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A model of a snorer's upper airway
Mathematical Biosciences 170 (2001) 79±90
www.elsevier.com/locate/mbs
A model of a snorer's upper airway Tero Aittokallio a,b,*, Mats Gyllenberg a,b, Olli Polo c a
b
Department of Mathematics, University of Turku, 20014 Turku, Finland Turku Centre for Computer Science TUCS, Lemmink aisenkatu 14A, 20520 Turku, Finland c Department of Physiology, University of Turku, 20520 Turku, Finland
Received 23 March 1999; received in revised form 31 October 2000; accepted 15 November 2000
Abstract In snorers, the physiologic decrease of postural muscle tone during sleep results in increased collapsibility of the upper airway. Measurement of nasal pressure changes with prongs is increasingly used to monitor ¯ow kinetics through a collapsing upper airway. This report presents a mathematical model to predict nasal ¯ow pro®le from three critical components that control upper airway patency during sleep. The model includes the respiratory pump drive, the stiness of the pharyngeal soft tissues, and the dynamic support of the muscles surrounding the upper airway. Depending on these three components, the proposed model is able to reproduce the characteristic changes in ¯ow pro®le that are clinically observed in snorers and nonsnorers during sleep. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Flow in collapsible tube; Flow limitation; Partial upper airway obstruction; Sleep apnea; Snoring; Upper airway muscles
1. Introduction Medical research has only recently discovered that breathing may become compromised during sleep. As soon as we fall asleep, there is a physiologic relaxation of all postural muscles, including those surrounding and supporting the upper airway. With less muscle tone during sleep, the upper airway of some individuals may collapse and cause signi®cant ¯ow limitation or complete cessation of air¯ow (sleep apnea). The upper airway patency during sleep is determined by the balance between collapsing and dilating forces. Inspiration creates a negative airway pressure which increases the tendency of the upper airway to collapse. Starling's resistor theory predicts that in a collapsible tube inspiratory *
Corresponding author. Tel.: +358-2 333 8668; fax: +358-2 333 8600. E-mail address: [email protected].® (T. Aittokallio).
0025-5564/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 6 2 - 6
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¯ow can increase with increasing driving pressure drop only up to a critical value for the transmural pressure, i.e., the pressure dierence between the inside and outside of the upper airway. With a higher driving pressure drop during inspiration a partial collapse occurs and there is no further increase in ¯ow despite greater inspiratory eort. In fact, with increased eort the upper airway is predisposed to further collapse that may lead to soft tissue ¯utter. The critical transmural pressure and the capability of the upper airway to resist collapse is individual and depends on the stiness of pharyngeal soft tissues and on the operation of the dilating muscles of the upper airway. Many factors may aect the subtle balance between the dilating and collapsing forces. Ethanol ingestion, sleep-promoting medication, or sleep deprivation predispose to collapse by reducing the muscle tone during sleep. Also fat deposits or ¯oppy connective tissue may weaken the ecacy of the dilator muscles. There are several clinical studies on upper airway pressure±¯ow relationship in patients with obstructive sleep apnea, see e.g., [1±3]. Some experimental investigations have applied a Starling's resistor setting to explain partial upper airway collapse [4] or snoring [5]. Mathematical models which describe collapsible tube behaviour can generally be classi®ed into two groups: lumped parameter models, which characterize the ¯ow by a number of time-dependent spatially invariant variables [6,7], and more complex models that allow the selected variables also to vary over one [8,9] or two [10,11], spatial dimensions. Recently, Pedley outlined some models and developments of the problem ®eld [12]. Theoretical models have also been applied to describe the mechanics of the upper airway during breathing disorders. These lumped parameter models assume that the upper airway can be represented by a single [13,14], or two [15,16], compliant segments. In this paper, we consider a physiologically more compatible model to the collapsible upper airway, which includes the longitudinal dimension, that is, we describe an extension to a one-dimensional model. While the previous theoretical studies focused mainly on the vibration of the upper airway wall during inspiration, our purpose is to predict the various ¯ow traces that commonly occur in genuine patient populations. The development of a new model of a collapsible upper airway is motivated by the increasing interest in using the nasal ¯ow pro®le to diagnose upper airway dysfunction during sleep. The nasal ¯ow signal is obtained by connecting nasal prongs to a pressure transducer. The nasal pressure changes induced by inspiration and expiration can be monitored throughout the sleep period. Several inspiratory ¯ow patterns have been identi®ed [17] but their pathophysiological signi®cance is not fully understood. Although many factors in¯uence the behavior of the upper airway during sleep, there are no simple diagnostic procedures to identify which factors are critical in each individual. Therefore, most patients today are being treated each night with a nasal continuous positive airway pressure (nCPAP) device, which eectively maintains upper airway patency irrespective of the cause of dysfunction. Development of more speci®c therapies requires speci®c information on individual key dysfunction. We believe that analysis of the nasal ¯ow pro®le could help to ®nd the treatment of choice for a given individual. Developing a mathematical model to predict the abnormal ¯ow pro®les that are produced by snorers during sleep studies is an important step in learning the pathophysiology of the snorer's disease. The proposed model consists of a set of partial dierential equations with supplementary boundary conditions. Our model is able to interpret the normal behaviour of the upper airway as well as the partial collapse and ¯ow oscillation. We compare the predictions of the model with data measured with nasal prongs from sleeping subjects, see Fig. 1. The nasal prong/pressure
Nasal flow
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T. Aittokallio et al. / Mathematical Biosciences 170 (2001) 79±90
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Nasal flow
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Fig. 1. Examples of ¯ow contours measured from sleeping subjects with nasal prongs. The signal below the zero-¯ow line (dotted line) represents inspiration, above is expiration. The ¯ow unit is arbitrary, and the sampling frequency is 50 Hz. The duration of a respiratory cycle is about 4 s. (a) Sinusoidal inspiratory ¯ow shape in a snorer. Note the pause at end-expiration. (b) Partial collapse with two peaks, and (c) with a plateau. (d) Flow oscillations during inspiration and expiration in a snorer with increased respiratory eorts. Partial expiration through the mouth is typical for most snorers. This manifests as smaller expiratory areas.
