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Theoretical modeling of the interaction between alveoli during inflation and deflation in normal and diseased lungs
ARTICLE IN PRESS Journal of Biomechanics 43 (2010) 1202–1207
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Theoretical modeling of the interaction between alveoli during inflation and deflation in normal and diseased lungs Kerstin Schirrmann a,n, Michael Mertens b,c, Ulrich Kertzscher a, Wolfgang M. Kuebler b,d, Klaus Affeld a a
´ - Universit¨ Biofluid Mechanics Laboratory, Charite atsmedizin Berlin, Germany Institute of Physiology, Charite´ - Universit¨ atsmedizin Berlin, Germany c Institute for Human Biology and Anthropology, Free University Berlin, Germany d Department of Surgery, University of Toronto, ON, Canada b
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 21 November 2009
Alveolar recruitment is a central strategy in the ventilation of patients with acute lung injury and other lung diseases associated with alveolar collapse and atelectasis. However, biomechanical insights into the opening and collapse of individual alveoli are still limited. A better understanding of alveolar recruitment and the interaction between alveoli in intact and injured lungs is of crucial relevance for the evaluation of the potential efficacy of ventilation strategies. We simulated human alveolar biomechanics in normal and injured lungs. We used a basic simulation model for the biomechanical behavior of virtual single alveoli to compute parameterized pressure–volume curves. Based on these curves, we analyzed the interaction and stability in a system composed of two alveoli. We introduced different values for surface tension and tissue properties to simulate different forms of lung injury. The data obtained predict that alveoli with identical properties can coexist with both different volumes and with equal volumes depending on the pressure. Alveoli in injured lungs with increased surface tension will collapse at normal breathing pressures. However, recruitment maneuvers and positive endexpiratory pressure can stabilize those alveoli, but coexisting unaffected alveoli might be overdistended. In injured alveoli with reduced compliance collapse is less likely, alveoli are expected to remain open, but with a smaller volume. Expanding them to normal size would overdistend coexisting unaffected alveoli. The present simulation model yields novel insights into the interaction between alveoli and may thus increase our understanding of the prospects of recruitment maneuvers in different forms of lung injury. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Simulation Lung injury Stability Surface tension Recruitment
1. Introduction Mechanical ventilation is necessary in patients with acute lung injury to assure sufficient oxygen supply and carbon dioxide removal. However, mechanical ventilation may lead to ventilatorinduced lung injury (VILI). Avoiding VILI is the aim of protective ventilation strategies. One of the central strategies follows the open lung approach (Lachmann, 1992) and includes recruitment maneuvers. Recruitment maneuvers involve the initial application of high ventilation pressures to recruit unaerated lung regions and subsequent positive endexpiratory pressures (PEEP) to keep these regions open. Still under discussion is whether to apply this method as routine in the clinical scenario (Kacmarek and Kallet, 2007). The mechanisms assumed to be contributing to VILI are high shear stress in the transition region between open and collapsed alveolar regions
n
Corresponding author. Tel.: + 49 30 450 553 798; fax: + 49 30 450 553 938. E-mail address: [email protected] (K. Schirrmann).
0021-9290/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2009.11.025
(Mead et al., 1970), the cyclic opening and collapse of individual alveoli (Carney et al., 2005) and the overdistension of alveoli (Hubmayr, 2002). Understanding the biomechanics of individual alveoli better may thus allow for an improved evaluation of ventilation strategies in different forms of lung injury and disease. Recently, alveolar models were presented for flow simulations (Tsuda et al., 2008; Sznitman et al., 2009; Kumar et al., 2009) and dynamic parenchymal behavior (Denny and Schroter, 2000). Regarding lung stability, several simulation models have been reported (Fung, 1975; Mead et al., 1970; Stamenovic and Smith, 1986; Stamenovic and Wilson, 1992; Wilson, 1981; Wilson et al., 2001). These studies showed that regional collapse is not a contradiction to structural integrity. The applied models allowed for the description of shear stress in adjacent open and closed lung regions (Mead et al., 1970) and of the stabilizing effect of the surrounding parenchyma on a lung region (Fung, 1975). However, the microstructural models considered regions larger than the individual alveoli (Mead et al., 1970). We studied the interaction of individual alveoli and the stability of the resulting systems. For
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¨ the stability analysis (Muller and Strehlow, 2004), we generated alveolar pressure–volume curves (pv curves) with a model similar to the one of Clements et al. (1961). However, Clements’ model does not include the interaction of connected alveoli and therefore excludes the stabilizing effects of parallel alveoli. Our approach describes the alveoli as interacting structures with equal pressures. Importantly, the use of theoretically derived pv curves of single alveoli allows for parametric studies to simulate effects of lung injury and disease at the alveolar level. With our model, we want to provide insights into the effects of changes in surface tension and tissue properties on the performance of interacting alveoli. This was studied during both normal breathing and recruitment maneuvers. 2. Material and Methods We consider the following as the main influences on the alveolar pressure– volume behavior: morphology, surface tension of the fluid lining, elasticity of the tissue and the embedding of the alveolus in the tissue, that is the connection to other alveoli via elastin and collagen fibers. Regarding morphology, we used a simple geometrical model (Clements et al., 1961). A rigid ring forms the alveolar mouth (diameter set at 100 mm) while the fluid–air interface of the alveolus itself is modeled as a spherical cap. For simplicity, we assumed a constant surface tension in the fluid lining in our model. The value of the surface tension was set to g =0.02 N/m in the normal model alveolus according to previously reported values for minimal equilibrium surface tension (Otis et al., 1994). The influences of tissue elasticity and the embedding of the alveoli in the tissue were combined to a tissue component and were approximated by a linear relationship in the pv curve as of a certain volume. This was motivated by pv curves of saline filled lungs, where these two effects produce an only slightly sigmoidal curve starting at a nonzero volume (Hubmayr, 2002). Two points are required to define the simplified pv curve (see Appendix): We used the volumes and pressures at the beginning and end of inspiration in a human lung during normal respiration. We assumed a total number of alveoli (na) of 300,000,000 (Weibel, 1963), a functional residual capacity (VFRC) of 3.2 l, a dead space volume (Vds) of 0.15 l, a tidal volume (Vt) of 0.5 l, an end-expiratory pressure of 500 Pa, and an end-inspiratory pressure of 700 Pa (Klinke and Silbernagel, 1996). Pressures were set to zero below the intersection with the volume axis to avoid negative pressures, which are related to compressive stress, since it was assumed that alveolar walls would not be able to sustain a compressive stress (Fung, 1975). The superposition of the influences of the fluid lining (pfl(V) = 2g/r(V), see Appendix) and of the tissue component (pti(V)= aV+ b for V 4Vsf, 0 else) resulted in the pv curve of the standard model alveolus (Eq. (1), Fig. 1). 8 2g b > > for V o Vsf ¼ > < a rðVÞ ð1Þ pðVÞ ¼ pfl ðV Þþ pti ðV Þ ¼ 2g b > > > þ aV þ b for V Z Vsf ¼ : a rðVÞ
