J. theor. Biol. (1991) 150, 547-560
Computer Simulation of Growth of Anastomosing Microvascular Networks MOHAMMAD F. K I A N l t § AND ANTAL G. HUDETZ:~
t Department of Biomedical Engineering, Louisiana Tech University, Ruston, Louisiana 71272 and ~. Department of Physiology, Medical College of Wisconsin, Milwaukee, Wisconsin 53226, U.S.A. (Received on 2 August 1989, Accepted in revised form on 10 December 1990) Stochastic growth of polygonal microvascular networks was simulated on computer by dichotomous terminal branching and bridging (anastomosing with an existing segment). The model was applied to describe microvascular growth into a rectangular plane from the sides when vessels bifurcate in a probabilistic manner. The angle of bifurcation was drawn from a normal distribution, the mean of which was varied between 40° and 80°. The resulting networks contained an average of 88-104 nodes of which 30-38% were due to bridging. Number of nodes, number of branches, number of vascular polygons and a fractal dimension representing the density of nodes were calculated for each simulated network. Capillary density increased when mean angle of bifurcation was increased between 40° and 80°. Distributions of normalized vessel lengths and polygon shapes were compared with those of a mesenteric vascular network. The distributions were not found to be significantly different (p < 0.05) for most values of the mean angle of bifurcation, matching best for the mean bifurcation angle of 50°. Vascular polygons had an average shape between pentagonal and hexagonal for the mesenteric network as well as for all values of the mean bifurcation angle used in this study.
Introduction An understanding of the significance and mechanisms of microvascular growth and adaptation is important not only for its intrinsic value, but also for the bearing it may have on our understanding of how the structure of the microvascular network relates to its function. The role of various factors and their interaction in microvascular growth and development in vivo is not fully understood (Hudlicka, 1984). Because o f the complexity o f vascular systems, model studies provide a useful tool for studying how different factors may influence the development o f vascular networks. The present work represents an example of such a model study. Microvascular networks are formed mainly by a continuous process of sprouting and joining of capillaries during growth and regeneration. From early on it was recognized that capillaries send out sprouts, which extend until they meet and anastomose with other sprouts (Clark, 1929). This observation has been confirmed by others (Chilingarian & Paravian, 1971; Dyson et al., 1976; Myrhage & Hudlicka, § Address for reprints: M. F. Kiani, Department of Biophysics, University of Rochester, Rochester, NY 14642, U.S.A. 547 0022-5193/91/120547+ 14 $03.00/0 ~) 1991 Academic Press Limited
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1978). In planar networks, such as the mesenteric vascular bed, the process of capillary growth and anastomosing leads to the formation of vascular polygons. Capillary polygons are thought to safeguard tissue perfusion when capillary segments become blocked by white cells (Plyley et al., 1976; Honig et aL, 1977). It also may serve as a way to equalize pressure throughout the capillary network during growth of the cerebral cortex (Barker, 1978). The most common mode of branching in the cardiovascular system is the dichotomous mode (Zamir & Phipps, 1988) in which each parent vessel bifurcates and produces only two daughter vessels. The angle of birfurcation (BA) is defined as the angle between the continuation of the parent vessel and its daughter vessel (see Fig. 1). A vascular bifurcation with two bifurcation angles of 60 ° each produces three equal angles of 120° between the parent vessel and the two daughter vessels. Prevalence of bifurcation angles close to 60° in vivo has been reported in the literature, e.g. in sartorius muscle of the frog (Plyley et al., 1976) and human brain cortex (Meencke & v o n Keyserlingk, 1979). However, some modeling studies in the literature (Ley et al., 1986) suggest that in addition to dichotomous terminal growth, segmental growth may also be involved in the formation of new vascular branches.
FIG. 1. Illustration of a vascular bifurcation. The angles /3~ and /32 are the angles of bifurcation for the daughter branches 1 and 2, respectively.
