# Numerical and Experimental Analysis of Shear Stress Influence on Cellular Viability in Serpentine Vascular Channels

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Numerical Analysis

^{3}, dynamic viscosity = 3.5 × 10

^{−3}Pa.s. Young’s Modulus = 0.4 × 10

^{6}Pa, Poisson’s ratio = 0.5. GelMA material properties (density = 1020 kg/m

^{3}, dynamic viscosity = 4.2 × 10

^{−3}Pa.s, Young Modulus 3.18 KPa) were used for the modeling of the serpentine wall. Simulations were performed at a reference temperature of 293.15 K. Navier–Stokes equation was used to represent the momentum of the fluid model. Time-dependent partial differential equation of incompressible Navier–Stokes (Equation (1)) was used for the simulation.

_{1}= 4.48 mm/min, V

_{2}= 4.62 mm/min and V

_{3}= 4.76 mm/min] and physiological flow-PF [V

_{4}= 4.48 mm/min, V

_{5}= 4.62 mm/min and V

_{6}= 4.76 mm/min] were considered in the study. The physiological functions (waveform) were derived from 4D-Laser Doppler data of blood flow across a cross-section of sub-clavicular artery of healthy subject. The study was approved by Institute Ethical Committee (NITRR/IEC/2021/12). Obtained tempero-spatial functions were interpolated thereafter using Curve-Fitting MATLAB toolbox version 2018a, Mathworks, India. Thereafter, the interpolated function with lowest RMSE value was selected for the study. The functions corresponding to inlet velocity models for V

_{1}to V

_{6}are given in Table 2. One complete cycle of both sinusoidal and physiological flow corresponds to one complete cardiac cycle (represented by T). Different study conditions were formulated based on the above boundary conditions: no-slip (wall) and sinusoidal inlet flow, slip (wall) and sinusoidal inlet flow, no-slip (wall) and physiological inlet flow, slip (wall) and physiological inlet flow (as tabulated in Table 3). A no-slip boundary condition will exclusively provide the influence of inlet flow type in the development of flow physics of media inside the serpentine. On the other hand, slip wall boundary condition will simulate the elasticity of the serpentine wall (similar to the natural blood vessel wall).

_{1}(4, −2.5), P

_{2}(5.8, 2.4) and P

_{3}(7.7, −2.5) on the neck, abdominal, and rear region, respectively, were identified on the serpentine channel. These three regions are the downstream region immediately after curvilinear orientation of ascending and descending phases of the serpentine structure, which are more prone to Coriolis forces within the serpentine model. Flow parameters (pressure, shear rate, stress) over the entire cardiac cycle were evaluated at an interval of T/3, T/2, and T cycles.

#### Grid Convergence Test

_{1}with no slip condition had been selected for three different element sizes. For all element sizes—pressure, shear rate and velocity at probe points P

_{1}and P

_{2}(as marked in Figure 1) were measured for the entire time cycle. In the current study, the grid refinement ratio r, which is equivalent to mean refinement ratio ${r}_{mean}$of ${r}_{12}$and ${r}_{23}$were calculated using Equation (3a)

_{1}and P

_{2}at different grid resolution at time T.

_{s}) selected for this study was considered to be 1.25.

_{1}and P

_{2}did not change significantly on further decreasing the mesh size to represent finer grid resolution.

#### 2.2. Experimental Analysis

_{2}was used as a chemical cross-linker during printing process of the serpentine. The printing environment parameters are presented in Table 6.

#### 2.2.1. Endothelization of Vascular Network

^{7}cells per mL) was injected via a micro-pipette to fill the channel. Simultaneously, the inlet and the outlet of the construct was sealed with a pinch-clamp. Channels were injected with HUVECs stained with CellTrace CFSE dye (ThermoFisher, Waltham, MA, USA). The device had been centrifuged at 40 rcf for 1 min with a slow acceleration and deceleration to let the cells settle to the bottom. Incubation at 37 °C facilitated the adhesion of cells to form the innermost layer of the vessels. After incubation for 30 min, the construct was placed over 180° for cell adhesion on the other side of the vessel. Finally, the cells were incubated for 5 h at 37 °C on a shaker.