transducer measuring equipment is a commercial system developed for this particular purpose. It is a DC-coupled sensor with no lower limit of frequency response and upper limit well beyond the frequencies focused on in the current report. The zero-¯ow lines were manually set under visual control. No drift was observed up to 8 h of recording. The examples in Fig. 1 are recorded during sleep in snorers with increased respiratory eorts and active expiration. Therefore, expiration is short and subject to ¯ow oscillations. 2. The model We model the behaviour of the collapsible part of the upper airway as a ¯exible tube of length L mounted between rigid pipes with a cross-sectional area of Au , see Fig. 2. The tube is enclosed in a sealed chamber at an external pressure pe x; t, which is in general a function of the longitudinal coordinate x and time t. The driving pressure upstream is p0 t and the outlet of the tube is at the pressure p1 t. The variables of interest are the cross-sectional area of the tube A x; t, and the cross-sectionally averaged internal pressure p x; t and velocity u x; t. Euler's equation for conservation of mass for a ¯uid in a horizontal tube is oA ouA 0 ot ox and the momentum equation, i.e., Newton's law of motion, is
1
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Fig. 2. Sketch of a one-dimensional collapsible tube model. The collapsible part of length L is enclosed in a chamber at an external pressure pe . The driving pressure at the inlet of the tube is p0 and the outlet of the tube is at pressure p1 . Constant Au is the uniform cross-sectional area of the unstretched tube. The ¯ow is characterized by the temporally and spatially varying cross-sectional area A, velocity u and internal pressure p.
ou ou u ot ox
1 op q ox
f A; uu:
2
Here, q is the density of air (assumed incompressible) and the function f represents viscous friction. We ignore the possibility of turbulence and of ¯ow separation in the collapsible part, and use the quasi-steady estimates of Cancelli [8] for laminar ¯ow, that is, we assume that 8pm= qA for A P Au ; f A 3 2 8pmAu = qA for A < Au : Here, m is the viscosity of air. The third equation required to complete the system (1) and (2) for the three unknown variables A, p and u is the so-called tube law, which represents the elastic properties of the tube by the equation p
pe ps g A
j
o2 A ; ox2
4
where function ps x; t and constant j are proportional to the bending stiness and longitudinal tension of the tube wall, respectively [12]. Because we lack measurements on the elastic behaviour of the upper airway wall in response to the transmural pressure applied to it, we follow the tube law of Bertram and Pedley [6], see Fig. 3 (dotted line), which is based on static experiments using rubber tubes: ( for A P Au ; k A=Au 1 5 g A 3=2 for A < Au : 1 A=Au
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Fig. 3. The tube law (4) computed using (5) with k 150 (dotted line), and (6) with m 20 (solid line). The bending stiness and longitudinal tension of the tube wall are ps 0:2 and j 0, respectively.
Here, k represents the compliance of the tube when it is distended. As a matter of fact, we take a slightly modi®ed form of this law, see Fig. 3 (solid line), with continuous derivative in order to avoid diculties in the numerical integration procedure. More precisely, we choose g A A=Au m
A=Au
3=2
:
6
The positive constant m has mainly an eect on the distension capacity of the tube. When the transmural pressure is positive, the tube has a circular cross-section and is rather sti. When A falls below Au , however, the tube tends to collapse and becomes very compliant, until eventually becomes very sti again when A is very small. The system governed by Eqs. (1), (2) and (4) requires four boundary conditions, which are obtained by ®xing the pressure conditions at the inlet and the outlet of the collapsible part: p 0; t p0 t;
p L; t p1 t;
7
and by requiring that the cross-sectional area must equal that of the rigid tubes at both ends: A 0; t A L; t Au :
8
In order to model the behaviour of the upper airway during sleep-related breathing disorders, we next include the three necessary components of the human respiratory system in the model. Gas exchange in the lungs is powered by the respiratory muscles, which operate under the control of the respiratory centers in the brain stem. We assume that the respiratory pump induces sinusoidal pressure into the inlet of the collapsible part p0 t
PD sin 2pt=T ;
9
where PD represents the amplitude of the drive and T is the duration of one respiratory cycle. The deeper the respiration, the higher the value of the parameter PD , see Fig. 4(a). Each cycle starts
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Fig. 4. The form of the respiratory system functions. (a) Respiratory pump drive p0 of normal and deep breath. (b) Stiness of the soft tissues ps in collapsible and uncollapsible upper airway. (c) Overall eect of the supporting muscles pe in sleep and wakefulness.
with inspiration (negative driving pressure) and ends with expiration (positive driving pressure). We further assume that the outlet of the collapsible part is at atmospheric pressure, i.e., p1 t 0. The capability of the upper airway to resist collapse depends mainly on the stiness of the pharyngeal soft tissues and on the functioning of the muscles surrounding and supporting the upper airway. We model the basic stiness of the upper airway via the function ps x; t of the form ps x S cos 2px=L PS ;
10
where S represents the dierence in stiness between the ends and the middle part of the tube, and PS is the average stiness of the tube having higher values for uncollapsible upper airway and smaller values for collapsible upper airway, see Fig. 4(b). The upper airway muscles provide two types of support to the upper airway. One is tonic muscle activity that is constant throughout the respiratory cycle, but decreases as we fall sleep. The other is phasic muscle activity that provides extra support during inspiration, when the airway is most likely to collapse. The operation of the supporting muscles is included in the model by a negative external pressure pe x; t, for which we assume a form pe t
M sin 2pt=T
PM :
11
Here, PM represents the tonic muscle activity and M the phasic muscle activity. We model the decrease of muscle tone during sleep by decreasing the value of the parameter PM , see Fig. 4(c). The muscles are regulated in such a way that they support the airway most powerfully at peak inspiration t T =4 (mod T), and after that the muscle support decreases smoothly when approaching peak expiration t 3T =4 (mod T). During expiration, some muscles may even constrict the upper airway, acting as a brake to expiration.
