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A model of ventilation of the healthy human lung
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 1 ( 2 0 1 1 ) 144–155
journal homepage: www.intl.elsevierhealth.com/journals/cmpb
A model of ventilation of the healthy human lung K.L. Steimle a , M.L. Mogensen a,∗ , D.S. Karbing a , J. Bernardino de la Serna b , S. Andreassen a a b
Center for Model-Based Medical Decision Support, Aalborg University, Aalborg, Denmark Center for Biomembrane Physics, University of Southern Denmark, Odense, Denmark
a r t i c l e
i n f o
a b s t r a c t
Article history:
This paper presents a model of the lung mechanics which simulates the pulmonary alveolar
Received 14 December 2009
ventilation. The model includes aspects of: the alveolar geometry; pressure due to the chest
Received in revised form
wall; pressure due to surface tension determined by surfactant activity; pressure due to lung
18 June 2010
tissue elasticity; and pressure due to the hydrostatic effects of the lung tissue and blood. The
Accepted 28 June 2010
cross-sectional area of the lungs in the supine position derived from computed tomography is used to construct a horizontally layered model, which simulates heterogeneous ventila-
Keywords:
tion distribution from the non-dependent to the dependent layers of the lungs. The model is
Physiological models
in agreement with experimentally measured hysteresis of the pressure–volume curve of the
Pulmonary ventilation
lungs, static lung compliance, changes in lung depth during breathing and density distribu-
Lung mechanics
tions at total lung capacity (TLC) and residual volume (RV). In the dependent layers of the
Static PV-curves
lungs, alveolar collapse may occur at RV, depending on the assumptions concerning lung tissue elasticity at very low alveolar volumes. The model simulations showed that ventilation increased with depth in the lungs, although not as pronounced as observed experimentally. The model simulates alveolar ventilation including all of the mentioned components of the respiratory system and to be validated against all the above mentioned experimental data. © 2010 Elsevier Ireland Ltd. All rights reserved.
1.
Introduction
Appropriate ventilator settings for intensive care patients with respiratory disorders reduce recovery time and lower the risk of ventilator induced lung injury [1]. However, it is not an easy task to determine the optimal ventilator settings, as changes in settings to secure gas exchange may increase the risk of ventilator induced lung injury. To get a better understanding of how the lung mechanics affect gas exchange in sick lungs, it is first required that the mechanics and gas exchange of healthy lungs are understood. Numerous physiological models of gas exchange and lung mechanics have been formulated, e.g. [2–9]. It has been shown
that gas exchange can be accurately described with a model identified from routine clinical data [4]. This model, however, is limited to describe gas exchange and do not describe the link between gas exchange and lung mechanics. Similarly, models have been formulated of varying complexity describing the mechanics of the respiratory system exclusively, e.g. [2,3,6,8,9]. The majority of these models have not been intended for use in clinical practice, and have instead described specific components of the respiratory system in high detail such as the elastic mechanical behavior of the lungs [2,9]. Alternatively, models have been developed describing the complete system from an empirical approach [3,8]. Some authors have built models to couple gas exchange and lung mechanics. Liu et al. [5] developed a nonlinear model
∗ Corresponding author at: Aalborg University, Center for Model-Based Medical Decision Support, Fredrik Bajersvej 7, DK 9220 Aalborg, Denmark. Tel.: +45 9940 8764. E-mail address: [email protected] (M.L. Mogensen). 0169-2607/$ – see front matter © 2010 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2010.06.017
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 1 ( 2 0 1 1 ) 144–155
including the mechanics of the airways, gas exchange, and blood flow of the human lungs. However, the model is a lumped single-compartment model unable to describe heterogeneity within the lungs. Smith et al. [7] presented a model intended to couple gas exchange and lung mechanics, assuming that over-distended alveoli and collapsed alveoli do not participate in gas exchange representing alveolar dead space and shunt, respectively. However, the model omits several components of the respiratory system including a hysteresis mechanism preventing the model from simulating both inspiration and expiration. This paper presents a physiological model that describes lung mechanics and distribution of ventilation in the healthy lungs. The model describes the lungs as stratified, which enables simulations of local pressure–volume curves (PVcurves) and ventilation at different depths, going from the non-dependent to the dependent layers of the lungs. The effect of surfactant is included in the model, since surfactant is the main contributor of hysteresis in the PV-curve [10]. In addition, the model includes the mechanical contributions of the chest wall, lung tissue elasticity, and hydrostatic effects of lung tissue and blood pressure. Furthermore, the model includes an anatomical description of the cross-sectional area of the lungs in supine position derived from computed tomography scans of healthy human subjects [11].
2.
Methods
2.1.
Model introduction
NA NLayers
;
i = [1 : NLayers ]
ventilation. PHydro,i can be calculated as stated in Eq. (3).
PHydro,i =
Lung,j · g · tj
(3)
j=1
where Lung,j is the density of layer j, g is the gravitational acceleration and tj is the thickness of layer j. The alveolar transmural pressure, PAlvTM,i , is defined as the difference between the extraalveolar pressure and the alveolar pressure, stated by Eq. (4). PAlvTM,i = PA − PEA,i
(4)
Since the transmural pressure is matched by the recoil pressures of the alveolus due to lung tissue elasticity and surface tension, the alveolar transmural pressure can also be calculated by Eq. (5). PAlvTM,i = PE,i + P,i
(5)
where PE,i is the pressure due to lung tissue elasticity and P,i is the pressure due to surface tension at layer i. Combining Eqs. (2), (4) and (5) gives the alveolar pressure as stated in Eq. (6). (6)
The total volume of air in the lungs, VAir , is calculated by Eq. (7).
(1)
where NA is the total number of alveoli and NLayers is the number of layers and i is the index that controls the layer depth (i = 1 at the most non-dependent layer). The chest wall pressure, PCW , hydrostatic pressure, PHydro and the pressure exerted by the muscles within thorax compose the extraalveolar pressure, PEA , in each layer as stated in Eq. (2). It is assumed that each alveolus in a layer has the same extraalveolar pressure independent of its position. (2)
where PHydro,i is the hydrostatic pressure at a given lung depth. The pressure exerted by muscles is assumed 0 kPa because the model simulates a paralyzed subject during mechanical
NLayers
VAir = VAD +
(NA,i · VA,i )
(7)
i=1
where VAD is the anatomical deadspace and VA,i is the volume of an alveolus. The alveolus is assumed to be spherical and inflated by a rigid ring. The volume of an alveolus in layer i can be calculated using Eq. (8) [12]. VA,i =
1 · bi 2 (3ri − bi ) 3
(8)
where b is the height of the alveolus from the opening and r is the radius of the alveolus. The surface area SA of the alveolus can be calculated as stated in Eq. (9) [12]. SA,i = 2 · ri bi = (bi 2 + a2 )
(9)
where a is the length of the opening radius at the rigid ring, which is assumed constant and the same in all alveoli. From Eq. (9), r can be isolated, as a function of a and b, as stated in Eq. (10). ri =
PEA,i = PCW + PHydro,i + PMus
i−1
PA = P,i + PE,i + PCW + PHydro,i
The presented model was implemented in Matlab (Mathworks, Natick, MA) to simulate the lung mechanics of a resting healthy subject during mechanical ventilation. All conditions were assumed to be static implying no airway resistance. Fig. 1 illustrates the overall structure of the model, where the input to the model is the airway pressure or alveolar pressure PA , which was assumed to be equal in all alveoli. The model describes the lungs as divided into horizontal layers distributed vertically from the non-dependent to the dependent part of the lungs. The number of alveoli in each layer is equal and constant and is defined by Eq. (1). NA,i =
145
a2 + bi 2 2bi
(10)
By defining the total lung capacity, VTLC , as the anatomical deadspace plus the maximal total amount of air in the alveoli, the maximal volume in a single alveolus is calculated by Eq. (11), by assuming that all alveoli are equally sized at total lung
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Fig. 1 – Illustration of the total lung model. A transverse slice obtained by computed tomography shows the lungs as divided into 10 horizontal layers [11]. PEA , extraalveolar pressure. PA , airway pressure. PAlvTM , alveolar transmural pressures. PCW , pressure exerted by the chest wall. P␥ , pressure due to surface tension. PE , pressure due to tissue elasticity. PHydro , hydrostatic pressure due to lung tissue and blood. i, index controlling layer depth measured from the top. a, opening radius of the alveoli. b, height of the alveolus from the opening. r, radius of the alveolus.
capacity.
VAMax =
defined by Eq. (13). VTLC − VAD NA
(11)
The following components need to be identified before the pulmonary ventilation can be determined: the pressure exerted by the chest wall, PCW ; the pressure due to surface tension, P ; the pressure due to lung tissue elasticity, PE ; the anatomy of the lungs.
Sr,i =
SA,i S0,i
where SA is the alveolar surface area and S0 is the alveolar surface area before compression and the state variable which controls the hysteresis. S0 must be updated through a breath according to Eq. (14) to give the proper amount of hysteresis.
S0new,i =
2.2.
The pressure exerted by the chest wall
A mathematical description of the pressure component, PCW , due to the chest wall as a function of lung volume is built on the experimental data measured by Konno and Mead [13]. They measured the static pressure–volume relation of the chest wall and abdomen in six healthy subjects based on an analysis of the changes in anterior–posterior diameters of the chest wall and abdomen. In Fig. 2A the data read from Konno and Mead [13] are shown with a fitted sigmoid curve for PCW as stated in Eq. (12).
PCW = 0.071 kPa − ln
2.3.
