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Modelling neutrophil transit through the human pulmonary circulation Wiggs, Barry James Ryder 1994

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MODELLING NEUTROPHIL TRANSIT THROUGH THE HUMAN PULMONARYCIRCULATIONbyBARRY JAMES RYDER WIGGSBSc (UBC) 1982, MSc (UBC) 1989A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTSFOR THE DEGREE OF DOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF EXPERIMENTAL MEDICINEWe accept this thesis as conforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust, 1993cBarry James Ryder Wiggs, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives, It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of c-\c r’;:The University of British ColumbiaVancouver, CanadaDate a2/3DE-6 (2/88)AbstractThe object of this thesis was to construct a model to simulate the arterial, capillary andvenous networks of the pulmonary circulation and to compute transit times of red blood cells andneutrophils through the human lung. A complete model of the arterial system was constructedusing existing anatomical data for the arterial branching system combined with a set ofprobabilities to describe the branching nature of the arterial network beyond the existing data.Flow continuity and energy balance equations were used to estimate the total pressure dropsthrough a wide range of possible pathways in this network. When flow had reached thecapillary bed, a strictly stochastic model of cell transit through a randomly generated gridnetwork was used to simulate capillary flow. Finally, the flow was returned to the left atriumby modifying the arterial network to represent the venous system.Studies in several species have shown that neutrophils have much longer transit times thanred blood cells in the pulmonary circulation. The model described above was used to test thehypothesis that this delay is due to the greater deformability of red blood cells with respect toneutrophils. The results show that erythrocyte transit times are accurately predicted fromphysical data using the model. However, the neutrophil transit times predicted from the delaywhich neutrophils experience as they deform to enter smaller capillary segments are shorter thancurrent experimental results. This suggests that factors additional to those that result from celldeformation times delay the neutrophils. These factors could include either receptor-mediatedneutrophil adhesion to endothelial cells or the time required for neutrophils to actively movethrough segments with very low driving pressures. It further suggests that these components areresponsible for a major time delay of neutrophils in the pulmonary microcirculation which could11not be modelled with currently available data. The results obtained with the model also suggeststhat the fall in pulmonary vascular resistance and rise in circulating neutrophil count associatedwith increased pulmonary blood flow could be related to the flushing of neutrophils out of thepulmonary capillary bed.111Table of ContentsAbstract iiTable of Contents ivList of Tables viList of Figures viiiAcknowledgements xi1 Objective 12 The Large Vessels 62.1 Introduction 62.2 Arterial Branching Pattern 112.3 Estimation of Flow in Arterial Vessels 172.4 Calculation of Arterial Pressure Drop 202.5 Pulmonary Venous Circulation 283 Capillary Bed Model 303.1 Capillary Dimensions and Their Relation to Neutrophil Size 303.2 Effect of Plugging in the Capillary Network 343.3 Modelling the Effect of Lung Height on Capillary Size 393.4 Neutrophil Deformation Time 41iv3.6 Capillary Bed Network Model 454 Total Pulmonary Transit Time 515 Discussion 545.1 Large Vessel Model 555.2 Capillary Network Model 665.3 Total Pulmonary Transit Time 75Summary and Conclusions 77References 191VList of Tables1 Number of branches and regions in Horsfield data 792 Probability of bifurcation or trifurcation in arterial tree 813 Branching probabilities in upper cast region 834 Branching probabilities in lower cast region 855 Branching pattern for orders 48 to 44 in arterial tree 876 Branching pattern for orders 44 to 1 in arterial tree 897 Ratio between arterial and venous Strahier ordering data 928 Relative increase in number of branches from largest tosmallest vessels in arterial and venous cast data 949 Venous branching pattern 9610 Node to ground resistances in electrical resistor grid networks 9911 Relative increase in node resistances in resistor grids with10% blockages 10112 Relative increase in node resistances in resistor grids with 25%blockages 10313 Internal resistor grid patterns scaled to total input resistance 10514 Relative increase in node resistances in resistor grids withunidirectional flow and 10% blockages 10715 Relative increase in node resistances in resistor grids withunidirectional flow and 25% blockages 10916 Capillary dimensions regionally in the lung 111vi17 Summary of capillary bed results 11318 Flow fractions to different lung regions 115viiList of Figures1 Pulmonary angiogram 3 seconds post injection 1172 Pulmonary angiogram 8 seconds post injection 1193 Pulmonary angiogram 20 seconds post injection 1214 Indicator dilution results for RBC 1235 Indicator dilution results for RBC and PMN 1256 Strahier ordering system 1277 Weibel symmetric ordering system 1298 Horsfield ordering system 1319 Rough Horsfield ordering 13310 Final Horsfield ordering 13511 Estimated number of vessels in arterial tree versus order 13712 Estimated cross-sectional area of arterial tree vs order 13913 Possible distribution of flow in vessels based on conservation of flow 14114 Average flow per vessel versus order 14315 Distribution of total arterial pressure drop 14516 Distribution of arterial path lengths 14717 Arterial model simulation of precapillary arteriole flow velocity 14918 Distribution of venous path lengths 15119 Distribution of total venous pressure drop 15320 Cumulative pressure drop through arterial and venous trees 155vm21 Frequency distribution of human neutrophil diametersand capillary segment diameters 15722 25m thick section of single human alveolar wall 15923 5x5 grid of representing alveolar wall model 16124 5x5 grid showing location of blockages in resistor network simulation 16325 Capillary diameters versus lung height 16526 Smooth curve fit to capillary diameters versus lung height 16727 Time for neutrophil to deform versus ratio of neutrophil diameter tovessel diameter 16928 Frequency distribution of number of capillary segmentstraversed from arteriole to venule 17129 Frequency distribution of total pathlength traversed from arterioleto venule 17330 Frequency distribution of red blood cell transit times throughcapillary bed 17531 Frequency distribution of neutrophil transit times throughupper and lower lung regions of capillary bed 17732 Frequency distribution of the number of stops made by a neutrophilas it traverses the capillary bed 17933 Frequency distribution of the average time per stop made bya neutrophil in the capillary bed 18134 Frequency distribution of transit times through the arterial tree 183ix35 Frequency distribution of transit times through the venous tree 18536 Frequency distribution of transit times through the entire pulmonarycirculation 18737 Frequency distribution of red blood cell transit times throughthe pulmonary vasculature in a normal human subjects 189xAcknowledgementsI wish to gratefully acknowledge the contributions that so many people havemade in conducting this research. It will be a strong foundation from which to builda future of scientific inquiry.The questions posed by this work are a start to a lifetime dedicated tounravelling the mysteries that are held, unsolved within the pulmonary vasculature.It is believed that the knowledge and skills learned through my program of studieswill provide me with the tools needed to navigate the tough journey ahead.To my committee, all the excellent researchers at the Pulmonary ResearchLab, and my scientific colleagues - thank you. Without your guidance, critique andassistance this would not have been possible. Most of all to my wife, Veronica, whohas been beside me through all the good times and the not so good times.Hey folks .... IT IS DONE!!!!!!!!!!!!!!!!!!!!xi1 ObjectiveThe modelIiig of biological systems has increased our understanding of structure andfunction at the molecular, cellular and organ level. This is well illustrated by Watson andCricks’ (94) structural model of DNA which revolutionized biology by providing new insightinto DNA function. Similar, if less dramatic, advances have been provided by modelling of thecellular functions of transport (97), volume control (86) and mobility (51). The understandingof organ function has also been advanced, particularly in the lung, by the model of pulmonaryblood flow introduced by West et al. (98), the model of ventilation distribution by Milic Emilet al. (67), and by the model of gas exchange across the alveolar capillary membrane providedby Wagner and associates (89).The introduction of the computer into biology enhanced the sophistication of themodelling process by providing the rapid calculation capability required when anatomically basedmodels are used to predict function. In a previous series of studies which formed the basis ofmy MSc degree (101, 103), a computational model of airway structure and function wasdeveloped which has been useful in predicting the effect of structural changes on airwayfunction. The model allowed quantitative anatomic data obtained from normal and diseasedairways to be entered and used to predict their effects on airways function. The informationobtained from the model was then used to construct new hypotheses that are being tested eitherin whole animals or in isolated in vitro systems.1The model presented in this thesis was developed to investigate cell traffic through thelung vasculature. It is based on studies in both animals (18,19,20,21,42,62) and man (8,59,60)where a double indicator dilution technique was used to compare erythrocyte (RBC), platelet andneutrophil (PMN) traffic during a single transit through the pulmonary vascular bed. Thesestudies showed that PMN were delayed with respect to RBC with only 20-30% of PMNappearing at the outflow with the RBC. They also showed that the lung capillary bed is uniquelydesigned to accommodate these difference in PMN and RBC transit time because it provides avast network of short interconnecting capillary segments that allow the faster moving RBC tostream around segments filled with slower moving PMN (41). Unfortunately, indicator dilutiontechnology does not allow a complete analysis of this system because the much longer transittime of PMN with respect to RBC allow substantial recirculation of REC before all of the PMNhave completed a single pass through the lung. Although this problem might have beenovercome by using an isolated organ system that prevented recirculation, construction of acomputer model based on existing anatomic and physiologic data was chosen in an attempt tounderstand the large body of experimental data.The complexity of the pulmonary vasculature can be appreciated by examining theangiograms shown in Figures 1, 2 and 3. These images, from a normal human subject, taken3, 8 and 20 seconds after dye injection, show the arterial tree filled at 3 seconds, the venous treeat 8 seconds with only a small amount of dye remaining in the pulmonary vessels. These imagesprovide an overview of the very complex branching network and range of transit in thepulmonary circulation. At least two approaches are available for obtaining the information2needed to construct a model of this complex system. The first involves the application of atheoretical knowledge of branching systems based on a knowledge of fractals. The second isbased on existing information about the anatomic structure of the lung where behaviour of thebranching system is represented by stochastic rules.The available morphometric data from the arterial trees of dogs, humans and cats havebeen used to graph the relationship between the number of vessels at any given order vs theaverage diameter of a vessel at that particular order (109). Typically, these plots are drawn ona log-log scale and show a remarkedly linear relationship between these two parameterssuggesting a scale independent nature of the branching structure. The use of fractal analysis todescribe physiological phenomena has been effectively employed by Bassingthwaite et al. (4,5)to investigate the spatial heterogeneity of blood flow in the heart and by Glenny et al (34,35) toinvestigate the relative contribution of gravity to regional variation in pulmonary flow. Whilethe use of fractals allows the very complex systems to be simplified to an extremely smallnumber of parameters it also loses some of the anatomical feel for the physiological systembeing investigated. Although it is possible to study the human vasculature using an analyticalapproach based on fractals, the present thesis is based on an anatomical approach to thisproblem. Principally, this decision was based on the fact that the pathologists, physiologists andclinicians with whom I collaborated to collect much of the data that stimulated the developmentof the model, found it easier to consider problems when posed in anatomical rather than abstractframeworks.3The starting point for construction of the model was the knowledge that: PMN aredelayed with respect to RBC in a single pass through the canine (10,41,57,62,63), rabbit (18,19,20) and human lungs (42,59,60,68). Using indicator dilution studies, Martin et al (62, 63)labelled both RBC and PMN in order to compare how these cells traverse the pulmonarycirculation. When RBC labelled with either Tc or 51Cr were simultaneously injected into theright atrium and collected at the aorta, there was no difference in how the cells traversed thepulmonary circulation (Figure 4). However, a similar injection where Tc RBC are comparedto 51Cr PMN showed that the PMN are markedly delayed in the lung with respect to the RBC(Figure 5). A series of studies subsequently showed that from 60 - 80% of PMN are delayedwith respect to RBC in the human, rabbit and dog respectively (42).Although PMN have similar maximum diameters to RBC, they deform much more slowly(13,23). This difference in deformability has been used to explain the observed difference inPMN and RBC transit time (40,41,42). However, the vast parallel arrangement of thepulmonary capillary segments means that the pressure drop across individual capillary segmentsmust be small and raises the question as to where the large forces required to move PMN intosingle capillaries (23) are generated. The fact that it is difficult, if not impossible to measurethe pressure across individual segments of the pulmonary capillary bed, led us to considercreating a model of the pulmonary vasculature to estimate the delay that PMN experience withrespect to RBC. The anatomic data for the model comes from several sources. The data forthe large vessels was generously provided by Dr. Keith Horsfield from existing published andunpublished information obtained from a human pulmonary vascular cast studied in his4laboratory. This information allowed us to obtain adequate data to construct a reasonable modelof the arterial supply and to infer the venous drainage of the pulmonary capillary bed. The dataon the PMN and capillary segment dimensions was based on Weibels’ (95,96) original reportsthat were supplemented by independent studies from our own laboratory (21).The goal was to construct a model of the arterial, capillary and venous systems of thehuman pulmonary vasculature that would provide realistic RBC and PMN transit times. Itinvolved the construction of a set of probabilities to describe the branching nature of the arterialnetwork that allowed the pressure drop through the system to be estimated using flow continuityand energy balance equations. A strictly stochastic model of cell transit was then used tosimulate capillary flow through a randomly generated grid network. The flow was then returnedto the left atrium by modifying the arterial network to represent the venous system.The results suggest that the model accurately predicts RBC transit times butunderestimates the delay which PMN experience with respect to the RBC during a single transitthrough the lung. This suggests that factors such as receptor mediated PMN endothelialinteractions and the ability of trapped PMN to actively move through restrictions provided bythe capillary bed may be important determinants of their transit time. The computational modelthat has been constructed provides a framework in which to think about the problem of celltransit through the pulmonary circulation and suggests new experiments to try in order to achievea better understanding of this problem.52. The Large Vessels2.1 IntroductionHorsfield and his colleagues (44,45,47,78,79) made detailed measurements of the arterialand venous branching system on a cast of the pulmonary vascular system obtained from thelungs of a 32-year old woman who died of uremia and was free from any known respiratorydisease. The complete process for preparing the cast was described in detail by Singhal et at(78). Briefly, the trachea and main pulmonary vessels were washed free of secretions and bloodclots and then cannulated. The airspaces were inflated with carbon dioxide momentarily andthen allowed to deflate by their own elastic recoil. This process of inflation and deflation wascontinued to replace the air in the lung with carbon dioxide. The lungs were then floated in atank of boiled air-free tap water and a plastic tube was attached to the bronchial cannula. Thefree end of the tube was left in that tank below water level and a deBakey roller pump was usedto pump water from the tank into the lungs. Water easily passed through the lung tissue andreturned to the tank via the pulmonary blood vessels and the pleural surface. By adjusting pumpspeed, any level of lung inflation could be obtained.The lungs were inflated to the top of their pressure-volume curve and then deflated downto a volume corresponding to functional residual capacity. Pumping was continued until thelungs sank to the bottom of the tank, implying that most of the carbon dioxide had been replacedby air-free tap water. Formaldehyde solution (40%) was added to the tank to make a 2%6formalin solution, which circulated through the lungs for 48 hours. During fixation, 0.9%sodium chloride solution was pumped through the arterial cannula at a constant pressure of 25cm H20. After fixation, the lung volume was 5.01±0. 