Uncertainty in Reference Physiology Across Populations

Why Reference Physiology Depends on the Population

A PBPK model may use organ volumes, blood flows, and tissue composition as if each were a single known number. In practice, these values are estimated from people sampled under particular conditions: an age range, a body-size distribution, a health status, and a measurement protocol. A value reported as a “reference” is therefore a summary of a defined population, not a universal constant. Its suitability depends on how well that population matches the people the model is intended to represent.

Two distinct questions follow. First, how uncertain is the estimated population summary because only a sample was measured? Second, how much do the physiological values vary among individuals in the target population? Those quantities are not interchangeable. A precisely estimated mean can coexist with substantial individual variability, and a large individual standard deviation does not by itself mean that the population mean is poorly known.

Definition: A population-specific reference value is a physiological summary, such as a mean organ volume, mean blood flow, or mean tissue fraction, defined for a stated target population and measurement context. Uncertainty in the summary describes uncertainty about that population quantity, often because it was estimated from a sample. Interindividual variability describes differences in physiological values among members of the population.

The target population should be described in terms relevant to the model input. Age, body size, sex, health status, and other measured characteristics can affect reference physiology. A broad label alone is not a sufficient explanation for a physiological difference, and population averages should not be used to assign a particular individual’s physiology without an appropriate basis. The distinctions between measurement error, biological variability, and knowledge uncertainty, and between epistemic uncertainty and interindividual variability, are essential here.

Estimating a Population Mean

Suppose a sample of \(n\) people is drawn independently from the target population and a physiological quantity is measured in each person. The sample mean \(\bar{x}\) estimates the population mean \(\mu\); the sample standard deviation \(s\) describes the spread of observed individual values. When the observations are independent and the sample is representative, the estimated standard error of the sample mean is \(SE=s/\sqrt{n}\). If the measurement distribution is approximately normal, a two-sided confidence interval for \(\mu\) is

\bar{x} \pm t_{n-1,\,1-\alpha/2}\frac{s}{\sqrt{n}},

where \(t_{n-1,\,1-\alpha/2}\) is the corresponding quantile of the \(t\) distribution with \(n-1\) degrees of freedom. This interval quantifies sampling uncertainty in the mean, under the stated assumptions. It does not describe the range in which most individuals’ values fall. For a skewed variable, a small sample, a clustered sample, or a design with unequal sampling probabilities, this simple interval may be unsuitable without an appropriate model or design-based calculation.

The standard error formula follows from the variance of a mean. If \(X_1,\ldots,X_n\) are independent measurements with variance \(\sigma^2\), then

\operatorname{Var}(\bar{X}) =\operatorname{Var}\left(\frac{1}{n}\sum_{i=1}^{n}X_i\right) =\frac{1}{n^2}\sum_{i=1}^{n}\operatorname{Var}(X_i) =\frac{\sigma^2}{n}.

The estimated standard deviation of \(\bar{X}\) is consequently \(s/\sqrt{n}\). Increasing the sample size reduces uncertainty in the mean when the sampling design and population remain comparable; it does not remove real differences among people.

Worked Example: Uncertainty in a Reference Liver Volume

In a hypothetical representative sample of 25 healthy adults from a defined reference population, mean liver volume is 1.42 L and the sample standard deviation is 0.24 L. Assume independent observations and an approximately normal distribution of liver volume in this population. Estimate the population mean and calculate a 95% confidence interval.

Here \(n=25\), \(\bar{x}=1.42\) L, and \(s=0.24\) L. The estimated standard error is

SE=\frac{0.24\ \mathrm{L}}{\sqrt{25}}=0.0480\ \mathrm{L}.

With \(df=24\), the 97.5th percentile of the \(t\) distribution is approximately 2.064. The interval is

1.42 \pm 2.064(0.0480) =1.42 \pm 0.0991 =(1.321,\ 1.519)\ \mathrm{L}.

Under repeated sampling using the same assumptions and method, 95% of intervals constructed this way would contain the population mean liver volume. The interval does not say that 95% of individuals have liver volumes between 1.321 and 1.519 L. The observed \(s=0.24\) L describes individual spread in the sample; the smaller \(SE=0.0480\) L describes the estimated precision of the sample mean.

Comparing Population-Specific Flow Estimates

A reference flow measured in one population may not be an appropriate mean for another. For two independent samples, a difference in sample means estimates the difference in population means. A Welch confidence interval uses the standard error

SE(\bar{x}_1-\bar{x}_2) =\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}.

This approach does not require equal population variances. Its interpretation still depends on independent, representative observations within each population and on comparable measurement definitions. If samples are paired, clustered, or differently selected, the independent-sample calculation is not justified as written.

Worked Example: Comparing Hepatic Blood Flow in Two Defined Populations

Suppose two hypothetical, independent samples use the same measurement protocol and have comparable adult age, sex, and body-size distributions. In population A, \(n_1=36\), mean hepatic blood flow is 1.18 L/min, and \(s_1=0.18\) L/min. In population B, \(n_2=36\), the mean is 1.05 L/min and \(s_2=0.15\) L/min. Estimate the difference, population A minus population B, and form a 95% Welch confidence interval.

The estimated difference is \(1.18-1.05=0.130\) L/min. Its estimated standard error is

SE=\sqrt{\frac{0.18^2}{36}+\frac{0.15^2}{36}} =\sqrt{0.000900+0.000625} =0.03905\ \mathrm{L/min}.

The Welch degrees of freedom are approximately 67.8, giving a 95% critical value of about 1.994. Therefore,

0.130 \pm 1.994(0.03905) =0.130 \pm 0.0779 =(0.0521,\ 0.2079)\ \mathrm{L/min}.

