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Network analysis of microcirculation of cat mesentery
MICROVASCULAR
Network
RESEARCH
7, 73-83 (1974)
Analysis H.
of Microcirculation
of Cat Mesentery’
H. LIPOWSKY’ AND B. W. ZWEIFACH AMES (Bioengineering), The University of California, San Diego, La Jolla, California 92037 Received May 25,1973
The modular configuration of the microcirculatory system in cat mesentery is subjected to a hydrodynamic network analysis assuming Poiseuillian dynamical behavior. Intravascular pressure, vessel pressure gradient, and wall shear stress are computed for an isolated module and presented as a function of vessel diameter, from arterial affluent to venous effluent. Computed and in vivo intravascular pressures show a marked disparity on the arterial side of the true capillaries and a fair agreement on the venous side. This is attributed to the effects of precapihary sphincter action and non-Newtonian rheological behavior. Computed pressure gradients based on a simple Poiseuillian relationship are approximately six times greater than those measured in vivo. By comparison of predicted and measured pressure gradients, the magnitude of maximum vessel wall shear stress is estimated to be on the order of 10 dyn/cm2.
INTRODUCTION The delineation of the functional behavior of the microcirculation requires a knowledge of the intimate relationships between geometric and hemodynamic parameters such as intravascular pressure, pressure gradient, velocities, and shear stress.Inasmuch as measurement of these parameters, either directly or indirectly for an entire capillary bed is precluded by the large number of vesselsinvolved, it becomesworthwhile to determine their magnitude and distribution as a function of vessel geometry and spatial orientation for a specified mode of dynamical behavior. To this end, several simplified analysesof the microvasculature as a network of serial and parallel elements whose conductance is dictated by Poiseuille flow have appeared in the literature (Chien, 1971). Renkin (1964) has drawn upon the early morphological studies of Chambers and Zweifach (1944) as a frame of reference of capillary clearance in a simplified model of a terminal vascular network composed of parallel and serial vesselsinterposed between arterioles and venules. A more realistic distribution of geometric parametersinfluencing the hemodynamics has been presented by Intaglietta and Zweifach (1971) for the microvasculature of rabbit omentum. In their work a serial network was characterized by the distribution of hydraulic hindrance from arteriole to venule in terms of the “order of branching” of vesselsbased on in vivo distribution data. This approach also represents an oversimplification of the hemodynamic dependency on microvascular geometry (vessellength and diameter). I Supported in part by USPHS Grant No. HL-10881-06. z Predoctoral Fellow, USPHS Training Grant No. HL-05946-02. Copyright 0 1974 by Academic Press, Inc. 73 All rights of reproduction in any form reserved. Printed in Great Britain
74
LIPOWSKY
AND ZWEIFACH
Further attempts to represent the terminal vascular bed by a simplified network of serial and parallel elements have been made recently by Nellis (1972) and Lee (1973). In these studies the basic network of Renkin was utilized to gain insight into the distribution of mean transit times which they were able to measure in vivo by indicator dilution studies in the cat mesentery. In order to allow for better agreement between experiment and theory, Lee went a step further by incorporating into his network model some of the characteristics of the mesenteric microvascular module described by Frasher and Wayland (1972). He proceeded to generalize the mesenteric module as a rectangular area of tissue bounded on four sides by paired arterial and venous vessels. The arterial and venous vesselson the perimeter are then connected acrossthe module interior by vesselsof different diameters, arranged like rungs of a ladder to represent capillaries, metarterioles, arterioles and venules. This model is still a crude representation of the equivalent network within the module interior. Schmid-Schiinbein (1972) has attempted on a larger scale to synthesize a hypothetical network from aorta to vena cava. He devised a composite seriesnetwork based on the morphometric studies of Mall (1888) for fixed canine mesentery and by Weidemann (1963) for the living circulation in the wing of the bat. He then used hydraulic hindrances of this geometrical composite to compute such dynamical parameters as intravascular pressure, pressure gradient, and wall shear stress.Although this type of analysis can be used to define the gross dynamical aspectsof the circulatory system, it fails to elucidate what is actually taking place on the microvascular level in the successivefunctional segmentsof the terminal vascular bed. In order to provide a more precise analytical description of microvascular hemodynamics, the analysis of blood flow is performed here based on a relatively exact in uivo distribution mapped out in detail for the mesenteric circulation of the cat. The techniques employed are those used in electrical network analysis for a linear resistive network whose conductances are determined by assuming Poiseuille flow. METHODS The circulation in the mesentery of the anesthetized cat (pentobarbital) was used for the construction of detailed maps. The preparation for microscopic observation has been described in detail (Zweifach and Richardson, 1970).Serial photographs were taken at 32x magnification and assembledto include one or more complete modules. Vessel diameters for the capillaries were determined at 400x magnification. The dimensions (length and diameter) of the arterioles and venules were recorded from the low power photographs. All of the capillaries visible under the microscope were introduced into the model, although our observations indicate that up to 25-30x of the capillaries may not have an active flow under control conditions. Several fixed preparations of large areas of the mesentery were used to construct a sector and to calculate the averagenumber of modules per unit area of mesentery. The animal was given an injection of 2 ml of Pelikan carbon suspensionto make it easier to identify the capillaries and the tissue was then fixed by suffusing it with 10% gluteraldehyde. The preparation was cleared in glycerine and photographed for the reconstruction of the large sector area of the mesentery.The module usedfor modelling representedthe in uivo state of the vascular bed.