1OL.The method for casting the vasculature tree was that of Tompsett (87). A mixture ofmonomer, resin, catalyst and accelerator was allowed to flow into the arterial cannula from aheight of 25 cm above the formalin level. During casting, the lungs remained submerged andthe density of the casting material was 1.125 gIml, similar to water, so that little if any pressuredifference existed between different regions of the fluid filled lung. The casting mixture flowedeasily through the arteries, capillary bed and venous system. After four minutes, the resin gelledand it was allowed to harden for eight days. The surrounding lung tissue was corroded fromthe cast with concentrated hydrochloric acid and the cast washed clean in tap water. Noshrinkage was detected during the setting or cleaning process.The data subsequently published from observations on this cast was useful but the goalsof the authors differed significantly from the objectives of this study. Horsfield et al wanted toprovide detailed anatomical information about the branching structure of the arterial and venoustrees in a normal human lung. They chose to order the complex branching network of thearterial and venous trees using a Strahier ordering system. This system has been widely usedby geomorphologists to describe the branching patterns of rivers (85) and has the uniqueproperty of producing a branching structure with the minimum number of orders.Figure 6 shows a simple network ordered according to Strahler rules. In this branchingscheme, all identifiable terminal endpoints are denoted as order 1. When two branches of the7same order meet at a junction, the resulting parent branch increases in order by one. If twobranches of different orders join, the order of the parent branch remains the order of the largestordered branch. Unfortunately, this branching scheme, while efficient, loses a great deal ofinformation. In particular, estimates of length and fluid flow from the main pulmonary arteryto the terminal arterioles are impossible when modelled using the Strahier ordering system sincemany divisions will occur where the length from the parent branch to the division is notrecorded.Alternative branching schemes are possible and the most frequently considered are theWeibel symmetric branching scheme and the Horsfield asymmetric scheme. Figure 7 shows theresult of symmetric ordering of the tree in Figure 6. In this method, the main pulmonary arterywould be labelled order 0 and at each division the two daughter branches would each bedesignated one order larger than the parent. This branching method obviously relies on a highlysymmetric branching pattern and cannot be reasonably considered as a viable scheme forordering the very asymmetric vascular tree shown in Figures 1-3. The system which Horsfieldand Cumming used to describe the branching network of the bronchial tree (46) is similar toStrahier ordering where each terminal branch is identified as an order 1 vessel. However, asany two vessels join, the parent vessel is ordered as one order larger than the largest daughter.Figure 8 shows the same tree as in Figures 6 and 7 with Horsfield orders.Several points can be made from the investigation of these different ordering methods.First, Weibel ordering is a “top down” while Horsfield and Strahler are “bottom up” methods.In this regard, Weibel ordering has clear advantages for recording cast information. It is not8necessary with a Weibel symmetric (96) structure to locate all the terminal endpoints which canbe extremely difficult when using casts. Secondly, Weibel structures are uniform bifurcatingsystems where Horsfield and Strahier systems need not be. The angiograms shown in Figures1-3 clearly show a high degree of disparity between the size of daughter branches from a singleparent. Third, Strahier ordering has a fundamental flaw which renders it unusable for anyinvestigations of transit time. Looking closely at Figure 6, we see that on the right side of thefigure, there is a single branch labelled “2” with a branch “1” offshoot. Since the entire vessel2 is recorded as a single unit rather than as a parent and a daughter, it is impossible to recordthe correct path length for any blood travelling down the branch “l” offshoot. The number ofsegments, and therefore bifurcations, traversed in a branching structure cannot be determinedusing this ordering scheme. Also, since Strahier ordering loses vital length information, it isnot possible to estimate the pressure drop along a length of vessel.Because of the unique nature of the casts prepared and studied by Dr. Horsfield, he wasapproached for permission to study the original data. The purpose was to completely reorderall cast data using the Horsfield branching system so that true vessel length and diameter of eachvessel segment could be modelled. Unfortunately, the original casts, now some 20 years old,had been destroyed but Dr. Horsfield was able to provide the data sheets showing the originalvessel identification and methods used for cataloguing the branches of the cast. He also allowedus to study copies of many drawings of the cast branches. Using both the drawings and thevessel identification charts, the casts were reconstructed in a computer that allowed any orderingscheme to be applied.9The available data was in four distinct regions and the number of individual branchesavailable in each region is summarized in Table 1. These four regions divided the available castdata into sets of vessels of decreasing diameter. The first task was to study the originaldrawings since these were the only visual link back to the original cast, and several observationswere made. First, it became apparent, both from the drawings and from angiographic images,that the pulmonary vasculature consisted not only of bifurcating but also trifurcating branches.The number of trifurcations were sufficient that they could not be ignored and would have to beincluded to obtain an accurate model. Second, within each zone, a large number of smallervessels were broken or lost at the end of the casts. This artifact tends to lower the numbers ofbranches recorded for the very small orders as they broke more easily than larger branches.Third, studies of the drawings in relation to the angiograms showed that it was possible fordaughter branches from a single parent to have very different diameters. Finally, the model wasfurther complicated by the presence of numerous 15 m vessels which appeared to extend outfrom many vessels at a nearly right angle. These tiny vessels were distributed throughout thecast and do not appear to be merely ruptures in the vessel walls giving fine strands of castingmaterial. These vessels were eliminated from the dataset and not considered in modellingpressure drops, flows and transit times.102.2 Arterial Branching PatternEach of the four data regions (Table 1) where information was available were orderedas follows. First, all terminal branches were labelled as order 1 vessels. Then a computerprogram was written to track each daughter pair to the appropriate parent and ordered the parentbranch one order higher than the largest daughter. Trifurcations could be easily handled usingthis branching scheme, as could the vast difference between the orders of daughter branchesfrom a single parent. This process required that all terminal branches be of a set size. Thesmallest remaining broken cast branches were assigned an’ order based on their diameter inrelation to the already ordered branches. The computer program then attempted to reorder asmuch of the remaining tree as possible using an iterative process, returning to order unbrokenbranches as required, until every available branch in each of the four regions was ordered. Thefinal branch ordering was determined to be the scheme with the smallest diameter coefficient ofvariation when data were represented on a log scale.Once each region was ordered, the individual regions were connected to generate a singleordering system from 15gm to 3cm vessels. The rough Horsfield ordering can be seen in Figure9 where each region has been artificially placed on a scale from 0 to 80 so that some gap is leftbetween regions. Note from this figure that as the smaller regions of each zone are approached,diameters appear to decline in a curvilinear manner. The reason for this curvilinear drop wasthat whole cast portions had been destroyed, resulting in biased estimates. It was very difficult,as Horsfield noted, to use the cast and not have whole sections break away. The remaining11unbroken sections had been preserved and tended to have a greater number of larger vesselssince portions with smaller vessels were trimmed off to aid counting. This loss artificiallyincreased the estimated vessel diameter. The original zones were carefully selected by Horsfieldand his colleagues to overlap each other by a few orders. The diameter of the smallest vesselsin one region were compared to the diameters of the largest vessels in the next region. The largevessels of this more distal region were given orders similar to the same diameter vessels in themore proximal region. Once it was established how one region’s orders related to all others, thecomplete dataset was reordered.The results using a modified Horsfield ordering scheme to order the arterial branches ofthe pulmonary circulation are shown in Figure 10. There were 48 orders from the mainpulmonary artery to the precapillary vessels of approximately l5jLm in diameter. Thelog(diameter) was related to vessel order using a weighted regression analysis (Systat, Evanston,IL) where the data at each order were weighted inversely by the variability about the mean ofthat order and the results fitted to the relationship:log(diameter) = A + B*Order [EQN 1]where:Estimate Value Standard ErrorA=Intercept 2.64 0.04B=Slope 0.15 0.0112The overall model had a fit, R2, of 0.995 for the relation between log(average diameter) versusorder. The estimates for slope and intercept correspond to a precapillary estimated diameter of15gm at order 1 and each order’s mean diameter increases by about 25% for each single orderincrease.Once the vessels had been ordered, further information regarding how the vesselsbranched was required to estimate path lengths, pressure drops and transit times. Horsfleldordering systems, as already mentioned, are “bottom up” ordering systems. However, “topdown” branching rules would be preferred so computations could begin at the main pulmonaryartery and terminate at precapillary vessels. To formulate a “top down” ordering scheme, twopieces of information were recorded for each branching site. First, whether the vessel bifurcatedor trifurcated (or divided an even greater number of times) was recorded. Second, the decreasesin order for each daughter from the parent branch (parent branch order minus daughter branchorder) was recorded. This second piece of information was recorded for every daughter whilekeeping track of which daughter had the largest, second largest, etc, diameter. The data fromthe four regions were combined to give two major cast regions, 3cm to 100gm, and 100gm to10jm, since two regions had relatively few branches (Table 1).Table 2 shows the results for bifurcations and trifurcations in these regions. In largevessels, 82% of the branches were bifurcations. There were a significant number oftrifurcations, and the fraction of branches that were trifurcations increased in the smallervessels. Table 3 shows how the daughters branched, relative to the parent, in large vessels andtable 4 shows the same results for the small vessels. In these tables, a bifurcation would have13a largest and a smallest daughter while a bifurcation would have a largest, second largest andsmallest daughter. The data for the bifurcations and trifurcations were combined as there wasno difference between their branching patterns, the difference in orders between the parent anddaughter branches was not affected by a division being a bifurcation or trifurcation. It is ofinterest to note that the largest daughter branched relatively slowly, dropping typically 1 or atmost 2 orders smaller than the parent, while the smaller daughters dropped several ordersrelative to the parent branch. These data show that the path lengths from the main pulmonaryartery to the 15m precapillary vessels vary considerably and are likely to result in the widerange of transit times through the vascular tree.The probabilities displayed in tables 2,3 and 4 provide a set of branching rules which canbe used to reconstruct a model of the arterial branching system. One very important piece ofinformation which will be required is the relationship between branch length and vessel order.Because the data for this work was reconstructed from drawings and old files, the valuablelength data contained in the original cast was largely lost. Only about 600 vessels, ordered byour new system, had length information available, and most of these were larger vessels.Singhal et al found a linear relationship between the log(length) and vessel diameter (78). Thisrelationship is likely to be a function of the branch ordering scheme they selected, and we couldnot accurately estimate any relationship between length and order. In this study the ratio oflength to diameter was plotted versus order and the relationship showed that the ratio was similarfor all orders. Realizing that this is a potential weakness, but without the original cast to eitherconsult or measure, we took all values for which both length and diameter were available and14estimated an average value for length relative to diameter, regardless of order to be140% ±30%.The actual large vessel model was constructed using the anatomical measurements fromthe cast data for all branches from order 45 to 48 (the largest vessels). The reason for this wasthat the estimated lengths, diameters and branching probabilities for these first few divisionswere not accurately represented by the prediction methods proposed. Table 5 shows thebranching scheme and dimensions for these larger vessels. Once order 44 was reached, theprobabilities from tables 2, 3 and 4 were continually applied until all vessels terminated in anorder 1 vessel. To terminate the tree, a vessel was not allowed to branch beyond order 1.Table 6 shows the resulting branching pattern for the orders from order 44 to 1. Figure 11shows the relationship between the model’s estimate of the number of branches at any givenorder from the main pulmonary artery to the l5m order 1 precapillary vessels. At order 1, themodel predicts 3x108 terminal arterioles.In Figure 12, the relationship between cross-sectional area available for blood flow andanatomical order from the main pulmonary artery to precapillary vessels is shown. There is avast increase in the available cross-sectional area as flow approaches the capillary bed. Thislarge increase is required to obtain a sufficient area for gas exchange between the alveolarairspaces and the capillary blood. Also of interest in Figure 12 is the reduction in cross-sectional area between orders 40 to 45. A corresponding reduction in the cross-sectional areaavailable for airflow has been noted by Weibel and Gomez (96) and Horsfield and Cumming(46). Unfortunately, since simultaneous cast information on both airways and pulmonary vessels15is not available, it is difficult to state with certainty that both cross-sectional area reductionsoccur at the same anatomical location.Using the branching probabilities estimated in this section, a realistic model of the arterialtree based on anatomical information can be computed.162.3 Estimation of Flow in Arterial VesselsThe calculation for available cross-sectional area in Figure 12 is based on an assumptionthat the entire cardiac output passed through each order so that all branches of a given ordercontributed to the available cross-sectional area. This is clearly incorrect because the cast datashow that at most branches one daughter branch is markedly smaller than the other. Thisinequality in daughter branches implies that some fraction of the flow arrived at a more distallocation in the arterial tree without flowing through a vessel of every order. The consequenceof this result is that estimations of the path length from main pulmonary artery to the capillarybed, the fraction of flow in a given branch and the expected pressure drop in the arterial treemust be made with consideration as to how flow divides at each bifurcation.As previously stated, the division of flow at a bifurcation can not be determined simplyby assuming that the flow to a given order branch is equal to the total cardiac output divided bythe total number of branches of that order. This technique has been used by others to mapairflow flow in the lung (72,102) and to map blood flow in the cat lung (110) but the markedasymmetry in the human pulmonary arterial tree makes this approach questionable. Instead,continuity equations were used to determine the possible range of flows that could exist in abranch of any given order. Consider the possibilities for bifurcating branching patterns of avessel with a given order. We will denote the division of a parent into its two daughters as2: (1,1), indicating that a parent of order 2 has divided into two daughters, each of order 1. Thisis the only possible choice for an order 2 branch as it has already been assumed that the arterial17tree terminates at order 1 vessels. Another assumption will be that the entire cardiac output isevenly distributed to the terminal arterioles so that each order 1 branch carries 1 unit of theflow. Continuity states that the total flow into the daughter vessels must equal the total flow inthe parent vessel. Therefore, each order 2 branch must carry 2 units of order 1 vessel flow. Thebranching choices for an order 3 branch are 3: (1,1), 3: (1,2), and 3: (2,1). Recall that the 3: (1,1)is possible since daughters may be up to 5 orders smaller than the parent branch and the largestdaughter can branch either one or two orders from the parent. Since each order 2 branch carries2 units of flow and each order 1 branch 1 unit, order 3 branch vessels can carry either 3 or 2units of flow. The possibilities at order 4 are: 4:(3,3), 4:(3,2), 4:(3,1), 4:(2,3), 4:(2,2) and4:(2, 1) and the resulting flows are either 2,3,4,5,6,7 or 8 units of flow. While these examplesdeal only with bifurcations, trifurcations can also be considered and every possible combinationfor any branch order can be determined.Figure 13 shows the possible distribution of flows in branches of given orders relativeto order 1 branches if all pathways are equally likely. As the branch increases in order, itcarries, on average, a higher flow rate but the variability of the flow rate quickly increases asthe number of available pathways increases. Using a computer to track the flows, it was possibleto map every possible pathway up to an order 20 vessel. Beyond this order, the number ofavailable pathways was so immense that they could no longer be stored for analysis usingavailable equipment.Figure 14 shows the average flow through each vessel for each order as a fraction ofthe total cardiac output. Using the average flow in a vessel and the total number of vessels of18each order, it is possible to estimate the average flow through an entire order. The estimatedaverage flow through each order was computed to be approximately 50% of the total cardiacoutput. Also, from the data in Figure 13 used to generate Figure 14 each order also has astandard deviation about the mean of the flow. The average flow and the standard deviation ofthe average flow through a branch of a given order were fit to an exponential equation for orders44 to 2:average flow (ml/s) = exp(-14.682 + O.397*Order) [EQN 2]SD Flow (ml/s) = exp(-15.682 + O.403*Order) [EQN 3]Based on estimates of the possible flow to branches of an arterial tree where eachterminal branch receives a uniform fraction of the total cardiac output, equations 2 and 3 willbe used to estimate the flow in a branch for any given order.192.4 Calculation of Arterial Pressure DropBernoulli’s equation states that, in conditions of steady flow for a frictionlessincompressible fluid, that the energy stored in the fluid is a constant (28,71). The equation is:+ Z + = constant [EQN 4]y 2gwhere:P/’y = energy stored in the fluid due to the pressure thefluid is under per unit weight of the fluidZ = vertical height of the fluidV2/(2g) = kinetic energy stored in the fluidP = pressure-y = specific weight of the fluid = pgp = density of the fluidg = acceleration of gravityV = average velocitya = kinetic energy coefficientThe height effect, Z, in this equation is negligible for a single bifurcation and can beeliminated allowing [EQN 4] to be rewritten as:20P/p + aV2/2 constant [EQN 5]At a single bifurcation or trifurcation the Bernoulli equation between any two points in a pipenetwork must be related according to the energy or head loss between these two points. As anexample of the calculation procedure consider a simple bifurcation (a trifurcation was handledin an analogous manner) where the parent branch is denoted by the subscript 1 and the daughtersby the subscripts 2 and 3. The energy balance equations are then:P V2 P V2(—1 + - (_! + a2--) = H [EQN 6]P V2 P V2(__I + c_L)- (_ + 3—) = H [EQN 7]where H is the total head loss for the route from the parent through the bifurcation to eitherdaughter 2 or daughter 3 and represents the unwanted conversion of mechanical energy tothermal energy.Head loss can be separated into a major, hL, and minor, hLM, component. The majorhead loss component is due to friction and if the change in height between the two networklocations is negligible, as we are assuming, then:21[EQNS]where L =pipe length; D =pipe diameter; V =velocity and is a nondimensional parameter usedfor describing fluid behaviour and is equal to 64/(Reynold’s Number) for laminar flow (28).Reynold’s Number is equal to:[EQN9]v flvDwhere:V = velocityD pipe diameterQ = flow ratev = viscosityp = densityThe minor losses are related to changes in pipe geometry, the most important factor inthis model being the bifurcation itself. These losses can then be calculated as:H, = f!