A Mathematical Physiological Model of the Pulmonary Capillary Perfusion

Proceedings of the 7th IFAC Symposium on Modelling and Control in Biomedical Systems Aalborg, Denmark, August 12-14, 2009

A Mathematical Physiological Model of the Pulmonary Capillary Perfusion Mads Lause Mogensen, Kristoffer Lindegaard Steimle, Dan Stieper Karbing and Steen Andreassen Department of Health Science and Technology, Aalborg University, Denmark Abstract: This study presents a stratified model that simulates the pulmonary capillary blood flow under influence of different lung volumes. The model includes capillary geometry, capillary wall elasticity, pressure exerted by the heart, blood viscosity, the effect of the chest wall and hydrostatic effects of the lung tissue and of the blood. The model simulates highly pulsatile blood flow with a heterogenous flow distribution down the lungs, in agreement with previous experimental studies. Moreover the model is in agreement with experimentally measured total capillary flow, total capillary volume, total capillary surface area and transition time of red blood cells passing through the pulmonary capillary network. The presented model is the first to describe the link between lung volume and perfusion. Keywords: physiological models, pulmonary perfusion, pulmonary capillaries, lung mechanics 1. INTRODUCTION

2. METHODS 2.1 Model introduction

Appropriate ventilator settings for intensive care patients with respiratory disorders are crucial for reducing recovery time and minimizing the probability of ventilator induced lung injury (VILI) (Fenstermacher and Hong, 2004) (Lumb, 2003). Finding the settings is difficult and requires a tradeoff between adequate gas exchange without stressing the lungs and damaging the alveoli. Positive end expiratory pressure (PEEP) and tidal volume affect both the lung mechanics and the gas exchange. It has been indicated that low tidal volumes reduce lung damage (ARDSNet, 2000). Furthermore high pressures by means of PEEP have proven to prevent alveolar collapse and improve gas exchange in the diseased lungs (Fenstermacher and Hong, 2004). It is still not clear how to improve gas exchange while reducing lung damage (Fenstermacher and Hong, 2004). In order to understand gas exchange, pulmonary ventilation and perfusion need to be understood. The perfusion has not been studied as intensely as the ventilation even though it is indicated that the lung volume has great effect upon the perfusion (Baile et al., 1982). No previous models of the pulmonary capillary perfusion has described how the perfusion is affected by the lung volume. This paper presents a physiological model that describes the pulmonary microcirculation, enabling simulation of capillary blood flow around the alveoli in the entire lung. The model describes the geometry, hemodynamics and blood rheology of a capillary. The model includes the extra alveolar pressure which is dependent on the lung height and the total lung volume. This enables a simulation of the flow distribution in the entire pulmonary microcirculation during mechanical ventilation of a simulated subject.

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The model has been implemented in MatLab (Mathworks, Natick, MA). The model simulates a healthy human subject at rest in the supine posture during mechanical ventilation. Fig. 1 illustrates the model representation of the pulmonary system. The lungs are modelled by 20 layers each reflecting a vertical height. A constant lung height of 16 cm is assumed (anterior to posterior). The height must be included in the model because the lung tissue weighs down on the layers below causing a hydrostatic gradient Phydro,tissue . The blood in the capillary network also imposes a hydrostatic gradient that increases the blood pressure down the lungs Phydro,blood . The hydrostatic pressure is calculated by (1). Phydro,i = ρ · g · hi

(1)

where ρ is the density of lung tissue (300 kg/m3 ) or blood (1060 kg/m3 ), g is the gravitational acceleration (9.81 m/s2 ) and hi is the height measured from the top of the lungs. The chest wall pressure and the hydrostatic pressure for tissue compose the extra alveolar pressure Pea as stated in (2). The chest wall can either exert a recoil pressure at high lung volumes or an opposite directed pressure at low lung volumes. This has been measured experimentally by (Konno and Mead, 1968). Fig. 2 shows the measured data points and a fitted curve. The curve fit is stated in (3). Pea,i = Pcw + Phydro,tissue,i Pcw = 0.71kP a − ln(

