Sleep-induced periodic breathing and apnea: a theoretical study MICHAEL C. K. KHOO, ALLAN GOTTSCHALK, AND ALLAN Biomedical Engineering Department, University of Southern California, California 90089; and Departments of Anesthesia and Medicine, University of Pennsylvania, Philadelphia, Pennsylvania 19104 KHOO,MICHAELC. K., ALLANGOTTSCHALK,ANDALLAN I. PACK. Sleep-induced periodic breathing and apnea: a theoretical study. J. Appl. Physiol. 70(5): 2014-2024, 1991.-To elucidate the mechanisms that lead to sleep-disordered breathing, we have developed a mathematical model that allows for dynamic interactions among the chemical control of respiration, changes in sleep-waking state, and changes in upper airway patency. The increase in steady-state arterial PCO, accompanying sleep is shown to be inversely related to the ventilatory response to CO,. Chemical control of respiration becomes less stable during the light stage of sleep, despite a reduction in chemoresponsiveness, due to a concomitant increase in “plant gain” (i.e., responsiveness of blood gases to ventilatory changes). The withdrawal of the “wakefulness drive” during sleep onset represents a strong perturbation to respiratory control: higher magnitudes and rates of withdrawal of this drive favor instability. These results may account for the higher incidence of periodic breathing observed during light sleep and sleep onset. Periodic ventilation can also result from repetitive alternations between sleep onset and arousal. The potential for instability is further compounded if the possibility of upper airway occlusion is also included. In systems with high controller gains, instability is mediated primarily through chemoreflex overcompensation. However, in systems with depressed chemoresponsiveness, rapid sleep onset and large blood gas fluctuations trigger repetitive episodes of arousal and hyperpnea alternating with apneas that may or may not be obstructive. Between these extremes, more complex patterns can arise from the interaction between chemoreflex-mediated oscillations of shorter-cycle-duration (a36 s) and longer-wavelength (=6080 s) state-driven oscillations. model; respiratory control; sleep state; arousal
stability;
ventilatory
oscillations;
SEVERAL THEORETICAL and experimental studies have demonstrated that instability in the chemoreflex control of ventilation, resulting from prolonged circulatory delays or enhanced chemoresponsiveness, can lead to frank periodic breathing (PB) and apnea (12). Although PB can be induced in awake subjects through hypoxic gas administration or during altitude acclimatization, the tendency to develop PB is enhanced during sleep (5,25). It is easy to understand why hypoxia, which increases both. hypoxic and hypercapnic chemosensitivities, would promote PB. However, the role of sleep in decreasing respiratory control stability is not so apparent. Studies employing special signal-processing techniques have demonstrated the existence of ventilatory oscillations in 2014
0161-7567/91
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Copyright
I. PACK Los Angeles,
awake subjects (8, 22). It is therefore possible that the periodicities in ventilation, which become more apparent in sleep, are simply masked by voluntary influences and nonspecific environmental stimulation of the respiratory centers during wakefulness (3). This would lead one to expect a progressively higher incidence of PB with increasing depth of sleep. However, in fact, several studies have noted that respiratory patterns are generally regular during slow-wave sleep and that PB occurs primarily during the light stages of sleep or at sleep onset (1,5,22). The ventilatory responses to hypercapnia and hypoxia are generally decreased during sleep (1,5,9,25). Because cardiac output does not decrease much with sleep, it is unlikely that there will be substantial prolongation of the lung-to-chemoreceptor delay (25). On the basis of these changes, one would expect respiratory control during all sleep stages to be highly stable, contrary to what is observed in light sleep. A further discrepancy arises from the finding that sleep-related PB is more prevalent in the elderly (22). This again appears to be at variance with the instability hypothesis, because hypercapnic and hypoxic sensitivities are known to decrease with aging (24). Passive hyperventilation, which leads to loss of chemical drive, has been shown to produce apnea in sleeping subjects once arterial PCO, (Pa& is reduced 3-6 Torr below initial levels; the same degree of hyperventilation rarely leads to apnea in awake subjects (28). This and other pieces of evidence suggest that the state of wakefulness, even in the absence of any voluntary influences on breathing, is characterized by tonic excitation of the respiratory centers (5, 25). This extraneural input has been termed the “wakefulness stimulus” (1). Support for the existence of this stimulus, which presumably is related to activity in the brain stem reticular formation, comes from neurophysiological studies (20). Variations in the depth of sleep or level of “vigilance” would be reflected as fluctuations in this extraneous drive to breathe. In this way, primary fluctuations in state could drive secondary oscillations in ventilation and blood gases (1, 25). A recent study has found oscillations in ventilation and those in frequency content of the electroencephalogram to be essentially synchronous (21). Furthermore, arousal from sleep is generally accompanied by transient hyperpnea, which apparently marks the rapid restoration of the wakefulness stimulus (25). This hyperpnea will tend to lead instability in the chemical feedback system. Thus, oscillations in ventilation in-
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Society
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duced by fluctuations in state and those induced by instability of the chemical feedback system may interact and indeed might be naturally reinforcing. The effects of these oscillatory processes may be further magnified by their relative effects on the action of upper airway muscles and on the diaphragm (27). Thus, oscillations in state may lead to repetitive obstructive apneas. Because such obstruction can produce large disturbances in blood gas tensions, again chemically induced instability may be superimposed (17). It is therefore likely that the pathogenesis of PB and repetitive apnea involves the interaction of several factors, the individual contributions of which may vary considerably depending on the subject. In preliminary studies (11,21), using mathematical models, we have explored the implications of coupling between sleep-waking state and the chemoreflex control of respiration. In this theoretical study, the previous results are consolidated into a more comprehensive model. Computer simulations with the model enable us to systematically investigate the interactions between and relative importance of chemical control, sleep-waking state changes, and upper airway patency to the overall stability of breathing during sleep. In so doing, our basic premise (14) as to how the respiratory system can produce sustained periodicities is expanded in scope to encompass a considerably larger variety of physiological and pathophysiological conditions.
