# PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Student Objectives for this Chapter PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models To draw the scheme and write the differential equations appropriate to a multi-compartment pharmacokinetic model To recognize and use the integrated equations for these models to calculate parameter values and for dosage regimen calculations To calculate the parameters of these models using the method of residuals So far we have talked about the pharmacokinetics of drugs in terms of a one compartment model. We have assumed that the drug, once administered is mixed instantaneously in the blood and that the drug distributes throughout the body rapidly reaching equilibrium throughout the tissue into which the drug enters. We have in essence considered that the body acts as a well mixed container. Figure Linear Plot of Cp Versus Time for a One-Compartment - IV Bolus With first order drug elimination we found that the plasma concentration will fall monoexponentially with time following IV bolus administration. Page 1 of 23

2 And the log of the plasma concentration will fall as a straight line. Figure Semi-Log Plot of Cp Versus Time Figure Semi-Log Plot of Cp Versus Time. Two-Compartment - IV Bolus. Note Fast and Slow Processes Commonly we find with real data, especially if we have a number of early data points, that the log Cp versus time plot is not a straight line. We see an initial early deviation from the straight line, followed by a log-linear phase. The initial phase is a more rapid drop in plasma concentration before settling into the log-linear fall in plasma concentration. This suggests that the body is not behaving as a single well mixed compartment. There appears, mathematically, to be distribution between two (or more) compartments. That is we don't have instantaneous equilibrium between the drug in all the various tissues of the body. In the next approximation we can consider that the body is behaving as two distinct compartments. These compartments can be called the central compartment and the peripheral compartment. Exact anatomical assignment to these compartments is not always possible. However, generally the rapidly perfused tissues often belong in the central compartment. This page ( was last modified: Wednesday 02 Oct 2013 at 12:03 PM Material on this website should be used for Educational or Self-Study Purposes Only Copyright David W. A. Bourne TP Page 2 of 23

3 Intravenous Administration Scheme or diagram PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models Figure Two Compartment Pharmacokinetic Model Clearance Model Equilibrium Model Differential equation The differential equation for drug in the central compartment following intravenous bolus administration is:- Equation Differential Equation for the Central Compartment The kel X1 term describes elimination of the drug from the central compartment, while the k12 X1 and k21 X2 terms describe the distribution of drug between the central and peripheral compartments. Writing differential equations can be reviewed in Chapter 2. Integrated equation Integration of this equation (using Laplace transforms) leads to a biexponential equation for plasma concentration as a function of time, Equation Equation Integrated Equation for Plasma Concentration versus Time with α > β and Equation Integrated Equation for Cp versus Time including k21 and V 1 Page 3 of 23

4 Equation Calculating values for A and B The A, B, α, and β terms were derived from the micro-constants during the integration process. They are functions of the micro-constant k12, k21, kel and V1 Using the substitutions for the sum and product of α and β. α + β = kel + k12 + k21 α β = kel k21 If we know the values of kel, k12 and k21 we can calculate α + β as well as α β. Substituting these values into Equation gives us values for α and β. Equation Converting from kel, k12 & k21 to α & β Note, in this equation, α is calculated when '+' is used in the numerator and β is calculated when '-' is used in place of the '±'. Thus α is greater than β. Once we have values for α and β we can calculate values for A and B using Equation An example calculation (from the homework) "A drug follows first order (i.e. linear) two compartment pharmacokinetics. After looking in the literature we find a number of parameter values for this drug. These numbers represent the micro constants for this drug. In order that we can calculate the drug concentration after a single IV bolus dose these parameters need to be converted into values for the macro constants. The kel and V1 for this drug in this patient (90.5 kg) are hr -1 and 0.39 L/kg, respectively. The k12 and k21 values this drug are 1.86 and 1.68 hr -1, respectively. What is the plasma concentration of this drug 1.5 hours after a 500 mg, IV Bolus dose. In order to complete this calculation first calculate the appropriate A, B, α and β values." Since α + β = kel + k12 + k21 = = and α x β = kel x k21 = x 1.68 = Now: α = [(a+b) + sqrt((a+b) 2-4xaxb)]/2 = [ sqrt( x )]/2 = [ ]/2 = hr -1 β = [ ]/2 = hr -1 A = Dose x (α - k21)/[v1 x (α - b)] = 500 x ( )/[90.5 x 0.39 x ( )] = 500 x 1.963/[ x ] = mg/l B = Dose x (k21 - b)/[v1 x (α - b)] = 500 x ( )/[ x ] = mg/l The last step is Cp = α x e (-α x t) + β x e (-β x t) = x e ( x 1.5) x e ( x 1.5) = x x = = 5.59 mg/l Page 4 of 23

5 Later in this chapter we will use equations for the reverse process of converting α, β, A and B into values for k12, k21, kel and V 1. Page 5 of 23