The present study was devised to simulate network growth bv the combination of dichotomous branching and bridging (anastomosing) as described above. The purpose of this approach was two-fold. First, we wanted to determine if dichotomous terminal growth of planar anastomosing networks as simulated by computer would result in networks with morphometrical and topological characteristics similar to those of observed microvascular systems. Second, we wanted to determine if changing the angle of bifurcation around 60 ° would have an influence on the morphometry and topology of the final network. Furthermore, an attempt was made in this study to introduce the fractal dimension of node density as an additional means of characterizing network patterns. Until
now, the most frequently used index in vascular growth has been the capillary density (Hudlicka, 1984), i.e. n u m b e r of vessels per area of the tissue, or per volume as the case may be. The distributions of vessel lengths and vessel diameters have also been c o m p a r e d to determine how similar various networks are (Pawlik et al., 1981). In planar networks, the n u m b e r o f vascular polygons and polygon shapes (number o f sides in a polygon) can also be used as indices for comparing vascular networks (Popel et al., 1988), as they were used in the present study. However, as will be discussed later these indices are not sufficient for the characterization of polygonal network patterns. The new field of fractal geometry may offer new ways o f classifying networks as will be shown later. For a discussion of the physiological applications of fractal geometry see Bassingthwaighte (1988).
Methods A rectangular growth field of 1000× 1000 units (relative units) was defined in each simulation. In this study growth was only simulated in two dimensions, i.e. planar growth. Growth of a microvascular network was started with one vessel at each edge of the square growth field. The location of each of the four vessels at the edge was chosen randomly. New branches were generated by terminal bifurcation. Terminal bifurcation is defined as the process by which two daughter vessels are formed at the terminal end of a parent vessel (Fig. 1). Growth was simulated in an iterative manner. At each iteration, the lengths of all growing vessels were increased by an equal amount, one relative unit, which resulted in synchronous growth. A growing branch that intersected an existing vessel was made to fuse with the existing vessel, forming a node, and terminated the growth of the branch. The latter anastomosing process was defined as bridging. The node thus formed was defined as a bridging node and marked as such in the simulation. As discussed earlier, this type o f anastomosing process is observed in vivo. T h e growing microvascular network could be observed on a c o m p u t e r graphic screen, which was updated after each iteration. Vascular polygons were formed by the process of bridging. In general, growth of a branch was terminated by its bifurcation, bridging, or crossing the outer boundary of the plane. When a growing vessel crossed the outer boundary of the growth field a node was formed at the outer boundary. The node thus formed was defined as a terminal node and marked as such in the simulation. It was assumed that the terminal nodes are the points at which the network connects to the rest of the vascular system o f the tissue. The simulation was terminated when all vessels ceased to grow by either bridging or crossing the boundary of the growth field. The process described above is similar to the growth process of blood vessels in planar networks. For example, such a process has long been observed by introducing a transparent c h a m b e r into the rabbit's ear after injuring the ear (Sandison, 1928). In such a process, new vessels start to form and grow from the edges. These vessels then bifurcate and give birth to the new vessels. Final network is formed when growing vessels anastomose with other vessels and form vascular polygons.
It has been reported that the number of anastomoses available per capillary is highly correlated with capillary length. If one tissue has a capillary length twice as much as another, the number of anastomoses observed will be twice as many (Honig et al., 1977). Therefore, in the present model branching probability of each growing vessel was increased in linear proportion with the vessel length, i.e. as a vessel grew longer its probability o f branching was increased. This resulted in mean vessel length not increasing without bounds, in agreement with the in vivo findings (Honig et al., 1977; Plyley et aL, 1976). In order to stochastically assign branching angles, bifurcation angles were randomly drawn from a normal distribution, the mean of which was varied. Prevalence o f bifurcation angles close to 60 ° in vivo was discussed earlier. Therefore, five sets of simulations were performed with the mean bifurcation angles of 40 °, 50 °, 60 °, 70 °, and 80 ° standard deviation, S.D. = 16"0, 20.0, 24"0, 28-0 and 32-0, respectively). For each value of the mean angle of bifurcation, 20 networks were generated. Simulated networks were grouped by the mean angle o f bifurcation. The results were analyzed using the typical indices (number of branches, nodes, terminal nodes, bridging nodes and vascular polygons) for vascular networks. The number of branches (Nb), number o f nodes ( N , ) and number of polygons (Np) in a planar network are not independent of each other and are related by the following equation (James & James, 1959):
