Abstract
A computational model of the pulmonary microcirculation is developed and used to examine blood flow from arteriole to venule through a realistically complex alveolar capillary bed. Distributions of flow, hematocrit, and pressure are presented, showing the existence of preferential pathways through the system and of large segmenttosegment differences in all parameters, confirming and extending previous work. Red blood cell (RBC) and neutrophil transit are also analyzed, the latter drawing from previous studies of leukocyte aspiration into micropipettes. Transit time distributions are in good agreement with in vivo experiments, in particular showing that neutrophils are dramatically slowed relative to the flow of RBCs because of the need to contract and elongate to fit through narrower capillaries. Predicted neutrophil transit times depend on how the effective capillary diameter is defined. Transient blockage by a neutrophil can increase the local pressure drop across a segment by 100–300%, leading to temporal variations in flow and pressure as seen by videomicroscopy. All of these effects are modulated by changes in transpulmonary pressure and arteriolar pressure, although RBCs, neutrophils, and rigid microspheres all behave differently.
 simulation
 hemodynamics
 pulmonary circulation
 neutrophil margination
 microcirculation
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 onehalf 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 >50fold 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 × 6square 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 tissuemembrane 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 neutrophilinduced 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
A detailed description of the model and the underlying equations is presented in the . 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.
Capillary compliance.
A new theory was developed to describe the changes in capillary crosssectional 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 (lengtha), 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 a _{0} andC _{0}, and wall stiffnessk _{c}, are selected to best fit the available experimental data of Fung and Sobin (12, 13) (see Fig.3).
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).
The time required for a neutrophil to stop and deform to a shape that would allow it to pass through a segment of a given dimension must also be considered. In addition, the pressure distribution in the entire network must be recomputed each time a neutrophil becomes lodged in a narrow segment to account for the subsequent redistribution of flow. The deformation and entry times were based on a combination of previous theoretical and experimental results. For small deformations (values of neutrophil radius to segment radius close to 1), we used the result from the linearized theory of Yeung and Evans (41) and for larger ratios, the experiments of Fenton et al. (8). The resulting expression for entrance time provided for a smooth transition between these two. It should be noted, however, that these results are all based on capillaries of circular cross section and need to be used with caution for capillaries that may have highly noncircular cross sections, such as those in the lung. This issue is discussed more at a later point. In addition to the entry times, the passing time through the segment and junction is also computed. For this purpose, we develop a model based on the prediction of Fenton et al. for the ratio of neutrophil velocity to bulk blood velocity in a capillary of circular cross section. The total time for entry and passage is summed for all segments traversed by a particular neutrophil and reported as the total transit time.
To simulate the transit times measured in vivo more realistically, it was necessary to link several septal networks together. We did this in serial fashion, letting all of the blood flow from one septum enter the next and so forth through to the venular end (Fig.4). We chose to use a number of septae consistent with the number of segments encountered by a typical neutrophil on its passage through the lung as estimated by videomicroscopy (3, 16, 17).
Parameter values.
To the extent possible, values for the model parameters were obtained from sources in the literature as indicated in Table1. 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
Changes in capillary height with pressures.
To evaluate this tissuemembrane 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 cmH_{2}O over a range of Ptm. Figure 4 demonstrates that the curve predicted by the tissuemembrane model, with parameter valuesa _{0} = 6 × 10^{−4} cm,C _{0} = 1.65 × 10^{−3} cm, andk _{c} = 16 cmH_{2}O, agrees very well with Fung and Sobin's experimental measurements. For the simulations presented in this paper, we chose to use slightly different values,a _{0} = 6 × 10^{−4} cm,C _{0} = 1.5 × 10^{−3} cm, andk _{c} = 15 cmH_{2}O. 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 , one can obtain the volumetric flow rate in each segment of a network generated by the methods described in Capillary networks inbrief 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.
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. Figure6 A 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 6 B 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. Figure7, A and B, gives the frequency distribution of hematocrits for the segments, which 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.
Figure 8, A and B, shows the pressure distribution in the capillary network with and without a segment blocked, respectively. For these calculations, pressures are 10 cmH_{2}O at the network entrance and 9.1 cmH_{2}O at the network exit. Figure9, A and B, corresponding to Fig. 8, A and B, gives the pressure drop across each segment. In this particular case, the pressure drop increases from 0.107 to 0.237 cmH_{2}O when the segment is blocked by a neutrophil. A single capillary obstruction typically results in a 100–300% increase in the pressure drop across the blocked segment. This estimate is considerably higher than previous estimates of ∼60% (1). The increase is primarily due to a more accurate representation of network geometry; in the present model, there are fewer capillary segments per septum, and, therefore, blockage of one induces a greater change in the blood flow distribution.
