**3. Application to the fluid experiment of a computational model: In case of a difference in viscoelasticity**

#### **3.1. Method**

#### *3.1.1. Experimental model*

#### *Experimental apparatus*

Figure 1 shows a schema of the experimental apparatus used in this study. The experiment tube consisted of two silicone tubes which was different form previous experiments (Kitawaki et al., 2003; Kitawaki & Shimizu 2005) with an inner diameter of 9 mm, a thickness of 0.5 mm, and a length of 1.45 m, connected by a rigid brass tube 5 cm in length and 9 mm in inner diameter. A piston pump was connected to one end of the tube via a 5-cm long brass tube, and a water tank with a valve was connected to the other end via another 5-cm long brass tube. The three brass tubes were fixed to a metal plate upon which the complete tube rested without longitudinal tension, allowing the silicone tubes to change shape freely. A pressure sensor (Nihon Koden, DX-100) was connected to the middle of each brass tube.

**Figure 1.** Experimental apparatus

formulations is as follows,

*i*

number *k* was 10, because *Δτ* was calculated to be 1.2x10-5.

*i*

**of a difference in viscoelasticity** 

**3.1. Method** 

*3.1.1. Experimental model* 

*Experimental apparatus* 

0

(12)

(13)

(15)

*i*

{ ( ) ( )}

*k*

() 4 2 () ()

() 0 ( 0)

( ) () ( 0)

We need to determine the value of term number *k* in the Eq. (12) from the value of *Δτ* with consideration of approximation accuracy. In present experimental condition, the term

The weight function of the convolution integral in the tube law of Eq. (7) can be expressed as a summation of the exponential function. Therefore, the tube law of the recursive

0 0

*pt p A A z Cs*

*z t t*

*fe At t At*

It is possible to determine the term number *n* in Eq. (14) from the viscoelastic characteristics.

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

Figure 1 shows a schema of the experimental apparatus used in this study. The experiment tube consisted of two silicone tubes which was different form previous experiments (Kitawaki et al., 2003; Kitawaki & Shimizu 2005) with an inner diameter of 9 mm, a thickness of 0.5 mm, and a length of 1.45 m, connected by a rigid brass tube 5 cm in length and 9 mm in inner diameter. A piston pump was connected to one end of the tube via a 5-cm long brass tube, and a water tank with a valve was connected to the other end via another 5-cm long brass tube. The three brass tubes were fixed to a metal plate upon which the complete tube rested without longitudinal tension, allowing the silicone tubes to change shape freely. A pressure sensor (Nihon Koden, DX-100) was connected to the middle of each brass tube.

() 0 ( 0)

( ) () ( 0) { ( ) ( )}

<sup>1</sup> () ( )

/ /2

*t*

*i i t i*

*zt t e zt*

*i i*

1

(14)

*t*

*n i i*

*t i*

*y t t*

*yt t e yt <sup>t</sup> me V t t V t*

( /2)

where *τ*=*νt/R2* and *Δτ=νΔt/R2* are normalized time and time step, respectively.

*i i*

*n i i n i*

 

*Ft Vt y t* 

> Figure 2(a) is a schema of the piston pump. The piston pump was driven by a computercontrolled stepping motor, and was capable of generating various waveforms with various flow volumes. The piston cylinder receives the backpressure from the water inside the tube. Therefore, displacement of the inner cylinder was measured by a laser displacement sensor (Keyence, LK-030).

**Figure 2.** Piston pump

#### *Experimental conditions*

The tube, piston pump, and water tank were filled with water. Baseline internal pressures were set by adjusting the water head of the tank. The piston pump generated a single impulse. The flow volume at the inlet of the experiment tube was determined as shown in Fig. 2(b); when the impulse time *ti* was set to 0.1 s, and the maximum flow volume *Qm* was 5.0 ml/s. As seen in Fig. 2(c) which shows the Fourier transformation of the flow volume when a single impulse was generated, the highest frequency component of the flow volume was 30 Hz. Signals from the three pressure sensors and the displacement sensor were recorded for 10 seconds by a PC at a sampling rate of 1 kHz. The trials, that is, generation of a single impulse by the pump, were performed at increasingly higher baseline pressures of 2.5, 5.0, 7.5 and 10.0 kPa at 90-minute intervals. The trials were generated more than 70 minutes after changing the baseline pressures because 30 minutes were necessary for the viscoelasticity to return to a steady state. The baseline pressures were established by opening the valve, and the valve was closed just before the start of each trial.

