### Fundamental theory

On the basis of Voigt and Maxwell models [26], a three-component mechanical model of an arterial segment was established and employed to characterize the viscoelastic properties of the arterial wall in the study. As shown in Fig. 1, the lumped arterial viscoelastic model consists of three parallel components which correspond to the mass (M) (or inertia), viscosity (η), and elastance (E) of the arterial wall, respectively. In this model, the elastance component was reasonably assumed to be a function of time or transmural pressure. When an external sinusoidal vibration force (*F*
_{
E
}), perpendicular to the axis of an arterial lumen, contacted with the outside of the arterial wall and created the arterial wall displacement, *x*, the arterial wall would generate a reactive force (*F*
_{
R
}) in response to the external force. According to the force balance, the reactive force should be equal to the summation of the inertia-related force, viscosity-related force and elastance-related force [27], as follows:

$$F_{R} (t) = \frac{{d^{2} x(t)}}{d(t)} + \eta \frac{dx(t)}{dt} + E(t)x(t).$$

(1)

When the external force was of the sinusoidal vibration, the displacement of the arterial wall also was a sinusoidal function. It can be written as:

$$x(t) = A_{m} \sin (\varpi t),$$

(2)

where *A*
_{
m
} is the peak amplitude and *ω* is the angular frequency. Furthermore, its first and second derivatives can be respectively expressed as:

$$\frac{dx(t)}{dt} = A_{m} \varpi \cos (\varpi t),$$

(3)

and

$$\frac{{d^{2} x(t)}}{dt} = - A_{m} \varpi^{2} \sin (\varpi t).$$

(4)

According to Eqs. (2), (3) and (4), Eq. (1) can be changed to

$$\frac{{F_{R} (t)}}{{A_{m} }} = [E(t) - M\varpi^{2} ]\sin (\varpi t) + \eta \varpi \cos (\varpi t).$$

(5)

In order to simplify Eq. (5), we know that

$$\begin{aligned} x(t) &= A_{m} \sin (\omega t) = A_{m} ,\quad \omega t = \pm \frac{(n + 1)\pi }{2},\quad n \in i, \\ \frac{dx(t)}{dt} &= A_{m} \cos (\omega t) = 0, \quad \omega t = \pm \frac{(n + 1)\pi }{2},\quad n \in i, \end{aligned}$$

where *i* is an integer. Then, we define *T*
_{
m
} as: \(T_{m} = \frac{\pi }{2\omega }.\) When *t* is *T*
_{
m
}, Eq. (5) can be simplified and *F*
_{
R
} is equal to the maximum value, *F*
_{
R_max
},

$$\frac{{F_{R} (T_{m} )}}{{A_{m} }} = \frac{{F_{R\_\text{max} } }}{{A_{m} }} = E(T_{m} ) - M\omega^{2}$$

(6)

According to Eq. (6), at the specific time *T*
_{
m
} the ratio of *F*
_{
R_max
} and *A*
_{
m
} is negatively and linearly proportional to the square of angular frequency (*ω*
^{2}). In this linear polynomial function, *E*(*T*
_{
m
}) is the intercept and *M* is the absolute slope.

### Measuring system

Figure 2 shows the schematic diagram of the designed measuring system in this study. The system consists mainly of a vibrator, a force sensor, a vibrator driver, a sinusoidal waveform generator, a sensor driver, and an MP100 system. A sinusoidal signal delivered by the sinusoidal waveform generator was used to trigger the vibrator driver (DPS-270, DiaMidical System Cor., Tokyo, Japan). The vibrator driver drove the vibrator and the reactive force was sensed by a force sensor placed at the tip of the vibrator. Then, the reactive force signal was converted to a digital signal by the MP100 system (BIOPAC System, Inc., USA). The resolution is 12 bits and the sampling rate is 1000 Hz. For practical applications, the peak displacement of the sinusoidal vibration was set to 0.15 mm and the contact area of the vibrator with the radial arterial surface was about 0.2 cm^{2}. The data processing was performed with the AcqKnowledge Software 3.9 (BIOPAC System, Inc., USA).

