The effect of arteriolar resistance on perfusion distribution in a model of the pulmonary perfusion

Transcription

1 The effect of arteriolar resistance on perfusion distribution in a model of the pulmonary perfusion Mads L. Mogensen*. Dan S. Karbing* and Steen Andreassen* *Center for Model-based Medical Decision Support, Department of Health Science and Technology, Aalborg University, Aalborg, DK-9220, Denmark (Tel: ; lause@hst.aau.dk). Abstract: Simulations of lung perfusion with a model including elasticity and flow resistance of the lung capillaries, viscosity of the blood and chest wall elasticity show a higher perfusion at the bottom of the lungs (dependent part) than at the top (non-dependent part) due to the effect of gravity. The simulated ratio of dependent/non-dependent perfusion was 6.1, which is higher than experimentally determined ratios of 3.0. This paper explores whether inclusion in the model of arteriolar resistance can reduce this ratio, under the restriction that the pressure drop cause by the arterioles and by the capillaries should be about equal to agree with experimental data. Under this restriction, alveolar resistance reduced the ratio to This leaves room for other mechanisms, either passive (anatomical) or active (hypoxic vasoconstriction) to further reduce this ratio. Keywords: Physiological models, pulmonary perfusion, arteriolar resistance, lung mechanics. 1. INTRODUCTION In the 1960s West and his colleagues demonstrated that ventilation and perfusion are not evenly distributed in the human lungs (e.g. West (1962)). However, important questions remain as to the underlying effects causing ventilation and perfusion to vary down the lungs as well as how these distributions are affected by changes for example in posture or in ventilation pressures during mechanical ventilation. Approaches to answer these questions include both experimental studies (e.g. Baile et al. (1982), Jones et al. (2001)) and studies with mathematical physiological models (e.g. Fung and Sobin (1969), Liu et al. (1998)). Mogensen et al. (2010a) recently presented a stratified mathematical model of the pulmonary perfusion to elucidate the mechanisms contributing to the distribution of perfusion in the healthy human lungs. This model was novel in that it included models of capillary elasticity and resistance, viscosity of the blood, chest wall elasticity, number and length of lung capillaries. In addition assumptions were made about the pressure profile at the proximal end of the lung capillaries. The distribution of lung tissue densities were taken from a model of pulmonary ventilation (Steimle et al. (2010), Mogensen et al. (2010b)). Although the model by Mogensen et al. (2010a) produced distributions of perfusion similar to that previously measured in healthy human subjects, the model overestimated the ratio, Q Ratio, between the perfusion at the bottom of the lungs (dependent part) and the top (non-dependent part). The model as originally presented included capillary and venous blood pressures but omitted the resistance of the pulmonary arterioles. The purpose of the present paper is therefore to explore the effects of including a constant pulmonary arteriolar resistance in the model of pulmonary perfusion by Mogensen et al. (2010a). The paper presents the modifications of the model equations as well as a sensitivity test illustrating the changes in simulated distributions of pulmonary perfusion with changes in the arteriolar resistance. Furthermore it will be investigated how capillary length affects the capillary blood volume, capillary transition time, ratio between the arteriolar and capillary blood pressure drop, capillary surface area and Q Ratio. 2. METHODS 2.1 Mathematical model of the pulmonary perfusion Fig. 1 illustrates the model representation of the pulmonary system. The model describes the lungs as divided into a number of horizontal layers, (N Layers 20) (Mogensen et al. (2010a, b), Steimle et al. (2010)). Fig. 1. Illustration of the model describing the pulmonary system. P CW : Pressure exerted by the chest wall. P pa : Pulmonary arterial pressure. P pv : Pulmonary venous pressure. P HydroBlood : Hydrostatic pressure due to blood. P HydroTissue : Hydrostatic pressure due to lung tissue i: Index controlling layer number measured from the top to the number of layers (N Layers ). CT-scan is by courtesy from a study by Lo et al. (2010). Copyright by the International Federation of Automatic Control (IFAC) 5018