(2)

95% − 1) · 0.58kP a (3) %T LC − 22%

The capillary transmural pressure is defined by (4). Pcap,tm = Pcap − Pea

(4)

10.3182/20090812-3-DK-2006.0020

7th IFAC MCBMS (MCBMS'09) Aalborg, Denmark, August 12-14, 2009

ea cap1,tm

cap2,tm

cap3,tm

ea cap,1

ven 

heart

hydro, blood

art cw

Fig. 1. Illustration of the total lung model. Pea extra alveolar pressure. Pa the airway pressure. Pcap capillary pressure. Pcap,tm and Pa,tm capillary and alveolar transmural pressures. Part and Pven arterial and venous blood pressure. Pheart pressure exerted by the heart. Pcw pressure exerted by the chest wall. Phydro,tissue and Phydro,blood hydrostatic pressure due to lung tissue and blood. i index controlling layer height measured from the top.

[kPa] cw

P

FRC

−1 Tidal breathing

−2

30

40

50

cap,4

ven

where Pheart is the pressure exerted by the heart measured at the level of the pulmonary valve which is assumed to be 5 cm down the lungs. Assuming that the blood flow is laminar and that the blood is a Newtonian fluid, the resistance Rcap,i in (5) can be determined by Poiseuille’s law stated in (7). Rcap,i =

TLC

1

−3

cap,3

where Part,i is the arterial capillary blood pressure, Pven,i is the venous capillary blood pressure and Rcap,i is the resistance to flow in the capillary. The venous blood pressure is assumed to be constant during systole and diastole, only changing with the hydrostatic pressure down the lungs. The venous pressure is 1.066 kPa measured at the level of the pulmonary valve (Despopoulos and Silbernagl, 2003). Part,i can be calculated by (6). Part,i = Pheart + Phydro,blood,i (6)

[Konno and Mead, 1968]

0

cap,2

Fig. 3. Schematic representation of the capillaries divided into three pieces each having a pressure drop. Pea extra alveolar pressure. Part and Pven arterial and venous blood pressure. Q˙ blood flow. Pcap,tm capillary transmural pressure. Pcap capillary pressure.

cap,tm cap

hydro, tissue

art

a,tm

a

60 70 Percent of TLC [%]

80

90

100

Fig. 2. The graph shows a curve fitted to data (Konno and Mead, 1968) (RMS=4.092 r2 = 0.9915). The chest wall pressure is identified for functional residual capacity (FRC) and total lung capacity (TLC). The pressure range during a normal tidal breathing of approximately 500 ml is also indicated. Assuming a TLC of 6.5 l, the lung volume at FRC and after tidal inflation of 500 ml comprise 34.3 and 42% of TLC in supine posture (Ibanez and Raurich, 1982). As indicated in Fig. 1, Pea affects both the capillary and alveolar transmural pressures. In this way the blood flow is coupled to the total lung volume. The lungs are assumed to contain 300 mio. alveoli (Despopoulos and Silbernagl, 2003). Each layer is modelled as containing an equal number of alveoli and an equal number of capillaries surrounding an alveolus regardless of layer number. However just one capillary per alveolus is depicted in Fig. 1 for the sake of simplicity. The blood flow through a capillary can be determined by (5). Part,i − Pven,i Q˙ cap,i = Rcap,i

(5)

158

8 · Lcap + ηblood (rcap,i ) 4 π · rcap,i

(7)

where Lcap is the length of a capillary, ηblood is the blood viscosity and rcap,i is the capillary radius. The radius of a capillary is dependent on the capillary transmural pressure, see (4). The capillary transmural pressure has so far been assumed to be uniform along the entire length of a capillary. However the capillary pressure should decrease along the capillary before finally reaching the venous pressure. This is approximated by modelling the capillaries as three segments (N=3) of equal lengths each accounting for a pressure drop as shown in Fig. 3. This causes the radius to decrease and the resistance to increase along the capillary length. This approach introduces the problem of identifying the pressure drops caused by each segment, however it is known that the flow in each of the three segments must be equal as stated in (8). This leaves three ˙ Pcap,2 and Pcap,3 . equations with three unknowns being Q, Part − Pven Pcap,n − Pcap,n+1 Q˙ cap = = Rcap Rcap,n

