Karunya Kandimalla, Ph.D. Associate Professor, Pharmaceutics Principles of Pharmacokinetics Pharmacokinetics of IV Administration, 1-Compartment Karunya Kandimalla, Ph.D. Associate Professor, Pharmaceutics Karunya.kandimalla@famu.edu CP1154565 Clarke,L FD/MB 06-04-2004

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

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

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 (kel, t½, Vd, AUC and clearance) To estimate the values of kel, t½, Vd, AUC and clearance from plasma or blood concentrations of a drug following intravenous administration. CP1154565 Clarke,L FD/MB 06-04-2004

Recommended Readings Chapter 3, p. 47-62 IV route of administration Elimination rate constant Apparent volume of distribution Clearance CP1154565 Clarke,L FD/MB 06-04-2004

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 (kel) and volumes of distribution (Vd) enable us to calculate doses or infusion rates that produce target (desired) concentrations CP1154565 Clarke,L FD/MB 06-04-2004

Disposition Analysis (Dose Linearity)

Disposition Analysis (Time Variance)

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

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

One Compartment Model (IV Bolus) Schematically, one compartment model can be represented as: Where Xp is the amount of drug in the body, Vd is the volume in which the drug distributes and kel is the first order elimination rate constant kel Drug Eliminated Drug in Body Xp = Vd • C The rate of change in systemic drug level is dependent on the balance between the rates of drug absorption and elimination For both zero and first order processes, the net rate of drug accumulation is equal to the rate of drug absorption minus the rate of drug elimination The absorption (ka) and elimination (kel) rate constants describe how quickly serum concentrations rise (ka) or decline (kel) after administration. The elimination rate of a drug can be computed by taking the product of the elimination rate constant (kel) and the amount of drug in the body (Xp) In general, low molecular weight, high lipid solubility and lack of charge encourage absorption. CP1154565 Clarke,L FD/MB 06-04-2004

One Compartment Data (Linear Plot)

One Compartment Data (Semi-log Plot)

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

Two Compartment Data (Linear Plot)

Two Compartment Data (Semi-log Plot)

One Compartment Model-Assumptions 1-Compartment—Intravascular drug is in rapid equilibrium with extravascular drug Intravascular drug [C] proportional to extravascular [C] Rapid Mixing—Drug mixes rapidly in blood and plasma First Order Elimination Kinetics: Rate of change of [C]  Remaining [C] Notice here that the log-transformed equation represents the equation of a straight line CP1154565 Clarke,L FD/MB 06-04-2004

Derivation-One Compartment Model Bolus IV Central Compartment (C) Kel This is the simplest of the intravascular administration models The entire dose enters the systemic circulation immediately and the body is depicted as a single homogenous unit Plasma concentrations are not necessarily equal to tissue concentrations, but they are proportional The advantage of this model is that it permits calculation of drug concentrations at any time (no absorption phase, no distribution phase) The disadvantage is that the kinetic parameters do not have physiologic meaning CP1154565 Clarke,L FD/MB 06-04-2004

Concentration versus time, semilog paper IV Bolus Injection: Graphical Representation Assuming 1st Order Kinetics C0 = Initial [C] C0 is calculated by back-extrapolating the terminal elimination phase to time = 0 C0 = Dose/Vd C0 = Dose/Vd Slope = -K/2.303 Slope = -Kel/2.303 Note that half-life can also be read from the graph Concentration versus time, semilog paper CP1154565 Clarke,L FD/MB 06-04-2004

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

Half-Life (t1/2) Time taken for the plasma concentration to reduce to half its original concentration Drug with low half-life is quickly eliminated from the body t/t1/2 % drug remaining 1 50 2 25 3 12.5 4 6.25 5 3.125

Change in Drug Concentration as a Function of Half-Life

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

Volume of Distribution—The Concept Plasma [C] Tissue [C] “Apparent” Vd • • •• • • • • • • • • • • • • • • • • • • • • • • • • Volume of distribution is important in that it determines the size of loading doses needed to achieve a particular steady state concentrations. Elimination rate constants and half-lives are known as dependent parameters, because their value depend on the clearance and volume of distribution of drugs. These parameters can change, either because of a change in clearance or a change in the volume of distribution. Because the values for clearance and volume of distribution depend solely on physiologic parameters and can vary independently of each other, there are known as independent parameters. • • • • • • • • • NB: For lipid-soluble drugs, Vd changes with body size and age (decreased lean body mass, increased fat) CP1154565 Clarke,L FD/MB 06-04-2004