NETWORK
ANALYSIS MICROCIRCULATION
75
Pressure measurementswere made with sharpened micropipettes containing 1.0 A4 saline and recorded by an electronic servo system similar to that used by Wiederhielm et al. (1967). Pressuregradients were established by inserting two micropipettes across the different vascular segments.The servo system is accurate to within kO.25 cm H,O. GEOMETRICAL
CONSIDERATIONS
The morphometric description of the microcirculation in the mesenteryof the cat has been described by the studies of Frasher and Wayland (1972) mentioned earlier. As demonstrated therein the microvasculature is characterized by three distinct orders of vessel size and branching. The first is that of the large artery-vein pairs which are distributed radially from the posterior line of attachment toward the bowel, with arterial diameter ranging from 150-300 pm in diameter. These vesselsbifurcate in the vicinity of the intestine to form an artery-vein pair running parallel to the bowel and from which the circulation to the bowel proper is formed. An approximately triangular sector of mesentery is thus circumscribed by these first order vessels, and shall be referred to herein as simply the sector (seeFig. 1). Within the sector, there is a second artery vein
5Oi24
l&O
FIG. 1. Schematic representation of a typical sector. Single lines within the interior represent con-
tinuous artery-vein pairs forming the perimeters of many contiguous modules.
76
LIPOWSKY AND ZWEIFACH
order of branching, characterized by artery-vein pairs (of diameter on the order of 20-40 pm) which traverse the interior and form a continuous interarcading network. This pattern in turn circumscribes areasof tissue referred to as modules by Frasher and Wayland (1972).
a. a
V a
b. FIG. 2. a. Photomontage of a typical module. b. Schematic representation of the module drawn from an
overlay tracing. Nodes labelled a and v represent boundary nodes for the hydrodynamic computation. c. A typical node illustrating the niathematical nomenclature.
The interior of a typical sector is shown schematically in Fig. 1. The continuous artery-vein pairs forming the perimeter of the several modulus are shown with average diameters indicated. Although Frasher and Wayland (1972)emphasizedthe appearance of a five-sided module, several geometric shapesare seen.It should also be noted that in someinstances the precise limits of a given module within any given sector is subject to the interpretation of the observer. It is fairly frequent to observe an artery-vein pair whose diameters are small enough (15-20 pm) to justify labelling them as interior
NETWORK
ANALYSIS MICROCIRCULATION
77
modular vessels.Yet their continuity (from a first order artery to another first order artery and similarly from first order vein to vein) justify their designation as perimeter vesselsof contiguous modules. The interior of each module is composed of an elaborate network of pre- and postcapillary vessels.Figure 2a presents a photomontage of a typical module. An overlay tracing of this module is shown schematically in Fig. 2b where eachvesselis represented by a tube of invariant diameter and specified length betweenjunctions (termed nodes) of two or more vessels.A typical node is shown in Fig. 2c to indicate the mathematical nomenclature utilized in the following hemodynamic analysis. As an example of the distribution of the number of vesselsin a module, Fig. 2 contains 165 vesselsand 115 nodes. The length and diameter of eachvesselwere scaledfrom the photomontage. Although measurementsat such low magnification (32x) are accurate only to within ?I ,um, this was deemed satisfactory for the present analysis. Vesselinternal diameters were measured to the nearest micron, except for the true capillaries, where high power magnifications (400x) showed diameters to range from 6 pm to 8 pm. An average diameter of 7 pm was used in the computations for these vessels. It is the objective of this study to assign specific dynamical boundary conditions (e.g. intravascular pressure) on the periphery of this module and then to compute the distribution of flow parameters within the interior. This necessitatesthe approximation that hydrodynamic transport between the interior of the module and its exterior occurs only through the large peripheral vessels originating and terminating at the arterial and venous nodes labelled a and v, respectively, in the network schematic.This appears to be a reasonable approximation since volumetric flow through side branches of the peripheral vesselsare comparatively small in comparison with that within the peripheral vesselsas they branch out to perfuse a number of contiguous modules. Within the interior of the module those areas of tissue with large fat deposits are richly endowed with capillary networks. Inasmuch as their vascular pattern is not readily mapped out and it is difficult to determine those with an active flow, only vessels which could be easily traced from the photomontage were included in this area. This was a reasonable approximation with an insignificant effect on the overall module interior. HYDRODYNAMIC
ANALYSIS
In order to determine the hemodynamic parameters for a given network configuration, the flow in eachbranch is assumedto follow Poiseuille’s law. For the low Reynolds number regime considered herein (approximately 10m2)it has been demonstrated by Vawter ef al., (1972) that for a Newtonian fluid behavior this is a valid approximation if one assumesthat (1) blood behaves as a homogeneous fluid of constant viscosity throughout the network; (2) the blood vesselsare representedasrigid tubes of invariant diameter and length between nodes; and (3) the flow is steady. Their work also shows that deviations from a fully developed Poiseiullian flow at the entrance of each vessel will contribute to a negligible error. The assumption of Newtonian fluid flow implies that the variation of hemodynamic parameters throughout the network are strictly representative of vascular geometry and peripheral vesselboundary conditions alone.