_{2}and N

_{2}delivery, etc., were achieved. Induction of media to the housing was made by using a pulsatile pump, as shown in Figure 4. It creates pulsatile hydrostatic pressure to the printed bioink construct, similar to the optimized inlet velocity profiles finalized from numerical study. The printed constructs were imaged using fluorescence microscopy to evaluate cell attachment and cell distribution within the channel.

#### 2.2.2. In-Vitro Testing

#### 2.2.3. Uncertainty Analysis

^{2}.

#### 2.2.4. Cellular Viability Assessment

#### 2.2.5. Sensitivity Analysis

_{6}. The sensitivity of the serpentine model was calculated using Equation (6)

^{3}. Change in the pressure due to change in density can be expressed by the Equation (7) (modified version of Equation (6))

^{3}) and 6% of blood density (997 kg/m

^{3}) at a point P

_{1,}respectively, as given in Equation (9)

## 3. Results

_{1}and V

_{3}, there was no variation in the pressure parameter over time. Maximum pressure generation for velocity models V

_{1}, V

_{2}and V

_{3}were recorded as 2–5 × 10

^{5}Pa. Similarly, minimum pressure generation for different velocity models V

_{1}, V

_{2}, and V

_{3}were obtained as 0.2 × 10

^{5}, −1 × 10

^{5}, and −1 × 10

^{5}Pa, respectively. In the velocity model V

_{2}, pressure generation on point P

_{1}at a time T was maximum, compared to other points, as shown in Figure 6a. It has been observed that the shear rate profile was globally constant for all velocity models, ranging from 0.5 × 10

^{4}to 4.5 × 10

^{4}1/s. The shear rate results for all velocity models, around points P

_{1}and P

_{3}at time T was maximum, as shown in the Figure 6b. For all velocity models, it can be concluded that the flow development was linearly correlated with the progression of time within the T cycle, as shown in Figure 6c. The minimum and maximum axial velocity of 2 mm/min at neck region (P

_{1}) and 20 mm/min in between the abdominal (P

_{2}) and rear regions (P

_{3}), respectively, were observed for condition I. From Figure 6d, it had been inferred that the local maximum stress having a magnitude of 18 × 10

^{−10}N/m

^{2}had been generated for the case of velocity model V

_{2}at the end of the cycle in between the abdominal (P

_{2}) and rear regions (P

_{3}).

_{1}, V

_{2}, and V

_{3}(Figure 7). A significant variation of pressure distribution was recorded for velocity model V

_{1}compared to the two other velocity models. For velocity models V

_{2}and V

_{3}, the local maximum pressure equal to 4 × 10

^{5}Pa was observed in the neck region (P

_{1}) of the channel at end of cycle time T, as shown in Figure 7a. Maximum pressure was generated at the time T/2 sec in the periphery of the rear region (P

_{2}). It was noticed that the value of axial velocity was similar for V

_{1}, V

_{2}, and V

_{3}velocity models. The maximum velocity was generated at the end of the T cycle for all inlet velocity models. The minimum and maximum axial velocities were observed as 0 and 20 mm/min, respectively, in the neck region (P

_{1}) and periphery of the rear region (P

_{3}), as shown in Figure 7c. Negligible variation had been observed in the shear rate for different velocity models in all regions of the serpentine at T/3, T/2, and T sec. The shear rate profiles for velocity models V

_{1}, V

_{2}, and V

_{3}were similar and its range of magnitude was 0.5–4.5 × 10

^{5}1/s, as shown in Figure 7b.

_{4}, V

_{5}, and V

_{6}at a measured time period T/3, T/2, and T sec was evaluated similarly for physiological flow with no slip wall boundary condition (condition III). For each velocity model, it was observed that local maximum pressure (3.5–4 × 10

^{5}Pa) developed at the T/3 and T/2 sec at the neck region of the channel, as shown in Figure 8a. For the same condition, the local maximum (at neck region) and minimum (rear and abdominal regions) shear rate value were predicted as 4.5 × 10

^{5}1/s and 0.5 × 10

^{5}1/s, respectively (Figure 8b). The range of local minimum and maximum axial velocity was calculated between 0 to 18 mm/min, respectively (Figure 8c). The velocity contour was found to be higher for periods T/3 and T/2 sec. Maximum velocity had been developed at the neck region of the serpentine channel.