95% − 1 · 0.58 kPa (VAir /VTLC ) − 22%
(12)
Pressure due to surface tension
A model of the pressure, P , due to surface tension is built upon data from Lu et al. [14]. The surface tension, , as a function of surface area was measured by Lu et al. [14] with captive bubble surfactometry with a compression speed of 25 s/cycle. The study showed an increase in hysteresis when the bubble surface area was compressed more than 61.7%. This effect is modelled with a second order polynomial and a state variable which describes the area before compression, S0 , and hereby controls the hysteresis. The relative alveolar surface area is
(13)
⎧ S ⎪ ⎨ A,i, ⎪ ⎩
if 1 < Sr,i
S0,i ,
if SMeta ≤ Sr,i ≤ 1
SA,i /SMeta ,
if Sr,i < SMeta
(14)
where SMeta is the point where the relative surface area reaches the minimum surface tension. SMeta is according to the data by Lu et al. [14] set to 61.7%. The second order polynomial used for calculation of the surface tension is stated in Eq. (15). i = 98.9 mN/m · Sr,i 2 − 89.0 mN/m · Sr,i + 18.3 mN/m
(15)
Fig. 2B shows an example of the surface tension, , as a function of the relative surface area with a minimum relative surface area of 55% along with data read from Lu et al. [14]. Having the surface tension as a function of the alveolar surface area and assuming spherical alveoli allows calculation of the pressure due to surface tension using the Laplace equation stated in Eq. (16). P,i =
2 · i ri
(16)
where i is the surface tension and ri is the radius of the alveolus, at layer i.
2.4.
Pressure due to lung tissue elasticity
The pressure, PE , due to lung tissue elasticity has been examined by Smith and Stamenovic [15]. They inflated
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Fig. 2 – (A) The relationship between the total lung volume and the pressure exerted by the chest wall. Along with data read from Konno and Mead [13] a fitted sigmoid curve (Eq. (12)) is shown. (B) The relationship between surface tension and relative surface area as modelled by Eqs. (13)–(15). Along a simulation with a minimum relative surface area of 55%, data read from Lu et al. [14] is shown. (C) Relationship between the elastic recoil pressure exerted by the lung tissue and the volume of the alveoli as modelled by Eq. (17). The data are adapted from Smith and Stamenovic [15]. Two model fits are included assuming either no or some degree of resistance to alveolar collapse.
excised rabbit lungs with saline and obtained quasi-static pressure–volume curves, making it possible to eliminate the effect of surface tension and surfactant. However, no information on how the lungs behave under negative pressure and lung volumes below 20% was given. Two different models of the lung tissue elasticity at low lung volumes were therefore developed. The first, model I, assumes that lung tissue cannot resist any negative transmural pressure. The second, model II, assumes that the tissue to some degree resists collapse, which means that the connecting fibers and tissue are assumed to produce a small negative pressure at low lung volumes. In both models it is assumed that the elastic properties of rabbit and human lungs are identical and that the tissue elasticity is the same throughout the lungs. This makes it possible to translate the relative total lung volume measured by Smith and Stamenovic [15] into a relative alveolar volume. In Fig. 2C the data read from Smith and Stamenovic [15] for PE are shown along with the two models fit to data. The data show that inflation and deflation of the lungs with saline reveal some degree of hysteresis, however, this effect is disregarded in the models. Eq. (17) states the sigmoid curve used for both models, the difference between models being that model I was fitted to data read from Smith and Stamenovic [15] plus a guessed point at zero pressure and volume.
PE,i = B − ln
1−ε −1 (VA,i /VAMax ) − ε
·C
k=
VTLC VTLCRef
(18)
where VTLC is the total lung capacity of a given subject and VTLCRef is the reference value of total lung capacity. k is used for scaling the reference values of the blood volume (VBloodRef ), number of alveoli (NARef ), lung tissue volume (VTissueRef ), and anatomical deadspace (VADRef ), to the individual subject by Eqs. (19)–(22). VBlood = k · VBloodRef
(19)
NA = k · NARef
(20)
VTissue = k · VTissueRef
(21)
VAD = k · VADRef
(22)
(17)
where ε = −0.8, B = 0.06 kPa and C = 0.3 kPa for model I and ε = −0.05, B = 0.36 kPa and C = 0.3 kPa for model II.
2.5.
by Ochs et al. [16] who found that the number of alveoli was closely related to total lung volume. The size of the lungs is linked with the blood volume, number of alveoli and lung tissue volume by a scaling factor, k, which is defined by Eq. (18).
Lung anatomy
In order to model different lung sizes, the relationship between total lung capacity (TLC), the pulmonary blood volume, number of alveoli, lung tissue volume and anatomical deadspace has to be determined. Assuming that an alveolus has the same maximum size and properties independent of lung size of the subject, a linear relationship between the above mentioned parameters is assumed. This was supported
In this way the model is initialized by a single parameter which is the total lung capacity, VTLC . The reference values are found in Table 1. It is assumed that the amount of blood and tissue is equally distributed between the alveoli. The volumes of the lung tissue per alveolus (VTissuePrAlv ), and blood per alveolus (VBloodPrAlv ), are therefore given by Eqs. (23) and (24), respectively.
VTissuePrAlv =
VBloodPrAlv =
VTissue NA VBlood NA
(23)
(24)
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Table 1 – Model parameters. Parameter a Tissue Blood VTLCRef VBloodRef VTissueRef NARef VADRef f t NLayers
Reference value
Description
References
100 m 1.0 g/cm3 1.06 g/cm3 6.8 l 250 cm3 450 cm3 375 mio. 150 ml 12 breaths/min 50 ms 100
Alveolar opening radius Tissue density Blood density Reference value of total lung capacity Reference value of blood volume Reference value of tissue volume Reference value of number of alveoli Reference value of deadspace Breathing frequency Incremental time between pressure steps Number of layers
[47] [18] [21] [20] [22] [22] [23] [47] [47]
plying with the pixel dimension (0.7813 mm × 1 mm), resulting in the profile shown in Fig. 4A (marked AScan ). The total volume of the lungs (air, blood and tissue) according to the CT-scan, VScan , is 7.9372 l, which is calculated by multiplying the total number of pixels with the volume of each lung pixel (0.7813 mm × 0.7813 mm × 1 mm). The depth–area curve must be scaled to match the current lung volume. To achieve this, the current relative lung volume is calculated from Eq. (25). Vr = Fig. 3 – (A) Slice 200 out of 394 slices obtained by computed tomography in the transverse plane [11]. (B) The same slice as in A, segmented [11] to obtain the cross-sectional area.
VLung VScan
(25)
where VLung is the total lung volume of air, tissue and blood calculated by Eq. (26). VLung = VAir + VTissue + VBlood
2.6.
Lung cross-section during breathing
A profile of lung cross-sectional areas, AScan , as a function of depth, DScan , was constructed from high resolution CT-scans taken at total lung capacity of a healthy subject in supine position provided and segmented by Lo et al. [11]. The profiles are shown in Fig. 3A and B. The CT-scans have a resolution of 512 × 512 pixels in each slice from apex to base. There are 394 slices separated by 1 mm. Since the subject is modelled in the supine position, a profile of the cross-sectional area has been calculated for the frontal plane by summing up the number of pixels at a given depth (ventral–dorsal) for all scans and multi-
(26)
where VTissue and VBlood are the total volume of tissue and blood in the lungs of the simulated subject. Assuming isotropic expansion the current lung profile can be calculated by Eqs. (27) and (28). DCurrent = DScan · Vr 1/3
ACurrent (Di ) = AScan
(27) Di
Vr 1/3
· Vr 2/3
(28)
where DScan and AScan are lung depths and the corresponding cross-sectional areas during the scanning and Di is the depth
Fig. 4 – (A) The relationship between the cross-sectional area, A, and lung depth, D, at total lung capacity of 6.8 l (solid line) and at VAir = 3 l (dashed line). (B) The distribution of volume in layers when the lungs are divided into 10 layers. The thickness, ti , and volumes, Vi , of each layer are also illustrated.
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from the top of the lungs to the top of layer i (D1 = 0), calculated by Eq. (29).
Di =
i−1
tj
(29)
j=1
where tj is the thickness of layer j. If the volume of air in the lungs decreases from a volume at TLC (6.8 l in this example) to a current lung volume of 3 l (VAir = 3 l), the cross-sectional area profile changes as illustrated in Fig. 4A. Fig. 4B shows the distribution of volume when the lungs are divided into 10 layers (NLayers = 10) at TLC.
149
where Tissue and Blood are the density of the tissue and blood, respectively. Having the density of the first layer only leaves the thickness of this layer, t1 , to be identified in order to calculate the hydrostatic pressure by Eq. (3). The thickness, t1 , depends on the shape of the lungs described by the profile of the cross-sectional area as a function of the depth. Having the profile of the lungs for the guessed lung volume, VAir , from Eqs. (25)–(28), the thickness of the layer, t1 , can be found by solving equation Eq. (33), where the only unknown is ti .
Di +ti
VLung,i =
ACurrent (Di ) dt
(33)
Di
2.7.
Numerical solution of the equations
In order to calculate the lung density, Lung,i , and thickness, ti , of the ith layer, the volume of air in the alveoli must be known. This causes a problem since the volume of air in the alveoli at layer i a given alveolar pressure, PA , depends on the density and the thickness of the higher layers, because they determine the hydrostatic pressure as stated in Eq. (3). This layer interaction problem is solved by introducing an iterative model inspired by Millar and Denison [17]. The model consists of a stack of slices that exert a pressure on the slices below due to gravitational forces. In this way the interaction problem is solved by starting at the top of the lungs where the hydrostatic pressure is zero (PHydro,1 = 0). In the following it is described how PHydro,1 enables calculations of the alveolar volume, lung density and layer thickness in layer 1, which in turn enables calculation of PHydro,2 . This can be repeated through all layers. The model is solved for a given input pressure, PA , by guessing a total volume of air, VAir , in the lungs. From this volume the pressure exerted by the chest wall can be calculated by Eq. (12) and the current lung profile, DCurrent and ACurrent , can be calculated from Eqs. (25)–(28). By rewriting Eqs. (6)–(30) and including the arguments of the different components, the right hand side can be calculated for the first layer since PHydro,1 = 0. Eq. (30) can then be solved numerically by identifying the alveolar height, b1 , for the first layer, allowing calculations of the alveolar radius, r1 , alveolar volume, VA,1 , and surface area, SA,1 (Eqs. (8)–(10)) and hence the pressures due to tissue elasticity and surface tension. P (bi , S0 ) + PE (bi ) = PA − PHydro,i − PCW (VAir )
(30)
The total volume of the first layer, VLung,1 , can then be calculated by Eq. (31). VLung,i = NA,i · (VA,i + VBloodPrAlv + VTissuePrAlv )
(31)
Knowing the volume of air in the alveoli and volume of tissue and blood per alveolus from Eqs. (23) and (24), enables calculation of the lung density in the layers by dividing the mass per alveolus with the volume per alveolus, as stated in Eq. (32). It is assumed that air density is zero. Lung,i =
Tissue · VTissuePrAlv + Blood · VBloodPrAlv VA,i + VTissuePrAlv + VBloodPrAlv
(32)
Having both the density and the thickness of the first layer, the hydrostatic pressure of layer two, PHydro,2 , can be calculated by Eq. (3). The hydrostatic pressure of layer two can then be used to solve Eq. (30) and the procedure to find the new alveolar height, b2 , radius, r2 , area, SA,2 , and volume, VA,2 , along with the density, Lung,2 , and thickness, t2 , for layer two is then repeated. This is done for all layers and the total lung volume in the lungs including air, blood and tissue, VLungNew , can then be calculated by Eq. (34).
