The vascular system is arguably the most important biological system in many organisms. Although the general principles of its architecture are simple, the growth of blood vessels occurs under extreme physical conditions. Optimization is an important aspect of the development of computational models of the vascular branching structures. This review surveys the approaches used to optimize the topology and estimate different geometrical parameters of the vascular system. The review is focused on optimizations using complex cost functions based on the minimum total energy principle and the relationship between the laws of growth and precise vascular network topology. Experimental studies of vascular networks in different species are also discussed.
A complex network of arteries, capillaries, and veins, the vascular system carries blood away from the heart to the rest of the body and back. The fact that the existence of higher organisms depends on the condition of the vascular system warrants a detailed study of its structure. The structure of such a system is not formed instantly but occurs relatively slowly under many constraints. For example, the space for growth is limited by the volume of the organ, and the amount of material available for vessel construction is fixed. However, the main functions of the circulatory system, such as metabolic transport, should be maximized for the effective functioning of the whole organism. Therefore, the application of an optimization approach to this system is valid. It should be noted that the condition of the vascular system can be understood as not only involving optimized structures but also their stability and integrity. Regarding the arterial network geometry, there is no universal cost function that can consider all factors that influence system growth. Notably, the simultaneous application of several optimization criteria is often impossible due to their formal inconsistency. However, the ‘synthetic’ cost optimization functions may not be intuitively clear, although they are able to represent the competitive nature of different physical laws.
The theoretical studies published by Murray were some of the first in the field regarding the optimality principles of vascular systems . The theoretical problem formulated in his work was the need to identify the best match between the criterion of optimization and the generalization of a quantitative statement. The arterial system was first considered in terms of the ‘principle of economy’, and it was found that there were two main opposing factors within this approach. In the case of small vessels, the determining factor is the work required to maintain a sufficient blood flow. In contrast, for large vessels, the blood volume is more essential. Therefore, the main goal was to estimate the efficiency of each factor and to analyse the relationship between them. This approach was subsequently adapted to describe the topology of arterial networks and oxygen metabolism in tissues . The Poiseuille equation was used as a fair approximation of the blood flow in a cylindrical tube. The ‘cost’ of blood volume per unit length was proportional to the cross section of the vessel. Thus, the minimization criterion was associated with the total energy involved in operating a vessel section. The sum of two terms that describe the friction power loss and blood maintenance energy has been postulated as the optimization cost function:
Accordingly, Murray found the simplest condition necessary to achieve the maximum efficiency of blood circulation to be the following: the blood flow should be proportional to the cube of the radius for each branch of the vessel. Thus, the optimality condition for the network in terms of geometry was
where r0, r1, and r2 are the radii of parent and offspring vessels, respectively, for one network segment. Based on the estimates obtained using this equation, it was concluded that the use of the total work as the main optimality criterion can reliably assess blood flow in small vessels. However, Murray later noted that this model is not suitable to describe an aorta because most of the work is required to produce intermittent acceleration of the blood, while only a small portion of it is necessary to overcome friction. The same principles were used to develop a theoretical law to describe the angles between the branches at the bifurcation point (Figure 1). The following relationship was obtained while minimizing the total work :
Another approach using optimality principles to construct a hypothetical system similar to the circulatory system of mammals was proposed by Cohn . The following biological parameters were used as the key factors in an optimization problem: (i) the size of the aorta; (ii) the capillary dimensions and the volume of tissue supplied by a capillary; (iii) the system connecting the aorta and capillaries; and (iv) the total resistance of the system to ﬂow. For this purpose, the optimal sizes of the different blood vessels were investigated, and they also attempted to estimate the relationship between the large and small vessels. The lower limit of the aortic radius was set in such a way that there was no turbulence in the blood flow, as the aortic blood flow resistance is negligible compared with the total resistance of the system. Based on the main function of the capillary network, it was assumed that, for maximum efficiency, it was necessary to maintain as large a surface area as possible for a single vessel. Because the amount of tissue supplied by capillaries is limited by the metabolic rate, the only way to avoid this restriction is to have a large number of small capillaries that have a radius that should be comparable to the size of an erythrocyte. In the case of symmetric bifurcation, the estimate of the relative radii of the branches based on minimizing the total system flow resistance was made based on the assumption that the flow is determined by the Poiseuille equation and that a fixed amount of material is available for the vessel walls. Using the Laplace relationship for the vessel wall constraint, Cohn derived an optimal relationship for vessel radii :
where ri is the radius of the ith offspring of the initial vessel with radius r0. This result is in good agreement with Murray's conclusions in the case of a symmetric bifurcation. The ability to assess the lengths of the vessels inside the network is another important capability of this technique. To identify the relationship between the lengths of the parent vessel and its offspring, an approach based on the partition of abstract space into smaller domains was used. For a system with sequential branching, Cohn found that the law describing the changes in the lengths is the same as that for radii:
Generally, the proposed approach had good agreement with the anatomical data with respect to the radii of the aorta and capillaries, the lengths of vessels and the number of generations. The main advantage of this method is that it can provide very precise values for the blood flow despite an essential increase in the number of optimization parameters compared with Murray's model.
