A Dynamic Morphometric Model of the Normal Lung for Studying Expiratory Flow Limitation in Mechanical Ventilation

Transcription

1 Annals of Biomedical Engineering, Vol. 33, No. 4, May 2005 ( 2005) pp DOI: /s A Dynamic Morphometric Model of the Normal Lung for Studying Expiratory Flow Limitation in Mechanical Ventilation PAOLO BARBINI, 1 CHIARA BRIGHENTI, 2 GABRIELE CEVENINI, 1 and GIANNI GNUDI 3 1 Dipartimento di Chirurgia e Bioingegneria, Università di Siena, Viale Bracci 2, Siena, Italy; 2 Dipartimento di Elettronica, Informatica e Sistemistica, Università di Bologna, Viale Risorgimento 2, Bologna, Italy; and 3 Dipartimento di Elettronica, Informatica e Sistemistica, Università di Bologna, Via Venezia 52, Cesena, Italy (Received 16 July 2004; 12 October 2004) Abstract A nonlinear dynamic morphometric model of breathing mechanics during artificial ventilation is described. On the basis of the Weibel symmetrical representation of the tracheobronchial tree, the model accurately accounts for the geometrical and mechanical characteristics of the conductive zone and packs the respiratory zone into a viscoelastic Voigt body. The model also accounts for the main mechanisms limiting expiratory flow (wave speed limitation and viscous flow limitation), in order to reproduce satisfactorily, under dynamic conditions, the expiratory flow limitation phenomenon occurring in normal subjects when the difference between alveolar pressure and tracheal pressure (driving pressure) is high. Several expirations characterized by different levels of driving pressure are simulated and expiratory flow limitation is detected by plotting the isovolume pressure flow curves. The model is used to study the time course of resistance and total cross-sectional area as well as the ratio of fluid velocity to wave speed (speed index), in conductive airway generations. The results highlight that the coupling between dissipative pressure losses and airway compliance leads to onset of expiratory flow limitation in normal lungs when driving pressure is increased significantly by applying a subatmospheric pressure to the outlet of the ventilator expiratory channel; wave speed limitation becomes predominant at still higher driving pressures. Keywords Lung mechanics, Nonlinear morphometric model, Dynamic conditions, Expiratory flow limitation, Mechanical ventilation. INTRODUCTION In the last few years, research into the mechanics of breathing has been directly responsible for major advances in the understanding, diagnosis, prevention, treatment, and rehabilitation of many major respiratory diseases. 25 Mathematical modeling has led to major insights into spontaneous breathing and mechanical ventilation. Model uses range from reproducing and interpreting certain mechanisms under normal and pathological conditions 6,15,20,36 to Address correspondence to Paolo Barbini, Dipartimento di Chirurgia e Bioingegneria, Università di Siena, Viale Bracci 2, Siena, Italy. Electronic mail: barbini@biolab.med.unisi.it /05/ /1 C 2005 Biomedical Engineering Society 518 estimating model parameters which have proved useful for diagnosis, monitoring, and treatment of lung diseases. 1,4,16 The modeling approach has proven particularly useful for studying expiratory flow limitation, 14 showing that there are two basic mechanisms involved in the flow-limiting process: the wave-speed mechanism predominant at medium and high lung volumes, 5,13 and the viscous mechanism predominant at low lung volumes. 13,35 These mechanisms were investigated by Lambert et al. 17,19 and Elad et al. 7 by simulation of forced expiration maneuvers in normal subjects, but only under quasi-static conditions. A nonlinear functional model of breathing mechanics was recently proposed to interpret, under dynamic conditions, the main nonlinear mechanisms determining expiratory flow limitation in subjects with chronic obstructive pulmonary disease during artificial ventilation. 2 It showed that pathological changes in the pressure volume characteristic of the intermediate airways promoting easier collapse cause expiratory flow limitation when combined with a significant increase in lower-airway resistance. This model, however, does not consider bronchial tree morphometry and therefore cannot assess the contributions of different airway generations to the phenomenon of flow limitation. A dynamic morphometric model has been used to evaluate the impact of different ventilator strategies and constrictive pathologies on the time course of acinar pressures and flows. 26 It is based on a simplified description of the original Horsfield representation of the human bronchial tree 10 and can be useful when asymmetrical branching is a key feature to consider in the system. 27 Models of this type can reproduce pathological conditions characterized by airway heterogeneity in a given generation, however the asymmetrical nature of branching makes them very complex and difficult to implement when the main nonlinear mechanisms producing flow limitation have to be considered under real dynamic conditions. Indeed, in its original formulation, the model neglects both wave speed and viscous flow limitation mechanisms, and a more recent model with Horsfield-like

2 A Dynamic Morphometric Model of the Normal Lung for Studying 519 geometry again investigates forced expiration under quasistatic conditions. 32 The assumption of quasi-static conditions may represent a considerable restriction when the purpose is to simulate dynamic experiments, such as maneuvers to detect flow limitation during mechanical ventilation. Moreover, since dynamic models provide time-dependent flows, volumes and pressures of the patient, these signals could be used to test algorithms for estimating significant respiratory parameters. Dynamic models are also suitable for predicting the effect of changes in ventilator setup on time-dependent clinical features important for treating, monitoring, and weaning flow-limited mechanically ventilated patients. Here we develop an anatomically consistent model of the human lung which is dynamic and, at the same time, takes various nonlinear properties of breathing mechanics into consideration. This model, based on the Weibel symmetrical morphometric description of the tracheobronchial tree, includes both wave speed and viscous mechanisms determining flow limitation and reproduces the respiratory mechanics of a subject undergoing mechanical