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3. Numerical analysis The scales used for non-dimensionalization are as follows: longitudinal coordinate L, time T, area Au , velocity L=T , and pressure qL2 =T 2 . We further eliminate internal pressure from the system by substituting (4) into (2) and (7). Denoting the dimensionless variables and parameters as before, the governing equations (1), (2), and (4) take the form oA ouA 0; ot ox ou ou oA u ps xg0 A ot ox ox
k
o3 A ox3
ps0 xg A
f Au:
Here, the functions (3) and (6) are given by lA 1 for A P 1; f A lA 2 for A < 1; g A Am
A
3=2
;
2
where k jAu T = qL4 and l 8pmT = qAu are two dimensionless positive constants. The boundary conditions (7) and (8) become A 0; t A 1; t 1; o2 A 0; t pe t p0 t; ox2 o2 A k 2 1; t pe t; ox
k
where the upper airway functions (9)±(11) take the form p0 t
PD sin 2pt;
ps x S cos 2px PS ; pe t
M sin 2pt
PM :
Note that the initial area A x; 0 must satisfy the boundary conditions at x 0 and x 1. In all the numerical calculations, we take A x; 0 1 PM x 1 x= 2k and u x; 0 0. Numerical integration was done using MacCormack's explicit ®nite-dierence scheme [18]. Simulation was halted as soon as the system converged to a stable cycle. The spatial grid size in each numerical calculation was Dx 30 1 and the choice for the time grid size Dt was made to assure the stability of the calculation. To analyze the above system numerically, we ®xed the dimensional values of those parameters that we assume to be subject-independent: q 1:293 kg/m3 , m 16:7 10 6 N s/m2 , T 4 s, Au 0:49 10 3 m2 , L 0:1 m, j 0:01 N/m2 , i.e., k 0:606 and l 2:65. We also ®x some of the values of the dimensionless parameters: m 20, S 0:01, and M 0:03. The rest of the dimensionless parameters (i.e., PD , PS , and PM ) are used in modeling some important features of the upper airway.
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The ®rst experiment was to compare the operation of the collapsible and uncollapsible upper airway during wakefulness and sleep. For a uncollapsible and collapsible upper airway we chose the stiness parameter values PS 1:0 and PS 0:15, respectively. Muscle tone was assumed to decrease from the wakefulness value PM 0:1 to the sleep value PM 0:02. Also, the respiratory pump drive was assumed to increase from PD 0:4 to PD 0:5, representing the slightly increased respiratory eorts during sleep. In Fig. 5, we show the results in the form of velocity at the outlet of the collapsible part during one-and-a-half respiratory cycles. Fig. 5(a) demonstrates that operation of the non-compliant upper airway is not severely aected by sleep. Both curves represent a normal ¯ow pro®le, see Fig. 1(a). However, when the muscle tone in a subject with a collapsible airway is decreased as a consequence of the subject falling asleep, marked alterations in the inspiratory ¯ow shape are observed, see Fig. 5(b). The velocity curve at the outlet of the collapsible part has lost its sinusoidal form, and a partial collapse is generated at maximum inspiratory ¯ow. This change is compatible with observations in sleep recordings, where partial collapse with two
Fig. 5. Simulation results of collapsible and uncollapsible upper airway during wakefulness and sleep. (a) Uncollapsible upper airway during wakefulness (dotted line) and during sleep (solid line), and (b) corresponding events in a collapsible upper airway.
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peaks was the most common one [17], see Fig. 1(b). Dierent patterns of partial collapses can be generated by altering the parameter values. Fig. 6 shows the operation of a collapsible upper airway during sleep as a function of the longitudinal coordinate at dierent phases of the respiratory cycle, i.e., t 0; T =4; T =2; 3T =4 (mod T). From Fig. 6(a) one can see the motion of the collapse, and Fig. 6(b) shows the velocities along the upper airway. Note the phase shift in the velocity between the inlet and outlet of the collapsible part. As mentioned above, one can generate dierent kinds of ¯ow abnormalities in the upper airway by changing the parameter values PD , PS , and PM . In the following four examples, the operation of the supporting muscles was kept in sleep position, i.e., PM 0:02. Another kind of partial collapse can be caused with the stiness value PS 0:1 and drive PD 0:5, see Fig. 7(a). Such a partial collapse, which causes a plateau in the inspiratory ¯ow contour, is often regarded as the only type of partial collapse [19], see Fig. 1(c). Also, the mirror shape of this pro®le has been detected in snorers, especially in post-menopausal women. One can generate it using parameter values PS 0:2 and PD 0:25, see Fig. 7(b). An interesting phenomenon of the upper airway is ¯ow
Fig. 6. Simulation results of a collapsible upper airway during sleep at dierent phases of the respiratory cycle. (a) Cross-sectional area, and (b) velocity.
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Fig. 7. Upper airway ¯ow pro®les predicted by the model. (a) and (b) Partial collapse with an inspiratory plateau. (c) and (d) Low- and high-frequency ¯ow oscillation.
oscillation, which often occurs with increased breathing eort. Fig. 7(c) shows a low-frequency ¯ow oscillation pattern when the drive is increased to PD 1:0 and the stiness is PS 0:18. The amplitude of ¯ow oscillation is higher during inspiration than expiration, which is also supported by sleep studies in snoring individuals, see Fig. 1(d). The frequency of oscillations can be increased by stiening the upper airway, see Fig. 7(d), where PS 0:33. High-frequency ¯ow oscillations in the upper airway are responsible for the snoring sound, which was not analyzed in this study. 4. Discussion We have described a model for a collapsible upper airway. The model incorporates a number of new features as compared with similar models that have appeared in the literature. Firstly, our model incorporates not only the temporal but also the spatial variation in the cross-sectional area and the velocity in the upper airway. Secondly, we include the three components of the respiratory system which have the most signi®cant eect on the operation of the upper airway: the respiratory pump drive (both the inspiratory and the expiratory phase), the stiness of the pharyngeal soft tissues, and the overall support of the muscles surrounding the upper airway. By changing the parameter values PD , PS , and PM , which correspond to the three components, we were able to generate some typical phenomena of the upper airway in non-snoring and snoring subjects. The inclusion of the individually speci®c parameters is based on current knowledge on the determinants of upper airway patency during sleep. These characteristics are extremely dicult to measure directly but using the model the individual parameters could be estimated from the measured signals. This could in future help to choose the right therapy for given patients. Our model in the present paper does not provide any suggestion about which particular anatomic structures of the upper airway would be responsible for the ¯ow shape abnormalities.
