The pharmacokinetics of nicotine can be approximated with the material balance:

$V\frac{dC}{dt}=k_{1}n-k_{2}C-k_{3}C$ when smoking; IC $C=C_0$ at $t=0$

$V\frac{dC}{dt}=-k_{2}C-k_{3}C$ when done smoking; with IC $C=C_0$ at $t=0$

where

$\backslash$table$[[V,40,1,$volume of distribution of nicotine in the body$],[C,,ngm{l}^{-1},$plasma concentration of nicotine$],[n,,,$number of cigarettes smoked per hour$],[k_1,800,000,ng,$amount of nicotine absorbed per cigarette$],[k_2,25,1h^{-1},$rate coefficient for hepatic metabolism$],[k_3,11,1h^{-1},$rate coefficient for renal clearance$],[t,,h,$time$]]$

$($a$)$ Label each term in the first balance with the letter$($s$)$ corresponding to its significance. Some terms may have more than one correct answer, in that case give both appropriate answers.

A$.$ Accumulation

B$.$ Flow in

C$.$ Flow out

D$.$ Generation

E$.$ Consumption

F$.$ Growth

G$.$ Metabolism

H$.$ Recycle

I. Mass transfer

J$.$ Excretion

$1$

$($b$)$ Circle each of the following that are true about this model

A$.$ Assumes the volume $V$ is well$-$mixed $($instantaneous distribution$)$

B$.$ Assumes zero$-$order kinetics of nicotine metabolism in the liver

C$.$ Assumes steady$-$state

D$.$ Assumes no loss of nicotine through exhaled breath

E$.$ Assumes gas$-$liquid equilibrium partitioning of nicotine in the lung

F$.$ Assumes first$-$order kinetics of nicotine elimination via the urine

G$.$ Assumes no generation of nicotine in the body

$($c$)$ What is the half$-$life of nicotine in the body after smoking stops?

$($d$)$ Suppose it is desired to design a transdermal anti$-$smoking aid patch that will deliver nicotine continuously to the body through the skin. The target blood concentration of nicotine is $12ngm{l}^{-1}.$ What is the required nicotine delivery rate $($in $ng{h}^{-1})$ to achieve this steady concentration? TIP: replace the term in the material balance that is associated with delivery of nicotine from cigarettes with a term for delivery from the patch.

$($e$)$ The solution to the first differential equation given above is:

$C=\frac{k_{1}n}{k_2+k_3}(1-e^{\frac{-(k_2+k_3)t}{V}})+C_0{e}^{-\frac{-(k_2+k_3)t}{V}}$

How long does it take after putting the patch on to reach $10ngm{l}^{-1}?$ Assume there is no nicotine in the blood initially.