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Principles of Pharmacokinetics Pharmacokinetics of IV Administration, 1-Compartment Karunya Kandimalla, Ph.D. Associate Professor, Pharmaceutics Karunya Kandimalla, Ph.D. Associate Professor, Pharmaceutics

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2 Pharmacokinetics & Pharmacodynamics Target organ R R R R R R ADME

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3 Kinetics From the Blood or Plasma Data Pharmacokinetics of a drug in plasma or blood Absorption (Input) Disposition Distribution Elimination Excretion Metabolism

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4 Objectives Be able to: To understand the properties of linear models To understand assumptions associated with first order kinetics and one compartment models To define and calculate various one compartment model parameters (k el, t ½, V d, AUC and clearance) To estimate the values of k el, t ½, V d, AUC and clearance from plasma or blood concentrations of a drug following intravenous administration. Be able to: To understand the properties of linear models To understand assumptions associated with first order kinetics and one compartment models To define and calculate various one compartment model parameters (k el, t ½, V d, AUC and clearance) To estimate the values of k el, t ½, V d, AUC and clearance from plasma or blood concentrations of a drug following intravenous administration.

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5 Recommended Readings Chapter 3, p IV route of administration Elimination rate constant Apparent volume of distribution Clearance Chapter 3, p IV route of administration Elimination rate constant Apparent volume of distribution Clearance

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6 Intravascular Administration IV administration (bolus or infusion): Drugs are injected directly into central compartment (plasma, highly perfused organs, extracellular water) No passage across membranes Population or individual elimination rate constants (k el ) and volumes of distribution (V d ) enable us to calculate doses or infusion rates that produce target (desired) concentrations IV administration (bolus or infusion): Drugs are injected directly into central compartment (plasma, highly perfused organs, extracellular water) No passage across membranes Population or individual elimination rate constants (k el ) and volumes of distribution (V d ) enable us to calculate doses or infusion rates that produce target (desired) concentrations

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7 Disposition Analysis (Dose Linearity)

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8 Disposition Analysis (Time Variance)

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9 Linear Disposition The disposition of a drug molecule is not affected by the presence of the other drug molecules Demonstrated by: a) Dose linearity Saturable hepatic metabolism may result in deviations from the dose linearity b) Time invariance Influence of the drug on its own metabolism and excretion may cause time variance The disposition of a drug molecule is not affected by the presence of the other drug molecules Demonstrated by: a) Dose linearity Saturable hepatic metabolism may result in deviations from the dose linearity b) Time invariance Influence of the drug on its own metabolism and excretion may cause time variance

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10 Disposition Modeling A fit adequately describes the experimental data A model not only describes the experimental data but also makes extrapolations possible from the experimental data A fit that passes the tests of linearity will be qualified as a model A fit adequately describes the experimental data A model not only describes the experimental data but also makes extrapolations possible from the experimental data A fit that passes the tests of linearity will be qualified as a model

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11 One Compartment Model (IV Bolus) Schematically, one compartment model can be represented as: Where X p is the amount of drug in the body, V d is the volume in which the drug distributes and k el is the first order elimination rate constant Schematically, one compartment model can be represented as: Where X p is the amount of drug in the body, V d is the volume in which the drug distributes and k el is the first order elimination rate constant Drug in Body Drug Eliminated X p = V d C k el

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12 One Compartment Data (Linear Plot)

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13 One Compartment Data (Semi-log Plot)

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14 Two Compartment Model (IV Bolus) For both 1- and 2-compartment models, elimination takes place from central compartment Drug in Central Compartment Drug Eliminated Drug in Peripheral Compartment kel Blood, kidneys, liver Fat, muscle K 12 K 21

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15 Two Compartment Data (Linear Plot)

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16 Two Compartment Data (Semi-log Plot)

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17 One Compartment Model-Assumptions 1-CompartmentIntravascular drug is in rapid equilibrium with extravascular drug Intravascular drug [C] proportional to extravascular [C] Rapid MixingDrug mixes rapidly in blood and plasma First Order Elimination Kinetics: Rate of change of [C] Remaining [C] 1-CompartmentIntravascular drug is in rapid equilibrium with extravascular drug Intravascular drug [C] proportional to extravascular [C] Rapid MixingDrug mixes rapidly in blood and plasma First Order Elimination Kinetics: Rate of change of [C] Remaining [C]

