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ISSN 0015-4628, Fluid Dynamics, 2011, Vol. 46, No. 1, pp. 16–23. © Pleiades Publishing, Ltd., 2011. Original Russian Text © A.I. Dyachenko, G.A. Lyubimov, I.M. Skobeleva, M.M. Strongin, 2011, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2011, Vol. 46, No. 1, pp. 20–28.

Generalization of the Mathematical Model of Lungs for Describing the Intensity of the Tracheal Sounds during Forced Expiration A. I. Dyachenko, G. A. Lyubimov, I. M. Skobeleva, and M. M. Strongin Received September 14, 2010

Abstract— The possibility to relate the nature of the forced expiration tracheal sounds with the sound radiation by a separated flow that arises in the region of dynamic trachea constriction during forced expiration is investigated. A mathematical model of forced expiration is used for estimating. The calculated form of the time dependence of the sound intensity during forced expiration qualitatively corresponds to the experimental dependence obtained experimentally for normal subjects. The results should be taken into account in the physical explanation of tracheal sound generation mechanisms and in the justification of using the tracheal sound characteristics in the diagnostics of human lung pathologies. DOI: 10.1134/S0015462811010029 Keywords: mathematical model, respiration mechanics, forced expiration test, flow-volume curve, lungs, respiratory tract, diagnostics, tracheal sounds.

During the forced expiration test, one of most widespread in the practice of respiratory examination, the patient performs a fastest exhalation after a maximal inhalation. The result of this maneuver is described by means of a flow-volume curve which represents a relationship between the volume expiration rate V˙ measured near the patient’s mouth and the relative expiration volume V /Vmax at the same time (V is the expired volume and Vmax is the maximal expired volume). The outward shape of this curve and a set of its numerical characteristics form a basis for diagnostic conclusions concerning the nature and localization of the lung pathology. The mathematical model of the forced expiration maneuver justified in [1, 2] makes it possible to estimate the effect of physical lung parameters on the shape of the flow-volume curve [3] and analyze the regularities of the coughing act [4]. On the basis of this model the dynamics of restoring the lung characteristics of a smoker that stopped smoking was described [5], etc. About 10 years ago, in the literature, studies concerning the investigation of acoustic effects observed during the human forced expiration began to appear systematically [6–11]. These investigations are practically stimulated by a desire to enhance the informativeness of the widely used forced expiration test by simultaneously recording the intensity and structure of the sounds that arise in the respiratory system. Different mechanisms that may be the source of the sound measured on the patient’s neck during forced expiration (tracheal sound) were analyzed and estimated [9, 12–14]. Simultaneously, medical investigations in which the correlation between different parameters characterizing the tracheal sounds and various lung pathologies was analyzed were carried out [12, 15, 16]. In spite of certain achievements, it is still not quite clear to which physical effects the recorded tracheal sound parameters are related and, hence, in which way changes in these parameters are related with specific lung pathologies and their localization. In this study, using a generalization of our mathematical model [1, 2], we attempt to describe how the tracheal sound intensity changes during forced expiration. Comparing the calculation results and the 16

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Fig. 1. Trachea passage cross-sectional area A vs. transmural pressure ptm [17].

experimental data obtained on normal subjects, we conclude that the forced expiration tracheal sounds are mainly related to the acoustic effects produced by the separated flow in the region of dynamic trachea constriction during the forced expiration. 1. DYNAMIC TRACHEA CONSTRICTION DURING FORCES EXPIRATION AND ITS EFFECT ON THE INTENSITY OF THE TRACHEAL SOUNDS The mathematical model of forced expiration we use in the present study takes into account the effect of a possible dynamic constriction of the passage cross-section of the trachea due to its compliant back wall if the transmural pressure ptm on this wall becomes smaller than a certain critical value. The transmural pressure is the difference between the tracheal p1 and the pleural p pl pressures, that is, ptm = p1 − p pl . This property of the trachea is characterized by the dependence of the passage cross-sectional area A on the transmural pressure, whose qualitative shape under static conditions is shown in Fig. 1 [17]. The data presented are obtained on excised trachea specimens inside which the pressure was varied. In [18], it was experimentally shown that during forced expiration the trachea cross-section changes significantly. In Fig. 2 taken from [18] the measured trachea cross-section shape is shown at different moments of the forced expiration process. The relative passage cross-sectional area α = A/A0 , where A0 is the trachea cross-sectional area at a positive transmural pressure, is plotted along the ordinate axis and the transmural pressure ptm along the abscissa axis. In [1, 2], the effect of dynamic trachea constriction was described by a maximally simple equation

α˙ = (ptm − ptm (α )st )/μ .

