The fractal dimension is the most important parameter to quantify the fractality of objects. And measuring the information dimension is an efficient way to estimate the fractal dimension in geometry (Pitsianis et al., 1989; Liu et al., 2018). For the calculation of information dimension, non-overlapping boxes are adopted to cover the image of the object and the mass probability of each box, which is defined as the ratio of the number of signal pixels in the box to that of the whole image, is obtained. And the information dimension D_I (Pitsianis et al., 1989) is estimated as:

where ∑i=1N⁢(L)-Pi⁢log⁡Pi is the total entropy of mass according to the information theory, P_i is the mass probability of the ith box and N(L) is the number of boxes needed to cover the image with size L.

The total mass, which is the number of signal pixels in the whole image, obeys the law of conservation regardless of the box size. And so does the blood flow. As shown in Figure 1, the total volume of blood flow at the same generation also follows the law of conservation no matter how many times the vascular tree bifurcates. Thus, with appropriate modification, the fractal methods in fractal geometry can be introduced to investigate the fractality of blood flow. By replacing the mass probability P_i in Eq. 1 with the flow probability p_i and 1/L with the number of vessel segments N(g) at generation g, the expression of the fractal dimension for the blood flow D_Q is derived as:

Practically, D_Q is estimated as the slope of the linear fitting curve of the discrete data pair (log⁡N(g), ∑−p_ilog⁡p_i) (Wang et al., 2019). The precondition of the fractal analysis is scale invariance. And the strong linearity over three orders of magnitude of the fitting curve can be taken as the criteria of the existence of scale invariance (Halley et al., 2004).

For the fractal analysis in geometry, the box-counting dimension (So et al., 2017; Nayak et al., 2019), also known as capacity dimension, is the most popular. This method also requires the image to be covered by non-overlapping boxes. However, the box-counting dimension method only considers the existence of signal pixels in the box but ignores the number of pixels. When applied to blood flow analysis, it produces fractal dimension about the vascular structure other than the distribution of blood flow. In contrast, the information dimension method takes the quantity of blood flow into account and thus can reflect the blood flow distribution.

Very few objects possess perfect mono-fractality exhibiting a single fractal dimension (Gould et al., 2011). In reality, objects with the subsets having different scaling properties are much more common and the estimation of multifractality is more desirable. For the multifractal measure of blood flow, the multifractal spectrum f(α)∼α of the blood flow is adopted (Chhabra and Jensen, 1989) and modified as:

in which

where q is the moment order. And the range of the spectrum Δα = α_max − α_min can be used to measure the degree of multifractality (Halsey et al., 1986).

The methods given in Section “Establishment of Fractal Methods for Blood Flow” are based on the premise that the total blood flow at the same generation in a vascular tree obeys the law of conservation. This premise is valid for a perfect binary tree, as shown in Figure 1, but not for real vascular trees as shown in Figure 2A. If the branch which stops bifurcating before reaching the maximal generation is regarded as a branch covering multiple generations, the vascular tree in Figure 2A can be thought of as the perfect binary tree as shown in Figure 2B. In this case, the proposed methods can be applied and a strategy for the calculation is raised: If a vessel segment stops bifurcating at generation n (n is smaller than the maximal generation of vascular tree), it will be involved in the calculation at all the generations greater than n.

That is to say, for the vascular tree in Figure 2A, V_2,1 will be included in the calculation of total entropy at generation 3 and V_1,2 will be included in the calculation at both generation 2 and generation 3. It should be noted that the number of vessel segments N in Eq. 2 should be 2ⁿ at generation n but not the number of vessel segments at this generation. The reason is explained below.

The blood flow distribution in the vascular tree in Figure 2B can be analogous to a one-dimension graph as shown in Figure 2C. The gray segments indicate the range of the graph and the black segments reflect the blood flow in vessel segments. Each bisection corresponds to a bifurcation of the vascular tree and divides this geometrical structure into smaller subsections. When adopting Eq. 1 for the estimation of fractal dimension for this one-dimension graph, L is corresponding to the length of the smallest subsection. And each bisection of the graph halves L no matter whether there is always a black segment in each subsection. The number of subsections should be 2ⁿ after n bisections. Similarly, the number of vessel segments N(g) in Eq. 2 should be 2ⁿ at generation n.

By now, the methods established in “Establishment of Fractal Methods for Blood Flow” and “Generalization of the Established Methods” can be used to characterize the blood flow distribution in arbitrary vascular trees. For validation, the established fractal methods were tested in a real arterial tree (Reglin et al., 2009, 2017; Wang et al., 2019) under normal and ischemic state. Besides, the capability of the derived fractal parameters was examined in two situations.