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In comparison with earlier models, our approach is dierent in the sense that we do not measure the snoring sound but changes in ¯ow shape. Although snoring has been claimed to occur typically during inspiration and only rarely during expiration [13], our recordings in genuine snorers indicate that ¯ow oscillations frequently occur both during inspiration and expiration (Fig. 1(d)). However, the oscillation frequencies and amplitudes are usually dierent during inspiration and expiration, which may explain why the snoring sounds, which represent frequencies much higher than the ¯ow oscillations we analyzed, are heard more frequently during inspiration. In our model, the proportions of inspiratory and expiratory ¯ow oscillations are controlled by the asymmetry of the tube law (Fig. 3). We have presented the numerical results in terms of the velocity at the outlet of the collapsible part (Figs. 5 and 7), which allows us to compare the predictions of the model with real data measured with nasal prongs (Fig. 1). A more detailed study of velocity and cross-sectional area along the collapsible part is also possible (Fig. 6) but harder to verify with measured data. The proposed model contains several physiological simpli®cations. First, we concentrated only on the collapsible part of the upper airway, although it also includes non-collapsible structures such as the nasal turbinates and the larynx. The impact of these structures could be modeled by including resistance and inertia in the rigid pipes at both ends of the collapsible part (Fig. 2). By denoting these values by RN ; RL ; IN , and IL , respectively, the boundary conditions for pressure in (7) would become p0 t
p 0; t RL u 0; t IL
ou 0; t; ot
p L; t
p1 t RN u L; t IN
ou L; t: ot
Here, the associated pressure changes in the rigid pipes are chosen to be linear [6]. Another simpli®cation is the assumption that each respiratory system function follows a sinusoidal form (Fig. 4). Although evidence from cats suggests that the sinusoidal form is a decent estimation of the driving pressure also during airway collapse [20], it is not known whether the central respiratory command remains sinusoidal through all stages of sleep. The model is also unable to predict the episodes of zero-¯ow that occur in patients with sleep apnea. This case could be considered by modifying the tube law. A major limitation of the current model is that it describes only one ¯ow shape at a time. This is due to lack of feedback information from chemoreceptors. In the intact control system, insucient ventilation results in increasing carbon dioxide (CO2 ) concentration in the alveoli, venous blood and tissues. The central chemoreceptor in the brain stem senses the increase of CO2 concentration, and transmits a stimulatory feedback signal to the respiratory muscles to decrease the CO2 concentration [21]. Depending on the degree of CO2 stimulation, secondary ¯ow pro®le changes will occur. Increasing stimulation to the respiratory pump muscles may lead to an increase of soft tissue ¯utter and worsening of ¯ow limitation. Increasing stimulation to the upper airway dilator muscles may improve ¯ow because of eective support against collapse. As a result, the ¯ow pro®le is under constant dynamic change from breath to breath. Despite its simpli®cations, the current model is capable of producing new insight into the behavior of the upper airway during sleep and wakefulness. By including some of the features discussed above, especially the chemoreceptor feedback, the model might explain some of the mechanisms by which the human respiratory system adapts to sleep-related upper airway narrowing in the long term. This is a challenge that we intend to tackle in the future.
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Acknowledgements The work of M.G. is supported by the Academy of Finland. The authors thank two anonymous reviewers for their constructive criticism of an earlier version of this paper, which led to an essential improvement of the model.
References [1] P.L. Smith, R.A. Wise, A.R. Gold, A.R. Schwartz, S. Permutt, Upper airway pressure-¯ow relationships in obstructive sleep apnea, J. Appl. Physiol. 64 (1988) 789. [2] S. Isono, T.R. Feroah, E.A. Hajduk, R. Brant, W.A. Whitelaw, J.E. Remmers, Interaction of cross-sectional area, driving pressure, and air¯ow of passive velopharynx, J. Appl. Physiol. 83 (1997) 851. [3] J.J. Hosselet, R.G. Norman, I. Ayappa, D.M. Rapoport, Detection of ¯ow limitation with a nasal cannula/ pressure transducer system, Am. J. Respir. Crit. Care Med. 157 (1998) 1461. [4] R. Condos, R.G. Norman, I. Krishnasamy, N. Peduzzi, R.M. Goldring, D.M. Rapoport, Flow limitation as a noninvasive assessment of residual upper-airway resistance during continuous positive airway pressure therapy of obstructive sleep apnea, Am. J. Respir. Crit. Care Med. 150 (1994) 475. [5] L. Huang, S.J. Quinn, P.D.M. Ellis, J.E. Ffowcs Williams, Biomechanics of snoring, Endeavour 19 (1995) 96. [6] C.D. Bertram, T.J. Pedley, A mathematical model of unsteady collapsible tube behaviour, J. Biomech. 15 (1982) 39. [7] J.P. Armitstead, C.D. Bertram, O.E. Jensen, A study of the bifurcation behaviour of a model of ¯ow through a collapsible tube, Bull. Math. Biol. 58 (1996) 611. [8] C. Cancelli, T.J. Pedley, A separated-¯ow model for collapsible-tube oscillations, J. Fluid Mech. 157 (1985) 375. [9] P. Morgan, K.H. Parker, A mathematical model of ¯ow through a collapsible tube ± I. Model and steady ¯ow results, J. Biomech. 22 (1989) 1263. [10] M. Heil, T.J. Pedley, Large post-buckling deformations of cylindrical shells conveying viscous ¯ow, J. Fluids Struct. 10 (1996) 565. [11] M. Heil, Stokes ¯ow in collapsible tubes: computation and experiment, J. Fluid Mech. 353 (1997) 285. [12] T.J. Pedley, X.Y. Luo, Modelling ¯ow and oscillations in collapsible tubes, Theoret. Comput. Fluid Dynamics 10 (1998) 277. [13] N. Gavriely, O. Jensen, Theory and measurements of snores, J. Appl. Physiol. 74 (1993) 2828. [14] L. Huang, J.E. Ffowcs Williams, Neuromechanical interaction in human snoring and upper airway obstruction, J. Appl. Physiol. 86 (1999) 1759. [15] R. Fodil, C. Ribreau, B. Louis, F. Lofaso, D. Isabey, Interaction between steady ¯ow and individualised compliant segments: applications to upper airways, Med. Biol. Eng. Comput. 35 (1997) 638. [16] S. Reisch, H. Steltner, J. Timmer, C. Renotte, J. Guttmann, Early detection of upper airway obstructions by analysis of acoustical respiratory input impedance, Biol. Cybern. 81 (1999) 25. [17] T. Aittokallio, O. Nevalainen, U. Pursiheimo, T. Saaresranta, O. Polo, Classi®cation of nasal inspiratory ¯ow shapes by attributed ®nite automata, Comput. Biomed. Res. 32 (1999) 34. [18] J.C. Strikwerda, Finite Dierence Schemes and Partial Dierential Equations, Wadsworth and Brooks/Cole Advanced Books and Software, Paci®c Grove, CA, 1989. [19] J.M. Montserrat, E. Ballester, H. Olivi, A. Reolid, P. Lloberes, A. Morello, R. Rodriguez-Roisin, Time-course of stepwise CPAP titration, Am. J. Respir. Crit. Care Med. 152 (1995) 1854. [20] N.M. Siafakas, H.K. Chang, M. Bonora, H. Gautier, J. Milic-Emili, B. Duron, Time course of phrenic activity and respiratory pressures during airway occlusion in cats, J. Appl. Physiol. 51 (1981) 99. [21] M.C.K. Khoo, A model-based evaluation of the single-breath CO2 ventilatory response test, J. Appl. Physiol. 68 (1990) 393.