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18 Derivation-One Compartment Model Bolus IV K el Central Compartment (C)

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19 IV Bolus Injection: Graphical Representation Assuming 1 st Order Kinetics C 0 = Initial [C] C 0 is calculated by back-extrapolating the terminal elimination phase to time = 0 C 0 = Dose/Vd Slope = -K/2.303 Slope = -K el /2.303 Concentration versus time, semilog paper

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20 Elimination Rate Constant (K el ) K el is the first order rate constant describing drug elimination (metabolism + excretion) from the body K el is the proportionality constant relating the rate of change of drug concentration and the concentration The units of K el are time -1, for example hr -1, min -1 or day -1 K el is the first order rate constant describing drug elimination (metabolism + excretion) from the body K el is the proportionality constant relating the rate of change of drug concentration and the concentration The units of K el are time -1, for example hr -1, min -1 or day -1

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21 Half-Life (t 1/2 ) Time taken for the plasma concentration to reduce to half its original concentration Drug with low half-life is quickly eliminated from the body Time taken for the plasma concentration to reduce to half its original concentration Drug with low half-life is quickly eliminated from the body t/t 1/2 % drug remaining

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22 Change in Drug Concentration as a Function of Half-Life

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23 Apparent Volume of Distribution (V d ) V d is not a physiological volume V d is not lower than blood or plasma volume but for some drugs it can be much larger than body volume Drug with large V d is extensively distributed to tissues V d is expressed in liters and is calculated as: Distribution equilibrium between drug in tissues to that in plasma should be achieved to calculate V d V d is not a physiological volume V d is not lower than blood or plasma volume but for some drugs it can be much larger than body volume Drug with large V d is extensively distributed to tissues V d is expressed in liters and is calculated as: Distribution equilibrium between drug in tissues to that in plasma should be achieved to calculate V d

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24 Volume of DistributionThe Concept Plasma [C] Tissue [C] Apparent Vd NB: For lipid-soluble drugs, Vd changes with body size and age (decreased lean body mass, increased fat)

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25 Area Under the Curve (AUC) AUC is not a parameter; changes with Dose Toxicology: AUC is used as a measure of drug exposure Pharmacokinetics: AUC is used as a measure of bioavailability and bioequivalence Bioavailability: criterion of clinical effectiveness Bioequivalence: relative efficacy of different drug products (e.g. generic vs. brand name products) AUC has units of concentration time (mg.hr/L) AUC is not a parameter; changes with Dose Toxicology: AUC is used as a measure of drug exposure Pharmacokinetics: AUC is used as a measure of bioavailability and bioequivalence Bioavailability: criterion of clinical effectiveness Bioequivalence: relative efficacy of different drug products (e.g. generic vs. brand name products) AUC has units of concentration time (mg.hr/L)

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26 Calculation of AUC using trapezoidal rule

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27 Clearance (Cl) The most important disposition parameter that describes how quickly drugs are eliminated, metabolized and distributed in the body Clearance is not the elimination rate Has the units of flow rate (volume / time) Clearance can be related to renal or hepatic function Large clearance will result in low AUC The most important disposition parameter that describes how quickly drugs are eliminated, metabolized and distributed in the body Clearance is not the elimination rate Has the units of flow rate (volume / time) Clearance can be related to renal or hepatic function Large clearance will result in low AUC

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28 ORGAN C initial C final elimination If C final < C initial, then it is a clearing organ Clearance -The Concept

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29 Practical Example IV bolus administration Dose = 500 mg Drug has a linear disposition IV bolus administration Dose = 500 mg Drug has a linear disposition Time (hr) Plasma Conc. (mg/L) ln (Plasma Conc.)

