INTRODUCTION

TOP ABSTRACT INTRODUCTION BRIEF METHODS RESULTS DISCUSSION APPENDIX REFERENCES

THE PULMONARY CAPILLARY NETWORK is a geometrically complex, distensible, and vastly interconnected structure, containing a high concentration of neutrophils available on demand for host defense. In passing from arteriole to venule, a typical blood cell encounters 40-100 segments (3, 16, 17, 38). Because the average capillary diameter is similar to or smaller than the average size of a neutrophil (34, 39), a neutrophil often encounters a capillary segment that requires it to pause and change shape before passing through (2). Consequently, in contrast to red blood cells (RBCs), which generally traverse the lung in a few seconds with little or no observable delay (29), about one-half of the neutrophils stop at least once in their passage through the lung (14, 16) and take a few seconds to >20 min to traverse the pulmonary capillary bed (24). Geometric changes in the capillary networks with changes in transpulmonary (Ptp) and transmural (transcapillary) pressure (Ptm) and the deformation of neutrophils during their transit are believed to play crucial roles in modulating neutrophil transit through, and capture within, the pulmonary microvasculature, although these effects have never been modeled. Because of their long transit times through the pulmonary capillaries, the concentration of neutrophils is >50-fold greater in the lung microvasculature than in the systemic circulation (23, 28, 31). This phenomenon, here termed neutrophil margination (not to be confused with the adhesion of neutrophils to venular walls), is thought to provide a means by which the body can maintain a reservoir of neutrophils in the lungs that can be recruited to combat infection.

Recently, our laboratory developed a discrete capillary model to describe blood flow through the pulmonary capillary bed (1). In that model, a single alveolar septum was represented by a network of interconnected capillaries arranged into a 6 × 6-square grid with a linear pressure distribution along the boundaries. While useful in many respects, this model is limited in the simulation of pulmonary blood flow by the idealized geometric arrangement in which the junction connects four perpendicular segments and by a fixed linear pressure distribution along the network boundaries. Also, although this model does account for capillary compliance to changing transcapillary pressure, it lacks the capability of simulating the effect of changes in lung volume with Ptp, which is crucial, more generally, in the evaluation of neutrophil transit time. Because of these limitations in the previous model, neutrophil transit was not considered.

In the present study, we develop new pulmonary flow and cellular transit models to address many of these complicating issues relating to RBC and neutrophil passage. The purpose of the model is both to provide further insight into existing experimental observations and to suggest new testable hypotheses concerning pulmonary microcirculatory transit. Geometrically realistic pulmonary capillary networks are generated computationally, based on measurements of alveolar septal structure. A tissue-membrane model is developed to describe the geometric changes in capillary cross section with Ptm and Ptp. We develop models for blood flow through a single capillary, an alveolar septum, and multiple septa that lie between arteriole and venule and calculate the local blood flow, pressure distribution, and hematocrit distribution. Much of the paper is devoted to model development, but several issues of physiological significance are also investigated and discussed. We simulate RBC transit and compare our predicted RBC transit times with other investigators' measurements. Capillary recruitment is also considered and compared with results from the literature. Neutrophil transit through the pulmonary microvasculature is simulated to study the effects of neutrophil-induced capillary blockage on the local resistance and pressure distribution, the delay of neutrophils during their transit, the venular aspect ratio of neutrophils, and neutrophil transit times. A sensitivity analysis is provided to show which of the various parameters exerts a strong influence on the results.

	BRIEF METHODS

TOP ABSTRACT INTRODUCTION BRIEF METHODS RESULTS DISCUSSION APPENDIX REFERENCES

A detailed description of the model and the underlying equations is presented in the APPENDIX. Here we present a brief overview and discuss the fundamental assumptions of the model.

Capillary networks. The capillary segments are placed randomly in a plane representing a single septum. Their number, length, diameter, and level of interconnectedness (represented by the number of capillary segments that join at a junction) are based on the available anatomic data. A computer algorithm randomly generates the networks that match the mean statistics of the measurements. Examples of the networks generated in this way are shown in Fig. 1. The septum is considered to be a square sheet for the purpose of these calculations, bounded by corner vessels, each of which is shared by three intersecting septae. All flow is assumed to enter the septal network at one corner and exit from the opposite one.

