Blood physiological and flow characteristics within coronary artery circulatory network for human heart based on vascular fractal theory

Abstract

To analyze the fractal form of the vascular network in the human circulatory system, the optimal transport effect has been achieved from the point of view of biological evolution. The blood flow mathematical models based on the fractal theory for capillary network and arteriole–capillary vascular fractal network were established using theory derivation, and the blood flow characteristics, dynamic flow resistance effects, and vascular fractal physiology property based on the fractal porous medium theory for the coronary artery circulatory network were analyzed under the consideration of some influencing factors, namely, non-Newtonian fluid characteristics of blood, hemocoagulation and embolization effect in capillaries, and plasma mass flow effects. Moreover, the Poiseuille flow equation is modified by introducing the correction function, and the flow model of blood in the vascular network is established. Obviously, the relationship characteristics between blood flow and bifurcation grade in fractal vascular, fractal dimension in the arteriole–capillary vascular network, fractal dimensions of the diameter of capillary tubular diameter, fractal dimensions of capillary blood vessels, blood Casson yield stress, and ratio of red blood cell radius to capillary diameter can be obtained. And the relationship characteristics between blood flow resistance and ratio of erythrocyte radius to capillary diameter, ratios of the distance between adjacent red blood cells to the radius of red cells, and bifurcation grade can be obtained. Finally, the clinical verification tests were accomplished to verify the theory is worthy of authenticity and rationality where the curve tendencies are very similar with those obtained by numerical simulations based on the theoretical models and experimental test showed that the theoretical calculation and simulation analysis of blood circulation system of fractal vascular network were reasonable and applicable by means of experimental relative method because of the maximum relative error is less than 10%, whatever fractal dimension m changes under different conditions.

Introduction

The vascular vessel is an important component of blood transport and material exchange in the circulatory system of the human body, and the distribution of blood vessels is mostly in the form of tree-branched bifurcation meshes based on the fractal theory within biological world; this mode of material transportation must be an “ideal” network transportation structure that can achieve maximum transport efficiency and minimum transmission resistance from the perspective of biological evolution theory.^1–3 The fractal theory of transportation network vessel for biological circulatory system has been emphasized by scientists and physiologists from the research institutions at home and abroad the last century, which includes the distribution precognition of fractal blood pipeline network, optimization of transportation efficiency models, and ideal distribution pattern of energy consumption for the biological blood circulation system.^4,5 The famous Murray’s law, a basic physical principle for transfer networks, was proposed based on the previous theoretical analysis and research results in 1926 that the required total energy cost for transport and maintenance blood circulation motion is minimized relative to other distribution reticular structures in the vascular and respiratory systems of animals and it is predicted that the optimal vascular radius/diameter is proportional to the cube of blood flow velocity for the local area. ⁶ Obviously, Murray’s law is also a powerful biomimetics design tool in engineering. Murray’s original analysis is based on the assumption that the radii inside a lumen-based system are such that the work for transport and upkeep is minimized. Larger vessels lower the expended energy for transport, but increase the overall volume of blood in the system; blood being a living fluid and hence requiring metabolic support. Murray’s law is therefore an optimization exercise to balance these factors. ⁷ It has been applied in the design of self-healing materials, batteries, photocatalysts, and gas sensors to exclude small trachea device for the larger error. ⁸ Subsequently, a large number of research results for fractal theory and extensibility studies based on Murray’s law were analyzed and proposed to design advanced materials, reactors, and industrial processes for maximizing mass or energy transfer to improve material performance and process efficiency. ⁹ For example, ER Weibel ¹⁰ considered the developments in understanding the quantitative anatomy of the lung in the correlation of anatomy with physiology and looked into the mode of distribution and geometric forms that should eventually facilitate mathematical and physical considerations regarding the function of the lungs. G Hooper ¹¹ and JB West ¹² established a mathematical model of the sheep bronchus to derive a relationship between the diameter of a bronchus and the diameters of its branches which is applicable to different types of branching and is consistent with principles of minimum energy loss within the bronchial tree; that is, the weight, and hence volume, of a cast of the bronchial tree of a sheep distal to any point was found to be a function of the diameter of the bronchus at that point. K Horsfield and Cumming ¹³ found that linear relationships were obtained when the number, mean diameter, and mean length of branches in each order are plotted semilogarithmically against order via the branches were ordered by the method of Strahler and the number of branches in each order counted for dogs. E Gabryś et al. ¹⁴ used the fractal dimension to study the effect of fractal network symmetry on the blood circulation system that is blood vessel models have been generated according to morphology and structural properties, such as segment radii and lengths, branching angle, and their relation to segment diameter and influence of asymmetry on creation circulatory system has been explored by means of the various numbers of morphological parameters have been tested for symmetrical and asymmetrical trees. A Dokoumetzidis and Macheras ¹⁵ developed a simple model for the heterogeneous transport of materials in the circulatory system of mammals, based on a single tube dispersion–convection system which is equivalent to the fractal network of the branching tubes. M Baptiste and Benjamin’s ¹⁶ study result showed that Murray’s original law also applies to fluids of im-undergo phase separation effect such as blood and an extended version of Murray’s law was derived when Fahræus-like phase separation effects occur, and the optimal geometries factors of fractal trees to mimic an idealized arterial based on Quemada’s fluid model were analyzed theoretically. The main contribution of D Quemada ¹⁷ is that extension to visco-elastic behavior has been obtained using a Maxwell model with instantaneous values of viscosity and elasticity, which both are functions of the structural variable phi p(t, gamma). CY Cheung et al. ¹⁸ found that the new score of retinal vascular optimality combining fractals and caliber showed a strong association with blood pressure, and retinal blood vessel fractal dimension has an important influence on the risk factors of the eye. A general survey of recent works on capillary blood flow is given and a method is offered for the calculation of pressure drop in the capillary as a function of various physical parameters according to YC Fung’s ¹⁹ study results. The overall hydrodynamics is described in terms of hydraulic conductivity coefficients for the arterial and venous flow rates whose functional form provides an explanation for the singular behavior of the vascular resistance observed in experiments, and results show that assumption of uniform interstitial pressure is not generally appropriate. ²⁰ DA Beard and Bassingthwaighte ²¹ found that the distributed models must account for diffusional as well as permeation processes to provide physiologically appropriate parameter estimates. Fumihiko Kajiya et al. ²² thought that flow velocity waveforms of coronary arterial inflow and venous outflow of myocardium are influenced by cardiac contraction and relaxation: arterial flow is exclusively diastolic; venous outflow is systolic; and an understanding of mechanoenergetic interaction is fundamentally important to an understanding of intramyocardial coronary circulation. Vittorio Cristini and Kassab ²³ provided an overview of the state-of-the-art computational methods for modeling of red blood cell (RBC) rheology and dynamics in the microcirculation to produce new algorithms capable of describing the motion and deformation of large systems of RBCs in microvessels at physiologically relevant volume fractions. Benjamin Kaimovitz et al. ²⁴ found the present model constitutes the first most extensive reconstruction of the entire coronary arterial system which will serve as a geometric foundation for future studies of flow in an anatomically accurate three-dimensional (3D) coronary vascular model, and the 3D tree structure was reconstructed initially in rectangular slab geometry using global geometrical optimization using parallel simulated annealing (SA) algorithm. The fluid is acted by an oscillating pressure gradient and an external magnetic field based on a mathematical model with Caputo fractional derivatives in terms of some practical problems, and the interaction between magnetic nanoparticles (NPs) and blood pressure gradient in the blood flow were analyzed by this mathematical method. ²⁵ SI Abdelsalam and colleagues^26,27 proposed a simplified mathematical model of long creep wave and peristaltic flow field to study the effects of magnetic field and endoscope on peristaltic blood flow of nanofluid in porous ring, that is composed of temperature, continuity, NP concentration, and equation of motion. A Korolj et al. ²⁸ discussed the examples of the implementation of fractal theory in designing novel materials, biomedical devices, diagnostics, and clinical therapies and suggest moving toward using fractal frameworks as a basis for the research and development of better tools for the future of biomedical engineering. N Abbas et al. ²⁹ and Alblawi et al. ³⁰ deliberated the flow of magnetized micropolar hybrid NPs fluid flow over the Riga curved surface, and comparison with the literature has been worked out and excellent agreement is found. S Nadeem and colleagues^31–33 also studied the numerical analysis of water-based carbon nanotubes (CNTs) flow of micropolar fluid, reduce atherosclerotic lesions method of bifurcated arteries with compliance walls and the finite volume method of mixed convection of nanofluids.

