Influence of Blood Compressibility on Pulse Wave Propagation Properties Based on Elastic Thin-Walled Tube Theory

Yicheng Lu; Bensen Li; Wenbo Gong; Xuehang Sun; Fuxing Miao

doi:10.5334/paah.214

Full Article

1 Introduction

Pulse wave refers to cyclical changes in pressure and volume in arteries during the cyclical contraction and relaxation of the heart; pulse wave causes cyclical expansion and contraction of the elastic vessel wall (Jing et al., 2019) and is rich in information about human physiology and pathology (Zhang, et al., 2019). Pulse wave propagation velocity is considered as an important indicator for assessing the stiffness of arteries and is an independent risk factor for a variety of cardiovascular diseases (Zhong, et al., 2018). Pulse wave has profound significance for human health, especially in the diagnosis, treatment, and prevention of cardiovascular diseases. At present, research on the clinical application of pulse wave by domestic and foreign scholars mostly focuses on pulse wave propagation speed, which is used to assess the cardiovascular status of patients with diseases, such as hypertension and coronary heart disease (Gajdova et al., 2017; Kim et al., 2014; Wang, et al., 2018; Yue, et al., 2012).

Scholars at home and abroad have done much work on pulse wave and blood compressibility. Robert et al. (Robert et al., 2019) uses an ideal model of the perivascular space to investigate how variation in the arterial pulse influences fluid flow; they demonstrated that Chiari patients without a syrinx maintained a significantly higher level of perivascular inflow over a physiologically likely range of pulse wave shapes. Ma et al. (Ma YJ et al., 2018) assumed that blood is an incompressible fluid and established a simple relationship between blood pressure and pulse wave velocity by designing in vitro experiments and theoretical derivation. Wang (Wang SH, et al., 2001) treated blood as a mixture of ultrafiltrate and concentrated protein and derived the linear relationship between compressibility and density of blood based on the compressibility of components and the additivity of density. Antanovskii (Antanovskii and Ramkissoon, 1997) studied the long wave creep propagation of a compressible viscous fluid in a finite pipe, which is an axisymmetric viscous flow in a long pipe driven by fluctuating pressure gradient and wall deformation. Khan (Khan and Ahmed, 2014) considered identifying the basic parameters of Newtonian fluid compressible viscous flow to determine the optimal set of compressible viscous fluid parameters; the developed corresponding model solutions matched the observed data. Considering the compressibility of blood, Marom (Marom G, et al., 2012) established a fluid structure coupling model of aortic valve and performed numerical analysis of hemodynamics in aortic valve models with different characteristics. Mnassri (Mnassri I, et al. 2017) proposed a detailed theoretical study on the 3D modal analysis of a compressible fluid in an elastic tube. The dispersion equations of bending, torsion, and longitudinal modes were derived using elastodynamics theory and unsteady coefficient equation. Luan et al. (Luan et al., 2021) presented a 3D microfluidic guided assembly method that bypasses these limitations to yield complex 3D microvascular structures from 2D precursors that exploit the full sophistication of 2D fabrication methods. Wang et al. (Wang, et al., 2016; Miao et al., 2020) put forward the research idea of holistic view of traditional Chinese medicine (TCM); by understanding the pulse wave system as the whole system in which life energy spreads in the blood in the form of wave and considering the compressibility of blood and the expansion and contraction of blood vessel wall, they proposed a series of models of fluid-solid coupling, which can provide richer information for the interpretation of pulse diagnosis in TCM. However, most of the current studies on blood flow are calculated as an incompressible viscous Newtonian fluid, and no in-depth analysis has reported on the influence of blood compressibility on pulse wave propagation characteristics.

Brachial artery is the most commonly used location for blood pressure measurement, and the pulse wave of this location has high research value. Brachial and ankle pulse wave velocity refers to the velocity of pulse wave propagation between brachial and ankle arteries. It is a new evaluation index of arteriosclerosis in recent years and is used to evaluate and predict mean arterial pressure, arterial stiffness, and cardiovascular and cerebrovascular diseases (Lowe, et al., 2019; Grillo, et al., 2020; Munakata, 2020). In the present work, the brachial artery was selected as the research object. According to the theory for thin-walled tube and fluid mechanics, a fluid-structure interaction and coupling model of the blood-vessel system was established and carried out by fully coupled algorithm using COMSOL MultiPhysics. The model was used to study the influence of blood compressibility on pulse wave propagation properties in the brachial artery with considering hemodynamic viscosity coefficients of blood and to explore the relationship between blood compressibility and pulse wave propagation law for health body.

