Mathematical Modeling of Oxygen Diffusion from Capillary to Tissues during Hypoxia through Multiple Points Using Fractional Balance Equations with Memory.

Srivastava, Vineet; Tripathi, Dharmendra; Srivastava, P K; Kuharat, Sireetorn; Bég, O Anwar

Abstract

The diffusion of oxygen through capillary to surrounding tissues through multiple points along the length has been addressed in many clinical studies, largely motivated by disorders including hypoxia. However relatively few analytical or numerical studies have been communicated. In this paper, as a compliment to physiological investigations, a novel mathematical model is developed which incorporates the multiple point diffusion of oxygen from different locations in the capillary to tissues, in the form of a fractional dynamical system of equations using the concept of system of balance equations with memory. Stability analysis of the model has been conducted using the well known Routh-Hurwitz stability criterion.Comprehensive analytical solutions for the differntial equation problem in the new proposed model are obtained using Henkel transformations. Both spatial and temporal variation of concentration of oxygen is visualized graphically for different control parameters. Close correlation with simpler models is achieved. Diffusion is shown to arise from different points of the capillary in decreasing order along the length of the capillary i.e. for the different values of z. The concentration magnitudes at low capillary length far exceed those further along the capillary. Furthermore with progressive distance along the capillary, the radial distance of diffusion decreases, such that oxygen diffuses only effectively in very close proximity to tissues. The simulations provide a useful benchmark for more generalized mass diffusion computations with commercial finite element and finite volume software including ANSYS FLUENT.

Citation

Srivastava, V., Tripathi, D., Srivastava, P. K., Kuharat, S., & Bég, O. A. (2024). Mathematical Modeling of Oxygen Diffusion from Capillary to Tissues during Hypoxia through Multiple Points Using Fractional Balance Equations with Memory. Critical Reviews in Biomedical Engineering, 52(6), 1-13. https://doi.org/10.1615/CritRevBiomedEng.2024053461

• Journal Article Type: Article
• Acceptance Date: May 23, 2024
• Online Publication Date: May 31, 2024
• Publication Date: Aug 16, 2024
• Deposit Date: May 24, 2024
• Publicly Available Date: Jun 1, 2025
• Journal: Critical reviews in biomedical engineering
• Print ISSN: 0278-940X
• Publisher: Begell House
• Peer Reviewed: Peer Reviewed
• Volume: 52
• Issue: 6
• Pages: 1-13
• DOI: https://doi.org/10.1615/CritRevBiomedEng.2024053461
• Keywords: Fractional Diffusion Equation; Routh-Hurwitz stability criterion; Hypoxia; capillary; Henkel Transformation; mathematical biology