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Multiple dosing: intravenous bolusadministration
Dr Mohammad Issa Saleh
Multiple dosing calculations using Superposition
Let:Dose 1 Conc. 1
and:Dose 2 Conc. 2
then the response system behaves according to the superposition principle if:
Dose 1 +Dose 2 Conc. 1 + Conc. 2
and in that case the response system is a linear response system
2
Multiple dosing calculations using Superposition
A patient is to be given 100 mg of a drug intravenously. Assuming that K = 0.10 hr-1 and a V = 15 L, estimate the following:1. The half life
hr hr K
t 93.61.0
)2ln()2ln(2/1
3
Multiple dosing calculations using Superposition
2. The concentration 2 hrs after the dose
3. The concentration 10 hrs after the dose
mg/L 5.46D
)2( )2( tKeV
tC
mg/L 2.45D
)10( )10( tKeV
tC
4
Multiple dosing calculations using Superposition
4. The concentration 18 hrs after the dose
mg/L 10.1D
)18( )18( tKeV
tC
5
6
Multiple dosing calculations using Superposition
5. Assuming that 100 mg of the drug is administered every 8 hrs, estimate the concentration 2 hrs after the third dose using the values calculated in parts 2-4. What property of the linear systems did you use to answer this question?
7
Conc. After the first dose)(1 tC
8
Conc. After the second dose)(2 tC
9
Conc. After the third dose)(3 tC
10
Total Conc.)'(3 tCn
11
)2()10()18()2'( 3213 tCtCtCtCn
t = 2 hrs after third dose = 10 hrs after second dose = 18 hrs after first dose
12
Multiple dosing calculations using Superposition
5. Assuming that 100 mg of the drug is administered every 8 hrs, estimate the concentration 2 hrs after the third dose using the values calculated in parts 2-4. What property of the linear systems did you use to answer this question?
mg/L 01.9)2'(
46.545.210.1)2'(
)2()10()18()2'(
3
3
3213
tC
tC
tCtCtCtC
n
n
n
13
Multiple dosing calculations using Superposition
The principle of superposition assumes that early doses of drug do not affect the pharmacokinetics of subsequent doses.
Therefore, the blood levels after the second, third, or nth dose will overlay or superimpose the blood level attained after the (n – 1)th dose
Multiple administration every 4 hrs
Dose Number
Time (hr)
Dose 1 Dose 2 Dose 3 Dose 4 Total
1 0 0 0
1 21.0 21.0
3 19.8 19.8
2 4 16.9 0 16.9
5 14.3 21.0 35.3
7 10.1 19.8 29.9
3 8 8.50 16.9 0 25.4
9 7.15 14.3 21.0 42.5
11 5.06 10.1 19.8 35.0
4 12 4.25 8.50 16.9 0 29.7
13 3.58 7.15 14.3 21.0 46.0
15 2.53 5.06 10.1 19.8 37.5
Multiple IV bolus administration
Concentration after n doses:
where r:
n: number of doses, T: dosing interval
)(D tK
n eV
rC
KT
nKT
ee
r
11
Multiple IV bolus administration
Concentration at steady state:
where R is the accumulation ratio:
T: dosing interval
)(D tK
ss eV
RC
KTeR
1
1
Multiple IV bolus administration: useful equations
Maximum concentration after n doses:
Maximum concentration at steady state:
VRC
SS
Dmax
VrC
N
Dmax
Multiple IV bolus administration: useful equations
Minimum concentration after n doses:
Minimum concentration at steady state:
KTe
VRC
SS
Dmin
KTe
VrC
N
Dmin
Conc time profile:
The AUC during a dosing interval at steady state is equal to the total AUC following a single dose (For linear PK)
AUC for a single dose is:
As explained in the previous slide,
Multiple IV bolus administration: useful equations
Average concentration at steady state:
0
.dtC
CSS
average
SS
KVd
XC average
SS
0
0
τ
0
SS dose).dt C(single.dtC
KVd
X 0
0
dose).dt C(single
Predicting average Css using single dose data
Time to reach steady state conc.
The time required to reach to a certain fraction of the steady-state level is given by:
Time required to achieve steady-state depends on the half-life and is independent of the rate of dosing and the clearance
To get to 95% of the steady-state: 5 half-lives are needed
To get to 99% of the steady-state: 7 half-lives are needed
)1ln(44.1 5.0 fsstn
Different doses regimen have the same average steady state conc: The same dosing rate (Dose/ T)
Multiple IV bolus dosing compared to IV infusion
Multiple IV bolus
IV infusion
Multiple IV bolus dosing compared to IV infusion
For IV infusion:
For multiple IV bolus (dosing rate = dose/ dosing interval):
The steady-concentration depends on the rate of dosing and the clearance
clearance
rate Dosing0 KVd
KC average
SS
clearance
rate dosing0 KVd
XC average
SS
Example 1
To a patient 250 mg penicillin with t½ of 1 h and Vd of 25 L is administered every 6 h intravenously
1. Estimate Cmax, Cmin and Cav at steady state
2. Has the objective of maintaining concentration above minimum inhibitory concentration (4 mg/L) been achieved in this therapy? Elaborate!
3. How long did it take to reach 95% of Css? 4. Is the idea of giving a bolus dose to
achieve Css in a shorter time feasible with regard to this drug?
Example 1
016.11
11
16*693.0
ee
RKT
1
0.5
hr 0.6931
0.693
t
0.693K
mg/L 10.1625
2501.016
V
DRCmax
SS
mg/L 0.16e25
2501.016e
V
DRC *60.693KTmin
SS
mg/L 2.416*25*0.693
250
KVdτ
XC 0average
SS
Example 1
Drug concentration cannot be maintained above the MIC if it is being administered every 6 h (6 x t½). Because almost 98% of the dose is out of the body at the time of the next administration. However, conventionally penicillins are given q.i.d. and it is known that they are effective. Therefore, there is no need for keeping the concentration above MIC during the entire therapy.
4.3 hrs are needed to get to 95% of Css (i.e. Css was obtained as a result of the first dose)
hr 4.30.95)ln(1*1.44
fss)ln(1t1.44
*1
0.5
nτnτ
Example 1
The steady-state is achieved very rapidly (after the first dose). Since there is no need for accumulation, there is little justification for giving a loading dose.
Example 2
A patient is receiving 1000 mg of sulfamethoxazole iv every 12 hours for the treatment of severe gram-negative infection. At steady state the maximum and minimum serum sulfamethoxazole concentrations were 81.5 mg/L and 40 mg/L, respectively. Estimate the values of K and VD
Example 2
T
CC
tt
CCK SSSS )ln()ln()ln()ln( minmax
12
21
1hr 0.05912
ln(40)ln(81.5)K
97.11
11
112*059.0
eeR
KT
L 24.281.5
10001.97
C
DRV
V
DRC
max
max
SS
SS
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