www.elsevier.com/locate/mbs
A model of a snorer's upper airway Tero Aittokallio a,b,*, Mats Gyllenberg a,b, Olli Polo c a
b
Department of Mathematics, University of Turku, 20014 Turku, Finland Turku Centre for Computer Science TUCS, Lemmink aisenkatu 14A, 20520 Turku, Finland c Department of Physiology, University of Turku, 20520 Turku, Finland
Received 23 March 1999; received in revised form 31 October 2000; accepted 15 November 2000
Abstract In snorers, the physiologic decrease of postural muscle tone during sleep results in increased collapsibility of the upper airway. Measurement of nasal pressure changes with prongs is increasingly used to monitor ¯ow kinetics through a collapsing upper airway. This report presents a mathematical model to predict nasal ¯ow pro®le from three critical components that control upper airway patency during sleep. The model includes the respiratory pump drive, the stiness of the pharyngeal soft tissues, and the dynamic support of the muscles surrounding the upper airway. Depending on these three components, the proposed model is able to reproduce the characteristic changes in ¯ow pro®le that are clinically observed in snorers and nonsnorers during sleep. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Flow in collapsible tube; Flow limitation; Partial upper airway obstruction; Sleep apnea; Snoring; Upper airway muscles
1. Introduction Medical research has only recently discovered that breathing may become compromised during sleep. As soon as we fall asleep, there is a physiologic relaxation of all postural muscles, including those surrounding and supporting the upper airway. With less muscle tone during sleep, the upper airway of some individuals may collapse and cause signi®cant ¯ow limitation or complete cessation of air¯ow (sleep apnea). The upper airway patency during sleep is determined by the balance between collapsing and dilating forces. Inspiration creates a negative airway pressure which increases the tendency of the upper airway to collapse. Starling's resistor theory predicts that in a collapsible tube inspiratory *
Corresponding author. Tel.: +358-2 333 8668; fax: +358-2 333 8600. E-mail address: [email protected].® (T. Aittokallio).
0025-5564/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 6 2 - 6
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¯ow can increase with increasing driving pressure drop only up to a critical value for the transmural pressure, i.e., the pressure dierence between the inside and outside of the upper airway. With a higher driving pressure drop during inspiration a partial collapse occurs and there is no further increase in ¯ow despite greater inspiratory eort. In fact, with increased eort the upper airway is predisposed to further collapse that may lead to soft tissue ¯utter. The critical transmural pressure and the capability of the upper airway to resist collapse is individual and depends on the stiness of pharyngeal soft tissues and on the operation of the dilating muscles of the upper airway. Many factors may aect the subtle balance between the dilating and collapsing forces. Ethanol ingestion, sleep-promoting medication, or sleep deprivation predispose to collapse by reducing the muscle tone during sleep. Also fat deposits or ¯oppy connective tissue may weaken the ecacy of the dilator muscles. There are several clinical studies on upper airway pressure±¯ow relationship in patients with obstructive sleep apnea, see e.g., [1±3]. Some experimental investigations have applied a Starling's resistor setting to explain partial upper airway collapse [4] or snoring [5]. Mathematical models which describe collapsible tube behaviour can generally be classi®ed into two groups: lumped parameter models, which characterize the ¯ow by a number of time-dependent spatially invariant variables [6,7], and more complex models that allow the selected variables also to vary over one [8,9] or two [10,11], spatial dimensions. Recently, Pedley outlined some models and developments of the problem ®eld [12]. Theoretical models have also been applied to describe the mechanics of the upper airway during breathing disorders. These lumped parameter models assume that the upper airway can be represented by a single [13,14], or two [15,16], compliant segments. In this paper, we consider a physiologically more compatible model to the collapsible upper airway, which includes the longitudinal dimension, that is, we describe an extension to a one-dimensional model. While the previous theoretical studies focused mainly on the vibration of the upper airway wall during inspiration, our purpose is to predict the various ¯ow traces that commonly occur in genuine patient populations. The development of a new model of a collapsible upper airway is motivated by the increasing interest in using the nasal ¯ow pro®le to diagnose upper airway dysfunction during sleep. The nasal ¯ow signal is obtained by connecting nasal prongs to a pressure transducer. The nasal pressure changes induced by inspiration and expiration can be monitored throughout the sleep period. Several inspiratory ¯ow patterns have been identi®ed [17] but their pathophysiological signi®cance is not fully understood. Although many factors in¯uence the behavior of the upper airway during sleep, there are no simple diagnostic procedures to identify which factors are critical in each individual. Therefore, most patients today are being treated each night with a nasal continuous positive airway pressure (nCPAP) device, which eectively maintains upper airway patency irrespective of the cause of dysfunction. Development of more speci®c therapies requires speci®c information on individual key dysfunction. We believe that analysis of the nasal ¯ow pro®le could help to ®nd the treatment of choice for a given individual. Developing a mathematical model to predict the abnormal ¯ow pro®les that are produced by snorers during sleep studies is an important step in learning the pathophysiology of the snorer's disease. The proposed model consists of a set of partial dierential equations with supplementary boundary conditions. Our model is able to interpret the normal behaviour of the upper airway as well as the partial collapse and ¯ow oscillation. We compare the predictions of the model with data measured with nasal prongs from sleeping subjects, see Fig. 1. The nasal prong/pressure
