Time Plasma Conc (hr) (mg/L) 1 19. 5 2 14.4 5 5.8 8 2. 4 12 0.7 18 Below limit of detection

1 Compartment IV Analysis & Renal Elimination 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection • What model best describes this profile? • Calculate AUC, K, Vd, T½ and Cl. • Write an equation that will calculate • concentrations at anytime after the 1st dose. • How much is in the body at time zero? • How much is in the body at 12 hours? • Where did the tobramycin go? • How much is in the urine at 12 hrs? • What is the renal clearance of tobramycin? • Is tobra a high or low extraction drug?

1 Compartment IV Analysis 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection • What model best describes this profile? • Calculate AUC, K, Vd, T½ and Cl. • Write an equation that will calculate • concentrations at anytime after the 1st dose. • How much is in the body at time zero? • How much is in the body at 12 hours? • Where did the tobramycin go? • How much is in the urine at 12 hrs? • What is the renal clearance of tobramycin? • Is tobra a high or low extraction drug?

Graph Patient Data 100 60 30 10 6 4 2 1 Using semi-log paper, graph the data following Iv bolus of 500 mg of tobramycin. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection 0 2 4 6 8 10 12 14

Graph Patient Data in Excel® Using semi-log paper, graph the data following IV bolus of 500 mg of tobramycin. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection What model best describes this profile?

Graph Patient Data 100 60 30 10 6 4 2 1 What model best describes this profile? … a 1 compartment model with iv bolus input 0 2 4 6 8 10 12 14

1 Compartment IV Analysis 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection • What model best describes this profile? • Calculate AUC, K, Vd, T½ and Cl. • Write an equation the will calculate • concentrations at anytime after the 1st dose. • How much is in the body at time zero? • How much is in the body at 12 hours? • Where did the tobramycin go? • How much is in the urine at 12 hrs? • What is the renal clearance of tobramycin? • Is tobra a high or low extraction drug? What do you calculate first? Cl T½ AUC Vd K

1 Compartment IV Analysis 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection • What model best describes this profile? • Calculate AUC, K, Vd, T½ and Cl. • Write an equation that will calculate • concentrations at anytime after the 1st dose. • How much is in the body at time zero? • How much is in the body at 12 hours? • Where did the tobramycin go? • How much is in the urine at 12 hrs? • What is the renal clearance of tobramycin? • Is tobra a high or low extraction drug? What do you calculate first? Cl Dose/AUC or K x Vd T½ - 0.693 K AUC –/ K Vd - [ ]0 by back extrapolation of K K …?

Graph Patient Data Slope based on log of the concentrations 100 60 30 10 6 4 2 1 Time Conc Log (hr) (mg/L) Conc 119.5 1.29 214.4 1.16 55.8 0.76 82.4 0.38 120.7 -0.15 18 BLOQ Slope = -K/2.303 Slope = (y2 – y1) / (x2 – x1) = 0 2 4 6 8 10 12 14

Graph Patient Data Slope based on log of the concentrations 100 60 30 10 6 4 2 1 Time Conc Log (hr) (mg/L) Conc 119.5 1.29 214.4 1.16 55.8 0.76 82.4 0.38 120.7 -0.15 18 BLOQ Slope = -K/2.303 Slope = (y2 – y1) / (x2 – x1) = (-0.15 – 1.29)/(12-1) = (- 1.44) / (11) = -0.1309 - K = slope x 2.303 0 2 4 6 8 10 12 14

Graph Patient Data Slope based on log of the concentrations 100 60 30 10 6 4 2 1 Time Conc Log (hr) (mg/L) Conc 119.5 1.29 214.4 1.16 55.8 0.76 82.4 0.38 120.7 -0.15 18 BLOQ Slope = -K/2.303 Slope = (y2 – y1) / (x2 – x1) = (-0.15 – 1.29)/(12-1) = (- 1.44) / (11) = -0.1309 - K = slope x 2.303 = - 0.1309 x 2.303 K = 0.3015 hr-1 0 2 4 6 8 10 12 14

