Corresponding author G. S. Longobardo: Department of Medicine, UMDNJ-New Jersey Medical School, 185 South Orange Avenue, MSB/I-510 Newark, NJ 07103, USA. Email: firstname.lastname@example.org
Increased loop gain (a function of both controller gain and plant gain), which results in instability in feedback control, is of major importance in producing recurrent central apnoeas during sleep but its role in causing obstructive apnoeas is not clear. The purpose of this study was to investigate the role of loop gain in producing obstructive sleep apnoeas. Owing to the complexity of factors that may operate to produce obstruction during sleep, we used a mathematical model to sort them out. The model used was based on our previous model of neurochemical control of breathing, which included the effects of chemical stimuli and changes in alertness on respiratory pattern generator activity. To this we added a model of the upper airways that contained a narrowed section which behaved as a compressible elastic tube and was tethered during inspiration by the contraction of the upper airway dilator muscles. These muscles in the model, as in life, responded to changes in hypoxia, hypercapnia and alertness in a manner similar to the action of the chest wall muscles, opposing the compressive action caused by the negative intraluminal pressure generated during inspiration which was magnified by the Bernoulli Effect. As the velocity of inspiratory airflow increased, with sufficiently large increase in airflow velocity, obstruction occurred. Changes in breathing after sleep onset were simulated. The simulations showed that increases in controller gain caused the more rapid onset of obstructive apnoeas. Apnoea episodes were terminated by arousal. With a constant controller gain, as stiffness decreased, obstructed breaths appeared and periods of obstruction recurred longer after sleep onset before disappearing. Decreased controller gain produced, for example, by breathing oxygen eliminated the obstructive apnoeas resulting from moderate reductions in constricted segment stiffness. This became less effective as stiffness was reduced more. Contraction of the upper airway muscles with hypercapnia and hypoxia could prevent obstructed apnoeas with moderate but not with severe reductions in stiffness. Increases in controller gain, as might occur with hypoxia, converted obstructive to central apnoeas. Breathing CO2 eliminated apnoeas when the activity of the upper airway muscles was considered to change as a function of CO2 to some exponent. Low arousal thresholds and increased upper airway resistance are two factors that promoted the occurrence and persistence of obstructive sleep apnoeas.
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An increase in loop gain, the product of controller and plant gain, is now recognized as an important cause of the instability in feedback control that leads to recurrent central apnoeas during sleep (periodic breathing; Badr, 2005; Cherniack & Longobardo, 2006).
However, the anatomy and mechanical properties of the upper airway are crucial in determining the occurrence of obstructive sleep apnoeas. Narrowed upper airway passages and decreased stiffness of pharyngeal walls have been reported in many studies of patients with obstructive sleep apnoea (Brown et al. 1985; Bradley et al. 1986; Avrahami & Englender, 1995; Ryan & Love, 1996; Ciscar et al. 2001). More negative intrapharyngeal pressures are required to block the flow of air in normal subjects than in persons with obstructive sleep apnoea syndrome (Remmers et al. 1978; Bradley et al. 1987). Positive pressure applied to the nasal pharynx (CPAP) is usually a successful treatment for patients with the syndrome (Ryan & Bradley, 2005; Chowdhuri, 2007). Surgical procedures and mechanical devices that widen the upper airways or buttress their walls reduce or prevent obstructive apnoeas during sleep in many cases (Jacobowitz, 2006).
In contrast, there are observations, such as the convertibility of obstructive to central apnoeas and the frequent occurrence of both in the same diseases, and even in the same patient, which suggest that neurochemical factors producing unstable regulation of the movement of the chest wall and upper airway muscles (such as increased loop gain) contribute significantly to airway obstruction during sleep (Longobardo et al. 1982; Onal & Lopata, 1982; Onal et al. 1986; Younes et al. 2001; White, 2005; Cherniack & Longobardo, 2006). Tracheotomy has been reported to convert intermittent obstructive apnoeas to recurrent central apnoeas. Hypoxia is reported to transform obstructive to central apnoeas, while breathing oxygen can change central to obstructive apnoeas (Gold et al. 1985; Warner et al. 1987). Obstructive sleep apnoeas and recurrent central apnoeas (Cheyne-Stokes breathing) occur in both cardiovascular and cerebrovascular disease, sometimes in the same patient (Tkacova et al. 2001). Increased controller gains are reported to predispose to both central and obstructive apnoeas (Younes et al. 2001; Younes, 2003). However, the interaction of changes in controller gain and upper airway stiffness in causing different kinds of apnoea remains unclear.
Theoretically, many factors may influence the architecture of both breathing and sleep and the number and kind of apnoeas. The complexity of the geometry and fluid mechanics of the upper airways, the diversity of influences that may affect the non-linear responses of the respiratory control system, and the intricate and incompletely understood interplay between the sleep–waking cycle and breathing make it virtually impossible to predict with any degree of certainty the effects of variations in any single factor on breathing during sleep without the use of a mathematical model.
