next up previous
Next: Lab Techniques for the Up: The In-Class Mega-Model - Previous: Modelling Glucose Transport and

A Useful Simplification

This model, unfortunately, is plenty complicated, and involves two quantities which are very difficult to measure ($g_i$ and $a$). One avenue of simplification is to assume that certain effects are not going to be important (for example, if one assumes that osmotic leakage is small, then essentially $\gamma = 0$). An additional simplifcation arises from the fact that equations (6 - 8) form a compartment model if $\omega = 0$, or alcohol inhibition is neglected. That is, at some level everything is accounted for, nothing lost, and therefore the derivative of some summed up quantity must be zero. To see this, set $\omega = 0$ and add up (6 - 8) in the following way:

\begin{eqnarray*}
\dot{g}_o \quad & \! = \! & -\alpha  g_o y + \gamma  ( g_i ...
... + \frac{\beta}{\epsilon} y \right) & \! = \! &
\hspace{1in} 0 ,
\end{eqnarray*}



and consequently we deduce

\begin{displaymath}
\frac{d}{dt} \left(g_o + g_i + \frac{\beta}{\epsilon} y \right) = 0,
\end{displaymath}

or

\begin{displaymath}
g_o + g_i + \frac{\beta}{\epsilon} y = C\mbox{onstant}.
\end{displaymath}

This allows us to eliminate one of the unmeasurable variables, $g_i$, through the relation

\begin{displaymath}
g_i = C -g_o -\frac{\beta}{\epsilon} y.
\end{displaymath}

This results in coupled equations for $y$ and $g_o$ when the above expression for $g_i$ is substituted into (6) and (8):
$\displaystyle \dot{g}_o$ $\textstyle = \!$ $\displaystyle -\alpha  g_o y + \gamma  y  \left[
C - 2 g_o -\frac{\beta}{\epsilon} y\right],$ (10)
$\displaystyle \dot{y}$ $\textstyle = \!$ $\displaystyle \epsilon  y  \left[ C -g_o
-\frac{\beta}{\epsilon} y\right] .$ (11)


next up previous
Next: Lab Techniques for the Up: The In-Class Mega-Model - Previous: Modelling Glucose Transport and
James Powell
2000-07-31