Elsevier

Journal of Theoretical Biology

Volume 242, Issue 1, 7 September 2006, Pages 237-256
Journal of Theoretical Biology

A reduced mathematical model of the acute inflammatory response II. Capturing scenarios of repeated endotoxin administration

https://doi.org/10.1016/j.jtbi.2006.02.015Get rights and content

Abstract

Bacterial lipopolysaccharide (LPS; endotoxin) is a potent immunostimulant that can induce an acute inflammatory response comparable to a bacterial infection. Experimental observations demonstrate that this biological response can be either blunted (tolerance) or augmented (potentiation) with repeated administration of endotoxin. Both phenomena are of clinical relevance. We show that a four-dimensional differential equation model of this response reproduces many scenarios involving repeated endotoxin administration. In particular, the model can display both tolerance and potentiation from a single parameter set, under different administration scenarios. The key determinants of the outcome of our simulations are the relative time-scales of model components. These findings support the hypothesis that endotoxin tolerance and other related phenomena can be considered as dynamic manifestations of a unified acute inflammatory response, and offer specific predictions related to the dynamics of this response to endotoxin.

Introduction

The initial response of the body to acute biological stress such as bacterial infection or tissue trauma is an acute inflammatory response. This response involves a cascade of events mediated by a large array of cells and molecules that locate invading pathogens or damaged tissue, alert and recruit other cells and effector molecules, eliminate the offending agents, and restore the body to equilibrium. Bacterial lipopolysaccharide (LPS; endotoxin) is a highly conserved, highly immunogenic, constituent molecule of the outer cell wall of Gram-negative bacteria. When bacteria are lysed by immune effector cells and molecules, surges of endotoxin may be released into the host, intensifying the inflammatory response and causing further activation of immune effector cells (Alexander and Rietschel, 2001; Janeway and Medzhitov, 2002). In fact, the administration of antibiotics can lead to pulses of endotoxin release from Gram-negative bacteria as the antibiotics kill the invading bacteria, confirming the clinical importance of this subject matter (Eng et al., 1993). Since direct endotoxin administration in animals and humans can induce an acute inflammatory response that reproduces many of the features of an actual bacterial infection, such as fever, it stands as a valid experimental model for investigating the inflammatory response (Copeland et al., 2005; Morrison and Ryan, 1987; Parrillo, 1993).

High doses of endotoxin can be lethal, even though this bacterial byproduct does not proliferate as a Gram-negative bacteria would (Senaldi et al., 1999). It has been observed, however, that in some instances repeated doses of endotoxin result in a considerably less vigorous immune response, a phenomenon referred to as endotoxin tolerance (Beeson, 1947). In fact, the induction of tolerance can greatly blunt the effect of a dose of endotoxin that would be lethal to a naïve animal. A variety of studies have followed up on Beeson's initial reports of endotoxin tolerance (for a historical perspective see Cross, 2002; Schade et al., 1999; West and Heagy, 2002). Experimentally, it is now possible to assess the activation status of inflammatory cells or the levels of signaling proteins, such as cytokines, in organs or the blood as direct measures of inflammation (Nathan, 2002; Nathan and Sporn, 1991). The cytokine Tumor Necrosis Factor-α (TNF) in blood serum, for instance, has become a prominent marker of inflammation (Janeway et al., 2001; Sanchez-Cantu et al., 1989). Thus, observing that the concentration of this cytokine is lower than levels normally observed after endotoxin administration suggests that inflammation is being suppressed.

Interestingly, the inverse phenomenon, called potentiation, has also been observed. In the extreme, an otherwise non-lethal dose of endotoxin rapidly following another non-lethal dose can result in death (Cavaillon, 1995). We hypothesized that a simple mathematical model of the acute inflammatory response could reconcile tolerance and potentiation, on the premise that the observed outcomes result from dynamic interactions between components of innate immunity. Accordingly, we adapted a recently developed computational model of the inflammatory response (Reynolds et al., 2006) and simulated various scenarios involving repeated endotoxin administration. We use actual experimental mouse scenarios to guide in silico experiments that recreate these scenarios qualitatively, including the phenomena of endotoxin tolerance and potentiation.

