index of simulations 1-compartment, iv bolus 1-compartment, iv infusion: steady- state1-compartment,...

Index of Simulations

• 1-Compartment, IV bolus

• 1-Compartment, IV infusion: Steady-State

• 1-Compartment, IV infusion: Non-Steady-State

• 1-Compartment, IV infusion: No Elimination Phase

1-Compartment, IV Bolus

• The following data were obtained after 100 mg of Drug X was administered to a healthy volunteer. Blood was collected starting at one-hour post-dose for a total of 12 hours. Calculate Cl, V and t1/2

Data (X0 = 100 mg)

Time (h) [Drug X] (ug/L)

1 314.7

1.5 280.0

2 246.5

2.5 219.5

3 194.1

4 152.8

6 94.1

8 59.0

10 35.6

12 22.1

Step 1) Graph on Semi-log Paper

Log scale Linear scale

10

100

1000

0 5 10 15

Time (h)

[Dru

g X

] (u

g/L

)

Step 2) Draw best fit line

10

100

1000

0 5 10 15

Time (h)

[Dru

g X

] (u

g/L

)

Step 3) Find V

• Use the relationship:

• Rearranged for V:

– We know X0 (100 mg) and C0 we can get from the graph

V

XC 0

0

0

0

C

XV

Step 3) Find V

10

100

1000

0 5 10 15

Time (h)

[Dru

g X

] (u

g/L

) C0

100

1000

0 1 2 3 4 5

Time (h)

[Dru

g X

] (u

g/L

)

C0 = 400 ug/L

200

300

400

Step 3) Find V

• We know X0 (100 mg) and C0 we can get from the graph (C0 = 400 ug/L) (please watch units)

• Now we have our volume (250 L)

L 250ug/L 400

ug 100000

C

XV

0

0

Step 4) Find t1/2 (and k)

• Half-life (t1/2) can be obtained directly from the graph by reading how long it takes for the concentration to be reduced by 50%

Step 4) Find t1/2

10

100

1000

0 5 10 15

Time (h)

[Dru

g X

] (u

g/L

) C0 = 400 ug/L

200

t1/2 = 3

Start with C0 which equals 400 ug/L

Half of 400 is 200. Draw a line from 200 across until it intersects your best fit line

At the intesection, draw a line down to the X-axis (time). Read the value the line intersects the axis…this is t1/2

Your t1/2 is ~3 hours

Step 5) Find Cl

• Clearance (Cl) can be calculated from k (Step 4) and V (Step 4) and using the following equation:

L/h 57.8L 2501/h 0.231VkCl

Summary

• Cl = 57.8 L/h

• V = 250 L

• t1/2 = 3 h

Return to Table of Contents Onto Steady-State Infusion

1-Compartment, IV Infusion: Steady-State

• The following data were obtained after 100 mg of Drug X was infused over 15 hours to a healthy volunteer. Blood was collected starting at one-hour post-dose for a total of 24 hours. Calculate Cl, V and t1/2

Data (X0 = 100 mg)


0 0

1 23.7

3 57.3

6 85.0

10 100.5

15 109.8

18 54.1

20 32.6

24 12.5



1

10

100

1000

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

)


• Half-life (t1/2) can be obtained directly from the graph by reading how long it takes for the concentration to be reduced by 50%. – For infusions, you must use the

terminal portion where concentrations are falling!!

• First however, draw a best fit line through the terminal portion

1

10

100

1000

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

)Step 2) Find t1/2

1

10

100

1000

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

)Step 2) Find t1/2

C ~ 110 ug/L

55

t1/2 = 18 - 15

Start with C which you know C at 15 h = 110 ug/L


At the intesection, draw a line down to the X-axis (time). Read the value the line intersects the axis…

Your t1/2 is the time you just read minus infusion time (18 h – 15 h = 3 hours)


• Half-life (t1/2) from the graph is 3 hours. We can find k by the following equation:

1/h 0.231h 3

0.693

t

0.693k

21

Step 3) Find Cl

• Clearance (Cl) can be calculated from the steady-state concentration (Css) and the infusion rate (k0) using the equation:

• Rearranged to:

Cl

kC 0

SS

ss

0

C

kCl

Step 3) Find Cl

• We know the dose (100 mg) and infusion time (T=15 h), therefore infusion rate is:

mg/h 6.67h 15

mg 100

T

Xk 0

0

Step 3) Find Cl

• We can obtain Css from the graph by looking to see when concentrations stop changing.