2 Fig. 2. Schematic representation of an arteriole and a capillary divided into segments each having a pressure drop. For simplicity only three segments are shown. P EA,i : Extraalveolar pressure. P a,i : Arterial blood pressure. P v,i : Venous blood pressure. Q i : Blood perfusion. P CapTM,,i : Capillary transmural pressure. P Cap,i,n : Capillary pressure at layer i and segment n. P Art,i : Arteriolar pressure. Fig. 1 is a CT-scan of a healthy person in supine position, hence the layers are in the frontal plane and numbered in the ventro-dorsal direction, such that the most dependent layer has the highest number. Lung tissue weighs down on the layers below causing an increasing hydrostatic pressure, P HydroTissue. The pulmonary blood also imposes a hydrostatic gradient that increases the blood pressure down the lungs, P HydroBlood. The hydrostatic pressures of tissue and blood in a layer are calculated from the gravitational acceleration, density of blood and lung tissue, and the height of the layer (Mogensen et al. (2010a)). In order to simulate a subject under mechanical ventilation a previous ventilation model is used (Mogensen et al. (2010b), Steimle et al. 2010)). This model uses the pressure exerted by the ventilator as input to calculate densities and thicknesses of the different layers. It is assumed that all layers contain the same number of alveoli and the same number of arterioles and capillaries surrounding an alveolus regardless of layer height. The chest wall pressure determined using the ventilation model (Mogensen et al. (2010b), Steimle et al. (2010)) and the hydrostatic pressure due to the lung tissue add up to the extraalveolar pressure outside the arterioles and capillaries, P EA, as stated in (1): P + EA,i PCW PHydroTissue,i (1) The perfusion through an arteriole and a capillary, Q i, at layer i can be determined by (2): PArt,i Pv,i Qi (2) R + R Art Cap,i where P Art,i is the arterial blood pressure, P v,i is the venous capillary blood pressure, R Art and R Cap,i is the resistance to flow in the arteriole and capillary at layer i. It is assumed that the arteriolar resistance is constant for all layers. R Art is therefore not indexed with i. P Art can be calculated by (3): P + Art,i Ppa PHydroBlood,i (3) where P pa is the pulmonary artery pressure measured at the level of the pulmonary valve which is assumed to be 5 cm down the lungs (Ferner and Staubesand (1982)). The pulmonary venous blood pressure, P v, is assumed to be constant during systole and diastole, only changing with the hydrostatic pressure down the lungs. The venous pressure is assumed to be 1.1 kpa measured at the level of the pulmonary valve (Despopulos and Silbernagl (2003)). Capillary transmural pressure, P CapTM, is defined by (4): P CapTM,i P P (4) Cap,i EA,i The capillary transmural pressure is, however, not uniform along the entire length of a capillary and the capillary pressure should decrease along the capillary before finally reaching the venous pressure, P v,i. This is approximated by modelling the capillaries in a number of segments, N S, of equal lengths each accounting for a pressure drop (Fig. 2). N S, 20 is used throughout all simulations. The perfusion in each segment must be equal as stated in (5). PArt,i Pv,i PArt,i PCap,i,1 Qi RArt + RCap,i RArt PCap,i,n PCap,i,n + 1 ; RCap,i,n n [ 1 : N ] Capillary resistance at segment n, R Cap,i,n can be determined by Poiseuille's law for tubes with elliptical cross sections at negative transmural pressure as described