# Analysis of a Model for the Morphological Structure of Renal Arterial

Authors: Aurora Espinoza-Valdez; Francisco C. Ordaz-Salazar; Edgardo Ugalde; Ricardo Femat
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

One of the fields of applied mathematics is related to model analysis. Biomedical
systems are suitable candidates for this field because of their importance in
life sciences including therapeutics. Here we deal with the analysis of a model
recently proposed by Espinoza-Valdez et al. (2010) for the kidney vasculature
developed via angiogenesis. The graph theory allows one to model quantitatively
a vascular arterial tree of the kidney in sense that (1) the vertex represents a
vessels bifurcation, whereas (2) each edge stands for a vessel including
physiological parameters. The analytical model is based on the two processes of
sprouting and splitting angiogeneses, the concentration of the vascular
endothelial growth factor (VEGF), and the experimental data measurements of the
rat kidneys. The fractal dimension depends on the probability of sprouting
angiogenesis in the development of the arterial vascular tree of the kidney,
that is, of the distribution of blood vessels in the morphology generated by the
analytical model. The fractal dimension might determine whether a suitable renal
vascular structure is capable of performing physiological functions under
appropriate conditions. The analysis can describe the complex structures of the
development vasculature in kidney.


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## Body

## 1. Introduction

The arterial structure of organs has been the subject of many studies [1–5]. However, not all systems have similar functions; for the organs the
purpose of arterial structure is to provide the blood required in the metabolic
process and other specific functions. The arterial structure is highly nonuniform
because it is determined by reasons of anatomy and local flow requirements [4]. The kidney is one of the most complicated
organs in terms of structure and physiology because it is highly vascularized and
constitutes the main organ for maintaining chemical balance in blood [6]. The kidney consists of three trees:
arterial, venous, and ureter. The arterial vascular tree of the kidney is structured
by the renal artery branches into interlobar arteries, arcuate arteries, and
interlobular arteries, which are formed by bifurcation [6]. The development of the arterial vascular tree of the kidney
can be formed mainly by angiogenesis [1]. The
process of angiogenesis is the formation of new blood vessels from preexisting
vessels and consists of two different processes: sprouting and splitting. Sprouting
refers to the case in which the new branch literally sprouts to some existing
branch. In splitting a branch is divided into two new branches [7]. The factors involved at the vascular
development are VEGF, renin-expression, ephrins A and B, platelet-derived growth
factor-B (PDGF-B), Ets family (as Ets-1 and TEL), and angiopoietins 1-2. The VEGF is
an essential regulatory factor for both processes of angiogenesis in the development
of renal arterial vascularization [1]. Then,
we only consider VEGF, because the other factors increase the complexity of the
analytical model.Zamir and Phipps in [8] studied the
morphological characteristics of the rat kidney and found that the branching rules
of these vessels are determined by considerations of the angiological function.
Subsequently, Zamir generated a model based on L-systems (Lindenmayer System) and
incorporated some physiological laws, that is, random parameters [4]. Zamir's results suggest that the arterial
structure of the tree is determined mainly by the growth rules of arterial tree
branching [4]. The structural morphologic
reconstruction of renal vasculature from microcomputed tomography (micro-CT) images
was presented by Nordsletten et. al. The arterial and venous trees of the rat kidney
were generated numerically using micro-TC [3].
These morphological data provide a statistical basis from which renal vascular
topology can be generated [5]. Graph theory
generates branching tree structures incorporating the physiological laws of the
renal artery branching through the process of angiogenesis [5]. Many pathological conditions, such as cancers, arteriovenous
malformations, and diabetes, induce changes to vessel's morphology or spatial
organization. The fractal analysis has been applied to a large variety of healthy or
pathological vascular networks [9–14]. Sabo et al. concluded in [15] that the microvessel fractal dimension as a
marker of tumor microvascular complexity might provide important pronostic
information as well as shed light on the complex interactions between tumor
angiogenesis and growth. Moreover, Cross et al. analyzed the fractal dimension of
renal angiograms: a normal kidney, a congenitally dysplastic kidney, a kidney with
renal artery stenosis, and a kidney with recurrent thromboembolism lesions (see
Figure 3 in [16]). The analysis of fractal
dimension may allow for characterization of the vascular morphology.In this paper we extend the results of [5] by
analysing mathematically the structure of the arterial vascular tree of the kidney
and by incorporating physiological parameters generated from a computational model.
In the implementation of the proposed model the branches are represented by the
edges of the graph and the ramification point is represented by the nodes [5]. We expect that this representation
implemented in computer methodology will help to understand the mechanism involved
in biological systems, specifically the development of the renal arterial
bifurcation. The analysis provides the mean of the length in each level, and the
average width of the arterial vascular tree of the rat kidney is in the range of
what is observed in previous experimental and computational models. The fractal
dimension depends on the sprouting angiogenesis that appears in the development of
the arterial vascular tree of the kidney. The complex structures of the renal
arterial tree are consistent with previous fractal analysis of vascular structures
[9, 11–14]. These study opens
the possibility of a new taxonomy for normal kidneys and for the pathological
injuries related to the vascular morphology of the kidneys.