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30 Linear Plot

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31 Natural logarithm Plot K el ln (C 0 )

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32 Half-Life and Volume of Distribution t 1/2 = / K el = hrs V d = Dose / C 0 = 500 / = ln (C 0 ) = C 0 = Inv ln (2.4155) = mg/L t 1/2 = / K el = hrs V d = Dose / C 0 = 500 / = ln (C 0 ) = C 0 = Inv ln (2.4155) = mg/L

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33 Clearance Cl = D/AUC Cl = V d K el Cl = = 9.73 L/hr Cl = D/AUC Cl = V d K el Cl = = 9.73 L/hr

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34 Home Work Determine AUC and Calculate clearance from AUC Determine AUC and Calculate clearance from AUC

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Principles of Pharmacokinetics Pharmacokinetics of IV Administration, 2-Compartment Karunya Kandimalla, Ph.D Karunya Kandimalla, Ph.D

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36 Objectives Be able to: Describe assumptions associated with multi- compartment models Describe processes that take place during distribution and terminal elimination Define and calculate, β, t ½, Vi, V d SS, Cl and AUC Understand influence of Volume of distribution on loading doses and toxicity Design appropriate experiments to determine proper modeling of drug disposition Be able to: Describe assumptions associated with multi- compartment models Describe processes that take place during distribution and terminal elimination Define and calculate, β, t ½, Vi, V d SS, Cl and AUC Understand influence of Volume of distribution on loading doses and toxicity Design appropriate experiments to determine proper modeling of drug disposition

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37 Recommended Readings Chapter 4, p , Multicompartment model assumptions (73-4) Two-compartment open model (75-9) Method of residuals (79-81) Digoxin simulation (81-84) Apparent volume of distribution (84-90) Drug in tissue compartment (90-91) Clearance and elimination constant (92) Determination of compartment models (95-7) Chapter 4, p , Multicompartment model assumptions (73-4) Two-compartment open model (75-9) Method of residuals (79-81) Digoxin simulation (81-84) Apparent volume of distribution (84-90) Drug in tissue compartment (90-91) Clearance and elimination constant (92) Determination of compartment models (95-7)

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38 Physiological Perspective One compartment Two compartments k 12 Quick Slow

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39 Notes on Two-Compartment Modeling Blood or Plasma Pharmacokinetics (2 compartment model) Blood or Plasma Pharmacokinetics (2 compartment model) Absorption (Input) Disposition Ideal model should mimic distribution and disposition Full set of rate processes seldom taken into account Tissue [C] often unknown Because tissue [C] correlates with plasma [C], response often (but not always) correlates with plasma [C] Invasive nature of tissue sampling limits sophistication DistributionDistribution Elimination Excretion Metabolism

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40 ViVi VtVt ViVi Vt3Vt3 Vt2Vt2 k 12 k 21 k 12 k 21 k 31 k 13 k 10 Two compartment model Three compartment model V i = Volume of central compartment V t 2 or 3 = Volume of peripheral compartments Multicompartment Modeling

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41 Assumptions (Two-Compartment Model) Drug in peripheral compartment (bone, fat, muscle etc.) equilibrates with drug in central compartment Plasma, highly perfused organs, extracellular water [C] in a given compartment is uniform Two-compartment drugs distribute into various tissues at different, first order rates Elimination follows a single 1 st order rate process only after distribution equilibrium is reached ViVi VtVt k 12 k 21 k 10

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42 Two-Compartment Model (Mathematical Perspective) C t is a bi-exponential decaying function that depends on 2 hybrid constants (A and B), which can be determined graphically, and the distribution ( ) and elimination (β) rate constants C t = A e - t + B e –βt A function of k 10, k 12 and k 21 Because >> than β, this term goes to zero at greater t values

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43 Clinical Pharmacology and Therapeutics. 1993;53: Compartment Data (Linear Plot) Concentration-Time Course of Caffeine IV Bolus

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44 Clinical Pharmacology and Therapeutics. 1993;53: Compartment Data (Semi-log Plot) Concentration-Time Course of Caffeine IV Bolus

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45 2-Compartment Data (Semi-Log Plot) Distribution or Alpha Phase Elimination or Beta Phase Slope = β/2.303 Concentration-Time Course of Caffeine IV Bolus Note the bi-exponential decline in drug concentration A B Slope = /2.303

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46 Calculation of Micro-constants k 21 = B + β A A + B k 10 = β k 21 k 12 = + β – k 10 – k 21 k 21 = B + β A A + B k 10 = β k 21 k 12 = + β – k 10 – k 21 ViVi VtVt k 12 k 21 k 10 Note: Micro-constants cannot be calculated by direct means