,n=[1:N ]

(8)

Summarizing in order to identify the pulmonary capillary perfusion the following needs to be identified: Capillary shape and radius, pressure exerted by the heart, blood viscosity and the geometry of the pulmonary capillaries being length and number. 2.2 Capillary shape and radius Scanning electron micrographs of the alveolar wall reveals a circular capillary cross section under zone III conditions (Part > Pa > Pven ), but that the capillaries flatten under

7th IFAC MCBMS (MCBMS'09) Aalborg, Denmark, August 12-14, 2009

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[Sobin, 1972] [Glazier, 1969]

2.6

Mean = 1.6839

2.4

5

Pheart [kPa]

Capillary radius [µm]

6

4 3

2.2 2 1.8 1.6

2

1.4 1 0 −2

1.2 −1

0 1 2 Capillary transmural pressure [kPa]

3

1

4

Fig. 4. The relation between capillary transmural pressure and capillary radius. Data from two studies are represented according to the legend. A sigmoidal curve has been fitted to the data points (RMS=0.3296 r2 = 0.9823). The shape of the capillaries is indicated at different transmural pressures. zone II conditions (Pa > Part > Pven ) (Weibel, 1986). However this way of dividing the lungs into zones, assumes that the airway pressure directly affects the capillaries, neglecting the effect of Pea . The capillary transmural pressure should be calculated as stated in (4) and not as Pa − Pcap . Since no measurements of the extra alveolar pressure have been performed the observations under different zones cannot be directly translated into transmural pressures. However it seems reasonable that the capillaries begin to flatten at negative transmural pressures. This motivates a model of the capillary cross section as being circular at positive transmural pressures and that they become elliptic at negative transmural pressures. It is assumed that the capillaries stretch at positive transmural pressures, but merely deform at negative transmural pressures. The deformation is modelled as the capillary circumference being constant at negative transmural pressures. The relationship between the capillary transmural pressure and the radius of a capillary has been described (Sobin et al., 1972) (Glazier et al., 1969). However the capillary transmural pressure was defined as Pa − Pcap in those two studies again neglecting the effect of Pea . Since these studies are performed on excised lungs the effect of the chest wall is removed and only the hydrostatic gradient due to the lung tissue changes the extra alveolar pressure. This implies that Pea can be calculated simply by (1). From the data on the transmural pressure provided in the studies the capillary pressure can be obtained if the airway or capillary pressure are provided along with the height of measurement. This is the case for Sobin et al. (1972) and some of the data points from Glazier et al. (1969). The points are plotted in Fig. 4 along with a fitted curve. The curve fit is stated in (9). rcap = 2µm +

11µm 1+e

P a −( cap,tm−2.11kP 0.8401kP a

)

(9)

2.3 Pressure exerted by the heart A pressure profile during a heart beat is not measurable in the capillaries, however such a profile has been measured in the pulmonary artery (Takala, 2003). The mean pressure from the pulmonary artery to the capillaries declines

159

0.1

0.2

0.3

0.4

0.5 0.6 Time [s]

