Maturation Functions for Pediatric Drug Clearance

Why clearance needs an age function

Body size is an important source of clearance differences in children, but it may not capture developmental change at a given size. A maturation function represents the additional age-related change in clearance. Its job is specific: it multiplies a reference clearance, usually alongside a body-size term, to describe how clearance approaches its mature level. The function is a model assumption to be evaluated against data, not a universal biological curve.

Let \(a\) denote age on a specified scale, such as postmenstrual age (PMA) or postnatal age (PNA), and let \(F_m(a)\) be a dimensionless maturation multiplier. If \(CL_{\mathrm{ref}}\) is reference clearance and \(S(WT)\) is the body-size scaling factor, a typical structure is

Here \(i\) indexes an individual. The size factor should be selected and parameterized consistently with the body-size model; the maturation function represents age-related change beyond that size term. This multiplicative structure assumes that size and maturation effects combine in this way. It does not establish that they are biologically independent.

Common maturation functions

A sigmoid function permits maturation to begin below its mature value, rise most rapidly over an intermediate age range, and approach an upper asymptote. A convenient form sets the mature multiplier to 1:

At age zero, \(F_m(0)=F_{\min}\); as age becomes large, \(F_m(a)\) approaches 1. At \(a=TM_{50}\), the fraction in the function equals one-half, so \(F_m(TM_{50})=(1+F_{\min})/2\). A larger \(TM_{50}\) shifts the transition to older ages. A larger \(h\) makes the transition steeper around \(TM_{50}\), while \(F_{\min}\) determines the predicted fraction of mature clearance at age zero.

A simpler alternative is an exponential approach to the mature value:

where \(k>0\) has units reciprocal to age. This function starts at \(F_{\min}\) and approaches 1, with a rate set by \(k\). Unlike the Hill form, it has no explicit half-maturation age or independently adjustable transition steepness. The exponential may be adequate when the data support a smooth rise but not the extra shape flexibility of a sigmoid.

Specify the age scale before fitting

PMA is gestational age plus time since birth, expressed in a consistent unit. PNA is elapsed time since birth. They are not interchangeable in the neonatal period: two infants with the same PNA can have different PMA if they were born at different gestational ages. The selected age covariate should match the developmental interpretation and the available observations. Using PNA when the intended relationship is indexed to PMA can assign different maturation predictions to individuals with the same PMA.

Age also needs a clearly defined origin. The Hill form above uses age zero as its lower-bound reference. In an application, state whether zero means birth (for PNA) or the chosen origin for another age scale, and ensure that the age values used for prediction are nonnegative. A model should not be extrapolated to ages or developmental groups unsupported by its data without an explicit justification.

From maturation to predicted exposure

For linear pharmacokinetics under conditions where bioavailability \(F\) and dose are fixed, steady-state average exposure over a dosing interval is related to clearance by \(AUC_{\tau}=F\cdot Dose/CL\). Consequently, changing only the maturation multiplier changes clearance in direct proportion to \(F_m(a)\), and changes AUC in inverse proportion. This interpretation requires the same dose, bioavailability, and other relevant model conditions for the comparison. It does not automatically apply if absorption, bioavailability, clearance mechanisms, or pharmacokinetics themselves change with age.

For the parameters in the preceding example, consider an individual whose body-size factor is held fixed. The maturation-related clearance ratio between 40 and 4 weeks is \(0.5750/0.1584=3.630\). Under the stated linear-PK assumptions, the corresponding AUC ratio at the same dose is the reciprocal, \(0.1584/0.5750=0.2755\). Thus the model predicts about 27.55% of the earlier-age AUC at 40 weeks, all else equal. Dose adjustment could alter that comparison, but selecting a dose is a separate decision requiring an exposure target and other clinical considerations.

Parameterization, estimation, and checks

A useful parameterization should have interpretable parameters, plausible boundary behavior, and enough data to inform its shape. Before fitting, verify that ages use a consistent unit and origin, that the chosen function is defined over the observed age range, and that \(F_m(a)\) stays positive and within the intended range. The fixed mature asymptote of 1 is a normalization choice: it makes \(CL_{\mathrm{ref}}\) the mature reference clearance after accounting for size. If the data do not support a mature asymptote at 1, that normalization must be reconsidered rather than treated as established fact.

The sigmoid parameters can be correlated in estimation. For example, a change in \(TM_{50}\) can be partly offset by a change in \(h\), particularly when observations cover only a narrow age range. If the data include few young children, \(F_{\min}\) may be weakly informed; if they do not extend toward the mature range, the upper plateau may be uncertain. A visually smooth fitted curve does not by itself resolve such uncertainty.

A practical model review should examine the following:

• Age coverage: Plot observed ages and individual clearance information against the fitted curve. Sparse or absent data near an asymptote make predictions there more dependent on assumptions.
• Age definition: Confirm that the recorded age variable is the one specified by the model, especially when PMA and PNA differ.
• Residual pattern: Look for systematic under- or overprediction by age. A smooth average curve may conceal age-specific bias.
• Parameter plausibility and uncertainty: Check estimates, uncertainty, and correlations. Boundary estimates or very broad uncertainty can indicate that the data do not identify the curve’s shape.
• Combined effects: Check whether the age trend remains after applying the size term. Otherwise an apparent maturation effect may be compensating for an inadequately represented size relationship.

These checks diagnose whether the selected relationship is compatible with the available data; they do not prove the assumed biological mechanism. The function also has limitations: a single smooth curve can average over meaningful heterogeneity, and a multiplicative age effect may not represent every clearance pathway or clinical condition. Predictions outside the observed ages should be treated as extrapolations, with sensitivity to plausible alternative functions reported where consequential.

Choosing and interpreting the function

The Hill and exponential forms are alternatives, not competing truths. The Hill form can represent a more localized transition and separately control its steepness, but requires more shape information from the data. The exponential is simpler, but its particular curvature may be inadequate if maturation is nearly flat at early ages and then rises sharply. A more flexible form is not automatically preferable: unnecessary parameters can be poorly estimated and make predictions unstable.

Compare candidate functions using fit diagnostics, parameter uncertainty, and predictions over the age range relevant to the development question. Evaluate whether conclusions about clearance and exposure are robust to reasonable function choices. Do not select a curve solely because it produces a desired dose or exposure result. The purpose is to represent the age relationship supported by the data and assumptions, while making uncertainty and extrapolation visible.