This paper is an attempt to develop the simplest model that can reproduce the essential features of obstructive sleep apnoea. We have used this model to simulate breathing following sleep onset in order to evaluate the hypothesis that both upper airway mechanics and the properties of the neurochemical feedback loop regulating breathing contribute to the occurrence of apnoeas. To do this, we added to a model for the neurochemical control system of ventilation (Longobardo et al. 2001, 2005) a description of the physical properties of the upper airway in order to simulate respiratory responses during central and obstructive apnoeas during sleep. The upper airway is considered to contain a compressible segment which has a smaller cross-sectional area than the two rigid segments of the upper airway which it connects, so that the velocity of airflow through it is greater than through the rigid segments. When the elastance (stiffness) of this segment is sufficiently reduced as it is during sleep, it can be collapsed completely by transmural pressure, thus preventing airflow. Dynamically, the transmural pressure changes are amplified in this segment via the Bernoulli Effect. The elastance of this segment is affected by both chemical drives and neural factors, such as arousals. ‘Arousal’ in our model occurs when ventilatory drive increases beyond a level tolerable to the subject during central apnoeas or obstructive apnoeas, and his arousal threshold is reached. Sleep does not return until ventilatory drive is reduced to below some comfort level, the subject's sleep threshold.
Using this integrated model, we have studied the impact of loop gain and elastance and their interaction on the characteristics of recurrent apnoea during sleep. The model is not intended to be comprehensive but rather is constructed so that the potential effects of variables that have been considered to be more important in producing or preventing apnoeas can be examined.
The upper airway
A schematic diagram of the model of the upper airway muscles and chest wall muscles is shown in Fig. 1. Nomenclature for symbols and equations is shown in Table A1.
Table A1. Nomenclature
Pharnygeal Compressible Segment
Functional Residual Capacity
Vdotlung (l min−1)
Airflow through the pharynx, into and out of the lung
Elung (mmHg cm−3)
Ec (mmHg cm−3)
Pharygeal Compressible Segment
Rlung (mmHg (l min−1)−1)
Rua (mmHg (l min−1)−1)
8 awake and asleep
Pharygeal Compressible Segment
1 asleep, rigid
Velua cm min−1
Velc cm min−1
Pharygeal Compressible Segment
awake and asleep
O2 Total Body (l min−1)
Brain (l min−1)
CO2 Total Body (l min−1)
Brain (l min−1)
The upper airway is considered to be a virtually incompressible chamber with a volume corresponding to the dead space and containing a compressible segment which, when its elastance is sufficiently reduced as it is during sleep, can be collapsed completely by transmural pressure, blocking airflow. Subatmospheric alveolar pressure during inspiration leads to decreased segmental cross-sectional area and, because of the Bernoulli Principle, produces additional pressure reductions and further area diminution. Transmural pressure increases are directly related to the square of the ratio of cross-sectional area of the upper airway to the cross-sectional area of the collapsible segment. Upper airway occlusion is prevented if the elastance of the collapsible segment is sufficiently high that it does not decrease very much during sleep. Obstruction, when present, can be relieved by increases in elastance, which occur either neurally by arousal or chemically by hypercapnia or hypoxia. The upper airway and lower airways have separate resistances.
The dynamics of both motion of the lung and airflow through the lung and the upper airway are calculated from mass balance and equations of motion. The equation for pressure and flow through the pharyngeal restriction is calculated using the fluid dynamics energy equation, the Bernoulli Equation. An incompressible, inviscid fluid is assumed.
Air is driven into the lung during inspiration as alveolar pressure falls and is forced to exit during passive expiration by the relaxation of the chest. We have written equations for airflow into and out of the lung using the equations of motion for the lung. Inspiration depends on the input to the diaphragm (and the intercostal muscles) from chemoreceptors in the blood and in the medulla and from supramedullary centres. We have represented this ventilatory drive as input to a respiratory pattern generator (RPG) in the medulla, which determines tidal volume and frequency of each breath and then drives the respiratory muscles. For the collapsible segment, we have written equations of flow for a convergent–divergent tube with a compressible segment using Bernoulli's Equation for conservation of energy for an inviscid fluid. We assumed an incompressible fluid and steady flow; both assumptions are widely accepted in the literature (Van Hirtum et al. 2005). Resistances for the upper airway and the lungs are included. During inspiration, as air enters the upper airway a pressure drop (Patm−Pua) occurs depending on the magnitude of upper airway resistance. Air passes into the compressible segment of the upper airway and, if the area in the compressible segment is smaller than the entry area, velocity increases and pressure is diminished by virtue of the interchange between kinetic energy, a function of velocity squared, and energy generated by pressure divided by density (flow work, but commonly called potential energy) as described by Bernoulli's Theorem. Since we have hypothesized an incompressible, frictionless fluid, the pressure Pua leaving the compressible segment is the same as the pressure entering. We have chosen to account for any possible loss in pressure caused through friction in the compressible segment by increases in the upper airway resistance, as explained in the Appendix. We have checked a posteriori the assumption of incompressibility and find that the absolute pressure in the compressible segment is never less than 0.53 of the entry absolute pressure, the thermodynamic criterion for choking.