ventilation. The aim of the present study was to use computer simulation to investigate the role of these mechanisms in the flow limitation phenomenon in sedated patients with normal lung mechanics, during controlled artificial ventilation. In order to reproduce a condition similar to a forced expiration maneuver, we simulate several expirations characterized by different levels of driving pressure and investigate dynamic airway compression and expiratory flow limitation by means of isovolume pressure flow curves. We also use the model to evaluate the time course of resistance, total cross-sectional area, and the ratio of fluid velocity to wave speed (speed index) in the different airway generations during the breathing cycle. METHODS Nonlinear Morphometric Model of the Human Bronchial Tree A clear understanding of structure is a necessary foundation for a mathematical model of a physiological system. A useful approach to the tracheobronchial tree is that of Weibel 40 who numbered successive generations of airways from the trachea (generation 0) down to the alveolar sacs (generation 23). The trachea is the root of the tree and forks into two main bronchi; each branch (parent) gives rise to two new branches (daughters) down to the alveolar sacs. The number of branches in each generation is double that in the previous generation and generation n has 2 n branches. In a symmetrical representation (regular dichotomy), each parent divides into two identical daughters. The airways are grouped into conductive zone (first 17 generations) and respiratory zone (last 7 generations). 28,42 FIGURE 1. Electrical analog of a branch in the bronchial tree. External pressure is equal to atmospheric pressure for generation 0 (endotracheal tube) and to intrapleural pressure for the other generations. In the current study, each branch of the conductive zone was modeled as a cylinder of constant length, the diameter of which varies with transmural pressure. The respiratory zone was represented as a single compartment. Figure 1 shows the electrical analog of a branch of generation n in the conductive zone. Since the wavelengths of breathing signals are much greater than anatomical branch lengths, the distributed nature of the system was ignored and flow resistance, inertance, and airway compliance were lumped in the branch parameters R bn,l bn and C bn, respectively. Each branch resistance was modeled as a nonlinear Rohrer resistor 34 R bn = 128ηl b n πdb 4 (1 + k bn V bn ) (1) n The first term represents resistive properties for laminar flow accounting for air viscosity (η) and tubular geometry of the branch; l bn and d bn are branch length and diameter, respectively. The additional resistive term, depending also on branch airflow ( V bn ), becomes significant at higher flow rates when turbulence may occur. The constant k bn was estimated in each generation using resistance values previously obtained for flows between 10 and 100 l/min. 31 Branch inertance, due primarily to the air mass, was modeled as follows 41 L bn = 4ρl b n πdb 2 (2) n where ρ is air density. Branch compliance was calculated from the following relationship C bn = dv b n (3) dp bn where V bn and P bn are branch volume and transmural pressure, respectively. The following expression 29 was used to describe the relationship between branch diameter and transmural pressure P bn = γ 1bn 100 d bn % γ 2 bn d bn % + γ 3 bn (4)

3 520 BARBINI et al. FIGURE 2. Branch diameter (d bn %), as a percentage of its maximum value plotted against transmural pressure (P bn )forairways of the conductive zone (normal lung). The numbers below each curve indicate the corresponding branch generation. where d bn % is branch diameter expressed as a percentage of its maximum value (d bn max), i.e. d bn % = d b n d bn max 100 (5) and γ 1bn, γ 2bn, and γ 3bn are constant airway generation coefficients. The relationship between V bn and P bn and therefore branch compliance are easily obtained from Eqs. (4) and (5) by replacing d bn with 4V bn /(πl bn ). The coefficients γ 1bn, γ 2bn, and γ 3bn were estimated in generations 2 and 3 to fit experimental data obtained by Hyatt. 12,17 Since no experimental data are available for lower generations or negative transmural pressure, we chose to match the curves proposed by Lambert et al. 19 for the first eight generations and assumed that all generations from 8 to 16 were described by the same pressure diameter relationship. Figure 2 shows the resultant branch characteristics used in the present paper. Note that γ 3bn represents horizontal curve shift, while γ 1bn and γ 2bn influence the slope of the upperright and lower-left parts, respectively, and consequently also the slope of the curve in the middle part. Table 1 shows the geometrical dimensions of a single airway in each generation of the conductive zone, the branch parameter k bn [see Eq. (1)], and parameters γ 1bn, γ 2bn, and γ 3bn modeling the branch elastic characteristic [see Eq. (4)]. The geometrical dimensions were obtained from Weibel data. 40 Air density and viscosity were taken to be gcm 3 and cmh 2 O s, respectively. The respiratory zone (generations 17 23) was modeled as a Voigt body, including viscoelastic properties of the lung parenchyma. 8 It was therefore characterized by two parameters R p and C p, representing the overall resistance and compliance of this compartment, respectively. Finally, the chest wall was modeled as constant compliance C cw, connecting the lung to the atmosphere. R p, C p, and C cw were set equal to 4 cmh 2 O s/l, l/cmh 2 O, and 0.1 l/cmh 2 O, respectively. 2,3 Because of regular branching dichotomy, it was possible to significantly simplify the morphometric model of the tracheobronchial tree. 9 Since all branches of a given generation are identical, and hence also their resistances, inertances, and compliances, all the 2 n airways in the nth generation were replaced with a single RLC cell having resistance, inertance, and compliance given by R n = R bn /2 n, L n = L bn /2 n, and C n = 2 n C bn. The airflow entering the nth generation is therefore V n = 2 n V bn and the transmural pressure of the nth generation is P n = P bn. TABLE 1. Airway dimensions and branch parameters in generations of the conductive zone. Generation number d bn max a (mm) l bn (mm) k bn (s/l) γ 1bn (cmh 2 O) γ 2bn (cmh 2 O) γ 3bn (cmh 2 O) a Computed from Weibel data 40 adjusted to total lung capacity.