For the first situation, the vascular tree was fixed to be a perfect binary tree and the blood flow distribution was adjusted by the split ratio. For a bifurcation with a parent vessel segment and two daughter branches, the split ratio r is defined as the ratio of the smaller blood flow to the larger one in the two daughter branches, ranging from 0 to 1. Assuming that r is constant throughout the perfect binary tree, the fractal dimension for blood flow can be obtained as shown in Eq. 6. The detailed derivation of the equation is given in the Appendix.

For the second situation, a series of vascular trees were constructed. The structures of these vascular trees were diverse while the blood flow distribution was estimated under the same boundary condition.

The successive dichotomous division is the most common branching pattern of the vascular tree, in which a parent vessel segment is divided into two daughter branches (Zamir, 2001). Based on this pattern, the construction of a vascular tree calls for the determination of the vessel diameter and length. For a bifurcation with the diameter of the parent vessel being d₀ and those of the two daughter branches being d₁ and d₂, a power-law relationship between the diameters is given as shown in Eq. 7 according to Murray’s law.

where d₀ > d₁ ≥ d₂, k is bifurcation exponent and λ is asymmetry ratio. It is reported that the k value varies from 2.33 to 3.0 (Gabrys et al., 2005). And a value above 0.6 is most commonly observed for λ (Schmidt et al., 2004; Cheung et al., 2011; Takahashi, 2014). Based on the power-law relationship, the diameters of all vessel segments in a vascular tree can be estimated with the given root diameter and cut-off diameter. In this study, the root diameters of all constructed vascular trees are set as 300 μm and all the terminal diameters are 10 μm, the size of capillaries. An empirical formula (Takahashi et al., 2009) is adopted to obtain the length l (μm) from the diameter d (μm):

With the estimated vessel diameters and lengths, a vascular tree can be constructed, and serves for the hemodynamic simulation. According to Hagen-Poiseuille’s law as shown in Eq. 9, the blood flow Q (μm³/s) in a vessel segment is proportional to the pressure drop ΔP (Pa) between the inlet and outlet. And the flow resistance R is determined by vessel diameter d (μm), vessel length l (μm), and blood viscosity μ (Pa ⋅ s).

Fahraeus and Lindqvist (1931) reported a decline in apparent blood viscosity with decreasing tube diameter, the so-called Fahraeus-Lindqvist effect. Among the models to describe the relationship between the blood viscosity μ (mPa ⋅ s) and the diameter of the vessel segment d (μm), the model proposed by Pries and Secomb (2005) matches well with the in vivo experimental data. In this model, the in vitro viscosity is firstly estimated as:

in which H_d (%) is the hematocrit and:

The estimation of the in vivo viscosity should take account of the effect of the endothelial surface layer. This will involve two parameters, the effective diameter d_eff = d − 2W_eff and the physical diameter d_ph = d − 2W_ph. The effective thickness of the layer W_eff and physical thickness of the layer W_ph are estimated as:

Based on the experimental data, E_HD = 1.18, E_peak = 0.6, E_amp = 1.1, E_width = 0.03, D_off = 2.4 μm, D_crit = 10.5 μm, D₅₀ = 100 μm, and W_max = 2.6μm. By replacing the d in Eqs 11 and 12 with d_ph, we can get the in vitro viscosity μ_vitro. And the in vivo viscosity μ_vivo can be obtained as below.

For the bifurcation as shown in Figure 3, the relationship between the hemodynamic parameters can be given by Eq. 18 based on Hagen-Poiseuille’s law and the conservation law of flow.

Without losing generality, in the hemodynamic simulation we prescribed the inlet pressure as 1 mmHg and the outlet pressure at all terminal branches as 0 mmHg (Yang and Wang, 2013). With each bifurcation of the vascular tree assigned an equation like Eq. 18, the blood pressure at each branch node can be obtained by solving these equations. Further, the blood flow in each vessel segment is estimated by Eq. 9 and finally the blood flow distribution in a tree can be acquired.

In this study, all the calculations and simulations were programmed by MATLAB R2019a (MathWorks Co., MA, United States). Firstly, the node information for each constructed vascular tree was obtained. Then, the blood flow in each vessel segment of the constructed vascular tree was captured by solving the equations set. Ultimately, the fractal dimension and multifractal spectrum were calculated. All the results about the fractal parameters were presented as Mean ± SD.

