ADD equations for washing steps

code/microcarrier_diffusion_washing.m

@@ -172,7 +172,7 @@ function [pl,ql,pr,qr] = bound(rl,cl,rr,cr,t)
     % assume that the concentration boundary
     % is the average of the initial and
     % theoretical final concentration in bulk
-    pr = cr - (C_Lbf + C_Lbf)/2;                     
+    pr = cr - (C_Lb0 + C_Lbf) / 2;
     qr = 0;
 end
 

tex/thesis.tex

@@ -1566,24 +1566,25 @@ the center of the microcarriers was reached. We also assumed that the pores in
 the microcarriers were large enough that the interactions between the \gls{stp}
 and surfaces would be small, thus the apparent diffusivity could be represented
 as a fraction of the diffusion coefficient of \gls{stp} in water. This model was
-given by \cref{eqn:stp_diffision_1,eqn:stp_diffision_2}:
+given by \cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2}:
 
 % TODO actually derive these equations, eg state the initial conditions and
 % governing equation
 \begin{equation}
-  \label{eqn:stp_diffision_1}
+  \label{eqn:stp_diffusion_1}
   \frac{dr}{dt} = \frac{-D_{app}C}{Br(1-r/R)}
 \end{equation}
 
 \begin{equation}
-  \label{eqn:stp_diffision_2}
+  \label{eqn:stp_diffusion_2}
   \frac{dC}{dt} = \frac{-4 \pi n D_{app} C}{V(1/r-1/R)}
 \end{equation}
 
 \noindent where
 \begin{itemize}[label={}]
 \item $D_{app}$ is the apparent diffusion rate which is equal to $D_{STP}\beta$
-\item $D_{STP}$ the diffusion rate of \gls{stp} in water
+\item $D_{STP}$ the diffusion rate of \gls{stp} (or \glspl{mab} for later
+  calculations) in water
 \item $\beta$ a fractional parameter representing the tortuousity and void
   fraction of the microcarriers (here called the `geometric diffusivity')
 \item $r$ is the interfatial radius of the unbound biotin within a microcarrier
@@ -1602,11 +1603,60 @@ equations were then used analogously to describe the reaction profile of
 \glspl{mab} assuming a diffusion rate of
 \SI{4.8e-7}{\cm\squared\per\second}\cite{Sherwood1992}.
 
-These same coefficients
+To model the washing of the microcarriers, they once again were assumed to be
-were used in determining the kinetics of the washing steps, and
+porous spheres filled with whatever amount of reagent was left unbound from the
-\SI{5.0e-6}{\cm\squared\per\second}\cite{Niether2020} was used as the diffusion
+previous step (which was assumed to be equal to concentration in the
-coefficient for free biotin (which should be the only species left in solution
+supernatent). The diffusion out of the microcarriers is given by the following
-after all the \gls{snb} has hydrolyzed).
+partial differential equation and boundary conditions:
+
+\begin{equation}
+  \label{eqn:stp_washing}
+  \frac{\partial C_i}{\partial t} = \frac{1}{r^2}\frac{\partial}{\partial
+    r}\left(r^2 D_{app} \frac{\partial C_i}{\partial r}\right)
+\end{equation}
+
+\begin{equation}
+  \label{eqn:stp_washing_left_bc}
+  C_i(r, 0) = C_{i,0}
+\end{equation}
+
+\begin{equation}
+  \label{eqn:stp_washing_left_bc}
+  N_i(0, t) = 0
+\end{equation}
+
+\begin{equation}
+  \label{eqn:stp_washing_right_bc}
+  C_i(R, t) = (C_{b,0}+C_{b,\infty}) / 2
+\end{equation}
+
+\noindent where (in addition to the variables given already for
+\cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2})
+\begin{itemize}[label={}]
+\item $N_i$ is the radial flux of the species in question inside the
+  microcarriers
+\item $C_i$ is the concentration of the species in question inside the
+  microcarriers
+\item $C_{i,0}$ is the initial concentration of the species in question inside
+  the microcarriers (which is assumed to be the concentration in the bulk before
+  the wash volume is added)
+\item $C_{b,0}$ is the initial bulk concentration of the species in question
+  outside the microcarriers after the initial wash volume has been added
+\item $C_{b,\infty}$ is the final bulk concentration of the species in
+  question outside the microcarriers
+\end{itemize}
+
+Note that in order to avoid solving a moving boundary value problem, the
+concentration at the boundary of the microcarriers was fixed at the average of
+the final and initial concentration expected to be observed in bulk. This should
+be a reasonable assumption given that the volume inside the microcarriers is
+tiny compared to the amount of volume added in the wash, thus the boundary
+concentration should change little.
+
+The same diffusion coefficients were used in determining the kinetics of the
+washing steps, and \SI{5.0e-6}{\cm\squared\per\second}\cite{Niether2020} was
+used as the diffusion coefficient for free biotin (which should be the only
+species left in solution after all the \gls{snb} has hydrolyzed).
 
 All diffusion coefficients were taken to be valid at \gls{rt} and in \gls{di}
 water, which is a safe assumption given that our reaction medium was 1X
@@ -1615,8 +1665,6 @@ water, which is a safe assumption given that our reaction medium was 1X
 See \cref{sec:appendix_binding} and \cref{sec:appendix_washing} for the MATLAB
 code (and output in the case of the washing steps) used.
 
-% METHOD add the equation governing the washing steps
-
 \subsection{Luminex Analysis}\label{sec:luminex_analysis}
 
 Luminex was performed using a \product{ProcartaPlex kit}{\thermo}{custom} for
@@ -1910,7 +1958,7 @@ this experimental binding data to fit a continuous model for the \gls{stp}
 binding reaction. Using the diffusion rate of the \gls{stp}
 (\SI{6.2e-7}{\cm\squared\per\second}), we then calculated the geometric
 diffusivity of the microcarriers to be 0.190 (see
-\cref{eqn:stp_diffision_1,eqn:stp_diffision_2}).
+\cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2}).
 
 % RESULT state how I calculated the number of mab/surface area
 Using this effective diffusivity and the known diffusion coefficient of a