Nb=N.+Np-1. Distributions o f vascular polygon shapes (number o f sides in a polygon) and capillary lengths in each network were also generated. Distributions of vascular polygon shapes and normalized vessel lengths were compared to those o f a mesenteric microvascular network (Lipowsky & Zweifach, 1974). All vessel lengths were normalized to the mean of the network they were in. This normalization was performed due to the inherent variability o f average vessel length (Honig et al., 1977) and the relative units employed in both the simulations and the measurement of the mesenteric network. The mesenteric microvascutar network was chosen for comparison because it is a typical polygonal network and it can be considered to be planar. The mesenteric tissue is essentially a sheet in which blood vessels very rarely cross each other in the third dimension. Fractal dimensions are scale ihdependent and fractals structures are characterized by the property o f self-similarity (Bassingthwaighte, 1988). As stated earlier, in this study relative units were used in simulating the growth of microvascular networks. Since fractals have no characteristic scales, a fractal dimension was used to ensure that the results obtained are not influenced by the use of relative units. Also the fact that the network growth was simulated in a repetitive and self-similar manner, makes them specially suitable for fractal analysis. The following node density functions discussed by Bedrosian & Jaggard (1987) was used to calculate a fractal dimension in each simulated network. p ( r ) = Constant
no. o f nodes
r = radial distance from the node Dy = fractal dimension d = topological dimension, equal to two for planar topologies, and the object is fractal provided Df is not equal to d. In each network, D y - d is the slope of the linear regression line of the log-log plot of p(r) vs. r for different nodes in that network. This node density function is a measure of n u m b e r of nodes per unit surface, or volume as the case might be, as a function of radius. Therefore, the network with a higher fractal dimension would have a higher node density and would be more space filling. This conclusion would be independent of the scaling system, or units, employed. The fractal dimensions thus obtained were then used to classify the simulated microvascular networks obtained in this study. The K o l m o g o r o v - S m i r n o v test ( K - S test) was used for comparing distributions such as vessel length with standard distributions (normal, Weibull, etc) and with distributions of the data from the literature. The K - S test allows one to determine if two samples come from the same distribution. This procedure calculates the m a x i m u m distance between the cumulative distribution of the two samples. If the distance is large enough, p < 0.05, the null hypothesis that the distributions are the same is rejected. Results Five sets of simulations were performed with the mean bifurcation angles (BA) of 40 °, 50 °, 60 °, 70 ° and 80 °. Typical networks resulting from the simulation are depicted in Fig. 2. The four initial nodes in each network, from which the growth was started, are marked as such in each figure. Resulting microvascular networks resemble naturally occurring planar microvascular networks such as that of the mesenteric vascular bed. O f course, naturally occurring microvascular networks are not c o m p o s e d of straight vessels (lines), as depicted here. The resemblance is in the general form and the overall appearance of the network. Also, if in a planar microvascular net capillaries are a p p r o x i m a t e d by straight lines the resemblance becomes more apparent; e.g. see Lipowsky & Zweifach (1974). The microvascular networks thus simulated were analyzed using the indices discussed earlier. The growth area in all cases was constant, 1000x 1000 relative units. Therefore, n u m b e r o f nodes and n u m b e r o f branches formed are representative of capillary density. It is expected that in a more compact network, the number of vascular polygons and the percentage of bridging nodes would increase while the percentage o f terminal nodes would decrease. The resulting networks contained an average o f 87.8-104.5 nodes. The nodes were classified as those due to bifurcation and those due to bridging. In the networks generated, on the average, 30-38.3% of the nodes were due to bridging. The statistical
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FIG, 2, Typical networks generated by simulating microvascular growth with m e a n bifurcation angles o f 40 ° (top left), 50 ° (top right), 70 ° (bottom left), and 80 ° (bottom right) degrees, The initial nodes, from which the growth was started, are marked by the letter "'I" in each panel.
