ODE Models

For pharmacokinetic models without analytical solutions (e.g., saturable elimination, target-mediated drug disposition), FeRx provides an ODE solver.

Structural Model Declaration

[structural_model]
  ode(obs_cmt=OBSERVABLE_COMPARTMENT, states=[state1, state2, ...])

• obs_cmt: The compartment whose concentration is observed (matched to DV)
• states: List of state variable names (compartments)

ODE Equations

The [odes] block defines the right-hand side of the ODE system:

[odes]
  d/dt(state_name) = expression

Expressions can reference:

• State variables by name
• Individual parameters defined in [individual_parameters]
• Arithmetic operators and functions (exp, log, sqrt, etc.)

Example: Michaelis-Menten Elimination

A one-compartment oral model with saturable (Michaelis-Menten) elimination:

[parameters]
  theta TVVMAX(10.0, 0.1, 1000.0)
  theta TVKM(2.0, 0.01, 100.0)
  theta TVV(10.0, 0.1, 500.0)
  theta TVKA(1.0, 0.01, 50.0)

  omega ETA_VMAX ~ 0.09
  omega ETA_V    ~ 0.04

  sigma PROP_ERR ~ 0.1

[individual_parameters]
  VMAX = TVVMAX * exp(ETA_VMAX)
  KM   = TVKM
  V    = TVV * exp(ETA_V)
  KA   = TVKA

[structural_model]
  ode(obs_cmt=central, states=[depot, central])

[odes]
  d/dt(depot)   = -KA * depot
  d/dt(central) = KA * depot / V - VMAX * central / (KM + central)

[error_model]
  DV ~ proportional(PROP_ERR)

Solver Details

FeRx uses a Dormand-Prince RK45 adaptive solver:

The solver automatically adapts step sizes based on local error estimates.

Dose Handling

• Bolus doses: Applied as instantaneous state changes at dose times. The dose amount is added to the target compartment: state[cmt] += amt
• Compartment indexing: Compartments are 1-indexed in the data file (CMT=1 corresponds to the first state in the states list)
• Multiple doses: The ODE is integrated in segments between dose events, with state discontinuities at each dose

Limitations

• Infusion doses (RATE > 0) are not currently supported for ODE models. Use bolus approximations or analytical models for infusions
• The observable compartment contains the amount (not concentration). Divide by volume in the ODE equations if needed
• Steady-state (SS=1) is not directly supported for ODE models