Effect of Viscoelasticity on Arterial-Like Pulsatile Flow Dynamics and Energy

Abstract

Time-dependent arterial wall property is an important but difficult topic in vascular mechanics. Hysteresis, which appears during the measurement of arterial pressure–diameter relationship through a cardiac cycle, has been used to indicate time-dependent mechanics of arteries. However, the cause–effect relationship between viscoelastic (VE) properties of the arterial wall and hemodynamics, particularly the viscous contribution to hemodynamics, remains challenging. Herein, we show direct comparisons between elastic (E) (loss/storage < 0.1) and highly viscoelastic (loss/storage > 0.45) conduit structures with arterial-like compliance, in terms of their capability of altering pulsatile flow, wall shear, and energy level. Conduits were made from varying ratio of vinyl- and methyl-terminated poly(dimethylsiloxane) and were fit in a mimetic circulatory system measuring volumetric flow, pressure, and strain. Results indicated that when compared to elastic conduits, viscoelastic conduits attenuated lumen distension waveforms, producing an average of 11% greater cross-sectional area throughout a mimetic cardiac cycle. In response to such changes in lumen diameter strain, pressure and volumetric flow waves in viscoelastic conduits decreased by 3.9% and 6%, respectively, in the peak-to-peak amplitude. Importantly, the pulsatile waveforms for both diameter strain and volumetric flow demonstrated greater temporal alignment in viscoelastic conduits due to pulsation attenuation, resulting in 25% decrease in the oscillation of wall shear stress (WSS). We hope these findings may be used to further examine time-dependent arterial properties in disease prognosis and progression, as well as their use in vascular graft design.

1. Introduction

The importance of arterial viscoelasticity may be best illustrated by comparison of healthy to both diseased and treated states. Viscoelastic (VE) arterial hysteresis has become a metric of health [1,2], coinciding with application of time-dependent materials in arterial replacements [3,4], such as vascular grafts [5,6]. During the progression of arterial diseases, a known phenomenon is increased pressure modulus (E_P) of the artery [7,8]. Additionally, recent studies show significant decreases in hysteresis loops during arterial diseases [2]. What remains unclear, however, is the importance of arterial viscoelastic properties to arterial flow dynamics and thus arterial health. Similar questions also apply to arterial disease treatments such as arterial grafting. In studies comparing the human femoral artery to graft materials including cryogenically preserved artery, saphenous vein, and polytetrafluoroethylene graft, the viscoelastic response of graft materials correlated with the success rate, though underlying mechanisms were unknown [9–11]. While viscoelastic biomaterials are now being explored in vascular grafting, the question on whether increasing viscous response of the material could improve mechanical and biological responses remains largely unexplored [6]. Therefore, unraveling the interaction between viscoelastic properties of blood vessels and flow dynamics will be important to advance understandings of vascular mechanobiology and improve vascular implant design.

Effects of arterial wall time-dependent properties on hemodynamics have been a difficult question for the past 60 years. Branching of the arterial tree, arterial integration into surrounding tissues, and hierarchical structure on the arterial wall all contribute to the complexity of measuring such effects in situ or ex vivo. Arterial tissue consists of multiple layers, with varying time-dependent stiffness responses and differing roles in arterial loading [12–14]. Stress–strain relationship of each tissue layer additionally changes with arterial geometry [15] as well as boundary conditions created by tethering of the arterial adventitia to the surrounding tissue [16]. Additionally, healthy, native arteries express a frequency-dependent viscoelastic response and a viscoelastic ratio (tan(δ) = E″/E′) of approximately 0.1–0.15 [9]. Through significant strain and hysteresis, arteries were shown to provide protective dampening for the circulatory system [2,9,17–19]. Though important, measurements taken in situ or ex vivo do not provide the cause–effect relationship between time-dependent mechanical properties of the arterial wall and hemodynamics.

In computational models, viscoelasticity has been shown to attenuate pulsatile flow waves over long distances [19–24]. Of the previous works that experimentally validated flow models, most did not independently characterize the tube material properties [22], and all used materials with viscous responses that were only a slight fraction (tan(δ) ∼ 0.02) of the overall stiffness response [22,24]. Also, the contributions of viscous and elastic (E) components to hemodynamics are yet to be separated experimentally to quantify the impact brought by changes in arterial viscoelasticity during disease or therapy. Additionally, though pressure wave attenuation was previously studied, the variation in stiffness with time-dependent loading, the time-lag of the wall strain due to the viscoelastic nature of artery, as well as their coupling with the flow waveforms has not been well understood [22,24,25]. As the wall shear stress (WSS), a primary signal to vascular endothelial cells and vascular health [26–29], is determined by volumetric flow and cross-sectional area, the phasic alignment of the strain and volumetric flow waveforms is important to determine impact of viscoelastic materials on maintaining healthy hemodynamics. However, that aspect has seldom been examined previously.

This study has examined the effects of conduit viscoelasticity on flow dynamics, including flow and pressure waveforms both upstream and downstream of the conduit. Different from previous studies, we have constructed conduits with arterial-like compliance and variable viscoelasticity, allowing a direct comparison of flow dynamics between elastic and viscoelastic conduits. Conduit compliance and viscoelastic material properties may be separated and compared, with the advent of materials in which overall stiffness and tan(δ) may be controlled [30,31]. Arterial mechanics and hemodynamics have both been shown to play important roles in disease pathology [2,8,9,28,29]. Understanding the interplay between flow dynamics and viscoelastic tubing may aid in establishing viscous wall response of arterial tissue as an important signal in arterial health, as well as defining this important parameter in arterial graft design and development.

2. Materials and Methods

2.1. Material Design, Fabrication, and Characterization

2.1.1. Material Design Overview.

Clinical measures including pressure modulus (E_P), arterial distensibility, and compliance all indicated an increase in the arterial stiffness during the process of arterial disease [2,7,8]. Viscoelasticity, as measured by the ratio of viscous to elastic response (tan δ = E″/E′) within the frequency range of 1–10 Hz, was previously measured in healthy femoral, carotid, and coronary arteries, ranging between 0.1 and 0.15, though some studies report values as high as 0.3 [9,32,33]. In the aged and diseased conditions, hysteresis area has shown a marked decline [2,18], which is supported by decreased tan(δ). Additionally, comparison of arterial graft viscoelasticity in limited examples available indicates synthetic materials such as polytetrafluoroethylene, polycaprolactone, and polylactic acid have negligible wall damping capabilities, or tan(δ) approaching to zero [17]. Therefore, the impact of viscoelasticity on vascular health may be studied by varying tan(δ) while maintaining a similar stiffness. In particular, to evaluate such impact on flow parameters and relate them to clinical measures, we determine dynamic compliance (E_P), inlet and outlet pressures, mean volumetric flow, and pulsatility index (PI) as readouts, which may further aid in future cell studies in simulated disease conditions. Therefore, we refer to previous studies on artery stiffness measures such as carotid artery and use these as guidelines for designing the material parameters of proposed conduit [7,34–36]. Specifically, the proposed viscoelasticity of fabricated artery-like conduits has E_P* = 800–1200 mmHg, while tan(δ) related directly to viscoelastic to elastic ratios takes tan(δ) < 0.1 for elastic model and tan(δ) > 0.45 for viscoelastic model. By exaggerating viscoelasticity, we may uncouple the time-dependent effects of both stiffness and viscoelasticity on fluid wave travel. Detailed material characterizations for the mechanics of thick-walled pressure vessel mechanics are given in Sec. A1 available in the Supplemental Materials on the ASME Digital Collection.