6 Calculator Calculate A, B, α and β Dose: 500 V 1 : 25 kel: 0.2 k21: 1.5 k12: 2.0 A is: α is: B is: β is: Calculate This page ( was last modified: Wednesday 02 Oct 2013 at 10:50 AM Material on this website should be used for Educational or Self-Study Purposes Only Copyright David W. A. Bourne Page 6 of 23

7 Parameter Determination Method of residuals PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models Values for kel, k12, k12 and other parameters can be determined by first calculating A, B, α, and β. For this we can use the method of residuals (in a similar fashion to determining ka and kel for the one compartment model after oral administration). Starting with the equation for Cp. Equation Concentration versus time after an IV Bolus Dose, Two Compartment Model By definition α is greater than β then as t approaches, e -α t approaches 0 faster than e -β t. Therefore if the ratio α/β is large enough (greater than 5) the terminal data points will fall on the line Equation Equation for Cp late versus time This equation is similar to the equation for the late plasma concentration values after oral administration with a one compartment model. This will be a straight line if plotted on semi-log graph paper. Figure Semi-Log Plot of Cp Versus Time Showing Cp late Extrapolated Back to B From the slope of this line a value of β can be determined. Equation Determining β from the Cp late Line The units for β and α, below, are reciprocal time, for example min -1, hr -1, etc. Page 7 of 23

8 Biological half-life or Terminal half-life The t 1/2 calculated as 0.693/β is often called the biological half-life or terminal half-life. It is the half-life describing the terminal elimination of the drug from plasma. [For the one compartment model the biological half-life was equal to 0.693/kel]. The difference between the Cp late values (red line) at early times and the actual data at early times is again termed the 'residual' Equation Equation for Residual versus time Figure Semi-Log Plot of Cp Versus Time Showing Residual Line and Cp late Line Click on the figure to view as an interactive graph or a Interactive graph as a Linear Plot The slope of the residual line (green line) will provide the value of α and A can be estimated as the intercept of the concentration axis (yaxis). A more accurate value for the α value can be determined by expanding the scale on the time axis (Figure ). Don't forget to use the new time values when calculating α from the equation Equation Determining α from the Residual Line Page 8 of 23

9 Figure Semi-Log Plot of Cp Versus Time Showing Residual Line and Cp Data - NOTE the expansion of the time axis (x axis) Converting macro constants to micro constants With A, B, α, and β determined using the method of residuals we can calculate the micro-constants from the equations. Equation Converting from A, B, α & β to kel, k12 & k21 Page 9 of 23

10 Calculator Calculate k10, k12, k21 and V 1 Dose: 500 A: 24 α: 1.5 B: 6 β: 0.25 kel is: k12 is: k21 is: V 1 is: Calculate This page ( was last modified: Wednesday 26 May 2010 at 08:59 AM Material on this website should be used for Educational or Self-Study Purposes Only Copyright David W. A. Bourne Page 10 of 23

11 PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models Effect of k12 and k21 on Drug Concentration versus Time Changing the Ratio of k12 to k21 Figure Plot of Cp versus Time Showing the Effect of Different k12/21 Ratio Values Click on the figure to view as an interactive graph or a Interactive graph as a Semi-log Plot From the k12 and k21 values we can assess the extent of distribution of drug into the peripheral compartment. The higher the ratio k12/k21 the greater the distribution of drug into the peripheral compartment. The larger the individual values of k12 and k21 the faster is the transfer between the central and peripheral compartments and the more the body behaves as a single compartment. As the ratio increases the distribution phase is more pronounced. Conversely with the ratio 1/4 there is very little distribution phase. Also note that the β value or the slope of the terminal phase is changing even though the kel is fixed at 0.2 hr -1. Changing the Magnitude of k12 and k21 with the Same Ratio With faster and faster distribution the initial drop in plasma concentration becomes quite rapid. If you were sampling every 30 minutes, the initial phase would be missed. The data would look just like a one compartment model. Redrawing the slow plot with k12/k21 (0.5/0.25) over 24 hours and gives a plot that is definitely still biexponential. Page 11 of 23

12 Figure Plot of Cp versus Time Showing the Effect of Different k12/21 Magnitudes Click on the figure to view as an interactive graph or a Interactive graph as a Semi-log Plot Simulate other concentration versus time curves after IV bolus administration with the two compartment pharmacokinetic model using macro constants (A, B, α and β) Linear plot - Semi-log plot or micro constants (kel, k12, k21, and V1) Linear plot - Semi-log plot This page ( was last modified: Wednesday 26 May 2010 at 08:59 AM Material on this website should be used for Educational or Self-Study Purposes Only Copyright David W. A. Bourne Page 12 of 23

