*4.2.1. Experimental results of pressure propagation*

As an example of experimental results, the internal pressure measured at three points in the experiment tube, and the flow rate at the proximal end are shown in Fig. 15, at the baseline pressure of 5.6 kPa. The change in the whole time experiment is shown in Fig. 15(a), and an enlarged view of the last but one full-cycle pulsation is shown in Fig. 15(b). The pressure value shown here was set to be a differential pressure from the baseline internal pressure.

As shown in Fig. 15(a), there is a rise in the mean pressure or waveform change up to the first three pulses, but then it converges in the steady state. Also, there are oscillating pressure waves at the proximal and distal ends of the initial pulsation, due to wave reflections at both ends of the experiment tube; however, these oscillatory waves disappeared after the second pulsation, because this phenomenon is caused by the remaining reflected wave of the previous pulse.

As shown in Fig. 15(b) of the steady state of the pressure waveform, part of the waveform measured at the proximal end becomes almost flat in its upper portion for about 0.3 seconds, and then the pressure exponentially decreases. It is thought that this flat portion is generated by the progressive wave, which is produced by the movement of the piston pump, overlapping the reflected wave coming from the distal end of the tube. At the same time, the pressure waveform measured at the distal end has a pointed peak shape and a small reflected wave at the shoulder. The maximum pressure gradually increases from the proximal end of the tube towards the distal end, and the phenomenon is similar to the Peaking phenomenon *in vivo*.

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

The calculated results in each tube law model were compared with experimental results in order to confirm the effect of the difference of the tube law model on the experimental results. Figure 16 shows examples of steady-state pulse waveforms at the baseline internal pressure of 5.6 kPa. Experimental results are shown by a bold black line, and calculated results are shown as a thin red line. In the Voigt model, calculated results in which the relaxation time parameter is multiplied by 4 (10.0ms) are also shown as a dotted line.

requires much computational time.

**4.2. Results and discussions** 

pressure.

*4.2.1. Experimental results of pressure propagation* 

remaining reflected wave of the previous pulse.

Peaking phenomenon *in vivo*.

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

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.

As an example of experimental results, the internal pressure measured at three points in the experiment tube, and the flow rate at the proximal end are shown in Fig. 15, at the baseline pressure of 5.6 kPa. The change in the whole time experiment is shown in Fig. 15(a), and an enlarged view of the last but one full-cycle pulsation is shown in Fig. 15(b). The pressure value shown here was set to be a differential pressure from the baseline internal

As shown in Fig. 15(a), there is a rise in the mean pressure or waveform change up to the first three pulses, but then it converges in the steady state. Also, there are oscillating pressure waves at the proximal and distal ends of the initial pulsation, due to wave reflections at both ends of the experiment tube; however, these oscillatory waves disappeared after the second pulsation, because this phenomenon is caused by the

As shown in Fig. 15(b) of the steady state of the pressure waveform, part of the waveform measured at the proximal end becomes almost flat in its upper portion for about 0.3 seconds, and then the pressure exponentially decreases. It is thought that this flat portion is generated by the progressive wave, which is produced by the movement of the piston pump, overlapping the reflected wave coming from the distal end of the tube. At the same time, the pressure waveform measured at the distal end has a pointed peak shape and a small reflected wave at the shoulder. The maximum pressure gradually increases from the proximal end of the tube towards the distal end, and the phenomenon is similar to the

The calculated results in each tube law model were compared with experimental results in order to confirm the effect of the difference of the tube law model on the experimental results. Figure 16 shows examples of steady-state pulse waveforms at the baseline internal pressure of 5.6 kPa. Experimental results are shown by a bold black line, and calculated results are shown as a thin red line. In the Voigt model, calculated results in which the

relaxation time parameter is multiplied by 4 (10.0ms) are also shown as a dotted line.

*4.2.2. Comparison between calculated results and experimental results* 

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

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).

Numerical Simulation Model with Viscoelasticity of Arterial Wall 209

0.2s

Experiment GVM(1)

In order to find out how many changes in the viscoelastic property of the tube were accompanied by baseline internal pressure change, and how this change influenced pulse wave propagation, the value of the viscoelastic parameter was determined under different conditions of baseline internal pressure. Initially, pressure waveforms at the baseline internal pressure of 8.4 kPa and 11.2 kPa were calculated using the value of the viscoelastic parameter at a baseline internal pressure of 5.6 kPa (hereinafter, this viscoelastic parameter is referred to as GVM(1)). The change in tube deformation compliance was incorporated into these calculations. As an example, a comparison of the calculated results and experimental results at the proximal and distal ends at a baseline internal pressure of 11.2 kPa is shown in Fig. 17. PP at the proximal end using GVM(1) was smaller than the experimental value, and it was proven that the viscoelastic effect of