_{4}, the pressure magnitude at T/3 and T/2 sec was minimum (0.2 × 10

^{10}Pa). The maximum pressure (1 × 10

^{10}Pa) was developed at approximately the periphery of the neck region and the rear region at the end of the T cycle, as shown in Figure 9a. At T/3 and T/2 secs, local maximum pressure was developed near the periphery of P

_{1}, as shown in Figure 9a. The contour plot of the shear rate was found similar for all velocity models of the physiological flow condition. The minimum shear rate corresponding to V

_{4}, V

_{5}, and V

_{6}were obtained as 1 × 10

^{7}, 0.2 × 10

^{5}, and 0 × 10

^{5}, respectively. Similarly, the maximum shear rate for V

_{4}, V

_{5}, and V

_{6}were 8 × 10

^{7}, 1.8 × 10

^{5}, and 2.5 × 10

^{5}, respectively, as shown in Figure 9b. A maximum axial velocity of 200 mm/min was developed on all measured points at T sec, which was extremely high as compared to other conditions shown in Figure 9c. For velocity models V

_{5}and V

_{6}, a maximum pressure of 2.5 × 10

^{5}Pa had been observed on the entire channel at T/3 and T/2 sec at the inlet region. For the same period, the local maximum velocity for model V

_{6}were found to be higher than that of V

_{5}in the neck as well as the periphery of rear region. For the results obtained in Figure 9d, it was observed that the stress decreases with an increase in velocity. Global values of stress for all velocities fluctuated between 2 and 25 × 10

^{−10}N/m

^{2}in the rear region of the serpentine channel.

## 4. Discussion

_{1}, P

_{2}, and P

_{3}. For condition I (as given in Table 3), the proposed architecture follows the Hagen–Poiseuille model for a Newtonian fluid where pressure and velocities are proportional to each other [44]. Variation in the axial velocity influences the shear rate distribution in a closed channel flow. The magnitude of the shear rate describes the flow behavior of working fluid inside the serpentine channel. A minimal variation in shear rate has been found due to a nominal variation in the axial velocity with respect to the radius of the serpentine because of considered rigid wall configuration [45]. Variation in axial velocities also influences the downstream generated pressure in serpentine structures. Localized maximal pressure was generated due to chaotic motion of fluid particles. When the axial velocity of a fluid increases, some of the energy used by the random motion particles to follow the fluid direction develops a lower downstream pressure [46]. The developing downstream pressure further gives rise to localized stress. It was also correlated that the velocity enhancement at the downstream region leads to formation of maximal stress. At higher axial velocity, fluid flow moved slowly near the wall due to diffusion and dispersion of fluid particles [47]. Hence, the inlet velocity V

_{2}was found to create higher stress than its counterpart velocity V

_{3}= Performing transient analysis, the maximum variation in the axial velocity profile was obtained at the end of the full cycle in condition II (see Table 3) due to positive acceleration of fluid particles [48].

_{4}), which occurred due to slip conditions at the end of the T cycle. When a fluid flow comes into contact with the curvature section of the serpentine, it creates Coriolis force, which causes a transversal slope in the flowing fluid. Resultant interaction between Coriolis force and transversal slope develops a secondary force on the flow cross-section. This secondary force had disseminated to the curvature section, producing a higher axial velocity magnitude [52]. Maximum axial velocity and presence of curvature of proposed serpentine structure were the effective parameters for developing maximum pressure near the neck and rear region of the serpentine structure at the end of T cycle.

_{2}at the end of T/3 cycle. Similarly, the minimum deviation was reported for the pressure on point P

_{3}at the end of the T cycle. As the flow approached near the wall, it was converted into a transitional flow due to elasticity of the wall of the serpentine. Hence, the velocity of flowing fluid increases, producing a maximum deviation of velocities [54].

_{3}in boundary condition IV (refer to Table 3) due to the physiological relevance flow in the presence of the elastic wall. Such a flow condition brought a non-stationary axial velocity profile over the period of time [59].

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Comparison of velocity variables and extrapolated values between two grid solutions and the Richardson extrapolation estimation.

