ADD equations for washing steps

This commit is contained in:
Nathan Dwarshuis 2021-08-03 19:53:51 -04:00
parent 75b1b9e588
commit 06431bd255
2 changed files with 65 additions and 17 deletions

View File

@ -124,11 +124,11 @@ diff = 1000000; % init to some huge number
while diff > tolerance
t = linspace(t_0, t_f, 50);
Y = pdepe(m,@pde,@init,@bound,r,t);
C = Y(:,:,1);
Y = pdepe(m, @pde, @init, @bound, r, t);
C = Y(:, :, 1);
% final concentration in center of carrier at final time
C_f_center = C(end,1);
C_f_center = C(end, 1);
% test to see how far off the center is from bulk
diff = C_f_center / C_Lbf - 1;
@ -166,13 +166,13 @@ function u0 = init(r)
u0 = C_Lc_pw;
end
function [pl,ql,pr,qr] = bound(rl,cl,rr,cr,t)
function [pl, ql, pr, qr] = bound(rl, cl, rr, cr, t)
pl = 0;
ql = 1;
% 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

View File

@ -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
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).
To model the washing of the microcarriers, they once again were assumed to be
porous spheres filled with whatever amount of reagent was left unbound from the
previous step (which was assumed to be equal to concentration in the
supernatent). The diffusion out of the microcarriers is given by the following
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