Single Compartment Model

Figure 4.1.The single compartment model

Figure 4.2.Graphical illustration of the quantity of tracer, q versus time for relatively high and low values of the fractional turnover, k.

Figure 4.1 shows a single compartment model that the flow of a tracer through a blood vessel follows an ideal bolus injection. The compartment in the system is closed except for the inflow and outflow of the trace, and the tracer is injected as illustrated. In this theory, the tracer will mix immediately and uniformly throughout the compartment according to its injection. And its quantity will reduce with time depending on the rate of outflow. The variables used in the system are:

If we define the fractional turnover, k, as the ratio of these two parameters, i.e.

${\displaystyle k={\frac {-dq/dt}{q}}}$

which can be rewritten as:

${\displaystyle {\frac {dq}{dt}}=-kq}$

And the solution to this equation is:

${\displaystyle q=q_{0}\ exp(-kt)\,\!}$

where q_o is the quantity of tracer present at time, t = 0. This equation is plotted in Figure 4.2.

Two Compartment Model-a closed system

Figure 4.3 Closed two compartment model

Figure 4.4 The change in the quantity of tracer in Compartments #1 and #2 versus time in a closed compartment model.

In a closed system the tracer simply moves between the two compartments without any overall loss or gain as shown in Figure 4.3. Therefore,

${\displaystyle {\frac {dq_{1}}{dt}}=k_{21}q_{2}-k_{12}q_{1}}$ and ${\displaystyle {\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{21}q_{2}}$.

Since there is no loss of tracer from the system,

${\displaystyle q_{1}+q_{2}={\text{constant}}=q_{0}\,\!}$

Therefore,

${\displaystyle {\frac {dq_{1}}{dt}}=-{\frac {dq_{2}}{dt}}\,,}$

As the quantity of tracer in Compartment #1 decreases, the quantity in Compartment #2 increases, and vice versa. And when the tracer is injected into Compartment #1 at time, t = 0,

${\displaystyle q_{1}=q_{0}\,\!}$

and

${\displaystyle q_{2}=0\,,\,\!}$

So, at the initial stage

${\displaystyle {\frac {dq_{1}}{dt}}=-k_{12}q_{0}}$

and

${\displaystyle {\frac {dq_{2}}{dt}}=k_{12}q_{0}}$

The solutions to this system are:

${\displaystyle q_{1}=q_{0}\left\lbrack 1-{\frac {k_{12}}{k_{12}+k_{21}}}\left\lbrace 1-\ {\text{exp}}\ -(k_{12}+k_{21})t\right\rbrace \right\rbrack }$

and

${\displaystyle q_{2}=q_{0}\left\lbrack {\frac {k_{12}}{k_{12}+k_{21}}}\left\lbrace 1-\ {\text{exp}}\ -(k_{12}+k_{21})t\right\rbrace \right\rbrack }$

Figure 4.4 shows one of these system if the volume of the compartment are the same.

Two Compartment Model - Open Catenary System

Two compartments connected like chain and the last compartment have a sink as shown in Figure 4.5.

Figure 4.5 Open catenary two compartment model.

Figure 4.6 The quantity of tracer versus time in the open catenary two compartment model.

In this model,

${\displaystyle {\frac {dq_{1}}{dt}}=-k_{12}q_{1}}$ and ${\displaystyle {\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{20}q_{2}}$

The solutions to these equations are:

${\displaystyle q_{1}=q_{0}\ {\text{exp}}(-k_{12}t)}$

and

${\displaystyle q_{2}=q_{0}{\frac {k_{12}}{k_{12}-k_{20}}}\lbrack {\text{exp}}(-k_{20}t)-{\text{exp}}(-k_{12}t)\rbrack \,,}$

and the behaviour of q₁ and q₂ is shown in Figure 4.6.

Two compartment Model- Open Mammillary System

The central compartment have a sink which is not related to the other compartment. Although these compartments do not necessarily have a physiological significance, common designations are:

• Comp 1 (central) - blood and well perfused organs, e.g. liver, kidney, etc.; "plasma"
• Comp 2 (peripheral) - poorly perfused tissues, e.g. muscle, lean tissue, fat; "tissue"

Figure 4.7 Open mamillary two compartment model.

Figure 4.8 The quantity of tracer versus time in the open mamillary two compartment model.

In this case,

${\displaystyle {\frac {dq_{1}}{dt}}=-k_{10}q_{1}-k_{12}q_{1}+k_{21}q_{2}}$ and ${\displaystyle {\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{21}q_{2}}$

At t = 0:

${\displaystyle q_{1}=q_{0}\,\!}$ and ${\displaystyle q_{2}=0\,\!}$

So,the solutions are:

${\displaystyle q_{1}=q_{0}\left\lbrack {\frac {k_{21}-a_{1}}{a_{2}-a_{1}}}{\text{exp}}(-a_{1}t)+{\frac {k_{21}-a_{2}}{a_{1}-a_{2}}}{\text{exp}}(-a_{2}t)\right\rbrack }$ and

${\displaystyle q_{2}={\frac {q_{0}k_{12}}{a_{2}-a_{1}}}\lbrack {\text{exp}}(-a_{1}t)-{\text{exp}}(-a_{2}t)\rbrack }$