1. Multiple IV Dosing Pharmacokinetic Research, Bioequivalence Studies,
Drug – Drug Interactions
Generally all evaluated
through single dose studies.
But patients receive and most drug therapy is given in a multiple dose regimen
Multiple doses generally result in accumulation of drug,
… but how much, how fast ?

2. Problem: A patient with renal dysfunction (CrCl~ 20 mL/min) is
given tobramycin IV. The half-life of tobramycin in this
patient is 8 hours. Although tobramycin is normally
infused over at least 30 minutes, we will administer
100 mg of tobramycin to our patient by IV bolus every
8 hours. Calculate the amount of tobramycin in the
body after each dose for the first 10 doses.
Dose Time Amount Amount Increase
(n) (hr) in the Eliminated in Body
Body During Store
(mg) Dose Int (mg)
1. Zero
8.00

3. Tobramycin, 100 mg administered by IV bolus every
8 hours. Half-life of of tobramycin in this patient is 8 hrs.
Calculate the amount in the body after each of the first 10 doses.
Dose Time Amount Amount Increase
(n) (hr) in the Eliminated in Body
Body During Store
(mg) Dose Int (mg)
1. Zero
8.00
16.0
24.00

4. Tobramycin, 100 mg administered by IV bolus every
8 hours. Half-life of of tobramycin in this patient is 8 hrs.
Calculate the amount in the body after each of the first 10 doses.
Dose Time Amount Amount Increase
(n) (hr) in the Eliminated in Body
Body During Store
(mg) Dose Int (mg)
1. Zero
Increase
is difference
between
dose and
what was
eliminated

5. Tobramycin, 100 mg administered by IV bolus every
8 hours. Half-life of of tobramycin in this patient is 8 hrs.
Calculate the amount in the body after each of the first 10 doses.
Dose Time Amount Amount Increase
(n) (hr) in the Eliminated in Body
Body During Store
(mg) Dose Int. (mg)
nth
zero

6. Eventually, at some point accumulation stops
Tobramycin, 100 mg administered by IV bolus every
8 hours. Half-life of of tobramycin in this patient is 8 hrs.
Calculate the amount in the body after each of the first 10 doses.
Dose Time Amount Amount Increase
(n) (hr) in the Eliminated in Body
Body During Store
(mg) Dose Int. (mg)
nth
zero
nth
zero
Eventually, at some point accumulation stops
This is called steady-state (ss)

7. At steady state the amount eliminated during an interval is
Tobramycin, 100 mg administered by IV bolus every
8 hours. Half-life of of tobramycin in this patient is 8 hrs.
Calculate the amount in the body after each of the first 10 doses.
Dose Amount Amount Increase
(n) in the Body Eliminated in Body
End of Interval During Store
(mg) Dose Int. (mg)
nth = Dose
At steady state the amount eliminated during an interval is
equal to the dose, or with oral dosing the amount of the dose absorbed.
IN = OUT

8. At steady state the amount eliminated during an interval is
Tobramycin, 100 mg administered by IV bolus every
8 hours. Half-life of of tobramycin in this patient is 8 hrs.
Calculate the amount in the body after each of the first 10 doses.
Dose Amount Amount Increase
(n) in the Body Eliminated in Body
End of Interval During Store
(mg) Dose Int. (mg)
nth = Dose
How did SS occur?
During each
dosing interval
(8 hour period)
the body appears
to eliminate
(metabolise, clear)
more drug…
How is this
possible?
At steady state the amount eliminated during an interval is
equal to the dose, or with oral dosing the amount of the dose absorbed.
IN = OUT

9. body stores increased by half as much as the previous dose… why?
Tobramycin, 100 mg administered by IV bolus every
8 hours. Half-life of of tobramycin in this patient is 8 hrs.
Calculate the amount in the body after each of the first 10 doses.
Dose Amount Amount Increase
(n) in the Body Eliminated in Body
End of Interval During Store
(mg) Dose Int. (mg)
nth = Dose
When did SS occur?
Increase in body stores continued with each dose, in fact with each dose
body stores increased by half as much as the previous dose… why?