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.
Using our tissuemembrane structure model of capillary cross section, we simulated nonrandom capillary recruitment in alveolar septa. We assumed that the h distribution in a septum is in the range of a mean value ±1 μm. We assumed that all values of height within this range were equally probable. When Ptp or capillary pressure is changed, the crosssectional shape of the capillary will change. If theh calculated using our model is not <2.7 μm, which is the minimum capillary diameter that a RBC can pass through, we consider it opened and capable of perfusion. Figure10 shows the percentage of perfused capillaries at different Ptp and capillary pressures. The parameter values used here are a _{0} = 6 μm,C _{0} = 12 μm, andk _{c} = 30 cmH_{2}O. Our results show that, for Ptp = 2 mmHg, ∼50 and 97% of capillaries are perfused at 3 and 8mmHg capillary pressure, respectively, and for Ptp = 10 mmHg, ∼10 and 100% of capillaries are perfused at 14 and 22mmHg capillary pressure, respectively. These results approximate those of Godbey et al. (15), who found that recruitment occurred over a range of capillary pressures from 2 to 10 mmHg when Ptp = 2 mmHg, and from 10 to 30 mmHg when Ptp = 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 and compared our predictions with the measurements of Presson et al. (29). For the network shown in Fig. 1 B, 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. Figure11 A, leftand 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 cmH_{2}O. The Ptm values are set to 10 cmH_{2}O in the arteriole and 2 cmH_{2}O in the venule. The entrancetoexit pressure drops are 0.9, 1.0, 1.1, 1.3, 1.6, and 2.1 cmH_{2}O, reflecting the progressive fall in internal pressure (and, therefore, capillary crosssectional area) from arteriole to venule. This calculation yields a mean value of 2.6 s for the mean RBC transit time through this network.
For the purpose of this calculation, we assumed that all RBCs enter the first network from the arteriole at time 0, whereas in experimental measurements the injected bolus of labeled RBCs enters over a brief but finite period of time. If we simulate the experimental RBC distribution at the entrance (29), the distribution of RBC transit times through the series of six networks is shown in Fig.12. Both this distribution and the mean RBC transit time of 3.1 s agree closely with the measurements of Presson et al. (29).
To evaluate the effects on RBC transit of changes in Ptp during breathing and the effects of different Ptm values, calculations were performed with an arteriolar Ptm of 10 cmH_{2}O, dropping to 2 cmH_{2}O in the venule, but keeping Ptp fixed at 3 cmH_{2}O (Fig. 11 B). Ptp was then fixed at 5 cmH_{2}O with the arteriolartovenular pressure drop still set at 8 cmH_{2}O, but the Ptm in the arteriole was raised to 14 cmH_{2}O (Fig. 11 C). In both cases, RBCs pass through the networks more quickly than at higher Ptp or lower arteriolar pressure (Fig. 11 A). This is because decreasing the Ptp reduces the septal dimension (thereby shortening the RBC pathway length), and increasing the Ptm will increase capillary crosssectional area and, therefore, decrease the resistance to RBC flow and increase flow velocities.
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.13 A, our model predicts that 46% of neutrophils pass through without stopping, 51% stop once, and 3% stop at least twice. In Fig. 13 A, the Ptp is 5 cmH_{2}O, Ptm in the arteriole is 10 cmH_{2}O, and the arteriolartovenular pressure drop is 8 cmH_{2}O. Figure13 B shows the effects on neutrophil transit time of reducing Ptp from 5 to 3 cmH_{2}O, whereas Fig. 13 C shows the effects of increasing arteriolar Ptm from 10 to 14 cmH_{2}O. Comparing Figs. 13 B and 13 A, decreasing Ptp had little effect on the number of stops that neutrophils made. However, comparing Figs. 13 C and13 A, 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.
Nondeformable microsphere transit through networks.
It has been assumed in these calculations that, if the neutrophil has a greater crosssectional 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 crosssectional 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.