NLayers
VLungNew =
VLung,i
(34)
i=1
The error between the initially guessed lung volume, VLung , calculated from Eq. (26) and the calculated VLungNew is stated in Eq. (35). VErr (VAir ) = VLung − VLungNew (VAir )
(35)
To minimize this error a numerical optimization algorithm (Newton–Raphson method) is applied [18]. The method changes the guess until the absolute error is below 1 ml. In order to initialize the state variable A0 the simulations were run for two consecutive breaths and only the simulation results for the last breath are used.
2.8.
Model parameters
In the following the parameters used in Eqs. (18)–(21), VTLCRef , Tissue , Blood , VBloodRef , VTissueRef and NARef will be identified. VTLCRef : Withers et al. [19] studied lung volumes in 162 healthy male non-smokers. The results showed a mean total lung capacity of 6.783 l with a range between 4.679 and 9.663 l. Therefore VTLCRef is set to 6.8 l. Tissue and Blood : The densities of lung tissue and blood were assumed to be Tissue = 1.0 g/cm3 and Blood = 1.06 g/cm3 [17,20]. VBloodRef and VTissueRef : The amount of blood, VBloodRef , and tissue, VTissueRef , in the lungs for a standard subject can be identified from density measurements at TLC. Millar and Denison [17] studied the density at TLC and residual volume (RV) in 12 healthy men in the supine position. The measurements were performed with Computed Tomography, where Hounsfield units where translated to densities in pixels of 1 cm × 1 cm from top to bottom of the lung in an arc at the lung periphery. The lung density at TLC is almost homogenous down the
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lungs at 0.1 g/cm3 . To achieve this value at TLC, the total lung weight must have been about 680 g assuming a mean TLC of 6.8 l for the 12 men. This is in good agreement with a study by Chen et al. [21]. They examined lung weights in post-mortem lungs and found that patients who died from blunt head injury had a lung weight between 330 and 1660 g with a median of 760 g including both blood and tissue. According to the data obtained by Millar and Denison [17] and Chen et al. [21] and the assumption that lung tissue roughly accounts for about 65% and blood accounts for 35% of the total lung weight, VBloodRef is set to 250 cm3 and VTissueRef is set to 450 cm3 . NARef : The last parameter to be identified is the number of alveoli, NARef . The number of alveoli was estimated in 32 adult men by Angus and Thurlbeck [22]. The results showed a large range between 212 and 605 mio. with a mean of 375 mio. alveoli, why NARef is set to 375 mio.
3.
Results
The first part of this section introduces simulations of pressures and volumes in different layers generated by applying different alveolar pressures, PA . The second part of this section is used for validation and estimation of model parameters. Fig. 5 shows pressure and volumes during a simulation with PA ranging between 4 and −1.5 kPa. Pressures and volumes are shown for layers number 1, 25, 50, 75 and 100 (100 being the most dependent or dorsal layer. This layer is indicated by a dashed line). The simulation is performed with model II of the lung tissue elasticity (resists some collapse) and all model parameters set to values found in Table 1. The points where the simulation reaches a lung volume of TLC, functional residual volume (FRC) and RV are indicated in Fig. 5E. FRC is by definition identified when alveolar pressure is zero (PA = 0 kPa). RV is reached at −1.5 kPa since a further decrease below −1.5 kPa would not have led to a pronounced change in alveolar volume, VA , as shown in Fig. 5E. Fig. 5E shows the alveolar volume in all layers also reach a plateau at an alveolar pressure of 4 kPa, in this case TLC. Fig. 5A shows the extraalveolar pressure, PEA , which is the sum of the hydrostatic pressure, PHydro , and the pressure exerted by the chest wall, PCW . The chest wall pressure can be identified as, PEA , of layer number 1, since the hydrostatic pressure at this layer is zero. It can be noticed that the chest exerts a pressure of about 1.7 kPa at TLC. At RV the chest wall pressure almost matches the pressure in the alveoli (PA = −1.5 kPa), since both the pressure, PE , due to lung tissue elasticity and pressure, P , due to surface tension are near zero at low lung volumes. The pressure, PE , due to lung tissue elasticity, shown in Fig. 5B, declines during expiration and increases again during inspiration. The simulation shows that the tissue exerts a small negative pressure at very low lung volumes in the most dependent layers, i.e. resisting collapse in these layers. The elastic pressure at TLC is about 2 kPa, where the pressure due to surface tension, shown in Fig. 5C, is only about 0.3 kPa. The pressure due to surface tension, P , declines during expiration. During inspiration, the most dependent layers have a more rapid increase in P than the non-dependent layers due to the properties of surfactant which depends on the level of alveolar surface area reduction. The transmural pressure, PAlvTm , in Fig. 5D is
Fig. 5 – Simulation results for an expiration from TLC to RV followed by an inspiration back to TLC using model II of the lung tissue elasticity. Results are plotted as a function of alveolar input pressure and shown for layers number 1, 25, 50, 75 and 100 (100 being the most dependent layer, dashed line) (A) Extraalveolar pressure (PEA ). (B) Pressure due to lung tissue elasticity (PE ). (C) Pressure due to surface tension (P ). (D) Alveolar transmural pressure (PAlvTM ). (E) alveolar volume (VA ). Vertical dotted lines indicate separation of expiration and inspiration.
given as the pressure difference between input airway pressure and the extraalveolar pressure and the sum between the pressures due to lung tissue elasticity and surface tension. At TLC the transmural pressure is around 2.3 kPa and at low lung volumes the dependent layers have near zero transmural pressure. The alveolar volume in Fig. 5E indicates an unequal distribution of air in the alveoli during the simulation, which is more pronounced at low lung volumes. At TLC (PA = 4.0 kPa) all alveoli are simulated to have a volume of about 18 nl. At FRC (PA = 0 kPa) alveoli in the most depend layer are simulated to have a volume of about 4 nl whereas the alveoli in the most non-dependent layer are 9 nl. Alveoli are simulated to have a volume of 2 and 8 nl at RV (PA = −1.5 kPa) in the most dependent layer and the most non-dependent layer, respectively. The results in Fig. 5E also indicate that no collapse occurs in the healthy lungs between RV and TLC.
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Fig. 6 – Alveolar volume during a simulation from TLC to RV and back to TLC using model I of the lung tissue elasticity. Dotted lines indicate separation of expiration and inspiration.
Using model I instead of model II of the lung tissue elasticity gives almost the same results as shown in Fig. 5A–D. However, since the alveoli are assumed not to resist collapse in model I, the alveolar volume shown in Fig. 6 differs. This model indicates that collapse occurs below around RV.
3.1.
Validation
This section will validate the model against experimentally measured data. Fig. 7 shows the results of the density calculations from the study by Millar and Denison [17]. The results imply that a vertical density gradient is present at RV. The simulated density distribution down the lungs calculated from Eq. (29) at TLC (PA = 4 kPa) and RV (PA = −1.5 kPa) is also shown in Fig. 7 using the two models of the lung tissue elasticity. No difference between the two models is observed at TLC and both simulations follow well the homogenous density around 0.1 g/cm3 . As indicated in Fig. 6, model I simulates alveolar collapse at RV. The collapse of the most dependent layers leads to a marked increase in density in these layers, not shown in the data by Millar and Denison [17]. The density simulated by using model II, however, is in good agreement with data.
Fig. 7 – The simulated density, Lung , related to lung depth, D, is shown at TLC (dashed line) and RV (solid line) for both models of the lung tissue elasticity. Data read from Millar and Denison [17] is shown for TLC and RV (crosses and dots, respectively).
151
Fig. 8 – The static pressure–volume relationship during a simulation with an alveolar pressure range between 0 and 2.66 kPa and a TLC of 5.8 l. pressure–volume measurements by Sharp et al. [25] (crosses) are also shown. Dashed and solid lines indicate inspiration and expiration, respectively.
Therefore it is indicated that lung tissue may to some degree resist collapse and model II is the most appropriate model to describe density distribution. Fig. 7 also indicates that the changes in lung depth from TLC to RV are simulated well in both cases when compared to data. The volume of air in the lungs, VAir , can be summed up according to Eq. (7) giving VAir = 2.19 l at FRC (PA = 0 kPa) and VAir = 1.57 l at RV (PA = −1.5 kPa). These results are in good agreement with experimental observations. The volume of air in the lungs at FRC has been measured in 50 men in the supine position with a mean ± SD of 2.23 ± 0.61 l by Ibanez and Raurich [23]. Baydur [24] studied the lung mechanics in 10 subjects in supine position at different lung volumes including RV and FRC. The RV value was found to be with a mean ± SD of 1.61 ± 0.77 l and the FRC value was 2.47 ± 0.88 l. The simulated residual volume of 1.57 l is also in good agreement with the data obtained by Konno and Mead [13], shown in Fig. 2A. Assuming that the subjects in the experiment had a total lung capacity of 6.8 l, the residual volume is found very near 23% of TLC shown in data. Sharp et al. [25] studied PV-curves in 28 healthy men in the supine position. Fig. 8 shows the mean PV-curve for the study by Sharp et al. [25] and a model simulation with the same alveolar pressure range from 0 to 2.66 kPa and VTLC = 5.8 l using model II of the lung tissue elasticity. No visible difference using models I and II of the lung tissue elasticity was found. The maximum hysteresis being the maximum pressure difference between the expiratory and inspiratory limbs at equal volumes was simulated to be 0.28 kPa, compared to the 0.212 kPa reported by Sharp et al. [25]. The compliance measured between the two extremes was 1.24 l/kPa in the simulation, compared to the 1.13 l/kPa reported by Sharp et al. [25]. Brudin et al. [26] used Positron Emission Tomography to study the relationship between regional ventilation in eight healthy subjects in the supine position. Measurements were performed during quiet breathing and calculated as ml of air ventilated per minute per cm3 of lung. Since it has not been possible to find data in absolute values in mechanically ventilated subjects these measurements from subjects breathing
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the model was found to reproduce experimentally consistent trends regarding the above mentioned data.