4 A Dynamic Morphometric Model of the Normal Lung for Studying 521 The pressure drop at the junction between two generations was described by the Bernoulli equation, assuming that the transition from one generation to the next takes place over such a short distance that friction losses can be considered negligible. 17 According to the hypotheses of Lambert et al., 17 we also assumed that there is no pressure recovery when the airflow goes into a generation with a larger total cross-sectional area. Thus the total pressure drop across the nth generation is P n 1 P n = 1 2 αξρ ( V 2 n A 2 n V 2 n 1 A 2 n 1 ) + R n V n + d(l n V n ) dt (6) where A n is the total cross-sectional area of the nth generation, ξ is a coefficient accounting for Lambert hypotheses, and α is the momentum correction factor for departure from a blunt velocity profile, 5 which was assumed to be equal to one. In particular, during expiration ξ was set equal to 1 for A n 1 < A n and equal to 0 for A n 1 A n, while during inspiration ξ was equal to 1 for A n 1 > A n and equal to 0 for A n 1 A n. The speed index, i.e., the ratio of local linear flow speed to local wave speed, in the nth generation was calculated as follows 30 SI n = V ( ) n ρ da 0.5 n (7) A n A n dp n Model of the Endotracheal Tube The patient ventilator coupling was described in order to reproduce typical conditions in intensive care units. Generation 0 was bypassed by a 8.0-mm-ID standard length endotracheal tube (ETT) modeled as a single RLC cell. ETT resistance was described with a Rohrer equation using data from Polese et al., 33 inertance was assumed to be according to Eq. (2), 38 and compliance was set equal to l/cmh 2 O to account for gas compressibility. 24 Model of the Mechanical Ventilator Operation of the mechanical ventilator was simulated to reproduce a mandatory mode of ventilation. During inspiration it was modeled by a pressure generator with high internal resistance (volume ventilation) and without external positive end-expiratory pressure (PEEP). The expiratory phase was assumed to be passive and a postinspiratory pause with mouth airflow equal to zero was introduced between inspiration and expiration. The resistance of the expiratory circuitry was modeled by a nonlinear Rohrer resistor R ec = k 1ec + k 2ec V, where k 1ec and k 2ec were assumed equal to 0.35 cmh 2 O s/l and 8.7 cmh 2 Os 2 /l 2, respectively. 39 In standard operating mode, the outlet of the expiratory channel was opened directly to the atmosphere (zero pressure). A tidal volume of about 570 ml and a respiratory frequency of 15 breaths/min were used, and the inspiratory time and postinspiratory pause were set at 0.9 and 0.1 s, respectively. Simulation Fifteen breathing cycles were simulated to allow the steady state to be reached. The whole system (nonlinear morphometric model of breathing mechanics and operation of ventilator) was implemented in MATLAB-SIMULINK software and simulated using a variable-step solver for stiff problems. Isovolume Pressure Flow Curves Expiratory flow limitation was investigated, as is customary, using isovolume pressure flow (IVPF) curves, each of which quantifies the relationship between driving pressure and tracheal flow at a given lung volume. 14,23,42 The driving pressure was calculated as the difference between alveolar and tracheal pressures. 22 Once a respiratory steady state was reached, a positive or negative constant pressure (ranging from 5 to 40 cmh 2 O) was applied to the outlet of the expiratory channel of the mechanical ventilator during early expiration in order to simulate expirations characterized by different levels of driving pressure. Driving pressure and flow rate were plotted at three fixed lung volumes (0.25, 0.35, and 0.45 l above functional residual capacity) to obtain a set of three IVPF curves. RESULTS Figure 3 shows the time courses of tracheal airflow and pressure and the corresponding changes in lung volume obtained for the simulated case during two consecutive breathing cycles, once a respiratory steady state was reached. Tracheal airflow was calculated inside the ETT and tracheal pressure was computed at its end. Operation of the mechanical ventilator was set as previously described. As customary in mechanical ventilation, tracheal airflow was taken as positive during inspiration. Flow in the final part of expiration was practically zero, indicating that expiration time was chosen long enough to allow lung volume to reach its functional residual capacity (FRC) and no intrinsic positive end-expiratory pressure (PEEPi) was present. In this case, the changes in lung volume ( v), obtained by digital integration of the flow signal, represent the difference between the lung volume and FRC. Figure 4 shows IVPF curves at volumes 0.45, 0.35, and 0.25 l above FRC. Points on the curves were obtained for different values of pressure applied to the expiratory circuit outlet, from 5 to 40 cmh 2 O, decreasing from left to right in steps of 5 cmh 2 O. Driving pressure (i.e. difference between alveolar and tracheal pressure) and expiratory airflow (i.e. the opposite of mouth airflow) were represented on the abscissa and ordinate, respectively. The IVPF

5 522 BARBINI et al. FIGURE 3. Model-generated tracheal airflow and pressure and lung volume changes during two breathing cycles under mechanical ventilation. FIGURE 4. Model-generated isovolume pressure flow curves at lung volumes 0.45, 0.35, and 0.25 liters above FRC. Continuous and broken lines, corresponding to zero pressure and 30 cmh 2 O applied to the expiratory circuit outlet, respectively, connect the three working points. curves obtained by the model are in line with the experimental curves for spontaneous breathing and mechanical ventilation. 14,39 In the curve corresponding to the greatest lung volume ( v = 0.45 l), expiratory flow increases with driving pressure and no defined limit to expiratory flow can be observed, indicating that maximum expiratory flow depends heavily on driving pressure (effort-dependent expiration) at this lung volume. On the contrary, at the lowest lung volume ( v = 0.25 l), expiratory flow increases with driving pressure up to a maximum, beyond which further increases in pressure do not produce any significant change in flow. This reproduces the expiratory flow limitation phenomenon observed at low lung volumes even in non artificially ventilated patients with healthy lung mechanics. 