#### *3.1.2. Tube wall properties*

#### *Static tube law*

Tube wall properties were determined for recent silicone tube. The static tube law and compliance of the silicone tube was obtained from the relationship between the volume of the piston pump and the internal pressure, as shown in Fig. 3, while performing one stroke of the piston pump over a period of about 30 minutes. The compliances, reference pressure, and reference cross-section of local tube deformation to the numerical calculation of each trial were determined from the collinear approximation of the pressure range of the experimental conditions. For example, a range of 2.5 - 4.3 kPa was used when the baseline pressure was 2.5 kPa, because the pressure pulse amplitude of a single impulse was about 1.8 kPa. These local compliances, assumed constant under the flow experimental conditions, are shown by squares in Fig. 3. These local compliances were used in all the wall viscoelastic models.

Numerical Simulation Model with Viscoelasticity of Arterial Wall 195

frequency(Hz)

0.1 1.0 10.0

**Figure 4.** Dynamic viscoelastic modules of silicone tube.(a) Ratio of dynamic and static modulus of

(b) <sup>l</sup> (a) Ratio of dynamic and static modulus of viscoelasticity (b) loss tangent

The procedure used to decide the value of the viscoelastic tube parameter for the experimental condition change (Kitawaki & Shimizu, 2006) is described below. (1) First, the relaxation time parameter *τi* was determined so that it could cover the frequency range 0.1- 30 Hz and it could increase between 0.01 and 200 Hz at regular intervals on a logarithmic scale. The term number *n* in Eq. (7) was set to 7 to keep the term number to a minimum and to show that the viscoelastic properties tended to increase smoothly. (2) The values of the dynamic viscoelasticity parameter *fi* were then determined to keep the difference between the experimental results and calculation results to a minimum from the low frequency term (*i*=1) to the high frequency term (*i*=7), using the different effect of each term on the numerical calculation results. (3) By changing the experimental conditions, the viscoelastic parameter was fitted by fixing the relaxation time parameter and changing only the dynamic

0

0.02 0.04 0.06 0.08 0.1

Loss tangent tan( )

Relaxation time *τV* of the Voigt model was determined as 2.5 ms from measurements of the delay time of the displacement of the outer diameter for the internal pressure change of the

The basic equations of the numerical calculation 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 with numerical friction as space was used (Jameson & Baker, 1983). Flow volume and cross-section of the next time step were obtained from an equation of continuity and momentum conservation, and then the pressure was calculated from the tube law as a function of time (Kitawaki et al., 2003). Convolution integrals appear in the viscous resistance term in Eq. (3) and in the viscoelastic term in Eq. (7). Since calculation of the convolution integrals requires a lot of computer memory to hold past velocity and cross-section values, and takes a lot of computational time, a high-speed calculation method for the viscous resistance term of a rigid tube (Kagawa et al., 1983) was

viscoelasticity

frequency(Hz) 0.1 1.0 10.0

1

1.1

1.2

Normalized storage modulus

1.3

1.4

viscoelasticity parameter.

*3.1.3. Numerical calculation* 

applied, as shown in *section 2.4*.

The following conditions were used in the calculation.

silicon tube.