The minimum area of the compressible segment, Ac, is given by:
where Aua is the area of the upper airway immediately before the compressible segment.
The elastance of the upper airway compressible segment, Ec, is considered to have two components. One is a constant component, E0, which represents the tonicity of the upper airway muscles and the passive properties of non-muscular tissue in the airway and is independent of chemical and neural drives. The other is a variable component caused by the contraction of the upper airway muscles in response to changes in chemical and neural inputs. Therefore, the variable component is a function of hypercapnia, hypoxia and alertness. Although in the past we have suggested a more complicated chemical formulation, the simplest formulation is to consider this variable component to change in proportion to ventilatory drive (Longobardo et al. 1982). Except when otherwise described, the effect of upper airway muscle contraction was simulated that way. Therefore, the upper airway elastance, Ec, becomes:
with the assumption of proportionality with ventilatory drive:
where kstiff is a constant coefficient.
The pressure in the compressible segment, Pc, is calculated from Bernoulli's Equation for the conservation of energy, which states that the sum of the energy generated by pressure and kinetic energy anywhere along the tube is a constant:
where Pc/D is the potential energy, D being equal to density, and (Velc)2/2 is the kinetic energy.
which shows that Pc decreases with decreases in Pua and Ac, and with increases in Velua.
The parameter Velua depends on the airflow to the lungs, , which is calculated from the equations of motion for the lung during inspiration and expiration.
where V= Lung volume.
where FRC = functional residual capacity.
Also dependent on is Pua:
Equation (7) shows that increases in both and Rua reduce Pua, which, according to eqn (5), lowers Pc. This in turn decreases Ac (see eqn (1)).
Also note from eqn (6a) that increases in Pphrenic, which correspond to increases in ventilatory drive, increase . According to eqn (7) this also reduces Pua, and so reduces Ac. Therefore, increased ventilatory drive can reduce Ac in two ways: one is by reducing Pua and the other by increasing Velc.
These equations show that the pressure in the compressible segment, Pc, is lowered as the square of any increase in airflow to the lungs, by eqn (4). Since Pc is negative and the segment is elastic, the cross-sectional area, Ac, decreases further, according to eqn (1). This results in an increase in the velocity of airflow through the compressible segment, and Ac is reduced again. If the elastance of the compressible segment is not great enough to halt the decrease in area, the compressible segment closes, causing an obstructive apnoea. Note that for the same partial pressures of CO2 (P) and O2 (P) and alertness, the ventilation, and therefore the airflow, increases with greater controller and plant gains so that obstruction occurs even sooner after the onset of a breath. These dynamic relationships are shown in Fig. 2.
Figure 2A and B shows airflow dynamics in the compressible segment and its effect on lung volume during inspiration. In the upper panel of Fig. 2A, airflow increases as the lung expands. The increase in airflow causes an increase in velocity in the compressible segment, decreasing the pressure there. The decrease in pressure, by virtue of the elastance of the compressible segment, reduces its area. The process proceeds until, at around 0.4 s, the increase in velocity is enough to accelerate the process of area reduction (the pressure can drop from −2 to −15 mmHg over a period of less than one thousandth of a second), and the compressible segment obstructs. Figure 2B illustrates the effects of increases in controller gain and shows how increases in controller gain cause obstruction to occur sooner. Alveolar pressures computed by the model during the awake state are in the range of generally accepted values (West, 2004; Morgan et al. 2005), and volume flow rates are similar in form to those shown in Fig. 1 of Mansour et al. (2003).
The neurochemical model
The neurochemical feedback control system for respiration used here has been previously described (Longobardo et al. 2002, 2005). Its components are the body gas stores for CO2 and oxygen and a controller. The body gas stores consist of the lung, the brain and a compartment for the rest of the body. These stores are repositories of peripheral (arterial) and central (brain) chemoreceptors, which in response to ventilation send as inputs to the controller arterial P and P and brain P. The brain compartment is connected to the rest of the body via the cerebral circulation. Cerebral blood flow varies with changes in P and O2 saturation. Neural drives from higher brain centres, such as the wakefulness drive, are also sent to the controller. The respiratory muscles are driven by neural input from the RPG, which determines tidal volume and frequency (Longobardo et al. 2005). The RPG is a five-respiratory neuronal network based on the model of Botros & Bruce (1990). Vagal effects are included in the dynamics of the RPG. The RPG not only allows lung mechanics to be integrated easily into the model but also allows differentiation between long expirations and apnoeas.