## 2. Theoretical Basis

The representation of the renal arterial vascularization in the computational model
is as follows:(1) the arterial vascular tree of the kidney is defined
by labeled and oriented graphs; we can identify two objects: the branches (edges)
and the branching point (nodes). That is, the edge represents a blood vessel and the
vertex is where the angiological stimulus appears. (2) The length and diameter of the new branches are
smaller than their parent branches [5, 6].The arterial vascular tree of the kidney down to the interlobular arteries can be
structured as follows: renal artery, level 0: interlobar arteries, levels 1-2:
arcuate arteries, levels 3-4: interlobular arteries, and levels 5–9. We based
in the experiments of Tomanek [3] for the
morphological structure of the renal arterial tree. In the model, we develop the
arterial vascular tree of the rat kidney. Therefore, the model based on the graph
theory for the arterial tree is a binary tree of ten levels and has a total of
vessels equal to 1023. The physiological parameters, and rules are based on parent
vessel branch level, the parameters, and the morphological structure of their child
branches, and this process deploys the structure of the model. The parameters
included in the edges of the binary tree model are s,  Cgf,  l,  d, and  θ as follows (for details see [5]).(i)
The Type of Angiogenesis: Sprouting or Splitting(s). Angiogenesis is identified by the
variable s which can take the values in the set
{ab,ap}, where ab denotes sprouting angiogenesis and
ap denotes splitting angiogenesis. Both
sprouting and splitting angiogeneeis generate branching tree structures,
which can be represented using graph theory. It is conjectured that the
sprouting and splitting processes occur with different probability in
the renal arterial tree for the morphology of the kidney. In the
proposed computational model this assumption is controlled
probabilistically. Sprouting angiogenesis is more probably than
splitting angiogenesis because it requires only reorganization of
existing endothelial cells (not migration), that is, having less energy
requirements [1].(ii)
Concentration of the VEGF in the Vessel(Cgf). Both processes, sprouting and
splitting angiogeneses, depend on the regulation of
Cgf in the pre-existing vessel [1]. While in sprouting angiogenesis
it is known that endothelial cells promotes the differentiation,
migration, proliferation, and assembly, the effect on splitting
angiogenesis is still in research. In the model
Cgf is generated randomly in a uniform way
within the range [0,35] ng/mL reported in [5, 17].(iii)
Length of the Vessel(l). Here, we focus on determining the
value of the lengths in the arterial bifurcations depending of the
concentration of the VEGF. Although the effect of
Cgf over length for the splitting
angiogenesis is unknown, in the sprouting angiogenesis we consider that
the length of the new vessels depends of the Cgf. We obtain the value of the
dimensionless length le:[Cgf_,Cgf¯]→[le_,le¯] from experimental data; the function
which can be approximated for VEGF121 is [5] le=0.00878Cgf3-0.51326Cgf2+8.52128Cgf+81.12064.
The  le is applied in this model; the length of
the vessel l depends on le and is compared with the length ranges
reported in the literature according to the level [3, 5]. The
l formed by splitting is defined by a
contraction factor. In the case that l is out of these ranges, the process is
repeated finite number of times. If l is still out of range, then it is
adjusted to the length of its parent branch.(iv)
Diameter of the Vessel(d). In this work, we do not discuss the
value of the diameters in the blood vessels. This is due to the fact
that we do not have sufficient information of Cgf in d for sprouting and splitting
angiogeneses. Then, in sprouting angiogenesis, the diameter of the
sprout is adjusted to a factor of the diameter of its parent branch. In
splitting angiogenesis the diameter of every new vessel is adjusted to
one-half of the diameter of its parent branch.(v)
Angle of Bifurcation(θ). In sprouting angiogenesis the angle of
bifurcation is adjusted within the range ±[60°,80°] with respect to its parent branch
[1, 18]. In splitting angiogenesis, one branch is
adjusted to +32.5° and the other branch to
-32.5° [18].The physiological parameters are included in the label of the edge and have the form(s,Cgf,l,d,θ). The arterial vascular tree of the kidney has
vertices with oriented edges in such a way that from each vertex one edge arrives
and two edges leave (the orientation symbolizes the circulation of blood flow in
arteries of the kidney). The renal arterial vascular tree of the kidney has 10
levels, because we based in the experiments of Tomanek. Therefore, the arterial
vascular tree of the kidney has 210 nodes. The set of integers to name every node is
[0,…,210-1].An algorithm generates step by step the physiological characteristics of every branch
and saves it as labeled edges with the format{s,Cgf,l,d,θ}. Just as the tree is generated, the other structure
is constructed in which we save the position of every node in
ℝ2. The algorithm calculates the position of every
node in function of the physiological characteristics of the branches and the
position of nodes of its parent branch. The physiological parameters of the root
(the initial condition) are determined: s=ab with probability Pab={0.1,0.2,0.3,0.4,0.5} and s=ap, where Pap=1-Pab, Cgf∈[0,35] ng/mL, l=5,d=1,θ=0, and they continue in numerical order with the
other branches using the information of parent branches. Some characteristics such
as s and Cgf are generated probabilistically in every step, and
the others depend on these parameters, their parent branch, and the rules that were
enumerated in the previous section.


