1 Compartment Model: IV Dosing

1. 1 Compartment Model: IV Dosing Dose = 1000 mg Time Conc (hr) (mg/L) 0 100.0 1 89.1 2 79.4 4 60.0 12 25.0 18 12.5 24 6.25 Calculate Vd, k, half-life, AUC (0-t & 0 –inf), and Clearance Use of Spreadsheets (Excel®) to Calculate Pharmacokinetic Parameters. Dose V -kt C = ---------- e

2. Brief Tutorial on the use of Spreadsheets (Excel®) A spread sheet is designed to complete numerical calculations which can be automated to allow data analysis, projections, reporting and construction of graphs and other visual aid plots.

3. Brief Tutorial on the use of Spreadsheets (Excel®) Spreadsheets consist of rows and columns In Excel the columns are given alphabetic labels A, B, C. D …. X, Y, Z, AA, AB… and the rows are numbered 1, 2, 3 … 6000+ Individual “cells” are identified by their row and column location, e.g. - A1 Cell A1

4. Brief Tutorial on the use of Spreadsheets (Excel®) Each cell can contain a variety of “types” of information Words Numbers Dates Formula Pictures… etc. Formula can be created by the user or the user may select a formula from a vast variety of formula in a pick-list.

5. Brief Tutorial on the use of Spreadsheets (Excel®) Creating Formula: Formula can be constructed by first initiating a formula with an equal sign ‘=‘ and then using the following operators: Addition: ‘+’ Subtraction: ‘-’ Division: ‘/ ’ Multiplication: ‘*’ such that (3x4)divided by 2 is would convert to: =(3*4)/2 Order of operations applies and so in the previous formula parenthesis were not required.

6. Brief Tutorial on the use of Spreadsheets (Excel®) Formula can also be created or chosen from the pick-list to complete any mathematical calculation: Financial functions Statistical functions Engineering functions Math/Trig functions Logical functions (IF statements)

7. Brief Tutorial on the use of Spreadsheets (Excel®) • Creating Formula: • Column ‘A’ contains • a series of numbers. • In cell A6 • we can create a • formula to add all of • the numbers in • column A together. • This can be done • at least two ways: • Brute Force: • = A1+A2+A3+A4+A5 • 2. Using the Pick-list • of formula: • =SUM(A1:A5)

8. Brief Tutorial on the use of Spreadsheets (Excel®) Enter the Concentration-Time data into Excel®. Draw a [ ]-time profile and Calculate: Volume of Distribution Half-life, Area Under the Curve Clearance

9. Brief Tutorial on the use of Spreadsheets (Excel®) Draw a [ ] - time profile 1. Select Chart Wizard

10. Brief Tutorial on the use of Spreadsheets (Excel®) • Draw • a [ ] - time profile • Select Chart Wizard • Select XY (Scatter)

11. Brief Tutorial on the use of Spreadsheets (Excel®) • Draw • a [ ] - time profile • Select Chart Wizard • Select XY (Scatter) • you should also • select for no line • although a chart • with data points • connected by a • smooth line • or a straight line • will be acceptable.

12. Brief Tutorial on the use of Spreadsheets (Excel®) • Draw • a [ ] - time profile • Select Chart Wizard • Select XY (Scatter) • Ensure the Series • is in columns.

13. Brief Tutorial on the use of Spreadsheets (Excel®) • Draw • a [ ] - time profile • Select Chart Wizard • Select XY (Scatter) • Ensure the Series • is in columns. • 4. Select the • data range and • ‘click’ the red arrow

14. Brief Tutorial on the use of Spreadsheets (Excel®) • Draw • a [ ] - time profile • Select Chart Wizard • Select XY (Scatter) • Ensure the Series • is in columns. • 4. Select the • data range and • ‘click’ the red arrow • 5. With the left mouse • button held down • select cells A4 – B10. • Click the red arrow to • return to the wizard

15. Brief Tutorial on the use of Spreadsheets (Excel®) Draw a [ ] - time profile 6. On return to the Wizard your graph should look like a [ ] –time profile.

16. Brief Tutorial on the use of Spreadsheets (Excel®) Draw a [ ] - time profile 6. On return to the Wizard your graph should look like a [ ] –time profile. 7. Enter labels for the x and y axis. Other tabs allow further refinements to the graph. You could eliminate the legend.

17. Brief Tutorial on the use of Spreadsheets (Excel®) • Draw • a [ ] - time profile • The final step selects • placement of the • graph in either the • current sheet • beside the data • or as a graph in a • separate sheet. Select “As Object in” and you will be able to see the [ ]-Time plot beside the data near where we will add in calculations.

18. Brief Tutorial on the use of Spreadsheets (Excel®) You have successfully completed construction of the concentration-time Graph.

