• 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

Good Modeling Practices (GMP)

Rational model building

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Determination of initial model

Structural model

- Preliminary data analysis would give an idea of the number of disposition phases for the drug or about the PD model.

- A simple structural model could be defined by NAD or by NPD approach.(NAD or NPD approach)

Structural model could be expressed in terms of primary PK or PD parameters, ie., clearances and volumes , rather than rate constants, as it would be easier to explore relationship between parameters and covariates like weight, sex, age or creatinine clearance.

Random effect model

1. Between subject variability (BSV) model

BSV models could be

o Additive (normal)

CL_i = CL_pop + η_CL where
CL_i     = individual clearance
CL_pop   = Population clearance
η_CL     = random between subject variability

o Proportional

CL_i = CL_pop • (1 + η_CL)

o Exponential (log normal)

CL_i = CL_pop • exp(η_CL)

Given that all PK parameters are usually considered to be positive, they cannot arise from a normal distribution. The above lognormal distribution model implies that the distribution of individual clearances are skewed to the right which is a commonly observed feature of distributions of pharmacokinetic parameters.

Note: Proportional and exponential BSV models are indistinguishable when a First order estimation (FO) method (in NONMEM) is used because of the approximations involved.

Caution: Use of an exponential error model for PD parameters like slope (linear effect) or E_max is not possible.

2. Between occasion variability (BOV) model

BOV models can also be additive, proportional or exponential as explained above in BSV models. Random between occasion variability (Variance) is assumed to come from the same distribution for a parameter for all occasions inorder to reduce the number of parameters.

3. Residual variability or Within-subject variability (WSV) model

WSV (also called as intra-individual variability) includes intra-individual error, measurement error and any model misspecification error.

In order to choose the correct appropriate residual error model, it is essential to understand the concepts of weighting. Consider a typical calibration curve for an analytical method with three standards (1, 10 and 100 ng/mL). The standard deviations (determined using say 3 samples) of these concentrations could be similar or dependent on the concentrations. If the standard deviation is constant (see Figure 1 below) the variance is considered homoscedastic. However, if it is not then the variance is considered heteroscedastic (Figure 2). For HPLC methods, generally, the standard deviation is proportional to the sample concentration.

Figure 1 Figure 2

If we consider plotting the standard deviation against the mean concentration from figure 1, it would result in a line similar to the flat line shown in Figure 3 below. Similarly when the heteroscedastic standard deviations from figure 2 are plotted against means, the plot will look like the inclined line as in figure below. The flat line represents the additive or constant error component. The slope of the inclined line would be equal to the coefficient of variation. Therefore, when the error is proportional to the concentration, such an error model is called a constant coefficient of variation error model or proportional error model.

Figure 3

Based on the above mentioned concepts, four different types of residual error models are described below.

- If the variance is assumed to be constant for all observations, then the weights are typically set to a constant value, usually 1. Such an error model is called an additive or constant error model. An additive error model is written as

y_ij = y_mij + ε_ij

where
y_ij = observed concentration in indiviual i at time j
y_mij= model (m) predicted concentrations in individual i at time j
ε_ij = independant, identically distributed statistical errors with ~N(0,σ²)

- Sample analyses in PK studies are normally performed using HPLC or LC-MS-MS techniques. With such analytical methods, it is observed that the precision often changes with concentrations. The experimental errors are considered to be heteroscedastic and can be written as:

A constant coefficient of variation (CCV) model or a proportional model

y_ij = y_mij •(1 + ε_ij)

Another way of writing a CCV model is the exponential error model (approximates constant coefficient of variation)

y_ij = y_mij •exp(ε_ij)

- In the case of PK concentrations, where wide range of concentrations are measured, the error in measurement based on the analytical method is usually a combination of additive and proportional or exponential error. Use of a combined error model would improve predictions at lower limit of assay precision where variance may be assumed a constant and a proportional error model at higher concentration range.

A combined error model can be written as

y_ij = y_mij + y_mij •ε₁+ ε₂

or

y_ij = y_mij•exp(ε₁)+ ε₂

 

In population analysis,

- a proportional error model could be used when the range of concentration covers more than one order of magnitude

- an additive error model could be used when the range of concentrations are narrow