Modeling Tumor Disease and Sepsis by Networks of Adaptively Coupled Phase Oscillators

Abstract

In this study, we provide a dynamical systems perspective to the modelling of pathological states induced by tumors or infection. A unified disease model is established using the innate immune system as the reference point. We propose a two-layer network model for carcinogenesis and sepsis based upon the interaction of parenchymal cells and immune cells via cytokines, and the co-evolutionary dynamics of parenchymal, immune cells, and cytokines. Our aim is to show that the complex cellular cooperation between parenchyma and stroma (immune layer) in the physiological and pathological case can be qualitatively and functionally described by a simple paradigmatic model of phase oscillators. By this, we explain carcinogenesis, tumor progression, and sepsis by destabilization of the healthy homeostatic state (frequency synchronized), and emergence of a pathological state (desynchronized or multifrequency cluster). The coupled dynamics of parenchymal cells (metabolism) and nonspecific immune cells (reaction of innate immune system) are represented by nodes of a duplex layer. The cytokine interaction is modeled by adaptive coupling weights between the nodes representing the immune cells (with fast adaptation time scale) and the parenchymal cells (slow adaptation time scale) and between the pairs of parenchymal and immune cells in the duplex network (fixed bidirectional coupling). Thereby, carcinogenesis, organ dysfunction in sepsis, and recurrence risk can be described in a correct functional context.

Critical parameters in dynamic network modeling of sepsis

“…In addition to the plethora of dynamical states, adaptivity also induces a high degree of multistability [49]. In this study, we build on the findings from [19] and extend these in order to understand certain parameter dependencies for the emergence of sepsis.…”

“…1 A. In [19], we have introduced a functional model to describe the dynamic interaction of parenchyma (organ tissue) and stroma (immune layer). The network layer of parenchymal cells (superscript 1) are represented by N phase oscillators φ 1 i , i = 1, .…”

“…The adaptation rates 0 < µ 1 separate the time scales of the slow dynamics of the coupling weights and the fast dynamics of the oscillatory system. The adaptation rate of the parenchymal layer 1 is assumed to be slow compared to the adaptation rate of the immune layer 2 , i.e., 1 2 to account for the faster reaction of the immune cells, see also [19]. Thus we have two classes of adaptive coupling weights modeling two different cytokine mechanisms on two different timescales.…”

“…In this paper, we employ a two-layer network model for sepsis based upon the interaction of parenchymal cells and immune cells via cytokines and the coevolutionary dynamics of parenchymal and immune cells and cytokines [19]. Parenchyma is the bulk of functional substance in an organ or structure, in contrast to the stroma, which refers to the structural tissue of organs or structures, namely, the unspecific connective tissues.…”

“…In terms of our proposed functional model, we investigate the emergence of synchronization and frequency synchronization in the parenchyma, which is related to a healthy and unhealthy condition of the organ system, respectively. In [19], it has been shown that an initially activated immune system may induce an activation of the parenchyma, i.e., emergence of frequency clusters, or leave the parenchyma unaffected depending on a patient's individual characteristics. The present study provides insight into the robustness of the emerging pathological state with respect to changes of parameters.…”