- [EQN1O]22where Le is the equivalent pipe length that would represent a frictional loss as severe as thegeometry change and is typically determined experimentally. Gelin (32) made several detailedinvestigations into the pressure drop in tubes past a trifurcation point. In his experiments, flowwith an entrance pressure of 160,000 dynes/cm2 was calculated to have a drop of 116,850dynes/cm2 in the lead vessel and 17,040 dynes/cm2 in the branch vessels. This left anunaccounted drop, from the input to the parent vessel to the output of each daughter, of 26,110dynes/cm2 (160,000-116,850-17,040). These pressures were modelled by Charm and Kurland(12) using laminar flow assumptions, and they obtained results very nearly identical to thosedetermined experimentally by Gelin. Charm and Kurland proposed that the remainingunaccounted for 26,110 dynes/cm2was due to the division in flow itself. This value wouldindicate that the bifurcation or trifurcation contributed at least as much to the pressure drop asdid the pressure drop along the daughter branch. Using Gelin’s results, a value of 1.5 was usedfor LeID in the equation for minor head losses (26,110/17,040). The calculated losses due toflow division were included in the calculation of pressure drops only if the estimated Reynold’sNumber exceeded 100, since pressure drops at a low Reynolds numbers will be determinedsolely by the viscous losses (72).Equations 6,8 and 10 can be combined to give:P V2 P V2 ;v2 L€V2(- + - (a + 2--) =f1 +f__!_- [EQN 11]23The ideal solution to this equation would be to compute the pressures and flows using thecombinations of the continuity equation and energy balance equation through the entirepulmonary arterial tree. This, however, is impossible because of the vast number of branchesinvolved. Instead, each order was considered as a complete unit in much the same manner asZhuang et al (109). Equation 11 can be rewritten as:LI’2 LeV2=- V12) ÷ 2 + fp P2 [EQN 12]ORP1 =AP÷P2Obviously, the pressure difference between any two orders, P, is made of three components:(p/2)(V2-V1)is the change in kinetic energy; fp(L2V)/(D) is the major head loss;fp(Le2V)/ D)is the minor head loss. Each order has approximately 50% of the total cardiacoutput flowing through it and this flow divides into the next 5 orders. The velocity of flow ineach order will be estimated as the total flow divided by the cross-sectional area available forflow. The major head loss will be then be computed directly using equation 8. The minor headloss is computed using equation 10 with LeID = 1.5. The kinetic energy change is estimated forthe difference in velocities between the parent order and each of the next five daughter orders.Each daughter order will then have 5 different possible values for this equation and a weightedaverage based on the probabilities in tables 2 to 4 will be used to estimate an average kineticenergy transfer.24It is well known (29) that the viscosity of whole blood is affected by the size of vesselthrough which it flows. The change in viscosity, commonly termed the Fahraeus-Lindquist effect(24) is a reduction in the viscosity as vessel diameter decreases. Kiani and Hudetz (50) havedeveloped an empirical model to estimate the dependence of blood viscosity on vessel diameterand hematocrit. The model they developed, for tubes larger than 10.cm is:sUapp = p[1(1!Lp/!.Lc)(12ô/d)4][1(Dm Id)4]-1 [EQN 13]where:ILapp = apparent viscosity of blood relative to plasma= viscosity of plasma= core apparent viscosity = exp(0.48 + 2.35 *hematocrit)= marginal layer width (microns) = 2.03 - 2*hematocritd = tube diameter in micronsDm = effective diameter of a single red blood cell in the vessel and wasassumed to be constant at 3 microns but this has little effect for vesselsof sizes greater than 15 microns.The pressure drops in the arterial tree were then computed as follows:1) The probabilities in tables 2, 3 and 4 were used to compute thearterial tree and determine the number of vessels of a given order.2) The diameters and lengths of vessels were assigned to branches ateach order with a coefficient of variation of 30%.253) The total flow through all branches at each order from 44 to 4 wasassumed to be 50% of the total cardiac output. Orders 48 to 44have 23 endpoints (Table 5), it was assumed that each of theseendpoints carried 1/23 of the cardiac output. Similarly the ordersfrom 4 to 1 were mapped assuming that all possible branchingpatterns to order 1 vessels were equally likely.4) Using the geometry in 2) and the flows in 3) the velocity of flowwas estimated.5) The viscosity was estimated using equation 13.6) The distribution of Reynolds number at each order was estimated7) The friction factor was estimated as 64/Reynold’s Number8) The major head loss due to friction was computed9) The loss of kinetic energy due to varying flow velocities wascomputed as: (V2 - V112)p/210) The entire pressure drop for each order was estimated11) The average pressure drop for the entire arterial tree was estimatedby summing the pressure drop for each order.The simulation program was written in Gauss VM (Aptech Systems, Kent WA) and allcalculations were made on an IBM 30386 based computer.Therefore, the following sequence is followed to estimate the pressure drop of aparticular pathway through the arterial tree. Starting at the main pulmonary artery, the26probabilities from tables 2 to 4 were used to predict the orders of the daughter branches. Thedaughters were assigned diameters and lengths using equation 1. Flows were purposely assignedto either one daughter in a bifurcation or 2 daughters in a trifurcation based on equations 2 and3. The continuity equation was used to compute the flow in the remaining daughter. Equation12 was then solved for the pressure drop from parent to each daughter. One daughter wasrandomly chosen as the parent for the next division and the process continued until an order 1vessel was reached. Using this technique, the average pressure drop from main pulmonary arteryto terminal arteriole averaged 1.8mmHg and ranged from 0.2 mmHg to 4 mmHg. Theinterquartile range of arterial pressure drops for a random path from the main pulmonary arteryto the precapilary terminal arterioles was 1.5 to 2.5 mmHg. The distribution of total pressuredrops is shown in Figure 15. Figure 16 shows the distribution of pathlengths using thesimulation results. The average pathlength is 5.3cm ranging from 2.1cm to 11.2cm. In Figure17 the estimated velocity of blood flow at the precapillary level is shown.Using equations to balance the energy contained in a volume of flowing fluid and themass of the fluid; accounting for pressure losses due to a division of flow; a changing viscosity;the total pressure drop through a model of the arterial tree can be calculated.272.5 Pulmonary Venous CirculationAlthough the remaining drawings of the cast of the venous tree were more limited thanthose available from the arterial system. Previous work by Horsfield and Gordon (47) includeda detailed Strahier ordering of the pulmonary veins in man. Using Strahier ordering, they foundthat the veins contained only 15 orders as compared to the 17 orders found in the arterial tree.Some of the difference between the number of orders is due to the venous return to the leftatrium actually arriving as 4 distinct vessels, effectively cutting off the top two orders of thevenous tree. Table 7 shows the ratio between arterial and venous trees for the number ofvessels, the diameter and the length for each Strahler order. There are 5 times as many terminalvenules as there are arterioles but the average diameter and length for vessels at each order iswithin the individual vessel variation recorded for the arterial tree (about ± 30%) (Figure 10).Table 8 shows the percent reduction in the number of branches for each order relativeto the order immediately distal to it. Both the arterial and venous trees appear to converge fromorder 1 to order 15 or 17 in a similar pattern. Since no actual information was available on howthe branches of the venous tree divided, no tables of probabilities could be constructed as forthe arterial system. However, because of the striking similarity between the Strahler orderingof both the arterial and venous trees, we decided to use tables 2 to 4 to describe the structureof the venous tree. The only minor difference was that the top 4 Horsfield orders in the treewere eliminated as was the single bottom order. By trimming the tree in this manner, 78 x 106terminal venules were recorded in contrast to 73x 106 actually estimated by Horsfield and Gordon28(47). Table 9 shows the branch orders, number of branches and average diameters for thevenous tree structure similar to Table 6.The calculations to estimate geometry and pressure drops was exactly as for the arterialtree. Figure 18 shows a histogram of the pathlengths through the venous tree. The distributionof the total pressure drop along these pathlengths is shown in Figure 19. The average venouspressure drop is estimated to be 1.2mmHg with a range from 0.01 to 4. lmmHg. The maximumcumulative pressure drop for both the arterial and venous systems is displayed in Figure 20. Thismaximum pressure drop was obtained from the longest possible pathway, one that involvesevery order through the arterial and venous tree. Also shown in Figure 20 is the averagepressure drop for each order. If the total pressure drop across the pulmonary circulation is 8mmHg then only about on half of this drop is due to pressure drops in each of the large vesselregions and the rest of the pressure drop must be located in the capillary bed.In this model it is was assumed that the branching structure of the venous tree is identicalto that of the arterial tree with exception of fewer orders being present in the venous system.293 The Capillary Bed3.1 Capillary Dimensions and Their Relation to Neutrophil SizeAs early as 1963, Weibel (95) reported measurements on the size of capillaries in thehuman lung. He found that the average capillary segment diameters were 8.36 ±2.96gm (range2-14jm) and the capillary segment lengths were 12.30 ±4.72 m (range 3-25gm). In additionSchmid-Schonbein et al (75) measured the diameter of leukocytes to be 7.25gm. Using atechnique similar to Schmid-Schonbein et al (75) and modified by Williams (104) a purepopulation of human neutrophils and a random sample of human capillaries were measured toobtain the dimensions of these structures and to verify published values from several differentlaboratories. The purpose was to provide direct dimensional comparisons between neutrophilsand capillaries so that these values could be used to develop a computational model of thecapillary network.Lung tissue from patients undergoing lung resection (59) for small peripheral tumourswas used for the measurement of capillary structure. Briefly, the surgery proceeded until thevasculature of the lobe to be resected was isolated. The lung was then inflated and ventilatedat a constant rate for approximately 10 minutes. The artery and vein supplying the lobe weretied off and the resection completed. The lobe was immediately inflated with 7.0%glutaraldehyde at a constant pressure of 25cm of fixative through the bronchus. The specimenswere submersed in fixative overnight, sliced in the gravitational plane and random tissue samplesfrom normal regions of lung were removed and processed for both light and electron microscopy.30Tissue sections were randomly selected for light microscopy from the available blocksand were dehydrated in graded alcohols and embedded in paraffin. The dimensions of the blockshad been previously recorded so that any shrinkage due to the dehydration could be correctedfor. Light microscope sections 25gm thick were cut from these blocks and stained with toluidineblue 0. Capillary segment diameters and lengths were measured at 400X magnification witha Nikon microscope equipped with a camera lucida and digitizing tablet interfaced to amicrocomputer. Only whole capillaries that were entirely within a single focal plane weremeasured. These measurements were then corrected for linear shrinkage by comparing thedimensions of the cut sections to the original block dimensions.The PMN dimensions were recorded on three human subjects undergoing lung resectionfor cancer. RBC were sedimented from the whole blood using dextran (molecular weight 100-200KD) and the leukocyte rich plasma was then removed and gently mixed with 7%glutaraldehyde for fixation. These fixed cells were suspended in 20% bovine serum albuminwhich was then cross-linked with 0.5% glutaraldehyde to form a rigid structure. Finally, thiscell block was cut into 1mm cubes, post fixed with 1 % osmium tetroxide, dehydrated withgraded alcohols and embedded in epoxy resin. Sections 60-9Onm thick were cut using adiamond knife on an ultramicrotome (Reichart Ultra Cut II) and stained with uranyl acetate andSatos lead for examination by electron microcopy (Philips 400). Photographs of 75-135 sectionswere sequentially obtained for each human sample at 10,000X magnification.Unlike the measurements of the capillary segments, direct measurements of the neutrophildiameters can not be made. It is, however, possible to formulate a set of probability rules31(75,104) that consider the neutrophils as a system of polydispersed spheres. Any single randomsection through a sphere will provide (if the section is thin enough) a single circular cross-sectioned profile with a measurable diameter. The profiles can be grouped by diameter into oneof perhaps 30 bins, each with a designated diameter range. These profiles in the largest diameterbin represent the population of neutrophils that are the largest and have been sectioned at ornear their exact centre. If 50 profiles of 500 measured profiles fall into this largest bin, theexpected number of profiles that should be found in the bin with the next smallest range ofdiameters can be estimated. If more profiles are found in the second bin than expected based onthe number found in the first bin, then a second population of neutrophils of a slightly smallermaximum diameter than the ones in the first bin must exist. If less profiles than expected arefound then not enough total profiles have been sampled. This process of matching expected toobserved number of profiles is continued until the last bin size is reached.Important assumptions for this technique are that the suspended and fixed neutrophils thatwere sectioned are spherical and that no neutrophil is sampled twice. While the exact 3-dimensional structure can not be determined the cells did in fact appear quite spherical. Thesections were guaranteed to never contain profiles from the same neutrophil by trimming theblocks between section cuts by a thickness larger than any expected neutrophil diameter.This process was verified by simulating random sections through a computer generatedsystem of spheres. A collection of random polydispersed spheres, diameter 6.39 ±0.70, wassectioned in the described way yielding a calculated diameter of 5.74±1.25 when only 300profiles were used. The technique appeared to systematically underestimate the diameters by3210% and neither increasing the number of profiles measured nor narrowing the dispersion of thesimulated spheres had any effect. Therefore, our measurement of PMN diameter is likely to bean underestimate of the actual value.The frequency distribution of neutrophil diameters in the pooled data of the three humansstudied is shown in Figure 21. Also shown in Figure 21 is the frequency distribution of capillarysegment diameters. The estimated human neutrophil diameter when the neutrophil assumed aspherical shape was 6.8±0.8jLm(N=308) with an estimated volume of 174 fL. In contrast to theneutrophil diameter, the capillary segment diameter was 7.48 ±2.31gm with a length of14.4 ±5.84m (N=352), a volume of 632.5fL. Approximately 38% of the capillary segmentswere smaller than the mean spherical neutrophil diameter. This fraction is probablyunderestimated because the neutrophil diameter is slightly underestimated. Finally, no bivariaterelationship between the capillary segment lengths and capillary segment diameters was evident.These data show that the PMN diameter is larger than the capillary segment diameter andthat some type of PMN deformation is required for the PMN to enter most capillary segments.Figure 22 shows an entire alveolar wall with its associated capillary network. What isimmediately obvious from the examination of this and other such sections is the immensenumber of potential pathways that a cell could travel while traversing the capillary bed.Therefore, it would be of interest to know the effect of obstruction of individual segments of thecapillary bed on the flow through those that remain open and on the pressure drop across thenetwork of segments containing the obstructed units.333.2 Effect of Plugging in the Capillary NetworkStaub and Schultz (83) found that between 10 and 20 alveolar walls of 5-8 alveoli arecrossed when travelling from a terminal arteriole to a venule in the canine lung. This means thatthe network of capillary segments in an individual alveolar wall such as that shown in Fig. 22must connect with a similar network in other alveolar walls to form an interconnecting grid of10-20 walls which provide passage from the terminal arterioles to venules. Figure 23 showsan example of how each wall was conceptualized for computational purposes. Each alveolar wallwas arranged as a small grid of interconnecting pipes and the walls themselves were consideredto be interconnected by corner vessels which had diameters larger than the neutrophils. The inputto each of these simulated walls was taken to be either an arteriole or corner vessel and the exitwas another corner vessel or venule. Two of the bordering sides were taken to be inputs and theremaining two were outputs, these sides were always randomly chosen for each alveolar wallin the model.The development of a computational model of the entire capillary network is obviouslyan impossible task. Instead, either small representative sections must be modelled or a single cellmust be able to pass through the pulmonary circulation without any need to worry about the‘history’ of previous cells. If the previous passage of cells has little or no effect on subsequentcells then a model can be designed that dynamically constructs the capillary bed in front of eachcell as it traverses the microcirculation.34Therefore, determining if a cell trapped within the alveolar wall would alter theeffective resistance between any site in the wall and the exit became a major focus of thesestudies. Examination of enface sections of alveolar walls similar to Figure 22 showed that eachalveolar wall contained between 40 and 80 segments. If the alveolar wall is constructed as asquare matrix of interconnecting pipes using