Maturation Plateaus and Adult Reference Values

Why the Adult Plateau Needs a Clear Definition

A maturation function describes how a pharmacokinetic parameter changes with developmental age and approaches a mature value. When pediatric and adult observations are modeled together, that mature value is not merely a convenient number: it defines the reference level toward which the age effect tends. The model should state whether this plateau is fixed, estimated, or represented by a reference parameter whose meaning depends on other covariates being held at specified values.

This tutorial uses the sigmoidal Hill function introduced in the earlier material on maturation functions. Let \(A\) denote the chosen developmental age, in a stated unit such as years, and let \(TM_{50}\) be the age at which the maturation fraction is half its plateau. The age metric must be defined consistently for pediatric and adult records; when relevant to the population, the earlier material on gestational, postnatal, and postmenstrual age provides the distinction. Here the focus is what the plateau and mature reference mean once that age metric has been chosen.

In this parameterization, the multiplier has a fixed upper limit of 1; it is not a separately estimated plateau parameter. The value \(P_{\mathrm{adult}}\) carries the units of the modeled parameter, while \(F(A)\) is dimensionless. For example, if \(P\) is clearance, \(P_{\mathrm{adult}}\) is expressed in units of volume per time. If \(P\) is a volume, the mature reference is expressed in units of volume. The age term describes the relative maturation level, and the reference parameter gives the mature scale.

What the Function Implies

The normalized function is useful because its asymptote is explicit and its parameters have direct interpretations. At \(A=TM_{50}\), the numerator and the two terms in the denominator are equal, so \(F(A)=1/2\). The exponent \(h\) controls the steepness of the transition around \(TM_{50}\); it does not change the limiting plateau.

For positive age, the function increases with age. Differentiating with respect to \(A\) gives

Thus, with a positive mature reference, the typical modeled parameter rises toward that reference. This is a property of the specified function, not a guarantee that every individual’s parameter rises with age: individual random effects and other model terms can also affect individual predictions. Nor does the limit mean the curve reaches exactly 1 at a particular finite age. For a Hill function, the plateau is an asymptote.

Joint Pediatric and Adult Modeling

In a joint analysis, pediatric and adult records can contribute to the same population model. A simplified nonlinear mixed-effects specification for an individual parameter is

Here \(i\) indexes individuals, \(A_i\) is the individual’s age, and \(\eta_i\) is a normally distributed random effect on the log scale with variance \(\omega^2\). This expression illustrates the role of the age function and mature reference; it does not replace the observation model for measured concentrations or other data. Under this lognormal random-effects specification, \(P_{\mathrm{adult}}\) is the typical value at the plateau, corresponding to \(\eta_i=0\). Because the random effect is exponentiated, it is also the median individual value at a fixed age under this model. The arithmetic mean across individuals is larger: at the plateau it is \(P_{\mathrm{adult}}\exp(\omega^2/2)\).

Adult observations help estimate the mature scale, while pediatric observations help characterize how values change below the plateau. Adult measurements are not necessarily observations at infinite age, and therefore need not equal the asymptotic value. If adult ages are well beyond the transition, the Hill multiplier may be close to 1; if they are not, the model still predicts a multiplier below 1. The adult reference remains defined by the model’s limit, not by the sample mean of the observed adult records.

Choosing and Checking a Plateau Parameterization

The normalized function \(F(A)\) makes the mature reference easy to interpret because the multiplier tends to 1. A different formulation can include an estimated upper limit, for example \(M(A)=M_0+(M_{\infty}-M_0)F(A)\). In that formulation, \(M_{\infty}\) is the plateau of the maturation multiplier, not automatically the mature value of a parameter measured in units such as L/hour. If the parameter is written as \(P(A)=P_{\mathrm{scale}}M(A)\), then its mature value is \(P_{\mathrm{scale}}M_{\infty}\). Calling \(P_{\mathrm{scale}}\) the adult reference would be incorrect unless the parameterization makes that identity true.

There is also a scale-identification issue. If the model instead writes \(P(A)=P_{\mathrm{scale}}M_{\infty}F(A)\), and both \(P_{\mathrm{scale}}\) and \(M_{\infty}\) are free, the data identify their product rather than the two factors separately. One must fix one factor, define the estimand differently, or use a parameterization that gives each parameter an independently interpretable role. When the scientific target is the mature value in the units of \(P\), a direct mature reference with a multiplier normalized to 1 at the plateau is often the clearest choice.

A free upper limit can be useful when there is a reason to estimate a plateau relative to a separate scale. Fixing the normalized plateau at 1 is simpler and gives the adult reference a direct meaning, but it assumes that this is the appropriate asymptotic normalization. A hard age cutoff that sets all ages beyond a selected boundary exactly to the plateau is another possibility, but it changes the curve’s form and can create an abrupt change unless the transition is carefully constructed. These are different modeling assumptions, not interchangeable labels for the same curve.

Check the plateau specification before interpreting estimates. In particular, confirm that:

• The age definition and age units are consistent across pediatric and adult records.
• The mature reference is reported at its stated reference values for other covariates and in the correct units.
• The fixed or estimated plateau in the multiplier is distinguished from the mature value of the modeled parameter.
• The observed ages include enough information about the transition and mature range to support the intended interpretation.
• The model predictions approach the claimed plateau over the age range relevant to the analysis.

If all observed ages lie well below the transition, the data may give little information about the adult reference. If nearly all observations are at mature ages, the data may anchor the mature value but provide little information about \(TM_{50}\) or \(h\). A model can still return estimates in either case; numerical convergence alone does not show that the data support the curve’s shape. Compare the fitted predictions with observed information across age, inspect parameter uncertainty and correlations, and consider whether a simpler or externally informed specification is needed. These checks concern the adequacy of the plateau model, not a proof that the chosen physiological mechanism is correct.

Common Pitfall: Treating “Adult” as a Model Value

A common mistake is to assume that every adult observation has maturation multiplier exactly 1, then describe an asymptotic Hill model as if it imposed that rule. The Hill function approaches 1 continuously; at finite ages its value is generally less than 1. The correction is to calculate or inspect \(F(A)\) at the actual ages and describe \(P_{\mathrm{adult}}\) as the limiting reference. If the scientific assumption is instead that maturation is complete at a specific age, that should be represented explicitly by a model that reaches a plateau there, with its continuity and interpretation stated.

A second mistake is to call the fitted adult reference the observed adult mean. The reference is a model parameter, conditional on the structural form, covariates, and random-effects assumptions. It may differ from a raw mean because observations vary between individuals, ages, covariate values, and residual error. The reference is not a causal effect of adulthood; it is the mature level specified by the model.