For the study of vascular metrology, blood transitive properties and blood vessel fractal characteristics for the organism body have been proposed and many models of approximation algorithms focusing on direct numerical solutions within circulatory system were proposed. However, the publications devoted to the effects of blood physiological properties on blood flow and flow resistance based on vascular fractal theory using theoretical research and physiological testing are relatively scarce. Therefore, the blood physiological and flow characteristics within coronary artery circulatory network for human heart based on vascular fractal theory were investigated using blood flow within the arterioles, and the complex branching system of capillaries is equivalent to the porosity media for osmotic fluids. The blood flow characteristics and dynamic flow resistance effects in the coronary artery circulatory network were analyzed by considering some influencing factors, namely, non-Newtonian fluid characteristics of blood, hemocoagulation and embolization effect in capillaries, and plasma mass flow effects.

Theoretical analysis of blood flow within fractal vascular network for heart and coronary network

Vascular fractal mechanism

The vascular network of the human circulatory system exists in the form of a tree bifurcation, and Murray’s law predicts the thickness of branches in transport networks, such that the cost for transport and maintenance of the transport medium are minimized. Murray’s law is therefore an optimization exercise to balance these factors. For n child branches splitting from a common parent branch, the law states that (Figure 1) ³⁴

$$d_0^3=d_1^3+d_2^3+d_3^3+\cdots +d_n^3$$

(1)

where d₀ is the diameter of the parent branch, and d₁, … , d_n are the diameters of the child branches.

Figure 1.

The simplified graphics of Murray’s law.

Obviously, the fractal dimension is an important parameter to describe fractal scale properties of porous medium in terms of fractal statistics. ³⁵ The theoretical analysis method of fractal scaling can be used to analyze the physio-physical properties of the vascular network according to the similarity mechanism between vascular network and porous medium in terms. The function between porosity and fractal scales for porous medium is obtained as follows

$$N(R\ge \kappa )={\left(\frac{{\kappa }_{max}}{\kappa }\right)}^{D_{\beta }}$$

(2)

where $\kappa$ is pore diameter, $N$ is the number of pores at $\kappa =R$ , ${\kappa }_{max}$ is the maximum pore diameter, and $D_{\beta }$ is the fractal dimension for porous medium.

