# Mathematical Modeling of Non-Small-Cell Lung Cancer Biology through the Experimental Data on Cell Composition and Growth of Patient-Derived Organoids

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Patients

#### 2.2. Generation of PDTOs

^{−1}of collagenase I (Life Technologies, Waltham, MA, USA). Afterward, the minced tissue was allowed to incubate at 37 °C for 1.5 h with gentle and slow agitation.

^{−1}of basic fibroblast growth factor (bFGF, 10014-HNAE, Sino Biological, Beijing, China), 50 ng mL

^{−1}of human epidermal growth factor (EGF, ab55566, Abcam, Cambridge, UK), N2 supplement (PanEco, Moscow, Russia), NeuroMax (PanEco, Moscow, Russia), 10 mM Glutamax (Gibco), 1 mM N-acetyl cysteine (Merck, Burlington, MA, USA), 10 mM nicotinamide (Sigma), 10 μM Y27631 (ab120129, Abcam), 15 μM HEPES (Merck), and 1% PenStrep. The total cell count was determined using a hemocytometer, and a portion of the collected cells was preserved by freezing for subsequent flow cytometry analysis.

_{2}. The growth medium was replaced every other day throughout the entire experiment. Over a period of 14 days, the organoids were subject to daily examination, during which their size and morphology were observed using bright-field microscopy with an AxioVert.A1 microscope (Zeiss, Oberkochen, Germany).

#### 2.3. Flow Cytometry

^{®}647 monoclonal anti-human antibodies directed to mannose receptor/CD206 (ab195192, Abcam), mouse monoclonal anti-human antibodies directed to CD8α (ab33786, Abcam), rabbit recombinant anti-human antibodies directed to PD-L1 (ab205921, Abcam), and rabbit recombinant anti-human antibody directed to α-smooth muscle actin/αSMA (ab124964, Abcam). After the incubation, the cells were washed twice in ice-cold Versene and resuspended in 1× PBS. Then, the samples were incubated with secondary antibodies for 30 min in the dark and on ice. The following secondary antibodies were used: goat anti-rabbit IgG H&L conjugated with Alexa Fluor

^{®}647, preadsorbed (ab150083, Abcam), and goat anti-mouse IgG H&L conjugated with Alexa Fluor

^{®}647 (ab150115, Abcam). After the incubation, the cells were washed with Versene and subjected to flow cytometry.

#### 2.4. Mathematical Approaches Used in This Study

## 3. Results

#### 3.1. General Work Flow

#### 3.2. Mathematical Model

_{c}—number of cytotoxic T cells. The values of γ, K, q

_{1}, k, q

_{3}, δ

_{M}

_{2}, and δ

_{Tc}were previously assessed by others (see Table 2). The remaining parameters, namely q

_{2}, q

_{4}, q

_{5}, q

_{6}, δ

_{CAF}, q

_{7}, q

_{8}, and q

_{9}, were assessed by us using the quantity of cells measured in PDTOs on days 7, 14, and 21 via flow cytometry for PDTOs developed from various tumors. Performing the calculations, we assumed that all assessed parameters had to be above zero. The results of calculations are represented in Table 3.

#### 3.3. Solving the System of Differential Equations

^{T}× A)

^{−1}× A

^{TB}

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The logic model of NSCLC organoid tumor microenvironment. Cancer cells (CAN) promote the activation of stromal cells into cancer-associated fibroblasts (CAF) and the polarization of macrophages toward the M2 phenotype (M2). These cells stimulate each other and suppress the anti-tumor activity of cytotoxic lymphocytes—CTL (red arrows). Contrarily, their anti-tumor activity is stimulated by specific antigens on the surface of some tumor cells (green arrows). Solid lines represent cell stimulation, dashed lines—cell suppression, dash-dotted lines dots—cell destruction.

**Figure 2.**Model-predicted dynamics of cell quantity in organoid from 7th to 14th day of incubation. (

**a**) Patient 8; (

**b**) Patient 9; (

**c**) Patient 13; (

**d**) Patient 14. PD-L1—cancer cells, CAF—cancer-associated fibroblasts, M2—M2 polarized macrophages, CD8—cytotoxic T-cells.