Nasal flow
Nasal flow
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0
0
1
2
5
1
2 Time
0
6
0
1
2
3 Time
0
1
2
3 Time
(b)
0
0
(c)
3 4 Time
Nasal flow
Nasal flow
(a)
3
81
4
5
0
4
(d)
4
5
Fig. 1. Examples of ¯ow contours measured from sleeping subjects with nasal prongs. The signal below the zero-¯ow line (dotted line) represents inspiration, above is expiration. The ¯ow unit is arbitrary, and the sampling frequency is 50 Hz. The duration of a respiratory cycle is about 4 s. (a) Sinusoidal inspiratory ¯ow shape in a snorer. Note the pause at end-expiration. (b) Partial collapse with two peaks, and (c) with a plateau. (d) Flow oscillations during inspiration and expiration in a snorer with increased respiratory eorts. Partial expiration through the mouth is typical for most snorers. This manifests as smaller expiratory areas.
transducer measuring equipment is a commercial system developed for this particular purpose. It is a DC-coupled sensor with no lower limit of frequency response and upper limit well beyond the frequencies focused on in the current report. The zero-¯ow lines were manually set under visual control. No drift was observed up to 8 h of recording. The examples in Fig. 1 are recorded during sleep in snorers with increased respiratory eorts and active expiration. Therefore, expiration is short and subject to ¯ow oscillations. 2. The model We model the behaviour of the collapsible part of the upper airway as a ¯exible tube of length L mounted between rigid pipes with a cross-sectional area of Au , see Fig. 2. The tube is enclosed in a sealed chamber at an external pressure pe x; t, which is in general a function of the longitudinal coordinate x and time t. The driving pressure upstream is p0 t and the outlet of the tube is at the pressure p1 t. The variables of interest are the cross-sectional area of the tube A x; t, and the cross-sectionally averaged internal pressure p x; t and velocity u x; t. Euler's equation for conservation of mass for a ¯uid in a horizontal tube is oA ouA 0 ot ox and the momentum equation, i.e., Newton's law of motion, is
1
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Fig. 2. Sketch of a one-dimensional collapsible tube model. The collapsible part of length L is enclosed in a chamber at an external pressure pe . The driving pressure at the inlet of the tube is p0 and the outlet of the tube is at pressure p1 . Constant Au is the uniform cross-sectional area of the unstretched tube. The ¯ow is characterized by the temporally and spatially varying cross-sectional area A, velocity u and internal pressure p.
ou ou u ot ox
1 op q ox
f A; uu:
2
Here, q is the density of air (assumed incompressible) and the function f represents viscous friction. We ignore the possibility of turbulence and of ¯ow separation in the collapsible part, and use the quasi-steady estimates of Cancelli [8] for laminar ¯ow, that is, we assume that 8pm= qA for A P Au ; f A 3 2 8pmAu = qA for A < Au : Here, m is the viscosity of air. The third equation required to complete the system (1) and (2) for the three unknown variables A, p and u is the so-called tube law, which represents the elastic properties of the tube by the equation p
pe ps g A
j
o2 A ; ox2
4
where function ps x; t and constant j are proportional to the bending stiness and longitudinal tension of the tube wall, respectively [12]. Because we lack measurements on the elastic behaviour of the upper airway wall in response to the transmural pressure applied to it, we follow the tube law of Bertram and Pedley [6], see Fig. 3 (dotted line), which is based on static experiments using rubber tubes: ( for A P Au ; k A=Au 1 5 g A 3=2 for A < Au : 1 A=Au
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Fig. 3. The tube law (4) computed using (5) with k 150 (dotted line), and (6) with m 20 (solid line). The bending stiness and longitudinal tension of the tube wall are ps 0:2 and j 0, respectively.
Here, k represents the compliance of the tube when it is distended. As a matter of fact, we take a slightly modi®ed form of this law, see Fig. 3 (solid line), with continuous derivative in order to avoid diculties in the numerical integration procedure. More precisely, we choose g A A=Au m
A=Au
3=2
:
6
The positive constant m has mainly an eect on the distension capacity of the tube. When the transmural pressure is positive, the tube has a circular cross-section and is rather sti. When A falls below Au , however, the tube tends to collapse and becomes very compliant, until eventually becomes very sti again when A is very small. The system governed by Eqs. (1), (2) and (4) requires four boundary conditions, which are obtained by ®xing the pressure conditions at the inlet and the outlet of the collapsible part: p 0; t p0 t;
p L; t p1 t;
7
and by requiring that the cross-sectional area must equal that of the rigid tubes at both ends: A 0; t A L; t Au :
8
In order to model the behaviour of the upper airway during sleep-related breathing disorders, we next include the three necessary components of the human respiratory system in the model. Gas exchange in the lungs is powered by the respiratory muscles, which operate under the control of the respiratory centers in the brain stem. We assume that the respiratory pump induces sinusoidal pressure into the inlet of the collapsible part p0 t
PD sin 2pt=T ;
9
where PD represents the amplitude of the drive and T is the duration of one respiratory cycle. The deeper the respiration, the higher the value of the parameter PD , see Fig. 4(a). Each cycle starts
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Fig. 4. The form of the respiratory system functions. (a) Respiratory pump drive p0 of normal and deep breath. (b) Stiness of the soft tissues ps in collapsible and uncollapsible upper airway. (c) Overall eect of the supporting muscles pe in sleep and wakefulness.