2 Coupling analysis model of blood vascular system

An artery is an extremely complicated, multiphase, and dynamic system and is located in a living body. Apparently, establishing a full congruent model to simulate the real artery is difficult. Therefore, we only analyze the structure part of artery and ignore the physiological parameters and information of a healthy body in this paper. Blood vessel and blood are the major structure parts, which were studied from the idea of structure, stress, and strain based on the theories of mechanics. The modeling and analysis process is as follows.

2.1 Governing equations

In this section, the brachial artery of the human body is taken as the research object. A bit segment of the brachial artery can be seen as a straight tube. Thus, the geometric model of a thin-walled circular tube is established. The red and green parts are regarded as blood and blood vessel wall, respectively. L, D₀, and h₀ are used to represent vessel length, vessel diameter, and vessel wall thickness, respectively. Taking the center of the upper end face of the vessel as the origin and the axis of the vessel as the X axis, the O-XYZ spatial Cartesian coordinate system is established, as shown in Figure 1.

Schematic diagram of blood vessel model.

Blood is assumed as uniform Newtonian fluid, and vascular walls are isotropic linear elastomers. According to fluid mechanics theory, blood flow in the blood vessels of the human body follows the law of conservation of mass, momentum, and energy. The blood volume force vector, flow velocity vector, and fluid pressure are designated as F_b, V_b and p, and subscripts b and v refer to blood and blood vessel, respectively. The conservation equations of mass, momentum, and energy of blood flow are shown in Equations. (1)–(3).

1

\frac{\partial ρ_{b}}{\partial t} + \nabla \cdot (ρ_{b} v_{b}) = 0

2

ρ_{b} \frac{\partial v_{b}}{\partial t} + ρ_{b} v_{b} \cdot \nabla v_{b} = \nabla (- p I + τ_{b}) + F_{b}

3

ρ_{b} (\frac{\partial E}{\partial t} + v_{b} \cdot \nabla E) = ρ_{b} Q - \nabla \cdot q + ρ_{b} F_{b} \cdot v_{b} - \nabla \cdot (p v_{b}) + \nabla \cdot (τ_{b} \cdot v_{b})

where τ_b is the viscous stress tensor, and it can be written as Equation (4),

4

τ_{b} = μ [\nabla v_{b} + {(\nabla v_{b})}^{T}]

In Equations (1)–(4), I is the unit tensor of second order, ρ_b is the blood density, μ is the hemodynamic viscosity coefficient, t represents time, E is the energy of the blood system, Q is the heat transmitted to the blood system by heat radiation or internal heat source per unit mass, and q is the heat flux. In this model, the effect of gravity and heat transfer on blood flow are not considered, and the temperature remains constant, i.e.,

F_{b} = 0, Q = 0, q = 0 .

Thus, Equation (3) can be simplified as Equation (5),

5

ρ_{b} (\frac{\partial E}{\partial t} + v_{b} \cdot \nabla E) = - \nabla \cdot (p v_{b}) + \nabla \cdot (τ_{b} \cdot v_{b})

According to thin-walled tube theory, the circumferential stress σ_θ and circumferential strain ε_θ of thin-walled vessels under internal pressure P are as follows:

6

σ_{θ} = \frac{P D_{0}}{2 h_{0}}

7

ε_{θ} = \frac{D - D_{0}}{D_{0}}

2.2 Boundary conditions for bidirectional fluid–solid coupling analysis

For the human artery system, blood and blood vessel can be modeled as solid and fluid, respectively. The dilation and contraction of the walls of blood vessels affect the state of blood flow; by contrast, blood flow affect the dilation and contraction of the walls of blood vessels. Thus, a bidirectional fluid–solid interaction analysis of blood and vessel wall is established and carried out. The bidirectional fluid–solid coupling analysis is realized by iterative calculation of data exchange between blood and blood vessel wall.

