Download Monte Carlo Simulation: Sense and Non-sense

Monte Carlo Simulation
Sense and Non-Sense
G.L. Drusano, M.D.
Co-Director
Ordway Research Institute &
Research Physician
New York State Department of Health
Monte Carlo Simulation
Sense and Non-Sense

Monte Carlo simulation was invented by
Metropolis and von Neumann
 This technique and its first cousin Markov
Chain Monte Carlo have been used since for
construction of distributions (Markov Chain
Monte Carlo was actually described as a
solution to the “simulated annealing
problem” in the Manhattan Project Metropolis, Metropolis, Teller and Teller)
Monte Carlo Simulation
Sense and Non-Sense

The first use of Monte Carlo simulation for
drug dose choice and breakpoint
determination was presented on October 15,
1998 at an FDA Anti-Infective Drug
Products Advisory Committee
 At this time, the drug was presented as
“DrugX” but was evernimicin
 The ultimate outcome was predicted by the
method (but the drug died)
Monte Carlo Simulation
Sense and Non-Sense
 What
is Monte Carlo simulation, as
applied to Infectious Diseases issues?
 What are the technical issues?
 For what is Monte Carlo simulation
useful?
Monte Carlo Simulation
Sense and Non-Sense
 What
is Monte Carlo simulation?
MC simulation allows us to make use of
prior knowledge of how a target population
handles a specific drug to predict how well
that drug will perform clinically at the dose
chosen for clinical trials
Monte Carlo Simulation
Sense and Non-Sense

How is this done?
Through use of the mean parameter vector and
covariance matrix, derived from a population PK
study, a sampling distribution is set up (think of
every body in the world in a bucket from which
you randomly select a large number of subjects,
each of whom knows their PK parameter values).
This allows the peak concentrations, AUC and
Time > threshold to be calculated for all the
subjects
Monte Carlo Simulation
Sense and Non-Sense
 How
do we use this to predict the clinical utility of
a specific drug dose?
1) Identify the goal of therapy (cell kill,
organism eradication resistance suppression)
2) Identify the sources of variability that
affect achieving the goal of therapy
a) PK variability (accounted for by MCS)
b) Variability in MIC’s (or EC95, etc)
c) Protein binding (only free drug is
active)
Monte Carlo Simulation
Sense and Non-Sense
 What
do we do?
As an example, for a drug that is AUC/MIC
driven in terms of goal of therapy (e.g. AUC/MIC
of 100 for a good microbiological outcome), we
can now take the 2000 (or 10000 or whatever)
simulated subjects and divide the AUC by the
lowest MIC in the distribution, then determine how
many achieve the target of 100. This is then
repeated with higher MIC values until the target
attainment is zero or some low number
Monte Carlo Simulation
Sense and Non-Sense
 How
does this help evaluate the utility of a
specific drug dose?
We have target attainment rates at each
MIC value in the organism population distribution.
A specific fraction of the organisms have a specific
MIC. A weighted average for the target attainment
rate (taking an expectation) can be calculated. This
value will be the overall “expected” target
attainment rate for the outcome of interest for that
specific dose.
Monte Carlo Simulation
Sense and Non-Sense
Technical Issues
Monte Carlo Simulation
Sense and Non-Sense

What are the factors that may affect the
simulation?
►Model mis-specification
►Choice of distribution
►Covariance matrix (full vs diagonal)
►Simulating the world from 6 subjects
Monte Carlo Simulation
Sense and Non-Sense
Model Mis-specification
Monte Carlo Simulation
Sense and Non-Sense

Model mis-specification
Sometimes, data are only available from
older studies where full parameter sets and
their distributions were not reported
 Some investigators have used truncated
models for simulation (1 cmpt vs 2 cmpt)
 This may have more effect for some drugs
relative to others (β lactams vs quinolones)
Monte Carlo Simulation
Sense and Non-Sense
Choice of Distribution
Monte Carlo Simulation
Sense and Non-Sense

There are many underlying distributions
possible for parameter values
 Frequently, there are insufficient numbers of
patients to make a true judgement
 One way to at least make the choice rational
is to examine how one distribution vs
another recapitulates the mean parameter
values and measure of dispersion
 A quinolone example follows (N vs Log-N)
Monte Carlo Simulation
Sense and Non-Sense
Param
Sim
Mean
22.80
Pop
SD
33.51
Sim
SD
30.15
Distr
Vol
Pop
Mean
23.32
Kcp
2.662
2.985
9.591
11.84
LN
Kpc
0.9327 0.7515 12.03
4.388
LN
SCL
6.242
4.303
LN
6.252
4.360
LN
Monte Carlo Simulation
Sense and Non-Sense
Param
Sim
Mean
36.82
Pop
SD
33.51
Sim
SD
24.23
Distr
Vol
Pop
Mean
23.32
Kcp
2.662
8.926
9.591
6.311
N
Kpc
0.9327 9.914
12.03
7.370
N
SCL
6.242
4.360
3.817
N
6.936
N
Monte Carlo Simulation
Sense and Non-Sense

