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Model components

Structural model

  • mathematical form of algebraic/differential equations
    • parameters, their relationships, or rate of change [📖] [📖] [📖]
    • parameters are considered as “true” values for the population with possible variability [📖]
  • describes the central tendency in the data
    • "e.g. the central tendency of the cortisol concentration-time profiles after administration of hydrocortisone"
  • to develop the simplest model, which still describes the data accurately [📖]
\[y_{ij}=f(𝜙_{i}, x_{ij})\]

\(i\) - certain individual;
\(j\) - certain timepoint;
\(f\) - a nonlinear function;
\(𝜙\) - vector of model parameters (\(CL\), \(V_c\));
\(x\) - study design variables (covariates, dose and sampling times).

Pharmacostatistical model

  • describes the variability, which can be subdivided into IIV and RUV models [📖] [📖]
  • several hierarchical levels of pharmacostatistical models

IIV

  • InterIndividual Variability
  • difference between individuals
  • allows for the individual parameter estimate to differ from the population estimate
  • \(𝜂_i\)
    • discrepancy between the population estimate and individual model parameter
    • Empirical bayes estimates, or EBE
    • independent of each other
    • normally or log-normally distributed around 0 [📖] [📖]
    • variance of \(ω^2\)
    • the same within an individual unless IOV is applied
    • can be added as:
      • additive
      • proportional
      • exponential
        • the most common as parameters are usually log-normally distributed and non-negative values
\[𝜙_{i}=g(𝜃, z_{i}) + 𝜂_i\]
\[𝜙_{i}=g(𝜃, z_{i}) \cdot (1+ 𝜂_i)\]
\[𝜙_{i}=g(𝜃, z_{i}) \cdot e^{𝜂_i}\]

\(i\) - certain individual;
\(𝜙\) - vector of model parameters (\(CL\), \(V_c\));
\(θ\) - population parameter estimates;
\(z_{i}\) - covariates.

IOV

  • InterOccasion Variability
  • variability between different occasions
  • depends on study design (different doses/days/study periods...)
  • does not describe the reason for the variability between occasions
  • should only be used if the model parameters change randomly between occasions
  • \(k_i\) [📖]
\[𝜙_{i}=g(𝜃, z_{i}) \cdot e^{𝜂_i+k_i}\]

RUV

  • Residual Unexplained Variability
  • to explain the difference between model-predicted values and observations in the form of distribution variance [📖] [📖]
  • unexplained variability
    • e.g. resulted by measurement error, model misspecification and errors in dosing
  • discrepancy between the observed and individually predicted
  • \(ε_{ij}\)
    • normally distributed around 0
    • variance of \(σ^2\)
    • can be added as
      • additive
        • If estimating parameters using log-transformed data an additive model is commonly applied, since it approximates an exponential or a proportional RUV model on a linear scale
      • proportional
      • combined
\[y_{ij}=f(𝜙_{i}, x_{ij}) + ε_{add,ij}\]
\[y_{ij}=f(𝜙_{i}, x_{ij}) \cdot (1+ ε_{prop,ij})\]
\[y_{ij}=f(𝜙_{i}, x_{ij}) \cdot (1+ ε_{prop,ij}) + {add,ij}\]

Covariate model

  • explain variability using observable factors between subjects [📖]
    • e.g. age, disease progression, height, weight, or interacting agents/drugs
  • whether any dose adjustments are needed in specific populations
  • potentially reducing some unexplained IIV [📖]
    • body size related covariates
    • creatinine clearance for drugs with renal elimination
    • time-varying covariates [📖]
\[𝜙_{i}=𝜃 + 𝜃_{cov} \cdot (z_{i}-z_{median})\]

\(θ\) - population parameter;
\(θ_{cov}\) - covariate effect;
\(z_{i}\) - individual covariate value;
\(z_{median}\) - median value of the covariate.

"Final model"

  • needs to have a scientific basis and descriptive and predictive power to address given clinical questions [📖]
  • based on components, mechanisms, and assumptions, which should be
    • credible;
    • reasonable;
    • comparable with existing system components.