P6-8, Fogler Tarzlon is a liquid antibiotic that is taken orally to treat ...

P6-8, FoglerTarzlon is a liquid antibiotic that is taken orally to treat infections of the spleen. It is effective only ifit can maintain a concentration in the blood-stream (based on volume of body fluid) above 0.4mg/dm 3 of body fluid. Ideally, a concentration of 1.0 mg/dm 3 in the blood would like to be realized.However, if the concentration in the blood exceeds 1.5 mg/dm 3 , harmful side effects can occur.Once the Tarzlon reaches the stomach it can proceed in two pathways, both of which are firstorder:(1) It can be absorbed into the blood stream through the stomach walls; (2) it can pass outthrough the gastrointestinal tract and not be absorbed into the blood. Both these processes are firstorder in Tarzlon concentration in the stomach. Once in the bloodstream, Tarzlon attacks bacterialcells and is subsequently degraded by a zero-order process Tarzlon can also be removed from theblood and excreted in the urine through a first-order process within the kidneys. In the stomach:Adsorption into blood k 1= 0.15 h -1Elimination through gastrointestine k 2= 0.6 h -1In the bloodstream:Degradation of Tarzlon k 3= 0.1 mg/dm 3 -hElimination through urine k 4= 0.2 h -1(a) Plot the concentration of Tarzlon in the blood as a function of time when 1 dose (i.e., one liquidcapsule) of Tarzlon is taken.(b) How should the Tarzlon be administered (dosage and frequency) over a 48-h period to be mosteffective?(c) Comment on the dose concentrations and potential hazards.(d) How would your answers change if the drug were taken on a full or empty stomach?One dose of Tarzlon is 250 mg in liquid form: Volume of body fluid = 40 dm 3 .SOLUTIONIn solving this problem we need to recognize a couple of facts. First we will need to draw adistinction between Tarzlon in the bloodstream and Tarzlon in other locations (where it is noteffective). Let A = Tarzlon in stomach, B = Tarzlon in the bloodstream, C = Tarzlon in thegastrointestine, D = Tarzlon degraded by attack on bacterial cells and E = Tarzlon in the urine. Thereaction scheme then looks likeA -----> B ------> D\ \\--> C \->EWe are interested in computing C B. The second thing to note is that when dealing with first ordersystems of constant volume (we'll assume constant volume for this since it is a liquid system) thevolume will cancel out of the equation. For example consider the balance on Adn A /dt = (-k 1 C A - k 2 C A )V = -(k 1 +k 2 )(n A /V)V = -(k 1 +k 2 )n AWe could also multiply this equation by the molecular weight of Tarzlon to convert the molar units tomass units, i.e.,dm A /dt = -(k 1 +k 2 )m ASimilarly for B we getdn B /dt = (k 1 C A - k 3 /MW A - k 4 C B )V

where the MW A term is necessary since k 3 is given in mass units. Multiplying by V and MW A givesdm B /dt = k 1 m A - k 3 V - k 4 m BThe other balances aredm C /dt = k 1 m Adm D /dt = k 3 Vdm E /dt = k 4 m BNow define the constants, set the initial values and integrate this set of five equations.(a)k 1 := 0.15k 2 := 0.6k 3 := 0.1k 4 := 0.2m Ao := 250V := 40y :=⎛⎜⎜⎜⎜⎜⎝D( t,y)m Ao0000:=⎞⎟⎟⎟⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣k 1 ⋅y 0( )− k 1 + k 2 ⋅y 0− k 3 ⋅V− k 4 ⋅y 1k 2 ⋅y 0k 3 ⋅Vk 4 ⋅y 1⎤⎥⎥⎥⎥⎥⎥⎦