14,42 In the flat part of the curve, expiratory flow is independent of driving pressure (effort-independent region). This effort independence suggests that resistance to airflow increases with driving pressure (dynamic compression). A similar pattern can also be observed in the IVPF curve at intermediate lung volume ( v = 0.35 l). To have an insight into system behavior, two different working conditions are shown in Fig. 4. The first (control condition) represents the standard operating mode of the mechanical ventilator, in which the outlet of the expiratory channel opens directly to the atmosphere (zero pressure applied). The second (test condition) corresponds to a pressure of 30 cmh 2 O applied to the expiratory circuit outlet, radically increasing driving pressure during expiration with respect to the control condition. Continuous and broken lines connect the three control and three test points. In control condition, it can be seen that all working points lie in the effort-dependent region of the IVPF curve for the lung volumes considered, suggesting an absence of flow limitation during expiration. On the other hand, in the test situation, the working points at v equal to 0.35 and 0.25 l lie in the effort-independent region of the IVPF curve, indicating flow limitation at these lung volumes. Figure 5 shows the time courses of tracheal airflow and pressure in two consecutive breathing cycles, corresponding to control and test conditions, respectively. The abrupt negative peak in the flow time course during the test cycle occured when subatmospheric pressure was suddenly applied to the outlet of the expiratory channel of the mechanical ventilator in early expiration (about 140 ms after the beginning of expiration). It can be observed that in test condition the flow at v = 0.45 l almost matched peak expiratory flow caused by sudden application of subatmospheric pressure. As proved by Valta et al. in flow-limited patients, 39 this sharp spike depends heavily on the subatmospheric pressure applied and consequently changes with driving pressure. This explains why in the IVPF curve for this lung volume the expiratory flow does not reach a plateau even at the highest value of external pressure. Figure 6 shows the resistance of generations 1 16 at the three lung volumes considered in control (closed circles)

6 A Dynamic Morphometric Model of the Normal Lung for Studying 523 FIGURE 5. Model-generated tracheal airflow and pressure in two consecutive breathing cycles, applying zero pressure and 30 cmh 2 O, respectively, to the expiratory circuit outlet.,, and indicate v = 0.45, 0.35, and 0.25 l, respectively. FIGURE 6. Resistance of the generations from 1 to 16 of the tracheobronchial tree at three different lung volumes. and indicate control condition (zero pressure) and test condition ( 30 cmh 2 O applied to the expiratory circuit outlet), respectively. and test condition (open circles). Generation 0 was not considered because it was bypassed by tracheal intubation to simulate mechanical ventilation conditions. In line with previous findings, 31,42 the major sites of airway resistance in the tracheobronchial tree are in bronchi, with small bronchioles contributing less resistance. Application of a high negative pressure at the expiratory circuit outlet produces an increase in resistance of all 16 generations, most evident at the lowest lung volume, where expiratory flow limitation occurs, as discussed above. In the latter situation, a very marked increase of airway resistance can be observed in generations 4 5. To investigate the phenomena involved in expiratory flow limitation, the time courses of total cross-sectional area and resistance during expiration were studied for different airway generations. We subdivided the conductive zone into three parts: upper airways (main bronchi), intermediate airways (lobar, segmental, and small bronchi), and lower airways (bronchioles and terminal bronchioles), and we analyzed three sample generations in detail, one for each part. We chose three sites where the differences in resistance between control and test conditions in Fig. 6 are very evident: generation 1 for the upper airways, generation 4 for the intermediate airways, and generation 12 for the lower airways. Figure 7 shows the expiratory time courses of total cross-sectional area for these generations in control (continuous line) and test condition (broken line). The same symbols as in Figs. 4 and 5 are used to mark the working points corresponding to lung volumes 0.45, 0.35, and 0.25 l above FRC. Arrows indicate when negative pressure was applied to the outlet of the expiratory channel in the test case. In control and test conditions, the area of each generation mostly decreased in the first part of expiration, while minor variations were observed in the final part of expiration. This decrease was more remarkable and rapid after subatmospheric pressure was applied. For a given generation, the percentage difference between the total cross-sectional area in the control and test cases increased when v decreased from 0.45 to 0.25 l. This proves that when driving pressure increases with respect to the control condition, greater percentage narrowing of the conductive airways occurs at lower lung volume, causing dynamic compression which may lead to expiratory flow limitation. For example, generation 4 showed big variations between the control and test case: under test condition, the area was about 70% that in control condition for v = 0.45 l, whereas it dropped to about 20% of the corresponding area in control condition for v = 0.25 l. In the first part of expiration, lung volumes also decreased more quickly under test condition. In fact, in this case, v went from 0.45 to 0.25 l in about one-half the time taken in control condition. Figure 8 shows the expiratory time courses of airflow resistance for the same generations in control (continuous line) and test (broken line) conditions. In control condition

7 524 BARBINI et al. FIGURE 7. Expiratory time courses of total cross-sectional area for generations 1, 4, and 12 in control (continuous line) and test condition (broken line).,,and indicate v = 0.45, 0.35, and 0.25 l, respectively. Arrows indicate application of 30 cmh 2 O pressure to the outlet of the expiratory channel of the mechanical ventilator in test condition. the resistances of generations 1 and 4 reach their maximum values in the first part of the expiratory phase, then they decrease and level off. At first sight this may seem to disagree with the previous time courses of the corresponding total cross-sectional areas, which do not increase at any time during expiration. Actually, the resistance of the first generations may also depend significantly on airflow, since turbulence may occur in these airways at high flow rates. The decrease in resistance time courses is therefore due to the reduction of flow rates in these airway generations, which go to zero at the end of expiration. On the other hand, the time course of resistance in generation 12 does not show any decrease during expiration, since laminar flow is essentially present in peripheral airways because of lower flow rates due to their larger total cross-sectional area. When these airways narrow, their resistance therefore increases. Much higher resistances were obtained in the test condition for all generations. After application of subatmospheric pressure, the resistances of generation 1 and 4 showed a sharp increase followed by a slowly decreasing trend. In contrast, the resistance of generations 12 remarkably FIGURE 8. Expiratory time courses of total resistance for generations 1, 4, and 12 in control (continuous line) and test condition (broken line).,,and indicate v = 0.45, 0.35, and 0.25 l, respectively. Arrows indicate application of 30 cmh 2 O pressure to the outlet of the expiratory channel of the mechanical ventilator in test condition. went on increasing during