Lognormal Distributions for Positive PBPK Parameters

Why Use a Lognormal Distribution?

Clearance, partition coefficients, and many other PBPK inputs must be positive. Their uncertainty is often more naturally described in relative terms than in absolute terms: an estimate might plausibly be 20% higher or lower, rather than a fixed number of units higher or lower. A lognormal distribution represents this kind of multiplicative variation while ensuring that simulated values remain positive.

The distribution is not automatically appropriate just because a parameter is positive. It is a modeling choice about the shape of uncertainty or variability. To use it well, distinguish the original parameter scale from the scale of its logarithm, and make sure any reported summary is interpreted on the scale on which it was calculated.

Definition: A positive quantity \(X\) has a lognormal distribution if its natural logarithm is normally distributed. Write \(\ln X \sim N(\mu_L,\sigma_L^2)\), or equivalently \(X\sim\operatorname{Lognormal}(\mu_L,\sigma_L^2)\). Here \(\mu_L\) is the mean of \(\ln X\), and \(\sigma_L\) is the standard deviation of \(\ln X\). The distribution of \(X\) is positive and typically right-skewed.

If \(X\) is a clearance, first state the units in which its numerical values are expressed, such as L/h. The logarithm in the formulas below is taken of the numerical value in those stated units. Changing the units shifts \(\mu_L\), so log-scale parameters must always be reported with the original parameter's units and convention made clear. The standard deviation \(\sigma_L\), being a scale of relative variation, does not change under a simple change of units.

Log-Scale Parameters and Arithmetic-Scale Summaries

Let \(\ln X\sim N(\mu_L,\sigma_L^2)\). The median of \(X\) is \(\exp(\mu_L)\), and its geometric standard deviation (GSD) is \(\exp(\sigma_L)\). A GSD describes multiplicative spread: values one log-scale standard deviation above and below the median are \(\exp(\mu_L)\exp(\sigma_L)\) and \(\exp(\mu_L)/\exp(\sigma_L)\), respectively.

The arithmetic mean and standard deviation are different summaries. For a lognormal quantity, they are

E[X]=\exp\left(\mu_L+\frac{\sigma_L^2}{2}\right), \qquad \operatorname{SD}(X)= \exp\left(\mu_L+\frac{\sigma_L^2}{2}\right) \sqrt{\exp(\sigma_L^2)-1}.

The arithmetic mean exceeds the median whenever \(\sigma_L>0\), because the right tail pulls the mean upward. The coefficient of variation, defined as the standard deviation divided by the mean, is \(\sqrt{\exp(\sigma_L^2)-1}\). These formulas describe the model distribution; they do not by themselves say whether its uncertainty represents between-person variability, measurement error, or knowledge uncertainty.

A percentile is also straightforward to calculate. If \(z_q\) is the \(q\)-quantile of a standard normal distribution, then the \(q\)-quantile of \(X\) is \(\exp(\mu_L+z_q\sigma_L)\). Thus the central 95% interval under the model is obtained using \(z_{0.975}\approx1.96\). It is an interval containing 95% of the model's probability, not a confidence interval for an unknown parameter based on a sampling procedure.

Converting Arithmetic Mean and Standard Deviation

Suppose the available summaries are the arithmetic mean \(m_A\) and standard deviation \(s_A\), both on the original positive scale. Define the coefficient of variation \(CV=s_A/m_A\). Matching the lognormal mean and variance to those summaries gives

\sigma_L^2=\ln(1+CV^2), \qquad \mu_L=\ln(m_A)-\frac{\sigma_L^2}{2}.

These equations are useful when assay or literature summaries report an arithmetic mean and standard deviation. Their use assumes \(m_A>0\), \(s_A\geq0\), and that a lognormal approximation is appropriate for the underlying positive quantity. If \(s_A=0\), the distribution is degenerate at \(m_A\); it has no log-scale spread.

Conversion result: Given a positive arithmetic mean \(m_A\) and positive arithmetic standard deviation \(s_A\), calculate \(CV=s_A/m_A\), then use \(\sigma_L=\sqrt{\ln(1+CV^2)}\) and \(\mu_L=\ln(m_A)-\sigma_L^2/2\). The resulting lognormal distribution matches the specified arithmetic mean and standard deviation.

Worked Example: Converting a Clearance Mean and Standard Deviation

Suppose a hypothetical source reports a clearance with arithmetic mean 3.00 L/h and arithmetic standard deviation 1.50 L/h. Construct a lognormal distribution and find its median and central 95% interval.

