Parametric and non‐parametric modeling of autonomous physiologic systems: applications and multi‐scale modeling of sepsis

2014-08-08Fundamental to physiology is the concept of homeostasis, the dynamic maintenance of an equilibrium that is, in general, in disequilibrium with the larger environment. Physiologic systems must, then, be able to respond to exogenous perturbations in the environment and to their internal state. That is, physiologic systems can be understood as a set of input‐to‐output transformations between system variables, both exogenous and endogenous. Such systems are quite complex, and knowledge must be summarized in the form of models, either informal or formal. Formal mathematical modeling approaches can be either parametric or non‐parametric. Non‐parametric modeling represents an inductive approach, wherein the model form is estimated directly from observed time‐series data. The fundamental advantage of the non‐parametric approach is that it can effectively summarize very complex systems when little is known about the internal workings. The Volterra series expansion is a canonical non‐parametric methodology for representing the input‐output functional relationship between any two variables. However, as classically formulated, the Volterra series cannot be applied to nonlinear oscillators or chaotic systems, nor can it capture autonomous systems behaviors not directly driven by an exogenous input. Moreover, even when a strict input‐output formulation is theoretically proper, internal system feedbacks or recursion can give rise to inefficiencies in the input‐output functional representation. By relaxing the strict input‐output requirement of the classical formulation and making the outputs recursive, we expand dramatically the class of systems that are amenable to Volterra‐style modeling, including those that operate strictly autonomously. We have demonstrated the method's efficacy on several test systems. In the single‐output setting, auto‐recursive Volterra models can capture autonomous behaviors, such as action potential firing in neural systems or population growth in ecological systems. In the multi‐output case, in which outputs of interest interact dynamically, the concept of recursive outputs allows ""closed"" or ""nested"" loop configurations to be captured in a Volterra‐style framework. ❧ Bacterial sepsis is characterized by a complex inflammatory response, and there has been significant interest in modeling this process as a dynamical system. It is believed that the pathology of sepsis is largely due to the immune response itself, which has motivated the clinical strategy of blocking pro‐inflammatory cytokines. However, this strategy has uniformly failed in clinical trials, motivating a systems approach to the disorder. We have shown that the recursive Volterra methodology can describe both autonomous growth, e.g. that of bacteria, and can capture the dynamics of simulated endotoxemia, a simplified model for actual sepsis. Therefore, we were hopeful that it could be applied to clinical time‐series data for actual sepsis patients. However, the temporal resolution of available data has proven too coarse, and thus the constraints imposed by the data have necessitated a parametric approach to modeling this system. Parametric modeling represents a deductive, hypothesis‐driven methodology, and we build our parametric representation of bacterial infection and the immune response in a hierarchical manner. This allows a careful examination of how the incorporation of different mechanisms, and different formal representations for these mechanisms, affects overall model dynamics and predictions. This approach has generated several predictions salient with respect to both future modeling efforts and to the clinical management of sepsis. We find that the interaction of local and systemic macrophage and neutrophil populations, respectively, is essential both for efficiently mobilizing the immune system and for resolving inflammation. Antibiotic treatment is modeled and found to interact synergistically with the immune response; however, this synergism is weak in immunocompromised hosts. Finally, we find that an exuberant inflammatory cytokine response is the natural consequence of a severe infection or compromised immune system, and while this may contribute to systems pathology, it is appropriate with respect to clearing infection. Thus, anti‐inflammatory interventions are likely to have a limited clinical role, and the mainstay of treatment must remain treating the cause, principally via antibiotics