10. Steady State is considered to have been achieved when conc. (amount)
Tobramycin, 100 mg administered by IV bolus every
8 hours. Half-life of of tobramycin in this patient is 8 hrs.
Calculate the amount in the body after each of the first 10 doses.
Dose Amount Amount Increase
(n) in the Body Eliminated in Body
End of Interval During Store
(mg) Dose Int. (mg)
nth = Dose
When did SS occur?
Steady State is considered to have been achieved when conc. (amount)
are within 10%. For this patient this occurs between the 3rd and 4th dose.

11. Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Amount in the Body
Body Following a First and SINGLE Dose
(mg)
1. Zero

12. Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body after the First Dose
(mg)
1. Zero
Amount = Dose
Amount = Dose1 * e-Kt = Dose1 * e-Kτ
τ (tau) represents dosing interval

13. What is the concentration immediately following
The second dose (8.01 hrs)?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body after the Second Dose
(mg)
Amount = Dose1 * e-Kt = Dose1 * e-Kτ
Amount = Dose2 + Dose1 * e-Kτ
= 8hr single dose +0 hr single dose =
= 150 mg
=  
Time Am’t in
(hr) the Body
(mg)  
3 = 2 + 1

14. 3 = 2 + 1 Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body after the Second Dose
(mg)
Amount = Dose1 * e-Kt = Dose1 * e-Kτ
Amount = Dose2 + Dose1 * e-Kτ
Amount = [Dose2 + Dose1 * e-Kτ] * e-Kτ
= Dose2 e-Kτ + Dose1 * e-2Kτ
= 8hr single dose +16hr single dose =
= 75 mg
= 
Time Am’t in
(hr) the Body
(mg)   
3 = 2 + 1

15. 6 = 1 + 2 + 3 Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body after the Third Dose
(mg)
Amount = Dose2 e-Kτ + Dose1 * e-2Kτ
Amount = Dose3 +Dose2 e-Kτ + Dose1 * e-2Kτ
= Dose (1 + e-Kτ + e-2Kτ)
= 0 hr + 8hr + 16 hr =
= 175 mg
=  
Time Am’t in
(hr) the Body
(mg)   
6 =

16. 6 = 1 + 2 + 3 Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body following the Third Dose
(mg)
Amount = Dose3 +Dose2 e-Kτ + Dose1 * e-2Kτ
= Dose (1 + e-Kτ + e-2Kτ)
Amount = Dose3 e-Kτ +Dose2 e-2Kτ +Dose1e-3Kτ
= Dose (e-Kτ + e-2Kτ +e-3Kτ)
= 8 hr + 16 hr + 24 hr
= = 12.5
= 87.5 mg
=  
Time Am’t in
(hr) the Body
(mg)   
6 =

17. Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body Second & Third Doses
(mg)
Amount = Dose2 + Dose1 * e-Kτ
Amount = [Dose2 + Dose1 * e-Kτ] * e-Kτ
= Dose2 e-Kτ + Dose1 * e-2Kτ
Amount = Dose3 +Dose2 e-Kτ + Dose1 * e-2Kτ
= Dose (1 + e-Kτ + e-2Kτ)
Amount = Dose3 e-Kτ +Dose2 e-2Kτ +Dose1e-3Kτ
= Dose (e-Kτ + e-2Kτ +e-3Kτ)

18. Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body Third Dose
(mg)
Amount = Dose3 +Dose2 e-Kτ + Dose1 * e-2Kτ
= Dose (1 + e-Kτ + e-2Kτ)
Amount = Dose3 e-Kτ +Dose2 e-2Kτ +Dose1e-3Kτ
= Dose (e-Kτ + e-2Kτ +e-3Kτ)
This is a geometric progression
for both the initial concentration
following each dose (Cmax)
and the end-of interval
concentration (Cmin).
But what does this show us?

19. Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body
(mg)
Amount = Dose3 e-Kτ +Dose2 e-2Kτ +Dose1e-3Kτ
= Dose (e-Kτ + e-2Kτ +e-3Kτ)
But what does this show us?
First, that the concentration in a
multiple dose profile can be
obtained by adding the conc.
from various times following
a single dose.
For example: [ ]8hr after dose 3 = [3]8hr + [2]16hr + [1]24hr

20. Can we develop an equation
which will predict these amounts / concentrations ?
Dose Time Amount
(n) (hr) in the Equation for Amount in the Body
Body
(mg)
Amount = Dose3 e-Kτ +Dose2 e-2Kτ +Dose1e-3Kτ
= Dose (e-Kτ + e-2Kτ +e-3Kτ)
Time Am’t in
(hr) the Body
(mg)
For example: [ ]8hr after dose 3 = [3]8hr + [2]16hr + [1]24hr
= = 87.5 mg

21. Can we develop an equation
which will predict these amounts / concentrations ?
Second that we have a Geometric Progression
Solution to the Geometric Progression
Since
Cminn = Cmaxn e-Kτ
These equations will calculate the concentration
for the Cmax and Cmin of any dose (n) in a multiple dosing regimen.
Cmax after doses where SS has not been achieved – blue arrows

22. Can we calculate the concentration at any time after any dose?
Can we develop an equation which will
predict concentrations at any time?
Can we calculate the
concentration at any time
after any dose?
Since
Cminn = Cmaxn e-Kτ
Then
Ct = Cmaxn e-Kt
and where n = 6 to concentrations
corresponding to the red arrows above.

23. Can we develop an equation which will
predict concentrations at any time?
Therefore the more general
equation which will calculate
concentrations at any time t after
any dose (1 through ) is:
and when n is large the term
e-nKτ approaches zero and then:
Predicts for concentrations at steady state.

24. Can we develop an equation which will
predict concentrations at any time?
Using our equation:
Example: Calculate the concentration
4 hrs after dose 3 in a patient.
Dose = 1000 mg, V=10 L; τ = 8 hr
T½ = 8 hr (K = hr-1)
1000 (1- e-3*0.0866* 8)
(1- e * 8)
Ct = e *4
Ct = 100*[( )/(1-0.50)]*0.7
= 100 * [(0.875)/(.50)] * 0.707
= 100 * 1.75 * 0.707
= 175 * 0.707
= mg/L
What is ??

25. With Multiple Doses Drugs Accumulate.
Can we predict the degree of accumulation quickly, without an equation?
We have seen that in a patient
given 1000 mg Q8H (τ)
that when the T½ is 8 hr
(K = hr-1) and
V = 10L, the degree of
accumulation
is a factor of 2.
Dose Amount Amount
in the Body in the Body
Start of Interval End of Interval
(n) (mg) (mg)
nth

26. Can we predict how much accumulation WILL occur?
Consider giving 100 mg of tobramycin IV, at different dosing
intervals, when half-life is 8 hours and Volume is 10 L.
Conditions : Dose =100; V = 10 L half-life = 8 hr
CMAX
Dose Q2H Q4H Q6H Q8H Q12H Q24H
1st
2nd.
3rd.
@ 24.
@ 48
@ 96
Using MD 1C-IV Bolus Excel Sheet determine degree of
accumulation for Q2, Q4, Q8, Q12, Q16 and Q24 hr regimens.
Conditions : Dose = 200; V = 20 L half-life = 8 hr

27. Can we predict how much accumulation WILL occur?
Accumulation depends on the rate (frequency)
at which we give the drug (dosing interval).
Consider giving 100 mg of tobramycin IV, at different dosing
intervals, when half-life is 8 hours and Volume is 10 L.
Conditions : Dose =100; V = 10 L half-life = 8 hr
CMAX
Dose Q2H Q4H Q6H Q8H Q12H Q24H
1st
2nd
3rd @ @ @

28. Can we predict how much accumulation WILL occur?
Accumulation depends on the rate (frequency)
at which we give the drug (dosing interval).
Consider giving 100 mg of tobramycin IV, at different dosing
intervals, when half-life is 8 hours and Volume is 10 L.
Initial concentration is : Dose/ Vd = 100/ 10 = 10 mg/L
Dose Q2H Q4H Q6H Q8H Q12H Q24H @ @ @
Notice that at
Q8H,
degree of
accumulation
is a factor of 2