First, the coefficient of variation is \(CV=1.50/3.00=0.5000\). Therefore,

\sigma_L^2=\ln(1+0.5000^2)=\ln(1.25)=0.22314, \qquad \sigma_L=\sqrt{0.22314}=0.47238.

The log-scale mean is

\mu_L=\ln(3.00)-\frac{0.22314}{2} =1.09861-0.11157=0.98704.

The median is \(\exp(0.98704)=2.683\) L/h. For the central 95% interval, calculate the two log-scale endpoints and exponentiate:

\left[ \exp(0.98704-1.96(0.47238)), \exp(0.98704+1.96(0.47238)) \right] \approx[1.063,\ 6.773]\ \mathrm{L/h}.

The interval is asymmetric around the median on the arithmetic scale, as expected for a lognormal distribution. By construction, the model's arithmetic mean and standard deviation are 3.00 L/h and 1.50 L/h; the median is lower than the mean because of right skew.

Converting Geometric Summaries and Using Percentiles

Some sources report the geometric mean and GSD. In a lognormal model, these directly specify \(\mu_L\) and \(\sigma_L\): the geometric mean is \(\exp(\mu_L)\), while the GSD is \(\exp(\sigma_L)\). Consequently, \(\mu_L\) is the logarithm of the geometric mean, and \(\sigma_L\) is the logarithm of the GSD. A reported percentile can provide another route to these parameters if a median or other percentile is also available.

Worked Example: Converting a Geometric Mean and GSD

Suppose a hypothetical tissue partition coefficient has geometric mean 4.00 and GSD 2.00. The coefficient is dimensionless. Find its log-scale parameters and arithmetic mean.

The log-scale parameters are

\mu_L=\ln(4.00)=1.38629, \qquad \sigma_L=\ln(2.00)=0.69315.

The arithmetic mean is

E[X]=\exp\left(\ln(4.00)+\frac{[\ln(2.00)]^2}{2}\right) =4.00\exp(0.24023) \approx5.086.

For context, the median remains 4.00. The arithmetic mean is higher because the distribution has a long upper tail. The arithmetic-scale coefficient of variation is \(\sqrt{\exp(0.69315^2)-1}\approx0.7854\). Multiplying this by the mean gives an arithmetic standard deviation of approximately \(5.086(0.7854)=3.995\). The large standard deviation is consistent with the broad multiplicative spread implied by a GSD of 2.

Worked Example: Inferring Spread from a Median and Percentile

Suppose a hypothetical clearance is summarized by a median of 2.00 L/h and a 95th percentile of 4.00 L/h. If the quantity is lognormal, estimate its log-scale parameters and arithmetic mean.

For a lognormal quantity, the median is \(\exp(\mu_L)\), so \(\mu_L=\ln(2.00)=0.69315\). The 95th percentile is \(\exp(\mu_L+z_{0.95}\sigma_L)\), where \(z_{0.95}\approx1.64485\). Solving for the standard deviation gives

\sigma_L= \frac{\ln(4.00)-\ln(2.00)}{1.64485} =\frac{\ln(2)}{1.64485} \approx0.42140.

The corresponding GSD is \(\exp(0.42140)\approx1.524\). The arithmetic mean is

E[X]=\exp\left(0.69315+\frac{0.42140^2}{2}\right) \approx2.186\ \mathrm{L/h}.

This result follows only under the lognormal assumption. The 95th percentile is not enough by itself to identify both \(\mu_L\) and \(\sigma_L\); here the median supplied the second summary. The median and percentile must also refer to the same target population or uncertainty distribution.

Justification of the Conversion Formulas

The conversions follow from the moments of a normal variable. Set \(Y=\ln X\), with \(Y\sim N(\mu_L,\sigma_L^2)\), so \(X=\exp(Y)\). The moment-generating function of \(Y\) is \(E[\exp(tY)]=\exp(t\mu_L+t^2\sigma_L^2/2)\). Taking \(t=1\) and \(t=2\) yields the first two moments of \(X\):

E[X]=\exp(\mu_L+\sigma_L^2/2), \qquad E[X^2]=\exp(2\mu_L+2\sigma_L^2).

Subtracting the square of the mean from the second moment gives the variance:

\operatorname{Var}(X) =E[X^2]-E[X]^2 =\exp(2\mu_L+\sigma_L^2)\bigl(\exp(\sigma_L^2)-1\bigr).

