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

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1 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 Data Understand the difference between absolute and relative bioavailability Fit IV and Oral Data Simultaneously Determine of ka Estimation Method of Residuals Wagner-Nelson Method Method of Inspection Fitting using non-linear regression analysis Chapter 18 1

2 Method of Residuals Starting with the Equation for Cp Cp = F Dose ka V ( ka kel) e kel t e ka t which can be written as [ ] where Cp = A e kel t A e ka t A = F Dose ka V ( ka kel) Method of Residuals IF ka/kel or kel/ka > 5 AND Both ka and kel are first order Cp = F Dose ka V ( ka kel) [ e kel t e ka t ] Cp at Later Times At later time the exponential with the faster rate constant approaches zero more quickly 1 IF ka > kel 0.8 Slower exponential e ka t 0 and Cp late = A e kel t e (-k' t) 0.6 Faster exponential 0.4 Difference Time (hr) Chapter 18 2

3 Semi-log Plot of Cp versus Time Concentration (mg/l) F Dose ka Intercept = V (ka-kel) Cp late kel (from slope) Time (hr) The Next Step Once we determine Cp late from the terminal slope Cp = A e kel t A e ka t since Cp late = A e kel t Cp = Cp late A e ka t or Cp late Cp = A e ka t { Residual Plot Residual versus Time Plot of Residual versus Time Concentration (mg/l) Feathering ka from slope Time (hr) Chapter 18 3

4 Requirements for the Method of Residuals One Rate Constant at least five times greater than the other Both Absorption and Elimination are first Order Processes Example Data Set Dose 250 mg Time (hr) Plasma Concentration (mg/l) Cp(late) (mg/l) Residual [Col 3 - Col 2] (mg/l) Residual = 5.5 e 2.05 t Cp late = 5.5 e 0.2 t Semi-log Plot 10 Concentration (mg/l) Cp (mg/l) Cp (late) (mg/l) Residual (mg/l) Time (hr) Chapter 18 4

5 Wagner-Nelson Method * Advantages: Absorption and Elimination can be quite similar in value Absorption doesn t need to be first order Disadvantages Need to have a value for kel - from IV Data Calculations are more involved * Wagner, J and Nelson, E J. Pharm. Sci., 53, 1392 Working Equation Amount Absorbed A Theory Amount In Body X = + Differentiating each term gives da dt or da dt = dx dt + du dt = V dcp + V kel Cp dt Amount Eliminated U Theory (contd) da = V dcp + kel V Cp dt Integrating gives: A = V Cp + kel V or A V = Cp + kel Cp dt t 0 t 0 Cp dt Chapter 18 5

6 Theory (contd) 2 A V = Cp + kel Cp dt t 0 Amount absorbed Up to time t Divided by V kel x AUC from t = 0 to t = t A max V At t = ; Cp = 0 = kel Cp dt = kel AUC 0 0 If Absorption is First Order (A max -A) is the Amount Remaining to be Absorbed A max V A V = Xg V = Xg0 V e ka t = F Dose e ka t V Plot ln(amax-a)/v versus Time Time (hr) Example Data - W-N Method Concentration (mg/l) kel = 0.2 hr -1 kel = 0.2 hr from I.V. Data -1 from IV data Col 3 AUC Col 4 AUC Col 5 kel AUC A/V [Col2+Col5] 2 + ] (Amax-A)/V (mg/l) Chapter 18 6

7 The Plot - Semi-log ka = hr -1 The Plot - Linear 5 (Amax-A)/V (mg/l) Time (hr) By Inspection Assuming ka somewhat larger than kel (>5 times) Assuming absorption complete (> 95%) at time of peak concentration Determine t peak = 5 x t 1/2 (absorption) That is, t 1/2 (absorption) = t peak /5 Chapter 18 7

8 Example t peak 1.5 hour 40 Concentration (mg/l) Time (hr) Example t peak 1.5 hour t 1/2 (absorption) 0.3 hour ka 2.3 hr -1 Actual value 2 hr -1 Calculation of F F/V from only Oral Data Need to have Reference Data Absolute Bioavailability Relative Bioavailability Using Plasma Data Using Urine Data Chapter 18 8

9 F - Bioavailability, Fraction Absorbed Cp = F Dose ka V ( ka kel) [ e kel t e ka t ] From Oral Data Alone Can calculate kel and ka Know Dose Cannot separate F from V We get V/F or F/V Bioavailability (F) Absolute Bioavailability Comparison with IV Dose Relative Bioavailability Comparison with another Extravascular Dosage form Bioavailability Study may use IV, Solution or Innovator s Product as Reference Using Plasma Data Equation for A max from Wagner-Nelson Derivation A max = F Dose = kel V AUC therefore F = kel V AUC Dose Chapter 18 9

10 Comparing Two Dosage Forms Since we don t know V unless we have prior information F A F B = kela V A AUC A Dose A Dose B kel B V B AUC B If Dose A = Dose B and we can assume that kel A = kel B and V A = V B F = FA F B = AUCA AUC B Example Calculation Data: AUC A = 12.4 mg.hr/l; Dose A = 250 mg; AUC B = 14.1 mg/l; and Dose B = 200 mg Question: Calculate F Equation: Result: F = AUCA Dose A Dose B AUC B F = x = 0.70 Using Urine Data Starting with the equation for fe (last semester) U fe = U Xg = 0 F Dose U F = fe Dose F A U A F = B fe A Dose feb Dose B A U B Chapter 18 10

11 Using Urine Data F A F B = U A fe A Dose feb Dose B A U B If Dose A = Dose B and fe A = fe B F = FA F B = U A U B Example Calculation Data: Dose = 250 mg; U A = 175 mg; U B = 183 mg Question: What is F Equation: F = U A U B Results: F = = 0.96 Data Analysis using Boomer Draw the Model - Numbering Components and Data Sets Create Table of Parameters and Constants Start the Program Chapter 18 11

12 Drug in G-I Tract The Model Dose #1 ka Drug in Body V/F #2 kel #1 Cp Versus Time Parameter Values From Earlier - Method of Residuals ka = 2.05 hr -1 kel = 0.2 hr -1 A = 5.5 mg/l = F Dose ka V ( ka kel) 5.5 = F V ( ) V F = = 50.4 L Create the Parameter Table Name Type Value To Name From Dose ka kel V/F Cp 2 Chapter 18 12

13 Fitting Oral Data Alone Using Boomer Open the.bat file Using the SAAM II Start SAAM II Load Program File Fitting Oral and IV Data Together The Model Chapter 18 13

14 Simultaneous Fit to IV and Oral data 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 Data Understand the difference between absolute and relative bioavailability Fit IV and Oral Data Simultaneously Chapter 18 14

PHAR 7633 Chapter 19 Multi-Compartment Pharmacokinetic Models

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