# American Institute of Mathematical Sciences

2015, 12(5): 1127-1139. doi: 10.3934/mbe.2015.12.1127

## Combining robust state estimation with nonlinear model predictive control to regulate the acute inflammatory response to pathogen

 1 Department of Mathematics, University of California, Irvine, 340 Rowland Hall, Bldg #400, Irvine, CA 92697-3875, United States 2 Electrical Engineering and Computer Science Department, Masdar Institute of Science and Technology, Masdar City, Abu Dhabi, United Arab Emirates 3 Department of Mathematics, University of Tennessee, 1403 Circle Dr, Ayres Hall 227, Knoxville, TN, 37996-2250, United States

Received  September 2014 Revised  March 2015 Published  June 2015

The inflammatory response aims to restore homeostasis by means of removing a biological stress, such as an invading bacterial pathogen. In cases of acute systemic inflammation, the possibility of collateral tissue damage arises, which leads to a necessary down-regulation of the response. A reduced ordinary differential equations (ODE) model of acute inflammation was presented and investigated in [10]. That system contains multiple positive and negative feedback loops and is a highly coupled and nonlinear ODE. The implementation of nonlinear model predictive control (NMPC) as a methodology for determining proper therapeutic intervention for in silico patients displaying complex inflammatory states was initially explored in [5]. Since direct measurements of the bacterial population and the magnitude of tissue damage/dysfunction are not readily available or biologically feasible, the need for robust state estimation was evident. In this present work, we present results on the nonlinear reachability of the underlying model, and then focus our attention on improving the predictability of the underlying model by coupling the NMPC with a particle filter. The results, though comparable to the initial exploratory study, show that robust state estimation of this highly nonlinear model can provide an alternative to prior updating strategies used when only partial access to the unmeasurable states of the system are available.
Citation: Gregory Zitelli, Seddik M. Djouadi, Judy D. Day. Combining robust state estimation with nonlinear model predictive control to regulate the acute inflammatory response to pathogen. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1127-1139. doi: 10.3934/mbe.2015.12.1127
##### References:
 [1] D. C. Angus and T. van der Poll, Severe sepsis and septic shock, New Eng J Med, 369 (2013), 840-851. [2] O. Bara, J. Day and S. Djouadi, Nonlinear state estimation for complex immune responses, Proceedings of the $52^{nd}$ IEEE Conference on Decision and Control, Florence, Italy, December 10-13 (2013), 3373-3378. [3] G. Conte, C. H. Moog and A. M. Perdon, Nonlinear Control Systems: An Algebraic Setting, Springer-Verlag London, Ltd., London, 1999. [4] J. M. Coron, Control and Nonlinearity, American Mathematical Society, Providence, RI, 2007. [5] J. Day, J. Rubin and G. Clermont, Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation, Math Biosci Eng, 7 (2010), 739-763. doi: 10.3934/mbe.2010.7.739. [6] M. de Waal, J. Abrams, C. Bennett, B. Figdor and J. de Vries, Interleukin 10(il-10) inhibits cytokine synthesis by human monocytes: An autoregulatory role of il-10 produced by monocytes, J Exp Med, 174 (1991), 1209-1220. [7] J. A. Florian Jr., J. L. Eiseman and R. S. Parker, Nonlinear model predictive control for dosing daily anticancer agents using a novel saturating-rate cell-cycle model, Comput. Biol. Med., 38 (2008), 339-347. [8] J. Hogg, G. Clermont and R. S. Parker, Acute inflammation treatment via particle filter state estimation and mpc, 9th International Symposium on Dynamics and Control of Process Systems, 9 (2010), 272-277. [9] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2101-0. [10] A. Reynolds, J. Rubin, G. Clermont, J. Day, Y. Vodovotz and G. B. Ermentrout, A reduced mathematical model of the acute inflammatory response. i. derivation of model and analysis of anti-inflammation, J Theor Bio, 242 (2006), 220-236. doi: 10.1016/j.jtbi.2006.02.016. [11] D. Simon, Optimal State Estimation: Kalman, H-infinity and Nonlinear Approaches, Wiley-Interscience, Hoboken, NJ, 2006. doi: 10.1002/0470045345. [12] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, Oxford, 2008. [13] J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser Boston, Inc, Boston, MA, 1992.

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##### References:
 [1] D. C. Angus and T. van der Poll, Severe sepsis and septic shock, New Eng J Med, 369 (2013), 840-851. [2] O. Bara, J. Day and S. Djouadi, Nonlinear state estimation for complex immune responses, Proceedings of the $52^{nd}$ IEEE Conference on Decision and Control, Florence, Italy, December 10-13 (2013), 3373-3378. [3] G. Conte, C. H. Moog and A. M. Perdon, Nonlinear Control Systems: An Algebraic Setting, Springer-Verlag London, Ltd., London, 1999. [4] J. M. Coron, Control and Nonlinearity, American Mathematical Society, Providence, RI, 2007. [5] J. Day, J. Rubin and G. Clermont, Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation, Math Biosci Eng, 7 (2010), 739-763. doi: 10.3934/mbe.2010.7.739. [6] M. de Waal, J. Abrams, C. Bennett, B. Figdor and J. de Vries, Interleukin 10(il-10) inhibits cytokine synthesis by human monocytes: An autoregulatory role of il-10 produced by monocytes, J Exp Med, 174 (1991), 1209-1220. [7] J. A. Florian Jr., J. L. Eiseman and R. S. Parker, Nonlinear model predictive control for dosing daily anticancer agents using a novel saturating-rate cell-cycle model, Comput. Biol. Med., 38 (2008), 339-347. [8] J. Hogg, G. Clermont and R. S. Parker, Acute inflammation treatment via particle filter state estimation and mpc, 9th International Symposium on Dynamics and Control of Process Systems, 9 (2010), 272-277. [9] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2101-0. [10] A. Reynolds, J. Rubin, G. Clermont, J. Day, Y. Vodovotz and G. B. Ermentrout, A reduced mathematical model of the acute inflammatory response. i. derivation of model and analysis of anti-inflammation, J Theor Bio, 242 (2006), 220-236. doi: 10.1016/j.jtbi.2006.02.016. [11] D. Simon, Optimal State Estimation: Kalman, H-infinity and Nonlinear Approaches, Wiley-Interscience, Hoboken, NJ, 2006. doi: 10.1002/0470045345. [12] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, Oxford, 2008. [13] J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser Boston, Inc, Boston, MA, 1992.
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