# A Multiwell Disc Appliance Used to Deliver Quantifiable Accelerations and Shear Stresses at Sonic Frequencies

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

^{2}, and shear stresses of 0.01–1.4 Pa. It is well-suited for studying cell function underlying vocal fold lamina propria homeostasis, inflammation, and wound healing under differential vibration conditions.

## 1. Introduction

_{2}conditions using a custom incubator fitted to the commercial rheometer [13].

## 2. Experimental Section

#### 2.1. Maximizing Vibration Regimens

^{−5}kg m

^{2}.

_{0}is defined in terms of the elastic modulus, radius, thickness of the recoil material h, and moment of inertia of the entire assembly I:

#### 2.2. Using SPH to Model Shear Stresses

_{Ω}f(x')δ(x − x′)dx'

_{1}, r

_{2},..., r

_{N}, the interpolation of quantity A(r) can be approximated by a summation interpolant,

#### 2.3. Recoil Material Evaluation

^{2}. To predict properties at these accelerations, we calculated G' and G" with analytic expressions [11]. Rewriting complex strain γ* as acceleration, we assume the small angle approximation rθ = rd where r is radial distance from the center of rotation, θ is angular displacement, and d is sample thickness. Using the second derivative of sinusoidal motion e

^{iωt}at angular frequency ω,

#### 2.4. Shear Stress Modeling

^{3}) was fixed. Fluid viscosity of cell culture medium, measured in triplicate with the rheometer, was 2.26 ± 0.13 mPa s. We also tested the model using a viscosity three times that of cell culture medium. Computations were repeated for several simulation cycles of vibration to ensure a time-independent solution. Shear stresses were then calculated for particle arrays.

**Figure 2.**Schematic of Stokes Second Problem; a semi-infinite surface begins oscillation at time t with a frequency n.

## 3. Results and Discussion

#### 3.1. Viscoelastic Properties of Recoil Materials

^{2}acceleration at 100–250 Hz (solid lines in Figure 3). Properties of the four tested recoil materials fell within these ranges. Their elastic and viscous moduli were 1.3–350 kPa and 1.1–140 kPa, respectively (symbols with dotted lines in Figure 3 top and middle). Direct measures were accurate up to 100 Hz and then extrapolated to 250 Hz using line fits (Table 1).

**Table 1.**Linear fit equations with highest R

^{2}and corresponding coefficients used to extrapolate viscoelastic values of recoil materials to 250 Hz.

Double-sided foam tape | Dimethyl silicon | Removable mounting squares | Hot melt pressure sensitive adhesive | |
---|---|---|---|---|

Elastic modulus | ||||

Freq range of data used in extrapolation (Hz) | 0.1–100 | 10–100 | 0.1–100 | 0.1–100 |

Line fit function | y = aln(x) + b | y = ae^{bx} | y = ax^{b} | y = ax + b |

a (Pa) | 34,669 | 279,045 | 30,709 | 6877 |

b (Pa) | 225,278 | 0.0016 | 0.078 | 0.1689 |

R^{2} | 0.9961 | 0.7644 | 0.9818 | 0.9952 |

Viscous modulus | ||||

Freq range of data used in extrapolation (Hz) | 0.1–100 | 32–100 | 0.1–100 | 0.1–100 |

Line fit function | y = aln(x) + b | y = ax + b | y = ax^{b} | y = ax + b |

a (Pa) | 4079.9 | −11.813 | 4187.9 | 112.83 |

b (Pa) | 54098 | 41086 | 0.1237 | 1899.6 |

R^{2} | 0.7611 | 0.0105 | 0.9607 | 0.9631 |

**Figure 3.**Predicted shear elastic moduli (

**top**) and viscous modulus (

**middle**) and loss tangent (viscous modulus/elastic modulus) (

**bottom**) achieve multiwell disc accelerations of 10–1000 m/s

^{2}(solid lines). Measured shear elastic modulus (

**top**) and viscous modulus (

**middle**), and loss tangent (

**bottom**) of four different recoil materials: hot melt pressure sensitive adhesive (arranged in a ring) (

**♦**,

**◊**), removable foam mounting squares (

**●**,

**○**), dimethyl silicon polymer (

**■**,

**□**), double-sided foam tape (

**▲**,

**Δ**).

#### 3.2. TRB-Multiwell Disc Accelerations

^{2}or more at frequencies over 100 Hz. The greatest accelerations were obtained using the two stiffest recoil materials. Peak accelerations were 336 and 303 m/s

^{2}at 151 Hz using the dimethyl silicon polymer (squares) and foam tape (triangles), respectively. Up to 250 Hz, the foam tape maintained slightly higher accelerations (285–170 m/s

^{2}) compared to the dimethyl silicon polymer (285–115 m/s

^{2}). At frequencies below 100 Hz, accelerations of the TRB-multiwell disc system were maximal when using the mounting squares (circles) as the recoil material. Accelerations of 100 m/s

^{2}were achieved and sustained beginning at 28 Hz through 250 Hz. The hot melt pressure sensitive adhesive (diamonds) was the least effective at maximizing accelerations, both below and above 100 Hz, compared to the other three recoil materials. Peak accelerations were 100 m/s

^{2}at 115 Hz and stayed above 78 m/s

^{2}through 250 Hz.