Graph Patient Data Slope based on log of the concentrations 100 60 30 10 6 4 2 1 Time Conc Log (hr) (mg/L) Conc 119.5 1.29 214.4 1.16 55.8 0.76 82.4 0.38 120.7 -0.15 18 BLOQ Slope = -K/2.303 K = 0.3015 hr-1 T½ = 0.693 / K = 0.693 / 0.3015 = 2.298 hr 0 2 4 6 8 10 12 14

1 Compartment IV Analysis K T½

1 Compartment IV Analysis 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection • What model best describes this profile? • Calculate AUC, K, Vd, T½ and Cl. • Write an equation the will calculate • concentrations at anytime after the 1st dose. • How much is in the body at time zero? • How much is in the body at 12 hours? • Where did the tobramycin go? • How much is in the urine at 12 hrs? • What is the renal clearance of tobramycin? • Is tobra a high or low extraction drug? Now calculate Volume. Since it was an IV dose, and appears to be a 1 compartment model, we will assume instantaneous distribution and so we will back-extrapolate to t=0 to determine the initial concentration.

Calculation of Volume • Back extrapolation can be • completed a number of ways • Graphically 100 60 30 10 6 4 2 1 Intercept 0 2 4 6 8 10 12 14

Calculation of Volume • Back extrapolation can be • completed a number of ways • Graphically • By equation 100 60 30 10 6 4 2 1 Intercept Time Plasma Conc (hr) (mg/L) 119.5 214.4 [ ]t = C0 e-Kt or [ ]0= C1 e+Kt = 0 2 4 6 8 10 12 14

Calculation of Volume • Back extrapolation can be • completed a number of ways • Graphically • By equation 100 60 30 10 6 4 2 1 Intercept Time Plasma Conc (hr) (mg/L) 119.5 214.4 [ ]t = C0 e-Kt or [ ]0= C1 e+Kt [ ]0 = 19.5 e(0.3015 x 1) = 26.36 mg/L Volume = Dose / [ ]0 = 0 2 4 6 8 10 12 14

Calculation of Volume • Back extrapolation can be • completed a number of ways • Graphically • By equation 100 60 30 10 6 4 2 1 Intercept Time Plasma Conc (hr) (mg/L) 119.5 214.4 [ ]t = C0 e-Kt or [ ]0= C1 e+Kt [ ]0 = 19.5 e(0.3015 x 1) = 26.36 mg/L Volume = Dose / [ ]0 = 500 / 26.36 = 18.97 L 0 2 4 6 8 10 12 14

Back extrapolation can be • completed a number of ways • Graphically • By equation • By Excel • Using Intercept function • =10^(INTERCEPT(C5:C9,A5:A9)) Calculation of Volume

1 Compartment IV Analysis 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection • What model best describes this profile? • Calculate AUC, K, Vd, T½ and Cl. • Write an equation the will calculate • concentrations at anytime after the 1st dose. • How much is in the body at time zero? • How much is in the body at 12 hours? • Where did the tobramycin go? • How much is in the urine at 12 hrs? • What is the renal clearance of tobramycin? • Is tobra a high or low extraction drug? Now calculate AUC. Since it was an IV dose, we must realise that the data set starting at 1 hour is missing concentration data from 0 to 1 hour. Therefore, AUC by trapezoidal rule must use a time zero concentration of 26.38 mg/L.

Trapezoidal Rule AUC =((C0+C1)/2)*(t1-t0) The value of 9.75 for the AUC is based on a noconcentration or a concentration of 0.00 at time zero. Was the Concentration zero? Or should we determine the time zero concentration Calculation of AUC Trapezoidal Rule (Excel) AUC = ((B4+B5)/2)*(A5-A4) The value of 9.75 in this cell is based on a noconcentration or a concentration of 0.00 at time zero in cell B4.

