Stochastic Differential Equations (SDE / Diffusion)

Maturity: experimental — see Feature Maturity for what this means.

The [diffusion] block adds continuous stochastic noise to ODE state variables. This models system noise — structural uncertainty that accumulates between observations — as opposed to measurement noise (sigma) or inter-individual variability (omega).

What is system noise?

In a standard ODE-based NLME model the trajectory of each subject is deterministic once the individual parameters are fixed. In an SDE model the ODE is replaced by an Itô stochastic differential equation:

dX = f(X, t, θ) dt  +  diag(σ_w) dW

where dW is a vector of independent Wiener increments. The diagonal entries σ_w are the diffusion standard deviations; their squares σ²_w are the diffusion variances declared in [diffusion].

ferx-core estimates the diffusion variances with an Extended Kalman Filter (EKF): the state covariance matrix P is propagated alongside the ODE trajectory and updated at each observation. The total observation variance is:

V_total = P_ekf[obs_cmt, obs_cmt]  +  V_residual

where V_residual is the variance from the [error_model] (sigma-based).

Declaring diffusion variances

[diffusion]
  STATE_NAME ~ initial_value
  STATE_NAME ~ initial_value FIX

STATE_NAME must be one of the state names listed in [structural_model].
The value after ~ is the initial estimate of the variance σ²_w (≥ 0).
FIX pins the parameter at its initial value (not estimated).
Each declared state gets a population parameter named DIFF_<STATE> (e.g., DIFF_CENTRAL). These appear in theta_names of the fit result alongside the regular structural parameters.

How diffusion differs from sigma and omega

	sigma	omega	diffusion
What it is	Measurement noise (assay/residual error)	Between-subject variability	Within-subject system noise
When it acts	At each observation time	At model initialisation (between subjects)	Continuously along the trajectory
What it affects	Observation residuals only	Individual parameter values	State trajectories between doses
NONMEM analogy	`EPS` / `SIGMA`	`ETA` / `OMEGA`	No direct analogy

A large estimated DIFF_CENTRAL relative to sigma suggests that the ODE structure is missing an important mechanism (e.g., a compartment, a feedback loop, or a covariate effect).

When to use SDE models

The primary signal is residual autocorrelation. If the IWRES Durbin-Watson statistic is low (< 1.5) in a fitted ODE model, adding a diffusion term on the slow-varying compartment often absorbs the systematic drift.

Unit convention (important)

The EKF propagates the variance of the ODE state in whatever units the user defines it. ferx adds dose.amt directly to the state (u[cmt-1] += amt) and returns the raw state value as ipred — there is no implicit amount → concentration normalisation. You are responsible for making the ODE’s state match the DV column.

Two common forms for a 1-cpt oral model (both equations shown so the expressions are self-contained):

# State = amount (mg).  DV must be in mg.
d/dt(depot)   = -KA * depot
d/dt(central) =  KA * depot - (CL / V) * central

# State = concentration (mg/L).  DV must be in mg/L.
d/dt(depot)   = -KA * depot
d/dt(central) =  KA * depot / V - (CL / V) * central

If the units don’t match (e.g. amount-form ODE with concentration DV), predictions and observations will differ by a factor of V and the fit will not converge. DIFF_<STATE> is then the variance of process noise in the same units as the state — e.g. (mg/L)² for the concentration form, (mg)² for the amount form. See examples/bioavailability_ode.ferx for a worked concentration-form ODE.

Worked example: 1-cpt IV with central diffusion

[parameters]
  theta TVCL(5.0, 0.1, 50.0)
  theta TVV(50.0, 1.0, 500.0)
  omega ETA_CL ~ 0.09
  sigma ADD ~ 1.0

[individual_parameters]
  CL = TVCL * exp(ETA_CL)
  V  = TVV

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

[odes]
  d/dt(central) = -(CL/V) * central

[diffusion]
  central ~ 0.5       # initial estimate for DIFF_CENTRAL

[error_model]
  DV ~ additive(ADD)

[fit_options]
  method = foce

The estimated DIFF_CENTRAL gives the variance of the stochastic driving noise on the central compartment amount per unit time. A value near zero indicates the ODE alone explains the data; a large value suggests model misspecification.

Constraints and incompatibilities

Situation	Behaviour
`[diffusion]` on an analytical PK model (`pk ...`)	Parse-time error
`method = saem` with `[diffusion]`	Hard error at fit time
Analytic gradients with `[diffusion]`	Not supported; use `gradient_method = fd`
State name not in `states = [...]`	Parse-time error
Negative initial value	Parse-time error

ferx-r usage note

When using ferx-core from R via the ferx-r package, the [diffusion] block is part of the .ferx model file and requires no special R-side argument. The estimated diffusion variances appear in the returned fit object alongside other theta parameters. See the ferx-r documentation (?ferx_fit) for details on accessing theta_names and interpreting uses_sde in the fit object.