartment that is perfused by arterial blood. Pbco, is related to Pa&, (i.e., the Pco2 of arterial blood as it enters the brain compartment) by

dpbco, Qb MRbco, -=- dt Vb

(P&0, - Pbco,) + KcoaVb

(12)

We have assumed here that the slope of the CO2 dissociation curve for brain tissue is the same as that for blood (i.e. Kc02 = 0.0065). The direct and instantaneous re-

sponse of the written as

controller to Pbco, can then be

. Vc = Gc

MRbco, Pbco, - .

QbXco2 - Ic (13)

Note that, in the steady state, when dPbco,/dt = 0, substituting Eq. 12 into Eq. 13 would yield Eq. 11.

ANALYSIS

The mathematical procedures by which we obtained expressions for the system loop gain (LG) and total phase lag (+) are described in APPENDIX II. These are derived by vectorial summation of the three components that make up the overall loop transfer function-the peripheral CO2 and 02 loop transfer functions and the central CO2 loop transfer functions (Eqs. Al3 and Al4 in APPEN-

DIX II). By calculating LG and + over a wide range of oscillation cycle times ( T), the Nyquist Plot can be constructed for a particular set of conditions, and the stability behavior of the system can be studied.

The values of the model parameters define the set of conditions under which the system is operating. Table 1 gives the values we have used to represent the typical conditions that characterize wakefulness, sleep, or exposure to high altitude in normal humans. Some of these parameter values are known with less certainty than others, and some may indeed vary widely across individuals. On the other hand, such large variances would be of little consequence if the associated parameters do not influence LG or + significantly. The normal physiological values of the following parameters are approxim$ely: T - 5, T2 - o.;, vcoz

100 (sea level) or 15 (high altitude), VA -

- 3.2, Vo2 - 2.5, TTl - 150, and TT2 - 60. Using these values, we find that certain terms are significantly smaller than others in Eqs. A7-Al2 for cycle times on the order of 20-30 s. We can thus simplify the equations into the following approximate forms (symbols are de- finedin APPENDIX I)

for peripheral CO2

LG zz GPfodho, - PICO,) 5 vco:! (14)

J TY .

[l + bTd2][l + bd2][l + (~372)~]

@Pl z tirp + tan-‘(oT1) + tan(cJ72) + tan-‘( wTJ (15)

for peripheral 02

LGp,= O.O5GPfo@aco, - IP)(PIO, - Pao,)

vo 2 (16)

(T&j2 .

[l + (wT~d~][l + (LJ~I)~][~ + (~37~)~]

$p2 z WP + tan-‘&) + tan-‘(c372) + tan(oTE2) (17)

for central CO2

650 KHOO,KRONAUER,STROHL,AND SLUTSKY

LG Gc(h202 - Pk0,) C z

vco2

J TT .

[l + WV2][l + (oTd2][l + (G.u~)~][I + ((c)T~)~]

(18)

+c Z c37c + tan(w71) + tan-$& (19)

+ tan-‘(oTl) + tan(wTb)

where c3 = 2n/T ( in radians per second), TEE = (VA/VO, + 1/T2)-l, fo2 = e-“.05pao2, and the component phase lags (GP,, +P,, +c) are given in radians.

The above expressions show that LG and + are relatively independent of the effects exerted by the body tissues compartment. In fact, it can be shown that this statement holds true for all values of TTl and TT,. An- other important point is that cardiac output affects LG and + only indirectly by altering the transport lags (up, ~c, 71, and 72) and the lung washout times ( T1 and T2). However, one should not ignore the fact that Q can alter the operating state of the system (as characterized by Pace, and Pace,), which in turn would affect LG and Q, significantly.

The first three terms of Eqs. 15 and 17 may be collec- tively labeled as c3Tp [= ~37~ + tan-‘(tiT1) + tan-‘(tiT2)], where Tp refers to the mean circulatory delay between the lungs and carotid body inclusive of mixing effects. Similarly in Eq. 19, we may define Tc as the mean lung- to-brain circulatory delay. Thus each of +p, and a)pY consists of the mean circulatory delay and a lag contril bution from the gas exchange process in the lungs. +c contains an additional component representing the washout of CO2 in the brain compartment. While the lung washout time constants for any individual are relatively easy to determine, the lung-to-chemoreceptor delays are impossible to measure noninvasively. This accounts for the lack of accurate values for these quantities in man in the existing literature. With the use of an ear-oximeter, we performed experiments to measure the lung-to-ear transport lag, which gives a close approximation to the delays in question.