Trapezoidal Rule If the initial concentration of 26.38 is used in the calculation of AUC from time 0 to 1 hr, a more appropriate AUC of 22.94 mg*hr/L is estimated. AUC0-12h is 88.69 mg*hr/L. Calculation of AUC You must also calculate AUC from 12 hr to . This can only be done using the PCK method: [ ]LP/K.  = 88.69

AUC from 12hr to  Recall that K was calculated as 0.3015 hr-1 using concentrations at 1 hr and 12 hrs. Using this K value, AUC12- is: [ ]LP/K. = 0.7/0.3015 = 2.32 mg*hr/L. Calculation of AUC The sum of the AUC (0-12hr) and AUC (12hr-). is: AUC (0-) = 88.69 + 2.32 = 91.01 mg*hr/L.  = 88.69

Other Estimates of AUC (0-) • Several AUC (0-) estimates • are possible based on the • estimate of K • and the method used. • Kinetic method • Using the determined • back-extrapolated t=0 • concentration of 26 .38 mg/L • AUC0-  can also be estimated as: • [ ]t=0/K. = 26.38/0.3015 • = 87.496 mg*hr/L. Calculation of AUC

Other Estimates of AUC (0-) Several AUC (0-) estimates are possible based on the estimate of K and the method used. 2. Estimates of K In the Excel sheet using the last 2 … or last 3 …or last 4… K values can be estimated ranging from of 0.30809 hr-1 0.30245 hr-1 0.30185hr-1 and 0.301806 hr-1 (these are effectively the same ~ differ by less than 0.007 hr-1) These variations will produce small differences in AUC Calculation of AUC

Other Estimates of AUC Excel sheet shows some of the possible estimates of K and using 0.0318 hr-1 and calculates AUC12h- As 2.32 mg*hr/L. Also shown are Calculation of AUC

Other Estimates of AUC Excel sheet shows some of the possible estimates of K and using 0.0318 hr-1 and calculates AUC12h- As 2.32 mg*hr/L. Also shown are two estimates of AUC0-: 1. Based on kinetic method from the t=0 determined concentration and 2. Based on sum of Trap Rule AUC & kinetic method from LP Calculation of AUC

Other Estimates of AUC AUC0- is 91.01 mg*hr/L calculated by trapezoidal rule. But if we use the PCK method from time zero (26.38 mg/L) we would calculate AUC0- as 87.41 mg*hr/L. Which is more correct? Calculation of AUC

1 Compartment IV Analysis 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection • What model best describes this profile? • Calculate AUC, K, Vd, T½ and Cl. • Write an equation the will calculate • concentrations at anytime after the 1st dose. • How much is in the body at time zero? • How much is in the body at 12 hours? • Where did the tobramycin go? • How much is in the urine at 12 hrs? • What is the renal clearance of tobramycin? • Is tobra a high or low extraction drug? • Now calculate Clearance. • This can be completed • by at least 2 methods: • Dose / AUC • K x Vd

Calculation of Clearance Estimates of Clearance Although a variety of estimates of K have been calculated all are very close (~0.3018 hr-1) Using the volume calculated from the t=0 concentration determined by back-extrap using this K value (Excel Sheet) Cl = K x Vd = 0.3018 x 18.95 L = 5.720 L/hr

Calculation of Clearance Estimates of Clearance Although a variety of estimates of K have been calculated all are very close (~0.3018 hr-1) Cl = 5.720 L/hr (K x V) When we use AUC calculated by the PCK method, clearance estimate is identical to estimate determined by K*Vd. Clearance is 5.72 L/hr and AUC0- of 87.41 mg*hr/L. is the more correct estimate.

Equation describing profile Profile is a 1 compartment model with bolus input. [ ]t = C0 e-Kt Where K = 0.3018 hr-1 and C0 can always be calculated from Dose/V Ct = (Dose / V) e-Kt and for our patient K = 0.3018 hr-1 V = 18.95 L

1 Compartment IV Analysis 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Time Plasma Conc (hr) (mg/L) 119.5 214.4 55.8 82.4 120.7 18Below limit of detection • What model best describes this profile? • Calculate AUC, K, Vd, T½ and Cl. • Write an equation that will calculate • concentrations at anytime after the 1st dose. • How much is in the body at time zero? • How much is in the body at 12 hours? • Where did the tobramycin go? • How much is in the urine at 12 hrs? • What is the renal clearance of tobramycin? • Is tobra a high or low extraction drug? At time zero, immediately following the dose, there should be 500 mg in the body. The entire dose is delivered into the body.

