Chapter 4: One Compartment Open Model: Intravenous Bolus Administration

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1 Home Readings Search AccessPharmacy Adv. Search Alied Bioharmaceutics & Pharmacokinetics, 7e > Chater 4 Chater 4: One Comartment Oen Model: Intravenous Bolus Administration avid S.H. Lee CHAPTER OBJECTIVES escribe a one comartment model, IV bolus injection. Provide the harmacokinetic terms that describe a one comartment model, IV bolus injection, and the underlying assumtions. Exlain how drugs follow one comartment kinetics using drug examles that follow onecomartment kinetics. Calculate harmacokinetic arameters from drug concentration time data using a onecomartment model. Simulate one comartment lasma drug level grahically using the one comartment model equation. Calculate the IV bolus dose of a drug using the one comartment model equation. htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

2 Relate the relevance of the magnitude of the volume of distribution and clearance of various drugs to underlying rocesses in the body. erive model arameters from sloe and intercet of the aroriate grahs. While the oral route of drug administration is the most convenient, intravenous (IV) administration is the most desirable for critical care when reaching desirable drug concentrations quickly is needed. Examles of when IV administration is desirable include antibiotic administration during setic infections or administration of antiarrhythmic drugs during a myocardial infarction. Because harmacokinetics is the science of the kinetics of drug absortion, distribution, and elimination, IV administration is desirable in understanding these rocesses since it simlifies drug absortion, essentially making it comlete and instantaneous. This leaves only the rocesses of drug distribution and elimination left to study. This chater will introduce the concets of drug distribution and elimination in the simlest model, the one comartment oen model. The one comartment oen model assumes that the body can be described as a single, uniform comartment (ie, one comartment), and that drugs can enter and leave the body (ie, oen model). The simlest drug administration is when the entire drug is given in a raid IV injection, also known as an IV bolus. Thus, the one comartment oen model with IV bolus administration is the simlest harmacokinetic model. It assumes that the drug is administered instantly into the body, it is instantaneously and raidly distributed throughout the body, and drug elimination occurs immediately uon entering the body. This model is a simlistic reresentation of the rocesses in the body that determine drug disosition, but nonetheless, it can be useful to describe and redict drug disosition. In reality, when a drug is administered intravenously, the drug travels through the bloodstream and distributes throughout the bloodstream in the body. While this rocess is not truly instantaneous, it is relatively raid enough that we can make this assumtion for most drugs. Through the bloodstream, the drug is distributed to the various tissue organs in the body. The rate and extent of distribution to the tissue organs deends on several rocesses and roerties. Tissues in the body are resented the drug at various rates, deending on the blood flow to that organ, and the drug may have different abilities to cross from the vasculature to the organ deending on the molecular weight of the drug. Tissues also have different affinity for the drug, deending on liohilicity and drug binding. Finally, large organs may have a large caacity for drugs to distribute to. While drug distribution is comlex, if these rocesses are raid enough, we can simlify our concetualization as if the drug uniformly distributes into a single (one) comartment of fluid. The volume of this single comartment is termed the aarent volume of distribution, V. The aarent volume of distribution is not an actual volume in the body, but is a theoretical volume that the drug uniformly distributes to immediately after being injected into the body. This uniform and instantaneous distribution is termed a well stirred one comartment model. The aarent volume of distribution is a roortion between the dose and the concentration of the drug in lasma,, at that time immediately after being injected. Most drugs are eliminated from the body by liver metabolism and/or renal excretion. All of the rocesses of drug elimination can be described by the elimination rate constant, k. The elimination rate constant is the roortion between the rate of drug elimination and the amount of drug in the body. htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