Determination of circulatory delays. In two healthy young adults, we measured ventilatory flow, expired Pco, (PEco,), and Po2 (PEo,), and ear capillary 02 saturation (Sao,). Each subject was asked to perform repeti- tive sequences of a breathing maneuver consisting of three large breaths followed by a breath hold. These were first performed under normoxic conditions and later under 12% 02. The top panel of Fig. 4 shows the breathing pattern of one of the subjects in 12% 02 during and after - a series of these maneuvers. Large swings in PE~:~,, PE+ and Sao, were produced (Fig. 4, second and third panels). By best fitting the predictions of a single-compartment gas exchange model to the measured PE~~~~,, and PEo.,, the corresponding alveolar (A) Pco2 and Po2 &me- courses were reconstructed from the data. These are represented by the saw-toothed waveforms (PA~:~, and PAN,) in the second, fourth, and fifth panels. [Further details of the technique are described in Khoo (17).] By converting PAQ into 02 saturation it was possible to

deduce the O2 saturation time course (SAG,) in blood just leaving the lungs. This is shown as the saw-toothed tracing in the third panel of Fig. 4.

By convolving the circulatory delav and mixing transfer function (as defined in Eq. A3) with the calculated SAG, waveform, we could deduce the Sao, waveform that would appear at any arterial site downstream from the lungs. As mentioned earlier, we used the values ~1 = 1 s and 72 = 2 s that were demonstrated by Lange et al. (19). The third panel of Fig. 4 shows the calculated Sao, waveform after the mixing effects have been imposed but with the pure delay (7J set to zero. By increasing 7e to a value that would best fit the calculated Sao, and measured Sao, waveforms, it was possible to deduce the absolute lung-to-ear delay. The fourth and fifth panels show the waveforms of Pace, and Pao,, respectively, that would be expected to appear at the ear.

Figure 5 shows the calculated absolute lung-to-ear delay (7e) as a function of mean Sao, (Sa.o, ) for the two subjects. 7e decreases from 6.4 s at 96% saturation to 5.4 s at 68% saturation. Pulmonary blood flow could be estimated independently from the large swings in expired gas tensions produced by the breathing maneuvers (17). Concomitant with the above decreases in Sao, and 7e we

found a doubling of pulmonary blood flow. However, there was a much stronger correlation between 7e and Q in normoxia; plotting 1/7e against predicted Q for each of the points at 96% saturation (Fig. 5) produced a straight line. Although we are still unclear about the nature of these effects, we have incorporated the results into our model for lack of similar data available in the literature.

It is probably reasonable to assume that 7e gives an upper bound to the lung-carotid delay and a lower bound to the lung-brain delay. We propose that

7p = 7e - 0.5 cw

7c = 7f3 + 0.5 (21)

for all Sao,.

RESULTS

Sea-level (awake) conditions. Under normal conditions, the respiratory control system in awake humans is found to have a relatively low loop gain (LG = 0.17 when + = 180”), as shown by curve AI in Fig. 6. However, the system can be made unstable by performing a hyperven- tilatory sequence of 2 min (12). This usually leads to an apnea lasting another 2 min before ventilation is re- sumed. PACO, is rapidly restored to near-normal levels, but PAN, remains low because of the long washout time for O2 at sea level. It is this factor of transient hypoxia that increases loop gain eightfold (LG = 1.35 at + = 180”; curve A2 in Fig. 6) and consequently induces periodic breathing. It should be noted that, with the resumption of ventilation after the apnea, PAO, will rise towards its normal levels. Thus, the locus A2 would tend to move back towards AI. Douglas and Haldane’s data (12) show a cycle time on the order of 30 s, which is compatible with our prediction of 26-27 s.

High-altitude (awake) conditions. Acute exposure to


0 IO 20 3:) 4cJ 50 63 7'3 80 90 100

Calculated

50. I I I , 1 I I 1 1 I I 0 10 20 30 40 50 6 L 70 80 90 100

i

0 10 20 30 I I40 50 60 70 80 90 100

FIG. 4. Phase relationships during “forced” periodic breathing. Top panel: tidal volumes (~01s) vs. time in seconds. Minute ventilation (\37~) calculated on a breath-to-breath basis is superimposed. Second panel: time courses of calculated alveolar (PACQ) and expired (PE~Q) and measured expired (PESO,) CO2 pressures. Predicted PECK, plateaus have been best- fitted to data-the high quality of fit makes it difficult to distinguish one from the other. Third panel: time courses of calculated end-capillary Sao,, calculated Sao, after mixing in heart and vasculature (but effect of transport delay not included), and Sao, measured by ear oximeter. Note that circulatory mixing re- moves intrabreath variations. Fourth panel: calculated alveolar Pco2 (PAN*, ) and arterial Pco:! appearing at peripheral controller (Pa&). Fifth panel: calculated alveolar Paz (PAo,) and arterial PO,! appearing at peripheral controller <Pa&).