13 PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models Apparent Volumes of Distribution The concentration of drug in the body is determined not only by the rate constant values but also by the apparent volume of distribution. In the case of the two compartment model a number of volume terms can been defined. V 1 The apparent volume of the central compartment, V 1 or V c, can be calculated as: Equation Apparent Volume of Central Compartment This parameter is important because it allows the calculation of the highest plasma concentration or Cp 0 after an IV bolus administration. This concentration may result in transient toxicity. V 1 can also be used in dose calculations. V area ( = V ß ) V area or V ß is defined as: Equation Apparent Volume, V area Because of the relationship with clearance and β and with V 1 and kel this parameter is quite useful in dosing calculations. This parameter can be readily calculated via AUC and β values from the 'raw' data and is therefore commonly quoted. V ss V ss, V steady state defined as: Equation Apparent Volume, Steady State This term relates the total amount of drug in the body at 'steady state' with the concentration in plasma or blood Page 13 of 23

14 Figure Plot of X1 (Plasma) and X2 (Tissue) Compartment Concentrations, Showing 'Steady State' with Both Lines Parallel The relationship between volume terms is that: V area > V ss > V 1 And for a one compartment model the values for all these parameters are equal. Example Calculation As an example we can look at the data in the table below. Time (hr) Concentration (mg/l) Cplate (mg/l) Residual (mg/l) Table Two Compartment Pharmacokinetics Page 14 of 23

15 Figure Plot of Cp versus Time Illustrating the Method of Residuals The first two columns are the time and plasma concentration which may be collected after IV bolus administration of 500 mg of drug. These data are plotted (n) in Figure above. At longer times, after 4 hours, out to 12 hours the data appears to follow a straight line on semi- log graph paper. Since α > β this terminal line is described by B e -β t. Following it back to t = 0 gives B = 10 mg/l. From the slope of the line β = 0.25 hr -1. Cp late values at early times are shown in column 3 and the residual in column 4. The residual values are plotted (o) also giving a value of A = 25 mg/l and α = 1.51 hr -1 Note that α/β = 6, thus these values should be fairly accurate. B = 10 mg/l, β = (ln 10 - ln 0.5)/12 = 2.996/12 = 0.25 hr -1 A = 25 mg/l, α = (ln 25 - ln 0.27)/3 = 4.528/3 = 1.51 hr -1 Therefore Cp = 25 e t + 10 e t We can now calculate the micro-constants. The AUC by the trapezoidal rule + Cp last /β = = 58.3 mg.hr.l -1, [Note the use of β] thus Page 15 of 23

16 Notice that V area > V ss > V 1 [34.3 > 26.7 > 14.3] Want more practice with this type of problem! This page ( was last modified: Wednesday 26 May 2010 at 08:59 AM Material on this website should be used for Educational or Self-Study Purposes Only Copyright David W. A. Bourne Page 16 of 23

17 Dosage Calculations PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models Dosage calculations are complicated by the extra terms in the equations however some calculations are still reasonably straightforward. The dose required for a particular initial plasma concentration can be calculated if V 1 is known. Thus: Equation Equation for Loading Dose To achieve an initial Cp of 20 mg/l given V 1 = 30 liter would require a DOSE = 20 * 30 = 600 mg. Alternately if a dose of 500 mg is given and the V 1 value is 16 L, the expected Cp 0 can be calculated. Cp 0 = 500/16 = 31.3 mg/l If the A, B, α, and β values are known or calculated, then the plasma concentration at any time after a single IV dose can be calculated. The plasma concentration achieved after a continuous IV infusion is given by the same equation described for the one compartment model, i.e.: Equation Equation for Maintenance Infusion Rate If a plasma concentration of 30 mg/l is required and V 1 = 15 L and kel is 0.2 hr -1 then the required infusion rate can be readily determined. k0 = 30 x 15 x 0.2 = 90 mg/hr Page 17 of 23

18 Figure Linear Plot of Cp Versus Time With IV Bolus and Infusion Click on the figure to view as an interactive graph or a Interactive graph as a Semi-log Plot Since the time to reach the steady state concentration is controlled by the β value this could mean a slow approach to the desired value, thus an IV bolus loading dose may be useful. Unfortunately this calculation is not straight forward as you will see if you explore the applet. With V 1 = 15 L, kel = 0.2 hr -1, and required Cp = 30 mg/l Bolus DOSE = 15 x 30 = 450 mg and Infusion Rate = k0 = 30 x 15 x 0.2 = 90 mg/hr Figure Linear Plot after an IV Infusion and a Higher or Lower Bolus Dose As you can see (Figure and above) this gives quite a dip in the Cp versus time curve. With Bolus DOSEs, either 600 or 300 mg (shown in Figure ) the curves may or may not be better depending on the therapeutic range of the drug. Page 18 of 23