29. Can we predict how much accumulation WILL occur?
Repeat the process, but this time use a half-life of 6 hours
(patient with a different CrCl).
Administer 100 mg of tobramycin IV, at different dosing
intervals, when half-life is 6 hours and Volume is 10 L.
Conditions : Dose =100; V = 10 L half-life = 6 hr
CMAX
Dose Q2H Q4H Q6H Q8H Q12H Q24H
1st
2nd.
3rd.
@ 24.
@ 48
@ 96
Using MD 1C-IV Bolus Excel Sheet determine degree of
accumulation for Q2, Q4, Q8, Q12, Q16 and Q24 hr regimens.
Conditions : Dose = 200; V = 20 L half-life = 6 hr

30. Can we predict how much accumulation WILL occur?
Repeat the process, but this time use a half-life of 6 hours
(patient with a different CrCl).
Administer 100 mg of tobramycin IV, at different dosing
intervals, when half-life is 6 hours and Volume is 10 L.
Conditions : Dose =100; V = 10 L half-life = 6 hr
Dose Q2H Q4H Q6H Q8H Q12H Q24H
1st
2nd
3rd @ @ @
When is the degree of accumulation of 2 observed now?

31. Can we predict how much accumulation WILL occur?
Repeat the process, but this time use a half-life of 6 hours
(patient with a different CrCl).
Administer 100 mg of tobramycin IV, at different dosing
intervals, when half-life is 6 hours and Volume is 10 L.
Conditions : Dose =100; V = 10 L half-life = 6 hr
CMAX
Dose Q2H Q4H Q6H Q8H Q12H Q24H
1st
2nd
3rd @ @ @
2 x

32. Can we predict how much accumulation WILL occur?
Accumulation depends on the rate at which we give the drug
(dosing interval τ) and the rate at which the drug is eliminated (K)
When the Dosing Interval and half-life are equal, accumulation is 2x
Half-life = 8 hours
Dose Q2H Q4H Q6H Q8H Q12H Q24H @ @ @

33. Can we predict how much accumulation WILL occur?
Accumulation depends on the rate at which we give the drug
(dosing interval τ) and the rate at which the drug is eliminated (K)
When Dosing interval is greater than the half-life,
more drug is eliminated during a dosing interval
and LESS accumulation occurs
Half-life = 8 hours
Dose Q2H Q4H Q6H Q8H Q12H Q24H @ @ @

34. Can we predict how much accumulation WILL occur?
Accumulation depends on the rate at which we give the drug
(dosing interval τ) and the rate at which the drug is eliminated (K)
When Dosing Interval is shorter than the half-life,
less drug is eliminated during a dosing interval (because it is short)
and MORE accumulation occurs
Half-life = 8 hours
Dose Q2H Q4H Q6H Q8H Q12H Q24H @ @ @

35. The ratio of the Peak Concentration at Steady State
and the Peak Concentration after the first dose
is a direct measure of the degree of accumulation.
Cmax ss
= Maximum Accumulation Factor
Cmax 1
Dose Q2H Q4H Q6H Q8H Q12H Q24H @ @ @
MAF =

36. -------- = MAF = -----------
The ratio of the Peak Concentration at Steady State
and the Peak Concentration after the first dose
is a direct measure of the degree of accumulation.
Cmax ss 1
= MAF =
Cmax e -K
Cmaxss = Cmax1 x MAF
Cminss = Cmin1 x MAF
MAF rules of thumb
if  = T½… then MAF = 2
if  > T½… then MAF < 2
if  < T½… then MAF > 2

37. -------- = MAF = -----------
The ratio of the Peak Concentration at Steady State
and the Peak Concentration after the first dose
is a direct measure of the degree of accumulation.
Cmax ss 1
= MAF =
Cmax e -K
MAF considers only
 … the dosing interval
K … the half-life

38. MAF = ----------- … predicts accumulation
Building the Multiple Dose Equation 1
MAF = … predicts accumulation
1 - e -K 1 x
1 – e -K
Dose V
-Kt
Ct = e
First Dose Conc. Accumulation Steady State Equation
and the Steady State equation predicts ONLY steady state concentrations
We need the other piece to calculate concentrations following ANY dose

39. Building the Multiple Dose Equation
Provides for all concentrations from first dose until steady state.
1- e -nK x
1 – e -K
Dose V
-Kt
C = e
In this equation what happens to the term
1- e -nk when n (number of doses) is large?

