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 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 Kpc 0.9327 0.7515 12.03 4.388 SCL 6.242 6.252 4.360 4.303

Monte Carlo Simulation Sense and Non-Sense 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 Kpc 0.9327 9.914 12.03 7.370 SCL 6.242 6.936 4.360 3.817

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 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

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 non-exhaustive 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 MIC (g/mL) MBC (g/mL) 2.35x10-6 2.2x10-6 0.8 1.6

+ [3] Bacteria (XT/R) [4] [5] [6] [1] [7] f(CcH)= [2] Y1=XT=XS+XR Peripheral (thigh) Compartment (Cp) Bacteria (XT/R) + kcp kpc f(c) Central Blood Compartment (Cc) dXS=KGS x XS x L - fKS(CcH ) x XS dt [4] IP injection [5] dXR= KGR x XR x L- fKR(CcH ) x XR dt L = (1- (XR + XS)/POPMAX) [6] ke Kmax  CcH  C H 50+CcH  dCa= -kaCa dt [1] , =K and  = S,R [7] f(CcH)= dCc= kaCa+kpcCp-kcpCc-keCc dt [2] Y1=XT=XS+XR Y2=XR [8] [3] dCp = kcpCc - kpc Cp dt [9]

Mean Parameter Estimates of the Model. KmaxGS 0.117 KmaxGR 0.163 KmaxKS 94.01 KmaxKR 12.16 HKS 6.26 HKR 2.37 C50KS 123.5 C50KR 129.8 Popmax = 3.6 x 1010 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

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 A-posteriori 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 A-posteriori 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 100 75 50 25 AUC/MIC <125 Percent of Patients Remaining Culture-positive AUC/MIC 125-250 AUC/MIC >250 0 2 4 6 8 10 12 14 Days of therapy Forrest et al AAC 1993;37:1073-1081

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 was determined Drusano et al Antimicrob Agents Chemother 46:913-916

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!

George→

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 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

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?

Required Factors for Rational Dose/Drug Comparison 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 80 70 ELF 60 Concentration (mg/L) 50 40 30 20 10 0 5 10 15 20 25 Time (hours)

Levofloxacin Pulmonary Penetration 95% Confidence Bounds - Plasma 30 25 20 15 10 5 0 5 10 15 20 25 Plasma Concentration (mg/L) Time (hours)

Levofloxacin Pulmonary Penetration Steady State for a 750-mg Dose Penetration Ratio (ELF/Plasma) = 1.161 Time (h) Concentration (µg/L) Plasma ELF 20 10 5 15 25 Derived from Population Mean Parameter Estimates.

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