19 Figure Linear Plot of Cp versus Time With Fast and Slow Infusion Click on the figure to view as an interactive graph or a Interactive graph as a Semi-log Plot Another alternative is to give a fast infusion followed by the maintenance infusion. Here 1200 mg was given over 4 hours (at 300 mg/hr) before switching to the slower 90 mg/hr maintenance rate. Try a dosing problem with a drug following a two compartment pharmacokinetic model. This page ( was last modified: Wednesday 26 May 2010 at 08:59 AM Material on this website should be used for Educational or Self-Study Purposes Only Copyright David W. A. Bourne Page 19 of 23

20 Oral Administration PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models Following oral administration of a drug with two compartment characteristics, Cp is described by an equation with three exponential terms. The model is shown in Diagram Diagram Scheme for Oral Two-Compartment Pharmacokinetic Model Differential equation Equation Differential Equation for Drug Amount in the Body after Oral Administration where A + B + C = 0 Equation Integrated Equation for Drug Amount in the Body after Oral Administration Page 20 of 23

21 Figure Semi-Log Plot Showing Pronounced Distribution Figure Semi-Log Plot Without Distribution Phase Evident Page 21 of 23

22 Bioavailability Bioavailability calculations are the same as for the one compartment model, i.e., by comparison of AUC or U. These apply for any linear system. Also if α, β, and ka are sufficiently separated the method of residuals can be applied (twice) to determine values for these three parameters. Average Plasma Concentration The average plasma concentration equation can also be used to calculate appropriate dosing regimens. For example if an average plasma concentration of 20 mg/l is required and V1 = 15 L, kel = 0.15 hr -1, F = 0.9 and a dosing interval of 12 hours is to be used then the required dose can be calculated from the equation for Cp average. Equation Equation for Average Plasma Concentration The required dose can be calculated using Equation and the data provided. Thus Page 22 of 23

### PHAR 7633 Chapter 21 Non-Linear Pharmacokinetic Models

Student Objectives for this Chapter PHAR 7633 Chapter 21 Non-Linear Pharmacokinetic Models To draw the scheme and write the differential equations for compartmental pharmacokinetic models with non-linear

### IV solutions may be given either as a bolus dose or infused slowly through a vein into the plasma at a constant or zero-order rate.

د.شيماء Biopharmaceutics INTRAVENOUS INFUSION: IV solutions may be given either as a bolus dose or infused slowly through a vein into the plasma at a constant or zero-order rate. The main advantage for

### Calculation of Bioavailability Parameters. Objectives. Determine of ka. 20 July Chapter Estimation

Calculation of Bioavailability Parameters Objectives Determine ka using the Method of Residuals Wagner-Nelson Method Method of Inspection as Appropriate to calculate ka Calculate F using Plasma or Urine

### Extravascular routes of drug administration

Copyright Pharmaceutical Press www.pharmpress.com 6 Extravascular routes of drug administration 6.1 Introduction 106 6.2 Drug remaining to be absorbed, or drug remaining at the site of administration 6.3

### PHAR 7633 Chapter 22 Non-Linear Regression Analysis of Pharmacokinetic Data Individual Data and Population Analysis

PHAR 7633 Chapter 22 Non-Linear Regression Analysis of Pharmacokinetic Data Individual Data and Population Analysis Student Objectives for this Chapter Understand the use of computer programs such as Boomer

### Drug Metabolism and Pharmacokinetics in Drug Discovery: A Primer For Bioanalytical Chemists, Part II

Drug Metabolism and Pharmacokinetics in Drug Discovery: A Primer For Bioanalytical Chemists, Part II Chandrani Gunaratna, Ph.D. Bioanalytical Systems, Inc. 2701 Kent Avenue West Lafayette, IN 47906-1382

### Absorption and Half-Life. Objectives. Absorption. Extent of Absorption. Rate of Absorption

Absorption and Half-Life Nick Holford Dept Pharmacology & Clinical Pharmacology University of Auckland, New Zealand Objectives Understand the physiological determinants of rate and extent of absorption

### Statistics and Pharmacokinetics in Clinical Pharmacology Studies

Paper ST03 Statistics and Pharmacokinetics in Clinical Pharmacology Studies ABSTRACT Amy Newlands, GlaxoSmithKline, Greenford UK The aim of this presentation is to show how we use statistics and pharmacokinetics

### Pharmacokinetics & Pharmacodynamics of Controlled Release Systems

Pharmacokinetics & Pharmacodynamics of Controlled Release Systems Presented By: Govardhan.P Dept. of pharmaceutics, University College of Pharmaceutical Sciences, Kakatiya University, Warangal-506009.