**Figure 4.**Measured accelerations of the torsional rheometer bioreactor (TRB)-multiwell system using maximum torque and specified recoil material: hot melt pressure sensitive adhesive (◊) and removable foam mounting squares (○), dimethyl silicon polymer (

**□**), and double-sided foam tape (

**Δ**) meet and exceed expectations over a broad frequency range.

#### 3.3. Stokes Second Equation Comparisons

**Table 2.**Approximate elastic moduli of recoil materials (in kPa) to achieve target accelerations at frequencies 2–250 Hz (e.g., to generate 100 m/s

^{2}at 200 Hz, the recoil material’s elastic modulus is between 50 and 400 kPa).

Acceleration (m/s^{2}) | 0.4 | 1 | 2 | 5 | 10 | 15 | 20 | 40 | 60 | 80 | 100 | 200 | 300 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Frequency (Hz) | Approximate Elastic Modulus of Recoil Materials (kPa) | ||||||||||||

2 | 30 | ||||||||||||

4 | 30 | ||||||||||||

6 | 30 | ||||||||||||

8 | 10–300 | 10–35 | 30 | ||||||||||

10 | 10–300 | 40 | |||||||||||

20 | 11–300 | 40 | |||||||||||

50 | 10–400 | 10–40 | 40 | ||||||||||

80 | 10–400 | 10–50 | 50 | ||||||||||

100 | 20–400 | 20–50 | |||||||||||

130 | 20–400 | 400 | |||||||||||

150 | 20–400 | 400 | |||||||||||

180 | 20–400 | 400 | |||||||||||

200 | 20–400 | 50–400 | 400 | ||||||||||

220 | 20–400 | 50–400 | 400 | ||||||||||

250 | 20–400 | 50–400 |

**Table 3.**Approximate viscous moduli of recoil materials (in kPa) to achieve target accelerations at frequencies 2–250 Hz (e.g., to generate 100 m/s

^{2}at 200 Hz, the recoil material’s viscous modulus is between 8 and 80 kPa).

Acceleration (m/s^{2}) | 0.4 | 1 | 2 | 5 | 10 | 15 | 20 | 40 | 60 | 80 | 100 | 200 | 300 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Frequency (Hz) | Approximate Viscous Modulus of Recoil Materials (kPa) | ||||||||||||

2 | 4 | ||||||||||||

4 | 5 | ||||||||||||

6 | 5 | ||||||||||||

8 | 3−70 | 3−5 | 5 | ||||||||||

10 | 4−60 | 5 | |||||||||||

20 | 5−60 | 6 | |||||||||||

50 | 7−70 | 7−8 | 7 | ||||||||||

80 | 8−70 | 8−10 | 8 | ||||||||||

100 | 8−70 | 8−10 | |||||||||||

130 | 8−70 | 70 | |||||||||||

150 | 8−70 | 40−70 | |||||||||||

180 | 8−70 | 40−70 | |||||||||||

200 | 8−80 | 80 | |||||||||||

220 | 8−80 | 80 | |||||||||||

250 | 8−80 | 80 |

Frequency and amplitude conditions | 2D SPH | 3D SPH | Stokes 2nd problem |
---|---|---|---|

Shear stress (Pa) | |||

126 Hz; 0.46 mm | 0.3 | 0.5 | 1.6 |

126 Hz; 0.069 mm | 0.05 | 0.06 | 0.5 |

220 Hz; 0.063 mm | 0.07 | 0.1 | 0.2 |

220 Hz; 0.0094 mm | 0.01 | 0.03 | 0.08 |

#### 3.4. Simulation of Shear Stress

**Figure 5.**Smooth particle hydrodynamics (SPH) computational shear stress solutions in two dimensions reach a steady state after one cycle. Largest stress occurs at the well bottom and diminishes with vertical distance from the moving surface.

**Figure 6.**Tunable shear stresses inside the partially filled wells of the multiwell disc span a 100-fold range, as determined by SPH simulations. Tunable features include frequency, amplitude, fluid volume, and fluid viscosity.

## 4. Conclusions

^{2}at 100–250 Hz by using different recoil materials with the TRB-multiwell system. These accelerations are much lower than predicted vocal fold accelerations using three-mass computer simulation or stroboscopy data (1000–4000 m/s

^{2}at 100–400 Hz) [20,21] but they are the highest values generated by any bioreactor to date [7,8,10,22,23,24,25,26]. Stiff recoil materials provide accelerations in the order of 300 m/s

^{2}above 100 Hz and more compliant materials provide 100 m/s

^{2}at 28–100 Hz vibrations.