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with r(V) volume-dependent curvature of the spherical cap, Vsf stress-free volume, a and b parameters defining the pti (see Appendix). This relationship was the basis for the following studies of interaction and stability and for parameter variations. Stability analyses were performed with the graphical method described by ¨ Muller and Strehlow (2004) for rubber balloons and exemplified here. This method requires the pv curves of all interacting structures in the system. Stable equilibriums of the structures are characterized by equal pressure and independence to small disturbances, and are studied graphically. Consider a system with two connected structures with identical simple pv curves at an arbitrary system volume V0 (Fig. 2): The solid line shows the pressure in structure one corresponding to its volume (V1); the dashed line represents the pressure in the second structure. States with equal pressure in both structures are represented by the crossing points but not all of them are stable. If some air is moved from structure one to structure two in point a, we move down on the curves. The pressure in structure two is higher and the air moves back to structure one, the situation is stable. If the same experiment is made at point b, the pressure in structure one becomes higher than the pressure in structure two, and even more air is shifted from the first to the second structure, this situation is not stable. To derive the pv curve of a system with two interacting alveoli, we calculated the volume distributions between the two alveoli [V1, V2] (V1, V2A[0,1.5 10 11 13 m ] in steps of 5 10 15 m3) that result in equal pressures in both alveoli: p1(V1) = p2(V2). For these pairs [V1, V2] we tested the stability as described above for the system volume V1 + V2 = V0. The stable states formed the quasi-static pv curve of the system. If two stable states existed at a fixed filling volume, we specified inflation (filling) and deflation (emptying) according to the previous states (Fig. 3). Pv curves for n alveoli were derived successively from the pv curves of single alveoli and of n-1 alveoli with the same procedures as described above. We simulated 15 interacting alveoli which are connected to the same alveolar duct, so that pressure equalization is possible. The actual spatial configuration was irrelevant, since we did not model dynamic effects which are dependent on the distance between the alveoli. To simulate lung injury and disease, we used a standard alveolus as described above and varied the model parameters in Eq. (1). Different types of lung injury and disease can be modeled if they can be described with the parameters concerning alveolar size, surface tension and tissue properties. We studied the principle effects of two parameters. We increased the surface tension of the alveolar fluid lining to the surface tension of water (g1 = 0.072 N/m), simulating the absence of surfactant. To simulate lung fibrosis or similar changes with decreased lung compliance and tissue remodeling, we changed the tissue component of Eq. (1) (a2 = 2a, Vsf,2 = 0.5Vsf). Both, surfactant abnormalities and fibrosis, are hallmarks of different disease stages during the course of the Acute Respiratory Distress Syndrome (Matute-Bello et al., 2008; Ware and Matthay, 2000). To study the secondary effects of lung injury and disease on intact alveoli, we studied systems with one standard model alveolus and one model alveolus with altered parameters as described above.
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pressure (Pa) Fig. 2. Evaluation of the stability of a system with two connected structures. Pressures in two connected structures are given corresponding to the volume of one structure (V1) at a constant overall volume (V0): solid line—pressure in structure 1, dotted line—pressure in structure 2 (volume of structure 2 is V0–V1). Line intersections mark the states with possible coexistence of both structures. Full circles represent stable states (a, c), open circle represents unstable state (b).
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Fig. 3. Typical plots for stability analysis of a system with two connected alveoli at progressively increasing (a-e) or decreasing (e-a) overall volume. Pressures in two connected alveoli are given corresponding to the volume of one alveolus (V1) at a constant overall volume (V0): solid line—pressure in alveolus 1, dotted line—pressure in alveolus 2 (volume of alveolus 2 is V0 V1). Line intersections mark the states with possible coexistence of both alveoli. Full circles represent stable states, open circles represent unstable states. In plot (c) four stable configurations exist (+ and #), always two of them equivalent. The distributions marked with + can be reached only in the filling process, distributions # can be reached only in the emptying process.
We derived pv curves of single alveoli and studied the influences of alveolar characteristics individually. Neither can be tested in animal experiments thus preventing the direct validation of the presented model. In the absence of direct validation, we discuss our results with relation to data from literature to support our model.
The stability analysis for two standard alveoli can be reduced to the cases shown in Fig. 3. The resulting pv curve of that system is given in Fig. 4, of 15 connected standard model alveoli in Fig. 5. The x-axis represents the transmural pressure, the y-axis represents the overall volume of the system and selected states are marked from a–k to illustrate the distribution of the system volume between the alveoli in Fig. 4. The opening of the alveoli takes place at the pressure maxima, from the second opening on it is accompanied by a pressure jump and a redistribution of the volume between the alveoli. The number of possible open alveoli at the different pressure ranges in these and the other simulated systems is listed in Table 1. Inflation and deflation processes differ slightly; a hysteresis occurs and is more evident in the system with 15 alveoli. Fig. 6 shows the pv curve of the system with one standard alveolus (left alveolus in the illustrations) and one alveolus with increased surface tension (0.072 N/m). Note, that at pressures between 500 and 700 Pa, representing the pressures during normal breathing, the alveolus with increased surface tension cannot be open (a, b in Fig. 6, Table 1). Point f marks the state where the surface areas of the two alveoli and of a system with two standard alveoli are equivalent at 700 Pa.
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Fig. 4. Pressure–volume curve of a system with two connected standard model alveoli, derived by combining the pressure–volume curves of single alveoli (Fig. 1) after the stability analysis. The y-axis shows the overall volume of the system. Volume distributions between the two alveoli are illustrated in small sketches: (a) both alveoli collapsed; (b–f) one alveolus collapsed, one open; (g–k) both alveoli open; (f, h) opening of the second alveolus; (g, e) closing of one alveolus. The grey part of the curve is valid only at deflation (h, g, e). Below 300 Pa both alveoli are collapsed, in the pressure range of 300–800 Pa alveoli of different size can coexist, above 800 Pa both alveoli are open.
Fig. 7 shows the pv curve of the system with one standard alveolus (left alveolus in the illustrations) and one alveolus with a reduced compliance and a shift of the stress-free volume. When
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both alveoli are open the stiff alveolus is smaller than the standard alveolus (b–d). Expanding the system to a surface area equivalent to that of two standard alveoli at 700 Pa, results in an increased volume of the standard alveolus (c).
4. Discussion In contrast to soap bubbles, where the smaller always fills the larger, alveoli can coexist both at equivalent and at different volumes in the intact lung. Which states of interacting alveoli are stable depends on the pressure and the filling state of the system.
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At very low and at large volumes alveoli need to have the same size to provide stability. In the medium range, alveoli can have different volumes and an equal volume distribution does not always result in a stable state (Fig. 3). This is evident in a system with two alveoli (Fig. 4) and remains valid with 15 alveoli (Fig. 5). More realistic heterogeneous geometry and tissue properties result in diverse volumes of the open alveoli in the model, as seen in vivo (Weibel, 1963; Mertens et al., 2009). We can derive the conditions for volume distributions with two open alveoli from the profile of the curve in Fig. 4. If the pressure was previously higher than 800 Pa (both alveoli were opened) and the pressure did not subsequently drop below 300 Pa, then both alveoli are open. The collapse of an alveolus and system volumes below 0.018 mm3 can occur only if the pressure minimum of 300 Pa is reached. The situation is comparable in a system with 15 alveoli. The transmural pressure usually provided by the human thorax is higher than the modeled minimum pressure of 300 Pa, so that cyclic opening and collapse of alveoli does not occur during the normal respiratory cycle. The hysteresis occurring in our model results from different stable states (Fig. 3). It separates alveolar opening and collapse. Such a system is insensitive to minor fluctuations in pressure and volume and thus avoids unnecessary
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Fig. 5. Pressure–volume curve of a system with 15 connected alveoli. The right curve shows inflation with the opening of alveoli marked by pressure maxima and subsequent pressure drops due to volume redistribution. The left curve shows deflation, the pressure jumps mark the redistribution of the volume at alveolar collapse. At 500 Pa either 0 (a), 1 (b, c), 2 (d, e), 3 (f, g), 4 (h) or 15 (i) alveoli are open, different alveolar sizes of open alveoli coexist at (b), (d) and (f). At 700 Pa from 0 to 15 alveoli can be open (all open alveoli with equal size).