In our simulations, we find that both the timing and magnitude of endotoxin doses, relative to each other and to the dynamical interplay between pro- and anti-inflammatory mediators, is the key to discriminating between the seemingly disparate phenomena of endotoxin tolerance and potentiation. Our results, derived from a mathematical model not constructed specifically to address the issue of preconditioning, support the perspective that endotoxin tolerance and related phenomena could be better explained and understood as “inflammatory-stimuli-induced” effects rather than specific, distinct phenomena (Cavaillon, 1995). This perspective is also supported by studies showing that various inflammatory stimuli (e.g. trauma, hemorrhage, cytokines) can act either to tolerize or to prime the host for subsequent homologous or heterologous stimuli (Bumiller et al., 1999; Cavaillon et al., 1994; Kariko et al., 2004; Keel et al., 1996; Leon et al., 1992; Mendez et al., 1999; Vogel et al., 1988; Zervos et al., 1999). The intent of this paper is not to carry out a detailed mathematical analysis of our model. Rather, we hope to argue convincingly that endotoxin tolerance, potentiation, and other phenomena related to repeated endotoxin administration are best viewed and understood via the acute inflammatory response (Copeland et al., 2005; Yadavalli et al., 2001) and to demonstrate this with a mathematical model of that response.

Section snippets

A mathematical model of the acute inflammatory response to endotoxin

To examine repeated endotoxin administration in the context of the acute inflammatory response, we use a mathematical model that incorporates the effects of key aspects of the immune system's response to an insult (Eqs. (1), (2), (3), (4)). The detailed derivation of this model, based on previous experimental findings, and a term-by-term explanation of its components are outlined by Reynolds et al. (2006). The model we use replaces the pathogen equation of Reynolds et al. with an endotoxin

Model simulations of experimental scenarios

For our in silico simulations, we emulate the scenarios below using the dynamical systems analysis software XPPAUT (Ermentrout, 2002). Eqs. (1), (2), (3), (4) are integrated numerically using the Runge–Kutta algorithm with step size 0.01 for 200 time units (hours), taking into account the simulated i.v. injections of PE at the specified times. Thus, the design of our in silico endotoxin simulations can closely resemble actual endotoxin experimental scenarios, which originally were carried out

The importance of the dynamics of the late pro-inflammatory and anti-inflammatory mediators to tolerance

A system of ordinary differential equations becomes complicated very rapidly as the number of equations increases. It can, therefore, be advantageous to attempt to reduce the number of equations to a manageable number by applying a steady state assumption. This strategy is most appropriately applied to variables that are transient, and is accomplished by setting their derivatives to zero; for example, if x=f(x,y), then we apply the steady state assumption to x by setting x=X(y) such that f(X(y)

Insight from the model's responses to endotoxin administration

Looking at these preconditioning phenomena from the point of view of the dynamics of a mathematical model of the acute inflammatory response, we are able to offer insight into why these disparate results are seen experimentally. It is important to note that the development of this model only took into account empirical observations about the interactions of somewhat abstracted immune effectors. However, none of the endotoxin administration results that we have reproduced was built into the

Discussion

The preconditioning phenomena of potentiation and tolerance characterize acute inflammation in both rodents and humans (Copeland et al., 2005; Yadavalli et al., 2001); in humans, the latter phenomenon is often referred to as “immune paralysis” or “immune exhaustion”, in which leukocytes–derived from patients with severe inflammation as measured by circulating pro-inflammatory cytokines—often produce low levels of these same inflammatory agents (Pinsky, 2001, Pinsky, 2004). In this paper, we

Acknowledgments

This work was supported under NIH grants R01-GM67240 and P50-GM-53789 and the Intramural Research Program at NIH, NIDDK (CCC).

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