previously (Mogensen et al. (2010a)). Poiseuille s law including a elliptic correction factor, M 0, which is a function of radii (r 1,i,n and r 2,i,n ) describing the shape of the elliptic tube, is stated in (6): 0.75 LCap ηblood ( r2,i,n ) RCap,i,n 3 r1,i,n ( 2 r2,i,n ) M o( r1,i,n,r2,i,n ) where L Cap is the length of a capillary and η Blood is the blood viscosity. r 2,i,n is dependent on the capillary transmural pressure (7): r2,i,n f ( PCapTM,i, n ) (7) r 1,i,n can be calculated using geometric equations. Equations (5), (6) and (7) can be solved numerically yielding all pressure P Cap,i,n, flows Q i and resistances R Cap,i,n. (Mogensen et al. (2010a)). S (5) (6) 5019

3 2.2 Pulmonary artery pressure at the pulmonary valve In the model by Mogensen et al. (2010a) the pressure proximal to the capillaries was estimated by scaling a pressure profile measured in the pulmonary artery (Takala et al. (2003)) (Despopulos and Silbernagl (2003)). In this study the same pressure profile was used as an estimate of the pulmonary arterial pressure by scaling it to a pressure range between kpa with a mean of 2.2 kpa (Lumb (2003)) (Fig. 3, dashed line). Simulated pressure in the capillaries (Fig. 3, full line) has a mean pressure of 1.6 and the ratio between the arteriolar and capillary blood pressure drop is ΔBP Art /ΔBP Cap 1.0. density and extraalveolar pressure, small fluctuations during the simulation are observed. Fig. 4-B shows that the pressure drop over the arterioles is largest in the dependent part of the lungs (dashed line), which is opposite to the pressure drop over the capillaries (Fig 4-C), which is largest in the nondependent part of the lung. Fig 4-D shows that the simulated arteriolar and capillary perfusion is highly pulsatile and that perfusion is completely stopped during diastole. Fig. 3. The pulmonary artery pressure profile at the pulmonary valve during one heartbeat (Takala (2003)) scaled into the pressure range at the proximal end of the pulmonary capillaries (full line) and in the pulmonary artery (dashed line). The heart rate is assumed to be 60 beats/min and R Art kpa s/nl. 3. RESULTS The first part of the results section introduces a simulation of capillary pressures and perfusion during mechanical ventilation of a tidal breath of 508 ml. The breathing frequency was 12 breaths/min with an inspiration time of 2 s and expiration time of 3 s for all simulations. All simulations were generated by simulating mechanical ventilation with ventilator pressure between kpa. The second and third part of the results section shows the results of the sensitivity analysis of R Art, L cap and N CapPerAlv. 3.1 Model simulations Fig. 4 shows the simulated extraalveolar pressure, P EA, pressure drop over arterioles and capillaries, and pulmonary perfusion at layer number 1, 7, 14 and 20. The simulation is performed using an arteriolar resistance of R Art 0.25 kpa s/nl. Fig. 4-A shows P EA, that is composed by the hydrostatic pressure due to tissue and P CW. Since hydrostatic pressure at layer number 1 (lowest solid line) is 0 kpa, the chest wall pressure can be identified at this layer. The simulation shows that the extraalveolar pressure increases during inspiration and declines during expiration as the lungs are inflated and deflated. As a result of the change in capillary blood volume during a heartbeat which affects lung Fig. 4. Simulation results of a tidal breath for R Art. 0.5 kpa s/nl and L Cap 350 μm of layers 1, 7, 14 and 20 (Most dependent, dashed line). A: Extraalveolar pressure, P EA. B: Pressure drop over the arterioles. C: Pressure drop over the capillaries. D: Capillary perfusion, Q. Vertical dashed line separates inspiration from expiration. 3.2 Arteriolar resistance sensitivity test In the following it will be investigated how arteriolar resistance, capillary length and number of capillaries per alveolus, N CapPrAlv affect model simulation. Fig. 5 shows the results from the sensitivity analysis of the arteriolar resistance on the distribution of perfusion down the lungs performed with L Cap 350 µm (Mogensen et al. (2010a)). In the data from Brudin et al. (1994) the ratio, Q Ratio, is 3.0 determined from a second order polynomial fit to the data (not shown in Fig. 5). The mean pulmonary perfusion, Q, in the study by Brudin et al. (1994) is 7.7 l/min estimated by the polynomial 5020