The elastic blood vessel is a solid material as shown in Table 1, which happen deformation under the action of pulsating blood flow as well as the pressure on the blood vessel changes. The dilation and contraction of blood vessels also change the blood flow field, thus affecting the blood flow characteristics. For this bidirectional fluid–solid coupling analysis model, the blood surface and the inner surface of the blood vessel are established as fluid–solid coupling interface, through which the blood and the blood vessel transfer data for coupling analysis. According to continuum mechanics theory, on the fluid–solid coupling interface, blood and blood vessels should agree the boundary conditions of continuous displacement and continuous stress, as shown in Equation (8).

Table 1

Material parameters of blood and vessel wall in simulation calculation.

	BLOOD MATERIAL PARAMETERS			VESSEL WALL MATERIAL PARAMETERS
	VOLUME COMPRESSION MODULUS	INITIAL DENSITY	DYNAMIC COEFFICIENT OF VISCOSITY	Density	YOUNG’S MODULUS	POISSON’S RATIO
Example 1	0.5GPa	1050 kg/m³	0.004Pažs	1150 kg/m³	2.5MPa	0.45
Example 2	2.5GPa
Example 3	4.5GPa
Example 4	∞

8

{\begin{cases} u_{b} = u_{v} \\ σ_{b} = σ_{v} \end{cases}

In Equation (8), u_b and u_v represent the displacement vectors of blood and blood vessel, respectively, while σ_b and σ_v are the blood and blood vessel stresses, respectively.

In addition, for the proposed bidirectional fluid–solid coupling analysis model, firstly, blood flow model is selected as laminar flow type without slippage between it and the inner surface of the blood vessel wall. Secondly, it is not be considered to the constraint conditions at both ends of the blood vessel wall. Thirdly, the outer surface of artery system is simulated as a free wall. Therefore, a fluid–solid interaction (FSI) based finite element analysis model of brachial artery was established to determine the influence of blood compressibility in the brachial artery on pulse wave propagation properties under the effect of pulse pressure upstream of blood vessels. The advantages of this proposed model are that, for the computing elements, the stresses of blood vessel elements can transmit to blood parts, the stresses of blood parts computing elements can response to blood vessel at the same time and computing step. It can improve the accuracy of coupling effect between blood and blood vessel. The reliability of the computing results is increased than that of single direction coupling model.

2.3 Blood compressibility

In general, human blood is a kind of special fluid. Quantitative analysis of the compressibility of human blood is of great significance to further understand the propagation properties of pulse wave. The compressibility of blood is reflected in the change of volume or density when pressure applied to a certain amount of blood increases. The relative variable of volume or density is in a certain proportion to the change of pressure, which is called volume compression modulus K_b as defined in Equation (9),

9

K_{b} = - \frac{d p}{d V / V_{0}}

where p, V₀, and ρ₀ is the pressure, initial blood volume, and initial blood density, respectively. The smaller the volume compression modulus is, the higher the compressibility is. When K_b = ∞, blood is regarded as incompressible fluid. The blood control volume follows the conservation of mass under dp pressure,

10

ρ_{0} V_{0} = (ρ_{0} + d ρ) (V + d V)

By combining Equations (9) and (10) and omitting higher-order terms, that is,

11

\frac{d p}{d ρ} = \frac{K_{b}}{ρ_{0}}

Where, blood density in the simulation analysis can be given by Equation (12),

12

ρ_{b} =(ρ_{0} + \frac{p}{d p / d ρ}) {kg/m}^{3} = (ρ_{0} + \frac{p}{K_{b} / ρ_{0}}) {kg/m}^{3}

2.4 Simulation process and grid sensitivity analysis

By referring to the average geometric size of human brachial artery (Westerhof, et al., 1969; Stergiopulos, et al., 1992), D₀ = 5 mm, h₀ = 0.5 mm, and D₀/h₀ = 10 were taken as the geometric model of blood vessel in this paper; l = 1 m is regarded as the computing length to avoid the influence of premature reflection on pulse wave shape during pulse wave propagation. The blood was assumed to be an adiabatic viscous Newtonian fluid with initial density ρ₀ = 1050 kg/m³ and hemodynamic viscosity μ = 0.004 Pažs (Liu et al., 2016). The pressure pulse wave history loaded is obtained from clinical measurement data, which is the pressure pulse wave shape of a diabetic volunteer, as shown in Figure 2. Time-dependent Solver is applied in simulation by using COMSOL MultiPhysics.