We simulated the transit of microspheres with diameters of 3.08, 4.62, or 5.85 μm. Figure 14,A–C, shows the percentages of the sizes of microspheres that can pass through the same three conditions of Ptp and Ptm as in Fig. 13, A–C, respectively. These results show that all of the 3.08μm and none of the 5.85μm microspheres passed through six networks under all circumstances, although the distribution of trapping within the network differed. This can be compared with the experimental observations of Wiggs and colleagues (40), where 86% of the 3.08μm beads and 15% of the 5.85μm beads passed through the lungs (14 ± 2 and 85 ± 9% retention within the lungs of 3.08 and 5.85μm microspheres, respectively).
In contrast, a large fraction (6–56%, depending on the conditions, Fig. 14) of microspheres that measure 4.62 μm in spherical diameter pass through the six networks. These data suggest that neutrophil deformability is critical in determining the transit time and/or that the crosssectional area rather than the minimum diameter is the major determinant of neutrophil transit. In addition, the transit of the 4.62μm microspheres depended on the physiological conditions. At Ptp of 5 cmH_{2}O and arteriolar Ptm of 10 cmH_{2}O, 6% of the 4.62μm microspheres and 46% of the neutrophils passed through without stopping; at Ptp of 3 cmH_{2}O and arteriolar Ptm of 10 cmH_{2}O, 28% of the 4.62μm microspheres and 42% of the neutrophils passed through without stopping; and at Ptp of 5 cmH_{2}O and arteriolar Ptm of 14 cmH_{2}O, 56% of the 4.62μm microspheres and 86% of the neutrophils passed through six networks without stopping. The increase in microsphere but not neutrophil passage with decreasing Ptp suggests that decreasing the size of the alveoli (and septae) may make the capillaries more circular in cross section without increasing their crosssectional area. The greater percentage of neutrophils than microspheres that pass through the network at higher pulmonary arteriolar pressures is also likely due to the effect of crosssectional shape. We acknowledge, though, that the complexity of the changes in diameter and pressure gradient across the segments that occurs on changing even one parameter weakens our conclusion, but the computational studies do lead to a testable hypothesis.
In another microsphere study, Ring et al. (33) observed that ∼0–20% of microspheres with diameter ≥8 μm could pass through the pulmonary circuit at higher capillary pressures. Our simulation of this experiment yielded 12% passage through the six networks at a mean transcapillary pressure of 18 mmHg (25 cmH_{2}O), consistent with the results of Ring et al. The differences between these results and those described above are attributable to the larger crosssectional areas at the higher capillary pressures.
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. Figure15 A 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 cmH_{2}O had little effect on the aspect ratio, whereas increasing arteriolar Ptm from 10 to 14 cmH_{2}O resulted in many more circular neutrophils.
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 crosssectional area as the segment as described in the , and 2) using theh as the effective diameter, which results in a smaller crosssectional 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 cmH_{2}O, the Ptm in the arteriole is 10 cmH_{2}O, and the pressure drop from arteriole to venule is 8 cmH_{2}O. Figs. 16 and17 give the neutrophil transit time distributions after passage through one and six capillary networks using the crosssectional area (Fig. 16) or h (Fig. 17) to determine the effective diameter. When crosssectional area is used, all neutrophils pass through the entire network in <20 s (Fig.16 B). 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. 17 B).
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 2in 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.
In the last row of Table 2, we consider the effect of using a different model for the viscosity of whole blood, one by Pries et al. (32) developed based on observations in vivo. This presumably takes into account factors present in vivo but not in glass capillary tubes and more than doubles the transit time for both RBCs and neutrophils.
DISCUSSION
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 twophase 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 cmH_{2}O and increasing pulmonary arterial pressure by 4 cmH_{2}O 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 bloodendothelial 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 2cmH_{2}O 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 4cmH_{2}O 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 crosssectional 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 crosssectional 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 coworkers (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 crosssectional 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 crosssectional area in determining its passage characteristics. It seems likely that, after entering a narrow capillary with an elliptical crosssectional area, the neutrophil will change its crosssectional 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.
Acknowledgments
This research was supported by National Heart, Lung, and Blood Institute Grants HL33009 and HL48160.