Erythrocyte embolization effect

From arteries to veins, the blood has to go through the “capillary” blood vessels. These blood vessels are so small that often their diameter is smaller than that of the RBCs. Intimate interactions occur, therefore, between the blood cells and the blood vessels, and hemodynamic behavior of blood is complicated due to the presence of RBCs and their motion in the capillary. The RBCs will deform to get into the blood vessels in the manner of being squeezed when a red cell moves in a capillary blood vessel. If the red cell is circular in planform and thin at the center, the relationship function between flow and pressure gradient in the absence of slip conditions is obtained ¹⁹

$$\{\begin{array}{c}Q=-\left(\frac{\pi a^4}{8{\eta }_a}\right)\frac{\mathrm{dp}}{\mathrm{dx}}\hfill \\ {\eta }_a=\frac{\eta }{(1-\frac{b^4}{a^4})}\hfill \end{array}$$

(3)

where ${\eta }_a$ is the apparent viscosity, $b$ is the radius of the RBC, and $a$ is the radius of the capillaries.

The effect and model of plasma mass flow between RBCs

There is also a phenomenon of plasma flow between two adjacent RBCs in the capillaries that the plasma between the adjacent two RBCs is mobilized on both sides of the blood vessel symmetrical center axis in Figure 2. According to Fung’s ³⁶ theory, the Navier–Stokes function based on the Poiseuille corrections coefficient theory is obtained as

$$-\nabla p+\eta {\nabla }^2\mathbf{V}=0$$

(4)

where $p$ is the pressure, $\eta$ is the viscosity of plasma, and $\mathbf{V}$ is the speed of plasma relative to RBCs.

Figure 2.

The plasma flow effect between closed red blood cells.

Non-Newtonian characteristics of blood

It can be seen that blood has non-Newtonian fluid properties due to the presence of blood cells, and the Casson fluid’s motion characters are similar to the relationship function between shear rate and shear stress in human blood according to Copley’s research results in Figure 2. Obviously, the stress and variability function of Casson fluid is obtained ³⁷

$$\stackrel{·}{\gamma }=f\left(\tau \right)=\{\begin{array}{c}0\mathrm{when}\tau <{\tau }_c\hfill \\ \frac{1}{{\eta }_c}{(\sqrt{\tau }-\sqrt{{\tau }_c})}^2\mathrm{when}\tau <{\tau }_c\hfill \end{array}$$

(5)

where ${\eta }_c$ is the Casson viscosity and ${\tau }_c$ is the Casson yield stress.

Blood flow model within the fractal vascular network

The blood flow model in an arteriole–capillary vascular network in muscle tissue was established using equivalent analogy method that arterial vessel network is equivalent to a random fractal dendritic bifurcation network, and the capillary network and tissue cells within the tissue are equivalent to the porous media matrix void to meet warping, bending, and fractals’ physical feature requirements for physiological circulatory system by the random fractal scaling law of porous medium.

Kinetic equation of blood bolus flow within capillaries net

The following derivation process was proved by applying Lew and Fung’s theoretical results which assume that there is no slip and infiltration of plasma between the capillary wall and erythrocyte spheres. ³⁷ The flow continuity equation for blood vessels is obtained

$$\frac{\partial u}{\partial x}+\frac{1}{r}\frac{\partial }{\partial r}\left(\mathrm{rv}\right)=0$$

(6)

and the boundary conditions for motion function are as follows

$$\{\begin{array}{c}{u(x,r)|}_{r=a}=-U\hfill \\ \multicolumn{1}{c}{{v(x,r)|}_{r=a}=0}\\ \multicolumn{1}{c}{{u(x,r)|}_{x=\pm L}=0}\\ \multicolumn{1}{c}{{v(x,r)|}_{x=\pm L}=0}\end{array}$$

(7)

To solve the above problem, the pressure equation $p(x,r)$ and speed decomposition equation $\mathbf{V}(x,r)$ for plasma are assumed as follows

$$\{\begin{array}{c}p(x,r)=\frac{\partial p_0}{\partial x}+p_1(x,r)\hfill \\ \mathbf{V}(x,r)={\mathbf{V}}_0+{\mathbf{V}}_1\hfill \end{array}$$

(8)

According to the Navier–Stokes equation and omitting the inertia term, the simultaneous equation between equations (4) and (7) can be obtained

$$\begin{array}{c}\{\begin{array}{c}\multicolumn{1}{c}{-\mathbf{x}\frac{\partial p_0}{\partial x}+\eta {\nabla }^2{\mathbf{V}}_0=0}\\ \multicolumn{1}{c}{-\nabla p_1+\eta {\nabla }^2{\mathbf{V}}_1=0}\end{array}\hfill \\ \Rightarrow \{\begin{array}{c}-\frac{\partial p_0}{\partial x}+\eta {\nabla }^2{u}_0=0\hfill \\ {\nabla }^2{v}_0=0\hfill \end{array}\hfill \\ \Rightarrow \{\begin{array}{c}{u}_0=-\left(\frac{a^2\partial p_0}{4\eta \partial x}\right)(C-\frac{r^2}{a^2})\hfill \\ v_0=0\hfill \end{array}\hfill \end{array}$$

(9)