• How do we know for sure this is steady-state? Remember steady-state is 3-5 half-lives.– Half-life from Step 2 = 3h– 3 x 5 (or 3 or 4) = 15 h– Infusion was stop at 15 hours therefore we are

at steady-state and this approach is valid

Step 3) Find Cl

1

10

100

1000

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

)

Css= 110 ug/L

Step 3) Find Cl

• We have k0 (6.67 mg/h), we have CSS (110 ug/L), now we can calculate Cl

L/h 60.6ug/L 110

ug/h 6670

C

kCl

ss

0

Step 4) Find V

• Volume (V) can be calculated from k (Step 3) and Cl (Step 1) and using the following equation:

• Rearrange and solve for VVkCl

L 2621/h 0.231

L/h 60.6

k

ClV

Summary

• Cl = 60.6 L/h

• V = 262 L

• t1/2 = 3 h

Return to Table of Contents Onto Non-Steady-State Infusion

1-Compartment, IV Infusion: Non-Steady-State


Data (X0 = 100 mg)


0 0

1 61.6

3 159.1

6 260.0

10 136.2

15 61.7

18 38.2

20 25.2

24 14.4



1

10

100

1000

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

)


• Half-life (t1/2) can be obtained directly from the graph by reading how long it takes for the concentration to be reduced by 50%. – For infusions, you must use the

terminal portion where concentrations are falling!!

• First however, draw a best fit line through the terminal portion

1

10

100

1000

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

)Step 2) Find t1/2

1

10

100

1000

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

)Step 2) Find t1/2

C ~ 260 ug/L

130

t1/2 = 10 - 6Start with C which you know. C at 6 h = 260 ug/L


At the intesection, draw a line down to the X-axis (time). Read the value the line intersects the axis…

Your t1/2 is the time you just read minus infusion time (10 h – 6 h = 4 hours)


• Half-life (t1/2) from the graph is 3 hours. We can find k by the following equation:

1/h 0.173h 4

0.693

t

0.693k

21

Step 3) Find Cl

• Clearance (Cl) can be calculated two-ways. Please select a method to calculate clearance

• AUC Method – More exact but more calculations

• Equation Method – Quicker but less exact

Clearance Via AUC

• To calculate clearance via the AUC, you must first calculate the AUC via the trapezoidal rule

Trapezoidal Rule

0

50

100

150

200

250

300

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

) For this method, we break the

curve into individual

trapezoids as shown here…

C1

C2

t1 t2

The area of the trapezoid (or this case a

triangle) is the average height

(C1+C2)/2 multiplied by the

base (t2-t1)

Step 2) Setup the tableA B A*B

Time [Drug X] (C2+C1 )/2 (t2-t1) AUC

0 0 --- --- ---

1 61.6 30.8 1 30.8

3 159.1 110.4 2 220.8

6 260.0 209.6 3 628.8

10 136.2 198.1 4 792.4

15 61.7 99.0 5 495.0

18 38.2 50.0 3 150.0

20 25.2 31.7 2 63.4

24 14.4 39.6 4 158.4

Tail --- --- 83.1

SUM 2622.7

k

CAUC Last

tail

Step 3: Calculate Cl• Since we now have AUC, using the

dose (100 mg), and the equation:

• Solve for Cl:

Cl

XAUC 0

L/h 38.1ug.h/L 2622.7

ug 100000

AUC

XCl 0

Go to Volume calculationSelect another Cl calculation

Clearance via Infusion Equation

• We can use the equation that describes an infusion and solve for Cl.– During Infusion (t = time during infusion)

– Solving for Cl

)e(1Cl

kC t)k(0

)e(1C

kCl t)k(0

Clearance via Equation• Now plug in the values we know

(infusion rate, C, t, k)

L/h 42.4

)e(1ug/L 159.1

6hug 100000

)e(1C

kCl

h) 31/h 0.173(

t)k(0

Go to Volume calculationSelect another Cl calculation

Step 3) Find Cl


mg/h 6.67h 15

mg 100

T

Xk 0

0

Step 3) Find Cl

• We can obtain Css from the graph by looking to see when concentrations stop changing.