Fig. 6. Pressure–volume curve of a system with two connected model alveoli, one standard alveolus and one with increased surface tension. Sketches illustrate the volume distribution between the standard alveolus with a surface tension of 0.02 N/m (left) and the alveolus with increased surface tension of 0.072 N/m (right) in the following states: (a) both alveoli closed at 700 Pa; (b) standard alveolus open at 700 Pa; (d) opening of the diseased alveolus at 2880 Pa with major parts of the volume in the standard alveolus, subsequent pressure jump to (g); (e, c) closing of the affected alveolus; (f) surface area is equivalent to the surface area of two standard alveoli at 700 Pa; (h) alveolus with increased surface tension has the size of the standard alveolus at 700 Pa, standard alveolus has increased volume; (i) increased volume in both alveoli.
Table 1 Number of open alveoli against the pressure range in the different simulations (compare Figs. 4–7). Pressure range in Pa
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pressure (Pa) Fig. 7. Pressure–volume curve of a system with two connected model alveoli, one standard alveolus and one with a reduced compliance, with an accompanying shift of the intersection point. Sketches illustrate volume distribution between the standard alveolus (left) and the alveolus with reduced compliance (right) at critical states as follows: (a) open standard alveolus, alveolus with reduced compliance is collapsed at 700 Pa; (b) both alveoli open at 700 Pa, stiff alveolus is smaller than the standard alveolus; (c) surface area of the system is equivalent to the surface area of two standard alveoli at 700 Pa; (d) stiff alveolus has the size of the standard alveolus at 700 Pa, standard alveolus has increased volume.
strain caused by numerous opening and collapse processes, which are assumed to be the cause of VILI (atelectrauma). This hysteresis, however, does not contribute to the lung hysteresis seen in the static lung pv curves during normal breathing cycles (500–700 Pa). In the 15 alveoli system, the individual alveolar volume does not always change continuously but also successively by either opening (volume range 0.004–1.9 mm3, Fig. 5) or closing (volume range 1.3–0.004 mm3). The results from the simulation of 15 alveoli also suggest that the successive opening or closing at the alveolar level is hardly reflected in the pv curves of whole lungs: The more alveoli are considered, the lower the pressure drops at the opening of single alveoli. Moreover, heterogeneous alveolar properties implicate heterogeneous opening pressures. These data provide an explanation for previous experimental observations that inflection points on the pv curve do not necessarily represent alveolar recruitment–derecruitment phenomena (DiRocco et al., 2007), which in fact occur successively over a wide range of lung volumes. Lung injury and disease typically affect not only one, but several characteristics of alveoli. This model simulation provides the opportunity to selectively study the effects of each of these parameters on alveolar interaction. Increased surface tension results in higher opening pressures and higher pressures below which alveoli are collapsed. The profile of the respective pv curve shows that alveoli with a high constant surface tension are not open at normal breathing pressures (applies to g 40.045 N/m). This is in agreement with experimental observations (Luecke et al., 2006) and with the results of the stability analysis for larger lung regions by Stamenovic and Wilson (1992). Our model also predicts that these alveoli can be opened by high pressure (p42880 Pa) and kept open if the pressure does not decrease below 1100 Pa. Alveoli with increased surface tension are themselves not overexpanded after recruitment, if the PEEP is approximately 1000 Pa (Fig. 6 point h). This finding demonstrates that recruitment maneuvers can be successful regarding the reestablishment of the alveolar surface area if an adequate PEEP is applied, a notion in line with data from clinical studies with
recruitment pressures of 3000–5000 Pa (Kacmarek and Kallet, 2007; Mols et al., 2002) and applied PEEPs of around 900 Pa (ARDS Network, 2000). However, the model predicts overdistension of parallel existing standard alveoli both during the recruitment maneuver as well as subsequently at the applied PEEP, which is again in agreement with respective clinical considerations (Gattinoni and Pesenti, 2005; Hubmayr, 2002; Kacmarek and Kallet, 2007). Changes in tissue characteristics modify the pv curves in a different manner. In the presented simulation the second minimum pressure moves to slightly higher values (from 300 to 390 Pa). These changes implicate that alveolar derecruitment is unlikely but possible, and high pressures are necessary to reach the equivalent surface area of a system with standard alveoli. In this scenario, the alveoli will remain small after the recruitment maneuver and the PEEP application, while coexisting standard alveoli will be overexpanded. We consider an optimal model to be as simple as possible and as complex as necessary to reasonably address the issue of interest. However, simplifications imply model limitations. The airspace geometry model utilized in this study is simple. Yet, using more complex models does not provide any further improvements concerning the scope of our study. The deformation of the entrance ring is small compared to the alveolar walls, since the fiber density is increased (Matsuda et al., 1987). At low volumes the geometry of the alveolar walls is not defined, only the fluid boundary surface is modeled as a cap. At higher volumes the geometry becomes less important, as long as the radius of curvature increases with volume. Modeling alveoli as spherical caps has recently been challenged (Prange, 2003): (1.) alveoli seem polygonal in histological lung slices, (2.) application of Laplace’s law would lead to incorrect conclusions and contradictions and (3.) alveoli are connected. However, (1.) alveoli look round in intravital microscopy, in optical coherence tomography (Mertens et al., 2009) and in confocal microscopy (Perlman and Bhattacharya, 2007; Namati et al., 2008). Importantly, Lindert et al. (2007) showed that the alveolar wall liquid is located predominantly in the corners of the alveolus, thus smoothing the alveolar structure. (2.) The proposed incorrect conclusions and contradictions occur only if the surface tension of the fluid lining is incorrectly supposed to be the sole effect on alveolar mechanics. (3.) The interdependence of the alveoli includes the connected lumen, which implies equal pressure, and a fiber network with shared alveolar septa, which implies geometrical interdependence of directly neighbored alveoli. Shared alveolar septa are not included in the presented approach. We assume the geometrical effects to be non-essential for alveoli that are not directly neighbored, since tissue elasticity and the effect of the fiber network are included in the model. Another simplification is the assumed constant surface tension in the alveolar fluid lining during inspiration and expiration, resulting in identical curves for single alveoli. Introducing a variability of surface tension with surface area changes the pv curve, but the most relevant characteristics remain: a low opening pressure, a descending branch and a minimum pressure for open alveoli below 500 Pa. The nonlinearity of tissue behavior is not included, since it is important only at higher volumes (Navajas et al., 1995). At high volumes we point out the risk of overdistension, which would be similarly present in a model with limited distension. Airway closure is not studied here, though it is assumed to be an important issue in the derecruitment of larger lung regions, it is of minor importance for the interaction of the individual alveoli addressed here. In conclusion, our parametric simulation model explains the interaction of alveoli under normal breathing conditions and the very different effects that recruitment maneuvers have on alveolar recruitment and distension depending on the underlying
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lung injury and disease. If alveolar damage is dominated by a failure of the surfactant system, recruitment maneuvers with adequate PEEP might be successful. In lung diseases with reduced alveolar compliance due to changes of biomechanical tissue properties, recruitment maneuvers are likely to be unsuccessful and may even be counterproductive. High pressures during recruitment maneuvers carry the risk of overdistension of unaffected and/or only partly affected alveoli. Conflict of interest statement None.
