Influence of low-frequency vibrations on blood flow improvement in human’s limbs

Abstract

The fundamental cause of diabetic limbs’ problem is insufficient blood supply. The aim of the current work was to experimentally and numerically investigate the blood flow velocity and pressure changes in the channel during vibrational excitation. The micro-scale Particle Image Velocimetry (μPIV) technique as well as corresponding numerical channel model in COMSOL Multiphysics software were used to investigate the influence of external vibrations. Momentum upstream flow were noted on the fluid that was influenced by vibrations. Furthermore, momentum flow velocity increased by more than 3 times in both experimentally and theoretically. These results show that suggested novel low-frequency vibrational excitation method should be investigated in clinical studies in case of improvement of blood circulation in human limbs.

Keywords

Vibrational exposure capillaries blood flow improvement diabetes mellitus

1. Background

International Diabetes Federation predicts that there will be 366 million of diabetics by 2030. About 90,000 are carried out of limbs amputation. Permanent blood stimulation and effective control of the problem can help protect the patient from further complications.

The scale of the circulatory problem is huge. This encourages the development of modern and innovative disease suppression tools to reduce patient disability, medical costs and to increase quality of life. The proposed methodology of low-frequency vibrational therapy, seeking to compensate the obstruction of blood circulation, especially in capillaries.

Vibrations’ influence on cardiovascular system is described in plenty of studies. Research made by Lythgo et al. (2009) [1] shows that blood flow in leg systematically increases during the vibration bouts. Osawa et al. (2011) [2] has identified that the whole body vibration therapy influences improvement of arterial blood flow and other cardiovascular parameters. Another study with twenty healthy adults [3] was performed on the vibrating platform imitating mechanical vibrations of 26 Hz. The mean blood flow velocity in the popliteal artery increased from 6.5 to 13.0 cm × s⁻¹. Baum et al. (2007) [4] conducted studies on type II diabetes patients, performing vibration exercise. They suggested that this type of exercise may be an effective and low time consuming tool to enhance glycaemic control. Investigation on vibration training on cardiovascular system has showed that capillaries are opened in order to keep a necessary level of cardiac output needed for the body [5]. The studies performed at Loma Linda University [6] indicated the significant increase in skin blood flow after five minutes duration vibrations of 30 Hz or 50 Hz. Another study at Loma Linda University [7] showed the increased skin blood flow after whole body vibration exercise at post intervention time intervals. However, the lack of investigations of the analysis of fluid parameters’ changes on affecting it by external vibrations were noted.

Papers of fluid velocity analysis methods on vibrational excitation were analyzed to select the most appropriate one. Hybrid experimental-numerical analysis techniques were used for investigation of micro spray systems. Laminar flow analysis in straight channel was made by Silva et al. (2009) [8] and Puccetti et al. (2014) [9]. The results of the studies have been compared to theoretical calculations and indicated a 4% difference between experimental and theoretical results. The accuracy of μPIV method was also determined by Wang et al. (2009) [10], who investigated different length sides of four square micro-channels with different Reynolds number (Re). μPIV measurements close to micro-channel wall with attached artificial thrombus were made by Tolouei et al. (2009) [11] and the results of similar study of Completo et al. (2014) [12] had showed that μPIV method could be used for the assessment of blood flow and thrombus geometry influence near the walls. In general, μPIV technology is the most appropriate on the investigations of fluid velocity parameters’ response to low frequency vibrations.

2. Objective

The objective of the current work was to experimentally and numerically investigate the blood flow velocity and pressure changes in the artificial blood vessel during low-frequency vibrational excitation.

3. Methods

Considering capillary as a hardly deformable structure, its displacement coincides with surrounding soft human tissue movement. Therefore, fluid velocity and pressure were measured, inside the artificial blood vessel that was induced by external vibrations of low-frequencies. A unique experimental setup has been assembled for this purpose. A numerical 2D model of the same parameters as the vibrating micro-channel has been designed for testing the influence of various parameters. The purpose of the numerical micro-channel model was to substitute and to reduce the demand of experimental setup for further studies.

3.1. Experimental setup

The majority of the previous studies where the vibrational training influence was analysed, were performed by using whole-body vibrating plates. Working regimes of these machines are more or less defined and limited. Amplitudes are ranging from 2 to 6 mm and frequencies ranging from 20 to 50 Hz [1–7]. However, it is noticed that further studies on influence of amplitude selection should be done. Martinez-Pardo et al. [13] had made a study investigating high and low amplitude effects on the development of strength, mechanical power of the lower limb, and body composition. They have found that higher amplitude (4 mm) of whole body vibrational training is essential when on purpose for improved fitness and a full workout. Concerning previous findings, experimental setup has been made focusing on the low-frequency vibrations (1–10 Hz) and higher amplitudes (1–20 mm).