According to the flow’s continuity equation, the speed equation ${\mathbf{V}}_1(x,r)$ for plasma is assumed as follows

$$\begin{array}{c}{\mathbf{V}}_1=\nabla \times \nabla \times \left[\mathbf{x}f(x,r)\right]=\nabla \frac{\partial f}{\partial x}-\mathbf{x}{\nabla }^{2}f\hfill \\ =-\mathbf{x}\frac{1}{r}\frac{\partial }{\partial r}r\frac{\partial f}{\partial r}+\mathbf{r}\frac{{\partial }^{2}f}{\partial r\partial x}\hfill \end{array}$$

(10)

Substituting equation (10) into equation (9), the following equation can be obtained

$$\begin{array}{c}-\nabla p_1+\eta (\nabla {\nabla }^2\frac{\partial f}{\partial x}-\mathbf{x}{\nabla }^{4}f)=0\hfill \\ \Rightarrow -\frac{\partial p_1}{\partial x}\mathbf{x}-\frac{\partial p_1}{\partial r}\mathbf{r}+\eta \frac{\partial }{\partial x}{\nabla }^2\frac{\partial f}{\partial x}\mathbf{x}\hfill \\ +\eta \frac{\partial }{\partial r}{\nabla }^2\frac{\partial f}{\partial x}\mathbf{r}-\mathbf{x}{\nabla }^{4}f=0\hfill \\ \Rightarrow \{\begin{array}{c}-\frac{\partial p_1}{\partial r}+\eta \frac{\partial }{\partial x}{\nabla }^2\frac{\partial f}{\partial x}+{\nabla }^{4}f=0\hfill \\ -\frac{\partial p_1}{\partial r}+\eta \frac{\partial }{\partial r}{\nabla }^2\frac{\partial f}{\partial x}=0\hfill \end{array}\hfill \\ \Rightarrow \{\begin{array}{c}-p_1+\eta {\nabla }^2\frac{\partial f}{\partial x}=0\hfill \\ {\nabla }^{4}f=0\hfill \end{array}\hfill \end{array}$$

(11)

Obviously, according to the boundary condition (27) and boundary value of the biharmonic equation f(x, r), the constant $C=1/2$ of equation (9) can be derived, and the boundary conditions for ${\mathbf{V}}_1(x,r)$ are as follows

$$\{\begin{array}{c}{u_1(x,r)|}_{r=a}=0\hfill \\ \multicolumn{1}{c}{{v_1(x,r)|}_{r=a}=0}\\ \multicolumn{1}{c}{{u_1(x,r)|}_{x=\pm L}=-U[1-2\left(\frac{r^2}{a^2}\right)]}\\ \multicolumn{1}{c}{{v_1(x,r)|}_{x=\pm L}=0}\end{array}$$

(12)

Admittedly

$$\begin{array}{c}f(x,r)=a^{2}U\sum _{n=1}^{\infty }{A}_n\{[-(1+\lambda k_n\mathrm{cth}\lambda k_n)\frac{\mathrm{ch}{k}_n\left(\frac{x}{a}\right)}{\mathrm{ch}{k}_n\lambda }\hfill \\ +\frac{k_{n}x}{a}\frac{\mathrm{sh}\left(\frac{k_{n}x}{a}\right)}{\mathrm{ch}{k}_n\lambda }]\frac{J_0\left(\frac{k_{n}r}{a}\right)}{k_n^{2}J\left(k_n\right)}-\sum _{m=1}^{\infty }{B}_{\mathrm{nm}}\cos \left(\frac{m\pi }{\lambda }\frac{x}{a}\right)\hfill \\ [\frac{I_0\left(\frac{m\pi r}{\lambda a}\right)}{{\left(\frac{m\pi }{\lambda }\right)}^2{I}_0\left(\frac{m\pi }{\lambda }\right)}\hfill \\ -\frac{\frac{r}{a}{I}_1\left(\frac{m\pi r}{\lambda a}\right)}{\frac{m\pi }{\lambda }[2I_0\left(\frac{m\pi }{\lambda }\right)+\left(\frac{m\pi }{\lambda }\right)I_1\left(\frac{m\pi }{\lambda }\right)]}]\}\hfill \end{array}$$

(13)

Substituting equation (13) into equation (10), the following equation can be obtained

$$\{\begin{array}{c}{u}_1=U\sum _{n=1}^{\infty }{A}_n\{[-(1+\lambda k_n\mathrm{cth}\lambda k_n)\frac{\mathrm{ch}\left(\frac{k_{n}x}{a}\right)}{\mathrm{ch}{k}_n\lambda }+\frac{k_{n}x}{a}\frac{\mathrm{sh}\left(\frac{k_{n}x}{a}\right)}{\mathrm{ch}{k}_n\lambda }]\hfill \\ \multicolumn{1}{c}{\frac{J_0\left(\frac{k_{n}r}{a}\right)}{J\left(k_n\right)}-\sum _{m=1}^{\infty }{B}_{\mathrm{nm}}\cos \left(\frac{m\pi x}{\lambda a}\right)[\frac{I_0\left(\frac{m\pi r}{\lambda a}\right)}{I_0\left(\frac{m\pi }{\lambda }\right)}}\\ \multicolumn{1}{c}{-\frac{2J_0\left(\frac{m\pi r}{\lambda a}\right)+\left(\frac{m\pi r}{\lambda a}\right)I_1\left(\frac{m\pi r}{\lambda a}\right)}{2I_0(m\pi /\lambda )+(m\pi /\lambda )I_1(m\pi /\lambda )}]\}}\\ \multicolumn{1}{c}{v_1=U\sum _{n=1}^{\infty }{A}_n\{[\lambda k_n\mathrm{cth}\left(\lambda k_n\right)\frac{\mathrm{sh}\left(\frac{k_{n}x}{a}\right)}{\mathrm{ch}\left(\lambda k_n\right)}-\frac{k_{n}x}{a}\frac{\mathrm{ch}\left(\frac{k_{n}x}{a}\right)}{\mathrm{ch}\left(\lambda k_n\right)}]}\\ \multicolumn{1}{c}{\frac{J_1\left(\frac{k_{n}r}{a}\right)}{J_1\left(k_n\right)}-\sum _{m=1}^{\infty }{B}_{\mathrm{nm}}\sin \left(\frac{m\pi x}{\lambda a}\right)[\frac{I_1\left(\frac{m\pi r}{\lambda a}\right)}{I_0\left(\frac{m\pi }{\lambda }\right)}}\\ \multicolumn{1}{c}{-\frac{\left(\frac{m\pi r}{\lambda a}\right)I_0\left(\frac{m\pi r}{\lambda a}\right)}{2I_0\left(\frac{m\pi }{\lambda }\right)+\left(\frac{m\pi }{\lambda }\right)I_1\left(\frac{m\pi }{\lambda }\right)}]\}}\end{array}$$