a 5 by 5 array (Figure 23) it contains 60 individualcapillary segments. Array sizes of 4 by 4 contain 40 segments and 6 by 6 contain 84 segments,the upper and lower limits for the number of segments in each alveolar wall. For each arraysize, blood flow can be modelled as current by considering each segment as a resistor, the inputsides (either the arteriole or corner vessels) as a voltage source and the output side (either cornervessel or venule) as ground. If a neutrophil was to become stuck, effectively eliminating asegment from the network for a finite period of time, then that segment would appear to havean infinite resistance.With each segment in Figure 23 replaced by a resistance, an electrical circuit model wassimulated using the electrical circuit simulator PSPICE (version 5.1, January 1992,, MicroSimCorp., Irvine CA). The exact values used for resistances and voltages are unimportant as theycan be easily scaled to realistic values, and our goal was to compare the matrix sizes andcharacteristics. Each resistance was modelled as a 10 ohm ideal resistor with a tolerance of 50%.This ensured that a random pattern of resistors between 5 and 15 ohm populated the circuit. Theinput voltage was set at 10 volts. In this model a node is the junction of individual capillarysegments within a single alveolar wall. Surprisingly, the different network sizes, 4x4 5x5 and6x6, had little effect on the total resistance across the network. The larger the grid size, the35smaller the total resistance, due to the vast number of parallel pathways, but the total resistanceacross the 5x5 grid was 90% of that across the 4x4 grid and the 6x6 was 83% of the 4x4 grid.Table 10 shows the resistances from each node to the output ground in each of the threeconfigurations. In all of the circuits, the location of the input “walls” and output “walls” in eachconfiguration has been held constant to allow for easier comparison of the tables but this has noeffect on the qualitative outcome. In all examples, the resistance within the network is amaximum of about 7 ohms in the upper left corner and falls to about 3 ohms in the lower rightcorner. Therefore, the actual network size has very little, if any, effect on the resistance acrossthe network.To model the effect of neutrophil plugging, multiple simulations were run in which 10%or 25% of all the resistors in the network were randomly eliminated and replaced with opencircuits. Figure 24 shows the location of the resistors eliminated in the 5x5 network. Multiplesimulations were performed in which the locations of the plugs were randomly shifted. Theresults showed that plug location had no effect on the qualitative pattern of changes inresistances across each segment. The relative increase in total resistance between a plugged andunplugged network was approximately 1.2 for the 4x4, 5x5 and 6x6 networks with 10% plugsand between 2.0 and 3.5 relative increase for 25% plugs. In this model the resistance from eachnode to the output has not reached infinity because while certain segments are occluded, manyalternative pathways still exist from each node to the output. Once again the network size hadno effect on total resistance and the wider range for 25% blockages merely reflects the greaterpossibilities to remove entire portions of the network. Table 11 shows the relative increase from36each node to the output for the three grid sizes for 10% blockages and Table 12 for 25%blockages. These tables show that, the relative increase in resistance for each segment is thesame throughout the grid, indicating that even with 25% blockages, the pattern of resistancedrops within the network has not been affected by blockages. To show this more clearly, table13 has scaled the values for the 5x5 network in the following manner:Rn = resistance from each node to the output without blockagesR* = total resistance without blockages from input to outputrn = resistance from each node to the output with blockagesr* = total resistance with blockages from input to outputRnIR* = resistance at each node (without blockages) scaled to total resistancern/r* = resistance at each node (with blockages) scaled to the total resistanceThe values shown in Table 13 are:(rn/r*)/(Rn/R*)If this value equal 1, then the pattern of resistance, with blockages, has not been altered relativeto the pattern without blockages. The values in table 13 are all between 0.8 and 1.1, indicatingthat 10% and even 25% blockages have virtually no affect on the resistance pattern within thenetwork.The direction of flow must also be considered when modelling the capillary network. Ateach node in the model shown in Figure 23, it is reasonable to assume that 2 vessels would flowinto the node and two would flow out from the node, at least on average across time and space.To model this situation, the model was designed to allow unidirectional flow through the37network. Obviously, this reduces the number of available pathways for current flow (blood flowby analogy) through the model. The relative total network resistance produced by adding thisconstraint was 1.5 times regardless of network size. The remarkable features of this newnetwork can be seen in table 14 which shows the relative increase in resistance between a modelwith 10% blockages and a model with no blockages. The node at position 3,3 in the 6x6 gridhas an infmite resistance because both paths leading from this node were blocked. This a worstcase example of a blockage. Notice that relative increase in the surrounding nodes has increasedby a factor of 1.7 or 1.8 and that 2 segments away from the plug have relative increases inresistance of 1.3 to 1.4. In the smaller grid, 4x4, the relative increases in resistance become2 to 3 times that without blockages. The corresponding result for 25% blockages is shown intable 15 where very high local resistances are seen around neutrophil blockages with little effecton areas a short distance away.These results suggest that while local areas of increased resistance occur in the region ofPMN plugs, most capillary segment resistances within a single alveolar wall are unaffected.Therefore, the presence of a PMN plug within a single alveolar wall can be ignored and it canbe assumed that moving PMN will flow around segments that are blocked.383.3 Modelling the Effect of Lung Height on Capillary SizeGlazier et a! (33) have measured the difference in capillary segment diameters at variouslung heights in dog lungs under Zone III conditions. His data, redrawn in Figure 25, clearlydemonstrates a plateau in the size of the capillary diameter at a location which corresponds tothe bottom of the dog lung. This relationship between height and capillary pathway diameter isparalleled by an increase in the margination of PMN in the upper lung regions relative to thelower lung regions (62,63). However, the available measurements of capillary segmentdiameters (Figure 21) were made on tissue taken from lungs where the arteries and venous bloodsupply was simultaneously occluded under Zone III conditions and the lung was filled with liquidfixative. In animals this procedure fixes the capillaries in a distended state but there is novariation with lung height as there is liquid pressure on both sides of the vessel wall. Thismeans that the capillary segment diameters from Figure 21 must be scaled to account for thedifferences in diameter in the gravitational plane observed in air filled lungs (33).In Figure 26, a smooth curve has been fit to Glazier’s data which has been rescaled andshown to plateau at 100%. The total vertical gradient studied by Glazier (Figure 25) was 47cm,larger than the vertical height of a normal human lung in the upright position. Assuming that anormal lung measures 30cm in height from the top to the bottom, five sections of 6cm each weretaken from the centre of Glazier’s data. The dotted lines on Figure 26 indicate the location ofthese even divisions to divide the regions from the top of the lung down towards the lung basein an upright human. The midpoint value of the curve in each of these regions was taken to39represent the percentage the average capillary segment in that region would be relative to fullydistended capillaries in the bottom of the lung. Taking values from the middle of Glazier’s datameant that the very small values and the very largest values for capillary segment diameters werenot used. Table 16 shows the values used for regional capillary sizes up and down the lungbased on Glazier’s data.When modelling flow through capillary networks in each of these five regions it will beassumed that all capillary segment diameters are from a uniformly distributed population witha mean and standard deviation as given in Table 16.403.4 Neutropliil Deformation TimeThe amount of flow through a network is based upon the pressure drop across the networkand the resistance of the vessels within the network. Due to the wide discrepancy in capillarysegment diameters and PMN diameters, many PMN will obviously encounter vessels too smallto pass through without deformation. The time required for the PMN to deform from its currentdiameter to a diameter equal to that of the capillary segment will delay the PMN relative tofaster deforming RBC. The actual time required for the PMN to deform is likely a function ofthe PMN diameter/capillary segment diameter and the pressure drop across the PMN.Warnke and Skalak (90,9 1) and Skalak (81) have made a network model of the systemiccapillary bed. Their model was based on a grid system of 260 nodes with a single feedingarteriole and a single collecting venule. In Warnke and Skalak’s work, they computed thepressure in each of the 260 nodes and estimated the added pressure across a single WBC whenit became plugged in a capillary segment. This study has used a similar approach to Warnkeand Skalak (92) and that of Lipowsky and Zweifach (58) to model the pulmonarymicrocirculation.Needham and Hochmuth (70) have calculated the duration a WBC is present in a pluggedsituation while entering a small diameter pipet as a function of pipet size, cell viscosity and thepressure drop across the cell to be:T&f (seconds) =R3(4/3)i1{1 - (1 + InR3)/R}/P EQN[14]where41R = WBC diameter I capillary segment diameter= viscosity of WBC= pressure drop across WBCTo confidently use this equation information must be available on the viscosity of PMN and thepressure drop across the PMN and PMN viscosity when plugging occurs. Warnke and Skalak(90) have made many measurements on WBC viscosity and found values ranging from 45 to3000Poise. Several other authors (14,22,80) have also attempted to measure WBC viscosity butactual published values vary considerably and Warnke and Skalak finally used a mean value ofl200Poise in their models. The pressure drop across a plugged PMN is also very difficult toestimate. Schmid-Schonbein (76) has estimated the required pressure drop across a granulocyteto clear the cell from a capillary segment to be in the range of 6-8cmH2O. Under theseconditions, where the available pressure is considerably less than the required pressure, theestimated pressure drop across the plugged cell would be equal to the pressure drop inneighbouring segments that remain unplugged (76).In section 2.5 the total pressure drop across all large vessels was estimated to be 3mmHgleaving a pressure drop of 5mmHg across the pulmonary capillary network if the total pressuredrop is SmmHg. Staub and Schultz (83) has shown that the minimum number of capillarysegments between an arteriole and a venule is equal to about 100 capillary segment lengths. Ifthe pressure drop across the capillary bed is uniformly distributed across all capillary segmentsthen the maximum pressure drop across each segment is 5mmHgIlOO segments =0.O5mmHg/segment. If, in equation 14, the pressure drop across the plugged cell is then just42the pressure drop in adjacent capillary segments, 0.O5mmHg, and the minimum estimated cellviscosity of 45Poise is assumed then extremely long plug durations of hours or days arepredicted.Fenton et al (26) have also investigated the time required for a WBC to be aspirated intoa pipet. In their studies the time required for WBC of various diameters to enter pipets rangingin diameter from 4-7gm was recorded. The aspiration pressures used ranged from 200 to 400Paor 1.5 to 3 mmHg. This pressure is likely high relative to the actual pressure drop across theplugged cell but this should then estimate plug durations which are shorter than expected. Theempirically derived equation proposed by Fenton et a! (26) was:time to deform (seconds) = (1/1.6)exp(3.68*R4.77) EQN [15]where:R = WBC diameter / pipet diameterIn this equation, a log relationship was found between the time to deform to enter an aspirationpipet (slope=3.68, intercept =4.77). A correction factor of 1/1.6 was also empirically derivedto correct for the temperature difference between the majority of the experiments (made at roomtemperature) and a smaller collection of studies at physiological temperatures (37 degrees).Figure 27 shows the estimated deformation times versus the WBC/pipet diameter ratio.It was also evident that the estimates of deformation times of Fenton et al (26) varied byapproximately 30% over the entire range of pressures and WBC/pipet diameters recorded. Nopressure effect was noted.43The model presented in this work uses equation 15 to estimate the required time to deforma PMN from a starting spherical cell diameter to the diameter of the capillary segment.443.5 Capillary Bed Network ModelUsing the preceding information, it is now possible to propose a model that can be usedto simulate the capillary network. The electrical circuit simulations provide reasonable evidencethat capillary plugs have little if any affect on resistances to most capillary segments within asingle alveolar wall. While local areas of high resistance likely occur, it is reasonable to assumethat these elevated resistances appear not to extend to other capillary segments. This modeltherefore, ignores the effect of neutrophil plugging and assumes that any plugs within individualcapillary segments merely redirects flow to unplugged regions of alveolar walls without anychange in capillary pathway resistance. The flow chart for modelling the capillary bed was asfollows:1) A PMN of random size based on the morphometric measurements was chosen.2) This PMN was assigned a fixed, constant velocity, between 75OumIs and 2000um/sthroughout the capillary bed. Previous work by Staub and Schultz (83) has shownthat this range of velocity of blood is reasonable. Work by Lien et al (56,57)using intravital videomicroscopy has demonstrated that the PMN move rapidlythrough the capillary bed, stop completely, then return to a relatively rapidvelocity. Figure 17 from the large vessel portion of this simulation also showprecapillary blood velocities within this range.453) A random number of alveolar walls between 10 and 20 was assigned to this PMNfor it to pass through based on the data of Staub and Schultz (83).4) Each wall was randomly assigned a size of 4x4, 5x5 or 6x6 with equal probability.5) Each alveolar wall had two sides randomly chosen as inputs and two as outputs.6) All capillary segments in each wall were randomly assigned a length based onmorphometric measurements.7) All capillary segments in each wall were assigned diameters such that the minimumpathway diameters approximated the beta distribution and the total capillarydiameter distribution matched morphometric results after they had been scaled toaccount for lung height. To force a beta distribution for the minimum capillarypathway diameter a uniform distribution of capillary segment diameters wasassumed ranging from the mean value in table 18 to ±2 standard deviations of themean. The distribution shown in Figure 21 for capillary segment diameter appearsquite broad so a uniform distribution does not seem unreasonable. This stepdetermines whether the simulation will represent a collection of pathways from theupper or lower lung regions.8) The PMN randomly entered the grid along either of the input walls.9) If the cell diameter was less than the capillary segment diameter then the lengthof the capillary segment traversed is recorded. If the PMN was larger than thecapillary segment then the time to deform was computed using [EQN 15] and thistime, and capillary segment length was recorded. Previous work by Evans (23) has46shown that the time for a granulocyte to recover back to its spherical shape afteraspiration into a pipet exceeds 12 seconds. Therefore, once a PMN had deformedto a size smaller than its full spherical diameter it remained at that size until a stillnarrower capillary segment was encountered.10) PMN travelled through the model in a unidirectional manner where only twocapillary segments from each node were possible exits.ii) Yen and Fung (108) have studied the separation of gelatin pellets in relatively large(0.32cm-0.64cm diameter) branching tubes. They found that the discharge tubehaematocrits of gelatin pellets could be related to the feed tube haematocrits andthe discharge tube velocities by the following equation (derived empirically):H1/H2-1 = a(vl/v2-1)where:Hi and H2 are the discharge tube haematocritsvi and v2 are the discharge tube velocitiesa is a nondimensional parameter meant to account for cell diameter/tube diameter ratio,feed tube haematocrit, cell shape and cell viscosity.47Yen and Fung (108) found that when the discharge tube flow velocity ratios were lessthan a critical ratio of 3:1, the above equation predicted the experimental results.Unfortunately, no experimental data are available which provide the simultaneous flowvelocity ratios in capillaries at a given time point. Therefore, we assigned a constantvelocity to each cell as it traversed the capillary bed. Furthermore, it was assumed thatthe velocity of adjacent capillaries was identical. Under these conditions, the data of Yenand Fung show that the discharge tube haematocrits of the capillaries would be identical.Unfortunately, it is not possible to infer from the work of Yen and Fung how a PMNwould behave in such a system. With these factors under consideration, RBC and PMNwould enter the branches of a bifurcation (where discharge tubes had identical lengths)in proportion to the ratio of the cross-sectional areas of the branches. Mayrovitz andRubin (65) have made a more physiologically appropriate investigation into thedistribution of leukocytes at small (5-2Oum diameter) branches by imaging cells flowingthrough systemic vessels. In their studies, only a very slight dependence was seenbetween flow and cell concentration and no critical threshold was observed as in thestudies of Yen and Fung.It was arbitrarily decided in the current study that for two vessels of cross-sectionalarea Al and A2, a cell would enter vessel Al with probability: A1A2/(A1A2+A2A2),which would divide cell flow relative to flow velocity. This equation likely overestimatesthe division of PMN into branch vessels but the true cell distributions into branches atdivision are not known. This equation is arbitrary but does not seem unreasonable. It is48also interesting to note, as indicated in the thesis, that all of these studies which estimateflow division at a bifurcation are only crude approximations of the true situations. In atrue branching network such as the pulmonary vasculature there is little relationshipbetween how flow divides at a bifurcation and the size of the vessels of the bifurcation.The dominant factor ruling how flow will divide at any division is the apparent resistanceof each pathway. This resistance is obviously a complex function of all distal branchesof the network at a particular bifurcation.12) Once a PMN had completely traversed a single alveolar wall it immediately arrivedat the next with no time delay.Using the above flow chart to control how cells travelled