# A Proof-of-Concept Study Using Numerical Simulations of an Acoustic Spheroid-on-a-Chip Platform for Improving 3D Cell Culture

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geometry and Model Description

#### 2.2. Governing Equations

#### 2.2.1. Microfluidic Flow

#### 2.2.2. Transport of Dilute Species

#### 2.2.3. Acoustic

#### 2.3. Boundary Conditions

#### 2.3.1. Microfluidic Flow

#### 2.3.2. Transport of Dilute Species

#### 2.3.3. Acoustic

#### 2.4. Numerical Method

#### 2.5. Mesh-Independent Study

#### 2.6. Validation of the Study

## 3. Results and Discussion

#### 3.1. Conventional Spheroid-on-Chip Platform (No Acoustic)

#### 3.2. Acoustic Spheroid-on-Chip Platform

#### 3.3. Boundary Displacement Amplitude

#### 3.4. Flow Rate

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Torre, L.A.; Bray, F.; Siegel, R.L.; Ferlay, J.; Lortet-Tieulent, J.; Jemal, A. Global cancer statistics, 2012. CA Cancer J. Clin.
**2015**, 65, 87–108. [Google Scholar] [CrossRef] [Green Version] - Young, E.W. Cells, tissues, and organs on chips: Challenges and opportunities for the cancer tumor microenvironment. Integr. Biol.
**2013**, 5, 1096–1109. [Google Scholar] [CrossRef] - Dhiman, N.; Shagaghi, N.; Bhave, M.; Sumer, H.; Kingshott, P.; Rath, S.N. Indirect co-culture of lung carcinoma cells with hyperthermia-treated mesenchymal stem cells influences tumor spheroid growth in a collagen-based 3-dimensional microfluidic model. Cytotherapy
**2020**, 23, 25–26. [Google Scholar] [CrossRef] [PubMed] - Moghadas, H.; Saidi, M.S.; Kashaninejad, N.; Kiyoumarsioskouei, A.; Nguyen, N.-T. Fabrication and characterization of low-cost, bead-free, durable and hydrophobic electrospun membrane for 3D cell culture. Biomed. Microdevices
**2017**, 19, 74. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mehta, G.; Hsiao, A.Y.; Ingram, M.; Luker, G.D.; Takayama, S. Opportunities and challenges for use of tumor spheroids as models to test drug delivery and efficacy. J. Control. Release
**2012**, 164, 192–204. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kashaninejad, N.; Nikmaneshi, M.R.; Moghadas, H.; Kiyoumarsi Oskouei, A.; Rismanian, M.; Barisam, M.; Saidi, M.S.; Firoozabadi, B. Organ-Tumor-on-a-Chip for Chemosensitivity Assay: A Critical Review. Micromachines
**2016**, 7, 130. [Google Scholar] [CrossRef] [PubMed] - Schmeichel, K.L.; Bissell, M.J. Modeling tissue-specific signaling and organ function in three dimensions. J. Cell Sci.
**2003**, 116, 2377–2388. [Google Scholar] [CrossRef] [Green Version] - Friedrich, J.; Seidel, C.; Ebner, R.; Kunz-Schughart, L.A. Spheroid-based drug screen: Considerations and practical approach. Nat. Protoc.
**2009**, 4, 309. [Google Scholar] [CrossRef] - Nguyen, N.T.; Schubert, S.; Richter, S.; Dötzel, W. Hybrid-assembled micro dosing system using silicon-based micropump/valve and mass flow sensor. Sens. Actuators A Phys.
**1998**, 69, 85–91. [Google Scholar] [CrossRef] - Dinh, T.; Phan, H.-P.; Dao, D.V.; Woodfield, P.; Qamar, A.; Nguyen, N.-T. Graphite on paper as material for sensitive thermoresistive sensors. J. Mater. Chem. C
**2015**, 3, 8776–8779. [Google Scholar] [CrossRef] [Green Version] - Altmann, B.; Grün, C.; Nies, C.; Gottwald, E. Advanced 3D Cell Culture Techniques in Micro-Bioreactors, Part II: Systems and Applications. Processes
**2021**, 9, 21. [Google Scholar] [CrossRef] - Huang, Y.L.; Ma, Y.; Wu, C.; Shiau, C.; Segall, J.E.; Wu, M. Tumor spheroids under perfusion within a 3D microfluidic platform reveal critical roles of cell-cell adhesion in tumor invasion. Sci. Rep.
