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A graph theory analysis of renal glomerular microvascular networks
Microvascular Research 67 (2004) 223 – 230 www.elsevier.com/locate/ymvre
A graph theory analysis of renal glomerular microvascular networks Eric M. Wahl, a,* Louis V. Quintas, a Lorraine L. Lurie, b and Michael L. Gargano c a
The New York Institute for Bioengineering and Health Science, New York, NY 10011, USA b Department of Mathematics, Pace University, New York, NY 10038, USA c Department of Computer Science, Pace University, New York, NY 10038, USA Received 31 July 2003 Available online 27 February 2004
Abstract A graph theory model and its invariants are used to compare previously published renal glomerular networks of six adult rats, one adult uremic rat, and one newborn rat. Invariants calculated include order, size, cycle rank, eccentricity, root distance, planarity, and vertex degree distribution. These invariants enabled the differentiation of six normal adult glomerular microvascular networks from that of the uremic glomerulus and from that of the normal newborn glomerulus. These invariants might then be used to differentiate between normal and pathological vascular networks. Also proposed are graph theory invariants that might be used to develop a quantitative model for angiogenesis. D 2003 Elsevier Inc. All rights reserved. Keywords: Renal glomerular network; Microvasculature; Glomerulus; Uremia; Vascular branching patterns; Angiogenesis; Graph theory
Introduction Many researchers have studied the branching pattern of the renal glomerular microvascular network. Early studies of the glomerular network were essentially hand-drawn illustrations of the vascular pattern of the glomerulus, the first of which was given by Malpighi in the 17th century (Andruecci, 1972). These drawings depicted an anastomosing and lobular nature of the glomerular network but left the exact quantitative nature of the branching pattern unexamined. The topology of a microvascular network is of interest as it pertains to its physical properties and its embryological development. Investigators have used various ordering and numbering schemes to analyze the topology of vascular networks in relation to their geometric and hemodynamic properties. Shea (1979) used graph theory to map and tabulate experimental measurements of vascular segments of the renal glomerulus. Fenton and Zweifach (1981) originally applied the Horton – Strahler ordering technique in the analysis of the * Corresponding author. The New York Institute for Bioengineering and Health Science, 30 Fifth Avenue, New York, NY 10011. E-mail addresses: [email protected] (E.M. Wahl), [email protected] (L.V. Quintas), [email protected] (L.L. Lurie), [email protected] (M.L. Gargano). 0026-2862/$ - see front matter D 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.mvr.2003.11.005
microvasculature by obtaining hierarchical bifurcation ratios of tree-like vascular structures and relating this information to vessel lengths and diameters. This concept was modified to include planar arteriolar networks as seen in the rat muscle (Engelson et al., 1985). Random branching algorithms used to generate frequency distributions of vessel branch segments with respect to their generation numbers further advanced the network analysis of the microvasculature (Ley et al., 1986). With the aid of automated microdissection and computer reconstruction from serial sections, investigators have been able to reproduce the branching pattern of the glomerulus (Antiga et al., 2001; Nyengaard and Marcussen, 1993; Remuzzi et al., 1992; Shea, 1979; Shea and Raskova, 1984; Wahl et al., 1984; Winkler et al., 1991). These investigators depicted blood vessels as edges and branch points as vertices, thereby constructing graph theoretical representations of the glomerular network. By reducing complex anastomosing networks to quantitative theoretical models, the graph theory has been able to illustrate the complex branching pattern of the renal glomerulus as well as provide such information as the number of vessels and branch points in its network. As a related branch of topology, the graph theory can also be used to assign numbers to both vessels and branch points and thereby correlate the geometric and hemodynamic properties of a network with the topological features of its associated graph.
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induced renal failure and a graph of a newborn rat glomerular network are analyzed and compared with graphs produced from the networks of normal adult rats. This article is meant to demonstrate how graph theory can be used to analyze the topology of renal glomerular networks even though the small number of subjects in this study does not allow for any definitive conclusions based on statistical analysis. The present study is an introduction to a systematic and uniform graph theory analysis of microvascular networks. The section on Graph theory background provides the definitions of all graph theory invariants discussed in this paper.
Graph theory background
Fig. 1. Invariants for this hypothetical graph are as follows: order n = 9; size t = 12; cycle rank b = 4; cycle basis: {cycle (2,3,5), cycle (3,4,6,7,5), cycle (4,6), cycle (6,7,8)} (note that cycle (3,4,6,7,5) + cycle (6,7,8) = cycle (3,4,6,8,7,5)); root distance rd = 5 (see path (1,2,5,7,8,9)); eccentricity distributiona: (5,4,4,3,3,4,3,4,5); radius R = 3 (see vertices 4, 5, and 7); diameter D = 5 (see vertices 1 and 9); vertex degree distributiona: (1,3,3,3,3,4,3,3,1); the edges defined by vertices 4 and 6 are parallel edges. Superscript a if for the vertex set: (1,2,3,. . .,9).