^{2}accelerations, could be realized using the more compliant drivers with the TRB-multiwell disc system, as exemplified in Figure 4. In the brain, endogenous neural mechanobiology is not well understood, nor is there good understanding of the cellular-molecular responses to traumatic brain injury [44,45]. A finding that the neuropeptide galanin has protective functions against shear stress of cortical neurons was determined using a microfluidics device [46]. The stresses with that device are a subset of the quantitative FADS parameters possible with the TRB-multiwell disc system. Controls on the rheometer permit single or short duration pulsing to simulate blunt trauma forces, or duty ratio-to-continuous low frequency motion to simulate endogenous conditions on multiple samples simultaneously. Particularly for systems such as the brain and vocal fold, forces are complex and knowledge of their mechanobiology is limited. A device such as the TRB-multiwell disc system can provide an invaluable tool, quantifying forces to gain understanding of cell behavior directly related to FADS parameters.

## Acknowledgments

## Conflicts of Interest

## Appendix

## 1. Supplemental Experimental Section: Fibroblast Adhesion

#### 1.1. Cell Culture and Adhesion Assay

_{2}atmosphere in Dulbecco’s modified Eagle medium (DMEM), high glucose with glutamine, 10% heat-inactivated fetal bovine serum (FBS). The day prior to experimentation, cells were lifted using trypsin and replated into flasks, then removed the following day using an enzyme-free dissociation solution known to minimize the disturbance of cell surface proteins (Gibco, Carlsbad, CA, USA). Cells were then treated with Calcein AM (calcein acetoxymethyl ester) (Invitrogen Corporation, Carlsbad, CA, USA) to fluorescently tag adherent cells. Twenty wells in two the multiwell discs (one experimental and one control) were coated with 0.5 µg/mL fibronectin with 100 µL per well and stored overnight.

^{6}cells/mL) were placed in each coated well using complete cell culture medium. Experimental conditions commenced 30 min later. Upon completion of a prescribed vibration regimen, the experimental and control multiwell discs were removed from their TRB. Each well was washed 3 times with PBS so only adhered cells remained, inverted for 5 min to dry, and replenished with 200 µL of PBS. Optical density (i.e., fluorescent intensity) was measured for each disc (absorption wave length = 490 nm; emission wave length = 525 nm) along with serial dilution wells (prepared similarly to the discs but maintained in a standard incubator during testing) using a FLUOstar Omega microplate reader (BMG Labtech; Durham, NC, USA).

#### 1.2. Vibration Regimens

^{2}). To reduce the four vibration parameters (frequency, duration, acceleration, stress) down to two, we used a metric proposed by Titze et al. for vocal fold vibration: cumulative distance (i.e., vocal dose)

_{rms}F

_{0}t

_{rms}is the root mean square amplitude of movement (converted from the acceleration), F

_{0}is the frequency and t is duration of voicing [21]. The three variables are measured by the high precision torque motor and sensing system of the rheometer, quantifying input and output vibrations every second for the duration of each experiment. Shear stress is determined computationally, as explained previously. A total of twelve cumulative distances covering a 10-fold range (47–550 m) were tested.

#### 1.3. Adhesion Ratio Calculation

## 2. Supplementary Results

#### 2.1. Adhesion Sensitivity to Vibration

**Figure A1.**Mean early adhesion ratio of human vocal fold fibroblast cells (within 7 h of initial plating on fibronectin) as a function of shear stress (

**top**) and cumulative distance (

**bottom**). Error bars correspond to standard error of the mean. Linear (

**top**) and exponential (

**bottom**) fits, their coefficients and goodness of fit are listed.

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## Share and Cite

**MDPI and ACS Style**

Klemuk, S.A.; Vigmostad, S.; Endapally, K.; Wagner, A.P.; Titze, I.R.
A Multiwell Disc Appliance Used to Deliver Quantifiable Accelerations and Shear Stresses at Sonic Frequencies. *Processes* **2014**, *2*, 71-88.
https://doi.org/10.3390/pr2010071

**AMA Style**

Klemuk SA, Vigmostad S, Endapally K, Wagner AP, Titze IR.
A Multiwell Disc Appliance Used to Deliver Quantifiable Accelerations and Shear Stresses at Sonic Frequencies. *Processes*. 2014; 2(1):71-88.
https://doi.org/10.3390/pr2010071

**Chicago/Turabian Style**

Klemuk, Sarah A., Sarah Vigmostad, Kalyan Endapally, Andrew P. Wagner, and Ingo R. Titze.
2014. "A Multiwell Disc Appliance Used to Deliver Quantifiable Accelerations and Shear Stresses at Sonic Frequencies" *Processes* 2, no. 1: 71-88.
https://doi.org/10.3390/pr2010071