0.7

0.8

0.9

1

Fig. 5. The pressure exerted by the heart (Pheart ) profile during one heartbeat. Note that the duration is around one second since the subject is at rest. The profile is measured in the pulmonary artery (Takala, 2003), and scaled into the pressure range for pulmonary capillaries (Despopoulos and Silbernagl, 2003). approximately 20%, resulting in a pressure range of 12.7 kPa at rest at the level of the pulmonary valve (Despopoulos and Silbernagl, 2003). Fig. 5 shows the pressure profile by Takala (2003) scaled into the pressure range for the capillaries. 2.4 Blood viscosity The blood viscosity is a function of the tube radius mainly due to the behavior of erythrocytes at different radii according to the Fahraeus-Lindqvist effect. Data from several studies of this phenomenon in glass tubes have led to a model of the capillary viscosity as a function of the diameter of the glass tubes and hematocrit (Pries and Secomb, 2003). This model has been adapted with the hematocrit set to a normal value of 0.45 (Despopoulos and Silbernagl, 2003). When the capillaries become elliptic, the smallest radius in the ellipse is used to determine the viscosity of blood in the model. 2.5 Model calibration The capillary length and the total number of capillaries are indeed decisive for the pulmonary perfusion. The total number of capillaries is described as a number of capillaries per alveolus, assuming that all alveoli are surrounded by capillaries in the same manner. However the capillary length and number of capillaries per alveolus is not well described in the literature. The capillary length has been reported to be approximately 600 µm for cat lungs and 800 µm for dog lungs (Staub and Schultz, 1968). The number of capillaries per alveolus has not been quantified. The model has therefore some degree of freedom regarding these two parameters, which have been adjusted in order to obtain the best possible description of a series of known output parameters. These parameters are total capillary flow, total capillary volume, total capillary surface area and transition time. This adjustment resulted in a capillary length of 1000 µm and 5 capillaries per alveolus. Since each capillary has been estimated to cross 5-12 alveolar walls (Staub and Schultz, 1968), this implies that a cross section of an alveolus should show 25-60

7th IFAC MCBMS (MCBMS'09) Aalborg, Denmark, August 12-14, 2009

capillaries. This seems reasonable by inspection of electron microscopic pictures of the alveolar wall (Weibel, 1986). 3. RESULTS Three model simulations have been performed for model validation. The three simulations have been performed at following lung volumes: 1) FRC 2) TLC 3) during a tidal breathing of 500 ml. The breath starts from a lung volume of 500 ml above FRC, then an expiration brings the lung volume to FRC followed by an inspiration back to 500 ml above FRC. These three scenarios cover two extremes supplying the total range of the output parameters and a normal breathing enabling a comparison with data found in the literature. The breath frequency is 17 breaths/min and the heart rate is 60 beats/min. The simulated extra alveolar pressure, capillary pressure, transmural pressure, capillary radius and capillary flow during a tidal breathing of 500 ml are shown as a function of time in Fig. 6. The simulation result of Fig. 6-D indicates that the radius varies 2 µm (systole-diastole) and approximately 2 µm (top-bottom). Fig. 6-E illustrates that the flow is highly pulsatile. It can further be identified that the flow is different from top-bottom. These results are in agreement with data from experimental studies (Lee, 1971), (Hughes et al., 1968). The simulation results describing flow distribution, capillary perfusion, capillary blood volume, capillary surface area and transition time in Figs. 7 - 11 can all be compared with data described in the literature. The mean values for the different simulations are listed in Table 1. The transition time is listed as a range with the lowest value being the mean value from the most dependent layer and the highest being the mean value from the most nondependent layer. Flow distribution The distribution of capillary flow at different lung heights during the tidal breathing is plotted in Fig. 7 against measured data from Kosuda et al. (2000). It is well known that the blood flow is heterogenously distributed according to lung height (Hughes et al., 1968). The flow increases from the non-dependent lung and downwards, except for the very bottom of the lungs, where flow decays again (Hughes et al., 1968). According to Fig. 7 the simulated flow does increase from the non-dependent lung and downwards, however the decrease in blood flow in the most dependent lung is not seen. Pulmonary capillary perfusion The total pulmonary perfusion in Fig. 8 is simulated with a mean of 6.24 l/min. The pulmonary capillary perfusion should equal the cardiac output which at rest is 5-6 l/min for a healthy human (Despopoulos and Silbernagl, 2003). The simulated mean pulmonary perfusion is 0.19 l/min at TLC almost completely diminishing blood flow. Pulmonary capillary blood volume The simulation result for the pulmonary capillary blood volume in Fig. 9 shows a mean value of 63.1 ml during normal tidal breathing. This value is low compared to experimentally measured values of 86 ml for men (Zanen et al., 2001). Even higher values have been estimated in the range 100-200 ml in the postmortem lungs (Glazier et al., 1969).

