Abstract
The fact that snoring and obstructive apnea only occur during sleep means that effective neuromuscular functioning of the upper airway during sleep is vital for the maintenance of unimpeded breathing. Recent clinical studies in humans have obtained evidence demonstrating that upper airway neural receptors sense the negative pressure generated by inspiration and “trigger,” with a certain delay, reflex muscle activation to sustain the airway that might otherwise collapse. These findings have enabled us to propose a model in which the mechanics is coupled to the neuromuscular physiology through the generation of reflex wall stiffening proportional to the retarded fluid pressure. Preliminary results on this model exhibit three kinds of behavior typical of unimpeded breathing, snoring, and obstructive sleep apnea, respectively. We suggest that the increased latency of the reflex muscle activation in sleep, together with the reduced strength of the reflex, have important clinical consequences.
 sleep reflex latency
 upper airway oscillation
 flowinduced vibration
 eigenvalues
snoring occurs in at least 15% of the adult (especially male) population and is easily recognized as an unpleasant lowfrequency noise that is accompanied by the vibration of the upper airways (2, 8). Wheezing is another example in which the airway vibrates during inspiration (3). The mechanics of both have been modeled with assumptions about the instability of the airway walls, with the wall tissue behaving like a piece of rubber. Mechanical modeling of this kind has contributed much to the understanding of physiological processes, but, in the present study, we show that it is unlikely to be sufficient for the understanding of snoring and upper airway obstruction during sleep. The present study argues the importance of neuromechanical coupling in the upper airway.
Inspiration is driven by neural drive to the inspiratory musculature creating a subatmospheric intrathorasic pressure; this pressure also tends to collapse the airway walls, predominantly at the oropharyngeal level. The system appears to counter this tendency by reflexly activating palatal and tongue muscles (that enlarge the upper airways), in addition to activating dilator muscles around the collapsing segments. The effectiveness of this mechanism is reduced in sleep. The reflexes have been known to exist in animals for some time, but have only been confirmed recently in humans. Horner and his colleagues (46) documented such reflexes in normal awake and sleeping subjects, resulting in activation of the tongue muscles (genioglossus). These authors found that local anaesthesia of the nose palate or pharynx reduced the power of the reflex, and they deduced that neural receptors in the mucosal surface of these structures served the local intraluminal pressure in the experimental paradigm used. Subsequently, Kobayashi and his colleagues (9) were able to show that activation of the genioglossus during inspiration enlarges the retroglossal space, at least in the anteroposterior diameter.
The palatoglossus muscle pulls the soft palate downward and forward to open the retropalatal space. Obstruction of the airway at the palatal level during sleep occurs commonly in obstructive sleep apnea and snoring. A reflex, similar to that which activates genioglossus, has now been described for palatoglossal activation with negative upper airway pressure (10); the power of this reflex is also decreased by upper airway surface anesthesia.
Horner et al. (5) showed that the magnitude of reflex activation of the genioglossus increases with the magnitude of the negative upper airway pressure; it is less effective during sleep. The onset of the muscle response had a latency of 40 ms in wakefulness and as much as 110 ms in sleep. These findings have now led us to propose a lumpedparameter model to describe the stability of airflow through the narrowest section of the upper airway. Solution of the eigenvalue problem suggests that, if the reflex latency is larger than a small critical level, then the reflex mechanism can always cause flutter, depending on what mechanical damping coefficients are assigned to the upper airway. We propose that this analysis contributes to the understanding of the flutter mechanism. We are able to define conditions for stable oscillation, flutter, and collapse in the upper airway; these states, we suggest, result in unobstructed breathing, snoring, and obstructive sleep apnea, respectively.
MODEL
Details of upper airway mechanics are complicated, and theoretical models of various sophistication have been tested. For example, Gavriely and Jensen (2) proposed a lumpedparameter model that aimed at explaining the process of airway collapse; Fodil et al. (1) studied the interaction of multiple discrete elements; whereas Huang (7) recently tested a continuum model. However, in all these models and in others relating to different aspects of snoring mechanisms the factor of neuromechanical coupling has been ignored. In an attempt to isolate the role of this factor, we believe that it is sufficient and illustrative to go back to the lumpedparameter model in which the vibrating part of the airway, such as the oropharynx during snoring, shown in Fig.1, is simplified as a “piston” with mass m, damping coefficientR, and spring stiffnessK, all measured over a unit surface area in contact with the airflow. The definitions of these parameters are for algebraic convenience and should not be confused with the modeling of a system with multiple degrees of freedom, as in Refs. 1and 7.
Air of density ρ_{0} flows over the piston at speed U. The pressure on the surface is a function of the piston displacement. If the piston moves into the passage by a small amount η, which simulates the partial collapse of the pharyngeal passage, the flow accelerates, and the steadyflow pressure perturbation (p′) is given by the Bernoulli equation
The dynamics of the piston system is governed by
Before we proceed to solve Eq. 1 , it is informative to discuss the range of values relevant to the realistic situations. For people who snore or suffer from obstructive sleep apnea, the upper airway routinely collapses, either partially or completely. That means thatK _{mech}, which takes account of the Bernoulli effect, is probably approaching zero or even becoming negative. The extent to which this is so depends on the structural stiffness K. For people with relatively limp upper airway walls,K is small, and air passage begins to collapse as soon as the inspiration reaches a critical level; that leads on to a higher local flow speedU and a diminishing value ofK _{mech}. The neuromuscular coefficient A and stiffness parameterK _{neuro} decrease as sleep becomes deeper and deeper. The neural stiffness termK _{neuro} also depends on the level of negative pressure experienced at the throat. At present, we have no clinical data on which an estimate ofK _{neuro} can be based. We must, therefore, content ourselves with the qualitative results of system behavior at high or low ratio ofK _{neuro}/K _{mech.}We also have very little idea of how the damping term ς should be specified. What we do know is that palatal snoring is typically ∼35 Hz, rising to some 100 Hz if part of the pharyngeal wall is involved.