Given the sampling assumptions, the data are consistent with a population mean flow in A that is about 0.0521 to 0.2079 L/min higher than in B. This comparison does not establish why the means differ, nor does it establish a difference for every individual. Matching important characteristics and measurement methods makes the reference comparison more interpretable, but does not prove the samples represent every relevant feature of their populations.

A pooled reference value can conceal a difference between defined populations. Conversely, a subgroup difference can reflect different age or body-size composition rather than a difference within comparable subgroups. The choice is not simply “use one mean” versus “use two means”: the modeler must first identify the intended target population and decide which characteristics should be represented or standardized.

Standardizing to the Target Population

When a physiological quantity differs across relevant strata, such as age groups, a target-population mean can be calculated using the target population’s stratum weights. If stratum \(g\) has target weight \(w_g\) and mean \(\mu_g\), with nonnegative weights summing to one, the standardized mean is \(\mu_T=\sum_g w_g\mu_g\). Using weights from the target population prevents the sample’s accidental subgroup proportions from determining the reference mean.

Worked Example: Target-Population Adipose Fraction

In hypothetical measurements, mean adipose fraction is 0.18 with \(s_1=0.06\) among 100 younger adults and 0.27 with \(s_2=0.07\) among 100 older adults. A target population is defined as 60% younger and 40% older. Assume independent, representative samples within the two age strata, and treat the target weights as known. Estimate the target mean and its sampling uncertainty.

The standardized estimate is

\hat{\mu}_T=0.60(0.18)+0.40(0.27)=0.108+0.108=0.216.

Thus the estimated target-population mean adipose fraction is 0.216, or 21.6%. Under independent stratum samples and fixed weights, the estimated variance of this weighted mean is the sum of the stratum mean variances multiplied by squared weights:

SE(\hat{\mu}_T) =\sqrt{0.60^2\frac{0.06^2}{100} +0.40^2\frac{0.07^2}{100}} =\sqrt{0.0000208} =0.00456.

Using 1.96 as a large-sample 95% critical value gives \(0.216\pm1.96(0.00456)=(0.2071,\ 0.2249)\). This interval estimates uncertainty in the target mean, not the range of individual fractions.

The distinction can be seen by estimating individual variability in the combined target population. The variance decomposition is the weighted within-stratum variance plus the weighted squared difference between each stratum mean and the target mean:

\begin{aligned} \widehat{\operatorname{Var}}(X_T) &=0.60\{0.06^2+(0.18-0.216)^2\}\\ &\quad+0.40\{0.07^2+(0.27-0.216)^2\}\\ &=0.006064, \qquad \hat{\sigma}_T\approx0.0779. \end{aligned}

This estimated individual standard deviation is about 0.0779, much larger than the mean’s standard error of 0.00456. The variance calculation uses sample standard deviations as estimates of within-stratum variability; it is not a confidence interval. If the target weights themselves are estimated rather than fixed, their uncertainty must also be considered. Furthermore, tissue fractions are compositionally constrained: changing one fraction can require changes in others, so this mean alone is not a complete set of mutually consistent tissue inputs.

Choosing and Checking Reference Inputs

A single reference mean is most defensible when the target population is clearly defined, the sample is representative of it, and a mean is an appropriate summary for the model’s purpose. A covariate-stratified or standardized reference is preferable when relevant population composition differs and credible stratum-specific data are available. A distribution can be more informative when the PBPK analysis needs to represent individual variability rather than only a reference subject. These choices answer different questions; a narrow confidence interval for a mean does not make the underlying individual distribution narrow.

Check the population description, sample size, recruitment and exclusion criteria, measurement method, units, and whether the reported quantity is absolute or normalized to body size. For example, a mean organ volume in a heavier sample should not automatically be transferred to a smaller target population, and a flow expressed per unit tissue mass must not be treated as a whole-organ flow. Confirm that the physiological inputs remain coherent with body-size scaling, organ-volume and blood-flow constraints, and tissue-composition bounds.

Warning signs include a reference sample with a markedly different age or body-size composition from the intended population; a reported “mean” with no population or measurement context; an implausibly small standard error compared with the reported individual spread; and an apparent subgroup contrast obtained under different methods or units. If the target population is not represented, do not disguise that knowledge gap as a precise population-specific estimate. Record the mismatch and treat the value as uncertain for the intended application.

Common pitfall: “The 95% confidence interval for mean organ volume is narrow, so almost every person has an organ volume near that mean.” This confuses uncertainty in a population summary with interindividual variability. The interval concerns the mean; the sample standard deviation or an appropriate individual-level distribution describes spread among people.

Key takeaway: Define the target population, estimate or select reference physiology that represents it, and label whether each uncertainty measure concerns the population summary or individual values. A precise estimate from the wrong population is not a suitable reference.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A representative sample has \(n=16\), mean organ volume 1.50 L, and \(s=0.20\) L. What is the estimated standard error of the mean?
What does a 95% confidence interval for a population mean describe, and what does it not describe?
A target population is 70% in stratum A and 30% in stratum B, with stratum means 0.20 and 0.30. Calculate the standardized mean.
Name two checks needed before applying a published flow reference to a PBPK target population.

Solutions. (1) \(SE=0.20/\sqrt{16}=0.0500\) L. (2) It describes the repeated-sampling coverage of the procedure for the population mean under its assumptions; it does not give the range of individual values or the probability that the fixed mean lies in this particular interval. (3) \(0.70(0.20)+0.30(0.30)=0.23\). (4) For example, check that the source and target populations have relevantly comparable age and body-size composition, and verify units and measurement definitions; also assess sampling representativeness and health status as relevant.