# Three-Dimensional Modeling of Avascular Tumor Growth in Both Static and Dynamic Culture Platforms

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Tumor Growth Model

#### 2.1.1. Nutrient Concentrations

#### 2.1.2. Growth Equation

#### 2.1.3. Initial and Boundary Conditions

#### 2.1.4. Applying the Modified Model to Tumor Spheroid in Microfluidic Devices

#### 2.2. Geometry

#### 2.3. Numerical Method

#### 2.4. Model Validation

^{6}cells/mL. The authors only provided the exact value of the proliferation rate constant of HT-29. Here, we applied the model for the proliferating phase of the HT-29 tumor spheroids in the dynamic microwells, Figure 3b. Although the general trend is achieved, there are some disagreements between the experimental and simulation results. This is directly due to the lack of knowledge about some necessary parameters of HT-29 cell line, such as the necrosis and apoptosis rates, the diffusion coefficient of the nutrients to the HT-29 tumor spheroids, as well as Michaelis-Menten parameters of the nutrients. The only known parameter was the proliferation rate constant, and the rest of the parameters were adopted from EMT6/Ro cell line. The better agreement could be achieved in future studies by replacing the exact parameters of HT-29.

## 3. Results and Discussion

#### 3.1. Tumor Growth in Static, Unlimited Culture Medium

#### 3.1.1. Effect of Nutrient Concentration on Final Tumor Volume

#### 3.1.2. Effect of Initial Tumor Radius on Tumor Growth

#### 3.2. Tumor Growth in a Microchannel Containing U-shaped Barrier

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^{2}is appropriate for cancer cell culture under continuous flow. Therefore, the shear stress throughout this study is in a suitable range for cancer cell culture.

#### 3.3. Comparison between Tumor Growth in Microchannels with U-Shaped Barrier and Microwell Trap

## 4. Limitations and Suggestions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Three main regions in a tumor. Proliferative zone, nutrients concentrations are above their critical value; Quiescent zone, the concentration of one of the nutrients fall below its critical value; Necrotic zone, nutrients concentrations are below their critical value.

**Figure 2.**Tumor spheroid (the objects in red) in a microchannel containing: (

**a**) A U-shaped barrier; (

**b**) A microwell trap. The dimensions are in micrometer.

**Figure 3.**(

**a**) Comparison of the tumor volumes from experiment data of the Sutherland and Freyer for EMT6/Ro spheroid [14] and simulation based on the proposed model. In both cases, the tumor is exposed to the unlimited static culture medium, and the glucose and oxygen concentrations in the culture medium are 0.8 mM and 0.28 mM, respectively. (

**b**) Applying the model for HT-29 tumor growth in proliferative phase in a microfluidic device with a microwell trap proposed by Ziółkowska et al. [31]. To better validate the results, on-chip measurements of specific parameters of the HT-29 cell line was necessary. These parameters included the necrosis and apoptosis rates, the diffusion coefficient of the nutrients to the HT-29 tumor spheroids, as well as Michaelis-Menten parameters of the nutrients.

**Figure 4.**(

**a**) Effect of nutrient concentrations on the tumor volume after 550 h. Tumors with 24.3 μm initial radius are exposed to unlimited static culture mediums varying in nutrient concentrations. (

**b**) Effect of the initial tumor radius on the growth. All of the tumors are exposed to the unlimited static culture medium with 0.8 mM glucose and 0.28 mM oxygen concentration.

**Figure 5.**(

**a**) Glucose; (

**b**) Oxygen concentration during the growth of a tumor (concentrations of glucose and oxygen in the culture medium are constant versus time and they are respectively 0.8 mM and 0.28 mM). The initial radius of the tumor is 24.3 μm. (

**c**) Volume distributions of the different regions of the tumor.

**Figure 6.**(

**a**) Tumor volume; (

**b**) Quiescent zone volume; (

**c**) Necrotic zone volume for three inlet flows (0.5, 5, and 50 µL/min) versus time (glucose and oxygen concentration at the inlet are respectively 0.8 mM and 0.28 mM). The initial radius of the tumors in both of the traps is 24.3 μm.

**Figure 7.**(

**a**) Tumor volume; (

**b**) Quiescent zone volume; (

**c**) Necrotic zone volume for U-shaped barrier and microwell trap versus time (glucose and oxygen concentration at the inlet of each case are respectively 0.8 mM and 0.28 mM). The initial radius of the tumors in both of the traps is 24.3 μm.

