Maturation Functions for Clearance in Infants and Children

Why Model Clearance Maturation?

Clearance can change substantially as drug-metabolizing and eliminating pathways develop. A single clearance value applied to every infant and child can therefore misrepresent expected drug exposure. A maturation function gives a continuous way to describe how clearance approaches a mature value with developmental age. It can support dose calculations at ages between those represented by observed data, provided the function is appropriate for the drug and population.

The age scale used here is postmenstrual age (PMA): gestational age at birth plus postnatal age. For example, a baby born at 32 weeks’ gestation and studied 8 weeks after birth has a PMA of 40 weeks. PMA is not the same as postnatal age; two infants of the same postnatal age can have different PMAs if they were born at different gestational ages.

The function developed below describes maturation of clearance on a fixed reference-size basis. It does not itself describe changes in body size, disease, or organ function. The earlier material on allometric scaling addresses size-based dose individualization; combining size and maturation effects requires care so that the same age-related change is not counted twice.

A Sigmoid Maturation Function

A common model represents maturation as a fraction that rises from near zero at very low PMA toward one at high PMA. Clearance is the mature clearance scale multiplied by this fraction.

Definition: Let $P$ denote PMA in weeks, $CL_{\mathrm{mat}}$ the mature clearance at a specified reference size, $\mathrm{TM}_{50}$ the PMA at which the maturation fraction is one-half, and $h$ a positive dimensionless Hill coefficient. The sigmoid maturation function is $$ M(P)=\frac{P^h}{P^h+\mathrm{TM}_{50}^h}, \qquad CL(P)=CL_{\mathrm{mat}}M(P). $$ Here $M(P)$ is dimensionless, lies between zero and one for $P>0$, and clearance has the same units as $CL_{\mathrm{mat}}$.

The parameters have direct interpretations. $CL_{\mathrm{mat}}$ is the clearance approached as PMA becomes large under this model, not necessarily the clearance of every older child or adult. At $P=\mathrm{TM}_{50}$, the numerator and the second denominator term are equal, so $M(P)=0.5$, and $CL(P)=0.5CL_{\mathrm{mat}}$. Larger $h$ makes the transition around $\mathrm{TM}_{50}$ steeper; smaller positive $h$ spreads it over a wider age range. The parameter $\mathrm{TM}_{50}$ has units of weeks, whereas $h$ has no units.

Why the Function Has These Properties

The sigmoid can be written in terms of the ratio of PMA to the midpoint:

M(P)=\frac{1}{1+\left(\frac{\mathrm{TM}_{50}}{P}\right)^h}.

This form shows that the function depends on the relative position of PMA and $\mathrm{TM}_{50}$. If $P=\mathrm{TM}_{50}$, the ratio is one and $M=1/(1+1)=0.5$. If $P$ is much smaller than $\mathrm{TM}_{50}$, the ratio raised to $h$ is large and $M$ is near zero. If $P$ is much larger, that ratio is near zero and $M$ is near one. Thus the model is monotone increasing and has an upper limit of one.

The local slope at the midpoint also explains the role of $h$. Differentiating $M(P)=P^h/(P^h+\mathrm{TM}_{50}^h)$ with respect to $P$ gives

\frac{dM}{dP} = \frac{hP^{h-1}\mathrm{TM}_{50}^h} {\left(P^h+\mathrm{TM}_{50}^h\right)^2}.

At $P=\mathrm{TM}_{50}$, this becomes $h/(4\mathrm{TM}_{50})$. Therefore, for a fixed midpoint, increasing $h$ increases the slope at the midpoint. This is a mathematical property of the model, not evidence that maturation in a particular drug pathway truly follows this shape.

Worked Example: Calculate Clearance at an Early PMA

Suppose a hypothetical drug has $CL_{\mathrm{mat}}=4.00$ L/h, $\mathrm{TM}_{50}=50.0$ weeks, and $h=3.00$. A baby was born at 32 weeks’ gestation and is assessed 8 weeks after birth. Calculate PMA, the maturation fraction, and predicted clearance.

First calculate PMA:

P=32\ \mathrm{weeks}+8\ \mathrm{weeks}=40\ \mathrm{weeks}.