**2020**, 10, 9648. [Google Scholar] [CrossRef] - Shao, C.; Chi, J.; Zhang, H.; Fan, Q.; Zhao, Y.; Ye, F. Development of cell spheroids by advanced technologies. Adv. Mater. Technol.
**2020**, 5, 2000183. [Google Scholar] [CrossRef] - Bourn, M.D.; Batchelor, D.V.; Ingram, N.; McLaughlan, J.R.; Coletta, P.L.; Evans, S.D.; Peyman, S.A. High-throughput microfluidics for evaluating microbubble enhanced delivery of cancer therapeutics in spheroid cultures. J. Control. Release
**2020**, 326, 13–24. [Google Scholar] [CrossRef] [PubMed] - Moshksayan, K.; Kashaninejad, N.; Warkiani, M.E.; Lock, J.G.; Moghadas, H.; Firoozabadi, B.; Saidi, M.S.; Nguyen, N.-T. Spheroids-on-a-chip: Recent advances and design considerations in microfluidic platforms for spheroid formation and culture. Sens. Actuators B Chem.
**2018**, 263, 151–176. [Google Scholar] [CrossRef] [Green Version] - Rostami, P.; Kashaninejad, N.; Moshksayan, K.; Saidi, M.S.; Firoozabadi, B.; Nguyen, N.-T. Novel approaches in cancer management with circulating tumor cell clusters. J. Sci. Adv. Mater. Devices
**2019**, 4, 1–18. [Google Scholar] [CrossRef] - Wu, L.Y.; Di Carlo, D.; Lee, L.P. Microfluidic self-assembly of tumor spheroids for anticancer drug discovery. Biomed. Microdevices
**2008**, 10, 197–202. [Google Scholar] [CrossRef] [PubMed] - Yahyazadeh Shourabi, A.; Kashaninejad, N.; Saidi, M.S. An integrated microfluidic concentration gradient generator for mechanical stimulation and drug delivery. J. Sci. Adv. Mater. Devices
**2021**, 6, 280–290. [Google Scholar] [CrossRef] - Lee, S.W.; Hong, S.; Jung, B.; Jeong, S.Y.; Byeon, J.H.; Jeong, G.S.; Choi, J.; Hwang, C. In vitro lung cancer multicellular tumor spheroid formation using a microfluidic device. Biotechnol. Bioeng.
**2019**, 116, 3041–3052. [Google Scholar] [CrossRef] - Weiswald, L.-B.; Bellet, D.; Dangles-Marie, V. Spherical cancer models in tumor biology. Neoplasia
**2015**, 17, 1–15. [Google Scholar] [CrossRef] [Green Version] - Fukuda, J.; Nakazawa, K. Orderly arrangement of hepatocyte spheroids on a microfabricated chip. Tissue Eng.
**2005**, 11, 1254–1262. [Google Scholar] [CrossRef] [PubMed] - Barisam, M.; Saidi, M.S.; Kashaninejad, N.; Vadivelu, R.; Nguyen, N.-T. Numerical simulation of the behavior of toroidal and spheroidal multicellular aggregates in microfluidic devices with microwell and U-shaped barrier. Micromachines
**2017**, 8, 358. [Google Scholar] [CrossRef] [Green Version] - Rocha, H.L.; Godet, I.; Kurtoglu, F.; Metzcar, J.; Konstantinopoulos, K.; Bhoyar, S.; Gilkes, D.M.; Macklin, P. A persistent invasive phenotype in post-hypoxic tumor cells is revealed by novel fate-mapping and computational modeling. bioRxiv
**2021**. [Google Scholar] [CrossRef] - Kim, C.H.; Ko, A.R.; Lee, S.Y.; Jeon, H.M.; Kim, S.M.; Park, H.G.; Han, S.I.; Kang, H.S. Hypoxia switches glucose depletion-induced necrosis to phosphoinositide 3-kinase/Akt-dependent apoptosis in A549 lung adenocarcinoma cells. Int. J. Oncol.
**2010**, 36, 117–124. [Google Scholar] [PubMed] - Wang, B.; Zhao, Q.; Zhang, Y.; Liu, Z.; Zheng, Z.; Liu, S.; Meng, L.; Xin, Y.; Jiang, X. Targeting hypoxia in the tumor microenvironment: A potential strategy to improve cancer immunotherapy. J. Exp. Clin. Cancer Res.
**2021**, 40, 24. [Google Scholar] [CrossRef] - Barisam, M.; Saidi, M.S.; Kashaninejad, N.; Nguyen, N.