## 3. Main Results

### 3.1. Length and Width

The length of each level into the kidney depends onab and ap; that is, the length for each level
j denoted by lj in the arterial vascular tree of the kidney is
analytically determined as follows: (1)lj=l00(Pabλb+Papλp)j, where the parameters are defined as in the
previous section. Then, the mean length for each segment
j is (2)l0=5mm,l1=3.095mm,l2=1.919mm,l3=1.191mm,l4=0.741mm,l5=0.462mm,l6=0.287mm,l7=0.179mm,l8=0.112mm,l9=0.070mm. These data are within the range of rat trees
reported in [3]; we compare the results
with Nordsletten from depth j=0 to j=9.Moreover, the analytical average width denoted byAGR is the root length (renal artery,
j=0) to the leaves of the tree (interlobular
arteries, j=9), and then (3)AGR=l00∑j=09‍(Pabλb+Papλp)j, where 0≤j≤9;  l00=5mm is the length of the renal artery [5]; Pab is defined in the set {0.1,0.2,0.3,0.4,0.5} and Pap=1-Pab; λb=0.5 is the average of the contraction factor for
ab; and λp=0.67 is the contraction factor for
ap. Then, (4)Pab=0.1,Pap=0.9,AGR=14.20mm,Pab=0.2,Pap=0.8,AGR=13.59mm,Pab=0.3,Pap=0.7,AGR=13.01mm,Pab=0.4,Pap=0.6,AGR=12.48mm,Pab=0.5,Pap=0.5,AGR=11.99mm.These data are within the range of rat trees reported in [3]; while we consider 10 levels [1, 5], Nordsletten
considers 11 levels.However, ifλb∈[0.3,0.7], the average with respect to
λb is as follows: (5)Eb((Pabλb+Papλp)j)=52Pab∫.3.7‍(Pabλb+Papλp)jPabdλb=52Pab(j+1)[(.7Pab+Papλp)j+1-(.3Pab+Papλp)j+1].Leta=(.7Pab+Papλp)j+1 and b=(.3Pab+Papλp)j+1, and then (6)AGR=5l002Pab∑j=09‍(aj+1j+1-bj+1j+1)≃5l002Pab∑j=0∞‍(aj+1j+1-bj+1j+1), where (7)∑j=0∞‍aj+1j+1=∑j=0∞‍(∫0a‍xjdx)=∫0a‍(∑j=0∞‍xj)dx(byuniformconvergence)=∫0a‍dx1-x=-ln(1-x)|0a=ln(11-a). Similarly, ∑j=0∞(bj+1/j+1)=ln(1/1-b), and then (8)AGR=5l002Pabln(1-b1-a)=5l002Pabln(1-(.3Pab+Papλp)1-(.7Pab+Papλp))±.711, where ±.711 mm is the error.Substituting the valuesl00,  Pab,  Pap, and  λp=0.67, (9)Pab=0.1,Pap=0.9,AGR=14.42±.711mm,Pab=0.2,Pap=0.8,AGR=13.79±.711mm,Pab=0.3,Pap=0.7,AGR=13.23±.711mm,Pab=0.4,Pap=0.6,AGR=12.73±.711mm,Pab=0.5,Pap=0.5,AGR=12.28±.711mm.We generate 5000 trees of renal arterial vasculature for different probabilities
inab and ap. In Figure 1 the mean ± SD (standard deviation) width of the kidney has an
intersection with these data and previous result where the mean width is
17.454±6.165 mm [3].Figure 1
The mean ± SD of width of 5000 trees generated for 5 different
probabilities ofab and ap.Therefore, theAGR depends on the process of angiogenesis; that
is, AGR is inversely proportional to the probability of
the ab.


### 3.2. Fractal Dimension of Kidney Vascular Tree

The fractal dimension(D) quantifies through dimension the ability of an
object to occupy a space. Different methods exist and are based on the different
definitions of fractal dimension [9, 11–14, 19].The analytical fractal dimension can be derived using the following result [19]. Let {ℝm;w1,w2,…,wN} be a hyperbolic iterative function system
(IFS), and let A denote its attractor. Suppose that
wn is a similitude of scaling factor
sn for each n∈{1,2,3,…,N}. If the IFS is totally disconnected or just
touching, the attractor has fractal dimension D(A), which is given by the unique solution of
(10)∑n=1N‍|sn|D(A)=1,D(A)∈[0,m]. If the IFS is overlapping, then
D-≥D(A), where D- is the solution of (11)∑n=1N‍|sn|D-=1,D-∈[0,∞).The morphology of arterial vascular tree of the kidney, produced by an IFS, is
composed ofn applications of contraction with a factor
sn. Thus, from the subsection of length and width,
the contraction factor for the splitting angiogenesis is
λp=0.67 while for sprouting angiogenesis the average
contraction factor is λb=0.5. Since we have PabλbD+PapλpD=1, the analytical fractal dimension of the renal
arterial tree is (12)D=ln(Pap)-ln((1/2)-PabλbD)-ln(λp), where λb is generated randomly within the range defined
by λb∈[0.3,0.7]. In other words, the fractal dimension depends
on the number of vessels, the spatial relationships between the vascular
components, and the surrounding environment.


