In Silico modeling of immune-cardiovascular-endocrine interactions

The immune system provides an intricate, balanced response to combat the effects of in ﬂ ammatory stimuli. It incorporates both positive and negative feedback from multiple physiological systems such as the cardiovascular and endocrine systems including mechanisms functioning on a variety of time scales. They have been studied individually via scienti ﬁ c experiments and using mathematical modeling. However, more analysis is needed to study the interactions between these three systems during an in ﬂ ammatory event. We present the ﬁ rst dynamical systems model studying immune, cardiovascular and endocrine responses to a 2 ng/kg bolus dose of endotoxin. The model is calibrated to experimental data from two endotoxin challenge studies and we use this model to investigate the effects of endotoxin dosage, administration timing and administration method. Our model shows that most repercussions of endotoxin administration clear the system within 24 hours, but effects can linger for up to 72 hours.


Introduction
It is well-known that the immune system, the cardiovascular system and the endocrine system interact during an immune event that incites an infl ammatory response, such as an infection or injury [1][2][3][4][5][6][7][8][9] These three complex systems function on varying time scales, ranging from seconds to days, which affects the progression and resolution of infl ammation and resulting symptoms accordingly. While numerous experimental in vitro and in vivo studies have been conducted to study the infl ammatory response and its bi-directional communication with the cardiovascular and endocrine systems [2,7,[10][11][12] in silico experiments have the capability to further investigate dynamics between systems, challenge the included concepts and mechanisms and extrapolate the model to study clinically relevant scenarios not captured in the traditional experimental setting. Because of the abundance of data from in vitro and in vivo studies, in silico experiments can make more accurate predictions by using the data to calibrate and validate the model. The use of mathematical modeling to improve our knowledge surrounding the infl ammatory response is critical to expedite diagnosis, develop and test treatment regimes and decrease mortality associated with infl ammatory diseases such as sepsis.

Methods
Here, we discuss our recent results of a mathematical model coupling immune, cardiovascular, and endocrine dynamics across various time scales during an endotoxin (lipopolysaccharide, LPS) challenge [13]. To study the effect of time scales on dynamics, we utilize our model to simulate variations in endotoxin dosage, administration timing, and administration method. We also gain insight into relevant but understudied experimental setups investigating infl ammatory diseases, which are of interest in the clinical environment. Our Citation: Windoloski  The model includes 17 ordinary differential equations and one delay differential equation. Individual model subsystems were derived in previous work [14][15][16], but the combined immunecardiovascular-endocrine model is the fi rst model examining the dynamics between these three subsystems integrating various time scales. The model schematic is shown in Figure 1. Stimulation of the hypothalamus both from the body's natural circadian rhythm as well as the immune system prompts the start of this hormone cascade [6,14]. There is extensive evidence of communication between these three submodels during an infl ammatory event [1][2][3][4][5][6][7][8][9]. These couplings in the model are highlighted in Figures 1,2.
The mathematical model described above was calibrated to mean experimental data from two separate studies, both administering 2 ng/kg of LPS to male volunteers. We used cytokine, temperature, and hormonal data from Clodi, et al. and cardiovascular and pain data from Janum et al. to calibrate the model [18,19]. Parameters used in [14][15][16] for the original individual submodels were scaled and recalibrated to fi t the experimental data. New parameters corresponding to couplings between the three systems were added sequentially into the model and were manually adjusted.

Results and discussion
The initial model calibration (Figure 3   kg. The immune response intensifi ed as the dose increased by activating at a faster rate and producing much larger concentrations of IL-6 and IL-10. A lower pain threshold, a slightly higher fever and a drop in blood pressure approaching hypotension also appeared as the LPS dose was elevated. In the HPA axis, initial hormone release was suppressed and oscillations were dampened in subsequent 24-hour cycles as the LPS dose increased. The method of endotoxin administration is also a pertinent topic within the scientifi c community. Repeated endotoxin dose studies simulate the recurrence of infection following an initial insult and have been done experimentally [21][22][23] and in silico [14,24]. Our in silico repeated dose study showed that immune, temporal, pain and heart rate states are exacerbated when the second dose is administered at or near peak levels from the fi rst dose (within 6 hours) but suppressed when the second dose is administered after recovery. Like these dynamics, blood pressure drops close to hypotension when the second dose is given at or just after peak levels but narrowly deviates from baseline if given after recovery. Effects on hormone release are seen up to 48 hours after the fi rst dose, including increased or damped oscillations and peaks. . The red lines denote system dynamics prior to endotoxin administration and the black lines denote dynamics after endotoxin administration. Data (mean ± SE) from [2] is represented by dark blue markers and data (mean ± SE) from [20] is represented by light blue markers. Another variation of endotoxin dose administration is a continuous infusion. While a bolus dose of LPS is the commonly accepted endotoxin challenge form, it is recognized that a continuous infusion of endotoxin over several hours better portrays immunostimulant exposure in infl ammatory diseases [13,17,25,26]. Therefore, we use our model to simulate a 2 ng/ kg continuous infusion of endotoxin over a 4-hour period ( Figure 4). As seen in data conducted from in vivo experiments [27], our model shows a delay in the activation of the immune system. This delay then cascades to delay cardiovascular, hormonal, temporal and pain reactions. Once activated, the system exhibits larger peaks in immune, temporal and nitric oxide states. A more extreme cardiovascular response is seen, too, where blood pressure drops to about 100mmHg nearing hypotension. Hormone release was moderately suppressed and oscillation frequency was somewhat increased in the following 24-hour cycle. For a more in-depth look at the results regarding in silico experiments conducted using our model, we refer to our publication [13].
Here, we presented a complex, dynamic systems model studying coupled immune, cardiovascular and endocrine dynamics during an endotoxin challenge. We incorporated various time scales into the model including both circadian and ultradian rhythms and studied endotoxin variations in administration timing, dose and method. The novelty of our model is that it is the fi rst model to investigate dynamics between the immune, cardiovascular and endocrine systems in response to endotoxin up to several 24-hour cycles following endotoxin administration. The strength of the couplings between the systems is of great interest in gaining an understanding of how the submodels interact. Our work builds the foundation for further investigation into these couplings, including mathematically rigorous sensitivity analysis and parameter estimation schemes to determine which parameters have the greatest infl uence on system dynamics. Because of the lack of knowledge associated with the coupling parameter values, it would also be appropriate to formally quantify the uncertainty of these parameters.