Red blood cells (RBCs) exhibit an interesting response to hydrodynamic flow, releasing adenosine triphosphate (ATP). Subsequently, these liberated ATP molecules initiate a crucial interaction with endothelial cells (ECs), thereby setting off a cascade involving the release of calcium ions (Ca2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{2+}$$\\end{document}). Ca2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{2+}$$\\end{document} exerts control over a plethora of cellular functions, and acts as a mediator for dilation and contraction of blood vessel walls. This study focuses on the relationship between RBC dynamics and Ca2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{2+}$$\\end{document} dynamics, based on numerical simulations under Poiseuille flow within a linear two-dimensional channel. It is found that the concentration of ATP depends upon a variety of factors, including RBC density, channel width, and the vigor of the flow. The results of our investigation reveals several features. Firstly, the peak amplitude of Ca2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{2+}$$\\end{document} per EC escalates in direct proportion to the augmentation of RBC concentration. Secondly, increasing the flow strength induces a reduction in the time taken to reach the peak of Ca2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{2+}$$\\end{document} concentration, under the condition of a constant channel width. Additionally, when flow strength remains constant, an increase in channel width corresponds to an elevation in calcium peak amplitude, coupled with a decrease in peak time. This implies that Ca2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{2+}$$\\end{document} signals should transition from relatively unconstrained channels to more confined pathways within real vascular networks. This notion gains support from our examination of calcium propagation in a linear channel. In this scenario, the localized Ca2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{2+}$$\\end{document} release initiates a propagating wave that gradually encompasses the entire channel. Notably, our computed propagation speed agrees with observations.
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