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47 Two-Compartment Elimination Rate Constants k 10 represents elimination from central compartment only Larger than β Not dependent on drug transfer into tissue compartment β represents elimination when distribution equilibrium attained Influenced by drug transfer into deep tissues Clinically more useful than k 10 k 10 represents elimination from central compartment only Larger than β Not dependent on drug transfer into tissue compartment β represents elimination when distribution equilibrium attained Influenced by drug transfer into deep tissues Clinically more useful than k 10

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48 Initial Concentration (Time = 0) Answer: The initial concentration at time = 0 is equal to the sum of the intercepts A and B Question 1: Based on the information gathered thus far, what is the drug concentration at time Zero?

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49 Half-Life Compounds demonstrating two compartment kinetics will have t 1/2 estimates for each exponential phases Distribution half-life t ½ Dist = ln2/ Elimination half-life t ½ Elim = ln2/β Terminal Half life is the elimination half life for most of drugs Compounds demonstrating two compartment kinetics will have t 1/2 estimates for each exponential phases Distribution half-life t ½ Dist = ln2/ Elimination half-life t ½ Elim = ln2/β Terminal Half life is the elimination half life for most of drugs

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50 Clinical Pharmacokinetics Concepts and Applications, Third edition, Lippincott Williams & Wilkins, Media, PA What is the Elimination Half-Life (Aspirin Vs. Gentamicin)? Aspirin Distribution phase accounts for 31% of the dose Elimination phase accounts for 69% of the dose Terminal half life is the elimination half-life for aspirin Gentamicin Distribution phase accounts for 98% of the dose Elimination phase accounts for 2% of the dose Distribution half life is the appropriate half-life for gentamicin Aspirin Distribution phase accounts for 31% of the dose Elimination phase accounts for 69% of the dose Terminal half life is the elimination half-life for aspirin Gentamicin Distribution phase accounts for 98% of the dose Elimination phase accounts for 2% of the dose Distribution half life is the appropriate half-life for gentamicin

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51 Volume of Distribution (V d ) One compartment model V d is constant: Two compartment model V d changes with time and reaches a plateau at the distribution equilibrium One compartment model V d is constant: Two compartment model V d changes with time and reaches a plateau at the distribution equilibrium

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52 Two Compartment Model (V d vs. Time) ViVi V dss VtVt

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53 Determination of V i, V dss and V d From Hybrid Constants V i = Dose A + B V d SS = V i [1 + k 12 ] k 21 V d SS = Dose β AUC 0 V i = Dose A + B V d SS = V i [1 + k 12 ] k 21 V d SS = Dose β AUC 0 V t = V i k 12 k 21 Note that V d SS is a function of transfer rate constants The more extensively a drug distributes (i.e., the higher k 12 ) the larger the volume of distribution

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54 V dss - The Concept V t is mostly influenced by the elimination rate and doesnt reflect distribution V dss is mostly influenced by distribution Volume term of the steady state when a drug is infused at a constant rate Lies between V i and V t Generally, difference between V dss and V t is small Aspirin V dss = 10.4 L, V t = 10.5 L Gentamicin V dss = 345 L, V t = 56 L Substantially eliminated before distribution equilibrium achieved V t is mostly influenced by the elimination rate and doesnt reflect distribution V dss is mostly influenced by distribution Volume term of the steady state when a drug is infused at a constant rate Lies between V i and V t Generally, difference between V dss and V t is small Aspirin V dss = 10.4 L, V t = 10.5 L Gentamicin V dss = 345 L, V t = 56 L Substantially eliminated before distribution equilibrium achieved

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55 Loading Doses Loading doses are designed to achieve therapeutic concentrations faster A: 45 mg/h constant IV infusion B: Plasma [C] C: Drug remaining from 530 mg IV loading dose D L = Cp target Vd F

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56 Two-Compartment Distribution, Loading Doses & Site of Action Some 2-compartment drugs exert their therapeutic and toxic effects on target organs located in the central compartment Lidocaine, quinidine, procainamide Some 2-compartment drugs exert their therapeutic and toxic effects on target organs located in the central compartment Lidocaine, quinidine, procainamide Question 2: How should loading doses for these drugs be handled?

