Why Estimate a Maturation Curve?
In pediatric pharmacokinetics, clearance can change with age as elimination pathways develop. A maturation function represents this age-related change as a fraction of mature clearance. Estimating its parameters from clearance data makes the assumptions about the age pattern explicit and allows predictions at ages that were not directly observed. The example below concerns hypothetical data only.
The clearance values used for fitting should first be put on a compatible size basis, for example by standardizing clearance to a reference weight using the allometric model already introduced in the earlier material on standardizing clearance. The curve fitted here describes the remaining maturation pattern; it does not by itself distinguish maturation from size effects if those effects have not been handled appropriately.
Age and \(T_{50}\) must use the same units. The fraction \(F\) is dimensionless and lies between 0 and 1. At ages much smaller than \(T_{50}\), the fraction is low; at ages much greater than \(T_{50}\), it approaches 1. The parameter \(h\) controls how quickly the transition occurs around \(T_{50}\), but it does not change the fraction at \(T_{50}\).
Fitting on the Log-Clearance Scale
A common fitting approach assumes that clearance has multiplicative error, or equivalently that log clearance has additive error: $$ \log(CL_i)=\log(CL_{\mathrm{mature}})+\log F(a_i;T_{50},h)+\epsilon_i, \qquad \epsilon_i\sim N(0,\sigma^2). $$ For ordinary nonlinear least squares on the log scale, the fitted parameters minimize the sum of squared log residuals. This formulation presumes that the clearances are positive, the chosen size standardization is appropriate, and the log-scale residuals are reasonably represented by a common-variance error model. A population PK analysis may use a more complete likelihood, but the same maturation function can be part of that model.
The parameters are nonlinear because \(T_{50}\) and \(h\) occur inside the sigmoid. They are typically estimated by numerical optimization. The optimizer should find parameter values that minimize the specified objective; a set of plausible-looking predictions is not, by itself, evidence that the parameters are a fit to the data.
Worked Example: Fitting a Hypothetical Pediatric Dataset
Worked Example: Fitting a Hypothetical Pediatric Dataset
Suppose six hypothetical size-standardized clearances were recorded at ages 0.1, 0.2, 0.5, 1, 2, and 4 years. For this demonstration, the values are generated from a sigmoid with \(CL_{\mathrm{mature}}=6.00\) L/h, \(T_{50}=0.500\) years, and \(h=2.00\). They therefore provide a transparent example in which the fitted curve can be checked directly.
| Age \(a\) (years) | \(F(a)\) | Standardized clearance (L/h) |
|---|---|---|
| 0.1 | 0.03846 | 0.2308 |
| 0.2 | 0.1379 | 0.8276 |
| 0.5 | 0.5000 | 3.000 |
| 1 | 0.8000 | 4.800 |
| 2 | 0.9412 | 5.647 |
| 4 | 0.9846 | 5.908 |
For example, at age 0.2 years, $$ F(0.2)=\frac{0.2^2}{0.2^2+0.5^2} =\frac{0.0400}{0.2900}=0.1379, \qquad CL(0.2)=6.00\left(\frac{0.0400}{0.2900}\right)\approx0.8276\text{ L/h}. $$ The same calculation at each listed age reproduces the table. Because the data were generated from the stated curve, minimizing the log-scale sum of squared residuals gives the displayed parameter values, with zero residuals before rounding. This is an exact-fit demonstration, not a claim that real pediatric observations will lie perfectly on a sigmoid.
In an empirical dataset, one would numerically minimize $$ S=\sum_i\left[\log(CL_i)-\log(CL_{\mathrm{mature}})-\log F(a_i;T_{50},h)\right]^2. $$ Starting values, parameter constraints, convergence, and the residual pattern should be checked. With these exact generated values, the stated parameters yield \(S=0\) before rounding, the smallest possible value, so they are a minimizer.
Worked Example: Predicting Mature Fractions
Worked Example: Predicting Mature Fractions
Use the fitted values from the hypothetical dataset to calculate the fraction at age 0.25 years and the predicted standardized clearance at age 0.75 years.
At 0.25 years, $$ F(0.25)=\frac{0.25^2}{0.25^2+0.50^2} =\frac{0.0625}{0.3125}=0.2000. $$ The model assigns a mature fraction of 0.2000 at this age. This is a model-based fraction, not a statement that every individual child has exactly 20% of mature clearance.
At 0.75 years, $$ F(0.75)=\frac{0.75^2}{0.75^2+0.50^2} =\frac{0.5625}{0.8125}=0.6923, \qquad CL(0.75)=6.00(0.6923)=4.154\text{ L/h}. $$ Thus, under the fitted model and the size standardization used, predicted standardized clearance at 0.75 years is 4.154 L/h.
Parameter Uncertainty and Its Consequences
A point estimate does not show how strongly the data determine \(T_{50}\), \(h\), or mature clearance. Uncertainty can be summarized using a likelihood-based interval or a bootstrap that simulates or resamples data under a specified model and refits the curve. A bootstrap interval describes variation in estimates under that procedure and its assumptions; it is not a probability statement that a fixed parameter lies in the observed interval.