4 fit and a profile of lung cross-sectional areas as a function of lung depth (Steimle et al. (2010)). During the sensitivity simulations Q was fixed at 7.7 l/min by adjusting N CapPrAlv. Table 1 summarizes the simulation results from the sensitivity analysis. The arteriolar and capillary pressure drops are calculated as a mean over the entire breath for all layers. As shown in table 2 there is no arteriolar pressure drop when using R Art 0 kpa s/nl. The perfusion ratio, Q Ratio, between the dependent and the non-dependent part of the lungs is 6.1 for the model without arteriolar resistance. Fig. 5 shows that introduction of an arteriolar resistance reduces this ratio from Q Ratio 6.1 for zero resistance to Q Ratio 3.8 for R Art 1.0 kpa s/nl. For R Art 0.5 kpa s/nl the Q Ratio 4.1, which is still higher than the ratio of 3.0 from data of Brudin et al. (1994). By using R Art 0.5 kpa s/nl, mean arteriolar and capillary pressure drops are 0.55 kpa 0.50 kpa respectively, (ΔBP Art /ΔBP Cap 0.91), this is quite close to the originally assumed ΔBP Art /ΔBP Cap 1.0. Fig. 5. Simulation results of the sensitivity analysis on R Art. Perfusion distribution is shown as a function of lung depth for arteriolar resistances of 0.1 kpa s/nl, 0.25 kpa s/nl, 0.5 kpa s/nl and 1.0 kpa s/nl. Thick line shows simulation of R Art 0 kpa s/nl similar to one described by Mogensen et al. (2010a). Data from Brudin et al. (1994) (dots). Table 1. Simulation results R Art (kpa s/nl) N CapPrAlv ΔP Art (kpa) ΔP Cap (kpa) Q Ratio (Fraction) Capillary length sensitivity test Previously the capillary length was assumed constant, L Cap 350 µm (Mogensen et al. (2010a)). It will now be investigated how the capillary length affects the ratio between the arteriolar and capillary blood pressure drop, ΔBP Art /ΔBP Cap, capillary blood volume, V Cap and capillary surface area, A Cap. The interval of the most appropriate range of R Art, L Cap and N CapPrAlv will be investigated in the following. Fig. 6. Simulation results of the sensitivity analysis on L Cap. A: Ratio between the arteriolar and capillary blood pressure drop, ΔBP Art /ΔBP Cap. B: Capillary blood volume, V Cap. C: Capillary surface area, A Cap. Along with the simulation results, also normal physiological ranges are shown as shaded areas. Dashed line represents simulation results from R Art 0 kpa s/nl. Other simulations are with R Art 0.1 kpa s/nl, 0.33 kpa s/nl, kpa s/nl, 0.52 kpa s/nl and 0.7 kpa s/nl. Fig. 6-A shows simulated ratio ΔBP Art /ΔBP Cap. In the methods it was assumed that arterioles and capillaries account for an equal pressure drop and the ratio ΔBP Art /ΔBP Cap is close to 1.0 (Lumb (2003)). Indicated on Fig. 6-A are therefore capillary lengths L Cap at which simulated arteriolar resistances give ΔBP Art /ΔBP Cap 1.0. As an example we will use R Art 0.7 kpa s/nl. In order to obtain equal pressure drop between arterioles and capillaries L Cap must be 584 μm (Fig. 6-A). From this value of L Cap the number of capillaries per alveolus N CapPrAlv can be determined by requiring that the total pulmonary blood flow Q remains at 7.7 l/min. From L Cap and N CapPrAlv both the total capillary blood volume V Cap and the total capillary surface area A cap can be calculated. Fig. 6-B shows V Cap calculated this way. In the literature V Cap has the range of ml (Lee (1971), Lewis et al. (1958), Weibel (1986)). To obtain a simulation within this range, L Cap must be between 130 and 383 μm, as indicated by the bracket in Fig. 6-B. A similar observation can be made for A cap, which has the range of m 2 in the literature (Levitzky (2003) and Weibel (1986)). This range gives an interval for L Cap between 184 and 374 μm when R Art 0.7 kpa s/nl is used. The ranges in Fig. 6-B and 6-C are not compatible with the 584 μm estimated in Fig. 6-A. R Art 0.7 kpa s/nl is therefore not a plausible estimate for the arteriolar 5021

