# In-Silico Modeling of Tumor Spheroid Formation and Growth

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

#### 2.1. Analytical Solution

**Proposition**

**1.**

#### 2.2. Model Simplification

#### 2.3. Numerical Solution

## 3. Results and Discussion

#### 3.1. Model Analysis

^{2}·s

^{−1}[52]. As can be seen, the shrinkage of the tumor is faster for cells with higher motility (K). The tumor spheroid decays further until the concentration of cells reaches the relaxed concentration where the diffusivity of cells and adhesion forces are in balance (minimum tumor radius). At this point, the proliferation continues to elevate cell concentration and breaks the balance. To reach a new balance, the tumor spheroid increases its radius to reduce the local concentration which finally leads to monotonic growth. Rate of proliferation for highly proliferative cancer cells is normally in the order of $\eta}_{0}={10}^{-2}\phantom{\rule{0.166667em}{0ex}$ h

^{−1}. This value was used as a reference number to non-dimensionalize the proliferation rates in our analysis. The effect of cell motility on formation of spheroids is illustrated in Figure 1a in which parameters are set as $\eta =1.8\times {\eta}_{0}$, $\frac{{C}_{0}}{{C}_{i}}=1.5$ and ${R}_{0}=0.01$ cm. The higher K in the figure corresponds to lower minimum radius and faster shrinkage. This result also shows that tumor spheroids with higher K grow faster since cells can rapidly respond to local proliferation and reach balance by moving the boundary of the tumor. Unlike many types of mammalian cells which have an intrinsic cell program that restricts their proliferation, most cancer cells are highly proliferative [2]. When cells proliferate, the local concentration increases, and the generated pressure moves cells away. The formation of a tumor spheroid is faster if cells have a high proliferation rate. To illustrate this effect, the formation of a tumor spheroid is depicted in Figure 1b for different values of $\eta}^{*}=\frac{\eta}{{\eta}_{0}$, holding other parameters fixed, i.e., ${K}^{*}=1$, $\frac{{C}_{0}}{{C}_{i}}=1.5$ and ${R}_{0}=0.01$ cm. As can be seen, a tumor spheroid with a higher proliferation rate assembles faster.

#### 3.2. Model Validation

^{−1}and ${\eta}_{U87}=0.026\pm 0.003\phantom{\rule{0.166667em}{0ex}}$ h

^{−1}, which are in the range of data reported in [53], i.e., ${\eta}_{U251}=0.038\phantom{\rule{0.166667em}{0ex}}$ h

^{−1}and ${\eta}_{U87}=0.033\phantom{\rule{0.166667em}{0ex}}$ h

^{−1}. Results of U251 spheroid formation over 210 h are shown in Figure 4. During the formation phase, intercellular interactions generate adhesion forces which pull cells together and increase the concentration of cells within the spheroid. The size of the spheroid reduces since the proliferation is not yet dominant. The tumor spheroid shrinks until the concentration reaches the relaxed concentration (${C}_{0}$). At this minimum radius, the driving forces are in balance, i.e., adhesion forces and forces due to high concentration of cells within the spheroid. This balance breaks once the proliferation of cells becomes dominant, elevating the local concentration above the relaxed concentration. To remove the produced force, the boundary of the tumor spheroid moves to increase the volume. This volume increment reduces the concentration of cells and equilibrates the forces inside the tumor spheroid.

## 4. Experimental Methodology

#### 4.1. Proliferation Rate

#### 4.2. Spheroid Culture

^{TM}Trypsin-EDTA (0.5%) and were centrifuged at 300× g for 5 min. After removing the supernatant and suspending the cell pellet in 1 mL of medium, the number of cells was counted using a Trypan blue assay. Afterwards, self-filling micro-well arrays (SFMAs) were used to produce uniform tumor spheroids [54]. The desired concentration of cells was loaded dropwise through guiding channels of SFMAs and were gently seeded into the wells. The microwells were kept in an incubator and imaging started 5 h after seeding to let the cells fully settle in the wells. Cells were supplemented with fresh medium every 24 h to maintain the concentration of nutrients. The formation of spheroids was imaged using optical microscopy (Axio Observer, ZEISS, Oberkochen, Germany) over 210 h. The size of spheroids was measured using ImageJ [55].

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Rate of Change of Spheroid’s Volume

#### Appendix A.2. Proof of Proposition 1

#### Appendix A.3. Solution of R_{1} (t)

^{2}/s, ${R}_{0}=0.01$ cm. We also assume that the initial phase of growth does not take longer than a few days, ${t}_{max}=$240 h, and $\frac{{C}_{0}}{{C}_{i}}=2$.

