Predicting efficacy of 5-fluorouracil therapy via a mathematical model with fuzzy uncertain parameters

Sajad Shafiekhani, Amir Homayoun Jafari, Leila Jafarzadeh, Vahid Sadeghi, Nematollah Gheibi

DOI: 10.4103/jmss.jmss_92_21


Background: Due to imprecise/missing data used for parameterization of ordinary differential equations (ODEs), model parameters are uncertain. Uncertainty of parameters has hindered the application of ODEs that require accurate parameters. Methods: We extended an available ODE model of tumor-immune system interactions via fuzzy logic to illustrate the fuzzification procedure of an ODE model. The fuzzy ODE (FODE) model assigns a fuzzy number to the parameters, to capture parametric uncertainty. We used the FODE model to predict tumor and immune cell dynamics and to assess the efficacy of 5-fluorouracil (5-FU) chemotherapy. Result: FODE model investigates how parametric uncertainty affects the uncertainty band of cell dynamics in the presence and absence of 5-FU treatment. In silico experiments revealed that the frequent 5-FU injection created a beneficial tumor microenvironment that exerted detrimental effects on tumor cells by enhancing the infiltration of CD8+ T cells, and natural killer cells, and decreasing that of myeloid-derived suppressor cells. The global sensitivity analysis was proved model robustness against random perturbation to parameters. Conclusion: ODE models with fuzzy uncertain kinetic parameters cope with insufficient/imprecise experimental data in the field of mathematical oncology and can predict cell dynamics uncertainty band.


5-fluorouracil, fuzzy, ordinary differential equation, uncertain

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Renardy M, Hult C, Evans S, Linderman JJ, Kirschner DE. Global sensitivity analysis of biological multiscale models. Curr Opin Biomed Eng 2019;11:109-16.

Eling N, Morgan MD, Marioni JC. Challenges in measuring and understanding biological noise. Nat Rev Genet 2019;20:536-48.

Tsimring LS. Noise in biology. Rep Prog Phys 2014;77:26601.

Casanova MP. Noise and synthetic biology: How to deal with stochasticity? Nanoethics 2020;14:113-22.

Moore N, Doty D, Zielstorff M, Kariv I, Moy LY, Gimbel A, et al. A multiplexed microfluidic system for evaluation of dynamics of immune-tumor interactions. Lab Chip 2018;18:1844-58.

Rihan FA, Hashish A, Al-Maskari F, Hussein MS, Ahmed E, Riaz MB, et al. “Dynamics of tumor-immune system with fractional-order.” Journal of Tumor Research 2, no. 1 (2016): 109-115.

Hara A, Iwasa Y. Coupled dynamics of intestinal microbiome and immune system – A mathematical study. J Theory Biol 2019;464:9-20.

Allahverdy A, Moghaddam AK, Rahbar S, Shafiekhani S, Mirzaie HR, Amanpour S, et al. An agent-based model for investigating the effect of myeloid-derived suppressor cells and its depletion on tumor immune surveillance. J Med Signals Sens 2019;9:15-23.

Pennisi M, Pappalardo F, Motta S. Agent Based Modeling of Lung Metastasis-Immune System Competition. In: International Conference on Artificial Immune Systems; 2009. p. 1-3. Back to cited text no. 9

Baldazzi V, Castiglione F, Bernaschi M. An enhanced agent based model of the immune system response. Cell Immunol 2006;244:77-9. Back to cited text no. 10

Gong C, Milberg O, Wang B, Vicini P, Narwal R, Roskos L, et al. A computational multiscale agent-based model for simulating spatio-temporal tumour immune response to PD1 and PDL1 inhibition. J R Soc Interface 2017;14:20170320.

Cosgrove J, Butler J, Alden K, Read M, Kumar V, Cucurull-Sanchez L, et al. Agent-based modeling in systems pharmacology. CPT Pharmacometrics Syst Pharmacol 2015;4:615-29.

da Silva JG, de Morais RM, da Silva IC, de Arruda Mancera PF. Mathematical models applied to thyroid cancer. Biophys Rev 2019;11:183-9.

Mahasa KJ, Ouifki R, Eladdadi A, Pillis L. Mathematical model of tumor-immune surveillance. J Theor Biol 2016;404:312-30.

de Pillis LG, Radunskaya AE, Wiseman CL. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res 2005;65:7950-8.

Pianosi, Francesca, Fanny Sarrazin, and Thorsten Wagener. “A Matlab toolbox for global sensitivity analysis.” Environmental Modelling & Software 70 (2015):80-85.

Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol 2008;254:178-96.