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The result of the Pulmonary capillary surface area simulation in Fig. 10 gives a total capillary surface area of 40.6 m2 . The pulmonary capillary surface area has been stated to be around 90 m2 (Lumb, 2003) (Hlastala and Robertson, 1998), which is a factor of two higher than the simulation. It has also been stated that the capillary surface area is 10 % less than the alveolar surface area. This ensures an almost maximal surface area for gas exchange. Assuming that the alveolar radius is 150 µm at FRC and that the lungs contains 300 mio. alveoli (Despopoulos and Silbernagl, 2003), this leads to an alveolar surface area of 84 m2 implying spherically alveoli. Transition time The pulmonary capillary transition time for layer number 1, 5, 10, 15 and 20 is shown for the tidal breathing simulation in Fig. 11-A and plotted against the lung height in Fig. 11-B. Model simulation resulted in transition times from 2.9-14.9 sec (bottom-top), see Table 1. This is slightly higher than the values estimated by Wagner et al. (1986) of 1.6-12.3 sec (bottom-top) in anesthetized dogs. Table 1. Mean simulation results

Surface area (m2 ): Blood volume (ml): Blood flow (l/min): Transition time (s):

FRC 37.9 72.4 7.75 1.9 - 7.4

Tidal breathing 35.4 63.1 6.24 2.9 - 14.9

TLC 17.4 13.2 0.19 47.7 - 157.0

4. DISCUSSION The model simulates the four presented output measures well, however the total capillary surface area and total capillary blood volume are lower than the presented data. Zone IV As shown in Fig. 7, the model does not simulate a decrease in flow at the bottom of the lungs. The decrease in flow at the bottom of the lungs has partly been explained by the effect of vascular tone (Nemery et al., 1983). This makes sense in the way that vascular tone would constrict blood flow during hypoxia at the bottom of the lung distributing the blood to better ventilated parts of the lungs. Another explanation for the flow decrease at the bottom of the lungs is due to extra capillary vessels being more compressed at the bottom of the lungs, increasing the flow resistance in this area (Hughes et al., 1968). These effects could be a topic for future development since the total pulmonary blood flow would become lower if vasoconstriction was included in the model and hence the number of capillaries per alveolus could be increased in order to maintain a total flow of around 6 l/min. The increase in number of capillaries per alveolus would increase the total capillary blood volume and total capillary surface area without affecting the transition time. Since the flow decreases at the very bottom of the lungs the transition time might also increase in this area. This increase in transition time has not been measured by Wagner et al. (1986). This disagreement could be a result of the transition time not being measured at the very bottom where the flow decreases, or that the capillary density is lower at the bottom, reducing the flow measured in this

7th IFAC MCBMS (MCBMS'09) Aalborg, Denmark, August 12-14, 2009

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A

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Pea [kPa]

0.5

0

−0.5

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[kPa] cap,tm

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3.5

D

5

Time [s]

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2.5

0.2

2

6

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3.5

Fig. 8. Simulation of the total capillary blood flow at functional residual volume (FRC), total lung capacity (TLC) and tidal breathing.

3.5

C

3

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Radius [µm]

20

2

0

P

B

30

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Blood volume (Total) [l]

P

cap

[kPa]

4

FRC Tidal breathing TLC

FRC Tidal breathing TLC

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1.5

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Time [s]

2

2.5

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3.5

2 0

0.5

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1.5

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Fig. 9. Simulation of the total capillary blood volume at functional residual volume (FRC), total lung capacity (TLC) and tidal breathing.