**Figure 3.** Static relation between pressure and cross section of the silicone tube

#### *Dynamic viscoelastic property of the tube*

Dynamic viscoelasticity of a strip of the silicone tube was measured by a device (TA Instruments DMA2980). The measurement conditions were determined as initial strain of 0.18% and amplitude of 0.145%. These values correspond to a baseline pressure of 0.28 kPa and a wave amplitude of 1.8 kPa, respectively. The measured viscoelastic properties are in Fig. 4. This figure shows the dynamic modules of viscoelasticity normalized against the static modules in Fig. 4(a) and the loss tangent in Fig. 4(b), respectively. These results mean that, in the measurement range 0.1~30 Hz, the dynamic viscoelasticity property of the silicon tube, both the real parts and loss tangent, tends to increase gradually with increase in the frequency.

**Figure 4.** Dynamic viscoelastic modules of silicone tube.(a) Ratio of dynamic and static modulus of viscoelasticity

The procedure used to decide the value of the viscoelastic tube parameter for the experimental condition change (Kitawaki & Shimizu, 2006) is described below. (1) First, the relaxation time parameter *τi* was determined so that it could cover the frequency range 0.1- 30 Hz and it could increase between 0.01 and 200 Hz at regular intervals on a logarithmic scale. The term number *n* in Eq. (7) was set to 7 to keep the term number to a minimum and to show that the viscoelastic properties tended to increase smoothly. (2) The values of the dynamic viscoelasticity parameter *fi* were then determined to keep the difference between the experimental results and calculation results to a minimum from the low frequency term (*i*=1) to the high frequency term (*i*=7), using the different effect of each term on the numerical calculation results. (3) By changing the experimental conditions, the viscoelastic parameter was fitted by fixing the relaxation time parameter and changing only the dynamic viscoelasticity parameter.

Relaxation time *τV* of the Voigt model was determined as 2.5 ms from measurements of the delay time of the displacement of the outer diameter for the internal pressure change of the silicon tube.

#### *3.1.3. Numerical calculation*

194 Viscoelasticity – From Theory to Biological Applications

Tube wall properties were determined for recent silicone tube. The static tube law and compliance of the silicone tube was obtained from the relationship between the volume of the piston pump and the internal pressure, as shown in Fig. 3, while performing one stroke of the piston pump over a period of about 30 minutes. The compliances, reference pressure, and reference cross-section of local tube deformation to the numerical calculation of each trial were determined from the collinear approximation of the pressure range of the experimental conditions. For example, a range of 2.5 - 4.3 kPa was used when the baseline pressure was 2.5 kPa, because the pressure pulse amplitude of a single impulse was about 1.8 kPa. These local compliances, assumed constant under the flow experimental conditions, are shown by squares in Fig. 3. These local compliances were used in all the wall viscoelastic

**Figure 3.** Static relation between pressure and cross section of the silicone tube

2.5

Initial pressure (kPa)

> 10.0 7.5 5.0

Dynamic viscoelasticity of a strip of the silicone tube was measured by a device (TA Instruments DMA2980). The measurement conditions were determined as initial strain of 0.18% and amplitude of 0.145%. These values correspond to a baseline pressure of 0.28 kPa and a wave amplitude of 1.8 kPa, respectively. The measured viscoelastic properties are in Fig. 4. This figure shows the dynamic modules of viscoelasticity normalized against the static modules in Fig. 4(a) and the loss tangent in Fig. 4(b), respectively. These results mean that, in the measurement range 0.1~30 Hz, the dynamic viscoelasticity property of the silicon tube, both the real parts and loss tangent, tends to increase gradually with increase in the

0 2.5 5 7.5 10 12.5 Internal Pressure (kPa)

0.1957 0.1856 0.1790 0.1883 Compliance (mm2/kPa)

0.16

0.18

0.2

0.22 0.24

Compliance(right)

Cross section (left)

Compliance (mm2/kPa)

0.26

0.28

0.3

*Dynamic viscoelastic property of the tube* 

62

62.5

63

63.5

Cross section of the tube (mm 2

)

64

64.5

65

*3.1.2. Tube wall properties* 

*Static tube law* 

models.

frequency.

