# Modeling Dendritic Cell Pulsed Immunotherapy for Mice with Melanoma—Protocols for Success and Recurrence

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Growth of Tumor Cells

#### 2.2. Growth of Dendritic Cells

#### 2.3. Dynamic of Activated CTLs

#### 2.4. Dynamic of $TGF-\beta $ Cytokine

## 3. Computer Simulation and Results

#### Sensitivity Analysis

## 4. From Exhaustive Search to Non-Linear Analysis

#### 4.1. Non-Dimensionalization

#### 4.2. Recurrence and Periodic Orbits Without Pulsed Immunotherapy

**Proposition**

**1.**

**Proof.**

## 5. A Model with Periodically Pulsed Immunotherapy

#### Recurrence with Pulsed Immunotherapy

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 6. Discussion and Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Supporting Information

Variable of the Model | Value | Time Interval | Reference |
---|---|---|---|

$T(t)$ | $6\times {10}^{4}$ | $[-\tau ,0]$ | Piñon-Zárate et al. [8] |

$D(t)$ | 0 | $[-\tau ,0]$ | |

${C}_{a}(t)$ | 0 | $[-\tau ,0]$ | |

${F}_{\beta}$ | 0 | $[-\tau ,0]$ |

Parameter | Name | Value | Observations |
---|---|---|---|

${r}_{T}$ | Maximum rate of tumor growth | $0.001$${h}^{-1}$ | Kronik et al. [18] and Castillo-Montiel E. et al. [17]. |

K | carrying capacity of tumor | $6.754\times {10}^{15}$$cell$${10}^{11}$$cell$ | Observing the biological results, the mice is sacrified when the tumor reached a diameter of 4.3 cm Castillo-Montiel E. et al. [17]; Kronik et al. [18] |

${a}_{T}$ | Maximum efficiency of cytotoxic cells | $0.12$${h}^{-1}$ | [19,30] and |

${h}_{T}$ | Saturation efficiency of CTLs to a large tumor size | $5\times {10}^{8}cell$ | [19,31] and |

${a}_{T\beta}$ | Maximum effect of reduction of the cytokine $TGF-\beta $ in the efficiency of CTLs | $0.69$ none | and [18,32] |

${e}_{T\beta}$ | Dependence of CTLs the efficiency in the efficiency of $TGF-\beta $ | ${10}^{4}$$pg$ | and [18,33] |

${r}_{e}$ | Rate of expansion of CTLs by CD | $3.5416$${h}^{-1}$ | [34] |

${\theta}_{D}$ | Limit the density of the CD by the maximum rate of proliferation of CTLs | 212 $cells$ | [34] |

${r}_{a}$ | Activation rate of CTLs by CD | $0.00004$${(cell*h)}^{-1}$ | [34] |

${\mu}_{{C}_{a}}$ | Death rate of CLTs activated at the tumor site | $0.01925$${h}^{-1}$ | [20] |

${r}_{T\beta}$ | Production rate TGF $TGF-\beta $ by a single tumor cell | $5.57\times {10}^{-6}$$pg{(cell\times h)}^{-1}$ | [18] |

${\mu}_{\beta}$ | Degradation rate of $TGF-\beta $ | $7{h}^{-1}$ | [18] |

${C}_{naive}$ | Number of naive CTL cells contributing to primary clonal expansion | 370 $cells$ | [20] (ah doc) |