through the capillary bed acomputer program was written in Gauss VM386 (Aptech Systems, WA) which generated acapillary network in front of a PMN as it passed from a terminal arteriole to a venule.Figure 28 shows a histogram of the number of segments a PMN, or RBC traversed ineach pathway and Figure 29 the total path length travelled. The distribution of transit times forRBC, which were assumed to require no time for deformation, is shown in Figure 30. Themedian transit time for RBC was 0.8s with an interquartile range of (0.6-1.1) seconds.As expected, the PMN transit times shown in Figure 31 are much longer than those forRBC (Figure 30). Figure 31 also shows the effect of regional differences in capillary segmentdiameters in the upper and lower lung regions. PMN taking longer than 10,000 seconds are notshown. Figure 32 shows the distribution of the number of stops for both the upper and lower49lung regions and Figure 33 the average time per stop. The values for time per stop, totalcapillary transit time and number stops for the five regions of the lung are summarized in table17.504 Total Pulmonary Transit TimeThe calculation of total pulmonary transit time requires that the three separate models,arterial, venous and capillary, be combined into a single system. The arterial and venous treeshave very fast transit times and PMN should have no difficulty traversing these regions becausethe vessels are much larger than the PMN. The distribution of RBC transit times for the arterialsystem is shown in Figure 34 and for the venous system in Figure 35. The median arterial andvenous transit times were about 0.7 and 0.8 seconds respectively. Once the distribution oftransit times through each of the three regions was known, the total pulmonary RBC transit timeswas taken to be the mathematical convolution of the large vessel transit time and the capillarybed transit time. It was also assumed that short arterial pathways are joined with short venouspathways but that no preference in capillary path length was seen for any length of arterial orvenous path.In order to determine the amount of flow that should be delivered to each of the fivegravitationally determined regions of the lung it is necessary to know the relative flow to eachlung region. Anthonisen and Milic-Emili (3) have mapped the regional disthbution of pulmonaryperfusion in upright humans. This data was used to partition the fraction of the cardiac outputto each of the five lung regions. Table 18 shows the proportion of flow to each of the lungregions.For computational purposes 10,000 RBC and PMN were passed through the arterial,capillary and venous beds and these cells were delivered into the five possible regions based on51the fraction of blood flow distribution described in table 18. Figure 36 shows the estimatedtransit time distribution of RBC and PMN through the entire pulmonary circulation where themedian RBC transit time is 2.1 seconds and the median PMN transit time is 8.6 seconds. Thedifferences in transit time for PMN in each of the lung regions is summarized in Table 17 wherethe median PMN transit time was 64 seconds at the top and 5.3 seconds at the bottom of thelung.The results of the model were compared to the experimental results reported by MacNeeet al from our laboratory (59) where we obtained an estimate of the distribution of RBC transittimes in the resected human lung specimens. These subjects, all of whom required lung surgeryfor lung cancer, were studied preoperatively where they received a bolus injection of Tclabelled RBC into the median basiic vein of the arm. The radiolabelled cells were observedpassing through the right and left ventricles using a Siemens ZLC 3700 gamma camera with awide field of view. The distribution of RBC transit times calculated by deconvolution of thetime-activity curves recorded over the right and left ventricles compared favourably to thosedetermined from measurements of blood volume and flow made on individual pieces of resectedlung. Figure 37 shows the pooled frequency distribution for 718 samples from eight lungs fromthis study where the mean and median transit times for RBC were 4.9 and 3.5s respectively.More recently, Hogg et al. (43) have used a similar approach to calculate the capillarytransit times of both RBC and PMN using quantitative histological techniques to determine thenumber of RBC and PMN/ml of capillary blood andT’MAA to measure regional blood flow.The number of RBC or PMN delivered to the capillaries of each lung sample was calculated52from the isotopic determination of blood flow and the peripheral blood RBC or PMN count.The regional transit time obtained by dividing cells/mi by cells/mi/sec) showed a distribution oftransit times that ranged from 0.03 to 14.5seconds with a median of 1.2 s for RBC and from3.3-935 seconds with a median value of 129s for PMN with 18% overlap of the values for eachcell type. These experimental values for RBC PMN capillary transit will be compared to thoseobtained by the model as well as others that are available in the literature in the discussionsection.535 DiscussionThe computational model described in the thesis allows the transit time of red blood cellsand neutrophils in the arterial, capillary and venous components of the human pulmonaryvascular bed to be estimated. A large vessel model based on anatomical data for the arterialbranching system was used to construct a set of probabilities which would describe the branchingnature of the arterial network. With the arterial network specified, flow continuity and energybalance equations were used to estimate the total pressure drops through a wide range of possiblepathways in this system. When flow had reached the capillary bed, a strictly stochastic modelof cell transit through a randomly generated grid network was used to simulate capillary flow.Finally the flow was returned to the left atrium by making a slight modification on the arterialnetwork to represent the venous system.The information required to develop the model was accumulated from several differentsources and the net result was based on anatomical geometry, fluid flow mechanics andprobability theory. The results obtained with the model were then compared to experimentalobservations from our own and other laboratories.545.1 Large Vessel ModelThe anatomical structure of the human pulmonary arterial tree shown in the angiogramin Figures 1, 2 and 3 is extremely complex. The model presented in this thesis is based on theassumption that a modified Horsfield branching structure could be used to represent this veryasymmetric tree. However, any number of asymmetric patterns could have been proposed thatwould have resulted in similar path length distributions and pressure drops.This model uses a set of branching probabilities which detail whether a division is abifurcation or trifurcation, and the relative size of the daughter vessels in relation to the parentvessel, based on anatomical measurements. The probabilities in tables 2,3 and 4 were estimatedfrom the available cast data and strictly determine the number of branches the model estimatesat a given order. If these probabilities are incorrect then the estimated number of vessels in theorders will be in error, also, the estimated relationship between parent and daughter branch sizeswill also be incorrect. Detailed Horsfleld ordering of the human arterial tree has not beenpreviously published so no direct comparison between this work and others can be made.However, an important feature of many naturally occurring branching systems is a linearrelationship between the log of the number of vessels and the order (78). Figure 10 clearlyshows that the proposed model has this important feature. Furthermore, the number of terminalarteriole endpoints (order 1 vessels) estimated from the model can be compared to availablemorphometric estimates made by Weibel (95). Even small shifts in the probabilities shown intables 2,3 and 4 result in dramatic changes in the number of terminal arterioles, the most55sensitive probability being the relative fraction of bifurcations in the arterial tree. Since thenumber of estimated precapilary arterioles closely match those of Weibel (95), it has beenassumed that these values provide a valid set of branching rules which accurately describe thearterial tree.The method used to represent the branching structure of the pulmonary vasculature is onewhich can be best described as a random model. Other investigators have made models of thepulmonary vasculature which have included detailed mathematical descriptions of the observedbranching network. Two alternative methods for representing the branching nature of this systemare Woldenberg’s spatial hierarchy (105, 106, 107) and the fractal descriptions of Krenz (52,53)and Glenny (36, 37).In Woldenberg’ s (105,106,107) representation of systems of rivers, or pulmonary vessels,he noted that streams (or areas) join to form larger rivers and that using the ordering schemesdescribed earlier in this thesis that as the stream, or vessel order, increase (larger vessels) thenumber of these vessels declines geometrically. Also, the ratio of areas between successiveorders can be related to the branching ratio of these orders. Hierarchial structures such as thesecan be described as stochastic models, random models such as used in this thesis, or, asWoldenberg uses, work related space filling models based on the hexagon theory of Christaller(15). Christaller developed a theory of central economic places (towns) and their associatedmarket areas and discovered that the optimum partitioning of the space was achieved usinghexagonal shapes. Larger areas, or structures, are then represented by larger groups orcollections of hexagons. Christaller proposed that a large hexagon may contain 3,4 or 7 smaller56hexagons but Woldenberg has extended this to much larger sizes. The best description of thehexagonal area is given by Woldenberg (105,107):Suppose we are given a hexagonal area. Let us divide the area into equalhexagonal areas. When the ratio of these areas, RA, equals three, space may befilled with hexagons. Ratios of less than three have been observed in the humanairway. RA can be less than three, if one cuts the surface and excises some area.This is possible in the lung, which can be though of as a cut surface, which isfolded into alveoli and connected to tubes. A riverflows on a bent, but continuous,surface. Thus no area is excluded, and the river must have RA 3.The value ofRA can never be less than iwo becausefor branching systems twobranches of equal order are required to elevate a branch to higher order. RA <7,because small outside branches tending to flow to the centre would eventually joinwith branches in the ring ofsix hexagons surrounding the centre and create higherorder branches before RA could exceed seven. For analogous reasons RAfor marketareas cannot be less than three (unless there are unobserved areas) nor greaterthan seven. In the latter case the intervening towns would become higher order.This model may be extended to organic branching systems. Rivers are threedimensional systems, although they appearplanar on the map. While rivers, trees,lungs etc., are three-dimensional, functionally each system operates on or throughsurfaces. For instance, operationally the lung is considered to be a surface through57which CO2 and 02 are exchanged and is served by channels which may becompared topologically to a river system. In the capillary bed in the lung, thenetwork vasculature is clearly hexagonal. At the centre ofeach hexagon is a pointmost completely removedfrom the surrounding hexagonal syncytium. Since we mustfill space, circular channels are not allowed and this length of capillary channelsurrounding each centre point is a minimum for the area enclosed; however, thesize of the perimeter is adjusted to allow the centre point to be perfissed withoxygen. Allow a branching network to be attached to the syncytium, only the smallbranches are connected to the synctium. These small branches join to form largebranches, uniting small hexagonal areas into larger ones, and this processcontinues until only one branch is left so that all flows to orfrom the whole areaare united in this one branch. The model resembles a tree trunk with its rootsexposed, the smallest rootlets touching the hexagonal capillary system.These remarks are still speculative, yetperhaps they are operationally correct,since the available evidence seems to support the hypothesis of hexagonalhierarchies in urban, fluvial and organic systems.The spatial analysis proposed by Woldenberg is based on a geometrical approach thatallows for the description of large structures by applying inter-connected series of shapes(hexagons). These shapes are considered to be self similar regardless of whether one is lookingat the structure as a whole or only a small piece of the structure. Fractal structures are those that58display a self similarity that is invariant to scaling and therefore a logical extension of the workof Woldenberg is to consider the pulmonary vasculature as a fractal structure. Glenny (36, 37)and Krenz (52,53) have utilized fractal analysis to look respectively at pulmonary blood flow andthe pulmonary arterial tree.Krenz et al (52,53) modelled the relationship between the number of vessels at a givenorder, Nj, and diameter of vessels at that order, Dj ,and related these two values through a thirdconstant term 6= a1where a1 is the dimension of the largest vessel. They were able to further define the resistanceat each generation and the volume at generation all in terms of the descriptive parameter j3. Intheir analysis they estimated values of j3 between 2 and 3, smaller than the value of 3 whichwould be used by implicating Murray’s law (69) yet slightly larger than previous values foundby Horsfield and Woldenberg. Unfortunately, as noted by Krenz et al. (52), the estimates theyobtained for resistance and volume of the pulmonary vasculature are very sensitive to theavailable morphometric data and they conclude, as this work has, that far more detailedinvestigations to acquire this information would be a great benefit.Other investigations of the arterial tree (78,110) have used Strahier ordering schemes.While Stralfier ordering is a convenient method of describing complex networks this method has,as already described, the drawback that individual pathway lengths can not be estimated.However, in the arterial cast data of Horsfield (44), much of the individual vessel length datawas also missing. The small number of vessels for which length was available were dispersed59over a range of orders located in the large vessel region. No obvious relationship betweenvessel length and order could be clearly identified. The relationship between the ratio of vessellength and diameter also showed no obvious pattern. It was decided, based on the little lengthinformation available, that an average ratio between length and diameter, and the standarddeviation of this ratio, would be used to estimate length from diameter throughout the arterialtree. This is clearly a simplification that will effect path length and pressure drop calculationsthroughout this work. A 10% error in vessel length, if it was always in either the positive ornegative direction from the mean value, would result in a 10% error in the pathway length. Asimilar error in pressure drops would also occur. It seems unlikely that all vessels of a particularpathway would be uniformly shorter or longer and the standard deviation of ±30% allowedvessels lengths to vary between 80% and 200% of the vessel diameter 95% of the time. Whilethe distribution of arterial, and venous, pathways is not known, Figures 16 and 18 providevalues that seem reasonable for a human lung 30cm tall.This model has made the assumption that the venous system can be represented by thesame branching probabilities as were used for the arterial system. No cast data was available onthe venous tree which could be used to compare to an equivalently ordered arterial system usinga Horsfield scheme. However, previously published results, for both arterial and venousnetworks are available using Strahler ordering (47,78) and a comparison of these networks showsa strong similarity between branching structures. As with the arterial tree proposed in this work,the exact system used to describe the venous network is likely unimportant given that at leastthe essential asymmetric pattern is maintained.60Future investigations are required to obtain more detailed anatomical data. Idealinformation for modelling would include vessel lengths, diameters, branching angles and truespatial position within the lung. A convenient, non-invasive technique may now be available toobtain this information. Hatabu (39) has shown that high-resolution magnetic resonance imagers(MR1) can be used to obtain extremely detailed images of the pulmonary vasculature. Theseimages have a high sensitivity to fluid flow and are therefore ideal for the investigation of thepulmonary vasculature. When these techniques have been perfected they should allow rapidanalysis of the pulmonary vasculature under many different conditions.Flow through vessels in this work was assumed to be Newtonian, steady andincompressible. Blood is not a simple fluid, but a complex two phase fluid of cells suspendedin plasma. An important feature of blood is that its apparent viscosity changes as vessel diameterchanges. In this work the equations of Kiana and Hudetz (50) were used to correct for thiseffect. Because of the relatively large vessel diameters used in this model (even at theprecapillary level) equation 13 had virtually no affect and could be eliminated completely fromthe methods. Time-varying, pulsatile flow patterns also exists in the pulmonary circulation (29)and extend into the capillary bed. In addition, while the blood can be considered incompressible,the vessels through which it flows are themselves distendable. These effects have not beenconsidered in this work as the major goal was to first develop a framework upon which theserefinements could be made in later investigations.In the current work flow was assumed to vary as a function of the radius to the fourthpower (7,9,17,29,30,31,77). Uylings (88) has expanded upon the earlier work of Murray (69)61to present an alternative proposal where flow more closely relates to flow to the third power.In these studies flow through a vessel is modelled as strictly Hagen-Poisseuille flow with a workcost equal to the flow (proportional to the fourth power of the radius) times the pressure drop.An arbitrary additional energy requirement for the “maintenance of the blood” is added whichMurray (69) argues as follows:To study the antagonism between the friction and the volume of blood the latterfactor must be multiplied by a dimensional constant. Let b, then, be the cost ofblood in ergs per second per cubic centimetre of whole blood of averagecomposition (and let B be the cost in calories per day per cc. of blood). There is,as far as I can see, nothing arbitraiy about this step: it is certain that themaintenance ofblood requiresfuel. (The cost ofblood may, however, be a complexaccount distributed among such factors as the small metabolism ofthe blood itseñthe cost of upkeep of all the constituents, perhaps especially of hemaglobin, thecost of containing vessels, and the burden placed upon the body in general by themere weight of blood).These issues are not of a fluid dynamic nature. Furthermore, the studies of Murray (69),Uylings (88), Woldenberg (105,106,107) and Fung (29) in this area are not supported byempirical evidence of actual pressure flow relationships in blood vessels. Both Murray and Fungthemselves comment that the use of these cost functions is arbitrary. In the modelling results62of Fung only flow proportional to radius to the fourth power