0 10 20 30 40 50 60 70 80 80 100

Time (sets) -

hypoxic conditions at high altitude increases loop gain. ency between our predictions and Waggener’s data.’ At 8,000 ft above sea level, loop gain is almost 0.9, while In Fig. 4, the subject simulated periodic breathing going up further to 14,000 ft raises loop gain to 1.6. It is under 12% Oz with an approximate cycle time of 21 s. interesting to note that Waggener et al. (37) reported a rise in the incidence of apneic oscillations (i.e., percentage of time spent breathing with a Cheyne-Stokes pattern)

’ Waggener et al. (37) also reported a systematic but small dependence of cycle time on oscillation strength. They defined a strength

from 24% at 8000 ft to 40% at 14,000 ft. They also found index (M) for nonapneic oscillations: M = (vmax - Vmin)/(Vmax +

1 negative correlation between oscillation cycle time and vmin) vmax and vmin being the respective maximum and minimum .1 7 . . . . . _- . _ _ altitude. Our analysis predicts a decrease in cycle time ventrlatrons within a cycle. Using the cycle time vs. M relationship,

‘ram 30 s at sea level through 23 s at 8,000 ft to 20 s at they adjusted the periodicity of each pattern to that expected for a

14,000 ft. Figure 8 demonstrates the remarkable consist- standard oscillation strength (M = 1). The “standard” cycle times are plotted in Fig. 8.


7.5-

7.0- ’

l Subject 1

0 Subject 2

a Mean value of . 0 each cluster of points

0

5.0 I 1 I I 1 I 1 I 100 95 90 85 80 75 70 65 60

Mean Arterial Saturation, Soo2 (Oh)

FIG. 5. Variation of absolute lung-to-ear delay (T& with mean arterial 02 saturation @a~,).

@= 270”

The consequent hypoxia produced by this level of inspired 02 is similar to the conditions at an altitude of 14,000 ft. The waveforms of Pace, and Pao, appearing at the carotid body (i.e., Pago, and Pa&, respectively) were predicted with a lung-to-ear delay deduced by the pro- cedure mentioned earlier. -By visual inspection, one can clearly see that Pap co, and Pa& are roughly 180" out of phase with TE. This provides a favorable check on the value of 20 s predicted by the model.

The calculations in A41 and Hl were made with a typical value of 2.5 liters for the FRC (Vo,) of a young male adult in the supine position. It is known that changing the subject’s posture to a standing position will increase FRC and also VCO~. The effect of such a postural change on system behavior is shown in Fig. 6 (MZ for 8,000 ft, HZ for 14,000 ft). In both cases, an increase in lung volumes of 28% leads to a decrease in loop gain of almost 10% and a lengthening of cycle time by 1 s.

\ \

@= 360° FIG. 6. Nyquist Plots for various conditions in awake state at sea level (Al, A2), at 8,000 ft (MI, M2), and at 14,000 ft (HI, H2). Hypoxia destabilizes the system significantly.


Conversely, this implies that the effect of changing pos- tures from upright to supine by itself contributes toward destabilizing the system. An accompanying effect, which is probably more important, especially in patients with chronic obstructive lung disease, is the increase in closing volume with decrease in FRC. Such a situation produces an increase in physiological shunting that destabilizes the system further with the subsequent hypoxia. Haldane and Priestly (14) noted that the recumbent position favored the development of PB, an observation that undoubtedly agrees with our model prediction.

Sea-level (sleep conditions). Studies of ventilatory response to CO2 in non-REM sleep have indicated a mod.est decrease in slope and a shift of the response line to the right by about 6 Torr (5,27). By contrast, the ventilatory response to hypoxia has been found to be roughly the same for both awake and sleep conditions (31). We can model this behavior by assuming that only the central

e= 270°

controller is affected by sleep while peripheral control remains unchanged. Our calculations show that for the overall controller gain to decrease by 20% and for the VE-

Pace, line to shift 6 Torr to the right, the central response must decrease in slope by 40% and shift 9 Torr to the right.

Even with a depression in central response, loop gain is increased fivefold from its normal level in the awake condition (Sir compared with Al in Fig. 7) as a consequence of the 2% drop in 02 saturation and a rise in Pace,. By focal cooling of the intermediate area of che- mosensitive zones on the ventral surface of the medulla, Cherniack et al. (9) were able to depress the central CO2 response in cats without affecting peripheral gain. This produced a depression in slope and shift to the right of the CO2 response line (as in the case of sleep). PB was induced in all the subjects.

If we assume a concomitant drop in cardiac output of

@=360° @=OO

FIG. 7. Nyquist Plots for range of physiological variations in parameters found in sleeping (Sl-SS) and awake (Al-A3) normals.