In graph theory, certain quantifiable parameters are called invariants (Harary, 1972). Specifically, an invariant is a property of a graph that is independent of labeling. Examples include the number of vertices or the number of edges. Quantifying microvascular networks with invariants may facilitate the differentiation between normal and abnormal networks, such as in uremia or malignancy. The graph theory may also provide a mathematical basis for modeling the growth and development of microvascular networks as seen in the renal glomerulus (Wahl et al., 1984). In this study, the graph theory is used to compare six previously published normal adult rat glomerular networks. The published graphs of a glomerulus of an adult rat with
A graph G = (V,E) consists of a nonempty set V, whose elements are called vertices (or nodes), and a set E of unordered pairs of distinct elements of V called edges. A subgraph of G is a graph whose vertices are contained in V and whose edges are contained in E. The order n of G is its number of vertices and the size t is its number of edges. The vertices that define an edge are said to be adjacent. An edge is said to be incident to each of its vertices. The degree of a vertex is the number of edges incident to it. A path is an alternating sequence of distinct vertices and edges, starting and ending with a vertex, such that consecutive vertices define the edge between them. A cycle is a path combined with an edge connecting the first and last vertex in the path. A cycle can also be visualized as a closed path. It is understood that, to say a graph contains a specified path or cycle, all edges must be in the graph. Parallel edges are defined as two or more edges connecting the same adjacent vertex pair. The number of edges in a shortest path connecting a pair of vertices u,v is the distance d(u,v). When the vertices u and v are fixed and defined as the source and sink vertices of a network, respectively, the distance between them is called the root distance rd. The eccentricity e(u) of a vertex u in a connected graph G is the maximum possible distance d(u,v) from u to any other vertex v in G, in other words the eccentricity of u is the length of the shortest path connecting u to a vertex furthest from it. The radius R and the diameter D are the minimum
Fig. 2. The nonplanar graphs of Kuratowski: K5 and K3,3.
E.M. Wahl et al. / Microvascular Research 67 (2004) 223–230
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Table 1 This table shows invariants calculated in this study: n = order, t = size, b = cycle rank, R = radius, D = diameter and rd = root distance Study Adult Shea (1979) Remuzzi et al. (1992) Nyengaard and Marcussen (1993) Antiga et al. (2001) Winkler et al. (1991) Wahl et al. (1984) Mean Standard deviation Uremic Shea and Raskova (1984) Newborn Nyengaard and Marcussen (1993)
n
t
b
R
D
rd
rd/n
rd/t
195 247 256 302 312 358 278.3 57.4
322 403 426 460 466 595 445.0 89.9
128 157 171 159 155 238 168.0 37.1
9 9 9 12 11 12 10.3 1.5
15 15 15 18 16 19 16.3 1.8
11 12 11 12 13 11 11.7 0.82
0.056 0.049 0.043 0.040 0.042 0.031 0.043 0.008
0.034 0.030 0.026 0.026 0.028 0.019 0.027 0.005
159
276
119
9
17
7
0.044
0.025
24
35
12
4
7
5
0.208
0.143
and maximum eccentricities among the set of all vertices of G, respectively. The cycle rank b is the number of independent cycles and is calculated by the relation: b = t n + 1, where t is the number of edges (size) and n is the number of vertices (order) of the graph. A subgraph consisting of the edges of two cycles and excluding any edges that appear in both cycles is called the sum of the two cycles. The vertex set of the sum is made up of vertices from the edges in its edge set. A set of cycles is independent if no cycle in the set can be expressed as a sum of the remaining cycles. A set of b
independent cycles is a basis. Any cycle in a graph can be expressed as a sum of cycles in any basis. Fig. 1 shows a hypothetical graph illustrating all of the invariants defined above. A graph is said to be planar if it can be drawn in the plane in such a way that none of its edges intersect except at a vertex. Two graphs are homeomorphic if both can be obtained from the same graph by replacing one or more edges with a path. Any graph containing a subgraph homeomorphic to either K5 or K3,3 (Fig. 2) is nonplanar (Kuratowski, 1930).
Fig. 3. Cycle rank plots show squares representing data from the normal adult glomerular networks. The triangles represent the uremic network and the circles represent the newborn network.
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Material and methods Graphs from six normal adult rat renal glomerular networks found in the literature were analyzed for graph theory invariants (Antiga et al., 2001; Nyengaard and Marcussen, 1993; Remuzzi et al., 1992; Shea, 1979; Wahl et al., 1984; Winkler et al., 1991). Also, a seventh graph representing the glomerular network of an adult rat with renal failure (uremia) by Shea and Raskova (1984) and an eighth graph published by Nyengaard and Marcussen (1993) representing the glomerular network of a 5-day-old newborn rat were evaluated. In all of these studies, investigators used a microtome and standard tissue staining and embedding techniques to obtain serial cross-sections of a renal glomerular vascular network. Blood vessels and branch points were identified by tracing and following the vascular contours
along the entire sequence of serial sections. These investigators created a graph theoretical representation of the glomerular network by replacing the vessels with edges and the branch points with vertices. To perform an analysis of a graph theoretic representation of a network, the vertices of the graph must be numbered (labeled) and then these numbers are listed as a complete set of labeled vertex pairs (edge list). By assigning an integer arbitrarily to every vertex in a graph, a unique pair of numbers will represent each edge. The set of all vertex pairs is the edge set of the graph. An edge set may then be entered into a computer for graph theory analysis. Five of the eight published glomerular networks had been depicted as unlabeled graphs (Remuzzi, Nyengaard (2), Winkler, Antiga). The vertices of these five graphs were labeled arbitrarily and five edge lists were thus generated.
Fig. 4. Eccentricity distributions of the glomerular networks analyzed.
E.M. Wahl et al. / Microvascular Research 67 (2004) 223–230
Two of the graphs (Shea and Shea/Raskova) were depicted as labeled networks and those numbers were used to create an edge list. One graph (Wahl) was directly represented as an edge list. The edge list for each of the eight graphs was entered into a Dell XPS-Z Computer (Pentium III/40 GB/256 MBRAM/930 MHz) for analysis of graph theory invariants. The invariants were calculated for all graphs using Mathematica (technical computational software by n2003 Wolfram Research, Inc.), then tabulated and compared.