-T. Prediction of necrotic core and hypoxic zone of multicellular spheroids in a microbioreactor with a u-shaped barrier. Micromachines
**2018**, 9, 94. [Google Scholar] [CrossRef] [Green Version] - Xu, J.; Vilanova, G.; Gomez, H. A mathematical model coupling tumor growth and angiogenesis. PLoS ONE
**2016**, 11, e0149422. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Grimes, D.R.; Kelly, C.; Bloch, K.; Partridge, M. A method for estimating the oxygen consumption rate in multicellular tumour spheroids. J. R. Soc. Interface
**2014**, 11, 20131124. [Google Scholar] [CrossRef] [Green Version] - Im, G.-B.; Kim, S.-W.; Bhang, S.H. Fortifying the angiogenic efficacy of adipose derived stem cell spheroids using spheroid compaction. J. Ind. Eng. Chem.
**2021**, 93, 228–236. [Google Scholar] [CrossRef] - Zhao, M.; Li, X.; Zhang, Y.; Wang, Y.; Wang, B.; Zheng, L.; Zhang, D.; Zhuang, S. Rapid quantitative detection of chloramphenicol in milk by microfluidic immunoassay. Food Chem.
**2021**, 339, 127857. [Google Scholar] [CrossRef] - Rousset, N.; Monet, F.; Gervais, T. Simulation-assisted design of microfluidic sample traps for optimal trapping and culture of non-adherent single cells, tissues, and spheroids. Sci. Rep.
**2017**, 7, 245. [Google Scholar] [CrossRef] - Patra, B.; Peng, C.-C.; Liao, W.-H.; Lee, C.-H.; Tung, Y.-C. Drug testing and flow cytometry analysis on a large number of uniform sized tumor spheroids using a microfluidic device. Sci. Rep.
**2016**, 6, 21061. [Google Scholar] [CrossRef] [Green Version] - Bruus, H.; Dual, J.; Hawkes, J.; Hill, M.; Laurell, T.; Nilsson, J.; Radel, S.; Sadhal, S.; Wiklund, M. Forthcoming Lab on a Chip tutorial series on acoustofluidics: Acoustofluidics—Exploiting ultrasonic standing wave forces and acoustic streaming in microfluidic systems for cell and particle manipulation. Lab Chip
**2011**, 11, 3579–3580. [Google Scholar] [CrossRef] [Green Version] - Ding, X.; Li, P.; Lin, S.-C.S.; Stratton, Z.S.; Nama, N.; Guo, F.; Slotcavage, D.; Mao, X.; Shi, J.; Costanzo, F. Surface acoustic wave microfluidics. Lab Chip
**2013**, 13, 3626–3649. [Google Scholar] [CrossRef] - Yeo, L.Y.; Friend, J.R. Surface acoustic wave microfluidics. Annu. Rev. Fluid Mech.
**2014**, 46, 379–406. [Google Scholar] [CrossRef] [Green Version] - Li, P.; Mao, Z.; Peng, Z.; Zhou, L.; Chen, Y.; Huang, P.-H.; Truica, C.I.; Drabick, J.J.; El-Deiry, W.S.; Dao, M. Acoustic separation of circulating tumor cells. Proc. Natl. Acad. Sci. USA
**2015**, 112, 4970–4975. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shilton, R.; Tan, M.K.; Yeo, L.Y.; Friend, J.R. Particle concentration and mixing in microdrops driven by focused surface acoustic waves. J. Appl. Phys.
**2008**, 104, 014910. [Google Scholar] [CrossRef] [Green Version] - Li, S.; Guo, F.; Chen, Y.; Ding, X.; Li, P.; Wang, L.; Cameron, C.E.; Huang, T.J. Standing surface acoustic wave based cell coculture. Anal. Chem.
**2014**, 86, 9853–9859. [Google Scholar] [CrossRef] [Green Version] - Greco, G.; Agostini, M.; Tonazzini, I.; Sallemi, D.; Barone, S.; Cecchini, M. Surface-acoustic-wave (SAW)-driven device for dynamic cell cultures. Anal. Chem.
**2018**, 90, 7450–7457. [Google Scholar] [CrossRef] - Wu, Y.; Ao, Z.; Chen, B.; Muhsen, M.; Bondesson, M.; Lu, X.; Guo, F. Acoustic assembly of cell spheroids in disposable capillaries. Nanotechnology
**2018**, 29, 504006. [Google Scholar] [CrossRef] [Green Version] - Jeger-Madiot, N.; Arakelian, L.; Setterblad, N.; Bruneval, P.; Hoyos, M.; Larghero, J.; Aider, J.