40. Building the Multiple Dose Equation
And the term
1- e -nK
1 – e -K
Can convert the single dose IV equation
to a multiple dose equation

41. How long does it take to get to steady state?
When did SS occur?
Steady State is considered to have been achieved when the
concentration or amount in the body is within 10% of the total
amount of drug in the body. For this patient total was 100mg.
Dose Amount Amount Increase Cumulative
(n) in the Body Eliminated in Body Increase
End of Interval During Storein Body Store
(mg) Dose Int. (mg) (mg)
nth = Dose
90% achieved
Between 3rd
& 4thdose

42. How long does it take to get to steady state?
When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
Compare patients with T½ of 6 and 8 hrs for time to reach SS.
T½ = 6 hr
Dose Time Q2H T½
T½ = 8 hr
Dose Time Q2H T½
90% of 96 hr [ ] =

43. How long does it take to get to steady state? When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
Sometimes, SS is not considered to have
been achieved until concentrations exceed
95% of the true steady state concentrations.
This occurs between 4 and 5 half-lives.
This gives rise to the statement that
steady state is achieved in 3-5 half-lives.
Each T½ reduces the gap between
current concentrations and SS by half.
Dosing interval (τ) does not affect time to SS

44. When is 90% of true (eventual) e-Kt determines proportion lost
How long does it take to get to steady state?
When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
When is 90% of true (eventual)
Steady State Achieved?
e-Kt determines proportion lost

45. How long does it take to get to steady state? When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
When is 90% of true (eventual)
Steady State Achieved?
e-Kt determines proportion lost
e-K# determines proportion lost
for a set number (#) of half-lives
Example:
If K = and # = 2 T½
= e ( x 2)
= 0.25

46. How long does it take to get to steady state? When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
When is 90% of true (eventual)
Steady State Achieved?
e-K# determines proportion lost
… then 1 - e-K# will determine
proportion of steady-state achieved.
Example:
If K = and # = 2 T½
= 1 -e ( x 2)
= 0.75
Or expressed as a %
75% of SS after 2 T½

47. How long does it take to get to steady state? When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
When is 90% of true (eventual)
Steady State Achieved?
e-K# determines proportion lost
and … 1 - e ( x #)
determines proportion of steady-state achieved.
and … 100 x (1 - e ( x #) )
determines percent of steady-state achieved,
where # is the number of half-lives.
% SS = 100 x (1 - e ( x #) )

48. When is 90% of true (eventual) Therefore, 90% of true SS is achieved…
How long does it take to get to steady state?
When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
When is 90% of true (eventual)
Steady State Achieved?
% SS = 100 x (1 - e ( x #) )
Therefore, 90% of true SS is achieved…
90 = 100 x (1 - e ( x #) )
0.9 = 1 - e ( x #)
e ( x #) = 0.1
ln(e ( x #) = ln(0.1)
x # =
# = 3.322

49. How long does it take to get to steady state? When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
When is 90% of true (eventual)
Steady State Achieved?
After half-lives
… frequently stated as 3.3 half lives
Or in hours: Time to SS (hr) = x T½
If the T½ for your drug is 10 hours
it will take (10 * 3.322) hours
… or hrs to achieve 90% of SS.

50. How long does it take to get to steady state? When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
When is 95% of true (eventual)
Steady State Achieved?
After how many half-lives (#)
is 95% of Steady State Achieved?
ln(e ( x #) = ln(0.05)
x # =
# = 4.322
(95% of SS achieved after half-lives)
or in hours: Time to SS (hr) = x T½