2.1.2. Conduit Design.

To study the effect of vessel lumen mechanics on flow dynamics, a simple design (Fig. 1(a)) was used to maintain material viscosity with controlled stiffness. The design employs an external rigid sheath of polypropylene to prevent radial strain. Strain response to this can be modeled through the equation for wall motion in thick-walled cylinders:

$$u_r\left(r\right)=\frac{1-\nu }{E^{\mathrm{*}}}\frac{(r_i^2{p}_i-r_o^2{p}_o)r}{r_o^2-r_i^2}+\frac{1+\nu }{E^{\mathrm{*}}}\frac{(p_i-p_o)r_i^2{r}_o^2}{\left(r_o^2-r_i^2\right)r}$$ (1)

Fig. 1.

Mock vessel design. (a) The design is illustrated with shaded molecular structure of swollen PDMS on the left: tetrafunctional crosslinker and elastic vinyl-PDMS (black), solvent methyl-PDMS. (b) The lumen radius was modeled by coupling pressure and material properties for both purely elastic and viscoelastic materials. Diameter strain was projected for independent effects of both viscoelastic properties (dynamic modulus: |G*|; viscoelastic ratio: tan(δ)) indicating both attenuate strain. Maximal attenuation was projected with combined viscoelastic properties (|G*|*tan(δ)) with |G*| have the greatest effect.

where r_i and r_o are the inner and outer radii, respectively, r is the radial coordinate, and p_i and p_o are the inner and outer pressures, respectively. Material properties are both the shear modulus G* and the Poisson's ratio ν. By setting $u_r\left(r_o\right)=0$ and ν to 0.5 (for poly(dimethyl siloxane) (PDMS)), we may then solve for p_o in terms of p_i, r_i, and r_o, and subsequently solve for $u_r\left(r_i\right)$

$$u_r\left(P\right)={\mathcal{F}}^{-1}\left\{\frac{\mathcal{F}\left\{P\right\}}{E^{\mathrm{*}}}\right\}\frac{r_i{\left(r_i^2-r_o^2\right)}^2}{2(r_o^2-r_i^2)(r_o^2+3r_i^2)}$$ (2)

We then have a geometric factor as a relationship between the inner (r_i) and outer (r_o) radii on the right side of the equation, as well as our pressure/modulus relationship $\left({\mathcal{F}}^{-1}\right\{\frac{\mathcal{F}\left\{\mathit{P}\right\}}{{\mathit{E}}^{*}}\left\}\right)$. We may then use target material properties (as described above) to project diameter strain from a pressure wave cycle. In the projected diameter strain from this relationship, we find a more gradual return to the strain minimum with any time-dependent property introduced (either |E*| or tan(δ), independently or combined: Fig. 1(b)). Increasing the total cross-sectional area throughout the cycle could decrease peak pressure and flow velocity, due to decreased impedance.

2.1.3. Material Synthesis and Conduit Fabrication.

Material used to cater both stiffness (E*) and viscoelasticity (tan δ) is PDMS swollen gels, previously described in the literature [30,31]. Different termination groups within PDMS are separated into the elastic network (vinyl-terminated) and swelling solvent (methyl-terminated). Vinyl-terminated PDMS (v-PDMS) is crosslinked through a tetrafunctional vinyl crosslinker, which additionally prevents methyl-terminated PDMS (m-PDMS) from bonding to the network. The elastic network stiffness may be catered through both crosslink density and chain entanglements, as illustrated in Fig. 1(a). Solvent (m-PDMS), which may be controlled by m-PDMS molecular weight, is designed to increase viscous dissipation within the material while contributing to stiffness through entanglements. Swollen gels were fabricated through methods outlined previously in the literature [30,31]. Herein, vinyl-terminated PDMS (M_W: 108 kDa) was from Siltech Corporation (Toronto, ON, Canada), methyl-terminated PDMS (Viscosity: 500 CSt and 1 M CSt) from Clearco Products, Inc., (Willow Grove, PA), tetrakis(dimethylsiloxy)silane and platinum catalyst (Pt(0)-2,4,6,8-tetramethyl-2,4,6,8-tetravinylcyclotetrasiloxane) from Sigma-Aldrich (St Louis, MO). Two groups of samples were created, elastic (58.8% 500 CSt m-PDMS: 108 kDa v-PDMS) and viscoelastic (70% 1 M CSt m-PDMS: 108 kDa v-PDMS). Because low molecular weight solvent is below the entanglement weight of PDMS, it has little contribution to the stiffness or viscous response. It may also be treated as a simple theta solvent within the swollen gel [30,31]. This additionally allows v-PDMS/m-PDMS fraction (ϕ) to be used in determining the material stiffness ($\left|G^{*}\right|\propto {\phi }^{2.3}$) catered to match viscoelastic PDMS material response.

Fabrication of the tube conduit starts with mixing v-PDMS and m-PDMS by differing weight percentage, i.e., viscoelastic (PDMS-70% (108k-1M CSt)) and elastic (PDMS-58.8% (108k-500 CSt)), at room temperature. Crosslinker was then added at a stoichiometric ratio of 4:1 crosslinker functional group to vinyl chain termination, and mechanically mixed for 15 min. Subsequently, platinum catalyst was added at a concentration of 800 ppm, and mechanically mixed for 10 min. The mixture was placed in vacuum and degassed until all visible gas bubbles were removed. Prepared mixture (∼45 g) was added into a 50 mL tube mold. Air was removed by centrifugation for 20 min. Then, a polymethylmethacrylate rod (R = 0.265 cm) was inserted into the material before curing it to allow the formation of a lumen structure. To prevent collapse of the material upon removing the rod, an elastic layer was created to reduce viscous shear on the rod during removal. To create this elastic layer, pure v-PDMS material was mixed as described above with the change of a stoichiometric ratio of vinyl crosslinker to v-PDMS end group of 3:1. This v-PDMS material was mixed in a 1:1 ratio by volume of n-heptane (Fisher Scientific, Co., Pittsburgh, PA) to decrease solution viscosity. Rods were then dip-coated and hung to dry in vacuum for 1 h to allow n-heptane evaporation. After adequate evaporation time, v-PDMS was then cured for 1 h at 50 °C, creating a fully cured elastic layer approximately 70–100 microns in thickness. Coated polymethylmethacrylate rods were then inserted into the material and centered using a PDMS washer. Conduits were then crosslinked cured in vacuum at 70 °C for 72 h. Finally, bottom polypropylene material of the tube mold was removed, and PDMS was capped using v-PDMS material. Rods were then removed using ethanol as a lubricant, and lumens were left intact. To finalize conduit construction, 3/8″ polypropylene luer connections were inserted in lumen ends, and again sealed using v-PDMS material. Rigid outer diameter material of the polypropylene tube was then sectioned along the outer circumference.

