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 SimulationMonte Carlo SimulationSense and Non-SenseSense and Non-Sense

G.L. Drusano, M.D.Co-Director

Ordway Research Institute &Research Physician

New York State Department of Health


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)


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)


What is Monte Carlo simulation, as applied to Infectious Diseases issues?

What are the technical issues?For what is Monte Carlo simulation

useful?


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


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


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)


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


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.


Technical Issues


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


Model Mis-specification


Model mis-specificationSometimes, 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)


Choice of Distribution


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)


Param Pop Mean

Sim Mean

Pop SD

Sim SD Distr

Vol 23.32 22.80 33.51 30.15 LN

Kcp 2.662 2.985 9.591 11.84 LN

Kpc 0.9327 0.7515 12.03 4.388 LN

SCL 6.242 6.252 4.360 4.303 LN


Param Pop Mean

Sim Mean

Pop SD

Sim SD Distr

Vol 23.32 36.82 33.51 24.23 N

Kcp 2.662 8.926 9.591 6.311 N

Kpc 0.9327 9.914 12.03 7.370 N

SCL 6.242 6.936 4.360 3.817 N


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)


Full vs Major DiagonalCovariance Matrix


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!)


Mean = 139.6

Median = 120.2

SD = 82.4

95% CI = 41.2-348.8

Mean = 140.4

Median = 121.4

SD = 83.5

95% CI = 40.7-351.4

0 200 400 600 800 1000Levofloxacin 750 mg AUC-Full Covariance Matrix

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Simulating the WorldFrom 6 Subjects


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


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


Utility of Monte Carlo simulation, a non-exhaustive list:

►Determination of drug dose to attain a specific endpoint

►Determination of a breakpoint


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


What About Emergence of Resistance as an Endpoint?

P. aeruginosaP. aeruginosa outcome outcome studiesstudies

Rf in vitro Rfin vivo MIC (g/mL) MBC (g/mL)

2.35x10-6 2.2x10-6 0.8 1.6

Peripheral (thigh)Compartment (Cp)

Central Blood Compartment (Cc)IP

injection

kcp kpc

+ Bacteria(XT/R)

f(c)

dCc= kaCa+kpcCp-kcpCc-keCc

dt

ke

dXS=KGS x XS x L - fKS(CcH ) x XS

dtdXR= KGR x XR x L- fKR(Cc

H ) x XR

dt

Kmax CcH

C H

50+CcH

f(CcH)=

Y1=XT=XS+XR

Y2=XR

[4]

[5]

[6]

[7]

[8]

, =K and = S,R

[2]

L = (1- (XR + XS)/POPMAX)

[9]

dCp = kcpCc - kpc Cp

dt

[3]

dCa= -kaCa

dt[1]

KmaxGS

0.117

KmaxGR

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KmaxKS

94.01

KmaxKR

12.16

HKS

6.26

HKR

2.37

C50KS

123.5

C50KR

129.8

KmaxG -maximum growth rate (hr-1) in the presence of drug

KmaxK -maximum kill rate (hr-1)

C50K -drug concentration (g/mL) to decrease kill rate by half

HK -rate of concentration dependent kill

Popmax -maximal population size

Mean Parameter Estimates of the Model.Mean Parameter Estimates of the Model.

Popmax = 3.6 x 1010


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 A-posteriori Probability (MAP) Bayesian step


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 A-posteriori Probability (MAP) Bayesian step



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


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?


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 Fluoroquinolone Pharmacodynamics: Duration of Pharmacodynamics: Duration of

TherapyTherapyPerc

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AUC/MIC 125-250

AUC/MIC >250

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Forrest et al AAC 1993;37:1073-1081

Suppression of Emergence of Resistance: Suppression of Emergence of Resistance:

A Pharmacodynamic SolutionA Pharmacodynamic SolutionIs 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


Breakpoint Determination


Breakpoint?

Monte Carlo SimulationMonte Carlo SimulationSense and Non-SenseSense and Non-SenseThe same type analysis can be

employed for antivirals(A Prospective Validation to Show

the Approach Works!)


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

was determinedDrusano et al Antimicrob Agents Chemother 46:913-916


Drusano et al Antimicrob Agents Chemother 46:913-916

Monte Carlo SimulationMonte Carlo SimulationSense and Non-Sense Sense and Non-Sense

Overall ConclusionsOverall 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


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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The Role of Monte Carlo The Role of Monte Carlo Simulations in Antimicrobial Simulations in Antimicrobial

PharmacologyPharmacology 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


PharmacologyPharmacologyAn 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%


PharmacologyPharmacology Conclusions (cont’d)Conclusions (cont’d)

The distribution range provides insight into why some patients may respond sub-optimally 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


PharmacologyPharmacology

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

Target Attainment to Evaluate Dose



What About Breakpoint Determination?

Role of Monte Carlo Simulation for Dose Choice Role of Monte Carlo Simulation for Dose Choice for Clinical Trials of Anti-Infectivesfor 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:Use of Simulation:Overall ConclusionsOverall 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


PharmacologyPharmacologyMonte 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


PharmacologyPharmacologyHowever, 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


Drug Penetration Distribution

Levofloxacin Penetration into Epithelial Levofloxacin Penetration into Epithelial Lining Fluid (ELF) as Determined by Lining Fluid (ELF) as Determined by

Population Pharmacokinetic Modeling and Population Pharmacokinetic Modeling and Monte Carlo SimulationMonte Carlo Simulation

G.L. Drusano, S.L. Preston, M.H. G.L. Drusano, S.L. Preston, M.H. Gotfried, L.H. Danziger and K.A. RodvoldGotfried, L.H. Danziger and K.A. Rodvold

AAC 2002;46:586-589AAC 2002;46:586-589


PharmacologyPharmacologyIt 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 Levofloxacin Pulmonary Penetration 95% Confidence Bounds - ELF95% Confidence Bounds - ELF

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Levofloxacin Pulmonary Penetration Levofloxacin Pulmonary Penetration 95% Confidence Bounds - Plasma95% Confidence Bounds - Plasma

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Derived from Population Mean Parameter Estimates.

Penetration Ratio (ELF/Plasma) = 1.161

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Levofloxacin Pulmonary Levofloxacin Pulmonary PenetrationPenetration

Levofloxacin PulmonaryPenetrationLevofloxacin PulmonaryPenetrationELF/Plasma RatioELF/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



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

Conclusions

Documents

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