Pressure pulse history loaded upstream of the blood vessel.

During simulation, two techniques are used: (i) a pressure pulse was applied at the upstream position of blood, and the relative pressure at the downstream position is controlled as 0 Pa for simulating the lengthening of blood vessels; and (ii) the loading waveform cycle was shortened by 10 times according to the dimensionless data analysis to improve the computing efficiency under the condition of limited hardware configuration. To ensure the high quality of the mesh, the blood and blood vessel walls adopt hexahedral mesh, with 786,000 hexahedral elements and 833,055 mesh vertices. Through grid independence verification, the number of grids can meet the requirements of computing accuracy.

3 Influence of blood compressibility on pulse wave propagation velocity

3.1 Comparison of compressible and incompressible under rigid models assumption

In this section, we first assume that blood vessel comply with rigid constitution relation, simplified as rigid model in the following. A rigid model of blood vessel is established for comparative analysis of the influence of blood compressibility in the elastic model to exclude the effect of the deformation of blood vessel wall. The effect of blood compressibility on pulse wave propagation velocity in the rigid model of blood vessel is studied, where the pulse wave propagation velocity in the rigid model of blood vessel is E_v = ∞, and the density of blood vessel is ρ_v = 1150 kg/m³.

Under the rigid model of blood vessels, as demonstrated in Figure 2, the loading time of the pressure pulse at the upstream position of blood vessels is reduced to 1/100 due to faster pulse wave propagation. Investigation of relative literature shows that the compression modulus of blood volume in healthy body is about 2.5 GPa (Urick, 1947; Laurent et al., 1994). That is, K_b = 2.5 GPa is used as the case of considering the compressibility of blood. When blood is incompressible, K_b = ∞ is used.

As shown in Figure 3, four different locations on the central axis of brachial artery were selected as monitoring points, with coordinates of (x₁,0,0), (x₂,0,0), (x₃,0,0), and (x₄,0,0), and marked as points A, B, C, and D, respectively. To avoid the influence of reflected waves and boundary conditions of downstream position of vessels, we let x₁ = l/20, x₂ = l/10, x₃ = 3l/20, and x₄ = l/5 and consider the propagation law of pressure pulse wave and axial velocity pulse wave, as shown in Figure 4. The peak value of pressure pulse wave and axial velocity pulse wave is attenuated with increasing propagation distance, and the attenuation amplitude is small when considering the compressibility of blood into the rigid model of blood vessels. The peak attenuation of pressure pulse wave from point A to point D is about 9.22%, and the peak attenuation of axial velocity pulse wave is 8.99%. The attenuation amplitudes of pressure pulse wave and axial velocity pulse wave are extremely comparable. The peak of each wave is utilized as a marker point for the propagation velocity of pulse waves, which can be computed using the distance between the monitoring locations and the time difference between the peak points. The propagation velocity of a pressure pulse wave is around 1754 m/s, while that of a velocity pulse wave is approximately 1636 m/s, with a difference of 6.73%. Obviously, pulse wave travel much faster than blood flow. Hence, the waveforms of pressure wave and velocity wave are very similar.

Propagational properties of (a) pressure, (b) axial velocity pulse wave at different monitoring points with a rigid model of blood vessels.

According to stress wave theory (WANG, 2005), the relationship between the variation of blood pressure dp and the variation of blood flow velocity dv follows Equation (13):

13

d p = ρ_{b} c d v

Where ρ_bc is the wave impedance and represents the proportional coefficient between the variation of blood pressure dp and the variation of blood flow velocity dv. In fact, for a rigid model of blood vessels with incompressible blood, the pressure at different sites changes with loading at time 0 and reaches the peak value at the same time, that is, the pulse wave velocity of blood vessels is c = ∞. When the blood fluid velocity increases to a certain value, it decays slowly, and the maximum velocity is about 0.0012 m/s.