### Mathematical Expressions of the Pharmacokinetic and Pharmacodynamic Models implemented in the PFIM software

Mathematical Expressions of the Pharmacokinetic and Pharmacodynamic Models implemented in the PFIM software Anne Dubois, Julie Bertrand and France Mentré UMR738, INSERM, University Paris Diderot Programmer:

### Pharmacokinetics in Toxicology Research

Pharmacokinetics in Toxicology Research Center for Human Health Assessment A Course on Physiologically Based Pharmacokinetic (PBPK) Modeling and Risk Assessment February 11 February 15, 2008 Copyright

### 1: CLINICAL PHARMACOKINETICS

: CLINICAL PHARMACOKINETICS General overview: clinical pharmacokinetics, 2 Pharmacokinetics, 4 Drug clearance (CL), 6 Volume of distribution (Vd), 8 The half-life (t½), 0 Oral availability (F), 2 Protein

### Plasma Drug Concentration Time Profile Plotting Data

UNIT 2 PHARMACOKINETICS-BASIC CONSIDERATIONS Plasma Drug Concentration Time Profile Plotting Data S. SANGEETHA., M.PHARM., (Ph.d) Department of Pharmaceutics SRM College of Pharmacy SRM University INTRODUCTION

### PHY 192 Absorption of Radiation Spring

PHY 192 Absorption of Radiation Spring 2010 1 Radioactivity II: Absorption of Radiation Introduction In this experiment you will use the equipment of the previous experiment to learn how radiation intensity

### Appendix E: Graphing Data

You will often make scatter diagrams and line graphs to illustrate the data that you collect. Scatter diagrams are often used to show the relationship between two variables. For example, in an absorbance

Exploring Radioactive Decay Use the accompanying interactive Excel spreadsheet to address the questions that follow. The yellow cells and sliders ( ) are adjustable variables. First let s consider the

### Questions and Answers for Health Care Providers: Renal Dosing and Administration Recommendations for Peramivir IV

Questions and Answers for Health Care Providers: Renal Dosing and Administration Recommendations for Peramivir IV The purpose of this document is to provide additional clarification to the existing information

### A Determination of g, the Acceleration Due to Gravity, from Newton's Laws of Motion

A Determination of g, the Acceleration Due to Gravity, from Newton's Laws of Motion Objective In the experiment you will determine the cart acceleration, a, and the friction force, f, experimentally for

### A short introduction to pharmacokinetics

European Review for Medical and Pharmacological Sciences 2002; 6: 33-44 A short introduction to pharmacokinetics R. URSO, P. BLARDI *, G. GIORGI Dipartimento di Farmacologia Giorgio Segre, University of

### Chemical Kinetics. 2. Using the kinetics of a given reaction a possible reaction mechanism

1. Kinetics is the study of the rates of reaction. Chemical Kinetics 2. Using the kinetics of a given reaction a possible reaction mechanism 3. What is a reaction mechanism? Why is it important? A reaction

### Functions and straight line graphs. Jackie Nicholas

Mathematics Learning Centre Functions and straight line graphs Jackie Nicholas c 004 University of Sydney Mathematics Learning Centre, University of Sydney 1 Functions and Straight Line Graphs Functions

### Section 14 Simple Linear Regression: Introduction to Least Squares Regression

Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship

### Graphing Linear Equations

Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

### Understanding and using logs and exponential equations

Essential Maths for Medics and Vets Page of 5 Module. Understanding and using logs and exponential equations M What is e? Why natural logs? e is just a number, in fact it is roughly.7888.. You know about

### WEEK #2: Cobwebbing, Equilibria, Exponentials and Logarithms

WEEK #2: Cobwebbing, Equilibria, Exponentials and Logarithms Goals: Analyze Discrete-Time Dynamical Systems Logs and Exponentials Textbook reading for Week #2: Read Sections 1.6 1.7 2 Graphical Analysis

### e = random error, assumed to be normally distributed with mean 0 and standard deviation σ

1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.

### Sample Exercise 14.1 Calculating an Average Rate of Reaction

Sample Exercise 14.1 Calculating an Average Rate of Reaction From the data given in the caption of Figure 14.3, calculate the average rate at which A disappears over the time interval from 20 s to 40 s.

### MATHEMATICAL NOTES #1 - DEMAND AND SUPPLY CURVES -

MATHEMATICAL NOTES #1 - DEMAND AND SUPPLY CURVES - Linear Equations & Graphs Remember that we defined demand as the quantity of a good consumers are willing and able to buy at a particular price. Notice

### IVIVC Methods and Applications in MR Product Development

IVIVC Methods and Applications in MR Product Development H. Rettig LLC www.ivivc.com Purpose of IVIVC and BCS Reduction of regulatory burden: IVIVC in lieu of additional in vivo experiments, leading to

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

### EXCEL Tutorial: How to use EXCEL for Graphs and Calculations.

EXCEL Tutorial: How to use EXCEL for Graphs and Calculations. Excel is powerful tool and can make your life easier if you are proficient in using it. You will need to use Excel to complete most of your

### How to Use Semilog Graph Paper

How to Use Semilog Graph Paper In semilog paper the C-axis is marked off so that the height represents the log of the value to be graphed. If you look at a semilog graph you'll notice that the horizontal