-L. Self-organization and culture of Mesenchymal Stem Cell spheroids in acoustic levitation. Sci. Rep.
**2021**, 11, 8355. [Google Scholar] [CrossRef] [PubMed] - Chen, B.; Wu, Y.; Ao, Z.; Cai, H.; Nunez, A.; Liu, Y.; Foley, J.; Nephew, K.; Lu, X.; Guo, F. High-throughput acoustofluidic fabrication of tumor spheroids. Lab Chip
**2019**, 19, 1755–1763. [Google Scholar] [CrossRef] - Muller, P.B.; Barnkob, R.; Jensen, M.J.H.; Bruus, H. A numerical study of microparticle acoustophoresis driven by acoustic radiation forces and streaming-induced drag forces. Lab Chip
**2012**, 12, 4617–4627. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Raghavan, R.V.; Friend, J.R.; Yeo, L.Y. Particle concentration via acoustically driven microcentrifugation: microPIV flow visualization and numerical modelling studies. Microfluid. Nanofluidics
**2010**, 8, 73–84. [Google Scholar] [CrossRef] - Cui, W.; Zhang, H.; Zhang, H.; Yang, Y.; He, M.; Qu, H.; Pang, W.; Zhang, D.; Duan, X. Localized ultrahigh frequency acoustic fields induced micro-vortices for submilliseconds microfluidic mixing. Appl. Phys. Lett.
**2016**, 109, 253503. [Google Scholar] [CrossRef] - Anada, T.; Fukuda, J.; Sai, Y.; Suzuki, O. An oxygen-permeable spheroid culture system for the prevention of central hypoxia and necrosis of spheroids. Biomaterials
**2012**, 33, 8430–8441. [Google Scholar] [CrossRef] - Bruus, H. Acoustofluidics 2: Perturbation theory and ultrasound resonance modes. Lab Chip
**2012**, 12, 20–28. [Google Scholar] [CrossRef] - Datta, M.; Via, L.E.; Chen, W.; Baish, J.W.; Xu, L.; Barry, C.E.; Jain, R.K. Mathematical model of oxygen transport in tuberculosis granulomas. Ann. Biomed. Eng.
**2016**, 44, 863–872. [Google Scholar] [CrossRef] [Green Version] - Jackson, A.R.; Huang, C.-Y.C.; Brown, M.D.; Yong Gu, W. 3D finite element analysis of nutrient distributions and cell viability in the intervertebral disc: Effects of deformation and degeneration. J. Biomech. Eng.
**2011**, 133, 091006. [Google Scholar] [CrossRef] - Alhourani, A.H.; Tidwell, T.R.; Bokil, A.A.; Røsland, G.V.; Tronstad, K.J.; Søreide, K.; Hagland, H.R. Metformin treatment response is dependent on glucose growth conditions and metabolic phenotype in colorectal cancer cells. Sci. Rep.
**2021**, 11, 10487. [Google Scholar] [CrossRef] - Das, P.K.; Snider, A.D.; Bhethanabotla, V.R. Acoustothermal heating in surface acoustic wave driven microchannel flow. Phys. Fluids
**2019**, 31, 106106. [Google Scholar] [CrossRef] - Shurbaji, S.; Anlar, G.G.; Hussein, E.A.; Elzatahry, A.; Yalcin, H.C. Effect of flow-induced shear stress in nanomaterial uptake by cells: Focus on targeted anti-cancer therapy. Cancers
**2020**, 12, 1916. [Google Scholar] [CrossRef] [PubMed] - Dual, J.; Schwarz, T. Acoustofluidics 3: Continuum mechanics for ultrasonic particle manipulation. Lab. A Chip
**2012**, 12, 244–252. [Google Scholar] [CrossRef] [PubMed] - Lien, S.-C.; Chang, S.-F.; Lee, P.-L.; Wei, S.-Y.; Chang, M.D.-T.; Chang, J.-Y.; Chiu, J.-J. Mechanical regulation of cancer cell apoptosis and autophagy: Roles of bone morphogenetic protein receptor, Smad1/5, and p38 MAPK. Biochim. Biophys. Acta BBA Mol. Cell Res.
**2013**, 1833, 3124–3133. [Google Scholar] [CrossRef] [Green Version] - Bizik, J.; Kankuri, E.; Ristimäki, A.; Taieb, A.; Vapaatalo, H.; Lubitz, W.; Vaheri, A. Cell–cell contacts trigger programmed necrosis and induce cyclooxygenase-2 expression. Cell Death Differ.
**2004**, 11, 183–195. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**(