Number (mean ±s.19., n = 2 0 in each group) of nodes, bridging nodes, terminal nodes and branches for each BA. BA is the mean angle of bifurcation and is given in degrees BA
40 50 60 70 80
87.8+ 14.1 89.5± 14.8 91.7 ± 15.6 96.2± 14.6 104.5+14-9
30.0±6.3 31.3±5.9 32,3 ± 5.4 33.5±6.5 38.3±6-0
17.9±2.8 17.5±2.6 17.5 ± 3.3 18,6±2-t 18,0±2-7
113.8±20-1 116.8±20.5 120,0 ± 20.8 125,7+20-9 138.8+20-6
information on number of nodes, bridging nodes, terminal nodes and branches are given in Table 1. As indicated in Table 2, vascular polygons had an average shape (number o f sides) between pentagonal and hexgonal for all values of the mean bifurcation angle as did the mesenteric vascular bed obtained from the literature (Lipowsky & Zweifach, 1974). As depicted graphically in Fig. 3, number of nodes, branches and vascular polygons increased with the mean angle of bifurcation. Figure 4 shows that the percentage of bridging nodes increased with the mean bifurcation angle while the percentage of terminal nodes decreased with the mean bifurcation angle. Despite the large values of standard deviation shown in Figs 3 and 4, significant (slope > 0 with a p < 0.05) positive linear regression was found in all cases (negative for the percentage of terminal nodes). The large scatter around the mean is in agreement with the in vivo findings (Zamir & Bigelow, 1984). These results indicate that more compact networks can be obtained by increasing the mean angle of bifurcation over the range considered here. The normalized vessel lengths for all values o f the mean bifurcation angle had a Weibull distribution, significance level for the Kolmogorov-Smirnov test=0.28, 0.99, 0.15, 0.27 and 0.07 for the bifurcation angles of 40 °, 50 °, 60 °, 70 ° and 80 °, respectively. As will be discussed later, a Weibull distribution of vessel lengths in vivo may be the outcome of a growth process with random branching. The distributions of polygon shapes and normalized vessel lengths were compared, using the Kolmogorov-Smirnov test, with those o f the microvascular bed of the cat mesentery (Lipowsky & Zweifach, 1974) and the results are presented in Table 3. For most values of mean bifurcation angle, the distributions obtained in the present study were not found to be significantly different from those of the mesenteric network. Polygon shapes matched best for the mean bifurcation angle of 40 ° and were not significantly different for other values of the mean angle of bifurcation. Normalized vessel lengths matched best for the mean bifurcation angle of 50 °. For the mean bifurcation angle of 50 ° both length and polygon shape distributions TABLE 2
Number (mean + S.D., n = 20 in each group) of polygons and polygon shapes (number of sides in a polygon) for each BA. BA is the mean angle of bifurcation and is given in degrees. The data for the mesentery are given for comparative purposes. Polygon shapes from the simulated networks are not significantly (p < 0.05) different from the mesenteric network for all BA BA
40 50 60 70 80 Mesentery
27.0±6-3 28.3 ± 5-9 29.3 ± 5.4 30.5±6-5 35.3±6.0 28
5-6±2.1 5-4 ± 1.6 5-6± 1.7 5-6± 1.7 5.5± 1.8 5.4 ± 1.9
150 140 ,c u c
110 I00 90
4O o gt
20 40 50 60 1"0 80 Bifurcation angle (degrees)
FIG. 3. N u m b e r of nodes, branches and vascular polygons (mean ± S.D., n = 20) for each mean angle of bifurcation.
agreed with those of the mesenteric microvascular network. Note that in the Kolmogorov-Smirnov test, distributions are considered to be significantly different only if the observed significance level, p, is less than 0.05. The fractal dimensions calculated for each mean angle o f bifurcation are shown in Fig. 5. As expected, all fractal dimensions are between the topological dimensions
FIG. 4. Percentage of bridging and terminal nodes (mean+s.D., n=20) for each mean angle of bifurcation. (O), Terminal nodes; (O), bridging nodes.