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AND ZWEIFACH
This is illustrated by the work of Benis and Lacoste (1968) in their analysis of the rheological effects in small blood vessels.Consideration of non-Newtonian fluid behavior is beyond the scope of the present analysis. For Poiseuille flow in a vessel between any two nodes, represented by i and j, the volumetric flow, Q,,, is given in terms of the pressuredifferential, AP,,, vesselconductance, Gl,, and viscosity ,u, by
Q,, = h’128d
APij Gj,
(1)
where APtj = P, - Pj, Gj = dVi,, and d,, and Ii, are the diameter and length, respectively, between nodes i and j. As shown in Fig. 2c there are two or more vesselswhich emanate from thejth node anywhere in the network. The number of vesselsconverging at the jth node being ml. By conservation of masswe must have
zl Q,, = 0,
(2)
where the volumetric flow into a node is considered positive and that out of a node is negative for any branch. From Eq. (1) and (2) we obtain a set of linear algebraic equations in pressure for N nodes in the network, viz. 2 (Pi - P,)Gij = 0. i-1
(3)
The set of equations representedby (3) reduce to a set of simultaneous linear algebraic terms for the nodal pressuresonce the conductances are evaluated from the geometry, and suitable boundary conditions are specified. The boundary conditions chosen are those of nodal pressureson the periphery of the module at arterial and venous nodes, labelled a and v respectively in Fig. 2b. The formulation of the problem in this fashion has two obvious advantages: (1) Since viscosity is a constant throughout, it and the other factors of proportionality cancel out in going from Eq. (2) to (3), and (2) a boundary condition involving intravascular volume flow is much more difficult to measure, whereas on the microvascular level pressure is easier to measure and can be specified with a greater degree of confidence. Hence the solution to Eq. (3) is sought with a knowledge of pressure at M nodes in the network and the pressuresat N-M nodes must be determined. This technique is well formulated for analogous problems in electrical network analysis where it is referred to as a node voltage analysis. Details of the method of solution may be found in Weinberg (1962). COMPUTATIONS The nodal pressuresfor the module of Fig. 2 have been computed for arterial and venous boundary pressuresof 61.Oand 21.Ocm H,O. These values are typical of those measured in vivo for a module situated in the central region of a sector. To display the
NETWORK
ANALYSIS
79
MICROCIRCULATION
pressure distribution graphically as a function of diameter and arterial and venous roles, the intravascular pressure,P, is computed from the nodal pressuresby taking the arithmetic average of upstream and downstream nodal pressures,i.e., p = (Pupstream + pdow”stremlw.o. The intravascular pressure is in turn averaged for all vesselsof a given diameter. The network may then be dichotomized into arterial and venous vesselsby comparing the mean intravascular pressureof eachdiscretevesseldiameter to the averageintravascular pressure for the vesselsof minimum diameter, which are the true capillaries. Vessels with pressuresabove this value are labelled here as arterial and those below are venous, primarily for purposes of graphic representation. Figure 3 shows the computed distribution of mean intravascular pressure for the module under consideration. In order to provide an accurate assessmentof the trend
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VESSEL DIAMETER _ micro
FIG. 3. Distribution of mean intravascular pressure. Horizontal lines to the left and right of the ordinate represent the assumed boundary pressures at the boundary nodes of Fig. 2b. Computed mean intravascular pressures, averaged at each discrete diameter are represented by the circular symbols bracketed by a band of + la (where e equals the standard deviation). Standard deviations less than 1 cm HZ0 are not shown. The solid line represents a cubic spline fit fairing of the computed pressures. The cross-hatched area represents the band of intravascular pressures measured in vim for 150 modules.
described by the computation points, particularly in the region of minimum diameter, a cubic spline fit fairing of the computation points is also presented (solid curve). The cubic spline fit curve was chosen as the best approach for smoothing out the curve, since it minimizes the square of the second derivative along the length of the curve. Such an approximation was believed necessarysincenonuniformities in the distribution arise becauseof (1) the paucity of vesselsat a particular diameter, and (2) the presence of shunting vesselswhose parameters are atypical when compared to the overall sample and show large dispersions about the mean of the parameter of interest, particularly when only one or two such vesselsare present. For comparison, Fig. 3 includes the band of intravascular pressure (cross hatched area) measured in vivo for 150 modules. It should be noted that the experimental measurementspresented in this paper are considered to be rough data since they represent a sampling of only one-fourth of the cat mesentery micropressure data acquired over the past 2 yr in our laboratory. The
80
LIPOWSKY
AND ZWEIFACH I lAP/Ll x IO2_ cm Hz0/micron
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VESSEL DIAMETERI micro
FIG. 4. Distribution of pressure gradient, AP/l. Circular symbols represent the computed pressure gradient averaged at each discrete diameter. The solid line represents the cubic spline fit. Standard deviations are not shown. The dashed line represents in viva measurements.
data has not beenpresentedherein for fine details, but only to indicate trends and orders of magnitude. A more precise presentation of this data has been given by Zweifach (1973). In addition to intravascular pressures,pressure gradient, API1 and wall shear stress z, have been computed. Figure 4 presents the distribution of AP/I given in cm H20/ micron which is shown in a manner similar to that of the intravascular pressure. Standard deviations have been omitted since their trends are similar to those of the intravascular pressure. Also shown in the figure (broken line) is the distribution of pressure gradient established by in viva measurements. Presentedin Fig. 5 is the distribution of vesselwall shearstress,z,, which is computed from the Poiseuillian flow formula, r,,, = (APd/41),and is given in dyn/cm*. In both Fig. 4 and 5 the cubic spline fit servesto define the computed parameter distribution as a
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. 26
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I6
-:A--. ,I4
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VESSEL DIAMETER - micra
FIG. 5. Distribution of vessel wall shear stress, presented in the same manner as the pressure gradient. The dashed segment of the cubic spline fit is intended to demonstrate an order of magnitude only, in the indicated range of vessel diameters.
