On Mathematical Analysis of Active Drug Transport Coupled With Flow-Induced Diffusion in Blood Vessels

Molecular Communications in Viral Infections Research: Modeling, Experimental Data, and Future Directions

“…However, if the virus crosses the endothelial cell, the virions are transported via intracellular trafficking and ejected from the basolateral (abluminal) membrane into the extracellular space. This leads to the following chemical kinetic model [55]:…”

“…These values do not take into account the dispersion in the blood and shall be considered in entering points to the blood vessels. This mechanism is modelled by a continuoustime Markov chain framework leading to the following general homogeneous boundary condition [55]:…”

“…The blood is assumed to have a laminar flow in the axial direction with uniform velocity profilev(r) = vâ z ms −1 , whereâ z is the axial unit vector. The fundamental characteristic function for advectiondiffusion called the concentration Green's function is analytically derived in terms of a convergent infinite series [55].…”

“…However, if the virus crosses the endothelial cell, the virions are transported via intracellular trafficking and ejected from the basolateral (abluminal) membrane into the extracellular space. This leads to the following chemical kinetic model [55] : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{align*} \frac {\partial V^{\left ({b}\right)}\left ({\bar {r},t}\right)}{\partial t}=&-\left ({c+k_{f}^{1}}\right)V^{\left ({b}\right)}\left ({\bar {r},t}\right) + k_{b}^{1}V^{\left ({b}\right)}_{\mathrm {BV}}\left ({\bar {r},t}\right) \tag{7}\\ \frac {\partial V^{\left ({b}\right)}_{\mathrm {BV}}\left ({\bar {r},t}\right)}{\partial t}=&k_{f}^{1} V^{\left ({b}\right)}\left ({\bar {r},t}\right) - \left ({k_{b}^{1}+k_{f}^{2}}\right)V^{\left ({b}\right)}_{\mathrm {BV}}\left ({\bar {r},t}\right) + \\&+\,\,k_{b}^{2}V^{\left ({b}\right)}_{\mathrm {EV}}\left ({\bar {r},t}\right) \tag{8}\\ \frac {\partial V^{\left ({b}\right)}_{\mathrm {EV}}\left ({\bar {r},t}\right)}{\partial t}=&k_{f}^{2}V^{\left ({b}\right)}_{\mathrm {BV}}\left ({\bar {r},t}\right) - k_{b}^{2}V^{\left ({b}\right)}_{\mathrm {EV}}\left ({\bar {r},t}\right) - \\&-\,\,\left ({k_{f}^{3}+k_{p}}\right)V^{\left ({b}\right)}_{\mathrm {EV}}\left ({\bar {r},t}\right) \tag{9}\\ \frac {\partial V^{\left ({b}\right)}_{\bar {\mathrm {V}}}\left ({\bar {r},t}\right)}{\partial t}=&k_{p} V^{\left ({b}\right)}_{\mathrm {EV}}\left ({\bar {r},t}\right)\tag{10}\\ \frac {\partial V^{\left ({b}\right)}_{\hat {\mathrm {V}}}\left ({\bar {r},t}\right)}{\partial t}=&k_{f}^{3} V^{\left ({b}\right)}_{\mathrm {EV}}\left ({\bar {r},t}\right),\tag{11}\end{align*} \end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$V^{(b)}$ \end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$V^{(b)}_{\mathrm {BV}}$ \end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$V^{(b)}_{\mathrm {EV}}$ \end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$V^{(b)}_{\bar {\mathrm {V}}}$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$V^{(b)}_{\hat {\mathrm {V}}}$ \end{document} represent the viral load at the extracellular space (luminal side), the viral load at the endothelial host cell membrane (host cell-bound virions), the viral load in the host cell endosomes, the viral load in the host cell cytosol that penetrated the endosomes, and the viral load at the extracellular space (abluminal side), at point …”

“…These values do not take into account the dispersion in the blood and shall be considered in entering points to the blood vessels. This mechanism is modelled by a continuous-time Markov chain framework leading to the following general homogeneous boundary condition [55] : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{align*}&D\left ({\frac {\partial ^{2}}{\partial t^{2}}+\left ({k_{b}^{1}+k_{f}^{2}+k'}\right)\frac {\partial }{\partial t} + k_{f}^{2}k_{f}^{3} + k_{f}^{2}k_{p} + k_{b}^{1}k'}\right) \\&\quad \,\, \times \,\, \nabla V^{\left ({b}\right)}\left ({\bar {r},t}\right)\cdot \hat {n} \\&\;= k_{f}^{1}\left ({\frac {\partial ^{2}}{\partial t^{2}} + \left ({k_{f}^{2}+k'}\right)\frac {\partial }{\partial t} + k_{f}^{2}k_{f}^{3} + k_{f}^{2}k_{p} }\right)V^{\left ({b}\right)}\left ({\bar {r},t}\right), \\ {}\tag{12}\end{align*} \end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${D}$ \end{document} is the diffusion coefficient in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\text{m}^{\textrm {2}}\text{s}^{-\textrm {1}}$ \end{document} of the virions in the blood, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$k' = k_{b}^{2} + k_{f}^{3} + k_{p}$ \end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\nabla $ \end{document} is the gradient operator, ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\cdot $ \end{document} ) is the inner multiplication operator, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\hat {n}$ \end{document} is the surface normal at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\bar {r} \in \partial \mathcal {D}$ \end{document} pointing towards the vessel luminal side. The virion advection-diffusion in the observed blood vessel can then be modelled by the Fick’s second law: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usep...…”