the whole expiration, reaching its maximum at the end of expiration. Generation 4 showed the most accentuated differences between control and test conditions. At the end of expiration, its resistance in test condition reached a value about one hundred-fold that observed in the control case. In Fig. 9 the time course of the speed index (SI) during expiration is plotted for generations 1, 4, and 12. Continuous and broken lines indicate control and test conditions, respectively, and the same symbols are employed to indicate v equal to 0.45, 0.35, and 0.25 l. As expected, in control condition, where flow limitation was absent, the speed index was always much less than one throughout expiration. However, it was also very low in generations 1 and 12 in test condition. In generation 4, SI increased greatly after application of subatmospheric pressure, reaching a maximum in the first part of expiration, but remaining below one; then it decayed markedly to the end of expiration. This behavior is different from experimental findings observed in normal subjects during a maximally forced vital capacity maneuver, where the speed index takes values equal to or greater than one over most of the expiratory phase. 30

8 A Dynamic Morphometric Model of the Normal Lung for Studying 525 predominant role in the onset of expiratory flow limitation when a normal subject is mechanically ventilated with standard values of tidal volume and respiratory frequency and a negative pressure of 30 cmh 2 O is applied to the expiratory circuit outlet. In fact, it is well known that, when SI 2 1, the pressure gradient in airway generations only depends on dissipative pressure losses. 18,32 Consequently, in healthy lungs, application of not very high values of subatmospheric pressure to the patient mouth can produce expiratory flow limitation during mechanical ventilation, despite the almost negligible contribution of wave-speed mechanism. DISCUSSION FIGURE 9. Expiratory time courses of speed index for generations 1, 4, and 12 in control (continuous line) and test condition (broken line).,, and indicate v = 0.45, 0.35, and 0.25 l, respectively. Arrows indicate application of 30 cmh 2 O pressure to the outlet of the expiratory channel of the mechanical ventilator in test condition. There may be several reasons for this, the first being that, in sedated patients undergoing ventilation, the increase in driving pressure is obtained by applying constant levels of subatmospheric pressure to the outlet of the expiratory channel of the mechanical ventilator. 39 Of course this approach does not reproduce the physiological mechanisms involved in the maximally forced vital capacity maneuver. A second reason is that in mechanical ventilation, the presence of the expiratory circuitry and endotracheal tube adds external resistances which greatly influence system behavior during forced expiration. 30 As a consequence, the peak expiratory flow obtained in our test condition was much lower than those observed during a forced expiration from total lung capacity. A third reason is that in our simulation, expiration starts from lung volumes far below total lung capacity. The IVFP curves demonstrate that flow limitation is achieved more easily at low lung volumes than near TLC and thus both maximal limited flow and peak expiratory flow decline with decreasing lung volume. Our simulation results indicate that the coupling between dissipative pressure losses and airway compliance plays a The purpose of the present study was to use a modeling approach to understand the role of different mechanisms that cause expiratory flow limitation in normal subjects undergoing mechanical ventilation when a suitable value of subatmospheric pressure is applied to the outlet of the ventilator expiratory circuit. The technique of negative expiratory pressure, formerly used by Smaldone and Smith, 37 was validated by Valta el al. 39 in a sample of patients with acute ventilatory failure and it is currently used to detect expiratory flow limitation during artificial ventilation in intensive care units. Because of different working conditions, major differences can be expected between the results observed during a maximally forced vital capacity maneuver in a laboratory of pulmonary pathophysiology and those obtained during expiration with negative mouth pressures in mandatory ventilation with standard values of tidal volume and respiratory frequency. In fact different variations in lung volume as well as the artificial devices and techniques used cause major changes in system behavior. The modeling approach proved useful since it allowed us to evaluate and interpret the influence of several mechanisms which may contribute to the onset of the expiratory flow limitation. First the model showed that application of sufficiently high negative pressures to the expiratory circuit outlet caused a peak in expiratory flow, after which expiratory flow limitation was even observed in subjects with normal lung mechanics. The peak in expiratory flow depended on driving pressure, in line with the experimental results of Valta el al. 39 who showed that in flow-limited patients, application of subatmospheric mouth pressures produced a transient spike in flow related to the negative pressure value. Our simulation results also showed that peak expiratory flows obtained by the negative pressure technique during mechanical ventilation were much less than those observed experimentally during a maximally forced vital capacity maneuver. 30 As already stated, this is due to the different working conditions of the two experiments. The constant value of negative expiratory pressure used in our test leads to a driving pressure less than those typically observed during a maximum forced expiratory maneuver. Furthermore,

9 526 BARBINI et al. in mechanically ventilated patients, the exhalation circuit and the endotracheal tube also cause a significant reduction in peak expiratory flow. 21 Under these conditions, the speed index time courses also differ greatly between negative pressure expiration and forced vital capacity maneuver. In particular, the results obtained demonstrate that the speed index was always much less than one in upper and lower airways throughout expiration, even when a subatmospheric pressure of 30 cmh 2 O was applied to the outlet of the expiratory circuit. It drastically increased in the intermediate airways, but only just after peak expiratory flow, i.e., in the first part of expiration after the expiratory valve was opened to negative pressure. For example, in generation 4 the speed index reached a maximum for v between 0.35 and 0.25 l, where flow limitation was present and it subsequently underwent a considerable decrease to the end of expiration (see Fig. 9). This suggests that the viscous mechanism (i.e. the coupling between dissipative pressure losses and airway compliance) 13 plays a very significant role in the expiratory flow limitation observed under standard conditions of mechanical ventilation for this range of negative expiratory pressure. It is illustrative to examine the influence of certain experimental conditions on the mechanisms causing the expiratory