(14)

Obviously, according to the boundary condition (12), the constant function $B_{\mathrm{nm}}$ is obtained as follows

$$\begin{array}{c}{B}_{\mathrm{nm}}=-\frac{4}{\lambda }\frac{k_n^2\mathrm{th}\lambda k_n}{{[k_n^2+{\left(\frac{m\pi }{\lambda }\right)}^2]}^2}\hfill \\ \frac{\left(\frac{m\pi }{\lambda }\right)\cos \left(m\pi \right)}{\frac{I_1\left(\frac{m\pi }{\lambda }\right)}{I_0\left(\frac{m\pi }{\lambda }\right)}-\frac{\left(\frac{m\pi }{\lambda }\right)I_0\left(\frac{m\pi }{\lambda }\right)}{2I_0\left(\frac{m\pi }{\lambda }\right)+\left(\frac{m\pi }{\lambda }\right)I_1\left(\frac{m\pi }{\lambda }\right)}}\hfill \end{array}$$

(15)

Obviously, group constant $A_n$ can be obtained according to the various values of r/a using configuration method. And to analyze the effects of the bolus flow during blood flow motion, the correction function of the Poiseuille flow equation $\beta (\lambda ,\zeta )$ is proposed as follows

$$\{\begin{array}{c}\frac{\partial \overline{p}}{\partial x}=-8\beta (\lambda ,{\lambda }_1){\eta }_a\left(\frac{U}{a^2}\right)\hfill \\ \multicolumn{1}{c}{\beta (\lambda ,{\lambda }_1)={\beta (\lambda ,\zeta )|}_{\zeta ={\lambda }_1}=-\frac{\partial \overline{p}}{\partial x}\frac{a^2}{8{\eta }_{a}U}}\\ \multicolumn{1}{c}{=-\left(\frac{{\overline{p}|}_{x=-L}-{\overline{p}|}_{x=L}}{2l}\right)\frac{a^2}{8{\eta }_{a}U}}\\ \multicolumn{1}{c}{\overline{p}=\eta \frac{U}{a}4\sum _{n=1}^{\infty }{A}_n\{\frac{\mathrm{sh}\left(\frac{k_{n}x}{a}\right)}{\mathrm{ch}\left(\lambda k_n\right)}}\\ \multicolumn{1}{c}{-\frac{4}{\lambda }{k}_n^2\mathrm{th}\left(\lambda k\right)\sum _{m=1}^{\infty }\frac{\left(\frac{m\pi }{\lambda }\right)\cos \left(m\pi \right)}{{[k_n^2+{\left(\frac{m\pi }{\lambda }\right)}^2]}^2}}\\ \multicolumn{1}{c}{\frac{\frac{I_1\left(\frac{m\pi }{\lambda }\right)}{2I_0\left(\frac{m\pi }{\lambda }\right)+\left(\frac{m\pi }{\lambda }\right)I_1\left(\frac{m\pi }{\lambda }\right)}\sin \left(\frac{m\pi x}{\lambda a}\right)}{\frac{I_1\left(\frac{m\pi }{\lambda }\right)}{I_0\left(\frac{m\pi }{\lambda }\right)}-\frac{\left(\frac{m\pi }{\lambda }\right)I_0\left(\frac{m\pi }{\lambda }\right)}{2I_0\left(\frac{m\pi }{\lambda }\right)+\left(\frac{m\pi }{\lambda }\right)I_1\left(\frac{m\pi }{\lambda }\right)}}\}}\end{array}$$

(16)

where $u$ is the plasma velocity at x-axis; $v$ is the plasma velocity at y-axis; $a$ is the radius of the capillaries; $U$ is the erythrocyte velocity; $2L$ is the distance between adjacent blood cells; x is the unit vector in the x-axis direction; $C$ is the coefficient to be determined; $A_n$ and $B_{\mathrm{nm}}$ are arbitrary constants, respectively; $J_i$ and $I_i$ are the ith order Bessel function and the correction Bessel function, respectively; $\lambda =L/a$ is the ratio of the distance between the blood cells and the radius of the blood vessel; $k_n$ is the nth root of the equation $J_0(x)=0$ ; m is an integer; and ${\lambda }_1$ is the ratio of the erythrocyte radius and of the blood vessel radius.