Significance levels ( p ) from the comparison of vessel length and polygon shape distributions with those of the mesenteric microvascular network. The significance levels are calculated by the Kolmogorov-Smirnov test. If p < 0-05 the null hypothesis (distributions are similar) is rejected. BA is the mean angle of bifurcation and is given in degrees BA
p value (polygon shapes)
p value (vessel length)
40 50 60 70 80
0"27 0"22 0-19 0.07 0"07
0"01 0"21 0"01 0' 16 0'02
I.G gn c:
E -o o o
I ....... 40
BifurcatiOn ongle (degrees}
FIG. 5. Fractal dimension (mean ± S.D., n = 20) obtained for each mean angle of bifurcation. of a line and a plane, i.e. between one and two. The fractal dimensions calculated here are representative of node density in a scale-independent fashion. In agreement with the previous findings, fractal dimension (i.e. node density) increased with the angle of bifurcation. Discussion
In the present work, dichotomous terminal growth of planar anastomosing microvascular networks was simulated by computer. The simulation resulted in networks with geometrical and topological characteristics similar to those o f observed vascular systems o f sheet tissues. The agreement of the distribution of polygon shapes and normalized vessel lengths between simulated networks and a mesenteric microvascular network of a cat (Lipowsky & Zweifach, 1974) was fairly good for a large range of bifurcation angles, although the best match was found for the microvascutar networks generated with the mean bifurcation agle of 50 °. The statistical distribution of normalized vessel lengths of simulated networks were found to follow the Weibuli distribution for all values of the mean bifurcation angle. This may have a physiological significance. The segment lengths o f the cat mesenteric net (Lipowsky & Zweifach, 1974) also had a Weibull distribution ( p > 0.99 with the K-S test). Segment lengths in the vessels of rat gracilis muscle (Honig
et al., 1977) and the cat cerebral cortex (Pawlik et al., 1981) were reported to have the same distribution. The distribution o f capillary lengths in sartorius muscle in frogs was reported to be exponential (Plyley et al., 1976). However, exponential distributions are a special case of the Weibull family of distributions (Walpole & Myers, 1978). It is hypothesized that random branching, as was simulated in the present model, in vivo can lead to a Weibull distribution o f capillary lengths (Plyley et al., 1976). Also, the combination of random sprouting and vessel elongation during growth in vivo is hypothesized to be the reason why a Weibull distribution of capillary lengths is seen in vivo (Pawlik et al., 1981). The average polygon shape was between pentagonal and hexagonal, for all values o f the bifurcation angle studied (40 °, 50 °, 60 °, 70 ° and 80°). The polygon shape for the mesenteric microvascular network used for comparison (Lipowsky & Zweifach, 1974) was also between pentagonal and hexagonal, 5 - 4 + l . 9 ( m e a n + s . D . ) . An average polygon shape between pentagonal and hexagonal is also reported for the arteriolar network o f the cat sartorius muscle (Popel et aL, 1988) and the cat cerebral cortex (Meencke & von Keyserlingk, 1979). Having a polygon shape close to hexagon may be significant. Amongst all polygons, hexagons allow the closest packing in an area. This close packing minimizes the movement cost to a given point from all the surrounding areas. Also, minimum work can be achieved only if all areas are hexagonal (Woldenberg, 1969). From the results o f the present study it would appear that regardless of the prevailing angle of bifurcation, the polygons still have a shape between pentagonal and hexagonal. This might be due to the fact that the angles of bifurcation studied here were close to 60 °. It is to be noted that in a regular hexagon all angles are equal to 60 °. Despite the good agreement o f predicted and experimental data for several mean bifurcation angles, the mean bifurcation angle had a consistent influence on the characteristics of computer generated networks. The number of nodes, branches, and vascular polygons increased with the mean angle of bifurcation. The percentage of bridging nodes increased with the mean angle of bifurcation while the percentage o f terminal nodes decreased with the mean angle of bifurcation. These findings suggest that in general more compact networks can be obtained with an increasing bifurcation angle. In other words, for a given size growth field, space filling will be more complete