Kolon kanseri ve bağışıklık sistemi ilişkisi: kesirli mertebeden bir modelleme yaklaşımı

Abstract

Bu çalışmada, kolon kanseri için detaylı analize yönelik yeni bir kesirli mertebeden diferansiyel denklem sistemi geliştirilmiştir. Modelde tümör hücreleri, makrofaj hücreleri, dendritik hücreler ve CD4+ T yardımcı hücreleri arasındaki etkileşim, Michaelis-Menten kinetiği kullanılarak oluşturulmuştur. Ayrıca pozitiflik ve sınırlılık gibi matematiksel analizler de yapılmaktadır. Kolon kanserinin hücreler arası seyrini gözlemlemeye yönelik sayısal sonuçlar elde edilmekte ve biyolojik yorumlara da yer verilmektedir.

Keywords

• Kesirli dereceli matematiksel model
• Kolon kanseri
• Kolorektal kanser
• Sayısal simülasyon
• Michaelis-Menten kinetiği

Supporting Institution

This research was supported by Scientific and Technological Research Council of Türkiye (TÜBİTAK) under the undergraduate research project.

The relationship between colon cancer and immune system: a fractional order modelling approach

Abstract

In this paper, a new fractional-order differential equation system is developed for colon cancer to address the detailed analysis. In the model, the interaction between tumor cells, macrophage cells, dendritic cells and CD4+ T helper cells is established using Michaelis-Menten kinetics. In addition, mathematical analyses such as positivity and boundedness are also carried out. Numerical results are obtained to observe the intercellular course of colon cancer and biological interpretations are also included.

Keywords

• Mathematical model of fractional-order
• Colon cancer
• Colorectal cancer
• Numerical simulation
• Michaelis-Menten kinetics

References

1. Grup Florence Nightingale. https://www.florence.com.tr/kolon-kanseri. (12.05.2024)
2. Memorial Sağlık Grubu. https://www.memorial.com.tr/hastaliklar/kolon-kalin-bagirsak-kanseri. (12.05.2024)
3. Johnston, M. D., Edwards, C. M., Bodmer, W. F., Maini, P. K., & Chapman, S. J., Mathematical modeling of cell population dynamics in the colonic crypt and in colorectal cancer. Proceedings of the National Academy of Sciences, 104(10), 4008-4013, (2007).
4. Li, L., Hu, Y., Xu, Y., & Tang, S., Mathematical modeling the order of driver gene mutations in colorectal cancer. PLOS Computational Biology, 19(6), e1011225, (2023).
5. Lo, W. C., Martin Jr, E. W., Hitchcock, C. L., & Friedman, A., Mathematical model of colitis-associated colon cancer. Journal Of Theoretical Biology, 317, 20-29, (2023).
6. Delitala, M., & Lorenzi, T., A mathematical model for progression and heterogeneity in colorectal cancer dynamics. Theoretical Population Biology, 79(4), 130-138, (2011).
7. Amilo, D., Sadri, K., Kaymakamzade, B., & Hincal, E., A mathematical model with fractional-order dynamics for the combined treatment of metastatic colorectal cancer. Communications in Nonlinear Science and Numerical Simulation, 130, 107756, (2024).
8. Paterson, C., Clevers, H., & Bozic, I., Mathematical model of colorectal cancer initiation. Proceedings of the National Academy of Sciences, 117(34), 20681-20688, (2020).