Type of Cells | Cell-Specific Biomarker |
---|---|

Cancer cells | PD-L1 |

Cancer-associated fibroblasts | αSMA |

M2-polarized macrophages | CD206 |

Cytotoxic lymphocytes | CD8 |

Parameter | Definition | Published Value | References | Adjusted Value |
---|---|---|---|---|

γ | Growth rate of cancer cells | 0.05–0.44 day^{−1} | [17,18] | 0.05 day^{−1} |

K | Final number of cancer cells | 10^{9}–3.3 × 10^{9} day^{−1} | [19] | 10^{6} day^{−1} |

q_{1} | Stimulation of cancer cells by M2-polarized macrophages | 0.4 day^{−1} | [19] | 4 × 10^{−5} day^{−1} |

q_{3} | Stimulation of M2 macrophages by cancer cells | 4 × 10^{−8} day^{−1} | [19] | 4 × 10^{−8} day^{−1} |

δ_{M}_{2} | Death rate of M2-polarized macrophages from natural causes | 0.2 day^{−1} | [18] | 0.2 day^{−1} |

k | Number of cancer cells eliminated by cytotoxic cells | 3.4 × 10^{−10}–1 × 10^{−3} cell^{−1} day^{−1} | [18] | 0.001 cell^{−1} day^{−1} |

δ_{Tc} | Death rate of cytotoxic cells | 2 × 10^{−3}–1 day^{−1} | [18] | 0.1 day^{−1} |

Parameter | Description | Calculated Values, Day^{−1} |
---|---|---|

q_{2} | Stimulation of cancer cells by cancer-associated fibroblasts | 0.0001–0.005 |

q_{4} | Stimulation of M2-polarized macrophages by cancer-associated fibroblasts | 0.0001–0.001 |

q_{5} | Stimulation of cancer-associated fibroblasts by cancer cells | 0–0.00001 |

q_{6} | Stimulation of cancer-associated fibroblasts by M2-polarized macrophages | 0.00001–0.001 |

q_{7} | Stimulation of cytotoxic T cells by cancer cells | 0.0009–0.0015 |

q_{8} | Suppression of cytotoxic T cells by M2-polarized macrophages | 0–0.00001 |

q_{9} | Suppression of cytotoxic T cells by tumor-associated macrophages | 0–0.00001 |

δ_{CAF} | Death rate of cancer-associated fibroblasts | 0.1 |

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**MDPI and ACS Style**

Sulimanov, R.; Koshelev, K.; Makarov, V.; Mezentsev, A.; Durymanov, M.; Ismail, L.; Zahid, K.; Rumyantsev, Y.; Laskov, I.
Mathematical Modeling of Non-Small-Cell Lung Cancer Biology through the Experimental Data on Cell Composition and Growth of Patient-Derived Organoids. *Life* **2023**, *13*, 2228.
https://doi.org/10.3390/life13112228

**AMA Style**

Sulimanov R, Koshelev K, Makarov V, Mezentsev A, Durymanov M, Ismail L, Zahid K, Rumyantsev Y, Laskov I.
Mathematical Modeling of Non-Small-Cell Lung Cancer Biology through the Experimental Data on Cell Composition and Growth of Patient-Derived Organoids. *Life*. 2023; 13(11):2228.
https://doi.org/10.3390/life13112228

**Chicago/Turabian Style**

Sulimanov, Rushan, Konstantin Koshelev, Vladimir Makarov, Alexandre Mezentsev, Mikhail Durymanov, Lilian Ismail, Komal Zahid, Yegor Rumyantsev, and Ilya Laskov.
2023. "Mathematical Modeling of Non-Small-Cell Lung Cancer Biology through the Experimental Data on Cell Composition and Growth of Patient-Derived Organoids" *Life* 13, no. 11: 2228.
https://doi.org/10.3390/life13112228