The basic equations of the numerical calculation 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 with numerical friction as space was used (Jameson & Baker, 1983). Flow volume and cross-section of the next time step were obtained from an equation of continuity and momentum conservation, and then the pressure was calculated from the tube law as a function of time (Kitawaki et al., 2003). Convolution integrals appear in the viscous resistance term in Eq. (3) and in the viscoelastic term in Eq. (7). Since calculation of the convolution integrals requires a lot of computer memory to hold past velocity and cross-section values, and takes a lot of computational time, a high-speed calculation method for the viscous resistance term of a rigid tube (Kagawa et al., 1983) was applied, as shown in *section 2.4*.

The following conditions were used in the calculation.

1. Baseline pressure was determined as an initial pressure by the water head of the tank.

Numerical Simulation Model with Viscoelasticity of Arterial Wall 197

0.2s

0.2s

0.2s

experimental tube are closed after the first pressure wave is generated by the piston pump. However, we can see an oscillatory wave when the pressure wave is returning back to the proximal end. This wave shows that the piston cylinder receives backpressure from the

Reynold's numbers and the wavelengths of the pressure waves were calculated from the measured value of the maximum velocity and pressure data from these experiments. Inspection of the Reynold's numbers (about 700) shows that laminar flow occurred under all experimental conditions. The wavelengths of the pressure waves (2.1 m) were sufficiently longer than the 4.5-mm radius of the tube, validating the long wavelength assumption and the assumptions of the Womersley model. Compared with the wavelength of the pressure wave, the length of the central rigid brass tube (5 cm) is sufficiently short, and because the rigid brass tube has a very small effect on the propagating pressure wave, it can be neglected in the calculation. The reproducibility of the pressure and flow volumes was good. Additionally, repeated measurements were virtually identical, so viscoelasticity

**Figure 6.** Calculated pressure waves (thin line) compared with the measured waves (bold line) at three

reflected pressure wave inside the tube.

changes such as memory effect did not happen.

2.0kPa

0m

(a) Elastic model

2.0kPa

0m

(b) Voigt model

2.0kPa

1.45m

2.9m

1.45m

2.9m

location in the tube. Difference in viscoelastic models.

0m

(c) Generalized viscoelastic model

1.45m

2.9m


Time step Δt and grid interval Δx were set at 0.5 ms and 0.05 m, respectively. The Courant number was 0.18~0.21 because the propagation velocity of the pressure wave was about 18~21 m/s, and the CFL condition (numerical stability condition) was satisfied. Actual calculation was performed on a workstation computer.

#### **3.2. Results and discussions**

#### *3.2.1. Difference between calculated and experimental results*

#### *Experimental results of pressure propagation*

The internal pressure waves measured at the 3 positions in the tube, and the flow volume are shown in Fig. 5, for a baseline pressure of 2.5 kPa. The flow volume was calculated from the measured displacement data of the piston pump by using LPF (FIR 25 Hz) and derivative filter.

**Figure 5.** Time profiles of the pressure waves at three positions in the tube. The distance between input piston pump and measurement position are indicated on the left. The flow volume into the tube are given in the lower curve.

We can see that the pressure wave of the single impulse, generated by the movement of the piston pump, propagates towards the distal end, from where it is reflected. Upon returning to the proximal end, it is reflected again, back towards the distal end. The amplitude of the propagating pressure wave gradually attenuates, and the width of the pressure wave gradually increases because of the viscosity of the fluid and the viscoelasticity of the tube. The mean pressure rises for the fluid pushed by the piston pump. Both ends of the experimental tube are closed after the first pressure wave is generated by the piston pump. However, we can see an oscillatory wave when the pressure wave is returning back to the proximal end. This wave shows that the piston cylinder receives backpressure from the reflected pressure wave inside the tube.

196 Viscoelasticity – From Theory to Biological Applications

4. No flow boundary condition was applied to the distal end.

*3.2.1. Difference between calculated and experimental results* 

calculation was performed on a workstation computer.

2. No flow in the initial state.

**3.2. Results and discussions** 

*Experimental results of pressure propagation* 

2kPa

0.0m

Flow Volume

1.45m

Pressures

2.9m

condition.

derivative filter.

given in the lower curve.

