e) If the concentration must stay between L and H, what is the appropriate dosage for this drug?

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1 Deermining he Proper Drug Dosage When you ake a pain-reliever o reduce he effecs of a sore knee, you have many choices, each wih a paricular amoun of acive ingredien o be aken a specific inervals Medicaion Size of Table Repea Every Max Daily Inake Aceaminophen 325 mg 4-6 hours 10 in 24 hours Aspirin 325 mg 4 hours 12 in 24 hours Ibuprofen 200 mg 4-6 hours 6 in 24 hours Naproxen Sodium 220 mg 12 hours 5 in 24 hours Why hose sizes and ime inervals How is he size of a pill deermined? The size and inervals are deermined o mee wo essenial crieria: 1) he dosage is high enough o provide he needed pain relief, and 2) he dosage is no so high as o become oxic when aken regularly according o he prescribed imeable The wo imporan concenraions are he minimum effecive concenraion (MEC) and he minimum oxic concenraion (MTC) Beween hese wo limis lies he herapeuic window conaining he range of concenraions for which he drug is effecive and safe Deermining he Dosage and Time Inerval A paien is given a dosage Q mg/ml of a drug a regular inervals of ime T hours Assume ha he drug eners he sysem immediaely upon ingesion and ha he decrease in he concenraion in he blood over ime is proporional o he concenraion iself wih decay rae k If a residual level above H is unsafe (H is he MTC) and below L is ineffecive (L is he MEC), find a dose schedule T for a safe and effecive concenraion of he drug in he blood a) If he firs dose is adminisered a 0, find he amoun remaining in he blood a ime T This is known as he residual, R 1 b) Find R 2, R 3, and he nh residual R n c) Find he limiing value R lim Rn of he residual concenraion for dose of size Q mg/ml n repeaed a inervals of T hours d) If a residual level above H is unsafe and below L is ineffecive, find a dose schedule T for a safe and effecive concenraion of he drug in he blood e) If he concenraion mus say beween L and H, wha is he appropriae dosage for his drug? f) Ibuprofen follows his model well The dose is 200 mg every 6 hours In he 6 hour ime period, he body has meabolized abou 50% of he drug in he body Use his informaion o find R for his dosage of ibuprofen 1

2 g) From he able above, esimae he values of MEC, MTC, and k for aspirin, ibuprofen, naproxen sodium, and Aceaminophen h) Proscribe a herapeuic regimen (dosage and ime inerval) ha is effecive and safe for a drug wih MEC = 320 mg/ml, MTC = 500 mg/ml, and which is eliminaed a a consan rae of 12% per hour To be safe and o accoun for he variaion in individual size and meabolic raes, you wan o say in he middle of he herapeuic region, so you design your dosage and inervals o be 15% above MEC and 20% below MTC Wha s Going on in There? A few years ago, I began aking wo Aggrenox each day, one in he morning and one in he evening approximaely 12 hours apar Aggrenox is a combinaion of wo drugs, 20 mg of Aspirin and 200 mg of Dipyridamole These wo drugs do no inerac, so he pharmacokineics in combinaion are he same as when each is adminisered in isolaion I presenly weigh 188 pounds and have approximaely 60 milliliers of blood per kilogram body weigh a) Use he pharmacokineic informaion below o develop a model using Euler s mehod ha will allow you o describe he concenraion in μg/ml of hese wo drugs in my body a any ime unil he process reaches is seady sae Iniially, assume he drugs are insanly absorbed and available You will need o use an if/hen saemen o allow for he pulse of medicaion followed by he exponenial decay unil he nex dose Once you are saisfied wih his model, modify i o include he linear absorpion Compare he wo models Is much los by assuming insananeous absorpion? a) Wha happens o he concenraion of each drug if I forge o ake a pill one evening? If I miss a pill in he morning, should I ake one or wo in he evening? b) Wha is he average value of he daily concenraion once seady sae has been reached? Pharmacokineics Aspirin: Peak levels of aspirin are achieved 063 hours afer ingesion Aspirin is 99% meabolized o salicylae and oher meabolies A low doses, he eliminaion is firs-order and he half-life remains consan a 2-3 hours; however, a higher doses, he enzymes responsible for meabolism become sauraed and he apparen half-life can increase o hours Because of his, 5-7 days may be required before a seady-sae concenraion is reached Salicylae and is meabolies are excreed primarily by he kidneys Almos all of an ingesed dose is excreed in he urine Source: MAGNUM: Migraine Awareness Group web-page hp://wwwmigrainesorg/ Dipyridamole: Peak plasma levels of dipyridamole are achieved 2 hours afer adminisraion Dipyridamole is meabolized in he liver and has a prolonged eliminaion phase, which, when adminisered in pill form, is firs order and resuls in a half-life of abou 136 hours In geriaric paiens (over 65), plasma concenraions (as deermined as AUC) were abou 40% higher han in younger subjecs Source: hp://wwwaggrenoxcom/bipi/pageindex 2