Acknowledgments Supported by the Deutsche Forschungsgemeinschaft (DFG) AF ¨ 3/33-1, KU1218/4-1 and Promotionsforderung des Landes Berlin.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2009.11.025.
References ARDS Network, 2000. Ventilation with lower tidal volumes as compared with traditional tidal volumes for acute lung injury and the acute respiratory distress syndrome. The New England Journal of Medicine 342, 1301–1308. Carney, D.E., DiRocco, J., Nieman, G.F., 2005. Dynamic alveolar mechanics and ventilator-induced lung injury. Critical Care Medicine 33 (3 [Suppl.]), S122– S128. Clements, J.A., Hustead, R.F., Johnson, R.P., Gribetz, I., 1961. Pulmonary surface tension and alveolar stability. Journal of Applied Physiology 16 (3), 444–450. Denny, E., Schroter, R.C., 2000. Viscoelastic behavior of a lung alveolar duct model. Journal of Biomechanical Engineering 122 (2), 143–151. DiRocco, J.D., Carney, D.E., Nieman, G.F., 2007. Correlation between alveolar recruitment / derecruitment and inflection points on the pressure-volume curve. Intensive Care Medicine 33, 1204–1211. Fung, Y.C., 1975. Stress, deformation, and atelectasis of the lung. Circulation Research 37, 481–496. Gattinoni, L., Pesenti, A., 2005. The concept of ‘‘baby lung’’. Intensive Care Medicine 31, 776–784. Hubmayr, R.D., 2002. Perspective on lung injury and recruitment. a skeptical look at the opening and collapse story. American Journal of Respiratory and Critical Care Medicine 165, 1647–1653. Kacmarek, R.M., Kallet, R.H., 2007. Should recruitment maneuvers be used in the management of ali and ards? Respiratory Care 52 (5), 622–631. Klinke, R., Silbernagel, S. (Eds.), 1996. Lehrbuch der Physiologie. Thieme. Kumar, H., Tawhai, M.H., Hoffman, E.A., Lin, C.L., 2009. The effects of geometry on airflow in the acinar region of the human lung. Journal of Biomechanics 42 (11), 1635–1642.
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Lachmann, B., 1992. Open up the lung and keep the lung open. Intensive Care Medicine 18, 319–321. Lindert, J., Perlman, C.E., Parthasarathi, K., Bhattacharya, J., 2007. Chloridedependent secretion of alveolar wall liquid determined by optical-sectioning microscopy. American Journal of Respiratory Cell and Molecular Biology 36 (6), 688–696. Luecke, T., Meinhardt, J.P., Herrmann, P., Weiss, A., Quintel, M., Pelosi, P., 2006. Oleic acid vs saline solution lung lavage-induced acute lung injury: effects on lung morphology, pressure-volume relationships, and response to positive end-expiratory pressure. Chest 130 (2), 392–401. Matsuda, M., Fung, Y.C., Sobin, S.S., 1987. Collagen and elastin fibers in human pulmonary alveolar mouths and ducts. Journal of Applied Physiology 63 (3), 1185–1194. Matute-Bello, G., Frevert, C.W., Martin, T.R., 2008. Animal models of acute lung injury. American journal of physiology. Lung cellular and molecular physiology 295 (3), L379–L399. Mead, J., Takishima, T., Leith, D., 1970. Stress distribution in lungs: a model of pulmonary elasticity. Journal of Applied Physiology 28 (5), 596–608. Mertens, M., Tabuchi, A., Meissner, S., Krueger, A., Schirrmann, K., Kertzscher, U., Pries, A.R., Slutsky, A.S., Koch, E., Kuebler, W.M., 2009. Alveolar dynamics in acute lung injury: heterogeneous distension rather than cyclic opening and collapse. Critical Care Medicine 37 (9), 2604–2611. Mols, G., Hermle, G., Fries, G., Benzing, A., Lichtwarck-Aschoff, M., Geiger, K., Guttmann, J., 2002. different strategies to keep the lung open: A study in isolated perfused rabbit lungs. Critical Care Medicine 30 (7), 1598–1604. ¨ Muller, I., Strehlow, P., 2004. Rubber and Rubber Balloons, Paradigms of Thermodynamics. of Lecture Notes in Physics, Vol. 637. Springer-Verlag, Berlin Heidelberg. Namati, E., Thiesse, J., de Ryk, J., McLennan, G., 2008. Alveolar dynamics during respiration: are the pores of Kohn a pathway to recruitment? American Journal of Respiratory Cell and Molecular Biology 38 (5), 572–578. Navajas, D., Maksym, G.N., Bates, J.H., 1995. Dynamic viscoelastic nonlinearity of lung parenchymal tissue. Journal of Applied Physiology 79 (1), 348–356. Otis, D.R., Ingenito, E.P., Kamm, R.D., Johnson, M., 1994. Dynamic surface tension experiments and theory. Journal of Applied Physiology 77 (6), 2681–2688. Perlman, C.E., Bhattacharya, J., 2007. Alveolar expansion imaged by optical sectioning microscopy. Journal of Applied Physiology 103 (3), 1037–1044. Prange, H.D., 2003. Laplace’s law and the alveolus: a misconception of anatomy and a misapplication of physics. Advances in physiology education 27 (1-4), 34–40. Stamenovic, D., Smith, J.C., 1986. Surface forces in lungs. ii. microstructural mechanics and lung stability. Journal of Applied Physiology 60 (4), 1351–1357. Stamenovic, D., Wilson, T.A., 1992. Parenchymal stabilitiy. Journal of Applied Physiology 73 (2), 596–602. ¨ Sznitman, J., Heimsch, T., Wildhaber, J.H., Tsuda, A., Rosgen, T., 2009. Respiratory flow phenomena and gravitational deposition in a three-dimensional spacefilling model of the pulmonary acinar tree. Journal of Biomechanical Engineering 131 (3), 031010. Tsuda, A., Henry, F.S., Butler, J.P., 2008. Gas and aerosol mixing in the acinus. Respiratory physiology & neurobiology 163 (1-3), 139–149. Ware, L.B., Matthay, M.A., 2000. The acute respiratory distress syndrome. The New England Journal of Medicine 342 (18), 1334–1349. Weibel, E.R., 1963. In: Morphometry of the Human Lung. Academic Press Inc. and Springer-Verlag. Wilson, T., 1981. Relations among recoil pressure, surface area, and surface tension in the lung. Journal of Applied Physiology: Respiratory, Environmental and Exercise Physiology 50 (5), 921–926. Wilson, T.A., Anafi, R.C., Hubmayr, R.D., 2001. Mechanics of edematous lungs. Journal of Applied Physiology 90, 2088–2093.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Theoretical modeling of the interaction between alveoli during inflation and deflation in normal and diseased lungs Kerstin Schirrmann a,n, Michael Mertens b,c, Ulrich Kertzscher a, Wolfgang M. Kuebler b,d, Klaus Affeld a a
´ - Universit¨ Biofluid Mechanics Laboratory, Charite atsmedizin Berlin, Germany Institute of Physiology, Charite´ - Universit¨ atsmedizin Berlin, Germany c Institute for Human Biology and Anthropology, Free University Berlin, Germany d Department of Surgery, University of Toronto, ON, Canada b
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 21 November 2009
Alveolar recruitment is a central strategy in the ventilation of patients with acute lung injury and other lung diseases associated with alveolar collapse and atelectasis. However, biomechanical insights into the opening and collapse of individual alveoli are still limited. A better understanding of alveolar recruitment and the interaction between alveoli in intact and injured lungs is of crucial relevance for the evaluation of the potential efficacy of ventilation strategies. We simulated human alveolar biomechanics in normal and injured lungs. We used a basic simulation model for the biomechanical behavior of virtual single alveoli to compute parameterized pressure–volume curves. Based on these curves, we analyzed the interaction and stability in a system composed of two alveoli. We introduced different values for surface tension and tissue properties to simulate different forms of lung injury. The data obtained predict that alveoli with identical properties can coexist with both different volumes and with equal volumes depending on the pressure. Alveoli in injured lungs with increased surface tension will collapse at normal breathing pressures. However, recruitment maneuvers and positive endexpiratory pressure can stabilize those alveoli, but coexisting unaffected alveoli might be overdistended. In injured alveoli with reduced compliance collapse is less likely, alveoli are expected to remain open, but with a smaller volume. Expanding them to normal size would overdistend coexisting unaffected alveoli. The present simulation model yields novel insights into the interaction between alveoli and may thus increase our understanding of the prospects of recruitment maneuvers in different forms of lung injury. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Simulation Lung injury Stability Surface tension Recruitment