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)

Calculation of AUC using trapezoidal rule

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

Clearance -The Concept Cinitial Cfinal ORGAN elimination If Cfinal < Cinitial, then it is a clearing organ

Practical Example Time (hr) Plasma Conc. (mg/L) ln (PlasmaConc.) 1 9.46 2.25 2 7.15 1.97 3 5.56 1.71 4 4.74 1.56 6 3.01 1.10 10 1.26 0.23 12 0.83 -0.19 IV bolus administration Dose = 500 mg Drug has a linear disposition

Linear Plot

Natural logarithm Plot Kel ln (C0)

Half-Life and Volume of Distribution t1/2 = 0.693 / Kel = 3.172 hrs Vd = Dose / C0 = 500 / 11.12 = 44.66 ln (C0) = 2.4155 C0 = Inv ln (2.4155) = 11.195 mg/L

Clearance Cl = D/AUC Cl = VdKel Cl = 44.66  0.218 = 9.73 L/hr

Home Work Determine AUC and Calculate clearance from AUC

Karunya Kandimalla, Ph.D Principles of Pharmacokinetics Pharmacokinetics of IV Administration, 2-Compartment Karunya Kandimalla, Ph.D kandimalla.karunya@mayo.edu CP1154565 Clarke,L FD/MB 06-04-2004

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, t½, Vi, VdSS, Cl and AUC Understand influence of Volume of distribution on loading doses and toxicity Design appropriate experiments to determine proper modeling of drug disposition CP1154565 Clarke,L FD/MB 06-04-2004

Recommended Readings Chapter 4, p. 73-92, 95-97 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) CP1154565 Clarke,L FD/MB 06-04-2004

Physiological Perspective One compartment Quick One compartment models are used when the drug appears to distribute into tissues instantaneously and uniformly. Multicompartment drugs are delivered concurrently to one or more peripheral compartments composed of groups of tissues with lower blood perfusion and different affinity for the drug. The one compartment model is the simplest of all multicompartment models. k12 Two compartments Quick Slow CP1154565 Clarke,L FD/MB 06-04-2004

Notes on Two-Compartment Modeling 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 Blood or Plasma Pharmacokinetics (2 compartment model) Absorption (Input) Disposition Distribution Elimination Excretion Metabolism

Multicompartment Modeling k12 Vi Vt Two compartment model k21 k10 Three compartment model k31 Vt3 Vi k12 Vt2 k13 k21 k10 Vi = Volume of central compartment Vt 2 or 3 = Volume of peripheral compartments

Assumptions (Two-Compartment Model) k12 Vi Vt k21 k10 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 1st order rate process only after distribution equilibrium is reached Different peripheral compartments will accumulate multicompartment drugs at different rates Distribution in different compartments may be uneven, i.e., drugs generally concentrate unevenly in different groups of tissues CP1154565 Clarke,L FD/MB 06-04-2004

Two-Compartment Model (Mathematical Perspective) Ct 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 Ct = A • e -t + B • e –βt Because  >> than β, this term goes to zero at greater t values A function of k10, k12 and k21

2-Compartment Data (Linear Plot) Concentration-Time Course of Caffeine IV Bolus Clinical Pharmacology and Therapeutics. 1993;53:6-14

2-Compartment Data (Semi-log Plot) Concentration-Time Course of Caffeine IV Bolus Clinical Pharmacology and Therapeutics. 1993;53:6-14

2-Compartment Data (Semi-Log Plot) Concentration-Time Course of Caffeine IV Bolus Distribution or Alpha Phase A Elimination or Beta Phase Slope = β/2.303 B Drug elimination and distribution occur concurrently and at varying degrees during the distribution phase, yet there is a net transfer of drug from the central compartment to the tissue compartment Notice that the plasma concentration time curve has 2 distinct slopes—one that is a hybrid constant made up of all compartment distribution constants, and another that describes both the dynamic process of drug transfer between tissue and central compartments and elimination from the central compartment Slope = /2.303 Note the bi-exponential decline in drug concentration CP1154565 Clarke,L FD/MB 06-04-2004