3.5

−13

E

4 3

60

2 1 0

0

0.5

1

1.5

2

2.5

3

3.5

Time [s]

Relative capillary perfusion [%]

Fig. 6. Simulation results during a tidal breath for layers 1, 5, 10, 15 and 20 (most dependent, dashed line). A: extra alveolar pressure (Pea ). B: capillary pressure (Pcap ). C: capillary transmural pressure (Pcap,tm ). D: capillary radius. E: capillary flow. Pea decreases during expiration and that Pea is higher at dependent part of the lungs. The capillary pressure is also highest at the dependent part of the lungs, due to the hydrostatic pressure. This results in varying transmural pressures and capillary radii down the lungs. Furthermore this leads to a highly pulsatile flow indicated. Simulation: Tidal breathing [Kosuda, 2000]

140 120 100 80 60 40 0

Dependent

0.1

0.2

0.3

0.4 0.5 0.6 Relative lung height

0.7

0.8

0.9

1

non−Dependent

Fig. 7. Comparison of simulated pulmonary capillary perfusion with data from Kosuda et al. (2000). The data is normalized to the maximal perfusion, contrary to the simulation, which is plotted in respect to the mean start of zone IV from the data (7 cm above the most dependent layer).

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Surface area (Total) [m2]

Flow [l/min]

x 10

FRC Tidal breathing TLC

50 40 30 20 10 0

0.5

1

1.5

Time [s]

2

2.5

3

3.5

Fig. 10. Simulation of the total capillary surface area at functional residual volume (FRC), total lung capacity (TLC) and tidal breathing. area without affecting the transition time. Supine posture The model assumes supine posture. It has been indicated that the flow distribution in the lungs is not only dependent on gravitational forces, and it is indicated that an anatomical difference from the anterior to posterior exists (Glenny et al., 1991). If a subject were to be simulated in other postures than supine, the model should include parameters reflecting the effects of posture. Constant lung height The lung height is assumed constant at 0.16 m. However during breathing the lungs expand. It has been reported that the height (anterior to posterior) of the human lungs varies from 10-16 cm with lung volumes at residual volume and total lung capacity, respectively (Millar and Denison, 1989). This effect would imply a smaller hydrostatic gradient and hence a smaller difference in extra alveolar pressure down the lungs. Blood viscosity In vivo measurements of the blood viscosity are poorly described and remains to be fully under-

7th IFAC MCBMS (MCBMS'09) Aalborg, Denmark, August 12-14, 2009

Transition time [s]

150 A 100 50 0 0

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Transition time [s]

15 B

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Fig. 11. Simulation of A: transition time during a tidal breath in layer number 1, 5, 10, 15 and 20 (most dependent, dotted line) and B: mean transition time as a function of the lung height. stood. However a few measurements of the blood viscosity have been performed in vivo indicating much higher values for viscosity at similar radii. When the capillaries become elliptic, the smallest radius in the ellipse is used to determine the viscosity. This could cause an overestimation of viscosity since the erythrocytes might be able to squeeze into a more elliptic shape. PEEP The recoil pressure exerted by the chest wall is determined by the volume of air in the lungs. PEEP results in a higher amount of air in the lungs and will increase the pressure from the chest wall and hence the extra alveolar pressure resulting in a reduced radius of the capillaries and decreased blood flow. Since the lung volume is input to the model, an exact value of PEEP can not directly be included. Steimle et al. (2009) have proposed a model of the mechanical properties describing the pressure volume relationship in the lungs. Combining these two models would enable the link between PEEP and blood flow. REFERENCES ARDSNet (2000). Ventilation with lower tidal volumes as compared with traditional tidal volumes for acute lung injury and the acute respiratory distress syndrome. N Engl J Med, 342(1301-1308). Baile, E.M., Pare, P.D., Brooks, L.A., and Hogg, J.C. (1982). Relationship between regional lung volume and regional pulmonary vascular resistance. J Appl Physiol, 52(4)(914-920). Despopoulos, A. and Silbernagl, S. (2003). Color Atlas of Physiology. Georg Thieme Verlag. ISBN: 3-13-545005-8. Fenstermacher, D. and Hong, D. (2004). Mechanical ventilation - what have we learned? Crit Care Nurs Q, 27(3)(258-294). Glazier, J.B., Hughes, J.M.B., Maloney, J.E., and West, J.B. (1969). Meassurements of capillary dimensions and blood volume in rapidly frozen lungs. J Appl Physiol, 26(1)(65-76). Glenny, R.W., Lamm, W.J.E., Albert, R.K., and Robertson, H.T. (1991). Gravity is a minor determinant of