flow limitation during mechanical ventilation, as well as bringing out some limitations of the model. First it is useful to investigate whether wave speed limitation also plays a key role during mechanical ventilation of normal lungs, when an experiment similar to a forced vital capacity maneuver is reproduced, for example by increasing tidal volume and driving pressure with respect to our test condition. It is also evident that the choice of model parameters can greatly influence system behavior, producing quite different results, for example, in pathological conditions. Finally, analysis of the assumptions is essential to underline the limitations of the model, which need to be taken into account when simulating different test conditions. our model, we inflated the lung about 1 l above FRC, then we compared the expirations obtained applying two significantly different levels of subatmospheric pressure to the expiratory circuit outlet. The difference between these levels of pressure was chosen equal to 50 cmh 2 O, as in the Smaldone and Smith experiment: in the first simulation we used a subatmospheric pressure of 30 cmh 2 O(asin our test condition), while in the second we set the external pressure at 80 cmh 2 O. In both cases the negative pressure was applied at the beginning of expiration, just after the postinspiratory pause. Simulation results corresponding to lower and higher levels of negative pressure are shown on the left and right sides of Fig. 10, respectively. Unlike our previous test condition, the speed index in generation 3 reached and exceeded 1 over most of tidal expiration when the applied pressure was 80 cmh 2 O, so that the driving pressure attained very high values, like those generated during a forced vital capacity maneuver. At higher lung volumes (i.e. when the speed index reached 1 with a negative pressure of 80 cmh 2 O) the lateral pressure in generation 4 (segmental bronchi) was about the same in the two simulated conditions. This indicates that expiratory flow limitation is present under both conditions at these lung volumes, and that the flowlimiting segment is in the segmental bronchi. 37 At lower lung volumes the lateral pressures in generation 4 become Partial Forced Expiration Maneuver The results obtained indicate that airway resistance shows the most accentuated differences between control and test conditions in the intermediate generations (see Figs. 6 and 8). In line with previous studies, the simulation model confirms that the behavior of these airways is essential for this phenomenon, even under mechanical ventilation. Smaldone and Smith reproduced a maneuver similar to a partial forced expiration in normal patients during mechanical ventilation. 37 They applied very high negative pressure to the outlet of the expiratory circuit of three subjects without acute lung parenchymal or airway disease, on ventilators because of intracranial hemorrhage, and demonstrated that flow-limiting segments were in the intermediate airways (lobar, segmental, or subsegmental bronchi). In order to replicate a similar experiment with FIGURE 10. Expiratory time courses of model generated signals during application of a negative expiratory pressure (NEP) of 30 cmh 2 O(left) and 80 cmh 2 O(right). Downwards, from the top: speed index of generation 3, lateral pressure of generations 3, 4, and 6 (cmh 2 O) and lung volume changes (l).

10 A Dynamic Morphometric Model of the Normal Lung for Studying 527 different in the two conditions, but remain about the same in generation 6 (subsegmental bronchi). In line with the results of Smaldone and Smith, this indicates that the flowlimiting segment shift to deeper airway generations when lung volume decreases during forced expiration. In conclusion, the above simulation results suggest that the proposed model also reproduces the wave speed limitation phenomenon observed during a partial forced expiration maneuver and describes the flow-limiting segment shift occurring during expiration. A Pathological Case (Preliminary Study) Recent simulation results obtained with a functional nonlinear model of mechanical ventilation showed that pathological changes in the characteristics of intermediate and lower airways can lead to expiratory flow limitation at standard values of tidal volume and respiratory frequency, even without any subatmospheric pressure applied at the mouth. 2 A similar test was carried out using the present anatomically consistent model in order to evaluate its ability to reproduce the expiratory flow limitation phenomenon in obstructed patients during standard mechanical ventilation. Obstructive pulmonary disease is characterized by high airflow resistance. Small airways may suffer early changes, later extending to most bronchial airways, with the following pathological alterations: 43 (a) decrease in airway section due to thickening of airway walls and/or partial occlusion of the airway lumen due to excessive secretions, (b) abnormality outside the airways due, for example, to partial destruction of lung parenchyma, which may cause loss of radial traction and consequent narrowing. To mimic such a scenario we significantly modified the characteristics of the simulation model with respect to the normal case from the small bronchi to the respiratory zone. Specifically, we halved the maximum airway diameter from generations 8 to 16 and we changed the transmural pressure diameter curves as shown in Fig. 11, to account for the decrease in section and the easier collapsibility of these airways. We also increased respiratory zone resistance setting R p equal to 6 cmh 2 O s/l. The presence of expiratory flow limitation was detected by the negative expiratory pressure (NEP) technique, 39 often used for this purpose in mechanically ventilated patients. It consists in applying a NEP of few cmh 2 O (usually 5 cmh 2 O) to the expiratory circuit outlet during expiration in a test breath and in comparing the expiratory flow volume curve of this test breath with the corresponding curve obtained during the preceding control breath, when the expiratory circuit outlet was at atmospheric pressure. Expiratory flow limitation is present when application of the subatmospheric pressure does not produce any increase in expiratory flow. FIGURE 11. Branch diameter (d bn %), as a percentage of its maximum value plotted against transmural pressure (P bn )for airways of the conductive zone (pathological lung). The numbers below each curve indicate the corresponding branch generation. The upper part of Fig. 12 shows the test and control expiratory flow volume curves in simulated normal and pathological cases (left and right side, respectively) again using the ventilation setup (tidal volume, respiratory frequency, inspiratory/expiratory ratio) illustrated in Methods. The test curve (broken line), obtained with a NEP of 5 cmh 2 O, was FIGURE 12. Expiratory flow and 8th generation speed index vs. lung volume changes in normal (left) and pathological (right) conditions. Control cycle (continuous line) is compared with a test cycle (broken line) obtained by applying a negative pressure of 5 cmh 2 O to the expiratory circuit outlet during expiration.