The reflex latency Δt measured by Horner et al. (5) using a sudden drop in airway pressure provides a clue to the kind of magnitude one should expect during sleep. There is as yet no experimental evidence to support our conjecture that the neuromuscular function is linearly correlated with the perturbation pressure. The latency is ∼100 ms during sleep, which is 3 to 10 times the period of the snoring oscillation cycle, depending on the type of snore. However, there may well be other neuromuscular functions at work during breathing that contribute to the dynamics of the upper airway, such as the central neural control that precedes the respiratory maneuver. The collective effect of all the neural factors may possibly produce a smaller latency than that suggested by the isolated reflex mechanism, although the latter is normally perceived as a very quick reaction.
EIGENVALUES
The solution to Eq.
1
determines the characteristic angular frequency of the system from which a judgement can be made as to whether the system is stable to small disturbances. The roots of the equation are called eigenvalues (for frequency ω). Denote the real (ω_{r}) and imaginary (ω_{i}) parts of the complex eigen frequency by
Before we calculate the eigenvalues by numerical means, it is instructive to solve for a special case analytically. When the latency is very small, the timedelay term exp(−iωΔt) in Eq.1
is simply (1 −iωΔt). So the eigen frequencies are then
When Δt is large, say, several times the eigen oscillation cycle, Eqs.
2
and
3
have to be solved numerically. Perhaps the easier way to find the eigenvalues is to fix ω_{i} first and solve for ω_{r} by canceling out Δt fromEqs.4
and
5
. Then Δt is found as a result instead of being assumed a priori. The final result should be a functional relationship between ω and Δt or, simply, ω(Δt), for which typical graphs are shown in Fig. 2. In preparing the graphs, most dimensional quantities have been normalized by using a reference frequency (ω_{ref}) defined as
Figure 2 C plots the growth factor, which is the ratio of amplitude of one cycle to that of the previous cycle, against the latency expressed in terms of the number of cycles delayed. It can be seen that the curves forn = 0,1,2,... roughly correspond to solutions around n cycles of delay, ω_{r}Δt≈ 2nπ. Finally, for very small Δt, there is a range in which no unstable eigen frequency exists. That latency is just enough to overcome the level of mechanical damping given by ς.
SNORING AND OTHER RESPIRATORY CONDITIONS
Snoring is caused by the vibration of the upper airway. In linear analysis, such vibration is seen as the manifestation of unstable eigen vibration modes. We, therefore, identify snoring by an eigen frequency with a nonzero ω_{r} and negative ω_{i}. Normal breathing, on the other hand, should correspond to the stable modes, which have positive ω_{i}. The borderline is ω_{i} = 0, for whichEqs.2
and
3
can be rewritten as
The above range and the whole solution are meaningful only if the square root is real, i.e.,K
_{mech} >K
_{neuro}. Otherwise, the earlier assumption of instability border, ω_{i} = 0, is invalid and there will be an unstable mode. This means that one of the necessary conditions for snoring is that the mechanical stiffness should be the dominating force for the maintenance of the upper airway patency. Also, the term (
Another bad respiratory condition is obstructive sleep apnea, which features the complete closure of the passage at some point in the upper airway. The fact that airway stays closed for a considerable period of time means zero angular frequency, ω_{r} = 0, in our analysis. However, an unstable mode also requires a positive rate of growth for small disturbances, ω_{i} < 0. The ω_{r} = 0 means thatEq.
3
is automatically satisfied. Equation2
then gives
We first notice that ω_{i} = 0 is always a solution to Eq.
7
. The other solution is normally positive as Δt increases from 0 until the positive root is merged with ω_{i} = 0. After that, negative roots appear. The condition for the double root forEq.
7
is
In summary, we conclude that a mechanical system that is not normally susceptible to total collapse in the absence of latency will become susceptible when the latency exceeds 2ς/K _{neuro}.
Conclusions
First of all, the importance of having a sufficiently rigid airway cannot be overstated. A low tissue (structural) stiffnessK will cause initial narrowing of the upper airway, so that the Bernoulli effect lowers the effective stiffness K _{mech}. If K _{mech} is insufficient to maintain the stability of the airway, neuromuscular functions become crucial. However, these functions are very much reduced during sleep, and the muscle reflex mechanism can have a time delay of several cycles of oscillations experienced during snoring. A delayed restoring force can cause flutter as part of the force is transformed into negative damping. The eigenvalue calculation shows that, depending on the level of the damping coefficient ς, there is always an eigen mode that is at least marginally unstable. However, there exists a very small margin of damping coefficient for which stable breathing is guaranteed. Sleep apnea occurs when the total upper airway stiffness,K _{mech} +K _{neuro}, becomes negative, and the airway collapses totally during inspiration. A time delay of 2ς/K _{neuro} will render neuromuscular function powerless to stop the airway from collapsing.
Should our theoretical model be representative of real human airways, it may provide an avenue for finding new treatments for snoring and sleep apnea. New clinical data are needed to test our hypothesis.
Acknowledgments
The authors thank Professor A. Guz, whose advice and help were very helpful in keeping the research in touch with clinical practice.
Footnotes

Address for reprint requests and other correspondence: L. Huang, Dept. of Mechanical Engineering, The Hong Kong Polytechnic Univ., Hunghom, Kowloon, Hong Kong (Email: mmlhuang{at}polyu.edu.hk).
 Copyright © 1999 the American Physiological Society