View larger version (45K): [in this window] [in a new window]   Fig. 1.   A: 3-dimensional structure of a computationally generated alveolar capillary network (distances in µm). B and C: top view of 2 networks generated by computer.

Capillary compliance. A new theory was developed to describe the changes in capillary cross-sectional area and shape as a function of transcapillary pressure (capillary minus alveolar gas pressure) and Ptp (or lung volume). In the model, the circumference of each capillary is assumed to be composed of two parts: one that is tethered to the septal fiber matrix and expands and contracts with changes in septal dimension (length a), and another that is relatively thin and compliant, separating capillary blood from alveolar gas (length C) (see Fig. 2). Three parameters of the model, initial lengths a₀ and C₀, and wall stiffness k_c, are selected to best fit the available experimental data of Fung and Sobin (12, 13) (see Fig. 3).

View larger version (9K): [in this window] [in a new window]   Fig. 2.   Tissue-membrane structure of a capillary cross section. Ptm, transmural pressure; a1, length of the tissue part of the circumference at Ptm > 0 for a given transpulmonary pressure (Ptp); C, length of the membrane segment of the circumference; d, distance between 2 capillaries; R and h, radius and the height, respectively, of the cross section; , angle.

View larger version (16K): [in this window] [in a new window]   Fig. 3.   Comparison between our model-predicted capillary height (a0 = 6 × 104 cm, C0 = 1.65 × 103 cm, and kc = 16 cmH2O) with Fung and Sobin's measurements of cat septal thickness at Ptp = 10 cmH2O as Ptm is varied (12, 13). See Glossary for definition of terms.

Capillary blood flow. The model for flow through a single capillary segment and its associated junctions are loosely based on a previous model from our group (1) in which the flow speed is computed based on the segment dimensions, the pressure drop, and the apparent viscosity of the blood. The latter is calculated by first determining the hematocrit of the blood and then using an empirical expression based on measured pressure drops for flow through glass pipettes. Because the hematocrit in a given segment is a function of the rate of blood flow through the entire network, this calculation is performed by an iterative procedure that ultimately converges to the distribution of blood flow, hematocrit, and pressure that satisfies the complete set of governing equations. The resistance of the junction is also taken into account on the assumption that the junctional contribution is proportional to the portion of the junction's surface area associated with a given segment.

Corner vessels. Corner vessels are treated differently because there are few data available on their dimensions or compliance. It is assumed that they remain circular and that each conveys a flow of blood equal to three times the average septal capillary blood flow. This is admittedly arbitrary but, having tried several different assumptions, we found that it has relatively little effect on the overall flow characteristics. As new data on septal flow characteristics become available, these could be incorporated into the model.

Hematocrit distribution. It was mentioned above that the apparent viscosity, and hence the pressure drop across each segment, depends on the blood hematocrit. This is determined by computing, at each junction, the division of RBCs between the two or three tributaries. For this, we use the model of Levin et al. (23) to estimate the hematocrit in each branch, based on the flow rates down the branches. Note that, for highly nonuniform distributions, when the majority of the flow passes down one of the branches, the hematocrit in the opposite branch can become zero.

Network flow model. Once the network has been constructed and the equations for compliance and flow resistance set, these can be cast into a set of equations that can be solved to yield the distributions of pressure, flow, and hematocrit throughout a given septum. The equations are those of mass conservation at each junction (the sum of the flows entering and leaving the junction equals zero) and the local relationship between blood flow rate through, and pressure drop across, each segment. The solution is based on pressures specified at the inlet and outlet to the network, alveolar gas pressure, and inlet hematocrit.

Transit time calculations. To estimate RBC transit time, we first calculate the time to pass through each individual segment and junction based on the computed flows and local geometry. It is assumed that the RBCs pass through at a speed equal to the mean speed of the blood multiplied by an empirically determined factor to account for the differences observed in transit time between the plasma and RBCs through the pulmonary capillary networks (29).