3 Coninuous Infusion One of he sandard glucose olerance ess infuses glucose coninuously ino he bloodsream a he rae of G mg/min Blood concenraions are measured a subsequen ime inervals unil seady sae is reached I is assumed ha he glucose is used by he body according o he differenial equaion dx kx G d, where x is he amoun of glucose, G is he consan infusion rae of glucose, and k is he urnover rae a) Explain he relaionship beween he urnover rae and he equilibrium soluion b) Daa from a ypical glucose olerance es are given below Use he daa o esimae he consans in he model and deermine he concenraion a seady sae Time (min) x() (mg/dl) Convoluion Inegral In modeling he concenraion of a medicaion ha is inroduced ino he body over a period of ime or he concenraion of a hormone in he human body which has a similar infusion rae, simple models like he differenial equaion dc C S d are ofen used In his model, C is he concenraion of he medicaion or hormone a ime (which is being meabolized a a rae, 0 S is he, proporional o he concenraion a ha ime), while rae a which he medicaion is being inroduced or he hormone is being secreed ino he sysem The wo processes, secreion and eliminaion happen simulaneously Of course, when S 0 we have he classic exponenial decay model, A more sophisicaed model for he concenraion is he convoluion inegral, C S x E x dx, dc C d where S is he rae of secreion and E he rae of eliminaion For mos hormone secreions, S is eiher a pulse funcion or a varian of a Gaussian funcion (shown above) Noe ha x is jus a dummy variable in he inegraion; boh S and E are funcions of ime 3

4 2 2 x 06x 1) Use echnology o skech a graph of he concenraion funcion C e e dx Compare he graph o ha of secreion funcion S e 2 2) For he inegral model, C SxExdx, find C if E e and S (consan) Does he differenial equaion C S x 3) Suppose C Sxe dx, wih 2 dc also represen his model? d S a non-consan secreion rae By using he definiion of derivaive, l Hopial s rule, and he 2 nd Fundamenal Theorem of Calculus, find dc d h e 1C a) h h C h C e x Firs, show ha Sxe dx h h h c) Ch C Find lim h0 h S The Origins of he Convoluion Inegral How is he convoluion inegral creaed? Where did i come from? Suppose you are required o ake 4 pills 8 hours apar Each pill conains 100 mg of medicaion and in he 8 hours, 5% of he medicaion is meabolized each hour Define a coninuous funcion of ime whose value a ime is he amoun of medicaion in your body hours afer aking he firs pill For simpliciy, we assume insananeous absorpion We can define such a funcion in many ways The mos basic comes from our earlier work Le 0 if 0 example, A1 represens he funcion for he firs 8 hour period The second 8 hour e if 0 0 if 8 period can be models wih A and he hird by e if 8 0 if 16 A 3, and he fourh by e e e if 16 4

5 0 if 24 A e e e e if 24 Our funcion, hen, is he very awkward funcion, F A1 A2 A3 A4 wih he graph shown a righ Bu we can do much beer han his 0 if 0 Define A, hen e if 0 F A A8 A16 A 24 More compacly sill, we can wrie 3 F 100A8n n0 Now, suppose insead of individual spikes of medicaion every 8 hours, he medicaion was delivered coninuously over an exended inerval of ime For example, suppose 180 mg of medicaion was adminisered according he funcion D 0 if if 0 6 We can approximae his process using he same mehod as before Suppose we compue values of D for 0,1, 2,3, 4,5, 6 We can creae an approximaion using hese values We can assume he value of D is consan over he inerval of widh one, so 6 F D A n wih 1 n0 The graph is shown a below a lef We can generally make a beer approximaion using a smaller inerval as shown a righ below As he size of he inerval shrinks o zero, he curve smoohs ou 5

6 6 F D n A n wih 1 n F D n A n wih 2 If we coninue o shrink he size of he inerval and increase he number of inervals o cover he inerval [0,6], we can creae he convoluion inegral graphed agains he wo approximaions below s 0 n0 C D s f s ds The convoluion is 6