with inspiration (negative driving pressure) and ends with expiration (positive driving pressure). We further assume that the outlet of the collapsible part is at atmospheric pressure, i.e., p1 t 0. The capability of the upper airway to resist collapse depends mainly on the stiness of the pharyngeal soft tissues and on the functioning of the muscles surrounding and supporting the upper airway. We model the basic stiness of the upper airway via the function ps x; t of the form ps x S cos 2px=L PS ;
10
where S represents the dierence in stiness between the ends and the middle part of the tube, and PS is the average stiness of the tube having higher values for uncollapsible upper airway and smaller values for collapsible upper airway, see Fig. 4(b). The upper airway muscles provide two types of support to the upper airway. One is tonic muscle activity that is constant throughout the respiratory cycle, but decreases as we fall sleep. The other is phasic muscle activity that provides extra support during inspiration, when the airway is most likely to collapse. The operation of the supporting muscles is included in the model by a negative external pressure pe x; t, for which we assume a form pe t
M sin 2pt=T
PM :
11
Here, PM represents the tonic muscle activity and M the phasic muscle activity. We model the decrease of muscle tone during sleep by decreasing the value of the parameter PM , see Fig. 4(c). The muscles are regulated in such a way that they support the airway most powerfully at peak inspiration t T =4 (mod T), and after that the muscle support decreases smoothly when approaching peak expiration t 3T =4 (mod T). During expiration, some muscles may even constrict the upper airway, acting as a brake to expiration.
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3. Numerical analysis The scales used for non-dimensionalization are as follows: longitudinal coordinate L, time T, area Au , velocity L=T , and pressure qL2 =T 2 . We further eliminate internal pressure from the system by substituting (4) into (2) and (7). Denoting the dimensionless variables and parameters as before, the governing equations (1), (2), and (4) take the form oA ouA 0; ot ox ou ou oA u ps xg0 A ot ox ox
k
o3 A ox3
ps0 xg A
f Au:
Here, the functions (3) and (6) are given by lA 1 for A P 1; f A lA 2 for A < 1; g A Am
A
3=2
;
2
where k jAu T = qL4 and l 8pmT = qAu are two dimensionless positive constants. The boundary conditions (7) and (8) become A 0; t A 1; t 1; o2 A 0; t pe t p0 t; ox2 o2 A k 2 1; t pe t; ox
k
where the upper airway functions (9)±(11) take the form p0 t
PD sin 2pt;
ps x S cos 2px PS ; pe t
M sin 2pt
PM :
Note that the initial area A x; 0 must satisfy the boundary conditions at x 0 and x 1. In all the numerical calculations, we take A x; 0 1 PM x 1 x= 2k and u x; 0 0. Numerical integration was done using MacCormack's explicit ®nite-dierence scheme [18]. Simulation was halted as soon as the system converged to a stable cycle. The spatial grid size in each numerical calculation was Dx 30 1 and the choice for the time grid size Dt was made to assure the stability of the calculation. To analyze the above system numerically, we ®xed the dimensional values of those parameters that we assume to be subject-independent: q 1:293 kg/m3 , m 16:7 10 6 N s/m2 , T 4 s, Au 0:49 10 3 m2 , L 0:1 m, j 0:01 N/m2 , i.e., k 0:606 and l 2:65. We also ®x some of the values of the dimensionless parameters: m 20, S 0:01, and M 0:03. The rest of the dimensionless parameters (i.e., PD , PS , and PM ) are used in modeling some important features of the upper airway.
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The ®rst experiment was to compare the operation of the collapsible and uncollapsible upper airway during wakefulness and sleep. For a uncollapsible and collapsible upper airway we chose the stiness parameter values PS 1:0 and PS 0:15, respectively. Muscle tone was assumed to decrease from the wakefulness value PM 0:1 to the sleep value PM 0:02. Also, the respiratory pump drive was assumed to increase from PD 0:4 to PD 0:5, representing the slightly increased respiratory eorts during sleep. In Fig. 5, we show the results in the form of velocity at the outlet of the collapsible part during one-and-a-half respiratory cycles. Fig. 5(a) demonstrates that operation of the non-compliant upper airway is not severely aected by sleep. Both curves represent a normal ¯ow pro®le, see Fig. 1(a). However, when the muscle tone in a subject with a collapsible airway is decreased as a consequence of the subject falling asleep, marked alterations in the inspiratory ¯ow shape are observed, see Fig. 5(b). The velocity curve at the outlet of the collapsible part has lost its sinusoidal form, and a partial collapse is generated at maximum inspiratory ¯ow. This change is compatible with observations in sleep recordings, where partial collapse with two
Fig. 5. Simulation results of collapsible and uncollapsible upper airway during wakefulness and sleep. (a) Uncollapsible upper airway during wakefulness (dotted line) and during sleep (solid line), and (b) corresponding events in a collapsible upper airway.