View larger version (21K): [in this window] [in a new window]   Fig. 4.   Serial connection of capillary networks (septae) simulating the pathway extending from arteriole to venule. A constant flow rate () is maintained at each connecting exit-entrance site.

Parameter values. To the extent possible, values for the model parameters were obtained from sources in the literature as indicated in Table 1. Values that could not be obtained directly were estimated. The effects of small variations in several of the more critical parameters are shown in the sensitivity analysis in Table 2.

	RESULTS

TOP ABSTRACT INTRODUCTION BRIEF METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Changes in capillary height with pressures. To evaluate this tissue-membrane model of the capillary cross section, our predicted relationship between capillary height (h) and Ptm was compared with Fung and Sobin's (12, 13) measurements of cat septal thickness at Ptp = 10 cmH₂O over a range of Ptm. Figure 4 demonstrates that the curve predicted by the tissue-membrane model, with parameter values a₀ = 6 × 10⁴ cm, C₀ = 1.65 × 10³ cm, and k_c = 16 cmH₂O, agrees very well with Fung and Sobin's experimental measurements. For the simulations presented in this paper, we chose to use slightly different values, a₀ = 6 × 10⁴ cm, C₀ = 1.5 × 10³ cm, and k_c = 15 cmH₂O. To generate the initial capillary dimensions, a <30% variation in these parameters around their mean values at the different locations of the network was randomly chosen.

Local volumetric flow rate. Solving Eqs. 23-25 in the APPENDIX, one can obtain the volumetric flow rate in each segment of a network generated by the methods described in Capillary networks in BRIEF METHODS. Figure 5 shows the flow rate distributions in two different capillary networks. Blood flows into the capillary networks from the lower left corner and out from the upper right corner. The arrows indicate the direction of flow and magnitude of the flow rate. These results are similar in character to those discussed at greater length by Dhadwal et al. (1) and are, consequently, not examined further here. Note, however, that, if any segment is blocked, the flow pattern will be changed. In contrast to the result of Dhadwal et al., who used a rectangular grid, blockage of one segment in this more realistic geometric model can, in some cases, even change the direction of flow.

View larger version (13K): [in this window] [in a new window]   Fig. 5.   A and B: flow rate distributions in the 2 networks of Fig. 1, B and C, respectively. Blood flows into the capillary networks from the lower left corner and flows out from the upper right corner. Entrance pressure is 10 cmH2O, and exit pressure is 9.1 cmH2O. The length of each arrow is proportional to the magnitude of the flow rate. The entrance flow rates are 8.3 × 104 µm3/s for the network in A and 5.8 × 104 µm3/s for the network in B.

Hematocrit distribution. The hematocrit distribution in the network with no segments blocked was examined by using a value at the entrance of 0.4. In the model, the nonuniform distribution arises due to the partitioning of RBCs at a bifurcation, with the segments receiving higher flows also having an increased hematocrit. Figure 6A shows that the hematocrits differ markedly in different capillaries, primarily as a result of variations in local flow rate caused by the combination of variations in segment diameter and network geometry. Some segments are almost void of RBCs, although plasma continues to flow, giving rise to hematocrits approaching zero. Figure 6B shows the hematocrit distribution in the same network but with one segment blocked. This blockage results in a significant change in the hematocrit distribution, especially in the vicinity of the blockage. Figure 7, A and B, gives the frequency distribution of hematocrits for the segments, which correspond to Fig. 6, A and B, respectively.

View larger version (56K): [in this window] [in a new window]   Fig. 6.   Hematocrit distributions in the network of Fig. 1B with no segments blocked (A), and with 1 segment blocked (X; B). The hematocrit at the entrance is 0.4. Scale indicates hematocrit ranging between 0 and 0.45.

View larger version (14K): [in this window] [in a new window]   Fig. 7.   Percentage of segments having different hematocrit values. A and B correspond to Fig. 6, A and B, respectively.

