Kinetic Models, Math Fundamentals.pdf

PHARMACOKINETIC MODELS AND MATHEMATIC FUNDAMENTALS IN PHARMACOKINETICS The RATE of a chemical reaction Is the velocity with which it occurs The ORDER of a reaction Is the way in which the concentration of a drug or reactant in a chemical reaction affects t...

PHARMACOKINETIC MODELS AND MATHEMATIC FUNDAMENTALS IN PHARMACOKINETICS The RATE of a chemical reaction Is the velocity with which it occurs The ORDER of a reaction Is the way in which the concentration of a drug or reactant in a chemical reaction affects the rate Classes: Zero-order rate process First-order rate process Pseudo-order rate process Zero-order Reaction First-order Reaction Apparent or Pseudo-first-order Reaction The drug concentration The drug concentration Describes a situation where one changes with respect to changes with respect to time of the reactants is present in time at a constant rate equal the product of the rate large excess or does not C= -K0t + C0 constant and the effect the overall reactions concentration of drug and can be held constant Where: remaining A situation where-in the C = drug concentration at any time C= C0e-kt second-order reaction behaves In C = - kt + In C0 like a first-order reaction K0 = zero-order rate log C = -kt/2.3 + log C0 constant (units of concentration per time) = is Where: the slope of the line C = drug concentration at any C0 = is the y intercept = drug time concentration, when time (t) k = first-order rate constant equals zero (units of reciprocal time, or time- Negative sign = indicates 1 ) that the slope is decreasing -k/2.3 = is the slope of the line C0 = is the y intercept = drug concentration, when time (t) equals zero Significance of Rate Constants (k) Characterize the change of drug concentration in a particular reference region Give the speed at which a drug: Enters the compartment (absorption rate constant, ka) Distributes between a central and peripheral compartments (distribution rate constant) Is eliminated from the systemic circulation (elimination rate constant, k) Zero-order elimination kinetics First-order elimination kinetics The Cp vs t profile during the elimination phase is A linear process linear rate of elimination is proportional to the drug Example: 1.2 mg are eliminated every hour, concentration independently of the drug concentration in the the elimination processes are not saturated and body. can adapt to the needs of the body, to reduce Zero Order elimination is rare accumulation of the drug mostly occurring when the elimination 95% of the drugs in use at therapeutic system is saturated concentrations are eliminated by first order An example is the elimination of Ethanol. elimination kinetics  HALF-LIFE (t½) Expresses the period of time required for the concentration of a drug to decrease by one half Is the time required to decrease the initial dose of drug by 50% (one half of original value) Units: time Significance of Half-life Determine the dosing interval necessary to obtain the desired Cp of the drug Generally, the dosing interval is the same as t½ Predict how long it will take a drug to reach steady-state levels Predict the accumulation of a drug in the body for a specific dosing interval During multiple dosing or continuous IV infusion it takes approximately 4-5 half-lives to reach steady-state levels Predict how long it will take a drug concentration to decrease to a lower concentration All drugs are decreased by 96% after 4 half-lives Zero-order Half-life First-order Half-life Is not constant for a zero-order process Is constant for a first-order process Is proportional to the initial amount or Is related to the first-order rate constant concentration of the drug and is inversely No matter what the initial amount or proportional to the zero-order rate concentration of the drug is, the time constant, k0 required for the amount to decrease by t½ = (0.5)(A0)/k0 one half is constant (t½)(k0) = (0.5)(A0) t½ = 0.693/k (t½)(k) = 0.693 Because the t½ changes periodically as drug concentration decline, this has little practical value NOTE: k (“rate constant”) k0 = initial k = kel = elimination —------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- NONLINEAR PHARMACOKINETICS also known as: Capacity-limited Dose-dependent Saturation pharmacokinetics DO NOT follow first-order kinetics as the dose increases Result from the saturation of an enzyme- or carrier-mediated system Characteristics of nonlinear pharmacokinetics The AUC is not proportional to the dose The amount of drug excreted in the urine is not proportional to the dose The elimination half-life may increase at higher doses The ratio of metabolites formed changes with increased dose  Michaelis-Menten Kinetics Describe the velocity of enzyme reactions Used to describe nonlinear pharmacokinetics NOTE: Drugs that follow nonlinear pharmacokinetics may: Show zero-order elimination rates at high drug concentration A mix zero- and first-order elimination rates at intermediate