Multi Compartment Model PDF

MULTI COMPARTMENT MODEL: INTRAVENOUS BOLUS ADMINISTRATION PHAR 7302 Rahmat M Talukder, Ph.D., R.Ph.  Objecti ves Describe the differences between the one- compartment, two-compartment, and multi-compartment pharmaco...

MULTI COMPARTMENT MODEL: INTRAVENOUS BOLUS ADMINISTRATION PHAR 7302 Rahmat M Talukder, Ph.D., R.Ph.  Objecti ves Describe the differences between the one- compartment, two-compartment, and multi-compartment pharmacokinetic models Describe the plasma concentration-time profile after a single i.v. bolus injection of drugs that follow the two-compartment pharmacokinetic model Explain all the parameters of the two- compartment pharmacokinetic model Calculate the plasma concentration at any time after a single i.v. administration of a drug that follows the two-compartment model. Determine the pharmacokinetic parameters of the two-compartment  Multi Compartment Model For many drugs given as bolus I.V., the concentration vs. time profile does not decline as a single exponential (first-order) process, but declines biexponentially (sum of two first order process) or triexponentially. Figure 1: Plasma concentration-time graph for drugs showing mono-exponential (solid line) and bi-exponential decline (dashed line). Note: In mono exponential process, we have a single decline in the concentration. The drug distributes instantaneously and gets eliminated. With biexponential process, we have a dominant distribution phase in which the drug distributes throughout the body and a dominant  Multi Compartment Model In multi-compartment model, the drug does not equilibrate instantaneously throughout the body (as assumed for one- compartment model) but distributes at different rate to different tissues due to different blood flow and affinity for the drug. These differences in distribution accounts for the appearance of a non- linear curve. In two-compartment model, the drug distributes rapidly into a central compartment (blood, highly blood perfused tissues, etc.) and more slowly into a second compartment, the tissue/peripheral compartment. Drug transfer between the two compartments are assumed to take place by first order processes. After equilibrium, the terminal portion of the plasma or serum concentration vs. time curve reflects first order elimination. Figure 2: Tissue and plasma drug concentrations for Two- compartment model. Tissue concentrations may be greater or less than plasma concentrations. Figure 3: Typical organ groups for central & peripheral compartments Note: With the two-compartment model, there is a central compartment and a peripheral compartment. The central compartment is made of organs such as heart, liver, lung, kidney, and blood. From there, it goes to the peripheral compartments, such as fat tissue, muscle tissue, and CSF. The movement of the drug is reversible. Figure 4: Stages of drug distribution & elimination after i.v. bolus injection for a drug following two-compartment model  Figure 5: Plasma level-time curve for drug that follows a two- compartment model, may be divided into two parts, a distribution phase and an elimination phase. After an IV bolus injection, drug equilibrate rapidly in the central compartment. The distribution phase represents the initial, more rapid decline of drug from central compartment into tissue compartment. In this phase distribution is dominant process with concurrent drug elimination. In the elimination phase drug concentrations in both the central and tissue compartments decline in parallel. Figure 6: There are several possible two-compartment model. The rate constants k12 and k21 represent the first-order rate transfer constant for drug movement between the two compartments. Most two-compartment model assume that elimination occurs from the central compartment as the major eliminating organs, kidney and liver (highly perfused by blood) are in the  Drugs That Follow Two Compartment Model Vancomycin Digoxin Aminoglycosi des  Two-Compartment Open PlasmaModel, IVcurve level-time bolus after