1. Baseline pressure was determined as an initial pressure by the water head of the tank.

3. Flow volume calculated by displacement sensor was used as an input boundary

Time step Δt and grid interval Δx were set at 0.5 ms and 0.05 m, respectively. The Courant number was 0.18~0.21 because the propagation velocity of the pressure wave was about 18~21 m/s, and the CFL condition (numerical stability condition) was satisfied. Actual

The internal pressure waves measured at the 3 positions in the tube, and the flow volume are shown in Fig. 5, for a baseline pressure of 2.5 kPa. The flow volume was calculated from the measured displacement data of the piston pump by using LPF (FIR 25 Hz) and

**Figure 5.** Time profiles of the pressure waves at three positions in the tube. The distance between input piston pump and measurement position are indicated on the left. The flow volume into the tube are

0.2s

2cm3 /s

We can see that the pressure wave of the single impulse, generated by the movement of the piston pump, propagates towards the distal end, from where it is reflected. Upon returning to the proximal end, it is reflected again, back towards the distal end. The amplitude of the propagating pressure wave gradually attenuates, and the width of the pressure wave gradually increases because of the viscosity of the fluid and the viscoelasticity of the tube. The mean pressure rises for the fluid pushed by the piston pump. Both ends of the Reynold's numbers and the wavelengths of the pressure waves were calculated from the measured value of the maximum velocity and pressure data from these experiments. Inspection of the Reynold's numbers (about 700) shows that laminar flow occurred under all experimental conditions. The wavelengths of the pressure waves (2.1 m) were sufficiently longer than the 4.5-mm radius of the tube, validating the long wavelength assumption and the assumptions of the Womersley model. Compared with the wavelength of the pressure wave, the length of the central rigid brass tube (5 cm) is sufficiently short, and because the rigid brass tube has a very small effect on the propagating pressure wave, it can be neglected in the calculation. The reproducibility of the pressure and flow volumes was good. Additionally, repeated measurements were virtually identical, so viscoelasticity changes such as memory effect did not happen.

**Figure 6.** Calculated pressure waves (thin line) compared with the measured waves (bold line) at three location in the tube. Difference in viscoelastic models.

#### *Comparison between calculated and experimental results*

The numerical calculations were performed for a baseline pressure of 2.5 kPa and then compared with the experimental values of measurements of the pressure at the three positions along the experimental tube. The calculation results of combinations of the Womersley model with the elastic model, Voigt model and generalized viscoelastic model were compared, as shown in Fig. 6.

Numerical Simulation Model with Viscoelasticity of Arterial Wall 199

0.2s

0.2s

0.2s

Because the experimental results for the baseline pressures of 5.0, 7.5 and 10.0 kPa are very similar, the optimized value of the viscoelastic parameter for baseline pressure of 5.0 kPa was used for the three experimental conditions, and comparisons of the calculated results using this viscoelastic parameter and the experimental results are shown in Fig. 8. As can be seen, the experimental and calculated results agreed well throughout the experimental period when the baseline pressure was 5.0 kPa. In contrast, at baseline pressures of 7.5 and 10.0 kPa, the agreement is good for the first 0.2 seconds, and gradually depreciates thereafter, because of the slight difference in their propagation velocities. Therefore, the experimental result for baseline pressure of 7.5 and 10.0 kPa cannot be simulated using the

**Figure 8.** Calculated pressure waves (thin line) compared with the measured waves (bold line) using

viscoelastic parameter optimized for a baseline pressure of 5.0 kPa.

viscoelastic parameter optimized for baseline pressure of 5.0 kPa.

