Using spline-enhanced ordinary differential equations for PK/PD model development

Yi Wang, Kent Eskridge, Shunpu Zhang, Dong Wang

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


A spline-enhanced ordinary differential equation (ODE) method is proposed for developing a proper parametric kinetic ODE model and is shown to be a useful approach to PK/PD model development. The new method differs substantially from a previously proposed model development approach using a stochastic differential equation (SDE)-based method. In the SDE-based method, a Gaussian diffusion term is introduced into an ODE to quantify the system noise. In our proposed method, we assume an ODE system with form dx/dt = A(t)x + B(t) where B(t) is a nonparametric function vector that is estimated using penalized splines. B(t) is used to construct a quantitative measure of model uncertainty useful for finding the proper model structure for a given data set. By means of two examples with simulated data, we demonstrate that the spline-enhanced ODE method can provide model diagnostics and serve as a basis for systematic model development similar to the SDE-based method. We compare and highlight the differences between the SDE-based and the spline-enhanced ODE methods of model development. We conclude that the spline-enhanced ODE method can be useful for PK/PD modeling since it is based on a relatively uncomplicated estimation algorithm which can be implemented with readily available software, provides numerically stable, robust estimation for many models, is distribution-free and allows for identification and accommodation of model deficiencies due to model misspecification.

Original languageEnglish (US)
Pages (from-to)553-571
Number of pages19
JournalJournal of Pharmacokinetics and Pharmacodynamics
Issue number5
StatePublished - Oct 2008


  • Model uncertainty
  • Nonparametric function
  • Stochastic differential equation

ASJC Scopus subject areas

  • Pharmacology


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