# One-compartment open model

Filed under:
Differential equations,
PK/DK

Fit mixed effects model to the classical "phenopharbital" model

Pinheiro
& Bates (2000, Ch. 6.4) fitted a so-called 'one-compartment open
model' to a dataset known as the 'phenopharbital data'. A patient is given
a dose

where

*D*at time*t*_{d}, giving rise to a phenopharbital concentration*c*at a later time_{t}*t*:*c*=

_{t}*D/V*exp[-

*Cl/V*(

*t*-

*t*

_{d})],

*V*and

*Cl*are parameters (the so-called 'Volume of concentration' and the 'Clearance', respectively). Doses given at different time points contribute additively to

*c*.

_{t}_{}Pinheiro & Bates (2000) fitted a model with a linear predictor (and a log-link) for each of the paramere

*V*and

*Cl*. Each of the linear predictors contained one covariate

*Wt*and one random effect. A full description of the model can be found here pheno.pdf (quite old, so timing results mentioned are outdated).

Note: In this model the underlying ODE has an analytical solution, but in more general models it will not.

### References

Pinheiro, J., Bates, D.M. (2000),

*Mixed-Effects Models in S and S-PLUS*. Statistics and Computing, Springer.