with a larger bifurcation angle in the range of bifurcation angles studied here, 40°-80 °. Since the growth area was kept constant for all simulations, the number of branches is a measure of capillary density. Therefore, in the range of mean bifurcation angles studied here, 400-80 °, capillary density should be higher if the prevailing angle of bifurcation is larger. A similar increase in the compactness of the networks with the angle of bifurcation was found when growth was initiated from a single branch in the center of the growth field (Kiani & Hudetz, 1989). The results of the present study suggest that realistic networks can be obtained by the process of terminal growth and anastomosis. However, segmental and terminal bifurcation of vessels have been shown to have different topological consequences in the simulation of growth of vascular trees. In terminal bifurcation, formation of new vessels is simulated only at the terminal end of pre-existing vessels while in segmental bifurcation, new vessels can be simulated at any point on the branch of
A. G . H U D E T Z
a pre-existing vessel. Vascular trees generated by segmental bifurcation have been shown to be more symmetric than those generated by terminal bifurcation (Ley et aL, 1986). Thus, it is not clear if terminal bifurcation, segmental bifurcation or a combination of the two may be at work in vivo. Although several vascular tree growth models have been developed (Horsfield et al., 1987), simulation of growth of anastomosing vascular networks has not been reported in the literature. A distinction should be made between vascular trees and vascular networks. Vascular trees are divergent in that they do not anastomose and do not form vascular polygons. Vascular networks, on the other hand, are anastomosing and do form vascular polygons when branches meet. Also, as the number of vessel segments increases, the topological pattern of the vascular networks change (Van Pelt et ai., 1986). In most cases, however, a distinction between the simulation of growth of vascular trees and vascular networks is not made (Woldenberg, 1986). Vascular networks are often described as a combination of two or more wellconnected vascular trees (Zamir & Phipps, 1988). This assumption is not physiological as growing capillaries do not form trees that connect to form vascular networks. It is realized that more comprehensive topological and spatial indices are required in order to adequately characterize anastomosing vascular networks (Popel et al., 1988). The traditional methods of ordering and recording the spatial structure of vascular and botanical trees follow the growth of trees and, therefore, are not appropriate for classifying anastomosing vascular networks. In these schemes the most parent vessel or the most daughter vessels are selected as the starting point for ordering. This selection is a subjective task in anastomosing networks unless the blood flow pattern is also known (Ley et al., 1986). This is probably the reason why topological analyses of circuit networks are very scarce (Woldenberg, 1986). A hierarchical classification scheme for ordering anastomosing vascular networks was proposed by Popel et al. (1988), although the latter scheme still relies on the subjective assignment of the zero-order nodes. The new field of fractal geometry may offer new ways of classifying vascular networks. The node density function discussed by Bedrosian & Jaggard (1987) was used here to calculate a fractal dimension in the simulated networks. Using the more traditional indices, it was concluded that networks simulated with the larger angles of bifurcation are more compact. In agreement with the latter finding, larger fractal dimensions for the simulated networks were calculated when the mean angle of bifurcation was increased. As stated earlier, fractal dimensions are scale-independent (Bassingthwaighte, 1988). Therefore, the increase in t h e compactness of the networks with the mean angle of bifurcation is valid regardless of the fact that relative units were employed. As expected, the fractal dimensions calculated were always between one and two. This method may offer a way in which networks can be identified and classified according to their fractal dimension. The use of fractals is specially suitable in this case because the model used here described the microvascular growth process in a repetitive or self-similar manner. Such processes often create fractal objects with features that adhere to the same rules through a succession of different scales. In such cases, the fractal dimension reflects the scaling characteristics of the structure and serves as a measure of irregularity and
