Nonlinear optimal control of the acute inflammatory response

doi:10.1016/j.bspc.2019.101631

Biomedical Signal Processing and Control

Volume 55, January 2020, 101631

https://doi.org/10.1016/j.bspc.2019.101631 Get rights and content

Highlights

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Optimal control of the infusion of anti-inflammatory and anti-bacterial medication is critical for achieving more effective and less costly pharmaceutical treatments.
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The model of the acute inflammatory response undergoes linearization around a temporary operating point which is computed at each iteration of the control algorithm.
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The linearization relies on Taylor series expansion and on the associated Jacobian matrices.
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To find the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method.
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The stability properties of the control scheme are proven through Lyapunov analysis.

Abstract

The article proposes a nonlinear optimal control approach for the dynamic model of the acute inflammatory response. This model describes the reaction of the human body to bacterial infection. Optimal control of the infusion of anti-inflammatory and anti-bacterial medication is critical for achieving more effective and less costly pharmaceutical treatments. To solve the related nonlinear optimal control problem the state-space description of the acute inflammatory response undergoes first approximate linearization around a temporary operating point which is recomputed at each iteration of the control algorithm. The linearization relies on Taylor series expansion and on the computation of the associated Jacobian matrices. For the approximately linearized model of the acute inflammatory response a stabilizing H-infinity feedback controller is designed. This controller stands for the solution of the optimal control problem under model uncertainties and external perturbations. To find the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The stability properties of the control scheme are proven through Lyapunov analysis.

Introduction

The reaction of the immune system to infections caused by bacteria populations has complicated nonlinear dynamics [1], [2], [3], [4]. The result of such infections is inflammations. Actually, after exposure to bacteria the immune system exhibits the acute inflammatory response [5], [6], [7], [8]. This is a pathogenic condition that is retrofitted by the invading bacteria and the secreted inflammatory mediators, such as activated phagocytes and pro-inflammatory cytokines [9], [10], [11], [12]. Inflammations cause damage to the infected tissues which can be further aggravated by the immune system's reaction. Sustained inflammations can lead to septic conditions. Anti-inflammatory mediators can help the immune system to suppress and finally eliminate inflammations [13], [14], [15], [16]. Therefore, the solution of the control problem for anti-inflammatory medication infusion is important for the effective treatment of infections and for reducing the cost of anti-inflammatory therapies [17], [18], [19], [20]. In this area one can note some results towards solving the control problem for the acute inflammatory response model. These results rely primarily on nonlinear model predictive control methods [21], [22], [23], [24].

In this article a novel solution to the nonlinear optimal control problem of the acute inflammatory response model is proposed. First the state-space model of the acute inflammatory response is introduced and is shown to consist of a set of differential equations that describe the evolution of the following variables: (i) the concentration of bacteria, (ii) the concentration of pro-inflammatory mediators, (iii) the damage of the tissue and (iv) the concentration of anti-inflammatory mediators. This model receives two control inputs (i) the pro-inflammatory factors and (ii) the anti-inflammatory medication. Next, the dynamic model of the acute inflammatory response undergoes approximate linearization around a temporary operating point which is re-computed at each time-step of the control algorithm. This operating point is defined by the present value of the system's state vector and by the last value of the control inputs vector that was applied on it. The linearization relies on Taylor series expansion and on the computation of the associated Jacobian matrices [25], [26], [27], [28]. The modeling error which is due to truncating higher-order terms in the Taylor series expansion is considered to be a perturbation that is asymptotically compensated by the robustness of the control method.

For the approximately linearized model of the acute inflammatory response a stabilizing H-infinity feedback controller is designed. This controller provides the solution to the system's optimal control problem under model uncertainty and external perturbations. Actually it represents a min-max differential game in which (i) the control inputs try to minimize a cost function that comprises a quadratic term of the state vector's tracking error, and (ii) the model uncertainty and perturbation inputs which try to maximize this cost function. To compute the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm [29], [30]. The stability properties of the control scheme are proven through Lyapunov analysis, First, it is shown that the control loop satisfies the H-infinity tracking performance criterion [25], [31]. This signifies elevated robustness against model uncertainties and external perturbations. Next, under moderate conditions, it is proven that the control system is globally asymptotically stable, which signifies that the infused anti-inflammatory medication achieves finally the elimination of the inflammation and the eradication of the infection's effects. Finally, to apply state estimation-based control for the model of the acute inflammatory response, the H-infinity Kalman Filter is used [32], [33].

The article’s results are one of the most efficient approaches towards solving the nonlinear multivariable control problem of the acute inflammatory response system [34], [35], [36], [37]. Comparing to other nonlinear control schemes the article’s approach exhibits specific advantages: (1) unlike global linearization control schemes the article's approach does not require complicated state variables transformations (diffeomorphisms) so as to bring the state-space model of the acute inflammatory response to the canonical form; (2) unlike global linearization control methods the article’s approach is applied directly on the initial nonlinear model of the system and does not require inverse transformations for the computation of the control signal. In this manner singularity problems are also avoided; (3) unlike MPC the application of the article’s control method is not limited to linear dynamical systems and its use in the nonlinear model of the acute inflammatory response does not impose the risk of destabilization; (4) unlike NMPC the performance of the article’s control method and the convergence of its iterative search for an optimum is not dependent on initialization and on parameters’ values selection; (5) unlike backstepping control the article’s control method does not require the state-space model of the acute inflammatory response to be a-priori found in the triangular (backstepping integral) form; (6) unlike sliding-mode control, the article’s control approach does not require the state-space model of the acute inflammatory response to be a-priori found in the canonical (Brunovsky) form so as to define a sliding surface; (7) unlike PID control, the article’s approach is of proven global stability and functions in a reliable manner under changes of operating points and under external perturbations. Besides, it avoids heuristics and ad hoc methods for the selection of the gains of the stabilizing feedback controller.