concentrations First-order elimination rates at low drug concentrations —------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- MODELS and Compartments ONE-COMPARTMENT MODEL TWO-COMPARTMENT MODEL MODEL Is a mathematical description of a biologic system Is used to express quantitative relations concisely COMPARTMENT A group of tissues with similar blood flow and drug affinity Not a real physiologic or anatomic region  OPEN ONE-COMPARTMENT MODEL If the drug entering the body (input) distributes (equilibrates) instantly between the blood and other body fluids or tissues Drug is not necessarily confined to the circulatory system Drug may occupy the entire extracellular fluid, soft tissue or the entire body Distribution occurs instantly Is not pooled in a specific area Characteristics of Open One-Compartment Characteristics of One-open Compartment Models by Intravascular Routes Models by Extravascular Routes No absorption Absorption proceeds according to drug liberation Rapid distribution of drug between bloodstream and absorption mechanism and tissue At time 0 no drug is in systemic circulation Equilibrium is instantly obtained As absorption proceeds drug concentration in Fall of drug concentration depends on excretion systemic circulation increases to peak and then and metabolism decreases according to elimination Not necessarily all of the administered drug is One-Compartment Model I.V. Bolus absorbed Drug elimination is a first-order process Input: Absorption = zero-order Output: DME = first-order The first order elimination rate constant (k) K= Ke + Km Apparent Volume of Distribution (VD) Vd= Db°/Cp° Db°= dose given by IV bolus Cp°= extrapolated drug concentration at zero time  Open Two-Compartment Model lf the drug entering the body does not instantly distribute between the blood and those other body fluids or tissues which it eventually reaches Distribution of the drug in blood and other soft tissues Occurs at different rates Eventually steady state will be reached which terminates the distribution phase NOTE: Central > Peripheral > Central > Eliminate Characteristics of Open-two Compartment Characteristics of Open-two Compartment Model: Intravascular Routes Model: Extravascular Routes No absorption Absorption proceeds according to drug Slow distribution of drug between liberation mechanism bloodstream and tissue At time 0, there is no drug in systemic Equilibrium is obtained some later time after circulation administration As absorption proceeds, drug concentration Steep fall of first part of blood level curve in systemic circulation rises to peak, followed due to distribution by a steep fall due to slow distribution until Decline of second part of blood level curve equilibrium is obtained depends on back distribution of drug from Mono-exponential decline of curve depends tissue to blood, excretion and metabolism on back distribution of drug from tissue to blood, excretion and metabolism BIOAVAILABILITY and BIOEQUIVALENCE BIOAVAILABILITY Is a measurement of the rate and extent to which the active ingredient or active moiety becomes available at the site of action Is also considered as a measure of the rate and extent of therapeutically active drug that is systemically absorbed For intravascular route, f = 1 For extravascular route, f < 1 Absolute Bioavailability Relative Bioavailability Is the extent or fraction of drug absorbed Is the extent of drug absorbed upon upon extravascular administration in extravascular administration in comparison to comparison to the dose size administered the dose size of a standard administered by May be measured by comparing the the same route respective AUCs after oral and IV Availability of drug in the formulation is administration, as long as V, and k are compared to the availability of drug in a independent of the route of administration standard dosage formulation, usually a Formula: solution of the pure drug evaluated in a [ ] crossover study Formula: [ ] = [ ] = [ ] Where: Drug product B = recognized reference standard Practice:  Bioequivalence Is achieved if its extent and rate of absorption are not statistically significantly different from those of the standard when administered at the same molar dose. BIOEQUIVALENT DRUG PRODUCTS PHARMACEUTICAL EQUIVALENTS A generic drug product is considered Are drug products that contain the same; bioequivalent to the Reference (branded) active ingredient(s), same salt, ester, or drug product if both products are; chemical form; are of the same dosage form; pharmaceutical equivalents and and are identical; its bioavailability do not show statistically ❖ in strength and concentration and significant difference when administered in; ❖ route of administration the same dose of the active ingredient in the same chemical form May differ in characteristics such as; in a similar dosage form Shape by the same route of administration and Scoring configuration Release mechanisms under the same experimental condition Packaging And excipients (colors, flavors, preservatives) —------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- BIOAVAILABILITY AND BIOEQUIVALENCE may be determined using; 1. Plasma drug concentration versus time profiles 2. Urinary drug excretion studies 3. Measurements of an acute pharmacological effect 4. Clinical studies Less precise than other methods Highly variable due to individual differences 5. In vitro studies In vitro dissolution correlates with drug bioavailability in vivo —------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Plasma Drug Concentration VS time curve Used to measure the systemic bioavailability of a drug from a drug product Parameters Time for peak plasma concentration (max) Peak plasma concentration (max) Area under the plasma drug concentration versus time curve (AUC) Time of Peak Plasma Concentration (tmax) The time needed to reach maximum concentration, Cmax, and is independent of dose and dependent on the rate constants for absorption (ka) and elimination (k) Integration (simplified): t = ( / ) − t = 2.3 ( / ) − PEAK PLASMA CONCENTRATION (Cmax) Relates to the intensity of pharmacological response. Represents the maximum plasma drug concentration obtained after oral administration of drug Units: e.g., ug/mL, ng/mL AREA UNDER THE CURVE (AUC) Relates the rate and extent of drug absorption. The amount of systemic drug absorption is directly related to the AUC Units are concentration time: ug.hr/mL Methods of Calculating AUC: Counting method Weighing method Trapezoidal rule method Blood level equations Trapezoidal Rule Method The more data available, the more accurate the estimate of the AUC. Scarcity of data during the absorption phase and around the peak time will result in an underestimate of the AUC Scarcity of data during the distribution and elimination phases will result in an overestimate of the actual AUC AUC Determination by Trapezoidal Rule AUC Determination by Trapezoidal Method: EV Rule Method: IV Lag time - occurs at the beginning of AUC from time 0 to the last blood level systemic drug absorption point determined is composed of trapezoids ○ For some individuals, systemic absorption is delayed after oral drug administration because of delayed stomach emptying or other factors  Remaining Area (rest area) ○ The remaining area is calculated by dividing the last blood level point by k or under the assumption that this point is beyond the absorptive and the distributive phase on the terminal slope of the blood level AUCtx - ∞ = C /k or x Blood Level Equation AUC can be calculated from blood level equation if: the general blood level equation in a certain dose size is available a blood level curve can be well fitted and the intercepts with the ordinate and the rate constants are known AUC Determination from Blood AUC Determination from Blood Level Equations: IV Level Equations: EV For open one-compartment model, IV For open one-compartment model, EV route of administration, the AUC0-∞ can be route of administration, the AUC° can be calculated from the dose size, elimination rate calculated similarly to that of IV route of constant and the volume of distribution, since: administration 2. Urinary drug excretion studies Most accurate method of determining bioavailability if the active moiety is excreted unchanged in significant quantity in the urine The cumulative amount of active drug excreted in the urine (Du∞) Directly related to the extent of systemic drug absorption The rate of drug excretion in the urine (dDu/ dt) Directly related to the rate of systemic absorption The time for the drug to be completely excreted (t∞) Corresponds to the total time for the drug to be systemically absorbed and completely excreted after administration 3. Measurements of an ACUTE PHARMACOLOGIC EFFECTS ONSET TIME The time from administration to the MEC INTENSITY Proportional to the number of receptors occupied by the drug May occur before, after, or at peak concentration DURATION OF ACTION The time for which the drug concentration remains above the MEC THERAPEUTIC WINDOW The drug concentration range between the MEC and MTC  Factors Modifying Bioavailability Physiologically Modified Bioavailability Dosage Form Modified Bioavailability Age Particle size Sex Polymorphic form Physical state of the patient Presence of solvate or a hydrate Time of administration Chemical presentation of salts, ester, ether, Stomach emptying rate complexes Type and amount of food pH of dosage forms and environment pH and enzyme variations in GIT Solubility characteristics Motility of GIT Type and amount of vehicle substances Blood flow present Liver and kidney function Manufacturing method employed Body weight —------------------------------------------------------THE END—------------------------------------------------------ MODULE 8.1. Bioavailability vs Bioequivalence Bioequivalence Studies in New Drug Development (NDA) During drug development, bioequivalence studies are used to compare   early and late clinical trial formulations;  