laws are considered. The data usedin this model, most notably those of Gelin (32) and Lipowsky (58) are experimental results thatsupport the use of flow as being proportional to flow to the fourth power and are consistent withcurrent findings in this area.The division of flow through an arbitrary bifurcation or trifurcation is extremely difficultto compute and usually empirical approaches are used. In this work flow was randomly chosenfor one of the daughter branches in a bifurcation (two of the branches in a trifurcation) and flowcontinuity (the flow in a parent branch must equal the sum of the flows in the daughter branches)was used to estimate the flow in the remaining branch. The distribution from which flows wererandomly selected was based on mapping all possible flow patterns in branches up to order 20.It was assumed that each terminal arteriole carried one unit of the total cardiac output, each unitbeing equal to the cardiac output divided by the number of terminal arterioles. This estimationobviously depends on each order 1 vessel having 1 unit of flow and no data is available on this.Previous studies (72,102,110) have taken a somewhat similar approach but they allowed onlya single flow to exist for branches at each order. This method, for the division of flow, issimplistic and has likely resulted in incorrect estimates of pressure drops. It is difficult to predictexactly how this has affected the results in the current work. It may be possible to select smallportions of the model presented here and solve exactly a set of equations which would show, atleast in small portions of the network, possible flow patterns. This may be another area whereMRI could provide useful quantitative results to assist in formulating a method for flow division.Work by Fei (25) and Ku (54) has shown that MRI can be used to map steady and non-steady63flow patterns both with and without pulsatile flows. While these kinds of studies have not yetbeen done on humans, their results show that very complex flow velocity profiles can beobtained and from these, pressure drops estimated.In the study of Yen and Fung (108) an optimal cost (minimal) is considered in relationto branch angle. Their studies find that there exists, given that all their assumptions hold(especially Murray’s (693) study), a set of angles that would minimize the energy loss throughthe bifurcation. Once again these studies are not supported by experimental evidence. In veryrecent study, Collins et a! (16) have investigated the pressure losses through a bifurcatingnetwork experimentally and theoretically. Their studies show that at the low Reynold’s numbersunder consideration in models of either the airways or pulmonary vasculature, virtually nopressure loss is associated with the bifurcation. This finding is completely consistent with fluiddynamic findings that pressure losses to divergent flow at low Reynolds numbers are negligible.This being the case, a minor pressure loss correction has been implemented for larger vesselsbased on the experimental study of Gelin (32).As previously mentioned, the lack of exact length information affects the accuracy of themodel predictions concerning pressure drops in the large vessels. However, this model estimatesthat only 3 to 4 mmHg of the total pulmonary pressure drop is located in the large vessels,approximately 50% of the pressure drop across the pulmonary circulation. Zhuang et a!. (110)used an asymmetric bifurcating model where flow was always divided 2/3 to one daughter and1/3 to another at each division but pathway length was estimated from a Strahier-orderedbranching scheme. In their study, they estimated that a total pressure drop 11.2 mmHg occurred64in the large vessels with very little remaining pressure drop across the capillary network. Directmeasurement of microvascular pressure obtained by Bhattacharya and Staub (6) show that at least46.4% of the total pulmonary vascular resistance was located across the capillary bed whichagrees well with results of the current study. The discrepancy between Zhuang et al and thiswork is likely due to how large vessel pathways are predicted. In Zhaung et al. (110) everypathway contained vessels of each order, while the study presented here estimates that each orderhas only a 1/3 probability of being located in a given path. If, at every order in Zhuang’smodel, the probability that any particular order was present in a given pathway is set to 1/3,then the total estimated large vessel pressure drop is between 3 and 4 mmHg, the exact valuefound in this work which agrees with Bhattacharya and Staub’s measurements (6).655.2 Capillary Network ModelThe model of the capillary bed presented in this thesis is based on the anatomicobservations that the lung parenchyma consists of an interconnecting network of alveolar wallswhere each wall contains a network of short capillary segments. The size of the individualcapillary segments and the numbers of these segments in each alveolar wall are based onWeibel’s (95) data which were recently confirmed in our laboratory (21). The minimumcapillary pathway lengths predicted by the model (Figure 32) roughly correspond to the distancebetween terminal arteriole and proximal venule reported by Staub and Schultz (83) whomeasured the linear distance between arterioles and venules in planar sections of the lung.The alveolar wall capillary network was assumed to be a square grid (Figure 23). Twosides of the grid were arbitrarily assigned as flow inputs (arterioles or corner vessels) and twosides as flow outputs (venules or corner vessels). Each intersection between capillaries wasalways assume to involve four individual capillary segments. Comparison of the grid and the realalveolar wall (Figure 22) shows that the grid is a simplification of the actual structure. Thethick sections examined showed that a single alveolar wail could easily have more than 4 sidesand that capillary segments meet in junctions of three as well as 4. Furthermore, the possibilityof many flow inputs around the outside of a single alveolar wall could allow local regions of asingle alveolar wall to have reduced or stopped flow with flow entering the alveolar wail fromadjacent arterioles or corner vessels. The presence of trifurcations in the alveolar wall also66increases the possibility of a single PMN temporarily blocking more than a single segment. Thiscould occur if two vessels flow into a third, and this third vessel became occluded by a PMN.It was also assumed that as either RBC or PMN traversed the capillary bed, no effect ofplugged capillary segments would be seen. Hogg (40) has estimated the size of the marginatingpool in man and has determined that even if the entire marginated pool was located within thelung capillary network then only 10% of all segments would contain a PMN. The resistornetwork simulations presented here indicate that blocking 10% of the capillary segments withina single alveolar, would result in very little increase in total resistance across the network ofsegments in that wall. Furthermore, the resistance around the plugged segments would notincrease dramatically due to the vast number of alternative pathways available. Even when flowis restricted so that each capillary segment carries flow in a single direction, the increasedresistance resulting from a plugged segment is transmitted to only a few other capillary segmentsin an alveolar wall. If the number of plugged segments approaches very high values, 25% ofall capillary segments, then more significant increases in resistances are possible. Based on thedata suggesting that only 10% of the capillary segments would be blocked at any point in time(39), it seemed reasonable to ignore the presence of plugged segments and assume that any cellstraversing an alveolar wall would avoid capillary segments filled with PMN.Lien et al. (57) have simulated the pulmonary capillary bed as a large network ofinterconnecting segments. They found that only 1 % of the segments were required to be blockedto trap nearly 50% of the neutrophils at least once. In their work, cells traversed the capillarybed by a system of random walks, however, cells were allowed to attempt to enter a plugged67capillary which would then impede the passage of the following PMN. In the present study, itwas found that ten times the blockages (10%) suggested by Lien et al (57) would not affect theresistances of surrounding capillary segments and therefore cells would not be impeded as theytravelled around occluded segments. This finding agrees with that of Warnke and Skalak (88)who also found that leukocyte plugging had little effect on blood flow resistance in a singlecapillary network. From the present model, we estimate that 25% of the segments would needto be occluded before sufficient areas of impediment would be created to delay following cells.This suggests that the delay of PMN in the lung is not affected by capillary plugging by otherneutrophils but is a function of the geometric difference in neutrophil and capillary segmentdiameter.The equation of Fenton et al (26) used to predict the time required for a PMN to deformsufficiently to enter smaller diameter capillary segments was determined experimentally and doesnot contain a pressure term. This makes the equation suitable for this study where no estimatesof pressure were available. Fenton developed this equation by aspirating WBC into pipets atpressures several times larger than those likely to be present in the pulmonary microcirculation,where the vast number of parallel pathways that exist in the capillary bed make it very unlikelythat any significant pressure difference exists across a capillary segment plugged by a PMN.Indeed, the calculations made by Schmid-Schonbein (76) show that the large pressure requiredto clear a PMN through the capillary segment that it has plugged must exceed the availablepressure drop across individual segments in the lung. For example, if the pressure drop acrossa single PMN plugged in a capillary segment is equal to the pressure drop in adjacent open68capillary segments (76) and the total pressure drop across the entire pulmonary capillary bed isonly 4 mmHg, then a typical pathway containing 100 capillary segments would have a pressuredrop of only 0.04 mmHg across each segment. This strongly suggests that the deformationwhich the PMN must undergo to negotiate narrow segments is an active rather than a passiveprocess.The time required for deformation is very sensitive to the ratio between neutrophildiameter and capillary diameter. The capillary and neutrophil sizes used in this study are inagreement with the previous measurements of Weibel (95) and Schmid-Schonbein (75) andextend that work by establishing the entire distribution of cell and capillary size fixed undersimilar conditions. Since the ratio of cell to vessel size is of prime importance to this model,the validity of the regional changes based on Glazier’s work (33) are also critical. Sobin (82)and Fung and Sobin (30) measured the changes in alveolar sheet thickness as a function ofcapillary-alveolar pressure differences in cat lungs and report changes that exceed thosedetermined by Glazier by approximately 15%. If Sobin’s data are correct, then the transit timedifferences due to gravitational effect in capillary size would be even greater than currentlyestimated, but the difference between the two data sets may be due to species effects.Unfortunately, no detailed information is available on gravitationally related differences inhuman capillary size.Using thoracic windows in dogs, Lien (57) has shown that about 40% of fluorescentlylabelled PMN pass through the lungs at approximately the same velocity as RBC and do notstop, while about 60% of the PMN are delayed by one or more stops. The results obtained with69the model show that the median number of stops for even the lower lung regions was 4. Whileit is difficult to determine why the results reported here differ from those of Lein et a!, a recentreport showing that the capillary network on the interior of the lung has narrower segments thanthose on the pleural surface (11) would be consistent with this difference. Another possibleexplanation is that the numerous short stops predicted by the model are related to anunderestimate of the rate of deformation provided by equation 15. Alternatively, the neutrophilcould rapidly deform down to some smaller diameter very quickly, maintain this diameter andtherefore encounter fewer restrictions that would cause it to stop. Indeed, allowing theneutrophil to deform rapidly down to a diameter of 4.6gm reduced the number of stops by 2 buthad little effect on transit time. With this rapid deformation, 30% of the cells were able totraverse the lower lung region with a single stop of only 1 second which may have been beyondthe measurement abilities of Lien et a!. (57).Harris and Heath’s (38) review of the data on total pulmonary blood volume and providesix separate estimates in humans where the group mean value is 250 and the range of 211-3 11ml/m2BSA. Dividing this mean by an average cardiac output of 2.5-3 IJmin/m provides a totalpulmonary transit times of between 5 and 6 sec. MacNee et al (59) from our laboratory reportedsimilar values when pulmonary transit times obtained with a gamma camera technique in the preoperative period were compared to measurements of blood volume and flow made on individualpieces of resected lung post-operatively. This data showed good agreement between the pre andpost operative measurements with both methods yielding mean values of between 4 and 5 sec.The post-operative measurements also provided a frequency distribution of transit times based70on 7.8 samples from 8 resected lungs with a mean of 4.9 and a median of 3.5 s. In an extensionof this approach, Hogg et al. (43) used quantitative histology to measure the concentration ofRBC and PMN/ml of capillary blood and estimated the transit times by dividing thisconcentration by the flow of cells. This method estimated the median capillary transit time forRBC to be 1.2 s with a range of 0.3 to 14.5 sec and PMN capillary transit times with a medianvalue of 129 sec with a range of 3.3 - 935 sec. These values for RBC transit time are slightlylarger than those obtained with the model (med 0.8 range 0.5-4.5 sec) but are reasonableconsidering the experimental results were obtained on elderly subjects with mild COPD whowere under general anesthesia. Also, the capillary blood volume estimates of Hogg et at. (43)are larger than those obtained by physiological techniques (49,55,64,66) and this would have theeffect of making the model results agreeing closer with experimental results. However, thedistribution of PMN capillary transit times estimated using the model is shifted much moremarkedly to the left of the experimentally determined results (median 7.8 vs 129 sec). Thissuggests that the model underestimates the time the majority of the PMN spend in the pulmonarycapillary bed because it does not consider all of the factors which control the time PMN spendin the capillaries.One possibility is that PMN must actively deform and become motile to pass through therestrictions they encounter in the capillary bed. In an investigation of the random migration ofPMN, Manderino et al. (61) showed that PMN can travel approximately 1600 jm in 2 hoursor 0.22 gm/second in the absence of a chemotactic gradient. If a PMN managed to force itselfpartially into a capillary segment before it occluded flow in that segment, at least half of the71capillary segment would remain to be traversed. Travelling at a rate of 0.22 sm/second, a PMNwould require approximately 34 sec to travel half the length of a 15 jm capillary segment whichwould bring the model predictions much closer to the experimental result.If active cell deformation and movement are required for PMN through capillaryrestrictions, any intervention in the motility of PMN would increase PMN margination.Preliminary evidence in support of the concept has been provided by Inano et al (46) whoreported studies using coichicine to inhibit the microtubule assembly of the PMN and found thatthis reduced their ability to move through filters and increased their margination in the lung.Under normal physiological conditions, the number of neutrophils entering the pulmonarycapillary bed must equal the number of neutrophils leaving it. Furthermore, while there is agreater retention of neutrophils through the upper portion of the lung, the actual number of cellsper gram of tissue retained is virtually constant. The volume of cells in each lung region canbe estimated by multiplying the average capillary transit time for each region (Table 17) by thenumber of cells contained in the fraction of the cardiac output delivered to that region (Table18). The units are in numbers of cells/percentage of blood flow and can be ignored because thefraction of PMNs in blood delivered to all lung regions is identical. The results from top tobottom of the lung are 2.4, 2.0, 2.0, 1.9, 1.9 and indicate that there is little difference in PMNconcentration in each region. This result agrees with the measurements of Martin et al (62,63)who showed the number of marginated PMN per gram of tissue changed little with lung height.Therefore, the estimates of regional concentration of marginated PMN obtained in different lungregions, are consistent with the available data.72The electrical network simulations reported here are very similar to earlier studies byWest (92) where they also used a resistor grid network to model flows, pressures and resistancein the capillary bed. As with West’s earlier work, our simulations show numerous “reverseflow” conditions where electrical current gradients temporarily reversed direction however, itwas not possible to determine if this was an artifact of the method used to solve the equationsor true reverse flow. West et al (100) and Warrel et al (93) also comment in their studies thatthe patchy perfusion pattern observed in lungs may be the result of capillary recruitment. Thepresent study suggests an alternative explanation for the mechanism of capillary recruitment.When a single neutrophil stopped within an alveolar wall, the input resistance to the entire wallincreased. This would tend to alter the pattern of blood flow away from that alveolar wall to theadjacent walls. Uniform flow would then resume when either the neutrophil plug left thealveolar wall or all alveolar walls in a region of lung developed equal resistances. Thismechanism predicts that entire alveolar walls, or portions thereof, could be quite easily recruitedor derecruited in relation to the movement of neutrophils. The longer transit time of neutrophilsin the upper lungs regions would imply more derecruited regions and possibly account for theknown ability of the upper lung regions to increase blood flow remarkably during exercise.Under conditions of exercise blood vessels become fully dilated as flow rates increase toa maximum (99). This makes neutrophil plugging difficult and prevents derecruitment of vesselsby plugging. Several studies (27,68,72) have shown that circulating PMN counts are elevatedby almost 42% and that pulmonary vascular resistance decreases by approximately 20% duringexercise. The network simulations presented in this study suggest that if the capillary segments73in the pulmonary microcirculation that were blocked, by marginating PMN, were sufficientlydilated to release these cells into the systemic circulation a 20% reduction in resistance wouldresult. A rapid increase in circulating PMN as a result of demargination of PMN from the lungis consistent with the data of Muir et al (68). Although others have suggested that theleukocytesis of exercise is due to demargination from the spleen (73), this seems less likely inour view because a similar WBC response to exercise can be observed