Propagating Uncertainty in an Interspecies PK Projection

Why Uncertainty Must Travel With an Interspecies Projection

An interspecies pharmacokinetic (PK) projection can produce a precise-looking human clearance or exposure estimate from animal data. But the estimate depends on uncertain quantities: the animal clearance estimate, the assumed body-size exponent, the projected human bioavailability, and the adequacy of the scaling relationship. If these uncertainties are discarded, a single exposure value can give a misleading impression of precision precisely when a first-in-human (FIH) decision needs to account for risk.

The task here is to carry uncertainty through a specified projection and interpret the result as a decision aid. The calculation does not establish that a proposed dose is safe. The projected range depends on the model and its input assumptions, and it does not automatically include every source of uncertainty relevant to people.

Definition: An interspecies PK projection uses PK information from one or more nonhuman species, together with stated scaling and bioavailability assumptions, to predict a human PK quantity such as clearance or exposure. Uncertainty propagation carries uncertainty in the inputs and model into a distribution or interval for the projected quantity. A projected exposure interval is conditional on the chosen model and assumptions; it is not a guaranteed range for every participant.

Specify the Projection Before Calculating

For an oral dose \(D\), a simple linear PK projection uses the relationship \(AUC = F_hD/CL_h\), where \(F_h\) is projected human bioavailability and \(CL_h\) is projected human clearance. Here, AUC means area under the plasma concentration-time curve over the relevant dosing interval; for a single dose with complete sampling, it is total exposure. The relationship assumes dose-proportional PK over the dose range considered and a consistent definition of AUC.

For the worked case, project human clearance from one animal species using a fixed allometric form:

CL_h=CL_a\left(\frac{W_h}{W_a}\right)^\beta

Here \(CL_a\) is animal clearance, \(W_a\) and \(W_h\) are animal and human body weights, and \(\beta\) is the allometric exponent. This calculation assumes that the selected animal clearance is appropriately measured and scaled, that the same clearance definition is used across species, and that the allometric relationship is suitable for extrapolation to the human body size. It does not prove those assumptions. A body-size equation also cannot, by itself, resolve species differences in metabolism, transport, or bioavailability.

The uncertainty calculation below is made on the natural-log scale. It assumes the specified log-scale input uncertainties are reasonable, and initially assumes they are independent. Under a normal approximation on the log scale, the projected exposure is log-normal. This is a modeling assumption, not a consequence of the allometric equation alone.

Worked Example: Scaling Animal Clearance to a Human Estimate

Suppose, hypothetically, that an animal weighs \(0.250\) kg and has clearance \(CL_a=0.0180\) L/h. Use a \(70.0\)-kg human and \(\beta=0.750\). These are illustrative values, not results from a real study.

The body-weight ratio is \(70.0/0.250=280\). Applying the stated allometric equation:

\begin{aligned} CL_h &=0.0180\left(\frac{70.0}{0.250}\right)^{0.750}\ \text{L/h}\\ &=0.0180(280)^{0.750}\ \text{L/h}\\ &=0.0180(68.4486)\ \text{L/h}\\ &=1.232\ \text{L/h}. \end{aligned}

The arithmetic is conditional on the assumed exponent and the suitability of the scaling model. It is not evidence that human clearance is known to four significant figures; uncertainty in the inputs and model must still be carried forward.

Propagating Input Uncertainty

Taking logarithms of the AUC equation makes the contributions easier to track. With a fixed dose, uncertainty in human exposure arises from uncertainty in bioavailability, animal clearance, the exponent, and any additional discrepancy assigned to the scaling model. For the numerical case, suppose the projected log-scale standard uncertainties are \(0.250\) for \(F_h\), \(0.200\) for \(CL_a\), and \(0.050\) for \(\beta\). Also include a log-scale scaling-discrepancy standard uncertainty of \(0.2105\). These values are hypothetical inputs to the uncertainty analysis, not universal defaults.

For independent inputs, the variance of a sum or difference of log quantities is the sum of their variances. The exponent enters multiplied by \(\ln(W_h/W_a)\), so its contribution to log clearance uncertainty is amplified by the log body-weight ratio. In this example, assume the weights and dose are fixed and treat the listed uncertainties as independent. The log-scale standard uncertainty for AUC is therefore:

\begin{aligned} s_{\log AUC} &=\sqrt{(0.250)^2+(0.200)^2+ \left[0.050\ln(280)\right]^2+(0.2105)^2}\\ &=\sqrt{0.062500+0.040000+0.079376+0.044310}\\ &=0.4756. \end{aligned}

The \(\ln(280)\) term matters: a small uncertainty in the exponent can have a material effect when extrapolating across a large body-size ratio. If the inputs are correlated, simply adding their variances is not appropriate; covariance terms must be included.