(1.1)

Here, μ is a generalized viscosity of the back trachea wall tissue and ptm (α )st is the function under static conditions (Fig. 1). In model [1, 2] the trachea constriction is modeled by a circular diaphragm whose passage cross-section A(t) is determined by Eq. (1.1). The qualitative and quantitative aspects of changing the trachea cross-sectional area during forced expiration are illustrated in Fig. 3, in which the dependences α = α (t), ptm = ptm (t), V˙ = V˙ (t), and α = α (ptm ) calculated from model [1, 2] for one of forced expiration maneuvers performed by one subject are presented (see the procedure of finding the lung model parameters for a specific maneuver, for example, in [5]). From Fig. 3d it can be seen that due to the viscosity of the back wall the dependence of the trachea passage crosssection on the transmural pressure α = α (ptm ) is substantially different from the same dependence under static conditions αst (points 1 in Fig. 3). Moreover, in the beginning of the maneuver the characteristic time of a change in the rate V˙ and the transmural pressure is much smaller than the characteristic time of a change in the trachea cross-section. The forced expiration maneuver is characterized by a significant duration of trachea constriction α ∼ 0.1–0.2 during which the flow-rate V˙ changes substantially. From the example considered, we can conclude that for a significant part of the forced expiration time a separated gas flow is realized near the trachea constriction. Since the flow compression and the velocity are FLUID DYNAMICS

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Fig. 2. Trachea cross-section shape at different moments of forced expiration (along the ordinate axis the relative passage cross-sectional area α = A/A0 and along the abscissa axis the transmural pressure ptm are plotted) [18].

high, we can anticipate a considerable sound generation in this region. Since during measuring the tracheal sounds the microphone is located near this high-power sound emitter, we can suggest that the characteristics of the tracheal sounds (their intensity and spectrum) will be mainly determined by the characteristics of the separated flow in the region of dynamic trachea constriction. In order to estimate the acoustic characteristics of the tracheal sounds under forced expiration conditions, the model [1, 2] should be supplemented with relations determining the sound characteristics in the complex flow near the constriction of a channel of complex shape. These relations can be either calculated or obtained by processing experimental data on blowing the corresponding constrictions with recording the corresponding acoustic characteristics. If such relations are known, then, using the calculation results, we will be able to estimate from changes in V˙ (t) and α (t) in which way the characteristics of the tracheal sounds change with time. Unfortunately, the only study in which the characteristics of the sound emitted by the constriction of the respiratory tract were investigated experimentally that we could find in the literature was paper [19] devoted to modeling the glottis and investigating the characteristics of the sound associated with this constriction. However, the data presented in that paper were insufficient to estimate the intensity of the tracheal sounds since the experiments described in [19] were carried out only at relatively low rates. Therefore, we performed special aerodynamic blows of circular diaphragms with various relative constrictions for various flow-rates typical of the trachea during forced expiration. 2. EXPERIMENTAL DETERMINATION OF THE INTENSITY OF THE SOUND EMITTED BY A RESPIRATORY TRACT CONSTRICTION A physical model was made with account for the geometric and physical parameters of the trachea segment. The model (Fig. 4) consisted of an entrance segment 1 in which a stabilizer 2 damping noises that arise in the main was mounted. Farther away a feeding pipe 3 of diameter D1 equal to 18 mm, which corresponds to the human trachea, was located. By means of a joint unit 4 pipe 3 was connected with a measuring pipe 5 whose diameter and length were equal to 18 and 95 mm, respectively. In the joint unit 4, between pipes 3 and 5, removable circular diaphragms 7 with holes of cross-sectional areas equal to 5, 10, 20, 30, 50, 70, and 90% of the cross-sectional area of the measuring pipe were mounted. On the inner surface of the measuring pipe, at a distance of 20 mm from the diaphragm, a microphone 6 was located. In the feeding pipe, ahead of the diaphragm, the pressure was measured using a pressure sensor 8. The air volume flow-rate through the model was measured using a float-type flowmeter (not shown in Fig. 4). The measurement setup was made on the basis of a computer station which included a 80-channel 16-digital analog-digital transducer (“National Instruments”) of USB 6255 type with a maximum signal transduction frequency of 1.00 MS/s. As a sensor of the sound signal an electret microphone with a transmission band of 20–15000 Hz was used. The pressure was measured using an absolute pressure sensor IKDTDa-1500 with a measurement limit of 1500 mm Hg. The air volume flow-rate through the model was measured using a float-type rotameter RM-40 UZ. FLUID DYNAMICS