2.1.4. Material Characterization.

Material mechanical properties were evaluated using the ARES TA rheometer (TA Instruments, New Castle, DE). Contact force was brought to 0.5 N, and material tested at room temperature, in ambient conditions. Three tests were completed: strain sweep, stress relaxation, and frequency sweep. Strain sweep was performed at a frequency of 1 Hz, with increasing strain over the range of 0.1–50% strain, indicating strain softening occurrence at approximately 10%. Based on strain sweep results, stress relaxation was measured at 1%, 5%, and 10% strain, and final modulus values recorded as steady-state moduli. Frequency sweep tests were also performed over the range 0.1 Hz–40 Hz, and complex modulus recorded. Elastic PDMS was further evaluated using the TA Q800 rheometer (TA Instruments) dynamic mechanical analysis. Briefly, samples were placed in 0.01 N of tension within the apparatus and cycled at 1% strain at 1 Hz for 6 min. Final modulus was recorded as the average over this time period.

2.2. Flow Dynamics

2.2.1. Fluid/Material Coupling.

Coupling of the fluid pressure and the material frequency response may be through the assumption of a mathematically linear relationship within the frequency domain. Complex waves may be separated through Fourier Transform of the given waveform and applied to known material frequency dependence. This separates the complex waveform into a summation of sine waves, with discrete amplitude and phase shift, represented by a complex number. This may then be coupled to material data consisting of a complex modulus, and reverted to the time domain

$$\epsilon \left(t\right)={\mathcal{F}}^{-1}\left\{\frac{\mathcal{F}\left\{P\right\}}{3\hspace{0.17em}{G}^{\mathrm{*}}}\right\}\hspace{0.17em}={\mathcal{F}}^{-1}\left\{\frac{\mathcal{F}\left\{P\right\}}{{3|G}^{\mathrm{*}}\left|\exp \right(-i\theta )}\right\}$$ (3)

This allows us to examine the dominant frequency range in considering material testing. The frequency spectrum of the waveform created by the pulsatile pump is shown in Sec. A2 available in the Supplemental Materials on the ASME Digital Collection. The dominant frequency is 1 Hz, and the harmonics created by the wave extend out to 9 Hz, which is typical of cardiac waveforms [16]. This is within the range measured by the frequency sweep of the material (0.1–40 Hz).

2.2.2. Applied Fluid Parameters.

Flow was tested using de-ionized water with red food dye within a benchtop flow system, terminating in an open reservoir. Pulsatile flow was applied through a pulsatile flow pump (Harvard Apparatus, Holliston, MA), which controls volumetric flow rate through applying a driving piston. Pulsatile pressure range and flow pulsatility are controlled through the introduction of a compliance chamber before the mock vessel, introducing a Windkessel pressure damping, as described previously [37]. Flow response under two pressure ranges were recorded: 30 mmHg (low pressure flow (LPF)) and 70 mmHg (high pressure flow (HPF)). Pressure ranges were changed through altering compliance chamber levels. Compliance chamber levels were normalized for each pressure range using Tygon B-44-4X tubing (Saint Gobain S.A., Courbevoie, France) as an effectively rigid conduit, applying 4.67 cc/s flow rate at a frequency of 1 Hz. Flow was connected through 0.265 cm radius silicone tubing, also treated as rigid.

Volumetric flow rate was calculated through the intended applied WSS of 1.0 Pa, using the equation for laminar shear at the wall of a tube

$$\mathrm{WSS}=-\mu \frac{\partial \stackrel{\rightharpoonup }{u}}{\partial r}=\mu \frac{3\hspace{0.17em}\stackrel{\rightharpoonup }{Q}}{\pi \hspace{0.17em}{R}^3}\to \stackrel{\rightharpoonup }{Q}=\frac{\mathrm{WSS}\hspace{0.17em}\pi \hspace{0.17em}{R}^3}{3\hspace{0.17em}\mu }$$ (4)

Here, μ is the dynamic viscosity, u is the velocity, r is the location along the radial axis, Q is the volumetric flow, and R is the lumen radius. Therefore, assuming laminar, fully developed flow, a radius of 2.65 mm, and viscosity of 4 × 10⁻³Pa·s (viscosity of blood), resulting volumetric flow (Q) is 4.67 cc/s at a frequency of 1 Hz. Following energy analysis does not focus on viscous loss in the fluid as μ is a controlled constant across experiments; therefore, blood viscosity was used in determining WSS to ensure volumetric flow rate was within physiologic levels. Flow rate, frequency, and WSS all match physiological levels as measured in large animal models of common carotid arteries described in the literature [38].

2.2.3. Fluid Measurement.

The test bench fluid circuit was designed to reduce wave reflection and allow for pressure and volumetric flow measures through multiple cycles (Fig. 2). Pressure (P) was measured using Omega Engineering sensors (PX209-015G15V, Omega Engineering, Inc., Norwalk, CT) and fitted with nylon (polyamine) tee-joints. To increase P-measure accuracy, sensors were placed nearest to conduit terminals to reduce head loss from connector tubing.

Fig. 2.

Schematic representation of the fluid system setup. Fluid travels between two open reservoirs, driven by a pulsatile pressure pump, with pressure ranges controlled by a Windkessel reservoir at the inlet. Pressure (P) and volumetric flow (Q) are measured at both the inlet and outlet of the conduit.

Volumetric flow rate (Q) was measured concurrently with pressure (Sonoflow CO 55/060, Sonotec USA, Islandia, NY). Due to constraints of the flow meter, smaller connective tubing diameter was required at the point of measurement, with inner diameter of 1/16″. To avoid wave reflection and increased impedance, volumetric flow was measured in parallel to the conduit, connected through nylon tee-junctions of 0.265 cm inner radius. The measured flow waveform perpendicular to the flow is not directly proportional to the flow in line with the conduit, due to the tee-junction head loss, and impedance change resulting from different diameter of the connection tubing and Q-sensor [16,39] (Paritosh et al. 2007). Therefore, Q-waveform amplitudes were corrected for, as available in the Supplemental Materials on the ASME Digital Collection. As volumetric flow rate is minimally affected by head pressures, inlet Q-sensor was placed before the inlet P-sensor and the outlet Q-sensor was placed after the outlet P-sensor. Finally, area under Q curves were determined to estimate total volume during the recorded data cycle. Both Q _In and Q _Out values were normalized to the area to reduce variance from the sensor between data recordings. This maintained the conservation of mass within the system between inlet and outlet.

Pressure and volumetric flow data were sampled at 225 Hz and recorded on a real-time target (RT-Target) board (sb-RIO 9392, National Instruments, Austin, TX), which then packaged and sent data to a personal computer central processing unit (CPU) (Dell, Inc., Round Rock, TX). Volumetric flow signal passed through a resistor/capacitor low-pass signal filter, with a cutoff frequency of 22 Hz, and resulting capacitor phase lag was corrected for. Upon startup, the RT-target internal clock was synchronized with the CPU clock. Resulting data were time-stamped during recording on the RT-target before being transferred to the CPU and recorded in a text file.