Z := Rkadapt( y, 0, 10, 100,D)To convert the mass of Tarzlon in the bloodstream (m B ) to concentration divide by the volume ofthe body fluid.ii := 0..100Z ( ii,2)Z ( ii,6):=VNow plot the results for C B .0.6Tarzlon Blood Concentration Profile0.4Tarzlon Concentration (mg/dm^3)0.200.20.40 2 4 6 8 10Time (hours)b) From the results for a single dose we see that the Tarzlon blood concentration peaks at about0.598 mg/dm 3 and drops below 0.4 mg/dm 3 after 4 hours. What we need to do is to increase theinitial dose to bring the peak concentration to 1.5 mg/dm 3 . Let's try twice the dose.m Ao := 500

y :=⎛⎜⎜⎜⎜⎜⎝D( t,y)m Ao0000:=⎞⎟⎟⎟⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣k 1 ⋅y 0( )− k 1 + k 2 ⋅y 0− k 3 ⋅V− k 4 ⋅y 1k 2 ⋅y 0k 3 ⋅Vk 4 ⋅y 1⎤⎥⎥⎥⎥⎥⎥⎦Z := Rkadapt( y, 0, 10, 100,D)Z ( ii,2)Z ( ii,6):=VNow plot the results for C B .1.5Tarzlon Concentration ProfileTarzlon Concentration (mg/dm^3)10.500 2 4 6 8 10Time (hours)The peak concentration is 1.363 mg/dm 3 . So for a 250 mg dose the peak concentration is 0.598mg/dm 3 and for an initial dose of 500 mg the peak concentration is 1.363 mg/dm 3 . Use a linearinterpolation to get an initial dose of 544 mg.m Ao := 544

y :=⎛⎜⎜⎜⎜⎜⎝D( t,y)m Ao0000:=⎞⎟⎟⎟⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣k 1 ⋅y 0−( k 1 + k 2 ) ⋅y 0− k 3 ⋅V− k 4 ⋅y 1k 2 ⋅y 0k 3 ⋅Vk 4 ⋅y 1⎤⎥⎥⎥⎥⎥⎥⎦Z := Rkadapt( y, 0, 20, 200,D)ii := 0..200Z ( ii,2)Z ( ii,6):=VNow plot the results for C B .1.5Tarzlon Concentration ProfileTarzlon Concentration (mg/dm^3)10.500.50 5 10 15 20 25Time (hours)

The peak concentration of Tarzlon is now 1.498 mg/dm 3 . The Tarzlon concentration does not dropbelow 0.4 mg/dm 3 until after 7.6 hours. So let's take the initial dose and wait 7.5 hours beforetaking a second. If the second dose is larger than 250 mg but less than 544 mg we should be ableto take the peak concentration back up to 1.5 mg/dm 3 . After some trials, with a second dose of 395mg we get the results shown below.Z := Rkadapt( y, 0, 7.5, 100,D)ii := 0..100Z ( ii,2)Z ( ii,6):=VZ ( 100,6)= 0.426second := 450m A1 := Z ( 100,1)+ secondy :=⎡⎢⎢⎢⎢⎢⎢⎣m A1Z ( 100,2)Z ( 100,3)Z ( 100,4)Z ( 100,5)⎤⎥⎥⎥⎥⎥⎥⎦D( t,y):=⎡⎢⎢⎢⎢⎢⎢⎣k 1 ⋅y 0( )− k 1 + k 2 ⋅y 0− k 3 ⋅V− k 4 ⋅y 1k 2 ⋅y 0k 3 ⋅Vk 4 ⋅y 1⎤⎥⎥⎥⎥⎥⎥⎦Z1 := Rkadapt( y, 7.5, 15, 100,D)ii := 0..100Z1 ( ii,2)Z1 ( ii,6):=jj := 0..6VZZ ( ii,jj):= Z ( ii,jj)