iv bolus injection shows Cp declines bi- exponentially as sum of two first-order processes, distribution and elimination Cp = Ae-αt + Be-βt α and β are hybrid first-order rate constants for distribution & elimination, respectively. α is greater than β, indicating that distribution into tissues proceeds at a greater rate than elimination A and B are intercepts on the y axis for each exponential segment of the curve. Method of residual is applied to get A and α  Elimination Rate Constant (β) The slope of the terminal line is β (terminal elimination rate constant) Using two of the last 4 concentrations given and their corresponding times, you can use the formula below to calculate β Tim Observe β = ln(C2/C1)/(t2 - t1) e d Cp Example: (hr (mg/L) ) Step 1: Identify C2, t2, C1, t1 0.25 43 using two of the last four 0.5 32 points → 1.0 20 C2: 0.52, t2: 16.0, C1: 1.2, t1: 12.0 1.5 14 β = 0.21 2.0 11 Step 2: Using the above hr-1 4.0 6.5 equation, pluguse NOTE: We can in your linear Last four values and in regression solve → calculator to 8.0 2.8 points β=ln(0.52/1.2)/(16 calculate accurately- 12) slope, β 12.0 1.2 & intercept, B of the terminal line using last four points (x  Terminal Elimination The intercept Line of the (B) terminal elimination line is B The B-intercept on the y axis represents an terminal elimination intercept. Using the last 4 concentrations given, it’s corresponding time, and β, you can calculate for B. CP = Be-βt Example: Time (hr) 0.25 0.5 1.0 1.5 2.0 4.0 8.0 12.0 16.0 Observed 43 32 20 14 11 6.5 2.8 1.2 0.52 CP (mg/L) last four points Step 1: Identify equation of terminal line → CP = Be-βt Step 2: Rearrange equation to solve for B → B=CP/e-βt Step 3: Identify one concentration and time from the last four points → CP = 0.52, t = 16.0 Step 4: Plug in values, including β (β = 0.21) from previous B B = and = 0.52/equestion - 15 solve → 0.21(16) mg/L NOTE: As mentioned earlier we can use linear regression to calculate accurately slope, β & intercept, B of the terminal line  Extrapolated Plasma p Concentration Using B, β, and t (any time(C ’) chosen from the distribution phase) you’re able to use the formula below to find the extrapolated concentration (C ’) at a given time C ’= Be-βt p Example: Time (hr) 0.25 0.5 1.0 1.5 2.0 4.0 8.0 12.0 16.0 Observed Cp 43 32 20 14 11 6.5 2.8 1.2 0.52 Use the times from distribution phase B: 15 Step 1: Identify the equation of the p t: extrapolated Cp (C ’) → p’ C = Be- 0.25 β: Step βt 2: Insert your values into the equation 0.21 beginning first time from 15ethe- given Step 3: → Solve →0.21(0.25) Cp’= 15e-0.21(0.25) = 14.2 mg/L Step 4: Continue for each time in the  Method of Residuals It is a useful procedure for fitting a curve to the experimental data of a drug when the drug does not follow a one- compartment model It attempts to separate the two process of distribution and elimination The linear terminal portion of the log or lnCp vs. time curve is extrapolated up to the y-axis.  Method of Residuals Subtract the corresponding extrapolated points (C ’) from the P actual points (CP) observed at the same time: (CP - CP’) Example: Time (hr) 0.25 0.5 1.0 1.5 2.0 4.0 8.0 12.0 16.0 Observed Cp 43 32 20 14 11 6.5 2.8 1.2 0.52 Extrapolated (Cp’) 14.5 13.5 12.3 11 10 Residuals for these values (One at a time) Step 1: Identify Cp’ and Cp → Cp’: 14.5; Cp: 43 28.5 Step 2: Subtract Cp from Cp’ to get mg/L Continue this residual → 43for each = - 14.5 value in the distribution phase  Distribution Rate Constant The slope of the (α) residual line is α (distribution rate constant) Using two of the calculated residual values and their corresponding times, you can calculate α. Time (hr) 0.25 α0.5 = ln[(C 1.0 p - C1.5 p’)2- (C p - Cp4.0 2.0 ’)1] / t8.0 2 - t1 12.0 16.0 Example: Observed Cp 43 32 20 14 11 6.5 2.8 1.2 0.52 Extrapolated (Cp’) 14.5 13.5 12.3 11 10 Residual (Cp - Cp’) 28.5 18.5 7.7 3.0 1.0 (Cp - Cp’)2 : Step 1: Identify the equation of the Distribution 18.5 Rate Constant (α) → α = ln[(Cp - Cp’)2- (Cp - Cp’)1] / (Cp - Cp’)1: (t2 - t1) 