NETWORK
ANALYSIS MICROCIRCULATION
81
function of vessel diameter, particularly in the region of minimum diameter. In all three distributions, mean intravascular pressure, pressure gradient, and shear stress, the spline fit was constrained to pass through the computed point at the maximum arterial and venous diameters. For this reason, the smoothed curve for wall shear stress appears to rise slightly from 36 to 45 pm on the venous side. This particular trend is not to be interpreted as an exact variation but only as an order of magnitude in this diameter range, and hence is represented by a dashed line. DISCUSSION In comparing the computed intravascular pressures to the values recorded in the living animal, several points of interest can be observed. To begin with, the computation does not demonstrate as rapid a decrease of intravascular pressure as that measured in vivo on the arterial side. In contrast to the measured data, the major computed pressure drop occurs between vessels of approximately 15 pm in diameter on the arteriolar side and 15 pm on the venular side. Intravascular pressures computed in the true capillaries (minimum diameter) appear to be higher than anticipated. This is reasoned by assuming that since the arterial and venous boundary conditions imposed on the module are at the lower bound of the measured data band, then the true capillary pressures should also be at the lower bound. On the venous side, computed and measured pressures agree fairly well with respect to the rate of decrease with increasing diameter for vessels greater than 20 pm in diameter. The disparity between computed and measured pressure gradients (M/I) is even greater than that noted for mean intravascular pressure. At vessel diameters below 20 pm, the computed pressure gradient is approximately six times as great as that measured within a module. However, for larger vessels, on both the arterial and venous sides, good qualitative agreement is achieved within an order of magnitude. The attendant modular distribution of vessel wall shear stress demonstrates a maximum value of approximately 60 dyn/cm’ at the minimum diameter. This corresponds to about half the value estimated by Schmid-Schonbein (1972) (110 dyn/cm2) at the minimum diameter of his hypothetical serial network. Inasmuch as the computed pressure gradient is approximately six times as great as for in vivo measurements, and since wall shear stress is directly proportional to AP/I, it is reasonable to assume that the maximum value of wall shear stress in vivo should be on the order of 10 dyn/cm2. The differences noted between computed and measured intravascular pressures and pressure gradients can be ascribed to three factors not accounted for in a Poiseuillian flow analysis. First, the effective cross-sectional area available for blood flow in some of the microvessels may presumably be decreased by precapillary sphincter action. Vessel diameters used in this analysis were taken as the average luminal diameter which could be observed from the module photomontage. Vawter et al. (1973) have demonstrated that substantial pressure drops may occur because of the constriction at the entrance of such vessels. In view of their analysis, it is conceivable that the discrepancies observed herein could be attributed to these entrance effects. Second, red blood cell and vessel wall interaction at points of branching and bifurcations may also contribute to an increase in flow resistance. The dynamics of red blood cells entering capillary vessels and the effect on hemodynamic resistance has as yet not
82
LIPOWSKY AND ZWEIFACH
been satisfactorily delineated. The third, and presumably the major contributor to the disparities herein, is the non-Newtonian behavior of blood within all vessels of the module. Further information concerning these three factors is necessary before we can make a proper analysis of microcirculatory hemodynamics. In reviewing the approach taken in this study, one must also consider the validity of comparing the distribution of hemodynamic parameters computed for a single module to the data measured for many modules in a large number of laboratory animals. The in vivo intravascular pressures represent modules whose proximity to major vessels bounding the mesenteric sector varied from animal to animal and as a consequence, the modular boundary pressures varied accordingly. The narrow width of the band and the fact that the computed intravascular pressures fall outside this band demonstrate that the aforementioned rheological and geometrical parameters are the primary contributors to these discrepancies. The question of statistical inference for an entire sector of tissue would indeed be of importance if the computed pressure distribution fell entirely within the band of measured data. What is needed at this stage is a detailed description of the statistical distribution of blood flow for a given vessel diameter. This would ensure that the hydrodynamic model is a valid representation of the in vivo processes. It is felt that the network analysis approach to microcirculatory hemodynamics is indeed the proper way to model the functional behavior of the microcirculation. Future attempts to delineate blood rheology on the microcirculatory level should be coupled with a suitable nonlinear (non-Newtonian) network analysis. Once a proper description of hemodynamic behavior is achieved, then the transport phenomena associated with blood-tissue exchange may be satisfactorily described and it should be possible to begin to unravel the mechanics of regulation of the microcirculation.
REFERENCES BENIS,A. M. AND LACOSTE,J. (1968). Distribution
of blood flow in vascular beds: Model study of geometric, rheological and hydrodynamic effects. BiorheoZogy 5,147-161. CHAMBERS, R., AND ZWEIFACH,B. W. (1944). Topography and function of the mesenteric circulation. Amer. J. Anat. 75,173-198. CHIEN, S. (1971). A theory for the quantification of transcapillary exchange in the presence of shunt flow. Circ. Res. 29,172-180. FRASHER,W. G., AND WAYLAND, H. (1972). A repeating modular organization of the microcirculation of cat mesentery. Micorvasc. Res. 4,62-76. INTAGLIETTA,M., AND ZWEIFACH,B. W. (1971). Geometrical model of the microvasculature of rabbit omentum from in oiuo measurements. Circ. Res. 28, 593-600. LEE, J. (1973). Distribution of mean transit time, vascular volume and flow in a repeating modular network of the microcirculation. In “Proceedings of the 4th Annual Meeting of Biomedical Engineering Society,” paper 2.1. Academic Press, Inc., New York, NY. MALL, F. (1888). Die blut- und lymphroege im dunndaim des hundes. Ber Stichs. Ges. Akad. Wiss. 14, 151. NELLIS, S. H. (1972). Application of a tracer dilution technique to cat mesenteric microcirctdation. Ph.D. Dissertation, Univ. of Virginia. RENKIN, E. M. (1964). Normal regulation of tissue circulation. In “Effects of Anesthetics On The Circulation” (H. L. Price and P. J. Cohen, eds.), pp. 171-181. C Thomas, Springfield, IL. SCHMID-SCHONBEIN, H. (1972). Blood rheology in the microcirculation. Z’fliigers Arch. 336, (Suppl) 8487.
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VAWTER,D., FUNG, Y. C., AND ZWEIFACH,B. W. (1973). Theoretical pressure and flow distribution at branching points in the microcirculation. Znt. J. Biomed. Eng., in press. WEINBERG,L. (1962). “Network Analysis and Synthesis.” McGraw-Hill, New York. WIEDEMANN,M. P. (1963). Dimensions of blood vessels from distributing artery to collecting vein. Circ. Res. 12, 375-381. WIEDERHIELM,C. A., WOODBURY,J. W., KIRK, S., AND RUSHMER,R. F. (1964). Pulsatile pressures in the microcirculation of the frog’s mesentery. Amer. J Phpiol. 207, 173-176. ZWEIFACH,B. W., AND RICHARDSON,D. R. (1970). Microcirculatory adjustments of pressure in the mesentery. In “Proceedings of the 6th European Conference on Microcirculation” (J. Ditzel and D. H. Lewis, eds.), pp. 248-253. S. Karger, New York. ZWEIFACH,B. W. (1973). Circ. Res., in press.