BREATHING
D p = GJ
102.4 - Sao,)[Pa,,,
- Ip]
(0
where D, represents peripheral drive, Gp the peripheral gain factor, and SaoZ arterial 0, saturation. The function enclosed in brackets is defined such that [x] = 0 if x 5 0 and [x] = x if x > 0. Thus, I, represents that value of Pace, below which peripheral chemoreflex drive becomes zero. In eq. 1, we assume that a residual peripheral drive exists in hyperoxia (i.e., when Sao, = 100%) (4). By use of nominal parameter values (Table 1) in the above expression, the contribution of peripheral chemosensitivity to total gain is 25% in normoxia and 70% in hypoxia (S a% = 75%). Central drive (D,) is related linearly to Pcop of the brain compartment (Pb,,J Dc=
w%o,
- ICI
(2)
where G, is the central gain and I, is threshold value for Pb co2 below which central chemoreflex drive is zero. Pbco, is, in turn, related to PacoZ in the following manner (26) = Qb(Pac,,
DESCRIPTION
To better distinguish the effects of sleep-waking state changes from those mediated solely by chemical control and those of upper airway obstruction, three models of ascending complexity are considered: model 1, which incorporates the common features of all three models, including chemoreflex control and the effects of wakefulness-to-sleep transitions (without arousal); model 2, which incorporates all features of model 1, including the capability of arousal if chemical drive exceeds a certain threshold during sleep; and model 3, in which model 2 is modified to include complete upper airway obstruction once ventilatory drive falls below a minimum level. The following descriptions refer to model 1, unless otherwise stated. The “plant. ” The lungs are modeled as a homogeneous alveolar chamber connected in series to a dead space. During inspiration and expiration, the alveolar chamber increases and decreases in volume, but the dead space volume remains fixed. The gas exchange equations are presented in detail in Khoo and Kronauer (13). Pulmonary end-capillary and alveolar PCO, and PO, are assumed to be completely equilibrated. The blood-gas dissociation relationships of Grodins et al. (lo), which include the Bohr-Haldane effect, are modified to establish reasonable consistency with the tables published by Olszowka et al. (18). A convective delay and two vascular mixing transfer functions are placed in series to simulate the passage of blood from the lungs to the chemoreceptors (14); these provide the transformation relationships between alveolar gas tensions (PACES and PAN,) and arterial blood gas tensions (Pa,oz and Pa,,). The me-
2015
APNEA
tabolic exchanges for CO, and 0, in the body tissues are modeled as single-compartment processes that determine the levels of the mixed venous blood gases (14). CentraE and peripheral chemoreflexes. Peripheral chemoflex drive is assumed to be dependent on the multiplicative interaction of hypercapnia and hypoxia
dPbcozldt MODEL
AND
- Pb,,,)K,,,/60Kb,,, + (MRbco,
- 0.0011)/60Kb,,,
(3)
where MRbco, is the brain metabolic CO, production rate, Qb is cerebral blood flow, and Kco, and Kco, are the dissociation slopes for the brain tissue and blood, respectively. The dependence of Qb on Paooa and Paon is modeled after empirical relationships obtained in several studies (10, 15). ' Integration of driues. Total ventilatory drive in wakefulness (Dawake,l/min) is given by D awake = [D, + D, - h] + h
(4)
where h = 14.4 - 0.0138 PaoZ. This represents a simple summation of central and peripheral drives with the inclusion of an idealized hypoxia-dependent “dog leg” in the hypocapnic region during wakefulness (4). Dawake implicitly includes a wakefulness stimulus component that progressively declines with increasing depth of sleep. Mathematically, we have found it convenient to model this hypothesis by subtracting the nonchemical component (S) from Dawake.Sleep is also accompanied by a decrease in responsiveness of ventilatory drive to chemical stimuli. Thus D sleep
=
G,[
Da=ke
-
S]
(5)
In Eq. 5 G, (11) modulates the slope of the CO, response line while S (>O during sleep) effectively shifts the CO, response line to the right. Generation of airflow and breathing pattern. At the end of the preceding breath, a new breath is generated if total ventilatory drive (D) is greater than a minimal threshold level (Dmin)
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2016
SLEEP-INDUCED
D = DaWake if
PERIODIC
DaWake> Dmin
(6 a )
or D
=
DSleeP
if
DSleeP
>
BREATHING
AND
sleep, we assume a simple linear dependence of E on time (with t = T, s marking the start of the transition) E(t)
Dmin
APNEA
= (t - TO)/7
for
t - TO 5 7
uw
7
Wb)
(W
= 1 for
Because the model simulates tidal breathing, D is represented in terms of mean inspiratory flow. Tidal volume (VT, in liters) is thus determined as
t - TO >
In the above expression, 7 represents the time required to first attain the stage of sleep in question, assuming that no arousal occurs during the transition. VT = D.T1/6O (7) Arousal mechanism. In model 2, if total chemical drive (D, + DP) exceeds the threshold level (Da,), an arousal is where TI is inspiratory duration (in s), assumed constant. A constant expiratory duration (TE) is also as- triggered and the wakefulness stimulus is fully restored (i.e., E reverts to zero) on the next breath. If this occurs sumed, so that minute ventilation (VE, in l/min) is given during a wakefulness-to-sleep (WS) transition, the bY arousal is assumed to be transient and is followed by a VE = VT.GO/(TI + TE) (8) new WS transition of duration 7 in subsequent breaths. However, if total ventilatory drive is less than Dmin at the Consistent with recent data (6), arousal has been asend of a breath, no new breath will be generated until the sumed to be triggered by total chemical drive irrespective of the kind of stimulus. For purposes of comparison with deterioration of blood gases raises drive to levels higher empirical studies involving arousal, the value of D, used than Dmin. in the model (assuming nominal chemoreflex gains) is Effect of changes in sleep-waking state. Because the equivalent to a hypercapnic threshold of ~53 Torr durchanges accompanying rapid-eye-movement (REM) sleep are highly variable and complicated (29, we have ing hyperoxia or a hypoxic threshold of 75% in Sa,, during normocapnia (cf. Ref. 25). limited the scope of this study to non-REM sleep. SleepUpper airway obstruction. In model 3, we assume that waking state is quantified in terms of the index E. E is defined such that the fully awake state is represented by when Dsleep is less than Dmi, at the end of a breath, paE = 0, stage 1 sleep by 0.25 5 E < 0.5, stage 2 by 0.5 5 E < tency of the upper airway is lost and any subsequent inspiratory efforts would result in an obstructive apnea. 0.85, stage 3 by 0.85 5 E < 1, and stage 4 by E = 1. These is terminated only when ventilatory values have been chosen so that the reduction in CO, The obstruction drive builds up to and exceeds Da,. This assumption is response slope with the different sleep stages are compatcompatible with the work of Parisi et al. (23), who ible with the observations of Bulow (1) showed that restoration of upper airway electromyogram G w = 1 - 0.4E (9) is correlated to an increase in phrenic activity above some threshold level. When apnea is not present, model 3 For instance, in slow-wave (or stage 4) sleep, the effective CO, response slope is 60% as large as that during wake- behaves in exactly the same way as modeZ 2; i.e., arousal can occur during any sleep stage if ventilatory drive exfulness, when G, is unity. ceeds. Consistent with Bulow’s findings, we have also asComputer simulation. The finite-difference forms for sumed that the withdrawal of the wakefulness stimulus or, equivalently, the shift of the CO, response line is the above equations are solved numerically on a MicroVAX (Digital Equipment, Bedford, MA) computer using completed by the time stage 1 sleep is attained a time step of 0.025 s. Starting from initial values, the S = SOaE/0.25 for 0 5 E < 0.25 uw model is allowed to attain a steady state (i.e., breaths = S, for 0.25 5 E 5 1 (10~) achieve constant VT) before any changes are imposed. The values of the plant parameters used in all simulawhere S, is the full magnitude of the wakefulness stimtions are similar to those employed in a previous model ulus. (20): cardiac output = 6 I/min, lung-to-chemoreceptor Effects of sleep on plant parameters. We have assumed (peripheral and central) delay = 8 s (or 2 breaths), vascuthat cardiac output, the lung-to-chemoreceptor translar mixing time constants = 1 and 2 s (2 cascaded firstport lags, and vascular mixing time constants do not order low-pass filters), equivalent lung volume for CO, = change from wakefulness to sleep. On the other hand, the 3 liters, FRC = 2.5 liters, dead space volume = 0.15 liter, metabolic CO, production rate (MR,,,) and 0, con- body tissue CO, stores = 15 liters, body tissue 0, stores = sumption rate (MR,,) are assumed to decrease with in6 liters. The nominal values of the other parameters used creasing sleep state, so that in stage 4 sleep they would be in the simulations are given in Table 1. 15% lower than during wakefulness Mb,,
= 0.21(0.375G,
+ 0.625)
MR,,
= 0.26(0.3756,
+
0.625)
(114 um
In the above expressions, we have assumed that in wakefulness MRcOz and MR,, take values of 0.21 and 0.26 l/min STPD, respectively. Wakefulness-to-sleep transitions. To simulate the time course of the transition from wakefulness to any stage of
RESULTS Steady-state responses. If nominal parameter values are assumed, the normoxic steady-state ventilatory responses of the model to CO, in different sleep-waking states are shown in Fig. 1. The magnitude of the wakefulness drive (S,) required to generate these results is 18 l/min; this and other values of drive are defined in terms of mean inspiratory flow. With the onset of sleep, mean
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BREATHING
AND
2017
APNEA
1. Parameter values used in simulations
TABLE
Parameter
Abbreviation
Peripheral gain factor Central gain Peripheral apneic threshold Central apneic threshold Blood Cp, dissociation slope Brain tissue CO2 dissociation slope Cerebral blood flow Brain CO2 production rate Inspiratory duration Expiratory duration Minimum drive for breath generation Minimum drive for ending obstruction Magnitude of wakefulness drive WS transition time Arousal threshold WS gain factor All values wakefulness;
Value
0.375 1 min-l Torr-’ % (nominal) 4.5 1. min-l Torr-’ (nominal) 38 Torr (nominal) 45 Torr (nominal) 0.0057 1 CO,. 1 blood-’ Torr-1 0.0036 1 kg brain tissue-’ Torr-’ 0.5 1. min-l kg brain tissue-’ (Paoo, 0.031 1 min-l kg brain tissue-’ 1.5 s 2.5 s 3 l/min 45 Vmin (nominal) 18 Vmin (nominal) 20-60 s (W to Sl) 45 Vmin (nominal) 0.6 (NREM-4) to 1 (W) l
l
l
l
l
l
l
= 40 Torr)
l
MRbco,
l
TI TE D min D obst so 7
Qu Gv
of drive reflect mean inspiratory airflow rather Sl, sleep stage 1; NREM-4, non-rapid-eye-movement
than
minute ventilation. (stage 4) sleep.
