# An Overview of Mathematical Modelling in Cancer Research: Fractional Calculus as Modelling Tool

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## Abstract

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## 1. Introduction

## 2. Mathematical Modelling in Oncology

#### 2.1. White-, Grey- and Black-Box Modelling

#### 2.2. The Spatial Scale

#### 2.3. Models of Ordinary and Partial Differential Equations

#### 2.4. Other Modelling Techniques

#### 2.5. General Modelling Assumptions

#### 2.6. Modelling Treatment Intervention

#### 2.7. General Modelling Limitations

#### 2.8. Mathematical Oncology or Oncological Mathematics?

## 3. Fractional Calculus in Mathematical Modelling

#### 3.1. Early Development

#### 3.2. The Memory Effect

#### 3.3. General Applications

#### 3.4. Modelling Biological Phenomena

#### 3.5. Toward a Fractional Mathematical Oncology

## 4. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Mapping of possible interdisciplinary approaches to Mathematical Oncology, with respective keywords for each convergence (adapted from [18]).

**Table 1.**Ranking of the ten leading types of cancer for both men and women, pertaining to estimated new cases and deaths, as projected for 2022 in the United States (adapted from [11]).

Estimated New Cases | |||||

Males | Females | ||||

Prostate | 268,490 | 27% | Breast | 287,850 | 31% |

Lung and bronchus | 117,910 | 12% | Lung and bronchus | 118,830 | 13% |

Colon and rectum | 80,690 | 8% | Colon and rectum | 70,340 | 8% |

Urinary bladder | 61,700 | 6% | Uterine corpus | 65,950 | 7% |

Melanoma of the skin | 57,180 | 6% | Melanoma of the skin | 42,600 | 5% |

Kidney and renal pelvis | 50,290 | 5% | Non-Hodgkin lymphoma | 36,350 | 4% |

Non-Hodgkin lymphoma | 44,120 | 4% | Thyroid | 31,940 | 3% |

Oral cavity and pharynx | 38,700 | 4% | Pancreas | 29,240 | 3% |

Leukaemia | 35,810 | 4% | Kidney and renal pelvis | 28,710 | 3% |

Pancreas | 32,970 | 3% | Leukaemia | 24,840 | 3% |

All Sites | 983,160 | 100% | All Sites | 934,870 | 100% |

Estimated Deaths | |||||

Males | Females | ||||

Lung and bronchus | 68,820 | 21% | Lung and bronchus | 61,360 | 21% |

Prostate | 34,500 | 11% | Breast | 43,250 | 15% |

Colon and rectum | 28,400 | 9% | Colon and rectum | 24,180 | 8% |

Pancreas | 25,970 | 8% | Pancreas | 23,860 | 8% |

Liver and intrahepatic bile duct | 20,420 | 6% | Ovary | 12,810 | 4% |

Leukaemia | 14,020 | 4% | Uterine corpus | 12,550 | 4% |

Esophagus | 13,250 | 4% | Liver and intrahepatic bile duct | 10,100 | 4% |

Urinary bladder | 12,120 | 4% | Leukaemia | 9980 | 3% |

Non-Hodgkin lymphoma | 11,700 | 4% | Non-Hodgkin lymphoma | 8550 | 3% |

Brain and other nervous system | 10,710 | 3% | Brain and other nervous system | 7570 | 3% |

All Sites | 322,090 | 100% | All Sites | 287,270 | 100% |

Year | Title | Type of Cancer | Treatment | Fractional Operator | Ref. |
---|---|---|---|---|---|

2014 | A fractional diffusion equation model for cancer tumour | Not specified | Not specified | Caputo | [125] |

2014 | Numerical simulation of fractional bioheat equation in hyperthermia treatment | Not specified | Hyperthermia | Caputo | [126] |

2016 | Dynamics of tumour-immune system with fractional-order | Not specified | Immunotherapy | Caputo | [127] |

2016 | Modelling doxorubicin effect in various cancer therapies by means of fractional calculus | Not specified | Chemotherapy | Grünwald–Letnikov | [128] |

2017 | Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy | Not specified | Hyperthermia | Caputo | [129] |

2018 | New observations on optimal cancer treatments for a fractional tumour growth model with and without singular kernel | Obesity-associated cancer | Chemotherapy, Immunotherapy and Combined | Caputo and Caputo–Fabrizio | [71] |

2018 | Variable order fractional derivatives and bone remodelling in the presence of metastases | Not specified | Chemotherapy | Grünwald–Letnikov | [130] |

2019 | A new fractional model for the cancer treatment by radiotherapy using the Hadamard fractional derivative | Not specified | Radiotherapy | Hadamard | [131] |

2019 | Numerical solutions for time-fractional cancer invasion system with nonlocal diffusion | Not specified | — | Caputo | [120] |

2019 | Stability analysis and numerical simulations via fractional calculus for tumour dormancy models | Not specified | — | Caputo | [132] |

2019 | A study on discrete and discrete fractional pharmacokinetics pharmacodynamics models of tumour growth and anti-cancer effects | Colon carcinoma | Not specified | Rieman–Liouville | [133] |

2020 | On multistep tumour growth models of fractional variable-order | Breast cancer | — | Caputo | [121] |

2020 | Can fractional calculus help improve tumour growth models? | Breast cancer | — | Caputo | [56] |

2020 | Mathematical modelling of cancer and hepatitis co-dynamics with non-local and non-singular kernel | Co-infection of cancer & hepatitis | — | ABC | [134] |

2020 | Mathematical modelling of radiotherapy cancer treatment using Caputo fractional derivative | Uterine cervical cancer | Radiotherapy | Caputo | [135] |

2021 | Memory effects on the proliferative function in the cycle-specific of chemotherapy | Breast & ovarian cancer | Chemotherapy | Caputo | [136] |

2021 | A mathematical model and numerical solution for brain tumour derived using fractional operator | Glioblastoma | Chemotherapy and Surgery | Caputo | [137] |

2021 | A fractional modelling of tumour-immune system interaction related to lung cancer with real data | Lung cancer | — | Caputo | [138] |

2021 | Comparison of fractional-order and integer-order cancer tumour growth models: an inverse approach | Prostate cancer | Chemotherapy | Caputo | [139] |

2022 | Modelling and analysis of breast cancer with adverse reactions of chemotherapy treatment through fractional derivative | Breast cancer | Chemotherapy | Caputo–Fabrizio | [124] |

2022 | The dynamics of a fractional-order mathematical model of cancer tumour disease | Not specified | Chemotherapy | Caputo | [123] |

2022 | Modelling the dynamics of tumour-immune cells interactions via fractional calculus | Not specified | Immunotherapy | Caputo | [140] |

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## Share and Cite

**MDPI and ACS Style**

Vieira, L.C.; Costa, R.S.; Valério, D.
An Overview of Mathematical Modelling in Cancer Research: Fractional Calculus as Modelling Tool. *Fractal Fract.* **2023**, *7*, 595.
https://doi.org/10.3390/fractalfract7080595

**AMA Style**

Vieira LC, Costa RS, Valério D.
An Overview of Mathematical Modelling in Cancer Research: Fractional Calculus as Modelling Tool. *Fractal and Fractional*. 2023; 7(8):595.
https://doi.org/10.3390/fractalfract7080595

**Chicago/Turabian Style**

Vieira, Lourenço Côrte, Rafael S. Costa, and Duarte Valério.
2023. "An Overview of Mathematical Modelling in Cancer Research: Fractional Calculus as Modelling Tool" *Fractal and Fractional* 7, no. 8: 595.
https://doi.org/10.3390/fractalfract7080595