Worked Example: A Log-Scale Exposure Range

Use the hypothetical human clearance \(1.232\) L/h, a projected \(F_h=0.350\), and a \(10.0\)-mg oral dose. The central projected AUC is:

AUC=\frac{F_hD}{CL_h} =\frac{0.350(10.0\ \text{mg})}{1.232\ \text{L/h}} =2.841\ \text{mg}\cdot\text{h/L}.

Assume the log AUC is approximately normal with standard uncertainty \(0.4756\). A central 90% uncertainty interval under this model uses \(1.645\) standard deviations on either side of the log-scale center. Exponentiating returns the interval to the AUC scale:

\begin{aligned} AUC_{\text{low}} &=2.841\exp\{-1.645(0.4756)\} =1.30\ \text{mg}\cdot\text{h/L},\\ AUC_{\text{high}} &=2.841\exp\{1.645(0.4756)\} =6.21\ \text{mg}\cdot\text{h/L}. \end{aligned}

Thus, under the stated model and input assumptions, the central 90% projected range is approximately \(1.30\) to \(6.21\) mg·h/L at \(10.0\) mg. It is not a statement that there is a 90% probability that every participant's AUC will fall in this range. In particular, this calculation represents the specified uncertainty in the projection; it does not automatically include interindividual variability in human PK.

Covariance Can Change the Decision-Relevant Tail

Independence is a substantive assumption, not a convenient default to use unquestioned. For example, if the uncertainty in two projected inputs tends to move together, their effects on AUC may partly offset or reinforce one another. Since \(\log AUC\) contains \(+\log F_h\) and \(-\log CL_a\), positive covariance between \(\log F_h\) and \(\log CL_a\) reduces the variance of their difference. Negative covariance increases it.

For two log-scale inputs \(U=\log F_h\) and \(V=\log CL_a\), the relevant variance is:

\operatorname{Var}(U-V) =\operatorname{Var}(U)+\operatorname{Var}(V)-2\operatorname{Cov}(U,V).

This identity follows by expanding \(\operatorname{Var}(U-V)\): the cross-product terms contribute \(-2\operatorname{Cov}(U,V)\). The other independent uncertainty contributions can then be added as before. The covariance should be based on the way the estimates were obtained or on a defensible model; it should not be selected merely to narrow the projected range.

Worked Example: Recalculating With Correlated Inputs

Keep the preceding hypothetical inputs, but suppose the correlation between \(\log F_h\) and \(\log CL_a\) is \(0.723\). With standard uncertainties \(0.250\) and \(0.200\), their covariance is \(0.723(0.250)(0.200)=0.03615\). The revised total log-scale variance is:

\begin{aligned} s_{\log AUC}^2 &=0.062500+0.040000-2(0.03615)+0.079376+0.044310\\ &=0.153886,\\ s_{\log AUC}&=0.3923. \end{aligned}

The revised central 90% projected range is:

\begin{aligned} 2.841\exp\{-1.645(0.3923)\} &=1.49\ \text{mg}\cdot\text{h/L},\\ 2.841\exp\{1.645(0.3923)\} &=5.417\ \text{mg}\cdot\text{h/L}. \end{aligned}

The interval is narrower because the assumed positive covariance makes the log bioavailability and log clearance contributions partly cancel in the AUC calculation. This is a conditional sensitivity result. It is justified only if the correlation represents the relevant uncertainty structure; it is not a reason to assume correlation without evidence.

Using the Upper Exposure Projection in an FIH Decision

An upper projected exposure can be compared with a prespecified exposure constraint as one part of dose planning. To illustrate the arithmetic, suppose a hypothetical review team has specified \(4.00\) mg·h/L as a comparison level for this projection. That value is an assumed decision input for the example, not a universal safety threshold. If AUC is dose proportional and the projected exposure scales linearly with dose, the dose corresponding to the upper end of the independent-input interval is:

Worked Example: Translating the Upper Bound Into a Dose Constraint

At \(10.0\) mg, the independent-input upper projected AUC is \(6.212\) mg·h/L (unrounded value underlying the reported \(6.21\)). Under dose proportionality, use the same ratio to calculate the dose whose upper projected AUC would equal \(4.00\) mg·h/L:

D_{\text{bound}} =10.0\ \text{mg} \left(\frac{4.00\ \text{mg}\cdot\text{h/L}} {6.212\ \text{mg}\cdot\text{h/L}}\right) =6.44\ \text{mg}.