9. Kirshtein, A., Akbarinejad, S., Hao, W., Le, T., Su, S., Aronow, R. A., & Shahriyari, L., Data driven mathematical model of colon cancer progression. Journal of Clinical Medicine, 9(12), 3947, (2020).
10. Anaya, D. A., Dogra, P., Wang, Z., Haider, M., Ehab, J., Jeong, D. K., ... & Cristini, V., A mathematical model to estimate chemotherapy concentration at the tumor-site and predict therapy response in colorectal cancer patients with liver metastases. Cancers, 13(3), 444, (2021).
11. Hesse, J., Martinelli, J., Aboumanify, O., Ballesta, A., & Relógio, A., A mathematical model of the circadian clock and drug pharmacology to optimize irinotecan administration timing in colorectal cancer. Computational and Structural Biotechnology Journal, 19, 5170-5183, (2021).
12. Sameen, S., Barbuti, R., Milazzo, P., Cerone, A., Del Re, M., & Danesi, R., Mathematical modeling of drug resistance due to KRAS mutation in colorectal cancer. Journal of Theoretical Biology, 389, 263-273, (2016).
13. Michaelis, L. and Menten, M. L., Die Kinetik der Invertinwirkung Biochem. Z. 49, 333-369, (1913).
14. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J., Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier, (2006).
15. DePillis, L. G., Savage, H., & Radunskaya, A. E., Mathematical model of colorectal cancer with monoclonal antibody treatments. arXiv preprint arXiv:1312.3023, (2013).
16. De Mattei, V., Flandoli, F., Leocata, M., Polito, M. C., & Ricci, C., A mathematical model for growth of solid tumors and combination therapy with an application to colorectal cancer. arXiv preprint arXiv:1607.08009, (2016).
17. Abernathy, K., Abernathy, Z., Brown, K., Burgess, C., & Hoehne, R., Global dynamics of a colorectal cancer treatment model with cancer stem cells. Heliyon, 3(2), (2017).
18. Raeisi, E., Yavuz, M., Khosravifarsani, M., & Fadaei, Y., Mathematical modeling of interactions between colon cancer and immune system with a deep learning algorithm. The European Physical Journal Plus, 139(4), 345, (2024).
19. Idrees, M., Alnahdi, A. S., & Jeelani, M. B., Mathematical Modeling of Breast Cancer Based on the Caputo–Fabrizio Fractal-Fractional Derivative. Fractal and Fractional, 7(11), 805, (2023).
20. Wei, H. C., Mathematical modeling of tumor growth and treatment: Triple negative breast cancer. Mathematics and Computers in Simulation, 204, 645-659, (2023).
21. Oke, S. I., Matadi, M. B., & Xulu, S. S., Optimal control analysis of a mathematical model for breast cancer. Mathematical and Computational Applications, 23(2), 21, (2018).
22. Solís-Pérez, J. E., Gómez-Aguilar, J. F., & Atangana, A., A fractional mathematical model of breast cancer competition model. Chaos, Solitons & Fractals, 127, 38-54, (2019).
23. Zhang, X., Fang, Y., Zhao, Y., & Zheng, W., Mathematical modeling the pathway of human breast cancer. Mathematical Biosciences, 253, 25-29, (2014).
24. Özköse, F., Yılmaz, S., Yavuz, M., Öztürk, İ., Şenel, M. T., Bağcı, B. Ş., ... & Önal, Ö., A fractional modeling of tumor–immune system interaction related to Lung cancer with real data. The European Physical Journal Plus, 137, 1-28, (2022).
25. Amilo, D., Kaymakamzade, B., & Hincal, E., A fractional-order mathematical model for lung cancer incorporating integrated therapeutic approaches. Scientific Reports, 13(1), 12426, (2023).
26. Lourenco Jr, E., Rodrigues, D. S., Antunes, M. E., Mancera, P. F., & Rodrigues, G., A Simple Mathematical Model of Non-Small Cell Lung Cancer Involving Macrophages and CD8+ T Cells. Journal of Biological Systems, 31(04), 1407-1431, (2023).
27. Kim, Y., Lee, D., Lee, J., Lee, S., & Lawler, S., Role of tumor-associated neutrophils in regulation of tumor growth in lung cancer development: A mathematical model. PloS one, 14(1), e0211041, (2019).
28. Beretta, E., Cavaterra, C., Fornoni, M., Lorenzo, G., & Rocca, E., Mathematical analysis of a model-constrained inverse problem for the reconstruction of early states of prostate cancer growth. arXiv preprint arXiv:2404.12198, (2024).