**Figure 17.** Calculated pressure waves (dotted red line) compared with the measured waves (bold black

Based on this result, the value of the viscoelastic parameter was adjusted. We changed the value of the dynamic viscoelastic parameter in the low frequency region (*i*=1) so that the dynamic modulus elasticity ratio increased with the increase in baseline internal pressure in the experiment tube, and the value of the dynamic viscoelastic parameter in other frequency regions (*i*=2-7) was adjusted to a fixed ratio. The special evaluation function was not used to decide parameters but we took it into consideration to agree with the pulse waveform and the PWV. The viscoelastic parameter value decided in this way (hereinafter referred to as GVM(2)) is described in Table 2 and Fig. 18. The squares in Fig. 18 are the measured

Agreement of the calculated results and experimental results using GVM(2), although not

*4.2.3. Difference in the viscoelastic parameter* 

GVM(1) was not sufficient.

1.0kPa

line) at two locations in the tube

0.02m

2.02m

viscoelastic values of Fig. 4.

shown, was the same level as that in Fig. 16 (c).

**Figure 16.** Calculated pressure waves (thin red line) compared with the measured waves (bold black line) at three locations in the tube

On the other hand, the calculated results using the generalized viscoelastic model almost completely agreed with the experimental results in the whole region. As mentioned above, it was proven that numerical simulation using the generalized viscoelastic model could express the experimental result accurately.

#### *4.2.3. Difference in the viscoelastic parameter*

208 Viscoelasticity – From Theory to Biological Applications

1.0kPa

1.0kPa

(a) Elastic model

2.02m

2.02m

2.02m

1.02m

0.02m

1.0kPa

(c) GVM

(b) Voigt model

1.02m

0.02m

1.02m

0.02m

line) at three locations in the tube

express the experimental result accurately.

**Figure 16.** Calculated pressure waves (thin red line) compared with the measured waves (bold black

0.2s

0.2s

0.2s

On the other hand, the calculated results using the generalized viscoelastic model almost completely agreed with the experimental results in the whole region. As mentioned above, it was proven that numerical simulation using the generalized viscoelastic model could In order to find out how many changes in the viscoelastic property of the tube were accompanied by baseline internal pressure change, and how this change influenced pulse wave propagation, the value of the viscoelastic parameter was determined under different conditions of baseline internal pressure. Initially, pressure waveforms at the baseline internal pressure of 8.4 kPa and 11.2 kPa were calculated using the value of the viscoelastic parameter at a baseline internal pressure of 5.6 kPa (hereinafter, this viscoelastic parameter is referred to as GVM(1)). The change in tube deformation compliance was incorporated into these calculations. As an example, a comparison of the calculated results and experimental results at the proximal and distal ends at a baseline internal pressure of 11.2 kPa is shown in Fig. 17. PP at the proximal end using GVM(1) was smaller than the experimental value, and it was proven that the viscoelastic effect of GVM(1) was not sufficient.

**Figure 17.** Calculated pressure waves (dotted red line) compared with the measured waves (bold black line) at two locations in the tube

Based on this result, the value of the viscoelastic parameter was adjusted. We changed the value of the dynamic viscoelastic parameter in the low frequency region (*i*=1) so that the dynamic modulus elasticity ratio increased with the increase in baseline internal pressure in the experiment tube, and the value of the dynamic viscoelastic parameter in other frequency regions (*i*=2-7) was adjusted to a fixed ratio. The special evaluation function was not used to decide parameters but we took it into consideration to agree with the pulse waveform and the PWV. The viscoelastic parameter value decided in this way (hereinafter referred to as GVM(2)) is described in Table 2 and Fig. 18. The squares in Fig. 18 are the measured viscoelastic values of Fig. 4.

Agreement of the calculated results and experimental results using GVM(2), although not shown, was the same level as that in Fig. 16 (c).

Numerical Simulation Model with Viscoelasticity of Arterial Wall 211

an error in propagation velocity by a quantize error in calculating the propagation time. PP

Figure 19(a) also shows the elastic tube theoretical velocity (Moens-Korteweg's theoretical velocity) calculated from the tube inner diameter and tube deformation compliance. The theoretical velocity decreased by 5.8% as a result of the influence of the increase in tube deformation compliance, when the baseline internal pressure increased from 5.6 kPa to 11.2 kPa. On the other hand, PWV calculated from the experimental results was bigger than the elastic tube theoretical velocity, and the increasing ratio against the elastic theoretical

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

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

was obtained from the pulse wave at the proximal end.

