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.)
Conditional logic with the same if (cond) { ... } else { ... } and inline if (cond) expr else expr syntax described in Individual Parameters. For example, you can switch between linear and saturable elimination based on the central amount:
```
[odes]
  d/dt(depot)   = -KA * depot
  if (central > KM_THRESHOLD) {
    d/dt(central) = KA * depot - VMAX * central / (KM + central)
  } else {
    d/dt(central) = KA * depot - CL_LIN * central
  }
```
Each d/dt(state) reachable from any branch counts as defined; states that aren't assigned in the firing branch this step receive a derivative of 0.

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:

Setting	Value
Method	Explicit Runge-Kutta 4(5)
Absolute tolerance	1e-6
Relative tolerance	1e-4
Max steps	10,000
Initial step size	0.1
Minimum step size	1e-12

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
Infusion doses (RATE > 0): Treated as a continuous zero-order input. The integrator's timeline is broken at the infusion's end (time + amt/rate), and +rate is added to the target compartment's derivative for every segment fully spanned by the infusion. Overlapping infusions on the same compartment sum their rates
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 bolus

Limitations

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

FeRx-NLME Documentation