Calculation of Micro-constants k21 =  • B + β • A A + B k10 =  • β k21 k12 =  + β – k10 – k21 k12 Vi Vt k21 k10 Note: Micro-constants cannot be calculated by direct means

Two-Compartment Elimination Rate Constants k10 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 k10

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

Half-Life Compounds demonstrating two compartment kinetics will have t1/2 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

What is the Elimination Half-Life (Aspirin Vs. Gentamicin)? 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 Clinical Pharmacokinetics Concepts and Applications, Third edition, Lippincott Williams & Wilkins, Media, PA 19063

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

Two Compartment Model (Vd vs. Time) Vt Vdss Vi

Determination of Vi, Vdss and Vd From Hybrid Constants Vi = Dose A + B VdSS = Vi [1 + k12] k21 VdSS = Dose β • AUC 0  ∞ Vt = Vi k12 k21 Note that VdSS is a function of transfer rate constants The more extensively a drug distributes (i.e., the higher k12) the larger the volume of distribution

Vdss- The Concept Vt is mostly influenced by the elimination rate and doesn’t reflect distribution Vdss is mostly influenced by distribution Volume term of the steady state when a drug is infused at a constant rate Lies between Vi and Vt Generally, difference between Vdss and Vt is small Aspirin Vdss = 10.4 L, Vt = 10.5 L Gentamicin Vdss = 345 L, Vt = 56 L Substantially eliminated before distribution equilibrium achieved

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 DL = Cp target • Vd F

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 Question 2: How should loading doses for these drugs be handled? In this instance, the concentration of drug delivered to target organs could be much higher than expected and produce toxicity if the loading dose is not adjusted appropriately. The problem can be circumvented by calculating the total loading dose based on the total volume of distribution (Vd). The loading dose should be administered at a rate slow enough to allow for drug distribution into Vt; or the total loading dose should be given in sufficiently small bolus doses such that Cp in Vi does not exceed some predetermined critical concentration. CP1154565 Clarke,L FD/MB 06-04-2004

Loading Doses for Two-Compartment Drugs Acting in Vi Answer: Slow administration to allow for drug distribution into Vt OR… Small bolus doses such that Cp does not exceed predetermined concentrations DL = VC • CSS

Two-Compartment Distribution, Loading Doses & Site of Action Some 2-compartment drugs exert their therapeutic and toxic effects on target organs located in Vt Digoxin has a myocardium distribution half-life of 35 min and requires 8 to 12 h to completely distribute In this instance, the concentration of drug delivered to target organs could be much higher than expected and produce toxicity if the loading dose is not adjusted appropriately. The problem can be circumvented by calculating the total loading dose based on the total volume of distribution (Vd). The loading dose should be administered at a rate slow enough to allow for drug distribution into Vt; or the total loading dose should be given in sufficiently small bolus doses such that Cp in Vi does not exceed some predetermined critical concentration. Question 3: How should loading doses for these drugs be handled? CP1154565 Clarke,L FD/MB 06-04-2004

Loading Doses for Two-Compartment Drugs Acting in Vt Answer: Quick administration is fine since the initially observed high Cps are not dangerous. These concentrations, however, cannot be used to predict therapeutic effects. DL = VSS • CSS

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

Tip If Vdss is unknown, use a value that falls between Vt and Vi

AUC by Trapezoidal Method Estimation of AUC From hybrid constants: AUC0∞ = A + B  β Area t2 t3 = ½ (t3 – t2)(C2 + C3) AUC by Trapezoidal Method

Clearance –Two Compartment Model Q.CA Q.Cv ORGAN elimination CA = arterial blood concentration; Cv= Venous blood concentration; Q = blood flow Clearance = Q(Ca-Cv) Ca

Clearance (Two Compartment Model) Question 4: Clearance (1- compartment Model): Vd • Kel Clearance (2- compartment model): ? Answer: Clearance is model independent. However we need to use different rate constants depending on the choice of volume term Example: Cltotal = k10 • Vi

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

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

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

Tips For Solving the Problem Set

Plot Cp against time on semilog paper Extrapolate terminal phase to t = 0 Intercept = B Slope = b/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