${\mu}_{D}$ | Rate death of D cells | $0.009625$${h}^{-1}$ | [34] (ah doc) |

## References

- Bol, K.F.; Schreibelt, G.; Gerritsen, W.R.; de Vries, I.J.M.; Figdor, C.G. Dendritic cell—Based immunotherapy: State of the art and beyond. Clin. Cancer Res.
**2016**, 22, 1897–1906. [Google Scholar] [CrossRef][Green Version] - Neves, H.; Kwok, H.F. Recent advances in the field of anti-cancer immunotherapy. BBA Clin.
**2015**, 3, 280–288. [Google Scholar] [CrossRef][Green Version] - Van der Sluis, R.M.; Egedal, J.H.; Jakobsen, M.R. Plasmacytoid Dendritic Cells as Cell-Based Therapeutics: A Novel Immunotherapy to Treat Human Immunodeficiency Virus Infection? Front. Cell. Infect. Microbiol.
**2020**, 10, 249. [Google Scholar] [CrossRef] [PubMed] - Raggi, F.; Bosco, M.C. Targeting Mononuclear Phagocyte Receptors in Cancer Immunotherapy: New Perspectives of the Triggering Receptor Expressed on Myeloid Cells (TREM-1). Cancers
**2020**, 12, 1337. [Google Scholar] [CrossRef] [PubMed] - Rollins, M.R.; Spartz, E.J.; Stromnes, I.M. T Cell Receptor Engineered Lymphocytes for Cancer Therapy. Curr. Protoc. Immunol.
**2020**, 129, e97. [Google Scholar] [CrossRef] [PubMed] - Bonam, S.R.; Kaveri, S.V.; Sakuntabhai, A.; Gilardin, L.; Bayry, J. Adjunct immunotherapies for the management of severely ill COVID-19 patients. Cell Rep. Med.
**2020**, 2020, 100016. [Google Scholar] [CrossRef] [PubMed] - Sciutto, E.; Rosas, G.; Hernández, M.; Morales, J.; Cruz-Revilla, C.; Toledo, A.; Manoutcharian, K.; Gevorkian, G.; Blancas, A.; Acero, G.; et al. Improvement of the synthetic tri-peptide vaccine (S3Pvac) against porcine Taenia solium cysticercosis in search of a more effective, inexpensive and manageable vaccine. Vaccine
**2007**, 25, 1368–1378. [Google Scholar] [CrossRef] [PubMed] - Piñon-Zárate, G.; Herrera-Enríquez, M.; Hernández-Téllez, B.; Jarquín-Yáñez, K.; Castell-Rodríguez, A.E. GK-1 Improves the Immune Response Induced by Bone Marrow Dendritic Cells Loaded with MAGE-AX in Mice with Melanoma. J. Immunol. Res.
**2014**, 2014, 1–12. [Google Scholar] [CrossRef] - Palucka, K.; Banchereau, J. Cancer immunotherapy via dendritic cells. Nat. Rev. Cancer
**2012**, 12, 265–277. [Google Scholar] [CrossRef] - Emens, L.A. Cancer vaccines: On the threshold of success. Expert Opin. Emerg. Drugs
**2008**, 13, 295–308. [Google Scholar] [CrossRef] - Schuler, G. Dendritic cells in cancer immunotherapy. Eur. J. Immunol.
**2010**, 40, 2123–2130. [Google Scholar] [CrossRef] [PubMed] - Overwijk, W.W.; Restifo, N.P. B16 as a Mouse Model for Human Melanoma. Curr. Protoc. Immunol.
**2001**, 39, 1–33. [Google Scholar] [CrossRef] [PubMed] - Butterfield, L.H. Dendritic cells in cancer immunotherapy clinical trials: Are we making progress? Front. Immunol.
**2013**, 4, 454. [Google Scholar] [CrossRef][Green Version] - Ya, Z.; Hailemichael, Y.; Overwijk, W.; Restifo, N.P. Mouse model for Pre-Clinical study of human cancer immunotherapy. Curr. Protoc. Immunol.
**2015**, 108, 1–20. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pejawar-Gaddy, S.; Finn, O.J. Cancer vaccines: Accomplishments and challenges. Crit. Rev. Oncol. Hematol.
**2008**, 67, 93–102. [Google Scholar] [CrossRef] - Yoshimura, A.; Muto, G. TGF-β function in immune suppression. In Negative Co-Receptors and Ligands; Springer: Berlin/Heidelberg, Germany, 2010; pp. 127–147. [Google Scholar]
- Castillo-Montiel, E.; Chimal-Eguia, J.C.; Tello, J.I.; Piñon-Zaráte, G.; Herrera-Enríquez, M.; Castell-Rodríguez, A. Enhancing dendritic cell immunotherapy for melanoma using a simple mathematical model. Theor. Biol. Med. Model.
**2015**, 12, 1. [Google Scholar] [CrossRef][Green Version] - Kronik, N.; Kogan, Y.; Vainstein, V.; Agur, Z. Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics. Cancer Immunol. Immunother.
**2008**, 57, 425–439. [Google Scholar] [CrossRef] [PubMed] - Kronik, N.; Kogan, Y.; Schlegel, P.G.; Wölfl, M. Improving T-cell immunotherapy for melanoma through a mathematically motivated strategy: Efficacy in numbers? J. Immunother.
**2012**, 35, 116–124. [Google Scholar] [CrossRef][Green Version] - DePillis, L.; Gallegos, A.; Radunskaya, A. A model of dendritic cell therapy for melanoma. Front. Oncol.
**2013**, 3, 1–13. [Google Scholar] [CrossRef] [PubMed][Green Version] - Eisenhauer, E.A.; Therasse, P.; Bogaerts, J.; Schwartz, L.H.; Sargent, D.; Ford, R.; Dancey, J.; Arbuck, S.; Gwyther, S.; Mooney, M.; et al. New response evaluation criteria in solid tumours: Revised RECIST guideline (version 1.1). Eur. J. Cancer
**2009**, 45, 228–247. [Google Scholar] [CrossRef] - Schättler, H.; Ledzewicz, U. Optimal Control for Mathematical Models of Cancer Therapies; Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering; Westview Press: Nashville, TN, USA, 2014. [Google Scholar]
- Panetta, J.C. A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment. Bull. Math. Biol.
**1996**, 58, 425–447. [Google Scholar] [CrossRef] [PubMed] - Cojocaru, L.; Agur, Z. A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs. Math. Biosci.
**1992**, 109, 85–97. [Google Scholar] [CrossRef] - Wei, H.C.; Lin, J.T. Periodically pulsed immunotherapy in a mathematical model of tumor-immune interaction. Int. J. Bifurcat. Chaos
**2013**, 23, 1350068. [Google Scholar] [CrossRef] - Kirschner, D.; Panetta, J.C. Modeling immunotherapy of the tumor—Immune interaction. J. Math. Biol.
**1998**, 37, 235–252. [Google Scholar] [CrossRef][Green Version] - Wei, H.C. A modified numerical method for bifurcations of fixed points of ODE systems with periodically pulsed inputs. Appl. Math. Comput.
**2014**, 236, 373–383. [Google Scholar] [CrossRef] - Haass, N.K.; Beaumont, K.A.; Hill, D.S.; Anfosso, A.; Mrass, P.; Munoz, M.A.; Kinjyo, I.; Weninger, W. Real-time cell cycle imaging during melanoma growth, invasion, and drug response. Pigment Cell Melanoma Res.
**2014**, 27, 764–776. [Google Scholar] [CrossRef] [PubMed] - Arciero, J.; Jackson, T.; Kirschner, D. A mathematical model of tumor-immune evasion and siRNA treatment. Discret. Contin. Dynam. Syst. Ser. B
**2004**, 4, 39–58. [Google Scholar] - Kruse, C.A.; Cepeda, L.; Owens, B.; Johnson, S.D.; Stears, J.; Lillehei, K.O. Treatment of recurrent glioma with intracavitary alloreactive cytotoxic T lymphocytes and interleukin-2. Cancer Immunol. Immunother.
**1997**, 45, 77–87. [Google Scholar] [CrossRef] - Thomas, D.A.; Massagué, J. TGF-β directly targets cytotoxic T cell functions during tumor evasion of immune surveillance. Cancer Cell
**2005**, 8, 369–380. [Google Scholar] [CrossRef][Green Version] - Peterson, P.K.; Chao, C.C.; Hu, S.; Thielen, K.; Shaskan, E.G. Glioblastoma, transforming growth factor-β, and Candida meningitis: A potential link. Am. J. Med.
**1992**, 92, 262–264. [Google Scholar] [CrossRef] - Ludewig, B.; Krebs, P.; Junt, T.; Metters, H.; Ford, N.J.; Anderson, R.M.; Bocharov, G. Determining control parameters for dendritic cell-cytotoxic T lymphocyte interaction. Eur. J. Immunol.
**2004**, 34, 2407–2418. [Google Scholar] [CrossRef] [PubMed]