**Figure 8.**(

**a**) Glucose concentration in a tumor in the U-shaped barrier; (

**b**) Glucose concentration in a tumor in the microwell trap; (

**c**) Oxygen concentration in a tumor in the U-shaped barrier; (

**d**) Oxygen concentration in a tumor in the microwell trap versus time (concentrations of the glucose and oxygen in the inflow of culture medium are respectively 0.8 mM and 0.28 mM). All the contours are depicted in the XZ plane (see Figure 2).

Parameters | Values | Units | Descriptions |
---|---|---|---|

$\rho $ | 993.3 | kg/${\mathrm{m}}^{3}$ | Density of water at 37 °C, [28] |

Μ | 6.92$\text{}\times {10}^{-4}$ | Pa·s | Viscosity of water at 37 °C, [28] |

${D}_{glucose-water}$ | 9.27$\text{}\times {10}^{-10}$ | ${\mathrm{m}}^{2}/\mathrm{s}$ | Diffusion coefficient of glucose through water [28] |

${D}_{{O}_{2}-water}$ | 2.6$\times {10}^{-9}$ | ${\mathrm{m}}^{2}/\mathrm{s}$ | Diffusion coefficient of oxygen through water [28] |

${D}_{1}$ | 4.22$\text{}\times {10}^{-11}$ | ${\mathrm{m}}^{2}/\mathrm{s}$ | Diffusion coefficient of glucose through EMT6/Ro cancerous tissue [26] |

${D}_{2}$ | 1.65$\times {10}^{-9}$ | ${\mathrm{m}}^{2}/\mathrm{s}$ | Diffusion coefficient of oxygen through EMT6/Ro cancerous tissue [26] |

${C}_{glucose-critical}$ | 0.06 | mM | Critical concentration of glucose for EMT6/Ro spheroid [26] |

${C}_{{O}_{2}-critical}$ | 0.02 | mM | Critical concentration of oxygen for EMT6/Ro spheroid [26] |

${V}_{1}$ | 4.36$\times {10}^{-2}$ | mol/${\mathrm{m}}^{3}/\mathrm{s}$ | Maximum consumption rate of glucose for EMT6/Ro spheroid [29] |

${V}_{2}$ | 2.74$\times {10}^{-2}$ | mol/${\mathrm{m}}^{3}/\mathrm{s}$ | Maximum consumption rate of oxygen for EMT6/Ro spheroid [29] |

${K}_{1}$ | 4$\times {10}^{-2}$ | mol/${\mathrm{m}}^{3}$ | Michaelis constant of glucose for EMT6/Ro spheroid [28] |

${K}_{2}$ | 4.64$\times {10}^{-3}$ | mol/${\mathrm{m}}^{3}$ | Michaelis constant of oxygen for EMT6/Ro spheroid, Casciari et al. [12] |

${A}_{p}$ | 1.2153$\times {10}^{-9}$ | 1/s | Apoptosis rate constant for EMT6/Ro spheroid, Mahmood et al. [30] |

${A}_{n}$ | 2$\times {10}^{-6}$ | 1/s | Necrosis rate constant for EMT6/Ro spheroid, Mahmood et al. [30] |

$A$ | $3.3\times {10}^{-2}$ * | 1/h | Proliferation rate constant for 0.8 mM glucose and 0.28 mM oxygen for EMT6/Ro spheroid [14] |

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**MDPI and ACS Style**

Taghibakhshi, A.; Barisam, M.; Saidi, M.S.; Kashaninejad, N.; Nguyen, N.-T.
Three-Dimensional Modeling of Avascular Tumor Growth in Both Static and Dynamic Culture Platforms. *Micromachines* **2019**, *10*, 580.
https://doi.org/10.3390/mi10090580

**AMA Style**

Taghibakhshi A, Barisam M, Saidi MS, Kashaninejad N, Nguyen N-T.
Three-Dimensional Modeling of Avascular Tumor Growth in Both Static and Dynamic Culture Platforms. *Micromachines*. 2019; 10(9):580.
https://doi.org/10.3390/mi10090580

**Chicago/Turabian Style**

Taghibakhshi, Ali, Maryam Barisam, Mohammad Said Saidi, Navid Kashaninejad, and Nam-Trung Nguyen.
2019. "Three-Dimensional Modeling of Avascular Tumor Growth in Both Static and Dynamic Culture Platforms" *Micromachines* 10, no. 9: 580.
https://doi.org/10.3390/mi10090580