The body gas stores The dynamics of the body gas stores are important because, as receptor sites, the body gas stores determine lengths of apnoeas. Also note that a measure of plant gain is the rate of change of P or P with change in ventilation. Smaller stores produce increased effective plant gain during transients. The dynamics of the CO2 muscle and brain body gas stores are based on experiments (Cherniack & Longobardo, 1970).
The circulation connects the various body stores. We have used a constant circulatory delay for all the stores.
The controller The respiratory controller generates ventilatory drive from input via the peripheral and central P receptors and neural drives from higher brain centres. The controller is shown in Fig. 3B. Although the controller can be shown here only in the steady state, the controller in reality is time variant by the dynamic action of neural drives, such as hypoxic ventilatory depression, postsynaptic ventilation and changes in alertness, as from wakefulness to sleep. We have not included postsynaptic potentiation or hypoxic ventilatory depression in the model, the former because it is generally believed to have no impact during sleep, the latter because its time constant of 2.5 min makes it a small, practically constant negative neural drive, −0.3 l min−1 after 10 min of recurrent apnoeas during sleep. This can be compared with the far greater effects of changes in alertness produced by sleep, which are of the order of −10 to −20 l min−1. We had tested several responses with and without hypoxic ventilatory depression and found insignificant changes, e.g. 1–2 s in 60 s, in the period of the response or the apneic time within a cycle of the response. The controller is designed after Nielsen & Smith (1951) and Longobardo et al. (2002). The essentials of the Nielsen & Smith (1952) work applicable here are that the ventilatory responses to hypoxia and CO2 are not additive but multiplicative, that the resulting ‘hypoxic fan’ has its convergence below the axis of zero ventilation, and that at normoxia during wakefulness the zero thresholds for both the peripheral ventilation and central ventilation are at the same arterial P. With this formulation, hypoxia can shift zero CO2 thresholds to lower values, thus shortening lengths of apnoeas. The controller converts ventilatory drive into excitatory drives suitable as input for the respiratory pattern generator (Longobardo et al. 2005). Shown in Fig. 3B are controller representations for two levels of alertness, one during wakefulness, the other during sleep. Equilibrium points during these states are shown by intersections with the metabolic hyperbola. The chemical controller, the controller with zero alertness, is assumed always to have an output, down to zero P. For the controller to generate ventilatory drive, and therefore ventilation, there must be some minimum of excitatory impulses. If alertness drives are sufficiently low, as during sleep, ventilatory thresholds result and apnoeas can occur. When an apnoea occurs, CO2 rises and oxygen falls until breathing resumes when the threshold is reached. If the apnoea is too long, the low P when breathing resumes can result in too high a ventilatory response and if P is driven once again below the threshold the cycle continues. The same can occur when the controller gain is too high.
In the model, varying alertness between wakefulness and sleep generates a sleep–waking cycle. The importance of a sleep–waking cycle as a mechanism for producing periodic breathing is a feature of other mathematical models (Khoo et al. 1991). In the present model, arousal from sleep occurs when ventilatory drive exceeds a threshold level. Sleep recurs when drive becomes 1 l min−1 less than the arousal threshold.
Figure 4 shows the progression of obstruction of the upper airway compressible segment leading to arousal. The figure shows four breaths during sleep as they progress from hypopnoea to total obstruction, ending in arousal. Ventilatory drive during the series of breaths increases from 7.7 to 17.2 l min−1, when arousal occurs. Obstruction occurs at approximately the same airflow, , of about 47 l min−1. The time to obstruct decreases from 0.4 to 0.07 s, reducing tidal volume from 176 to 30 ml.
In the simulations illustrated, regardless of overall controller gain, the steady-state conditions for ventilation and arterial partial pressures of CO2 (P), and O2 (P) are made the same during wakefulness and sleep in order to negate the effects of variation of plant gain when assessing the effects of changes in controller gain. The steady-state design point for the normal chemical controller with no neural drives is shown in Table A2.
Table A2. . Steady-state conditions for various controller gains
All controller gains
Ventilation (l min−1)
Tidal volume (ml)
Arterial O2 saturation
Frequency (breaths min−1)
Normal controller gain
CO2 gain (l min−1 (mmHg P)−1)
O2 gain (l min−1) (% decrease in saturation)−1
Effects of controller gain changes at different levels of elastance
The rigid upper airway Simulations were made at sleep onset in which the entire upper airway was considered to be rigid (i.e. there was no compressible segment). These are shown in Fig. 5. In these simulations the transition to sleep was 6 s.
At normal controller gain, Fig. 5A shows, following sleep onset, steady breathing after a brief period during which tidal volume cycles with only one central apnoea. Increasing controller gain to 11/2 times normal with the same arousal threshold, shown in Fig. 5B, gives rise to recurrent periods of central apnoeas, each followed by arousal and hyperpnoea. If the arousal threshold is raised so that no arousals occur, as in Fig. 5C, then central apnoeas disappear much sooner.