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peaks was the most common one [17], see Fig. 1(b). Dierent patterns of partial collapses can be generated by altering the parameter values. Fig. 6 shows the operation of a collapsible upper airway during sleep as a function of the longitudinal coordinate at dierent phases of the respiratory cycle, i.e., t 0; T =4; T =2; 3T =4 (mod T). From Fig. 6(a) one can see the motion of the collapse, and Fig. 6(b) shows the velocities along the upper airway. Note the phase shift in the velocity between the inlet and outlet of the collapsible part. As mentioned above, one can generate dierent kinds of ¯ow abnormalities in the upper airway by changing the parameter values PD , PS , and PM . In the following four examples, the operation of the supporting muscles was kept in sleep position, i.e., PM 0:02. Another kind of partial collapse can be caused with the stiness value PS 0:1 and drive PD 0:5, see Fig. 7(a). Such a partial collapse, which causes a plateau in the inspiratory ¯ow contour, is often regarded as the only type of partial collapse [19], see Fig. 1(c). Also, the mirror shape of this pro®le has been detected in snorers, especially in post-menopausal women. One can generate it using parameter values PS 0:2 and PD 0:25, see Fig. 7(b). An interesting phenomenon of the upper airway is ¯ow
Fig. 6. Simulation results of a collapsible upper airway during sleep at dierent phases of the respiratory cycle. (a) Cross-sectional area, and (b) velocity.
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Fig. 7. Upper airway ¯ow pro®les predicted by the model. (a) and (b) Partial collapse with an inspiratory plateau. (c) and (d) Low- and high-frequency ¯ow oscillation.
oscillation, which often occurs with increased breathing eort. Fig. 7(c) shows a low-frequency ¯ow oscillation pattern when the drive is increased to PD 1:0 and the stiness is PS 0:18. The amplitude of ¯ow oscillation is higher during inspiration than expiration, which is also supported by sleep studies in snoring individuals, see Fig. 1(d). The frequency of oscillations can be increased by stiening the upper airway, see Fig. 7(d), where PS 0:33. High-frequency ¯ow oscillations in the upper airway are responsible for the snoring sound, which was not analyzed in this study. 4. Discussion We have described a model for a collapsible upper airway. The model incorporates a number of new features as compared with similar models that have appeared in the literature. Firstly, our model incorporates not only the temporal but also the spatial variation in the cross-sectional area and the velocity in the upper airway. Secondly, we include the three components of the respiratory system which have the most signi®cant eect on the operation of the upper airway: the respiratory pump drive (both the inspiratory and the expiratory phase), the stiness of the pharyngeal soft tissues, and the overall support of the muscles surrounding the upper airway. By changing the parameter values PD , PS , and PM , which correspond to the three components, we were able to generate some typical phenomena of the upper airway in non-snoring and snoring subjects. The inclusion of the individually speci®c parameters is based on current knowledge on the determinants of upper airway patency during sleep. These characteristics are extremely dicult to measure directly but using the model the individual parameters could be estimated from the measured signals. This could in future help to choose the right therapy for given patients. Our model in the present paper does not provide any suggestion about which particular anatomic structures of the upper airway would be responsible for the ¯ow shape abnormalities.
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In comparison with earlier models, our approach is dierent in the sense that we do not measure the snoring sound but changes in ¯ow shape. Although snoring has been claimed to occur typically during inspiration and only rarely during expiration [13], our recordings in genuine snorers indicate that ¯ow oscillations frequently occur both during inspiration and expiration (Fig. 1(d)). However, the oscillation frequencies and amplitudes are usually dierent during inspiration and expiration, which may explain why the snoring sounds, which represent frequencies much higher than the ¯ow oscillations we analyzed, are heard more frequently during inspiration. In our model, the proportions of inspiratory and expiratory ¯ow oscillations are controlled by the asymmetry of the tube law (Fig. 3). We have presented the numerical results in terms of the velocity at the outlet of the collapsible part (Figs. 5 and 7), which allows us to compare the predictions of the model with real data measured with nasal prongs (Fig. 1). A more detailed study of velocity and cross-sectional area along the collapsible part is also possible (Fig. 6) but harder to verify with measured data. The proposed model contains several physiological simpli®cations. First, we concentrated only on the collapsible part of the upper airway, although it also includes non-collapsible structures such as the nasal turbinates and the larynx. The impact of these structures could be modeled by including resistance and inertia in the rigid pipes at both ends of the collapsible part (Fig. 2). By denoting these values by RN ; RL ; IN , and IL , respectively, the boundary conditions for pressure in (7) would become p0 t
p 0; t RL u 0; t IL
ou 0; t; ot
p L; t
p1 t RN u L; t IN
ou L; t: ot
Here, the associated pressure changes in the rigid pipes are chosen to be linear [6]. Another simpli®cation is the assumption that each respiratory system function follows a sinusoidal form (Fig. 4). Although evidence from cats suggests that the sinusoidal form is a decent estimation of the driving pressure also during airway collapse [20], it is not known whether the central respiratory command remains sinusoidal through all stages of sleep. The model is also unable to predict the episodes of zero-¯ow that occur in patients with sleep apnea. This case could be considered by modifying the tube law. A major limitation of the current model is that it describes only one ¯ow shape at a time. This is due to lack of feedback information from chemoreceptors. In the intact control system, insucient ventilation results in increasing carbon dioxide (CO2 ) concentration in the alveoli, venous blood and tissues. The central chemoreceptor in the brain stem senses the increase of CO2 concentration, and transmits a stimulatory feedback signal to the respiratory muscles to decrease the CO2 concentration [21]. Depending on the degree of CO2 stimulation, secondary ¯ow pro®le changes will occur. Increasing stimulation to the respiratory pump muscles may lead to an increase of soft tissue ¯utter and worsening of ¯ow limitation. Increasing stimulation to the upper airway dilator muscles may improve ¯ow because of eective support against collapse. As a result, the ¯ow pro®le is under constant dynamic change from breath to breath. Despite its simpli®cations, the current model is capable of producing new insight into the behavior of the upper airway during sleep and wakefulness. By including some of the features discussed above, especially the chemoreceptor feedback, the model might explain some of the mechanisms by which the human respiratory system adapts to sleep-related upper airway narrowing in the long term. This is a challenge that we intend to tackle in the future.
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Acknowledgements The work of M.G. is supported by the Academy of Finland. The authors thank two anonymous reviewers for their constructive criticism of an earlier version of this paper, which led to an essential improvement of the model.
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