Pressure distribution. An estimate of the pressure drop across a capillary segment is critical for understanding neutrophil transit and calculating its transit time. For a neutrophil to enter a narrow segment, the pressure difference across it must be greater than the "critical pressure," which is here assumed to be that sufficient to draw a neutrophil into a micropipette of the same diameter (6). In many cases, the pressure drop across a patent capillary segment is insufficient to push a neutrophil through. The increase in segmental pressure drop due to blockage by a neutrophil, however, is often sufficient to raise the driving pressure above the critical level, thus allowing the neutrophil to pass. It is, therefore, crucial to evaluate the new pressure distribution in the network when a capillary is blocked.

View larger version (58K): [in this window] [in a new window]   Fig. 8.   Pressure distribution in the capillary network without (A) and with (B) a single segment blocked (X). The pressures are 10 cmH2O at the network entrance and 9.1 cmH2O at the network exit. Scale bar is pressure in cmH2O.

View larger version (56K): [in this window] [in a new window]   Fig. 9.   Pressure drop across each segment. A and B correspond to Fig. 8, A and B, respectively. Scale bar is pressure in cmH2O.

Capillary recruitment. Presson et al. (30) investigated the stability of capillary segmental opening in single alveolar septae by observing the reproducibility of the perfusion pattern when the same perfusion pressure was repeatedly applied to the networks. Their results show that, although there is variability in which pathways open, it is not a random process. The observation of nonrandom capillary recruitment suggests that the opening of a capillary is largely dependent on its initial diameter and the pressure distributions in the septum. Variability of the perfusion pattern in a septum is probably due to the changes in local pressure and flow pattern. For example, when a neutrophil blocks one segment, flow and pressure distributions in the network will be altered accordingly. In a separate study, Godbey et al. (15) evaluated the effect of capillary pressure and lung distension on capillary recruitment and found that capillaries in the excised lobes opened over a narrow capillary pressure range at low airway pressure, whereas, at higher airway pressures, recruitment occurred over a wider range of pressures.

View larger version (16K): [in this window] [in a new window]   Fig. 10.   Percentage of the capillaries perfused at different capillary pressures. Ptp values are 2 and 10 mmHg.

RBC transit. Presson and colleagues (29) showed that RBCs travel from arteriole to venule in 4.1 ± 0.4 s, which includes ~0.56 s during which the RBC resides in the arteriolar or venular networks. Capillary transit time is, therefore, estimated from these measurements to be ~3.5 s. Other results obtained with radiolabeled RBCs suggest mean transit times of 3.0 s (18) in resected lobes of human lungs. We calculated the RBC transit time using our multiple network model described above and in the APPENDIX and compared our predictions with the measurements of Presson et al. (29). For the network shown in Fig. 1B, there are 27 major pathways (entrance to exit), which transport >99% of the RBCs. Of these, five pathways convey >50% of the flow; these are the "preferred pathways" described in Ref. 1 and observed in Ref. 27. Figure 11A, left and right, shows the RBC transit time distributions after passage through one and six networks, respectively. Network 1 connects to the arteriole, and network 6 connects to the venule. The Ptp used in this calculation is 5 cmH₂O. The Ptm values are set to 10 cmH₂O in the arteriole and 2 cmH₂O in the venule. The entrance-to-exit pressure drops are 0.9, 1.0, 1.1, 1.3, 1.6, and 2.1 cmH₂O, reflecting the progressive fall in internal pressure (and, therefore, capillary cross-sectional area) from arteriole to venule. This calculation yields a mean value of 2.6 s for the mean RBC transit time through this network.

View larger version (20K): [in this window] [in a new window]   Fig. 11.   Red blood cell (RBC) transit time distributions. A: Ptp = 5 cmH2O and Ptm in the arteriole = 10 cmH2O. B: Ptp = 3 cmH2O and Ptm in the arteriole = 10 cmH2O. C: Ptp = 5 cmH2O and Ptm in the arteriole = 14 cmH2O. The pressure drop from the arteriole to the venule for all cases is 8 cmH2O. Left: distributions after passage through 1 of 6 networks in Fig. 3; right: distributions after passage through all 6 networks. All RBCs enter the first network at time 0.