4.1.
Fig. 9 – Simulation of the ventilation distribution down the lungs using model I (solid line) and model II (dashed line) along with the data read from Brudin et al. [26] (dots).
voluntarily were used for validation of simulated ventilation. In order to reproduce the data from patients breathing quietly, PA was varied between 0 and 0.4 kPa, resulting in a tidal volume of 437 ml. The ventilation was calculated by Eq. (36). V˙ i = (VA,Max,i − VA,Min,i ) · f · NAlvPrcm3 ,i
1 VA,Min,i + VBloodPrAlv + VTissuePrAlv
The pleural pressure, PPl , within the thin liquid film that lubricates the pleural surfaces and couples the lungs to the rib and diaphragm is omitted in this model. Despite of some evidence that pleural pressure is the pressure that sets regional lung volume (reviewed by Lai-Fook [27]) and common occurrence in the literature (e.g. [28]) the assumption that PEA = PPl is simply wrong. The extra alveolar pressure is always higher than or equal to the pleural pressure in healthy lungs. If this were not the case the pleural space would displace the lumen of the dependent regions of the lungs. If the pleural space displaces the lumen it will affect the lung volume and should therefore have been included in the model. Clinically this happens in patients with pleural effusion. However, in the healthy lungs the alveoli and capillaries will always be “exposed to see” the pressure within the lung parenchyma, which is the sum between chestwall pressure, hydrostatic pressure and pressure exerted by the muscles (Eq. (2)).
4.2.
(37)
Fig. 9 shows the simulated ventilation with models I and II of the lung tissue elasticity along with the data read from Brudin et al. [26]. In both cases, the models underestimated the increase in ventilation from the non-dependent to the dependent part of the lung, observed by Brudin et al. [26]. Even though, model I of the lung tissue elasticity had a higher gradient than model II, the whole model is unable to reproduce the ventilation observed by Brudin et al. [26] with the current parameters.
The simulated pressure–volume curve in Fig. 8 shows similar shape and hysteresis compared to data from Sharp et al. [25]. The main contributor to hysteresis is the effect of surfactant [10]. However, experimental results indicate that the tissue properties also induce a slight degree of hysteresis [15]. This implies that the model simulated hysteresis may be underestimated. Furthermore, the hysteretic behavior due to surfactant has been shown to depend on the pH, surfactant composition and speed of compression [29–32], effects which are not included in the present model, and could be a topic for further development. The number of alveoli has been shown to be highly variable among individual subjects [22]. Changing the number of alveoli in the model also affects the hysteresis. For instance an increase will redistribute the amount of air to a higher number of alveoli leading to relatively smaller alveoli. During breathing smaller alveoli will be more compressed in area, resulting in a higher amount of hysteresis, according to Eqs. (13)–(16).
4.3.
4.
Pressure–volume curve
(36)
where VA,Max,i and VA,Min,i are the maximal and minimal volume of the alveolus at layer i during the simulation, f is the breathing frequency (12 breaths/min) and NAlvPrcm3 is the number of alveoli per cm3 lung in layer i, calculated from Eq. (37). NAlvPrcm3 ,i =
The pleural pressure
Models I and II of the lung tissue elasticity
Discussion
A stratified model of the pulmonary ventilation derived from computed tomography has been presented which describes lung mechanics and includes aspects of: the alveolar geometry; pressure due to the chest wall; pressure due to surface tension determined by surfactant activity; pressure due to lung tissue elasticity; and pressure due to the hydrostatic effects of the lung tissue and blood. The presented model is the first to be validated against experimentally measured data on the density distribution, lung volumes (RV, FRC and TLC), PV-curve compliance and hysteresis and ventilation distribution of a healthy human subject in the supine position. Even though the model is based on many different studies on both humans and animals with many unknown parameters,
Two models of pressure due to lung tissue elasticity were developed due to the lack of information regarding tissue elastic properties at low lung volumes and negative transmural pressure, both models therefore represent postulated properties of the tissue at low lung volumes. Model I assumes that lung tissue cannot resist any negative transmural pressure. Model II assumes that the connecting fibers and tissue can produce a small negative pressure at low lung volumes. Both models of the tissue elasticity simulate a heterogeneous ventilation distribution down the lungs and indicate that no alveolar collapse occurs at FRC. At RV, model I however, simulates a collapse in the most dependent layers of the lung. The density measurements by Millar and Denison [17] showed no collapse at RV, which was consistent with simulations using
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 1 ( 2 0 1 1 ) 144–155
model II indicating that this is an appropriate model for the lung tissue elasticity. It has been discussed whether the surface area of the alveoli changes during quiet breathing [33,34]. In a model by Wilson et al. [9,35,36] the surface area of the alveoli is allowed to change without changing the lung volume since the volume in the alveolar ducts are assumed to change depending on surface tension. The alveolar ducts are not included in the present model and the alveolar surface area and volume are directly linked through the alveolar radius. Furthermore, the assumption of a rigid ring at the opening of the alveoli still needs further investigation. In a model by Kitaoka et al. [37] the opening radius was modelled as changing with the alveolar volume. In this way it is possible for the alveoli to close without collapsing. Air trapping in closed but not collapsed alveoli may explain the moderate density observed in the dependent layers by Millar and Denison [17]. However, it is still not clear in which way the alveoli or bronchi may close or collapse and in which way the alveolar duct and opening are participating in resisting collapse at very low lung volumes. How they generally affect the lung mechanics is therefore a topic for future investigations.
4.4.
Ventilation distribution
The change in density from RV to TLC, shown in Fig. 7, reflects the ventilation at different layers of the lungs. Using model II of the lung tissue elasticity results in a good fit to measured ventilation described by Millar and Denison [17], indicating that the model properly simulates ventilation distribution when breathing from RV to TLC. The simulated ventilation distribution in Fig. 9, however, does not match the measured data described by Brudin et al. [26] very well. The measurements by Brudin et al. [26] show an increase in ventilation from the non-dependent to dependent part of the lungs by a factor of two, whereas using models I and II only results in simulated increases by a factor of 1.5 and 1.2, respectively. The measurements by Brudin et al. [26] were performed during quiet breathing and not during mechanical ventilation as would be more appropriate for comparison. Measurements obtained during mechanical ventilation are therefore needed before further conclusions can be drawn regarding model simulations of ventilation at small tidal volumes. A number of methods for measuring regional ventilation with high resolution are found in the literature (e.g. SPECT, PET, CT, MRI, EIT, microsphere methods) [26,38–44]. However, none of these methods have been used to quantify the ventilation in absolute values as a function of lung depth for humans under mechanical ventilation in supine position. Most of the methods have however found that ventilation distribution cannot be accounted for by gravity alone since a large heterogeneity of ventilation within isogravitational planes was observed [26,39–41]. Other factors such as the geometry and pathway of the alveolar tree including the effects of alveolar ducts may have a complex influence on the distribution of the ventilation. The present model assumes static conditions and does not include airway resistances and viscoelastic effects for the sake of simplicity. This is not a problem when simulating static PV-curves or recruitment
153
maneuvers, but limits the model in some aspects regarding the ventilation distribution, e.g. at large flows or in patients with COPD [45]. Another mechanism that theoretically could affect the ventilation is hypoxia-induced airway dilation, or hypoxic bronchodilation, which would decrease airway resistance and facilitate gas flow to hypoxic areas of the lungs [46]. In this way the dependent areas of the lungs would be more ventilated and the simulated gradient shown in Fig. 9 would be increased. These factors are topics for further development since the only difference between an alveolus at the top and bottom of the lungs in the presented model is the gravitational impact of the weight of the above layers. Another assumption that may influence the ventilation is the homogenous distribution of blood. It is well known that a gravitational perfusion gradient exists [47]. Including a model of the pulmonary perfusion enables a more correct simulation of the hydrostatic pressures and densities down the lungs that may affect the ventilation. The effects of assuming that the lungs are expanding isotropically and all subjects have the same lung profile are also topics for future development. The presented model is validated against experimentally measured data on the ventilation distribution, density distribution, lung volumes, PV-curve compliance and hysteresis of healthy human subjects in the supine position. Two models of lung tissue elasticity have been developed. According to measurements of density distribution at RV it appears that the connecting fibers and tissue exert some negative pressure at very low lung volumes resisting collapse. The model simulates a heterogeneous ventilation distribution down the lungs and indicates no alveolar collapse at FRC and RV. The model is a step towards understanding how the lung mechanics change during mechanical ventilation, the next step being to understand how the pulmonary perfusion is affected simultaneously in response to mechanical ventilation. Combined, the two models could describe the ventilation–perfusion (V/Q) distribution in the lungs. The model could then potentially be used to predict how the V/Q relationship changes in the human healthy lungs during various ventilation strategies, closing the theoretical gap between lung mechanics and gas exchange.
Conflict of interest None.
Acknowledgements This work was partially supported by the Programme Commission on Neuroscience, Biotechnology and under the Danish Council for Strategic Research. JBS was supported by a Lundbeck Foundation personal fellowship and acknowledges taking advantage of MEMPHYS-Center for Biomembrane Physics and related grants (Danish National Research Foundation Center of Excellence). The authors would like to thank the Danish Lung Cancer Screening Trial (DLCST) for providing the chest CT-scan and Dr. Marleen de Bruijne of University of Copenhagen for performing the lung segmentations.
