Lie Symmetries as a Mathematical Methodology to Identify Conservation Laws in Physiological Systems

Systems Medicine aims to understand the dynamics of physiological systems and the differences between healthy and disease states, to then bring the latter back to health. To this aim, it is critical to identify the states that allow modifying the phenotype of a model system and are robust to perturbations. Indeed, for these changes to be sustained in time, the system’s robustness shall be investigated through various analyses and their emerging results. Lie symmetry analysis—a study of fixed variable relations in a differential equations model—uncovers the model’s hidden robustness through its conservation laws. The emerging conservation laws can then be used as a series of robust invariant characteristics of the system under a specific type of perturbations. Although it holds much predictive potential for robustness investigations, the application of Lie symmetry-based conservation law analysis to physiological systems is currently unexplored. Here, we propose a novel application of Lie symmetry-based conservation law analysis to identify the conservation laws—and their existence conditions—influencing the dynamics of a system towards robust remission or relapse. This methodology is used to analyse a minimal model of rheumatoid arthritis with the aim to: (i) investigate the existence and extent of robust disease characteristics as conservation laws of the model, (ii) clinically interpret their biological viability, and (iii) inform model plausibility, testing, and selection. This novel application of the Lie symmetry analysis can retrieve the robust characteristics of physiological conditions, thus providing a new analytical contribution to the Systems Medicine field.