To invert the moments, divide the variance by the squared mean. This gives \(CV^2=\exp(\sigma_L^2)-1\), hence \(\sigma_L^2=\ln(1+CV^2)\). Substituting into the mean equation and taking logarithms gives \(\mu_L=\ln(m_A)-\sigma_L^2/2\). This establishes the conversion formulas rather than treating them as separate rules.

Model Checks, Alternatives, and Limitations

The lognormal model is most defensible when the parameter is strictly positive and its variation is plausibly multiplicative, or when the log-transformed observations are reasonably consistent with a single normal distribution. For observed data, useful checks include plotting the log values, inspecting a histogram or normal quantile plot on the log scale, and checking for clusters or unusually influential high values. With only summary statistics, these shape checks cannot be performed; the distribution should then be described as an approximation rather than as empirically confirmed.

A lognormal distribution has no probability mass at zero and has an unbounded upper tail. It can therefore be unsuitable when zeros are genuine possible values, when the parameter has a known finite upper limit, or when data show separate subgroups or pronounced tail behavior inconsistent with the model. A very large arithmetic mean relative to the median, or an arithmetic standard deviation that is highly sensitive to a few large observations, is a warning to examine the scale and data more carefully.

Representation	Useful when	Main caution
Lognormal	The parameter is positive and multiplicative variation is plausible.	Cannot represent zero; permits arbitrarily large values.
Normal on the original scale	Variation is approximately additive and negative values are not problematic for the modeled quantity.	Assigns probability to negative values, which can be impossible for clearances and partition coefficients.
Empirical distribution	There are enough relevant observations to represent their observed shape directly.	May not adequately represent unobserved tail values or a different target population.

A normal distribution on the original scale may be a poor choice for a strictly positive parameter because it can assign probability to impossible negative values. An empirical distribution avoids imposing a lognormal shape, but it depends on having suitable data and does not resolve uncertainty beyond the observed values. Other positive distributions can also be considered; selecting among them requires evidence and a clear account of the parameter's bounds and data. A lognormal model should not be used to disguise uncertainty about those features.

In PBPK work, an additional diagnostic is whether the implied values remain physiologically and physicochemically plausible over the distribution's relevant range. A mathematically valid positive draw is not automatically a plausible parameter value. If the tails imply implausible values, report that limitation and reconsider the distributional representation rather than interpreting the tails as established biological possibilities.

Common Mistakes

Calling \(\mu_L\) the arithmetic mean. It is the mean of the logarithm; the arithmetic mean is \(\exp(\mu_L+\sigma_L^2/2)\).
Using the arithmetic mean as the median. For nonzero log-scale spread, the median is \(\exp(\mu_L)\), generally below the arithmetic mean.
Confusing standard deviation with GSD. The log-scale standard deviation is \(\sigma_L\); the geometric standard deviation is \(\exp(\sigma_L)\).
Reporting a percentile interval as a confidence interval. A model-based central 95% interval contains 95% of the modeled distribution; it does not express a 95% probability that a fixed parameter lies in an interval.
Ignoring units or the uncertainty target. State the parameter units and whether the distribution represents individual variability or uncertainty in a parameter estimate.

Key takeaway: Define the lognormal distribution through \(\ln X\), not by confusing its log-scale parameters with original-scale summaries. Convert reported summaries explicitly, check the implied range for plausibility, and state what kind of uncertainty the distribution is intended to represent.

Check Your Understanding

Work through each question, then compare your reasoning with the solutions below.

A positive parameter has \(\mu_L=0.50\) and \(\sigma_L=0.20\). What are its median and GSD?
A hypothetical parameter has arithmetic mean 10 units and arithmetic standard deviation 5 units. Find \(\sigma_L^2\) and \(\mu_L\).
Why can a lognormal central 95% interval be asymmetric around its median on the original scale?
A parameter can genuinely equal zero. Is a lognormal distribution suitable without modification? Explain.

Solutions. (1) The median is \(\exp(0.50)\approx1.649\) in the parameter's stated units, and the GSD is \(\exp(0.20)\approx1.221\). (2) \(CV=5/10=0.5000\), so \(\sigma_L^2=\ln(1.25)=0.2231\), and \(\mu_L=\ln(10)-0.2231/2\approx2.191\). (3) The log-scale interval is symmetric around \(\mu_L\), but exponentiation is nonlinear: equal additive distances on the log scale become unequal additive distances on the original scale. (4) No. A lognormal variable is strictly positive and assigns zero probability to exactly zero, so a model allowing genuine zeros needs a representation that includes them.