Sampling Continuous Physiological Variability

Why the Distribution of a PBPK Input Matters

A virtual population needs more than a typical enzyme abundance or organ volume: it needs a plausible representation of how those values vary between subjects. Two distributions can have the same mean and standard deviation yet assign very different probabilities to unusually low or high values. When a PBPK output responds nonlinearly to an input, those differences can carry through to the simulated exposure distribution, particularly in its tails.

This tutorial concerns between-subject variability in continuous inputs such as enzyme abundance, body size, organ volume, or a physiological fraction. It does not treat variability as measurement error or as uncertainty about a population mean. Those are different sources of uncertainty and should not be combined without a clear model. The earlier material on building a virtual population with pharmacogenotypes and sampling discrete genotypes supplies the context for combining continuous inputs with genotype-specific parameters.

The distribution is a modeling choice informed by physiological plausibility and data. A normal distribution allows any real value; a lognormal or gamma distribution is positive; a beta distribution is confined to the interval from zero to one. The choice should follow the input’s support and observed shape, rather than convenience alone.

Common Distributions for Continuous Inputs

A normal model \(X\sim N(\mu,\sigma^2)\) has mean \(\mu\) and standard deviation \(\sigma\). It is symmetric around its mean, but its support is the entire real line. This can be appropriate for quantities whose observed values are approximately symmetric and whose negative values are possible or negligible for the intended use. It is often inappropriate for a strictly positive quantity if the model assigns meaningful probability to values below zero.

A lognormal model is defined by \(\ln X\sim N(\mu_L,\sigma_L^2)\), so \(X\) is strictly positive. Its median is \(\exp(\mu_L)\), and its mean and coefficient of variation (CV) are:

If the desired arithmetic mean is \(m\) and the desired CV is \(c\), the log-scale parameters are \(\sigma_L^2=\ln(1+c^2)\) and \(\mu_L=\ln(m)-\sigma_L^2/2\). Thus, a stated mean and CV can determine a lognormal distribution, but they do not imply that this distribution is physiologically correct.

A gamma distribution with shape \(k>0\) and scale \(\theta>0\) is another positive model. Its mean is \(k\theta\), and its variance is \(k\theta^2\). Given mean \(m\) and CV \(c\), \(k=1/c^2\) and \(\theta=m/k\). The gamma shape affects skewness and tail behavior, so it need not produce the same simulated extremes as a lognormal with matching mean and CV.

For a continuous proportion or fraction, a beta distribution is often a useful bounded model. If \(X\sim\operatorname{Beta}(a,b)\), with \(a,b>0\), then \(0<X<1\), \(E(X)=a/(a+b)\), and \(\operatorname{Var}(X)=ab/[(a+b)^2(a+b+1)]\). A beta distribution can represent a range of shapes, but the parameters should be chosen to reflect the observed or justified variation.

From Input Distributions to Exposure Tails

A PBPK model maps inputs to outputs. Even when the mapping is simple enough to write as a formula, the output distribution generally does not have the same shape as the input distribution. For example, in a deliberately simplified model with a fixed dose and all other determinants held constant, suppose an exposure metric \(E\) is inversely proportional to enzyme abundance \(A\): \(E=1/A\), in normalized units. The inverse function is decreasing. Low enzyme abundance therefore maps to high exposure, and high enzyme abundance maps to low exposure.

For a continuous \(A>0\), the quantile transformation follows directly from this monotonicity. Write \(Q_A(p)\) for the \(p\)th quantile of \(A\). Then:

The final equality also requires the CDF of \(A\) to be strictly increasing for positive arguments (as for the lognormal and gamma distributions here); continuity alone is not sufficient under the generalized-inverse quantile definition. It shows why the upper exposure tail in this example is controlled by the lower enzyme-abundance tail. In a full PBPK model, the input-output relationship may not be exactly reciprocal or monotonic across the whole range. Exposure tails should then be evaluated by simulating the PBPK model, not by applying this simplified equation as though it were universal.

Equal Means and CVs Do Not Ensure Equal Tails

A common comparison holds the mean and CV fixed while changing the distribution family. This isolates one consequence of distributional shape, although it does not establish which family is more appropriate. The next example compares gamma and lognormal inputs under the same simplified inverse relationship. Both distributions have positive support; both have mean \(1.00\) relative unit and CV \(0.500\).

Bounded Inputs and Support Checks

Some physiological covariates are fractions, proportions, or normalized quantities that cannot fall below zero or exceed one. A normal distribution may generate impossible values for such inputs, even if its mean and standard deviation appear reasonable. A beta distribution respects the bounds by construction, although a beta model can still be a poor description of the actual data.

Diagnostics, Alternatives, and Limitations

Before drawing a virtual cohort, check that the distribution matches the input’s support and units, and that its parameters reproduce the intended moments or quantiles. After sampling, inspect the simulated values: look for impossible values, unexpected concentrations near a boundary, or a tail that is inconsistent with the evidence used to specify the model. Then propagate the samples through the PBPK model and examine exposure quantiles as well as central summaries. A plausible input distribution does not guarantee plausible exposure outputs.

Distributional alternatives answer different modeling needs. A normal distribution is symmetric and easy to parameterize, but can generate nonpositive values for a positive input. A lognormal distribution guarantees positivity and often represents right-skewed measurements, but its tail may be too large or too small for a particular population. A gamma distribution also guarantees positivity and offers a different range of shapes. A beta distribution respects unit-interval bounds, but does not apply to an unbounded quantity without a justified transformation. Truncating a normal distribution can enforce a bound, but changes its moments; one should not assume the truncated variable retains the original normal mean and standard deviation.

When adequate individual-level data are available, useful checks include comparing empirical and fitted quantiles, inspecting the lower and upper tails, and assessing whether a chosen family reproduces the observed support and skewness. Sparse data may not identify extreme quantiles reliably. In that case, tail results should be presented as conditional on the assumed distribution and examined under defensible alternatives. Sampling a marginal distribution also says nothing by itself about how that covariate varies jointly with other subject characteristics; realistic relationships among covariates require additional modeling.

A frequent mistake is to conclude that matching a mean and CV makes two distributions interchangeable. It does not: the worked comparison showed different 95th exposure percentiles under the same simplified exposure mapping. Another mistake is to treat every high simulated exposure as evidence of a coding error. An extreme value may be a consequence of a plausible low input, but it may also signal a misspecified distribution, an unsupported tail, or an incorrect PBPK response. Trace unusual outputs back to their inputs and model calculations before interpreting them.