2.0kPa

0m

(a) 5.0kPa

2.0kPa

0m

(b) 7.5kPa

2.0kPa

0m

(c) 10.0kPa

1.45m

2.9m

1.45m

2.9m

1.45m

2.9m

*Determination of viscoelastic properties* 

When the Womersley model was combined with all models, the initial pressure waves were close to the experimental value. However, when the Womersley model was combined with the elastic or Voigt models, the difference between the calculated and experimental results gradually increased because the calculated propagation velocities and attenuation level were underestimated. On the other hand, when combined with the generalized viscoelastic model together with the optimized viscoelastic parameter, the calculated results agreed well with the experimental results for all the experimental period. In the elastic model or Voigt model, when the tube deformation compliance or relaxation time *τv* was changed, the calculation result did not agree with the experimental result. These results show that the one-dimensional model using Womersley model combined with the generalized viscoelastic model are necessary in order for the numerically calculated result to agree with the experimental result.

#### *3.2.2. Difference of viscoelasticity by the change of baseline pressure*

#### *Difference between pressure propagation experiments of baseline pressure difference*

The flow volume of the inlet and the pressure waveform at the inlet position with changes in the baseline pressure are shown in Fig. 7. The flow volume in Fig. 7(a) are well controlled, and the movements of the piston are almost the same. As seen in Fig. 7(b), the pressure waveform changed differently when the baseline pressure was 2.5 kPa compared with other baseline pressures, because the propagation velocity was different. At baseline pressures of 5.0, 7.5 and10.0 kPa, the propagation is very similar during the first 0.8 seconds.

**Figure 7.** Flow volume and pressure waves at beginning of the tube. Difference in initial pressures

#### *Determination of viscoelastic properties*

198 Viscoelasticity – From Theory to Biological Applications

were compared, as shown in Fig. 6.

experimental result.

(a) Flow volume

Flow volume (ml/s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time(s)

*Comparison between calculated and experimental results* 

The numerical calculations were performed for a baseline pressure of 2.5 kPa and then compared with the experimental values of measurements of the pressure at the three positions along the experimental tube. The calculation results of combinations of the Womersley model with the elastic model, Voigt model and generalized viscoelastic model

When the Womersley model was combined with all models, the initial pressure waves were close to the experimental value. However, when the Womersley model was combined with the elastic or Voigt models, the difference between the calculated and experimental results gradually increased because the calculated propagation velocities and attenuation level were underestimated. On the other hand, when combined with the generalized viscoelastic model together with the optimized viscoelastic parameter, the calculated results agreed well with the experimental results for all the experimental period. In the elastic model or Voigt model, when the tube deformation compliance or relaxation time *τv* was changed, the calculation result did not agree with the experimental result. These results show that the one-dimensional model using Womersley model combined with the generalized viscoelastic model are necessary in order for the numerically calculated result to agree with the

*3.2.2. Difference of viscoelasticity by the change of baseline pressure* 

*Difference between pressure propagation experiments of baseline pressure difference* 

5.0, 7.5 and10.0 kPa, the propagation is very similar during the first 0.8 seconds.

2.5kPa 5.0kPa 7.5kPa 10.0kPa

**Figure 7.** Flow volume and pressure waves at beginning of the tube. Difference in initial pressures


(b) Differential pressure waves

Differential pressure P=P-P0 (kPa)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time(s)

2.5kPa 5.0kPa 7.5kPa 10.0kPa

The flow volume of the inlet and the pressure waveform at the inlet position with changes in the baseline pressure are shown in Fig. 7. The flow volume in Fig. 7(a) are well controlled, and the movements of the piston are almost the same. As seen in Fig. 7(b), the pressure waveform changed differently when the baseline pressure was 2.5 kPa compared with other baseline pressures, because the propagation velocity was different. At baseline pressures of Because the experimental results for the baseline pressures of 5.0, 7.5 and 10.0 kPa are very similar, the optimized value of the viscoelastic parameter for baseline pressure of 5.0 kPa was used for the three experimental conditions, and comparisons of the calculated results using this viscoelastic parameter and the experimental results are shown in Fig. 8. As can be seen, the experimental and calculated results agreed well throughout the experimental period when the baseline pressure was 5.0 kPa. In contrast, at baseline pressures of 7.5 and 10.0 kPa, the agreement is good for the first 0.2 seconds, and gradually depreciates thereafter, because of the slight difference in their propagation velocities. Therefore, the experimental result for baseline pressure of 7.5 and 10.0 kPa cannot be simulated using the viscoelastic parameter optimized for baseline pressure of 5.0 kPa.