View larger version (13K): [in this window] [in a new window]   Fig. 12.   RBC transit time distributions before and after passage through 6 networks (Fig. 3). RBC concentration at the arteriole is as observed in Ref. 29. Ptp is 5 cmH2O, and the Ptm in the arteriole is 10 cmH2O. Pressure drop from arteriole to venule is 8 cmH2O. Vertical scale is determined such that the integral under each curve is unity.

Neutrophil stops during transit. In an attempt to be consistent with the interpretation of in vivo experiments observing neutrophil motion in capillaries, a neutrophil was defined as "stopped" if it required 1 s to enter a segment. Using a similar definition, Gebb et al. (14) observed that 46% of neutrophils pass through the capillary network from arteriole to venule without stopping. As shown in Fig. 13A, our model predicts that 46% of neutrophils pass through without stopping, 51% stop once, and 3% stop at least twice. In Fig. 13A, the Ptp is 5 cmH₂O, Ptm in the arteriole is 10 cmH₂O, and the arteriolar-to-venular pressure drop is 8 cmH₂O. Figure 13B shows the effects on neutrophil transit time of reducing Ptp from 5 to 3 cmH₂O, whereas Fig. 13C shows the effects of increasing arteriolar Ptm from 10 to 14 cmH₂O. Comparing Figs. 13B and 13A, decreasing Ptp had little effect on the number of stops that neutrophils made. However, comparing Figs. 13C and 13A, increasing arteriolar pressure caused many more neutrophils to pass through the capillary networks without stopping because of the relatively large capillary diameters at the higher Ptm values.

View larger version (10K): [in this window] [in a new window]   Fig. 13.   Predicted no. of times a neutrophil stops during its transit from arteriole to venule. A: Ptp = 5 cmH2O, Ptm in the arteriole = 10 cmH2O; B: Ptp = 3 cmH2O, Ptm in the arteriole = 10 cmH2O; C: Ptp = 5 cmH2O, Ptm in the arteriole = 14 cmH2O. Pressure drop from arteriole to venule is 8 cmH2O for all 3 cases.

Nondeformable microsphere transit through networks. It has been assumed in these calculations that, if the neutrophil has a greater cross-sectional area than the capillary segment, it is delayed while it deforms to fit through and then retains that reduced cross section for the duration of its trip through the capillary network. A microsphere is rigid, however, and will be arrested when it first encounters a capillary with a major or minor diameter smaller than its own, even if the cross-sectional area of the capillary happens to be greater. Moreover, once the microsphere stops, it remains so because, unlike a neutrophil, it cannot deform under such low pressures. Therefore, the minimum diameter rather than the area of the capillary cross section determines microsphere passage.

View larger version (25K): [in this window] [in a new window]   Fig. 14.   Predicted percentages of microspheres [with diameters of 3.08 µm (solid bars), 4.62 µm (hatched bars), and 5.85 µm (open bars)] that pass through different nos. of septa. As in Fig. 12, A: Ptp = 5 cmH2O, Ptm in the arteriole = 10 cmH2O; B: Ptp = 3 cmH2O, Ptm in the arteriole = 10 cmH2O; C: Ptp = 5 cmH2O, Ptm in the arteriole = 14 cmH2O. Pressure drop from arteriole to venule is 8 cmH2O for all 3 cases.

Final aspect ratios of neutrophils. The study by Gebb et al. (14) shows that 94% of neutrophils in arterioles have aspect ratios 1.25, which means that they are nearly circular (aspect ratio of a circle is 1.0) as they reach the pulmonary capillaries from the feeding pulmonary arterioles. Passage through the pulmonary capillaries elongates the neutrophils, which is measurable in the first postcapillary venules. Gebb and colleagues showed that, in venules, only 47% of neutrophils have aspect ratios 1.25, 17% are between 1.25 and 1.5, 10% are between 1.5 and 1.75, 10% are between 1.75 and 2.0, and 16% are elongated with aspect ratios >2. Figure 15A shows our predictions for the final neutrophil aspect ratio distribution after passage through six septal networks, which gives 46% of neutrophils with aspect ratios 1.25, 35% with the aspect ratios between 1.25 and 1.5, 8.5% with the aspect ratios between 1.5 and 1.75, and 9.5% with aspect ratios between 1.75 and 2. These results compare favorably with those of Gebb et al. As in previous sections, we also provide Fig. 15, B and C, to illustrate the effects of Ptp and Ptm on the final aspect ratios of neutrophils. Similar to the observations of neutrophil stops in Fig. 13, decreasing Ptp from 5 to 3 cmH₂O had little effect on the aspect ratio, whereas increasing arteriolar Ptm from 10 to 14 cmH₂O resulted in many more circular neutrophils.

