INTRODUCTION TO BAYESIAN THEORY

1. INTRODUCTION TO BAYESIAN THEORY K. Arvind Nag Pharm. D Final Year DEPARTMENT OF PHARMACY PRACTICE JSS COLLEGE OF PHARMACY Mysore

2. INTRODUCTION… • Traditional method of dose adjustment • This process of repeated visits for dose adjustment continues until the drug behaves acceptably or a new candidate is chosen (if one exists) and the process is repeated • Not only is this process expensive and cumbersome, but it is also easy to overmedicate an individual especially if the side effects are less pronounced

3. INTRODUCTION… • This can be especially challenging when drugs are prescribed for the elderly • An article by Steed (2008) suggests that in Canada “85- and 90-year old overmedicated seniors are clogging emergency departments, blocking hospital beds and are sicker than they have any reason to be” • For the dose regimen to be truly optimized for an individual (i.e. personalized dosage regimen) it is necessary to understand how the drug level changes with time and whether or not this is within the therapeutic window

4. INTRODUCTION… Therapeutic Window for Two Different Drug Regimen (10 mg Once Daily and 5 mg Twice in a Day)

5. INTRODUCTION… • A typical study to generate well defined pharmacokinetic / Pharmacodynamic parameters requires a minimum of seven or more tests over time • For example, the following simulated data, shows the blood plasma concentrations for eight different individuals following a single oral dose where the subjects differ only in their pharmacokinetic parameters • Note the difference in uptake and elimination between individuals

6. INTRODUCTION… • Collecting such data for each patient is expensive, time-consuming and unpleasant for an individual • However, with this data it would be possible to estimate the PK/PD parameters for the individual and predict the drug concentration levels as a function of time for a variety of dosage regimens • If the therapeutic window is known then an optimization procedure is needed to search among the possible drug doses available and candidate dose intervals convenient to the individual to find a regimen that maximizes the time in the therapeutic window

7. THE NEED FOR MODELS • Population Pharmacokinetics / Pharmacodynamic models provide the means to store past experience with the behavior of the drugs, and to apply it to the care of future patients • The dosage regimen to achieve the target goal is computed and given • The patient is then monitored both clinically and by measuring serum concentrations • The serum concentrations are used not only to note if they are within a therapeutic range, but also to make a specific model of the behavior of the drug in that individual patient • One can see what the probable serum concentrations were at all other times when they where not measured

8. THE NEED FOR MODELS… • One can also see the computed concentrations of drugs in a peripheral nonserum compartment or in various effect compartments • These cannot be seen or inferred at all without such models • By comparing the clinical behavior of the patient with the behavior of the patient’s model, one can evaluate the patient’s clinical sensitivity to the drug, and can adjust the target goal appropriately • Digoxin, for example, the inotropic effect of the drug correlates best with the behavior of the drug in the peripheral compartment rather than with the serum concentrations

9. PARAMETRIC POPULATION MODELS • The usual parameter values are either the means or medians as measures of central tendency and the standard deviations as the measures of dispersions • The single most likely values for each parameter (volume of distribution, rate constants, clearance, etc.) are then used to compute the dosage regimen to achieve the desired response (usually a selected target serum concentration), which is best individualized for each patient according to his / her perceived need for the drug and the risk of toxicity which is felt to be acceptable in order to obtain the most benefit from the drug • The regimen to achieve and maintain the target goal is computed and the future concentrations are predicted using these parameter values

10. PARAMETRIC POPULATION MODELS • Examples of such Parametric Population Modeling Approaches • Standard Two Stage Approach • Bayesian Approach • Parametric EM (Expectation and Maximization) Method • Nonlinear Mixed Effects Modeling • Other variations on these approaches • Other Population Modeling Approach • Semi Non Parametric Approach • Non Parametric Maximum Likelihood Method • Non Parametric Expectation-Maximization Method

11. BAYESIAN APPROACH • The Bayesian approach incorporates TWO SETS OF DATA for estimating the patient's pharmacokinetic parameters • It uses the a priori pharmacokinetic parameters of the population model as the starting estimate for an individual • It then adjusts these estimates based on the patient's measured drug levels, taking into consideration the variability of the population parameters and the variability of the serum level measurement • Our population model is not discarded, rather it is incorporated into the estimation procedure • The serum level data is interpreted in light of both the variability of the population model the variability of the serum level measurement itself

12. BAYESIAN APPROACH… • Technically, • Initial beliefs about some unknown quantity are represented by a prior distribution • Information in the data is expressed by the likelihood function • The prior distribution and the likelihood function are then combined to obtain the posterior distribution for the quantity of interest • The posterior distribution expresses our revised uncertainty in light of the data, in other words an organized appraisal in the consideration of previous experience

13. BAYESIAN APPROACH… Posterior = Prior x Likelihood • The posterior distribution, a reflection of a parameter's uncertainty, is influenced by the strength of the prior knowledge • Irrespective of the current data, the posterior resembles a strong prior distribution • Whereas, the posterior resembles the current data more, when the prior is relatively uninformative Bayesian estimation Prior distribution of the parameter estimates and the actual data are used to estimate the posterior distribution of parameters

14. BAYESIAN APPROACH… • The common types of Bayesian Approaches are 1. Maximum Aposteriori Probability (MAP) Bayesian Approach 2. Sequential MAP Bayesian Approach 3. Sequential Multiple Model (MM) Bayesian Approach 4. Interacting Multiple Model (IMM) Bayesian Approach

15. BAYESIAN APPROACH… MAP Bayesian Approach : where Cobs is the collection of observed serum concentrations Var (Cobs) is the collection of their respective variances Cmod is the model estimate of each serum concentration at the time it was obtained Ppop is the collection of the various population model parameter values (Maximum Aposteriori Probability –MAP) Var (Ppop) is the collection of their respective variances Pmod is the collection of the Bayesian posterior model parameter values

16. BAYESIAN APPROACH… Characteristics • Could be applied to data sets with sparse data per individual • Effective use of prior knowledge • Population models in which there is greater diversity, and therefore greater variance, contribute less to the individualized model than do more uniform models having smaller variances • Similarly, a precise assay will draw the fitting procedure more closely to the observed concentrations, and a less precise assay will do the opposite • The more serum data are obtained, the more that information dominates the determination of the MAP Bayesian posterior parameter values (Pmod) in the patient's individualized pharmacokinetic model

17. BAYESIAN APPROACH… Limitations of MAP Bayesian Approach • The parameter values used to describe the behavior of the drug are assumed to be either normally or log-normally distributed • This is often not so, many drugs have rapid and slow metabolizers within the population and have parameter distributions for the elimination rate constant which may be multimodal (This is largely overcome by non parametric models) • There is no tool in MAP Bayesian Approach for evaluating the precision with which a desired dosage regimen developed to hit a desired target goal actually will do so

18. CONCLUSION • Bayesian approach to the determination of individual drug dosage requirements performs better • However, it should be emphasized that the population model must be appropriatefor the patient • It is wrong to use a drug model derived from a dissimilar patient population (E.g.: we should never use a model based on data from otherwise healthy adults in a frail elderly patient) • Likewise, outlying patientsin a population (i.e., those patients whose pharmacokinetic parameters lie outside of the 95th percentile of the population) may be put at risk • And, bad datawill corrupt the analysis • As is always the case, the computerized algorithms outlined below can only assist in the decision-making process and should never become a substitute for rational clinical judgment

19. Thank You