Footnotes

Address for reprint requests and other correspondence: R. D. Kamm, MIT, Rm. 3–260, 77 Massachusetts Ave., Cambridge, MA 02139 (Email: rdkamm{at}mit.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
 Copyright © 2001 the American Physiological Society
Appendix
Glossary
 a
 Length of tissue portion of the circumference of a capillary cross section at zero Ptm
 a*
 Major axis of an elliptical capillary cross section
 A_{c}
 Local crosssectional area of capillary segment
 A_{s}
 Average septal area
 a_{0}
 Length of tissue portion of the circumference of a capillary cross section at zero Ptp and zero Ptm
 a_{1}
 Length of tissue portion of the circumference of a capillary cross section at nonzero Ptp and Ptm
 b
 Parameter in the hematocrit distribution model
 b*
 Minor axis of an elliptical capillary cross section
 C
 Length of membrane segment of the circumference of a capillary cross section
 c_{i} _{,j}
 Percentage of RBCs passing through the jth pathway of the ith network
 C_{i} _{,m}
 RBC fraction passing through the mth pathway from the entrance of the first network to the exit of ith network
 C_{tt}
 Ratio of RBC transit time to plasma transit time in capillaries
 C_{0}
 Length of membrane segment of the circumference of a capillary cross section at zero Ptm
 d
 Distance between two capillaries
 D
 Diameter of a circular capillary cross section
 D_{cv}
 Diameter of corner vessel
 D_{h}
 Hydraulic diameter
 D_{m}
 Diameter of the smallest vessel that a RBC can pass through
 D_{mean}
 Mean hydraulic diameter of the capillaries
 E
 Elastic coefficient of septal wall
 f_{d}
 Darcy friction factor
 h
 Height of capillary cross section
 H_{d}
 Vessel hematocrit
 H_{d(out)i}
 Hematocrit in the ith daughter vessel
 k_{c}
 Stiffness of the capillary wall
 l
 Capillary length
 L
 Side length of alveolar septum; cell length inside the vessel at timet_{e}
 L*
 Dimensionless cell length inside the vessel at timet_{e}
 L_{m}
 Maximum cell length after enterance into a narrow vessel
 L_{m}*
 Dimensionless maximum cell length after entrance into a narrow vessel
 l_{0}
 Length between a capillary entrance to the position at which the capillary has the same radius as that of the cell
 L_{0}
 Side length of an alveolar septum at zero Ptp
 m
 Coefficient in linearized theoretical model of neutrophil deformation
 N_{s}
 Average number of septa between arteriole and venule
 N_{seg}
 Number of segments in an average pathway
 P_{cr}
 Minimum pressure required to aspirate a neutrophil into a micropipette
 P_{i}
 Local pressure in the ith junction
 P_{in}
 Pressure at the entrance of the capillary network
 P_{j}_{1}, P_{j}_{2}
 Pressures at the junctions connecting the jth segment
 P_{out}
 Pressure at the exit of capillary network
 Ptm
 Transmural (or transcapillary) pressure
 Ptp
 Transpulmonary pressure
 Q˙
 Flow rate
 Q˙_{cv}
 Flow rate in corner vessel
 Q˙_{i}_{,j,n}
 Flow rate in the nth capillary of the jth pathway in the ith network
 Q˙_{(in)}
 Total inflow rate in a junction
 Q˙_{ip}
 Flow rate in the pth segment connecting the ith junction
 Q˙_{j}
 Flow rate in the jth capillary segment
 Q˙_{mean}
 Average flow rate in septal capillaries
 Q˙_{n}
 Flow rate in the nth capillary
 Q˙_{oi,j,n}
 Total flow rate entering the upstream junction of the nth capillary of the jth pathway in the ith network
 Q˙_{(out)i}
 Outflow rate from the ith daughter vessel
 Q˙_{RBC(in)}
 Inflow volumetric flux of RBCs in a junction
 Q˙_{RBC(out)i}
 Outflow volumetric flux of RBCs in the ith daughter vessel
 r
 Flux cutoff parameter, which defines the minimal fractional blood flow required to draw RBCs into a daughter branch
 R
 Resistance per unit length of capillary
 R
 Radius of capillary cross section (see Fig. 2); effective radius when a noncircular cross section is approximated by a circular one
 R*
 Dimensionless r_{c}
 r_{c}
 Instantaneous radius of the portion of the cell lying outside the vessel when a cell is partly drawn into a vessel
 R_{c}
 Radius of cell
 R_{c(in)}
 Radius of the cell inside a capillary
 R_{cv}
 Resistance per unit length of corner vessels
 Re_{D}_{h}
 Reynolds number based on hydraulic diameter
 R_{j}
 Resistance of the jth capillary segment
 R_{junc1}, R_{junc2}
 Resistances of the two junctional regions of a capillary
 R_{mean}
 Average resistance per unit length of septal capillaries
 R_{ms}
 Radius of microsphere
 R_{p}
 Radius of suction pipette
 R_{seg}
 Flow resistance of capillary segment
 R^{*}_{0}
 Initial value of dimensionless cell radius before the cell enters a capillary segment
 R_{1,}R_{2}
 Radii at the entrance and exit of a capillary segment, respectively
 S_{seg}
 Surface area of capillary segment
 S_{junc1}, S_{junc2}
 Surface areas of the two junctional regions of a capillary
 t
 Neutrophil transit time