formulations used in clinical trials and stability studies, if different;  clinical trial formulations and to-be-marketed drug products, if different; and  product strength equivalence, as appropriate. Purpose of Bioavailability and Bioequivalence Studies Bioavailability and bioequivalence studies are important in the process of approving pharmaceutical products for marketing. Bioavailability is defined as the rate and extent to which the active ingredient or active moiety is absorbed from a drug product and becomes available at the site of action. On the ther hand, Bioequivalence is defined as the absence of a significant difference in the rate and extent to which the active ingredient or active moiety becomes available at the site of drug action when administered at the same molar dose under similar conditions in an appropriately designed study. Bioequivalence studies are used to compare the bioavailability of the same drug (same salt or ester) from various drug products. Bioavailability and bioequivalence can be considered as performance measures of the drug product in vivo. Relative and Absolute Availability A drug product’s bioavailability provides an estimate of the relative fraction of the administered dose that is absorbed into the systemic circulation. The AUC is considered the most reliable measure of a drug’s bioavailability, as it is directly proportional to the total amount of unchanged drug that reaches the systemic circulation. Absolute bioavailability compares the bioavailability of the active drug in the systemic circulation following extravascular administration with the bioavailability of the same drug following intravenous administration. Intravenous drug administration is considered 100% absorbed. In a relative bioavailability study, the systemic exposure of a drug in a designated formulation (generally referred to as treatment A or reference formulation) is compared with that of the same drug administered in a reference formulation (generally referred to as treatment B or test formulation).  Used in drug development include studies to characterize food effects and drug–drug interactions.  Used in developing new formulations of existing immediate-release drug products, such as new modified-release versions or new fixed-dose combination formulations.  Used for bridging formulations during drug development Methods for Assessing Bioavailability and Bioequivalence The FDA’s regulations list the following approaches to determining bioequivalence, in descending order of accuracy, sensitivity, and reproducibility:  In vivo measurement of active moiety or moieties in biological fluid (ie, a pharmacokinetic study)  In vivo pharmacodynamic (PD) comparison  In vivo limited clinical comparison  In vitro comparison  Any other approach deemed acceptable (by the FDA) MODULE 8.2. Biopharmaceutic Factors Influencing Bioavailability Biopharmaceutic Factors and Rationale For Drug Product Design In broad terms, the factors affecting drug bioavailability may be related to the formulation of the drug product or the biological constraints of the patient. Drugs are not usually given as pure chemical drug substances, but are formulated into finished dosage forms (ie, drug products). These drug products include the active drug substance combined with selected additional ingredients (excipients) that make up the dosage form. Although excipients are considered inert with respect to pharmacodynamic activity, excipients are important in the manufacture of the drug product and provide functionality to the drug product with respect to drug release and dissolution. Some common drug products include liquids, tablets, capsules, injectables, suppositories, transdermal systems, and topical creams and ointments. These finished dosage forms or drug products are then given to patients to achieve a specific therapeutic objective. The design of the dosage form, the formulation of the drug product, and the manufacturing process require a thorough understanding of the biopharmaceutic principles of drug delivery. Considerations in the design of a drug product to deliver the active drug with the desired bioavailability characteristics and therapeutic objectives include (1) the physicochemical properties of the drug molecule, (2) the finished dosage form (eg, tablet, capsule, etc), (3) the nature of the excipients in the drug product, (4) the method of manufacturing, and (5) the route of drug administration. Biopharmaceutics allows for the rational design of drug products and is based on:  The physical and chemical properties of the drug substance  The route of drug administration, including the anatomic and physiologic nature of the application site (eg, oral, topical, injectable, implant, transdermal patch, etc)  Desired pharmacodynamic effect (eg, immediate or prolonged activity)  Toxicologic properties of the drug annd Safety of excipients  Effect of excipients and dosage form on drug product performance  Manufacturing processes