2.2.4. Conduit Strain Measurement.

Due to the optical transparency of the conduit material, conduit distension was measured by color, high speed CCD camera (Basler Ace, Ahrensburg, Germany). Video imaging occurred concurrently with pressure and flow recording. Images were captured at length resolution of 17.6 microns/pixel and time resolution of 80 Hz, recorded by PXI board (National Instruments, Austin, TX). Frames were then time-stamped by the CPU and packaged into audio video interleave files. Fluid used was de-ionized water with red food coloring, and backlit using a white light-emitting diode array. This combination of red fluid coloration backlit to prevent light reflections within the conduit allowed for defined conduit edges within the images. Representative results may be seen in Sec. A2.2.3 available in the Supplemental Materials on the ASME Digital Collection.

2.3. Data Analysis and Calculation

2.3.1. Image Analysis.

Recorded audio video interleave files from imaging CCD camera were converted into graphics interchange format, imported into image analysis software written in-house in Mathematica (Wolfram Research, Champaign, IL). Analysis software filtered images by red coloration, binarized these images to clearly define lumen edge, and then measured diameter perpendicular to the lumen axis. Average lumen diameter within a single frame was then paired with the corresponding time-stamp for records.

2.3.2. Data Smoothing and Combination.

All recorded waveforms underwent discrete Fourier transforms, and magnitude and phases were recorded. Resulting Fourier harmonic phase values were normalized to their corresponding P _In phase, to allow for comparison between waveforms. Resulting harmonic moduli and phases were averaged and combined into one-dimensional functions for comparison, represented here by g(x, t)

$$g\left(x,t\right)=A_0+\sum _{f=1}^N\left|A_f\right|\mathrm{E}\mathrm{xp}\left[-i\hspace{0.17em}\left(2\pi f\left(t-\frac{x}{c_f}\right)-\hspace{0.17em}{\phi }_f\right)\right]$$ (5)

where A_f is the modulus of the complex Fourier harmonic constant, c_f is the wave speed of the harmonic, and φ_f is the phase shift of the corresponding harmonic. Only the first nine harmonics were used in resulting functions, as described in Sec. 2.2.1.

2.3.3. Conduit Stiffness.

We integrated clinical-relevant metrics for arterial stiffness, including pressure modulus (E_P) and impedance. Here we refer E_P as the quotient of peak-to-peak pressure range (ΔP) and strain range (Δɛ_R) over the loading cycle, using the mean value between pressure inlet (ΔP _In) and pressure outlet (ΔP _Out) to account for head loss

$$E_P=\frac{\Delta P}{\Delta {\epsilon }_R}=\frac{\mathrm{Mean}\left(\Delta P_{\mathrm{In}},\Delta P_{\mathrm{Out}}\right)}{\Delta {\epsilon }_R}$$ (6)

According to the mathematical relationship as derived by Womersley for pulsatile flow in a conduit, flow impedance is related through fluid density, viscosity, harmonic, initial diameter, and cross-sectional area. As the fluid, input waveform, and initial diameter are all control variables in our flow setup, impedance may only change through cross-sectional area change through diameter strain of conduit

$$Z\propto \frac{1}{R_0^2{\lambda \left(t\right)}^2}\propto \frac{1}{{\epsilon }_R{\left(t\right)}^2}$$ (7)

Therefore, we infer a direct effect between changes in diameter waveforms and pressure waveforms. Changes in impedance were determined through average cross-sectional area (ε_r ²) over the cycle. Values were determined through integrating radial strain over the cycle using a Riemann sum, using time intervals below the image sampling frequency. This allows a comparison between impedances over the flow cycle.

2.3.4. Fourier Harmonic Wave Speed.

Wave speeds were calculated from locating the wave foot and the maximum of the double derivative of the waveform with respect to time. This locates the acceleration of the waveform, and initial base of the recorded waveform. Wave speed (c) was then calculated as the length (x) of the conduit divided by the time (t) between inlet and outlet waveforms for total waveform using the foot-to-foot method, and each individual harmonic

$$c=\frac{\Delta x}{\Delta t}=\frac{x_{\mathrm{Out}}-x_{\mathrm{In}}}{t_{\mathrm{Out}}-t_{\mathrm{In}}}=\omega \frac{x_{\mathrm{Out}}-x_{\mathrm{In}}}{{\phi }_{\mathrm{Out}}-{\phi }_{\mathrm{In}}}$$ (8)

Using the conduit length, the wave speed can be mostly influenced by local conduit stiffness. Wave speed results are reported in Sec. A3 available in the Supplemental Materials on the ASME Digital Collection.

2.3.5. Mechanical Energy.

The relationships between the conduit material properties and hemodynamics relationships range from the mechanical energy of flow to clinical-relevant parameters such as the PI. It is possible to derive the relationship between material strain and flow from the mechanical energy of flow equation

$$\frac{\rho }{2}\frac{D}{\mathit{Dt}}\left(u^2\right)=\rho \mathit{gh}+\nabla \left(u·\sigma \right)+\left(\nabla ·u\right)P-\phi$$ (9)

where the material derivative of the kinetic energy (u²) may be equated with the gravitational potential (g), volumetric potential (P), viscous loss (φ), and applied stress (σ) on the fluid. Here, time and spatial derivatives of the kinetic energy may be directly related to the applied stress on the system (σ), potential energy (P) within the volume, and a summation of viscous loss throughout the volume (φ). Therefore, hysteresis resulting from the material strain would reduce applied stress and returned energy as the lumen contracts, leading to less energy head loss and reduced kinetic energy. To determine effects of viscoelastic material properties, the material derivative of kinetic energy is determined at both the inlet and outlet.

By isolating both the time-derivative and spatial-derivative in the mechanical energy of flow, we may directly relate it to PI and wave speed, respectively,

$$\begin{array}{c}{\frac{\partial }{\partial t}\left({u\left(x,t\right)}^2\right)]}_{x=0}\to {\frac{\partial }{\partial t}\left(Q_f^2\mathrm{E}\mathrm{xp}\left[i2\omega \hspace{0.17em}t-\frac{0}{c_Q}\right]\right)]}_{t_{\min }}^{t_{\max }}=i2\omega \left(u{\left(t_{\max }\right)}^2-u{\left(t_{\min }\right)}^2\right)\propto \frac{u_{\max }-u_{\min }}{\overline{u}}=\mathrm{PI}\end{array}$$ (10)

where Q_f is the Fourier harmonic amplitude, c_Q is the harmonic wave speed as described in Eq. (8), and the kinetic energy measured is located at fixed location (x = 0) at the point of measurement, therefore simplifying the equation. Herein, the PI, clinically measured to determine flow pulsation states within the arterial tree, is a simplified measure of the mechanical energy by the time-derivative of flow. Additionally, the mechanical energy is related to both frequency and wave speed by the spatial derivative of flow. By comparing mechanical energy between elastic and viscoelastic conduits, we may relate viscoelastic damping to the decrease in PI.

