Chapter 6.1, Problem 13P

An initial amount $\alpha$ of tracer (such as a dye or a radioactive isotope) is injected into Compartment $1$ of the two - compartment system shown in Figure $\mathrm{6.1.10}$. At time $t>0$, Let $x_1(t)$ and $x_2(t)$ denote the amount of tracer in Compartment $1$ and Compartment $2$, respectively. Thus under the condition stated $x_1(0)=\alpha$ and $x_2(0)=0$. The amounts are related to the corresponding concentrations ${\rho }_1(t)$ and ${\rho }_2(t)$ by the equations

$x_1={\rho }_1{V}_1$ and $x_2={\rho }_2{V}_2$ $(i)$

Where $V_1$ and $V_2$ are the constant respective volumes of the compartments.

A closed two – compartments system.

The differential equations that describe the exchanges of tracer between the compartments are

$\frac{d{x}_1}{dt}=-k_{21}{\rho }_1+k_{12}{\rho }_2$ $(ii)$

$\frac{d{x}_2}{dt}=k_{21}{\rho }_1-k_{12}{\rho }_2$,

Or, using the relations in $(i)$

$\frac{d{x}_1}{dt}=-L_{21}{x}_1+L_{12}{x}_2$ $(iii)$

$\frac{d{x}_2}{dt}=L_{21}{x}_1-L_{12}{x}_2$,

Where $L_{21}=k_{21}/V_1$ the fractional turnover rate of compartment $1$ with respect to $2$ and $L_{12}=k_{12}/V_2$ is fractional turnover rate Compartment $2$ with respect $1$.