3.2 Comparison of calculation results of compressible and incompressible elastic models

In this section, the blood vessel wall is regarded as an isotropic linear elastic material (Di Martino, et al., 2001) characterized by E_v = 2.5 MPa, ρ_v = 1150 kg/m³, and Poisson’s ratio υ = 0.45. Blood volume compression moduli 0.5, 2.5, and 4.5 GPa and ∞ were taken respectively to carry out fluid–solid coupling analysis model of blood and vessel wall and study the influence of blood compressibility on pulse wave propagation. Blood flow seems to pulse when pulsing pressure acts upstream of blood vessels. Figure 5 depicts the propagation law of pressure and axial velocity pulse waves at four monitoring stations along the central axis of the brachial artery.

Propagational properties of (a) pressure, (b) axial velocity pulse wave at different monitoring points with considering blood compressibility into an elastic model of blood vessels.

The pressure pulse wave displays apparent ascending and descending branches in its propagation, as seen in Figure 5(a). As the heart contracts and the left ventricle pumps blood into the artery, the blood pressure rises, causing a rising branch. Descending branches are formed during late ejection when pressure drops as the flow of blood into the artery is lower than the flow of blood out due to slow ejection. In this model, the reflection effect of pulse wave is not considered, and the waveform is almost the same as the loading waveform. The pressure pulse wave curves exhibit a high degree of coincidence at the same monitoring point for different blood volume compression moduli of 0.5, 2.5, and 4.5 GPa, and the pressure pulse wave curves almost entirely coincide. If the blood volume compression modulus is ∞, then the waveform at the ascending stage is obviously faster than that at the three other volume compression moduli, but no obvious difference occurs at the descending stage. In addition, the wave peak amplitude without considering the compressibility of blood is slightly higher than that in the case considering the compressibility of blood. Along the direction of blood flow, the peak attenuation of pressure pulse wave caused by blood compressibility increases with decreasing compression modulus of blood volume and increasing pulse wave propagation distance. For the case without considering the compressibility of blood, the peak value of which is taken as the benchmark, and the peak value of pressure wave at monitoring points A, B, C, and D when the compression modulus of blood volume is 0.5 GPa decreases by about 0.34%, 0.58%, 0.42%, and 0.79%, respectively.

Figure 5(b) shows that the axial velocity pulse wave has obvious characteristics of ascending branch and descending branch. At the same monitoring site, the axial velocity pulse wave curves of blood volume compression modulus of 0.5, 2.5, and 4.5 GPa almost coincide. In the ramping stage of the waveform, the waveform with the blood volume compression modulus ∞ is obviously faster than that of the cases with the three other volume compression moduli. The velocity pulse curve of the four volume compression moduli at monitoring points A, B, and C shows a high degree of coincidence in the descending branch stage, however the velocity pulse curve at monitoring point D decreases gradually when the blood volume compression modulus is ∞. The peak amplitude of the wave without considering the compressibility of blood is also slightly higher than that of the cases considering the compressibility of blood. For example, for the blood volume compression modulus 0.5 GPa, the peak value of the velocity wave at monitoring points A, B, C, and D falls by around 0.37%, 1.39%, 1.21%, and 0.96%, respectively.

During the propagation of pulse wave, the propagation rule of the radial displacement wave on the inner and outer surfaces of the blood vessel wall on the cross section of the four monitoring points with changing time is shown in Figure 6.

Propagational properties of radial displacement waves on inner and outer surfaces of vascular wall at different sections (a) x₁ = l/20, (b) x₂ = l/10, (c) x₃ = 3l/20, (d) x₄ = l/5 with considering blood compressibility into elastic model of blood vessels.

As shown in Figure 6, the waveform of radial displacement wave has obvious characteristics of ascending and descending branches. The ascending stage of the waveform represents the outward expansion of the vessel wall under the action of pulse pressure in the blood, and the peak represents the maximum expansion of the vessel wall. Due to blood pulse pressure, the descending branch stage of the waveform is the unloading stage, causing blood vessel distortion and contraction after the wave peak. The vessel wall continues to contract inward as the pulse wave travels further under the condition of negative radial displacement. The radial displacement of the inner surface is larger than that of the outer surface with the same compression modulus of blood volume.