To determine mechanical energy, we integrated the fluid velocity over the cross-sectional area at the point of measurement. Therefore, the material derivative of kinetic energy passing through the cross-sectional area of measurement is determined by inserting Eq. (5) as Q

$$\begin{array}{c}\pi \rho \frac{D}{\mathit{Dt}}{\int }_0^{R_0}\left(u{\left(x,t\right)}^2\right)\to {\left.\frac{4\left(8\right)\hspace{0.17em}\rho }{27\pi D_0^2}\left(\frac{\partial }{\partial t}\left(Q{\left(x,t\right)}^2\right)+4\frac{Q\left(x,t\right)}{\pi \hspace{0.17em}{D}_0^2}\frac{\partial }{\partial x}\left(Q{\left(x,t\right)}^2\right)\right)\right]}_{x=0}\\ =\frac{4\left(16\right)\hspace{0.17em}\rho }{27\hspace{0.17em}\pi \hspace{0.17em}{D}_0^2}\left({\left(Q{\left(0,t\right)}^2\right)}^{\prime }+\left(1-\frac{4}{\pi D_0^2}\frac{Q(0,\hspace{0.17em}t)}{c_Q}\right){\left(Q{\left(0,t\right)}^2\right)}^{\prime }\right)\end{array}$$ (11)

where ρ is fluid density, D ₀ is the original inner diameter of the conduit, Q is the volumetric flow, x is length in the axial direction parallel to flow, t is time, and c_Q is the harmonic wave speed. This suggests that the mechanical energy is directly proportional to the heart rate (ω), flow amplitude (Q) proportional to PI, and wave speed (c), all of which can be associated with clinical measures of cardiovascular disease.

2.3.6. Statistical Methods.

Measured response variables were compared using multifactor analysis of variance (ANOVA). Sample size for each measure was limited to three for each group, and all groups passed the Anderson–Darling test for normality. Main effects were determined to be conduit type (E and VE), applied flow type (LPF and HPF), and point of measurement (inlet and outlet), and subsequently compared in three-way ANOVA. In cases where there was not separation between Inlet and Outlet, such as diameter strain and pressure modulus (E_P), two-way ANOVA was performed. Main (single-factor) and interactive (two-factor) effects were examined, with special focus on conduit material type and inlet/outlet comparison, with a significance of α < 0.05. If results indicated a trend though not significant (0.05 < α < 0.10), results were reported with the associated p-value explicitly.

Additional comparisons between conduit types were completed using linear regression (viscoelastic = β′(elastic) + β ₀ ′). Through plotting the data derived from Fourier analysis, we compare differences in waveforms between elastic and viscoelastic conduit types. This additionally allows for decoupling of wave amplitude (|A_f|) and phase shift (φ_f) in comparisons of P and Q attenuation to diameter strain waveforms. Therefore, to couple two independent variables from separate data sets into a single regression analysis, values were initially grouped by Fourier harmonic. Harmonic values from comparative sets (i.e., 1 Hz LPF-E and LPF-VE) were subsequently paired within harmonic sets with ascending values and linear regression performed. Harmonic amplitudes were weighted by the inverse harmonics (f/f _Max) to establish equal variance throughout the regression. Both harmonic phase shift and wave speed were weighted by elastic amplitudes (|A_f|/|A₁|) to increase values of phase shift closer to the primary harmonic.

Validation of regression analysis was completed through comparison of each resulting slope (β′) and intercept (β ₀ ′) to zero through student's t-test. Student's t-test was performed to quantitatively compare regression, reliant on both correlation coefficient (R), sample size (n), and differentiation from null hypotheses. Further comparison was only between β′-values, because majority of β ₀ ′-values were determined to be null by Student's t-test. Resulting β′-values were compared to one by Student's t-test to determine equivalence between elastic and viscoelastic conduit harmonic values (E(f) = VE(f)). Regression slope (β′) of both P and Q were additionally compared to diameter by Student's t-test, combining standard error and variance under the assumption of equal sample size and variance.

3. Results

3.1. Material Properties and Conduit Stiffness.

Our results from the frequency sweep measurements (Sec. A1 available in the Supplemental Materials on the ASME Digital Collection) correspond well with those from the literature [30,31], indicating direct relationship between viscous loss and m-PDMS weight percentage. Values for both elastic (58.8% 500 CSt m-PDMS: 108 kDa v-PDMS) and viscoelastic (70% 1 M CSt m-PDMS: 108 kDa v-PDMS) materials are listed in Table 1. Strain sweep results indicate strain softening at approximately 10% for all samples, well within maximum projected strain.

Table 1.

Frequency-dependent material properties

	E (58.8% 500 CSt : 108 kDa PDMS)	VE (70% 1M CSt : 108 kDa PDMS)
Material	E′ = 68.9 kPa ± 0.210	E′ = 67.1 kPa ± 0.006
Properties at 1 Hz	E″ = 4.09 kPa ± 0.411	E″ = 32.3 kPa ± 0.012
	|E*| = 69.0 kPa ± 0.462	|E*| = 74.5 kPa ± 0.013
	tan(δ) = 0.060 ± 0.006	tan(δ) = 0.482 ± 4.00 × 10−4

E and VE material properties of conduit materials are briefly summarized and compared, with storage modulus (E′), loss modulus (E″), dynamic modulus amplitude (|E*|), and viscoelastic ratio (tan(δ)). Both E and VE materials are described in Sec. 2.1.3 and mechanical testing is further described in Sec. A1.1 available in the Supplemental Materials on the ASME Digital Collection.

Conduit stiffness, as described by the pressure modulus (E_P) in Eq. (6), showed no significant differences among the groups, with pressure moduli of approximately 1200 mmHg, which matched the values to arteries with low compliance (Fig. 3) [9,40].

Fig. 3.

Measured pressure modulus (E_P). No significant differences were seen among measured pressure modulus, defined as the ratio between the change in pressure and the change in strain (E_P = ΔP/Δɛ). Measurements were grouped by applied flow (LPF and HPF) and conduit material type (E and VE).

3.2. Pressure and Diameter Waveforms Are Attenuated in Viscoelastic Conduits.

Pulsatile pressure waveforms and diameter strain were compared to evaluate the contributions of material property to pressure wave attenuation (Fig. 4). Compared to elastic conduits, the attenuation and smoothing of the diameter curve in viscoelastic conduits were found for both LPF and HPF. We additionally found a slight phase lag for viscoelastic conduits when compared to elastic conduits. This is in agreement with previous studies of artery responses using hysteresis of pressure-strain relationship for viscoelastic characterizations [2,9,17].

Fig. 4.

Representative pressure waveforms at inlet and outlet and corresponding diameter strain curves. Elastic conduit waveforms are illustrated with solid lines, while viscoelastic waveforms are dashed lines. Differences between elastic and viscoelastic waveforms are shown with shading.

Low pressure flow waves (Fig. 4(a)) are shown to change between elastic and viscoelastic conduits, with both upstream and downstream pressure waves attenuated with viscoelastic conduit walls. Viscoelasticity of conduits showed less influences on pressure waves for HPF waves (Fig. 4(b)). Diameter strain curves indicated greater smoothing and phase delay in viscoelastic conduits over the cycle, with increased diameter strain later in the wave cycle (black, dashed line).

Statistical analysis of the pressure wave results, using three-way ANOVA comparisons on three independent variables, i.e., conduit material type (E versus VE), flow pressure range (LPF versus HPF), and measurement location (inlet versus outlet). Results trend toward attenuation from the main effect of material property between elastic and viscoelastic conduits (p = 0.083), though results are not significant. Importantly, viscoelastic conduits showed an average decrease of 3.9% in pressures compared to elastic ones. Also, the outlet pressures were significantly reduced compared to the inlet. Statistical analysis of diameter strain curves, using two-way ANOVA comparisons on conduit material type and flow pressure range showed 12.0% increase in the area summed over the cycle (ɛ² _Max − ɛ² _Min) with viscoelastic conduits. No significant interactions were found of conduit material (E versus VE) with either applied flow type (LPF versus HPF) or measurement location (inlet versus outlet).

