Gestational Trajectories for Maternal PBPK Parameters

Why Represent Maternal Parameters as Trajectories?

A pregnancy PBPK model needs physiological inputs that can change as gestation progresses. A single value for plasma volume, body composition, cardiac output, or glomerular filtration rate (GFR) cannot represent that time dependence. A trajectory makes the change explicit: it maps gestational time to a parameter value that can be supplied to the model at each simulation time.

The trajectory is not a claim that every pregnant person follows the same curve. It is a model representation of a chosen population pattern or individual profile. Its usefulness depends on the population, measurements, endpoint values, and assumptions behind it. The numbers below are deliberately hypothetical and illustrate the calculations; they are not clinical reference estimates.

For any trajectory, state the reference value, the domain, units, and how the function was selected. The examples use total plasma volume in liters, maternal fat-free and fat mass in kilograms, cardiac output in liters per minute, and GFR in milliliters per minute. In a PBPK model, these are not interchangeable compartments: plasma volume is contained within broader body water and body-composition descriptions, so it must not be added a second time when those compartments are constructed.

A Smooth Endpoint-Based Trajectory

One simple continuous representation begins with a baseline value and a fractional change at the end of gestation. Define the normalized time \(x=g/40\), and use the smooth increasing function \(f(x)=2x-x^2\) on \(0\leq x\leq1\). It satisfies \(f(0)=0\), \(f(1)=1\), and \(f'(x)=2-2x\geq0\) throughout the interval. Thus, it rises monotonically, with a steeper slope early and a flattening slope near the endpoint.

Here \(Y_0\) is the pre-pregnancy reference value and \(r\) is the proportional change at \(g=40\). For example, \(r=0.45\) represents a 45% increase relative to \(Y_0\) at the endpoint. The formula is a convenient interpolation assumption, not a physiological law. In particular, it assumes the same normalized shape for whichever parameters are assigned this curve. Independent data may support different shapes or different endpoint changes.

Piecewise Trajectories From Gestational Knots

A piecewise trajectory can be preferable when measurements or a physiological model provide values at selected gestational ages but do not justify a single smooth curve. Between adjacent knots \((g_i,Y_i)\) and \((g_{i+1},Y_{i+1})\), linear interpolation is

This construction reproduces every knot exactly and is continuous when adjoining pieces share the same value at their common knot. Its slope is constant within each interval and may change abruptly at a knot. A continuous value therefore does not necessarily have a continuous rate of change.

Rates of Change and Their Interpretation

A trajectory determines not only the parameter value but also its rate of change. For the smooth endpoint-based curve, differentiation with respect to gestational time gives

This derivative has units of \(Y\) per week. It describes the local rate implied by the mathematical representation; it does not prove that the underlying physiology changes at exactly that rate. For a piecewise linear curve, the corresponding rate within an interval is \((Y_{i+1}-Y_i)/(g_{i+1}-g_i)\), and a single derivative is not defined at a knot where the slopes differ.

Choosing, Checking, and Using a Trajectory

A constant value is the simplest alternative: it assumes no gestational change and can serve as a reference simulation, but it cannot represent a changing physiological input. A linear trajectory from baseline to endpoint assumes a constant absolute rate. The smooth function above permits a changing rate while using only a baseline and endpoint. A piecewise function can represent observed knots or different interval-specific rates, but requires enough information to support each segment. These alternatives should be compared as modeling assumptions, not treated as interchangeable calculations.

The parameter trajectories can alter the physiological state supplied to the PBPK model at early, middle, and late gestation. Changes in plasma volume and body composition affect the modeled distribution spaces; cardiac output is a total systemic input; and GFR is a renal physiological input. This tutorial specifies time-varying inputs, not their complete downstream consequences for drug clearance, regional blood-flow allocation, placental exchange, or exposure. Those require additional model structure and evidence.

Before use, check that all functions start at the stated reference, reach the intended endpoint, preserve units, and behave sensibly over the modeled domain. Compare predicted values with the measurements or physiological constraints used to justify the profile. Inspect slopes as well as values: an apparently plausible endpoint can conceal an implausible abrupt rate change. If a trajectory is fitted or selected from sparse data, uncertainty in its endpoints, knot values, and shape should be considered rather than presenting the curve as exact.

The introductory material on pregnancy physiology provides the broad context; trajectory construction adds a time-indexed mathematical representation. Its main limitation is that a simple curve can hide individual variation and differences among populations. A shared normalized shape across several parameters is especially restrictive. Where data support distinct shapes, parameter-specific trajectories are more defensible than forcing all inputs to change in lockstep.