654

15% and a consequent 15% rise in circulatory delays, loop gain is increased further to 0.8 (SZ). It has already been shown (Fig. 3) that halving cardiac output and thus doubling circulatory delays can produce a loop gain much higher than unity, even without additional hypoxia. Such a situation, applicable to the patient with congestive heart failure, probably does not occur in normal humans, although Bulow’s data (5) suggest otherwise (see DISCUS- SION). The cycle time of the oscillations induced here is about 60 s, consistent with periodicities observed in CSR of cardiac patients.

For those individuals under the same conditions as in SZ, but who have smaller lung volumes in the supine position, loop gain will be higher-now being almost unity (S3). If O2 saturation were to be slightly lower by another 1 percent, there would be an additional 25% increase in loop gain (S4). Finally, if all these factors were present in subjects with slightly higher controller gains [Gp and Gc values from Severinghaus (34)], loop gain would be about 1.5. Thus, under conditions of sleep, small physiological variations can lead to a wide range of loop gains (Sl to Sir), whereas the same physiological variations do not affect system behavior so significantly in the awake condition (AI to A3). On the other hand, there is much less variation in cycle time, the average value being 30 s. Similar cycle times have been recorded by Lugaresi et al. (22) and Specht and Fruhmann (35). Extrapolation of the observations of Waggener et al. (37) to sea level gives a cycle time of 32 s (Fig. 8).

From the above results, it may be safely deduced that the effect of sleep at high altitude would be to make the system more unstable than as shown in A& and HI (Fig. 6). Again, such a conclusion is consistent with observations that periodic breathing occurs even more frequently in mountaineers during sleep (14).

Periodic breathing in infants. Allometric data on mammals of different species and sizes show that FRC,, tidal volume, and stroke volume are proportional to body weight (W). Flow rates, such as VA, VE, and MRo2, are proportional to W”*75 (11). It follows that in general cardiopulmonary time constants should scale according to W”.“’ and thus by a factor of two from neonate to adult

X Mode I n Specht & Fruhmann 1972 A Lugoresl 1977 A Waggener 1979 o Rigotto 8 Brady 1972

jjTer et a”y’g77

k( 15% 02)

(170) 0

(1?4)

5000 (192) (*y ho2 (torr)

10000 15000 Altitude (ftl FIG. 8. Variation of oscillation cycle time with altitude (or equiva-

lent hypoxia). Dark circles (with corresponding error bars) represent data from study of Waggener et al. (37); extrapolation ( straight line) of these results to sea level gives a cycle time of 32 s. Predicted values are consistent with data from this and other studies.

KHOO, KRONAUER, STROHL, AND SLUTSKY

(Table 2). Applying this scaling factor to the infant data of Waggener (36) and Riggato and Brady (32) produces close agreement with both experimental and theoretical results obtained on adults (Fig. 8). These are also consistent with the average cycle time of 18 s reported in cats (W - 3 kg), in which PB was induced under normoxic conditions by artificially raising controller gain (9).

Behavior of individual loop gain components. Since the overall system loop gain consists of three components-central COZ, peripheral COZ, and peripheral Oz- it is important to recognize the nature and extent of their individual contributions. Figure 9 shows the components of total loop gain at Q, = 180” for conditions at Sl and HI. In each case, only the peripheral components are shown since the central contribution is much smaller, being 0.038 in Sl and 0.018 in Hl. The long brain tissue washout time ( Tb = 80 s) is primarily responsible for the high attenuation of central controller gain. Data from

TABLE 2. Adult-infant scaling ratio (time) ~-~

Important Time Constants, s Infant Adult (3.2 kg wt) (65 kg wt) Ratio

Inverse heart rate 0.45 0.8 1.8 Lung washout ( FRC/QL) 9.1 19.5 2.1 Lung exchange time (FRC/VA) 13.6 28.0 2.1

Typical adult-infant ratio = 2. FRC, functional residual capacity; &L, pulmonary flow; VA, alveolar ventilation.

(overal 1 Loop Gain) i’ LG,, =0.37

LG= 1.60 ’ (Peripheral 0,)

J--- --- _----

+= 180”---‘-1- -4 7 ,& = 1.25

‘i (Peripheral

\co2) \

\ \

\ \ \

+ = 360”

$=OO

BCOND/T/O/v SI fS/eep uf Set7 Leve/j

. . I

/

/

/ +=270;\\,

\

FIG. 9. Components of loop gain at + = 180” for sleep condition at sea level (SI ) and awake condition at 14,000 ft (HI). Angles and relative lengths approximately to scale. In both cases, central component is not shown, being much smaller than peripheral CO2 and Oz components. Broken circle represents all points where LG = 1.