Here, it is clear that the Log-normal distribution
better recaptures the mean parameter values and,
in general, the starting dispersion (except Kpc)
 However, for AUC distribution generation, it is
clear that Log-normal is preferred because it
performs better for the parameter of interest
(SCL) for both mean value and dispersion
 We have seen examples where there is no
substantive difference (N vs Log-N)
Monte Carlo Simulation
Sense and Non-Sense
Full vs Major Diagonal
Covariance Matrix
Monte Carlo Simulation
Sense and Non-Sense

Sometimes, only the population standard deviations
are available and only a major diagonal covariance
matrix can be formed
 Loss of the off-diagonal terms will generally cause
the distribution to become broader (see example)
 One can obtain an idea of the degree of impact if the
correlation among parameters is known (of course if
this is known it is likely one would also have the full
covariance matrix!)
Monte Carlo Simulation
Sense and Non-Sense
0.10
1000
900
0.08
800
Count
0.06
600
500
0.04
400
300
Proportion per Bar
700
0.02
200
100
0
0.00
0
200
400
600
800 1000
Levofloxacin 750 mg AUC-Full Covariance Matrix
Mean
= 139.6
Mean
= 140.4
Median = 120.2
Median = 121.4
SD
SD
= 82.4
95% CI = 41.2-348.8
= 83.5
95% CI = 40.7-351.4
Monte Carlo Simulation
Sense and Non-Sense
Simulating the World
From 6 Subjects
Monte Carlo Simulation
Sense and Non-Sense

Obviously, the robustness of the conclusions are
affected by the information from which the
population PK analysis was performed
 If the “n” is small, there may be considerable
risk attendant to simulating the world
 One of the underlying assumptions is that the PK
is reflective of that in the population of interest –
care needs to be taken and appropriate
consideration given to the applicability of the
available data to the target population
Monte Carlo Simulation
Sense and Non-Sense

But, in the end, something is probably better
than nothing, so simulate away, but interpret
the outcomes conservatively
 How many simulations should be done?
- Answer: as always, it depends
 To stabilize variance in the far tails of the
distribution (> 3 SD), it is likely that one
would require > 10000 simulations
Monte Carlo Simulation
Sense and Non-Sense

Utility of Monte Carlo simulation, a nonexhaustive list:
►Determination of drug dose to attain a
specific endpoint
►Determination of a breakpoint
Monte Carlo Simulation
Sense and Non-Sense
Required Factors for Rational Dose/Drug
Comparison/Breakpoint Determination
1.
Pharmacodynamic Target
2.
Population Pharmacokinetic
Modeling
3.
Target Organism(s) MIC, EC50 (or
EC90) Distribution
4.
Protein Binding Data
Monte Carlo Simulation
Sense and Non-Sense
What About Emergence of
Resistance as an Endpoint?
P. aeruginosa outcome
studies
Rf in vitro
Rfin vivo
2.35x10-6
2.2x10-6
MIC (g/mL)
0.8
MBC (g/mL)
1.6
Peripheral (thigh)
Compartment (Cp)
kcp
IP
injection
kpc
Central Blood
Compartment (Cc)
ke
[1]
[2]
[3]
dCa= -kaCa
dt
dCc= kaCa+kpcCp-kcpCc-keCc
dt
dCp = kcpCc - kpc Cp
dt
+
Bacteria
(XT/R)
f(c)
dXS=KGS x XS x L - fKS(CcH ) x XS
dt
dXR= KGR x XR x L- fKR(CcH ) x XR
dt
L = (1- (XR + XS)/POPMAX)
Kmax  CcH 
f(CcH)=
, =K and  = S,R
[4]
[5]
[6]
[7]
C H 50+CcH 
Y1=XT=XS+XR
[8]
Y2=XR
[9]
Mean Parameter Estimates of the Model.
KmaxGS
KmaxKS
HKS
C50KS
0.117
94.01
6.26
123.5
KmaxGR
KmaxKR
HKR
C50KR
0.163
12.16
2.37
129.8
Popmax = 3.6 x 1010
KmaxG
KmaxK
C50K
HK
Popmax
-maximum growth rate (hr-1) in the presence of drug
-maximum kill rate (hr-1)
-drug concentration (g/mL) to decrease kill rate by half
-rate of concentration dependent kill
-maximal population size
Monte Carlo Simulation
Sense and Non-Sense