Amount in the body at 12 hours Ct = (Dose / V) e-Kt and for our patient K = 0.3018 hr-1 Dose = 500 mg V = 18.95 L So the concentration at 12 hr would be: C12hr =

Amount in the body at 12 hours Ct = (Dose / V) e-Kt and for our patient K = 0.3018 hr-1 Dose = 500 mg V = 18.95 L So the concentration at 12 hr would be: C12hr = (500 / 18.95) e(-0.3018 x 12) = 28.38 e(-0.0267) = 0.7054 mg/L and the concentration reported at 12 hours was 0.7 mg/L

Amount in the body at 12 hours C12hr = (500 / 18.95) e(-0.3018 x 12) = 28.38 e(-0.0267) = 0.7054 mg/L and the concentration reported at 12 hours was 0.7 mg/L Since the volume is 18.95 L, the amount in the body at 12 hours is: Amount = [ ] x V = 0.7 mg/L x 18.95 L = 13.265 mg Only 13.265 mg of the initial 500 mg dose remains.

Amount in the body at 12 hours We could have also calculated the amount in the body directly from the equation: Instead of Ct = (Dose / V) e-Kt We would have Amtt =

Amount in the body at 12 hours We could have also calculated the amount in the body directly from the equation: Instead of Ct = (Dose / V) e-Kt We would have Amtt = Dose e-Kt and for our patient K = 0.3018 hr-1 Dose = 500 mg So the amount in the body at 12 hr would be: Amt12hr= (500) e(-0.3018 x 12) = 500 e(-0.0267) = 13.36 mg

Where did the tobramycin go? 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed.

Where did the tobramycin go? 500 mg of tobramycin is administered by iv bolus to Mr BB, a 68 yr old male who weighs 75 kg. Blood samples were drawn following the dose and the plasma concentration determined. It is known that tobramycin is excreted primarily unchanged in the urine. The following plasma concentrations were observed. Amount = [ ] x V = 0.7 mg/L x 18.95 L = 13.26 mg Amt12hr= (500) e(-0.3018 x 12) = 500 e(-0.0267) = 13.36 mg Only 13.3 mg of the initial 500 mg dose remains. Does this mean that 486.7 mg is in the urine in the first 12 hours following the dose?

Since renal elimination appears to be the only way tobramycin is cleared from the body, Total Body Clearance should equal renal clearance…? Clearance is 5.7 L/hr Calculation of Clearance

Renal Clearance of tobramycin At constant concentration, clearance can be estimated from blood flow and the difference between arterial and venous concentrations. ClH = Q [ (Ca – Cv) / Ca ] But in our situation the concentration changing. In the first hour concentration started at 26.38 mg/L and declined to 19.51 mg/L and we collected 130.26 mg of tobramycin in the urine. In the second hour concentration started at 19.51 mg/L and declined to 14.43 mg/L and we collected 96.33 mg of tobramycin in the urine. Equations Conc = Dose / V V = Dose/Conc Cl = Q x ER ER = Cl / Q

Renal Clearance of tobramycin As the concentration declines we recover less in the urine and because urinary excretion is the only method of elimination, drug excretion into the urine is tied to elimination from the body. Rate of appearance of drug in the urine is proportional to the amount in the body Therefore: Where X is the amount of drug in the body at time t and Xu is the amount of drug eliminated in the urine to time t. Equations Conc = Dose / V V = Dose/Conc Cl = Q x ER ER = Cl / Q dXu dt --- = ke X

Renal Clearance of tobramycin Since X, the amount drug in the body at time t changes according to X = X0 e (-Kt) or as we have previous written Amtt = Dose e (-Kt) Then … and a semi-log plot of excretion rate of drug in the urine vs. time should be linear and have a slope of –K/2.303. This is the same slope as for a plasma concentration curve. dXu dt --- = ke X0 e(-Kt) Equations Conc = Dose / V V = Dose/Conc Cl = Q x ER ER = Cl / Q

Time Plasma Conc (hr) (mg/L) 1 19. 5 2 14.4 5 5.8 8 2. 4 12 0.7 18 Below limit of detection