### An Area Correction Method To Reduce Intrasubject Variability In Bioequivalence Studies

An Area Correction Method To Reduce Intrasubject Variability In Bioequivalence Studies Hisham Y. Abdallah 1 Division of Biopharmaceutics, Center for Drug Evaluation and Research, United States Food and

### [ A] 2. [ A] 2 = 2k dt. [ A] o

Chemistry 360 Dr Jean M Standard Problem Set 3 Solutions The reaction 2A " P follows second-order kinetics The rate constant for the reaction is k350 0 4 Lmol s Determine the time required for the concentration

### Dosing Proton Pump Inhibitors (PPIs) in Infants and Young Children

Dosing Proton Pump Inhibitors (PPIs) in Infants and Young Children This document is intended to be a bridge between parents and physicians to improve the treatment of acid related disorders in children

### Study (s) Degree Center Acad. Period. 1201 - Grado de Farmacia FACULTY OF PHARMACY 3 Annual 1211 - PDG Farmacia-Nutrición Humana y Dietética

COURSE DATA Data Subject Código 34081 Name Biopharmacy and Pharmacokinetics Cycle Grade ECTS Credits 10.5 Curso académico 2014-2015 Study (s) Degree Center Acad. Period year 1201 - Grado de Farmacia FACULTY

### 7.6 Approximation Errors and Simpson's Rule

WileyPLUS: Home Help Contact us Logout Hughes-Hallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration

### Introduction to PK/PD modelling

Introduction to PK/PD modelling with focus on PK and stochastic differential equations Stig Mortensen, Anna Helga Jónsdóttir, Søren Klim and Henrik Madsen November 19, 2008 DTU Informatics DTU Informatics

### Chemical Kinetics. Reaction Rate: The change in the concentration of a reactant or a product with time (M/s). Reactant Products A B

Reaction Rates: Chemical Kinetics Reaction Rate: The change in the concentration of a reactant or a product with time (M/s). Reactant Products A B change in number of moles of B Average rate = change in

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

Radioactivity Lab Introduction In radioactive decay, a 'parent' nucleus is transformed into a 'daughter' nucleus while emitting radiation (alpha, beta, or gamma rays). Alpha particles are helium nuclei,

### What is Bioavailability and Bioequivalence?

What is Bioavailability and Bioequivalence? All generic medicines in New Zealand are approved by Medsafe and have been shown to be bioequivalent to innovator medicines, according to internationally accepted

### GRAPHING WITH LOGARITHMIC PAPER

Page 1 of 5 GRAPHING WITH LOGARITHMIC PAPER Department of Physics, University of Guelph Guelph, Ontario, Canada http://www.physics.uoguelph.ca Go to this site to print free graph paper (en francais aussi):

### POLYNOMIAL AND MULTIPLE REGRESSION. Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model.

Polynomial Regression POLYNOMIAL AND MULTIPLE REGRESSION Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model. It is a form of linear regression

### BIOAVAILABILITY & BIOEQUIVALENCE TRIALS

BIOAVAILABILITY & BIOEQUIVALENCE TRIALS Shubha Rani,, Ph.D. Technical Director & Head-Biometrics and Data Management Synchron Research Services Pvt. Ltd. Ahmedabad 380 054 drshubha@synchronresearch.com

### Electrochemical Kinetics ( Ref. :Bard and Faulkner, Oldham and Myland, Liebhafsky and Cairns) R f = k f * C A (2) R b = k b * C B (3)

Electrochemical Kinetics ( Ref. :Bard and Faulkner, Oldham and Myland, Liebhafsky and Cairns) 1. Background Consider the reaction given below: A B (1) If k f and k b are the rate constants of the forward

### AP CHEMISTRY CHAPTER REVIEW CHAPTER 11: RATE OF REACTION

AP CHEMISTRY CHAPTER REVIEW CHAPTER 11: RATE OF REACTION You should understand the definition of reaction rate, as well as how rates might be measured in a laboratory setting. You should know the difference

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### In order to describe motion you need to describe the following properties.

Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1-D path speeding up and slowing down In order to describe motion you need to describe the following properties.

### Introduction to PK/PD modelling - with focus on PK and stochastic differential equations

Downloaded from orbit.dtu.dk on: Jan 06, 2017 Introduction to PK/PD modelling - with focus on PK and stochastic differential equations Mortensen, Stig Bousgaard; Jónsdóttir, Anna Helga; Klim, Søren; Madsen,

### Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2).