1. Introduction Mechanical ventilation is necessary in patients with acute lung injury to assure sufficient oxygen supply and carbon dioxide removal. However, mechanical ventilation may lead to ventilatorinduced lung injury (VILI). Avoiding VILI is the aim of protective ventilation strategies. One of the central strategies follows the open lung approach (Lachmann, 1992) and includes recruitment maneuvers. Recruitment maneuvers involve the initial application of high ventilation pressures to recruit unaerated lung regions and subsequent positive endexpiratory pressures (PEEP) to keep these regions open. Still under discussion is whether to apply this method as routine in the clinical scenario (Kacmarek and Kallet, 2007). The mechanisms assumed to be contributing to VILI are high shear stress in the transition region between open and collapsed alveolar regions
n
Corresponding author. Tel.: + 49 30 450 553 798; fax: + 49 30 450 553 938. E-mail address: [email protected] (K. Schirrmann).
0021-9290/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2009.11.025
(Mead et al., 1970), the cyclic opening and collapse of individual alveoli (Carney et al., 2005) and the overdistension of alveoli (Hubmayr, 2002). Understanding the biomechanics of individual alveoli better may thus allow for an improved evaluation of ventilation strategies in different forms of lung injury and disease. Recently, alveolar models were presented for flow simulations (Tsuda et al., 2008; Sznitman et al., 2009; Kumar et al., 2009) and dynamic parenchymal behavior (Denny and Schroter, 2000). Regarding lung stability, several simulation models have been reported (Fung, 1975; Mead et al., 1970; Stamenovic and Smith, 1986; Stamenovic and Wilson, 1992; Wilson, 1981; Wilson et al., 2001). These studies showed that regional collapse is not a contradiction to structural integrity. The applied models allowed for the description of shear stress in adjacent open and closed lung regions (Mead et al., 1970) and of the stabilizing effect of the surrounding parenchyma on a lung region (Fung, 1975). However, the microstructural models considered regions larger than the individual alveoli (Mead et al., 1970). We studied the interaction of individual alveoli and the stability of the resulting systems. For
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¨ the stability analysis (Muller and Strehlow, 2004), we generated alveolar pressure–volume curves (pv curves) with a model similar to the one of Clements et al. (1961). However, Clements’ model does not include the interaction of connected alveoli and therefore excludes the stabilizing effects of parallel alveoli. Our approach describes the alveoli as interacting structures with equal pressures. Importantly, the use of theoretically derived pv curves of single alveoli allows for parametric studies to simulate effects of lung injury and disease at the alveolar level. With our model, we want to provide insights into the effects of changes in surface tension and tissue properties on the performance of interacting alveoli. This was studied during both normal breathing and recruitment maneuvers. 2. Material and Methods We consider the following as the main influences on the alveolar pressure– volume behavior: morphology, surface tension of the fluid lining, elasticity of the tissue and the embedding of the alveolus in the tissue, that is the connection to other alveoli via elastin and collagen fibers. Regarding morphology, we used a simple geometrical model (Clements et al., 1961). A rigid ring forms the alveolar mouth (diameter set at 100 mm) while the fluid–air interface of the alveolus itself is modeled as a spherical cap. For simplicity, we assumed a constant surface tension in the fluid lining in our model. The value of the surface tension was set to g =0.02 N/m in the normal model alveolus according to previously reported values for minimal equilibrium surface tension (Otis et al., 1994). The influences of tissue elasticity and the embedding of the alveoli in the tissue were combined to a tissue component and were approximated by a linear relationship in the pv curve as of a certain volume. This was motivated by pv curves of saline filled lungs, where these two effects produce an only slightly sigmoidal curve starting at a nonzero volume (Hubmayr, 2002). Two points are required to define the simplified pv curve (see Appendix): We used the volumes and pressures at the beginning and end of inspiration in a human lung during normal respiration. We assumed a total number of alveoli (na) of 300,000,000 (Weibel, 1963), a functional residual capacity (VFRC) of 3.2 l, a dead space volume (Vds) of 0.15 l, a tidal volume (Vt) of 0.5 l, an end-expiratory pressure of 500 Pa, and an end-inspiratory pressure of 700 Pa (Klinke and Silbernagel, 1996). Pressures were set to zero below the intersection with the volume axis to avoid negative pressures, which are related to compressive stress, since it was assumed that alveolar walls would not be able to sustain a compressive stress (Fung, 1975). The superposition of the influences of the fluid lining (pfl(V) = 2g/r(V), see Appendix) and of the tissue component (pti(V)= aV+ b for V 4Vsf, 0 else) resulted in the pv curve of the standard model alveolus (Eq. (1), Fig. 1). 8 2g b > > for V o Vsf ¼ > < a rðVÞ ð1Þ pðVÞ ¼ pfl ðV Þþ pti ðV Þ ¼ 2g b > > > þ aV þ b for V Z Vsf ¼ : a rðVÞ
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with r(V) volume-dependent curvature of the spherical cap, Vsf stress-free volume, a and b parameters defining the pti (see Appendix). This relationship was the basis for the following studies of interaction and stability and for parameter variations. Stability analyses were performed with the graphical method described by ¨ Muller and Strehlow (2004) for rubber balloons and exemplified here. This method requires the pv curves of all interacting structures in the system. Stable equilibriums of the structures are characterized by equal pressure and independence to small disturbances, and are studied graphically. Consider a system with two connected structures with identical simple pv curves at an arbitrary system volume V0 (Fig. 2): The solid line shows the pressure in structure one corresponding to its volume (V1); the dashed line represents the pressure in the second structure. States with equal pressure in both structures are represented by the crossing points but not all of them are stable. If some air is moved from structure one to structure two in point a, we move down on the curves. The pressure in structure two is higher and the air moves back to structure one, the situation is stable. If the same experiment is made at point b, the pressure in structure one becomes higher than the pressure in structure two, and even more air is shifted from the first to the second structure, this situation is not stable. To derive the pv curve of a system with two interacting alveoli, we calculated the volume distributions between the two alveoli [V1, V2] (V1, V2A[0,1.5 10 11 13 m ] in steps of 5 10 15 m3) that result in equal pressures in both alveoli: p1(V1) = p2(V2). For these pairs [V1, V2] we tested the stability as described above for the system volume V1 + V2 = V0. The stable states formed the quasi-static pv curve of the system. If two stable states existed at a fixed filling volume, we specified inflation (filling) and deflation (emptying) according to the previous states (Fig. 3). Pv curves for n alveoli were derived successively from the pv curves of single alveoli and of n-1 alveoli with the same procedures as described above. We simulated 