• How do we know for sure this is steady-state? Remember steady-state is 3-5 half-lives.– Half-life from Step 2 = 3h– 3 x 5 (or 3 or 4) = 15 h– Infusion was stop at 15 hours therefore we are

at steady-state and this approach is valid

Step 3) Find Cl

1

10

100

1000

0 10 20 30

Time (h)

[Dru

g X

] (u

g/L

)

Css= 110 ug/L

Step 3) Find Cl


L/h 60.6ug/L 110

ug/h 6670

C

kCl

ss

0

Step 4) Find V



L 2201/h 0.173

L/h 38.1

k

ClV

1-Compartment, IV infusion: No elimination phase


Data (X0 = 100 mg)


0 0

1 23.7

3 57.3

6 85.0

10 100.5

12 104.9

14 106.8

15 108.9



10

100

1000

0 5 10 15

Time (h)

[Dru

g X

] (u

g/L

)


• Since we do not have an elimination phase, we must find another way to estimate half-life. We will use the approach to steady-state method.

• So first we need to estimate CSS

kt)Cln(C tSS

10

100

1000

0 5 10 15

Time (h)

[Dru

g X

] (u

g/L

)Step 2) Find CSS

We can estimate CSs either taking the average of the last few concentrations or use a best fit line

106.5 ug/L

Just read CSS from the intercept of the Y-axis

Step 2) Find CSS

• Our CSS = 106.5 ug/L

• We can find K by plotting– CSS-Ct versus time on semi-log paper…

but first lets find CSs-Ct

Data (X0 = 100 mg)


(CSs-Ct)/CSS

0 0 (106.5-0)/106.5 = 1

1 23.7 (106.5-23.7)/106.5 = 0.78

3 57.3 (106.5-57.3)/106.5 = 0.46

6 85.0 (106.5-85.0)/106.5 = 0.20

10 100.5 (106.5-100.5)/106.5 = 0.06

12 104.9 ~ 0

14 106.8 ~ 0

15 108.9 ~ 0

0.01

0.1

1

0 5 10 15

Time (h)

(Css

-C)/

Css

Step 2) Find t1/2

Start with (Css-0)/Css which you know 1

Half of 1 is 0.5. Draw a line from 0.5 across until it intersects your best fit line

At the intersection, draw a line down to the X-axis (time). Read the value the line intersects the axis…

Your t1/2 is the time you just read (2.5 hours)

~ 1

t1/2 = 2.5

0.5


• Half-life (t1/2) from the graph is 2.5 hours. We can find k by the following equation:

1/h 0.277h 2.5

0.693

t

0.693k

21

Step 3) Find Cl

• Clearance (Cl) can be calculated from the steady-state concentration (Css) and the infusion rate (k0) using the equation:

• Rearranged to:

Cl

kC 0

SS

ss

0

C

kCl

Step 3) Find Cl


mg/h 6.67h 15

mg 100

T

Xk 0

0

Step 3) Find Cl

• We can obtain Css from the graph as we just did and had a value of 106.5 ug/L

Step 3) Find Cl


L/h 62.6ug/L 106.5

ug/h 6670

C

kCl

ss

0

Step 4) Find V



L 2261/h 0.277

L/h 62.7

k

ClV

Summary

• Cl = 62.6 L/h

• V = 226 L

• t1/2 = 2.5 h

Return to Table of Contents

Lesson Done!

index of simulations 1-compartment, iv bolus 1-compartment, iv infusion: steady- state1-compartment,...

Documents

v step

fit line slide

steadystate infusion

halflife t

graph c

terminal portion slide

infusion time

elimination phase slide