**Figure 3.**Hypothetical therapies results. They represent the tumor growth with the immunotherapy treatment and blue curves represent the growth of tumor cells without treatment. In Figure 3 all simulations implement a therapy using ${10}^{9}$ DCs infused every 72 h by 3 weeks using 100 different initial conditions. Subfigures (

**A**,

**C**) shows the result of one simulation without treatment; likewise, subfigures (

**B**,

**D**) shows also the result of one simulation but with immunotherapy treatment. Additionally, subfigures (

**E**,

**F**) show the results but for 100 different simulations showing the same behavior as before.

**Figure 4.**Hypothetical therapies results. The red curves represent the growth of tumor cells with immunotherapy and blue curves represent the growth of tumor cells without immunotherapy. Figure 4 all simulation result implements a therapy with infusion size of ${10}^{7}$ DCs injected every 168 h by 3 weeks. The subfigures (

**A**,

**B**) using different carrying capacity with $K={10}^{6}$ and $K={10}^{11}$ respectively. For cases (

**C**,

**D**) is varied the rate tumor growth parameter in $0.001$ and $0.1$. The subfigures (

**E**,

**F**) the dendritic cells infused is varied in ${10}^{3}$ and ${10}^{9}$ every 168 h by 3 weeks.

Protocol | Type of Cancer | Infusion | Size | Type of Mice |
---|---|---|---|---|

Subcutaneous melanoma | B16 melanoma | ${10}^{5}$ cells/mouse | 1 cm${}^{3}$ from 14 to 21 days | $C57BL/6$ |

**Table 2.**Hypothetical immunotherapy for ODEs model with different number of Dendritic cells (DCs) injected and different periods of infusion.