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Fig. 3. Time t dependence during forced expiration of the relative trachea passage cross-sectional area α (a), the transmural pressure ptm (b), and the volume flow-rate V˙ (c); dependence α (ptm ) (d).

The electric signals from the microphone and the pressure sensor were transmitted through amplifiers to the analog-to-digital converter and written down in a data file. Negative microphone signal values were then multiplied by −1 and the average total signal value was determined. This value is called the sound intensity S. The measurement software was made in a graphic LabVIEW environment. The experiment was organized as follows: the setup was speeded up to the given flow-rate, the pressure was measured, and the sound level measurement channel was switched on. The time sampling rate was equal to 0.1 μ s (10000 Hz) and the total number of data samples was equal to 16384 (the time of microphone signal recording was 1.6384 s). The experiment makes it possible to obtain arrays of data on the dependence of the sound intensity on the passage hole area of a circular diaphragm and the blown air volume flow-rate (Fig. 5). Since the microphone and the measurement system were not calibrated, in Fig. 5 the sound intensity S is shown at a certain conventional scale. We note that we used circular diaphragms, not models of real trachea passage cross-sections during forced expiration which could be taken, for example, from [18]. There were two reasons: (a) the individual shape of the real passage cross-sections cannot be known and (b) in the forced expiration model [1, 2] the flow hydraulics were modeled using formulas for circular diaphragms. Thus, the circular diaphragms were taken in our acoustic investigation for a uniformity of model elements. Supplementing model [1, 2] with the experimental dependence of the sound intensity on the flow-rate ˙ V and the relative tract constriction α (Fig. 5) makes it possible to estimate in each specific maneuver the lung model parameters for any specific patient, unmeasurable directly, and the parameters characterizing a specific maneuver (forces developed by respiratory muscles), as well as the time dependence of the tracheal sound intensity measurable independently. Obviously, using this generalization of the model, it is impossible to obtain an information about the spectrum of the tracheal sounds. FLUID DYNAMICS

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Fig. 4. Diagram of the physical model and the measurement setup.

Fig. 5. Sound intensity S vs. volume flow-rate of blown air V˙ for various diaphragm passage cross-sectional areas (family of curves).

3. EXAMPLE OF COMPARISON OF THE CALCULATED AND EXPERIMENTAL DATA ON THE INTENSITY OF THE TRACHEAL SOUNDS IN A SPECIFIC FORCED EXPIRATION MANEUVER For comparison, one forced expiration maneuver of a normal subject was taken. Simultaneously with the standard measurement of the flow-volume curve the tracheal sounds were recorded using a microphone fixed on the neck. Moreover, the patient performed the standard spirographic test and the airway resistance during quiet breathing was also measured. The microphone signal was processed by the same method as in the experimental measurement of the sound intensity described in Section 2. The data were averaged over the time interval Δt = 0.1 s. The experimental curve of the tracheal sound intensity S so obtained is presented in Fig. 6 for the maneuver considered at each measurement point. As mentioned above, using the standard method of determining the individual lung model parameters [5], for the maneuver considered the functions presented in Fig. 3 were, in particular, determined: α = α (t) (a), ptm = ptm (t) (b), V˙ = V˙ (t) (c), and α = α (ptm ) (d). In Fig. 3c, the individual points V˙ = V˙ (t) representing FLUID DYNAMICS

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Fig. 6. Experimental time t dependence of the tracheal sound intensity S (points 1 calculated tracheal sound intensities after calibration).

Fig. 7. Calculated dependence of the tracheal sound intensity S on time t.