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references
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journal homepage: www.intl.elsevierhealth.com/journals/cmpb
A model of ventilation of the healthy human lung K.L. Steimle a , M.L. Mogensen a,∗ , D.S. Karbing a , J. Bernardino de la Serna b , S. Andreassen a a b
Center for Model-Based Medical Decision Support, Aalborg University, Aalborg, Denmark Center for Biomembrane Physics, University of Southern Denmark, Odense, Denmark
a r t i c l e
i n f o
a b s t r a c t
Article history:
This paper presents a model of the lung mechanics which simulates the pulmonary alveolar
Received 14 December 2009
ventilation. The model includes aspects of: the alveolar geometry; pressure due to the chest
Received in revised form
wall; pressure due to surface tension determined by surfactant activity; pressure due to lung
18 June 2010
tissue elasticity; and pressure due to the hydrostatic effects of the lung tissue and blood. The
Accepted 28 June 2010
cross-sectional area of the lungs in the supine position derived from computed tomography is used to construct a horizontally layered model, which simulates heterogeneous ventila-
Keywords:
tion distribution from the non-dependent to the dependent layers of the lungs. The model is
Physiological models
in agreement with experimentally measured hysteresis of the pressure–volume curve of the
Pulmonary ventilation
lungs, static lung compliance, changes in lung depth during breathing and density distribu-
Lung mechanics
tions at total lung capacity (TLC) and residual volume (RV). In the dependent layers of the
Static PV-curves
lungs, alveolar collapse may occur at RV, depending on the assumptions concerning lung tissue elasticity at very low alveolar volumes. The model simulations showed that ventilation increased with depth in the lungs, although not as pronounced as observed experimentally. The model simulates alveolar ventilation including all of the mentioned components of the respiratory system and to be validated against all the above mentioned experimental data. © 2010 Elsevier Ireland Ltd. All rights reserved.
1.
Introduction
Appropriate ventilator settings for intensive care patients with respiratory disorders reduce recovery time and lower the risk of ventilator induced lung injury [1]. However, it is not an easy task to determine the optimal ventilator settings, as changes in settings to secure gas exchange may increase the risk of ventilator induced lung injury. To get a better understanding of how the lung mechanics affect gas exchange in sick lungs, it is first required that the mechanics and gas exchange of healthy lungs are understood. Numerous physiological models of gas exchange and lung mechanics have been formulated, e.g. [2–9]. It has been shown
that gas exchange can be accurately described with a model identified from routine clinical data [4]. This model, however, is limited to describe gas exchange and do not describe the link between gas exchange and lung mechanics. Similarly, models have been formulated of varying complexity describing the mechanics of the respiratory system exclusively, e.g. [2,3,6,8,9]. The majority of these models have not been intended for use in clinical practice, and have instead described specific components of the respiratory system in high detail such as the elastic mechanical behavior of the lungs [2,9]. Alternatively, models have been developed describing the complete system from an empirical approach [3,8]. Some authors have built models to couple gas exchange and lung mechanics. Liu et al. [5] developed a nonlinear model
∗ Corresponding author at: Aalborg University, Center for Model-Based Medical Decision Support, Fredrik Bajersvej 7, DK 9220 Aalborg, Denmark. Tel.: +45 9940 8764. E-mail address: [email protected] (M.L. Mogensen). 0169-2607/$ – see front matter © 2010 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2010.06.017
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including the mechanics of the airways, gas exchange, and blood flow of the human lungs. However, the model is a lumped single-compartment model unable to describe heterogeneity within the lungs. Smith et al. [7] presented a model intended to couple gas exchange and lung mechanics, assuming that over-distended alveoli and collapsed alveoli do not participate in gas exchange representing alveolar dead space and shunt, respectively. However, the model omits several components of the respiratory system including a hysteresis mechanism preventing the model from simulating both inspiration and expiration. This paper presents a physiological model that describes lung mechanics and distribution of ventilation in the healthy lungs. The model describes the lungs as stratified, which enables simulations of local pressure–volume curves (PVcurves) and ventilation at different depths, going from the non-dependent to the dependent layers of the lungs. The effect of surfactant is included in the model, since surfactant is the main contributor of hysteresis in the PV-curve [10]. In addition, the model includes the mechanical contributions of the chest wall, lung tissue elasticity, and hydrostatic effects of lung tissue and blood pressure. Furthermore, the model includes an anatomical description of the cross-sectional area of the lungs in supine position derived from computed tomography scans of healthy human subjects [11].
2.
Methods
2.1.
Model introduction
NA NLayers
;
i = [1 : NLayers ]
ventilation. PHydro,i can be calculated as stated in Eq. (3).
PHydro,i =
Lung,j · g · tj
(3)
j=1
where Lung,j is the density of layer j, g is the gravitational acceleration and tj is the thickness of layer j. The alveolar transmural pressure, PAlvTM,i , is defined as the difference between the extraalveolar pressure and the alveolar pressure, stated by Eq. (4). PAlvTM,i = PA − PEA,i
(4)
Since the transmural pressure is matched by the recoil pressures of the alveolus due to lung tissue elasticity and surface tension, the alveolar transmural pressure can also be calculated by Eq. (5). PAlvTM,i = PE,i + P,i
(5)
where PE,i is the pressure due to lung tissue elasticity and P,i is the pressure due to surface tension at layer i. Combining Eqs. (2), (4) and (5) gives the alveolar pressure as stated in Eq. (6). (6)
The total volume of air in the lungs, VAir , is calculated by Eq. (7).
(1)
where NA is the total number of alveoli and NLayers is the number of layers and i is the index that controls the layer depth (i = 1 at the most non-dependent layer). The chest wall pressure, PCW , hydrostatic pressure, PHydro and the pressure exerted by the muscles within thorax compose the extraalveolar pressure, PEA , in each layer as stated in Eq. (2). It is assumed that each alveolus in a layer has the same extraalveolar pressure independent of its position. (2)
where PHydro,i is the hydrostatic pressure at a given lung depth. The pressure exerted by muscles is assumed 0 kPa because the model simulates a paralyzed subject during mechanical
NLayers
VAir = VAD +
(NA,i · VA,i )
(7)
i=1
where VAD is the anatomical deadspace and VA,i is the volume of an alveolus. The alveolus is assumed to be spherical and inflated by a rigid ring. The volume of an alveolus in layer i can be calculated using Eq. (8) [12]. VA,i =
1 · bi 2 (3ri − bi ) 3
(8)
where b is the height of the alveolus from the opening and r is the radius of the alveolus. The surface area SA of the alveolus can be calculated as stated in Eq. (9) [12]. SA,i = 2 · ri bi = (bi 2 + a2 )
(9)
where a is the length of the opening radius at the rigid ring, which is assumed constant and the same in all alveoli. From Eq. (9), r can be isolated, as a function of a and b, as stated in Eq. (10). ri =
PEA,i = PCW + PHydro,i + PMus
i−1
PA = P,i + PE,i + PCW + PHydro,i
The presented model was implemented in Matlab (Mathworks, Natick, MA) to simulate the lung mechanics of a resting healthy subject during mechanical ventilation. All conditions were assumed to be static implying no airway resistance. Fig. 1 illustrates the overall structure of the model, where the input to the model is the airway pressure or alveolar pressure PA , which was assumed to be equal in all alveoli. The model describes the lungs as divided into horizontal layers distributed vertically from the non-dependent to the dependent part of the lungs. The number of alveoli in each layer is equal and constant and is defined by Eq. (1). NA,i =
145
a2 + bi 2 2bi
(10)
By defining the total lung capacity, VTLC , as the anatomical deadspace plus the maximal total amount of air in the alveoli, the maximal volume in a single alveolus is calculated by Eq. (11), by assuming that all alveoli are equally sized at total lung
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Fig. 1 – Illustration of the total lung model. A transverse slice obtained by computed tomography shows the lungs as divided into 10 horizontal layers [11]. PEA , extraalveolar pressure. PA , airway pressure. PAlvTM , alveolar transmural pressures. PCW , pressure exerted by the chest wall. P␥ , pressure due to surface tension. PE , pressure due to tissue elasticity. PHydro , hydrostatic pressure due to lung tissue and blood. i, index controlling layer depth measured from the top. a, opening radius of the alveoli. b, height of the alveolus from the opening. r, radius of the alveolus.
capacity.
VAMax =
defined by Eq. (13). VTLC − VAD NA
(11)
The following components need to be identified before the pulmonary ventilation can be determined: the pressure exerted by the chest wall, PCW ; the pressure due to surface tension, P ; the pressure due to lung tissue elasticity, PE ; the anatomy of the lungs.
Sr,i =
SA,i S0,i
where SA is the alveolar surface area and S0 is the alveolar surface area before compression and the state variable which controls the hysteresis. S0 must be updated through a breath according to Eq. (14) to give the proper amount of hysteresis.
S0new,i =
2.2.
The pressure exerted by the chest wall
A mathematical description of the pressure component, PCW , due to the chest wall as a function of lung volume is built on the experimental data measured by Konno and Mead [13]. They measured the static pressure–volume relation of the chest wall and abdomen in six healthy subjects based on an analysis of the changes in anterior–posterior diameters of the chest wall and abdomen. In Fig. 2A the data read from Konno and Mead [13] are shown with a fitted sigmoid curve for PCW as stated in Eq. (12).
PCW = 0.071 kPa − ln
2.3.
