• Terms and Definitions 1
  • Variable
  • Parameter
  • Model
  • Linear and Nonlinear model
  • Fixed effect parameter
  • Random effect parameter
  • Mixed effects modeling
  • Structural model
  • Random effect model
  • Population PK and PD
• Terms and Definitions 2
  • Maximum likelihood
  • Objective function value
  • Log-likelihood ratio
  • Optimization procedure
  • Point estimate
  • Confidence intervals
  • Maximum a posteriori/post hoc estimates
  • Simulation
  • Clinical trial design
  • Pharmacodynamic terms
• Population analysis methods
  • Naive pooled data approach (NPD)
  • Naive averaged data approach (NAD)
  • Standard two-stage approach (S2S)
  • Three stage analysis (3S)
  • Bayesian estimation
  • One stage analysis
• Estimation methods
  • Ordinary least squares (OLS)
  • Weighted least squares (WLS)
  • Extended least squares (ELS)
• Good Modeling Practices (GMP)
  • Data management
  • Rational modeling process
  • Report

 

Population analysis methods

Naive pooled data approach (NPD)¹

A model without between subject variability (BSV) and between occasion variability (BOV) is fitted to the pooled data from all individuals.

Features

-Specialized software not necessary.

-Covariate models can be tested, however this aspect is not well discussed in the literature.

Disadvantages

-All variability is lumped into residual variability

-Imbalance in data per individual could lead to biased estimates.

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Naive averaged data approach (NAD)

A model without BSV and BOV is fitted to the mean data from all individuals.

Features

-Specialized software not necessary.

Disadvantages

-Does not distinguish between BSV and WSV.

-Inappropriate means lead to biased parameter estimates.

-May produce model distortion i.e., 2 mono exponential equations averaged together can yield a biexponential.

-Covariate modeling cannot be performed.

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Standard two-stage approach (S2S)¹

1. Stage-I

Each individual's PK and/or PD parameters are estimated from that individual's dense concentration-time data.

2. Stage-II

-Relationships between covariates and the parameters are explored.

-Population mean and variance of each parameter is derived.

Three stage analysis (3S)

The first two stages are identical to the standard two stage analysis.

3. Stage-III

The best random distribution model for BSV is selected based on difference in objective function value

Features

-Specialized software not necessary.

-Covariate modeling could be done.

Disadvantages

-Dense data for each individual is required to get individual parameter estimates.

-BSV will tend to be overestimated. For e.g., let us consider an individual with 3 data points and another individual with 10 data points. The uncertainty in the point estimate of clearance will be higher in the individual with 3 data points than the individual with 10 data points. However, the second step of S2S considers only the mode of the parameter estimate and ignores the uncertainty. This will lead to over estimation of BSV/BOV.

-Suitable only for balanced rich studies without missing data. Data sets with missing data could lead to biased parameter estimates.

-Not well suited for dose-disproportional PKPD data except in the case of a full crossover study. Because, for e.g.: Consider a drug which follows Michaelis-Menten kinetics. A parallel design where 1,10, and 100mg doses were studied. Doses that produce concentration less than km cannot reliably estiamte Vmax and km without systematic bias.

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Bayesian Estimation

A brief and concise description of bayesian theory is given before proceeding to bayesian estimation.

Bayesian theory

Bayesian statistics provide a conceptually simple process for updating uncertainty in the light of evidence.²

- 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.

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

1. Prior distribution of the parameter estimates and the actual data are used to estimate the posterior distribution of parameters.

Features

- Could be applied to data sets with sparse data per individual.

- Effective use of prior knowledge.

Challenges

- Concensual prior estimates of parameters and variability needed.

- Fits may depend on priors (depending on uncertainty in prior)

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One-stage analysis (Linear and Non-linear mixed effects modeling)

1. All individuals are analyzed simultaneously.
2. Relationships between covariates and the parameters are explored.
3. Subsequently, individual parameters can be obtained using empirical bayes estimates.

   Features

   - BSV, WSV and BOV of the parameters can be estimated.

   - Both sparse and dense data can be used.

   - Imbalance in data per individual can be handled.

   - Flexibility of models (i.e., different models for different individuals (mixture models), if necessary)

   Challenges

   - Needs advanced training for using specialized software.

   - Fitting complicated models can be time-consuming.

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References

1. Sheiner LB, Beal SL., "Evaluation of methods for estimating population pharmacokinetic parameters II. Biexponential model and experimental pharmacokinetic data," Journal of PK & Biopharm 1981: 9(5): 635-651.

2. Bayesian theory, Accessed March 20,2005

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