To test the validity of the proposed methods, the fractal, and multifractal analysis were conducted on a real arterial tree under normal and ischemic state. The blood flow distribution in these two states is as shown in Figures 4A,B. The fractal dimension for the normal and ischemic state are 0.53 and 0.40, respectively. As for the multifractal spectrum for the blood flow, the results are shown in Figures 4C,D. We can observe that the multifractal spectrums for both states appear as curves indicating the existence of multifractality. While the maximal values of the multifractal spectrum for the two states are the same, the range of the multifractal spectrum Δα for the ischemic state is wider than that for the normal state which indicates a higher degree of multifractality for the ischemic state. The results verified the ability of the proposed methods in distinguishing different blood flow distribution in a real vascular tree.

The blood flow distribution in the perfect binary vascular tree was evaluated by both the fractal parameters and the CV, which is defined as the standard deviation divided by the mean value. For a perfect binary vascular tree with the identical split ratio for each bifurcation, the fractal dimension for blood flow was obtained based on Eq. 6. Figure 5A shows the trends of fractal dimension and the coefficient of variation with the change of split ratio. It is noticed that the fractal dimension increases monotonically from 0 to 1 with the increment of the split ratio. In the case r = 1, the distribution of blood flow has the highest value of fractal dimension. And the CV decreases from 10.1 to 4.4 with the increment of the split ratio.

As for the multifractal characteristic, the multifractal spectrums of the blood flow are shown in Figure 5B. With the decrement of the split ratio, the range of the multifractal spectrum Δα expands while the maximal value remains unchanged. It is worth noting that the multifractal spectrum is presented as a point when r = 1, implying the absence of multifractality.

We also examined the fractal dimension and the CV of the blood flow in perfect binary vascular trees with different maximal generations. As shown in Figure 6, the fractal dimension holds steady with the change of maximal generation while the CV varies greatly.

Vascular trees with diverse structures were constructed. To reflect the heterogeneity in the real vascular tree, the bifurcation exponent k, and asymmetry ratio λ of the bifurcations in each constructed vascular tree were set following the normal distribution. And ten vascular trees were constructed for each pair of k and λ. The statistical characteristics of these vascular trees are shown in Table 1. In these constructed vascular trees, the hemodynamics was simulated and the results are shown in Figure 7. We can see that there exists a strong linear relationship between the logarithmic values of the diameter and the blood flow rate.

The fractal dimensions for blood flow in the constructed vascular trees are shown in Figure 8. For the vascular trees with the mean k value of 2.7, the fractal dimension increases monotonically from 0.74 ± 0.01 to 1.00 ± 0.00 with λ rising from 0.60 to 1. While for the vascular trees with the mean λ value of 0.8, the fractal dimension fluctuates between 0.91 and 0.95 with the increment of k.

The multifractal spectrums of the blood flow for these vascular trees are presented in Figures 9A,C. It is observed that the multifractality of blood flow exists in all vascular trees but the multifractal spectrums fluctuate. With the increment of λ, the range of the multifractal spectrum Δα narrows from 0.89 ± 0.04 to 0.04 ± 0.01 and the maximal value grows from 0.93 ± 0.01 to 1.00 ± 0.00. And with the increment of k, Δα expands from 0.32 ± 0.04 to 0.65 ± 0.03 and the maximal value grows from 0.95 ± 0.01 to 0.99 ± 0.00. The features of the multifractal spectrums are shown in Figures 9B,D. Compared with the fractal dimensions, the difference of multifractal spectrums among these vascular trees is more striking.

The hemodynamic simulation was conducted to investigate the variation of fractal parameters with varying blood flow distribution. To make sure that the obtained blood flow distribution is reasonable, a quantitative comparison of the hemodynamic simulation with the existing physiological studies is necessary. For avoiding losing the generality, the boundary condition in the present work was prescribed with an inlet pressure of 1 mmHg and an outlet pressure of 0 mmHg. However, the pressure drop between the inlet and outlet may vary in different studies. Thus, for quantitative comparison, it is more appropriate to examine the relative indices.

As shown in Figure 7, a strong linear relationship exists between the logarithmic values of the diameter and the blood flow rate. This is consistent with the assertion in Huo and Kassab (2016) that there is a scaling law between the blood flow rate and diameter. The slope of the fitting line indicates the relation between the blood flow rate and vascular diameter thus can be an indicator for quantitative comparison. Within a similar diameter range, the slope with a value of 2.40 produced in our work is comparable with the reported work of 1.97 (Wang et al., 2009), 2.0 ± 0.2 (Pijewska et al., 2020), 2.33 (Huo and Kassab, 2012), and 2.49 ± 0.09 (Haindl et al., 2016). Thus, it can be concluded that the simulation results of our work are reasonable.

Two fractal parameters, i.e., fractal dimension and multifractal spectrum, were obtained in this study to investigate the fractality and multifractality of blood flow.

