**4. Application to the fluid experiment of a computational model: In case of periodic pulsatile flow**

#### **4.1. Method**

#### *4.1.1. Experimental model*

#### *Experimental apparatus*

Figure 11 shows a schema of the experimental apparatus used in this study in order to generate the periodic pressure pulse wave. A silicone tube with an inner diameter of 9.0 mm, 0.5 mm thick, and a 2.04 m long (*L*) was connected by three pressure sensors (Nihon Koden, DX-200), one each 2 cm from both ends of the tube and one in the center. These sensors were connected at intervals of 1.0 m. The experiment tube and each pressure sensor were connected by a narrow connecting tube made of silicone with an inner diameter of 3.0 mm, 0.5 mm thick, and 3 cm long. The frequency characteristics of each pressure sensor system, including these connecting tubes were sufficiently higher than those of the frequency component of the periodic pulsatile flow rate described later; hence, the sensor system can accurately measure the internal pressure of the experiment tube. A piston pump was connected to the proximal end of the experiment tube through the flow sensor (Nihon Koden, MFV-1100) and a tank was connected to the distal end of the tube through the terminal resistance. The experiment tube was placed on a metal plate without longitudinal tension, allowing the silicone tube to change shape freely. Figure 11(b) is a scheme of the piston pump connected to the proximal end of the experiment tube. The piston pump was driven by a computer-controlled stepping motor and was capable of generating various waveforms with various flow rates. Actual flow rate was measured at the proximal end of the tube by the flow sensor, as shown in Fig. 11.

The terminal resistance installed at the distal end of the tube simulated the peripheral arterioles *in vivo*. This terminal resistance, shown in Fig. 12, was filled with a bundle of around 200 thin stainless capillary tubes (26 G: with an outer diameter of 0.51 mm, inner diameter of 0.3 mm, and 50 mm long) in a rigid brass tube with an inner diameter of 8.0 mm.

**Figure 12.** Terminal resistance

#### *Experimental method*

202 Viscoelasticity – From Theory to Biological Applications

change independently.

**of periodic pulsatile flow** 

the tube by the flow sensor, as shown in Fig. 11.

Resistance

**P2 P3**

Pressure sensors

2.04m silicone tube Flow sensor

**Figure 11.** Experimental apparatus

**P1**

Piston pump

*4.1.1. Experimental model* 

*Experimental apparatus* 

**4.1. Method** 

analysis of the viscoelastic tube even their deformation compliance and viscoelasticity

2. The relationship between the deformation compliance and viscoelasticity, which depends on the internal pressure, can be analyzed using this numerical model by

**4. Application to the fluid experiment of a computational model: In case** 

Figure 11 shows a schema of the experimental apparatus used in this study in order to generate the periodic pressure pulse wave. A silicone tube with an inner diameter of 9.0 mm, 0.5 mm thick, and a 2.04 m long (*L*) was connected by three pressure sensors (Nihon Koden, DX-200), one each 2 cm from both ends of the tube and one in the center. These sensors were connected at intervals of 1.0 m. The experiment tube and each pressure sensor were connected by a narrow connecting tube made of silicone with an inner diameter of 3.0 mm, 0.5 mm thick, and 3 cm long. The frequency characteristics of each pressure sensor system, including these connecting tubes were sufficiently higher than those of the frequency component of the periodic pulsatile flow rate described later; hence, the sensor system can accurately measure the internal pressure of the experiment tube. A piston pump was connected to the proximal end of the experiment tube through the flow sensor (Nihon Koden, MFV-1100) and a tank was connected to the distal end of the tube through the terminal resistance. The experiment tube was placed on a metal plate without longitudinal tension, allowing the silicone tube to change shape freely. Figure 11(b) is a scheme of the piston pump connected to the proximal end of the experiment tube. The piston pump was driven by a computer-controlled stepping motor and was capable of generating various waveforms with various flow rates. Actual flow rate was measured at the proximal end of

Water tank

(a) Conceptual scheme of experiment (b) Cross-section of piston pump

Stepping Motor

Piston pump Flow sensor

(Controlled by PC) Displacement sensor

Pressure sensor (P1)

appropriately choosing the value of the viscoelastic parameter.

The experiment tube and piston pump were filled with water. Baseline internal pressure was set by adjusting the water head of the tank. Experimental trials were performed at increasingly higher baseline internal pressures of 5.6, 8.4 and 11.2 kPa (40, 60, 80 mmHg) respectively, because the deformation compliance and viscoelasticity of the experiment tube changed depending on the baseline internal pressure of the tube, as described later. The trials were generated at more than 60-minute intervals after changing the internal pressure, because 30-60 minutes were necessary for the viscoelasticity of the experiment tube to return to a steady relaxation state. Thus, the effect of a change in tube viscoelasticity on pulse wave propagation was examined.

