Integrated Inflammatory Stress (ITIS) Model

Abstract

During the last decade, there has been an increasing interest in the coupling between the acute inflammatory response and the Hypothalamic–Pituitary–Adrenal (HPA) axis. The inflammatory response is activated acutely by pathogen- or damage-related molecular patterns, whereas the HPA axis maintains a long-term level of the stress hormone cortisol which is also anti-inflammatory. A new integrated model of the interaction between these two subsystems of the inflammatory system is proposed and coined the integrated inflammatory stress (ITIS) model. The coupling mechanisms describing the interactions between the subsystems in the ITIS model are formulated based on biological reasoning and its ability to describe clinical data. The ITIS model is calibrated and validated by simulating various scenarios related to endotoxin (LPS) exposure. The model is capable of reproducing human data of tumor necrosis factor alpha, adrenocorticotropic hormone (ACTH) and cortisol and suggests that repeated LPS injections lead to a deficient response. The ITIS model predicts that the most extensive response to an LPS injection in ACTH and cortisol concentrations is observed in the early hours of the day. A constant activation results in elevated levels of the variables in the model while a prolonged change of the oscillations in ACTH and cortisol concentrations is the most pronounced result of different LPS doses predicted by the model.

Appendix

1.1 Parameter Values

The parameter values used for the simulations of the ITIS model (1)–(2) presented in Sect. 2 and the biological interpretation are shown in Table 1.

Table 1 Table of biological interpretation and values of parameters in the ITIS model presented in Sect. 2

1.2 Sensitivity Analysis

The relative sensitivities of the parameters in the ITIS model are calculated to investigate the quantitative sensitivity and the robustness of the results of the model output. The relative sensitivity of a model output $y_i$ to the model parameters $\theta _j$ where $j=1,\ldots ,q$ can be calculated from the sensitivity matrix

$$\begin{aligned} S^{\mathrm{relative}}_{i} = \left[ \begin{array}{ccc} \frac{\theta _1}{y_i} \frac{\partial y_i}{\partial \theta _1}(t_{i1}) &{} \cdots &{} \frac{\theta _q}{y_i} \frac{\partial y_i}{\partial \theta _q}(t_{i1}) \\ \vdots &{} \ddots &{}\vdots \\ \frac{\theta _1}{y_i} \frac{\partial y_i}{\partial \theta _1}(t_{ik_i})&{} \cdots &{} \frac{\theta _q}{y_i} \frac{\partial y_i}{\partial \theta _q}(t_{ik_i}) \end{array}\right] \end{aligned}$$

(3)

for each of the variables i in the model, where $t_{ij}$ is the $k_i$ instance of the jth measurement and $y_i \ne 0$. To compare the sensitivities, the two-norm of each column can be calculated and used as a time independent measure for each of the parameters. A histogram stacking the relative sensitivities for the variables in the ITIS model is shown in Fig. 9.

Fig. 9

Histogram of the relative sensitivities of the parameters in the ITIS model