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57 Loading Doses for Two- Compartment Drugs Acting in V i Answer: Slow administration to allow for drug distribution into V t OR… Small bolus doses such that Cp does not exceed predetermined concentrations D L = V C C SS

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58 Two-Compartment Distribution, Loading Doses & Site of Action Some 2-compartment drugs exert their therapeutic and toxic effects on target organs located in V t Digoxin has a myocardium distribution half-life of 35 min and requires 8 to 12 h to completely distribute Some 2-compartment drugs exert their therapeutic and toxic effects on target organs located in V t Digoxin has a myocardium distribution half-life of 35 min and requires 8 to 12 h to completely distribute Question 3: How should loading doses for these drugs be handled?

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59 Loading Doses for Two- Compartment Drugs Acting in V t Answer: Quick administration is fine since the initially observed high Cps are not dangerous. These concentrations, however, cannot be used to predict therapeutic effects. D L = V SS C SS

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60 Loading Doses: The Case of Lidocaine A: Loading dose + Infusion using V i (volume of central compartment) B: Loading dose + Infusion using V SS Doted line: Constant infusion with no loading dose Dashed line: Loading dose using V i, no infusion Seizures Hypotension

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61 Tip If V dss is unknown, use a value that falls between V t and V i

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62 Estimation of AUC From hybrid constants: AUC 0 = A + B β From hybrid constants: AUC 0 = A + B β Area t2 t3 = ½ (t 3 – t 2 )(C 2 + C 3 ) AUC by Trapezoidal Method

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63 ORGAN Q.C A Q.C v elimination Clearance –Two Compartment Model C A = arterial blood concentration; C v = Venous blood concentration; Q = blood flow Clearance = Q(C a -C v ) C a

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64 Clearance (Two Compartment Model) Answer: Clearance is model independent. However we need to use different rate constants depending on the choice of volume term Example: Cl total = k 10 V i Answer: Clearance is model independent. However we need to use different rate constants depending on the choice of volume term Example: Cl total = k 10 V i Question 4: Clearance (1- compartment Model): V d K el Clearance (2- compartment model): ?

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65 Model-Independent Calculation of Clearance Cl = Dose AUC 0 No modeling consideration needed, but requires accurate measurement of AUC Early & frequent sampling essential Units = Volume/time Theoretical volume of blood or plasma completely cleared of drug per unit time Cl = Dose AUC 0 No modeling consideration needed, but requires accurate measurement of AUC Early & frequent sampling essential Units = Volume/time Theoretical volume of blood or plasma completely cleared of drug per unit time

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66 One vs. Two Compartment Dilemma Distribution phase may be missed entirely if blood is sampled too late or at wide intervals after drug administration

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67 Use of One Compartment Modeling for Two-Compartment Drugs If no concentration-time data points lie above back-extrapolated terminal line (semilog paper), assume one-compartment kinetics One-compartment modeling can be used in place of two-compartment modeling provided: Duration of distribution is small compared with elimination half-life Elimination is minimal during distribution Referred to as non-significant 2-compartment kinetics Pharmacokinetic parameters must be computed after distribution is over If no concentration-time data points lie above back-extrapolated terminal line (semilog paper), assume one-compartment kinetics One-compartment modeling can be used in place of two-compartment modeling provided: Duration of distribution is small compared with elimination half-life Elimination is minimal during distribution Referred to as non-significant 2-compartment kinetics Pharmacokinetic parameters must be computed after distribution is over

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68 Tips For Solving the Problem Set

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69 Plot Cp against time on semilog paper Extrapolate terminal phase to t = 0 Intercept = B Slope = /2.303 Read at least 3 extrapolated [C]s during distribution Calculate residual [C]s Measured – extrapolated Plot residuals against time (semilog paper) Intercept of feathered line = A Slope = /2.303 Plot Cp against time on semilog paper Extrapolate terminal phase to t = 0 Intercept = B Slope = /2.303 Read at least 3 extrapolated [C]s during distribution Calculate residual [C]s Measured – extrapolated Plot residuals against time (semilog paper) Intercept of feathered line = A Slope = /2.303

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