Network
RESEARCH
7, 73-83 (1974)
Analysis H.
of Microcirculation
of Cat Mesentery’
H. LIPOWSKY’ AND B. W. ZWEIFACH AMES (Bioengineering), The University of California, San Diego, La Jolla, California 92037 Received May 25,1973
The modular configuration of the microcirculatory system in cat mesentery is subjected to a hydrodynamic network analysis assuming Poiseuillian dynamical behavior. Intravascular pressure, vessel pressure gradient, and wall shear stress are computed for an isolated module and presented as a function of vessel diameter, from arterial affluent to venous effluent. Computed and in vivo intravascular pressures show a marked disparity on the arterial side of the true capillaries and a fair agreement on the venous side. This is attributed to the effects of precapihary sphincter action and non-Newtonian rheological behavior. Computed pressure gradients based on a simple Poiseuillian relationship are approximately six times greater than those measured in vivo. By comparison of predicted and measured pressure gradients, the magnitude of maximum vessel wall shear stress is estimated to be on the order of 10 dyn/cm2.
INTRODUCTION The delineation of the functional behavior of the microcirculation requires a knowledge of the intimate relationships between geometric and hemodynamic parameters such as intravascular pressure, pressure gradient, velocities, and shear stress.Inasmuch as measurement of these parameters, either directly or indirectly for an entire capillary bed is precluded by the large number of vesselsinvolved, it becomesworthwhile to determine their magnitude and distribution as a function of vessel geometry and spatial orientation for a specified mode of dynamical behavior. To this end, several simplified analysesof the microvasculature as a network of serial and parallel elements whose conductance is dictated by Poiseuille flow have appeared in the literature (Chien, 1971). Renkin (1964) has drawn upon the early morphological studies of Chambers and Zweifach (1944) as a frame of reference of capillary clearance in a simplified model of a terminal vascular network composed of parallel and serial vesselsinterposed between arterioles and venules. A more realistic distribution of geometric parametersinfluencing the hemodynamics has been presented by Intaglietta and Zweifach (1971) for the microvasculature of rabbit omentum. In their work a serial network was characterized by the distribution of hydraulic hindrance from arteriole to venule in terms of the “order of branching” of vesselsbased on in vivo distribution data. This approach also represents an oversimplification of the hemodynamic dependency on microvascular geometry (vessellength and diameter). I Supported in part by USPHS Grant No. HL-10881-06. z Predoctoral Fellow, USPHS Training Grant No. HL-05946-02. Copyright 0 1974 by Academic Press, Inc. 73 All rights of reproduction in any form reserved. Printed in Great Britain
74
LIPOWSKY
AND ZWEIFACH
Further attempts to represent the terminal vascular bed by a simplified network of serial and parallel elements have been made recently by Nellis (1972) and Lee (1973). In these studies the basic network of Renkin was utilized to gain insight into the distribution of mean transit times which they were able to measure in vivo by indicator dilution studies in the cat mesentery. In order to allow for better agreement between experiment and theory, Lee went a step further by incorporating into his network model some of the characteristics of the mesenteric microvascular module described by Frasher and Wayland (1972). He proceeded to generalize the mesenteric module as a rectangular area of tissue bounded on four sides by paired arterial and venous vessels. The arterial and venous vesselson the perimeter are then connected acrossthe module interior by vesselsof different diameters, arranged like rungs of a ladder to represent capillaries, metarterioles, arterioles and venules. This model is still a crude representation of the equivalent network within the module interior. Schmid-Schiinbein (1972) has attempted on a larger scale to synthesize a hypothetical network from aorta to vena cava. He devised a composite seriesnetwork based on the morphometric studies of Mall (1888) for fixed canine mesentery and by Weidemann (1963) for the living circulation in the wing of the bat. He then used hydraulic hindrances of this geometrical composite to compute such dynamical parameters as intravascular pressure, pressure gradient, and wall shear stress.Although this type of analysis can be used to define the gross dynamical aspectsof the circulatory system, it fails to elucidate what is actually taking place on the microvascular level in the successivefunctional segmentsof the terminal vascular bed. In order to provide a more precise analytical description of microvascular hemodynamics, the analysis of blood flow is performed here based on a relatively exact in uivo distribution mapped out in detail for the mesenteric circulation of the cat. The techniques employed are those used in electrical network analysis for a linear resistive network whose conductances are determined by assuming Poiseuille flow. METHODS The circulation in the mesentery of the anesthetized cat (pentobarbital) was used for the construction of detailed maps. The preparation for microscopic observation has been described in detail (Zweifach and Richardson, 1970).Serial photographs were taken at 32x magnification and assembledto include one or more complete modules. Vessel diameters for the capillaries were determined at 400x magnification. The dimensions (length and diameter) of the arterioles and venules were recorded from the low power photographs. All of the capillaries visible under the microscope were introduced into the model, although our observations indicate that up to 25-30x of the capillaries may not have an active flow under control conditions. Several fixed preparations of large areas of the mesentery were used to construct a sector and to calculate the averagenumber of modules per unit area of mesentery. The animal was given an injection of 2 ml of Pelikan carbon suspensionto make it easier to identify the capillaries and the tissue was then fixed by suffusing it with 10% gluteraldehyde. The preparation was cleared in glycerine and photographed for the reconstruction of the large sector area of the mesentery.The module usedfor modelling representedthe in uivo state of the vascular bed.