**Figure A1.**The relative error in the solution of ${R}_{1}\left(t\right)$ in Equation (A7) using the upper bound, ${R}_{1}{\left(t\right)}_{ub}$, and lower bound, ${R}_{1}{\left(t\right)}_{lb}$, of Equation (A9). Parameters are set as $K={10}^{-10}$, ${R}_{0}=0.01$ cm, ${t}_{max}=$240 h and $\frac{{C}_{0}}{{C}_{i}}=2$.

#### Appendix A.4. Solution of Full RD Equation

#### Appendix A.5. Model Simplification

#### Appendix A.6. Numerical Method

^{®}[56] was used to perform iterations to obtain unknown concentrations at each node. Please note that we used explicit value of $R\left(t\right)$ in Equation (17a), i.e., ${R}^{i}\left(t\right)$, such that we could solve for the concentrations first and plug them into Equation (17b) to obtain ${R}^{i+1}\left(t\right)$.

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**Figure 1.**Formation of tumor spheroids with different values of (

**a**) cell motility, ${K}_{1}^{*}=1$, ${K}_{2}^{*}=2$ and ${K}_{3}^{*}=3$ (with ${\eta}^{*}=1.8$, $\frac{{C}_{0}}{{C}_{i}}=1.5$ and ${R}_{0}=0.01$ cm), (

**b**) proliferation rate, ${\eta}_{1}^{*}=1.6$, ${\eta}_{2}^{*}=1.8$ and ${\eta}_{3}^{*}=2$ (with ${K}^{*}=1$, $\frac{{C}_{0}}{{C}_{i}}=1.5$ and ${R}_{0}=0.01$ cm). A tumor spheroid with higher cell motility grows faster since cells can rapidly respond to local proliferation and reach balance by moving the boundary of the tumor. Additionally, a tumor spheroid with a higher proliferation rate assembles faster.

**Figure 2.**Formation of tumor spheroid obtained by analytical solution and numerical prediction, using two sets of parameters; set 1: ${\eta}^{*}=1.6,$$\frac{{C}_{0}}{{C}_{i}}=1.5,\phantom{\rule{0.166667em}{0ex}}{K}^{*}=1,\phantom{\rule{0.166667em}{0ex}}{R}_{0}=0.01$ cm, and set 2: ${\eta}^{*}=2,$$\phantom{\rule{0.166667em}{0ex}}\frac{{C}_{0}}{{C}_{i}}=2,\phantom{\rule{0.166667em}{0ex}}{K}^{*}=1,\phantom{\rule{0.166667em}{0ex}}{R}_{0}=0.02$ cm. In the contraction phase the analytical and numerical solutions reasonably match. The analytical solution loses accuracy once growth becomes dominant. This is a result of the simplification we made in Equation (14).

**Figure 3.**Proliferation rates of U251 and U87 cells cultured in DMEM supplemented with 10% (v/v) Fetal Bovine Serum (FBS) and 1% (v/v) Penicillin/Streptomycin, and incubated at 37 °C in 5% CO${}_{2}$. Rates were calculated by counting cells using Trypan blue assay over 24 h (n = 3).

**Figure 4.**In vitro formation of U251- spheroids over 210 h. Spheroids undergo an initial shrinkage and subsequent growth due to the competition between adhesion forces and proliferation pressure. Scale bars show $500\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m.

**Figure 5.**Formation of a tumor spheroid over time, obtained by spheroid culture of hGB cancer cell lines (n = 3), (

**a**) U251 and (

**b**) U87, and compared with both analytical and numerical predictions. The model is able to predict the formation of tumor spheroids and the minimum diameter, but loses accuracy after approximately 160∼180 h. This divergence from experimental results is started when the tumor spheroids lose their homogeneity due to hypoxia and/or necrosis initiation.

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**MDPI and ACS Style**

Amereh, M.; Edwards, R.; Akbari, M.; Nadler, B.
*In-Silico* Modeling of Tumor Spheroid Formation and Growth. *Micromachines* **2021**, *12*, 749.
https://doi.org/10.3390/mi12070749

**AMA Style**

Amereh M, Edwards R, Akbari M, Nadler B.
*In-Silico* Modeling of Tumor Spheroid Formation and Growth. *Micromachines*. 2021; 12(7):749.
https://doi.org/10.3390/mi12070749

**Chicago/Turabian Style**

Amereh, Meitham, Roderick Edwards, Mohsen Akbari, and Ben Nadler.
2021. "*In-Silico* Modeling of Tumor Spheroid Formation and Growth" *Micromachines* 12, no. 7: 749.
https://doi.org/10.3390/mi12070749