Modeling of Tumor-Immune System Interaction with Stochastic Hybrid Systems with Memory: A Piecewise Linear Approach

Abstract

In this work, we benefit from hybrid systems that are advantageous because of their analytical and computational usefulness in the case of inferential modeling. In fact, many biological and physiological systems exhibit historical responses such that the system and its responses depend on the whole history rather than a combination of historical events. In this work, we use and improve hybrid systems with memory (HSM) in the subclass of piecewise linear differential equations. We also include stochastic calculus to our model to exhibit uncertainties and random perturbations clearly, and we call this model stochastic hybrid systems with memory (SHSM). Finally, we choose tumor-immune system data from the literature and show that the model is capable to model history dependent behavior.

Keywords

• hybrid systems
• functional differential equations
• pattern memorization
• multistationarity
• regulatory dynamical systems

Kaynakça

1. [1] J.A. Adam, N. Bellomo, A survey of Models for Tumor-Immune System Dynamics, Birkhäuser, Boston, MA, 1996.
2. [2] L.J.S. Allen, An introduction to stochastic processes with applications to biology. Second edition. CRC Press, Boca Raton, FL, 2011.
3. [3] U. Bastolla, G. Parisi, Attractors in fully asymmetric neural networks, J. Phys. A: Math. Gen., 30, 5613--5631, 1997.
4. [4] G.A. Bocharov, F.A. Rihan, Numerical modeling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125, 183-199, 2000.
5. [5] N. Bellomo, Modeling the hiding-learning dynamics in large living systems, Appl. Math. Lett., 23, 907-911, 2010.
6. [6] C.G. Cassandras, J. Lygeros, Stochastic Hybrid Systems, CRC Press, FL, 2006.
7. [7] C. Cattani, A. Ciancio, Hybrid two scales mathematical tools for active particles modeling complex systems with learning hiding dynamics, Mathematical Models and Methods in Applied Sciences Volume 17, Issue 2, Pages 171-187, February 2007.
8. [8] L. Chen, Stability of Genetic Regulatory Networks With Time Delay, IEEE Transactions on Circuits and Systems I: Fundemental Theory and Applications, Vol. 49, No. 5, May 2002.