Lambertsen (18) suggest that the hypercapnea in Sl and the hypoxia in HI may increase cerebral blood flow (Qb) by 14 and 2176, respectively, and thus lower Tb accord- ingly. In an extreme case, halving Tb would double the central component, but its absolute contribution to overall LG would still be small. If we also assume corresponding decreases in 7p and ~c, total LG at Q, = 180" would actually be reduced by 23%. Thus, unless the central controller gain factor (Gc) is abnormally high, PB is mediated primarily by the peripheral controller. Cher- niack et al. (9) have found that peripheral chemodener- vation in fact eliminates PB in cats. This observation also implies that even if the central controller could somehow respond rapidly to arterial pH changes [as suggested by Borison et al. (2)] its contribution to overall LG would probably be unimportant. The assumption we made in deriving Eq. 12 is therefore not crucial for establishing the validity of our model.

In hypoxic conditions at high altitude (HI ), the CO2 loop gain component clearly predominates, whereas during sleep at sea level (SI ), 02 provides a slightly greater contribution to overall loop gain (Fig. 9). If the control of respiration is thought of as being shared between the 02 and CO2 subsystems, the above results imply counterin- tuitively that Pco2 is the major controlling variable in moderate to strong hypoxia, whereas in normoxia both POT and PCO~ are important. An accompanying feature is the phase difference between the peripheral CO2 and 02 components. The 02 component lags the CO2 component by about 30” at sea level and by 20” at 14,000 ft (awake). For a cycle time of 30 s at sea level and 20 s at 14,000 ft, these lags translate to about 2.5 and 1 s, respectively. In Fig. 4 (hypoxic conditions), a comparison of the Pa&., waveform with the Pa& waveform shows a clear difference of l-l.5 s (note vertical dashed lines in 4th and 5th panels). A consequence of the longer lung washout time for 02 relative to COZ, this phase lag explains the observations of Miller et al. (24) that the ventilatory response to an inspired bolus of CO2 in air occurs approximately 2 s faster than if an equivalent bolus of 15% 02 is inspired.

DISCUSSION

The present analysis leads us to view the respiratory control system as being relatively stable in the normal awake state. However, this situation can be drastically altered by a combination of mild hypoxia and hypercapnia (as occurs in sleep) or by transient moderate hypoxia (following Haldane’s hyperventilation experiment). Al- though Haldane associated the initiation of PB with hypocapnia, the latter plays a causative role only insofar \ as it produces the apnea which leads to hypoxia and subsequent destabilization of the system.

The critical variable in determining system stability in our model is loop gain. Table 3 summarizes the effects of changes in the major parameters on loop gain at + = 180" and on predicted cycle time. Abnormally high controller gain factors (Gp and Gc), which lead to very steep VE-

Pace, response curves, can destabilize the system, as in the case of patients with neurologic disorders. In other words, the controller tends to overcompensate for changes in state, making the system respond in a highly volatile fashion to small disturbances. The way in which

TABLE 3. Effect of changes in parameters on system stability

Parameter

Gp and G<l Circulatory delays (or

cardiac output) Vcoz and Voz PICO,

PIO,

Change

+25% Doubled (halved)

-28% From 0 to 15

Torr From 93 to 150

Torr

Approx Approx Change Change in in Loop Cycle Gain, % Time, s

+25 -0.5 +86 Doubled

+21 -0.5 -60 +3

-89 +9

GE) and Gc:, controller gain factors; Vc02 PICO, and PIO,, inspired CO:! and 02 pressures.

and Voz, lung volumes;

a prolonged circulatory delay promotes the onset of instability is quite different. In this case, there is a loss of effectiveness of control resulting from the fact that information about the current state of the system is received by the controller only at a much later time. Consequently, the controller tends to produce “the wrong response at the wrong time,” reinforcing rather than dampening the effects of disturbances on the system. Loop gain is also dependent on the operating state of the system, as defined by Pa co, and Pao,. Figure 10 illustrates the effects of hypoxia and hypercapnia on the normal, awake system breathing air at sea level. This dependence onPaco, and Pao, makes it possible for a sufficiently strong disturbance to “tip” a high-gain system into the unstable regime, even when the original loop gain is below unity.