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in other words, vertices of degree 3. The average percentage of degree 3 vertices in adult glomerular networks is 82.4% (range: 71.8 – 100%). The average percentage of degree 4 vertices in the adult networks is 14.2%, and of degree 5 vertices and higher is 3.4%. The percentage of degree 3 vertices in the newborn glomerular network of Nyengaard is 95.5%. The Shea/Raskova graph depicting a network of a glomerulus induced with chronic renal disease shows a shift in the vertex degree distribution. Here, the percentage of degree 3 vertices is only 64.1%.
Results Discussion Graph theory invariants for all eight glomerular networks studied are presented in Table 1. These invariants include order n, size t, cycle rank b, radius R, diameter D, and root distance rd (see section on Graph theory background). Cycle rank plots of all networks are given in Figs. 3a and b. The eccentricity distributions and vertex degree distributions associated with the networks are shown in Figs. 4 and 5, respectively. The order of the six adult glomeruli analyzed averages 278.3 vertices (range: 195 – 358). The order is 159 vertices for the uremic and 24 vertices for the newborn network. The size of the six adult glomeruli analyzed averages 445.0 edges (range: 322– 595). The size is 276 edges for the uremic and 35 edges for the newborn glomerular network. The cycle rank, indicating the number of independent cycles, averages 168.0 (range: 128 – 238) for the adult glomerular networks. The cycle rank is 118 for the uremic and 12 for the newborn glomerular networks. The relationships of cycle rank b vs. order n and cycle rank b vs. size t are shown in Figs. 3a and b, respectively. The radius (minimum eccentricity) of the six normal glomerular networks averages 10.3 (range: 9– 12) and the diameter (maximum eccentricity) of the six normal glomerular networks averages 16.3 (range: 15– 1 9). The radius and diameter of the uremic network are 9 and 17, respectively. The radius and diameter of the newborn network are 4 and 7, respectively (Table 1). The eccentricity distribution for each of the glomerular networks is shown in Figs. 4a– h. The eccentricity distribution illustrates the variation in the number of eccentricities from the radius to the diameter in terms of percentage of vertices for a given eccentricity. The root distance represents the smallest possible number of vessels that may be traversed from the afferent to the efferent arteriole. The root distance averages 11.7 (range: 11– 13) for adult glomerular networks and measures 7 and 5 for the uremic and newborn glomerular networks, respectively (Table 1). All graphs are nonplanar with the exception of the newborn glomerular network of Nyengaard. The vertex degree distributions for the glomerular networks are shown in Fig. 5. These distributions show that the majority of vertices in the renal glomerulus are bifurcations,
Quantifying the complexity of the renal glomerular microvascular network is technically demanding and mathematically challenging. A glomerular network, as noted previously, can be defined as a graph when the vessels are identified as edges and the branch points as vertices. These graphs can then be examined mathematically by calculating and comparing their invariants. This study represents the first attempt to compare several renal glomerular network graphs previously published by independent investigators (Table 1). Although all of these investigators constructed graphs, Shea (1979) having been the first, they focused mostly on the physical properties of the networks, such as capillary dimensions and blood flow. Shea and Raskova (1984) depicted a graph of a glomerular network taken from a rat with renal failure (uremia). Nyengaard and Marcussen (1993) compared the graph of an adult glomerular network with that of the glomerular network of a newborn (5 days old) rat. Wahl et al. (1984) showed that an adult glomerular network is nonplanar and that the majority of the branch points are bifurcations (vertices of degree 3). The application of cycle rank to the glomerular microcirculation has been previously introduced (Shimzu et al., 1988), but this investigator was able to analyze only the peripheral capillaries of the glomerular network. Most of the investigators mentioned have also described the glomerular network as having ‘‘lobules’’, implying that the network is divided (but not actually disconnected) into a separate grouping of parts of the total network. It is not clear whether lobules are true quantifiable separate parts of the network or just the appearance of a lobular arrangement of vertices and edges as a result of drawing the graph in a particular way. By investigating the location of large cycles in the graph of a glomerular network, one might be able to describe and quantify the perceived or real separation of the network into lobules. In this study, the networks presented are compared using the graph invariants: order, size, cycle rank (independent cycles), radius (minimum eccentricity), diameter (maximum eccentricity), root distance (from afferent to efferent arteriole), eccentricity distribution, vertex degree distribution, and planarity. Detailed discussions of these invariants
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Fig. 5. Vertex degree distribution for a newborn glomerular network, a uremic adult glomerular network, and a group of six normal adult glomerular networks averaged. Note that the newborn network has no vertices of degree 4.
have been given by Harary (1972) and Buckley and Harary (1990). Order and size are invariants that measure how many vessels and branch points are in a given glomerular network. The data analyzed in this study, although limited, suggest that an average adult glomerular network has approximately 278 branch points and 445 capillaries. The cycle rank b (number of independent cycles), which is obtained via the relation b = t n + 1, is linear with respect to t n, or with either t or n whenever n or t is fixed, respectively. Note, however, that for the data in this study, both n and t are variable. The plot of cycle rank relative to order n calculated for the known normal adult glomerular networks is shown in Fig. 3a, and for cycle rank relative to size t, in Fig. 3b. Also plotted are the corresponding values
for the uremic and newborn glomerular networks. For the data in this study, the cycle rank of a network increases with increasing size t (number of vessels) relative to order n (number of branch points). The root distance rd appears to vary only slightly (rd = 11 –13) for all adult renal glomerular networks analyzed, regardless of the order and size of the network. The root distance calculated in this study represents the smallest number of connected blood vessels starting with the afferent arteriole and ending with the efferent arteriole. This suggests that in the growth and development of glomerular networks, there are a limited and slightly varying number of vessels, creating a shortest path from afferent to efferent arteriole regardless of the ultimate number of total capillaries in the network.
