Individual Parameters

The [individual_parameters] block defines how population parameters (theta), random effects (eta), and covariates combine to produce individual PK parameters.

Syntax

PARAM = expression

Each line assigns a PK parameter using an arithmetic expression that can reference:

• Theta parameters -- names defined in [parameters] (e.g., TVCL, TVV)
• Eta random effects -- names defined as omega parameters (e.g., ETA_CL, ETA_V)
• Covariates -- column names from the data file (e.g., WT, CRCL)
• Constants -- numeric literals (e.g., 70, 0.75)

Supported Operators and Functions

Conditional Logic (if / else)

Two forms are supported and may be combined freely.

Block form

[individual_parameters]
  if (WT > 70) {
    CL = TVCL * (WT / 70)^0.75 * exp(ETA_CL)
  } else if (SEX == 1) {
    CL = TVCL * 1.2 * exp(ETA_CL)
  } else {
    CL = TVCL * exp(ETA_CL)
  }
  V = TVV * exp(ETA_V)

The block form is appropriate when the body contains multiple statements or when several alternative branches are needed. Conditions support comparison operators (<, <=, >, >=, ==, !=) and logical operators (&&, ||, !); parentheses group sub-conditions.

Inline (ternary) form

[individual_parameters]
  CL = if (SEX == 1) TVCL * 1.5 else TVCL
  V  = TVV  * exp(ETA_V)

The inline form produces a value and can appear anywhere an expression is allowed. Both then and else branches are required.

Interaction with mu-referencing

When the assignment to a parameter is wrapped in an if block, the (ETA → THETA) relationship is no longer unconditional, so ferx skips mu-reference detection for that parameter. Unconditional assignments in the same block continue to be detected normally.

Tip: if you want mu-referencing for a covariate-adjusted parameter, keep the assignment unconditional and bury the conditional inside the covariate term (e.g. CL = TVCL * (if (WT > 70) (WT / 70)^0.75 else 1.0) * exp(ETA_CL)).

Common Parameterizations

Exponential (log-normal) random effects

The standard approach for PK parameters that must be positive:

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

Allometric scaling with covariates

[individual_parameters]
  CL = TVCL * (WT/70)^0.75 * exp(ETA_CL)
  V  = TVV  * (WT/70)^1.0  * exp(ETA_V)

Estimated covariate effects

Use additional theta parameters for covariate coefficients:

[parameters]
  theta TVCL(0.134, 0.001, 10.0)
  theta THETA_WT(0.75, 0.01, 2.0)
  theta THETA_CRCL(0.5, 0.01, 2.0)

[individual_parameters]
  CL = TVCL * (WT/70)^THETA_WT * (CRCL/100)^THETA_CRCL * exp(ETA_CL)

Logit-normal bioavailability

Use inv_logit(logit(THETA_F) + ETA_F) to constrain bioavailability to (0, 1). The starting value for THETA_F is set directly on the (0, 1) scale — whatever fraction you specify is what the optimiser uses as the typical F:

[parameters]
  theta THETA_F(0.70, 0.001, 0.999)  # typical bioavailability = 70%
  omega ETA_F ~ 0.10                 # BSV on the logit scale

[individual_parameters]
  F = inv_logit(logit(THETA_F) + ETA_F)

• When ETA_F = 0, F_i = THETA_F exactly (the logit and inv_logit cancel).
• ETA_F shifts each individual's F on the logit scale, symmetrically around the typical value.
• The estimated THETA_F in the output is directly interpretable as the typical bioavailability.
• omega ETA_F is variance on the logit scale — not the variance of F itself.

The alternative form inv_logit(THETA_F + ETA_F) is also supported, where THETA_F is on the logit scale (e.g., logit(0.70) ≈ 0.847). This is less readable but may be useful when comparing with NONMEM models.

Automatic MU-referencing

When a line matches one of the patterns

PARAM = THETA * exp(ETA)              # multiplicative / log-normal
PARAM = THETA * <anything> * exp(ETA) # multiplicative with covariate terms
PARAM = exp(log(THETA) + ETA)         # canonical MU form
PARAM = THETA + ETA                   # additive

ferx records the (ETA → THETA) mapping and uses it to re-centre the inner-loop ETA search at each outer iteration — reproducing NONMEM / nlmixr2's MU-referencing behaviour without requiring you to write an explicit MU_i line. See the FAQ for details on which patterns are and are not detected, and for the mu_referencing = false escape hatch.

Covariate Detection

Any uppercase identifier in the expression that does not match a theta name or eta name is automatically treated as a covariate. The covariate value is read from the corresponding column in the data file.

For example, in CL = TVCL * (WT/70)^0.75 * exp(ETA_CL):

• TVCL matches a theta parameter
• ETA_CL matches an omega parameter
• WT matches neither, so it is treated as a covariate column

PK Parameter Names

The parameter names on the left side of each assignment must map to recognized PK parameter names:

For ODE models, the parameter names are user-defined and passed as a flat vector to the ODE right-hand side function.