**Figure 8.** Calculated pressure waves (thin line) compared with the measured waves (bold line) using viscoelastic parameter optimized for a baseline pressure of 5.0 kPa.

According to this result, the optimized value of the viscoelastic parameter for baseline pressure of 7.5 and 10.0 kPa were obtained respectively and calculations were conducted again. As shown in Fig. 9, the calculated and experimental results agree well. In the first 0.4 seconds, a clear difference between the two results can be seen, especially at higher baseline pressures. The differences could not be decreased by changing the viscoelastic parameters. Even with these differences, the agreement between the two results is good.

Numerical Simulation Model with Viscoelasticity of Arterial Wall 201

2.5kPa 5.0kPa 7.5kPa 10.0kPa

*<sup>i</sup>τi*(s) *f*(Hz) *fi*

**Table 1.** Optimized viscoelastic parameters for each baseline pressure

**Figure 10.** Change in dynamic viscoelastic module.

frequency(Hz)

0.1 1.0 10.0

(a) Ratio of dynamic and static modulus of viscoelasticity

viscoelastic properties change independently.

**3.3. Conclusion of this section** 

1

1.2

1.4

Normalized storage modulus

1.6

1.8

2.5kPa 5.0kPa 7.5kPa 10.0kPa Experiment

1 17.0 0.009 0.089 0.165 0.130 0.112

2 3.75 0.042 0.031 0.033 0.058 0.087

3 0.7 0.227 0.057 0.061 0.086 0.110

4 0.15 1.061 0.068 0.073 0.098 0.119

5 0.03 5.305 0.090 0.096 0.121 0.138

6 0.005 31.83 0.170 0.170 0.183 0.190

7 0.0008 198.9 0.220 0.220 0.223 0.224

Accordingly, the one-dimensional numerical model, which takes into account unsteady viscosity and the generalized viscoelastic model, is good for simulating the propagation of small pressure waves in silicone tubes even when their deformation compliance and

0

(b) loss tangent

frequency(Hz)

0.1 1.0 10.0

2.5kPa 5.0kPa 7.5kPa 10.0kPa Experiment

0.05

Loss tangent tan( )

0.1

0.15

For the numerical analysis of the viscoelastic tubes, a nonlinear one-dimensional numerical model was investigated by including the unsteady viscous resistance and the effect of the tube wall viscoelasticity. By comparing the calculated results using these models with experimental results of a viscoelastic silicone tube, we can make the following conclusions.

1. The approximation error of the numerical simulation model is satisfactory small when the Womersley model is combined with the generalized viscoelastic model for the flow

**Figure 9.** Calculated pressure waves (thin line) compared with the measured waves (bold line) using optimized viscoelastic parameter for each experimental condition.

These results show that the one-dimensional model using the Womersley model combined with the generalized viscoelastic model can accurately simulate the effect of changes in the silicon tube's viscoelasticity due to changes in the internal pressure when the viscoelastic parameter is appropriately determined. Additionally, the viscoelastic properties express the viscoelastic property change which depend the internal pressure of viscoelastic tube.

#### *3.2.3. Relationship between viscoelasticity and static elasticity*

The viscoelastic properties, both of the dynamic viscoelasticity parameters and the relaxation time parameters shown in Table 1 including the frequency *f* calculated from the relaxation time parameters, and the viscoelastic properties calculated from optimized values are plotted in Fig. 10. The squares in Fig. 10 indicate the measured values of the silicone tube viscoelasticity, and are close to the values obtained when the baseline pressure was 2.5 kPa, though a small error in the high frequency area of the loss tangent. This may be because the corresponding baseline pressure of the dynamic viscoelasticity measurement (0.28 kPa) is closer to the experimental baseline pressure of 2.5kPa than to the other experimental conditions.

