Edited by: Dingchang Zheng, Coventry University, United Kingdom
Reviewed by: Haipeng Liu, Coventry University, United Kingdom; Lisheng Xu, Northeastern University, China; Michal Strzelecki, Lodz University of Technology, Poland
This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology
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Blood perfusion is an important index for the function of the cardiovascular system and it can be indicated by the blood flow distribution in the vascular tree. As the blood flow in a vascular tree varies in a large range of scales and fractal analysis owns the ability to describe multi-scale properties, it is reasonable to apply fractal analysis to depict the blood flow distribution. The objective of this study is to establish fractal methods for analyzing the blood flow distribution which can be applied to real vascular trees. For this purpose, the modified methods in fractal geometry were applied and a special strategy was raised to make sure that these methods are applicable to an arbitrary vascular tree. The validation of the proposed methods on real arterial trees verified the ability of the produced parameters (fractal dimension and multifractal spectrum) in distinguishing the blood flow distribution under different physiological states. Furthermore, the physiological significance of the fractal parameters was investigated in two situations. For the first situation, the vascular tree was set as a perfect binary tree and the blood flow distribution was adjusted by the split ratio. As the split ratio of the vascular tree decreases, the fractal dimension decreases and the multifractal spectrum expands. The results indicate that both fractal parameters can quantify the degree of blood flow heterogeneity. While for the second situation, artificial vascular trees with different structures were constructed and the hemodynamics in these vascular trees was simulated. The results suggest that both the vascular structure and the blood flow distribution affect the fractal parameters for blood flow. The fractal dimension declares the integrated information about the heterogeneity of vascular structure and blood flow distribution. In contrast, the multifractal spectrum identifies the heterogeneity features in blood flow distribution or vascular structure by its width and height. The results verified that the proposed methods are capable of depicting the multi-scale features of the blood flow distribution in the vascular tree and further are potential for investigating vascular physiology.
The microcirculation is the end destination of the cardiovascular system and the patency of microvascular perfusion is essential for the maintenance of tissue metabolism (
To depict the scale-independent characteristic of objects, the fractal theory provides an efficient approach for multi-scale analysis (
The fractal analysis has been widely used to investigate the geometrical characteristics of the vasculatures (
In this study, the primary aim is to establish fractal methods for analyzing the blood flow distribution which is potential to be applied to real vascular trees. To achieve this goal, we firstly modified the fractal methods in geometry to accommodate the situation of blood flow and then applied the established methods on experimental data to test the validity. Further, to explore the physiological significance of the yielded fractal parameters, the blood flow distribution in vascular trees with fixed structure or with varying structures were examined in which the hemodynamics was simulated based on a hemodynamic model (
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 (
where
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
The schematic of the conservation of flow at different generations. Q is the blood flow and
Practically,
For the fractal analysis in geometry, the box-counting dimension (
Very few objects possess perfect mono-fractality exhibiting a single fractal dimension (
in which
where
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
That is to say, for the vascular tree in
The blood flow distribution in the vascular tree in
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 (
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
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 (
where
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
in which
The estimation of the
Based on the experimental data,
For the bifurcation as shown in
The schematic of the hemodynamic parameters at a bifurcation.
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 (
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
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.
The change of different parameters of the blood flow for different split ratios in a perfect binary vascular tree.
As for the multifractal characteristic, the multifractal spectrums of the blood flow are shown in
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
The change of different parameters of the blood flow in the perfect binary vascular trees with different maximal generations.
Vascular trees with diverse structures were constructed. To reflect the heterogeneity in the real vascular tree, the bifurcation exponent
The characteristics of the constructed vascular trees.
λ | Vessel number | Max generation | λ | Vessel number | Max generation | ||
2.7 | 0.60 | 7471 ± 97 | 33.8 ± 1.7 | 2.3 | 0.80 | 2822 ± 334 | 15.3 ± 0.6 |
0.65 | 8261 ± 130 | 28.2 ± 1.2 | 2.4 | 3525 ± 281 | 15.8 ± 0.6 | ||
0.70 | 9012 ± 83 | 24.9 ± 0.8 | 2.5 | 5102 ± 469 | 16.7 ± 0.5 | ||
0.75 | 9777 ± 73 | 21.5 ± 0.7 | 2.6 | 6987 ± 887 | 17.4 ± 0.7 | ||
0.80 | 10692 ± 99 | 19.5 ± 0.7 | 2.7 | 10210 ± 1389 | 18.8 ± 0.9 | ||
0.85 | 11387 ± 112 | 17.8 ± 0.4 | 2.8 | 14672 ± 1038 | 20.2 ± 0.7 | ||
0.90 | 12189 ± 93 | 16.4 ± 0.5 | 2.9 | 19149 ± 2544 | 21.2 ± 0.4 | ||
0.95 | 12818 ± 110 | 15.3 ± 0.5 | 3.0 | 27222 ± 1774 | 22.2 ± 0.9 | ||
1.00 | 16234 ± 102 | 13.0 ± 0.0 |
The log-log plot of the blood flow rate versus the vessel diameter for all constructed vascular trees. The solid line is the best fit result of linear regression.
The fractal dimensions for blood flow in the constructed vascular trees are shown in
The fractal dimension for blood flow of the vascular trees with varying λ
The multifractal spectrums of the blood flow for these vascular trees are presented in
The multifractal spectrums for blood flow of the vascular trees with varying λ
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
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
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
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 (
There are also some other quantitative or semi-quantitative methods for characterizing the blood flow heterogeneity (
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 (
In this study, the fractal methods were introduced, with appropriate modification, to characterize the multi-scale properties of blood flow. The application of the methods to the real physiological data verified its ability in distinguishing the variety of blood flow distribution. The yielded parameters, as the fractal dimension and the multifractal spectrum for blood flow, can quantify the degree of blood flow heterogeneity. With the increase of blood flow heterogeneity, the fractal dimension decreases and the multifractal spectrum expands. And the investigation on various constructed vascular trees suggests that both the vascular structure and the blood flow distribution influence the fractal parameters. With the aid of the fractal dimension, it is possible to look into the change of blood flow heterogeneity in a specific vascular tree. While the multifractal spectrum can be utilized to assess the blood flow heterogeneity for different vascular trees by considering the blood flow distribution and the structure of vascular trees separately. It can be concluded that the proposed methods provide efficient tools to describe the multi-scale properties of the blood flow distribution and has the potential to assist the study of multi-scale vascular physiology.
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.
PL and QP: conceptualization and methodology. PL: algorithm and writing (original draft). SJ and GN: writing (review and editing). MY and JY: discussion of the results and their relevance. GN and JY: supervision and project administration. All authors approved the manuscript.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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For the perfect binary vascular tree where the split ratio of each bifurcation is identical, if we normalize the blood flow in the main vessel at generation 0 as 1 and denote the split ratio by
Assuming
which is the expression of the expectation of a binomial distribution
And the fractal dimension of blood flow for the perfect binary vascular tree with split ratio