Blood flow model within capillary network

Obviously, the intercellular capillary network meets the requirements of curved capillary channels structure and quantity of fractal scale law for porous according to the physical similarity rule, the following equation can be obtained ³⁸

$$\{\begin{array}{c}N(L\ge r)={\left(\frac{r_{max}}{r}\right)}^{D_{f1}}\hfill \\ D_{f1}=2-\frac{\ln {\phi }_{f1}}{\ln \left(\frac{r_{min}}{r_{max}}\right)}\hfill \end{array}$$

(17)

If correction function $\beta (\lambda ,\zeta )$ is considered, the flow equation that considered pressure difference effect for the single twisted fractal capillary is obtained ³⁹

$$\{\begin{array}{c}\mathrm{dN}=-D_{f1}{r}_{max}^{D_{f1}}{r}^{-(D_{f1}-1)}\mathrm{dr}\hfill \\ q\left(r\right)=\beta (\lambda ,{\lambda }_1)\frac{\pi r^4(\mathrm{\Delta }p+\mathrm{\Delta }{p}_c)}{8{\eta }_a{D}_T{L}_0}{\left(\frac{L_0}{2r}\right)}^{1-D_T}\hfill \\ \mathrm{\Delta }{p}_c=\frac{\left(2\sigma \cos {\theta }_0\right)}{r}\hfill \end{array}$$

(18)

Certainly, the blood flow model within the capillary network through integral operation is given as follows

$$\begin{array}{c}{Q}_1=-\underset{r_{min}}{\overset{r_{max}}{\int }}q\left(r\right)\mathrm{dN}\left(r\right)\hfill \\ =\frac{\beta (\lambda ,{\lambda }_1)\pi D_{f1}{r}_{max}^{3+D_T}}{128{\eta }_a{L}_0^{D_T-1}(3+D_T-D_{f1})}\frac{\mathrm{\Delta }p+\mathrm{\Delta }{p}_c}{L_0}\hfill \end{array}$$

(19)

where D_f1 is the capillary dimension of the capillary tube; r and r_max are the pore radius and the maximum pore radius, respectively; D_T is the twisted fractal dimension of the capillary; Δp_c is the pressure difference between import and export for capillary; σ is the liquid surface tension; and θ₀ is the contact angle.

The blood flow model within arteriole–capillary vascular fractal network

The arteriole–arteriole dendritic fractal vascular network is considered equivalent to a simulation tree derivation of random fractal bifurcation network for porous medium. The flow equation of single-parent pipe for fractal bifurcation network traffic is obtained as follows ⁴⁰

$$q\left(d_0\right)=\frac{\pi d_0^4}{128\eta }\frac{1-\frac{\gamma }{N_0{\beta }^4}}{1-{\left(\frac{\gamma }{N_0{\beta }^4}\right)}^{m+1}}\frac{\mathrm{\Delta }p}{L_0}[1+\frac{\gamma (1-{\gamma }^m)}{1-\gamma }\cos \theta ]$$

(20)

Casson’s flow equation for blood with correction functions is obtained as follows ³⁶

$$\{\begin{array}{c}Q=F\left(\zeta \right)\left(\frac{\pi a^4}{8\eta }\right)\left(\frac{\mathrm{dp}}{\mathrm{dx}}\right)\hfill \\ \multicolumn{1}{c}{F\left(\zeta \right)=1-\left(\frac{16}{7}\right){\zeta }^{\frac{1}{2}}+\left(\frac{4}{3}\right)\zeta -\left(\frac{1}{21}\right){\zeta }^4}\\ \multicolumn{1}{c}{\zeta =\left(\frac{2{\tau }_c}{a}\right){\left(\frac{\mathrm{dp}}{\mathrm{dx}}\right)}^{-1}}\\ \multicolumn{1}{c}{{\tau }_c=-\left(\frac{r_c}{2}\right)\left(\frac{\mathrm{dp}}{\mathrm{dx}}\right)}\end{array}$$

(21)

Obviously, the blood flow mathematical model within arteriole–capillary vascular fractal network is obtained as follows

$$\{\begin{array}{c}{Q}_2=-\underset{{d_0}_{min}}{\overset{{d_0}_{max}}{\int }}{q}_2\left(d_0\right)\mathrm{dN}\left(d_0\right)\hfill \\ \multicolumn{1}{c}{=\left(\frac{\pi }{128\eta }\right)\left[\frac{D_f}{(4-D_f)}\right]d_{0max}^4[1-{\left(\frac{{d_0}_{min}}{{d_0}_{max}}\right)}^{4-D_f}]}\\ \multicolumn{1}{c}{\frac{(1-\frac{\gamma }{N_0{\beta }^4})}{1-{\left(\frac{\gamma }{N_0{\beta }^4}\right)}^{m+1}}[1+\frac{(\gamma -{\gamma }^{m+1})}{(1-\gamma )\cos \theta }]}\\ \multicolumn{1}{c}{[\frac{\mathrm{\Delta }p}{L_0}-\frac{16}{7}{\left(\frac{2{\tau }_c}{a}\right)}^{\frac{1}{2}}{\left(\frac{\mathrm{\Delta }p}{L_0}\right)}^{\frac{1}{2}}+\frac{4}{3}\left(\frac{2{\tau }_c}{a}\right)-\frac{1}{21}{\left(\frac{2{\tau }_c}{a}\right)}^4{\left(\frac{\mathrm{\Delta }p}{L_0}\right)}^{-3}]}\\ \multicolumn{1}{c}{D_f=2-\frac{\ln {\phi }_f}{\ln \left(\frac{{d_0}_{min}}{{d_0}_{max}}\right)}}\end{array}$$

(22)

where $F(\zeta )$ is a correction function of Casson’s fluid, ${\tau }_c$ is the blood yield stress, $r_c$ is the core flow radius, and $D_f$ is the number of fractal dimension for main pipe diameter within the arteriole–capillary vascular network.