All regimens were
simultaneously fit in a
large population model
 The displayed graph is
the predicted-observed
plot for the total
population after the
Maximum Aposteriori Probability
(MAP) Bayesian step
Monte Carlo Simulation
Sense and Non-Sense

All regimens were
simultaneously fit in a
large population model
 The displayed graph is
the predicted-observed
plot for the resistant
population after the
Maximum Aposteriori Probability
(MAP) Bayesian step
Monte Carlo Simulation
Sense and Non-Sense
Monte Carlo Simulation
Sense and Non-Sense



In this experiment, a dose
was selected to generate an
exposure that would
prevent emergence of
resistance
As this was at the limit of
detection, the measured
population sometimes had
“less than assay detectable”
for the colony count
These were plotted at the
detection limit
Monte Carlo Simulation
Sense and Non-Sense

We were able to determine how the overall (sensitive plus
resistant) population responds to pressure from this
fluoroquinolone
 More importantly, we were able to model the resistant
subpopulation and choose a dose based on simulation to
suppress the resistant mutants
 The prospective validation demonstrated that the doses
chosen to encourage and suppress the resistant mutants
did, indeed, work
 The identified AUC/MIC breakpoint was 157 – is this
value predictive for the clinic?
Monte Carlo Simulation
Sense and Non-Sense

For P aeruginosa and the suppression of resistance
target, 750 mg of levofloxacin achieves the goal
with a 61.2% probability
 The levofloxacin nosocomial pneumonia trial
cannot be examined for validation-a second drug
was added for Pseudomonas aeruginosa
 Simulation from Alan Forrest’s data for cipro (400
mg IV Q8h) shows a target attainment of 61.8%
and 24.8% for 200 mg IV Q12h
Fluoroquinolone
Pharmacodynamics: Duration of
Therapy
Percent of Patients Remaining
Culture-positive
100
75
AUC/MIC <125
50
25
AUC/MIC 125-250
AUC/MIC >250
0
0
2
4
6
8
10
Days of therapy
Forrest et al AAC 1993;37:1073-1081
12
14
Suppression of Emergence of Resistance:
A Pharmacodynamic Solution
Is Monte Carlo Simulation Predictive?
Peloquin studied 200 mg IV Q 12 h of ciprofloxacin in
nosocomial pneumonia - P aeruginosa resistance rate 70% (7/10
- pneumonia only) - 77% (10/13 - all respiratory tract)
Monte Carlo simulation with a resistance suppression target
(AUC/MIC = 157) predicts suppression in 24.8%
Fink et al studied ciprofloxacin in nosocomial pneumonia at a
dose of 400 mg IV Q 8 h - P aeruginosa resistance rate 33%
(12/36)
Monte Carlo simulation at this dose predicts suppression in
61.8% & resistance emergence in 38.2%
Peloquin et al Arch Int Med 1989;1492269-73
Fink et al AAC 1994;38:547-57
Monte Carlo Simulation
Sense and Non-Sense
Breakpoint Determination
Monte Carlo Simulation
Sense and Non-Sense
Breakpoint?
Monte Carlo Simulation
Sense and Non-Sense
The same type analysis can be
employed for antivirals
(A Prospective Validation to Show the
Approach Works!)
Monte Carlo Simulation
Sense and Non-Sense





GW420867X is a NNRTI
Population PK was
performed for each dose
Three 1000-subject MC
simulations were
performed
EC50 values were
corrected for protein
binding and the difference
between EC50 and EC90
Fraction of patients with
trough free drug > EC90
Drusano et al Antimicrob Agents Chemother 46:913-916
was determined
Monte Carlo Simulation
Sense and Non-Sense
Drusano et al Antimicrob Agents Chemother 46:913-916
Monte Carlo Simulation
Sense and Non-Sense
Overall Conclusions





MCS is useful for rational breakpoint determination
MCS allows insight into the probability that a specific
dose will attain its target
This has been prospectively validated
The technique rests upon certain assumptions and is as
reliable as the assumptions
Care needs to be taken when applying the method,
particularly as regards applicability of the population
studied and population size, among other issues
Monte Carlo Simulation
Sense and Non-Sense

WE CAN DO BETTER AND WE SHOULD!
– As an aside, I have trying since the early 1980’s to
interest the infectious diseases community in
pharmacodynamic modeling, notably WITHOUT
SUCCESS!
– WELL!
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D
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The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology




The AUC of levofloxacin
in ELF is slightly more
variable than in plasma
Target attainment is higher
in Plasma than in ELF up
to 0.5 mg/L
After 1.0 mg/L, ELF rates
are higher
This may explain some
“unexpected” successes
The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology
An Expectation taken over the MIC
distribution of 404 strains of Pseudomonas
aeruginosa for the ELF AUC distribution
yielded an overall target attainment of
65.6%
An Expectation taken over the MIC
distribution of 404 strains of Pseudomonas
aeruginosa for the Plasma AUC
distribution yielded an overall target
attainment of 67.1%
The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology
Conclusions (cont’d)
 The distribution range provides insight into
why some patients may respond suboptimally to a specific dose of drug while
others respond when they “should not”
 The effect of inflammation on drug
penetration is not accounted for in this
model
 Examination of ELF penetration with active
inflammation needs to be undertaken
The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology
Target Attainment to Evaluate Dose
 Let
us examine target attainment with a
robust population PK analysis (n = 272)
 Preston et al published a 272 patient study of
levofloxacin (500 mg QD) use in
community-acquired infections (JAMA
1998;279:125-129)
 This Population PK analysis was employed
The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology
What About Breakpoint
Determination?
Role of Monte Carlo Simulation for Dose Choice
for Clinical Trials of Anti-Infectives
Required Factors for Rational Dose/Drug
Comparison
1.
Pharmacodynamic Goals of Therapy
2.
Population Pharmacokinetic
Modeling
3.
Target Organism(s) MIC
Distribution
4.
Protein Binding Data in Animal and
Man
Use of Simulation:
Overall Conclusions

Simulation can also allow insight into drug
penetration into specialized spaces
 The variability in penetration may help
explain therapeutic failures at an “adequate”
drug dose
 Simulation is a powerful technique that
should be used more widely
The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology

Monte Carlo simulation for a suppression of
resistance for a 750 mg once-daily levofloxacin dose
demonstrates target attainment 61.2% of the time,
when an expectation is taken over the 404 strains of
P aeruginosa shown previously
 We cannot use the levofloxacin nosocomial
pneumonia study for validation, because in this
study, a second drug was added when P aeruginosa
was detected
The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology

However, simulations were also performed from the
data of Forrest et al (Antimicrob Agents Chemother
1993:37:1065-1072) for ciprofloxacin
 These data were derived from patients with
nosocomial pneumonia
 Doses of 200 mg Q 12 h and 400 mg Q 12 h were
simulated
 Target attainments for suppression of resistance
were 24.8% and 61.8%, respectively
Monte Carlo Simulation
Sense and Non-Sense
Drug Penetration Distribution
Levofloxacin Penetration into Epithelial
Lining Fluid (ELF) as Determined by
Population Pharmacokinetic Modeling and
Monte Carlo Simulation
G.L. Drusano, S.L. Preston, M.H.
Gotfried, L.H. Danziger and K.A. Rodvold
AAC 2002;46:586-589
The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology

It is important to ascertain the ability of
drugs to penetrate to their site of action, in
this case, the ELF
 Determination by penetration ratio often
provides a biased estimate of penetration
because of system hysteresis (penetration
ratio changes with time)
 We wished to employ population modeling
and Monte Carlo Simulation to examine the
penetration of levofloxacin into ELF
Levofloxacin Pulmonary Penetration
95% Confidence Bounds - ELF
100
90
Concentration (mg/L)
80
ELF
70
60
50
40
30
20
10
0
0
25
5
10
15
Time (hours)
20
Levofloxacin Pulmonary Penetration
95% Confidence Bounds - Plasma
30
Concentration (mg/L)
25
Plasma
20
15
10
5
0
0
5
10
15
Time (hours)
20
25
Levofloxacin Pulmonary
Penetration
Steady State for a 750-mg Dose
20
Concentration (µg/L)
Penetration Ratio (ELF/Plasma) = 1.161
Plasma
ELF
10
0
0
5
10
15
Time (h)
Derived from Population Mean Parameter Estimates.
20
25
Levofloxacin PulmonaryPenetration
ELF/Plasma Ratio
Penetration Ratio Distribution by
Monte Carlo Simulation
Mean
3.18
Median
1.43
St. Dev.
5.71
95% CI
0.143 - 19.12
61% > 1.0
The Role of Monte Carlo
Simulations in Antimicrobial
Pharmacology
Conclusions
 Population modeling avoids the issue of
system hysteresis and should be the
preferred method of analysis for penetration
studies
 Levofloxacin penetrates well into the ELF
with a mean penetration ratio exceeding 1.0
 Use of Monte Carlo simulation displays the
variability in penetration