95% − 1 · 0.58 kPa (VAir /VTLC ) − 22%
(12)
Pressure due to surface tension
A model of the pressure, P , due to surface tension is built upon data from Lu et al. [14]. The surface tension, , as a function of surface area was measured by Lu et al. [14] with captive bubble surfactometry with a compression speed of 25 s/cycle. The study showed an increase in hysteresis when the bubble surface area was compressed more than 61.7%. This effect is modelled with a second order polynomial and a state variable which describes the area before compression, S0 , and hereby controls the hysteresis. The relative alveolar surface area is
(13)
⎧ S ⎪ ⎨ A,i, ⎪ ⎩
if 1 < Sr,i
S0,i ,
if SMeta ≤ Sr,i ≤ 1
SA,i /SMeta ,
if Sr,i < SMeta
(14)
where SMeta is the point where the relative surface area reaches the minimum surface tension. SMeta is according to the data by Lu et al. [14] set to 61.7%. The second order polynomial used for calculation of the surface tension is stated in Eq. (15). i = 98.9 mN/m · Sr,i 2 − 89.0 mN/m · Sr,i + 18.3 mN/m
(15)
Fig. 2B shows an example of the surface tension, , as a function of the relative surface area with a minimum relative surface area of 55% along with data read from Lu et al. [14]. Having the surface tension as a function of the alveolar surface area and assuming spherical alveoli allows calculation of the pressure due to surface tension using the Laplace equation stated in Eq. (16). P,i =
2 · i ri
(16)
where i is the surface tension and ri is the radius of the alveolus, at layer i.
2.4.
Pressure due to lung tissue elasticity
The pressure, PE , due to lung tissue elasticity has been examined by Smith and Stamenovic [15]. They inflated
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147
Fig. 2 – (A) The relationship between the total lung volume and the pressure exerted by the chest wall. Along with data read from Konno and Mead [13] a fitted sigmoid curve (Eq. (12)) is shown. (B) The relationship between surface tension and relative surface area as modelled by Eqs. (13)–(15). Along a simulation with a minimum relative surface area of 55%, data read from Lu et al. [14] is shown. (C) Relationship between the elastic recoil pressure exerted by the lung tissue and the volume of the alveoli as modelled by Eq. (17). The data are adapted from Smith and Stamenovic [15]. Two model fits are included assuming either no or some degree of resistance to alveolar collapse.
excised rabbit lungs with saline and obtained quasi-static pressure–volume curves, making it possible to eliminate the effect of surface tension and surfactant. However, no information on how the lungs behave under negative pressure and lung volumes below 20% was given. Two different models of the lung tissue elasticity at low lung volumes were therefore developed. The first, model I, assumes that lung tissue cannot resist any negative transmural pressure. The second, model II, assumes that the tissue to some degree resists collapse, which means that the connecting fibers and tissue are assumed to produce a small negative pressure at low lung volumes. In both models it is assumed that the elastic properties of rabbit and human lungs are identical and that the tissue elasticity is the same throughout the lungs. This makes it possible to translate the relative total lung volume measured by Smith and Stamenovic [15] into a relative alveolar volume. In Fig. 2C the data read from Smith and Stamenovic [15] for PE are shown along with the two models fit to data. The data show that inflation and deflation of the lungs with saline reveal some degree of hysteresis, however, this effect is disregarded in the models. Eq. (17) states the sigmoid curve used for both models, the difference between models being that model I was fitted to data read from Smith and Stamenovic [15] plus a guessed point at zero pressure and volume.
PE,i = B − ln
1−ε −1 (VA,i /VAMax ) − ε
·C
k=
VTLC VTLCRef
(18)
where VTLC is the total lung capacity of a given subject and VTLCRef is the reference value of total lung capacity. k is used for scaling the reference values of the blood volume (VBloodRef ), number of alveoli (NARef ), lung tissue volume (VTissueRef ), and anatomical deadspace (VADRef ), to the individual subject by Eqs. (19)–(22). VBlood = k · VBloodRef
(19)
NA = k · NARef
(20)
VTissue = k · VTissueRef
(21)
VAD = k · VADRef
(22)
(17)
where ε = −0.8, B = 0.06 kPa and C = 0.3 kPa for model I and ε = −0.05, B = 0.36 kPa and C = 0.3 kPa for model II.
2.5.
by Ochs et al. [16] who found that the number of alveoli was closely related to total lung volume. The size of the lungs is linked with the blood volume, number of alveoli and lung tissue volume by a scaling factor, k, which is defined by Eq. (18).
Lung anatomy
In order to model different lung sizes, the relationship between total lung capacity (TLC), the pulmonary blood volume, number of alveoli, lung tissue volume and anatomical deadspace has to be determined. Assuming that an alveolus has the same maximum size and properties independent of lung size of the subject, a linear relationship between the above mentioned parameters is assumed. This was supported
In this way the model is initialized by a single parameter which is the total lung capacity, VTLC . The reference values are found in Table 1. It is assumed that the amount of blood and tissue is equally distributed between the alveoli. The volumes of the lung tissue per alveolus (VTissuePrAlv ), and blood per alveolus (VBloodPrAlv ), are therefore given by Eqs. (23) and (24), respectively.
VTissuePrAlv =
VBloodPrAlv =
VTissue NA VBlood NA
(23)
(24)
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Table 1 – Model parameters. Parameter a Tissue Blood VTLCRef VBloodRef VTissueRef NARef VADRef f t NLayers
Reference value
Description
References
100 m 1.0 g/cm3 1.06 g/cm3 6.8 l 250 cm3 450 cm3 375 mio. 150 ml 12 breaths/min 50 ms 100
Alveolar opening radius Tissue density Blood density Reference value of total lung capacity Reference value of blood volume Reference value of tissue volume Reference value of number of alveoli Reference value of deadspace Breathing frequency Incremental time between pressure steps Number of layers
[47] [18] [21] [20] [22] [22] [23] [47] [47]
plying with the pixel dimension (0.7813 mm × 1 mm), resulting in the profile shown in Fig. 4A (marked AScan ). The total volume of the lungs (air, blood and tissue) according to the CT-scan, VScan , is 7.9372 l, which is calculated by multiplying the total number of pixels with the volume of each lung pixel (0.7813 mm × 0.7813 mm × 1 mm). The depth–area curve must be scaled to match the current lung volume. To achieve this, the current relative lung volume is calculated from Eq. (25). Vr = Fig. 3 – (A) Slice 200 out of 394 slices obtained by computed tomography in the transverse plane [11]. (B) The same slice as in A, segmented [11] to obtain the cross-sectional area.
VLung VScan
(25)
where VLung is the total lung volume of air, tissue and blood calculated by Eq. (26). VLung = VAir + VTissue + VBlood
2.6.
Lung cross-section during breathing
A profile of lung cross-sectional areas, AScan , as a function of depth, DScan , was constructed from high resolution CT-scans taken at total lung capacity of a healthy subject in supine position provided and segmented by Lo et al. [11]. The profiles are shown in Fig. 3A and B. The CT-scans have a resolution of 512 × 512 pixels in each slice from apex to base. There are 394 slices separated by 1 mm. Since the subject is modelled in the supine position, a profile of the cross-sectional area has been calculated for the frontal plane by summing up the number of pixels at a given depth (ventral–dorsal) for all scans and multi-
(26)
where VTissue and VBlood are the total volume of tissue and blood in the lungs of the simulated subject. Assuming isotropic expansion the current lung profile can be calculated by Eqs. (27) and (28). DCurrent = DScan · Vr 1/3
ACurrent (Di ) = AScan
(27) Di
Vr 1/3
· Vr 2/3
(28)
where DScan and AScan are lung depths and the corresponding cross-sectional areas during the scanning and Di is the depth
Fig. 4 – (A) The relationship between the cross-sectional area, A, and lung depth, D, at total lung capacity of 6.8 l (solid line) and at VAir = 3 l (dashed line). (B) The distribution of volume in layers when the lungs are divided into 10 layers. The thickness, ti , and volumes, Vi , of each layer are also illustrated.
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from the top of the lungs to the top of layer i (D1 = 0), calculated by Eq. (29).
Di =
i−1
tj
(29)
j=1
where tj is the thickness of layer j. If the volume of air in the lungs decreases from a volume at TLC (6.8 l in this example) to a current lung volume of 3 l (VAir = 3 l), the cross-sectional area profile changes as illustrated in Fig. 4A. Fig. 4B shows the distribution of volume when the lungs are divided into 10 layers (NLayers = 10) at TLC.
149
where Tissue and Blood are the density of the tissue and blood, respectively. Having the density of the first layer only leaves the thickness of this layer, t1 , to be identified in order to calculate the hydrostatic pressure by Eq. (3). The thickness, t1 , depends on the shape of the lungs described by the profile of the cross-sectional area as a function of the depth. Having the profile of the lungs for the guessed lung volume, VAir , from Eqs. (25)–(28), the thickness of the layer, t1 , can be found by solving equation Eq. (33), where the only unknown is ti .
Di +ti
VLung,i =
ACurrent (Di ) dt
(33)
Di
2.7.
Numerical solution of the equations
In order to calculate the lung density, Lung,i , and thickness, ti , of the ith layer, the volume of air in the alveoli must be known. This causes a problem since the volume of air in the alveoli at layer i a given alveolar pressure, PA , depends on the density and the thickness of the higher layers, because they determine the hydrostatic pressure as stated in Eq. (3). This layer interaction problem is solved by introducing an iterative model inspired by Millar and Denison [17]. The model consists of a stack of slices that exert a pressure on the slices below due to gravitational forces. In this way the interaction problem is solved by starting at the top of the lungs where the hydrostatic pressure is zero (PHydro,1 = 0). In the following it is described how PHydro,1 enables calculations of the alveolar volume, lung density and layer thickness in layer 1, which in turn enables calculation of PHydro,2 . This can be repeated through all layers. The model is solved for a given input pressure, PA , by guessing a total volume of air, VAir , in the lungs. From this volume the pressure exerted by the chest wall can be calculated by Eq. (12) and the current lung profile, DCurrent and ACurrent , can be calculated from Eqs. (25)–(28). By rewriting Eqs. (6)–(30) and including the arguments of the different components, the right hand side can be calculated for the first layer since PHydro,1 = 0. Eq. (30) can then be solved numerically by identifying the alveolar height, b1 , for the first layer, allowing calculations of the alveolar radius, r1 , alveolar volume, VA,1 , and surface area, SA,1 (Eqs. (8)–(10)) and hence the pressures due to tissue elasticity and surface tension. P (bi , S0 ) + PE (bi ) = PA − PHydro,i − PCW (VAir )
(30)
The total volume of the first layer, VLung,1 , can then be calculated by Eq. (31). VLung,i = NA,i · (VA,i + VBloodPrAlv + VTissuePrAlv )
(31)
Knowing the volume of air in the alveoli and volume of tissue and blood per alveolus from Eqs. (23) and (24), enables calculation of the lung density in the layers by dividing the mass per alveolus with the volume per alveolus, as stated in Eq. (32). It is assumed that air density is zero. Lung,i =
Tissue · VTissuePrAlv + Blood · VBloodPrAlv VA,i + VTissuePrAlv + VBloodPrAlv
(32)
Having both the density and the thickness of the first layer, the hydrostatic pressure of layer two, PHydro,2 , can be calculated by Eq. (3). The hydrostatic pressure of layer two can then be used to solve Eq. (30) and the procedure to find the new alveolar height, b2 , radius, r2 , area, SA,2 , and volume, VA,2 , along with the density, Lung,2 , and thickness, t2 , for layer two is then repeated. This is done for all layers and the total lung volume in the lungs including air, blood and tissue, VLungNew , can then be calculated by Eq. (34).