Assembled experimental setup (Fig. 1) consists of μPIV system (Fig. 2), vibrating cantilever and measurement equipment. μPIV system (Fig. 2) consists of Nd: YAG laser (Dantec Dynamics), laser control system LPU 450 (Dantec Dynamics), 2048 × 2048 pixels FlowSense EO CCD camera (Dantec Dynamics) that was fixed on inverted Leica DM ILM microscope (Leica Microsystems). Prescribed fluid flow was generated by syringe pump (Aladdin AL4000, World Precision Instruments) with connected medical elastic tubing lines. Tubing line length starting from the syringe pump to microchannel, where measurements were performed, was 1.5-meter long. Micro-channel’s height (a) is 0.4 mm, and width (h) – 2 mm. The length of microchannel is 40 mm. About 30 cm away from the syringe pump, tubing line was fixed by oscillating cantilever where electromechanical actuator was used to generate the beating phenomenon and sufficient displacement. As it was defined on our earlier study, electric motors could generate force up to 41.48 N during beating stage [14]. Oscillations and movement amplitudes of the cantilever were measured with inductive displacement sensor IFM IG6084. The output signal was collected with oscilloscope Picoscope 3424. Measurements of the fluid parameters were made one minute after the start of vibrational excitation.

Fig. 1.

(a) Experimental setup: (1) – Leica DM ILM microscope; (2) – Syringe pump; (3) – YAG laser; (4) – DC power supply; (5) – Vibrating beam with two electric motors and attached unbalanced masses; (6) – Medical tubes; (7) – Oscilloscope Picoscope 3424; (8) – Inductive displacement sensor IFM IG6084. (b) Scheme of measurement setup.

Fig. 2.

Schematic drawing of μPIV system.

The artificial blood vessel that was used for experiments, was made from special vascular graft which is made of an expanded polytetrafluoroethylene vascular prosthesis and is produced by USA GORE Company.

Velocity of the fluid flow is measured by applying spatial flow lightning and registering the movement of indicating particles of 1.0 μm diameter. Velocity vectors’ field of the particles were determined by the μPIV measurements. Knowing the time interval between taking pictures $Δ t$ and the displacement of the particles $Δ x$ , the velocity could be calculated by the following equation: $\begin{matrix} (1) & v = \frac{Δ x}{Δ t} . \end{matrix}$

The principal scheme of image analysis is shown below (Fig. 3). Two successive particle images are recorded within known time interval. Image field is divided into interrogation regions of a certain size. Each region is transformed from real to complex domain using fast Fourier transform. Later, cross-correlation function (2) [15,16] is calculated for each region and product is transformed back to the real domain using inverse fast Fourier transform. In this manner, the displacement of all the particles within interrogation region is calculated.

Fig. 3.

Principal scheme of μPIV correlation establishment.

Calculating the Φ value, $\begin{matrix} (2) & Φ (x, y) = \sum_{x = 1}^{p} \sum_{y = 1}^{q} f (x, y) \cdot g (x + m, y + n), \end{matrix}$ where p and q – net window dimensions x and y direction, $f (x, y)$ and $g (x, y)$ – the image intensity functions on the first and the second frames, m and n – the displacement coordination, $Φ (x, y)$ – the correlation function between two windows of the net ( $p \times q$ size) with mutual displacement $(x, y)$ [15].

Furthermore, the investigation of blood flow pressure response on the vibrational excitation was performed. The pressure gauge Mpx5010gp was selected for the measurements because of the accuracy (4.413 mV/mm H₂O) and measuring range (0–10 kPa).

3.2. Computer modelling

Operations with coupled systems of partial differential equations were performed with Comsol Multiphysics software. At each computational step, the fluid flow field and the structure have been evolved as a coupled system. Flow and structure interaction forces were immediately accounted and their resultant motions enforced in each step. Literature of similar micro-channels’ models [8–10,16,17] were overviewed before the beginning of designing process.