in splenectomizedsubjects (84).745.3 Total Pulmonary Transit TimesThe estimate of total time that PMN and RBC take to traverse the model of the pulmonaryvasculature shown in Fig. 36 was obtained by combining the individual models of the capillaries(Fig. 30 and 31), arteries (Fig. 34) and veins (Fig. 35). Fig. 36 also illustrates the difficulty ofstudying PMN transit using indicator dilution technology because many PMN have theopportunity to recirculate before the remainder complete a single transit through the lung.Comparison of the model results to experimental values obtained in dogs with labelled PMN andRBC (Fig. 37) up to the time of recirculation of the RBC shows important differences. Theexperimental data shows that the PMN return to baseline whereas the model data shows a shiftin the two curves with steadily rising PMN values at the time one might expect RBCrecirculation. This suggests that the PMN transit through the lung has a biphasic nature whichis not accurately represented by a simple rightward shift of the PMN transit times by the model.Studies of PMN transit through the pulmonary circulation of humans has been attemptedusing a gamma camera technique. Although the initial studies (59) suggested a very lowretention of PMN during the first transit through the lung, it has now been established that thiswas the result of an error in technique (42). The correct analysis shows that the extraction andretention obtained using the gamma camera technique in humans are much closer to thoseobserved in the animal studies where blood was sampled directly. Therefore, if it were possibleto compare RBC and PMN transit using the indictor dilution technique with direct sampling ofblood in humans, the results would be very similar to those obtained in animals. This means75that a simple shift in the distribution of the PMN transit times to the right of the RBC transittimes predicted by the model in Fig. 36 is an oversimplification.The model also predicts a greater proportion of the PMN coming through with the RBCthan has been observed experimentally. Recent estimates of RBC and PMN transit times in thehuman lung (43) suggest that approximately 18% of PMN have transit times similar to RBC.The model predicts values that are much higher than this because there is far less delay in thecapillary bed. This means that the model underestimates the time that PMN spend in capillariesfor reasons that could relate to the time they need to become actively motile or to a delayproduced by adherence between PMN and endothelial cells not taken into account by themodelling process.766 Summary and ConclusionsThe fact that the vascular space contains one pool of white blood cells that circulate andanother that marginate along vessel walls was established early in this century (1, 2). The lunghas long been recognized as a major site of the marginated pool but in contrast to the systemiccirculation where marginated cells are found in post capillary venules, the majority of the cellsthat marginate in the lung are contained in the capillary bed (20,4 1). Studies from ourlaboratory have shown that the pulmonary capillary bed is capable of concentrating PMN 60-to 70-times with respect to the peripheral blood and suggest that this concentration is based ondifferences in the RBC and PMN capillary transit times (41). The realization that there weredifferences between erythrocyte and PMN transit of this magnitude led us to consider thepossibility of developing a computational model, based on anatomic data, to provide aframework in which to think about the problem of PMN kinetics.The data presented in this thesis used a set of probabilities to construct a complete modelof the arterial vessels from existing information on arterial casts and obtained a model ofpulmonary venous system by modifying the process used for the arterial vessels. Manyimportant assumptions were made in order to calculate the pressure drop in the large vessels andthe ones that were chosen result in a model that is consistent with the observation that the bulkof the pressure drop occurs across the capillary bed (6) rather than in the large vessels (110).The distances from the arterioles to the venules and the interconnections betweenindividual segments of the capillary network were modelled assuming that each alveolar wall was77a square grid. Because of the complexity of the system of interconnecting segments joiningarterioles and venules, a resistor network simulation was used to investigate the effect ofplugging individual segments on the pressure drop across the network of capillaries immediatelysurrounding an occluded segment. This showed that the pressure drop across individualsegments are small and little affected by plugging segments in the network with PMN. Indeed,the fact that the magnitude of the predicted pressure drop across a segment plugged by a PMNis smaller than the required pressure to move individual PMN through micropipettes of capillarysizes (22,76).The model also allowed us to test the hypothesis that the delay which PMN experiencewith respect to RBC is solely related to differences in their deformability. The analysis showsthat although the erythrocyte transit times were reasonably accurately predicted by the model,the PMN transit times were shorter than those obtained by either direct observation (56,57) indogs or from the more indirect information available in humans (59,60). This means that allof the mechanisms responsible for delaying PMNs have not been considered in the modellingprocess and that factors such as the need for PMN to become actively motile to negotiatecapillary restrictions or overcome adhesive force between their membranes and the endotheliumneed to be included to account for the observed delay in PMN traffic. Therefore, the modelpresented here provides a method for simulating PMN traffic and suggests new avenues ofinvestigation that will provide a better understanding of how circulating cells move through thepulmonary microvessels.78Table 1Shows data from the four regions of the arterial cast studied by Horsfield and hiscolleagues. The different regions were chosen with overlapping vessel diameters to assistin the formulation of a Strahier ordering system. As can be seen from this table, theregions from 600jm to 100gm and 90gm to 10gm had relatively few branches recorded.Due to the sparse nature of the data in these two regions the model presented in thisthesis consisted of only two regions, 3cm to 10Om and 3OOm to 10gm.79Table 1Summary Description of Horsfield Cast DataHorsfield’s Label Approximate Range Number of VesselsProximal 3cm to 8OOm 2998Intermediate 60Om to 10Om 765Upper Distal 300pm to lOOm 3284Lower Distal 90gm to lOm 61680Table 2Shows the number of daughter branches from all parents in either the large vessel orsmall vessel regions. The NONE column indicates either a terminal branch or brancheswhich are broken and have no daughters. While it is clear that the majority of divisionsare bifurcations, a significant fraction are trifurcations and a few parents have more than3 daughter branches. There are less bifurcations in the small vessel region as comparedto the large vessel region.81Table 2Number of Bifurcations and TrifurcationsNumber of Daughter Branches From ParentZone NONE 2 3 4+3cm to lOOm 1640 1223 230 30100gm to 10gm 490 1251 371 47Zone %Bifurcations %Trifurcations3cm to 1OOm 1223/1483 = 82% 18%10Om to 1Om 1251/1669 = 75% 25%82Table 3Shows the branching probability for daughters from a parent branch in the large vesselrange, 3cm to 100gm. Bifurcations have a largest daughter and a smallest daughterwhile trifurcations have an additional second (or middle) daughter. The largest daughterbranches to one order smaller than the parent with a probability of 0.89 and 2 orderssmaller with a probability of 0.11. The table shows that 58% of the differences betweenthe parent order and the smallest daughter order, are at 5 orders indicating that thedaughter branches are quickly approaching the capillary bed. No difference in branchingprobabilities was detected between bifurcations and trifurcations.83Table 3Branching Probabilities For 3cm to lOOm Sized VesselsFraction of Daughters Branching ‘N’ Orders Smaller Than Parent1 2 3 4 5Largest Daughter 0.890 0.110 0.000 0.000 0.000Second Daughter 0.231 0.169 0.133 0.123 0.344Smallest Daughter 0.056 0.067 0.157 0.140 0.58084Table 4Shows the branching probability for daughters from a parent branch in the small vesselrange, 100pm to 10gm. Bifurcations have a largest daughter and a smallest daughterwhile trifurcations have and additional second (or middle) daughter. The largest daughterbranches to one order smaller than the parent with a probability of 0.885 and 2 orderssmaller with a probability of 0.115. As seen in table 3 the difference between the parentorder and the smallest daughter order is typically 5 orders. No difference in branchingprobabilities was detected between bifurcations and trifurcations.85Table 4Branching Probabilities For lOOm to lOjLm Sized VesselsFraction of Daughters Branching ‘N’ Orders Smaller Than Parent1 2 3 4 5Largest Daughter 0.885 0.115 0.000 0.000 0.000Second Daughter 0.173 0.133 0.114 0.098 0.482Smallest Daughter 0.030 0.052 0.068 0.143 0.70786Table 5The largest vessels of the arterial cast data in the regions 3cm to lOOjLm are not wellrepresented by the proposed branching probabilities in this work. Therefore, the first fourdivisions of the large vessel region were recorded exactly as they occured in the castdata. This table shows how each parent in the first four divisions, branched. Within eachdivison, the largest vessels are displayed first. As an example, the first parent, order 48,divides into two order 47 branches. The first order 47 vessel is the largest and istherefore the first line of data from the 2nd divison values. The second order 47 vesselis the next line of data in the 2nd divison region.87Table 5Branching Pattern of Large Arterial BranchesDaughtersRegion Parent 1st 2nd 3rd Diameter(um)1st Division48 47 47 300002nd Division47 44 46 1900047 46 42 185003rd Division44 43 39 1000046 45 44 41 1550046 45 40 1800042 41 40 38 80004th Division43 42 40 850039 38 35 700045 44 42 42 1100044 43 42 850041 40 40 38 650045 44 41 1350040 39 39 700041 40 34 500040 39 35 450038 37 36 400088Table 6Shows the estimates for the number of bifurcations, trifurcations and diameter for vesselsordered 44 to 1 obtained using the model. While not shown in this table, the variabilityabout the mean diameter (displayed in Figure 10) is about 30%.89Table 6 (Part 1 of 2)Branching Pattern for Arterial Tree Orders 44 to 1Orders Bifurcations Trifurcations Diameter(um)(# of vessels) (# of Vessels)44 2 0 903643 2 1 784042 6 1 675941 7 1 569740 11 3 501539 20 4 435138 30 6 367137 42 9 316336 59 13 273235 89 20 235934 134 29 203833 196 43 176032 285 63 152031 418 92 131330 613 135 113429 900 198 97928 1324 291 84627 1945 427 73126 2853 626 63125 4188 919 54524 6147 1349 47123 9021 1980 40722 13241 2906 35121 19433 4266 30320 26088 8696 26219 40160 13386 22618 62067 20689 19517 96394 32131 16916 149632 49877 14615 228133 76044 12614 349232 116410 10990Table 6 (Part 2 of 2)Branching Pattern for Arterial Tree Orders 44 to 1Orders Bifurcations Trifurcations Diameter(um)(# of vessels) (# of Vessels)13 535204 178401 9412 820817 273606 8111 1259282 419760 7010 1930967 643655 609 2960802 986934 528 4539909 1513303 457 6961409 2320470 396 10674757 3558252 345 16368734 5456244 294 25099811 8366604 253 38487988 12829328 222 59017388 19672462 191 1391Table 7Shows the ratio of the arterial vessel number, diameter and length to the venous values.Clearly a larger number of precapillary arterioles exist than postcapillary venules. Ingeneral, the ratio of arterial diameter to capillary diameters tends to be within thevariation of arterial diameter (30%).92Table 7Ratio Between Number of Vessels, Vessel Diameter and Vessel Length for Arterialand Venous Cast Data Using Strahier OrderingAll values are Arteries/VeinsOrder #Vessels Diameter Length1 411.4% 100.0% 100.0%2 53.1% 110.5% 104.2%3 57.7% 117.2% 102.5%4 60.8% 125.6% 105.3%5 64.1% 134.4% 105.3%6 67.5% 143.8% 100.0%7 71.1% 160.0% 103.0%8 73.6% 159.5% 106.1%9 77.7% 134.6% 124.4%10 124.3% 139.3% 146.6%11 136.1% 109.9% 60.0%12 128.5% 110.0% 56.8%13 124.5% 125.9% 70.5%14 142.9% 111.3% 53.1%15 200.0% 58.1% 29.7%93Table 8Shows the percent decrease in the number of vessels in either the arterial or venous treefrom the smallest pre or post capillary vessels to the larger proximal vessels. Except forthe discrepancies between values from order 1 to 2, the rate of reduction of vesselsappears quite similar in both the arterial and venous trees when a Strahier scheme isused.94Table 8Percent Decrease in Number of Vessels Between Adjacent Orders for Arterial andVenous Trees Using Strahier OrderingArteries Veinslto2 95.9% 83.2%2to3 65.5% 68.3%3to4 66.6% 68.3%4to5 66.6% 68.3%5to6 66.6% 68.3%6to7 66.6% 68.3%7to8 66.6% 67.7%8to9 66.6% 68.4%9 to 10 60.9% 75.6%lOtoll 70.5% 73.1%11 to 12 69.9% 68.1%12 to 13 67.5% 66.5%13 to 14 69.7% 73.6%14 to 15 60.0% 71.4%15 to 16 62.5%16 to 17 66.7%95Table 9Shows the number of vessels and predicted diameters for a venous network modelobtained using the branching probabilities from tables 2,3 and 4 which were determinedfrom arterial cast data. The number of orders, and the starting number of vessels, werechosen to agree with previously published results (45).96Table 9 (Part 1 of 2)Summary Data for Venous Tree Branching Orders 43 to 1Orders Number of Vessels Diameter(um)43 4 784042 4 675941 8 569740 10 501539 16 435138 25 367137 37 316336 51 273235 72 235934 110 203833 163 176032 239 152031 348 131330 510 113429 748 97928 1098 84627 1615 73126 2372 63125 3479 54524 5107 47123 7496 40722 11001 35121 16147 30320 23699 26219 34784 22618 53546 19517 82756 16916 128525 14697Table 9 (Part 2 of 2)Summary Data for Venous Tree Branching Orders 43 to 1Orders Number of Vessels Diameter(um)15 199509 12614 304177 10913 465642 9412 713605 8111 1094423 7010 1679042 609 2574622 528 3947736 457 6053212 396 9281879 345 14233009 294 21824978 253 33466415 222 51317314 191 78689850 1398Table 10Shows the resistance in a network of resistors from each node to ground. Each grid isintended to represent a single alveolar wall where nodes represent the junction ofcapillary segments and capillary segments are replaced by resistors. Arteriole flow inputis replaced by a voltage source and venule outflow by ground. These tables show that theresistances within each network have similar values and the total resistance across eachgrid is wealdy related to the size of the grid.99Table 10Resistance from Node to Output for Electhcal Grids4x4 Grid Total Resistance 4.76.9 6.5 5.7 4.26.5 6.0 5.2 3.85.7 5.2 4.6 3.54.2 3.8 3.5 3.15x5 Grid Total Resistance = 4.26.7 6.7 6.2 5.5 4.16.7 6.5 6.0 5.2 3.86.2 6.0 5.5 4.8 3.65.5 5.2 4.8 4.4 3.54.1 3.8 3.6 3.5 3.06x6 Grid Total Resistance= 3.96.6 6.7 6.4 6.0 5.3 4.06.7 6.8 6.4 5.9 5.1 3.86.4 6.4 6.1 5.6 4.9 3.76.0 5.9 5.6 5.2 4.6 3.65.3 5.1 4.9 4.6 4.3 3.54.0 3.8 3.7 3.6 3.5 3.0— indicates input borders (arterioles)indicates output borders (venules)100Table 11Shows the relative increase in resistance at each node in a resistor grid if 10% ofthe resistors have been replaced by open circuits (which are meant to representcapillary blockages by PMN). In these tables infinite resistances are not seen dueto the vast number of alternative pathways. It can be seen in these tables thatwhile the resistance very close to capillary blockages increases, this increasedresistance is not transmitted back into the network grid more than one or twosegments. Blockages in this model have a greater affect in the 4x4 grid than thelarger 6x6 grid due the relatively fewer number of alternative pathways.101Table 11Relative Increase in Resistance at Each Node with 10% Blockages4x4 Grid Total Resistance = 1.21.3 1.2 1.3 1.11.3 1.3 2.0 1.11.2 1.5 1.4 1.11.2 2.2 1.8 1.15x5 Grid Total Resistance 1.21.3 1.3 1.2 1.0 1.21.3 1.3 1.3 1.0 1.01.1 1.1 1.2 1.0 1.01.1 1.0 1.0 1.0 1.01.1 1.0 1.0 1.0 1.06x6 Grid Total Resistance = 1.21.1 1.1 1.1 1.0 1.0 1.01.2 1.1 1.1 1.1 1.0 1.01.2 1.2 1.8 1.2 1.1 1.11.1 1.1 1.3 1.1 1.1 1.61.0 1.1 1.4 1.1 1.1 1.11.0 1.0 1.2 1.0 1.0 1.0,indicates input borders (arterioles)indicates output borders (venules)102Table 12Shows the relative increase in resistance at each node in a resistor grid if 25% ofthe resistors have been replaced by open circuits (which are meant to representcapillary blockages by PMN). In these tables infinite resistances are not seen dueto the vast number of alternative pathways. Although the resistance very closeto capillary blockages increases, this increased resistance is not transmitted backinto the network grid more than one or two segments but with this large numberof open circuits resistances throughout the grid networks have increased relativeto table 10.103Table 12Relative Increase at Each Node with 25% Blockages4x4 Grid Total Resistance = 3.23.9 2.5 2.7 2.93.0 3.1 3.8 1.51.8 2.4 2.3 1.21.8 1.6 1.3 1.05x5 Grid Total Resistance 2.02.0 1.5 1.4 1.3 1.21.8 1.6 1.5 1.6 1.32.3 1.5 1.3 1.3 1.72.2 1.4 1.2 1.5 1.92.6 1.3 1.1 1.2 2.06x6 Grid Total Resistance = 2.62.1 2.3 2.0 2.0 2.0 1.62.0 2.3 2.6 2.1 2.2 2.12.4 2.4 4.4 2.3 2.6 1.52.4 2.5 2.5 2.4 2.9 2.43.8 5.9 3.7 29 3.0 1.55.0 2.6 4.9 3.8 3.0 1.8_____indicates input borders (arterioles)indicates output borders (venules)104Table 13Shows how the pattern of resistance varies when blockages are present in theresistor grid network. From table 12 each node resistance was divided by the totalnetwork resistance, then a corresponding table of these relative resistancse werecomputed for 10% or 25% blockages. The two tables shown here display theresults for the scaled resistance with no blockages to those with either 10% or25% blockages. Values of 1.0 indicate no differnce in the resistance at a givennode relative to ground when blockages were present, values greater than 1indicate an increased relative resistance. These data show that when only 10% ofthe capillary segments within a single alveolar wall are blocked that no effect onthe pattern of pathway resistance can be seen. Also, when a larger number ofpathways are blocked, 25%, only those nodes at or near the blockages areaffected.105Table 13Resistance Pattern for 5x5 Grid Scaled to Total Grid Resistance10% Blockages1.1 1.1 1.0 0.8 1.01.1 1.0 1.1 0.8 0.80.9 0.9 1.0 0.8 0.80.9 0.8 0.8 0.8 0.80.9 0.8 0.8 0.8 0.820% Blockages1.0 1.3 1.4 1.5 1.71.1 1.2 1.3 1.3 1.50.9 1.3 1.5 1.5 1.20.9 1.4 1.7 1.3 1.10.8 1.5 1.8 1.7 1.0— indicates input border (arterioles)— indicates output border (venule)106Table 14Shows the relative increase in resistance in an electrical grid network, similar totable 13, except that current (flow by analogy) is limited to travel in aunidirectional manner in each segment. This network simulates the condition oftwo vessels flowing into each network node and two out from each node. Valuesof INF indicate an infinite resistance, meaning that both vessels leading from thatnode are plugged. Note that with 10% blockages, the infinite resistance at acompletely occluded node is only transmitted back one or at most two segmentsinto the capillary network.107Table 14Relative Increase in Resistance at Each Node with 10% Blockages Relative toNo Blockages: This Model Allows Unidirectional Flow in Each Segment4x4 Grid Total Resistance (Relative Change) = 1.22.2 1.4 1.6 1.01.5 3.2 INF 1.11.3 2.4 1.3 1.01.3 4.7 2.6 1.05x5 Grid Total Resistance (Relative Change) = 1.31.4 1.9 1.2 1.0 1.01.7 1.1 1.7 1.0 1.01.0 1.0 1.0 1.0 1.01.0 1.0 1.0 1.0 1.01.0 1.0 1.0 1.0 1.06x6 Grid Total Resistance (Relative Change) = 1.01.4 1.1 1.2 1.0 1.0 1.0.0 1.3 1.7 1.0 1.0 1.01.2 1.8 INF 1.1 1.1 1.21.1 1.2 1.5 1.1 1.4 2.81.0 1.1 2.2 1.0 1.0 1.01.0 1.0 1.0 1.0 1.0 1.0INF indicates an INFINiTE resistance, all pathways