According to Womersley’s theory [13], the end effects can be neglected. The wall of blood vessel is very thin. The radial displacement is much smaller than the radius of the blood vessel. Therefore, the flow volume and average velocity can be computed by the following equations in a free elastic blood vessel:

The equation of flow volume:

Q = ∑ m = 0 ∞ π R 2 A n ρ C n [ 1 + χ n F 10 ( α n ) ] e i ω n ( t − x / C n ) (1)

The average velocity can be described by the following equation:

U = ∑ m = 0 ∞ A n ρ C n [ 1 + χ n F 10 ( α n ) ] e i ω n ( t − x / C n ) (2)

in which the nth amplitude of pressure is A n , F 10 ( α n ) = 2 J 1 ( α n j 3 / 2 ) α j 3 / 2 J 0 ( α n j 3 / 2 ) , J 0

and J 1 are the 0th and 1th Bessel functions, j 2 = − 1 , α n = R ω n / ν , ν is the dynamic coefficient of viscosity, ω n is the nth angular frequency of pressure pulsation, R is the radius of blood vessel, C n is the wave speed corresponding to the nth pressure gradient, R is the present radius. And

χ = C 2 ( 1 − μ 2 ) c 0 2 ( F 10 − 2 μ ) − 1 − 2 μ F 10 − 2 μ , c 0 = E h 2 ρ R 0 , c = E h 2 ρ ( 1 − μ 2 ) R

μ is the Poisson’s ratio of blood vessel, c₀ and c are the initial and present wave speeds, h is the wall thickness of blood vessel, ρ is the blood density, R₀ is the initial radius of blood vessel. Take c₀, R₀ and ρ to normalize the above equations.

The equation of blood flow is

Q C 0 R 0 2 = ∑ m = 0 ∞ π R 2 A n ρ C n C 0 R 0 2 [ 1 + χ n F 10 ( α n ) ] e i ω n ( t − x / C n ) (3)

The equation of average velocity is

U C 0 = ∑ m = 0 ∞ A n ρ C 0 C n [ 1 + χ n F 10 ( α n ) ] e i ω n ( t − x / C n ) (4)

It can be seen that the controlling dimensionless parameters for blood flow are π R 2 A n ρ C n C 0 R 0 2 and χ n F 10 . The dimensionless controlling parameters for average velocity are A n ρ C 0 C n and χ n F 10 . Generally, χ n F 10 can be taken as a constant,

so there is only one controlling parameter for either blood flow or average velocity and their difference is that the diameter of blood vessel should be considered in the blood flow. The controlling parameters for wave shape are ω n , C n , respectively. In simulation, we can change the value of C n (which is related with the modulus.) to change the controlling parameters.

Take the same model of blood vessel network as that built by Xu et al. [14] which is an effective model for studying the dynamic of blood flow (Figure 1). The parameters of cardiovascular system presented by Xiao et al. [11] were adopted in the simulation (Table 1). The variation of blood flow and pulse wave was simulated by changing the parameters of the main organs. A sine wave signal simulating the blood pressure is output from the heart first and then transports

into the whole system. The signal is shown in Figure 2.

The parameters used in simulation are as follows [15]:

h R 0 ~ 0.1 , ρ = 1000 kg/m³, The Poison’s ratio 0.1, f₁₀(1) = 0.5939, f₁₀(2) = 0.3925, f₁₀(3) = 0.3165, f₁₀(4) = 0.2731, f₁₀(5) = 0.2564

Take the corresponding wave velocity c₀ = 0.3535 m/s of the modulus e = 0.025 MPa, the density ρ = 1000 kg/m³ and the radius R₀ = 0.05 m of the 19th

segment blood vessel to normalize the problem, then the dimensionless parameters are p ¯ = p ρ c 0 2 , t ¯ = t c 0 R 0 , x ¯ = x R 0 .

When the modulus of each segment is 2.5 MPa, the variation tendency of blood pressure at each segment is similar. But the amplitude of variation is different due to the effects of wave and the resistance (Figure 3). The variations of blood pressure at ascending aorta and thoracoabdominal aorta are the most obvious. The variations are several times of the other segments. Observing the variation of blood pressure after changing the dimensionless modulus E / ( ρ c 0 2 ) at different segment at a designed time (0.18 s in this paper), it can be seen that the variation of blood pressure caused by changing the parameter E / ( ρ c 0 2 ) at thoracoabdominal aorta is the most largest, while the effects of variation of the parameter E / ( ρ c 0 2 ) of liver is small. The variation of the parameter at each segment causes the largest blood pressure response of the nearest segment (Figure 4).

Because the radii of the ascending aorta and the thoracoabdominal aorta are large, the variation of blood pressure due to the change of dimensionless modulus is large at the same blood flow. If the modulus changes the same degree, the difference of blood pressure can be over 10 times at the segment with small

changes of the parameter π R 2 A n ρ C n C 0 R 0 2 , accordingly the responses is large.

The variation of the parameters at the head only changes the pressure of the near segments. The following figure is the result at time of 0.18 s. The changes in the parameters of the gastric artery and capillary network have great influences on the blood pressure of other segments (Figure 5 and Figure 6). The reason is that the radius of this segment is small. The blood flows through a short distance before returning to the heart.

Obvious changes can be caused by the variation of the parameters of the hepatic portal vein, especially the blood pressures at the entrance of the hepatic portal vein and the output port of the heart (Figure 7 and Figure 8). Because the liver is connected with the main organs in the abdomen and its radius and length of the blood vessel are both large.

Generally, the liver and the blood network in the abdomen have the function of regulating the total blood flow in the vessel system, i.e., storing or releasing blood into the blood vessels according to the status of the body. Numerical results show that the regulation of the liver to the blood flow affects the blood pressure and distribution of every segment. If the blood stored in the liver is released into the network of blood vessels, the pressure in the network will increase. The variation increases with the rise of the modulus of the liver (Figure 9). The influences on the heart and the segments connected with the liver are large. This may be the reason that the blood pressure will increase when one is angry when the liver will shrink.

Similarly, when the blood is released from the vessel network in the abdomen, the pressure increases in the whole blood vessel network. The effects of modulus are smaller than the blood release (see 3, 4 conditions in Figure 10). However, the relative changes are smaller than that of the variation of the blood released from the liver.

A little change in the blood flow at segment 18 (portal vein) can cause obvious variations in the blood pressure in the cardiovascular system. The variation in the parameters of the whole cardiovascular system is obviously larger than that in one segment. For example, when the modulus of the whole cardiovascular system increases 10 times, the maximum pressure increases 12 times, while the maximum pressure increases 5 times when the modulus of the liver’s capillary network increases 10 times (Figure 11 and Figure 12).