Admittedly, the blood flow rate in the vascular network is given as follows

$$Q=Q_1+Q_2$$

(23)

Analysis and discussion

Simulation parameters determination

The physiologic simulation parameters of the vascular network based on physiology data of human body and the above theoretical derivation are shown in Table 1.

Table 1.

Main parameters of fractal vascular network.

No.	Main parameters	Value
1	Maxillary diameter of main pipeline within the arteriole–capillary vascular fractal network d0max	3 mm
2	Minimum diameter of main pipeline within the arteriole–capillary vascular fractal network d0min	10 μm
3	Maximum radius of equivalent capillary pathway within porous media material rmax	1.5 mm
4	Minimum radius of equivalent capillary pathway within porous media material rmin	15 μm
5	Fractal dimension of the twisting path for fractal capillary network DT	1.1
6	Plasma viscosity coefficient in the human blood circulation system η	1 mPa s
7	Length of equivalent fractal vessel network within porous media material L0	100 mm
8	Bifurcation diameter ratio within the arteriole–capillary vascular fractal network β	2−1/3
9	Branch length ratio within the arteriole–capillary vascular fractal network γ	2−1/3
10	The number of single branching within the arteriole–capillary vascular fractal network N0	2
11	Bifurcation grade no. within the arteriole–capillary vascular fractal network m	30
12	Vascular radius of Casson equivalent fluid a	10 mm
13	Porous ratio of arteriole–capillary vascular network φf	0.1
14	Inner surface contact angle for the blood vessel θ0	2π/3
15	Surface tension of plasma σ	0.09 N/m
16	Porous ratio in capillary network φf1	0.1

Characteristic analysis and discussion

In this article, the MATLAB software is used for theoretical analysis and image acquisition. Figures 3 and 4 show the relationships between blood flow and bifurcation grade m under different pressure and bifurcation angle conditions, respectively. It is seen from Figure 3 that the blood flow in the fractal vascular network decreases as the number of bifurcation stages increases, the speed of which approaches to 0 when the stage number is high enough. Under different pressure conditions, the blood flow increases following the rise of pressure, which performs significantly in low bifurcation progression. And the differences in blood flow under different pressures decrease with increasing the bifurcation progression. It is seen from Figure 4 that as the bifurcation angle of the fractal vascular network increases, the blood flow decreases gradually, where the differences of blood flow first increase and then gradually decrease with the increase of bifurcation progression.

Figure 3.

The relationship between blood flow and bifurcation grade in fractal vascular network m under different pressures.

Figure 4.

The relationship between blood flow and bifurcation grade in fractal vascular network m under different bifurcation angles.

Figures 5 and 6 show the characteristics of blood flow and flow resistance affected by different diameters of the arteriole–capillary vascular network D_f under different pressure and bifurcation angle conditions, respectively. In Figures 5 and 6, it is seen that with the increase of the diameters of the arteriole–capillary vascular network, the blood flow rises while the flow resistance reduces.

Figure 5.

The relationship between the blood flow and the fractal dimension of diameter of the arteriole–capillary vascular network D_f in the different pressures.

Figure 6.

The relationship between the blood flow resistance and the fractal dimension D_f in the different bifurcation angles.

Figures 7 and 8 show the characteristics of blood flow and flow resistance affected by different fractal dimensions of the diameter of capillary tubular diameter $D_{f1}$ under different pressure and bifurcation angle conditions, respectively. The slowly increasing curves in Figure 7 suggest that the blood flow is less affected by fractal dimension of the diameter. The curves in Figure 8 show that blood flow resistance reduces with increase in the fractal dimensions, which also suggests that the flow resistance is less affected by fractal dimensions except the condition at $\theta =\pi /2$ .

Figure 7.

The relationship between blood flow and fractal dimensions of the diameter of capillary tubular diameter $D_{f1}$ under different pressure conditions.

Figure 8.

The relationship between blood flow resistance and fractal dimensions of the diameter of capillary tubular diameter $D_{f1}$ under different bifurcation angles.

Figure 9 shows the relationships between blood flow and fractal dimensions of capillary blood vessels D_T under different pressure conditions, in which the slowly decreasing curves suggest the blood flow reduces as the fractal dimensions increase. Figure 10 shows the effect on blood flow caused by blood Casson yield stress, and the decreasing curve in the figure suggests that the non-Newtonian rheological properties of blood do have an effect on blood flow.

Figure 9.

The relationship between blood flow and fractal dimensions of capillary blood vessels D_T under different pressure conditions.

Figure 10.

The relationship between blood flow and blood Casson yield stress ${\tau }_c$ .

Figures 11 and 12 show the characteristics of blood flow and flow resistance affected by different ratios of RBC radius to capillary diameter ${\lambda }_1$ under different pressure and bifurcation angle conditions, respectively. The curves in Figure 11 suggest that the blood flow reduces slowly at ${\lambda }_1<0.5$ , while the flow decreases rapidly at ${\lambda }_1=0.6-0.9$ . The curves in Figure 12 show that the blood flow resistance first keeps stable and then grows distinctly as the ratio increases, which suggests that the effect on blood flow and flow resistance caused by the embolization effect of the RBC is significant when the radius of the RBC is close to that of the blood vessel.