Substitute the values into the maturation function:

M(40)=\frac{40^3}{40^3+50^3} =\frac{64{,}000}{64{,}000+125{,}000} =\frac{64{,}000}{189{,}000} =0.3386.

Then multiply by the mature clearance:

CL(40)=4.00\ \mathrm{L/h}\times0.3386 =1.354\ \mathrm{L/h}.

Under this hypothetical model, predicted clearance at 40 weeks PMA is 1.354 L/h on the stated reference-size basis. The value is below half the mature clearance because 40 weeks is below the 50-week midpoint and the function is relatively steep with $h=3.00$.

From Predicted Clearance to a Maintenance Dose

For linear pharmacokinetics at steady state, the average concentration over a dosing interval is related to dose rate by clearance. For an intravenous dose given every $\tau$ hours, the maintenance dose that targets a specified average steady-state concentration is $CL$ times that concentration times the interval. For an oral or other extravascular route, include bioavailability $F$, the fraction of the administered dose reaching systemic circulation.

Key result: Under linear pharmacokinetics, for a target average steady-state concentration $C_{\mathrm{avg,ss}}$, dosing interval $\tau$, and bioavailability $F$, $$ \mathrm{Dose}=\frac{CL\,C_{\mathrm{avg,ss}}\,\tau}{F}. $$ For intravenous dosing, $F=1$. This is a maintenance-dose relationship; it does not calculate a loading dose. It assumes clearance is adequately represented by the model, the target is appropriate, and the drug’s relevant exposure is approximately linear in dose.

Worked Example: Calculate an Oral Maintenance Dose

For a hypothetical drug, let $CL_{\mathrm{mat}}=3.20$ L/h, $\mathrm{TM}_{50}=42.0$ weeks, and $h=2.00$. An infant has a PMA of 42 weeks. The proposed target is $C_{\mathrm{avg,ss}}=8.00$ mg/L, the interval is 12.0 h, and oral bioavailability is $F=0.800$. Calculate the model-based maintenance dose per interval.

At the midpoint, the maturation fraction is one-half:

M(42)=\frac{42^2}{42^2+42^2}=0.500.

Predicted clearance is therefore

CL(42)=3.20\ \mathrm{L/h}\times0.500 =1.60\ \mathrm{L/h}.

Apply the oral maintenance-dose equation, keeping the units visible:

\mathrm{Dose} =\frac{(1.60\ \mathrm{L/h})(8.00\ \mathrm{mg/L})(12.0\ \mathrm{h})}{0.800} =192\ \mathrm{mg}.

The units reduce to milligrams because $(\mathrm{L/h})(\mathrm{mg/L})(\mathrm{h})=\mathrm{mg}$. The result is a hypothetical model-based dose per 12-hour interval, not a clinical recommendation. Its use would require checking the model’s population, uncertainty, route-specific bioavailability, and whether the target and interval are clinically suitable.

Compare Predictions Across PMA

The function can quantify how its predicted clearance and a clearance-based dose change as PMA changes. This comparison holds the mature clearance scale, reference size, target concentration, interval, and bioavailability constant. It isolates the mathematical effect of the maturation function; it does not imply that an individual child’s size or clinical condition stays constant with age.

Worked Example: Compare Clearance and Dose at Two Ages

Consider a hypothetical intravenous drug with $CL_{\mathrm{mat}}=5.00$ L/h, $\mathrm{TM}_{50}=40.0$ weeks, and $h=2.00$. Compare predictions at 20 and 80 weeks PMA for a target average steady-state concentration of 6.00 mg/L and a 24.0-hour interval.

At 20 weeks:

M(20)=\frac{20^2}{20^2+40^2} =\frac{400}{2{,}000}=0.200, \qquad CL(20)=5.00\times0.200=1.00\ \mathrm{L/h}.

At 80 weeks:

M(80)=\frac{80^2}{80^2+40^2} =\frac{6{,}400}{8{,}000}=0.800, \qquad CL(80)=5.00\times0.800=4.00\ \mathrm{L/h}.