NETWORK
ANALYSIS MICROCIRCULATION
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Pressure measurementswere made with sharpened micropipettes containing 1.0 A4 saline and recorded by an electronic servo system similar to that used by Wiederhielm et al. (1967). Pressuregradients were established by inserting two micropipettes across the different vascular segments.The servo system is accurate to within kO.25 cm H,O. GEOMETRICAL
CONSIDERATIONS
The morphometric description of the microcirculation in the mesenteryof the cat has been described by the studies of Frasher and Wayland (1972) mentioned earlier. As demonstrated therein the microvasculature is characterized by three distinct orders of vessel size and branching. The first is that of the large artery-vein pairs which are distributed radially from the posterior line of attachment toward the bowel, with arterial diameter ranging from 150-300 pm in diameter. These vesselsbifurcate in the vicinity of the intestine to form an artery-vein pair running parallel to the bowel and from which the circulation to the bowel proper is formed. An approximately triangular sector of mesentery is thus circumscribed by these first order vessels, and shall be referred to herein as simply the sector (seeFig. 1). Within the sector, there is a second artery vein
5Oi24
l&O
FIG. 1. Schematic representation of a typical sector. Single lines within the interior represent con-
tinuous artery-vein pairs forming the perimeters of many contiguous modules.
76
LIPOWSKY AND ZWEIFACH
order of branching, characterized by artery-vein pairs (of diameter on the order of 20-40 pm) which traverse the interior and form a continuous interarcading network. This pattern in turn circumscribes areasof tissue referred to as modules by Frasher and Wayland (1972).
a. a
V a
b. FIG. 2. a. Photomontage of a typical module. b. Schematic representation of the module drawn from an
overlay tracing. Nodes labelled a and v represent boundary nodes for the hydrodynamic computation. c. A typical node illustrating the niathematical nomenclature.
The interior of a typical sector is shown schematically in Fig. 1. The continuous artery-vein pairs forming the perimeter of the several modulus are shown with average diameters indicated. Although Frasher and Wayland (1972)emphasizedthe appearance of a five-sided module, several geometric shapesare seen.It should also be noted that in someinstances the precise limits of a given module within any given sector is subject to the interpretation of the observer. It is fairly frequent to observe an artery-vein pair whose diameters are small enough (15-20 pm) to justify labelling them as interior
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ANALYSIS MICROCIRCULATION
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modular vessels.Yet their continuity (from a first order artery to another first order artery and similarly from first order vein to vein) justify their designation as perimeter vesselsof contiguous modules. The interior of each module is composed of an elaborate network of pre- and postcapillary vessels.Figure 2a presents a photomontage of a typical module. An overlay tracing of this module is shown schematically in Fig. 2b where eachvesselis represented by a tube of invariant diameter and specified length betweenjunctions (termed nodes) of two or more vessels.A typical node is shown in Fig. 2c to indicate the mathematical nomenclature utilized in the following hemodynamic analysis. As an example of the distribution of the number of vesselsin a module, Fig. 2 contains 165 vesselsand 115 nodes. The length and diameter of eachvesselwere scaledfrom the photomontage. Although measurementsat such low magnification (32x) are accurate only to within ?I ,um, this was deemed satisfactory for the present analysis. Vesselinternal diameters were measured to the nearest micron, except for the true capillaries, where high power magnifications (400x) showed diameters to range from 6 pm to 8 pm. An average diameter of 7 pm was used in the computations for these vessels. It is the objective of this study to assign specific dynamical boundary conditions (e.g. intravascular pressure) on the periphery of this module and then to compute the distribution of flow parameters within the interior. This necessitatesthe approximation that hydrodynamic transport between the interior of the module and its exterior occurs only through the large peripheral vessels originating and terminating at the arterial and venous nodes labelled a and v, respectively, in the network schematic.This appears to be a reasonable approximation since volumetric flow through side branches of the peripheral vesselsare comparatively small in comparison with that within the peripheral vesselsas they branch out to perfuse a number of contiguous modules. Within the interior of the module those areas of tissue with large fat deposits are richly endowed with capillary networks. Inasmuch as their vascular pattern is not readily mapped out and it is difficult to determine those with an active flow, only vessels which could be easily traced from the photomontage were included in this area. This was a reasonable approximation with an insignificant effect on the overall module interior. HYDRODYNAMIC
ANALYSIS
In order to determine the hemodynamic parameters for a given network configuration, the flow in eachbranch is assumedto follow Poiseuille’s law. For the low Reynolds number regime considered herein (approximately 10m2)it has been demonstrated by Vawter ef al., (1972) that for a Newtonian fluid behavior this is a valid approximation if one assumesthat (1) blood behaves as a homogeneous fluid of constant viscosity throughout the network; (2) the blood vesselsare representedasrigid tubes of invariant diameter and length between nodes; and (3) the flow is steady. Their work also shows that deviations from a fully developed Poiseiullian flow at the entrance of each vessel will contribute to a negligible error. The assumption of Newtonian fluid flow implies that the variation of hemodynamic parameters throughout the network are strictly representative of vascular geometry and peripheral vesselboundary conditions alone.