15 interacting alveoli which are connected to the same alveolar duct, so that pressure equalization is possible. The actual spatial configuration was irrelevant, since we did not model dynamic effects which are dependent on the distance between the alveoli. To simulate lung injury and disease, we used a standard alveolus as described above and varied the model parameters in Eq. (1). Different types of lung injury and disease can be modeled if they can be described with the parameters concerning alveolar size, surface tension and tissue properties. We studied the principle effects of two parameters. We increased the surface tension of the alveolar fluid lining to the surface tension of water (g1 = 0.072 N/m), simulating the absence of surfactant. To simulate lung fibrosis or similar changes with decreased lung compliance and tissue remodeling, we changed the tissue component of Eq. (1) (a2 = 2a, Vsf,2 = 0.5Vsf). Both, surfactant abnormalities and fibrosis, are hallmarks of different disease stages during the course of the Acute Respiratory Distress Syndrome (Matute-Bello et al., 2008; Ware and Matthay, 2000). To study the secondary effects of lung injury and disease on intact alveoli, we studied systems with one standard model alveolus and one model alveolus with altered parameters as described above.
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pressure (Pa) Fig. 2. Evaluation of the stability of a system with two connected structures. Pressures in two connected structures are given corresponding to the volume of one structure (V1) at a constant overall volume (V0): solid line—pressure in structure 1, dotted line—pressure in structure 2 (volume of structure 2 is V0–V1). Line intersections mark the states with possible coexistence of both structures. Full circles represent stable states (a, c), open circle represents unstable state (b).
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Fig. 3. Typical plots for stability analysis of a system with two connected alveoli at progressively increasing (a-e) or decreasing (e-a) overall volume. Pressures in two connected alveoli are given corresponding to the volume of one alveolus (V1) at a constant overall volume (V0): solid line—pressure in alveolus 1, dotted line—pressure in alveolus 2 (volume of alveolus 2 is V0 V1). Line intersections mark the states with possible coexistence of both alveoli. Full circles represent stable states, open circles represent unstable states. In plot (c) four stable configurations exist (+ and #), always two of them equivalent. The distributions marked with + can be reached only in the filling process, distributions # can be reached only in the emptying process.
We derived pv curves of single alveoli and studied the influences of alveolar characteristics individually. Neither can be tested in animal experiments thus preventing the direct validation of the presented model. In the absence of direct validation, we discuss our results with relation to data from literature to support our model.
The stability analysis for two standard alveoli can be reduced to the cases shown in Fig. 3. The resulting pv curve of that system is given in Fig. 4, of 15 connected standard model alveoli in Fig. 5. The x-axis represents the transmural pressure, the y-axis represents the overall volume of the system and selected states are marked from a–k to illustrate the distribution of the system volume between the alveoli in Fig. 4. The opening of the alveoli takes place at the pressure maxima, from the second opening on it is accompanied by a pressure jump and a redistribution of the volume between the alveoli. The number of possible open alveoli at the different pressure ranges in these and the other simulated systems is listed in Table 1. Inflation and deflation processes differ slightly; a hysteresis occurs and is more evident in the system with 15 alveoli. Fig. 6 shows the pv curve of the system with one standard alveolus (left alveolus in the illustrations) and one alveolus with increased surface tension (0.072 N/m). Note, that at pressures between 500 and 700 Pa, representing the pressures during normal breathing, the alveolus with increased surface tension cannot be open (a, b in Fig. 6, Table 1). Point f marks the state where the surface areas of the two alveoli and of a system with two standard alveoli are equivalent at 700 Pa.
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Fig. 4. Pressure–volume curve of a system with two connected standard model alveoli, derived by combining the pressure–volume curves of single alveoli (Fig. 1) after the stability analysis. The y-axis shows the overall volume of the system. Volume distributions between the two alveoli are illustrated in small sketches: (a) both alveoli collapsed; (b–f) one alveolus collapsed, one open; (g–k) both alveoli open; (f, h) opening of the second alveolus; (g, e) closing of one alveolus. The grey part of the curve is valid only at deflation (h, g, e). Below 300 Pa both alveoli are collapsed, in the pressure range of 300–800 Pa alveoli of different size can coexist, above 800 Pa both alveoli are open.
Fig. 7 shows the pv curve of the system with one standard alveolus (left alveolus in the illustrations) and one alveolus with a reduced compliance and a shift of the stress-free volume. When
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both alveoli are open the stiff alveolus is smaller than the standard alveolus (b–d). Expanding the system to a surface area equivalent to that of two standard alveoli at 700 Pa, results in an increased volume of the standard alveolus (c).
4. Discussion In contrast to soap bubbles, where the smaller always fills the larger, alveoli can coexist both at equivalent and at different volumes in the intact lung. Which states of interacting alveoli are stable depends on the pressure and the filling state of the system.
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At very low and at large volumes alveoli need to have the same size to provide stability. In the medium range, alveoli can have different volumes and an equal volume distribution does not always result in a stable state (Fig. 3). This is evident in a system with two alveoli (Fig. 4) and remains valid with 15 alveoli (Fig. 5). More realistic heterogeneous geometry and tissue properties result in diverse volumes of the open alveoli in the model, as seen in vivo (Weibel, 1963; Mertens et al., 2009). We can derive the conditions for volume distributions with two open alveoli from the profile of the curve in Fig. 4. If the pressure was previously higher than 800 Pa (both alveoli were opened) and the pressure did not subsequently drop below 300 Pa, then both alveoli are open. The collapse of an alveolus and system volumes below 0.018 mm3 can occur only if the pressure minimum of 300 Pa is reached. The situation is comparable in a system with 15 alveoli. The transmural pressure usually provided by the human thorax is higher than the modeled minimum pressure of 300 Pa, so that cyclic opening and collapse of alveoli does not occur during the normal respiratory cycle. The hysteresis occurring in our model results from different stable states (Fig. 3). It separates alveolar opening and collapse. Such a system is insensitive to minor fluctuations in pressure and volume and thus avoids unnecessary
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Fig. 5. Pressure–volume curve of a system with 15 connected alveoli. The right curve shows inflation with the opening of alveoli marked by pressure maxima and subsequent pressure drops due to volume redistribution. The left curve shows deflation, the pressure jumps mark the redistribution of the volume at alveolar collapse. At 500 Pa either 0 (a), 1 (b, c), 2 (d, e), 3 (f, g), 4 (h) or 15 (i) alveoli are open, different alveolar sizes of open alveoli coexist at (b), (d) and (f). At 700 Pa from 0 to 15 alveoli can be open (all open alveoli with equal size).