The compressible airway Additional simulations were obtained after sleep onset as the constant portion of the elastance of the compressible segment was reduced, ignoring the effects of the contraction of the upper airway muscles on elastance for the sake of clarity.
As the constant portion of elastance was decreased, obstructed breaths began to appear in clusters at sleep onset. Each cluster of obstructed breaths was terminated by an arousal. These clusters ultimately disappeared as arterial CO2 levels rose. Raising the arousal threshold tended to diminish the time required for apnoeas to subside and steady breathing to appear.
When the elastance was reduced to 2.35 mmHg cm−3 (a moderate decrease in elasticity), cycles of obstruction and arousal became continuous, as illustrated in Fig. 6A. The thick line represents obstructed apnoeas. The thin line represents hypopnoeas that result from partly obstructed breaths. The truncated breaths comprising the obstructive apnoeas in this case are less than 200 ml, and concomitantly during those apnoeas the P rises monotonically. The other non-obstructed small breaths are not rigorously mathematical central apnoeas, but because of dead space their effect on CO2 dynamics is the same as though they were. Halving controller gain, in Fig. 6B, eliminated obstructive apnoeas. As shown in Fig. 6C, so did breathing oxygen instead of air which was begun during wakefulness and continued into sleep. Breathing oxygen reduced controller gain by eliminating the peripheral chemoreceptor contributions to drive.
When elastance was decreased even further, the effects of changes in controller gain were weaker. Figure 7 shows that with a severely reduced elastance of 0.175 mmHg cm−3, neither halving controller gain (Fig. 7B) nor breathing oxygen (Fig. 7C) eliminated the occurrence of obstructed apnoeas. It is of interest though that breathing oxygen lengthened the period of obstructive breathing, concordant with observations reported in some patients with obstructive sleep apnoea (Longobardo et al. 1982). Doubling controller gain when the elastance was 0.175 mmHg cm−3 changed obstructive apnoeas to largely central apnoeas, suggesting that the mechanisms for both kinds of apnoeas can coexist (Fig. 7D). Large increases in controller gains in simulations in which the compressible segment was stiffer also converted obstructive to central apnoeas.
Changes in plant gain had similar effects to changes in controller gain at all elastance levels.
Effects of upper airway muscle contraction on obstructive apnoeas
The simulations depicted in Figs 8 and 9 show the effects of upper airway muscle contraction as a function of P, P and alertness on the occurrence of obstructive apnoeas. In these instances, the effects on the variable portion of elastance caused by muscle contraction are considered to increase linearly with ventilatory drive.
Compare Fig. 8A with Fig. 6A. Figure 6A shows that when the constant portion of the elastance is reduced to 2.35 mmHg cm−3 and controller gain is normal, cycles of obstruction and arousal become continuous. In Fig. 8A, for the same conditions, but now adding upper airway muscle contraction of 0.304 times the ventilatory drive to the constant elastance of 2.35, the occurrence of obstructive apnoeas after sleep onset is prevented. If the constant of the elastance, 2.35, is markedly reduced to 0.175 mmHg cm−3 as shown in Fig. 8B, then upper airway muscle contractions of the same strength cannot eradicate obstruction. Only when controller gain is reduced to half normal, in Fig. 8C, are the obstructions eliminated. Not shown is that increased controller gain restores obstructive apnoeas despite the contraction of the upper airway muscles.
These simulations demonstrate that increases in elastance, whether or not elastance varies with ventilatory drive, will reduce the occurrence of obstructive apnoeas. Also, greater increases in controller gain are needed to produce recurrent obstructive apnoeas as the stiffness of the upper airways increase.
In Fig. 9, inhalation of 14% O2 (at the 22 min mark) converts obstructive apnoeas to central apnoeas. In the simulation, the elastance Ec= 0.175 +(0.304 × Ventilatory drive) mmHg cm−3, and controller gain is normal. These are the same conditions as in Fig. 8B, where room air is breathed, except it is later after sleep onset. Hypoxia increases overall controller gain by increasing peripheral chemoreceptor output.
It has been reported that inhalation of CO2 can eliminate obstructive apnoeas in patients without producing arousal (Younes, 2003). This result could not be reproduced when we simulated the effects of inhaling 3% CO2 in air with the initial conditions of normal controller gain and elastance Ec= 0.175 + (0.304 × Ventilatory drive) so that the effect of upper airway muscle contractions on elastance were proportional to ventilatory drive, the same as in Fig. 8B. However, if the response of the upper airway muscles to CO2 was raised to some exponent rather than linear, obstructive apnoeas disappeared, as illustrated in Fig. 10. The elastance both prior to and after CO2 inhalation was Ec= 0.175 + (0.304 × Ventilatory drive) × (0.00001 ×P3). At the steady state which ultimately followed CO2 inhalation, where P was about 50 mmHg, stiffness was 25% greater than it would have been if the upper airway muscle effect was Ec = 0.175 + (0.304 × Ventilatory drive). This was sufficient to eliminate obstructions without producing arousals.