**Figure 19.** PWV and PP change with internal pressure change

(b) PP change

velocity gradually increased with the rise in baseline internal pressure.

Experiment Theoretical Value Elastic model Voigt model GVM(1) GVM(2)

Experiment Elastic model Voigt model GVM(1) GVM(2)

(a) PWV change

*Effect of dynamic elasticity modulus ratio* 

0.8

0.9

1

1.1

PP(kPa)

1.2

1.3

PWV(m/s)

**Figure 18.** Change in viscoelastic properties of silicone tube (a) Dynamic modulus viscoelasticity ratio (b) loss tangent


**Table 2.** Optimized viscoelastic parameters for each baseline pressure

#### *4.2.4. Effect of difference in viscoelasticity on pulse wave propagation*

The effect of the difference in viscoelastic properties on pulse wave propagation was considered by analyzing the calculated results in which the viscoelastic model and the viscoelastic parameter were changed. PWV and PP were obtained from the calculated and experimental results. The relationship of these indexes with the baseline internal pressure is shown in Fig. 19. These indexes were obtained from mean values in four pulsations in the latter half in the steady state. Defining the maximum value of the second order differential waveform of the pulse waveform as an initial rise of the pulse wave, PWV was calculated by dividing the distance between calculation points (at the proximal and distal ends) by the time difference in the initial rise of these pulse waves. The error range of PWV is shown as an error in propagation velocity by a quantize error in calculating the propagation time. PP was obtained from the pulse wave at the proximal end.

#### *Effect of dynamic elasticity modulus ratio*

210 Viscoelasticity – From Theory to Biological Applications

1

1.2

1.4

Dynamic modulus elasticity ratio

1.6

1.8

5.6kPa 8.4kPa 11.2kPa Experiment

**Figure 18.** Change in viscoelastic properties of silicone tube

frequency(Hz)

0.1 1.0 10.0

*<sup>i</sup>τi* (s) *f*(Hz) *fi*

(a) Dynamic modulus viscoelasticity ratio (b) loss tangent

1 17.0 0.009 0.095 0.169 0.227

2 3.75 0.042 0.033 0.034 0.036

3 0.7 0.227 0.061 0.063 0.066

4 0.15 1.061 0.073 0.076 0.079

5 0.03 5.305 0.096 0.100 0.104

6 0.005 31.83 0.170 0.177 0.184

7 0.0008 198.9 0.220 0.229 0.238

The effect of the difference in viscoelastic properties on pulse wave propagation was considered by analyzing the calculated results in which the viscoelastic model and the viscoelastic parameter were changed. PWV and PP were obtained from the calculated and experimental results. The relationship of these indexes with the baseline internal pressure is shown in Fig. 19. These indexes were obtained from mean values in four pulsations in the latter half in the steady state. Defining the maximum value of the second order differential waveform of the pulse waveform as an initial rise of the pulse wave, PWV was calculated by dividing the distance between calculation points (at the proximal and distal ends) by the time difference in the initial rise of these pulse waves. The error range of PWV is shown as

**Table 2.** Optimized viscoelastic parameters for each baseline pressure

*4.2.4. Effect of difference in viscoelasticity on pulse wave propagation* 

5.6kPa 8.4kPa 11.2kPa

frequency(Hz)

0.1 1.0 10.0

5.6kPa 8.4kPa 11.2kPa Experiment

0

0.05

Loss tangent tan()

0.1

Figure 19(a) also shows the elastic tube theoretical velocity (Moens-Korteweg's theoretical velocity) calculated from the tube inner diameter and tube deformation compliance. The theoretical velocity decreased by 5.8% as a result of the influence of the increase in tube deformation compliance, when the baseline internal pressure increased from 5.6 kPa to 11.2 kPa. On the other hand, PWV calculated from the experimental results was bigger than the elastic tube theoretical velocity, and the increasing ratio against the elastic theoretical velocity gradually increased with the rise in baseline internal pressure.

**Figure 19.** PWV and PP change with internal pressure change

PWV using the elastic model approximately agreed with the theoretical elastic tube velocity, since the viscosity of the tube was not included. Using the Voigt model, the small effect by the viscoelasticity on the increase in PWV and PWV was slightly larger than the theoretical elastic tube velocity; however, the calculated and experimental results agreed well with the generalized viscoelastic model in GVM(2).