29. Nasresfahani, F., & Eslahchi, M. R., Numerical convergence and stability analysis for a nonlinear mathematical model of prostate cancer. Numerical Methods for Partial Differential Equations, 39(4), 3064-3088, (2023).
30. Colli, P., Gomez, H., Lorenzo, G., Marinoschi, G., Reali, A., & Rocca, E., Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. Mathematical Models and Methods in Applied Sciences, 30(07), 1253-1295, (2020).
31. Valle, P. A., Coria, L. N., & Carballo, K. D., Chemoimmunotherapy for the treatment of prostate cancer: Insights from mathematical modelling. Applied Mathematical Modelling, 90, 682-702, (2021).
32. Kohandel, M., Sivaloganathan, S., & Oza, A., Mathematical modeling of ovarian cancer treatments: sequencing of surgery and chemotherapy. Journal of Theoretical Biology, 242(1), 62-68, (2006).
33. Cheng, F. H., Aguda, B. D., Tsai, J. C., Kochańczyk, M., Lin, J. M., Chen, G. C., ... & Chan, M. W., A mathematical model of bimodal epigenetic control of miR-193a in ovarian cancer stem cells. PloS One, 9(12), e116050, (2014).
34. Le Sauteur-Robitaille, J., Crosley, P., Hitt, M., Jenner, A. L., & Craig, M., Mathematical modeling predicts pathways to successful implementation of combination TRAIL-producing oncolytic virus and PAC-1 to treat granulosa cell tumors of the ovary. Cancer Biology & Therapy, 24(1), 2283926, (2023).
35. Belay, C., Effects of Diabetes on Ovarian Cancer: Data Analysis and Modeling Study. (2018).
36. Campbell, A., Sivakumaran, T., Davidson, M., Lock, M., & Wong, E., Mathematical modeling of liver metastases tumour growth and control with radiotherapy. Physics in Medicine & Biology, 53(24), 7225, (2008).
37. Weens, W., Mathematical modeling of liver tumor (Doctoral dissertation, Université Pierre et Marie Curie-Paris VI). (2012).
38. Green, J. E. F., Waters, S. L., Shakesheff, K. M., & Byrne, H. M., A mathematical model of liver cell aggregation in vitro. Bulletin of Mathematical Biology, 71, 906-930, (2009).
39. Blauth, S., Hübner, F., Leithäuser, C., Siedow, N., & Vogl, T. J., Mathematical modeling and simulation of laser-induced thermotherapy for the treatment of liver tumors. Modeling, Simulation and Optimization in the Health-and Energy-Sector (pp. 3-23). Cham: Springer International Publishing. (2022).
40. Bolaji, B., Onoja, T., Agbata, C., Omede, B. I., & Odionyenma, U. B., Dynamical analysis of HIV-TB co-infection transmission model in the presence of treatment for TB. Bulletin of Biomathematics, 2(1), 21-56, (2024).
41. Fatima, B., Yavuz, M., ur Rahman, M., Althobaiti, A., & Althobaiti, S., Predictive modeling and control strategies for the transmission of Middle East respiratory syndrome coronavirus. Mathematical and Computational Applications, 28(5), 98, (2023).
42. Rahman, M., Yavuz, M., Arfan, M., & Sami, A., Theoretical and numerical investigation of a modified ABC fractional operator for the spread of polio under the effect of vaccination. AIMS Biophysics, 11(1), (2024).
43. Ouaziz, S. I., & El Khomssi, M. Mathematical approaches to controlling COVID-19: optimal control and financial benefits. Mathematical Modelling and Numerical Simulation with Applications, 4(1), 1-36, (2024).
44. Mustapha, U. T., Maigoro, Y. A., Yusuf, A., & Qureshi, S., Mathematical modeling for the transmission dynamics of cholera with an optimal control strategy. Bulletin of Biomathematics, 2(1), 1-20, (2024).
45. Paul, S., Mahata, A., Mukherjee, S., Das, M., Mali, P. C., Roy, B., ... & Bharati, P., Study of fractional order SIR model with MH type treatment rate and its stability analysis. Bulletin of Biomathematics, 2(1), 85-113, (2024).
46. Podlubny, I., Fractional Differential Equations, Academic Press, USA. (1999).
47. W. Lin, Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications, 332, 709-726, (2007).