#### 3.2.1. Box CountingD of the Arterial Vascular Tree of the
Kidney

Additionally, the box counting theorem by Barnsley [19] states that, denoting with
Nn(GR) as the number of boxes of side
1/2n, (13)D=limn→∞ln(Nn(GR))ln(2n), where the arterial vascular tree of the
kidney has fractal dimension D. The box counting results allow one to
compute the rate of change in complexity with scale as well as a measure of
hetreogeneity.Here, we used the box counting theorem for determining the fractal dimension.
Figure2 shows the average of the
fractal dimension in each probability for the arterial vascular tree of the
kidney. The fractal dimension decreases with the increase in the probability
of sprouting in the development of the arterial vascular tree of the kidney,
whereas the mean width of the kidney decreases. Then, the fractal dimension
of the arterial vascular tree of the kidney depends on the probability of
ab and ap, that is, the distribution of blood vessels
in the morphology generated by graph theory model.Figure 2
The mean ± SD of fractal dimension for different probabilities
ofab, where ap has the probability
Pap=1-Pab.The results suggest that the fractal dimension is inversely proportional toPab. This behavior is congruent in the sense
that, by symmetry structure, the arterial vascular tree is bigger for small
values of Pab; that is, the renal space is covered more
efficiently. The complex structures of the vasculature in kidney are
consistent with previous studies of vascular structures about their fractal
nature [9–16].Examples derived from graph model using different probabilities inab and ap, mean width and fractal dimension, are
shown in Figure 3. The analysis of
these responses may allow for characterization of the vascular morphology on
the arterial vascular tree of the kidney.Figure 3
Trees generated with different probabilities inab and ap (where Pap=1-Pab), mean width
AGR and fractal dimension
D of the kidney.


## 3.1. Length and Width

The length of each level into the kidney depends onab and ap; that is, the length for each level
j denoted by lj in the arterial vascular tree of the kidney is
analytically determined as follows: (1)lj=l00(Pabλb+Papλp)j, where the parameters are defined as in the
