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I need answer of question 10 to 18. I uploaded the question. Just remember please Question 10 to Question 18.
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PK-PD Group Assignment: Fun with Vancomycin and Piperacillin-Tazobactam!
For this assignment, treat the pharmacokinetics as one-compartment with first-order
elimination unless otherwise noted.
Show your work for all questions!
Our patient is 70 years old, male, 80 kg, serum creatinine 1.5 mg/dL, history of COPD,
admitted to the hospital for COPD exacerbation. He has been in the hospital for three
days, and now he has gram-positive cocci in the blood with symptoms concerning for
hospital-acquired pneumonia. He has been admitted to the ICU.
1. Calculate the patient’s creatinine clearance. Use the Cockcroft-Gault equation, and
use his actual body weight. Use appropriate units in your answer.
The patient is going to be started on vancomycin and piperacillin-tazobactam. We’ll get
to the vancomycin in a minute, but first we’ll think about piperacillin-tazobactam.
2. List the population-based, i.e., package-insert, dosing for hospital-acquired
(nosocomial) pneumonia, including dose, duration of infusion, elimination half-life,
volume of distribution, and protein binding of piperacillin-tazobactam. You can find this
information in Micromedex. Hint: Look up “Zosyn” to get the appropriate data. Ranges
for any of these values are appropriate. If multiple dosing strategies are listed, including
dose adjustment in renal disease, list those as well. Lastly, if values are different
between piperacillin and tazobactam, use values for the piperacillin component.
3. Based upon the data you reported in your answer to question 2, what is the dose you
would administer this patient based on his potential nosocomial pneumonia and renal
function?
4. After administering the dose you suggested in your previous answer, describe the
free drug plasma concentration of piperacillin-tazobactam at 0.5 hours, i.e., immediately
post-infusion, 1 hour, 2 hours, 3 hours, 4 hours, 5 hours, and 6 hours, i.e., immediately
prior to the next dose. Regardless of what you listed in question 2, assume a volume of
distribution of 12.5 L and an elimination half-life of 1 hour. Treat the plasma as a onecompartment model for mathematical purposes. Be sure to list free drug concentrations.
Show your work!
See equations 4.7, 9.7, 9.28, 9.29.
After your first-dose kinetics are complete, you are curious about what the drug
concentrations will look like at steady state.
5. What is meant by the concept of steady state? What factors influence the steady state
concentrations of a drug present in the body?
6. What are the free steady state peak concentration (fCmaxss) and free steady state
trough concentration (fCminss) of piperacillin-tazobactam as administered in your
previous answer? How would these concentrations compare to the concentrations
during the first-dose? Show your work!
See equations 9.30, 9.31.
Antibiotic classes can be described by the pharmacokinetic parameter most associated
with their antimicrobial efficacy. Typically, antibiotics are either area-under-the-curve
(AUC) over minimum inhibitory concentration (MIC) driven, time over MIC drive, or
concentration over MIC driven, and maximizing these parameters allows for maximum
antimicrobial efficacy. Piperacillin-tazobactam is in the beta-lactam class.
7. Which of the three pharmacokinetic parameters listed above is associated with
piperacillin-tazobactam?
Micromedex lists an alternative dosing strategy for piperacillin-tazobactam in which it is
administered over 4 hours rather than 0.5 hours.
8. Similar to question 4, list the free plasma concentration of piperacillin-tazobactam at 1
hour, 2 hours, 3 hours, 4 hours, 5 hours, and 6 hours from the start of infusion. Be sure
to take the rate and duration of infusion into account when constructing your answer.
Show your work!
See equations 4.7, 9.7, 9.28, 9.29.
9. Similar to question 6, list the free steady state peak concentration (fCmaxss) and free
steady state trough concentration (fCminss) of piperacillin-tazobactam as administered by
this regimen. Take care with this problem. Remember, the residual is what is left from
the previous dose at the end of the next infusion, not at the beginning of the next
infusion. Show your work!
See equations 9.30, 9.31.
10. Although this patient is not ill with a gram-negative organism (spoiler alert!), suppose
that he was ill with a pathogenic Pseudomonas aeruginosa isolate in his bloodstream.
The piperacillin-tazobactam MIC of this particular P. aeruginosa is 16 mg/L. We have now
reached steady state kinetics for this drug. For the 0.5-hour infusion regimen, calculate
the percentage of the dosing interval in which free plasma piperacillin-tazobactam is at
or above 16 mg/L. Also, calculate the value of the fCmax/MIC for each regimen. For the
purposes of this question, use the fCmaxss from your answer to question 6, and consider
the concentration to be above the MIC as soon as the infusion starts. Show your work!
See equation 4.7.