The bifurcation of a single vessel based on the total energy optimization criterion.
Polyterminal trees are branched trees that are similar in their terminal parameters but have a different vessel topology. Thus, to determine the optimal structure, one needs to know the sequence of vessel connections in addition to the boundary conditions (Figure 2). The entire topology should be taken into account, as the flow velocity also depends on the full resistance [6,7]. It was shown that any tree with the terminal pressure determined by the criterion of minimum work has a volume comparable to that of the tree corresponding to the minimum volume criterion [8,9]. Therefore, the optimization problem of the global cost of a network yields the same result as that for a local energy minimization under surface and volume constraints .
The topological variability in the design of a polyterminal tree.
An extension of the principle described above for the various optimization functions was suggested by Zamir [11,12]. He proposed four hypotheses of optimality, which led to a different network geometry. An arterial junction is in an optimal state in any of the following four cases: when the total lumen surface of the arteries, the total lumen volume of the arteries, the power required to pump blood through that junction, or the total shear force acting on its lumen walls is a minimum. To use the principles of optimality for the branches forming a bifurcation, the vessel connection point is the main component of the vascular system. Therefore, the optimal state of every node leads to overall system optimality. The most important result of that study is the observation that when the branch angle is close to 90°, the parent vessel does not change its shape or direction. It should be noted that these results were obtained for all optimization functions, and this finding did not allow one of them to be selected as more effective. It was shown that, in the case of symmetric bifurcation, the optimum state corresponding to the minimum value of each of the optimization functions is achieved when the total angle ranged from 75° to 102°, which is consistent with experimental data. For asymmetrical bifurcations, the radius of the one offspring is close to that of the parent in most cases, while the total value of the bifurcation angle coincides with that in the case of symmetric bifurcation . Meanwhile, the flow resistance in asymmetric cases can be lower due to changes in the inner surface of the vessel .
Murray's law establishes the functional relationship between the vessel radius, volume flow, flow velocity, and wall shear stress for all vessels in an optimal system. Accordingly, it can be applied to both living and non-living systems, as evidenced by significant biological data . These relationships can be used to predict the properties of the vascular system at its different levels. Furthermore, it is possible to estimate the factors that lead to growth of the vascular system of animals as their size increases. For example, to maintain a constant capillary density during the growth of an organ, the number of capillaries should increase linearly with the mass or volume of the organ. The radius of the parent branch must also increase proportionally to minimize energy costs. Murray's law has a fundamental limitation, namely it can only be used for branching systems characterized by a flow proportional to r4, i.e. a laminar flow of liquid. However, because turbulent flow has a higher effective resistance, the optimal geometric patterns obtained from turbulence models are very similar to Murray's patterns . Another deviation from the standard case may occur when the non-Newtonian properties of blood are taken into account [17,18]. The topology can differ in terms of the size and radii of vessels, but in the presence of additional restrictions (e.g. inner surface , pulsatile flow with constant shear force ), the classical relationship is applicable. Therefore, such approaches have great practical significance for predicting the physiological parameters involved in the development of the vascular system during the growth of an organism.
Experimental study of the vascular networks in different species
The theoretical models described above should be checked for accuracy in experimental studies, and this verification is a significant part of investigations of optimality principles. However, due to the complexity of these experimental techniques, the direct relationship between the physical laws and precise vascular network topology is still unclear.