19. Brief Tutorial on the use of Spreadsheets (Excel®) If you place the cursor on the values of the y-axis and ‘double click’ you can further modify the appearance of your profile. Under tab “scale” you may select ‘Logarithmic scale’ and obtain a semi-log plot which should show a linear decline in the [ ] – Time profile.

20. Brief Tutorial on the use of Spreadsheets (Excel®) Your sheet should now look similar to this, although relative size may be different.

21. Brief Tutorial on the use of Spreadsheets (Excel®) Calculate: Volume of Distribution Half-life, Area Under the Curve Clearance In the following slides, you will be given instructions on how to complete the procedure and the ‘answer’ will be shown on the next slide.

22. Brief Tutorial on the use of Spreadsheets (Excel®) Calculate: Volume of Distribution Half-life, Area Under the Curve Clearance In cell A12 create a label: ‘Volume of Distribution’ and In cell B12 create a formula for Volume. • Several possibilities exist to calculate Volume: • Based on the initial concentration of 100 mg/L and a dose of 1000 mg.( V = Dose/ [ ] ) • Estimate a back-extrapolated ‘intercept’ using the function ‘intercept’ in the wizard. • This method requires that the concentrations be converted to log values first.

23. Brief Tutorial on the use of Spreadsheets (Excel®) Calculate: Volume of Distribution Half-life, Area Under the Curve Clearance In cell A16 create a label: ‘Half-life’ and In cell B16 create a formula for T½. The best method for calculation of half-life is to place a straight line through the log-linear portion of the concentration-time profile. This is done using the ‘slope’ function in the wizard and requires that the concentrations be converted to log values first. … Visual inspection of the graph & data will also indicate that the half-life is about 6 hrs.

24. Brief Tutorial on the use of Spreadsheets (Excel®) A Note on Half-life: What is the correct half –life? The choice of points to be included in the terminal phase is more obvious in a 1C model with IV bolus administration than following oral absorption or any situation where the first points are uncertain. Establish the choice of the number of points in the terminal phase using linear regression and the best r-value. Start with the last two points in the terminal phase. Place a straight line through the points using linear regression and the ‘slope’ function. Also calculate the correlation coefficient [‘correl’] using the wizard for this set of points. Repeat this process, each time increasing the number of points in the terminal phase. Greatest r-value & greatest # point is best.

25. Brief Tutorial on the use of Spreadsheets (Excel®) A Note on Half-life: What is the correct half –life? Q1. Please Explain slide 24, what is the correct half-life in the Excel Tutorial slide set? A. This slide involves some statistics and pharmacokinetics and is included as a more complete evaluation of the kinetics of this data set. Also, it should be pointed out, that the "answer" for the procedure discussed on one slide is shown on the next slide. Visual inspection of the graph & data from 12 to 18 hours and 18 to 24 hours will indicate that the half-life is about 6 hrs. Equations will also yield a half-life of 6 hours If you use any two concentrations based on times of time zero, 12, 18 or 24 hours. However, in this data set, and in concentrations drawn from patients, rounding and or inaccuracy in sample timing or sample analysis will result in concentrations which may not all fall exactly on the line. For example, it would appear that I developed this concentration-time profile using the 6 hour half-life (k=0.1155 hr-1) just calculated. Using a half-life of 6 hours, concentrations at 1 and 2 hours can be calculated. C(1) = C(0)x e(-kt) = 100 e(-.1155 x 1) = 89.09 mg/L The actual concentration in the table is 89.1 mg/L. This is a minor deviation but serves to show that not all concentrations will lie directly on the line.

26. Brief Tutorial on the use of Spreadsheets (Excel®) A Note on Half-life: What is the correct half –life? Similarly for the concentration at 2 hours, a concentration of 79.37 mg/L is calculated and the excel file lists a concentration of 79.4 mg/L. Deviations like this and actually much greater than this will occur all the time in pharamcokinetics, but you are usually unable to identify any particular concentration as being in error. Therefore, you must assume that the deviations occur with equal likelihood across all concentrations. As a result, analysis of the half-life is often done using more than just two selected concentrations. Half-life is usually calculated by finding the "average" or most likely slope running through a number of concentrations. This involves linear regression and the use of the Excel function "SLOPE" When you are using the SLOPE function you may not realise that you are using linear regression. As slide 24 points out, start with the last two points in the terminal phase (time 18 and 24 hr). Place a straight line through the points using linear regression and the ‘slope’ function from Excel. Also calculate the correlation coefficient [‘correl’ using Excel] using the wizard for the same set of points. Repeat this process, each time increasing the number of points in the terminal phase. In doing this you must first convert concentrations to their log values. The choice of points to be included in the terminal phase is more obvious in a 1C model with IV bolus administration than following oral absorption or any situation where the first points are uncertain. In this data set and in most 1C model data sets we have started with the last 2 point because these two points are in the terminal phase. While all points are obviously in the terminat phase in this example, following and IV infusion, oral absorption or IV bolus with distribution, the number of points which could potentially be included will vary depending on the observed kinetic results.