**Figure 4.**Experimental set up details (

**a**) syringe pump setup, (

**b**) magnifying image of microfluidic chip.

**Figure 6.**Contour Plot of (

**a**) pressure, (

**b**) shear rate, (

**c**) axial velocity, (

**d**) wall stress for sinusoidal flow, slip wall boundary condition for velocity models V1, V2 and V3.

**Figure 7.**Contour Plot of (

**a**) pressure, (

**b**) shear rate, (

**c**) axial velocity for sinusoidal flow, no-slip wall boundary condition for velocity models V1, V2, and V3.

**Figure 8.**Contour Plot of (

**a**) pressure, (

**b**) shear rate, (

**c**) axial velocity for physiological flow, no-slip wall boundary condition for velocity models V4, V5 and V6.

**Figure 9.**Contour Plot of (

**a**) pressure, (

**b**) shear rate, (

**c**) axial velocity, (

**d**) wall stress for physiological flow, slip wall boundary condition for velocity models V4, V5 and V6.

**Figure 10.**(

**a**) Sample bioprinted tissue construct with embedded serpentine channel, (

**b**) fluorescent imaging after 14 days, (

**c**) confocal imaging after 14 days by staining with LIVE-DEAD assay of serpentine channel seeded with HUVECs at inlet flow velocity = 4.62 mm/min, (

**d**) 4.48 mm/min, (

**e**) 4.76 mm/min.υ.

Parameters | Slip | No Slip |
---|---|---|

Mesh vertices | 2220 | 1562 |

Element type | All elements | |

Triangles | 2608 | 1578 |

Quads | 712 | 608 |

Edge elements | 762 | 328 |

Vertex elements | 40 | 38 |

Number of elements | 3320 | 2186 |

Minimum element quality | 0.2758 | 0.3608 |

Average element quality | 0.7663 | 0.7733 |

Element area ratio | 0.0761 | 0.04016 |

Mesh area | 20.31 mm^{2} | 21.33 mm^{2} |

Inlet Velocity | Model | Parameters Value | Function |
---|---|---|---|

Sinusoidal Flow (SF) | V_{1} | f = 1.25 Hz, t = 0 to 1 s | V_{1}(t) = $4.48\times $$\mathrm{sin}\left[2\times \pi \times f\times t\right]$ |

V_{2} | V_{2}(t) = $4.62\times $$\mathrm{sin}\left[2\times \pi \times f\times t\right]$ | ||

V_{3} | V_{3}(t) = $4.76\times $$\mathrm{sin}\left[2\times \pi \times f\times t\right]$ | ||

Physiological Flow (PF) | V_{4} | t = 0 to 1 s, a_{1} = 3.04, b_{1} = 0.8634, c_{1} = −0.07247, a_{2} = 1.24, b_{2} = 14.02, c_{2} = 2.164, a_{3} = 2.619, b_{3} = 5.81, c_{3} = −0.5393, a_{4} = 0.4497, b_{4} = 24.25, c_{4} = 1.474, a_{5} = 0.3847, b_{5} = 32.5, c_{5} = −2.181, a_{6} = 0.4739, b_{6} = 20.77, c_{6} = 0.4904, a_{7} = 0.2708, b_{7} = 37.65, c_{7} = 2.495, a_{8} = 0.1386, b_{8} = 52.47, c_{8} = −0.7126 | V_{4}(t) = ${a}_{1}\times sin\left[\left({b}_{1}\times t\right)+{c}_{1}\right]+\phantom{\rule{0ex}{0ex}}{a}_{2}\times sin\left[\left({b}_{2}\times t\right)+{c}_{2}\right]+{a}_{3}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{0.17em}sin\left[\left({b}_{3}\times t\right)+{c}_{3}\right]+{a}_{4}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{0.17em}sin\left[\left({b}_{4}\times t\right)+{c}_{4}\right]+{a}_{5}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{0.17em}sin\left[\left({b}_{5}\times t\right)+{c}_{5}\right]+{a}_{6}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{0.17em}sin\left[\left({b}_{6}\times t\right)+{c}_{6}\right]+{a}_{7}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{0.17em}sin\left[\left({b}_{7}\times t\right)+{c}_{7}\right]+{a}_{8}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}sin\left[\left({b}_{8}\times t\right)+{c}_{8}\right]$ |