Fig. 6. Pressure–volume curve of a system with two connected model alveoli, one standard alveolus and one with increased surface tension. Sketches illustrate the volume distribution between the standard alveolus with a surface tension of 0.02 N/m (left) and the alveolus with increased surface tension of 0.072 N/m (right) in the following states: (a) both alveoli closed at 700 Pa; (b) standard alveolus open at 700 Pa; (d) opening of the diseased alveolus at 2880 Pa with major parts of the volume in the standard alveolus, subsequent pressure jump to (g); (e, c) closing of the affected alveolus; (f) surface area is equivalent to the surface area of two standard alveoli at 700 Pa; (h) alveolus with increased surface tension has the size of the standard alveolus at 700 Pa, standard alveolus has increased volume; (i) increased volume in both alveoli.
Table 1 Number of open alveoli against the pressure range in the different simulations (compare Figs. 4–7). Pressure range in Pa
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pressure (Pa) Fig. 7. Pressure–volume curve of a system with two connected model alveoli, one standard alveolus and one with a reduced compliance, with an accompanying shift of the intersection point. Sketches illustrate volume distribution between the standard alveolus (left) and the alveolus with reduced compliance (right) at critical states as follows: (a) open standard alveolus, alveolus with reduced compliance is collapsed at 700 Pa; (b) both alveoli open at 700 Pa, stiff alveolus is smaller than the standard alveolus; (c) surface area of the system is equivalent to the surface area of two standard alveoli at 700 Pa; (d) stiff alveolus has the size of the standard alveolus at 700 Pa, standard alveolus has increased volume.
strain caused by numerous opening and collapse processes, which are assumed to be the cause of VILI (atelectrauma). This hysteresis, however, does not contribute to the lung hysteresis seen in the static lung pv curves during normal breathing cycles (500–700 Pa). In the 15 alveoli system, the individual alveolar volume does not always change continuously but also successively by either opening (volume range 0.004–1.9 mm3, Fig. 5) or closing (volume range 1.3–0.004 mm3). The results from the simulation of 15 alveoli also suggest that the successive opening or closing at the alveolar level is hardly reflected in the pv curves of whole lungs: The more alveoli are considered, the lower the pressure drops at the opening of single alveoli. Moreover, heterogeneous alveolar properties implicate heterogeneous opening pressures. These data provide an explanation for previous experimental observations that inflection points on the pv curve do not necessarily represent alveolar recruitment–derecruitment phenomena (DiRocco et al., 2007), which in fact occur successively over a wide range of lung volumes. Lung injury and disease typically affect not only one, but several characteristics of alveoli. This model simulation provides the opportunity to selectively study the effects of each of these parameters on alveolar interaction. Increased surface tension results in higher opening pressures and higher pressures below which alveoli are collapsed. The profile of the respective pv curve shows that alveoli with a high constant surface tension are not open at normal breathing pressures (applies to g 40.045 N/m). This is in agreement with experimental observations (Luecke et al., 2006) and with the results of the stability analysis for larger lung regions by Stamenovic and Wilson (1992). Our model also predicts that these alveoli can be opened by high pressure (p42880 Pa) and kept open if the pressure does not decrease below 1100 Pa. Alveoli with increased surface tension are themselves not overexpanded after recruitment, if the PEEP is approximately 1000 Pa (Fig. 6 point h). This finding demonstrates that recruitment maneuvers can be successful regarding the reestablishment of the alveolar surface area if an adequate PEEP is applied, a notion in line with data from clinical studies with
recruitment pressures of 3000–5000 Pa (Kacmarek and Kallet, 2007; Mols et al., 2002) and applied PEEPs of around 900 Pa (ARDS Network, 2000). However, the model predicts overdistension of parallel existing standard alveoli both during the recruitment maneuver as well as subsequently at the applied PEEP, which is again in agreement with respective clinical considerations (Gattinoni and Pesenti, 2005; Hubmayr, 2002; Kacmarek and Kallet, 2007). Changes in tissue characteristics modify the pv curves in a different manner. In the presented simulation the second minimum pressure moves to slightly higher values (from 300 to 390 Pa). These changes implicate that alveolar derecruitment is unlikely but possible, and high pressures are necessary to reach the equivalent surface area of a system with standard alveoli. In this scenario, the alveoli will remain small after the recruitment maneuver and the PEEP application, while coexisting standard alveoli will be overexpanded. We consider an optimal model to be as simple as possible and as complex as necessary to reasonably address the issue of interest. However, simplifications imply model limitations. The airspace geometry model utilized in this study is simple. Yet, using more complex models does not provide any further improvements concerning the scope of our study. The deformation of the entrance ring is small compared to the alveolar walls, since the fiber density is increased (Matsuda et al., 1987). At low volumes the geometry of the alveolar walls is not defined, only the fluid boundary surface is modeled as a cap. At higher volumes the geometry becomes less important, as long as the radius of curvature increases with volume. Modeling alveoli as spherical caps has recently been challenged (Prange, 2003): (1.) alveoli seem polygonal in histological lung slices, (2.) application of Laplace’s law would lead to incorrect conclusions and contradictions and (3.) alveoli are connected. However, (1.) alveoli look round in intravital microscopy, in optical coherence tomography (Mertens et al., 2009) and in confocal microscopy (Perlman and Bhattacharya, 2007; Namati et al., 2008). Importantly, Lindert et al. (2007) showed that the alveolar wall liquid is located predominantly in the corners of the alveolus, thus smoothing the alveolar structure. (2.) The proposed incorrect conclusions and contradictions occur only if the surface tension of the fluid lining is incorrectly supposed to be the sole effect on alveolar mechanics. (3.) The interdependence of the alveoli includes the connected lumen, which implies equal pressure, and a fiber network with shared alveolar septa, which implies geometrical interdependence of directly neighbored alveoli. Shared alveolar septa are not included in the presented approach. We assume the geometrical effects to be non-essential for alveoli that are not directly neighbored, since tissue elasticity and the effect of the fiber network are included in the model. Another simplification is the assumed constant surface tension in the alveolar fluid lining during inspiration and expiration, resulting in identical curves for single alveoli. Introducing a variability of surface tension with surface area changes the pv curve, but the most relevant characteristics remain: a low opening pressure, a descending branch and a minimum pressure for open alveoli below 500 Pa. The nonlinearity of tissue behavior is not included, since it is important only at higher volumes (Navajas et al., 1995). At high volumes we point out the risk of overdistension, which would be similarly present in a model with limited distension. Airway closure is not studied here, though it is assumed to be an important issue in the derecruitment of larger lung regions, it is of minor importance for the interaction of the individual alveoli addressed here. In conclusion, our parametric simulation model explains the interaction of alveoli under normal breathing conditions and the very different effects that recruitment maneuvers have on alveolar recruitment and distension depending on the underlying
ARTICLE IN PRESS K. Schirrmann et al. / Journal of Biomechanics 43 (2010) 1202–1207
lung injury and disease. If alveolar damage is dominated by a failure of the surfactant system, recruitment maneuvers with adequate PEEP might be successful. In lung diseases with reduced alveolar compliance due to changes of biomechanical tissue properties, recruitment maneuvers are likely to be unsuccessful and may even be counterproductive. High pressures during recruitment maneuvers carry the risk of overdistension of unaffected and/or only partly affected alveoli. Conflict of interest statement None.