ZZ ( ii+ 100,jj):= Z1 ( ii,jj)ij := 0..2001.5Tarzlon Concentration ProfileTarzlon Concentration (mg/dm^3)10.500 2 4 6 8 10 12 14 16Time (hours)Let's try a third dose of 450 mg.m A2 := Z1 ( 100,1)+ secondy :=⎡⎢⎢⎢⎢⎢⎢⎣m A2Z1 ( 100,2)Z1 ( 100,3)Z1 ( 100,4)Z1 ( 100,5)⎤⎥⎥⎥⎥⎥⎥⎦D( t,y):=⎡⎢⎢⎢⎢⎢⎢⎣k 1 ⋅y 0−( k 1 + k 2 ) ⋅y 0− k 3 ⋅V− k 4 ⋅y 1k 2 ⋅y 0k 3 ⋅Vk 4 ⋅y 1⎤⎥⎥⎥⎥⎥⎥⎦Z2 := Rkadapt( y, 15, 22.5, 100,D)

Z2 ( ii,2)( ):=( Z2) ii,6VZZ ( ii+ 200,jj):= Z2 ( ii,jj)ij := 0..3001.5Tarzlon Concentration ProfileTarzlon Concentration (mg/dm^3)10.500 5 10 15 20 25Time (hours)We could continue this but the cycles are fairly constant. Thus the recommended dosage would bean initial dose of 544 mg followed by 450 mg dosage every 7.5 hours. The 7.5 hour cycle doesn't fitinto a 24 hour day so for patient compliance it might be better to take a sslightly larger dosage every8 hours (the maximum will be exceeded for a short time)..(c) The different dosages would make manufacture and patient compliance more difficult. A betterdosage level would be one that was the same for the initial and all subsequent doses. The otherdrawback to the procedure above is that the cycles go from maximum safe concentration tominimally effective concentration. If a patient missed taking a dose the effectiveness would be zero.If they took a dose too soon, they would exceed the maximum allowable concentration. In terms ofsafety a dosage regime with smaller cycles would be better. A still better procedure would be togive an initial dose then start a constant feed of Tarzlon. For example, if a feed of Tarzlon equal to60 mg/hr is started 5.0 hours after an initial dose of 544 mg is given the response in the body wouldbe

y :=⎛⎜⎜⎜⎜⎜⎝D( t,y)m Ao0000:=⎞⎟⎟⎟⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣k 1 ⋅y 0−( k 1 + k 2 ) ⋅y 0− k 3 ⋅V− k 4 ⋅y 1k 2 ⋅y 0k 3 ⋅Vk 4 ⋅y 1Z := Rkadapt( y, 0, 5.0, 50,D)ii := 0..50⎤⎥⎥⎥⎥⎥⎥⎦Z ( ii,2)Z ( ii,6):=Vy :=⎡⎢⎢⎢⎢⎢⎢⎣Z ( 50,1)Z ( 50,2)Z ( 50,3)Z ( 50,4)Z ( 50,5)⎤⎥⎥⎥⎥⎥⎥⎦D( t,y):=⎡⎢⎢⎢⎢⎢⎢⎣−( k 1 + k 2 ) ⋅y 0 + 60k 1 ⋅y 0− k 3 ⋅V− k 4 ⋅y 1k 2 ⋅y 0k 3 ⋅Vk 4 ⋅y 1⎤⎥⎥⎥⎥⎥⎥⎦Z1 := Rkadapt( y, 5.0, 100, 100,D)k 4 = 0.2ZZ ( ii,jj):= Z ( ii,jj)ij := 0..100

Z1 ( ij,2)Z1 ( ij,6):=VZZ ( ij+ 55,jj):= Z1 ( ij,jj)ik := 0..1501.5Tarzlon Concentration ProfileTarzlon Concentration (mg/dm^3)10.5ZZ ( 100,2)= 39.99600 20 40 60 80 100Time (hours)(d) If the drug were taken on a full stomach some of the drug would be absorbed by the food. Thiswould slow the release of the drug into the blood. If the drug could be absorbed through theintestine the effect of the food would be to lower the peak height but spread out the release.