Figure 11.

The relationship between blood flow and ratio of red blood cell radius to capillary diameter ${\lambda }_1$ under different pressure conditions.

Figure 12.

The relationship between blood flow resistance and ratio of erythrocyte radius to capillary diameter ${\lambda }_1$ under different bifurcation angles.

Figures 13 and 14 show the characteristics of blood flow and flow resistance affected by different ratios of the distance between adjacent RBCs to the radius of red cells $\lambda$ under different pressure and bifurcation angle conditions, based on the effect of plasma flow between RBCs, respectively. The slowly increasing curves in Figure 13 suggest that the blood flow is less affected by the ratio $\lambda$ . The curves in Figure 14 show that blood flow resistance reduces with increasing the ratio $\lambda$ , which also suggests that the flow resistance is less affected by the ratio except the condition at $\theta =\pi /2$ . As $\lambda$ increases, the reducing plasma flow effect results in reducing flow resistance.

Figure 13.

The relationship between blood flow and ratios of the distance between adjacent red blood cells to the radius of red cells $\lambda$ under different pressure conditions.

Figure 14.

The relationship between blood flow resistance and ratios of the distance between adjacent red blood cells to the radius of red cells $\lambda$ under different bifurcation angles.

Figure 15 shows the characteristics of blood flow resistance caused by different bifurcation progression $m$ under different ratios of the distance between adjacent RBCs to the radius of red cells $\lambda$ , which suggests that the flow resistance is less affected by ratio $\lambda$ at lower $m$ while the effect turns distinctly as m rises with higher $\lambda$ .

Figure 15.

The relationship between blood flow resistance and bifurcation grade m under different ratios of the distance between adjacent red blood cells to the radius of red cells.

Experimental verification

Figure 16(a)–(d) shows the contrastographic pictures of cardiac coronary artery network for the male healthy volunteer (56 years old) at the Bethune First Hospital of Jilin University on 4 May 2018. Tables 2 and 3 show the main physical parameters and physiological parameters of cardiovascular network system for volunteer, respectively.

Figure 16.

The contrastographic pictures of cardiac coronary artery network: (a) cardiac coronary angiography image with case A, (b) cardiac coronary angiography image with case B, (c) cardiac coronary angiography image with case C, and (d) cardiac coronary angiography image with case D.

Table 2.

The main physical parameters by measuring method.

Name	Case A	Case B	Case C	Case D
Inlet pressure (mm Hg)	124	122	131	116
Outlet pressure (mm Hg)	13	15	14	12
Inlet diameter (mm)	2	2	2	2
Outlet diameter (μm)	10	10	10	10
Average flow (mL/min)	107	74	87	85
Equivalent length (mm)	111	114	117	118

Table 3.

Main physiological parameters.

	Parameters	Value
1	The maximum diameter d0max	2 mm
2	The smallest diameter d0min	10 μm
3	Plasma viscosity η	1 mPa s
4	Coronary network bifurcated diameter ratio β	2−1/3
5	Coronary network bifurcation length ratio γ	2−1/3
6	The number of single branching N0	2
7	Bifurcation grade m	50
8	Radius of Casson equivalent fluid α	10 mm
9	Contact angle on the inner surface θ0	2π/3
10	Plasma surface tension σ	0.09 N/m

Table 4 shows the number of fractal dimension and the average flow of cardiac coronary artery network for the volunteer. It can be seen that the fractal dimension m compared to Figure 16 (Case A/B/C/D) is 1.67, 1.44, 1.52, and 1.58 using the Box-counting dimension method to the coronary artery network, respectively. Moreover, the relative error ξ increases with fractal dimension m decreasing, when relative error ξ is almost minimal value at m = m_max, and experimental test showed that the theoretical calculation and simulation analysis of blood circulation system of fractal vascular network were reasonable and applicable using experimental relative method because of the maximum relative error ξ is less than 10%, whenever fractal dimension m changes under different conditions.

Table 4.

Comparative data.

Name	Case A	Case B	Case C	Case D
Fractal dimension m	1.64	1.44	1.52	1.58
		Min
Theoretical value (mL/min)	106.407	80.769	93.850	87.949
Measurements (mL/min)	107	74	87	85
Relative error ξ	0.554%	9.45%	7.88%	3.47%
		Max

Conclusion

The following conclusions can be drawn:

1. The blood flow models within the fractal vascular network including the kinetic equation of blood bolus flow within capillaries network, capillary network, and arteriole–capillary vascular fractal network were established.

2. The characteristics of blood flow and flow resistance were analyzed, which caused by the bifurcation progression $m$ , the fractal dimension of the diameter of different blood vessels networks D_f and D_f1, the fractal dimensions of capillary blood vessels D_T, the ratio of RBC radius to capillary diameter λ₁, and the ratio of the distance between adjacent RBCs to the radius of red cells λ.

3. The clinical verification tests were accomplished, where the curve tendencies are very similar to those obtained by numerical simulations based on the theoretical models.

4. The experimental test showed that the theoretical calculation and simulation analysis of blood circulation system of fractal vascular network were reasonable and applicable using experimental relative method because the maximum relative error ξ is less than 10%, whenever fractal dimension m changes under different conditions.