V_{5} | t = 0 to 1 s, a_{1} = 2.721, b_{1} = 1.588, c_{1} = −0.4734, a_{2} = 1.364, b_{2} = 13.48, c_{2} = 2.332, a_{3} = 3.081, b_{3} = 5.052, c_{3} = −0.1051, a_{4} = 0.3837, b_{4} = 24.37, c_{4} = 1.263, a_{5} = 0.4042, b_{5} = 32.44, c_{5} = −2.133, a_{6} = 0.4619, b_{6} = 19.567, c_{6} = 0.9404, a_{7} = 0.2744, b_{7} = 37.67, c_{7} = 2.482, a_{8} = 0.1435, b_{8} = 52.48, c_{8} = −0.7228 | V_{5}(t) = ${a}_{1}\times sin\left[\left({b}_{1}\times t\right)+{c}_{1}\right]+\phantom{\rule{0ex}{0ex}}{a}_{2}\times sin\left[\left({b}_{2}\times t\right)+{c}_{2}\right]+{a}_{3}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{3}\times t\right)+{c}_{3}\right]+{a}_{4}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{4}\times t\right)+{c}_{4}\right]+{a}_{5}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{5}\times t\right)+{c}_{5}\right]+{a}_{6}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{6}\times t\right)+{c}_{6}\right]+{a}_{7}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{7}\times t\right)+{c}_{7}\right]+{a}_{8}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}sin\left[\left({b}_{8}\times t\right)+{c}_{8}\right]$ | |

V_{6} | t = 0 to 1 s, a_{1} = 2.864, b_{1} = 1.405, c_{1} = −0.3165, a_{2} = 1.322, b_{2} = 13.96, c_{2} = 2.122, a_{3} = 3.008, b_{3} = 5.506, c_{3} = −0.3209, a_{4} = 0.7938, b_{4} = 22.63, c_{4} = 2.259, a_{5} = 0.4209, b_{5} = 31.95, c_{5} = −1.982, a_{6} = 0.8909, b_{6} = 20.65, c_{6} = 0.3462, a_{7} = 0.3141, b_{7} = 37.4, c_{7} = 2.577, a_{8} = 0.1474, b_{8} = 52.51, c_{8} = −0.7326 | V_{6}(t) = ${a}_{1}\times sin\left[\left({b}_{1}\times t\right)+{c}_{1}\right]+\phantom{\rule{0ex}{0ex}}{a}_{2}\times sin\left[\left({b}_{2}\times t\right)+{c}_{2}\right]+{a}_{3}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{3}\times t\right)+{c}_{3}\right]+{a}_{4}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{4}\times t\right)+{c}_{4}\right]+{a}_{5}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{5}\times t\right)+{c}_{5}\right]+{a}_{6}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{6}\times t\right)+{c}_{6}\right]+{a}_{7}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}sin\left[\left({b}_{7}\times t\right)+{c}_{7}\right]+{a}_{8}\times \phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}sin\left[\left({b}_{8}\times t\right)+{c}_{8}\right]$ |

Condition | Wall Boundary Condition | Inlet Boundary Condition | Inlet Velocity |
---|---|---|---|

I | Slip (S) | Sinusoidal Flow (SF) | V_{1} |

V_{2} | |||

V_{3} | |||

II | No-Slip (NS) | Sinusoidal Flow (SF) | V_{1} |

V_{2} | |||

V_{3} | |||

III | No-Slip (NS) | Physiological Flow (PF) | V_{4} |

V_{5} | |||

V_{6} | |||

IV | Slip (S) | Physiological Flow (PF) | V_{4} |

V_{5} | |||

V_{6} |

Probe | Grid | Number of Elements | Pressure at Time T | The Shear Rate at Time T | Velocity at Time T |
---|---|---|---|---|---|

P_{1} | Fine | 2366 | 210,698.8 | 42,091.81 | 12.64 |

Normal | 1578 | 215,248.60 | 43,115.82 | 12.71 | |

Coarse | 1106 | 230,802.2 | 44,968.63 | 12.99 | |

P_{2} | Fine | 2366 | 125,274.69 | 10,222.53 | 12.66 |

Normal | 1578 | 123,677.93 | 10,938.72 | 12.25 | |

Coarse | 1106 | 136,541.84 | 16,772.07 | 12.09 |

Variable | r | p | F_{s} | $\mathbf{G}\mathbf{C}{\mathbf{I}}_{23}(\%)$ | $\mathbf{G}\mathbf{C}{\mathbf{I}}_{12}(\%)$ |
---|---|---|---|---|---|