previous section. Then, the mean length for each segment
j is (2)l0=5mm,l1=3.095mm,l2=1.919mm,l3=1.191mm,l4=0.741mm,l5=0.462mm,l6=0.287mm,l7=0.179mm,l8=0.112mm,l9=0.070mm. These data are within the range of rat trees
reported in [3]; we compare the results
with Nordsletten from depth j=0 to j=9.Moreover, the analytical average width denoted byAGR is the root length (renal artery,
j=0) to the leaves of the tree (interlobular
arteries, j=9), and then (3)AGR=l00∑j=09‍(Pabλb+Papλp)j, where 0≤j≤9;  l00=5mm is the length of the renal artery [5]; Pab is defined in the set {0.1,0.2,0.3,0.4,0.5} and Pap=1-Pab; λb=0.5 is the average of the contraction factor for
ab; and λp=0.67 is the contraction factor for
ap. Then, (4)Pab=0.1,Pap=0.9,AGR=14.20mm,Pab=0.2,Pap=0.8,AGR=13.59mm,Pab=0.3,Pap=0.7,AGR=13.01mm,Pab=0.4,Pap=0.6,AGR=12.48mm,Pab=0.5,Pap=0.5,AGR=11.99mm.These data are within the range of rat trees reported in [3]; while we consider 10 levels [1, 5], Nordsletten
considers 11 levels.However, ifλb∈[0.3,0.7], the average with respect to
λb is as follows: (5)Eb((Pabλb+Papλp)j)=52Pab∫.3.7‍(Pabλb+Papλp)jPabdλb=52Pab(j+1)[(.7Pab+Papλp)j+1-(.3Pab+Papλp)j+1].Leta=(.7Pab+Papλp)j+1 and b=(.3Pab+Papλp)j+1, and then (6)AGR=5l002Pab∑j=09‍(aj+1j+1-bj+1j+1)≃5l002Pab∑j=0∞‍(aj+1j+1-bj+1j+1), where (7)∑j=0∞‍aj+1j+1=∑j=0∞‍(∫0a‍xjdx)=∫0a‍(∑j=0∞‍xj)dx(byuniformconvergence)=∫0a‍dx1-x=-ln(1-x)|0a=ln(11-a). Similarly, ∑j=0∞(bj+1/j+1)=ln(1/1-b), and then (8)AGR=5l002Pabln(1-b1-a)=5l002Pabln(1-(.3Pab+Papλp)1-(.7Pab+Papλp))±.711, where ±.711 mm is the error.Substituting the valuesl00,  Pab,  Pap, and  λp=0.67, (9)Pab=0.1,Pap=0.9,AGR=14.42±.711mm,Pab=0.2,Pap=0.8,AGR=13.79±.711mm,Pab=0.3,Pap=0.7,AGR=13.23±.711mm,Pab=0.4,Pap=0.6,AGR=12.73±.711mm,Pab=0.5,Pap=0.5,AGR=12.28±.711mm.We generate 5000 trees of renal arterial vasculature for different probabilities
inab and ap. In Figure 1 the mean ± SD (standard deviation) width of the kidney has an
intersection with these data and previous result where the mean width is
17.454±6.165 mm [3].Figure 1
The mean ± SD of width of 5000 trees generated for 5 different
probabilities ofab and ap.Therefore, theAGR depends on the process of angiogenesis; that
is, AGR is inversely proportional to the probability of
the ab.


