Example: Steady-state dosing (SS=1)

This example demonstrates fitting a model to data collected at steady state. When subjects have been dosed repeatedly before the observation window, the pharmacokinetic system is at (or near) steady state and the single-dose initial conditions are no longer valid. ferx handles this via the SS and II columns in the dataset — no changes to the model file are needed.

Dataset columns

Column	Purpose
`SS`	1 = pre-dose steady-state condition; 0 (or absent) = normal
`II`	Dosing interval (h) for the preceding repeated-dose regimen

A dose row with SS=1 and II=24 tells ferx: compute the steady-state initial conditions for a 24-hour repeated dosing regimen, then add the current dose on top before integrating forward.

Example CSV layout:

ID,TIME,DV,EVID,AMT,CMT,SS,II,MDV
1,0,.,1,10,1,1,24,1
1,1,18.2,0,.,1,0,0,0
1,4,25.8,0,.,1,0,0,0
1,8,22.1,0,.,1,0,0,0

Model file

The model file is identical to the single-dose warfarin example. No SS or II handling is needed in the .ferx file itself:

[parameters]
  theta TVCL(0.134, 0.001, 10.0)
  theta TVV(8.1, 0.1, 500.0)
  theta TVKA(1.0, 0.01, 50.0)

  omega ETA_CL ~ 0.07
  omega ETA_V  ~ 0.02
  omega ETA_KA ~ 0.40

  sigma PROP_ERR ~ 0.01 (sd)

[individual_parameters]
  CL = TVCL * exp(ETA_CL)
  V  = TVV  * exp(ETA_V)
  KA = TVKA * exp(ETA_KA)

[structural_model]
  pk one_cpt_oral(cl=CL, v=V, ka=KA)

[error_model]
  DV ~ proportional(PROP_ERR)

[fit_options]
  method     = foce
  maxiter    = 300
  covariance = true

Running

library(ferx)

ex  <- ferx_example("warfarin_ss")
fit <- ferx_fit(ex$model, ex$data, method = "foce")
fit

# Compare parameter estimates with the single-dose example
fit$theta

At steady state with τ = 24 h and the warfarin half-life of ~42 h (ke = CL/V ≈ 0.0165 /h) there is roughly 3-fold accumulation. Observed DV values are therefore substantially higher than in the single-dose dataset, but the underlying PK parameters should be consistent.

How ferx computes steady-state conditions

For analytical PK models (one_cpt_oral, two_cpt_iv_bolus, etc.), ferx derives the closed-form steady-state concentrations from the current individual parameters and dosing interval. No iterative pre-simulation is required.

For ODE models with SS=1, ferx runs an iterative pre-dose simulation: it simulates repeated doses at interval II until the state vector converges (relative tolerance 1e-6), then uses that state as the initial condition for the observation period.

SS=2 (additive steady-state)

Setting SS=2 on a dose row adds the computed steady-state concentration to the current state rather than replacing it. This is useful for modelling drugs given at steady state where there is also a residual concentration from a prior non-SS period.

Tips

If II is absent or zero on an SS=1 row, ferx will emit a warning and fall back to zero initial conditions.
For ODE models, convergence of the iterative SS computation can be slow for drugs with very long half-lives. Consider increasing inner_maxiter via settings if the fit reports unconverged steady-state computations.