NLayers
VLungNew =
VLung,i
(34)
i=1
The error between the initially guessed lung volume, VLung , calculated from Eq. (26) and the calculated VLungNew is stated in Eq. (35). VErr (VAir ) = VLung − VLungNew (VAir )
(35)
To minimize this error a numerical optimization algorithm (Newton–Raphson method) is applied [18]. The method changes the guess until the absolute error is below 1 ml. In order to initialize the state variable A0 the simulations were run for two consecutive breaths and only the simulation results for the last breath are used.
2.8.
Model parameters
In the following the parameters used in Eqs. (18)–(21), VTLCRef , Tissue , Blood , VBloodRef , VTissueRef and NARef will be identified. VTLCRef : Withers et al. [19] studied lung volumes in 162 healthy male non-smokers. The results showed a mean total lung capacity of 6.783 l with a range between 4.679 and 9.663 l. Therefore VTLCRef is set to 6.8 l. Tissue and Blood : The densities of lung tissue and blood were assumed to be Tissue = 1.0 g/cm3 and Blood = 1.06 g/cm3 [17,20]. VBloodRef and VTissueRef : The amount of blood, VBloodRef , and tissue, VTissueRef , in the lungs for a standard subject can be identified from density measurements at TLC. Millar and Denison [17] studied the density at TLC and residual volume (RV) in 12 healthy men in the supine position. The measurements were performed with Computed Tomography, where Hounsfield units where translated to densities in pixels of 1 cm × 1 cm from top to bottom of the lung in an arc at the lung periphery. The lung density at TLC is almost homogenous down the
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lungs at 0.1 g/cm3 . To achieve this value at TLC, the total lung weight must have been about 680 g assuming a mean TLC of 6.8 l for the 12 men. This is in good agreement with a study by Chen et al. [21]. They examined lung weights in post-mortem lungs and found that patients who died from blunt head injury had a lung weight between 330 and 1660 g with a median of 760 g including both blood and tissue. According to the data obtained by Millar and Denison [17] and Chen et al. [21] and the assumption that lung tissue roughly accounts for about 65% and blood accounts for 35% of the total lung weight, VBloodRef is set to 250 cm3 and VTissueRef is set to 450 cm3 . NARef : The last parameter to be identified is the number of alveoli, NARef . The number of alveoli was estimated in 32 adult men by Angus and Thurlbeck [22]. The results showed a large range between 212 and 605 mio. with a mean of 375 mio. alveoli, why NARef is set to 375 mio.
3.
Results
The first part of this section introduces simulations of pressures and volumes in different layers generated by applying different alveolar pressures, PA . The second part of this section is used for validation and estimation of model parameters. Fig. 5 shows pressure and volumes during a simulation with PA ranging between 4 and −1.5 kPa. Pressures and volumes are shown for layers number 1, 25, 50, 75 and 100 (100 being the most dependent or dorsal layer. This layer is indicated by a dashed line). The simulation is performed with model II of the lung tissue elasticity (resists some collapse) and all model parameters set to values found in Table 1. The points where the simulation reaches a lung volume of TLC, functional residual volume (FRC) and RV are indicated in Fig. 5E. FRC is by definition identified when alveolar pressure is zero (PA = 0 kPa). RV is reached at −1.5 kPa since a further decrease below −1.5 kPa would not have led to a pronounced change in alveolar volume, VA , as shown in Fig. 5E. Fig. 5E shows the alveolar volume in all layers also reach a plateau at an alveolar pressure of 4 kPa, in this case TLC. Fig. 5A shows the extraalveolar pressure, PEA , which is the sum of the hydrostatic pressure, PHydro , and the pressure exerted by the chest wall, PCW . The chest wall pressure can be identified as, PEA , of layer number 1, since the hydrostatic pressure at this layer is zero. It can be noticed that the chest exerts a pressure of about 1.7 kPa at TLC. At RV the chest wall pressure almost matches the pressure in the alveoli (PA = −1.5 kPa), since both the pressure, PE , due to lung tissue elasticity and pressure, P , due to surface tension are near zero at low lung volumes. The pressure, PE , due to lung tissue elasticity, shown in Fig. 5B, declines during expiration and increases again during inspiration. The simulation shows that the tissue exerts a small negative pressure at very low lung volumes in the most dependent layers, i.e. resisting collapse in these layers. The elastic pressure at TLC is about 2 kPa, where the pressure due to surface tension, shown in Fig. 5C, is only about 0.3 kPa. The pressure due to surface tension, P , declines during expiration. During inspiration, the most dependent layers have a more rapid increase in P than the non-dependent layers due to the properties of surfactant which depends on the level of alveolar surface area reduction. The transmural pressure, PAlvTm , in Fig. 5D is
Fig. 5 – Simulation results for an expiration from TLC to RV followed by an inspiration back to TLC using model II of the lung tissue elasticity. Results are plotted as a function of alveolar input pressure and shown for layers number 1, 25, 50, 75 and 100 (100 being the most dependent layer, dashed line) (A) Extraalveolar pressure (PEA ). (B) Pressure due to lung tissue elasticity (PE ). (C) Pressure due to surface tension (P ). (D) Alveolar transmural pressure (PAlvTM ). (E) alveolar volume (VA ). Vertical dotted lines indicate separation of expiration and inspiration.
given as the pressure difference between input airway pressure and the extraalveolar pressure and the sum between the pressures due to lung tissue elasticity and surface tension. At TLC the transmural pressure is around 2.3 kPa and at low lung volumes the dependent layers have near zero transmural pressure. The alveolar volume in Fig. 5E indicates an unequal distribution of air in the alveoli during the simulation, which is more pronounced at low lung volumes. At TLC (PA = 4.0 kPa) all alveoli are simulated to have a volume of about 18 nl. At FRC (PA = 0 kPa) alveoli in the most depend layer are simulated to have a volume of about 4 nl whereas the alveoli in the most non-dependent layer are 9 nl. Alveoli are simulated to have a volume of 2 and 8 nl at RV (PA = −1.5 kPa) in the most dependent layer and the most non-dependent layer, respectively. The results in Fig. 5E also indicate that no collapse occurs in the healthy lungs between RV and TLC.
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Fig. 6 – Alveolar volume during a simulation from TLC to RV and back to TLC using model I of the lung tissue elasticity. Dotted lines indicate separation of expiration and inspiration.
Using model I instead of model II of the lung tissue elasticity gives almost the same results as shown in Fig. 5A–D. However, since the alveoli are assumed not to resist collapse in model I, the alveolar volume shown in Fig. 6 differs. This model indicates that collapse occurs below around RV.
3.1.
Validation
This section will validate the model against experimentally measured data. Fig. 7 shows the results of the density calculations from the study by Millar and Denison [17]. The results imply that a vertical density gradient is present at RV. The simulated density distribution down the lungs calculated from Eq. (29) at TLC (PA = 4 kPa) and RV (PA = −1.5 kPa) is also shown in Fig. 7 using the two models of the lung tissue elasticity. No difference between the two models is observed at TLC and both simulations follow well the homogenous density around 0.1 g/cm3 . As indicated in Fig. 6, model I simulates alveolar collapse at RV. The collapse of the most dependent layers leads to a marked increase in density in these layers, not shown in the data by Millar and Denison [17]. The density simulated by using model II, however, is in good agreement with data.
Fig. 7 – The simulated density, Lung , related to lung depth, D, is shown at TLC (dashed line) and RV (solid line) for both models of the lung tissue elasticity. Data read from Millar and Denison [17] is shown for TLC and RV (crosses and dots, respectively).
151
Fig. 8 – The static pressure–volume relationship during a simulation with an alveolar pressure range between 0 and 2.66 kPa and a TLC of 5.8 l. pressure–volume measurements by Sharp et al. [25] (crosses) are also shown. Dashed and solid lines indicate inspiration and expiration, respectively.
Therefore it is indicated that lung tissue may to some degree resist collapse and model II is the most appropriate model to describe density distribution. Fig. 7 also indicates that the changes in lung depth from TLC to RV are simulated well in both cases when compared to data. The volume of air in the lungs, VAir , can be summed up according to Eq. (7) giving VAir = 2.19 l at FRC (PA = 0 kPa) and VAir = 1.57 l at RV (PA = −1.5 kPa). These results are in good agreement with experimental observations. The volume of air in the lungs at FRC has been measured in 50 men in the supine position with a mean ± SD of 2.23 ± 0.61 l by Ibanez and Raurich [23]. Baydur [24] studied the lung mechanics in 10 subjects in supine position at different lung volumes including RV and FRC. The RV value was found to be with a mean ± SD of 1.61 ± 0.77 l and the FRC value was 2.47 ± 0.88 l. The simulated residual volume of 1.57 l is also in good agreement with the data obtained by Konno and Mead [13], shown in Fig. 2A. Assuming that the subjects in the experiment had a total lung capacity of 6.8 l, the residual volume is found very near 23% of TLC shown in data. Sharp et al. [25] studied PV-curves in 28 healthy men in the supine position. Fig. 8 shows the mean PV-curve for the study by Sharp et al. [25] and a model simulation with the same alveolar pressure range from 0 to 2.66 kPa and VTLC = 5.8 l using model II of the lung tissue elasticity. No visible difference using models I and II of the lung tissue elasticity was found. The maximum hysteresis being the maximum pressure difference between the expiratory and inspiratory limbs at equal volumes was simulated to be 0.28 kPa, compared to the 0.212 kPa reported by Sharp et al. [25]. The compliance measured between the two extremes was 1.24 l/kPa in the simulation, compared to the 1.13 l/kPa reported by Sharp et al. [25]. Brudin et al. [26] used Positron Emission Tomography to study the relationship between regional ventilation in eight healthy subjects in the supine position. Measurements were performed during quiet breathing and calculated as ml of air ventilated per minute per cm3 of lung. Since it has not been possible to find data in absolute values in mechanically ventilated subjects these measurements from subjects breathing
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the model was found to reproduce experimentally consistent trends regarding the above mentioned data.