## 3.2. Fractal Dimension of Kidney Vascular Tree

The fractal dimension(D) quantifies through dimension the ability of an
object to occupy a space. Different methods exist and are based on the different
definitions of fractal dimension [9, 11–14, 19].The analytical fractal dimension can be derived using the following result [19]. Let {ℝm;w1,w2,…,wN} be a hyperbolic iterative function system
(IFS), and let A denote its attractor. Suppose that
wn is a similitude of scaling factor
sn for each n∈{1,2,3,…,N}. If the IFS is totally disconnected or just
touching, the attractor has fractal dimension D(A), which is given by the unique solution of
(10)∑n=1N‍|sn|D(A)=1,D(A)∈[0,m]. If the IFS is overlapping, then
D-≥D(A), where D- is the solution of (11)∑n=1N‍|sn|D-=1,D-∈[0,∞).The morphology of arterial vascular tree of the kidney, produced by an IFS, is
composed ofn applications of contraction with a factor
sn. Thus, from the subsection of length and width,
the contraction factor for the splitting angiogenesis is
λp=0.67 while for sprouting angiogenesis the average
contraction factor is λb=0.5. Since we have PabλbD+PapλpD=1, the analytical fractal dimension of the renal
arterial tree is (12)D=ln(Pap)-ln((1/2)-PabλbD)-ln(λp), where λb is generated randomly within the range defined
by λb∈[0.3,0.7]. In other words, the fractal dimension depends
on the number of vessels, the spatial relationships between the vascular
components, and the surrounding environment.


