A Study of the Synergies Between Control Mechanisms in the Immune System and the Variable Structure Control Paradigm

This thesis argues that variable structure control theory finds application in immunology. The immune system maintains a healthy state by using feedback to switch on and off immune responses. Experimental and mathematical work has analysed the dynamics of the immune response of T cells, relatively little attention has been paid to examine the underlying control paradigm. Besides, in modelling and simulation studies, it is necessary to evaluate the impact of uncertainty and perturbations on immunological dynamics. This is important to deliver robust predictions and insights. These facts motivate considering variable structure control techniques to investigate the control strategy and robustness of the immune system in the context of immunity to infection and tolerance. The results indicate that the dynamic response of T cells following foreign or self-antigen stimulation behaves as a naturally occurring switched control law. Further, the reachability analysis from sliding mode control highlights dynamical conditions to assess the performance and robustness of the T cell response dynamics. Additionally, this approach delivers dynamical conditions for the containment of Human Immunodeficiency Virus (HIV) infection by the HIV-specific CD8+ T cell response and antiretroviral therapy by enforcing a sliding mode on a manifold associated with the infection-free steady-state. This condition for immunity reveals particular patterns for early diagnosis of eventual success, marginal and failure cases of antiretroviral therapy. Together, the findings in this thesis evidence that variable structure control theory presents a useful framework to study health and disease dynamics as well as to monitor the performance of treatment regimes

Modelling and Simulation of the Dynamics of the Antigen-Specific T Cell Response Using Variable Structure Control Theory.

Experimental and mathematical studies in immunology have revealed that the dynamics of the programmed T cell response to vigorous infection can be conveniently modelled using a sigmoidal or a discontinuous immune response function. This paper hypothesizes strong synergies between this existing work and the dynamical behaviour of engineering systems with a variable structure control (VSC) law. These findings motivate the interpretation of the immune system as a variable structure control system. It is shown that dynamical properties as well as conditions to analytically assess the transition from health to disease can be developed for the specific T cell response from the theory of variable structure control. In particular, it is shown that the robustness properties of the specific T cell response as observed in experiments can be explained analytically using a VSC perspective. Further, the predictive capacity of the VSC framework to determine the T cell help required to overcome chronic Lymphocytic Choriomeningitis Virus (LCMV) infection is demonstrated. The findings demonstrate that studying the immune system using variable structure control theory provides a new framework for evaluating immunological dynamics and experimental observations. A modelling and simulation tool results with predictive capacity to determine how to modify the immune response to achieve healthy outcomes which may have application in drug development and vaccine design

Simulation of the time evolution of secondary CD8+ T cells with CD4+ T cell help <i>u</i>_<i>h</i>(<i>t</i>) = 0.5 and <i>t</i>_<i>off</i> = 10 during chronic infection.

<p>Top: (a) Time evolution of the population dynamics of LCMV specific memory CD8+ T cells following chronic infection and CD4+ T cell help. Bottom: (b) Time evolution of the level of CD4+ T cell help. The action of CD4+ T cell help increases the number of CD8+ T cells as well as the duration of the expansion. Further, the magnitude of CD4+ T cell help is proportional to the number of secondary effector CD8+ T cells. The achieved level of CD4+ T cell help is higher than that required throughout the simulation.</p

Simulation of the scenario of T cell population dynamics following acute infection.

<p>Left: Time evolution of the population dynamics of the antigen-specific T cell response. Right: the time evolution of the antigen-dependent activation function <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0166163#pone.0166163.e032" target="_blank">Eq (14)</a>. Following infection, the magnitude of the activation function is at a maximum. This induces the reduction of the state <i>N</i> and an increase of the state <i>A</i>. The state variable <i>N</i> decreases because naive T cells become activated T cells. The state variable <i>A</i> increases due to the expansion of the population of activated T cells. When the magnitude of the immune response function falls to zero, the expansion phase is interrupted. There is production of memory T cells, the state <i>M</i> increases whilst the activated T cells undergo contraction i.e the state <i>A</i> decreases. Consequently, the activation function prescribes the variation over time of the total population of the antigen -specific T cells (<i>N</i> + <i>A</i> + <i>M</i>) following infection.</p

Phase portrait of two candidate fixed structure control strategies for the oscillator system with <i>ξ</i> = 0.7.

<p>Left: (a) Stable dynamic (poles at −0.35 ± 0.69<i>j</i>). Right: (b) Unstable dynamic (poles at 0.5 and −1.2). In (a), the trajectories move in a clockwise motion to reach the origin. In (b), the trajectories move away from the origin except along the asymptotes of the hyperbolic structure in the quadrants with .</p

Time evolution of the sliding surface and reachability condition for the immune system Eqs (1)–(4) with the antigen-dependent activation function (14).

<p>Time evolution of the sliding surface and reachability condition for the immune system Eqs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0166163#pone.0166163.e001" target="_blank">(1)</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0166163#pone.0166163.e004" target="_blank">(4)</a> with the antigen-dependent activation <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0166163#pone.0166163.e032" target="_blank">function (14)</a>.</p

Time evolution of the equivalent control signal and the perturbation signal.

<p>This shows the relationship between the applied control signal and the external perturbation signal once the sliding mode is reached. The response of the equivalent control signal at earlier time points is not provided because by definition, the equivalent control occurs only after the sliding mode takes place. Importantly, this graph shows that when a sliding mode is enforced, the equivalent control can be used to reconstruct the dynamics of the perturbation signals implicit in the channel where the control signal is applied, the so-called matched uncertainty.</p

Phase plane portrait showing the response of the double integrator (<i>f</i>(<i>t</i>) = 0) and the perturbed system (<i>f</i>(<i>t</i>) = −<i>sin</i>(<i>y</i>)) with initial conditions <i>y</i>(0) = 1, .

<p>At first, there is a transient phase, known as the reaching phase, in which the dynamics of the system move towards the selected sliding manifold. During the reaching phase, the system dynamics are affected by the perturbation signal. Following this reaching phase, a sliding mode takes place. The trajectories of the system are confined to a vicinity of the sliding manifold and move in a straight line towards the origin. It is seen that during this sliding motion, the traces of the system with and without perturbation are indistinguishable. This shows that the action of the switched control in the sliding mode cancels the effects of the perturbation signal.</p