4.1.
Fig. 9 – Simulation of the ventilation distribution down the lungs using model I (solid line) and model II (dashed line) along with the data read from Brudin et al. [26] (dots).
voluntarily were used for validation of simulated ventilation. In order to reproduce the data from patients breathing quietly, PA was varied between 0 and 0.4 kPa, resulting in a tidal volume of 437 ml. The ventilation was calculated by Eq. (36). V˙ i = (VA,Max,i − VA,Min,i ) · f · NAlvPrcm3 ,i
1 VA,Min,i + VBloodPrAlv + VTissuePrAlv
The pleural pressure, PPl , within the thin liquid film that lubricates the pleural surfaces and couples the lungs to the rib and diaphragm is omitted in this model. Despite of some evidence that pleural pressure is the pressure that sets regional lung volume (reviewed by Lai-Fook [27]) and common occurrence in the literature (e.g. [28]) the assumption that PEA = PPl is simply wrong. The extra alveolar pressure is always higher than or equal to the pleural pressure in healthy lungs. If this were not the case the pleural space would displace the lumen of the dependent regions of the lungs. If the pleural space displaces the lumen it will affect the lung volume and should therefore have been included in the model. Clinically this happens in patients with pleural effusion. However, in the healthy lungs the alveoli and capillaries will always be “exposed to see” the pressure within the lung parenchyma, which is the sum between chestwall pressure, hydrostatic pressure and pressure exerted by the muscles (Eq. (2)).
4.2.
(37)
Fig. 9 shows the simulated ventilation with models I and II of the lung tissue elasticity along with the data read from Brudin et al. [26]. In both cases, the models underestimated the increase in ventilation from the non-dependent to the dependent part of the lung, observed by Brudin et al. [26]. Even though, model I of the lung tissue elasticity had a higher gradient than model II, the whole model is unable to reproduce the ventilation observed by Brudin et al. [26] with the current parameters.
The simulated pressure–volume curve in Fig. 8 shows similar shape and hysteresis compared to data from Sharp et al. [25]. The main contributor to hysteresis is the effect of surfactant [10]. However, experimental results indicate that the tissue properties also induce a slight degree of hysteresis [15]. This implies that the model simulated hysteresis may be underestimated. Furthermore, the hysteretic behavior due to surfactant has been shown to depend on the pH, surfactant composition and speed of compression [29–32], effects which are not included in the present model, and could be a topic for further development. The number of alveoli has been shown to be highly variable among individual subjects [22]. Changing the number of alveoli in the model also affects the hysteresis. For instance an increase will redistribute the amount of air to a higher number of alveoli leading to relatively smaller alveoli. During breathing smaller alveoli will be more compressed in area, resulting in a higher amount of hysteresis, according to Eqs. (13)–(16).
4.3.
4.
Pressure–volume curve
(36)
where VA,Max,i and VA,Min,i are the maximal and minimal volume of the alveolus at layer i during the simulation, f is the breathing frequency (12 breaths/min) and NAlvPrcm3 is the number of alveoli per cm3 lung in layer i, calculated from Eq. (37). NAlvPrcm3 ,i =
The pleural pressure
Models I and II of the lung tissue elasticity
Discussion
A stratified model of the pulmonary ventilation derived from computed tomography has been presented which describes lung mechanics and includes aspects of: the alveolar geometry; pressure due to the chest wall; pressure due to surface tension determined by surfactant activity; pressure due to lung tissue elasticity; and pressure due to the hydrostatic effects of the lung tissue and blood. The presented model is the first to be validated against experimentally measured data on the density distribution, lung volumes (RV, FRC and TLC), PV-curve compliance and hysteresis and ventilation distribution of a healthy human subject in the supine position. Even though the model is based on many different studies on both humans and animals with many unknown parameters,
Two models of pressure due to lung tissue elasticity were developed due to the lack of information regarding tissue elastic properties at low lung volumes and negative transmural pressure, both models therefore represent postulated properties of the tissue at low lung volumes. Model I assumes that lung tissue cannot resist any negative transmural pressure. Model II assumes that the connecting fibers and tissue can produce a small negative pressure at low lung volumes. Both models of the tissue elasticity simulate a heterogeneous ventilation distribution down the lungs and indicate that no alveolar collapse occurs at FRC. At RV, model I however, simulates a collapse in the most dependent layers of the lung. The density measurements by Millar and Denison [17] showed no collapse at RV, which was consistent with simulations using
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model II indicating that this is an appropriate model for the lung tissue elasticity. It has been discussed whether the surface area of the alveoli changes during quiet breathing [33,34]. In a model by Wilson et al. [9,35,36] the surface area of the alveoli is allowed to change without changing the lung volume since the volume in the alveolar ducts are assumed to change depending on surface tension. The alveolar ducts are not included in the present model and the alveolar surface area and volume are directly linked through the alveolar radius. Furthermore, the assumption of a rigid ring at the opening of the alveoli still needs further investigation. In a model by Kitaoka et al. [37] the opening radius was modelled as changing with the alveolar volume. In this way it is possible for the alveoli to close without collapsing. Air trapping in closed but not collapsed alveoli may explain the moderate density observed in the dependent layers by Millar and Denison [17]. However, it is still not clear in which way the alveoli or bronchi may close or collapse and in which way the alveolar duct and opening are participating in resisting collapse at very low lung volumes. How they generally affect the lung mechanics is therefore a topic for future investigations.
4.4.
Ventilation distribution
The change in density from RV to TLC, shown in Fig. 7, reflects the ventilation at different layers of the lungs. Using model II of the lung tissue elasticity results in a good fit to measured ventilation described by Millar and Denison [17], indicating that the model properly simulates ventilation distribution when breathing from RV to TLC. The simulated ventilation distribution in Fig. 9, however, does not match the measured data described by Brudin et al. [26] very well. The measurements by Brudin et al. [26] show an increase in ventilation from the non-dependent to dependent part of the lungs by a factor of two, whereas using models I and II only results in simulated increases by a factor of 1.5 and 1.2, respectively. The measurements by Brudin et al. [26] were performed during quiet breathing and not during mechanical ventilation as would be more appropriate for comparison. Measurements obtained during mechanical ventilation are therefore needed before further conclusions can be drawn regarding model simulations of ventilation at small tidal volumes. A number of methods for measuring regional ventilation with high resolution are found in the literature (e.g. SPECT, PET, CT, MRI, EIT, microsphere methods) [26,38–44]. However, none of these methods have been used to quantify the ventilation in absolute values as a function of lung depth for humans under mechanical ventilation in supine position. Most of the methods have however found that ventilation distribution cannot be accounted for by gravity alone since a large heterogeneity of ventilation within isogravitational planes was observed [26,39–41]. Other factors such as the geometry and pathway of the alveolar tree including the effects of alveolar ducts may have a complex influence on the distribution of the ventilation. The present model assumes static conditions and does not include airway resistances and viscoelastic effects for the sake of simplicity. This is not a problem when simulating static PV-curves or recruitment
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maneuvers, but limits the model in some aspects regarding the ventilation distribution, e.g. at large flows or in patients with COPD [45]. Another mechanism that theoretically could affect the ventilation is hypoxia-induced airway dilation, or hypoxic bronchodilation, which would decrease airway resistance and facilitate gas flow to hypoxic areas of the lungs [46]. In this way the dependent areas of the lungs would be more ventilated and the simulated gradient shown in Fig. 9 would be increased. These factors are topics for further development since the only difference between an alveolus at the top and bottom of the lungs in the presented model is the gravitational impact of the weight of the above layers. Another assumption that may influence the ventilation is the homogenous distribution of blood. It is well known that a gravitational perfusion gradient exists [47]. Including a model of the pulmonary perfusion enables a more correct simulation of the hydrostatic pressures and densities down the lungs that may affect the ventilation. The effects of assuming that the lungs are expanding isotropically and all subjects have the same lung profile are also topics for future development. The presented model is validated against experimentally measured data on the ventilation distribution, density distribution, lung volumes, PV-curve compliance and hysteresis of healthy human subjects in the supine position. Two models of lung tissue elasticity have been developed. According to measurements of density distribution at RV it appears that the connecting fibers and tissue exert some negative pressure at very low lung volumes resisting collapse. The model simulates a heterogeneous ventilation distribution down the lungs and indicates no alveolar collapse at FRC and RV. The model is a step towards understanding how the lung mechanics change during mechanical ventilation, the next step being to understand how the pulmonary perfusion is affected simultaneously in response to mechanical ventilation. Combined, the two models could describe the ventilation–perfusion (V/Q) distribution in the lungs. The model could then potentially be used to predict how the V/Q relationship changes in the human healthy lungs during various ventilation strategies, closing the theoretical gap between lung mechanics and gas exchange.
Conflict of interest None.
Acknowledgements This work was partially supported by the Programme Commission on Neuroscience, Biotechnology and under the Danish Council for Strategic Research. JBS was supported by a Lundbeck Foundation personal fellowship and acknowledges taking advantage of MEMPHYS-Center for Biomembrane Physics and related grants (Danish National Research Foundation Center of Excellence). The authors would like to thank the Danish Lung Cancer Screening Trial (DLCST) for providing the chest CT-scan and Dr. Marleen de Bruijne of University of Copenhagen for performing the lung segmentations.
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