Table 3 shows that the inhalation of CO2 leads to a reduction of loop gain, consistent with the observation that PB can be eliminated by breathing a high CO:! mixture (12). This follows from two, and possibly three, effects. First, it can be demonstrated that raising PIED), does not increase Pace, by the same amount in the steady state, thus (Pace, - PICO,) decreases. Secondly, the rise in Pace, would also lead to an increase in ?A and the consequent reduction of the hypoxic stimulus. Thirdly, the inhalation of CO2 may also produce an increase in cardiac output which would decrease circulation time. All these effects work in the same direction to lower loop gain and increase system stability (refer to Eqs. A7 and fW

The inhalation of 02 at high altitude lowers loop gain significantly and thus acts in a direction to abolish PB, as Douglas and Haldane (12) have observed. On the other hand, Bulow (5) found that 02 administration did not affect the incidence of PB during sleep at sea level. However, since Pao, was not measured in any of his subjects, it is not possible to draw firm conclusions from these results. In any case, 02 administration is much more effective in lowering loop gain in moderate to severe hypoxia (high altitude conditions) than in mild hypoxia (sleep at sea level) by virtue of the nonlinear 02 dissociation curve for blood. Furthermore, raising PIO, also increases the cycle time of an oscillation and therefore lengthens the period of the associated apneas (26). The latter would act to oppose the reduction in loop gain. Such competing factors probably account for the observation (13) that, although 02 inhalation does not elimi-


& s 15- - -G-

pOo2 =60

2 IO-

$ 04 305 ,’

00 I I I 1 I I 1 I 1

36 38 40 42 44 46 48 50 52 54 pace (torr)

2

FIG. 10. Dependence of loop gain at + = 180" on Face, and Pao,, calculated for the awake normal breathing room air (PI~:~,, = 0, pro., = 150 Torr). Combined effects of hypoxia and hypercapnea can raise loop gain dramatically.

nate CSR in patients with cardiovascular disease, t,he same data shows a weakening in strength of the PB and a prolongation of the corresponding apneas.

We have shown that, in the absence of abnormally high central controller sensitivity or abnormally long circulatory delays, the stability of the respiratory system is determined predominantly by peripheral control. All PB phenomena in normals may therefore be considered as being mediated by the peripheral chemoreceptors. Since the controller in Milhorn’s model responded only to tissue Pco~, it is similar to our central controller which responds to brain tissue Pco~. This undoubtedly explains why the controller gain had to be increased to 13 times normal or the lung-to-controller delay increased to 3.5 min before PB could be induced. On the other extreme, Longobardo et al. (21) attributed the total ventilatory response to what was essentially the peripheral controller. Had it not been for the inclusion of the entire volume of arterial blood (2.4 liters) as a component of lung CO:! and 02 storage, their model would have been highly variable in all normal circumstances. In fact, the effective lung volumes in Longobardo’s model were larger than ours by a factor greater than 4 for CO2 and 2 for 02. This accounts for the greater stability inherent in their model, as exemplified by the need for a large ventilatory disturbance (40 s by hyperventilation followed by 45 s of apnea) to initiate PB even at 14,000 ft. A further consequence is the longer cycle time (28 s) they obtained for conditions at 14,000 ft, even with a shorter mean lung-to-controller delay of 6 seconds. This contrasts with our prediction of 20 s, which is consistent with observed values at that altitude.

Contrary to the ideas of Cherniack and Longobardo , (6), we have found that loop gain, and therefore system stability, are virtually independent of tissue CO2 and 02 stores. Thus, whether one constructs a complicated model with multicompartmental body stores or a simple model without a body tissues compartment, similar predictions for cycle time and loop gain would be obtained. The lung storage volumes for CO2 (Vco,) and 02 (Vo,) provide the primary source of “damping” for the respiratory control system.

Our analysis has predicted that PB in normals during sleep occurs with cycle times on the order of 30 s agreeing

closely with the observations of Lugaresi et al. (22) and Specht and Fruhmann (35). On the other hand, Bulow (5) in the same study as that mentioned earlier, found the cycles times of his subjects to be between 40 and 60 s. Webb (38) reported episodes of CSR in sleeping adults that had cycle times of between 1 and 1.5 min. The apparent conflict between these sets of observations may be resolved when formed by Bulow

one considers on some of his

the experiments per- subjects to determine

the time taken for changes in inspired CO2 content to be first detected as subsequent changes in ventilation. He found values ranging from 21 to 27 s. Allowing for mixing in the heart and vasculature, this would represent a total lag (consisting of the mean lung-to-carotid delay and a lag due to lung washout) of 25-30 s. Thus oscillations resulting from high-gain instability may be predicted to have cycle times of 50-60 s, which fall within the range of the observed values. The long cycle times of about 1 min reported by Bulow (5) and Webb (38) imply a mean lung-to-carotid delay of roughly 15 s, which is 67% higher than our average value for Tp. We can only speculate at this point as to whether sleep lowers cardiac output markedly or a major regional redistribution of blood flow occurs.