11. Although this patient is not ill with a gram-negative organism (spoiler alert!), suppose
that he was ill with a pathogenic Pseudomonas aeruginosa isolate in his bloodstream.
The piperacillin-tazobactam MIC of this particular P. aeruginosa is 16 mg/L. We have now
reached steady state kinetics for this drug. For the 4-hour infusion regimen, calculate
the percentage of the dosing interval in which free plasma piperacillin-tazobactam is at
or above 16 mg/L. Also, calculate the value of the fCmax/MIC for each regimen. For the
purposes of this question, use the fCmaxss from your answer to question 9, and consider
the concentration to be above the MIC as soon as the infusion starts. Show your work!
See equation 4.7.
12. Based upon what you deemed to be the most important pharmacodynamic parameter
associated with piperacillin-tazobactam efficacy, which regimen is more appropriate?
Great work with the piperacillin-tazobactam! The patient has also been started on
vancomycin, which will require a bit more creativity on your part.
13. List the population-based, i.e., package-insert, dosing for hospital-acquired
pneumonia (lower respiratory tract infection), including dose, duration of infusion,
elimination half-life, volume of distribution, and protein binding of vancomycin. You can
find this information in Micromedex.
14. Regardless of your answer in question 13, the physician has asked you about dosing
vancomycin at 1,000 mg every 12 hours in this patient. Assuming a 1-hour infusion time,
an elimination half-life of 6 hours, a volume of distribution of 0.7 L/kg, and protein
binding of 55%, calculate the total drug concentration after infusion (Cmax) and the total
drug concentration immediately before the next infusion (Cmin).
15. What are the Cmaxss and Cminss of vancomycin with this dosing strategy?
See equations 9.30, 9.31.
Being the intelligent pharmacist that you are, you know that vancomycin is rarely dosed
based upon the package insert, and that vancomycin levels are routinely monitored.
16. Would this Cminss (trough) value be considered therapeutic?
Given our patient’s clinical picture, do you think he would be exposed to more or less
vancomycin than what you’ve calculated here? Remember, these parameters are based
on a healthy group of volunteers in a pharmacokinetic study. Does our patient look like
that? Think about how vancomycin is primarily cleared from the body and what might
differentiate our patient from healthy subjects in terms of organ performance.
17. Would our patient likely be exposed to more, less, or the same vancomycin
concentrations as what you calculated in question 15 based on his disease?
Now we are given pharmacokinetic equations specifically tailored to vancomycin. Refer
to the attached vancomycin dosing sheet for the following questions.
18. Now that we have a more clinical tool for dosing vancomycin, calculate Ke (k), Vd, t1/2,
and tau assuming a Cmax of 30 mg/L and a Cmin of 15 mg/L. You can assume an infusion
time of one hour for the purposes of this question.
Knowing what you know about clocks, you round tau to the nearest multiple of either 8
or 12 hours.
19. Using your rounded estimate of tau, what is X0 (dose) for this patient?
20. Is this a larger or smaller dose of vancomycin than what you calculated in question
15?
21. By using your rounded dosing regimen, what would be the actual values you obtain
for Cmax and Cmin?
22. How do these values compare to what you calculated in question 15? What variables
in our patient have contributed to these different values from those expected from
general population pharmacokinetics?
Because with the first-dose, the patient is not yet at steady state, the physician inquires
as to whether it might be a good idea to give a larger first-dose followed by what you
calculated in question 19.
23. Which of the following effects would a large first-dose given over the same infusion
time have on the vancomycin steady state concentration? It would raise the steady state
plasma concentration, it would achieve steady state more quickly, it would do both, or it
would do neither.
24. Using the t1/2 you calculated in question 18, and knowing that after roughly 6 halflives you have reached ~99% of steady state, how many doses will have been
administered before you’ve reached steady state and should draw the first vancomycin
trough level?
The patient is now at steady state, and, although this is not routinely practiced clinically,
you have obtained two vancomycin plasma levels. The levels were 20 mg/L 2 hours postinfusion and 10 mg/L 11 hours post-infusion, immediately prior to the next dose.
25. Calculate the patient’s actual Ke based on these values.
26. Based on your new Ke value from question 25, calculate a more appropriate dosing
regimen for this patient using the same parameters for desired Cmax and Cmin from
question 18. Be sure to double-check what your new dosing regimen would achieve
once you’ve rounded. Sometimes, you have to change both interval and dose!
Great work thus far! At this point, the organism has been identified as S. aureus, and it
has a vancomycin MIC of 1 mg/L. You know that vancomycin activity is best defined by
AUC/MIC over a 24-hour period. In fact, that target happens to be 400. Knowing that
clearance = Ke*Vd, you can calculate the vancomycin clearance rate. Also knowing that
AUC = dose/clearance, you can calculate a 24-hour AUC! Remember, you need the entire
amount of drug given over 24 hours.