Fig. 6. Eccentricity distribution of a newborn, a uremic, and a typical normal adult glomerular network. In this illustration, the typical adult glomerular network is from Antiga et al. (2001).
E.M. Wahl et al. / Microvascular Research 67 (2004) 223–230
The root distance rd of the uremic glomerular network of Shea and Raskova is 7 and the root distance rd of the newborn network is 5. These values are compatible with their smaller networks. To compare adult and newborn renal glomerular networks, we introduce in Table 1 the following invariant ratios: root distance to order (rd/n) and root distance to size (rd/t). The normal adult glomeruli and the uremic glomeruli show little difference when these invariant ratios are compared. The newborn glomerular network, however, shows significantly larger rd/n and rd/t ratios compared to the adult networks. This reflects a larger proportion of branch points and vessels in the path from the afferent arteriole to the efferent arteriole in the newborn network and suggests that the glomerular vascular network develops from and along a shortest path between the afferent and efferent arterioles. The radius (minimum eccentricity) and the diameter (maximum eccentricity), both parameters based on graph distance, vary slightly among all adult glomerular networks. Eccentricities of both adult and newborn glomerular networks, shown as a percentage of total vertices, demonstrate a characteristic unimodal distribution pattern (Fig. 4). The uremic glomerular network, however, shows a distribution of eccentricities that is wider and tends toward a uniform pattern. Fig. 6 illustrates how the eccentricity distribution varies among three networks: newborn, uremic, and a typical adult glomerular network (Antiga et al., 2001). These eccentricity distributions suggest that in a normal adult network the eccentricities tend to cluster around a mode, whereas in the uremic network the eccentricities appear to be distributed more uniformly from the minimum to the maximum eccentricity. The statistical moment calculations of skewness and kurtosis might quantify a difference between the shape of a normal and a uremic eccentricity distribution. The small sample size of the data in this study, however, precludes a determination of statistical significance. A network that is nonplanar indicates that it cannot be embedded in a plane and therefore suggests a structure of higher dimensional complexity. The only graph in this study that demonstrates planarity is that of the newborn glomerular network of Nyengaard. It has been shown previously that fetal glomerular networks are planar and that adult glomerular networks are nonplanar (Wahl et al., 1984). This study confirms that finding and suggests that nonplanarity is an invariant associated with mature glomerular networks. All graphs analyzed demonstrate that their networks consist primarily of vertices of degree 3. The percentage of degree 3 vertices in the sample of normal glomerular networks ranges from 71.8% to 100%. Compared to normal adult glomerular graphs, the Shea and Raskova graph of a uremic glomerular network demonstrates a shift in the vertex degree frequency to vertices of a higher degree (Fig. 5). Of course, more studies of uremic glomeruli would be needed to determine a statistical difference and suggest a pathologic cause.
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The variation in the amount of vertices of degree 3 might represent a technical artifact rather than an actual pathological or normal anatomical difference. It should be noted that Winkler used the smallest serial microtome spacing compared to all other investigators. Winkler used 1/2-Am sections whereas all other investigators used microtome sections of 1– 2 Am. Wider microtome sectioning might group more edges into one focal point and thereby create a vertex of higher degree. The most narrow microtome spacing would most likely result in a graph with the highest percentage of degree 3 vertices. A vertex of degree higher than 3 might be possible if a vessel branched into more than three vessels at precisely the same identical point. This suggests that if vertex degree is to be used to compare vascular networks, a consistent method of microtome sectioning and a precise anatomical definition of a branch point should be established. This study has shown that the renal glomerular network may be described by graph theory invariants. Graph theory invariants were used to differentiate among networks of normal adult glomeruli, a uremic glomerular network, and a newborn glomerular network. Because invariants are able to represent any network, the graph theory could be used to explore vascular networks other than in the renal glomerulus. Thus, information obtained from studies of angiogenesis (Risau, 1997), which describes the growth and development of blood vessels, may be useful in developing a graph theory model of angiogenesis in normal or abnormal microvascular networks. The ability to differentiate among various microvascular networks might establish a foundation for a new technology in the diagnosis of pathological angiogenesis.
Acknowledgments This work was supported in part by The Dyson College of Arts and Sciences and The School of Computer Science and Information Systems, Pace University.
References Andruecci, V.E., 1972. About the glomerulus. Kidney Int. 2, 349 – 351. Antiga, L., Ene-Iordache, B., Remuzzi, G., Remuzzi, A., 2001. Automatic generation of glomerular capillary organization. Microvasc. Res. 62, 346 – 354. Buckley, F., Harary, F., 1990. Distance in Graphs. Addison-Wesley, Reading, MA. Engelson, E.T., Skalak, T.C., Schmid-Schonbein, G.W., 1985. The microvasculature in skeletal muscle: I. Arteriolar network in rat spinotrapezius muscle. Microvasc. Res. 30, 29 – 44. Fenton, B.M., Zweifach, B.W., 1981. Microcirculatory model relating geometrical variation to changes in pressure and flow rate. Ann. Biomed. Eng. 9, 303 – 321. Harary, F., 1972. Graph Theory. Addison-Wesley, Reading, MA. Kuratowski, K., 1930. Sur le probleme des courbes gauches en topologie. Fundam. Math. 15, 271 – 283.
