Monte Carlo Simulation

Monte Carlo Simulation
G.L. Drusano, M.D. Co-Director Ordway Research Institute & Research Physician New York State Department of Health Professor of Medicine & Pharmacology Albany Medical College

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 et al)

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)

Role of Monte Carlo Simulation for Dose Choice for Clinical Trials of Anti-Infectives

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

Role of Monte Carlo Simulation for Dose Choice for Clinical Trials of Anti-Infectives
Drusano GL, SL Preston, C Hardalo, et al. Antimicrob Agents Chemother. 2001;45:13-22.

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 and to rationally set breakpoint values for susceptibility

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. 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, resistance suppression, etc) 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 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-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)

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 Kpc 0.9327 0.7515 12.03 4.388 SCL 6.242 6.252 4.360 4.303

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

Here, it is clear that the Log-normal distribution better recaptures the mean parameter values and, in general, the starting dispersion (except Kpc) And 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 Diagonal Covariance 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, one could calculate the full covariance matrix!)

Mean = 139.6 Median = 120.2 SD = 82.4 95% CI = Mean = 140.4 Median = 121.4 SD = 83.5 95% CI =

Simulating the World From 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 It is also important to examine the SD’s, as drawing inferences on drug dose from volunteer studies, where CV%’s are sometimes circa 10% may be risky 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 > simulations

Utility of Monte Carlo simulation, a non-exhaustive list: ► Determination of drug dose to attain a specific endpoint ► Determination of a breakpoint ► Examine variability in drug penetration

Some New Stuff: Effect simulations for combinations Use of estimated GFR in simulations Identification of a resistance-counterselective dose

Hope W et al. J Infect Dis 2005;192:

Greco Model for Combination Chemotherapy Hope W et al. J Infect Dis 2005;192:

Hope W et al. J Infect Dis 2005;192:

5-FC 30 mg/Kg/day Amphotericin B 1 mg/Kg/day 5-FC 30 mg/Kg/day Amphotericin B 0.6 mg/Kg/day 5-FC 30 mg/Kg/day Amphotericin B 0.3 mg/kg/day

It is straightforward to model combinations of agents Our laboratory has also done so for anti-retrovirals For Amphotericin B/5-FC, it is clear that the current dose of 5-FC is far too large (at least for C. albicans) and only adds toxicity Monte Carlo simulation shows that use of 30 mg/Kg 5-FC with Ampho B doses as low as 0.3 mg/Kg gives up little effect, but would have significantly diminished toxicity

Population Pharmacokinetic Parameter Values for Ceftobiprole
Kh Vc K23 K32 CLsl CLint Units h-1 L L/h Mean 51.8 7.65 3.05 1.10 0.510 2.35 Median 59.9 7.05 1.20 0.960 0.484 2.46 S.D. 17.5 3.89 5.14 0.951 0.318 1.98

Observed vs. Predicted Plot after the Bayesian Step
Observed = x Predicted ; r2 = 0.947; p << 0.001

Target Attainment Probabilities for a 500 mg dose of ceftobiprole administered as a 1
hour, constant rate intravenous infusion every 12 hours. Target was maintaining free drug concentrations in excess of the MIC for 30% of the dosing interval. Estimated creatinine clearances were held constant for each analysis at the indicated values between 20 ml/min and 120 ml/min.

Gumbo et al. J Infect Dis 2004;190:1642-1651.

The model system delineated above was applied to all the data simultaneously
Gumbo et al. J Infect Dis 2004;190:

Moxifloxacin Concentrations Total Population Resistant Population Gumbo et al. J Infect Dis 2004;190:

Monte-Carlo Simulation and Moxifloxacin in Mtb Therapy
Therapeutic target; moxifloxacin AUC/MIC of 53 in patients for resistance suppression Moxifloxacin doses of 400 mg a day, 600 mg a day, and 800 mg a day taken by 10,000 simulated patients Prior information: published population pharmacokinetic parameters

Moxifloxacin 400 mg a day. Target attainment=59.3%
The target here and in the next two slides is suppression of the resistant population

Moxi 600 mg

Moxi 800 mg

Moxifloxacin and M.tuberculosis Conclusion
Moxifloxacin resistance in sub-therapeutic exposure occurs early during 2nd week of therapy. Drug doses associated with excellent microbial kill may amplify resistant population. Drug exposure associated with suppression of resistance is an AUC0-24/MIC of 53. Moxifloxacin daily dose of 800 mg may be better for MDRTB as opposed to current 400 mg a day dose recommended by CDC/IDSA/ATC because of resistance issues. Such a dose would need careful clinical evaluation because of QTc prolongation

Monte Carlo Simulation 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 (and granting agencies) in pharmacodynamic modeling, notably WITHOUT SUCCESS! WELL!

George

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