**a**) 3D format and (

**b**) 2D format of the schematic of the proposed acoustic-microfluidic system for 3D cell culture The geometry of a spheroid by its nature is 3D (i.e., a sphere). When modeling it in a 2D domain as a circle, we are only observing the midplane of the system and neglecting the effect of the third dimension on the system. Theoretically, this 2D model represents more quantitatively an infinite cylindrical cell aggregate and more qualitatively a spherical cell aggregate. Using a 3D model was computationally prohibitive in this work, as the number of mesh elements increases non-linearly with the frequency, and for our MHz-range simulation, a 3D solution was not affordable. While the simulation results are not exactly the same for a 3D model, they provide a good qualitative solution for the proof-of-concept study. Additionally, we hold the conviction that our simulation results are much more comparable to a spherical cell aggregate model rather than a finite cylindrical one. Consider a finite cylinder. For this model, there is a sharp difference in the flow field between the middle of the domain and its ends. However, there is no such difference between the middle and the end sections for a spherical model. The flow cell consists of a perfusion channel and microwell in which the spheroid is cultured. Beneath the flow cell, a piezo transducer is located.

**Figure 2.**Mesh study graph for laminar flow and mass transport equations. As the grid size reduced in a step-by-step manner, both maximum shear stress and average oxygen concentration tend to be constant and independent of the number of the grids.

**Figure 3.**The geometry of the model simulated in [43], regenerated here for the sake of validation. The orange arrows represent the actuated velocity. The number adjacent to each boundary specifies its boundary condition which is described in Equation (24) with the same boundary numbers. The line graphs of Figure 4 are plotted on the white dashed line shown in this figure.

**Figure 4.**Comparison between the results obtained in this study for the geometry and boundary conditions of the problem in [43] and present results. (

**a**) Comparison between the vertical velocities, (

**b**) comparison between the vertical velocities in the near-wall region, and (

**c**) comparison between the horizontal velocities. All the sub-figures (

**a**–

**c**) are plotted on the white dashed line shown in Figure 3. Sub-figure (

**d**) depicts the pressure contour for the problem, which is identical to its counterpart in [43] (Figure 4a in that study).

**Figure 5.**(

**a**) Velocity contour and streamlines without acoustics. Velocity distribution is approximately similar to the laminar flow pattern inside a rectangular domain. (

**b**,

**c**) Oxygen and glucose concentration distribution and their fluxes. Here, the flow rate was set to be 1 μL/min.

**Figure 6.**(

**a**) Streamlines in the presence of the acoustic field. Near the microwell, the flow is mostly affected and forms a pattern similar to a vortex. This also leads to convection enhancement inside the well and around the spheroid. (

**b**,

**c**) Oxygen and glucose distribution and their fluxes, respectively, when the acoustic field is present. Acoustic field improved the proliferation zone significantly inside the spheroid.