Our understanding of the human coronary network structure has been improved by comparisons of different types of coronary arteries . According to the functional classification, they can be divided into two groups: distributing arteries and delivering vessels. The function of the first group is to provide interzonal blood transport, while the function of the second group is to deliver the blood to myocardial zones. Due to their functions, distributing arteries have a lower branching rate than delivering arteries. Moreover, the vessels can be comparable in size, but their rates of branching will be different, which was also proved by theoretical models . According to experimental data, a decrease in the length:diameter ratio may indicate peripheral vascular disease . Moreover, deviation from the optimality condition for the network in terms of geometry is a risk factor for coronary heart disease  and may also be associated with proliferative diabetic retinopathy .
Another network description of the coronary arterial tree was based on direct observations of arterial branching patterns. It was shown that the relationship between the diameters of parent and daughter segments at arterial nodes, as well as the unbranched length and diameter of segments, has a quasi-stochastic nature. Thus, the logarithmic representation of the mean vessel diameter and length versus bifurcation order is clearly linear . However, in studies of symmetry, only a correlation between these values can be found, and there may be deviations from Murray's angle law for large coronary arteries . In the case of hypertension, these deviations can reach a value of −16° from the normal bifurcation angle . Because the retinal and cerebral microcirculation are sufficiently homologous, a bifurcation analysis of the retinal microcirculation is a good indicator of the presence of cognitive diseases due to its correlation with the state of the vascular system. It was shown that deviation from optimality of the angle at arteriolar bifurcations was signiﬁcantly associated with logical memory .
While the human arterial network is complicated due to the presence of different vessel types and the large scale, the arterial tree of a rat is a good model for comparing the network characteristics with the classical tree models. An experiment on the classification of different types of vessels was performed by Zamir and Phipps . The entire network was divided into segments, and level parameters were defined for each segment. Moreover, the distributions of the average diameter and length were presented as fundamental branching properties of the system. On this basis, it was shown that the largest decrease in the vessel radius occurs at the first bifurcation of the aorta. Similar results were also obtained from an experimental study of rat pial microcirculation . The assessed geometric characteristics indicate the fractal nature of the vessel distribution, which was also confirmed by numerous studies [32–34]. The analysis of the fractal dimension is extremely useful for predicting diseases associated with vascular network dysfunction [35,36]. Comparing such experimental data with the results of mathematical modelling makes it possible to reconstruct the most applicable criterion of optimization.
The functionality of the circulatory system largely depends on its structure and topology. An effective supply of metabolites to the organ is achieved due to the branching of blood vessels and their optimal spatial distribution. With the improvement of numerical simulation methods and the development of visualization algorithms for experimental research, it is becoming possible to comprehensively study vascular networks while taking into account their complex structure and the difference in the scale of vessels. However, several questions remain in such research. Building a more realistic model of the vascular tree requires both the geometric properties and the processes of blood vessel formation to be accounted for (e.g. vasculogenesis and angiogenesis). For example, in the angiogenesis model of the arterial system, vascular growth was modelled as a chemotactic process with respect to vascular endothelial growth factor (VEGF) using optimality principles based on Murray's laws . However, the secretion of VEGF is up-regulated by the Wnt signalling pathway and is highly dependent on the average oxygen level in the tissue [38,39]. Although the intra-tissue level of metabolites is a subject for mathematical modelling , accurate estimates are still impossible in some cases.
The major difficulty associated with applying the optimization approach to the construction of arterial network computer models (e.g. Constrained Constructive Optimization and Global Constructive Optimization [41,42]) is that it does not require the resolution of a single optimization problem; rather, the complexity is determined by the minimization criterion. An attempt to build a vascular network on the scale of an organ makes it necessary to consider the vessels, with the structures being defined by different physical laws. The main reason for this is that an organ has a larger simulation space compared with the size of a single vessel. For such models, this limitation can be partly avoided by introducing additional layers of cost optimization. Nevertheless, this does not solve the problem of the direct relationship between the laws of growth and observable topology. Comparisons with experimental data make it possible to explain many of these issues. However, due to the stochastic nature of the system, it is essentially possible only at the level of the corresponding distributions of physical parameters.
The Authors declare that there are no competing interests associated with the manuscript.