View larger version (11K): [in this window] [in a new window]   Fig. 15.   Predicted final aspect ratio distribution of neutrophils after passage through 6 networks from arteriole to venule. As in Fig. 13, A: Ptp = 5 cmH2O, Ptm in the arteriole = 10 cmH2O; B: Ptp = 3 cmH2O, Ptm in the arteriole = 10 cmH2O; C: Ptp = 5 cmH2O, Ptm in the arteriole = 14 cmH2O. Pressure drop from arteriole to venule is 8 cmH2O for all 3 cases.

Neutrophil transit time. Neutrophil transit times are more difficult to evaluate than RBC transit time, in part because there is some ambiguity in determining how a neutrophil will change its shape when it enters a capillary with an elliptical cross section or what should be considered the critical effective diameter of a capillary. We took two approaches to calculate the effective diameter of the segment: 1) using the diameter of a circle with the same cross-sectional area as the segment as described in the APPENDIX, and 2) using the h as the effective diameter, which results in a smaller cross-sectional area than that of the original elliptical cross section. These two approaches represent approximate maximum and minimum effective segment diameters. In both simulations, the Ptp is 5 cmH₂O, the Ptm in the arteriole is 10 cmH₂O, and the pressure drop from arteriole to venule is 8 cmH₂O. Figs. 16 and 17 give the neutrophil transit time distributions after passage through one and six capillary networks using the cross-sectional area (Fig. 16) or h (Fig. 17) to determine the effective diameter. When cross-sectional area is used, all neutrophils pass through the entire network in <20 s (Fig. 16B). When h is used, 2.4% of neutrophils take longer than 1 min to pass, and 5.5% become entrapped and never pass through (Fig. 17B).

View larger version (14K): [in this window] [in a new window]   Fig. 16.   Predicted neutrophil transit time distribution after passage through 1 (A) and 6 (B) capillary networks. Effective capillary diameter is determined by its original cross-sectional area. Ptp = 5 cmH2O, Ptm in the arteriole = 10 cmH2O. Pressure drop from the arteriole to the venule is 8 cmH2O.

View larger version (17K): [in this window] [in a new window]   Fig. 17.   Predicted neutrophil transit time distribution after passage through 1 (A) and 6 (B) capillary networks. Effective diameter is determined by the capillary height. Ptp = 5 cmH2O, Ptm in the arteriole = 10 cmH2O. Pressure drop from the arteriole to the venule is 8 cmH2O. X, neutrophils that become trapped and do not pass through.

Sensitivity analysis. One of the useful functions of a model is that it allows one to ascertain which of the various features of the real system exert a significant effect on its behavior or, conversely, which features are largely irrelevant. We have selected several parameters from the many in Table 1 that are of particular interest and performed a sensitivity analysis varying each by ±10%. The results are summarized in Table 2 in terms of the parameters of greatest interest, namely, overall flow, RBC transit time, and neutrophil transit time. Of the parameters varied, the mean h exhibits the strongest influence, although neutrophil radius is also influential in determining neutrophil transit times. Thus it appears that the dimensions of the structure and the cells are most critical and that the effects of compliance, either in the model for capillary distension or the effects on capillary dimension arising from variations in pressure, are secondary.

