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. Google Scholar

[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. Google Scholar

[3]

G. Conte, C. H. Moog and A. M. Perdon, Nonlinear Control Systems: An Algebraic Setting, Springer-Verlag London, Ltd., London, 1999.  Google Scholar

[4]

J. M. Coron, Control and Nonlinearity, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Simon, Optimal State Estimation: Kalman, H-infinity and Nonlinear Approaches, Wiley-Interscience, Hoboken, NJ, 2006. doi: 10.1002/0470045345.  Google Scholar

[12]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, Oxford, 2008.  Google Scholar

[13]

J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser Boston, Inc, Boston, MA, 1992.  Google Scholar

show all references

References:
[1]

D. C. Angus and T. van der Poll, Severe sepsis and septic shock, New Eng J Med, 369 (2013), 840-851. Google Scholar

[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. Google Scholar

[3]

G. Conte, C. H. Moog and A. M. Perdon, Nonlinear Control Systems: An Algebraic Setting, Springer-Verlag London, Ltd., London, 1999.  Google Scholar

[4]

J. M. Coron, Control and Nonlinearity, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Simon, Optimal State Estimation: Kalman, H-infinity and Nonlinear Approaches, Wiley-Interscience, Hoboken, NJ, 2006. doi: 10.1002/0470045345.  Google Scholar

[12]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, Oxford, 2008.  Google Scholar

[13]

J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser Boston, Inc, Boston, MA, 1992.  Google Scholar

[1]

Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021097

[2]

Judy Day, Jonathan Rubin, Gilles Clermont. Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation. Mathematical Biosciences & Engineering, 2010, 7 (4) : 739-763. doi: 10.3934/mbe.2010.7.739

[3]

Divyang G. Bhimani. The nonlinear Schrödinger equations with harmonic potential in modulation spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5923-5944. doi: 10.3934/dcds.2019259

[4]

Jin-Won Kim, Amirhossein Taghvaei, Yongxin Chen, Prashant G. Mehta. Feedback particle filter for collective inference. Foundations of Data Science, 2021, 3 (3) : 543-561. doi: 10.3934/fods.2021018

[5]

Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193

[6]

Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337

[7]

Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037

[8]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521

[9]

Antonio Marigonda. Second order conditions for the controllability of nonlinear systems with drift. Communications on Pure & Applied Analysis, 2006, 5 (4) : 861-885. doi: 10.3934/cpaa.2006.5.861

[10]

Andrea Arnold, Daniela Calvetti, Erkki Somersalo. Vectorized and parallel particle filter SMC parameter estimation for stiff ODEs. Conference Publications, 2015, 2015 (special) : 75-84. doi: 10.3934/proc.2015.0075

[11]

Sahani Pathiraja, Wilhelm Stannat. Analysis of the feedback particle filter with diffusion map based approximation of the gain. Foundations of Data Science, 2021, 3 (3) : 615-645. doi: 10.3934/fods.2021023

[12]

Qifeng Cheng, Xue Han, Tingting Zhao, V S Sarma Yadavalli. Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter. Journal of Industrial & Management Optimization, 2019, 15 (1) : 177-198. doi: 10.3934/jimo.2018038

[13]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014

[14]

Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021009

[15]

Ali Wehbe, Marwa Koumaiha, Layla Toufaily. Boundary observability and exact controllability of strongly coupled wave equations. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021091

[16]

Vahagn Nersesyan. Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension. Mathematical Control & Related Fields, 2021, 11 (2) : 237-251. doi: 10.3934/mcrf.2020035

[17]

Daliang Zhao, Yansheng Liu. Controllability of nonlinear fractional evolution systems in Banach spaces: A survey. Electronic Research Archive, , () : -. doi: 10.3934/era.2021083

[18]

Laura Martín-Fernández, Gianni Gilioli, Ettore Lanzarone, Joaquín Míguez, Sara Pasquali, Fabrizio Ruggeri, Diego P. Ruiz. A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system. Mathematical Biosciences & Engineering, 2014, 11 (3) : 573-597. doi: 10.3934/mbe.2014.11.573

[19]

Omar Saber Qasim, Ahmed Entesar, Waleed Al-Hayani. Solving nonlinear differential equations using hybrid method between Lyapunov's artificial small parameter and continuous particle swarm optimization. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 633-644. doi: 10.3934/naco.2021001

[20]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]