Computational model is described by the fluid-structure interaction problem which can be defined by Ω, including structural domain $Ω_{s}$ and the fluid domain $Ω_{f}$ , with an external boundary Γ. The fluid-structure interaction is defined by $Γ_{s} = Ω_{s} \cap Ω_{f}$ . Fluid and structure dynamics can be defined as a result of the D’Alembert’s principle: $\begin{matrix} (3) & ρ {\dot{v}}_{i} - σ_{i j, j} + f_{i} = 0, \end{matrix}$ where $f_{i}$ is the body force, ρ is the mass density and $σ_{i j, j}$ represents the stress component. In the Structural domain, the equation is defined as $\begin{matrix} (4) & ρ^{s} {\dot{v}}_{i j, j}^{s} + f_{i}^{s} = 0, in {\overline{Ω}}_{s}, \end{matrix}$ where the superscript, s, labels the amount linked with the structure. The velocity, $v_{i}^{s}$ is the material time derivative of the displacement field $u_{i}^{s}$ , i.e., $v_{i}^{s} = {\dot{u}}_{i}^{s}$ . Equation (4) is usually used to describe the Lagrangian theorem. The first term of Equation (4) is linked with inertia and the second one – with internal stresses. When the material is defined as linear elastic, the structural stress will be described by Hooke’s law; i.e., $\begin{matrix} (5) & σ_{i j}^{s} = λ δ_{i j} ε_{l l} + 2 G ε_{i j}, \end{matrix}$ where the structural stress $σ_{i j}^{s}$ is a function of the strains, $ε_{i j}$ and the Lame constants λ and G, which are determined by $\begin{array}{l} (6) & ε_{i j} = \frac{1}{2} (u_{i, j} + u_{j, i}), \\ (7) & G = \frac{E}{2 (1 + v)}, \\ (8) & λ = \frac{E v}{(1 + v) (1 - 2 v)}, \end{array}$ where E is the Young’s modulus and v is the Poisson’s ratio. In the fluid domain, the equation is written as: $\begin{matrix} (9) & ρ^{f} {\dot{v}}_{i}^{f} - σ_{i j, j}^{f} + f_{i}^{f} = 0, in {\overline{Ω}}_{f} \end{matrix}$ which is usually standing for the Eulerian description. In the inertia term, equation is given by: $\begin{matrix} (10) & {\dot{v}}_{i}^{f} = \frac{d v_{i}^{f}}{d t} = \frac{\partial v_{i}^{f}}{\partial t} + v_{j}^{f} v_{i, j}^{f} . \end{matrix}$ In the case of incompressible Newtonian fluid model, the fluid stress $σ_{i j}^{f}$ is given by $\begin{matrix} (11) & σ_{i j}^{f} = - p δ_{i j} + τ_{i j}, \end{matrix}$ where $\begin{array}{l} (12) & τ_{i j} = 2 μ (e_{i j} - \frac{δ_{i j} e_{k k}}{3}), \\ (13) & e_{i j} = (v_{j, i}^{f} + v_{i, j}^{f}) . \end{array}$

The p is the static pressure used to enforce the incompressibility condition, $v_{i, i}^{f} = 0$ .

The no-slip state on the fluid-structure interaction $Γ_{s}$ is maintained by defining Dirichlet and Neumann conditions as: $\begin{array}{l} (14) & v_{i}^{s} = v_{i}^{f}, on Γ_{s}, \\ (15) & σ_{i j}^{s} n_{i} = σ_{i j}^{f} n_{i}, on Γ_{s} . \end{array}$ Equation (15) is in fact the differentiation of the displacement condition that both fields share the same interface, $\begin{matrix} (16) & x_{i}^{s} = x_{i}^{f}, on Γ_{s} . \end{matrix}$ In some cases, Equation (16) is being used instead of (14).

2D model (Fig. 4(a)) was created to investigate alterations of fluid properties while affecting it on various vibrational action. The channel model was built with reference to vascular graft tubes used for experimental setup. Part of the 0.15 m length channel was placed in human tissue properties’ imitating model.

Oscillations of a human tissue imitational model were defined by sine waveform function. Displacement of 0.1 ÷ 8 mm on the Y-axis and frequencies from 1 to 6 Hz have been determined. Variability of oscillations of the material, imitating the human tissue, was enabled by changing angular frequency values of sine waveform. The fluid material was imposed with reference to the experimental setup. Prescribed mesh displacement of 0 mm on the X-axis for the inlet and the outlet of the channel have been set. Prescribed mesh displacement of 0 mm on the Y-axis was imposed for longitudinal channel boundaries excluding fluid-solid interface boundaries. Roller movement restriction of the oscillating part of the model was defined.

Fig. 4.

2D (a) COMSOL Multiphysics software models of fluid filled micro-channel placed in human tissue, (b) schematic drawing.

Variations of fluid velocity were monitored at the core of the micro-channel. Inlet normal mean fluid velocity was equal to 0.4 mm/s. No slip boundary condition was designated on the all walls. The displacement at the Y-axis with the waveform function were defined to imitate oscillations. 10 seconds duration time dependent study with the step of 0.1 second was specified for all the calculations. The structure imitating horizontal human tissue was a flexible material with the following parameters: the density of 30 kg/m³, Young’s modulus of 25 MPa and Poisson’s ratio of 0.5 [18,19].