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Ley, K., Pries, A., Gaehtgens, P., 1986. Topological structure of rat mesenteric microvascular networks. Microvasc. Res. 32, 315 – 332. Nyengaard, J.R., Marcussen, N., 1993. The number of glomerular capillaries estimated by an unbiased and efficient stereological method. J. Microsc. 171 (1), 27 – 37. Remuzzi, A., Brenner, B., Pata, V., Tebaldi, G., Mariano, R., Belloro, A., Remuzzi, G., 1992. Three-dimensional reconstructed glomerular capillary network: blood flow distribution and local filtration. Am. J. Physiol. 263, F562 – F572. Risau, W., 1997. Mechanisms of angiogenesis. Nature 386, 671 – 674 (17 April). Shea, S.M., 1979. Glomerular hemodynamics and vasculature. Microvasc. Res. 18, 129 – 143.
Shea, S.M., Raskova, J., 1984. Glomerular hemodynamics and vasculature structure in uremia: a network analysis of glomerular path lengths and maximal blood transit times computed for a microvascular model reconstructed from subserial ultrathin sections. Microvasc. Res. 28, 37 – 50. Shimzu, H., Shinohara, N., Yokoyama, T., 1988. Topological analysis of the three-dimensional structure of the human renal glomerulus using a computer-aided reconstruction system. Microvasc. Res. 36, 130 – 139. Wahl, E.M., Daniels, F.H., Leonard, E.F., Levinthal, C., Cortell, S., 1984. A graph theory model of the glomerular capillary network and its development. Microvasc. Res. 27, 96 – 109. Winkler, D., Elger, M., Sakai, T., Kriz, W., 1991. Branching and confluence pattern of glomerular arterioles in the rat. Kidney Int. 39 (Suppl. 32), S2 – S8.
A graph theory analysis of renal glomerular microvascular networks Eric M. Wahl, a,* Louis V. Quintas, a Lorraine L. Lurie, b and Michael L. Gargano c a
The New York Institute for Bioengineering and Health Science, New York, NY 10011, USA b Department of Mathematics, Pace University, New York, NY 10038, USA c Department of Computer Science, Pace University, New York, NY 10038, USA Received 31 July 2003 Available online 27 February 2004
Abstract A graph theory model and its invariants are used to compare previously published renal glomerular networks of six adult rats, one adult uremic rat, and one newborn rat. Invariants calculated include order, size, cycle rank, eccentricity, root distance, planarity, and vertex degree distribution. These invariants enabled the differentiation of six normal adult glomerular microvascular networks from that of the uremic glomerulus and from that of the normal newborn glomerulus. These invariants might then be used to differentiate between normal and pathological vascular networks. Also proposed are graph theory invariants that might be used to develop a quantitative model for angiogenesis. D 2003 Elsevier Inc. All rights reserved. Keywords: Renal glomerular network; Microvasculature; Glomerulus; Uremia; Vascular branching patterns; Angiogenesis; Graph theory
Introduction Many researchers have studied the branching pattern of the renal glomerular microvascular network. Early studies of the glomerular network were essentially hand-drawn illustrations of the vascular pattern of the glomerulus, the first of which was given by Malpighi in the 17th century (Andruecci, 1972). These drawings depicted an anastomosing and lobular nature of the glomerular network but left the exact quantitative nature of the branching pattern unexamined. The topology of a microvascular network is of interest as it pertains to its physical properties and its embryological development. Investigators have used various ordering and numbering schemes to analyze the topology of vascular networks in relation to their geometric and hemodynamic properties. Shea (1979) used graph theory to map and tabulate experimental measurements of vascular segments of the renal glomerulus. Fenton and Zweifach (1981) originally applied the Horton – Strahler ordering technique in the analysis of the * Corresponding author. The New York Institute for Bioengineering and Health Science, 30 Fifth Avenue, New York, NY 10011. E-mail addresses: [email protected] (E.M. Wahl), [email protected] (L.V. Quintas), [email protected] (L.L. Lurie), [email protected] (M.L. Gargano). 0026-2862/$ - see front matter D 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.mvr.2003.11.005
microvasculature by obtaining hierarchical bifurcation ratios of tree-like vascular structures and relating this information to vessel lengths and diameters. This concept was modified to include planar arteriolar networks as seen in the rat muscle (Engelson et al., 1985). Random branching algorithms used to generate frequency distributions of vessel branch segments with respect to their generation numbers further advanced the network analysis of the microvasculature (Ley et al., 1986). With the aid of automated microdissection and computer reconstruction from serial sections, investigators have been able to reproduce the branching pattern of the glomerulus (Antiga et al., 2001; Nyengaard and Marcussen, 1993; Remuzzi et al., 1992; Shea, 1979; Shea and Raskova, 1984; Wahl et al., 1984; Winkler et al., 1991). These investigators depicted blood vessels as edges and branch points as vertices, thereby constructing graph theoretical representations of the glomerular network. By reducing complex anastomosing networks to quantitative theoretical models, the graph theory has been able to illustrate the complex branching pattern of the renal glomerulus as well as provide such information as the number of vessels and branch points in its network. As a related branch of topology, the graph theory can also be used to assign numbers to both vessels and branch points and thereby correlate the geometric and hemodynamic properties of a network with the topological features of its associated graph.
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induced renal failure and a graph of a newborn rat glomerular network are analyzed and compared with graphs produced from the networks of normal adult rats. This article is meant to demonstrate how graph theory can be used to analyze the topology of renal glomerular networks even though the small number of subjects in this study does not allow for any definitive conclusions based on statistical analysis. The present study is an introduction to a systematic and uniform graph theory analysis of microvascular networks. The section on Graph theory background provides the definitions of all graph theory invariants discussed in this paper.