Although there is insufficient data to confirm or refute the suggestion, some of the longer and more variable periodicities noted by Bulow and Webb may have been associated with upper airway obstruction. It is quite possible that periods of nonobstructive apnea associated with PB of strong magnitude to upper airway obstruction,

may and

predispose the system thus set the stage for

the initiation of recurring obstructive apneas (7). The cycle times of this kind of PB would be significantly longer than those our model has predicted. Since the prolonged apneas can lead to serious pathophysiological consequences, the interaction between an obstructive component and episodes of PB forms an important issue that has yet to be addressed in detail.

The way in which we have modeled the controller function during sleep may be extended to describe the paradoxical ventilatory response CO, in steady-state hypoxia (33).

of preterm infants to If we assume that only

the central controller is depressed by hypoxia, loop gain would increase in much the same way we have shown for sleep and thus predispose the infant to periodic breathing. The view that peripheral controller gain is not depressed by hypoxia is supported by the results of Alber- sheim et al. (1). Using the appropriate scaling laws, we have shown that the mechanisms that initiate PB in infants are probably the same as those that induce PB in adult humans.

In our simplistic characterization of sleep, we have ignored the important issues about the various stages of sleep and the transition to wakefulness. We can only speculate at present that the cycling between wakefulness and deep sleep may bring about a corresponding cycling of loop gain between low and high values. The cyclic withdrawal and return of the wakefulness stimulus may therefore occur concomitantly with the alternation of periodic lipson (27).

with regular breathing, as suggested by Phil-

657 A MODEL OF PERIODIC BREATHING IN HUMANS

APPENDIX I

Glossary of Symbols

Caco,, Cao2

cvco,, cvo,

GP, Gc

Ir, Ic

Kco~, Ko2

Pace,, Pa0,

p&o,, PEo,

PIcoz, PIO,

fo2

& . & s:,

T

L T2

TE1 TE2

Tb

Pb co,, Pbo,

71, 72

w, rc

Tr, Tc

re

MRco,, MRo,

Vb vco2, vo2

LG I&>,, LG,, LG<

arterial blood CO2 and 02 concentrations, ml of gas/ml of whole blood mixed venous blood CO2 and 02 concentrations, ml of gas/ml of whole blood gain factors associated with the peripheral and central controllers, respectively apneic thresholds for the peripheral and central responses to COP, respectively, Torr respective slopes of the CO2 and 02 dissociation curves, Torr-’ arterial blood CO2 and 02 partial pressures, Torr expired CO2 and 02 partial pressures, Torr

CO2 and 02 partial pressures of inspired air, Torr

e -O.MPa(~,

cardiac output, l/s cerebral blood flow, l/s arterial 02 saturation, oscillation cycle time, s lung washout time constants for CO2 and 02, respectively, s (~/TT, + l/T1 + QA/VCOy)-'

(1/Tr, + 1/T, + \~A/VOJ ’ washout time constant for brain compartment (=Vb/Qb), s partial pressures of brain compartment, Torr

mixing time constants in heart and vasculature, S

absolute delays (time to first appearance) from lungs to carotid body, s mean circulation times for lungs to carotid body and brain, respectively, s absolute lung-to-ear delay, s metabolic production rate for CO? and consumption rate for 02, respectively, l/s volume of brain tissue compartment, liters lung storage volumes for CO:! and 02, respectively, liters body tissue storage volumes for CO, and 02, respectively, liters Wo,/Q

W,,/Q alveolar ventilation, l/s minute ventilation, l/s

dead space ventilation, l/s dead space volume, liters total phase lag of system, radians or degrees phase lag contributions of peripheral COZ, peripheral 02, and central CO? components, respectively total loop gain loop gain contributions of peripheral CO?, peripheral 02, and central CO2 components, respectively complex angular frequency, rad/s frequency, rad/s over any variable represents small time variant perturbation in that variable

APPENDIX II

Derivation of Expressions for Loop Gain and Total Phase Lag

The system is assumed to be operating under steady-state conditions . with alveolar ventilation (GA) [and corresponding driving ventilation ( VE)] and arterial blood gasJensions (Pace, and ]Tjao, ). A small time- varying perturbatizn in VA [\j~( t)] is imposed. Corresponding pertur- bations in PacoT [Pa,,&)] and Pao [Pao2 ( t)] are produced, evoking the controller response VE( t). If the magnitudes of these fluctuations are small, the nonlinear system will behave linearly over the range of its perturbed states.

Applying the perturbation analysis Jo Eqs. 1-7 ignoring second- order terms [for instance, paco,( t ) and VA( t )], and taking the Laplace Transforms of the resulting expressions, we obtain

h0,(s, y=

(Pk0, - Pac0J (1 + TTIS)

VA(S) vco2 l

(Al) TT& + l/e,)

Pao,(s) (PIo~ - Pa0,) -= (1 + TT$)

.