11 528 BARBINI et al. compared with the corresponding control curve (continuous line). Expiratory flow volume curves in the normal case confirmed that expiratory flow limitation was absent during tidal breathing. On the contrary, in the obstructed case, application of NEP did not change the expiratory flow with respect to that of the previous control breath, except for a rapid initial flow transient. This indicated that flow limitation was present over most of the expiration. The significant likeness between the present simulated curves and those experimentally obtained by other authors 23,39 suggests that correct choice of model parameters enabled an obstructed lung scenario to be adequately reproduced. Figure 12 also shows the time course of the speed index in generation 8 (bottom part of the figure), i.e. in the first generation with different simulated characteristics between normal and pathological cases. It shows that the speed index exceeds 1 in the obstructed subject not only in the test expiration, but also over much of the control expiration, suggesting that the wave-speed mechanism plays a predominant role in limiting expiratory flow during a tidal breath in this situation. Of course, proper study of system behavior under pathological conditions requires a thorough examination of the scenarios to be reproduced and accurate setting up of the model. The choice of model characteristics can be a critical point when simulating obstructive pulmonary diseases, since this pathology can involve different parts of the bronchial tree and in different ways, thus influencing the onset of expiratory flow limitation phenomenon in different ways. The present formulation of the model is based on a symmetrical representation of the bronchial tree and does not consider lung heterogeneity, which may be important in determining expiratory flow limitation in pathological conditions. 32 The results obtained in our simulation of an obstructed patient must therefore only be considered as a preliminary analysis of model features, though the situation considered mimics realistic changes caused by this disease. Further investigations are necessary to more fully understand the complex phenomenon of expiratory flow limitation often observed in these patients. Remarks on the Model Assumptions One of the main potentials of modeling and simulation is to test hypotheses and possible scenarios quantitatively. This is very useful for complex systems such as breathing mechanics, for which it is difficult to formulate a comprehensive theory. With respect to previous papers 7,17,19,32 the present model enables the flow limitation phenomenon to be investigated under dynamic conditions. Since our model was set up to study normal subjects mechanically ventilated with standard tidal volume and respiratory frequency, some simplifying assumptions were reasonable to limit complexity, making it much easier to evaluate the effects of different nonlinear mechanisms involved in the flow limitation phenomenon under dynamic conditions. These assumptions of course limit the model, which like all models must be applied judiciously. Analysis of the assumptions can lead to improvements and/or modifications for different test conditions. The symmetrical description of the bronchial tree assumed in the morphological lung representation may be a limit of the model. A recent paper proved that airway heterogeneity can help to understand flow limitation during maximum forced expiratory maneuvers. 32 This aspect proved to be crucial in the presence of heterogeneity, when certain evidence cannot be explained with a symmetrical bronchial tree. However, in our study we were concerned with simulating healthy lung status, when airway heterogeneity plays a minor role. 40 Another assumption of the model was to represent each conducting airway as cylinder, the radius of which varies as a function of transmural pressure during a breathing cycle, and to concentrate convective pressure losses at the junctions between adjoining generations. This of course implies that time-varying changes in total cross-sectional area in a given airway generation are the same along its length, so that a whole airway generation may collapse but not just a subsegment. A more detailed description of the system could be obtained by subdividing each conducting airway into a number of segments, increasing model complexity. Since the trachea was bypassed by the endotracheal tube in our simulation, preventing collapse from taking place, and the lengths of the other airways were much less than the wavelengths of breathing signals, we considered each conducting airway as a unit. The computational model likewise does not consider changes in intrapleural airway length due to time variations in lung volume during breathing. Previous papers indicate that airway length changes with the cube root of lung volume. 11 Variations in airway length are therefore negligible (only a few percent) over a whole breath at tidal volumes of about 570 ml. It can also be observed that the viscoelastic behavior of airway tissue was ignored in the present representation of breathing mechanics. This may be a limit of the model when airway diameters change rapidly, in which case airway wall tissue should also be considered a viscoelastic material. Finally, the respiratory zone and chest wall were described by three constant parameters R p, C p, and C cw.of course more complex models must be used to account for nonlinear viscoelastic behavior of this part of the system, although in normal subjects our simplifying assumption is usually made when small variations in lung volume are considered. CONCLUSION The results obtained indicate that the present nonlinear morphometric model of breathing mechanics provides a satisfactory interpretation of the main dynamic phenomena