The effect of upper airway resistance
Increase in upper airway resistance requires a greater constant elastance or greater effect on elastance of upper airway muscle contraction to prevent obstructive apnoeas. Upper airway resistance reduces the pressure Pua at the entrance of the flexible segment and, for a given airflow, decreases the cross-sectional area of the flexible segment. The smaller the cross-sectional area of the compressible airway segment, the more likely are total obstructions. Figure 11A shows that as upstream resistance increases at the same elastance, from half normal to twice normal, the ventilatory response progresses from stable to continuous cycling with obstructive apnoeas. Controller gain is normal in all cases.
Figure 11B shows the increase in elastance required to maintain stable breathing during sleep as upper airway resistance grows greater. The boundary separates two zones, one of stable sleep, which might include obstructions and arousals but which eventually subsides as in the middle panel of Fig. 11A. The other zone is one of constant cycling. The boundary is at normal controller gain. Higher controller gain requires higher elastance for the same upper airway resistance to maintain sleep stability. With lower controller gain, upper airway resistance can be higher for the same elastance before constant cycling in breathing occurs.
Pathogenesis of obstructive sleep apnoea
This study supports the idea that obstructive sleep apnoeas can occur through different mechanisms, including increased loop gain, sleep–waking abnormalities and abnormalities in upper airway mechanics (Hudgel et al. 1988; Younes et al. 2001; Younes, 2003, 2004; White, 2005; Tkacova et al. 2006). Using a mathematical model, we investigated the potential interactions of gain changes in the neurochemical control system, and the mechanical properties of the upper airways in producing central and obstructive sleep apnoeas. The mathematical model developed was based on our previously described model of the neurochemical regulation of breathing but included a model emphasizing the dynamic properties of the upper airways (Longobardo et al. 2005). The mechanical properties of the conduits of the respiratory pump were simulated by in-series connections of alveoli and bronchi (lower airways) with the upper (pharyngeal–laryngeal) airways. The upper airways were considered to contain a compressible tubular segment connected at either end to rigid segments. The stiffness of the compressible segment depended on two factors. The first was constant and arose from passive properties of the tissue and the tonicity of the muscular elements comprising the upper airway walls, while the second varied directly with ventilatory drive, a function of arterial P, arterial P, brain P and alertness, and was generated from the tethering action produced by the inspiratory contraction of the upper airway dilating muscles. While the model of this segment of the upper airway resembles a Starling resistor structurally, as it is deformed by changing transmural pressures it operates more precisely as shown by the development of the equations in the Methods, as a convergent–divergent segment like a Venturi Tube (Smith et al. 1988; Leiter, 1992; Fajdiga, 2005). Thus, the degree and rate of narrowing and the time of closure of this compressible segment depend not only on its elastance, cross-sectional area and static distributions of pressure, but also on the velocity of airflow entering it and thus on ventilatory drive. The general effects of viscosity and flow separation are handled by increasing upper airway resistance.
In this model, obstruction results from conversion of the potential energy (pressure/D) entering the flexible tube to kinetic energy, which occurs as air flows through the narrowed and compressible segment of the upper airway (Kamm & Pedley, 1989). Since the airflow is the same into and out of the constricted area, airflow velocity in the constricted region must increase, hence the kinetic energy also increases. Since total energy is conserved, the pressure of the air inside the constricted region decreases. The loss of pressure is inversely related to the square of the cross-sectional area and is greater, the larger the airflow. The pressure reduction leads to further constriction and increases in airflow velocity, causing an exponential decrease in pressure so that the airway collapses completely for a portion of a breath or for an entire breath. An increase in the velocity of the airflow entering into the compressible segment promotes more rapid closure. This increase can occur as a result of changes in hypercapnia or hypoxia but also by larger controller gains so that more ventilation is demanded for the same change in neural and chemical drives.
With this model, we explored the role of controller gain in producing central and obstructed apnoeas under conditions in which the stiffness of the upper airway compressible segment was varied. Our findings support the idea that similar mechanisms underlie the occurrence of obstructive and central apnoeas (Cherniack, 1981; Onal & Lopata, 1982; Chapman et al. 1988). We found that in rigid and nearly rigid airways, increases in controller gain caused instability in feedback control and eventually produced recurrent central apnoeas which were intensified and prolonged by occurrence of arousals. The results from the model thus support the view of Younes (2004) that arousal can be destabilizing. When elasticity was low, increases in controller gain produced obstructive apnoeas but when increased still more could even give rise to recurrent central apnoeas. This is consistent with observations on patients by Onal & Lopata (1982), Hudgel et al. (1998) and Younes et al. (2001) on the role of loop gain in the production of obstructed apnoeas. It is also consistent with observations on patients that apnoeas that are apparently central may conceal an underlying airway obstruction (Issa & Sullivan, 1986; Tkacova et al. 2001). Moreover, in the simulations, increasing the elasticity of the upper airways could convert obstructive to central apnoea. This is consistent with the observation that in some patients with obstructive sleep apnoea, if the upper airway obstruction is bypassed then the obstructive sleep apnoea changes to central apnoea (Onal et al. 1981; Onal & Lopata, 1982). Hyperoxia increased the stability by reducing controller gain, thus doing away with obstructed breathing if the airway was not too compressible (Gold et al. 1985). However, as reported in patients, hyperoxia prolonged apnoeic cycles when controller gains were too high or elastance very low (Longobardo et al. 1982; Gold et al. 1985; Farney et al. 1992; Sakakibara et al. 2005). In contrast, hypoxia by increasing controller gain could convert obstructed apnoeas to central apnoeas, in agreement with observations reported in humans (Warner et al. 1987).