Pressure waveform Fourier harmonics were compared between E and VE conduit material types using linear regression (Fig. 5). Fourier harmonic coefficients were decoupled into both amplitude (|P_f|) and phase shift (φ_f). This allowed wave comparison to both measured and predicted diameter strain (Fig. 5), decoupling effects of time-dependent material properties (|G*| and δ). Regression slope (β′) for pressure harmonics (Fig. 5(a), |P_f|) of LPF-P indicated significant decreases in wave amplitudes with viscoelastic conduit materials, at both the inlet (blue) and outlet (yellow). HPF-P _In, green at the inlet does not show significant differences between E and VE conduits, though significant decreases are seen in HPF-P _Out (red).

Fig. 5.

Pressure Fourier harmonic regression comparison between VE and E conduit materials. All pressure regression comparisons show significant differences in β′ to diameter strain (LPF-D and HPF-D, black and gray) regressions. Regression coefficients and comparisons are shown in Table 2.

Regression analysis of Fourier harmonics resulted in strong, positive correlation in all comparisons. All waveforms marked by attenuation in Fig. 5 have statistically significant decreases (β′ < 1) for harmonic amplitudes |P_f|. These include all LPF waveforms (LPF-P _In, LPF-P _Out, LPF-D), as well as HPF-P _Out and HPF-D. Statistically significant decreases (β′ < 1) in φ_f were also shown in these waveforms, except for HPF-D. These results strongly indicate attenuation in pressure waves with viscoelastic conduits, especially in LPF compared to HPF. While |P_f| regressions indicate reductions, which correspond to diameter strain attenuation, they also indicate significant differences from corresponding diameter strain β′ values. Results are shown in Table 2.

Table 2.

Regression results (β′) for Fourier harmonic amplitude (|P_f|) and phase (φ_f)

	Amplitude |Pf|					Phase φf
Waveform	Slope β′		β = 1	β = D		Slope β′		β = 1	β = D
LPF-P In	β′ = 0.781	p < 0.001	p < 0.001	p = 0.024	LPF-P In	β′ = 0.960	p < 0.001	p = 0.002	p = 0.001
R 2 = 0.926	β 0 ′ = 0.003	p = 0.085			R 2 = 0.996	β 0 ′ = 0.008	p = 0.514
LPF-P Out	β′ = 0.869	p < 0.001	p = 0.007	p = 0.008	LPF-P Out	β′ = 0.952	p < 0.001	p = 0.032	p = 0.010
R 2 = 0.861	β 0 ′ = 0.003	p = 0.488			R 2 = 0.983	β 0 ′ = 0.070	p = 0.117
LPF-D	β′ = 0.662	p < 0.001	p < 0.001	—	LPF-D	β′ = 0.890	p < 0.001	p < 0.001	—
R 2 = 0.776	β 0 ′ = 0.007	p = 0.089			R 2 = 0.978	β 0 ′ = 0.081	p = 0.017
HPF-P In	β′ = 0.997	p < 0.001	p = 0.195	p < 0.001	HPF-P In	β′ = 0.997	p < 0.001	p < 0.001	p = 0.035
R 2 = 1.00	β 0 ′ = 0	p = 0.802			R 2 = 1.00	β 0 ′ = 0.001	p < 0.007
HPF-P Out	β′ = 0.961	p < 0.001	p = 0.030	p < 0.001	HPF-P Out	β′ = 0.984	p < 0.001	p < 0.001	p = 0.089
R 2 = 0.983	β 0 ′ = 0.000	p = 0.694			R 2 = 1.00	β 0 ′ = 0.002	p = 0.344
HPF-D	β 0 ′ = 0.731	p < 0.001	p < 0.001	—	HPF-D	β′ = 0.958	p < 0.001	p = 0.097	—
R 2 = 0.921	β 0 ′ = 0	p = 0.930			R 2 = 0.973	β 0 ′ = 0.017	p = 0.302

Resulting regression slopes (β′) were compared by t-test to values of zero, unity, and diameter (D). Resulting slope values significantly different and less than unity indicate attenuation.

Comparisons of regression constants indicate significant decreases in both |P_f| and φ_f for LPF waveforms, indicating wave attenuation. Pressure at both the inlet and outlet (LPF-P _In and LPF-P _Out) indicate significant increases for both |P_f| and φ_f results to diameter (LPF-D). Alternately, HPF indicates similar |P_f| between elastic and viscoelastic conduits (β′ = 1), which is significantly greater than diameter (HPF-D). Additionally, φ_f results indicate significant decreases, though resulting values also approach unity, indicating minimal shift in viscoelastic conduits.

These results indicate pressure wave attenuation at both the inlet and outlet for LPF-P, and HPF-P _Out, corresponding with diameter strain attenuation resulting from both time-dependent stiffness and phase shift of the material indicated. This additionally indicates upstream cushioning is important in attenuation of upstream pressure waves, seen in HPF-P _In regressions approaching unity.

3.3. Wall Shear Stress Waveforms Are Attenuated With Viscoelastic Wall Properties.

Results on volumetric flow (Q) waveforms over the cycle (Fig. 6) showed attenuation of Q waves in viscoelastic conduits, which exhibited decreased peak-to-peak amplitude indicating decreased pulsatility or PI. This decreased PI value corresponded well with attenuated distension and pressure waveforms. Low pressure flow waves (Fig. 6(a)) are shown to change between elastic and viscoelastic conduits, with both upstream and downstream Q waves attenuated with viscoelastic conduit walls. Viscoelasticity of conduits showed less influences on Q waves p for high pressure flow (Fig. 6(b)). Importantly, results also showed greater peak alignment between diameter strain curves and Q waves in viscoelastic conduits compared to elastic ones.

Fig. 6.

Representative volumetric flow (Q) waveforms at inlet and outlet and corresponding diameter strain curves. Elastic conduit waveforms are illustrated with solid lines, while viscoelastic waveforms are dashed lines. Diameter waveforms of viscoelastic conduits showed better peak phase alignment with Q (black, dashed line).

Fourier harmonics of Q-waveforms were decoupled into amplitude (|Q_f|) and phase (φ_f), and compared between E and VE conduits through linear regression (Fig. 7). Resulting β′ values were compared to unity to determine if waveform amplitudes and phases were equivalent. All waveforms, which attenuate with viscoelastic conduit material in Fig. 7, indicate statistically significant decrease in β′ for |Q_f|. Regression slope (β′) for pressure harmonics (Fig. 7(a), |Q_f|) of LPF-Q indicated significant decreases in wave amplitudes with viscoelastic conduit materials, at both the inlet (blue) and outlet (yellow). High pressure flow at the inlet (HPF-P _In, green) does not show significant differences between E and VE conduits, though significant decreases are seen in HPF-P _Out (red). Of these waveforms, only LPF-Q _In and LPF-D indicate significant decreases in φ_f. Results are shown in Table 3.