Pressure (P_{1}) | 1.209 | 6.48 | 1.25 | 3.72 | 1.11 |

Shear Rate (P_{1}) | 3.12 | 6.6 | 3.7 | ||

Velocity (P_{1}) | 6.41 | 1.13 | 0.3 | ||

Pressure (P_{2}) | 10 | 22.9 | 0.28 | ||

Shear Rate (P_{2}) | 9.39 | 13.48 | 1.77 | ||

Velocity (P_{2}) | 27 | 0.003 | 0.24 |

Parameters | Specification |
---|---|

Printing house temperature | 37 °C |

Extrusion pressure of the coaxial nozzle | 0.1 to 1 Mpa |

Nozzle diameter | 1 mm |

Needle size | 20 gauge |

Extrusion rate | 5 mm/s |

Dispensing speed | 4 mm/s |

Printing bed temperature and material | 2 °C to 40 °C and Aluminum/Glass |

Pulsatile pump speed, pressure and flow rate | 5 to 36 rpm, 80 to 120 mmHg; and 1–4 mL/min |

pH of media | 7.2 to 7.4 |

Configuration | Parameter | Point | Time | Deviation (%) | |
---|---|---|---|---|---|

S_SF_V1, V2, V3 | Pressure | P3 (V2) percent [V2, V3] | 0.5 | Min | 0.24937 |

P2 (V2) percent [V2, V3] | 0.3 | Max | 231.1239 | ||

Shear rate | P2 (V2) percent [V2, V3] | 0.5 | Min | 0.19267 | |

P2 (V1) percent [V2, V1] | 0.3 | Max | 19.17148 | ||

Velocity | P2 (V3) percent [V2, V3] | 0.5 | Min | 0.324655 | |

P1(V3) percent [V1, V2] | 0.5 | Max | 36.753 | ||

NS_SF_ V1, V2, V3 | Pressure | P1 (V3) percent [V2, V3] | 0.3 | Min | 3.603224 |

P3 (V1) percent [V3, V1] | 1 | Max | 14.14604 | ||

Shear rate | P1 (V3) percent [V2, V3] | 0.3 | Min | 1.854657 | |

P2 (V1) percent [V3, V1] | 0.5 | Max | 9.254331 | ||

Velocity | P2 (V3) percent [V2, V3] | 0.3 | Min | 1.775892 | |

P1 (V1) percent [V3, V1] | 1 | Max | 6.57622 | ||

NS_PF_V4, V5, V6 | Pressure | P1 (V6) percent [V6, V5] | 1 | Min | 3.075083 |

P3 (V4) percent [V6, V4] | 0.5 | Max | 14.18746 | ||

Shear rate | P2 (V5) percent [V4, V5] | 1 | Min | 2.104332 | |

P2 (V4) percent [V6, V4] | 0.5 | Max | 8.072614 | ||

Velocity | P3 (V6) percent [V6, V4] | 1 | Min | 2.813501 | |

P1 (V4) percent [V6, V4] | 0.5 | Max | 6.5163 | ||

S_PF_ V4, V5, V6 | Pressure | P3 (V6) percent [V6, V5] | 0.3 | Min | 2.485286 |

P3 (V6) percent [V6, V4] | 1 | Max | −3.9 × 10^{8} | ||

Shear rate | P3 (V6) percent [V6, V4] | 1 | Min | 2.959357 | |

P1 (V6) percent [V6, V5] | 0.3 | Max | 3229923 | ||

Velocity | P3 (V4) percent [V4, V5] | 0.5 | Min | 0.7726 | |

P3 (V5) percent [V4, V5] | 1 | Max | 291,407 |

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## Share and Cite

**MDPI and ACS Style**

Deshmukh, K.; Gupta, S.; Mitra, K.; Bit, A.
Numerical and Experimental Analysis of Shear Stress Influence on Cellular Viability in Serpentine Vascular Channels. *Micromachines* **2022**, *13*, 1766.
https://doi.org/10.3390/mi13101766

**AMA Style**

Deshmukh K, Gupta S, Mitra K, Bit A.
Numerical and Experimental Analysis of Shear Stress Influence on Cellular Viability in Serpentine Vascular Channels. *Micromachines*. 2022; 13(10):1766.
https://doi.org/10.3390/mi13101766

**Chicago/Turabian Style**

Deshmukh, Khemraj, Saurabh Gupta, Kunal Mitra, and Arindam Bit.
2022. "Numerical and Experimental Analysis of Shear Stress Influence on Cellular Viability in Serpentine Vascular Channels" *Micromachines* 13, no. 10: 1766.
https://doi.org/10.3390/mi13101766