Representing Pediatric Growth in Whole-Body PBPK Models

Why Growth Representation Matters

A pediatric whole-body PBPK model needs physiological parameters that reflect the child being simulated. Organ volumes and blood flows change as children grow, but a model must also specify how those values change during the simulation. A model can assign one set of parameters to an age band, evaluate a continuous growth curve, or use information about an individual child’s body size. These choices can produce different predicted concentrations even when the drug and the underlying PBPK structure are unchanged.

The goal is not to make every physiological parameter vary continuously merely for its own sake. The representation should match the available data, the intended prediction, and the timescale of the simulation. A fixed parameter set may be adequate for a short study in a narrowly defined age group. A continuous trajectory is useful when a simulation crosses a meaningful growth interval. Individual inputs can represent a particular child more closely, provided the necessary measurements are available and their use is justified.

This tutorial focuses on organ volumes and blood flows. It does not replace the physiological scaling relationships themselves: it describes how to implement a chosen relationship over age or body size and how to check that the implementation behaves coherently.

Three Ways to Represent Growth

Let \(a\) denote age, \(W\) body weight, \(V_i\) the volume of organ \(i\), and \(Q_i\) the blood flow to that organ. In any of the three approaches, volumes must have units of volume, such as litres, and flows must have units of volume per time, such as litres per minute. A parameter is not made physiologically meaningful just by labelling it “pediatric”; its value and its relationship to age or size must be specified.

In an age-band model, the parameter is constant within a band and changes when the subject crosses a band boundary. In a continuous model, one evaluates a function such as \(V_i(a)\) or \(Q_i(a)\) at the current age. In an individual-size model, one might use the child’s measured organ volume directly, or apply an explicitly stated size relationship to that child’s weight or other measured characteristics. Individualization is a choice of inputs, not a guarantee that every physiological quantity has been measured.

Implementing the Three Representations

For a fixed-band implementation, define the age intervals before simulation and assign each interval its parameter values. The intervals should cover the intended ages without gaps or overlaps, and the rule at each boundary must be explicit. For instance, defining bands as 2 to less than 4 years and 4 to less than 6 years assigns a child aged exactly 4 years to one band unambiguously. A model that changes parameters abruptly at that boundary may be reasonable for a simulation using separate age-specific cohorts, but that abrupt transition is a feature of the representation, not evidence that the child’s organ suddenly grew at that instant.

For a continuous representation, specify the function and its domain. A simple function can be convenient, but it should not be extrapolated outside the age or size range for which it is defensible. Interpolation between measured ages should avoid implausible jumps or reversals. A smooth mathematical curve is not automatically a valid physiological curve: it should be checked against plausible organ volumes and flows over the full simulated interval.

For an individual-size representation, state which inputs are child-specific and how each is obtained. If weight is used to scale an organ volume, specify the reference value and the scaling relationship. If a measured organ volume is used directly, do not also apply a second size adjustment to that same volume unless the model explicitly calls for it. Individual inputs can reduce mismatch between a child and an age-average child, but measurement error and missing measurements remain relevant.

Blood flows require a further consistency check. If organ flows are defined as fractions of cardiac output, the fractions for the included flow destinations should sum to one, subject to any explicitly modelled shunts or omitted pathways. When cardiac output changes with age or body size, the flows allocated from it should change consistently too. Assigning each organ a separate growth curve without checking the whole-body total can create a set of organ flows that exceeds, or falls short of, the model’s cardiac output.

Here \(Q_{\mathrm{CO}}\) is cardiac output and \(f_i\) is the fraction allocated to organ \(i\). This allocation is one modelling choice; an alternative is to specify organ flows directly and check their sum against the chosen cardiac output. In either case, flows used in the organ equations must be internally consistent.

Changing Volume and Drug-Mass Balance

When organ volume changes during a simulation, the concentration equation must remain consistent with drug-mass balance. Let \(A_i\) be the amount of drug in organ \(i\), and let \(C_i=A_i/V_i\) be its concentration. If net drug input to that organ is \(R_i\) amount per time, then the amount balance is \(dA_i/dt=R_i\). Since \(A_i=V_iC_i\), the product rule gives the concentration equation below.

Rearranging shows why simply substituting a changing volume into a fixed-volume concentration equation can be wrong:

The second term accounts for the relationship between amount, concentration, and changing volume. It does not create or remove drug amount: the amount equation remains \(dA_i/dt=R_i\). For a well-stirred tissue with blood flow \(Q_i\), arterial concentration \(C_{\mathrm{art}}\), tissue-to-blood partition coefficient \(K_i\), and intrinsic elimination clearance \(CL_i\), one possible net-input specification is \(R_i=Q_i(C_{\mathrm{art}}-C_i/K_i)-CL_iC_i\). The selected organ model determines the appropriate terms in \(R_i\); the product-rule correction applies whenever concentration is calculated from amount divided by a changing volume.

Worked Representations

Choosing an Approach and Checking It

The approaches are not interchangeable in every application. Age bands can be stable and easy to audit, but may poorly represent a child near a boundary. Continuous functions avoid artificial band jumps, but their smoothness cannot compensate for weak data or an unjustified function. Individual inputs can represent between-child differences, but require suitable measurements or a justified rule for deriving physiology from size. A model may also combine approaches—for example, individual weight with an age-dependent organ-flow fraction—provided each parameter’s source and dependencies are explicit.

Useful checks include plotting each \(V_i(a)\) and \(Q_i(a)\) over the full simulated range, checking that volumes and flows stay positive, and confirming that units remain consistent. Inspect values immediately before and after every age-band boundary. For flow allocation, calculate \(\sum_i Q_i\) and compare it with \(Q_{\mathrm{CO}}\), or verify that the fractions sum to one where that is the model definition. For continuous or individual-size models, check that values are plausible at the smallest and largest simulated sizes, not just at a reference age.

A final check is numerical and conceptual: if a volume changes with time, confirm that the implementation updates the amount balance and derives concentration consistently. A sudden concentration change at an age boundary may arise from an abrupt parameter switch, a numerical implementation issue, or both. Inspecting parameter trajectories alongside drug amount and concentration helps distinguish these possibilities. No one representation eliminates uncertainty in the physiological inputs; the representation only makes the assumed pattern of change explicit.

Common Pitfall