These authors contributed equally to this work.
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
To mimic
When speaking or singing, the extracellular matrix (ECM) and cells of the vocal fold lamina propria experience forces from oscillation and collision [
The above investigations demonstrate a high degree of versatility to configure the TRB for single sample experimentation but leave unanswered the challenge to process multiple samples simultaneously. A multiwell disc is evaluated in the current study. It accommodates up to 20 samples in a single experiment. The purposes of this study were to test the TRB’s ability to vibrate this multiwell disc at phonation frequencies with accelerations covering a 100fold range and to computationally determine the range of shear stresses achievable within the partially fluidfilled wells.
The multiwell disc is a commercially manufactured and sterilized 96well cellculture plate (BD Biosciences, San Jose, CA, USA) cut into a disc (Medical Instruments, The University of Iowa). The disc diameter is 57 mm, retaining eight usable wells at each of two outer radial positions and four wells at the innermost radial position (
Schematics of the multiwell disc assembly, side (
Three acceleration conditions occur for a given experiments. All are at the same frequency and occur simultaneously because of the radial well locations relative to the rotation axis.
Moment of inertia of the multiwell disc assembly is calculated using the equation for a cylinder rotating about a central axis (Equation (1))
Material density ρ (stainless steel, polycarbonate, acrylic, and rubber), radius
TRBmultiwell system vibration is achieved by securing the multiwell assembly to a torsional rheometer (Malvern Instruments), then lowering it onto a threedimensional recoil material (
The magnitude of displacement is limited by the effective resonance peak of the system as determined by the recoil material’s viscoelasticity [
With the torque (maximum = 0.15 N m), inertia, and radius as fixed parameters, we manipulated the recoil material. Different viscoelastic properties allowed exploitation of the effective resonance of the TRBmultiwell system. Four recoil materials were tested in the present study.
Fluid shear stresses within the partially fluidfilled wells of the multiwell disc are not directly measurable. Rather, computational methods were used to quantify this vibration parameter. Smoothed Particle Hydrodynamics (SPH) is a computational technique for calculating the fluid dynamics using a Lagrangian framework. This approach has been validated against many benchmark cases, and is particularly well suited for freesurface flows, such as the TRBmultiwell system. SPH was initially proposed by Lucy [
As opposed to meshbased computational fluid dynamics (CFD) approaches, SPH is a meshfree method based on interpolation theory. In this method, matter is divided into a set of interpolation points or particles. Material properties such as density and viscosity, as well as the field variables velocity, pressure, and stress, are assigned to the particles. Integral interpolation among these particles approximates the field variables,
If A(r′) is known only at a discrete set of N points r_{1}, r_{2},..., r_{N}, the interpolation of quantity A(r) can be approximated by a summation interpolant,
The final equation used in the present investigation is a cubic spline kernel that is frequently used in SPH simulations. The compact support of this kernel function is equal to 2
Selection of recoil materials for empirical studies was based on predicted viscoelastic properties that could generate TRBmultiwell system accelerations of 10–1000 m/s^{2}. To predict properties at these accelerations, we calculated G' and G" with analytic expressions [
Shear stresses within each well of the multiwell disc were computed during vibration conditions using the SPH model described above. The fluid within a well (radius = 3.25 mm, height = 12.1 mm), was modeled as a 2D particle array—60 × 129 for 100 μL or 30 × 129 for 50 μL of fluid—to oscillate with a given frequency and amplitude. Fluid density (1 g/cm^{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 timeindependent solution. Shear stresses were then calculated for particle arrays.
To validate simulation results, we analytically determined a classical fluidmechanics benchmark, the Stokes Second Problem. Stokes Second Problem describes oscillatory movement of a fluid that is bounded on the bottom by an infinitely long plane, freely bound on the top, and has no side walls. Oscillatory flow begins at time
Schematic of Stokes Second Problem; a semiinfinite surface begins oscillation at time
To predict optimal properties of potential recoil materials, we calculated viscoelastic moduli for specific accelerations using Equations (10) and (11) with a nominal phase shift
Linear fit equations with highest
Doublesided 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 





34,669  279,045  30,709  6877 

225,278  0.0016  0.078  0.1689 

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 





4079.9  −11.813  4187.9  112.83 

54098  41086  0.1237  1899.6 

0.7611  0.0105  0.9607  0.9631 
Predicted shear elastic moduli (
Loss tangent values were usually less than 0.2 across frequency. Exceptions to this dominance of elasticity over viscosity were for the dimethyl silicon polymer at frequencies less than 2.5 Hz and the hot melt pressure sensitive adhesive at frequencies greater than 100 Hz.
Accelerations of the TRBmultiwell disc system are shown in
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 (
A total of four different combinations of frequency and amplitude were determined analytically from Stokes second equation and compared to twodimensional and threedimensional SPH simulations. Shown in
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)  

30  

30  

30  

10–300  10–35  30  

10–300  40  

11–300  40  

10–400  10–40  40  

10–400  10–50  50  

20–400  20–50  

20–400  400  

20–400  400  

20–400  400  

20–400  50–400  400  

20–400  50–400  400  

20–400  50–400 
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)  

