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Introduction: After myocardial infarction (MI), the heart undergoes necrosis, inflammation, scar formation, and remodeling. While restoring blood flow is crucial, it can cause ischemia-reperfusion (IR) injury, driven by reactive oxygen species (ROSs), which exacerbate cell death and tissue damage. This study introduces a mathematical model capturing key post-MI dynamics, including inflammatory responses, IR injury, cardiac remodeling, and stem cell therapy. The model uses nonlinear ordinary differential equations to simulate these processes under varying conditions, offering a predictive tool to understand MI pathophysiology better and optimize treatments. Methods: After myocardial infarction (MI), left ventricular remodeling progresses through three distinct yet interconnected phases. The first phase captures the immediate dynamics following MI, prior to any medical intervention. This stage is mathematically modeled using the system of ordinary differential equations: The second and third stages of the remodeling process account for the system dynamics of medical treatments, including oxygen restoration and subsequent stem cell injection at the injury site. Results: We simulate heart tissue and immune cell dynamics over 30 days for mild and severe MI using the novel mathematical model under medical treatment. The treatment involves no intervention until 2 h post-MI, followed by oxygen restoration and stem cell injection at day 7, which is shown experimentallyand numerically to be optimal. The simulation incorporates a baseline ROS threshold (Rc) where subcritical ROS levels do not cause cell damage. Conclusion: This study presents a novel mathematical model that extends a previously published framework by incorporating three clinically relevant parameters: oxygen restoration rate (ω), patient risk factors (γ), and neutrophil recruitment profile (δ). The model accounts for post-MI inflammatory dynamics, ROS-mediated ischemia-reperfusion (IR) injury, cardiac remodeling, and stem cell therapy. The model’s sensitivity highlights critical clinical insights: while oxygen restoration is vital, excessive rates may exacerbate ROS-driven IR injury. Additionally, heightened patient risk factors (e.g., smoking, obesity) and immunodeficiency significantly impact tissue damage and recovery. This predictive tool offers valuable insights into MI pathology and aids in optimizing treatment strategies to mitigate IR injury and improve post-MI outcomes. 

Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems. 

This work presents a new framework for a competitive evolutionary game between monoclonal antibodies and signalling pathways in oesophageal cancer. The framework is based on a novel dynamical model that takes into account the dynamic progression of signalling pathways, resistance mechanisms and monoclonal antibody therapies. This game involves a scenario in which signalling pathways and monoclonal antibodies are the players competing against each other, where monoclonal antibodies use Brentuximab and Pembrolizumab dosages as strategies to counter the evolutionary resistance strategy implemented by the signalling pathways. Their interactions are described by the dynamical model, which serves as the game’s playground. The analysis and computation of two game-theoretic strategies, Stackelberg and Nash equilibria, are conducted within this framework to ascertain the most favourable outcome for the patient. By comparing Stackelberg equilibria with Nash equilibria, numerical experiments show that the Stackelberg equilibria are superior for treating signalling pathways and are critical for the success of monoclonal antibodies in improving oesophageal cancer patient outcomes.