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. Posterior = Prior x Likelihood
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 appropriate for 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 patients in 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 data will 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