The physical mechanism of the influence of vibrating wall to fluid is well known and analogous velocity profiles have been presented by other authors [20–25]. Vibrations’ induced transversal pressure waves, which amplitude rapidly decreases receding from vibrating surface. It means that vibrations’ influence changes across channel’s height (cross-section). Also, fluid flow is an inertial system, which velocity response to pressure variation delays. Womersley number (Wo) (17) defines flow non-stationarity caused by pressure pulsation. $\begin{matrix} (17) & Wo = L \sqrt{\frac{ω}{ν}} . \end{matrix}$ Here, L – appropriate length scale (hydraulic diameter), m; ω – angular frequency, rad/s; ν – kinematic viscosity, m²/s.

Shintaku and others [21] have described vibrating flow by using the following boundary conditions: $\begin{matrix} (18) & ψ_{1}^{*} = - \frac{1}{2} {(\frac{R_{1}}{R_{2}})}^{2} [1 - u_{z, tube}^{*} (t^{*})], \end{matrix}$ where $u_{z, tube}^{*}$ indicates vibrating tube’s velocity: $\begin{matrix} (19) & u_{z, tube}^{*} (t) = \frac{2 π A}{R_{0}} f^{*} sin (2 π f^{*} t^{*}), \end{matrix}$ where $f^{*}$ is frequency of the vibration: $\begin{matrix} (20) & f^{*} = R_{0} f / U . \end{matrix}$

$ω^{*}$ at uniform surface concentration, vibrating tube wall and zero flux condition are calculated as following: $\begin{matrix} (21) & ω^{*} = \frac{u_{z, tube}^{*} (t^{*})}{r^{*}} + \frac{1}{r^{*}} \frac{\partial^{2} ψ^{*}}{\partial r^{* 2}} . \end{matrix}$

At the inlet and the outlet of the blood, the fully developed flow, which can be described as $\frac{\partial u_{r}^{*}}{\partial z^{*}} = \frac{\partial u_{z}^{*}}{\partial z^{*}} = 0$ , is assumed and the condition can be described as: $\begin{matrix} (22) & \frac{\partial ψ^{*}}{\partial z^{*}} = \frac{\partial ω^{*}}{\partial z^{*}} = 0 . \end{matrix}$

4. Results

First, the fluid pressure changes in the effect of external vibrations were analysed. The highest rise of fluid pressure mean, using one motor gained a 3 kPa increase in value. Furthermore, two motors supplied by 5.1 V and 4.9 V respectively induced beating phenomenon and generated a rise of 8 kPa (Fig. 5). The most influential frequencies of the beating phenomenon were noted in the diapason of 2 to 5 Hz. The displacement amplitudes, ranging from 4 to 20 mm influence pressure alterations the most.

Fig. 5.

Results of using artificial blood vessel: two motors’ induced beating phenomenon and 8 kPa pressure rise (upper graph); one motor vibrations and 3 kPa pressure rise (lower graph).

Generated flow rate in microchannel was equal to 0.04 ml/min, Re_Dh based on hydraulic diameter of microchannel, was Re_Dh ∼ 0.5, which corresponds to blood flow in arterioles.

A certain amount of measurements at different vibration frequencies and amplitudes were analysed. In the case of stationary flow, parabolic flow velocity profile was obtained at Re_Dh = 0.5. Mean velocity increase was observed at vibration frequencies of 4.3 Hz, 4.9 Hz, 5.4 Hz and 5.7 Hz, and amplitudes from 1 to 6 mm. The decrease of mean velocity was observed at 6.3 Hz and higher frequencies. At low vibration frequencies (2–4 Hz) and high amplitudes (> 8.5 mm) no significant changes in mean velocity were observed.

Analysis of the μPIV experimental results revealed velocity magnitude and direction changes at different instants of one pulse period (Fig. 6). In the case of 4.3 Hz, Wo ≈ 3.5. Wo indicates that pulsation frequency in channel is high and the shift (lag) between vibration induced pressure and velocity in channel is approximately $π / 2$ [20,23]. Because of the low flow velocity and certain pulsation amplitude, reverse flow and recirculation zone at near-wall region can occur.

Fig. 6.

Instant velocity vectors’ fields at different cycle phases. 4.3 Hz, 6 mm.

Fig. 6.

(Continued.)