Paoo,,
l
arterial
Pco~;
WS,
wakefulness-to-sleep;
W,
25
0
35
40
45 ARTERIAL
50
Pco2
(mm
55
t3 2
0.0
1 0.0
a
AWAKE VENTUATORY
fig)
FIG. 1. Steady-state ventilatory responses to CO, during wakefulness (W) and in the different sleep states (Sl-S4) generated by the model. l , Steady-state operating minute ventilation and arterial PCO~ at each of these states.
l
L
2.0 RESPONSE
I
1
I
3.0
4.0
T O CO2 (L/min/mm
Hg)
FIG. 2. Difference between steady-state arterial PCO, (Pace,) during stage 2 sleep and steady-state Paoo, during wakefulness is plotted against corresponding ventilatory response to CO, for different values of S, (magnitude of wakefulness stimulus). l as defined in Fig. 1; i.e., for nominal magnitude of state change (S, = 18) and normal CO, response (2 25 14 min-’ Torr-‘), increase in Pko, is 3.6 Torr. l
Pa co is raised by 3-4 Torr and mean VE is decreased roug fi ly 20%, consistent with what is generally observed during sleep (5, 25). In Fig. 1, the awake ventilatory response to hypercapnia is 2.25 1 min-’ 4Torr-‘, with the peripheral chemoreflex contributing 25% of total ventilatory gain. In Fig. 2, the increase in PacOz associated with the change in state from wakefulness to sleep stage 2 is plotted against the awake ventilatory response to hypercapnia. For any fixed value of S,, the change in PacOz with sleep decreases with increasing CO, response in a hyperbolic manner. This behavior agrees closely with data obtained by Gothe et al. (9, Fig. 4). For a given CO, response, the magnitude of the Pa,,, change increases with increasing values of S, (vertical dashed line in Fig. 2). It should be noted that at the higher values of CO, response (>3.5 1. mine1 . Torr-l) the model exhibits sustained oscillations. The values of steady-state Paoq, change under such conditions are obtained by stabilizing the model with decreased convective lung-to-chemoreceptor delays. Reponse to transient airflow obstruction for model 1. To test the stability of the model in different sleep-waking
I 1 .o
l
states, the following intervention is applied. In wakefulness and each of the sleep stages, the model is first allowed to attain a steady state. Subsequently, at the end of a breath, airflow is obstructed for the next three consecutive breaths (i.e., 12 s). The ensuing response in breathing pattern is shown in Fig. 3 for wakefulness (A), stages 1 and 2 sleep (B), and stages 3 and 4 sleep (C). The subsequent fluctuations in VE are rapidly damped out in wakefulness and in the deeper stages of sleep. However, the response is clearly more oscillatory in light sleep, indicating a more unstable system, despite a lo-20% decrease in chemoresponsiveness. The reason for this somewhat surprising result is fully elucidated in the DISCUSSION. The present procedure emulates the protocol adopted in a recent experimental study (7) in which the authors found a significant correlation betyeen hypercapnic sensitivity and the initial increase in VE after the release of inspiratory airway obstruction. Although the results are not presented here, our model predicts a strong dependence of the postobstruction ventilatory overshoot on controller gain. VW transitions for model 1. The transition from one
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2018
SLEEP-INDUCED
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gLL 0.6
3E
AND
0.4
!& 3 0.2 P 0.0 0 TIME (seconds) 6
1 .o
APNEA
medium values of controller gain, slow WS transitions produce negligible dynamic effects (not shown in Fig. 4). On the other hand, in the presence of high controller gain, even a slow WS transition can spawn a growing oscillation that leads rapidly to a sustained episode of Cheyne-Stokes breathing, with apnea occuring at the hypopneic phase of each cycle (Fig. 4C). In all three cases, the cycle time of the sleep-induced oscillation is ~36 s. Performing the simulations over a large variety of WS transition times (7) and controller gains, we have found that the stability properties of model 1 are determined primarily by chemoresponsiveness and S, but are independent of 7 over the range investigated. The stability
A
l.Or
BREATHING
A Wq N u&i
z 08. 3
I$
‘1
\
s.
\ -----
- ----
- --------------
----qo()$$ U
& 0.6 zi E 0.4
--
rw 93 0.2
N
8 a” 30
0.0 50
100 TIME (seconds)
150
200
-
-
-
-
-
TIME (sets)
C
1 .o
W
r N E m&z
f
-
S
c \ -------------
-----------------a
loos 8 - 90 m”
0.6
8 a” 3050
100
150
-1
.
200
100
TIME (seconds)
200
250
=8 0 ,’
300
INAE (sets)
FIG. 3. Breathing patterns after 12 s of airway occlusion (applied at the end of the 1st breath) in wakefulness (A), light sleep (Sl-S2; B), and deep sleep (S3-S4; C). FRC, functional residual capacity.
sleep-waking state to another represents a dynamic disturbance that the chemoreflexes aim to correct. The ensuing response elicits fluctuations that may persist long after the transition has been completed. A faster transition produces a larger effect. The effect of the transition from wakefulness to stage 1 sleep (WS) on model 1 is shown in Fig. 4 for different values of CO, response. In Fig. 4A, a rapid transition (7 = 20 s) leads easily-to apnea, which raises Pacoz and lowers Sa,,. A very small-amplitude oscillation in VE and blood gases is set up, but this is damped out after a few cycles. With normal values of controller gain, a rapid WS transition produces larger oscillatory changes in VE and blood gases, which may-be classified as PB with periods of hypopnea but no apnea (Fig. 4B). However, the PB occurs only transiently for a few minutes and is eventually damped out. For low and
150
T
W N 6 u&i
f
.4
C bL \
S
1 \
x
--------------
--L-m------
100
150
200
-1
250
TIME (asca)
of wakefulness-to-sleep transition (dashed line) on conditions of low ventilatory response to COz min-’ . Torr-‘) and fast transition rate (7 = 20 s; A), moderate (2.25 1 min-’ Torr-‘) and fast transition rate (7 = 20 s; B), and gain (4 1. min-’ l Torr-‘) and slow transition rate (7 = 60 s; C). arterial O2 saturation; VT, tidal volume; S, sleep; W, wakefulness.
FIG.
model (0.51. gain high Sa,,,
4. Effect
1 under l
l
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SLEEP-INDUCED
PERIODIC
BREATHING
AND
2019
APNEA
from 10 to 20 s. Figure 6C illustrates a situation with high gain (4 1 min-l Torr-l) and slow WS transition (7 = 60 s). The slow transition provokes a small oscillation that is rapidly amplified by the high controller gain. The blood gas swings trigger an arousal at around t = 70 s. The hyperpnea accompanying arousal rapidly decreases chemical drive, and this, coupled with the subsequent onset of sleep, leads to hypopnea. The resulting buildup in chemical drive produces the next hyperpneic phase but is insufficient to trigger arousal. In this way, arousals occur ~80 s apart, but the hyperpneic phases recur every 40 s. Figure 6, B and C, demonstrate two forms of interl
z
iii
0’
05.