Lebedeva G, Sorokin A, Faratian D, Mullen P, Goltsov A, Langdon SP, et al. Model-based global sensitivity analysis as applied to identification of anti-cancer drug targets and biomarkers of drug resistance in the ErbB2/3 network. Eur J Pharm Sci 2012;46:244-58.

Cândea D, Halanay A, R?dulescu R, T?lmaci R. Parameter estimation and sensitivity analysis for a mathematical model with time delays of leukemia. AIP Conf Proc 2017;1798:20034.

Poleszczuk J, Hahnfeldt P, Enderling H. Therapeutic implications from sensitivity analysis of tumor angiogenesis models. PLoS One 2015;10:e0120007.

Alam M, Deng X, Philipson C, Bassaganya-Riera J, Bisset K, Carbo A, et al. Sensitivity analysis of an ENteric immunity SImulator (ENISI)-based model of immune responses to Helicobacter pylori infection. PLoS One 2015;10:e0136139.

Wu Y, Gan Y, Yuan H, Wang Q, Fan Y, Li G, et al. Enriched environment housing enhances the sensitivity of mouse pancreatic cancer to chemotherapeutic agents. Biochem Biophys Res Commun 2016;473:593-9.

Puri, Madan L, Ralescu DA, Zadeh L. “Fuzzy random variables.” In Readings in fuzzy sets for intelligent systems, Morgan Kaufmann, 1993. pp. 265-271.

Shafiekhani S, Poursheykhani A, Rahbar S, Jafari AH. Simulating ATO mechanism and EGFR signaling with fuzzy logic and petri net. J Biomed Phys Eng 2021;11:325-36.

Liu, Fei, Monika Heiner, and David Gilbert. “Fuzzy Petri nets for modelling of uncertain biological systems.” Briefings in bioinformatics 21, no. 1 (2020): 198-210.

Liu F, Sun W, Heiner M, Gilbert D. Hybrid modelling of biological systems using fuzzy continuous Petri nets. Brief Bioinform 2021;22:438-50.

Park, Inho, Dokyun Na, Doheon Lee, and Kwang H. Lee. “Fuzzy continuous Petri Net-based approach for modeling immune systems.” In Neural Nets, Springer, Berlin, Heidelberg, 2005. pp. 278-285.

Liu F, Heiner M, Yang M. Fuzzy stochastic petri nets for modeling biological systems with uncertain kinetic parameters. PLoS One 2016;11:e0149674.

Liu F, Chen S, Heiner M, Song H. Modeling biological systems with uncertain kinetic data using fuzzy continuous Petri nets. BMC Syst Biol 2018;12:42.

Shariatpanahi SP, Shariatpanahi SP, Madjidzadeh K, Hassan M, Abedi-Valugerdi M. Mathematical modeling of tumor-induced immunosuppression by myeloid-derived suppressor cells: Implications for therapeutic targeting strategies. J Theor Biol 2018;442:1-10.

Serre R, Benzekry S, Padovani L, Meille C, André N, Ciccolini J, et al. Mathematical modeling of cancer immunotherapy and its synergy with radiotherapy. Cancer Res 2016;76:4931-40.

Lai X, Friedman A. Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model. PLoS One 2017;12:e0178479.

Milberg O, Gong C, Jafarnejad M, Bartelink IH, Wang B, Vicini P, et al. A QSP model for predicting clinical responses to monotherapy, combination and sequential therapy following CTLA-4, PD-1, and PD-L1 checkpoint blockade. Sci Rep 2019;9:11286.

Worldwide Cancer Statistics | Cancer Research UK. Available from: [Last accessed on 2020 Apr 03].

Abbas, Abul K., Andrew H. Lichtman, and Shiv Pillai. Cellular and molecular immunology E-book. Elsevier Health Sciences, 2014.

Gajewski TF, Schreiber H, Fu YX. Innate and adaptive immune cells in the tumor microenvironment. Nat Immunol 2013;14:1014-22.

Guillerey C, Huntington ND, Smyth MJ. Targeting natural killer cells in cancer immunotherapy. Nat Immunol 2016;17:1025-36.

Parker KH, Beury DW, Ostrand-Rosenberg S. Myeloid-derived suppressor cells: Critical cells driving immune suppression in the tumor microenvironment. Adv Cancer Res 2015;128:95-139.

Liu C, Workman CJ, Vignali DA. Targeting regulatory T cells in tumors. FEBS J 2016;283:2731-48.

Groth C, Hu X, Weber R, Fleming V, Altevogt P, Utikal J, et al. Immunosuppression mediated by myeloid-derived suppressor cells (MDSCs) during tumour progression. Br J Cancer 2019;120:16-25.

Munder M, Schneider H, Luckner C, Giese T, Langhans CD, Fuentes JM, et al. Suppression of T-cell functions by human granulocyte arginase. Blood 2006;108:1627-34.