**Figure 7.**(

**a**) Effect of ${d}_{0}$ on glucose and oxygen concentrations inside the spheroid. With an increase in the magnitude of ${d}_{0}$, more oxygen and glucose will be found inside the cell aggregate, and the better cells in the inner side and the core will be nourished. (

**b**) While increasing ${d}_{0}$ is beneficial due to enhancement of oxygen and glucose concentrations, it is not also causing any side effects such as high magnitudes of fluid shear stresses or lift forces. Here, ${\tau}_{max}$ represents the maximum value of shear stress which is exerted to the peripheral boundary of the spheroid. (

**c**) Graphical illustration of oxygen’s proliferation zone after applying the acoustic field. Before applying acoustic to the microchip, the proliferation zone had only a 32.7% share of the spheroid. With the acoustic field, this share is increased to 66.8%.

**Figure 8.**(

**a**) Effect of flow rate on oxygen/glucose concentration with/without acoustics. As shown in the figure, the acoustic field enhanced concentrations greatly without the need for any increases in the flow rate. (

**b**) A study of the effect of flow rate and acoustic integration to the system on maximum fluid shear stress and the lift force. The acoustic wave is amplifying these two parameters, but they are still in the safe range for this application. (

**c**) A comparison in necrotic/quiescent shrinkage with a flow rate between acoustic and non-acoustic modes. Increasing the flow rate helps the growth of the proliferation zone. The acoustic field solely can perform better without the need for any increases in the flow rate. Approximately, acoustics improves the proliferating zone by 100% at each flow rate.

**Figure 9.**Distribution of glucose and oxygen with and without acoustics. This figure aims to show how oxygen (left column) and glucose (middle column) solely contribute to the formation of necrotic and quiescent zones. Without acoustics, the shortage of oxygen leads to a 32.7% proliferation zone (bottom-left), while this is only 11.2% for glucose (bottom-center). The huge loss of the proliferation zone is mainly due to the large quiescent zone (66.8%) caused by the lack of glucose. The combinatorial effect is 11.2% of proliferating zone, 43.8% quiescent zone, and 45% necrotic zone (bottom-right). With acoustics, the necrotic zone from glucose shortage is omitted completely and is decreased to 13.3% from the lack of oxygen. The quiescent zones of oxygen and glucose are also reduced in size, and consequently, a noticeable improvement in the combinatorial share of the proliferating zone is observed (54.9%, top-right).

Parameter | Value |
---|---|

Spheroid diameter | $300\mathsf{\mu}\mathrm{m}$ |

Well height | $380\mathsf{\mu}\mathrm{m}$ |

Well width | $450\mathsf{\mu}\mathrm{m}$ |

Channel height | $1000\mathsf{\mu}\mathrm{m}$ |

Channel length | $2000\mathsf{\mu}\mathrm{m}$ |

**Table 2.**Detailed values and descriptions of all the parameters used in the governing equations used in this study.

Parameters | Descriptions | Values | References |
---|---|---|---|

$Q$ | Inflow | $1\u201312\mathsf{\mu}\mathrm{L}/\mathrm{min}$ | [26] |

${c}_{0}{}_{{O}_{2}}$ | Inlet concentration of oxygen | $0.2\mathrm{mM}$ | [22] |

${c}_{0}{}_{Gl}$ | Inlet concentration of glucose | $5\mathrm{mM}$ | [50] |

${D}_{{O}_{2}-{H}_{2}O}$ | Diffusion coefficient of oxygen through H_{2}O | $2.6\times {10}^{-9}{\mathrm{m}}^{2}/\mathrm{s}$ | [22] |

${D}_{{O}_{2}-Sph}$ | Diffusion coefficient of oxygen through the cell aggregate | $1.83\times {10}^{-9}{\mathrm{m}}^{2}/\mathrm{s}$ | [22] |