Graph theory background
Fig. 1. Invariants for this hypothetical graph are as follows: order n = 9; size t = 12; cycle rank b = 4; cycle basis: {cycle (2,3,5), cycle (3,4,6,7,5), cycle (4,6), cycle (6,7,8)} (note that cycle (3,4,6,7,5) + cycle (6,7,8) = cycle (3,4,6,8,7,5)); root distance rd = 5 (see path (1,2,5,7,8,9)); eccentricity distributiona: (5,4,4,3,3,4,3,4,5); radius R = 3 (see vertices 4, 5, and 7); diameter D = 5 (see vertices 1 and 9); vertex degree distributiona: (1,3,3,3,3,4,3,3,1); the edges defined by vertices 4 and 6 are parallel edges. Superscript a if for the vertex set: (1,2,3,. . .,9).
In graph theory, certain quantifiable parameters are called invariants (Harary, 1972). Specifically, an invariant is a property of a graph that is independent of labeling. Examples include the number of vertices or the number of edges. Quantifying microvascular networks with invariants may facilitate the differentiation between normal and abnormal networks, such as in uremia or malignancy. The graph theory may also provide a mathematical basis for modeling the growth and development of microvascular networks as seen in the renal glomerulus (Wahl et al., 1984). In this study, the graph theory is used to compare six previously published normal adult rat glomerular networks. The published graphs of a glomerulus of an adult rat with
A graph G = (V,E) consists of a nonempty set V, whose elements are called vertices (or nodes), and a set E of unordered pairs of distinct elements of V called edges. A subgraph of G is a graph whose vertices are contained in V and whose edges are contained in E. The order n of G is its number of vertices and the size t is its number of edges. The vertices that define an edge are said to be adjacent. An edge is said to be incident to each of its vertices. The degree of a vertex is the number of edges incident to it. A path is an alternating sequence of distinct vertices and edges, starting and ending with a vertex, such that consecutive vertices define the edge between them. A cycle is a path combined with an edge connecting the first and last vertex in the path. A cycle can also be visualized as a closed path. It is understood that, to say a graph contains a specified path or cycle, all edges must be in the graph. Parallel edges are defined as two or more edges connecting the same adjacent vertex pair. The number of edges in a shortest path connecting a pair of vertices u,v is the distance d(u,v). When the vertices u and v are fixed and defined as the source and sink vertices of a network, respectively, the distance between them is called the root distance rd. The eccentricity e(u) of a vertex u in a connected graph G is the maximum possible distance d(u,v) from u to any other vertex v in G, in other words the eccentricity of u is the length of the shortest path connecting u to a vertex furthest from it. The radius R and the diameter D are the minimum
Fig. 2. The nonplanar graphs of Kuratowski: K5 and K3,3.
E.M. Wahl et al. / Microvascular Research 67 (2004) 223–230
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Table 1 This table shows invariants calculated in this study: n = order, t = size, b = cycle rank, R = radius, D = diameter and rd = root distance Study Adult Shea (1979) Remuzzi et al. (1992) Nyengaard and Marcussen (1993) Antiga et al. (2001) Winkler et al. (1991) Wahl et al. (1984) Mean Standard deviation Uremic Shea and Raskova (1984) Newborn Nyengaard and Marcussen (1993)
n
t
b
R
D
rd
rd/n
rd/t
195 247 256 302 312 358 278.3 57.4
322 403 426 460 466 595 445.0 89.9
128 157 171 159 155 238 168.0 37.1
9 9 9 12 11 12 10.3 1.5
15 15 15 18 16 19 16.3 1.8
11 12 11 12 13 11 11.7 0.82
0.056 0.049 0.043 0.040 0.042 0.031 0.043 0.008
0.034 0.030 0.026 0.026 0.028 0.019 0.027 0.005
159
276
119
9
17
7
0.044
0.025
24
35
12
4
7
5
0.208
0.143
and maximum eccentricities among the set of all vertices of G, respectively. The cycle rank b is the number of independent cycles and is calculated by the relation: b = t n + 1, where t is the number of edges (size) and n is the number of vertices (order) of the graph. A subgraph consisting of the edges of two cycles and excluding any edges that appear in both cycles is called the sum of the two cycles. The vertex set of the sum is made up of vertices from the edges in its edge set. A set of cycles is independent if no cycle in the set can be expressed as a sum of the remaining cycles. A set of b
independent cycles is a basis. Any cycle in a graph can be expressed as a sum of cycles in any basis. Fig. 1 shows a hypothetical graph illustrating all of the invariants defined above. A graph is said to be planar if it can be drawn in the plane in such a way that none of its edges intersect except at a vertex. Two graphs are homeomorphic if both can be obtained from the same graph by replacing one or more edges with a path. Any graph containing a subgraph homeomorphic to either K5 or K3,3 (Fig. 2) is nonplanar (Kuratowski, 1930).
Fig. 3. Cycle rank plots show squares representing data from the normal adult glomerular networks. The triangles represent the uremic network and the circles represent the newborn network.