27. Calculate the 24-hour AUC/MIC of vancomycin against this organism.
Well done, gumshoes! You’re on your way to becoming clinical pharmacokineticists!
Useful Pharmacokinetic Equations
Symbols
D = dose
W = dosing interval
CL = clearance
Vd = volume of distribution
ke = elimination rate constant
ka = absorption rate constant
F = fraction absorbed (bioavailability)
K0 = infusion rate
T = duration of infusion
C = plasma concentration
General
Elimination rate constant
§C ·
ln¨ 1 ¸
© C 2 ¹ ln C1 ln C 2
CL
ke
Vd
t 2 t1
t 2 t1
Half-life
0.693 Vd
t1 / 2
CL
ln( 2 )
ke
0.693
ke
Intravenous bolus
Initial concentration
D
C0
Vd
Plasma concentration (single dose)
C C 0 e k e t
Plasma concentration (multiple dose)
C 0 e k e t
C
1 e k e W
Peak (multiple dose)
C0
C max
1 e k e W
Trough (multiple dose)
C0 e k e W
C min
1 e k e W
Average concentration (steady state)
D
Cp ss
CL W
Oral administration
Plasma concentration (single dose)
F D ka
e k e t e k a t
C
Vd k a k e
Time of maximum concentration (single
dose)
§k ·
ln¨ a ¸
© ke ¹
t max
ka k e
Plasma concentration (multiple dose)
§ e k e t
F D ka
e k a t ·¸
C
¨
Vd k a k e ¨© 1 e k e W
1 e k a W ¸¹
Time of maximum concentration (multiple
dose)
§ k a 1 e k e W ·
¸
ln¨
¨ k 1 e k a W ¸
© e
¹
t max
ka ke
Average concentration (steady state)
F D
C
CL W
Clearance
Dose F
Cl
AUC
Cl
Equations/Useful_pharmacokinetic_equ_5127
ke Vd
1
Constant rate infusion
Plasma concentration (during infusion)
k0
C
1 e k e t
CL
Plasma concentration (steady state)
k0
C
CL
Calculated clearance (Chiou equation)
2 Vd C1 C 2
2 k0
CL
C1 C 2
C1 C 2 t 2 t1
Short-term infusion
Peak (single dose)
D
C max(1)
1 e k e T
CL T
Calculated peak
C max
C max
e k e t
with Cmax* = measured peak, measured at time
t* after the end of the infusion
Calculated trough
C min C min e k e t
with Cmin* = measured trough, measured at
time t* before the start of the next infusion
Calculated volume of distribution
Vd
D
k e T [C
1 e k e T
max (C min e
§ C max(desired ) ·
¸
ln¨
© C min(desired ) ¹
T
ke
W
Peak (multiple dose)
Calculated recommended dose
C max
Trough (multiple dose)
C min
C max e k e W T
Calculated elimination rate constant
§C ·
ln¨ max ¸
© C min ¹
ke
‘t
*
with Cmax = measured peak and Cmin* =
measured trough,
measured over the time interval ‘t
)]
Calculated recommended dosing interval
Trough (single dose)
C min(1) C max(1) e k e W T
1 e k e T
D
CL T 1 e k e W
k e T
D
C max( desired ) k e V T
1 e k e W
1 e k e T
Two-Compartment-Body Model
C
a x e Dt b x e Et
AUC f
a /D b/ E
Vd area ! Vd ss ! Vc
Creatinine Clearance
CL creat ( male)
CL creat ( female)
(140 age) x weight
72 x Cp creat
(140 age) x weight
85 x Cp creat
With weight in kg, age in years, creatinine plasma conc.