The case of 4.3 Hz frequency and 6 mm amplitude was analysed in detail. During single vibration period, five μPIV images were obtained. At the beginning of positive phase of induced pressure gradient (Fig. 6(a)) flow velocity profile is parabolic. Instant maximum velocity on the channel axis equals to ∼1 mm/s and Re_Dh ∼ 0.15. At the later time moment (Fig. 6(b)), pressure gradient and velocity is still increasing, and flattening of velocity profile is observed. At the end of positive pressure gradient phase (Fig. 6(c)), flow is the most accelerated and obtained velocity profile is flat, which represents disturbed flow. The instant maximum velocity in the channels axis increases up to 3 times compared to initial momentum maximum velocity and about up to 3.5 times compared to average maximum velocity (Fig. 7).

During the phase of induced negative pressure gradient in microchannel (Fig. 6(d)), the change of velocity direction is observed. The shape of velocity profile can be explained by high pressure gradients across the microchannel and the lag of fluid in axial plane, which leads to larger velocity magnitudes at near-wall region. At the last phase of cycle (Fig. 6(e)), flow composed of both downstream and upstream flows are observed. High velocity gradients occurs between these flows and recirculation zone with vortices is formed. Individual vortices moving downstream are observed.

Comparison of mean velocity profiles of vibrating and stationary flows presents velocity increase up to 1.4 times and changes of Re_Dh from 0.5 in the case of stationary flow to 0.9 at vibrating flow.

Velocity variations were observed on the COMSOL Multiphysics computational model after affecting the fluid with different external frequencies. The peak fluid velocity of oscillating model on 4.3 Hz vibrations was equal to 2.74 mm/s, while the maximum value of all the gathered data reached 3.23 mm/s and the average velocity during the 4.3 Hz frequency was 0.98 mm/s. It was also noted that in some moments the direction of the fluid flow was upstream. The studies where the frequencies of higher than 20 Hz were used together with low displacement amplitude (up to 1 mm) showed non-significant velocity changes. Instant velocity vectors’ fields at different cycle phases were very similar with experimental ones (Fig. 8).

Fig. 7.

(a) Average velocity vector field on 4.3 Hz and 6 mm. (b) Average velocity profile at 4.3 Hz and stationary flow.

Fig. 8.

Modelling results of instant velocity vectors’ fields at different cycle phases. 4.3 Hz, 6 mm.

5. Discussion

The uniqueness of this study was to investigate the influence on blood perfusion of various type of vibrational excitation which hasn’t been done before. Furthermore, the previous studies of vibrational excitation on human tissue were conducted by monitoring the influence on physiological parameters without depth investigation on processes inside blood vessels or capillaries. The number of studies [1–7] indicate positive effect of whole body vibrations exercises while performing them in the range of higher frequencies and lower amplitudes. However, our proposed method is based on lower frequencies (up to 5.8 Hz) and higher amplitudes (up to 20 mm) and could be particularly concentrated on blood perfusion in arterioles and capillaries. Furthermore, our results show the importance of the amplitude of the oscillating part and the frequency value. Our findings show why corresponding vibrations enhance blood circulation on separate vessels of human limbs and likely could be used as a novel method to increase blood flow on diabetic limbs. Moreover, local excitations by using our proposed method could eliminate negative effect of whole body vibrations.

Ability to increase blood flow velocity could be essential in solving blood circulation problems caused by diabetes or other circulatory diseases. Results of computer modelling and experiment showed adequacy of the mathematical model to physical one. These novel computer models enable to simplify investigations by shifting parameters as less time consuming method. In further investigations of the flowing erythrocyte in the vibrating arteriole and capillary, different material properties of erythrocyte should be used with reference to biochemical changes in membrane structure in type 2 diabetes [26].

6. Conclusions

Obtained results of experimental and numerical studies indicated the increase of momentum fluid flow velocity of more than 3 times during oscillations of 4.3 Hz and displacement of 6 mm. The increase in fluid velocity was noted on major amplitudes from 3.4 mm and up to 8 mm. During the experiment the highest velocity changes were obtained on measurements of 4.3 Hz and 6 mm, 5 Hz and 5.4 mm, 5.4 Hz and 8 mm, 4.8 Hz and 3.4 mm, 5.8 Hz and 6 mm. However, no significant change was recorded when frequencies higher than 49 Hz occurred with displacement amplitude of <1 mm. The margin of experimental and modelling results was 3.1%.

Footnotes

Acknowledgements

This research was funded by a grant (No. SEN-10/15) from the Research Council of Lithuania. The author also wants to express his/her appreciation to Lithuanian Energy Institute.

Conflict of interest

The authors have no conflict of interest to report.

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