Numerical Simulation Model with Viscoelasticity of Arterial Wall 213

3. It is necessary to measure the dynamic modulus elasticity ratio with accuracy below 4%

In this chapter, a nonlinear one-dimensional numerical model was constructed by including the effect of tube wall viscoelasticity and unsteady viscous resistance for numerical analysis of the viscoelastic tubes. This model can analyze the effect of viscoelasticity on intravascular flow that few previous papers have considered. Using a one-dimensional numerical simulation model of a viscoelastic tube, the effect of tube viscoelasticity on a pulsatile wave and periodic pulse wave propagation were examined by comparing the experimental results

Clinical blood vessel stiffness indexes vary with viscoelasticity change, because the elastic modulus is more effective than the energy dissipation effect by viscoelasticity change. This result showed that the viscoelasticity of the vessel wall plays an important role in the form of a pulsatile wave; therefore, it is important to consider the viscoelastic effect accurately in

Anliker, M., & Rockwall, R. L. (1971) Nonlinear Analysis of Flow Pulses and Shock Waves in

Avolio, A. P. (1980) Multi-Branched Model of the Human Arterial System, *Med. & Biol. Eng.* 

Hayashi, K., Handa, H., Nagasawa, S. (1980) Okumura, A., & Moritake, K., Stiffness and Elastic Behavior of Human Intracranial and Extracranial Arteries, *J. Biomech.*, Vol.13, pp.

Jameson, A., & Baker, T. J. (1983) Solution of the Euler Equations for Complex

Kagawa, T., Lee, I., Kitagawa, A., & Takenaka, T. (1983) High Speed and Accurate Computing Method of Frequency-Dependent Friction in Laminar Pipe Flow for Characteristics Method, *Trans. Jpn. Soc. Mech. Eng.*, Vol.B49, pp. 2638-44, ISSN 0387-5016 Kitawaki, T., Shimizu, M., Himeno, R., & Liu, H. (2003) One-Dimensional Numerical Simulation of Visco-Elastic Tube, *Trans. Jpn. Soc. Mech. Eng.*, Vol.A69-677, pp. 55-61,

Kitawaki, T., & Shimizu, M. (2005) Effect of the Blood Vessel Viscoelasticity on the Blood Pressure Wave Propagation, *Trans. Jpn. Soc. Mech. Eng.*, Vol.B71-707, pp. 1768-1774,

Configurations, *AIAA paper*, 83-1929, pp. 293-302, ISSN 0146-3705

in order to estimate the values of PWV and PP with accuracy below 5%..

with the calculated results using a viscoelastic silicone tube.

quantitative investigations using intravascular flow analysis.

*Graduate School of Health Sciences, Okayama University, Japan* 

Arteries, *ZAMP*, Vol.22, pp. 217-246, ISSN 0044-2275

*& Comput.*, Vol.18, pp. 709-18, ISSN 0140-0118

175-184, ISSN 0021-9290

ISSN 0387-5008

ISSN 0387-5016

**5. Conclusion** 

**Author details** 

Tomoki Kitawaki

**6. References** 

PP in the calculated results of the elastic model and the Voigt model was 90% of that in the experimental results. In the generalized viscoelastic model, the calculated results of GVM(2) agreed well with the experimental results.

This results show that, with the influence of viscoelastic change shown in Fig. 18, PWV increased by around 20-30% more than theoretical elastic tube velocity and PP increased by approximately 10% because increasing the elastic modulus by the effect of the low frequency component of viscoelasticity is more effective than the energy dissipation effect by the high frequency component of viscoelasticity. Furthermore, it can be said that the increase of viscoelasticity compensated for the influence caused by the change in tube deformation compliance.

#### *Effect of the viscoelastic parameter*

The effect of the viscoelastic parameter was examined from the difference in the calculated results between GVM(2) and GVM(1) for the baseline internal pressure of 11.2 kPa. The dynamic modulus elasticity ratio of GVM(1) was nearly 10.1% lower than that of GVM(2) within 1–10 Hz, as shown in Fig. 18. By this effect, PWV using GVM(1) was 6.6% lower than that using GVM(2), as shown in Fig. 19, and PP with GVM(1) was 6.4% lower than that with GVM(2). Considering the results with a baseline internal pressure of 8.4kPa, the difference of PWV and PP caused by the viscoelastic parameters was almost proportional to the change of baseline internal pressure; therefore, it might be necessary to determine the dynamic modulus elasticity ratio with accuracy below 4% in order to keep the accuracy of PWV and PP below 5%.

As described above, it was clarified that tube viscoelasticity played an important role in pulse wave propagation under periodic pulsatile conditions and had a significant effect on clinical arterial stiffness indexes, including PWV and PP.
