Home

Services

About Us

Projects

Contact Us

Disclaimer

Example: A two compartment vascular/tissue model for drug concentration

Please allow scripts to run on this page to format the mathematical notation.

For this example, an optimal  drug profile for the vascular compartment of a two compartment  vascular/tissue model is desired. The differential equations for a two compartment vascular/tissue model of drug metabolism with exchanges between the two compartments and loss from the vascular compartment are
$$\frac{dC_1}{dt} = -k_{12}C_1(t) + k_{21} C_2(t) -k_{10}C_1(t)$$
$$\frac{dC_2}{dt} = k_{12}C_1(t) - k_{21} C_2(t) $$
with rate constants $k_{ij}>0$ and the initial conditions $C_1(0) = C_{10}$ and  $C_2(0) = C_{20}$. The concentration $C_1$ represents the vascular drug compartment and the concentration $C_2$ represents the tissue drug compartment, with $k_{12}$ representing the rate of exchange from compartment $C_1$ to compartment $C_2$, $k_{21}$ representing the  rate of exchange from compartment $C_2$ to compartment $C_1$, and $k_{10}$ representing the rate of  loss from the compartment $C_1$.

	The two compartment model for drug concentration over time from an initial dose of one unit of a drug.
	The desired drug input profile for the vascular compartment. This profile achieves a vascular drug concentration of 20 units over a 12 hour time period, maintaining the 20 unit concentration for the next 36 hours, after which the drug input is stopped.
	The computed drug input profile with the desired profile over a 72 hour time period. The computed profile matches the desired profile over the first 48 hours, after which the drug input was halted and the vascular and tissue drug concentrations decrease.
	The constant drug input rates for each one hour time interval over the 48 hour time period for the desired profile.
	The drug concentration deviations for the computed profile relative to the desired drug profile. The deviations were computed as the desired profile minus the computed profile. The deviations observed for the computed profile during the first 12 hours are due to the nonlinearity of the drug concentration caused by the loss, exchanges between the compartments, and the relatively steep slope of the drug input profile. After 12 hours the deviations are due to the corrections necessary to achieve a constant vascular drug concentration after a rapid ramp up. A shallower slope and a longer ramp-up period for the drug concentration would reduce these deviations.
	Computed drug profile for a 48 hour ramp up time to a 20 unit vascular drug concentration and a 24 hour constant concentration.
	The constant drug input rates for each one hour time interval over the 72 hour time period for the 48 hour drug concentration ramp up  profile.
	The drug concentration deviations for the 48 hour drug concentration ramp up  profile. Again, there are nonzero deviations, but they are significantly smaller due to the long ramp-up time.