51. How long does it take to get to steady state? When did SS occur?
Steady State is considered to have been achieved when the
Concentration is within 10% of the true steady state concentration.
90% of true steady state will occur at ~ 3.3 half-lives.
Number Cumulative
Of Increase
T½ Body Store
(mg) SS
When is 90% of SS Achieved?
Shown slightly differently, Recall that:
the time to eliminate 50% of body stores is
T½ = 0.693/K
And is ln(0.5) (negative sign omitted)
Therefore, the time to reduce the amount
of drug in the body to 10% would be
Time in hours = ln(0.1)/K
= 2.302/K
or since K = 0.693/T½, then
Time (hr) = (ln(0.1)*T½) / or = (3.322)*(T½)

52. Multiple IV Doses & Concentrations at Steady State
In practice, some multiple dose
regimens are initiated with
a loading dose. Why do
we use loading doses?
To achieve steady state faster..?
Or to achieve a therapeutic concentration sooner?
Is a loading dose more useful for a drug
with a longer or shorter half-life?

53. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring
It is known that for many drugs, optimal effect is achieved with a
certain minimum concentration. It might also be known that
toxicity is more likely to occur with concentrations that exceed
a certain concentration. Therefore, it becomes the intent of a
multiple dose regimen to have peak and trough concentrations
fall within this therapeutic range.
Half-life is 8 hours, drug is administered every 8 hours.
Dose is 400 mg. VD = 40 L.
What dose will achieve a SS peak concentration of 30 mg/L?

54. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring
Half-life is 8 hours, drug is administered every 8 hours.
Dose is 400 mg. VD = 40 L.
What dose will achieve a steady state concentration of 30 mg/L?
Dose of 400 mg produces a peak of 400 mg/ 40 L = 10 mg/L
Since half-life and Dosing interval are equal … MAF = ???
And the dosing regimen of 400 mg q8h will produce SS [ ] of ?
Cpmax ss = Cp1 * MAF
= 10 mg/L * 2
= 20 mg/L

55. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring
Half-life is 8 hours, drug is administered every 8 hours.
Dose is 400 mg. VD = 40 L.
What dose will achieve a steady state concentration of 30 mg/L?
A dosing regimen of 400 mg q8h will produce SS [ ] of ?
Cpmax ss = Cp1 * MAF
= 10 mg/L * 2
= 20 mg/L
To acheive a SS peak of 30 mg/L we will need to administer
600 mg every 8 hours in this patient.

56. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring
Half-life is 8 hours, drug is administered every 8 hours.
Dose is 400 mg. VD = 40 L.
If 600 mg is administered IV every 8 hours, what
will the steady-state trough (Cmin) concentrations be?
Given Cmaxss = 30 mg/L
Since dosing interval and half-life are equal …
half the concentration is eliminated during a dosing interval.
Cpmaxss = 30 mg/L * 0.5
= 15 mg/L
At SS [ ] fluctuate between 15 mg/L and 30 mg/L

57. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring … tougher Question
Following the first 80 mg dose of an aminoglycoside administered
by IV bolus*, the concentrations at 2 & 6 hours were observed to
be 4 and 2 mg/L. At steady state the desired peak is 8 mg/L and
the trough is 1 mg/L. What dosing regimen (dose and interval)
will yield these desired concentrations?
What do you need to know?
What predicts degree of accumulation
What is the Half-life?
What should the dosing interval be?
* Aminoglycosides normally administered by IV infusion over min

58. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring … tougher Question
Following the first 80 mg dose of an aminoglycosisde administered
by IV bolus, the concentrations at 2 & 6 hours were observed to be
4 and 2 mg/L. At steady state the desired peak is 8 mg/L and the
trough is 1 mg/L. What dosing regimen (dose and interval) will
yield these desired concentrations?
MAF is based on τ and T½
Time [mg/L]
T½ = ??? K = hr-1
What do you need to know?
What predicts degree of accumulation
What is the Half-life?
What should the dosing interval be?

59. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring … tougher Question
Following the first 80 mg dose of an aminoglycosisde administered
by IV bolus, the concentrations at 2 & 6 hours were observed to be
4 and 2 mg/L. At steady state the desired peak is 8 mg/L and the
trough is 1 mg/L. What dosing regimen (dose and interval) will
yield these desired concentrations?
MAF is based on τ and T½
Dosing interval based on
Peak 8 mg/L
Trough 1 mg/L
3 T½’s (8,4,2,1) = 12 hr.
MAF = 1.143
What do you need to know?
What predicts degree of accumulation
What is the Half-life?
What should the dosing interval be?

60. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring … tougher Question
Following the first 80 mg dose of an aminoglycosisde administered
by IV bolus, the concentrations at 2 & 6 hours were observed to be
4 and 2 mg/L. At steady state the desired peak is 8 mg/L and the
trough is 1 mg/L. What dosing regimen (dose and interval) will
yield these desired concentrations?
Peak would be the time zero concentration …
What was the concentration at time zero following the first dose?
K= hr-1 C2 = 4;
C0 = C2e(-Kt)
= 4 e( x 2)
= 5.66 mg/L

61. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring … tougher Question
Following the first 80 mg dose of an aminoglycosisde administered
by IV bolus, the concentrations at 2 & 6 hours were observed to be
4 and 2 mg/L. At steady state the desired peak is 8 mg/L and the
trough is 1 mg/L. What dosing regimen (dose and interval) will
yield these desired concentrations?
What would the current dose (80 mg) yield for a Steady State Peak?
CSS-peak = 5.66 mg/L x MAF
= 5.66 x
= 6.46 mg/L

62. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring … tougher Question
Following the first 80 mg dose of an aminoglycosisde administered
by IV bolus, the concentrations at 2 & 6 hours were observed to be
4 and 2 mg/L. At steady state the desired peak is 8 mg/L and the
trough is 1 mg/L. What dosing regimen (dose and interval) will
yield these desired concentrations?
What dose is necessary to produce a peak of 8 mg/L (simple ratio)?
Dose = (8.0 / 6.46) x (80 mg)
= x 80
= mg
This dose is unreasonable and so should be rounded up to 100 mg.

63. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring … tougher Question
Following the first 80 mg dose of an aminoglycosisde administered
by IV bolus, the concentrations at 2 & 6 hours were observed to be
4 and 2 mg/L. At steady state the desired peak is 8 mg/L and the
trough is 1 mg/L. What dosing regimen (dose and interval) will
yield these desired concentrations?
Best Regimen is 100 mg administered every 12 hours.
Calculate the SS peak. (use either complete equation or peak1 x MAF
CmaxSS = (dose/volume) x MAF
= (100 mg / 14.14L) x 1.143
= 8.08 mg/L

64. Concentrations at Steady State
Multiple IV Doses &
Concentrations at Steady State
Therapeutic Drug Monitoring … tougher Question
Following the first 80 mg dose of an aminoglycosisde administered
by IV bolus, the concentrations at 2 & 6 hours were observed to be
4 and 2 mg/L. At steady state the desired peak is 8 mg/L and the
trough is 1 mg/L. What dosing regimen (dose and interval) will
yield these desired concentrations?
Best Regimen is 100 mg administered every 12 hours.
Calculate the SS Trough (use CmaxSS x e(-Kτ)
CmaxSS = 8.08 e(-Kτ)
= 8.08 e ( x 12)
= 1.01 mg/L

65. Calculation of Parameters at SS – IV dosing
Evaluate the following concentrations determined in an patient
following the 1st, 2nd, 6th and 15th 100mg doses of a drug given Q8H.
Dose Dose Dose Dose
Time [ ] Time [ ] Time [ ] Time [ ]
(hr) (mg/L) hr) (mg/L) (hr) (mg/L) (hr) (mg/L)
Steady State??
First Dose Half way % ~ True SS
1 T½ T½ T½

66. Calculation of Parameters at SS – IV dosing
Evaluate the following concentrations determined in an patient
following the 1st, 2nd, 6th and 15th 100mg doses of a drug given Q8H.
Dose Dose Dose Dose
Time [ ] Time [ ] Time [ ] Time [ ]
(hr) (mg/L) hr) (mg/L) (hr) (mg/L) (hr) (mg/L)
Does the half-life change?
Does the volume change?
Does clearance change?
Does AUC change?

