Sufficient conditions for the existence of asymptotically stable homeostatic equilibrium points in negative feedback physiological systems

While the adoption of mathematical models to represent and describe physiological phenomena is widely applied in bioengineering, medicine, and biochemistry research, the large variety of physio- and bio-logical systems, as well as the complex dynamics involved in homeostatic control, prevent the definition of a generalized biofidelic modeling approach for the analysis of systems’ dynamics. Leveraging the formalism and principles of control theory, this manuscript lays the theoretical foundations for the study of self-regulatory mechanisms in multivariable negative feedback physiological systems. Starting from the definition of a system-theoretic oriented mathematical modeling framework, general assumptions and model-specific theorems, tailored to the order of the model, are derived and demonstrated to define sufficient conditions for the existence of unique and asymptotically stable equilibrium points within the system's operating regions. Then, the proposed methodological approach is translated into practice through application to relevant case studies representing physiological control systems at increasing degrees of complexity (from 2nd- to 4th-order models), namely: (i) the regulation of thyroid hormones circulation in the blood; (ii) the prey–predator model describing the dynamics of tumor progression; (iii) the cortisol dynamics in response of pain/stress stimuli. Finally, the capability of the proposed framework to effectively capture the behavior of additional physiological systems (from 2nd- up to 7th-order) available in the literature is also demonstrated, thus shaping a promising theoretical and methodological route for the analysis of uniqueness and stability of homeostatic equilibrium in both physiological and pathological conditions.