4  

5  

5  

3−70  3−5  5  

4−60  5  

5−60  6  

7−70  7−8  7  

8−70  8−10  8  

8−70  8−10  

8−70  70  

8−70  40−70  

8−70  40−70  

8−80  80  

8−80  80  

8−80  80 
Comparison of 2D, 3D and Stokes Second Problem results.
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 
Smooth particle hydrodynamics (SPH) results showed that shear stresses reached steady state after a single vibration cycle (
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.
Tunable shear stresses inside the partially filled wells of the multiwell disc span a 100fold range, as determined by SPH simulations. Tunable features include frequency, amplitude, fluid volume, and fluid viscosity.
The multiwell disc appliance facilitates exploration of a frequency, acceleration, duration, shear (FADS) parameter space that has been rendered too time consuming and expensive with previously designed vibrating bioreactors. Our data demonstrate maximum accelerations of 9.2–334 m/s^{2} at 100–250 Hz by using different recoil materials with the TRBmultiwell system. These accelerations are much lower than predicted vocal fold accelerations using threemass computer simulation or stroboscopy data (1000–4000 m/s^{2} at 100–400 Hz) [
The large range of accelerations of the multiwell disc is made possible due to one main factor. The viscoelastic properties of the added recoil materials allow a mechanical resonance of the oscillating motor over a broad frequency range. The factorydesigned torsional rheometers require the test sample stiffness to generate a recoil force for oscillatory motion. For low stiffness test materials, particularly when operating at higher frequencies, the motor shaft and plate spin instead of oscillate. Previously, we showed that the effective resonance of a rheometersample system shifts along the frequency axis depending on the sample radius, thickness, and the unknown elastic modulus [
By adding a recoil material with known stiffness, we showed here that a system resonance could be exploited to provide a greater frequency range over which desired accelerations could be achieved. The viscoelastic properties of four recoil materials were tested to provide a two order of magnitude range of accelerations.
Shear stress is the last component in FADS parameter space we took into account when simulating vocal fold vibration conditions in the TRB. The partially filled wells of the multiwell disc exposed cells to shear stress, but measuring the stresses directly was not possible. Rather, using smooth particle hydrodynamics (SPH), we computed shear stresses in two dimensions—values comparable to those in threedimensional simulations but with less computational expense. The resulting shear stresses of 0.01–1 Pa were tunable by altering frequency, amplitude, viscosity, or fluid volume in the TRBmultiwell disc system.
Direct measures of shear stress within vibrating vocal fold tissue are not currently possible. Furthermore, until now, shear stresses within a bioreactor have not been a controllable parameter from which to systematically investigate vocal fold cell function or tissue remodeling. Target shear stresses of 100–1000 Pa in previous appliance designs were consistent with stresses reported in physical and computational models and with constitutive relationships of bulk vocal fold deformation [
While this outcome might appear to be a significant limitation to the TRBmultiwell disc system, preliminary results from our lab show vocal fold fibroblast sensitivities to these ranges (see Appendix Materials). We tested effects of FADS when vocal fold fibroblast cells first established ECM connectivity, a time scale that corresponds to events in early wound healing and in cell migration [
Future knowledge from studies using the TRBmultiwell disc system could extend beyond that of vocal fold cellular and tissue function. Skin and brain are two organ systems with known exposures to forces comparable to that of vocal fold tissue—forces transferred to embedded cells with comparable magnitudes due to similar viscoelastic properties of the tissues [
This work was supported by NIH Grant No. DC010275 and No. DC008047 from the National Institute on Deafness and Other Communication Disorders. We wish to acknowledge Mehrdad Hosnieh Farahani for his help with simulation coding and Susan Thibeault for her gracious gift of the immortalized vocal fold fibroblast cells.
The authors declare no conflict of interest.
An immortalized vocal fold fibroblast cell line was used for adhesion studies, originating from a 21 y.o. male (young adult cell line, hVFFCs) [
The day of the experiment, the fibronectincoated wells were washed with PBS. 100 µL of Calcein AMtreated cells (10^{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 (
Selection of vibration regimens mimicked human voice use. Frequencies were male and female fundamental frequencies (126 and 220 Hz respectively). ON/OFF vibration duty ratios were 0.55 (0.55 s ON and 0.45 s OFF), corresponding to voicedunvoiced characteristics of spoken American English and recorded during daily voice use of public school teachers [
Adherence changes due to vibration regimens were reported as the adhesion ratio. Measured optical density, from each well containing cells, was divided by the mean optical density of the control wells (
Human vocal fold fibroblast cells were tested for their early adhesion abilities (within 7 h of initially plating cells on fibronectin) in the multiwell disc under a variety of vibration conditions. The adhesion ratio (experimental/control emitted optical density) diminished linearly with calculated shear stress and diminished exponentially as a function of cumulative distance (
Mean early adhesion ratio of human vocal fold fibroblast cells (within 7 h of initial plating on fibronectin) as a function of shear stress (