3 Because the amount of drug in the body changes over time, the rate of drug elimination changes, but the elimination rate constant remains constant for first order elimination. This makes it convenient to summarize drug elimination from the body indeendent of time or the amount of drug in the body. However, because it s difficult to measure the amount of drug in the body,, harmacokineticists and harmacists also refer to convert drug amounts to drug concentrations in the body using the aarent volume of distribution. Thus, the elimination rate constant also describes the roortion between the rate of change of drug concentration and drug concentration in the comartment. The one comartment oen model with IV bolus dosing describes the distribution and elimination after an IV bolus administration and is shown in Fig The fluid that the drug is directly injected into is the blood, and generally, drug concentrations are measured in lasma since it is accessible. Therefore, this model redicts the concentrations in the lasma, but does not redict the concentrations in tissues. However, using this model, which assumes distribution to tissues is raid, we can assume the declines in drug concentration in the lasma and tissues will be roortional. For these reasons, the onecomartment oen model is useful for redicting concentrations in the lasma, and declines in lasma concentrations will be roortional to declines in tissue concentrations. B FIGURE 4 1 Pharmacokinetic model for a drug administered by raid intravenous injection. = drug in body; V = aarent volume of distribution; k = elimination rate constant. B View Full Size Favorite Figure ownload Slide (.t) View ELIMINATION RATE CONSTANT Tools Search Book To APPARENT VOLUME OF ISTRIBUTION CLEARANCE CLINICAL APPLICATION CALCULATION OF κ FROM URINARY EXCRETION htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

4 ATA PRACTICE PROBLEM PRACTICE PROBLEM CLINICAL APPLICATION CHAPTER SUMMARY LEARNING QUESTIONS 1. A 70 kg volunteer is given an intravenous dose of an antibiotic, and serum drug concentrations were determined at 2 hours and 5 hours after administration. The drug concentrations were 1.2 and 0.3 μg/ml, resectively. What is the biologic half life for this drug, assuming first order elimination kinetics? 2. A 50 kg woman was given a single IV dose of an antibacterial drug at a dose level of 6 mg/kg. Blood samles were taken at various time intervals. The concentration of the drug (C ) was determined in the lasma fraction of each blood samle and the following data were obtained: Favorite Table Print t (hours) C (µg/ml) a. What are the values for V, k, and t for this drug? b. This antibacterial agent is not effective at a lasma concentration of less than 2 μg/ml. What is htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

5 the duration of activity for this drug? c. How long would it take for 99.9% of this drug to be eliminated? d. If the dose of the antibiotic was doubled exactly, what would be the increase in duration of activity? 3. A new drug was given in a single intravenous dose of 200 mg to an 80 kg adult male atient. After 6 hours, the lasma drug concentration of drug was 1.5 mg/100 ml of lasma. Assuming that the aarent V is 10% of body weight, comute the total amount of drug in the body fluids after 6 hours. What is the half life of this drug? 4. A new antibiotic drug was given in a single intravenous bolus of 4 mg/kg to 5 healthy male adults ranging in age from 23 to 38 years (average weight 75 kg). The harmacokinetics of the lasma drug concentration time curve for this drug fits a one comartment model. The equation of the curve that best fits the data is etermine the following (assume units of μg/ml for C and hours for t): a. What is the t? b. What is the V? c. What is the lasma level of the drug after 4 hours? d. How much drug is left in the body after 4 hours? e. Predict what body water comartment this drug might occuy and exlain why you made this rediction. f. Assuming the drug is no longer effective when levels decline to less than 2 μg/ml, when should you administer the next dose? 5. efine the term aarent volume of distribution. What criteria are necessary for the measurement of the aarent volume of distribution to be useful in harmacokinetic calculations? 6. A drug has an elimination t of 6 hours and follows first order kinetics. If a single 200 mg dose is given to an adult male atient (68 kg) by IV bolus injection, what ercent of the dose is lost in 24 hours? 7. A rather intoxicated young man (75 kg, age 21 years) was admitted to a rehabilitation center. His blood alcohol content was found to be 210 mg%. Assuming the average elimination rate of alcohol is 10 ml of ethanol er hour, how long would it take for his blood alcohol concentration to decline to less than the legal blood alcohol concentration of 100 mg%? (Hint: Alcohol is eliminated by zero order kinetics.) The secific gravity of alcohol is 0.8. The aarent volume of distribution for alcohol is 60% of body weight. 8. A single IV bolus injection containing 500 mg of cefamandole nafate (Mandol, Lilly) is given to an htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

