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**PHARMACOKINETIC MODELS**

CHAPTER 3 PHARMACOKINETIC MODELS

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**PHARMACOKINETIC MODELING**

A Model is a hypothesis using mathematical terms to describe quantitative relationships MODELING REQUIRES: Thorough knowledge of anatomy and physiology Understanding the concepts and limitations of mathematical models. Assumptions are made for simplicity

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OUTCOME The development of equations to describe drug concentrations in the body as a function of time HOW? By fitting the model to the experimental data known as variables. A PK function relates an independent variable to a dependent variable.

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**FATE OF DRUG IN THE BODY ADME Excretion Oral Administration G.I. Tract**

Intravenous Injection Circulatory System Intramuscular Injection Tissues Metabolic Sites Subcutaneous Injection

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**Complexity of PK model will vary with:**

1- Route of administration 2- Extent and duration of distribution into various body fluids and tissues. 3- The processes of elimination. 4- Intended application of the PK model. We Always Choose the SIMPLEST Model

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**Types of PK Models 1- Physiologic (Perfusion) Models**

2- Compartmental Models 3- Mammillary Models

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**PHYSILOGIC PK MODELS Models are based on known physiologic**

and anatomic data. Blood flow is responsible for distributing drug to various parts of the body. Each tissue volume must be obtained and its drug conc described. Predict realistic tissue drug conc Applied only to animal species and human data can be extrapolated.

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**PHYSILOGIC PK MODELS Can study how physiologic factors may change**

drug distribution from one animal species to another No data fitting is required Drug conc in the various tissues are predicted by organ tissue size, blood flow, and experimentally determined drug tissue-blood ratios. Pathophysiologic conditions can affect distribution.

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**Physiological Model Simulation**

Perfusion Model Simulation of Lidocaine IV Infusion in Man Blood Metabolism RET Muscle Adipose Lung Time Percent of Dose

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COMPARTMENTAL MODELS The body is represented by a series of compartments that communicate reversibly with each other. 1 2 3 k12 k21

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**COMPARTMENTAL MODELS A compartment is not a real physiologic or**

anatomic region, but it is a tissue or group of tissues having similar blood flow and drug affinity. Within each compartment the drug is considered to be uniformly distributed. Drug move in and out of compartments Compartmental models are based on linear differential equations. Rate constants are used to describe drug entry into and out from the compartment.

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**COMPARTMENTAL MODELS The model is an open system since drug is**

eliminated from the system. The amount of drug in the body is the sum of drug present in the compartments. Extrapolation from animal data is not possible because the volume is not a true volume but is a mathematical concept. Parameters are kinetically determined from the data.

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MAMMILLARY MODELS Is the most common compartmental model used in PK. The model consists of one or more compartments connected to a central compartment 2 1 3 ka kel k12 k21

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**Intravenous and Extravascular Administration**

IV, IM, SC

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**Intravenous and extravascular Route of Administration**

Difference in plasma conc-time curve Time Cp Time Cp Intravenous Administration Extravascular Administration

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**One Compartment Open Model Intravenous Administration**

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**One Compartment Open Model Intravenous Administration**

The one compartment model offers the simplest way to describe the process of drug distribution and elimination in the body. Blood (Vd) i.v. Input kel Output When the drug is administered i.v. in a single rapid injection, the process of absorption is bypassed

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**One Compartment Open Model Intravenous Administration**

The one-compartment model does not predict actual drug levels in the tissues, but does imply that changes in the plasma levels of a drug will result in proportional changes in tissue drug levels.

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**FIRST-ORDER KINETICS The rate of elimination for most drugs is a**

first-order process. kel is a first-order rate constant with a unit of inverse time such as hr-1.

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Semi-log paper Plotting the data

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INTEGRATED EQUATIONS The rate of change of drug plasma conc over time is equal to: This expression shows that the rate of elimination of drug from the body is a first-order process and depends on kel

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**INTEGRATED EQUATIONS Cp = Cp0e-kelt ln Cp = ln Cp0 kelt**

DB = Dose . e-kelt ln DB = ln Dose kelt

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**Elimination Half-Life (t1/2)**

Is the time taken for the drug conc or the amount in the body to fall by one-half, such as Cp = ½ Cp0 or DB = ½ DB0 Therefore,

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**ESTIMATION OF half-life from graph**

A plot of Cp vs. time t1/2 = 3 hr

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**Fraction of the Dose Remaining**

The fraction of the dose remaining in the body (DB /Dose) varies with time. The fraction of the dose lost after a time t can be then calculated from:

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**Volume of Distribution (Vd)**

Is the volume in which the drug is dissolved in the body. Example: 1 gram of drug is dissolved in an unknown volume of water. Upon assay the conc was found to be 1mg/ml. What is the original volume of the solution? V = Amount / Conc = 1/1= 1 liter Also, if the volume and the conc are known, then the original amount dissolved can be calculated Amount = V X Conc= 1X1= 1 gram

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Apparent Vd It is called apparent because it does not have any physiological meaning. Drugs that are highly lipid soluble, such as digoxin has a very high Vd (600 liters), drugs which are lipid insoluble remain in the blood and have a low Vd. For digoxin, if that were a physiological space and I were all water, that would weigh about 1320 lb (599 kg).

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Apparent Vd Vd is the ratio between the amount of drug in the body (dose given) and the concentration measured in blood or plasma. Therefore, Vd is calculated from the equation: Vd = DB / CP where, DB = amount of drug in the body Cp = plasma drug concentration

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**For One Compartment Model with IV Administration:**

With rapid IV injection the dose is equal to the amount of drug in the body at zero time (DB). Where Cp is the intercept obtained by plotting Cp vs. time on a semilog paper.

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**For One Compartment Model with IV Administration:**

Cpo

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**Calculation of Vd from the AUC**

Since, dDB/dt = -kelD = -kelVdCp dDB = -kelVdCpdt dDB = -kelVd Cpdt Since, Cpdt = AUC Then, AUC = Dose / kelVd Model Independent Method

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**Significance of Vd Drugs can have Vd equal, smaller or greater than**

the body mass Drugs with small Vd are usually confined to the central compartment or highly bound to plasma proteins Drugs with large Vd are usually confined in the tissue Vd can also be expressed as % of body mass and compared to true anatomic volume Vd is constant but can change due to pathological conditions or with age

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Apparent Vd Example: if the Vd is 3500 ml for a subject weighing 70 kg, the Vd expressed as percent of body weight would be: The larger the apparent Vd, the greater the amount of drug in the extravascular tissues. Note that the plasma represents about 4.5% of the body weight and total body water about 60% of body weight.

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CLEARANCE (Cl) Is the volume of blood that is cleared of drug per unit time (i.e. L/hr). Cl is a measure of drug elimination from the body without identifying the mechanism or process. Cl for a first-order elimination process is constant regardless of the drug conc.

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INTEGRATED EQUATIONS

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**ESTIMATION OF PK PARAMETERS**

A plot of Cp vs. time Cpo kel

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Author: Julia Richards and R. Scott Hawley

Author: Julia Richards and R. Scott Hawley

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