By definition, the fractal dimension is determined by the total entropy of blood flow. And the total entropy is calculated by considering the existence as well as the quantity of blood flow in the vessel segment. The existence and the quantity of blood flow are corresponding to the vascular structure and the blood flow distribution, respectively. Thus, the fractal dimension characterizes the combination of the features of vascular structure and blood flow distribution. When the vascular structure is fixed, the lower entropy is obtained from the more heterogeneous distribution according to the information theory. That is to say, the fractal dimension reflects the degree of the blood flow heterogeneity for a specific vascular tree and the lower fractal dimension comes from the blood flow distribution with a higher degree of heterogeneity. The results in Figure 5A that the lower fractal dimension is corresponding to the lower split ratio also support this conclusion.

For a fractal object, the multifractal spectrum describes the scaling properties in different subsets. And when multifractality presents, the subsets of this object will be scaled by different multiples at the same q order moment during the calculation of the multifractal spectrum. Therefore, the degree of multifractality, which is measured by the width of the multifractal spectrum, essentially describes the degree of heterogeneity within the fractal object and it rises with the increment of the blood flow heterogeneity. This judgment is consistent with the results as shown in Figure 5B. The multifractal spectrum f(α(q)) reaches its maximal value when q = 0. In this case, the quantity of the blood flow volume no longer has an effect on the value of f(α(0)). In other words, the height of the multifractal spectrum reflects the heterogeneity or asymmetry of the vascular structure. The higher the multifractal spectrum the closer the vascular tree is to the perfect binary tree. And this makes clear why the height of the multifractal spectrums in Figures 4C,D or Figure 5B is the same. By means of the width and height, the multifractal spectrum separates the information about the vascular structure and blood flow distribution. In this sense, the multifractal spectrum makes the evaluation of the blood flow distribution in different vascular trees possible. As shown in Figure 9D, the heterogeneity of the vascular structure decreases with the increment of the bifurcation exponent while the heterogeneity of the blood flow distribution increases. The interaction of these two opposite trends may explain why the fractal dimension changes slightly with the bifurcation exponent as shown in Figure 8B.

Both the fractal dimension and the multifractal spectrum reflect the blood flow heterogeneity. Physiologically speaking, the change of blood flow heterogeneity is usually associated with pathological conditions. For microcirculation, the increment of blood flow heterogeneity can be an early indicator of diseases, such as sepsis and shock (Ince, 2005; Dubin et al., 2018; Ince et al., 2018) as well as peripheral vascular disease (Butcher et al., 2013). And the increase of blood flow heterogeneity can be depicted by the decrease of fractal dimension and the broadening of the multifractal spectrum. The multifractal spectrum can separate the information about the vascular structure and blood flow distribution. Thus, the multifractal spectrum is also able to distinguish the causes responsible for the change in blood flow heterogeneity, either by hemodynamic problems or by structural alteration due to the diseases such as large vessel stenosis (Kharche et al., 2018).

There are also some other quantitative or semi-quantitative methods for characterizing the blood flow heterogeneity (Bassingthwaighte et al., 1989; Pries and Secomb, 2009; Ince et al., 2018). In this study, the blood flow heterogeneity is also evaluated by CV. The results in Figure 5A confirm the availability of this statistical parameter. However, the CV treats different vessel segments in a vascular tree as independent components ignoring the connection of blood flow along the whole tree. This would make this parameter less accurate in some cases as discussed below. When the split ratio for each bifurcation in the perfect binary vascular tree is 1, the blood flow is evenly distributed for each generation in the vascular tree. The degree of blood flow heterogeneity should remain unchanged no matter what the maximal generation of the vascular tree is. And this property holds for the other values of the split ratio. As shown in Figure 6A, the fractal dimension stays the same with the change of the maximal generation. However, the values of CV for the vascular trees with different maximal generations are quite different. Considering both the connection and difference of blood flow in different vessel segments, the fractal parameters can provide a more accurate description of the blood flow heterogeneity for the tree-like vasculature.

It should be pointed out the hemodynamic simulation in this study was simplified. Nowadays, the RCL model has been developed for hemodynamic simulation in which the resistance (R), capacitance (C), and inductance (L) elements were used to mimic the effects of vessel resistance, vessel compliance, and blood inertia, respectively (Muller and Toro, 2014; Zhang et al., 2014). And models from 0D to 3D were established (Arciero et al., 2017; Liu et al., 2020). In the present hemodynamic model of microcirculation, only the resistance element for a vessel segment was considered. Although in the microcirculation the resistance element plays a dominant role in hemodynamics (Katanov et al., 2015; Nichols et al., 2015; Secomb, 2017) and the results show that the model is sufficient for produce varying blood flow distribution in a tree, a comprehensive model is worth being introduced in the future study.