6 adult female atient (63 years, 55 kg) for a seticemic infection. The aarent volume of distribution is 0.1 L/kg and the elimination half life is 0.75 hour. Assuming the drug is eliminated by first order kinetics and may be described by a one comartment model, calculate the following: a. The C 0 b. The amount of drug in the body 4 hours after the dose is given c. The time for the drug to decline to 0.5 μg/ml, the minimum inhibitory concentration for stretococci 9. If the amount of drug in the body declines from 100% of the dose (IV bolus injection) to 25% of the dose in 8 hours, what is the elimination half life for this drug? (Assume first order kinetics.) 10. A drug has an elimination half life of 8 hours and follows first order elimination kinetics. If a single 600 mg dose is given to an adult female atient (62 kg) by raid IV injection, what ercent of the dose is eliminated (lost) in 24 hours assuming the aarent V is 400 ml/kg? What is the exected lasma drug concentration (C ) at 24 hours ostdose? 11. For drugs that follow the kinetics of a one comartment oen model, must the tissues and lasma have the same drug concentration? Why? 12. An adult male atient (age 35 years, weight 72 kg) with a urinary tract infection was given a single intravenous bolus of an antibiotic (dose = 300 mg). The atient was instructed to emty his bladder rior to being medicated. After dose administration, the atient saved his urine secimens for drug analysis. The urine secimens were analyzed for both drug content and sterility (lack of bacteriuria). The drug assays gave the following results: Favorite Table Print t (hours) Amount of rug in Urine (mg) a. Assuming first order elimination, calculate the elimination half life for the antibiotic in this atient. b. What are the ractical roblems in obtaining valid urinary drug excretion data for the determination of the drug elimination half life? ANSWERS htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

7 Frequently Asked Questions What is the difference between a rate and a rate constant? A rate reresents the change in amount or concentration of drug in the body er time unit. For examle, a rate equal to 5 mg/h means the amount of drug is decreasing at 5 mg/h. A ositive or negative sign indicates that the rate is increasing or decreasing, resectively. Rates may be zero order, first order, or higher orders. For a first order rate, the rate of change of drug in the body is determined by the roduct of the elimination rate constant, k, and the amount of drug remaining in the body, that is, rate = k, where k reresents the fraction of the amount of drug in the body that is eliminated er hour. If k = 0.1 h and = 10 mg, then the rate = 0.1 h 1 10 mg = 1 mg/h. The rate constant in this examle shows that one tenth of the drug is eliminated er hour, whatever amount of drug is resent in the body. For a first order rate, the rate states the absolute amount eliminated er unit time (which changes with the amount of drug in the body), whereas the first order rate constant, k, gives a constant fraction of drug that is eliminated er unit time (which does not change with the amount of drug in the body). Why does k always have the unit 1/time (eg, h 1 B ), regardless of what concentration unit is lotted? The first order rate constant k has no concentration or mass units. In the calculation of the sloe, k, the unit for mass or concentration is canceled when taking the log of the number: 1 B If a drug is distributed in the one comartment model, does it mean that there is no drug in the tissue? The one comartment model uses a single homogeneous comartment to reresent the fluid and the vascular tissues. This model ignores the heterogeneity of the tissues in the body, so there is no merit in redicting recise tissue drug levels. However, the model rovides useful insight into the mass balance of drug distribution in and out of the lasma fluid in the body. If V is larger than the hysiologic vascular volume, the conclusion is that there is some drug outside the vascular ool, that is, in the tissues. If V is small, then there is little extravascular tissue drug storage, excet erhas in the lung, liver, kidney, and heart. With some knowledge about the liohilicity of the drug and an understanding of blood flow and erfusion within the body, a ostulation may be made as to which organs are involved in storing the extravascular drug. The concentration of a biosy samle may suort or refute the ostulation. How is clearance related to the volume of distribution and k? Clearance is the volume of lasma fluid that is cleared of drug er unit time. Clearance may also be derived for the hysiologic model as the fraction of drug that is eliminated by an organ as blood flows through it. The former definition is equivalent to Cl = kv and is readily adated to dosing since V is the volume of distribution. If the drug is eliminated solely by metabolism in the liver, then Cl = Cl. Cl is usually estimated by the difference between Cl and Cl. Cl is directly estimated by the roduct of the heatic blood flow and the extraction ratio. If we use a hysiologic model, are we dealing with actual volumes of blood and tissues? Why do H H R H htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