The structure of the article is as follows: in Section 2 the state-space model of the acute inflammatory response is introduced. In Section 3 approximate linearization of this state-space model is performed through Taylor series expansion and through the computation of the associated Jacobian matrices. In Section 4 an H-infinity feedback controller is designed for the linearized model of the acute inflammatory response. In Section 5 the global stability properties of the control method are proven through Lyapunov analysis. In Section 6 the robustness and state-estimation issues of the control loop are analyzed. Actually, the H-infinity Kalman Filter is introduced as a robust observer which allows for implementing state estimation-based control. In Section 7 the performance of the control scheme is further confirmed through simulation tests. Finally, in Section 8 concluding remarks are stated.

Section snippets

Dynamics of the acute inflammatory response

The dynamic model of the acute inflammatory response describes interaction between the following state-variables: (i) the concentration of bacteria, (ii) the concentration of pro-inflammatory mediators, (iii) the damage of the tissue and (iv) the concentration of anti-inflammatory mediators. The dynamic model of the acute inflammatory response is shown in Fig. 1 and comprises the following differential equations [2], [3] $\dot{P} = K_{pg} P (1 - \frac{P}{P_{\infty}}) - \frac{K_{pm} s_{m} P}{μ_{m} + K_{mp} P} - K_{pm} f (N) P$ $\dot{N} = \frac{s_{nr} R}{μ_{nr} + R} - μ_{N} N + U_{p} (t)$ $\dot{D} = K_{dn} \frac{f {(N)}^{6}}{x_{dn}^{6} + f (N}$

Approximate linearization of the acute inflammatory response model

The dynamic model of the acute inflammatory response of Eq. (16) undergoes approximate linearization around the temporary operating point (x*, u*), where x* is the present value of the system’s state vector and u* is the last value of the control inputs vector that was applied to it. The linearization relies on Taylor series expansion and on the computation of the associated Jacobian matrices. The linearized model is $\dot{x} = Ax + Bu + \tilde{d}$ where $\tilde{d}$ is the aggregate disturbance term that is due to the

Equivalent linearized dynamics of the acute inflammatory response

As previously explained the use of optimal control tools for administering anti-inflammatory drug infusion is imperative for achieving better therapeutic results and for reducing the therapy’s cost. However, the solution of such a type nonlinear optimal control problem is a non-trivial procedure which in certain cases becomes computationally complicated or even intractable. For instance the use of popular optimal control approaches, such as MPC or NMPC suffers from serious drawbacks. In

Lyapunov stability analysis

Through Lyapunov stability analysis it will be shown that the proposed nonlinear control scheme assures H_∞ tracking performance for the acute inflammatory response dynamics, and that in case of bounded disturbance terms asymptotic convergence to the reference setpoints is achieved. The tracking error dynamics for the acute inflammatory response is written in the form $\dot{e} = Ae + Bu + L \tilde{d}$ where in the acute inflammatory response case L = I ∈ R^4×4 with I being the identity matrix. Variable $\tilde{d}$ denotes model

Riccati equation coefficients in H_∞ control robustness

Parameter ρ in Eq. (30), is an indication of the closed-loop system robustness. If the values of ρ > 0 are excessively decreased with respect to r, then the solution of the Riccati equation is no longer a positive definite matrix. Consequently there is a lower bound ρ_min of ρ for which the H_∞ control problem has a solution. The acceptable values of ρ lie in the interval [ρ_min, ∞). If ρ_min is found and used in the design of the H_∞ controller, then the closed-loop system will have increased

Simulation tests

The performance of the proposed nonlinear optimal control method for the model of the acute inflammatory response has been further tested through simulation experiments. The parameters of the state-space model used in the simulation tests are in accordance to [2], [3]. The control inputs where computed by solving at each time-step of the control method the algebraic Riccati equation that appears in Eq. (41). State estimation-based control has been implemented with the use of the H-infinity

Conclusions

Optimal control of the acute inflammatory response is important for achieving more effective treatment of bacterial infections and for reducing the cost of the related therapies. In this article a novel nonlinear optimal control method for the model of the acute inflammatory response has been proposed. First, the state-space model of the immune system’s response has undergone approximate linearization around a temporary operating point which is recomputed at each time-step of the control

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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View full text

Nonlinear optimal control of the acute inflammatory response

Highlights

Abstract

Introduction

Section snippets

Dynamics of the acute inflammatory response

Approximate linearization of the acute inflammatory response model

Equivalent linearized dynamics of the acute inflammatory response

Lyapunov stability analysis

Riccati equation coefficients in H∞ control robustness

Simulation tests

Conclusions

Declaration of Competing Interest

J. Theoret. Biol.

J. Theoret. Biol.

Math. Biosci.

J. Theoret. Biol.

Math. Comput. Simul.

Math. Biosci.

Math. Biosci.

Curr. Opin. Syst. Biol.

Fuzzy Sets Syst.

J. Theoret. Biol.

Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons

Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation

Math. Biosci. Eng.

Model-free immune therapy: a control approach to acute inflammation

Modelling and optimal control of immune response of renal transplant recipients

J. Biol. Dyn.

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

Math. Biosci. Eng.

Activation of immune response in disease dynamics via controlled drug scheduling

IEEE Trans. Autom. Sci. Eng.

Optimal control of innate immune response

Optim. Control Appl. Methods

Systems engineering medicine: engineering the inflammation response to infections and traumatic challenges

J. R. Soc. Interface

Riccati equation coefficients in H_∞ control robustness