Analysis of a Model for the Morphological Structure of Renal Arterial Tree: Fractal Structure

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.

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,  𝐶_𝑔𝑓,  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 {𝑎_𝑏,𝑎_𝑝}, where 𝑎_𝑏 denotes sprouting angiogenesis and 𝑎_𝑝 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 (𝐶_𝑔𝑓). Both processes, sprouting and splitting angiogeneses, depend on the regulation of 𝐶_𝑔𝑓 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 𝐶_𝑔𝑓 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 𝐶_𝑔𝑓 over length for the splitting angiogenesis is unknown, in the sprouting angiogenesis we consider that the length of the new vessels depends of the 𝐶_𝑔𝑓. We obtain the value of the dimensionless length 𝑙_𝑒∶[𝐶_𝑔𝑓,𝐶_𝑔𝑓]→[𝑙_𝑒,𝑙_𝑒] from experimental data; the function which can be approximated for VEGF₁₂₁ is [5] 𝑙_𝑒=0.00878𝐶³_𝑔𝑓−0.51326𝐶²_𝑔𝑓+8.52128𝐶_𝑔𝑓+81.12064. The  𝑙_𝑒 is applied in this model; the length of the vessel 𝑙 depends on 𝑙_𝑒 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 𝐶_𝑔𝑓 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,𝐶_𝑔𝑓,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,𝐶_𝑔𝑓,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=𝑎_𝑏 with probability 𝑃_𝑎𝑏={0.1,0.2,0.3,0.4,0.5} and 𝑠=𝑎_𝑝, where 𝑃_𝑎𝑝=1−𝑃_𝑎𝑏, 𝐶_𝑔𝑓∈[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 𝐶_𝑔𝑓 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 on 𝑎_𝑏 and 𝑎_𝑝; that is, the length for each level 𝑗 denoted by 𝑙_𝑗 in the arterial vascular tree of the kidney is analytically determined as follows: 𝑙_𝑗=𝑙₀₀𝑃_𝑎𝑏𝜆_𝑏+𝑃_𝑎𝑝𝜆_𝑝^𝑗, (1) where the parameters are defined as in the previous section. Then, the mean length for each segment 𝑗 is 𝑙₀=5mm,𝑙₁𝑙=3.095mm,₂=1.919mm,𝑙₃𝑙=1.191mm,₄=0.741mm,𝑙₅𝑙=0.462mm,₆=0.287mm,𝑙₇𝑙=0.179mm,₈=0.112mm,𝑙₉=0.070mm. (2) 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 by 𝐴_𝐺𝑅 is the root length (renal artery, 𝑗=0) to the leaves of the tree (interlobular arteries, 𝑗=9), and then 𝐴_𝐺𝑅=𝑙⁹_00𝑗=0𝑃_𝑎𝑏𝜆_𝑏+𝑃_𝑎𝑝𝜆_𝑝^𝑗, (3) where 0≤𝑗≤9;  𝑙₀₀=5mm is the length of the renal artery [5]; 𝑃_𝑎𝑏 is defined in the set {0.1,0.2,0.3,0.4,0.5} and 𝑃_𝑎𝑝=1−𝑃_𝑎𝑏; 𝜆_𝑏=0.5 is the average of the contraction factor for 𝑎_𝑏; and 𝜆_𝑝=0.67 is the contraction factor for 𝑎_𝑝. Then, 𝑃_𝑎𝑏=0.1,𝑃_𝑎𝑝=0.9,𝐴_𝐺𝑅𝑃=14.20mm,_𝑎𝑏=0.2,𝑃_𝑎𝑝=0.8,𝐴_𝐺𝑅𝑃=13.59mm,_𝑎𝑏=0.3,𝑃_𝑎𝑝=0.7,𝐴_𝐺𝑅𝑃=13.01mm,_𝑎𝑏=0.4,𝑃_𝑎𝑝=0.6,𝐴_𝐺𝑅𝑃=12.48mm,_𝑎𝑏=0.5,𝑃_𝑎𝑝=0.5,𝐴_𝐺𝑅=11.99mm. (4)

These data are within the range of rat trees reported in [3]; while we consider 10 levels [1, 5], Nordsletten considers 11 levels.

However, if 𝜆_𝑏∈[0.3,0.7], the average with respect to 𝜆_𝑏 is as follows: 𝐸_𝑏𝑃_𝑎𝑏𝜆_𝑏+𝑃_𝑎𝑝𝜆_𝑝^𝑗=52𝑃_𝑎𝑏^.7_.3𝑃_𝑎𝑏𝜆_𝑏+𝑃_𝑎𝑝𝜆_𝑝^𝑗𝑃_𝑎𝑏𝑑𝜆_𝑏=52𝑃_𝑎𝑏𝑗+1.7𝑃_𝑎𝑏+𝑃_𝑎𝑝𝜆_𝑝^𝑗+1−.3𝑃_𝑎𝑏+𝑃_𝑎𝑝𝜆_𝑝^𝑗+1. (5)