to exit blocked_____indicatesinput border (arteriole)indicates output border (venule)108Table 15Shows the relative increase in resistance in an electrical grid network, similar totable 14, except that current (flow by analogy) is limited to travel in a singledirection in each segment. This network simulates the condition of two vesselsflowing into each network node and two out from each node. Values of INFindicate an infinite resistance, meaning that both vessels leading from that nodeare plugged. In contrast to table 16, when 25% of the segments are pluggedrelatively large increases in vessel resistances are seen throughout the network.In this simulation the greater increase in total grid resistance for the 4x4 modelis merely a reflection of the close proximity of the blocked segments. In general,the total network resistance was similar for all network sizes.109Table 15Relative Increase in Resistance at Each Node with 25% Blockages Relative toNo Blockages: This Model Allows Unidirectional Flow in Each Segment4x4 Grid Total Resistance (Relative Change) = 3.54.0 3.3 2.9 2.9INF INF INF 1.01.3 3.0 1.9 1.01.5 1.0 1.0 1.05x5 Grid Total Resistance (Relative Change) = 2.31.9 1.2 1.2 1.2 1.02.7 1.9 1.6 2.9 1.2INF 1.1 1.2 1.6 3.42.9 1.0 1.2 2.0 1.62.9 1.0 1.0 1.1 2.06x6 Grid Total Resistance (Relative Change) = 2.610.7 2.5 1.9 1.3 1.7 1.28.1 7.9 INF 2.1 2.3 3.17.4 6.5 INF 4.3 6.6 1.27.9 5.7 4.1 3.8 5.0 2.9INF INF 6.6 3.7 3.8 1.1INF 1.5 &2 5.6 3.5 2.0INF indicates an INFINfl’E resistance, all pathways to exit blocked______indicatesinput border (arteriole)indicates output border (venule)110Table 16Shows the values used for regional capillary diameters based on the data ofGlazier (33) (figures 25 and 26). The first column shows the relative amount thecapillary segment diameters are decreased compared to fully distended values.Also shown are the mean and standard deviation of the capillary segmentdiameters using the fully distended values estimated in section 3.1 of the thesis.111Table 16Regional Capillary Diameters%Maximum Mean SDLffl LffiUpper Lung 67% 5.01 1.55Slice 2 75% 5.61 1.73Slice 3 83% 6.21 1.92Slice 4 89% 6.66 2.06Lower Lung 94% 7.03 2.17112Table 17Shows the results of the model of the capillary bed. The different regionalresponses were estimated by varying only the capillary segment diameters basedon values from table 16. These data show that relatively little effect on thenumber of stops (number of vessels encountered with diameters less than that ofthe PMN) is seen regionally. However, the median time per stop and the mediantransit time is significantly increased when the PMN is forced to travel throughsmall diameter vessels.113Table 17Summary of Capillary Bed SimulationsRegion Median Median Time Median TransitStops per stop Time(#) (secs) (see)Upper Lung 5 12.8 63.9Slice 2 5 4.0 18.2Slice 3 4 2.3 10.2Slice 4 4 1.5 6.8Lower Lung 4 1.1 5.3114Table 18Shows the fraction of total pulmonary flow delivered to different lung regionsbased on the data of Anthonisen and Milic-Emili (3). When estimating totalpulmonary transit times these values were used to determine what fraction of thecells traversed each of the different lung regions where capillary segmentdiameter varied.115Table 18Fraction of Cardiac Output Delivered to Each RegionUpper Lung 3.7%Slice 2 11.1%Slice 3 19.9%Slice 4 28.5%Lower Lung 36.8%116Figure 1Shows a radiographic image of the human pulmonary arterial tree obtained 3 secondsafter an injection of dye into the right ventricle. Comparison of this picture to those inFig. 2 and 3 provides an overview of the complexity of the pulmonary vasculature andthe diversity of pathlengths from the pulmonary artery to pulmonary veins.117Figure 2From the same series as Figure 1 showing a second radiographic image taken 8 secondsafter the injection of dye in the right ventricle. The venous circulation is shown toadvantage in this image.119o.z’Figure 3Shows a third radiographic image from the same series as Figs. 1 and 2 taken 20 secondsafter the dye was injected m the right ventricle It demonstrates that the dye has almostcompletely cleared from the venous circulation at this phase.121Figure 4Shows the results of an indicator dilution study from reference 62 which compares MTCRBC to 51Cr RBC during a single pass through the lung. The labelled cells were injectedinto the right ventricle and sampled at the aorta and the data show that the twoprocedures used to label the RBC had no effect on their transit through the pulmonarycirculation.123Cowa)(‘54-’0a)C‘4-CC0CaU-Figure 4Time (seconds)124Figure 5Data from reference 62 comparing Tc RBC to 51Cr PMN with the same injection andsampling sites as in Figure 4. It shows that only about 20% of the PMN came throughwith the RBC which means that 80% were delayed in the lung.125Figure 51:()a)c3-2LlPulmonary Transit Time (seconds)126Figure 6This figure shows a branching network labelled using Strahier ordering rules. In thislabelling scheme, the smallest vessels (at the ends of the tree) are labelled as order 1.When two vessels meet the parent vessel is labelled one order larger than the daughtersif both daughters have the same order. If the two daughter branches have different ordersthen the parent branch is given the order of the largest daughter branch. This branchingscheme makes it impossible to track vessel length for a random path through the tree andtherefore is not practical for a fluid mechanics approach to studing the pulmonaryvasculature.1271Figure 61331211111128Figure 7This figure shows the same tree as in Figure 6 ordered using a symmetrical branchingsystem. In this scheme, the largest branch at the top of the tree is labelled an order 0vessel. Each daughter at each division is then assigned an order, one less than the parentvessel. This symmetric branching method cannot accurately represent the veryasymmetric branching structure present in the pulmonary angiogram (see Figures 1-3)and is not practical for ordering a pulmonary vasculature tree.129U)rJL4D0CDFigure 8Shows a Horsfield ordering scheme for the same tree as in Figures 6 and 7. In Horsfieldordering each terminal end branch is given an order 1 as in Strahier ordering. When two(or more daughters) meet at a division, the parent branch is assigned an order one largerthan the largest daughter. Unlike the Strahier ordering, this method makes it possible toaccurately record total path lengths and allows a very asymmetric branching pattern tobe modelled. This system was used to order the pulmonary vessels in this thesis.1311Figure 81541211111132Figure 9Shows the results of ordering each of the four regions of the available cast data (fable1) using the Horsfield scheme in Figure 8. Each region of cast data has been artificiallyseparated for display purposes. The curvilinear drop in each region is due to brokenbranches providing biased estimates of vessel diameter.1331Figure 9main pulmonary arteryC’)0IC)E1oooC)•1-’C)6C)0130 40 50 60 70Rough Horstield Ordering134Figure 10Shows the complete cast data ordered by the Horsfield branching scheme. The separatecast regions have been aligned by matching vessels of similar diameters in the proximalend of one region with those in the distal end of another. These data show a log-linearrelationship between diameter and order which is a common characteristic of naturallyoccurring branching patterns. The variability in diameters at each order is about 30%.135C,)0I—C-)Ea).1ci)2CuUFigure 10136Horsfield OrderFigure 11Shows an estimate of the number of branches in a given order for a simulated arterialtree. The probabilities in tables 2,3 and 4 were used to estimate these values. Thenumber of terminal arterioles in this model is 3x10’1 agrees with previous estimates (95).137Figure 1110oooooo00.1000000o0.10000O0:> 100000!I10001001010 5 10 15 20 25 30 35 40 45 50Horsfield Order138Figure 12Shows the cross-sectional area available for blood flow at each order in the simulatedarterial tree. These values are based on the estimated vessel diameter (figure 10) andthe estimated number of vessels (figure 11) at each order. It is interesting that there isa reduction in cross-sectional area at approximately order 44. A similar reduction incross-sectional area has been recorded in the tracheobronchial tree (see references 46,96)but as the estimates are made on casts from different lungs, the location of theseconstrictions cannot be compared.139C)CCIFigure 1210 15 20 25 30 35Horsfield Order140Figure 13Using the concept of flow continuity, which states that all the fluid flowing into abifurcation or trifurcation must flow out of the division we can estimate the possibleamount of flow that a particular order vessel could carry. If it is assumed that eachterminal arteriole carries 1 unit of flow then each order 2 arteriole must carry 2 units (fora bifurcation) or 3 units (for a trifurcation) since this model has assumed that allbranching terminates at order 1 vessels. With flow now solved for all possible order twovessels, the same logic can be extended to predict possible flows in order 3 vessels etc.This pattern of tracking was computed up to order 20 vessels. These data show that asvessels become larger they can carry more flow but that the variation in the amount offlow per vessel also increases.141Figure 13CoI0.0.01C04-.CCuILI100 200 300 400 500 600 700 800 900Units of Order 1 Vessel Flow142Figure 14This figure shows the estimated average flow in a vessel of a given order as a fractionof the total cardiac output. The average flow for orders 1-20 was estimated as describedin Figure 13. Average flow in vessels from order 21 to 48 are extrapolated based on thevalues for the smaller vessels. While not evident from this figure, by multiplying theaverage flow per vessel shown here by the number of vessels of a given order (figure 11)it is estimated that each order carries approximately 50% of the total cardiac output.This means that because the branching is not symmetrical, all of the flow does not gothrough each order of branching.143CUa)cIa)a)Ca)a)Figure 14Horsfield Order144Figure 15Shows the distribution of pressure drops for numerous simulated pathways from the mainpulmonary artery to the precapillary vessels. The different pressures reflect the differentpathways that the flow can take from the main pulmonary artery to the capillary network.While pressure drops as large as 4 or 5 mmHg can occur, the average pressure drop isonly 1.8 mmllg.Panel B shows the average pressure drop for the vessels of each arterial order.145—‘4Ibt3 0.1E0.010.001Figure 15AFigure 15BArterial Pressure Drop (mmHg)110 15 210 25 310 5 4o 45 DOHorsfield Order146Figure 16Shows the distribution of arterial vessel path lengths from the main pulmonary artery tothe precapillary vessels. As is apparent from the angiographic images of the vessels (seeFigures 1-3), a wide distribution of pathways should be expected. The maximumpredicted pathways are approximately 10cm in length which are reasonable for a humanlung.147Figure 164.543.52.521.510.5Arterial Path Length (cm)148Figure 17Shows an estimate of precapilary blood velocity based on the computational estimatesdescribed in the text. The majority are included in the range from 500 to 2000gm/swhich is in good agreement with published results (83). This indicates that the rules usedto divide flow at divisions and the computed pressure drops in the arterial system arereasonable.149Figure 171.5ci)Flow Velocity (umis)150Figure 18Shows the estimated distribution of path lengths for the simulated venous tree. The slightdifferences from the arterial system (Figure 16) are due to the fact that the arterial treewas trimmed to represent the venous system.1511)aci)1U-Figure 184.0 6.0Venous Path Length (cm)152Figure 19Shows the distribution of pressure drops in the venous network. The average pressuredrop of about 1.3 mmHg is slightly less than that of the arterial tree due to selectivepruning of the arterial data to obtain this model of the pulmonary veins.153Figure 19Cci)aci)Ii-Pressure Drop (mmHg)154Figure 20Compares the cumulative pressure drop in the large vessels (arteries and veins). The datahave been arbitrarily placed at an initial pressure of l5mmHg to match that of Zhuang(110). The solid line represents the maximum possible pressure drop calculated whenblood is allowed to pass through every order of both the arterial and venous tree. Thedotted line represents the estimate obtained using the model system described in thisthesis. The distance between these lines graphically displays the potential range ofpressures at every order. The pressure drop between orders can be infered by the slopeof the lines but this should be done with caution as it is clear that most of the blood doesnot flow through sequential orders. The discrepancy between the pressure at the terminalarteriole and post capillary venules provides an estimate of the pressure drop across thecapillary bed.155Figure 20:e7Capillaries1o -E0I Order156Figure 21Shows the distribution of human capillary segment diameters (solid bars) and humanPMN diameters (open bars) obtained using morphometric techniques. The capillarysegment diameters have a wider distribution than the PMN and many segments, at least58%, have diameters smaller than the average PMN diameter. This difference in PMNsize and pulmonary capillary size forces PMN to stop and deform as they attempt totraverse the pulmonary microvasculature.157>C.)Dci)ILIFigure 21o 1 2 3 4 5 6 8 9 10 11Diameter (microns)Capillary Segments Neutrophils158Figure 22This photograph is a 25jLm thick section of a human alveolar wall. Several RBC andPMN can be identified within the segments.159Figure 23This diagram shows the network design used to simulate a single alveolar wall. Twowalls were randomly chosen (not necessarily adjacent walls) to represent arteriole orcorner vessel input and the remaining two walls venule or corner vessel outputs.Capillary segments were taken to be each small line segment, there are 60 segments ofline in this 5x5 grid, and nodes are where four capillary segments meet. Blood isassumed to flow along the solid black lines in this diagram and the open white spaceswould represent the posts in a sheet flow model. No attept has been made to draw thegrid to anatomical scale. In the electrical simulations computed, the arterial inputs werevoltage sources, the venous return were grounds. Each capillary segment was replacedby a resistor and flow was allowed in a single direction through each segment.161Figure 23arterialI IRI 111111 liii———————————inputI I I I I I I I I I I I I I iii I I I I I I I Ivenous——.———..——‘IFcapillary1IiIIIIR1II iIiIiII.II.returnsegment162Figure 24Shows an example where capillary segments blocked by PMN were modelled byreplacing them with open circuits. The 0 indicates the situation with 10% of segmentsrandomly plugged. The X denote the additional segments that are removed from thecircuit when 25% capillary segments are plugged.163Figure 24--arterialI liii 11111 liii————————.———inputI I I I I I I II II I I I II I I I I I I II--PL———.——BBBBBBBBBBBIIIIIIIIIIIIIIF--.4-evenous returno locationX locationof 10% blockagesof 25 % blockages------0I I I I I I I I II I I I I I I I I I I I I I164Figure 25Data modified from Glazier et al showing capillary segment diameters in differentgravitational planes. The large distance down Zone ifi was achieved by controllingarterial and venous pressures to keep the entire lung under Zone ifi conditions.165Figure 25—6.56— uBoftomil of Lung“Top” of LungI I I5 10 15 20 25 30 35 40 45 50 55Distance Down Zone III (cm)166Figure 26Data from figure 25 which has been scaled to show capillary segment diameter as apercent of its maximum. The average capillary segment diameter, relative to themaximally dilated values, was determined for each of 5 six cm sections selected torepresent a 30 cm human lung. The numerical values for each of these regions areshown in Table 16.16722E1a)4-600C) 40Figure 26Relative Lung Height (%)168Figure 27Shows a line derived from the equation, empirically determined by Fenton et al (26),for the time required for a WBC to deform and enter a pipet. This equation was chosenbecause it was imperically derived without a pressure term which allowed it to be usedto model events across individual capillary segments where no pressure data is available.169Figure 27.F0E1PMN Diameter/Capillary Diameter170Figure 28Shows the estimated number of capillary segments traversed by cells moving from aterminal arteriole to a post capillary venule. On average each PMN passes through 90segments, with the majority of pathways containing between 70 and 120 segments.171CG)D0a)LLFigure 2818161412108642050 60 70 80 90 100 110 120 130’140Number of Segments Traversed172Figure 29Shows the estimate of the total capillary path length that a PMN travels from a terminalarteriole to a post capillary venule. This length can not be measured directly usingcurrent experimental techniques but Staub and Schultz (83) have measured the minimumarteriole to venule distance in a straight line between an arteriole and venule on a planarsection. The values shown in this figure are slightly larger than those recorded by Stauband Schultz (83) in dogs, cats and rabbits.173Figure 291614100 uu bUU 700 Oo 1100 1300 1500Arteriole to Venule Length (microns)174121o.Ca)6L1420•Figure 30Shows the RBC transit time estimated using the model where the median value of 0.8seconds compares well with estimated capillary transit times in normal seated subjectsusing physiologic techniques.175Figure 30.J.J45.4035— 3025Transit Time (seconds)176Figure 31This figure shows the estimated transit time for PMN travelling through the pulmonarycapillary bed where the median value is 7.8 sec a mean of 145.7 sec with a range of 0.6-10000 sec. The PMN transit times are much longer than those obtained for RBC (Fig.30) because the PMN were required to stop (based on equation 15) and deform beforethey could enter a restriction imposed by narrow capillaries.177C0D0SL1Figure 31PMN Capillary Transit Time (seconds)178Figure 32Shows the number of stops made by a PMN as it traversed through the model of thecapillary bed. The median number of stops in each of 5 lung regions is shown in Table17.179Figure 32Ca)Number of Stops180Figure 33Shows the average length of time per PMN stop as they traverse the capillary network.These values are the ratio of the total length of time required for the PMN to traversethe bed divided by the total number of stops.181itFigure 33Average Time Per Stop (seconds)182Figure 34Shows the distribution of arterial transit times where the median value was 0.7 seconds183a)IFigure 34Arterial Transit Time (seconds)184Figure 35Shows the estimated distribution of venous transit times where the median value was 0.8seconds.185Figure 3500IVenous Transit Time (seconds)186Figure 36Shows the total pulmonary transit times obtained by connecting the arterial path lengthsand venous path lengths and summing their two transit times. This resultant “largevessel” transit time was then convolved with the distribution of capillary transit times foreither RBC or a flow weighted disthbution of PMN transit times. The median RBCtransit time is 2.1 seconds with a range of 0.8-15.4 sec and median PMN transit timesof 8.6 seconds with a range of 1.1-9862 sec. The majority of the PMN transit times fallwithin the range of the RBC transit times.187Cci0SLL 0.Figure 36Pulmonary Transit Time (seconds)188Figure 37Compares the time activity curve from reference 62 obtained by direct sampling ofcentral arterial blood following an injection of 51Cr PMN and RBC into the centralvenous circulation (panel A) to the data obtained for PMN and RBC using the model(panel B). Note that in the experimental data, the PMN appearing prior to therecirculation of RBC fall to 0, whereas in the PMN data obtained using the modelremains high. 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