27. Brief Tutorial on the use of Spreadsheets (Excel®) A Note on Half-life: What is the correct half –life? Explanation of correlation When you have completed this process, inspect your results. Correl will give you a number between 0 and 1. When all points lie on the lie, r = 1, and error = 0:[Error = 1-r^2 (r squared)]. The best choice for the terminal phase is the set of points with the greatest r-value & the greatest # of points. Obviously the using only 2 points gives you an r value of 1 and so balancing greatest r value and greatest number of points may seem subjective. For this data set an r value of 0.9999 is observed for 7 points. On slide 25 this is shaded in. I, therefore, select a K value of 0.11512, which is equivalent to a half-life of 6.02 hours. It seems that the very minor rounding procedure that appeared in time points 1 and 2 hours resulted in a deviation of 0.02 hr from the estimated 6 hr half-life (~1 minute!).

28. Brief Tutorial on the use of Spreadsheets (Excel®) Calculate: Volume of Distribution Half-life, Area Under the Curve Clearance In cell A24 create a label: ‘AUC(0-t)’ and In cell B24 create a formula for AUC(0-t). The best method for calculation of AUC involves creating a column (D) where the AUC from one time point to the next is calculated. This may be done using either raw conc. values or log concentrations values. In this problem the method using the log of the concentrations will have less error. The sum of the ‘partial’ areas can be reported in A24.

29. Brief Tutorial on the use of Spreadsheets (Excel®) Note on Repeating Formula: In cell D5, the formula for AUC from time zero to 1 hour should be placed. However, this formula is essentially the same for all cells in D6 trough D10. Rather than re-type the formula you can copy the formula cells by a copy-paste procedure or a drag procedure. Drag procedure. Place the cursor in cell D5. The formula appears in the ‘edit formula’ section (green arrow). In the lower right corner of cell D5 there is a small square. Place your cursor on this square. Your cursor will turn to a smaller plus (+) sign. Depress the left mouse button. Drag the square to A10. The formula has been copied.

30. Brief Tutorial on the use of Spreadsheets (Excel®) Calculate: Volume of Distribution Half-life, Area Under the Curve Clearance Calculation of the AUC from 24 hours to infinity [AUC ([LP*]  )] can be calculated by the pharmacokinetic method. AUC = [ ]24/k * Where LP means Last Point Since the estimate of K or T½ affects this calculation, complete this calculation for all estimated elimination rate constants in cells F17 F22. Total AUC (0) can be calculated by adding AUC(0-t) [B24] and the best estimate of AUC (0).

31. Brief Tutorial on the use of Spreadsheets (Excel®) Note on ‘fixing’ a cell. When calculating the AUC ([LP]  ) The concentration of the LP or the concentration at 24 hours will be used in 6 calculations. When drag procedure is used to copy, the concentration used in the formula changes UNLESS the cell is ‘FIXED’ using a ‘$’ The placement of the $ in the formula is important. The formula for AUC (LP) in call F17 would be: ‘=B10/K17’ Dragging the formula to F18 yield: ‘=B11/K18. Placing the $ between B and 10 fixes the cell at line 10. [ =B$10/K17]

32. Brief Tutorial on the use of Spreadsheets (Excel®) Calculate: Volume of Distribution Half-life, Area Under the Curve Clearance Clearance can be calculated two ways with this data. 1. Clearance = Dose / AUC 2. Clearance = k * Vd Since we have a variety of estimates for (i) volume (B12 & B14) and we could have obtained additional estimates using different numbers of points and ; (ii) k; we can obtain a large number of estimates of clearance. Limit choices to 7 points.

33. Brief Tutorial on the use of Spreadsheets (Excel®) • Additional • Information & Tips • Converting • Raw Concentrations • to log concentration • and back again. • (ii) Using • Natural Logs • (Ln) for kinetic Analysis • (iii) Back Extrapolation

34. Brief Tutorial on the use of Spreadsheets (Excel®) (i) Converting Raw Concentrations to log concentration and back again. The log of a concentration can be obtained using the Excel function ‘LOG(##)’. The value in parenthesis (##) may be either an actual number or a cell reference. Using a Cell Reference allows the formula to be copied more easily.

35. Brief Tutorial on the use of Spreadsheets (Excel®) (i) Converting Raw Concentrations to log concentration and back again. If you have the log of a number and wish to convert it back to the ‘raw’ concentration, this can be done by computing the value of 10x where x is the log value you wish to convert. To do this in Excel the format is 10^x Where ‘^’ is the Excel operator for power.