Increases in elastance of the compressible segment caused by neurochemical drives to the upper airway muscles might also eliminate obstructive apnoeas. If the airway was not too flaccid, the increase in elastance induced by the tethering response of the upper airway muscles to these drives was sufficient to maintain airway stability.
Younes (2004) has shown that in many patients obstructive apnoeas may be overcome by inhalation of CO2 without incurring arousal. As shown in Fig. 10, we could also reproduce these observations when the increase in elastance produced by the upper airway muscles grew as a function of P to some exponent so that it countered the reduction in pressure inside the compressible segment as a result of the increased ventilatory drive caused by the inhalation of CO2. Only small increases in the effects of CO2 on elastance were needed to do this.
In addition, in the model, as seen in humans, recurrent obstructive sleep apnoeas were more frequent when upper airway resistance was increased (Lofaso et al. 2006).
More than 30 years ago, we proposed a mathematical model of obstructive sleep apnoea that was based on the opposing actions of upper airway and chest wall muscles (Longobardo et al. 1982). In that model, however, neither arousals nor the mechanical properties and configuration of the upper airway were explicitly considered. Obstructive apnoeas in that model occurred whenever the activities of upper airway and pump muscles were out of balance, which was more likely to occur when instability of chemical control was present. As far as we know, the present study is the first time that both the dynamic and the static mechanical properties of the upper airway have been included in a model of the regulation of breathing during sleep. The present model also includes changes in alertness, as well as changes in chemical drives (Longobardo et al. 2001). There are other mathematical models of the pressure variation in the pharynx but to our knowledge they do not include the effects of chemical drive or arousal and are static; that is, they do not consider effects of dynamically varying the size of the flexible segment (Gavriely & Jensen, 1993; Aittokallio et al. 2001; Huang et al. 2005; Sung et al. 2006; Jeong et al. 2007). This is further discussed in the Appendix, subsection on the Pharyngeal compressible segment.
Limitations of the model
Expiratory effects of obstructionMartin et al. (1980) described six obese subjects who had obstructive apnoeas in expiration preceded by decreases in central drives. This kind of occurrence was simulated by our earlier model, in which the upper airway was either open or closed depending on the balance between the collapsing force produced by the inspiratory pump muscles on the laryngopharynx and the dilating action of the upper airway muscles. The present model has an elastic segment, which has a range of diameters and reproduces better the directly observed changes in the upper airway during a breath. In patients with obstructive sleep apnoea, upper airway diameter changes much more during a breath than in healthy subjects, with airway collapse occurring during inspiration and widening during expiration (Morrell & Badr, 1998). However, much remains unknown about the increase in expiratory resistance that has been observed in some patients with sleep apnoea (Sanders et al. 1985).
Upper airway mechanics The upper airway is more complicated than the flexible tube model used in our mathematical simulations. It is complexly folded, which could cause it to behave differently from our flexible segment (Kamm & Pedley, 1989; Van Hirtum et al. 2005; Huang et al. 2005). In addition, we have made several simplifying assumptions. For example, we have assumed that there are no frictional losses as air flows through the compressible segment. We try to account for these through increases in upper airway resistance, which lowers pressure before the compressible segment and also after the segment. Any flow limitation we have during obstructions because of the obstruction per se is the result of stoppage of airflow. There are multiple sites of narrowing in the upper airway (Hudgel, 1986). The convolutions of the larynx and pharynx will affect pressure and flow, and these have not been taken completely into account though we do consider the aggregated resistance such structures may produce. These resistance changes may be magnified during sleep. Changes from nasal to mouth breathing alter resistance to upper airway flow much more during sleep, while lowering the nasal resistance with vasoconstrictive nose drops eliminates obstructive apnoeas in some patients (Kerr et al. 1992; Ayappa & Rapoport, 2003).