For intravenous dosing, $F=1$. The predicted maintenance doses are

\mathrm{Dose}_{20} =(1.00\ \mathrm{L/h})(6.00\ \mathrm{mg/L})(24.0\ \mathrm{h}) =144\ \mathrm{mg},

\mathrm{Dose}_{80} =(4.00\ \mathrm{L/h})(6.00\ \mathrm{mg/L})(24.0\ \mathrm{h}) =576\ \mathrm{mg}.

The 80-week prediction is four times the 20-week prediction because the maturation fraction rises from 0.200 to 0.800, also a fourfold change. This comparison follows from the assumed model and fixed target conditions; it is not evidence that a fourfold dose increase is appropriate for a real child.

Assumptions, Diagnostics, and Alternatives

Before using a fitted maturation function, check that its assumptions are defensible for the intended use. The basic sigmoid assumes a nonnegative PMA, monotone maturation, a single upper asymptote, and a clearance scale that does not otherwise change within the modeled comparison. The observed clearance data must support the relevant parts of the curve. If observations cover only low PMAs, the mature asymptote may be weakly informed; if all observations are well above the midpoint, $\mathrm{TM}_{50}$ and $h$ may be difficult to distinguish.

Useful diagnostics include plotting observed clearance against PMA with the fitted curve, examining residuals across the age range, and checking whether systematic underprediction or overprediction remains at particular developmental ages. Look for influential observations, sparse coverage near the transition, and implausible parameter estimates. Uncertainty in $CL_{\mathrm{mat}}$, $\mathrm{TM}_{50}$, and $h$ should be reflected in uncertainty for predicted clearance and dose; a point prediction alone can conceal substantial uncertainty.

The sigmoid is not the only possible description. Age categories are simple to communicate but create abrupt changes at category boundaries and discard within-category age information. A linear function may be adequate over a narrow observed range, but it can predict negative clearance or grow without limit when extrapolated. An exponential approach-to-plateau function can also describe monotone maturation, but has a different shape and parameter interpretation. These are alternatives to evaluate against data and intended use, not interchangeable parameterizations.

Common pitfall: Treating $\mathrm{TM}_{50}$ as the PMA at which clearance is mature. It is the PMA at which this model predicts one-half of $CL_{\mathrm{mat}}$. Also avoid using postnatal age in the function when it was defined for PMA: the resulting predictions can differ substantially, especially for infants born preterm.

A maturation function is a compact summary of a developmental pattern, not a universal biological law. It may fail if clearance has a nonzero lower level at very low PMA, if the relevant pathway does not mature monotonically, or if the data combine populations with different clearance mechanisms. In such cases, a model with a lower asymptote, additional covariates, or a different functional form may be warranted, provided the available data can identify its parameters. Predictions outside the PMA range represented by the data require particular caution.

Check Your Understanding

Answer each question, then compare with the solutions.

A child was born at 35 weeks’ gestation and is 10 weeks old. What is the child’s PMA?
For $M(P)=P^h/(P^h+\mathrm{TM}_{50}^h)$, what maturation fraction is predicted at $P=\mathrm{TM}_{50}$, and what does $h$ affect?
A hypothetical drug has $CL_{\mathrm{mat}}=6.00$ L/h, $\mathrm{TM}_{50}=30.0$ weeks, and $h=1$. Calculate clearance at 30 weeks PMA.
Using the clearance in question 3, calculate the intravenous maintenance dose for $C_{\mathrm{avg,ss}}=5.00$ mg/L and $\tau=8.00$ h.
Give one reason a fitted maturation curve should not be extrapolated uncritically beyond the observed PMA range.

Solutions. (1) $35+10=45$ weeks PMA. (2) The fraction is $0.5$; $h$ controls the steepness of the transition around the midpoint. (3) At the midpoint, $M=0.5$, so $CL=6.00\times0.5=3.00$ L/h. (4) With $F=1$, $\mathrm{Dose}=(3.00\ \mathrm{L/h})(5.00\ \mathrm{mg/L})(8.00\ \mathrm{h})=120$ mg per interval. (5) For example, the mature asymptote may be weakly informed if the data do not include older PMAs, making extrapolated clearance uncertain.