space filling regardless of the length of the measuring stick (Bassingthwaighte, 1988). For natural fractals, the system cannot be divided infinitely and this renders the system "pseudofractal" (Bassingthwaighte, 1988) in that the structure is fractal within some finite range. We are grateful to D r M a r g a r e t Maxfield ( D e p a r t m e n t o f M a t h e m a t i c s , L o u i s i a n a Tech University) for h e r help with the statistical analysis o f the data. V a l u a b l e advise o f D r R i c h a r d G i b b s ( D e p a r t m e n t o f Physics, L o u i s i a n a T e c h University) o n fractal g e o m e t r y is appreciated. We also w o u l d like to t h a n k D r J u l i a n H. L o m b a r d ( D e p a r t m e n t o f Physiology, Medical College o f W i s c o n s i n ) for reviewing t h e m a n u s c i p t . This work was s u p p o r t e d in part by the N a t i o n a l Science F o u n d a t i o n g r a n t n u m b e r CBT-8822851 a n d BCS 9001425. REFERENCES BARKER, J. N. (1978). Regulation and stunting of cerebral microvasculogenesis by blood flow and endothelial nutrition during the fetal/premature period and birth. In: Fetaland Newborn Cardiovascular Physiology Vol. 2 (Longo, L. D. & Reneau, D. D., eds) p. 135. New York: Garland STPM Press. BASSINGTHWAIGHTE, J. B. (1988). Physiological heterogeneity: fractal link determinism and randomness in structures and functions. NIPS, 3, 5-10. BEDROSIAN, S. O. & JAGGARD, D. L. (1987). A fractal-graph approach to large networks. Proc. I E E E 75(7), 966-068. CHILINGARIAN, A. M. ~,~ PARAVIAN, E. N. (1971). A study of brain capillary proliferation in various periods of postnatal life by means of the lead method. Brain Res. 28, 550-552. CLARK, E. R. (1929). Studies of the growth of blood-vessels in the tail of the frog larva by observation and experiment on the living animal. Am. J. Anat. 23(1), 37-88. DYSON, S. E., JONES, D. G. & KENDRICK, W. L. (1976). Some observations on the ultrastructure of developing rat cerebral capillaries. Cell Tiss. Res. 173, 529-542. HONIG, C. R., FELDSTEIN, M. L. & FRIERSON, J. L. (1977). Capillary lengths, anastomoses, and estimated capillary transit times in skeletal muscle. Am. J. Physiol. 21(1), H122-H129. HORSFIELD, K., WOLDENBERG, M. J. & BOWLS, C. L. (1987). Sequential and synchronous growth models related to vertex analysis and branching ratios. Bull. Math Biol. 49, 413-429. HUDLICKA, O. (1984). Development of microcirculation: capillary growth and adaptation. Handbook o f Physiology Vol. I'V, p. 165. JAMES, G. & JAMES, R~ C. (1959). Mathematics Dictionary, 2nd edn. New York: Van Nostrand. pp. 148. KIANI, M. F. & HUDETZ, A. G. (1989). Computer simulation of growth of anastomosing vascular networks. F A S E B J. 3(4), A1406 (Abstract). LEY, K., PRIES, A. R. & GAEHTGENS, P. (1986). Topological structure of rat mesenteric microvessel networks. Microvasc. Res. 32, 315-332. LIPowsKY, H. H. & ZWEIFACH, B. W. (1974). Net work analysis of microcirculation of cat mesentery. Microvasc. Res. 7, 73-83. MEENCKE, H. J. & VON KEYSERLINGK, D. G. (1979). Stereologische Untersuchung der Gef'~issarchitektur der Grosshirnrinde des Kaninchens. Acta Anat. 103, 365-373. MYRHAGE, R. & HUDLICKA, O. (1978). Capillary growth in chronically stimulated adult skeletal muscle as studied by intravital microscopy and histological methods in rabbits and rats. Microvasc. Res. 16, 73-90. PAWLIK, G., RACKL, A. & BING, R. J. (1981). Quantitative capillary topography and blood flow in the cerebral cortex of cats: an in vivo microscopy study. Brain Res. 208, 35-58. PLYLEY, M. J., SUTHERLAND, G. J. t¢~ GROOM, A. C. (1976). Geometry of the capillary network in skeletal muscle. Microvasc. Res. 11, 161-173. POPEL, A. S., TORRES F1LHO, l. P., JOHNSON, P. C. & BOUSKELA, E. (1988). A new scheme for hierarchiaI classification of anastomosing vessels. Int. Z Microcirc: Clin. Exp. 7, 131-138. SANDISON, J. C. (1928). Observations on the growth of blood vessels as seen in the transparent chamber introduced into the rabbit*s ear. Am. J. Anat. 41, 475-496. VAN PELT, J., VERWER, R. W. H. ¢~. UYLINGS, H. B. M. (1986). Application of growth models to the topology of neuronal branching patterns. J. neurosc. Meth. lg, 153-165. WALPOLE, R. ~,~ MYERS, R. H. (1978). Probability and Statistics for Engineers and Scientists. New York: Macmillan.
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