Upper airway muscle function In most simulations, we have assumed that the effects of the upper airway muscle contractions on elastance are linear. This assumption is far from certain. There are multiple muscles that can affect the cross-sectional area of the upper airways. Studies of EMG activity of single upper airway muscles, usually the genioglossus, often show a linear change in muscle electrical activity over a restricted range of chemical stimulation (e.g. Onal et al. 1981). The data in humans are limited, and it is not certain whether a more non-linear response would be seen over a wider range of stimulation. We do not know the relationship between upper airway muscle shortening or force production and elastance changes (Series et al. 1999). It is likely that the relationship is different for different upper airway muscles. All of these possibilities could produce curvilinear effects on elastance with upper airway muscle contraction.
The observation that large increases in oxygen levels could sometimes produce obstructive breathing could be explained if hypoxia had a greater stimulating effect on upper airway muscles than on chest wall pump muscles. Differences in the response of some upper airway muscles and the diaphragm to hypoxia have been described (Bruce et al. 1982). An exponential effect by the upper airway muscles as a group might also occur if each muscle had a different threshold of drive (P or P) for activation and their activity summed with increasing hypercapnia and hypoxia.
Inhalation of CO2 increases the velocity of flow through the upper airways and thereby also makes more negative the pressure inside the compressible segment in the upper airway. The present model does not include mechanoreceptor effects on the upper airway. It has been shown that the activity of some upper airway muscles rises linearly with increasing negative pressure via a reflex effect (Malhotra et al. 2000). If an overall increase in activity continued with very negative pressure, this by itself could explain stabilization of breathing with inhalation of CO2.
Responses to reflexes might be different in patients with obstructive sleep apnoea compared with normal subjects. Genioglossus activity is greater in awake patients with obstructive sleep apnoea than in normal subjects but decreases more with sleep (Fogel et al. 2005). Several studies have demonstrated a blunted response of the upper airway muscles to obstruction or negative pressure in patients with obstructive sleep apnoea compared with normal subjects (Ryan & Bradley, 2005; White, 2005).
Sleep–arousal dynamics Simulations with the model show that the dynamics of the arousal response may be quite important in setting the stage for central and obstructive apnoeas and in determining their duration. The model suggests that apnoeas are less likely to be recurrent when wakefulness after arousal is short. Arousal thresholds are also not well known. It is clear that they vary with sleep stage and even in the same sleep stage may show a cyclic alternating pattern.
Very little is known about the length of such awakenings. Current methods of evaluating the architecture of sleep may have to be refined in order to obtain the data needed. There is also considerable evidence that arousals may be graded and that changes in, for example, cardiovascular indices may occur without changes in the α-rhythm of the EEG (Togo et al. 2006). Whether respiratory changes occur in the absence of altered α-rhythm, as others have proposed, is unknown but needs further investigation. It is also likely that parameters such as upper airway elastance vary over a night's sleep as body and neck position change. Changes in controller gain may occur as alertness and arousability cycle. The mathematical model described in this paper should be useful in predicting the impact of these changes on breathing during sleep.
The symbols used in the mathematical model described in this paper are shown in Table A1.
Steady-state conditions for all controller gains are shown in Table A2.
Pharyngeal compressible segment
The shape of the pharynx has been compared to a ‘Venturi Tube’ by Fajdiga, who suggested that Bernoulli's Theorum plays a major role in snoring and in obstructive sleep apnoea (Fajdiga, 2005), to a ‘turbulent jet’ (Jeong et al. 2007), to a flexible collapsible tube to model a snorer's upper airway (Aittokallio et al. 2001), to a Starling resistor (Ayappa & Rapoport, 2003) and to a semicircular convergent–divergent passage (Van Hirtum et al. 2005; Chouly, 2005). Van Hirtum et al. (2005) found that the ‘flow theories used in the case of a Starling resistor provided poor agreement’ with their pressure measurements. The purpose of the Van Hirtum model was to calculate pressure distribution within the upper airway so that upper airway deformation could be assessed (but this was not done in their paper). They did not include deformation of the upper airway by neural inputs or fluid dynamics. Huang et al. (2005), using data from imaging studies, developed anatomically realistic models of the upper airway which will be useful in surgical approaches to the treatment of obstructive sleep apnoea.
In our model, the pharynx in the upper airway is considered to contain a convergent–divergent compressible flexible segment like a Venturi Tube (see Fig. 1). We have written equations of flow using Bernoulli's Equation, assuming an inviscid fluid. Friction, in the form of viscosity and flow separation, would decrease pressure into the flexible segment minimal area and reduce the pressure recovery out of the flexible segment in the model. Corrections for viscosity or flow separation cannot be determined analytically. Although Bernoulli's Equation is for an inviscid fluid, its usefulness is extended to practical situations by including an energy loss term or by expressing the pressure drop due to friction or separation in nozzles, orifices and Venturi Tubes in terms of a coefficient of discharge. We do not correct for viscosity or flow separation by coefficients of discharge in our equations, since any statements about these factors in the pharynx would be guesses. Decreases in pressure in the flexible segment can be handled in our model by increasing upper airway resistance, as shown in eqn (7) for Pua. As shown in Fig. 11A and B, increased upper airway resistance increases the likelihood of obstruction.