1
1
I
I
I
1
I
1.0
1.5
2.0
2.5
3.0
3.5
4.0
VENTIIATORY
RESPONSE
TO CO2
(L/min/mm
A sI\
Hg)
45
l
‘1 \
FIG. 5. Stability characteristics for model 1. Solid line separates conditions under which sustained oscillations will be produced after sleep onset (region above) from conditions under which model remains stable (region below). Dashed line separates cases with nonapneic oscillations (region below) from cases with repetitive apnea (region above).
diagram in Fig. 5 represents a compilation of all the simulations conducted with modeZ 1. The solid curve represents the minimum value of S, required to produce a sustained periodicity for any given value of CO, sensitivity. The region below the solid curve therefore represents the range of conditions in which the model is inherently stable, while in the region above, the model exhibits sustained oscillations. The dashed curve further separates the unstable region into oscillations with apnea (above curve) and periodicities without apnea (below curve). It should be noted that the dashed curve is coincident with the solid curve at all CO, sensitivities < 2.7 1. min-’ . Torr-l; i.e., sustained nonapneic oscillations are found only at gains higher than this level. A further point to note is that, for S, = 18 l/min (the nominal value), sleep onset will lead to sustained oscillations only when CO, sensitivity > 3.4 1 min-’ Torr? Stability properties of model 2. Figure 6 shows responses of the model to WS transition under circumstances in which an arousal mechanism is incorporated. When controller gain is very low (0.5 1. min-’ Torr-l), even rapid changes in state do not provoke an arousal; in fact, model 2 behaves no differently from model 1 in this case (Fig. 6A). When CO, sensitivity is moderate (2.25 1 min-l . Torr-l), only sufficiently rapid WS transition will lead to the propagation of a sustained oscillation. In Fig. 6B, 7 is assumed to be 20 s. The fast transition produces a subsequent large increase in Pacoz and fall in Sao,. As a consequence, chemical drive builds up to such high levels as to trigger an arousal at t = 35 s. The transient arousal leads to hyperpneic breaths, which quickly improve blood gases. As the stimulus for arousal is removed, sleep sets in again. The rapid reduction of chemical drive, coupled with the second onset of sleep, accelerates the withdrawal of total ventilatory drive and leads subsequently to central apnea. With apnea, blood gases rapidly deteriorate and the entire cycle of arousal and sleep onset is repeated. In this case, the fluctuations are large enough to perpetuate the periodicity. Cycle times are on the order of 45 s, and the apneas range in duration
l
vdi
w-
\,,,,,,,,-,,-,--,------------,,
‘OQ 8 90 t#
3 c
0 &100
150
200
250
300
TIME (sets)
3
50
E 540 cy 8
I
I ‘f c
l
k.s . . . . , . . . . , .
:J
Sa
100
, 1 - - - , - 150
200
250
. 0 9300
TIME (aecs)
l
l
h 501 -g
90 f#
E k
40
3 af
30 'F .5
0
50
100
. . . . . . . . . . (0 200 250 300
150
g
TIME (sets)
FIG. 6. Effects of wakefulness-to-sleep transition on model 2 under conditions of low gain (0.5 1. min-’ Torr-‘) and fast transition rate (7 = 20 s; A), moderate ventilatory response to CO, (2.25 1. min-’ . Torr-‘) and fast transition rate (T = 20 s; B), and high gain (4 1 min-’ Torr-‘) and slow transition rate (T = 60 s; C). In B and C, note that repetitive arousals (dashed line) coincide with start of hyperpneic breaths. l
l
l
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2020
SLEEP-INDUCED
PERIODIC
action between state-driven fluctuations and chemically mediated oscillations. In B, state oscillates in synchrony with chemoreflex drive. However, in C, the chemically mediated oscillation of shorter cycle duration rides on top of a state-driven oscillation of longer wavelength. The stability characteristics of model 2 are summarized in Fig. 7. In contrast to mode2 I, the stability properties of this model depend more on 7 in addition to chemoresponsiveness and S,. Thus, in Fig. 7, S, is plotted against 7, with low (0.5 1 min-’ Torr-l), moderate (2.25 1 min-’ . Tori?), and high (4 1 min-l Torr-l) levels of chemosensitivity represented by A, B, and C, respectively. As in Fig. 5, the solid curves represent the minimum magnitude of S, required to produce a sustained periodicity, whereas the dashed curves represent the minimum S, required to produce sustained apneic episodes. Two features are noteworthy. First, regardless of 7, the minimum S, required to generate sustained periodicities is generally lower than that in model 1 for the same CO, sensitivity. This implies that model 2 is less stable than model 1, particularly at the lower gains. For instance, for a CO, sensitivity of 0.5 1 min-’ .Torr-‘, a 35% increase in S, over its nominal value is sufficient to produce sustained PB with apnea in modeZ 2; in modeZ 1, an increase of 80% is required to achieve the same effect. Second, model 2 exhibits sensitivity to 7, particularly where sustained apneic oscillations are concerned. Thus, rapid WS transitions can be highly effective in promoting instability. Stability properties of model 3. Some samples of the variety of responses produced by model 3 to WS transitions are shown in Fig. 8. In Fig. SA, a low value (0.5 1 min-l Torr-‘) of CO, sensitivity is assumed. The fast WS transition (7 = 20 s) leads to rapid loss of chemical drive, which affects upper airway patency as well as ventilatory drive. Airflow obstruction occurs as inspiratory efforts are applied against a collapsible airway, leading to a prolonged apnea of -40 s. The long apnea produces a substantial degree of hypoxia and hypercapnia, and chemical drive builds up correspondingly to a level that triggers arousal (at t = 58 s). The rapid restoration of l
BREATHING
AND
APNEA
-50 3 E 540 cy
8 = 30
1-
l
l
l
4 zi
l
0 0
50
100
250
150
g
360
TIME (sets) B 1
WR
g --‘\ b,- ‘\_--------------7
-------- -------- >‘*E 8 -90 df
l
l
150
200
230
TIME (sets>
l
z
40
c d .’ z P 30 4 6 El fi 20 5 bi 2
A
-
B / I
/
I
0
/
150
200
250
300
TIME (sets)
l
/
/
/ j-
-
/0
10 -
0
1c -
/
VP fo,
100
l
/
/
’
50
l
/
-
0
FIG. 8. Effects of wakefulness-to-sleep transition on model 3 (with S, = 15 Vmin) under conditions of low ventilatory response to CO2 (0.5 1. min-l Torr-‘) and fast transition rate (r = 20 s; A), moderate gain (2.25 1. min-’ Torr-‘) and fast transition rate (7 = 20 s; B), and high gain (4 1. min-’ Torr-‘) and slow transition rate (60 s; C). In C, note interaction between state-driven oscillation and chemoreflex-mediated oscillation.