36. Brief Tutorial on the use of Spreadsheets (Excel®) (ii) Using Natural Logs (Ln) for kinetic Analysis. Kinetic data may also be analysed using natural logarithms. The natural log of a concentration can be obtained using the Excel function ‘LN(##)’.

37. Brief Tutorial on the use of Spreadsheets (Excel®) (ii) Using Natural Logs (Ln) for kinetic Analysis If you have the natural log of a number and wish to convert it back to the ‘raw’ concentration, this can be done by using the Excel function ‘EXP(##)’. Where ## is value to be converted.

38. Brief Tutorial on the use of Spreadsheets (Excel®) (ii) Using Natural Logs (Ln) for kinetic Analysis If you use Natural logs all calculation are exactly the same (see Worksheet ‘Analysis of IV Bolus using LN) except that the slope is the elimination rate constant (do not multiply by 2.303) and the intercept – Initial concentration must be converted to concentration using ‘EXP’ rather than ’10^## Done properly both methods yield identical results.

39. Brief Tutorial on the use of Spreadsheets (Excel®) • (iii) Back Extrapolation • In many situations • in pharmacokinetic analysis • you may need to back extrapolate • to determine the concentration • at earlier times. • For example, if a drug • was given by IV bolus • but time zero concentrations • are not available, • back extrapolation is necessary • to determine volume. • There are at least three ways • to do this. • Using the Excel • ‘INTERCEPT’ function • (b) Using the Excel ‘SLOPE’ function • (c) Using the formula: • [ ]t2 = [ ]t1 * e(k(t2 –t1))

40. Brief Tutorial on the use of Spreadsheets (Excel®) • (iii) Back Extrapolation • Using the Excel • ‘INTERCEPT’ function • When selecting the “appropriate” • number of points in the terminal • phase to determine ‘SLOPE’ • you can also determine • the intercept using the • ‘INTERCEPT’ function • and the same pairs • of conc. & time values. • In the worksheet on the left the • Initial intercept value of 100 was • obtained using the equation in Excel: • =10^INTERCEPT(C$9:C$10,A$9:A$10) • for the last 2 points.

41. Brief Tutorial on the use of Spreadsheets (Excel®) (iii) Back Extrapolation (b) Using the Excel ‘SLOPE’ function. In Excel when the slope is calculated on log-conc. & time data, and the line is straight we can estimate the concentration anywhere on the line as it is in the form of y = mx = b. A concentration at any time (t1)can be used and the concentration at another time (t2) can be determined. LOG [ ]t2 = LOG [ ]t1 + SLOPE * (t2 – t1) The log of concentration at t2 (LOG [ ]t2) can be convert to a raw concentration.

42. Brief Tutorial on the use of Spreadsheets (Excel®) (iii) Back Extrapolation (b) Using the Excel ‘SLOPE’ function. For example, if the concentration at time zero was to be calculated from the given data, t2 would = 0. t1 could be any other given time. We will use 18 hours. The concentration at 18 hours is 12.5 mg/L (as a log:1.097). LOG [ ]t2= LOG [ ]t1 + SLOPE * (t2 – t1) = 1.097 + (-0.050172 * (0-18) = 1.097 + (0.90309) = 2.00 and converting to raw concentration [ ]t2=0 = 10^2.00 = 100.00 Deviation of the concentration from the line of best fit may result in small deviations from the expected value of 100 if other concentrations and times are used. This method can be used to calculate a concentration at any time on the extrapolated line.

43. Brief Tutorial on the use of Spreadsheets (Excel®) (iii) Back Extrapolation (c) Using the formula: [ ]t2 = [ ]t1 * e( - k(t2 –t1)) Similarly, the concentration at any time on a line taken from the terminal phase can be calculated using the estimated elimination rate constant (k). A concentration at any time (t1)can be used and the concentration at another time (t2) can be determined. The difference in time (t2 – t1) is computed and if t2 occurs before t1 then difference in time is negative and this negative sign cancels the negative sign in the formula e –kt. If t2 occurs after t1 then the negative sign remains.

44. Brief Tutorial on the use of Spreadsheets (Excel®) (iii) Back Extrapolation (c) Using the formula: [ ]t2 = [ ]t1 * e( - k(t2 –t1)) For example, if the concentration at time zero was to be calculated from the given data, t2 would = 0. t1 could be any other given time. We will use 18 hours. The concentration at 18 hours is 12.5 mg/L. [ ]t2 = [ ]t1 * e(k(t2 –t1)) [ ]t2 = 12.5 * EXP( - 0.11554*(0-18) [ ]t2 = 12.5 * EXP( - 0.11554*(-18) [ ]t2 = 12.5 * EXP(2.0798) [ ]t2 = 12.5 * 8.003 [ ]t2 = 100.04 Deviation in the concentration from the line of best fit may result in small deviations from the expected value of100. This method can be used to calculate a concentration at any time on the extrapolated line.