CA(S) W')

Voa TT& + 1/m2)

where

TT, = (ho&)

Tr, = Wo,/Q)

T1 = (Vco2/863Kco2@ T2 = (Vo4863Ko4)

TE* = (~/TT, + l/T1 + \~A/VCO$’ TE2= (~/TT~ + l/T2 + i7A/VO2)-'

Eqs. Al and A2 represent the CO2 and 02 transfer functions for the controlled system. It is interesting to note that TEE and TEE represent the effective overall washout times for CO;! and 02, respectively, in the controlled system. Each comprises a component reflecting washout in the body tissues compartment, lung washout by perfusion, and the lung exchange time.

We employ the expression derived by Lange et al. (19) for the transfer function that describes the effects of circulatory mixing and delay associated with the passage of arterial blood from lungs to chemoreceptors

Pa&),(s) Pai’,, e - 7 . , . \

.-.z-= Pa,0,(s) Pa&) (1 + rls)(l + 72s)

(A3)

where the superscript or subscript x represents P for the lung-to-carotid transfer function and C-for the lung-to-brain transfer function.

A similar perturbation analysis applied to the controller responses

. V,,(s) = -0.05Gilfo2 (Pace, _ - I,@a&( s) + GI>fo#a&,( s) (A41

where fo2 = e -().():ih)i

- . Vds) = * f&0,(s) (A5)

where Tb = Vb/Qb, and

$E(S) = T&G) + i&.(s) L46)

A reasonable assumption is that since the fluctuatio_ns GE and ?A are small, VD would remain constant at a given value of VE. Thus, from E]qs. A&-A& an expression for the overall loop transfer function [i.e., VE(S)/VA( s)] can be found. This would consist of three vector components-the peripheral loop CO? and 02 transfer functions, and the central loop CO2 transfer function for a sinusoidal perturbation of cycle time T (or frequency ti = h/T). The loop gain and phase lag contributions from each of these components are

for peripheral CO2

LG,), = Gt,fo:! (Pace - PI(-~+) 2

VCOI, - (A7)

1 + (cJTT,)’ .

+, = UT: + G + tan-‘(oT&) + tan-‘(UT,) L

648) + tan-’ (tin) - tan-*(cc,TTJ (in radians)


for peripheral 02 -

LGp, = O.O5Gi>foz (Pa(:o, - I~~)(Pr0, - Pa0,)

vo2 1 + (WTTJ 2

. [1 + (&%#][l + (cJ#][l + ((37#]

LW

7l h, = c37p + -

2 + tan(tiTE2) + tan’(C37,)

(AlO) + tan’(c372) - tan- * (~3 TT,) (in radians)

for central CO2 -

LGc = Gc (Pwo, - PICO,)

vco2

. ’ [l + (dt%)2][l + (tiTF,#](l + (c3#][1 + (LJ#]

(All)

77 k = WI-(: + -

2 + tan-‘(cJTb) + tczn?(wTF,~) + tan’(w71)

(A12) + tan- ’ (~372) - tan ’ (~3 T7$ (in radians)

REFERENCES

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2. BORISON, H. L., S. F. GONSAIXES, S. P. MONTGOMERY, AND L. E. MCCARTHY. Dynamics of respiratory VT response to isocapnic pHa forcing in chemodenervated cats. J. AppZ. Physiol.: Respirat. En- viron. Exercise Physiol. 45: 502-511, 1978.

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The overall system loop gain (LG) and phase lag (+) are obtained by vectorial addition of the three component functions

LG = [LGk,,cos @is1 + LGl,,~os +, + LGccos 0~)~ (A13)

+ (LGp,sin +, +LGp,sin +p, + LGcsin QC:)“]“”

Cp = tan’ LG+sin +a, + LGp,sin +p, + LGcsin #x:

LGr+os +, + LGI-‘,cos +r, + LGccos &: (A14)

Note that LG and + are functions of the perturbation cycle time T (or frequency ~3). Therefore, a self-sustained oscillation of cycle time To (or frequency ~30) can occur if LG (tic)) L 1 and +(a(,) = 7~ radians.

This work was supported in part by the National Heart, Lung, and Blood Institute Grant HL-16325-05.

K. P. Strohl was supported by a National Institutes of Health Institutional Award, by HL-25830 and GM-07560.

A. S. Slutsky was supported in part by the Medical Research Council (Canada) and the Veterans Administration.

Received 19 June 1981; accepted in final form 19 March 1982.

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helium. In: DriZZ’s Pharmacology of Medicine, edited by J. R. DiPalma, New York: McGraw, 1965.

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