C
I
h ” 20
40
” 60
” 20
WAKEFULNESS-TO-SLEEP
40
” 60 TRANSITION
” 20
40
” 60
TIME (sea)
FIG. 7. Stability characteristics of model 2 with low ventilatory sponse to COz (0.5 1. min-‘~Torr-‘; A), moderate gain 1 min-’ . Torr-‘; B), and high gain (4 1. mine1 . Torr-‘; C). In each region above solid line encompasses all conditions that lead to tained oscillation, while dashed line separates nonapneic (below from apneic oscillations (above line). l
100
re(2.25 case, susline)
wakefulness drive leads to termination of the obstructive apnea. Large hyperpneic breaths follow, rapidly removing the chemical stimuli. At the same time, sleep sets in again and this, coupled with the loss of chemical drive, leads to the next obstructive apnea. In this way, a sustained periodicity is produced, with cycle and apnea durations of ~62 and 32 s, respectively. In Fig. 8Z3, CO, sensitivity is moderate (2.25 1. min-‘~Torr-l). A fast WS transition (7 = 20 s) leads to a transient period of oscillation (cycle time = 36 s), which is eventually damped out. However, ventilatory drive does not fall to sufficiently low values to produce apnea at any point. In comparing Fig. 8, A and B, with the corresponding cases for model 2
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SLEEP-INDUCED A
f I
0
I
B
a
’
(
20
40
60
’
WAKEFULNESS-TO-SLEEP
PERIODIC C
1
”
20
40
“‘I
60
J
20
TRANSITION
40
60
TIME (sets)
FIG. 9. Stability characteristics of model 3 with low ventilatory response to CO, (0.5 l*min-’ . Torr-‘; A), moderate gain (2.25 1. min-’ Torr-‘; B), and high gain (4 1 min-’ Torr-l; C). Solid lines distinguish stable region from conditions that lead to sustained oscillations; dashed lines distinguish stable region from conditions that lead to sustained oscillations; dashed lines distinguish nonapneic oscillations from oscillations with apnea. l
l
l
in Fig. 6, it is interesting to note that, with the inclusion of airflow obstruction, a lower controller gain seems to promote sleep-disordered breathing. Figure 8C represents a high-gain (4 1 mine1 . Torr-l) situation in which a slow WS transition (7 = 60 s) occurs. The result is similar to that shown in Fig. 6C in which arousals and termination of airflow obstruction coincide with every other hyperpneic phase. As before, a shorter-duration oscillation (-42 s) appears to ride on top of a slower oscillation (-82 s). However, the apneas produced here are longer and range from 10 to 15 s. The stability diagram for model 3 is presented in Fig. 9. Comparing the results with those in Fig. 7, it is clear that the stability boundaries for the high-gain case (C) are virtually unaltered. Although the moderate- gain case (B) is somewhat affected, it is the low-gain case (A) that shows the largest changes. For transition times
stability;
ventilatory
oscillations;
SEVERAL THEORETICAL and experimental studies have demonstrated that instability in the chemoreflex control of ventilation, resulting from prolonged circulatory delays or enhanced chemoresponsiveness, can lead to frank periodic breathing (PB) and apnea (12). Although PB can be induced in awake subjects through hypoxic gas administration or during altitude acclimatization, the tendency to develop PB is enhanced during sleep (5,25). It is easy to understand why hypoxia, which increases both. hypoxic and hypercapnic chemosensitivities, would promote PB. However, the role of sleep in decreasing respiratory control stability is not so apparent. Studies employing special signal-processing techniques have demonstrated the existence of ventilatory oscillations in 2014
0161-7567/91
$1.50
Copyright
I. PACK Los Angeles,
awake subjects (8, 22). It is therefore possible that the periodicities in ventilation, which become more apparent in sleep, are simply masked by voluntary influences and nonspecific environmental stimulation of the respiratory centers during wakefulness (3). This would lead one to expect a progressively higher incidence of PB with increasing depth of sleep. However, in fact, several studies have noted that respiratory patterns are generally regular during slow-wave sleep and that PB occurs primarily during the light stages of sleep or at sleep onset (1,5,22). The ventilatory responses to hypercapnia and hypoxia are generally decreased during sleep (1,5,9,25). Because cardiac output does not decrease much with sleep, it is unlikely that there will be substantial prolongation of the lung-to-chemoreceptor delay (25). On the basis of these changes, one would expect respiratory control during all sleep stages to be highly stable, contrary to what is observed in light sleep. A further discrepancy arises from the finding that sleep-related PB is more prevalent in the elderly (22). This again appears to be at variance with the instability hypothesis, because hypercapnic and hypoxic sensitivities are known to decrease with aging (24). Passive hyperventilation, which leads to loss of chemical drive, has been shown to produce apnea in sleeping subjects once arterial PCO, (Pa& is reduced 3-6 Torr below initial levels; the same degree of hyperventilation rarely leads to apnea in awake subjects (28). This and other pieces of evidence suggest that the state of wakefulness, even in the absence of any voluntary influences on breathing, is characterized by tonic excitation of the respiratory centers (5, 25). This extraneural input has been termed the “wakefulness stimulus” (1). Support for the existence of this stimulus, which presumably is related to activity in the brain stem reticular formation, comes from neurophysiological studies (20). Variations in the depth of sleep or level of “vigilance” would be reflected as fluctuations in this extraneous drive to breathe. In this way, primary fluctuations in state could drive secondary oscillations in ventilation and blood gases (1, 25). A recent study has found oscillations in ventilation and those in frequency content of the electroencephalogram to be essentially synchronous (21). Furthermore, arousal from sleep is generally accompanied by transient hyperpnea, which apparently marks the rapid restoration of the wakefulness stimulus (25). This hyperpnea will tend to lead instability in the chemical feedback system. Thus, oscillations in ventilation in-
0 1991 the American
Physiological
Society
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SLEEP-INDUCED
PERIODIC
duced by fluctuations in state and those induced by instability of the chemical feedback system may interact and indeed might be naturally reinforcing. The effects of these oscillatory processes may be further magnified by their relative effects on the action of upper airway muscles and on the diaphragm (27). Thus, oscillations in state may lead to repetitive obstructive apneas. Because such obstruction can produce large disturbances in blood gas tensions, again chemically induced instability may be superimposed (17). It is therefore likely that the pathogenesis of PB and repetitive apnea involves the interaction of several factors, the individual contributions of which may vary considerably depending on the subject. In preliminary studies (11,21), using mathematical models, we have explored the implications of coupling between sleep-waking state and the chemoreflex control of respiration. In this theoretical study, the previous results are consolidated into a more comprehensive model. Computer simulations with the model enable us to systematically investigate the interactions between and relative importance of chemical control, sleep-waking state changes, and upper airway patency to the overall stability of breathing during sleep. In so doing, our basic premise (14) as to how the respiratory system can produce sustained periodicities is expanded in scope to encompass a considerably larger variety of physiological and pathophysiological conditions.
BREATHING
D p = GJ
102.4 - Sao,)[Pa,,,
- Ip]
(0
where D, represents peripheral drive, Gp the peripheral gain factor, and SaoZ arterial 0, saturation. The function enclosed in brackets is defined such that [x] = 0 if x 5 0 and [x] = x if x > 0. Thus, I, represents that value of Pace, below which peripheral chemoreflex drive becomes zero. In eq. 1, we assume that a residual peripheral drive exists in hyperoxia (i.e., when Sao, = 100%) (4). By use of nominal parameter values (Table 1) in the above expression, the contribution of peripheral chemosensitivity to total gain is 25% in normoxia and 70% in hypoxia (S a% = 75%). Central drive (D,) is related linearly to Pcop of the brain compartment (Pb,,J Dc=
w%o,
- ICI
(2)
where G, is the central gain and I, is threshold value for Pb co2 below which central chemoreflex drive is zero. Pbco, is, in turn, related to PacoZ in the following manner (26) = Qb(Pac,,
DESCRIPTION