For the exact-fit illustration above, the residual sum of squares is zero, so estimating residual variance from those six observations would give no useful uncertainty estimate. To demonstrate uncertainty honestly, suppose an additional hypothetical analysis prespecifies a log-scale residual standard deviation of 0.08 based on external assay and model information. A parametric bootstrap can then simulate log-clearances at the six ages using this residual SD, refit each simulated dataset, and summarize the resulting parameter estimates. For illustration, suppose 2,000 successful refits give percentile intervals \(T_{50}=0.39\) to \(0.64\) years, \(h=1.55\) to \(2.55\), and \(CL_{\mathrm{mature}}=5.55\) to \(6.49\) L/h. These intervals are hypothetical bootstrap output, conditional on the assumed error model and the six ages; they are not calculated from the zero-residual fit alone.
Parameter uncertainty also affects predictions. For each bootstrap refit, calculate \(F(a)\) or \(CL(a)\) at the age of interest; percentiles of those predictions summarize uncertainty in the curve at that age. This is preferable to changing one parameter at a time when parameters are correlated, because bootstrap refits preserve the joint variation among parameter estimates. The prediction interval for an individual’s observed clearance would additionally need to account for residual and, where relevant, between-individual variability.
Worked Example: Interpreting a Bootstrap Interval
Suppose the hypothetical bootstrap interval for \(T_{50}\) is 0.39 to 0.64 years, with point estimate 0.500 years. The interval is about the age at which the modeled maturation fraction reaches one half; it is not an interval for an individual child’s age or clearance.
At the point estimate, \(F(0.500)=0.500\). Using the interval endpoints while holding \(h=2.00\) fixed gives $$ F(0.500;T_{50}=0.39,h=2) =\frac{0.25}{0.25+0.39^2}=0.6217, $$ and $$ F(0.500;T_{50}=0.64,h=2) =\frac{0.25}{0.25+0.64^2}=0.3790. $$ These calculations show the effect of varying \(T_{50}\) alone at this age. They are not a joint confidence interval for \(F(0.500)\), because \(h\) is held fixed and its uncertainty and covariance with \(T_{50}\) are omitted. A joint bootstrap prediction interval should instead be computed from the paired parameter estimates in each refit.
Diagnostics, Alternatives, and Limitations
Plot observed log-clearances and fitted values against age, and inspect residuals against age and fitted clearance. A systematic residual pattern—for example, consistently positive residuals at young ages and negative residuals at intermediate ages—suggests that the chosen sigmoid is not capturing the observed trend. Also check convergence from multiple starting values, boundary estimates, and whether the objective changes meaningfully near the reported optimum. Profile likelihoods or bootstrap distributions can reveal flat or skewed uncertainty that a local standard error may obscure.
The data need information across the transition to estimate both shape and timing. If all observed ages are much younger than \(T_{50}\), the data may mainly inform a combination of \(CL_{\mathrm{mature}}\), \(T_{50}\), and \(h\), rather than each parameter separately. If all ages are well beyond \(T_{50}\), fractions are near one and the curve provides little information about the transition. Sparse data, narrow age coverage, and age-related size patterns can therefore make parameters weakly identified even when the optimizer reports convergence.
A sigmoid is not the only option. A fixed maturation function may be appropriate when external evidence supports its parameters; a flexible spline can describe a less constrained age pattern but may need more data and careful regularization. A model with a different maturation function may be preferable if the sigmoid leaves systematic residual trends. The relevant comparison is not simply which curve fits best in-sample: biological plausibility, parameter identifiability, predictive performance, and the available age range all matter.
The curve approaches its asymptotes but does not literally reach zero or one at finite positive ages. Predictions far outside observed ages are consequently driven mainly by the assumed functional form. A good fit within the observed range does not validate such extrapolation. Moreover, the fitted curve is an average structural relationship on the chosen clearance scale; it does not establish a causal effect of age or account automatically for other developmental covariates.
Apply the stated size standardization and retain the age units and clearance units consistently.
Estimate \(CL_{\mathrm{mature}}\), \(T_{50}\), and \(h\) under a stated error model, checking convergence and residuals.
Use an appropriate interval or bootstrap, inspect parameter dependence, and avoid treating extrapolated predictions as well supported.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- What is the modeled mature fraction at age \(T_{50}\), regardless of the value of \(h\)?
- For \(T_{50}=0.5\) years and \(h=2\), calculate \(F(1)\).
- Why can data from only ages far above \(T_{50}\) provide little information about \(T_{50}\)?
- What does a bootstrap interval for \(T_{50}\) describe, and what does it not mean?
Solutions. (1) \(F(T_{50})=0.5\), since the numerator and the two terms in the denominator are equal. (2) \(F(1)=1/(1+0.5^2)=1/1.25=0.8000\). (3) At ages far above \(T_{50}\), \(F(a)\) is close to one, so changes in \(T_{50}\) have little effect on predicted clearance; the transition is poorly observed. (4) It summarizes the variability of estimates under the specified bootstrap procedure and model assumptions. It is not a probability statement that the fixed parameter lies in that particular interval.