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BioTransport. BIOEN $4001.$ Spring $2024$

Homework $1.$ Due Jan. $23.$

Pharmacokinetics. Intramuscular injection.

The term pharmacokinetics refers to movement of a pharmaceutical within the body and the time$-$course of delivery to the target tissue. The diagram below is a simple $3$ compartment model for drug delivery to a target tissue following intramuscular $($IM$)$ injection. After a bolus IM injection $($compartment $2),$ the drug moves to the central blood compartment over time $($compartment $1).$ Some of the drug is delivered from the blood to the target tissue $($compartment $3)$ and part is eliminated by the liver and kidneys. Once in the target tissue, some of the drug is consumed through binding and metabolism and some returns to the blood.

Fig. $1.$ Three compartment pharmacokinetic model of IM drug injection.

a$)$ Based on the $3-$compartment model in Fig. $1,$ derive differential equations describing how the concentrations in the blood and in the tissue change over time and write your equations in matrix form

$\frac{d(vec(C))}{dt}+$Mvec$(C)=$vec$(F)$

Give explicit expressions for the matrix $M$ and vector vec$(F)$ in terms of mass transfer coefficients and other parameters in the diagram.

b$)$ What is the mass transfer matrix for the case when $V_1=1,V_2=\frac{V_1}{15},V_3=\frac{V_1}{20}$ with units $m^3$ and mass transfer coefficients $k_{13}=0\mathrm{.}3,k_{31}=0\mathrm{.}006,k_{12}=1k_{21}=0\mathrm{.}1$;$k_m=0\mathrm{.}005,k_e=0\mathrm{.}001$ with units $m^{3}h{r}^{-1}?$ Use a numerical differential equation solver to find the concentrations predicted by Eq$.1$ as functions of time for a bolus injection in the muscle using concentration just after the injection of $C_2(0)=)=(0.$ Plot the concentration vs$.$ time for all $3$ compartments over a $1$ day period of time.