28.5 t2: 0.5 Step 2: Insert your values into the equation using t1: 0.25 any Residual point and its corresponding values Step → 3: Solve for α → α= ln[18.5 - 28.5] / 0.5 - 0.25 = 1.72 hr -1 ln[18.5 - 28.5] / 0.5 - 0.25  Distribution Constant The intercept of the (A) residual line is A Using a calculated residual point, it’s corresponding time and α, you can calculate A. (Cp - Cp’) = Ae-αt Example Time (hr) 0.25 0.5 1.0 1.5 2.0 4.0 8.0 12.0 16.0 Observed Cp 43 32 20 14 11 6.5 2.8 1.2 0.52 Extrapolated Cp’ 14.5 13.5 12.3 11 10 Residual (Cp - Cp’) 28.5 18.5 7.7 3.0 1.0 Step 1: Identify equation of Distribution Constant → (Cp (Cp - Cp’): - Cp’) = Ae-αt 28.5 α: 1.72 Step 2: Rearrange equation to solve for A → A = (Cp - t: 0.25 Cp’) / e-αt Step Step 3: 4: Insert your Solve for A values into the → A = 28.5 equation using any / 43 Residual e-1.72(0.25)=point and its corresponding time → 28.5 / e- mg/L 1.72(0.25)  Two-Compartment Open Model, Volume of Distribution Different Volume of Distributions 1. Central compartment volume, Vp Useful for determining drug concentration directly after injection into the body Cp = Ae-αt + Be-βt at time zero, no drug is eliminated, in the central compartment → C p0 = A + B therefore Vp = D0/ (A + B) alternatively, using AUC, Vp = D0/(k*AUC) where k = ⍺ꞵ(A+B)/Aꞵ+B⍺ Example: Administration of 108 mg of griseofulvin to 70 kg patient resulted in the following plasma concentration-time equation: Cp = 0.701e-0.372t + 0.492e-0.0492t Vp = D0 / (A + B) = 108 mg/(0.701+0.492 mg/L)=90.5 L  Two-Compartment Open Model, Volume of Distribution 2. Extrapolated volume of distribution, (VD)exp (VD)exp = D0/B -oversimplifies by disregarding distribution phase -may not provide useful volume term Example: Administration of 108 mg of griseofulvin to 70 kg patient resulted in the following plasma concentration-time equation: Cp = 0.701e-0.372t + 0.492e-0.0492t (VD)exp= D0/B = 108 mg/0.492 mg/L = 219.5 L  Two-Compartment Open Model, Volume of Distribution 3. Volume of distribution by Area, (VD)area aka (VD)β In one compartment→ VD= D0/ (k*AUC) β is used instead of k in two-compartment → (VD)β = D0/ (β*AUC) Relates amount of drug & Cp in post-distributive phase Example: Administration of 108 mg of griseofulvin to 70 kg patient resulted in the following plasma concentration-time equation: Cp = 0.701e-0.372t + 0.492e-0.0492t AUC= A/⍺ + B/ꞵ = (0.701 mg/L)/0.372 hr-1 + (0.492 mg/L)/(0.0492 hr-1) = 11.884 mg*hr/L (VD)β= D0/ (β*AUC) = 108 mg/(0.0492 hr-1 * 11.884 mg*hr/L) = 184.7L  Figure 9: Plasma concentration-time curves after IV 500 mg of a drug If appropriate design and sampling during pharmacokinetic studies is not done, a two-compartment model drug may be erroneously labelled as one compartment model drug. This error will affect the determination of pharmacokinetic parameters such as initial concentration, volume of distribution, AUC, etc.  Clinical Correlate: Digoxin is an inotropic drug used primarily for congestive heart Digoxin failure and atrial fibrillation. The use of pharmacokinetics to adjust the dosing regimen can reduce digoxin toxicity. Well described by two compartments. Plasma concentration rapidly decline as drug distributes out of plasma and in to muscle tissue. The plasma would be the central compartment and the muscle tissue would be the peripheral compartment. -1 - α= 1 (normal, 0.593 hr (renal - - β = hr 1 N) 1.331 failure,1RF) 0.019 Vp hr L/kg = 0.78 (N) (N) 0.007 hr (RF) (VD)β = 7.3 L/kg (N) 0.73 L/kg (RF) 4.7 L/kg (RF) Plasma concentration declines rapidly during the initial distributive phase. 3-4 hours to accumulate in tissue. Therapeutic Drug Monitoring (TDM): plasma samples taken at least 3-4 hours after i.v. dosing since the equilibrated level is more representative of myocardium digoxin level. If terminal t1/2 is calculated, must make sure that distribution  Practice Problem A patient was given an intravenous 1 injection of 50 mg of meperidine for post- operative pain. The following plasma concentrations were obtained. Determine A, α, B, β, Vp, [VD]β [VD]SS & ClT. Time (hr): 1.0 1.5 2. 