Let 𝑎=(.7𝑃_𝑎𝑏+𝑃_𝑎𝑝𝜆_𝑝)^𝑗+1 and 𝑏=(.3𝑃_𝑎𝑏+𝑃_𝑎𝑝𝜆_𝑝)^𝑗+1, and then 𝐴_𝐺𝑅=5𝑙₀₀2𝑃⁹_{𝑎𝑏𝑗=0}𝑎^𝑗+1−𝑏𝑗+1^𝑗+1≃𝑗+15𝑙₀₀2𝑃^∞_{𝑎𝑏𝑗=0}𝑎^𝑗+1−𝑏𝑗+1^𝑗+1,𝑗+1 (6) where ^∞_𝑗=0𝑎^𝑗+1=𝑗+1^∞_𝑗=0^𝑎₀𝑥^𝑗=𝑑𝑥_{𝑎0∞𝑗=0}𝑥^𝑗=𝑑𝑥byuniformconvergence_𝑎0𝑑𝑥(1−𝑥=−ln1−𝑥)^𝑎₀1=ln.1−𝑎 (7) Similarly, ∑^∞_𝑗=0(𝑏^𝑗+1/𝑗+1)=ln(1/1−𝑏), and then 𝐴_𝐺𝑅=5𝑙₀₀2𝑃_𝑎𝑏ln1−𝑏=1−𝑎5𝑙₀₀2𝑃_𝑎𝑏ln1−.3𝑃_𝑎𝑏+𝑃_𝑎𝑝𝜆_𝑝1−.7𝑃_𝑎𝑏+𝑃_𝑎𝑝𝜆_𝑝±.7¹¹, (8) where ±.7¹¹ mm is the error.

Substituting the values 𝑙₀₀,  𝑃_𝑎𝑏,  𝑃_𝑎𝑝, and  𝜆_𝑝=0.67, 𝑃_𝑎𝑏=0.1,𝑃_𝑎𝑝=0.9,𝐴_𝐺𝑅=14.42±.7¹¹𝑃mm,_𝑎𝑏=0.2,𝑃_𝑎𝑝=0.8,𝐴_𝐺𝑅=13.79±.7¹¹𝑃mm,_𝑎𝑏=0.3,𝑃_𝑎𝑝=0.7,𝐴_𝐺𝑅=13.23±.7¹¹𝑃mm,_𝑎𝑏=0.4,𝑃_𝑎𝑝=0.6,𝐴_𝐺𝑅=12.73±.7¹¹𝑃mm,_𝑎𝑏=0.5,𝑃_𝑎𝑝=0.5,𝐴_𝐺𝑅=12.28±.7¹¹mm. (9)

We generate 5000 trees of renal arterial vasculature for different probabilities in 𝑎_𝑏 and 𝑎_𝑝. 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 of 𝑎𝑏 and 𝑎𝑝.

Therefore, the 𝐴_𝐺𝑅 depends on the process of angiogenesis; that is, 𝐴_𝐺𝑅 is inversely proportional to the probability of the 𝑎_𝑏.

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 {ℝ^𝑚;𝑤₁,𝑤₂,…,𝑤_𝑁} be a hyperbolic iterative function system (IFS), and let 𝐴 denote its attractor. Suppose that 𝑤_𝑛 is a similitude of scaling factor 𝑠_𝑛 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 ^𝑁_𝑛=1𝑠_𝑛^𝐷(𝐴)=1,𝐷(𝐴)∈[0,𝑚]. (10) If the IFS is overlapping, then 𝐷≥𝐷(𝐴), where 𝐷 is the solution of ^𝑁_𝑛=1𝑠_𝑛^𝐷=1,𝐷∈[0,∞). (11)

The morphology of arterial vascular tree of the kidney, produced by an IFS, is composed of 𝑛 applications of contraction with a factor 𝑠_𝑛. Thus, from the subsection of length and width, the contraction factor for the splitting angiogenesis is 𝜆_𝑝=0.67 while for sprouting angiogenesis the average contraction factor is 𝜆_𝑏=0.5. Since we have 𝑃_𝑎𝑏𝜆^𝐷_𝑏+𝑃_𝑎𝑝𝜆^𝐷_𝑝=1, the analytical fractal dimension of the renal arterial tree is 𝑃𝐷=ln_𝑎𝑝−ln(1/2)−𝑃_𝑎𝑏𝜆^𝐷_𝑏−ln𝜆_𝑝, (12) where 𝜆_𝑏 is generated randomly within the range defined by 𝜆_𝑏∈[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 Counting 𝐷 of the Arterial Vascular Tree of the Kidney

Additionally, the box counting theorem by Barnsley [19] states that, denoting with 𝑁_𝑛(𝐺_𝑅) as the number of boxes of side 1/2^𝑛, 𝐷=lim_𝑛→∞ln𝑁_𝑛𝐺_𝑅ln(2^𝑛), (13) 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. Figure 2 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 𝑎_𝑏 and 𝑎_𝑝, 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 of 𝑎𝑏, where 𝑎𝑝 has the probability 𝑃𝑎𝑝=1−𝑃𝑎𝑏.

The results suggest that the fractal dimension is inversely proportional to 𝑃_𝑎𝑏. This behavior is congruent in the sense that, by symmetry structure, the arterial vascular tree is bigger for small values of 𝑃_𝑎𝑏; 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 in 𝑎_𝑏 and 𝑎_𝑝, 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 in 𝑎𝑏 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 on 𝑎_𝑏 and 𝑎_𝑝. 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.