Chapter Section : Equations of Lines Answers to Problems For problems -, put our answers into slope intercept form..* Find the equation of the line with slope, and passing through the point (,0).. Find

### Drug Prescribing in Kidney Disease: Initiative for Improved Dosing

Drug Prescribing in Kidney Disease: Initiative for Improved Dosing Effects of impaired kidney function on drug pharmacokinetics and pharmacodynamics Section Leaders: William Bennett and Domenic Sica Passage

### Volume of Distribution

1 Volume of Distribution Nick Holford Dept Pharmacology & Clinical Pharmacology University of Auckland, New Zealand 2 Objectives Learn the definition of volume of distribution Understand the physiological

### 2. Simple Linear Regression

Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

### 2. Order: Nipride 500 mg IV in 250 ml D5w at 2 mcg/kg/min for a patient weighing 125 lb. Administer at ml/hr

1 Pharmacology Review Quiz: 1. Order: Xylocaine 1 g IV in 500 ml D5W at 3 mg/min. Set the infusion pump at ml/hr 2. Order: Nipride 500 mg IV in 250 ml D5w at 2 mcg/kg/min for a patient weighing 125 lb.

Today we re going to talk about Radioactive Processes 1 In this lecture, we will talk about Radioactive Equilibrium. We re going to define what we mean by Radioactive Equilibrium and identify when it occurs.

### NONMEM A Programmer point of view

Paper PP12 NONMEM A Programmer point of view Raghu Kishor Koukuntla, Cytel, Hyderabad, India ABSTRACT: Non Linear Mixed Effects Modeling software called as NONMEM is a leading tool used for population

### DATA INTERPRETATION AND STATISTICS

PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

### DOSE-EFFECT RELATIONSHIP

DOSE-EFFECT RELATIONSHIP A fundamental principle of pharmacology is that the intensity of effect produced by a drug is a function of the quantity of drug administered (or the concentration of the drug

### Intraosseous Vascular Access and Lidocaine

Intraosseous Vascular Access and Lidocaine Intraosseous (IO) needles provide access to the medullary cavity of a bone. It is a technique primarily used in emergency situations to administer fluid and medication

### Population Pharmacokinetics and Pharmacodynamics of Cisplatinum During Intraperitoneal Chemohyperthermia Using a Closed Abdominal Procedure

Population Pharmacokinetics and Pharmacodynamics of Cisplatinum During Intraperitoneal Chemohyperthermia Using a Closed Abdominal Procedure Eddy Cotte, Brigitte Tranchand, Annie-Claude Beaujard, François-Noël

### Read the following information carefully before attempting the examples and exercises. TABLETS Formula method. = No. of tablets to be given

drug calculations Read the following information carefully before attempting the examples and exercises. TABLETS Required dose No. of tablets to be given Required dose : Required tablet : Stock tablet

### Univariate Regression

Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

### Day 3 Exercise 2 Parameter Optimization and Sensitivity Analysis: An Example Using a PBPK Model for Warfarin. April 19-23, 2010.

Day 3 Exercise 2 Parameter Optimization and Sensitivity Analysis: An Example Using a PBPK Model for Warfarin A Course on Physiologically Based Pharmacokinetic (PBPK) Modeling and In Vitro to In Vivo Extrapolation

### language Vectors, Scalars, Distance, Displacement, Speed, Velocity, Acceleration

I. Mechanics the study of the motion of objects introduction to the language Vectors, Scalars, Distance, Displacement, Speed, Velocity, Acceleration 1 Describing motion is a mathematical science. The underlying

### Simple linear regression

Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

### Regulation of Bioavailability and Bioequivalence Studies

Regulation of Bioavailability and Bioequivalence Studies (Amended Date: 2015-03-06) 1.Promulgated on April 02, 2009 2.Amended on January 31, 2013 3.Amended on March 6, 2014. Revised articles include Article

### Study Resources For Algebra I. Unit 1C Analyzing Data Sets for Two Quantitative Variables

Study Resources For Algebra I Unit 1C Analyzing Data Sets for Two Quantitative Variables This unit explores linear functions as they apply to data analysis of scatter plots. Information compiled and written

### Appendix C: Graphs. Vern Lindberg

Vern Lindberg 1 Making Graphs A picture is worth a thousand words. Graphical presentation of data is a vital tool in the sciences and engineering. Good graphs convey a great deal of information and can

### Determination of g using a spring

INTRODUCTION UNIVERSITY OF SURREY DEPARTMENT OF PHYSICS Level 1 Laboratory: Introduction Experiment Determination of g using a spring This experiment is designed to get you confident in using the quantitative

### Slope-Intercept Equation. Example

1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

### Algebra Based Physics Radioactivity & Half-life Lab

Algebra Based Physics Radioactivity & Half-life Lab I. Radioactive Decay Analogy The purpose of this activity is to provide a model for the process of radioactive decay using objects that can be seen easily.

### Francis Micheal et al /J. Pharm. Sci. & Res. Vol.4(8), 2012,

Concepts of Bioequivalence and it s Impact on Truncated Area Under Curve (AUC) of Drugs with Long Half Life in point estimate and intra-subject variability Francis Micheal*, S. Saranya, N. Aparna, Dr Sridevi,

### Lesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2 Method 2: Rewriting the equation in slope intercept form Use the same strategies that were used for solving equations: 1. 2. Your goal is to solve

### 2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

### Scientific Graphing in Excel 2010

Scientific Graphing in Excel 2010 When you start Excel, you will see the screen below. Various parts of the display are labelled in red, with arrows, to define the terms used in the remainder of this overview.