Fig. 7.

Volumetric flow Fourier harmonic regression comparison between E and VE conduit material. All pressure regression comparisons show significant differences in β′ to diameter strain (LPF-D and HPF-D, black and gray) regressions. Regression coefficients and comparisons are shown in Table 3.

Table 3.

Regression results (β′) for Fourier harmonic amplitude (|Q_f|) and phase (φ_f)

	Amplitude |Qf|					Phase φf
Waveform	Slope β′		β = 1	β = D		Slope β′		β = 1	β = D
LPF-Q In	β′ = 0.752	p < 0.001	p < 0.001	p = 0.061	LPF-Q In	β′ = 0.906	p < 0.001	p < 0.001	p = 0.266
R 2 = 0.944	β 0 ′ = 0.002	p = 0.214			R 2 = 0.984	β 0 ′ = 0.002	p = 0.823
LPF-Q Out	β′ = 0.812	p < 0.001	p < 0.001	p = 0.006	LPF-Q Out	β′ = 0.988	p < 0.001	p = 0.352	p = 0.001
R 2 = 0.968	β 0 ′ = 0.001	p = 0.488			R 2 = 0.974	β 0 ′ = 0.001	p = 0.966
LPF-D	β′ = 0.662	p < 0.001	p < 0.001	—	LPF-D	β′ = 0.890	p < 0.001	p < 0.001	—
R 2 = 0.776	β 0′= 0.007	p = 0.089			R 2 = 0.978	β 0 ′ = 0.081	p = 0.017
HPF-Q In	β′ = 1.001	p < 0.001	p = 0.464	p < 0.001	HPF-Q In	β′ = 0.999	p < 0.001	p = 0.419	p = 0.080
R 2 = 0.994	β 0 ′ = 0	p = 0.410			R 2 = 1.00	β 0 ′ = −0.001	p =0.592
HPF-Q Out	β′ = 0.973	p < 0.001	p = 0.011	p < 0.001	HPF-Q Out	β′ = 0.979	p < 0.001	p = 0.145	p = 0.240
R 2 = 0.998	β 0 ′ = 0.001	p = 0.037			R 2 = 0.990	β 0 ′ = 0.002	p = 0.687
HPF-D	β′ = 0.731	p < 0.001	p < 0.001	—	HPF-D	β′ = 0.958	p < 0.001	p = 0.097	—
R 2 = 0.921	β 0 ′= 0	p = 0.930			R 2 = 0.973	β 0 ′ = 0.017	p = 0.302

These results indicate significant changes between elastic and viscoelastic conduits, with greatest wave attenuation resulting in viscoelastic LPF.

Resulting Q-wave β′ were additionally compared to diameter β′ to determine diameter effects on wave attenuation. Results indicate significant decreases in both |Q_f| and φ_f for LPF waveforms, indicating wave attenuation. Volumetric flow (Q) at both the inlet and outlet (LPF-Q _In and LPF-Q _Out) indicate significant increases for both |Q_f| and φ_f results to diameter (LPF-D). Resulting comparisons indicate significant differences between LPF-Q _Out and LPF-D for both |Q_f| and φ_f. Additionally, LPF-Q _In did not show significant difference from LPF-D for either |Q_f| or φ_f.

Alternately, HPF indicates similar |Q_f| between elastic and viscoelastic conduits (β′ = 1), which is significantly greater than diameter (HPF-D). Additionally, while HPF-Q _In and HPF-Q _Out do not show the decrease that HPF-D indicates for |Q_f|, there is no significant difference between φ_f values for HPF-Q and HPF-D. Lastly, φ_f results for Q indicate significant decreases, though resulting values also approach unity, indicating minimal shift in viscoelastic conduits. Results for HPF φ_f. values all indicate unity, and therefore indicate minimal phase difference between waveforms or conduit material types.

Wall shear stress was determined through applying the formula relating volumetric flow rate as shown in Eq. (4), including radial strain as a function of time in the radius term R. The diameter increase during peak Q values for both inlet and outlet can contribute to the reduction in WSS oscillation (Eq. (4)). Similar to changes in the quantitative PI for Q waves (Q-PI), statistically significant decreases in the PI for WSS (WSS-PI) were found in viscoelastic conduits when compared to elastic ones (Fig. 8). In addition to the reduced WSS oscillation, the average WSS significantly decreased in viscoelastic conduits when compared to elastic ones.

Fig. 8.

Comparison of PI as measured by flow rate (Q) and WSS. Differences in waveform shapes and pulsatility (PI) present in comparison of E–VE conduits. (a) Measured PI of WSS waveforms comparing E and VE conduits. Significant differences were measured using Bonferroni post-test analysis (α < 0.05, labeled with +). (b) Phase alignment of waveforms in LPF at the outlet, LPF-Q at the outlet (Q-E: solid line, Q-VE: dashed) matched with conduit diameter values (E-diameter: solid line, VE-diameter: dashed line), with respective LPF-WSS shown below.

Main effect comparisons of volumetric flow waveforms showed a decrease of peak-to-peak amplitude in viscoelastic conduits compared to elastic ones, with 6% decrease in the PI, and 9.1% decrease in PI from inlet to outlet. More importantly, resulting measures of WSS indicate significant decreases of 22.8% (Fig. 8). Within the ANOVA comparison, we see no interactive effect of conduit material (E versus VE) with either applied flow type (LPF versus HPF), or measurement location (inlet versus outlet).

In the case of LPF, viscoelastic conduits caused a decrease in Q-PI by ∼15.9%, but decrease in WSS-PI by 29.1%. These results were also found in HPF flow, though less drastic, with decreases of 2.0% in Q-PI, and 20.0% decrease in WSS-PI. This indicated is that phase alignment between diameter and Q-waves had a significant effect on WSS. Greater phase alignment in viscoelastic conduits, as shown in Fig. 7, reduced WSS-PI response. Increased WSS-PI at the outlet in elastic conduits indicates that resulting phase misalignment exacerbates high pulsatility in flow. Therefore, phase lag due to viscoelastic material properties has significant effects on improving fluid shear stress mechanical signaling.

3.4. Mechanical Energy of Flow.

Peak mechanical energy decreased by 4.5% (p = 0.073) in viscoelastic compared to elastic conduits, though results were not significant. Mechanical energy also showed significant decreases of 6.2% occurred at the outlet compared to the inlet, following a similar trend as Q-wave comparisons Significant differences were found between LPF-E and LPF-VE in a t-test. Additionally, interaction effects of conduit material type (E versus VE) and flow type (LPF versus HPF) indicate that there might be more decrease (p = 0.098) of mechanical energy in viscoelastic compared to elastic conduits for applied LPF than HPF (Fig. 9). Importantly, no significant changes were found in interactive effects of conduit material (E versus VE) and upstream or downstream location (inlet versus outlet). The energy results exhibited a similar trend as Q-PI, which can be supported by the relationship between mechanical energy and PI described in Eq. (10).

Fig. 9.

Mechanical energy of flow was calculated as the material derivative of the kinetic energy. This required time derivative and divergence of the kinetic energy. The bar graph on the left shows decreased mechanical energy with viscoelastic conduits (VE) compared to elastic conduits. Example mechanical energy curves for LPF are illustrated on the right. Peak mechanical energy aligns with peak volumetric flow, but waveforms do not match.