	DISCUSSION

TOP ABSTRACT INTRODUCTION BRIEF METHODS RESULTS DISCUSSION APPENDIX REFERENCES

In this study, we developed a unique method to computationally generate geometrically realistic pulmonary capillary networks that have complex interconnected structures. The important ideas and methods developed here can also be used in other physiological network generations. Based on these structurally realistic capillary beds, we developed a computational model of pulmonary blood flow, which was used to simulate RBC and neutrophil transit through the pulmonary microvasculature. Using this model, we can estimate blood flow, distribution of hematocrit within the capillary segments, distribution of pressure gradients through the segments, capillary blockage, and cellular transit, as well as evaluate the effects of Ptp and Ptm on network structure, flow behavior, and cellular transit. The model can provide insight into the bases for previous observations and suggest new experiments to broaden our understanding of these critical phenomena.

Fung and Sobin (9, 12, 13) developed a model of blood flow in the pulmonary capillaries in which blood flow is viewed as passing between two compliant sheets, with the effects of individual capillary geometry and the two-phase nature of blood represented by an increase in the effective resistance to flow. It was reassuring to us that, despite the completely different basis for our model, which is based on the interconnected tubular anatomy, we found strong agreement between the two approaches in cases in which a direct comparison could be made (e.g., in the relationship between h and the Ptm). In terms of its underlying basis, the present model is more closely related to other models of the systemic microcirculation (28, 31, 36, 37). The main differences lie in the structure of the microvascular beds, with the pulmonary capillaries being much shorter and more highly interconnected, factors that account for neutrophil margination. The model confirmed and extended the results of others, including our laboratory's previous model, which did not incorporate the anatomy of the capillary bed to the degree to which the present model does (1), namely that blood flow, hematocrit, and pressure gradients along the capillary segments are highly variable between segments. This variability leads to the formation of preferential pathways within the alveolar septa, due to variability in resistance to blood flow and to the geometric arrangement of the capillary bed.

This study also evaluated the effect of blocking a capillary segment on the flow, hematocrit distribution, and the distribution of pressure gradients along segments. When neutrophils reach a capillary segment that is smaller than their diameter, they temporarily obstruct the segment while they deform and elongate. This blockage has been suggested to lead to changes in blood flow and pressure gradients that may facilitate their passage through the pulmonary capillaries. This study demonstrates that neutrophils temporarily blocking segments while undergoing deformation are very likely to induce large changes in the distribution of blood flow among segments, in the variability of hematocrit, and in the distribution of pressure gradients. In particular, the pressure gradient across a capillary segment is thought to be the major driving force for neutrophils to deform and pass through segments that are narrower than neutrophils when spherical. In many cases, the pressure gradient across unobstructed capillary segments is insufficient to cause neutrophil deformation and passage. However, an obstructing neutrophil appears able to increase the local pressure gradient by 100-300%, to a level that facilitates its passage through the segment. Thus relatively small changes in physiological parameters, for example, arteriolar pressure, can significantly alter the lung's tendency to capture or release neutrophils.

The results of our studies of pulmonary capillary RBC transit times show that the computed transit times closely mimic the RBC transit times obtained during in vivo studies using video microscopy. This comparison suggests that this model may simulate the realities of capillary blood flow within the lungs. Our studies extend previous work to demonstrate that both decreasing Ptp by 2 cmH₂O and increasing pulmonary arterial pressure by 4 cmH₂O each lead to decreased RBC transit times to similar degrees. They suggest that different mechanisms may underlie these changes in resistance and blood flow; changes in alveolar size, particularly shortening the capillary segments, underlie the decreased RBC transit times induced by decreased Ptp, whereas increased diameters may be more important when pulmonary arterial pressures are increased. In that no experimental data exist to either confirm or refute these predictions, they constitute a testable hypothesis that can be used to further assess the validity of the model and, if inconsistencies are found, to suggest ways in which the model might be made more realistic.

It should be noted that our predictions of RBC transit are based on estimates using the expression for viscosity obtained by Kiani and Hudetz (20) from in vitro measurements. However, when a different equation is used, based on in vivo measurements by Pries et al. (32), blood flow rates are markedly reduced and the transit times for RBCs and neutrophils are consequently increased. In fact, the estimates of RBC transit times are considerably longer than those observed in vivo. This raises important issues concerning the nature of blood-endothelial interactions in the lung microcirculation, compared with the systemic microcirculation, that need to be addressed by further experiments in vivo.