From Fig. 10, it seems that the viscoelastic property change may be related to the baseline pressure. For example, when the baseline pressure increases from 2.5 to 5.0 kPa, the increase in the dynamic modules is steady at all frequencies. On the other hand, when the baseline pressure is 5.0 kPa or greater, the dynamic modules increase with increasing baseline pressure by increasing the frequency. Because the normalized dynamic modulus in the high frequency region changes with baseline pressure, the viscoelastic change can be said to be independent of the deformation compliance.


Numerical Simulation Model with Viscoelasticity of Arterial Wall 201

**Table 1.** Optimized viscoelastic parameters for each baseline pressure

**Figure 10.** Change in dynamic viscoelastic module.

Accordingly, the one-dimensional numerical model, which takes into account unsteady viscosity and the generalized viscoelastic model, is good for simulating the propagation of small pressure waves in silicone tubes even when their deformation compliance and viscoelastic properties change independently.

#### **3.3. Conclusion of this section**

200 Viscoelasticity – From Theory to Biological Applications

According to this result, the optimized value of the viscoelastic parameter for baseline pressure of 7.5 and 10.0 kPa were obtained respectively and calculations were conducted again. As shown in Fig. 9, the calculated and experimental results agree well. In the first 0.4 seconds, a clear difference between the two results can be seen, especially at higher baseline pressures. The differences could not be decreased by changing the viscoelastic parameters.

> 2.0kPa

0m

(d) 10.0kPa

1.45m

2.9m

**Figure 9.** Calculated pressure waves (thin line) compared with the measured waves (bold line) using

(c) 7.5kPa 0.2s

0.2s

These results show that the one-dimensional model using the Womersley model combined with the generalized viscoelastic model can accurately simulate the effect of changes in the silicon tube's viscoelasticity due to changes in the internal pressure when the viscoelastic parameter is appropriately determined. Additionally, the viscoelastic properties express the

The viscoelastic properties, both of the dynamic viscoelasticity parameters and the relaxation time parameters shown in Table 1 including the frequency *f* calculated from the relaxation time parameters, and the viscoelastic properties calculated from optimized values are plotted in Fig. 10. The squares in Fig. 10 indicate the measured values of the silicone tube viscoelasticity, and are close to the values obtained when the baseline pressure was 2.5 kPa, though a small error in the high frequency area of the loss tangent. This may be because the corresponding baseline pressure of the dynamic viscoelasticity measurement (0.28 kPa) is closer to the experimental baseline pressure of 2.5kPa than to the other experimental

From Fig. 10, it seems that the viscoelastic property change may be related to the baseline pressure. For example, when the baseline pressure increases from 2.5 to 5.0 kPa, the increase in the dynamic modules is steady at all frequencies. On the other hand, when the baseline pressure is 5.0 kPa or greater, the dynamic modules increase with increasing baseline pressure by increasing the frequency. Because the normalized dynamic modulus in the high frequency region changes with baseline pressure, the viscoelastic change can be said to be

viscoelastic property change which depend the internal pressure of viscoelastic tube.

Even with these differences, the agreement between the two results is good.

optimized viscoelastic parameter for each experimental condition.

*3.2.3. Relationship between viscoelasticity and static elasticity* 

independent of the deformation compliance.

conditions.

2.0kPa

0m

1.45m

2.9m

For the numerical analysis of the viscoelastic tubes, a nonlinear one-dimensional numerical model was investigated by including the unsteady viscous resistance and the effect of the tube wall viscoelasticity. By comparing the calculated results using these models with experimental results of a viscoelastic silicone tube, we can make the following conclusions.

1. The approximation error of the numerical simulation model is satisfactory small when the Womersley model is combined with the generalized viscoelastic model for the flow analysis of the viscoelastic tube even their deformation compliance and viscoelasticity change independently.

Numerical Simulation Model with Viscoelasticity of Arterial Wall 203

Bundle of thin capillary tubes

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

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

Length

Rigid brass tube

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

8.0 mm.

**Figure 12.** Terminal resistance

Flow into terminal resistance

propagation was examined.

reproducibility was high.

*Experimental method* 

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