5 resistance. To achieve overlap between the value of L Cap estimated from Fig. 6-A and the ranges determined from Figs. 6-B and 6-C, R Art must be between 0.33 kpa s/nl and 0.52 kpa s/nl (mean kpa s/nl) and L Cap must be between 270 μm and 432 μm (mean 351 μm). This range is indicated as an overlap in Fig. 6-A. Fig. 7. A: Simulated number of capillaries per alveolus. Dot represents simulation results using R Art and L Cap 351 μm. B: ratio between the perfusion at the dependent part and the non-dependent part of the lungs. Dot represents simulation results using R Art 0.425, L Cap 351 μm and N CapPrAlv Dashed line represents simulation results from R Art 0.1 kpa s/nl. Other simulations are with R Art 0.33 kpa s/nl, kpa s/nl, 0.52 kpa s/nl and 0.7 kpa s/nl. During the capillary sensitivity analysis N CapPrAlv was determined by requiring that Q remained at 7.7 l/min by adjusting the number of capillaries per alveolus. Fig. 7-A shows the resulting values of N CapPrAlv. The value L Cap 351 μm corresponds to the middle of the overlap range, and for this value of L Cap, R Art will have the value R Art kpa s/nl. For these values of L Cap and R Art the number of capillaries per alveolus is N CapPrAlv 27.4 as indicated on Fig. 7-A. Fig. 7-B shows how Q Ratio is affected by L Cap. For L Cap 351 μm and R Art kpa s/nl, Q Ratio 4.25, which is a substantial improvement from a ratio of 6.1 for model simulations without arteriolar resistance. However, 4.25 is still higher than the ratio of 3.0 from data of Brudin et al. (1994). 6. DISCUSSION Mathematical modeling represents a valuable tool allowing quantitative modeling of the effect of the factors which determine the distributions of perfusion in human lungs. The aim of this study was to investigate how inclusion of the resistance of pulmonary arterioles in a stratified mathematical model of the pulmonary perfusion affects simulated perfusion distributions. As shown previously by Mogensen et al (2010a), the perfusion model overestimates the perfusion ratio, Q Ratio, compared to the measurements in healthy human subjects in supine position reported by Brudin et al. (1994). When a constant arteriolar resistance is included in the model, increases in arteriolar resistance leads to a flattening of the perfusion distribution curve, lowering flow mainly at the dependent part of the lungs, where pulmonary perfusion is highest, thus reducing Q Ratio. The pressure drops caused by arterioles and capillaries are assumed to be equally. Under this assumption the arteriolar resistance, capillary length and number of capillaries per alveolus were estimated. The estimated values were R Art kpa s/nl, L Cap 351 μm and N CapPrAlv These value of arteriolar resistance gives a perfusion ratio the Q Ratio 4.25 and although this remains higher than the experimentally observed ratio of 3.0 the results indicate that including an arteriolar resistance significantly improves the capability of the model to fit measured perfusion distributions. A limitation of the present model is that it does not produce the phenomenon termed zone 4. Zone 4 is the most dependent layers and in this zone many workers (e.g. Jones et al. (2001)) have reported a reduced perfusion although the zone 4 phenomenon is not clearly present in the data from Brudin et al. (1994). Arteriolar resistance is responsive to both neural and vasoactive stimuli, and plays a key role in physiologic and pathophysiologic control mechanisms of the cardiovascular system (Sobin et al. (1977)). One of the often suggested physiological mechanisms in this context is hypoxic pulmonary vasoconstriction (HPV). HPV leads to an increased vascular tone in response to hypoxia redistributing perfusion to better ventilated regions of the lungs (Nemery