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This is illustrated by the work of Benis and Lacoste (1968) in their analysis of the rheological effects in small blood vessels.Consideration of non-Newtonian fluid behavior is beyond the scope of the present analysis. For Poiseuille flow in a vessel between any two nodes, represented by i and j, the volumetric flow, Q,,, is given in terms of the pressuredifferential, AP,,, vesselconductance, Gl,, and viscosity ,u, by
Q,, = h’128d
APij Gj,
(1)
where APtj = P, - Pj, Gj = dVi,, and d,, and Ii, are the diameter and length, respectively, between nodes i and j. As shown in Fig. 2c there are two or more vesselswhich emanate from thejth node anywhere in the network. The number of vesselsconverging at the jth node being ml. By conservation of masswe must have
zl Q,, = 0,
(2)
where the volumetric flow into a node is considered positive and that out of a node is negative for any branch. From Eq. (1) and (2) we obtain a set of linear algebraic equations in pressure for N nodes in the network, viz. 2 (Pi - P,)Gij = 0. i-1
(3)
The set of equations representedby (3) reduce to a set of simultaneous linear algebraic terms for the nodal pressuresonce the conductances are evaluated from the geometry, and suitable boundary conditions are specified. The boundary conditions chosen are those of nodal pressureson the periphery of the module at arterial and venous nodes, labelled a and v respectively in Fig. 2b. The formulation of the problem in this fashion has two obvious advantages: (1) Since viscosity is a constant throughout, it and the other factors of proportionality cancel out in going from Eq. (2) to (3), and (2) a boundary condition involving intravascular volume flow is much more difficult to measure, whereas on the microvascular level pressure is easier to measure and can be specified with a greater degree of confidence. Hence the solution to Eq. (3) is sought with a knowledge of pressure at M nodes in the network and the pressuresat N-M nodes must be determined. This technique is well formulated for analogous problems in electrical network analysis where it is referred to as a node voltage analysis. Details of the method of solution may be found in Weinberg (1962). COMPUTATIONS The nodal pressuresfor the module of Fig. 2 have been computed for arterial and venous boundary pressuresof 61.Oand 21.Ocm H,O. These values are typical of those measured in vivo for a module situated in the central region of a sector. To display the
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ANALYSIS
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MICROCIRCULATION
pressure distribution graphically as a function of diameter and arterial and venous roles, the intravascular pressure,P, is computed from the nodal pressuresby taking the arithmetic average of upstream and downstream nodal pressures,i.e., p = (Pupstream + pdow”stremlw.o. The intravascular pressure is in turn averaged for all vesselsof a given diameter. The network may then be dichotomized into arterial and venous vesselsby comparing the mean intravascular pressureof eachdiscretevesseldiameter to the averageintravascular pressure for the vesselsof minimum diameter, which are the true capillaries. Vessels with pressuresabove this value are labelled here as arterial and those below are venous, primarily for purposes of graphic representation. Figure 3 shows the computed distribution of mean intravascular pressure for the module under consideration. In order to provide an accurate assessmentof the trend
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FIG. 3. Distribution of mean intravascular pressure. Horizontal lines to the left and right of the ordinate represent the assumed boundary pressures at the boundary nodes of Fig. 2b. Computed mean intravascular pressures, averaged at each discrete diameter are represented by the circular symbols bracketed by a band of + la (where e equals the standard deviation). Standard deviations less than 1 cm HZ0 are not shown. The solid line represents a cubic spline fit fairing of the computed pressures. The cross-hatched area represents the band of intravascular pressures measured in vim for 150 modules.
described by the computation points, particularly in the region of minimum diameter, a cubic spline fit fairing of the computation points is also presented (solid curve). The cubic spline fit curve was chosen as the best approach for smoothing out the curve, since it minimizes the square of the second derivative along the length of the curve. Such an approximation was believed necessarysincenonuniformities in the distribution arise becauseof (1) the paucity of vesselsat a particular diameter, and (2) the presence of shunting vesselswhose parameters are atypical when compared to the overall sample and show large dispersions about the mean of the parameter of interest, particularly when only one or two such vesselsare present. For comparison, Fig. 3 includes the band of intravascular pressure (cross hatched area) measured in vivo for 150 modules. It should be noted that the experimental measurementspresented in this paper are considered to be rough data since they represent a sampling of only one-fourth of the cat mesentery micropressure data acquired over the past 2 yr in our laboratory. The
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LIPOWSKY
AND ZWEIFACH I lAP/Ll x IO2_ cm Hz0/micron
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FIG. 4. Distribution of pressure gradient, AP/l. Circular symbols represent the computed pressure gradient averaged at each discrete diameter. The solid line represents the cubic spline fit. Standard deviations are not shown. The dashed line represents in viva measurements.
data has not beenpresentedherein for fine details, but only to indicate trends and orders of magnitude. A more precise presentation of this data has been given by Zweifach (1973). In addition to intravascular pressures,pressure gradient, API1 and wall shear stress z, have been computed. Figure 4 presents the distribution of AP/I given in cm H20/ micron which is shown in a manner similar to that of the intravascular pressure. Standard deviations have been omitted since their trends are similar to those of the intravascular pressure. Also shown in the figure (broken line) is the distribution of pressure gradient established by in viva measurements. Presentedin Fig. 5 is the distribution of vesselwall shearstress,z,, which is computed from the Poiseuillian flow formula, r,,, = (APd/41),and is given in dyn/cm*. In both Fig. 4 and 5 the cubic spline fit servesto define the computed parameter distribution as a
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FIG. 5. Distribution of vessel wall shear stress, presented in the same manner as the pressure gradient. The dashed segment of the cubic spline fit is intended to demonstrate an order of magnitude only, in the indicated range of vessel diameters.