To better distinguish the effects of sleep-waking state changes from those mediated solely by chemical control and those of upper airway obstruction, three models of ascending complexity are considered: model 1, which incorporates the common features of all three models, including chemoreflex control and the effects of wakefulness-to-sleep transitions (without arousal); model 2, which incorporates all features of model 1, including the capability of arousal if chemical drive exceeds a certain threshold during sleep; and model 3, in which model 2 is modified to include complete upper airway obstruction once ventilatory drive falls below a minimum level. The following descriptions refer to model 1, unless otherwise stated. The “plant. ” The lungs are modeled as a homogeneous alveolar chamber connected in series to a dead space. During inspiration and expiration, the alveolar chamber increases and decreases in volume, but the dead space volume remains fixed. The gas exchange equations are presented in detail in Khoo and Kronauer (13). Pulmonary end-capillary and alveolar PCO, and PO, are assumed to be completely equilibrated. The blood-gas dissociation relationships of Grodins et al. (lo), which include the Bohr-Haldane effect, are modified to establish reasonable consistency with the tables published by Olszowka et al. (18). A convective delay and two vascular mixing transfer functions are placed in series to simulate the passage of blood from the lungs to the chemoreceptors (14); these provide the transformation relationships between alveolar gas tensions (PACES and PAN,) and arterial blood gas tensions (Pa,oz and Pa,,). The me-
2015
APNEA
tabolic exchanges for CO, and 0, in the body tissues are modeled as single-compartment processes that determine the levels of the mixed venous blood gases (14). CentraE and peripheral chemoreflexes. Peripheral chemoflex drive is assumed to be dependent on the multiplicative interaction of hypercapnia and hypoxia
dPbcozldt MODEL
AND
- Pb,,,)K,,,/60Kb,,, + (MRbco,
- 0.0011)/60Kb,,,
(3)
where MRbco, is the brain metabolic CO, production rate, Qb is cerebral blood flow, and Kco, and Kco, are the dissociation slopes for the brain tissue and blood, respectively. The dependence of Qb on Paooa and Paon is modeled after empirical relationships obtained in several studies (10, 15). ' Integration of driues. Total ventilatory drive in wakefulness (Dawake,l/min) is given by D awake = [D, + D, - h] + h
(4)
where h = 14.4 - 0.0138 PaoZ. This represents a simple summation of central and peripheral drives with the inclusion of an idealized hypoxia-dependent “dog leg” in the hypocapnic region during wakefulness (4). Dawake implicitly includes a wakefulness stimulus component that progressively declines with increasing depth of sleep. Mathematically, we have found it convenient to model this hypothesis by subtracting the nonchemical component (S) from Dawake.Sleep is also accompanied by a decrease in responsiveness of ventilatory drive to chemical stimuli. Thus D sleep
=
G,[
Da=ke
-
S]
(5)
In Eq. 5 G, (11) modulates the slope of the CO, response line while S (>O during sleep) effectively shifts the CO, response line to the right. Generation of airflow and breathing pattern. At the end of the preceding breath, a new breath is generated if total ventilatory drive (D) is greater than a minimal threshold level (Dmin)
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2016
SLEEP-INDUCED
D = DaWake if
PERIODIC
DaWake> Dmin
(6 a )
or D
=
DSleeP
if
DSleeP
>
BREATHING
AND
sleep, we assume a simple linear dependence of E on time (with t = T, s marking the start of the transition) E(t)
Dmin
APNEA
= (t - TO)/7
for
t - TO 5 7
uw
7
Wb)
(W
= 1 for
Because the model simulates tidal breathing, D is represented in terms of mean inspiratory flow. Tidal volume (VT, in liters) is thus determined as
t - TO >
In the above expression, 7 represents the time required to first attain the stage of sleep in question, assuming that no arousal occurs during the transition. VT = D.T1/6O (7) Arousal mechanism. In model 2, if total chemical drive (D, + DP) exceeds the threshold level (Da,), an arousal is where TI is inspiratory duration (in s), assumed constant. A constant expiratory duration (TE) is also as- triggered and the wakefulness stimulus is fully restored (i.e., E reverts to zero) on the next breath. If this occurs sumed, so that minute ventilation (VE, in l/min) is given during a wakefulness-to-sleep (WS) transition, the bY arousal is assumed to be transient and is followed by a VE = VT.GO/(TI + TE) (8) new WS transition of duration 7 in subsequent breaths. However, if total ventilatory drive is less than Dmin at the Consistent with recent data (6), arousal has been asend of a breath, no new breath will be generated until the sumed to be triggered by total chemical drive irrespective of the kind of stimulus. For purposes of comparison with deterioration of blood gases raises drive to levels higher empirical studies involving arousal, the value of D, used than Dmin. in the model (assuming nominal chemoreflex gains) is Effect of changes in sleep-waking state. Because the equivalent to a hypercapnic threshold of ~53 Torr durchanges accompanying rapid-eye-movement (REM) sleep are highly variable and complicated (29, we have ing hyperoxia or a hypoxic threshold of 75% in Sa,, during normocapnia (cf. Ref. 25). limited the scope of this study to non-REM sleep. SleepUpper airway obstruction. In model 3, we assume that waking state is quantified in terms of the index E. E is defined such that the fully awake state is represented by when Dsleep is less than Dmi, at the end of a breath, paE = 0, stage 1 sleep by 0.25 5 E < 0.5, stage 2 by 0.5 5 E < tency of the upper airway is lost and any subsequent inspiratory efforts would result in an obstructive apnea. 0.85, stage 3 by 0.85 5 E < 1, and stage 4 by E = 1. These is terminated only when ventilatory values have been chosen so that the reduction in CO, The obstruction drive builds up to and exceeds Da,. This assumption is response slope with the different sleep stages are compatcompatible with the work of Parisi et al. (23), who ible with the observations of Bulow (1) showed that restoration of upper airway electromyogram G w = 1 - 0.4E (9) is correlated to an increase in phrenic activity above some threshold level. When apnea is not present, model 3 For instance, in slow-wave (or stage 4) sleep, the effective CO, response slope is 60% as large as that during wake- behaves in exactly the same way as modeZ 2; i.e., arousal can occur during any sleep stage if ventilatory drive exfulness, when G, is unity. ceeds. Consistent with Bulow’s findings, we have also asComputer simulation. The finite-difference forms for sumed that the withdrawal of the wakefulness stimulus or, equivalently, the shift of the CO, response line is the above equations are solved numerically on a MicroVAX (Digital Equipment, Bedford, MA) computer using completed by the time stage 1 sleep is attained a time step of 0.025 s. Starting from initial values, the S = SOaE/0.25 for 0 5 E < 0.25 uw model is allowed to attain a steady state (i.e., breaths = S, for 0.25 5 E 5 1 (10~) achieve constant VT) before any changes are imposed. The values of the plant parameters used in all simulawhere S, is the full magnitude of the wakefulness stimtions are similar to those employed in a previous model ulus. (20): cardiac output = 6 I/min, lung-to-chemoreceptor Effects of sleep on plant parameters. We have assumed (peripheral and central) delay = 8 s (or 2 breaths), vascuthat cardiac output, the lung-to-chemoreceptor translar mixing time constants = 1 and 2 s (2 cascaded firstport lags, and vascular mixing time constants do not order low-pass filters), equivalent lung volume for CO, = change from wakefulness to sleep. On the other hand, the 3 liters, FRC = 2.5 liters, dead space volume = 0.15 liter, metabolic CO, production rate (MR,,,) and 0, con- body tissue CO, stores = 15 liters, body tissue 0, stores = sumption rate (MR,,) are assumed to decrease with in6 liters. The nominal values of the other parameters used creasing sleep state, so that in stage 4 sleep they would be in the simulations are given in Table 1. 15% lower than during wakefulness Mb,,
= 0.21(0.375G,
+ 0.625)
MR,,
= 0.26(0.3756,
+
0.625)
(114 um
In the above expressions, we have assumed that in wakefulness MRcOz and MR,, take values of 0.21 and 0.26 l/min STPD, respectively. Wakefulness-to-sleep transitions. To simulate the time course of the transition from wakefulness to any stage of
RESULTS Steady-state responses. If nominal parameter values are assumed, the normoxic steady-state ventilatory responses of the model to CO, in different sleep-waking states are shown in Fig. 1. The magnitude of the wakefulness drive (S,) required to generate these results is 18 l/min; this and other values of drive are defined in terms of mean inspiratory flow. With the onset of sleep, mean
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SLEEP-INDUCED
PERIODIC
BREATHING
AND
2017
APNEA
1. Parameter values used in simulations
TABLE
Parameter
Abbreviation
Peripheral gain factor Central gain Peripheral apneic threshold Central apneic threshold Blood Cp, dissociation slope Brain tissue CO2 dissociation slope Cerebral blood flow Brain CO2 production rate Inspiratory duration Expiratory duration Minimum drive for breath generation Minimum drive for ending obstruction Magnitude of wakefulness drive WS transition time Arousal threshold WS gain factor All values wakefulness;
Value
0.375 1 min-l Torr-’ % (nominal) 4.5 1. min-l Torr-’ (nominal) 38 Torr (nominal) 45 Torr (nominal) 0.0057 1 CO,. 1 blood-’ Torr-1 0.0036 1 kg brain tissue-’ Torr-’ 0.5 1. min-l kg brain tissue-’ (Paoo, 0.031 1 min-l kg brain tissue-’ 1.5 s 2.5 s 3 l/min 45 Vmin (nominal) 18 Vmin (nominal) 20-60 s (W to Sl) 45 Vmin (nominal) 0.6 (NREM-4) to 1 (W) l
l
l
l
l
l
l
= 40 Torr)
l
MRbco,
l
TI TE D min D obst so 7
Qu Gv
of drive reflect mean inspiratory airflow rather Sl, sleep stage 1; NREM-4, non-rapid-eye-movement
than
minute ventilation. (stage 4) sleep.