Figure 13 shows the waveform of the flow rate (*Q(t)*) by the piston pump and the frequency component of the flow rate. In order to simulate periodic pulsation from the heart, we generated a pulsatile flow rate with 0.4-second ejection time (*ti*) and about 2.8 ml/s maximum flow rate (*Qm*) eight times in a 1.0-second period (*tp*) at the proximal end of the experiment tube. In this case, the laminar flow condition was satisfied, as Reynold's number and Womersley's number were 400 and 11.3, respectively. By the movement of the piston pump, the filling fluid from the piston pump flowed into the tank through the terminal resistance. The tank has a hole at the height of the baseline internal pressure head; thereby, the water head pressure always equaled the baseline internal pressure during each experimental trial. In each trial, signals from the three pressure sensors and the flow sensor were recorded for 10 seconds by a PC at a sampling rate of 1 kHz. The experiment was performed several times under the same conditions, and it was confirmed that reproducibility was high.

Numerical Simulation Model with Viscoelasticity of Arterial Wall 205

(16)

(17)

Viscoelasticity of a strip of the silicone tube was measured by a dynamic viscoelasticity measuring device. The measured viscoelastic properties are same as shown in section 3.1.2. This result indicates that both the dynamic modulus viscoelasticity ratio and loss tangent, tends to increase gradually with the increase in frequency. This trend is similar to the viscoelasticity property of blood vessels (Learoyd et al., 1966), and the dynamic modulus viscoelasticity ratio of the blood vessels is similar to or slightly more than that of the experiment tube. The procedure used to decide the values of the viscoelastic tube parameter

The viscous resistance of flow in the capillary tube of terminal resistance can be approximated by Poiseuille flow, as the terminal resistance occurred from rigid pipe flow and its inside diameter was sufficiently small. The ratio of the flow rate in the capillary tube to the pressure difference between both ends of the capillary tube (terminal resistance) seemed to be constant regardless of the flow rate; therefore, terminal resistance *RT* (1.80 kPa/ml/s) was determined from the pressure difference and the flow rate obtained by moving the piston pump at a

As initial conditions, baseline internal pressure and a cross-section which corresponded to the baseline internal pressure were used. It was assumed that there was no flow in the initial

> ( ,0) ( ,0) ( ,0) 0

 

*px p A x A Q x*

The flow rate measured by a flow sensor was used as the input boundary condition, and terminal resistance *RT* was used as the output boundary condition. Hence, the boundary

> (0, ) ( ) ( ,) ( ,)

*pLt QLt <sup>R</sup>*

We used the same numerical simulation methods as in a previous paper (Kitawaki & Shimizu, 2006), including calculation schemes, computational algorithms, and fast calculation methods. The basic equations of the numerical simulation were digitized using a staggered grid system in space. For the calculation, Jameson-Baker's 4th order 4-step method as a time differential and 4th order central differential as space with numerical friction was used (Jameson & Baker, 1983). The flow rate and cross-section of the next time step were obtained from equations of continuity and momentum conservation, and then pressure was calculated from the tube law as a function of time. Convolution integrals

*Q t Qt*

 

0 0

*T*

constant speed so as to generate a constant flow (about 1.1 ml/s).

state inside the tube. Hence, the initial conditions can be shown as follows,

are same as shown in section 3.1.2.

*Numerical simulation method* 

conditions can be shown as follows,

*Terminal resistance* 

**Figure 13.** Movement of piston pump

#### *4.1.2. Numerical simulation*

#### *Mechanical properties of experiment tube*

The static tube law and compliance of the experiment tube were obtained from the relationship between the increasing volume of the tube and the internal pressure while performing one slow stroke of the piston pump over a period of about 2.5 hours. This relationship between the static cross-section and the pressure, as well as the calculated tube deformation compliance are shown in Fig. 14. The static tube law was not exactly linear at any pressure. The measurement results in a range of pressures in each experimental trial were approximated to a linear model, because the nonlinear effects of tube law were not incorporated to the calculation in the present numerical simulation model. Tube deformation compliance at each baseline internal pressure is shown by squares in Fig. 14. At over 5 kPa, the tube deformation compliance increased with the increase in the internal pressure, showing countertrend to the tube law of blood vessels (Hayashi et al., 1980).

**Figure 14.** Static relation between pressure and cross-section of the silicone tube

Viscoelasticity of a strip of the silicone tube was measured by a dynamic viscoelasticity measuring device. The measured viscoelastic properties are same as shown in section 3.1.2. This result indicates that both the dynamic modulus viscoelasticity ratio and loss tangent, tends to increase gradually with the increase in frequency. This trend is similar to the viscoelasticity property of blood vessels (Learoyd et al., 1966), and the dynamic modulus viscoelasticity ratio of the blood vessels is similar to or slightly more than that of the experiment tube. The procedure used to decide the values of the viscoelastic tube parameter are same as shown in section 3.1.2.