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Material and methods Graphs from six normal adult rat renal glomerular networks found in the literature were analyzed for graph theory invariants (Antiga et al., 2001; Nyengaard and Marcussen, 1993; Remuzzi et al., 1992; Shea, 1979; Wahl et al., 1984; Winkler et al., 1991). Also, a seventh graph representing the glomerular network of an adult rat with renal failure (uremia) by Shea and Raskova (1984) and an eighth graph published by Nyengaard and Marcussen (1993) representing the glomerular network of a 5-day-old newborn rat were evaluated. In all of these studies, investigators used a microtome and standard tissue staining and embedding techniques to obtain serial cross-sections of a renal glomerular vascular network. Blood vessels and branch points were identified by tracing and following the vascular contours
along the entire sequence of serial sections. These investigators created a graph theoretical representation of the glomerular network by replacing the vessels with edges and the branch points with vertices. To perform an analysis of a graph theoretic representation of a network, the vertices of the graph must be numbered (labeled) and then these numbers are listed as a complete set of labeled vertex pairs (edge list). By assigning an integer arbitrarily to every vertex in a graph, a unique pair of numbers will represent each edge. The set of all vertex pairs is the edge set of the graph. An edge set may then be entered into a computer for graph theory analysis. Five of the eight published glomerular networks had been depicted as unlabeled graphs (Remuzzi, Nyengaard (2), Winkler, Antiga). The vertices of these five graphs were labeled arbitrarily and five edge lists were thus generated.
Fig. 4. Eccentricity distributions of the glomerular networks analyzed.
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Two of the graphs (Shea and Shea/Raskova) were depicted as labeled networks and those numbers were used to create an edge list. One graph (Wahl) was directly represented as an edge list. The edge list for each of the eight graphs was entered into a Dell XPS-Z Computer (Pentium III/40 GB/256 MBRAM/930 MHz) for analysis of graph theory invariants. The invariants were calculated for all graphs using Mathematica (technical computational software by n2003 Wolfram Research, Inc.), then tabulated and compared.
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in other words, vertices of degree 3. The average percentage of degree 3 vertices in adult glomerular networks is 82.4% (range: 71.8 – 100%). The average percentage of degree 4 vertices in the adult networks is 14.2%, and of degree 5 vertices and higher is 3.4%. The percentage of degree 3 vertices in the newborn glomerular network of Nyengaard is 95.5%. The Shea/Raskova graph depicting a network of a glomerulus induced with chronic renal disease shows a shift in the vertex degree distribution. Here, the percentage of degree 3 vertices is only 64.1%.
Results Discussion Graph theory invariants for all eight glomerular networks studied are presented in Table 1. These invariants include order n, size t, cycle rank b, radius R, diameter D, and root distance rd (see section on Graph theory background). Cycle rank plots of all networks are given in Figs. 3a and b. The eccentricity distributions and vertex degree distributions associated with the networks are shown in Figs. 4 and 5, respectively. The order of the six adult glomeruli analyzed averages 278.3 vertices (range: 195 – 358). The order is 159 vertices for the uremic and 24 vertices for the newborn network. The size of the six adult glomeruli analyzed averages 445.0 edges (range: 322– 595). The size is 276 edges for the uremic and 35 edges for the newborn glomerular network. The cycle rank, indicating the number of independent cycles, averages 168.0 (range: 128 – 238) for the adult glomerular networks. The cycle rank is 118 for the uremic and 12 for the newborn glomerular networks. The relationships of cycle rank b vs. order n and cycle rank b vs. size t are shown in Figs. 3a and b, respectively. The radius (minimum eccentricity) of the six normal glomerular networks averages 10.3 (range: 9– 12) and the diameter (maximum eccentricity) of the six normal glomerular networks averages 16.3 (range: 15– 1 9). The radius and diameter of the uremic network are 9 and 17, respectively. The radius and diameter of the newborn network are 4 and 7, respectively (Table 1). The eccentricity distribution for each of the glomerular networks is shown in Figs. 4a– h. The eccentricity distribution illustrates the variation in the number of eccentricities from the radius to the diameter in terms of percentage of vertices for a given eccentricity. The root distance represents the smallest possible number of vessels that may be traversed from the afferent to the efferent arteriole. The root distance averages 11.7 (range: 11– 13) for adult glomerular networks and measures 7 and 5 for the uremic and newborn glomerular networks, respectively (Table 1). All graphs are nonplanar with the exception of the newborn glomerular network of Nyengaard. The vertex degree distributions for the glomerular networks are shown in Fig. 5. These distributions show that the majority of vertices in the renal glomerulus are bifurcations,
Quantifying the complexity of the renal glomerular microvascular network is technically demanding and mathematically challenging. A glomerular network, as noted previously, can be defined as a graph when the vessels are identified as edges and the branch points as vertices. These graphs can then be examined mathematically by calculating and comparing their invariants. This study represents the first attempt to compare several renal glomerular network graphs previously published by independent investigators (Table 1). Although all of these investigators constructed graphs, Shea (1979) having been the first, they focused mostly on the physical properties of the networks, such as capillary dimensions and blood flow. Shea and Raskova (1984) depicted a graph of a glomerular network taken from a rat with renal failure (uremia). Nyengaard and Marcussen (1993) compared the graph of an adult glomerular network with that of the glomerular network of a newborn (5 days old) rat. Wahl et al. (1984) showed that an adult glomerular network is nonplanar and that the majority of the branch points are bifurcations (vertices of degree 3). The application of cycle rank to the glomerular microcirculation has been previously introduced (Shimzu et al., 1988), but this investigator was able to analyze only the peripheral capillaries of the glomerular network. Most of the investigators mentioned have also described the glomerular network as having ‘‘lobules’’, implying that the network is divided (but not actually disconnected) into a separate grouping of parts of the total network. It is not clear whether lobules are true quantifiable separate parts of the network or just the appearance of a lobular arrangement of vertices and edges as a result of drawing the graph in a particular way. By investigating the location of large cycles in the graph of a glomerular network, one might be able to describe and quantify the perceived or real separation of the network into lobules. In this study, the networks presented are compared using the graph invariants: order, size, cycle rank (independent cycles), radius (minimum eccentricity), diameter (maximum eccentricity), root distance (from afferent to efferent arteriole), eccentricity distribution, vertex degree distribution, and planarity. Detailed discussions of these invariants
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Fig. 5. Vertex degree distribution for a newborn glomerular network, a uremic adult glomerular network, and a group of six normal adult glomerular networks averaged. Note that the newborn network has no vertices of degree 4.