8 volumes of distribution emerge for drugs that often are greater than the real hysical volume? Since mass balance (ie, relating dose to lasma drug concentration) is based on volume of distribution rather than blood volume, the comartment model is used in determining dose. Generally, the total blood concentrations of most drugs are not known, since only the lasma or serum concentration is assayed. Some drugs have an RBC/lasma drug ratio much greater than 1, making the alication of the hysiologic model difficult without knowing the aarent volume of distribution. Learning Questions 1. The C decreased from 1.2 to 0.3 μg/ml in 3 hours. Favorite Table Print t (hours) C (µg/ml) These data may also be lotted on a semilog grah and t obtained from the grah. 2. ose (IV bolus) = 6 mg/kg 50 kg = 300 mg a. (1) Plot the data on semilog grah aer and use two oints from the line of best fit. htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

9 Favorite Table Print t (hours) C (µg/ml) (2) t (from grah) = 4 hours b. Alternatively, time t may be found from a grah of C versus t. c. Time required for 99.9% of the drug to be eliminated: (1) Aroximately 10 t (2) With 0.1% of drug remaining, d. If the dose is doubled, then will also double. However, the elimination half life or firstorder rate constant will remain the same. Therefore, htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

10 Notice that doubling the dose does not double the duration of activity. 3. At 6 hours: 4. C = 78e (the equation is in the form ) 0.46t a. htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

11 b. c. (1) (2) d. At 4 hours: e. V = 3846 ml The aarent V aroximates the lasma volume. f. C = 2 μg/ml Find t. Alternate Method htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

12 6. For first order elimination kinetics, one half of the initial quantity is lost each t. The following table may be develoed: Favorite Table Print Time Number of Amount of rug in Body Percent of rug in Percent of rug (hours) t (mg) Body Lost Method 1 From the above table the ercent of drug remaining in the body after each t is equal to 100% n times (), where n is the number of half lives, as shown below: Favorite Table Print Number of t Percent of rug in Body Percent of rug Remaining in Body after n t N 100 () n Percent of drug remaining, where n = number of t htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

13 Percent of drug lost At 24 hours, n = 4, since t = 6 hours. Percent of drug lost Method 2 The equation for a first order elimination after IV bolus injection is where = amount of drug remaining in the body B = dose = 200 mg 0 k = elimination rate constant 7. The zero order rate constant for alcohol is 10 ml/h. Since the secific gravity for alcohol is 0.8, Therefore, the zero order rate constant, k, is 8 g/h. rug in body at t = 0: 0 rug in body at time t: For a zero order reaction: htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

14 8. a. b. c The total drug concentration in the lasma is not usually equal to the total drug htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

15 concentration in the tissues. A one comartment model imlies that the drug is raidly equilibrated in the body (in lasma and tissues). At equilibrium, the drug concentration in the tissues may differ from the drug concentration in the body because of drug rotein binding, artitioning of drug into fat, differences in H in different regions of the body causing a different degree of ionization for a weakly dissociated electrolyte drug, an active tissue utake rocess, etc. 12. Set u the following table: Favorite Table Print Time (hours) (mg) d /t mg/h t* u / / u The elimination half life may be obtained grahically after lotting mg/h versus t*. The t obtained grahically is aroximately 2 hours. REFERENCE BIBLIOGRAPHY htt://accessharmacy.mhmedical.com/content.asx?bookid=1592&sectionid= /16

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