Super-twisting sliding mode control approach for tumor growth by immunotherapy


Cancer is the second leading cause of death after heart disease in the world and the third leading cause of death after heart disease and accidents in Iran. In general, cancer is a disorder of the rate of proliferation and cell differentiation that can occur in any tissue of the body and at any age and by invading healthy tissues may exacerbate the disease and eventually cause death. In a word, one of the most commonly used treatments for cancer is the use of chemotherapy. The drugs of the aforementioned chemotherapy are transported through the blood to cancer cells and all parts of the body. In addition to cancer cells, these drugs also have a detrimental effect on healthy cells, which can be seen as side effects. It is to note that they are temporary and can stop at the end of treatment. The subject behind this research is to propose super-twisting sliding mode control approach without chattering for mathematical model of cancer by immunotherapy with the aim of stabilizing the closed-loop system, as long as determining the optimal drug dose is taken into consideration as the innovation of this study to conclude which controller has the better performance in the presence of uncertainties and disturbances.

This is a preview of subscription content, access via your institution.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5


\( \gamma \) :

Fixed and positive parameter in effective cell rotation

\( u\left( t \right) \) :

Effective cell external source rate

\( \tau_{g} \) :

Delay in interleukin concentration

\( V_{g} \) :

Cell distribution rate

\( V_{i} \) :

Interleukin concentration rate

\( K_{xgi} \) :

Interleukin-dependent tumor cell rate

\( T_{gh} \) :

Effective cell and tumor cell communication index

\( T_{igmax} \) :

The maximum rate of interleukin secretion in phase II

\( G_{b} \) :

Target mass index


  1. 1

    Swanson K R, Alvord E C and Murray J D 2000 A quantitative model for differential motility of gliomas in grey and white matter Cell Proliferation 33: 317–329

    Article  Google Scholar 

  2. 2

    Gerlee P and Anderson A R 2007 An evolutionary hybrid cellular automaton model of solid tumour growth Journal of Theoretical Biology 246: 583–603

    MathSciNet  Article  Google Scholar 

  3. 3

    Anderson A R and Chaplain M A J 1998 Continuous and discrete mathematical models of tumor-induced angiogenesis Bulletin of Mathematical Biology 60: 857–899

    Article  Google Scholar 

  4. 4

    De Pillis L G, Gu W and Radunskaya A E 2006 Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations Journal of Theoretical Biology 238: 841–862

  5. 5.

    Bhabani Shankar D, Kumar Bera M and Krishna Roy B 2018 Super twisting sliding mode control of cancer chemotherapy IEEE International Workshop on Variable Structure Systems

  6. 6

    Colli P 2019 Sliding mode control for a phase field system related to tumor growth Applied Mathematics and Optimization 79: 647–670

    MathSciNet  Article  Google Scholar 

  7. 7

    Sharifi M, Jamshidi A and Namazi Sarvestani N 2018 An adaptive robust control strategy in a cancer tumor-immune system under uncertainties IEEE/ACM Transactions on Computational Biology and Bioinformatics 16: 865–873

    Article  Google Scholar 

  8. 8

    Cristian L, Argeseanu A. and Blaabjerg F 2019 Super-twisting sliding mode direct torque and flux control of induction machine drives IEEE Transactions on Power Electronics 35: 5057–5065

  9. 9

    Machado J C 2001 Interleukin 1B and interleukin 1RN polymorphisms are associated with increased risk of gastric carcinoma Gastroenterology 121: 823–829

  10. 10

    Muhammad Umer S 2018 Mathematical model based assessment of the cancer control by chemo-immunotherapy Pure and Applied Biology 7: 678–683

    Google Scholar 

  11. 11

    Motulo Firmansyah R Trisilowati T and Abdul R 2018 Optimal control of tumor growth model with dendritic cells as immunotherapy The Journal of Experimental Life Science 8: 103–108

Download references

Author information



Corresponding author

Correspondence to A H MAZINAN.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

AZARBAKHSH, G., MAZINAN, A.H. Super-twisting sliding mode control approach for tumor growth by immunotherapy. Sādhanā 45, 112 (2020).

Download citation


  • Super-twisting sliding mode control approach
  • cancer treatment
  • tumor modeling
  • immunotherapy