### 3.2.1. Box CountingD of the Arterial Vascular Tree of the
Kidney

Additionally, the box counting theorem by Barnsley [19] states that, denoting with
Nn(GR) as the number of boxes of side
1/2n, (13)D=limn→∞ln(Nn(GR))ln(2n), where the arterial vascular tree of the
kidney has fractal dimension D. The box counting results allow one to
compute the rate of change in complexity with scale as well as a measure of
hetreogeneity.Here, we used the box counting theorem for determining the fractal dimension.
Figure2 shows the average of the
fractal dimension in each probability for the arterial vascular tree of the
kidney. The fractal dimension decreases with the increase in the probability
of sprouting in the development of the arterial vascular tree of the kidney,
whereas the mean width of the kidney decreases. Then, the fractal dimension
of the arterial vascular tree of the kidney depends on the probability of
ab and ap, that is, the distribution of blood vessels
in the morphology generated by graph theory model.Figure 2
The mean ± SD of fractal dimension for different probabilities
ofab, where ap has the probability
Pap=1-Pab.The results suggest that the fractal dimension is inversely proportional toPab. This behavior is congruent in the sense
that, by symmetry structure, the arterial vascular tree is bigger for small
values of Pab; that is, the renal space is covered more
efficiently. The complex structures of the vasculature in kidney are
consistent with previous studies of vascular structures about their fractal
nature [9–16].Examples derived from graph model using different probabilities inab and ap, mean width and fractal dimension, are
shown in Figure 3. The analysis of
these responses may allow for characterization of the vascular morphology on
the arterial vascular tree of the kidney.Figure 3
Trees generated with different probabilities inab and ap (where Pap=1-Pab), mean width
AGR and fractal dimension
D of the kidney.


## 3.2.1. Box CountingD of the Arterial Vascular Tree of the
Kidney

Additionally, the box counting theorem by Barnsley [19] states that, denoting with
Nn(GR) as the number of boxes of side
1/2n, (13)D=limn→∞ln(Nn(GR))ln(2n), where the arterial vascular tree of the
kidney has fractal dimension D. The box counting results allow one to
compute the rate of change in complexity with scale as well as a measure of
hetreogeneity.Here, we used the box counting theorem for determining the fractal dimension.
Figure2 shows the average of the
fractal dimension in each probability for the arterial vascular tree of the
kidney. The fractal dimension decreases with the increase in the probability
of sprouting in the development of the arterial vascular tree of the kidney,
whereas the mean width of the kidney decreases. Then, the fractal dimension
of the arterial vascular tree of the kidney depends on the probability of
ab and ap, that is, the distribution of blood vessels
in the morphology generated by graph theory model.Figure 2
The mean ± SD of fractal dimension for different probabilities
ofab, where ap has the probability
Pap=1-Pab.The results suggest that the fractal dimension is inversely proportional toPab. This behavior is congruent in the sense
that, by symmetry structure, the arterial vascular tree is bigger for small
values of Pab; that is, the renal space is covered more
efficiently. The complex structures of the vasculature in kidney are
consistent with previous studies of vascular structures about their fractal
nature [9–16].Examples derived from graph model using different probabilities inab and ap, mean width and fractal dimension, are
shown in Figure 3. The analysis of
these responses may allow for characterization of the vascular morphology on
the arterial vascular tree of the kidney.Figure 3
Trees generated with different probabilities inab and ap (where Pap=1-Pab), mean width
AGR and fractal dimension
D of the kidney.


## 4. Conclusions

The arterial vascular tree of kidney development was analyzed through angiogenesis.
The renal arterial vascular tree of the kidney goes down into the interlobular
arteries. We generate 5000 trees of renal arterial vasculature for different
probabilities onab and ap. The analytical mean length in each level and the
average width of the arterial vascular tree of the kidney have an intersection with
previous studies. Then, the graph theory allows the vascular tree to model the
vascular growth; that is, it generates the ramification of structures arborescent
incorporating physiological laws of arterial branching.In conclusion, the analytical expression of the fractal dimension depends on the
number of vessels, the spatial relationships between the vascular components, and
the surrounding environment. The arterial vascular arterial tree of the kidney has a
fractal dimension which is inversely proportional to the probability of the
occurring sprouting angiogenesis. The fractal dimensions determined for the
development of the arterial vascular tree of the kidney by box counting may allow
for characterization of the vascular morphology. As a conjecture, the fractal
dimension might determine whether a suitable renal vascular structure is capable of
performing physiological functions under appropriate conditions (hemodynamics).
These studies could be expanded to include those pathologies originating from
arterial alterations.



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*Source: 396486-2013-07-02.xml*