Paoo,,
l
arterial
Pco~;
WS,
wakefulness-to-sleep;
W,
25
0
35
40
45 ARTERIAL
50
Pco2
(mm
55
t3 2
0.0
1 0.0
a
AWAKE VENTUATORY
fig)
FIG. 1. Steady-state ventilatory responses to CO, during wakefulness (W) and in the different sleep states (Sl-S4) generated by the model. l , Steady-state operating minute ventilation and arterial PCO~ at each of these states.
l
L
2.0 RESPONSE
I
1
I
3.0
4.0
T O CO2 (L/min/mm
Hg)
FIG. 2. Difference between steady-state arterial PCO, (Pace,) during stage 2 sleep and steady-state Paoo, during wakefulness is plotted against corresponding ventilatory response to CO, for different values of S, (magnitude of wakefulness stimulus). l as defined in Fig. 1; i.e., for nominal magnitude of state change (S, = 18) and normal CO, response (2 25 14 min-’ Torr-‘), increase in Pko, is 3.6 Torr. l
Pa co is raised by 3-4 Torr and mean VE is decreased roug fi ly 20%, consistent with what is generally observed during sleep (5, 25). In Fig. 1, the awake ventilatory response to hypercapnia is 2.25 1 min-’ 4Torr-‘, with the peripheral chemoreflex contributing 25% of total ventilatory gain. In Fig. 2, the increase in PacOz associated with the change in state from wakefulness to sleep stage 2 is plotted against the awake ventilatory response to hypercapnia. For any fixed value of S,, the change in PacOz with sleep decreases with increasing CO, response in a hyperbolic manner. This behavior agrees closely with data obtained by Gothe et al. (9, Fig. 4). For a given CO, response, the magnitude of the Pa,,, change increases with increasing values of S, (vertical dashed line in Fig. 2). It should be noted that at the higher values of CO, response (>3.5 1. mine1 . Torr-l) the model exhibits sustained oscillations. The values of steady-state Paoq, change under such conditions are obtained by stabilizing the model with decreased convective lung-to-chemoreceptor delays. Reponse to transient airflow obstruction for model 1. To test the stability of the model in different sleep-waking
I 1 .o
l
states, the following intervention is applied. In wakefulness and each of the sleep stages, the model is first allowed to attain a steady state. Subsequently, at the end of a breath, airflow is obstructed for the next three consecutive breaths (i.e., 12 s). The ensuing response in breathing pattern is shown in Fig. 3 for wakefulness (A), stages 1 and 2 sleep (B), and stages 3 and 4 sleep (C). The subsequent fluctuations in VE are rapidly damped out in wakefulness and in the deeper stages of sleep. However, the response is clearly more oscillatory in light sleep, indicating a more unstable system, despite a lo-20% decrease in chemoresponsiveness. The reason for this somewhat surprising result is fully elucidated in the DISCUSSION. The present procedure emulates the protocol adopted in a recent experimental study (7) in which the authors found a significant correlation betyeen hypercapnic sensitivity and the initial increase in VE after the release of inspiratory airway obstruction. Although the results are not presented here, our model predicts a strong dependence of the postobstruction ventilatory overshoot on controller gain. VW transitions for model 1. The transition from one
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2018
SLEEP-INDUCED
PERIODIC
gLL 0.6
3E
AND
0.4
!& 3 0.2 P 0.0 0 TIME (seconds) 6
1 .o
APNEA
medium values of controller gain, slow WS transitions produce negligible dynamic effects (not shown in Fig. 4). On the other hand, in the presence of high controller gain, even a slow WS transition can spawn a growing oscillation that leads rapidly to a sustained episode of Cheyne-Stokes breathing, with apnea occuring at the hypopneic phase of each cycle (Fig. 4C). In all three cases, the cycle time of the sleep-induced oscillation is ~36 s. Performing the simulations over a large variety of WS transition times (7) and controller gains, we have found that the stability properties of model 1 are determined primarily by chemoresponsiveness and S, but are independent of 7 over the range investigated. The stability
A
l.Or
BREATHING
A Wq N u&i
z 08. 3
I$
‘1
\
s.
\ -----
- ----
- --------------
----qo()$$ U
& 0.6 zi E 0.4
--
rw 93 0.2
N
8 a” 30
0.0 50
100 TIME (seconds)
150
200
-
-
-
-
-
TIME (sets)
C
1 .o
W
r N E m&z
f
-
S
c \ -------------
-----------------a
loos 8 - 90 m”
0.6
8 a” 3050
100
150
-1
.
200
100
TIME (seconds)
200
250
=8 0 ,’
300
INAE (sets)
FIG. 3. Breathing patterns after 12 s of airway occlusion (applied at the end of the 1st breath) in wakefulness (A), light sleep (Sl-S2; B), and deep sleep (S3-S4; C). FRC, functional residual capacity.
sleep-waking state to another represents a dynamic disturbance that the chemoreflexes aim to correct. The ensuing response elicits fluctuations that may persist long after the transition has been completed. A faster transition produces a larger effect. The effect of the transition from wakefulness to stage 1 sleep (WS) on model 1 is shown in Fig. 4 for different values of CO, response. In Fig. 4A, a rapid transition (7 = 20 s) leads easily-to apnea, which raises Pacoz and lowers Sa,,. A very small-amplitude oscillation in VE and blood gases is set up, but this is damped out after a few cycles. With normal values of controller gain, a rapid WS transition produces larger oscillatory changes in VE and blood gases, which may-be classified as PB with periods of hypopnea but no apnea (Fig. 4B). However, the PB occurs only transiently for a few minutes and is eventually damped out. For low and
150
T
W N 6 u&i
f
.4
C bL \
S
1 \
x
--------------
--L-m------
100
150
200
-1
250
TIME (asca)
of wakefulness-to-sleep transition (dashed line) on conditions of low ventilatory response to COz min-’ . Torr-‘) and fast transition rate (7 = 20 s; A), moderate (2.25 1 min-’ Torr-‘) and fast transition rate (7 = 20 s; B), and gain (4 1. min-’ l Torr-‘) and slow transition rate (7 = 60 s; C). arterial O2 saturation; VT, tidal volume; S, sleep; W, wakefulness.
FIG.
model (0.51. gain high Sa,,,
4. Effect
1 under l
l
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SLEEP-INDUCED
PERIODIC
BREATHING
AND
2019
APNEA
from 10 to 20 s. Figure 6C illustrates a situation with high gain (4 1 min-l Torr-l) and slow WS transition (7 = 60 s). The slow transition provokes a small oscillation that is rapidly amplified by the high controller gain. The blood gas swings trigger an arousal at around t = 70 s. The hyperpnea accompanying arousal rapidly decreases chemical drive, and this, coupled with the subsequent onset of sleep, leads to hypopnea. The resulting buildup in chemical drive produces the next hyperpneic phase but is insufficient to trigger arousal. In this way, arousals occur ~80 s apart, but the hyperpneic phases recur every 40 s. Figure 6, B and C, demonstrate two forms of interl
z
iii
0’
05.