have been given by Harary (1972) and Buckley and Harary (1990). Order and size are invariants that measure how many vessels and branch points are in a given glomerular network. The data analyzed in this study, although limited, suggest that an average adult glomerular network has approximately 278 branch points and 445 capillaries. The cycle rank b (number of independent cycles), which is obtained via the relation b = t n + 1, is linear with respect to t n, or with either t or n whenever n or t is fixed, respectively. Note, however, that for the data in this study, both n and t are variable. The plot of cycle rank relative to order n calculated for the known normal adult glomerular networks is shown in Fig. 3a, and for cycle rank relative to size t, in Fig. 3b. Also plotted are the corresponding values
for the uremic and newborn glomerular networks. For the data in this study, the cycle rank of a network increases with increasing size t (number of vessels) relative to order n (number of branch points). The root distance rd appears to vary only slightly (rd = 11 –13) for all adult renal glomerular networks analyzed, regardless of the order and size of the network. The root distance calculated in this study represents the smallest number of connected blood vessels starting with the afferent arteriole and ending with the efferent arteriole. This suggests that in the growth and development of glomerular networks, there are a limited and slightly varying number of vessels, creating a shortest path from afferent to efferent arteriole regardless of the ultimate number of total capillaries in the network.
Fig. 6. Eccentricity distribution of a newborn, a uremic, and a typical normal adult glomerular network. In this illustration, the typical adult glomerular network is from Antiga et al. (2001).
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The root distance rd of the uremic glomerular network of Shea and Raskova is 7 and the root distance rd of the newborn network is 5. These values are compatible with their smaller networks. To compare adult and newborn renal glomerular networks, we introduce in Table 1 the following invariant ratios: root distance to order (rd/n) and root distance to size (rd/t). The normal adult glomeruli and the uremic glomeruli show little difference when these invariant ratios are compared. The newborn glomerular network, however, shows significantly larger rd/n and rd/t ratios compared to the adult networks. This reflects a larger proportion of branch points and vessels in the path from the afferent arteriole to the efferent arteriole in the newborn network and suggests that the glomerular vascular network develops from and along a shortest path between the afferent and efferent arterioles. The radius (minimum eccentricity) and the diameter (maximum eccentricity), both parameters based on graph distance, vary slightly among all adult glomerular networks. Eccentricities of both adult and newborn glomerular networks, shown as a percentage of total vertices, demonstrate a characteristic unimodal distribution pattern (Fig. 4). The uremic glomerular network, however, shows a distribution of eccentricities that is wider and tends toward a uniform pattern. Fig. 6 illustrates how the eccentricity distribution varies among three networks: newborn, uremic, and a typical adult glomerular network (Antiga et al., 2001). These eccentricity distributions suggest that in a normal adult network the eccentricities tend to cluster around a mode, whereas in the uremic network the eccentricities appear to be distributed more uniformly from the minimum to the maximum eccentricity. The statistical moment calculations of skewness and kurtosis might quantify a difference between the shape of a normal and a uremic eccentricity distribution. The small sample size of the data in this study, however, precludes a determination of statistical significance. A network that is nonplanar indicates that it cannot be embedded in a plane and therefore suggests a structure of higher dimensional complexity. The only graph in this study that demonstrates planarity is that of the newborn glomerular network of Nyengaard. It has been shown previously that fetal glomerular networks are planar and that adult glomerular networks are nonplanar (Wahl et al., 1984). This study confirms that finding and suggests that nonplanarity is an invariant associated with mature glomerular networks. All graphs analyzed demonstrate that their networks consist primarily of vertices of degree 3. The percentage of degree 3 vertices in the sample of normal glomerular networks ranges from 71.8% to 100%. Compared to normal adult glomerular graphs, the Shea and Raskova graph of a uremic glomerular network demonstrates a shift in the vertex degree frequency to vertices of a higher degree (Fig. 5). Of course, more studies of uremic glomeruli would be needed to determine a statistical difference and suggest a pathologic cause.
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The variation in the amount of vertices of degree 3 might represent a technical artifact rather than an actual pathological or normal anatomical difference. It should be noted that Winkler used the smallest serial microtome spacing compared to all other investigators. Winkler used 1/2-Am sections whereas all other investigators used microtome sections of 1– 2 Am. Wider microtome sectioning might group more edges into one focal point and thereby create a vertex of higher degree. The most narrow microtome spacing would most likely result in a graph with the highest percentage of degree 3 vertices. A vertex of degree higher than 3 might be possible if a vessel branched into more than three vessels at precisely the same identical point. This suggests that if vertex degree is to be used to compare vascular networks, a consistent method of microtome sectioning and a precise anatomical definition of a branch point should be established. This study has shown that the renal glomerular network may be described by graph theory invariants. Graph theory invariants were used to differentiate among networks of normal adult glomeruli, a uremic glomerular network, and a newborn glomerular network. Because invariants are able to represent any network, the graph theory could be used to explore vascular networks other than in the renal glomerulus. Thus, information obtained from studies of angiogenesis (Risau, 1997), which describes the growth and development of blood vessels, may be useful in developing a graph theory model of angiogenesis in normal or abnormal microvascular networks. The ability to differentiate among various microvascular networks might establish a foundation for a new technology in the diagnosis of pathological angiogenesis.
Acknowledgments This work was supported in part by The Dyson College of Arts and Sciences and The School of Computer Science and Information Systems, Pace University.
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