HM | NDC | PI | %G | RT | ST | SuT | RECIST |
---|---|---|---|---|---|---|---|

1 | ${10}^{3}$ | 48 | 0.99 | 100 | 0 | 0 | PD |

2 | ${10}^{3}$ | 96 | 0.99 | 100 | 0 | 0 | PD |

3 | ${10}^{3}$ | 144 | 0.99 | 100 | 0 | 0 | PD |

4 | ${10}^{3}$ | 192 | 0.99 | 100 | 0 | 0 | PD |

5 | ${10}^{7}$ | 48 | 0.49 | 50 | 0 | 50 | PR |

6 | ${10}^{7}$ | 96 | 0.53 | 54 | 0 | 46 | PR |

7 | ${10}^{7}$ | 144 | 0.55 | 56 | 0 | 44 | PR |

8 | ${10}^{7}$ | 192 | 0.50 | 51 | 0 | 49 | PR |

9 | ${10}^{9}$ | 48 | 0.011 | 0 | 0 | 100 | CR |

10 | ${10}^{9}$ | 96 | 0.007 | 0 | 0 | 100 | CR |

11 | ${10}^{9}$ | 144 | 0.010 | 0 | 0 | 100 | CR |

12 | ${10}^{9}$ | 192 | 0.009 | 0 | 0 | 100 | CR |

Param | PR1 | S1 | PR2 | S2 |
---|---|---|---|---|

r | $2.10\times {10}^{-15}$ | 2 | $9.99\times {10}^{10}$ | $2.44\times {10}^{-4}$ |

K | $4.99\times {10}^{10}$ | $1.00$ | $1.49\times {10}^{11}$ | $1.00$ |

${\mu}_{D}$ | $9.95\times {10}^{10}$ | $0.008$ | $9.99\times {10}^{10}$ | $6.07\times {10}^{-4}$ |

${D}_{inf}$ | $9.99\times {10}^{10}$ | $-0.0034$ | $9.99\times {10}^{10}$ | $0.0034$ |

${\mu}_{Ca}$ | $9.98\times {10}^{10}$ | $0.003$ | $9.99\times {10}^{10}$ | $3.41\times {10}^{-4}$ |

${a}_{T,\beta}$ | $9.99\times {10}^{10}$ | $-3.34\times {10}^{-4}$ | $9.99\times {10}^{10}$ | $-3.34\times {10}^{-4}$ |

**Table 4.**Specific numerical results for the stable zone using Equation (25).

$\mathit{\tau}$ | d | Stability |
---|---|---|

20 | 50,000 | Unstable |

20 | 200,000 | Stable |

20 | 150,000 | Stable |

50 | 100,000 | Unstable |

50 | 600,000 | Stable |

50 | 100,000,000 | Stable |

100 | 50,000 | Unstable |

100 | 600,000 | Stable |

100 | 1,000,000 | Stable |

144 | 1000 | Unstable |

144 | 100,000,000 | Stable |

150 | 100,000 | Unstable |

150 | 200,000 | Stable |

150 | 100,000,000 | Stable |

168 | 10,000,000 | Stable |

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**

Chimal-Eguia, J.C.; Castillo-Montiel, E.; Rangel-Reyes, J.C.; Paez-Hernández, R.T.
Modeling Dendritic Cell Pulsed Immunotherapy for Mice with Melanoma—Protocols for Success and Recurrence. *Appl. Sci.* **2021**, *11*, 3199.
https://doi.org/10.3390/app11073199

**AMA Style**

Chimal-Eguia JC, Castillo-Montiel E, Rangel-Reyes JC, Paez-Hernández RT.
Modeling Dendritic Cell Pulsed Immunotherapy for Mice with Melanoma—Protocols for Success and Recurrence. *Applied Sciences*. 2021; 11(7):3199.
https://doi.org/10.3390/app11073199

**Chicago/Turabian Style**

Chimal-Eguia, Juan Carlos, Erandi Castillo-Montiel, Julio Cesar Rangel-Reyes, and Ricardo Teodoro Paez-Hernández.
2021. "Modeling Dendritic Cell Pulsed Immunotherapy for Mice with Melanoma—Protocols for Success and Recurrence" *Applied Sciences* 11, no. 7: 3199.
https://doi.org/10.3390/app11073199