The mechanical energy of flow was solved for by using Eq. (11) listed in Sec. 2.3.5. This data was then plotted against time (Fig. 9) to indicate differences seen during the wave cycle. This equation again assumes fully developed, laminar flow. Mechanical energy was quantified at each of the inlet and outlet sensors, using the static cross-sectional area of the connective tubing at the point of measurement.

4. Discussion

Healthy arterial mechanics reduce energy of the cardiac output as circulation branches down the arterial tree. Therefore, wall mechanics are important in disease prediction and prognosis, as well as vascular grafting treatment design. Time-dependent arterial mechanics have been studied for the previous 60 years [9,32,41]. However, difficult to measure and to evaluate in vivo, effect on mechanotransductive signals such as wall shear stress and pressure has not been fully elucidated. Using in vitro models, our results have demonstrated several unique findings, related to the contribution and mechanism of fluid energy attenuation in viscoelastic conduits with artery-like mechanical properties. More specifically, we showed that: (1) viscoelastic conduit properties attenuated WSS over the cycle, as well as pressure and flow waves upstream and downstream of the conduit; (2) decreased peak-to-peak and harmonic amplitudes of both P and Q waves corresponded to attenuation of diameter distension wave in viscoelastic conduits; (3) extension of strain through the cycle created greater phase alignment of diameter with Q-waveforms, thereby decreasing WSS pulsatility through increased cross-sectional area, as shown in Fig. 10. Additionally, attenuated strain waves indicated greater mean area over the cycle, thereby decreasing impedance, affecting upstream and downstream flow (Fig. 4). Finally, energy damping from viscous energy loss within the viscoelastic material could contribute to decreased PI over the length of the conduit. While greater Q-wave attenuation occurs with viscoelastic conduits, no relation is evident between PI decrease over the conduit length and conduit material properties. Therefore, the effects of impedance change may be the greater contributing factor to flow attenuation, though a longer conduit may show greater energy damping effects. We hope these results may be used to highlight the importance of time-dependent arterial mechanics in disease prediction and prognosis, as well as vascular graft design.

Fig. 10.

Phase alignment of the volumetric flow with the conduit distension for both low pressure (left) and high pressure (right) conditions. Viscoelastic conduits exhibit a phase shift and increased strain later in the pressure cycle, shown by the shading. This results in greater cross-sectional area during peak volumetric flow (Q), decreasing shear stress at the wall, indicated by the arrows.

Shape of the strain curve was predicted in projected mechanics of the lumen (Fig. 10), resulting from changing |G*| and δ-values across the frequency spectra of the material interacting with harmonics of the pressure wave. Resulting P, Q, and WSS attenuation is supported by previous computational methods [23], but has not been discussed experimentally with direct comparison between viscoelastic and elastic conduits. Wave attenuation has also not been directly attributed to frequency-dependent material response instead of viscous energy dissipation of the material.

This study has provided evidence showing that phase alignment is a major contributing factor in altering an important mechanical signal, WSS, whose influences on vascular endothelial and smooth muscle cells and on vascular inflammation have been well known [28,29,40]. This is a new finding, given that existing studies have not examined the alignment of area and flow phase. Previously, Q-waveforms were computationally shown to attenuate due to viscoelastic wall properties [23], but effects are more drastic when WSS is coupled to conduit distension. The final amplitude and shape of the measured pulse pressure wave or flow wave are determined by the phase relationship (timing). In the case of reduced arterial viscoelasticity, the increased transmission of the pulsatile energy into the peripheral microcirculation such as lung, kidney or brain circulations has detrimental effects on these peripheral circulations with organs highly perfused with low resistance [42,43].

While both viscoelastic conduit diameter and pressure indicate attenuation, regression coefficients (β′) indicate significant differences due to interdependence of P and D. Upstream P-wave attenuation indicates direct response to downstream impedance of the viscoelastic conduit, most clearly indicated by increased summation of cross-sectional area. Resulting pressure attenuation would additionally affect diameter strain within the conduit. Therefore, pressure and diameter create a dynamic feedback, resulting in eventual steady-state response

$$E\hspace{0.17em}\epsilon \left(t\right)=P\left(t\right)\propto \frac{Q\left(t\right)}{\epsilon {\left(t\right)}^2}$$ (12)

Upstream attenuation is most prominent in LPF, indicating greater upstream cushion significantly affects available pressure attenuation. This closely follows the hypotheses of the aorta as an energy reservoir, and aortal stiffening increasing downstream pulsatility [43–45]. Due to the interactions of pressure, impedance, and diameter strain, it becomes difficult to uncouple the effects, leaving further study to be done.

Mechanical energy of flow decreased with viscoelastic wall properties and was characterized by instantaneous change in kinetic energy within the volume. Within Eq. (10), we directly relate this to PI, a clinical measure of vascular fibrosis [43,44], indicating decreased pulsatility with mechanical energy loss. While mechanical energy relates to clinical and experimental metrics within the literature, it has been rarely associated with cell response or disease progression. Nevertheless, increased mechanical energy would directly relate to many parameters important to cardiovascular diseases, as illustrated by the Eq. (11).

This study, through comparing fluid responses of viscoelasticity conduit materials with elastic materials of similar stiffness, highlights time-dependent wall effects on pulse wave responses. This study differs from previous studies in the area [30,31] by providing experimental comparisons of viscoelastic to elastic conduits, with well-defined material properties and flow wave parameters within physiological conditions. Most previous studies have focused on computational modeling, with minimal material characterization for experimental validation [22–24]. Also, in the experimental validation of these studies, flow consisted of single harmonic waveform perturbations of the system, which could not show frequency-dependent material effects on pulse pressure and flow waves, as demonstrated here. Effects of viscoelastic wall properties on wave travel and attenuation have been a subject of interest over the past 40 years, with some debate over the contribution to pulse wave attenuation. Previous measurements of frequency-dependent arterial material properties showed tan(δ) of approximately 0.10–0.25 [9]. These values might be irrelevant to the cardiovascular system [46], and highly dependent upon smooth muscle contraction within the cycle [41,47].

Interestingly, the study results suggest that viscoelastic wall properties could have beneficial damping properties on flow pulsatility. Greater viscoelasticity may reduce the detrimental pulsatility in the upstream as well as downstream by smoothing out high pulse waves, which could improve mechanotransductive signals in the arterial lumen. Therefore, changes in viscoelastic properties of the artery may have use in arterial disease prognosis and progression, as arterial viscoelasticity has already been shown to decrease in some disease states [2]. It additionally presents a new avenue in vascular graft development, improving mechanotransductive signals in the graft lumen during or after endothelialization [6,17]. Further work should be done to fully develop these concepts as a metric for disease progression and vascular grafting design criteria.

Supplementary Material

Supplementary Material

Supplementary Files

Funding Data

• NIH (Grant No. NHLBI R01HL119371; Funder ID: 10.13039/501100000272).

• Children's Hospital at Cincinnati (Grant No. U1704149; Funder ID: 10.13039/100007172).

• Chinese National Natural Science Foundation (Grant No. 21471046; Funder ID: 10.13039/501100001809).