ADD derivations to binding kinetics stuff

tex/thesis.tex

@@ -4659,7 +4659,91 @@ hosted using \gls{aws} using their proprietary Aurora implementation.
 
 The code is available here: \url{https://github.gatech.edu/ndwarshuis3/mdma}.
 
-\chapter{BINDING KINETICS CODE}\label{sec:appendix_binding}
+\chapter{BINDING KINETICS}\label{sec:appendix_binding}
+
+\newcommand{\lig}{\textit{ligand}}
+\newcommand{\rcp}{\textit{receptor}}
+\newcommand{\ligs}{\textit{ligands}}
+\newcommand{\rcps}{\textit{receptors}}
+
+To model binding kinetics of either \gls{stp} or \glspl{mab} (here called
+\ligs{}), each microcarrier was assumed to be a porous sphere with a given
+number of binding sites for the \ligs{} (here called \rcps{}). The \rcp{}/\lig{}
+reaction was assumed to be instantaneous (which is reasonable given that these
+are reactions between biotin and \gls{stp} which are extremely strong). From
+this, we further assumed a spherical interface within each microcarrier and
+aligned at the center wherein all \rcps{} in the interior were unbound and all
+on the exterior were bound. At $t=0$ this interface was assumed to start with a
+radius equal to that of the microcarrier, and shrunk down to radius of zero as
+\ligs{} flowed into the porous microcarriers and bound. We assumed the
+concentration of \lig{} to be zero at the interface and equal to the bulk
+concentration at the exterior surface of the microcarrier. Furthermore, we
+assumed that the interface move slowly relative to the diffusion of \lig{} into
+the microcarriers, and thus we used a quasi-steady-state model to avoid solving
+a boundary value problem with two movable boundaries (the interface radius and
+the concentration in bulk).
+
+The concentration profile of \lig{} in the microcarriers is given by Fick's
+Second Law in spherical coordinates assuming only radial flux and steady state.
+This with the boundary conditions as stated is:
+
+\begin{equation}
+  \label{eqn:binding_ficks}
+  0 = \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dC_L}{dr}\right)
+\end{equation}
+
+\begin{equation}
+  \label{eqn:binding_bc_left}
+  C_L\rvert_{r_i} = 0
+\end{equation}
+
+\begin{equation}
+  \label{eqn:binding_bc_right}
+  C_L\rvert_R = C_{L,b}
+\end{equation}
+
+Solving \cref{eqn:binding_ficks} we find the a relation for the concentration
+profile in terms of the interfacial radius:
+
+\begin{equation}
+  \label{eqn:binding_conc}
+  C_L = \frac{C_{L,b}}{r(1/r_i - 1/R)}\left(\frac{1}{r_i} - \frac{1}{r}\right)
+\end{equation}
+
+Solving \cref{eqn:binding_conc} for flux, the molar flow rate into the
+microcarriers is given by:
+
+\begin{equation}
+  \label{eqn:binding_molar_flow}
+  Q = 4\pi R^2N\rvert_R = \frac{-4\pi DC}{1/r_i - 1/R}
+\end{equation}
+
+Using the quasi-steady-state assumption, we can now find time-dependent
+equations for the interfatial radius and the bulk concentration. The interfacial
+volume in terms of molar flow rate is given by:
+
+\begin{equation}
+  \label{eqn:binding_volume_change}
+  C_{R,0}\frac{dV_i}{dt} = -Q
+\end{equation}
+
+Substituting volume of a sphere and applying the chain rule:
+
+\begin{equation}
+  \label{eqn:radial_radial_change}
+  \frac{dr_i}{dt} = \frac{-Q}{4\pi r_i^2C_{R,0}}
+\end{equation}
+
+The change in bulk concentration is simply given by:
+
+\begin{equation}
+  \label{eqn:radial_conc_change}
+  \frac{dC_{L,b}}{dt} = \frac{-nQ}{V}
+\end{equation}
+
+Substituting \cref{eqn:binding_molar_flow} into \cref{eqn:radial_radial_change}
+and \cref{eqn:radial_conc_change} yields \cref{eqn:stp_diffusion_1} and
+\cref{eqn:stp_diffusion_2}.
 
 The \gls{stp} binding kinetic profile was fit and calculated using the following
 MATLAB code. Note that the \inlinecode{geometry} parameter was varied so as to