Selecting Marginal Distributions for PBPK Inputs

Why Marginal Distribution Choice Matters

A PBPK model may require uncertain inputs such as systemic clearance, fraction unbound, or tissue-to-plasma partition coefficients. A marginal distribution describes the uncertainty assigned to one such input considered individually. The choice affects which values are plausible and how much probability is placed near the center, boundaries, and tails. Those features matter because PBPK predictions can respond nonlinearly to inputs, and a small amount of probability in an extreme region may influence a prediction summary.

The earlier material on mapping input uncertainty to PBPK prediction distributions establishes why uncertain inputs are represented probabilistically. Here the task is more specific: choose a defensible distribution for each input and express its parameters in interpretable terms. No distribution is correct solely because it is familiar, and a physical range alone does not determine the shape within that range.

Begin With the Input and Its Evidence

Before selecting a family, establish what the input represents, its units, its physical domain, and the source of uncertainty. A clearance is positive and has units of volume per time. Fraction unbound is a proportion between zero and one. A tissue partition coefficient is usually positive and dimensionless. These constraints narrow the available choices, but do not by themselves establish whether the distribution should be symmetric, skewed, or concentrated near a boundary.

Separate the uncertainty being represented from biological variability or study-to-study heterogeneity. A distribution intended to represent uncertainty about a population parameter need not be the distribution of individual patient values. The target population, experimental conditions, and parameter definition should remain consistent with the PBPK scenario. If the evidence supplies only a point estimate, the analyst must justify any uncertainty scale added around it rather than presenting that scale as measured fact.

A useful selection sequence is to record the domain, summarize available evidence on central tendency and spread, inspect asymmetry or boundary accumulation, then assess candidate distributions against those facts. When data are available, plots and quantiles can reveal features that a mean and standard deviation conceal. When evidence is sparse, a transparent, interpretable approximation is preferable to a highly flexible distribution whose extra parameters cannot be supported.

Positive Inputs: Lognormal and Gamma Options

A lognormal distribution is a common candidate for a positive input with right skew. By definition, \(X\) is lognormal when \(\ln X\) is normally distributed: \(\ln X\sim N(\mu_{\ln},\sigma_{\ln}^2)\). Here \(\mu_{\ln}\) and \(\sigma_{\ln}\) describe the natural logarithm of the input, not its arithmetic mean and standard deviation. The support is \([0,\infty)\); the distribution assigns probability one to positive values and no probability mass to zero.

Its median is \(\exp(\mu_{\ln})\), and its arithmetic mean is \(\exp(\mu_{\ln}+\sigma_{\ln}^2/2)\). If a median \(m\) and geometric standard deviation \(g\) are supplied, then \(\mu_{\ln}=\ln(m)\) and \(\sigma_{\ln}=\ln(g)\). The geometric standard deviation describes multiplicative spread: under this model, the interval from \(m/g\) to \(mg\) is one standard deviation on the log scale, not an arithmetic interval around the mean.

A gamma distribution is another positive-support option. Using shape \(k\) and rate \(\lambda\), its mean is \(k/\lambda\) and its variance is \(k/\lambda^2\). Consequently, given a positive arithmetic mean \(m\) and coefficient of variation \(c\), the parameters are \(k=1/c^2\) and \(\lambda=k/m\). The rate has units reciprocal to the input; the shape is dimensionless. A gamma distribution can represent a range of right-skewed shapes, but it has a different tail form from the lognormal.

Bounded Inputs: Beta Distribution for a Fraction

For a fraction that is strictly between zero and one, the beta distribution is a flexible candidate. With shape parameters \(a>0\) and \(b>0\), its mean and variance are \(\operatorname{E}(X)=a/(a+b)\) and \(\operatorname{Var}(X)=ab/[(a+b)^2(a+b+1)]\). Its standard support is \([0,1]\), and it assigns probability one to \((0,1)\), with no point mass at either endpoint. Thus it can approach zero or one, but cannot represent a positive probability of an exact endpoint value.

Given a target mean \(m\) and variance \(v\), both with \(0<m<1\) and \(0<v<m(1-m)\), define \(q=m(1-m)/v-1\). Then \(a=mq\) and \(b=(1-m)q\). This follows by setting \(a+b=q\), using \(m=a/(a+b)\), and substituting into the beta variance formula. The condition on \(v\) ensures \(q>0\), so both shape parameters are positive.

If exact values of zero or one are scientifically possible with positive probability, a beta distribution alone is insufficient. A model with explicit endpoint probability may be needed, depending on the meaning of those values. Do not infer an endpoint probability just because measurements rounded to zero or one; rounding, censoring, and a true boundary value are distinct explanations. A beta distribution also need not be unimodal: depending on its shapes, it can be concentrated near a boundary or have other forms. Inspect the implied distribution rather than assuming that “beta” means a single generic shape.

Approximately Symmetric Inputs and the Normal Approximation

A normal distribution may be reasonable for an approximately symmetric input when its support does not conflict materially with the input’s physical domain. It is defined by \(X\sim N(\mu,\sigma^2)\), with mean \(\mu\), variance \(\sigma^2\), and support over the entire real line. For a positive parameter, this means a normal model always assigns some probability to negative values. It is defensible only when that probability is negligible for the intended use and the observed or elicited shape is approximately symmetric.

Diagnostics, Alternatives, and Limitations

A distribution should be checked against both scientific constraints and evidence. Useful diagnostics include comparing its median and mean with the supplied summaries, plotting its density or cumulative distribution, checking central and tail quantiles against observed or elicited ranges, and calculating the probability it assigns to implausible regions. For a positive input, examine how much mass lies near zero and in the upper tail. For a fraction, inspect concentration near zero or one and whether either exact endpoint is actually possible.

The main alternatives are not interchangeable. A lognormal and gamma both enforce positivity but can imply different shapes and tail probabilities. A beta distribution respects finite bounds but requires a separate treatment of genuine endpoint masses. A normal distribution is simple and often interpretable for near-symmetric inputs, but its unbounded support can violate physical constraints. Truncating a normal distribution changes its distribution and generally changes its mean and variance; parameters from the untruncated model should not automatically be reported as the moments of the truncated one.

When the evidence is insufficient to choose between plausible families, document the uncertainty and compare the resulting PBPK predictions under alternatives. This is a model-assumption check, not proof that one family is true. A marginal distribution also says nothing about dependence between inputs; coherent marginal choices do not guarantee a coherent joint distribution. Finally, a marginal fitted to one population or experimental condition may not transfer to another without justification.

A Compact Selection Checklist

• State the input definition, units, scenario, and physical domain.
• Use a positive-support family for positive quantities unless a normal approximation has negligible invalid mass and is otherwise justified.
• Use a bounded family for an interior fraction, and model true endpoint masses explicitly if they are present.
• Translate the available summaries into distribution parameters and verify the implied moments or quantiles numerically.
• Check boundary behavior and tails, not only the mean and standard deviation.
• Record assumptions and compare plausible alternatives when evidence is weak.