The radial displacement curves of blood volume compression moduli of 0.5, 2.5, and 4.5 GPa are almost identical. At the cross section x₁ = l/20, the radial displacement of blood with compressibility considered is slightly larger than that of the cases without compressibility. The radial displacement is higher without considering the compressibility of blood at the x₂ = l/10 cross section, x₃ = 3l/20 cross section, and x₄ = l/5 cross section. The attenuation amplitudes of the radial displacement peak on the inner and outer sides of the vessel wall assuming blood compressibility are 6.53% and 6.39%, respectively, when the radial displacement wave propagates from the x₁ = l/20 cross section to the x₄ = l/5 cross section. The attenuation amplitudes of the radial displacement peak of the inner and outer surface of the vessel wall are 5.74% and 5.48%, respectively, without considering the compressibility of blood. Therefore, the deformation degree of blood vessel wall is high, and the attenuation amplitude of peak value is small when the compressibility of blood is not considered.

The elastic blood vessel wall dilates and contracts in response to blood flow and pulse pressure, and this dilation and contraction deformation spreads along the blood vessel wall as the pressure wave propagates through the blood. The axisymmetric problem may be reduced to a 1D problem in cylindrical dimensions, and the degree of deformation of the blood artery wall can be described by measuring the magnitude of the radial displacement. The model is enlarged 100 times at t = 0.04 s. Figure 7 depicts the displacement waves of the outer surface of the vascular wall after deformation with various hemodynamic viscosity coefficients. Based on the bidirectional fluid–solid coupling model, the greatest radial displacements of the vessel wall for four distinct blood volume compression moduli of 0.5, 2.5, and 4.5 GPa and ∞ are 21.3, 21.3, 21.3, and 21.4 μm, respectively. Blood compressibility has a minor effect on the maximum radial displacement of the vessel wall, with the maximum radial displacement without compressibility increasing by 0.47% when compared with the case with compressibility. The blood flow diagram at t = 0.01 s is illustrated in Figure 8, and the analysis object for different blood volume compression moduli is l/5 length upstream of blood vessels. At t = 0.01 s, the upper end face pressure of blood vessels is loading, and the associated blood pulse waveform is increasing.

The radial displacement wave of artery at t = 0.04s. (a) K_b = 0.5GPa, (b) K_b = 2.5GPa, (c) K_b = 4.5GPa, (d) K_b = ∞.

Blood streamline diagram at t = 0.01s. (a) K_b = 0.5GPa, (b) K_b = 2.5GPa, (c) K_b = 4.5GPa, (d) K_b = ∞.

When the blood volume compression modulus is 0.5, 2.5, or 4.5GPa, the blood flow diagrams are nearly identical, and the maximum flow velocity is 0.23 m/s. The maximum flow velocity is 0.22 m/s with the blood volume compression modulus ∞. Furthermore, the circulation distribution is obviously greater for the blood volume compression modulus ∞ than that of the blood volume compression modulus of 0.5, 2.5, or 4.5 GPa. The waveform increases quicker in the rising stage for the compression modulus of blood volume ∞, as seen in Figure 8.

4. Analysis and discussion

To further analyze the influence of hemodynamic viscosity coefficient on pulse wave propagation, the peak distribution of pressure pulse wave and velocity pulse wave at each monitoring point under four different compression modulus of blood volume is shown in Figure 9. The influence of blood compressibility on pulse wave propagation is relatively minimal and can be neglected when considering the compressibility of blood, and the volume compression modulus swings 2 GPa over and below the usual range of the human body. However, the influence of blood compressibility on pulse wave propagation is more obvious than that without considering blood compressibility. Taking the peak amplitude of pressure pulse wave as an example, the peak value of pressure wave in the case considering the compressibility of blood decreases by approximately 0.34%, 0.58%, 0.42%, and 0.79%, respectively, at monitoring points A, B, C, and D by comparison with the peak value of pressure pulse wave without considering the compressibility of blood.

The relationship between blood compressibility and the peak amplitude of (a) pressure, (b) axial velocity pulse wave at different monitoring points.