in mg/dl and CLcreat in ml/min
Equations/Useful_pharmacokinetic_equ_5127
2
Ke for aminoglycosides
Ke = 0.00293(CrCL) + 0.014
Metabolic and Renal Clearance
EH
=
Cl int fu b
QH Cl int fu b
ClH
=
EH QH =
FH
=
QH
Q H Cl int fu b
Clren =
RBFE = GFR
Clren
rate of excretion
plasma concentration
=
QH Cl int fu b
QH Cl int fu b
C in C out
C in
ª Rate of secretion – Rate of reabsorption º
Clren = fu GFR «
»
Plasma concentration
¬
¼
Clren =
Urine flow urine concentration
Plasma concentration
Ideal Body Weight
Volume of Distribution
V VP VT K P
fu
V V P VT
fu T
Male
IBW = 50 kg + 2.3 kg for each inch over 5ft in
height
Female
IBW = 45.5 kg + 2.3 kg for each inch over 5ft in
height
Obese
ABW = IBW + 0.4*(TBW-IBW)
Equations/Useful_pharmacokinetic_equ_5127
Clearance
Cl
Dose
AUC
Cl
ke Vd
3
For One Compartment Body Model
If the dosing
involves the use
of I.V. bolus
administration:
For a single I.V. bolus administration:
C0
D
V
For multiple I.V. bolus administration:
Cn(t )
C
C0 e k e t
at peak: t = 0; at steady state nof
at trough: t = W
For a single short-term I.V. infusion:
Since W = t for Cmax
If the dosing
involves the use
of I.V. infusion:
D 1 e nkeW
e ket
V 1 e keW
Cmax
Cmin
C max ss
D
1
V ( 1 e keW )
Cmin ss
Cmax ss e k eW
For multiple short-term I.V. infusion at steady state:
D
1 e k eT
VkeT
Cmax
D 1 e k eT
VkeT 1 e k eW
k e (W T )
Cmin
Cmax e k e (W T )
Cmax e
Last modified 2010
C:Current Datapha5127_Dose_Opt_Iequations5127-28-equations.doc
D
e k eT 1 e ket (most general eq.)
during infusion t = T so,
VkeT
D
Ct
1 e k et (during infusion)
at steady state t o f, e , t o 0 so,
VkeT
k0
k0
D
D
Cpss
(steady state)
remembering k 0
and
VkeT Vke CL
T
CL V ke
Ct
If the dosing
involves a I.V.
infusion (more
equations):
-ket
For a single oral dose:
C
If the dosing
involves oral
administration:
F D ka
e ket e k a t
V k a ke
t max
ª ka º
1
ln « »
¬ ke ¼ k a ke
For multiple oral doses:
C
F D k a ª e ket
ekat
«
V k a ke «¬ 1 e k eW
1 eka
t max
ª k 1 e k eW º
1
ln « a
»
«¬ ke 1 e k aW »¼ k a ke
Last modified 2010
C:Current Datapha5127_Dose_Opt_Iequations5127-28-equations.doc
VANCOMYCIN DOSING TOOL
Variables
CrCl = creatinine clearance (mL/min)
Vd = volume of distribution (L)
X0 = dose (mg)
Ke = elimination constant (hr -1)
t1/2 = half-life (hrs)
τ = dosing interval (hrs)
t’ = infusion time (hrs) (usually 1 hr for each 1 g)
TBW = total body weight (kg)
Cmin = minimum concentration or “trough” (mcg/mL)
Cmax = maximum concentration or “peak” (mcg/mL)
INITIAL DOSING
Step
Instruction
1
Calculate the initial vancomycin dose and
estimate the Cmin and Cmax.
We need the patient’s CrCl and TBW.
2
Calculate the Ke and the Vd.
3
Calculate the dosing τ.
Here, the Cmin and Cmax represent the desired
Cmin and Cmax. The t’ is usually assumed to be
1 hr in this equation. Half-life t1/2 may also be
used to guide an appropriate τ.
4
5
Round to a convenient τ.
This τ goes in the equation to calculate the
dose (X0).
Now pick a dose (X0) and interval (τ) and
check what the estimated Cmin and Cmax are.
Play around with this to make sure that you
get the desired Cmin and Cmax.
Formula
Ke = 0.00083 X CrCl + 0.0044=____ hr −1
Vd = 0.7 X TBW =___L
τ=
ln(Cmin / Cmax)
+ t ‘ =___hrs
Ke
ln2
t1/2 = Ke =___hrs
-Keτ
Xo =(Ke Vd Cmax t) ( 1 – e-Ket’ ) =___mg
1-e
‘
Xo/t’
1 – e-Ket’
X
=___mcg/mL
KeVd
1 – e-Keτ
( τ-t )
Cmin = Cmax (e -Ke ) =___mcg/mL
Cmax =
‘
Notes on initial dosing: Step #4 calculates the suggested dose. If you prefer another way to choose the dose, skip to Step #5 to check the estimated Cmin (trough) and Cmax
(peak) with your chosen dose and interval. Note that these equations are only for steady state.
DOSE CORRECTION WITH ONE TROUGH LEVEL
After collecting a trough level, simply use the
collected trough level (Cmin(drawn)) to calculate
a new Ke and subsequently a new t1/2, τ, and
projected Cmin and Cmax levels.
You will need the X0, τ, t’, and Vd used earlier.
Ke =
In (1 + Cmin(drawn) x Vd/Xo)
=___hr -1
τ – t’
Source: Deanna Wung, PharmD
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