### Nursing 113. Pharmacology Principles

Nursing 113 Pharmacology Principles 1. The study of how drugs enter the body, reach the site of action, and are removed from the body is called a. pharmacotherapeutics b. pharmacology c. pharmacodynamics

### Modern Physics Laboratory Geiger-Muller Tube Characteristics and Radioactive Decay

Modern Physics Laboratory Geiger-Muller Tube Characteristics and Radioactive Decay Josh Diamond and John Cummings Fall 2009 Abstract In this experiment the influence of tube voltage and resolving time

### Experiment 8 RC Circuits

Experiment 8 ircuits Nature, to be commanded, must be obeyed. F. Bacon (1561-1626) OBJETIVE To study a simple circuit that has time-dependent voltages and current. THEOY ircuits with steady currents and

### Resistance and Ohm s Law - MBL

Resistance and Ohm s Law - MBL In this experiment you will investigate different aspects of Ohm s Law, which relates voltage, current, and resistance. A computer will be used to collect, display, and help

### The Cartesian Plane The Cartesian Plane. Performance Criteria 3. Pre-Test 5. Coordinates 7. Graphs of linear functions 9. The gradient of a line 13

6 The Cartesian Plane The Cartesian Plane Performance Criteria 3 Pre-Test 5 Coordinates 7 Graphs of linear functions 9 The gradient of a line 13 Linear equations 19 Empirical Data 24 Lines of best fit

### CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

### Non-Linear Regression 2006-2008 Samuel L. Baker

NON-LINEAR REGRESSION 1 Non-Linear Regression 2006-2008 Samuel L. Baker The linear least squares method that you have een using fits a straight line or a flat plane to a unch of data points. Sometimes

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)

CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

### Statistical estimation using confidence intervals

0894PP_ch06 15/3/02 11:02 am Page 135 6 Statistical estimation using confidence intervals In Chapter 2, the concept of the central nature and variability of data and the methods by which these two phenomena

### Procedure In each case, draw and extend the given series to the fifth generation, then complete the following tasks:

Math IV Nonlinear Algebra 1.2 Growth & Decay Investigation 1.2 B: Nonlinear Growth Introduction The previous investigation introduced you to a pattern of nonlinear growth, as found in the areas of a series

### Medication Removal by Apheresis. Yanyun Wu, M.D., Ph.D. Yale University School of Medicine

Medication Removal by Apheresis Yanyun Wu, M.D., Ph.D. Yale University School of Medicine 1 Objectives Review basic pharmacokinetics and its relevance in drug removal by therapeutic apheresis (TPE) Review

### Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

### 1/17/2013. Dose Response and Concentration Response Analysis of Drug Effects

Dose Response and Concentration Response Analysis of Drug Effects Juan J. L. Lertora. M.D., Ph.D. IH Clinical Center January 17, 213 1 DOSE-EFFECT RELATIOSHIP The intensity and duration of a drug s effect(s)

### 2-1 Position, Displacement, and Distance

2-1 Position, Displacement, and Distance In describing an object s motion, we should first talk about position where is the object? A position is a vector because it has both a magnitude and a direction:

### 6.5. Log-linear Graphs. Introduction. Prerequisites. Learning Outcomes. Learning Style

Log-linear Graphs 6.5 Introduction In this block we employ our knowledge of logarithms to simplify plotting the relation between one variable and another. In particular we consider those situations in

### or, put slightly differently, the profit maximizing condition is for marginal revenue to equal marginal cost:

Chapter 9 Lecture Notes 1 Economics 35: Intermediate Microeconomics Notes and Sample Questions Chapter 9: Profit Maximization Profit Maximization The basic assumption here is that firms are profit maximizing.

### Guideline for Bioequivalence Studies of Generic Products

English translation of Attachment 1 of Division-tification 0229. 10 of the Pharmaceutical and Food Safety Bureau, dated February 29, 2012 Guideline for Bioequivalence Studies of Generic Products Index

### Error Analysis. Table 1. Capacity Tolerances for Class A Volumetric Glassware.

Significant Figures in Calculations Error Analysis Every lab report must have an error analysis. For many experiments, significant figure rules are sufficient. For a brush up on significant figure rules,

Lecture 5: Linear least-squares Regression III: Advanced Methods William G. Jacoby Department of Political Science Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Simple Linear Regression

### CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS

CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS Specific Expectations Addressed in the Chapter Collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology

### Net ionic equation: 2I (aq) + 2H (aq) + H O (aq) I (s) + 2H O(l)

Experiment 5 Goals To determine the differential rate law for the reaction between iodide and hydrogen peroxide in an acidic environment. To determine the activation energy and pre-exponential factor for