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the values measured directly in the experiment are shown. The calculation results describe the experiment rather well and, hence, we can suggest that the other functions presented in Fig. 3 well describe the real variation with time of the presented parameters. Using the functions α (t) and V˙ (t) presented in Fig. 3a, 3c and the data shown in Fig. 5 we can estimate the change in the intensity of the tracheal sound, that is, to calculate the time dependence of the tracheal sound intensity S(t). The intensities of the tracheal sounds calculated for the same maneuver (Fig. 6) are given in Fig. 7. In order to compare the experimental and calculated dependences, we must find a relationship between the scales of the measuring circuits used in recording the tracheal sounds during forced expiration and the model experiments described in Section 2. To do this, we will assume that the calculated and experimental data coincide at one point, for example, at t = 0.3 s. The calculated intensities of the tracheal sounds after this calibration are shown in Fig. 6 (points 1). The fairly good quantitative agreement between the calculated and experimental distributions of the tracheal sound intensity testifies to the conclusion that the main mechanism of generation of these sounds may be the separated flow in the region of dynamic trachea constriction during forced expiration. However, certain details of the experimental time dependence of the tracheal sound intensity in the initial stage of the maneuver, in particular, its nonmonotonicity, cannot be described by the calculated curve. In order to find out if this effect is related to the insufficiency of the model or physical processes that are not taken in this study into account, further investigations are required. Summary. The performed investigation of the forced expiration tracheal sounds shows that one of the most important mechanisms of their generation may be the separated flow in the region of dynamic trachea constriction during forced expiration. The work was in part supported financially by the Russian Foundation for Basic Research (project No. 10-07-00486). REFERENCES 1. G.A. Lyubimov and I.M. Skobeleva, “Mathematical Model of Forced Expiration,” Fluid Dynamics 26 (4), 477–483 (1991). 2. G.A. Lyubimov, “Justification of a Model of an Inhomogeneous Lung for Forced Expiration,” Fluid Dynamics 34 (5), 632–640 (1999). 3. V.K. Kuznetsova, G.A. Lyubimov and I.M. Skobeleva, “Analysis of Certain Qualitative Effects Associated with Forced Expiration,” Fiziologiya Cheloveka 19 (5), 72–79 (1993). 4. G.A. Lyubimov and I.M. Skobeleva, “A Mathematical Model of Coughing for a Homogeneous Lung,” Fluid Dynamics 35 (5), 627–634 (2000). 5. G.A. Lyubimov, I.M. Skobeleva, G.M. Sakharova, and A.V. Suvorov, “On the Informativeness of the “FlowVolume” Curve of Forced Expiration,” Pulmonologiya, No. 2, 91–97 (2008). 6. H. Pasterkamp, S. Kraman, and G. Wodicka, “Respiratory Sounds. Advances beyond the Stethoscops,” Am. J. Respiratory and Critical Care Medicine 156 (3), 974–987 (1997). 7. S.S. Kraman, “The Forced Expiratory Wheeze. Its Site of Origin and Possible Association with Lung Compliance,” Respiration 44 (3), 189–196 (1983). 8. N. Gavriely, K.B. Kelly, J.B. Grotberg, and S.H. Loring, “Forced Expiratory Wheezes are a Manifestation of Airway Flow Limitation,” J. Appl. Physiol. 62 (6), 2398–2403 (1987). 9. N. Gavriely, T.R. Shee, D.W. Cugell, and J.B. Grotberg, “Flutter in Flow-Limited Collapsible Tubes: a Mechanism for Generation of Wheezes,” J. Appl. Physiol. 66 (5), 2251–2261 (1989). 10. V.I. Korenbaum, Yu.V. Kulakov, and A.A. Tagiltsev, “Acoustic Effects in the Human Respiratory System during Forced Expiration,” Akust. Zhurn. 43 (1), 78–86 (1997). 11. J.C. Hardin and J.L. Patterson, “Monitoring the State of the Human Airways by Analysis of Respiratory Sound,” Acta Astronaut. 6 (9), 1137–1151 (1979). 12. V.I. Korenbaum and I.A. Pochekutova, Acousto-Biomechanical Relationships in the Formation of Human Forced Expiration Noises (Dalnauka, Vladivostok, 2006) [in Russian]. FLUID DYNAMICS

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