48. Hellerstein, M., Hanley, M. B., Cesar, D., Siler, S., Papageorgopoulos, C., Wieder, E., ... & McCune, J. M., Directly measured kinetics of circulating T lymphocytes in normal and HIV-1-infected humans. Nature Medicine, 5(1), 83-89, (1999).
49. Joshi, H., & Yavuz, M. Numerical Analysis of Compound Biochemical Calcium Oscillations Process in Hepatocyte Cells. Advanced Biology, 2300647, (2024).
50. Işık, E., & Daşbaşı, B., A compartmental fractional-order mobbing model and the determination of its parameters. Bulletin of Biomathematics, 1(2), 153-176, (2023).
51. Joshi, H., Yavuz, M., & Stamova, I. Analysis of the disturbance effect in intracellular calcium dynamic on fibroblast cells with an exponential kernel law. Bulletin of Biomathematics, 1(1), 24-39, (2023).
52. Kar, N., & Özalp, N., A fractional mathematical model approach on glioblastoma growth: tumor visibility timing and patient survival. Mathematical Modelling and Numerical Simulation with Applications, 4(1), 66-85, (2024).
53. K. Diethelm, K., Ford, N. J., Freed, A. D. A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29 3-22, (2002).
54. Naik, P. A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S., Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135, 1-42, (2020).
55. Joshi, H., Yavuz, M., Townley, S., & Jha, B. K., Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate. Physica Scripta, 98(4), 045216, (2023).
56. Moustafid, A. Set-valued stabilization of reaction-diffusion model by chemotherapy and or radiotherapy. Fundamental Journal of Mathematics and Applications, 6(3), 147-156, (2023).
57. Yildiz, B., Haspolat, E., Yilmaz, A., & Önes, Z. Stability analysis and numerical simulation of dynamic and fractional SEIRD models for spread of nCOVID-19 in Turkey. Asian-European Journal of Mathematics, 15(12), 2250226, (2022).
58. Çakan, Ü. Stability analysis of a mathematical model SIuIaQR for COVID-19 with the effect of contamination control (filiation) strategy. Fundamental Journal of Mathematics and Applications, 4(2), 110-123, (2021).
59. Nguefack, D. J. T. Mathematical modeling of schistosomiasis transmission using reaction-diffusion equations. Fundamental Journal of Mathematics and Applications, 7(2), 118-136, (2024).
60. Joshi, H., Yavuz, M., & Özdemir, N. Analysis of novel fractional order plastic waste model and its effects on air pollution with treatment mechanism. Journal of Applied Analysis & Computation, 14(6), 3078-3098, 2024).
61. Evirgen, F., Uçar, S., Özdemir, N., & Jajarmi, A. (2024). Enhancing maize foliar disease management through fractional optimal control strategies. Discrete and Continuous Dynamical Systems-S, (2024).
62. Bhatter, S., Kumawat, S., Bhatia, B., & Purohit, S. D. Analysis of COVID-19 epidemic with intervention impacts by a fractional operator. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 14(3), 261-275, (2024).
63. Evirgen, F., Uçar, E., Uçar, S., & Özdemir, N. Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates. Mathematical Modelling and Numerical Simulation with Applications, 3(1), 58-73, (2023).
64. Yavuz, M., Akyüz, K., Bayraktar, N. B., & Ozdemir, F. N. Hepatitis-B disease modelling of fractional order and parameter calibration using real data from the USA. AIMS Biophysics, 11(3), 378-402, (2024).
65. Evirgen, F., Özköse, F., Yavuz, M., & Özdemir, N. Real data-based optimal control strategies for assessing the impact of the Omicron variant on heart attacks. AIMS Bioengineering, 10(3), 218-239, (2023).
66. Salih, R. I., Jawad, S., Dehingia, K., & Das, A. The effect of a psychological scare on the dynamics of the tumor-immune interaction with optimal control strategy. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 14(3), 276-293, (2024).
67. Naik, P. A., Yavuz, M., Qureshi, S., Owolabi, K. M., Soomro, A., & Ganie, A. H. Memory impacts in hepatitis C: A global analysis of a fractional-order model with an effective treatment. Computer Methods and Programs in Biomedicine, 254, 108306, (2024).