${D}_{Gl-{H}_{2}O}$ | Diffusion coefficient of Glucose through H_{2}O | $9.27\times {10}^{-10}{\mathrm{m}}^{2}/\mathrm{s}$ | [22] |

${D}_{Gl-Sph}$ | Diffusion coefficient of glucose through cell aggregate | $2.7\times {10}^{-10}{\mathrm{m}}^{2}/\mathrm{s}$ | [22] |

${S}_{{O}_{2}-Sphvs.{H}_{2}O}$ | Solubility coefficient of oxygen in the cell aggregate vs. H_{2}O | $4.81$ | [22] |

${S}_{Gl-Sphvs.{H}_{2}O}$ | Solubility coefficient of glucose in the cell aggregate vs. H_{2}O | $1$ | [22] |

${V}_{max}{}_{{O}_{2}}$ | Maximum reaction rate of oxygen | $0.0203\mathrm{mM}/\mathrm{s}$ | [22] |

${V}_{max}{}_{Gl}$ | Maximum reaction rate of glucose | $0.01076\mathrm{mM}/\mathrm{s}$ | [22] |

${K}_{m}{}_{{O}_{2}}$ | Michaelis-Menten constant of oxygen | $0.00463\mathrm{mM}$ | [22] |

${K}_{m}{}_{Gl}$ | Michaelis-Menten constant of glucose | $0.04\mathrm{mM}$ | [22] |

${f}_{0}$ | Actuation frequency | $1\mathrm{MHz}$ | - |

$\rho $ | Fluid density | $993.3\mathrm{kg}/{\mathrm{m}}^{3}$ | [22] |

$\mu $ | Fluid dynamic viscosity | $6.92\times {10}^{-4}\mathrm{Pa}\xb7\mathrm{s}$ | [22] |

${\mu}_{B}$ | Fluid bulk viscosity | $0.0024\mathrm{Pa}\xb7\mathrm{s}$ | [43] |

$Cp$ | Fluid specific heat at constant pressure | $4.18\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\mathrm{K}}$ | [43] |

${\alpha}_{0}$ | Fluid thermal expansion | $2.75\times {10}^{-4}1/\mathrm{K}$ | [43] |

${\beta}_{0}$ | Fluid isentropic compressibility | $4.48\times {10}^{-10}1/\mathrm{Pa}$ | [43] |

${d}_{0}$ | Wall displacement amplitude (Equation (21)) | $0.1-0.5\mathrm{nm}$ | [43,51] |

$C$ | Sound velocity in the fluid | $1502\mathrm{m}/\mathrm{s}$ | [43] |

**Table 3.**Parameters of the model used in Ref. [43], which have been re-simulated here.

Parameter | Value |
---|---|

$W$ | $380\mathsf{\mu}\mathrm{m}$ |

$H$ | $160\mathsf{\mu}\mathrm{m}$ |

$f$ | $1.97\mathrm{MHz}$ |

${T}_{0}$ | $25\xb0\mathrm{C}$ |

${d}_{0}$ | $0.1\mathrm{nm}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yahyazadeh Shourabi, A.; Salajeghe, R.; Barisam, M.; Kashaninejad, N.
A Proof-of-Concept Study Using Numerical Simulations of an Acoustic Spheroid-on-a-Chip Platform for Improving 3D Cell Culture. *Sensors* **2021**, *21*, 5529.
https://doi.org/10.3390/s21165529

**AMA Style**

Yahyazadeh Shourabi A, Salajeghe R, Barisam M, Kashaninejad N.
A Proof-of-Concept Study Using Numerical Simulations of an Acoustic Spheroid-on-a-Chip Platform for Improving 3D Cell Culture. *Sensors*. 2021; 21(16):5529.
https://doi.org/10.3390/s21165529

**Chicago/Turabian Style**

Yahyazadeh Shourabi, Arash, Roozbeh Salajeghe, Maryam Barisam, and Navid Kashaninejad.
2021. "A Proof-of-Concept Study Using Numerical Simulations of an Acoustic Spheroid-on-a-Chip Platform for Improving 3D Cell Culture" *Sensors* 21, no. 16: 5529.
https://doi.org/10.3390/s21165529