9. [9] D. Dee, M. Ghil, Boolean difference equations, i: Formulation and Dynamic behavior, SIAM J. Appl. Math., 44:111-126, 1984.
10. [10] M. Delbruck. Discussion, In Unites biologiques douees de continuite genetique, Editions du Centre National de la Recherche Scientifique, Paris. pp. 33-35, 1949.
11. [11] C.A. Dinarello, Biologic basis for interleukin-1 in disease. Blood, 87, 2095-2147, 1996.
12. [12] A. D'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy. Mathematical and Computer Modelling Volume 47, Issues 5-6, Pages 614-637, March 2008.
13. [13] T. Dvorkin, X. Song, S. Argov, R. M. White, M. Zoller, S. Segal, C. A. Dinarello, E. Voronov, R. N. Apte, Immune phenomena involved in the in vivo regression of brosarcoma cells expressing cell-associated IL-1alpha, J Leukoc Biol.; 80(1):96-106, 2006.
14. [14] R. Edwards, L. Glass, Combinatorial explosion in model gene networks, Journal of Chaos, volume 10, number 3, pp. 691-704, September 2000.
15. [15] R. Edwards, H.T. Siegelmann, K. Aziza, L. Glass, Symbolic dynamics and computation in model gene networks, Journal of Chaos, volume 11, number 1, pp. 160-169, March 2001.
16. [16] J. Gebert, H. Öktem, S.W. Pickl, N. Radde, G.-W. Weber, F.B. Yilmaz, Inference of Gene Expression Patterns by Using a Hybrid System Formulation An Algorithmic Approach to Local State Transition Matrices, Anticipative & Predictive Models in Systems Science 1, pp. 63-66, 2004.
17. [17] J. Gebert, N. Radde, G. W. Weber. Modeling gene regulatory networks with piecewise linear differential equations, Euro- pean Journal of Operational Research 181, 1148-1165, 2007.
18. [18] D. Godbole, J. Lygeros, and S. Sastry, Hierarchical hybrid control: A case study, Lecture Notes in Computer Science, vol. 999, pp. 166-190, 1995.
19. [19] R. Goebel, R. G. Sanfelice, A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press, 2012.
20. [20] N. Gökgöz. Development of Tools For Modeling Hybrid Systems With Memory, Msc. Thesis, Scientific Computing, Institute of Applied Mathematics, Middle East Technical University, 2008.
21. [21] N. Gökgöz, Modeling Stochastic Hybrid Systems With Memory With an Application to Immune Response of Cancer Dynamics, PhD Thesis, Scientific Computing, Institute of Applied Mathematics, Middle East Technical University, 2014.
22. [22] N. Gökgöz , H. Öktem, G. Weber , Modeling of Tumor-Immune Nonlinear Stochastic Dynamics with Hybrid Systems with Memory Approach, Results in Nonlinear Analysis, vol. 3, no. 1, pp. 24-34, 2020.
23. [23] B. Hancioglu, D. Swigon, G.A. Clermont, A dynamical model of human immune response to influenza A virus infection, J Theor Biol., 246(1):70-86, 2007.
24. [24] T.A. Henzinger. The theory of hybrid automata, Proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS), pages 278-292, 1996.
25. [25] M. Kahraman, H. Öktem, G. W. Weber, M. Akhmet, Using Piecewise Linear Systems with Delay to Grab the Functional Dynamics in Biological Systems, HIBIT08 - International Symposium on Health Informatics and Bioinformatics, 2008.
26. [26] S. A. Kaufman, The Origins of Order: Self Organization and Selection in Evolution. Oxford University Press, New York, 1993. [27] A. Khan, J.F. Gomez-Aguilar, T. Abdeljawad, H. Khan. Stability and numerical simulation of a fractional order plant-nectarpollinator model. Alexandria Engineering Journal, 59(1):49-59, 2020.
27. [28] H. Khan., A. Khan, W. Chen, K. Shah. Stability analysis and a numerical scheme for fractional Klein-Gordon equations. Mathematical Methods in the Applied Sciences, 30;42(2):723-32, 2019.
28. [29] X. D. Koutsoukos, D. Riley. Computational Methods for Verification of Stochastic Hybrid Systems. IEEE Transactions on Systems, Man and Cybernetics, Part A, Volume 38, Issue 2, Page(s):385 - 396, 2008.
29. [30] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcations analysis, Bull Math Biol., 56(2):295-321, 1994.
30. [31] J. Liu, A. R. Teel, Hybrid systems with memory: modeling and stability analysis via generalized solutions, IFAC World Congress, 2014.
31. [32] J. Lygeros, M. Prandini. Stochastic Hybrid Systems: A Powerful Framework for Complex, Large Scale Applications. European Journal of Control, 6: 583-594, 2010.
32. [33] N. Lynch, R. Segala, F. Vaandrager, H. B. Weinberg. Hybrid I/O automata,Lecture notes in computer science,pp. 496-510, 2000.
33. [34] A. Mantovani, M. Muzio, P. Ghezzi, C. Colotta, M. Introna, Regulation of inhibitory pathways of the interleukin-1 system, Ann. N. Y. Acad. Sci. 840, 338-351, 1998.
34. [35] H. Öktem, R. Pearson, K. Egiazarian, An Adjustable Aperiodic Model Class of Genomic Interactions Using Continuous Time Boolean Networks (Boolean Delay Equations), Chaos 13, 1167-1175, 2003.
35. [36] H. Öktem. Dynamic information handling in continuous time Boolean Network model of gene interactions, Nonlinear Analysis: Hybrid Systems, Volume 2, Issue 3, Pages 900-912, 2008.
36. [37] H. Öktem, A survey on piecewise-linear models of regulatory dynamical systems, Nonlinear Analysis, 63, 336-349,2005.
37. [38] M. Sainz-Trapága, C. Masoller, H. A. Braun, M. T. Huber, Influence of time-delayed feedback in the ring pattern of thermally sensitive neurons, Physical Review E, 70, 031904, 2004.
38. [39] A. J. van der Schaft, J. M. Schumacher, An Introduction to Hybrid Dynamical Systems, Lecture Notes in Control and Information Sciences, vol. 251 Springer, London, 2000.
39. [40] A. M. Selçuk, Hakan Öktem. An improved method for inference of piecewise linear systems by detecting jumps using derivative estimation. in: Nonlinear Analysis: Hybrid Systems, 3:3(277-287), 2009.
40. [41] P. Smolen, D. A. Baxter, and J. H. Byrne. Mathematical modeling of gene networks review, Neuron, vol. 26, no. 3, pp. 567--580, 2000.
41. [42] R. Thomas, Laws for the dynamics of regulatory networks, Int. J. Dev. Biol. 42, 479-485 (1998).