Furthermore, when the compressibility of blood is not taken into account, the attenuation amplitude of the peak value of the blood pressure pulse wave during the propagation of the pulse wave from monitoring point A to monitoring point D is about 5.93%. When the compressibility of blood is taken into account, the attenuation amplitude is about 6.36%. The influence law of blood compressibility on the peak value of axial velocity pulse wave is similar to that of pressure pulse wave. The attenuation amplitude of axial velocity pulse wave peak value is about 6.48% when the blood compressibility is not considered but is about 6.73% when the blood compressibility is considered. Thus, the compressibility of blood is one of the reasons for the attenuation of pulse wave peak value. Blood particles collide during compression and extrusion, resulting in energy consumption. The compressibility of blood is relatively small, so the influence on the attenuation of pulse wave peak value is relatively weak.

The attenuation amplitude of the peak value of the blood pressure pulse wave during the propagation of the pulse wave from monitoring point A to monitoring point D is about 5.93% when the compressibility of blood is not taken into account. Meanwhile, the attenuation amplitude is about 6.36% when the compressibility of blood is taken into account. Blood compressibility has a comparable effect on the peak value of an axial velocity pulse wave, similar to the peak value of a pressure pulse wave. The attenuation amplitude of the axial velocity pulse wave peak value is about 6.48% when blood compressibility is not taken into account but about 6.73% when blood compressibility is considered. One of the reasons for the attenuation of pulse wave peak value is the compressibility of blood. During compression and extrusion, blood particles collide, resulting in energy consumption. The ability to compress is nice.

For the propagation velocity of pulse wave, if the peak of each wave is used as a marker point, then the distance between the monitoring points and the time difference between the peak points can be calculated. Whether the compressibility of blood is taken into account or not, the average velocity values of the pressure pulse wave are 12.50 and 12.40 m/s, respectively, as shown in Figure 9. The average velocities of pressure pulse waves are about 13.76 and 12.93 m/s, respectively. Apparently, the velocity pulse wave propagates faster than that of the pressure pulse wave in the blood vascular system, and the velocity pulse wave reaches its peak at each monitoring point slightly later than the pressure pulse wave. Under the same condition, when considering the compressibility of blood, the average velocity levels of pressure and velocity pulse wave are 0.81% and 6.42% faster than that without considering the compressibility of blood.

5. Conclusions

The influence of blood compressibility on the propagation characteristics of pulse wave is investigated by fluid–solid interaction coupling computing model based on elastic assumption and elastic thin-walled tube theory. A blood-vessel interacted fluid–solid coupling finite element model is proposed. It is found that, the influence of blood compressibility on pulse wave propagation is very weak, even it can be ignored on considering the compressibility of blood and the volume compression modulus fluctuates 2 GPa above and below the normal range of human body. The effect of blood compressibility on the peak value of pulse wave is also weak, and the attenuation amplitude of pulse wave peak value is higher under considering blood compressibility than that of the cases of blood incompressibility. The difference of the results between compressible and incompressible models is less than 1%. The blood vessel is reduced as rigid wall, the propagation speed of the pulse wave depends only on the compressibility of blood. The pulse wave speed is remarkably large, and the blood flow speed is small. It can be seen the propagation characteristics of the pulse wave are agree with the stress wave propagation theory.

The influence of blood compressibility on pulse wave velocity is obvious. The average velocity amplitude levels of pressure and velocity pulse waves is respectively 0.81% and 6.42% faster than that of the models of incompressible blood. On the view of blood vessel deformation, the dilation of blood vessels is obvious when the pulse wave arrives any location. It is really that the velocity of pulse wave is much bigger than that of blood flow. These findings will be valuable to investigate deeply for evaluating pressure and velocity pulse waves, as well as in providing a waveform-based preliminary diagnosis of symptoms in human body.

Ethics and consent

This work has been reviewed and approved by Ethical Committee of Ningbo Traditional Chinese Medicine Hospital, the document No. AF/SQ-01/03.1. This study do not involve animal subjects.

Funding information

This work is supported by National Natural Science Foundation of China (Grant No. 11872218, U21A20502), Zhejiang Province Traditional Chinese Medicine Science and Technology Foundation (Grant No. 2022ZB317), the first batch of Medical and Health Brand Discipline Foundation in Ningbo (Grant No. PPXK2018-07).

Competing Interests

The authors have no competing interests to declare.