### Experimental protocol

Twenty-nine healthy college volunteers (17 M and 12 F, age: 23 ± 3 years, systolic pressure: 113 ± 14 mm Hg, diastolic pressure: 70 ± 10 mm Hg, heart rate: 73 ± 9 beats/min) participated in this study. The clinical trial was approved by the Institutional Review Board of the E-DA Hospital, Kaohsiung, Taiwan (no. EMRP61101 N), and an informed consent was obtained from each participant prior to the initiation of the study. The temperature in the research lab was maintained at 25 °C with air-conditioners. Each subject was put in a sitting position and was asked to take a 5-min rest before the measurement. We measured the blood pressure before and after cold and hot stress tests by OMRON blood pressure monitor. During the measurement, each subject was asked to sit on an adjustable-height chair. The subject’s left hand was placed on a table at the same horizontal height as the heart with the palm pointing upwards and the wrist resting on a soft pillow. The superficial part of the radial artery of the left hand was forced up by the vibrator with a peak displacement of 0.15 mm. The frequency of the vibration was increased from 40 to 85 Hz at a step of 5 Hz. In the experiment, there were three phases, a baseline test, a cold stress test, and a hot stress test. In the baseline test, subjects were measured at room temperature. In the cold stress test, a plastic bag containing a mixture of ice and water (around 4 °C) was put on the inside surface part of the left forearm. Subjects were measured after the 5-min cold stress. Then, subjects were asked to take a 5-min rest in order to make the subjects’ hemodynamic variables stable. Another plastic bag filled with hot water at 42 °C was placed on the left forearm of the subject lasting for 5 min, and subjects were measured again.

### Data processing

For different sinusoidal frequencies, 12 specific time sections (indicated by T1–T12) within one cardiac cycle were selected based on the ECG and the reactive force signals. Each time section was a vibration cycle. The fifth time section (T5) was set at the maximum amplitude cycle of the reaction force signal within the systolic duration, and T1 and T12 were set at the beginning and end cycles of the reactive force signal, respectively. The duration between T1 and T5 and between T5 and T12 were both equally divided. T2, T3, and T4 were selected within T1–T5, and T6, T7, T8, T9, T10, and T11 were chosen within T5–T12. Then, the peak amplitude of the reactive force signal in each time section was defined as the *F*
_{
R_max
}. As explanation, Fig. 3 demonstrates how to determine the *F*
_{
R_max
} of the reactive force at the specific time section. A sinusoidal vibration with 40 Hz forced the radial artery, as shown in Fig. 3a. Meanwhile, a reactive force signal from the arterial wall was measured and displayed in Fig. 3b. The reactive force signal within one cardiac cycle was shown in Fig. 3c. Then, the one-cycle reactive force within the T8 was displayed in Fig. 3d. The peak amplitude of the reactive force was found within this time section. Therefore, each time section had one *F*
_{
R_max
}. Since it was difficult to control human body movement in the measuring duration, five values of *F*
_{
R_max
} extracted from five-cycle reactive force signal were averaged to yield one typical *F*
_{
R_max
}. Similarly, other wall elastances corresponding to different time sections were determined using their average. We did this measurement again after the frequency of the vibrator was changed.

According to Eq. (6), *F*
_{
R_max
} and ω^{2} have a linear relation which is illustrated in Fig. 4. Because we only considered the systolic duration, the five specific time sections (T1, T2, … and T5) were used to estimate the different elastances of the arterial wall (E1, E2,… and E5) under the different transmural pressure. It is worthwhile noting that all lines in Fig. 4 are in parallel and have the same slope that equals to the effective mass.

### Statistical analysis

The quantitative data are expressed as mean ± SD. To compare the changes of the elastance in the cold stress and hot stress with those at room temperature, a 2-tailed paired *t* test was used. A p value of 0.05 or less was considered statistically significant. Also, the degree of linear relationship between the two variables was represented by correlation coefficient in linear regression analysis using Sigma Plot 11.0 (Systat Software, Inc., USA).