(370b) A Stochastic Ensemble Model of Human Endotoxemia

In the systemic inflammatory response, interacting divisions (immune cells, HPA/SNS, SCN, cardiac system) are composed of cell populations which act upon external signals. Pathogen recognition receptors in macrophages recognize LPS, leading to the activation of signal transduction cascades which induce activation and release of transcription factors. These transcription factors regulate the expression of pro- and anti-inflammatory cytokines, which are subsequently released into circulation and activate the HPA axis leading to the eventual release of stress hormones. The released hormones further interact with other distal organizational subunits eventually leading to the manifestation of systemic emergent changes, i.e. cardiac output. Therefore, emergence characterizes a wide range of structural units ranging from mounting inter-cellular activating signals, to circulating hormones and cytokines to changes in circadian patterns to depression of host-level phenotypes. The fundamental similarity of these systems is that their aggregate response is produced by the combined effort of a large population of individual cells with heterogeneous behaviors. A similar observation was made during the development of models of NF-kB regulation. Initially, it was observed that in response to a stimulus, NF-kB activity oscillates, but the oscillations quickly damp out [1]. Although the data leading to this assumption was not taken at the scale of a single cell, it was assumed that the aggregate behavior corresponds to the dynamics in each individual cell. Thus, a model was fit so that NF-kB behaved like a damped oscillator in an individual cell. Later, when single cell experiments were performed, it was observed that the oscillations in individual cells persisted, but they quickly lose their phase synchronization due to biological noise [2]. Therefore, the damped oscillations that are observed in the aggregate response are really the result of a large number of persistent oscillators falling out of phase due to random variability. This type of behavior can be modeled by introducing stochasticity into previously-deterministic models and simulating a population of noisy cells to capture the overall response of the system resulting in a more mechanistic model of cellular dynamics [3].

We extend our previous work using ODEs to model inflammation [4] by appropriately adding noise to the system and replacing aggregate variables with the output of ensemble of cells. This is done by translating the ODESs into stochastic differential equations (SDEs) to model the variability within the populations of cells.

This concept is applied in a number of areas in the model. It is clearly of importance in immune cells, as described above in the description of models of NF-kB activation. Thus, instead of assuming homogeneity in the responses of all immune cells, an ensemble of cells each with different stochastic dynamics has been considered. The outputs of these cells, pro- and anti-inflammatory cytokines, are combined to assess their impact on hormone production.

In addition to considering an ensemble of immune cells, the same concept is applied at other levels as well so that there are four separate compartments as described above: immune cells, HPA/SNS, SCN, cardiac system. At the lower level measured mRNA abundance quantifies the dynamics of ensembles of cellular events. Translation products activate signaling cascades leading to activated nuclear complexes further driving expression of circulating proteins which serve as inter-compartment signals eventually giving rise to macroscopic phenotypes. The CNS is linked to peripheral immune tissues by a bidirectional communication network that functions through the release of neuroendocrine hormones by the CNS and the secretion of cytokines by immune cells [5]. Reciprocal communication pathways between the immune and central nervous systems are major components of the integrated homeostatic network of the host. In response to circulating cytokines, the production rates of the hormones are modulated. The aggregate hormonal signal is released into the blood where it can interact with the other compartments. Our model of inflammation considers the effects of the immunomodulatory hormones cortisol, epinephrine, and melatonin. Each of these three hormones is produced in a spatially segregated subcompartment and will be modeled as such. First, cortisol production is dependent on the hypothalamic-pituitary-adrenal (HPA) axis, which has previously been modeled by considering each of the three components of the HPA as separate subcompartments [6]. Cortisol produced in the adrenal cortex then directly interacts with the adrenal medulla, stimulating epinephrine production [7]. Melatonin is produced by pinealocytes in the pineal gland, which will be modeled as a separate subcompartment of the neuroendocrine system. Both cortisol and melatonin production are directly regulated by the central circadian clock in the suprachiasmatic nucleus (SCN).

Circadian signals from the SCN strongly influence neuroendocrine secretory and autonomic activities [8], and there is also a considerable diurnal variation in the human host response to endotoxin, as described in SA1. When activated by light/dark signals from the retina, thousands of neurons in the SCN oscillate and their aggregate output is a critical physiological modulator. In order to model such complex dynamics, an ensemble of coupled cells must be used [9]. Cell to cell variability plays a constructive role in generating periodic oscillations and is an inherent feature of biological systems in general and the SCN in particular given the irregular exposure to light throughout the day. Thus, the SCN should be modeled as a group of coupled noisy oscillating neurons [9]. The outputs of these neurons are be combined to serve as an input into the circadian production of hormones. This allows for the complex dynamics concerning the interplay between inflammation and circadian rhythms to be modeled. The population of immune cells can communicate with the SCN via the secretion of cytokines while the SCN sends signals to immune cells via circadian regulation of hormones. In both directions, the signal is generated by the stochastic behavior of a population of cells and communicated only through their aggregate response.

Heart rate variability (HRV) has been extensively used as a predictor of trauma patient outcome [10]. One of the key hypotheses made in the development of the original model is that changes in HRV during inflammation occur due to conversion of the neuronal firing rate into neuromediator concentrations which affect changes in pacemaker cycle length, driving beat to beat variations as quantified by HRV [11]. In an effort to model such neuronal firing rate, an integrate-and-fire model is be used for the impulse generation process of nerve cells. This transforms a continuous-time input signal into discrete-time series ? a point process signal that in our case is the neuronal spike train. A point process is a useful mathematical representation of a signal which consists of repeated similar events. The significance of the event is attached to the time at which it occurs rather than the detailed properties of the event itself. This can be extended, similar to the discussion of immune cells and neurons in the SCN above, by considering the neurons as an ensemble of stochastic processes. In our model, since the pro-inflammatory response can signal to the autonomic nervous system via afferent sensory neurons [12], the integrate-and-fire model will transform the continuous-time signal into a train of spiking firing rates. Assuming that an increase in spiking firing rate is associated with increased release of neuromediator concentrations affecting heart rate, HRV is thereby be described by the time intervals where cardiac nerves have fired a pulse.

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