 t*
 Dimensionless entrance time of neutrophil
 T_{av}
 Time RBCs spend outside of capillaries (in arteriole and venule)
 T_{c}
 Hoap tension
 t_{e}
 Time required for a neutrophil to enter a narrow capillary (entrance time)
 t_{i} _{,j}
 Total RBC transit time for the jth pathway of theith network
 t_{i} _{,j,n}
 RBC transit time through the nth capillary segment of thejth pathway in the ith network
 T_{i} _{,m}
 RBC transit time for the mth pathway from the entrance of the first network to the exit of ith network
 t_{n}
 RBC transit time in the nth capillary
 t_{p}
 Passing time, the time neutrophil takes to pass through the capillary from one end to the other
 T_{s}
 Force in the septal wall per unit length
 u
 Local mean velocity
 u_{blood}
 Bulk blood velocity
 u_{neu}
 Neutrophil velocity
 V
 Alveolar volume
 V_{n}
 Volume of the nth capillary
 V_{0}
 Alveolar volume at zero Ptp
 ΔP
 Pressure difference between the entrance and the exit of one segment; aspiration pressure
 ΔP_{av}
 Ptm drop between arterial and venule
 θ
 Angle (see Fig. 2)
 μ
 Fluid viscosity
 μ_{app}
 Apparent viscosity
 μ_{c}
 Cytoplasmic viscosity
 μ_{p}
 Viscosity of plasma
 ρ
 Fluid density
 τ_{0}
 Average tension in cell cortex
Construction of the capillary networks.
To perform the simulation in a geometrically realistic structure, we developed a method and software to generate pulmonary capillary networks loosely based on measured data supplemented by plausible assumptions concerning alveolar septal structure. A capillary network is composed of many junctions and segments. Among the parameters needed to generate a network are the size of the alveolar septum, the number of junctions or segments, the probability of threeway or fourway junctions, and the lengths and crosssectional dimensions of capillaries. From photomicrographs of a typical microscopic field in dog lungs (24), one can roughly estimate that there are six septa between the arteriole and venule. In vivo observations have shown considerable interspecies variability in the number of segments in an average pathway: 60 in dogs (16), 45 in rabbits (3), and 90 in humans (17). If we consider the average number of segments in the blood pathways between the arteriole and venule to be 50, it gives the average number of segments in each individual flow pathways within one septum to be about eight. This number can be used to determine the number of junctions in a network. Lamm et al. (22) show that a rectangular region of the lung (400 × 500 μm) contains an average of 10 alveolar septa, which gives an average septal area of 2 × 10^{4}μm^{2}. This value is used in our simulation as the septal area at a physiological Ptp, which yields, using Eq. 2 below, the length of one side of a septum at zero Ptp to beL _{0} = 70 μm. Based on observations of the rat pulmonary capillary bed, we set bounds on the minimum and maximum capillary length, twothirds and fourthirds the mean capillary length, respectively. Experimental observations show that most junctions connect three segments, whereas a few connect four.
Based on these anatomic data, we can computationally generate many geometrically realistic capillary networks. Figure 1 Ashows the threedimensional structure of a network. Figure 1,B and C, shows the top views of two networks. These capillary networks are used in our pulmonary blood flow and cellular transit simulations.
Model for the dimensional changes of the capillary cross section.
A tissuemembrane structure model (Fig. 2) is developed to describe the geometric changes of capillary cross section with Ptp and Ptm. The following assumptions are made. 1) The circumference of a pulmonary capillary cross section can be divided into two parts: one adjacent to septal wall tissue (bottom) and one composed of the membrane (top). 2) Ptp is assumed to influence capillary crosssectional area only through its effect on the length of the bottom (septal wall). 3) Changes in length of the bottom due to Ptp are proportional to the change in overall septal dimension L. 4) Top (membrane) segment length increases linearly with increasing hoop tension T_{c}. To estimate changes in septal length and stress under a Ptp, we approximate a single septum as a square plane and an alveolus as a cube. First, consider the pure Ptp effects on the capillary crosssectional area by setting Ptm = 0. When transpulmonary pressure increases from zero to Ptp, alveolar volume increases from V_{0} to V, the side length from L
_{0} toL, and the length of any given membrane segment of a capillary from a
_{0} to a (=a
_{0}
L/L
_{0}). We use measurements of Mercer et al. (25) of rat alveolar volumes during inflation and deflation to obtain the relationship between V and Ptm
Equations 7
,
8
, and
11
yield
Flow model in a single capillary.