#### *Terminal resistance*

204 Viscoelasticity – From Theory to Biological Applications

**Figure 13.** Movement of piston pump

Time (s)

2 2 2

{8( / ) } (mod( , ) / 4) {1 2(2 / 1) } ( / 4 mod( , ) 3 / 4) ( ) {8( / 1) } (3 / 4 mod( , ) ) 0 ( mod( , ))

*i p*

1.0 1.4 -0.2

*t tt*

*m i p i m i i p i m i i p i*

*Q tt tt t Q tt t tt t Q t Q tt t tt t*

 

*Mechanical properties of experiment tube* 

62

62.5

63

63.5

Cross section of the tube (mm 2

)

64

64.5

65

The static tube law and compliance of the experiment tube were obtained from the relationship between the increasing volume of the tube and the internal pressure while performing one slow stroke of the piston pump over a period of about 2.5 hours. This relationship between the static cross-section and the pressure, as well as the calculated tube deformation compliance are shown in Fig. 14. The static tube law was not exactly linear at any pressure. The measurement results in a range of pressures in each experimental trial were approximated to a linear model, because the nonlinear effects of tube law were not incorporated to the calculation in the present numerical simulation model. Tube deformation compliance at each baseline internal pressure is shown by squares in Fig. 14. At over 5 kPa, the tube deformation compliance increased with the increase in the internal pressure, showing countertrend to the tube law of blood vessels (Hayashi et al., 1980).

(a) Flow rate profile (b) Normalized Fourier component

0 0.2 0.4 0.6 0.8 1

Normalized Fourier component

**Figure 14.** Static relation between pressure and cross-section of the silicone tube

11.2 8.4 5.6

Initial pressure (kPa)

0 2 4 6 8 10 12 14 Internal Pressure (kPa)

0.2063 0.1922 0.1804

Compliance (mm2/kPa)

0.16

0.18

0.2

0.22

Compliance(right)

Cross section (left)

Compliance (mm2/kPa)

0.24

0.26

0.28

<sup>0</sup> 5 10 15 20 Frequency (Hz)

*4.1.2. Numerical simulation* 

0.1 0.3 0.2 0.4

Flow (ml/s)

2.8 1.4

> The viscous resistance of flow in the capillary tube of terminal resistance can be approximated by Poiseuille flow, as the terminal resistance occurred from rigid pipe flow and its inside diameter was sufficiently small. The ratio of the flow rate in the capillary tube to the pressure difference between both ends of the capillary tube (terminal resistance) seemed to be constant regardless of the flow rate; therefore, terminal resistance *RT* (1.80 kPa/ml/s) was determined from the pressure difference and the flow rate obtained by moving the piston pump at a constant speed so as to generate a constant flow (about 1.1 ml/s).

#### *Numerical simulation method*

As initial conditions, baseline internal pressure and a cross-section which corresponded to the baseline internal pressure were used. It was assumed that there was no flow in the initial state inside the tube. Hence, the initial conditions can be shown as follows,

$$\begin{cases} p(\mathbf{x},0) = p\_0 \\ A(\mathbf{x},0) = A\_0 \\ Q(\mathbf{x},0) = 0 \end{cases} \tag{16}$$

The flow rate measured by a flow sensor was used as the input boundary condition, and terminal resistance *RT* was used as the output boundary condition. Hence, the boundary conditions can be shown as follows,

$$\begin{cases} Q(0,t) = Q(t) \\ Q(L,t) = \frac{p(L,t)}{R\_T} \end{cases} \tag{17}$$

We used the same numerical simulation methods as in a previous paper (Kitawaki & Shimizu, 2006), including calculation schemes, computational algorithms, and fast calculation methods. The basic equations of the numerical simulation were digitized using a staggered grid system in space. For the calculation, Jameson-Baker's 4th order 4-step method as a time differential and 4th order central differential as space with numerical friction was used (Jameson & Baker, 1983). The flow rate and cross-section of the next time step were obtained from equations of continuity and momentum conservation, and then pressure was calculated from the tube law as a function of time. Convolution integrals appear in the viscous resistance term in Eq. (3) and in the viscoelastic term in Eq. (7). A highspeed calculation method was applied, since calculation of the convolution integrals requires significant computer memory to hold past velocity and cross-section values and requires much computational time.

Numerical Simulation Model with Viscoelasticity of Arterial Wall 207


0

1

2

3

4

5

6

Pressure 0.02m Pressure 1.02m Pressure 2.02m Flow rate 系列4

7

1.0s

2cm3/s

Flow rate ml/s

**Figure 15.** Time profiles of pressure waves at three positions in the tube and flow volume into the tube.

7 7.2 7.4 7.6 7.8 8 8.2 8.4

In the calculated results using the elastic model and the Voigt model, the calculated maximum pressure is lower than that of the experimental results, and the calculated minimum pressure is higher than that of the experimental results; therefore, the difference between the calculated results and experimental results is large. Also, it is shown that the difference between the elastic model and the Voigt model is small, indicating that in the periodic change of waveforms, the Voigt model is not sufficiently effective. In addition, the calculated results did not agree with the experimental results in the Voigt model, because the effect is too small even when the value of the relaxation time parameter is varied as

shown in Fig. 16(b).


(b) Enlarged view in a period



0

0.5

1

1.5

2

2.5

(a) All experimental result (10 seconds)

2kPa

Pressure kPa

Flow rate

2.02m

1.02m

Pressures

0.02m

Time step Δt and grid interval Δx were set at 0.5 ms and 0.04 m, respectively. The Courant number was 0.325 because the propagation velocity of the pressure wave was about 26 m/s at maximum. As a result, the CFL condition (numerical stability condition) was satisfied.