12 A Dynamic Morphometric Model of the Normal Lung for Studying 529 occurring in mechanical ventilation. The model not only describes system behavior under mandatory ventilation, but also reproduces the expiratory flow limitation observed at high driving pressures even in subjects with healthy lung mechanics. It provides quantitative information on time variations in airflow, pressure, resistance, diameter, and speed index in the various conductive airways, enabling detailed analysis of the role played by different airway generations in this phenomenon. Simulation results show that the coupling between dissipative pressure losses and airway compliance predominantly affect the flow limitation phenomenon produced by application of a negative pressure of 30 cmh 2 Oattheexpiratory circuit outlet, when a normal subject is artificially ventilated with standard values of tidal volume and respiratory frequency. This suggests that in healthy lungs not very high values of subatmospheric pressure to the patient mouth can produce expiratory flow limitation during mechanical ventilation, although the contribution of wave-speed mechanism is nearly negligible. Wave speed limitation becomes important when reproducing experiments more similar to a forced vital capacity maneuver, i.e. when tidal volume and driving pressure are significantly greater than in the above test condition. Moreover, the mechanism may be not negligible during tidal breathing in patients with obstructive lung diseases. Pathological condition, however, need further investigation and will be the subject of future research. ACKNOWLEDGMENT This work was supported by the Italian Ministry of Education, University and Research (MIUR). REFERENCES 1 Avanzolini, G., and P. Barbini. A versatile identification method applied to analysis of respiratory mechanics. IEEE Trans. Biomed. Eng. 31: , Barbini, P., G. Cevenini, and G. Avanzolini. Nonlinear mechanisms determining expiratory flow limitation in mechanical ventilation: A model-based interpretation. Ann. Biomed. Eng. 31: , Barbini, P., G. Cevenini, F. Bernardi, M. R. Massai, G. Gnudi, and G. Avanzolini. Effect of compliant intermediate airways on total respiratory resistance and elastance in mechanical ventilation. Med. Eng. Phys. 23: , Barbini, P., G. Cevenini, K. R. Lutchen, and M. Ursino. Estimating respiratory mechanical parameters of ventilated patients: A critical study in the routine intensive-care unit. Med. Biol. Eng. Comput. 32: , Dawson, S. V., and E. A. Elliott. Wave-speed limitation on expiratory flow A unifying concept. J. Appl. Physiol. 43: , D Angelo, E., E. Calderini, G. Torri, F. M. Robatto, D. Bono, and J. Milic-Emili. Respiratory mechanics in anaesthetized paralysed humans: Effects of flow, volume and time. J. Appl. Physiol. 67: , Elad, D., R. D. Kamm, and A. H. Shapiro. Choking phenomena in a lung-like model. J. Biomech. Eng. 109:1 9, Fung, Y. C. Biomechanics: Mechanical Properties of Living Tissues. New York: Springer-Verlag, Ginzburg, I., and D. Elad. Dynamic model of the bronchial tree. J. Biomed. Eng. 15: , Horsfield, K., G. Dart, D. E. Olson, G. F. Filley, and G. Cumming. Models of the human bronchial tree. J. Appl. Physiol. 31: , Hughes, J. M., F. G. Hoppin, and J. Mead. Effect of lung inflation on bronchial length and diameter in excised lungs. J. Appl. Physiol. 32:25 35, Hyatt, R. E., T. A. Wilson, and Y. Bar. Prediction of maximal expiratory flow in excised human lungs. J. Appl. Physiol. 48: , Hyatt, R. E., J. R. Rodarte, T. A. Wilson, and R. K. Lambert. Mechanisms of expiratory flow limitation. Ann. Biomed. Eng. 9: , Hyatt, R. E. Expiratory flow limitation. J. Appl. Physiol. 55:1 7, Kaczka, D. W., E. P. Ingenito, E. Israel, and K. R. Lutchen. Airway and lung tissue mechanics in asthma. Effects of albuterol. Am. J. Respir. Crit. Care Med. 159: , Kaczka, D. W., E. P. Ingenito, S. C. Body, S. E. Duffy, S. J. Mentzer, M. M. DeCamp, and K. R. Lutchen. Inspiratory lung impedance in COPD: Effects of PEEP and immediate impact of lung volume reduction surgery. J. Appl. Physiol. 90: , Lambert, R. K., T. A. Wilson, R. E. Hyatt, and J. R. Rodarte. A computational model for expiratory flow. J. Appl. Physiol. 52:44 56, Lambert, R. K. Bronchial mechanical properties and maximal expiratory flows. J. Appl. Physiol. 62: , Lambert, R. K. A new computational model for expiratory flow from nonhomogeneous human lungs. J. Biomech. Eng. 111: , Lorino, A. M., H. Lorino, and A. Harf. A synthesis of Otis, Mead, and Mount mechanical respiratory models. Respir. Physiol. 97: , Lourens, M. S., B. van den Berg, H. C. Hoogsteden, and J. M. Bogaard. Flow-volume curves as measurement of respiratory mechanics during ventilatory support: the effect of the exhalation valve. Intensive Care Med. 25: , Lourens, M. S., B. van den Berg, A. F. M. Verbraak, H. C. Hoogsteden, and J. M. Bogaard. Effect of series of resistance levels on flow limitation in mechanically ventilated COPD patients. Respir. Physiol. 127:39 52, Lourens, M. S., B. van den Berg, H. C. Hoogsteden, and J. M. Bogaard. Detection of flow limitation in mechanically ventilated patients. Intensive Care Med. 27: , Lutchen, K. R. Optimal selection of frequencies for estimating parameters from respiratory impedance data. IEEE Trans. Biomed. Eng. 35: , Macklem, P. T. The mechanics of breathing. Am.J.Respir.Crit. Care Med. 157:S88 S94, Nucci, G., S. Tessarin, and C. Cobelli. A morphometric model of lung mechanics for time-domain analysis of alveolar pressures during mechanical ventilation. Ann. Biomed. Eng. 30: , Nucci, G., B. Suki, and K. R. Lutchen. Modeling airflow-related shear stress during heterogeneous constriction and mechanical ventilation. J. Appl. Physiol. 95: , Nunn, J. F. Applied Respiratory Physiology. London: Butterworths, Pardaens, J., K. P. van de Woestijne, and J. Clément. A physical model of expiration. J. Appl. Physiol. 33: , 1972.