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ANALYSIS MICROCIRCULATION
81
function of vessel diameter, particularly in the region of minimum diameter. In all three distributions, mean intravascular pressure, pressure gradient, and shear stress, the spline fit was constrained to pass through the computed point at the maximum arterial and venous diameters. For this reason, the smoothed curve for wall shear stress appears to rise slightly from 36 to 45 pm on the venous side. This particular trend is not to be interpreted as an exact variation but only as an order of magnitude in this diameter range, and hence is represented by a dashed line. DISCUSSION In comparing the computed intravascular pressures to the values recorded in the living animal, several points of interest can be observed. To begin with, the computation does not demonstrate as rapid a decrease of intravascular pressure as that measured in vivo on the arterial side. In contrast to the measured data, the major computed pressure drop occurs between vessels of approximately 15 pm in diameter on the arteriolar side and 15 pm on the venular side. Intravascular pressures computed in the true capillaries (minimum diameter) appear to be higher than anticipated. This is reasoned by assuming that since the arterial and venous boundary conditions imposed on the module are at the lower bound of the measured data band, then the true capillary pressures should also be at the lower bound. On the venous side, computed and measured pressures agree fairly well with respect to the rate of decrease with increasing diameter for vessels greater than 20 pm in diameter. The disparity between computed and measured pressure gradients (M/I) is even greater than that noted for mean intravascular pressure. At vessel diameters below 20 pm, the computed pressure gradient is approximately six times as great as that measured within a module. However, for larger vessels, on both the arterial and venous sides, good qualitative agreement is achieved within an order of magnitude. The attendant modular distribution of vessel wall shear stress demonstrates a maximum value of approximately 60 dyn/cm’ at the minimum diameter. This corresponds to about half the value estimated by Schmid-Schonbein (1972) (110 dyn/cm2) at the minimum diameter of his hypothetical serial network. Inasmuch as the computed pressure gradient is approximately six times as great as for in vivo measurements, and since wall shear stress is directly proportional to AP/I, it is reasonable to assume that the maximum value of wall shear stress in vivo should be on the order of 10 dyn/cm2. The differences noted between computed and measured intravascular pressures and pressure gradients can be ascribed to three factors not accounted for in a Poiseuillian flow analysis. First, the effective cross-sectional area available for blood flow in some of the microvessels may presumably be decreased by precapillary sphincter action. Vessel diameters used in this analysis were taken as the average luminal diameter which could be observed from the module photomontage. Vawter et al. (1973) have demonstrated that substantial pressure drops may occur because of the constriction at the entrance of such vessels. In view of their analysis, it is conceivable that the discrepancies observed herein could be attributed to these entrance effects. Second, red blood cell and vessel wall interaction at points of branching and bifurcations may also contribute to an increase in flow resistance. The dynamics of red blood cells entering capillary vessels and the effect on hemodynamic resistance has as yet not
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been satisfactorily delineated. The third, and presumably the major contributor to the disparities herein, is the non-Newtonian behavior of blood within all vessels of the module. Further information concerning these three factors is necessary before we can make a proper analysis of microcirculatory hemodynamics. In reviewing the approach taken in this study, one must also consider the validity of comparing the distribution of hemodynamic parameters computed for a single module to the data measured for many modules in a large number of laboratory animals. The in vivo intravascular pressures represent modules whose proximity to major vessels bounding the mesenteric sector varied from animal to animal and as a consequence, the modular boundary pressures varied accordingly. The narrow width of the band and the fact that the computed intravascular pressures fall outside this band demonstrate that the aforementioned rheological and geometrical parameters are the primary contributors to these discrepancies. The question of statistical inference for an entire sector of tissue would indeed be of importance if the computed pressure distribution fell entirely within the band of measured data. What is needed at this stage is a detailed description of the statistical distribution of blood flow for a given vessel diameter. This would ensure that the hydrodynamic model is a valid representation of the in vivo processes. It is felt that the network analysis approach to microcirculatory hemodynamics is indeed the proper way to model the functional behavior of the microcirculation. Future attempts to delineate blood rheology on the microcirculatory level should be coupled with a suitable nonlinear (non-Newtonian) network analysis. Once a proper description of hemodynamic behavior is achieved, then the transport phenomena associated with blood-tissue exchange may be satisfactorily described and it should be possible to begin to unravel the mechanics of regulation of the microcirculation.
REFERENCES BENIS,A. M. AND LACOSTE,J. (1968). Distribution
of blood flow in vascular beds: Model study of geometric, rheological and hydrodynamic effects. BiorheoZogy 5,147-161. CHAMBERS, R., AND ZWEIFACH,B. W. (1944). Topography and function of the mesenteric circulation. Amer. J. Anat. 75,173-198. CHIEN, S. (1971). A theory for the quantification of transcapillary exchange in the presence of shunt flow. Circ. Res. 29,172-180. FRASHER,W. G., AND WAYLAND, H. (1972). A repeating modular organization of the microcirculation of cat mesentery. Micorvasc. Res. 4,62-76. INTAGLIETTA,M., AND ZWEIFACH,B. W. (1971). Geometrical model of the microvasculature of rabbit omentum from in oiuo measurements. Circ. Res. 28, 593-600. LEE, J. (1973). Distribution of mean transit time, vascular volume and flow in a repeating modular network of the microcirculation. In “Proceedings of the 4th Annual Meeting of Biomedical Engineering Society,” paper 2.1. Academic Press, Inc., New York, NY. MALL, F. (1888). Die blut- und lymphroege im dunndaim des hundes. Ber Stichs. Ges. Akad. Wiss. 14, 151. NELLIS, S. H. (1972). Application of a tracer dilution technique to cat mesenteric microcirctdation. Ph.D. Dissertation, Univ. of Virginia. RENKIN, E. M. (1964). Normal regulation of tissue circulation. In “Effects of Anesthetics On The Circulation” (H. L. Price and P. J. Cohen, eds.), pp. 171-181. C Thomas, Springfield, IL. SCHMID-SCHONBEIN, H. (1972). Blood rheology in the microcirculation. Z’fliigers Arch. 336, (Suppl) 8487.
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VAWTER,D., FUNG, Y. C., AND ZWEIFACH,B. W. (1973). Theoretical pressure and flow distribution at branching points in the microcirculation. Znt. J. Biomed. Eng., in press. WEINBERG,L. (1962). “Network Analysis and Synthesis.” McGraw-Hill, New York. WIEDEMANN,M. P. (1963). Dimensions of blood vessels from distributing artery to collecting vein. Circ. Res. 12, 375-381. WIEDERHIELM,C. A., WOODBURY,J. W., KIRK, S., AND RUSHMER,R. F. (1964). Pulsatile pressures in the microcirculation of the frog’s mesentery. Amer. J Phpiol. 207, 173-176. ZWEIFACH,B. W., AND RICHARDSON,D. R. (1970). Microcirculatory adjustments of pressure in the mesentery. In “Proceedings of the 6th European Conference on Microcirculation” (J. Ditzel and D. H. Lewis, eds.), pp. 248-253. S. Karger, New York. ZWEIFACH,B. W. (1973). Circ. Res., in press.