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pulmonary blood flow distribution. J Appl Physiol, 71(2)(620-629). Hlastala, M.P. and Robertson, H.T. (1998). Complexity in structure and function of the lung. Marcel Dekker. ISBN: 0-8247-9879-1. Hughes, J.M.B., Glazier, J.B., Maloney, J.E., and West, J.B. (1968). Effect of lung volume on the distribution of pulmonary blood flow in man. Respiratory physiology, 4(58-72). Ibanez, J. and Raurich, J.M. (1982). Normal values of functional residual capacity in the sitting and supine position. Intensive Care Med, 173(8)(173-177). Konno, K. and Mead, J. (1968). Static volume-pressure characteristics of the rib cage and abdomen. J Appl Physiol, 24(544-548). Kosuda, S., Kobayashi, H., and Kusano, S. (2000). Change in regional pulmonary perfusion as a result of posture and lung volume assessed using technetium-99m macroaggregated albumin spet. European Journal of Nuclear Medicine, 27(5)(529-535). Lee, G.J. (1971). Regulation of the pulmonary circulation. British Heart Journal, 33(15-26). Lumb, A.B. (2003). Nunn’s Applied Respiratory Physiology. Butherworth Heinemann. ISBN: 0-7506-3107-4. Millar, A.B. and Denison, D.M. (1989). Vertical gradients of lung density in healthy supine men. Thorax, 44(6)(485-490). Nemery, B., Wijns, W., Piret, L., Caujwe, F., Brasseur, L., and Frans, A. (1983). Pulmonary vascular tone is a determinant of basal lung perfusion in normal seated subjects. J Appl Physiol, 54(1)(262-266). Pries, A.R. and Secomb, T.W. (2003). Rheology of the microcirculation. Clinical Hemorheology and microcirculation, 29(143-148). Sobin, S.S., Fung, Y.C., Tremer, H.M., and Rosenquist, T.H. (1972). Elasticity of the pulmonary alveolar microvascular sheet in the cat. Circulation Research, 30(440-450). Staub, N.C. and Schultz, E.L. (1968). Pulmonary capillary length in dog cat and rabbit. Respir Physiol, 5(371-378). Steimle, K.L., Mogensen, M.L., Karbing, D.S., Smith, B.W., Vacek, O., and Andreassen, S. (2009). A mathematical physiological model of the pulmonary ventilation. 7th IFAC Symposium on Modelling and Control in Biomedical Systems (including Biological Systems), August 12-14th, Aalborg, Denmark. Takala, J. (2003). Pulmonary capillary pressure. Intensive Care Med, 29(890893). Wagner, W.W., Latham, L.P., Hanson, W.L., Hofmeister, S.E., and Capen, R.L. (1986). Vertical gradient of pulmonary capillary transit times. J Appl Physiol, 61(4)(1270-1274). Weibel, E.R. (1986). The pathway for oxygen. President and Fellows if Harvard College. ISBN: 0-674-65791-8. Zanen, P., Lee, I., Mark, T., and Bosch, J. (2001). Reference values for alveolar membrane diffusion capacity and pulmonary capillary blood volume. European Respiratory Journal, 18(764-769).