Although the model, in its present form, considers flows at a single, static lung volume, the results can also be used to infer the behavior during breathing. Because the capillaries are relatively rigid compared with the neutrophil, as lung volume changes with each breath, the capillary dimensions will change accordingly with little effect due to the presence of neutrophils. As a consequence, neutrophils encountering a narrow capillary segment near end inspiration will have a slightly easier time passing than those arriving at end expiration. However, even if it is trapped at a time of low lung volume, a neutrophil may be released as lung volume increases. Large excursions in lung volume would have a more noticeable effect, potentially causing a surge of neutrophil release at the end of a deep inspiration. These predictions suggest important studies to be pursued in vivo.

Finally, this paper presents the first simulated results for neutrophil transit through the pulmonary capillaries, incorporating flows, resistances, pressure gradients, and transit times. The agreement between the computed predictions and the experimental observations is excellent for the percentage of neutrophils that do not stop and the final aspect ratio of neutrophils after passage through the capillary bed. Interestingly, with the use of the same model and the same parameters as for RBC transit, the percentage of neutrophils that stop and their final aspect ratio do not change with a 2-cmH₂O decrease in Ptp, in contrast to the predicted decrease in RBC transit times. These observations also suggest that capillary length is a less important determinant of neutrophil transit than pressure drops or capillary segment diameters. The observation that a 4-cmH₂O increase in pulmonary arterial pressure does decrease neutrophil transit supports the importance of capillary diameters in this process.

Reliable estimates of neutrophil transit times are difficult to make, as there are few data on which to base predictions of neutrophil deformation. In particular, estimating the diameter (or the cross-sectional shape) to which the neutrophil must deform is not possible based on current knowledge. In addition, the in vivo data examining neutrophil transit times are limited and, with only one exception, are not available as a transit time distribution, as in the case of RBCs. There have been several estimates of the average transit times for neutrophils, however, to which our predictions can be compared. We used two approaches that are based on the minimum and the maximum limits to which the neutrophil must deform. When a neutrophil is assumed to fill the whole cross-sectional area as it enters a narrow capillary with an elliptical cross section, the transit times are shorter than predicted, based on the available literature. When the neutrophil is assumed to remain circular in cross section (perpendicular to the capillary axis) and to deform until its diameter is equal to the height of the capillary segment, the transit times are long, that is, somewhat longer than the experimental values. Furthermore, very few neutrophils pass through the network without stopping when this assumption is made. These results can be compared with those of Lien and co-workers (24), who determined the distribution of transit times in subplural capillary beds using videomicroscopy. Their distribution of transit times differed from those presented in Fig. 17 in that a higher fraction of neutrophils in the experiment passed through in very short (<2 s) times. Their median transit time of 26.1 s (mean, 366 s) was, however, quite close to that of our second model (median, 20.1 s). Mean neutrophil transit times have also been determined, by the use of radiolabeled neutrophils, to be in the range of 190 s (mean) (18) and have also been reported to average 5.4 times that required for RBC transit. The first two measurements lie closer to our estimates when the segment height is used rather than the cross-sectional area (Fig. 17), although there is some ambiguity in how our infinite time blockages should be treated in computing mean transit times. These results do demonstrate, however, that the shape of the capillary cross section can be nearly as important as its cross-sectional area in determining its passage characteristics. It seems likely that, after entering a narrow capillary with an elliptical cross-sectional area, the neutrophil will change its cross-sectional shape to be an ellipse whose minor axis equals the h and whose major axis is larger than the h and smaller than the capillary width. This would lead to predicted transit times between the two cases depicted in Fig. 17 and somewhat shorter than the reported measurements.

These simulations reveal the key roles played by structures and mechanical properties of capillaries and neutrophils in modulating neutrophil transit and lead to a better understanding of the mechanisms of blood flow and cellular transit through the lungs. They also demonstrate the complexities of neutrophil transit through the lungs and point out a role for neutrophils in regulating the flow patterns of RBC and plasma, which has been poorly described to date. Finally, they point out the importance of the nature of the shape change that occurs in neutrophils in preparation for their transit through the many narrow capillary segments of the lungs.