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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 ( R c ) 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Nonstandard finite difference methods (NSFD) are used in physical sciences to approximate solutions of ordinary differential equations whose analytical solution cannot be computed. Traditional NSFD methods are elementary stable but usually only have first order accuracy. In this paper, we introduce two new classes of numerical methods that are of second order accuracy and elementary stable. The methods are modified versions of the nonstandard two-stage explicit Runge-Kutta methods and the nonstandard one-stage theta methods with a specific form of the nonstandard denominator function. Theoretical analysis of the stability and accuracy of both modified NSFD methods is presented. Numerical simulations that concur with the theoretical findings are also presented, which demonstrate the computational advantages of the proposed new modified nonstandard finite difference methods. [ABSTRACT FROM AUTHOR]

A mathematical model is presented to study the regulatory effects of growth factors in osteoblastogenesis. The model incorporates the interactions among mesenchymal stem cells, osteoblasts, and growth factors. The resulting system of nonlinear ordinary differential equations is studied analytically and numerically. Mathematical conditions for successful osteogenic differentiation and optimal osteoblasts population are formulated, which can be used in practice to accelerate bone formation. Numerical simulations are also presented to support the theoretical results and to explore different medical interventions to enhance osteoblastogenesis. [ABSTRACT FROM AUTHOR]

We study the phenomenon of slow convergence in families of discrete dynamical systems where the iteration function has a Puiseux series representation. Such occurrence consists in the slow convergence of orbits near non-hyperbolic parametric periodic points. We provide a precise new definition of the slowness of convergence which is based on literature results for the critical exponents associated with parametric periodic points. Such exponents establish a general classification for slow systems and provide a measure of rates of convergence. For dynamical systems whose iteration functions have Taylor series expansions, the new definition is natural with wider applicability. However, it can be also used for iteration functions where a more sophisticated approach, such as a Lagrange expansion, is needed. In addition, we show that even for such iteration functions, the critical exponent can be easily computed. The presented theoretical results are illustrated by numerical examples having different rates of convergence. [ABSTRACT FROM AUTHOR]