Abedi-Valugerdi M, Wolfsberger J, Pillai PR, Zheng W, Sadeghi B, Zhao Y, et al. Suppressive effects of low-dose 5-fluorouracil, busulfan or treosulfan on the expansion of circulatory neutrophils and myeloid derived immunosuppressor cells in tumor-bearing mice. Int Immunopharmacol 2016;40:41-9.

Umansky V, Blattner C, Gebhardt C, Utikal J. The role of myeloid-derived suppressor cells (MDSC) in cancer progression. Vaccines (Basel) 2016;4:E36.

Srivastava MK, Zhu L, Harris-White M, Kar UK, Huang M, Johnson MF, et al. Myeloid suppressor cell depletion augments antitumor activity in lung cancer. PLoS One 2012;7:e40677.

Si Y, Merz SF, Jansen P, Wang B, Bruderek K, Altenhoff P, Mattheis S, et al. Multidimensional imaging provides evidence for down-regulation of T cell effector function by MDSC in human cancer tissue. Sci Immunol 2019;4:eaaw9159.

Wilson S, Levy D. A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy. Bull Math Biol 2012;74:1485-500.

Twyman-Saint Victor C, Rech AJ, Maity A, Rengan R, Pauken KE, Stelekati E, et al. Radiation and dual checkpoint blockade activate non-redundant immune mechanisms in cancer. Nature 2015;520:373-7.

Werthmöller N, Frey B, Rückert M, Lotter M, Fietkau R, Gaipl US. Combination of ionising radiation with hyperthermia increases the immunogenic potential of B16-F10 melanoma cells in vitro and in vivo. Int J Hyperthermia 2016;32:23-30.

Orecchioni S, Talarico G, Labanca V, Calleri A, Mancuso P, Bertolini F. Vinorelbine, cyclophosphamide, and 5-FU effects on the circulating and intratumoral landscape of immune cells improve anti-PD-L1 efficacy in preclinical models of breast cancer and lymphoma. Br J Cancer 2018;118:1329-36.

Chen XL, Ciren SZ, Zhang H, Duan LG, Wesley AJ. Effect of 5-FU on modulation of disarrangement of immune-associated cytokines in experimental acute pancreatitis. World J Gastroenterol 2009;15:2032-7.

Tüysüz F, Kahraman C. Modeling a flexible manufacturing cell using stochastic Petri nets with fuzzy parameters. Expert Syst Appl 2010;37:3910-20.

Chen KC, Wang TY, Tseng HH, Huang CY, Kao CY. A stochastic differential equation model for quantifying transcriptional regulatory network in Saccharomyces cerevisiae. Bioinformatics 2005;21:2883-90.

Manninen T, Linne ML, Ruohonen K. Developing Itô stochastic differential equation models for neuronal signal transduction pathways. Comput Biol Chem 2006;30:280-91.

Bogle G, Dunbar PR. Agent-based simulation of T-cell activation and proliferation within a lymph node. Immunol Cell Biol 2010;88:172-9.

Castro C, Flores DL, Cervantes-Vásquez D, Vargas-Viveros E, Gutiérrez-López E, Muñoz-Muñoz F. An agent-based model of the fission yeast cell cycle. Curr Genet 2019;65:193-200.

Zhang SQ, Ching WK, Ng MK, Akutsu T. Simulation study in Probabilistic Boolean Network models for genetic regulatory networks. Int J Data Min Bioinform 2007;1:217-40.

Trairatphisan P, Wiesinger M, Bahlawane C, Haan S, Sauter T. A Probabilistic Boolean Network approach for the analysis of cancer-specific signalling: A case study of deregulated PDGF signalling in GIST. PLoS One 2016;11:e0156223.

Shafiekhani S, Kraikivski P, Gheibi N, Ahmadian M, Jafari AH. Dynamical analysis of the fission yeast cell cycle via Markov chain. Curr Genet 2021;67:785-97.

Shafiekhani S, Shafiekhani M, Rahbar S, Jafari AH. Extended robust Boolean network of budding yeast cell cycle. J Med Signals Sens 2020;10:95-105.

Tajmirriahi M, Amini Z, Hamidi A, Zam A, Rabbani H. Modeling of retinal optical coherence tomography based on stochastic differential equations: Application to Denoising. IEEE Trans Med Imaging 2021;40:2129-41.

Arnold, Ludwig. Stochastic differential equations. New York 1974.

Shafiekhani, Sajad, Rahbar S, Akbarian F, Jafari AH. “Fuzzy stochastic petri net with uncertain kinetic parameters for modeling tumor-immune system.” In 2018 25th National and 3rd International Iranian Conference on Biomedical Engineering (ICBME), pp. 1-5. IEEE, 2018.


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