et al. (1983)). HPV could be included in the model, however, this would require the model to be expanded to describe also gas exchange and oxygen contents of the blood. Further expansion of the model is part of our current work where we are combining the presented perfusion model with a model of pulmonary ventilation by Steimle et al. (2010), and adding equations for describing gas exchange and the acid-base chemistry of blood. In addition to enabling description of HPV and its effect on perfusion, combining these models would allow a description of both ventilation and perfusion distributions in the lung potentially allowing an investigation of how changes in ventilation pressures and volumes will affect matching of ventilation and perfusion in the lungs. REFERENCES 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), Brudin, L.H., Rhodes, C.G., Valind, S.O, Jones, T. and Hughes J.M. (1994) Interrelationships between regional blood flow, blood volume, and ventilation in supine humans. J Appl Physiol, 76, Despopulos, A. and Silbernagl, S. (2003). Color atlas of physiology. Georg Thieme Verlag, Stuttgart, New York. Ferner, H. and Staubesand, J. (1982). Sobotta Atlas of Human Anatomy. Urban & Schwarzenberg, Boston, Burr Ridge, Il Dubuque, IA Madison, WI New York, San Francisco, ST. Louis, Bangkok, Bogotá, Caracas, Kuala Lumpur, Lisbon, London, Madrid, Mexico City, Milan, Montreal, New Delhi, Santiago, Seoul, Singapore, Sydney, Taipei, Toronto. 5022

6 Fung, Y.C. and Sobin, S.S. (1969). Theory of sheet flow in lung alveoli. J Appl Physiol, 26(4), Jones, A.T., Hansell D.M. and Evans, T.W. (2001). Pulmonary perfusion in supine and prone positions: an electron-beam computed tomography study. J Appl Physiol, 90(4), Lee, G.de.J. (1971) Regulation of the pulmonary circulation, Br Heart J, 33, Levitzky, M.G. (2003) Pulmonary Physiology Sixth edition ed. Crawsford Indiana: McGraw Hills. Lewis, B.M., Lin, T.H., Noe, F.E. and Komisaruk, R. (1958) The measurement of pulmonary capillary blood volume and pulmonary membrane diffusing capacity in normal subjects; the effects of exercise and position, J Clin Invest, 37, Liu, C.H., Niranjan, S.C., Clark Jr., J.W., San, K.Y., Zwischenberger, J.B. and Bidani A. (1998). Airway mechanics, gas exchange, and blood flow in a nonlinear model of the normal human lung. J Appl Physiol, 84(4), Lo, P., Sporring, J., Ashraf, H., Pedersen, J.J. and de Bruijne, M. (2010). Vessel-guided airway tree segmentation: A voxel classification approach Med Image Anal, (14), Lumb, A.B. (2003). Nunn s Applied Respiratory Physiology. Butterworth Heinemann, Edinburgh, London, New York, Oxford, Philadelphia, St Louis, Sydney, Toronto. Mogensen, M.L., Steimle, K.S., Karbing, D.S. and Andreassen S. (2010a). A model of perfusion of the healthy human lung. Comput Methods Programs Biomed. doi: /j.cmpb Mogensen, M.L., Thomsen, L.P., Karbing, D.S., Steimle, K.L., Zhao, Y., Rees, S.E. and Andreassen, S. (2010b). A Mathematical Physiological Model of the Dynamics of the Pulmonary Ventilation. UKACC International Conference on CONTROL 2010 September 7-10th, Coventry, UK. Nemery, B., Wijns, W., Piret, L., Cauwe, 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), Sobin, S.S., Lindal, R.G. and Bernick, S. (1977). The pulmonary arteriole. Microvascular Research, 14, Steimle, K.L., Mogensen, M.L., Karbing, D.S., Bernardino de la Serna, J. and Andreassen S. (2010). A model of ventilation of the healthy human lung. Comput Methods Programs Biomed. doi: /j.cmpb Takala, J. (2003). Pulmonary capillary pressure. Intensive Care Med, 29 (6), Wagner, W.W., Jr., Latham, L.P., Gillespie, M.N., Guenther, J.P. and Capen, R.L. (1982). Direct measurement of pulmonary capillary transit times, Science, 218, Wagner, W.W., Jr., 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, Weibel, E.R. (1986) The pathway for oxygen: President and Fellows of Harvard College. West, J.B. (1962). Regional differences in gas exchange in the lung of erect man. J Appl Physiol, 17(6),