A “segment” is defined as the portion of a capillary between two “junctions.” Therefore, a capillary consists of one segment with a junction at either end. We first establish the flow model in the segment and then estimate the effects of junctions on flow resistance. As in Dhadwal et al. (1), an approximate equation is used to evaluate the flow resistance of a capillary segment (7), R_{seg}
Model for corner vessels.
Corner vessels are located along the boundaries of each alveolar septum. Unfortunately, we know of no published anatomic data on their dimensions or on the flow rate they carry. We, therefore, make the following plausible assumptions: that all corner vessels are circular in cross section, and that each corner vessel conveys an amount of blood flow equal to the average flow rate in the septal capillaries (Q˙_{mean}) multiplied by the number of adjoining septae. Typically, three septa share one corner vessel so that
Network flow model.
For a network with m junctions and n segments of septal capillaries and corner vessels, there exist m +n unknowns; these are the junction pressures P_{i},i = 1, 2,…, m and the segment flow rates Q˙_{j}, j = 1, 2, …, n. Therefore, we require m +n governing equations as well as boundary conditions at the network entrance and exit. If the number of branches at theith junction is q(i), mass conservation at each junction requires that
Hematocrit distribution.
At each bifurcation, RBCs preferentially enter the daughter branches with higher flow (10, 35), leading to nonuniform hematocrit (H_{d}) distributions. The nonuniform distribution of H_{d} will, in turn, alter the local flow resistance and thereby the pressure and flow rate distributions. To calculate the hematocrit distribution and take into account its effect on flow resistance, we introduce the model described in Levin et al. (23), from which we obtain
RBC transit time.
RBCs travel from arteriole to venule through several networks (septae), with each possessing many alternative pathways. One can evaluate RBC transit time in a single capillary by calculating the ratio of capillary volume V_{n} to flow rateQ˙_{n}
To estimate RBC transit time from arteriole to venule, the networks are assumed to be connected in serial fashion as shown in Fig. 4. To satisfy mass conservation, we maintain a constant flow rate at each connecting exitentrance site by adjusting the pressure drop across each network in an iterative manner. Note that, because capillary pressure falls monotonically in the direction of flow, each successive septum has a lower mean capillary pressure and is, therefore, less distended and more highly resistive than those upstream. In this multiple network structure, if there are jmax(i) pathways for the ith network, the total number of pathways from the first network connecting the arteriole to the exit ofith network, J_{i}
, should be the product of total number of pathways of the first i−1 networks, J_{i − 1} and jmax(i). The RBC transit time and the fraction for the mth (1 ≤m ≤ J_{i}
) pathway from the entrance of the first network to the exit of the ith network can be calculated by
Neutrophil transit time.
The estimation of neutrophil transit time is more difficult. While all RBCs behave in the same way within a given capillary and, therefore, experience the same transit time under a given flow condition, neutrophil transit time depends on its entrance diameter, which may vary depending on the diameters of capillaries that it previously encountered. That is, neutrophils with different entrance diameters, relating to their earlier “transit history,” will have different transit times in the same capillary under the same pressure gradient.
In this simulation, neutrophil transit time is computed in two steps: the entrance time t
_{e}, which is the time for neutrophil deformation to enter a narrow capillary, and passing timet
_{p}, which is the time it takes to pass through the capillary from one end to the other. The total transit timet is the sum of the entrance time and passing times
Considering a capillary with a radius of R
_{1} at the entrance and R
_{2} at the exit, we assume that the radius varies linearly in between so that
For R
_{c} > R
_{min}, there are two cases. First, if R
_{1} >R
_{2}
Neutrophilinduced segment blockage.
When a neutrophil diameter is greater than the segment it encounters, the segment will become blocked, at least temporarily. This neutrophilinduced blockage will change the flow pattern in the network and increase the pressure drop across the blocked segment. Considering the resistance of a blocked segment to be infinite, one can recalculate the local volume flow rate Q˙, hematocrit H_{d}, and pressure P for all other segments using Eqs. 2328 . The higher pressure drop across this blocked segment is then used to compute neutrophil entrance time.