67. Calculation of Parameters at SS– IV dosing
Evaluate the following concentrations determined in an patient
following the 1st, 2nd, 6th and 15th 100mg doses of a drug given Q8H.
Dose Dose Dose Dose
Time [ ] Time [ ] Time [ ] Time [ ]
(hr) (mg/L) hr) (mg/L) (hr) (mg/L) (hr) (mg/L)
Does the half-life change?
Does the volume change?
Does clearance change?
Does AUC change?
Protein binding.

68. Calculation of Parameters at SS – IV dosing
Evaluate the following concentrations determined in an patient
following the 1st, 2nd, 6th and 15th 100mg doses of a drug given Q8H.
Dose Dose Dose Dose
Time [ ] Time [ ] Time [ ] Time [ ]
(hr) (mg/L) hr) (mg/L) (hr) (mg/L) (hr) (mg/L)
Does the half-life change?
Does the volume change?
Does clearance change?
Does AUC change?
Dose / AUC or K*V

69. Calculation of Parameters at SS – IV dosing
Evaluate the following concentrations determined in an patient
following the 1st, 2nd, 6th and 15th 100mg doses of a drug given Q8H.
Dose Dose Dose Dose
Time [ ] Time [ ] Time [ ] Time [ ]
(hr) (mg/L) hr) (mg/L) (hr) (mg/L) (hr) (mg/L)
Does the half-life change?
Does the volume change?
Does clearance change?
Does AUC change?
Let’s calculate AUC

70. Calculation of Parameters at SS – IV dosing
Evaluate the following concentrations determined in an patient
following the 1st, 2nd, 6th and 15th 100mg doses of a drug given Q8H.
Dose Dose Dose Dose
Time [ ] Time [ ] Time [ ] Time [ ]
(hr) (mg/L) hr) (mg/L) (hr) (mg/L) (hr) (mg/L)
Does this
make sense?
In = Out
@ SS
AUC (mg*hr/L)
AUC(0-t)
AUC(0-)

71. Calculation of Parameters at SS – IV dosing
Evaluate the following concentrations determined in an patient
following the 1st, 2nd, 6th and 15th 100mg doses of a drug given Q8H.
Elimination of 100 mg following the first dose produced an
AUC(0-) of mg*hr/L.
At SS, a dose (100 mg) is eliminated during a dosing interval,
in this case 8 hours. At steady state we are eliminating a dose.
Therefore, the AUC(0-τ) at SS should be equivalent to the
AUC(0-) following the first dose.
1st Dose 2nd Dose 6th Dose 15th Dose
Half way % ~ True SS
1 T½ T½ T½
AUC (mg*hr/L)
AUC(0-t)
AUC(0-)

72. Calculation of Parameters at SS – IV dosing
Bioequivalence
Just as AUC(0-) following the 1st dose can be use to estimate
bioequivalence so can AUC(0-τ), as long as you are at steady state.
Steady state will require at least 3 T½ of dosing, and the greater
the time allowed to achieve steady, the smaller the difference
between AUC1(0-) and AUCSS(0-τ).
1st Dose 2nd Dose 6th Dose 15th Dose
Half way % ~ True SS
1 T½ T½ T½
AUC (mg*hr/L)
AUC(0-t)
AUC(0-)

73. No change in other parameters
Calculation of Parameters at SS – IV dosing
No change in other parameters

74. Multiple Dosing Summary … so far – IV dosing
We have seen that:
1. [ ] following single doses can be Summed to create MD profile
(Additivity of [ ])
2. At Steady State In (Dose) = Out (elimination)
3. MAF, predicts the degree of accumulation (K and τ)
4. (1/1-e(-Kτ)), converts a single dose equation to multiple dose
5. (1-e(-nKτ))/(1-e(-Kτ)) allows Calculation of [ ] at any time
6. Time to Reach steady state determine by K (Cl/V) -3.3 T½
AUC(0-τ) over a dosing SS = AUC(0-) 1st dose.
Just as AUC(0-) 1st dose can be use to estimate bioequiavalence
so can AUC(0-τ), as long as you are at steady state (min3 T½ dosing) .
9. Using MAF we can design an IV dosing regimen that will achieve
desired peak and trough concentrations at SS.
At Steady state all other parameters remain unchanged (ClR,ClH,
proportion metabolised.)