1
1
I
I
I
1
I
1.0
1.5
2.0
2.5
3.0
3.5
4.0
VENTIIATORY
RESPONSE
TO CO2
(L/min/mm
A sI\
Hg)
45
l
‘1 \
FIG. 5. Stability characteristics for model 1. Solid line separates conditions under which sustained oscillations will be produced after sleep onset (region above) from conditions under which model remains stable (region below). Dashed line separates cases with nonapneic oscillations (region below) from cases with repetitive apnea (region above).
diagram in Fig. 5 represents a compilation of all the simulations conducted with modeZ 1. The solid curve represents the minimum value of S, required to produce a sustained periodicity for any given value of CO, sensitivity. The region below the solid curve therefore represents the range of conditions in which the model is inherently stable, while in the region above, the model exhibits sustained oscillations. The dashed curve further separates the unstable region into oscillations with apnea (above curve) and periodicities without apnea (below curve). It should be noted that the dashed curve is coincident with the solid curve at all CO, sensitivities < 2.7 1. min-’ . Torr-l; i.e., sustained nonapneic oscillations are found only at gains higher than this level. A further point to note is that, for S, = 18 l/min (the nominal value), sleep onset will lead to sustained oscillations only when CO, sensitivity > 3.4 1 min-’ Torr? Stability properties of model 2. Figure 6 shows responses of the model to WS transition under circumstances in which an arousal mechanism is incorporated. When controller gain is very low (0.5 1. min-’ Torr-l), even rapid changes in state do not provoke an arousal; in fact, model 2 behaves no differently from model 1 in this case (Fig. 6A). When CO, sensitivity is moderate (2.25 1 min-l . Torr-l), only sufficiently rapid WS transition will lead to the propagation of a sustained oscillation. In Fig. 6B, 7 is assumed to be 20 s. The fast transition produces a subsequent large increase in Pacoz and fall in Sao,. As a consequence, chemical drive builds up to such high levels as to trigger an arousal at t = 35 s. The transient arousal leads to hyperpneic breaths, which quickly improve blood gases. As the stimulus for arousal is removed, sleep sets in again. The rapid reduction of chemical drive, coupled with the second onset of sleep, accelerates the withdrawal of total ventilatory drive and leads subsequently to central apnea. With apnea, blood gases rapidly deteriorate and the entire cycle of arousal and sleep onset is repeated. In this case, the fluctuations are large enough to perpetuate the periodicity. Cycle times are on the order of 45 s, and the apneas range in duration
l
vdi
w-
\,,,,,,,,-,,-,--,------------,,
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0 &100
150
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250
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TIME (sets)
3
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E 540 cy 8
I
I ‘f c
l
k.s . . . . , . . . . , .
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, 1 - - - , - 150
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. 0 9300
TIME (aecs)
l
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90 f#
E k
40
3 af
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. . . . . . . . . . (0 200 250 300
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g
TIME (sets)
FIG. 6. Effects of wakefulness-to-sleep transition on model 2 under conditions of low gain (0.5 1. min-’ Torr-‘) and fast transition rate (7 = 20 s; A), moderate ventilatory response to CO, (2.25 1. min-’ . Torr-‘) and fast transition rate (T = 20 s; B), and high gain (4 1 min-’ Torr-‘) and slow transition rate (T = 60 s; C). In B and C, note that repetitive arousals (dashed line) coincide with start of hyperpneic breaths. l
l
l
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2020
SLEEP-INDUCED
PERIODIC
action between state-driven fluctuations and chemically mediated oscillations. In B, state oscillates in synchrony with chemoreflex drive. However, in C, the chemically mediated oscillation of shorter cycle duration rides on top of a state-driven oscillation of longer wavelength. The stability characteristics of model 2 are summarized in Fig. 7. In contrast to mode2 I, the stability properties of this model depend more on 7 in addition to chemoresponsiveness and S,. Thus, in Fig. 7, S, is plotted against 7, with low (0.5 1 min-’ Torr-l), moderate (2.25 1 min-’ . Tori?), and high (4 1 min-l Torr-l) levels of chemosensitivity represented by A, B, and C, respectively. As in Fig. 5, the solid curves represent the minimum magnitude of S, required to produce a sustained periodicity, whereas the dashed curves represent the minimum S, required to produce sustained apneic episodes. Two features are noteworthy. First, regardless of 7, the minimum S, required to generate sustained periodicities is generally lower than that in model 1 for the same CO, sensitivity. This implies that model 2 is less stable than model 1, particularly at the lower gains. For instance, for a CO, sensitivity of 0.5 1 min-’ .Torr-‘, a 35% increase in S, over its nominal value is sufficient to produce sustained PB with apnea in modeZ 2; in modeZ 1, an increase of 80% is required to achieve the same effect. Second, model 2 exhibits sensitivity to 7, particularly where sustained apneic oscillations are concerned. Thus, rapid WS transitions can be highly effective in promoting instability. Stability properties of model 3. Some samples of the variety of responses produced by model 3 to WS transitions are shown in Fig. 8. In Fig. SA, a low value (0.5 1 min-l Torr-‘) of CO, sensitivity is assumed. The fast WS transition (7 = 20 s) leads to rapid loss of chemical drive, which affects upper airway patency as well as ventilatory drive. Airflow obstruction occurs as inspiratory efforts are applied against a collapsible airway, leading to a prolonged apnea of -40 s. The long apnea produces a substantial degree of hypoxia and hypercapnia, and chemical drive builds up correspondingly to a level that triggers arousal (at t = 58 s). The rapid restoration of l
BREATHING
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APNEA
-50 3 E 540 cy
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FIG. 8. Effects of wakefulness-to-sleep transition on model 3 (with S, = 15 Vmin) under conditions of low ventilatory response to CO2 (0.5 1. min-l Torr-‘) and fast transition rate (r = 20 s; A), moderate gain (2.25 1. min-’ Torr-‘) and fast transition rate (7 = 20 s; B), and high gain (4 1. min-’ Torr-‘) and slow transition rate (60 s; C). In C, note interaction between state-driven oscillation and chemoreflex-mediated oscillation.
C
I
h ” 20
40
” 60
” 20
WAKEFULNESS-TO-SLEEP
40
” 60 TRANSITION
” 20
40
” 60
TIME (sea)
FIG. 7. Stability characteristics of model 2 with low ventilatory sponse to COz (0.5 1. min-‘~Torr-‘; A), moderate gain 1 min-’ . Torr-‘; B), and high gain (4 1. mine1 . Torr-‘; C). In each region above solid line encompasses all conditions that lead to tained oscillation, while dashed line separates nonapneic (below from apneic oscillations (above line). l
100
re(2.25 case, susline)
wakefulness drive leads to termination of the obstructive apnea. Large hyperpneic breaths follow, rapidly removing the chemical stimuli. At the same time, sleep sets in again and this, coupled with the loss of chemical drive, leads to the next obstructive apnea. In this way, a sustained periodicity is produced, with cycle and apnea durations of ~62 and 32 s, respectively. In Fig. 8Z3, CO, sensitivity is moderate (2.25 1. min-‘~Torr-l). A fast WS transition (7 = 20 s) leads to a transient period of oscillation (cycle time = 36 s), which is eventually damped out. However, ventilatory drive does not fall to sufficiently low values to produce apnea at any point. In comparing Fig. 8, A and B, with the corresponding cases for model 2
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SLEEP-INDUCED A
f I
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B
a
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40
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WAKEFULNESS-TO-SLEEP
PERIODIC C
1
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20
40
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J
20
TRANSITION
40
60
TIME (sets)
FIG. 9. Stability characteristics of model 3 with low ventilatory response to CO, (0.5 l*min-’ . Torr-‘; A), moderate gain (2.25 1. min-’ Torr-‘; B), and high gain (4 1 min-’ Torr-l; C). Solid lines distinguish stable region from conditions that lead to sustained oscillations; dashed lines distinguish stable region from conditions that lead to sustained oscillations; dashed lines distinguish nonapneic oscillations from oscillations with apnea. l
l
l
in Fig. 6, it is interesting to note that, with the inclusion of airflow obstruction, a lower controller gain seems to promote sleep-disordered breathing. Figure 8C represents a high-gain (4 1 mine1 . Torr-l) situation in which a slow WS transition (7 = 60 s) occurs. The result is similar to that shown in Fig. 6C in which arousals and termination of airflow obstruction coincide with every other hyperpneic phase. As before, a shorter-duration oscillation (-42 s) appears to ride on top of a slower oscillation (-82 s). However, the apneas produced here are longer and range from 10 to 15 s. The stability diagram for model 3 is presented in Fig. 9. Comparing the results with those in Fig. 7, it is clear that the stability boundaries for the high-gain case (C) are virtually unaltered. Although the moderate- gain case (B) is somewhat affected, it is the low-gain case (A) that shows the largest changes. For transition times