Acknowledgments Supported by the Deutsche Forschungsgemeinschaft (DFG) AF ¨ 3/33-1, KU1218/4-1 and Promotionsforderung des Landes Berlin.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2009.11.025.
References ARDS Network, 2000. Ventilation with lower tidal volumes as compared with traditional tidal volumes for acute lung injury and the acute respiratory distress syndrome. The New England Journal of Medicine 342, 1301–1308. Carney, D.E., DiRocco, J., Nieman, G.F., 2005. Dynamic alveolar mechanics and ventilator-induced lung injury. Critical Care Medicine 33 (3 [Suppl.]), S122– S128. Clements, J.A., Hustead, R.F., Johnson, R.P., Gribetz, I., 1961. Pulmonary surface tension and alveolar stability. Journal of Applied Physiology 16 (3), 444–450. Denny, E., Schroter, R.C., 2000. Viscoelastic behavior of a lung alveolar duct model. Journal of Biomechanical Engineering 122 (2), 143–151. DiRocco, J.D., Carney, D.E., Nieman, G.F., 2007. Correlation between alveolar recruitment / derecruitment and inflection points on the pressure-volume curve. Intensive Care Medicine 33, 1204–1211. Fung, Y.C., 1975. Stress, deformation, and atelectasis of the lung. Circulation Research 37, 481–496. Gattinoni, L., Pesenti, A., 2005. The concept of ‘‘baby lung’’. Intensive Care Medicine 31, 776–784. Hubmayr, R.D., 2002. Perspective on lung injury and recruitment. a skeptical look at the opening and collapse story. American Journal of Respiratory and Critical Care Medicine 165, 1647–1653. Kacmarek, R.M., Kallet, R.H., 2007. Should recruitment maneuvers be used in the management of ali and ards? Respiratory Care 52 (5), 622–631. Klinke, R., Silbernagel, S. (Eds.), 1996. Lehrbuch der Physiologie. Thieme. Kumar, H., Tawhai, M.H., Hoffman, E.A., Lin, C.L., 2009. The effects of geometry on airflow in the acinar region of the human lung. Journal of Biomechanics 42 (11), 1635–1642.
1207
Lachmann, B., 1992. Open up the lung and keep the lung open. Intensive Care Medicine 18, 319–321. Lindert, J., Perlman, C.E., Parthasarathi, K., Bhattacharya, J., 2007. Chloridedependent secretion of alveolar wall liquid determined by optical-sectioning microscopy. American Journal of Respiratory Cell and Molecular Biology 36 (6), 688–696. Luecke, T., Meinhardt, J.P., Herrmann, P., Weiss, A., Quintel, M., Pelosi, P., 2006. Oleic acid vs saline solution lung lavage-induced acute lung injury: effects on lung morphology, pressure-volume relationships, and response to positive end-expiratory pressure. Chest 130 (2), 392–401. Matsuda, M., Fung, Y.C., Sobin, S.S., 1987. Collagen and elastin fibers in human pulmonary alveolar mouths and ducts. Journal of Applied Physiology 63 (3), 1185–1194. Matute-Bello, G., Frevert, C.W., Martin, T.R., 2008. Animal models of acute lung injury. American journal of physiology. Lung cellular and molecular physiology 295 (3), L379–L399. Mead, J., Takishima, T., Leith, D., 1970. Stress distribution in lungs: a model of pulmonary elasticity. Journal of Applied Physiology 28 (5), 596–608. Mertens, M., Tabuchi, A., Meissner, S., Krueger, A., Schirrmann, K., Kertzscher, U., Pries, A.R., Slutsky, A.S., Koch, E., Kuebler, W.M., 2009. Alveolar dynamics in acute lung injury: heterogeneous distension rather than cyclic opening and collapse. Critical Care Medicine 37 (9), 2604–2611. Mols, G., Hermle, G., Fries, G., Benzing, A., Lichtwarck-Aschoff, M., Geiger, K., Guttmann, J., 2002. different strategies to keep the lung open: A study in isolated perfused rabbit lungs. Critical Care Medicine 30 (7), 1598–1604. ¨ Muller, I., Strehlow, P., 2004. Rubber and Rubber Balloons, Paradigms of Thermodynamics. of Lecture Notes in Physics, Vol. 637. Springer-Verlag, Berlin Heidelberg. Namati, E., Thiesse, J., de Ryk, J., McLennan, G., 2008. Alveolar dynamics during respiration: are the pores of Kohn a pathway to recruitment? American Journal of Respiratory Cell and Molecular Biology 38 (5), 572–578. Navajas, D., Maksym, G.N., Bates, J.H., 1995. Dynamic viscoelastic nonlinearity of lung parenchymal tissue. Journal of Applied Physiology 79 (1), 348–356. Otis, D.R., Ingenito, E.P., Kamm, R.D., Johnson, M., 1994. Dynamic surface tension experiments and theory. Journal of Applied Physiology 77 (6), 2681–2688. Perlman, C.E., Bhattacharya, J., 2007. Alveolar expansion imaged by optical sectioning microscopy. Journal of Applied Physiology 103 (3), 1037–1044. Prange, H.D., 2003. Laplace’s law and the alveolus: a misconception of anatomy and a misapplication of physics. Advances in physiology education 27 (1-4), 34–40. Stamenovic, D., Smith, J.C., 1986. Surface forces in lungs. ii. microstructural mechanics and lung stability. Journal of Applied Physiology 60 (4), 1351–1357. Stamenovic, D., Wilson, T.A., 1992. Parenchymal stabilitiy. Journal of Applied Physiology 73 (2), 596–602. ¨ Sznitman, J., Heimsch, T., Wildhaber, J.H., Tsuda, A., Rosgen, T., 2009. Respiratory flow phenomena and gravitational deposition in a three-dimensional spacefilling model of the pulmonary acinar tree. Journal of Biomechanical Engineering 131 (3), 031010. Tsuda, A., Henry, F.S., Butler, J.P., 2008. Gas and aerosol mixing in the acinus. Respiratory physiology & neurobiology 163 (1-3), 139–149. Ware, L.B., Matthay, M.A., 2000. The acute respiratory distress syndrome. The New England Journal of Medicine 342 (18), 1334–1349. Weibel, E.R., 1963. In: Morphometry of the Human Lung. Academic Press Inc. and Springer-Verlag. Wilson, T., 1981. Relations among recoil pressure, surface area, and surface tension in the lung. Journal of Applied Physiology: Respiratory, Environmental and Exercise Physiology 50 (5), 921–926. Wilson, T.A., Anafi, R.C., Hubmayr, R.D., 2001. Mechanics of edematous lungs. Journal of Applied Physiology 90, 2088–2093.