3. 4. 6. 8. 0.5 2.0 5 0 0 0 0 Conc. 290 220 15 12 96 60 38 (ng/ml):420 180 0 5 Semi-log plot of meperidine plasma concentration vs. time 1000 Concentration 100 (ng/mL) 10 0 2 4 6 8 10 Time Practice Problem 1: Extrapolated line, determination of beta 3(β) & 4B t (hr): 6 8 Cp (mg/L): 125 96 60 38 Linear regression of the ln of the last 3 conc. vs. time in the calculator, we get the extrapolated line with intercept, B as 242 ng/ml and β as 0.232 hr-1 (r = - 0.9999). Using the equation of the line (Cp = Be-βt) we can calculate the extrapolated values at the corresponding time of plasma sample collection. Practice Problem 1: Residual Observed line, Extrapolated Residu al determination of alpha (α) & A 1.0 290 191.9 98.1 Time (hr) Concentrations (ng/ml) 1.5 220 170.9 49.1 2.0 180 152.2 27.8 By linear regression of natural log of residual conc. vs. time in calculator: intercept, A = 339.1 ng/ml and slope, α = 1.26 hr-1, (r = 0.9984) CP= 339e-1.26t + 242e-0.232t  Practice Problem Vp = Dose / (A + B) = 50 mg / (339 + 242) = ng/ml 86.1 L Vβ = (Dose * α) / ((B *α) + (A * β)) = (50 * 1.26) / [(242 * = 1.26)+(339*0.232 )] 164.2 ClT = Vβ * β L = 164.2 L * = 38.1 0.232 h-1 L/h  Extra Resources https://www.youtube.com/watch?v=ds4yB b55MGw  STUDY 1. GUIDEcentral compartment and Know the difference between peripheral compartment 2. Understand distribution, distribution equilibrium, and elimination in multi- compartment models 3. Know which drugs follow a two-compartment model 4. Understand how two-compartment models apply to different drugs in clinical situations 5. Know definition and be able to identify α, β, A, B, clearance in a given equation or graph 6. Know how to calculate α, β, A, B, clearance and when to use method of residuals to calculate A and α 7. Understand and calculate the different Volume of Distribution 8. Be able to identify one, two, and three compartment graphs  Learning Questions: 1. Which organs are a part of the central compartment? 2. Which of the phase (distribution or elimination phase) has a more rapid decline of drug concentration for drug following a two compartment model? 3. α and β are hybrid rate constants for and phase. 4. In two compartment models, do subtracting extrapolated points from actual points yields plasma concentrations. 5. The y-intercept of the residual line is 6. Patient GB weighs 198 pounds. What is the weight of GB in  Learning Questions 7. Label the following graphs as one, two, or three-compartment, and identify the phases in each graph.  Referen ces Applied Biopharmaceutics and Pharmacokinetics: Leon Shargel, S. Wu-Pong and Andrew Yu. 7th edition, 2016. McGraw-Hill, New York. Concepts in Clinical Pharmacokinetics: J. T. DiPiro. 4th edition, 2005. ASHSP, Bethesda, MD.  ASSIGNMENT 3 Please do the following PP 2-6 as a Team assignment and submit it by Friday, 9/16, EOB. (Total 10 points) PP 2: Patient RS was given 96 mg of a drug HM231 that follows the two- compartment model. The equation for the plasma concentration of HM231 in RS is given as: Cp(mg/L) = 0.701e-0.372t + 0.492e-0.0492t What is the Vp of the drug HM231 in RS? (1 point) PP 3: A patient was given an intravenous injection of 100 mg of a drug for postoperative pain. The following plasma concentrations were obtained. Calculate the elimination rate constant, β. (2 points)  ASSIGNME NT-0.123t PP4: Given: Cp = 0.739e 3 + 0.324e-0.087t, identify distribution rate constant, α (1 point) PP5: Calculate B of a drug following the two compartment model using these values: Cp : 1.7 mcg/L, t: 21.0 hr, β: 0.14 -hr , given Cp = Be-βt (Round to the nearest whole number) (1 point) PP6: Compute the equation that describes the following plasma concentration data obtained after rapid intravenous administration of a single 100 mg bolus dose of a drug that follows two compartment model. (5 points) Time 0.25 0.50 0.75 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10 12 14 (hr) Conc. 61.1 53.6 47.2 41.7 26.6 18.1 13.2 10.1 7.9 6.6 5.3 4.4 3.7 2.6 1.9 (µg/mL )