The units of AUC cancel, leaving a dose in milligrams. This is a model-based exposure constraint under the stated assumptions, not a recommended FIH dose by itself. If the correlated-input scenario were supported instead, its upper AUC of \(5.417\) mg·h/L would give \(10.0(4.00/5.417)=7.38\) mg. The difference shows why uncertainty structure can matter to a decision. Neither calculation establishes that the corresponding dose is safe.

A dose decision also requires consideration of relevant nonclinical findings, pharmacology, uncertainty in human bioavailability and clearance, and other applicable safeguards. In particular, this projection should be compared with—not substituted for—the MABEL and toxicology-based starting-dose considerations addressed in earlier material. A conservative starting dose may be lower than an exposure-based bound for other reasons. The value of propagation is that it makes assumptions and uncertainty visible, so they can be weighed rather than hidden behind a point estimate.

Diagnostics, Alternatives, and Limitations

Before using a projected range, check whether its assumptions are credible and whether the result is stable enough to inform the decision. Important checks include:

Check units and definitions. Clearance must be expressed in compatible units, weights must use the same units, and the bioavailability and AUC definitions must match the dose and sampling context.
Inspect influential inputs. The exponent uncertainty is multiplied by \(\ln(W_h/W_a)\). A wide species-to-human weight gap can make exponent uncertainty influential even when its standard uncertainty looks small.
Question the scaling model. A numerical fit or familiar exponent does not establish that the relationship applies to the compound or species involved. Large extrapolations and species-specific disposition can undermine the projection.
Check dependence. If inputs share data, methods, or estimation steps, independence may be implausible. Ignoring covariance can either understate or overstate uncertainty.
Distinguish uncertainty from variability. Uncertainty about the projected typical human exposure is not the same as variation among individuals. A range that includes only parameter and model uncertainty may not describe the exposure distribution in an FIH cohort.
Check dose proportionality. The dose rescaling calculation assumes exposure changes proportionally with dose. If that assumption is unsupported over the range, the calculated dose bound is not reliable.

For a small number of approximately normal log-scale inputs, the variance calculation shown here is transparent and easy to audit. A delta-method calculation can approximate uncertainty by linearizing a nonlinear function near the input estimates, but may be inadequate when uncertainty is large or the function is strongly nonlinear. A simulation approach can instead draw jointly from specified input distributions, apply the projection to each draw, and summarize the resulting exposure distribution. Simulation can represent nonlinear transformations and dependence, but it does not repair unjustified input distributions or a misspecified scaling model. Whichever approach is used, its assumptions and the meaning of its interval must be stated.

A projected interval is also limited by what was included in the model. Unmodeled species differences, uncertain dose proportionality, or a poor choice of animal data can make a narrow interval misleading. Conversely, a wide interval may be a useful finding: it can show that the available evidence does not support a precise exposure claim and that a cautious decision or further evidence may be warranted.

Common pitfall: “The projected mean AUC is below the comparison level, so the dose is acceptable.” This ignores uncertainty and treats a point estimate as if it were a protective bound. Instead, inspect a clearly defined upper projected range, verify the assumptions used to obtain it, and assess it alongside other FIH evidence. An upper uncertainty bound is still conditional on the model; it is not a guarantee against higher exposure or harm.

Check Your Understanding

Answer each question using the assumptions stated. Brief solutions follow the questions.

If animal clearance is \(0.0200\) L/h, \(W_h/W_a=200\), and \(\beta=0.750\), write the human clearance projection. What must be true for its units to remain L/h?
Why does uncertainty in \(\beta\) contribute in proportion to \(\ln(W_h/W_a)\) on the log-clearance scale?
In the covariance example, why does positive correlation between \(\log F_h\) and \(\log CL_a\) reduce uncertainty in \(\log AUC\)?
At a dose of \(8.00\) mg, suppose the upper projected AUC is \(5.00\) mg·h/L. Under dose proportionality, what dose corresponds to an upper projected AUC of \(3.00\) mg·h/L?
Does a central 90% uncertainty interval for projected typical AUC necessarily cover 90% of individual participant AUC values? Explain.

Solutions. (1) \(CL_h=0.0200(200)^{0.750}\) L/h, provided animal clearance is in L/h and the weight ratio is dimensionless. (2) Taking logs gives \(\log CL_h=\log CL_a+\beta\log(W_h/W_a)\), so a change in \(\beta\) is multiplied by that log ratio. (3) Since \(\log AUC\) contains \(+\log F_h-\log CL_a\), its variance includes minus twice their covariance; positive covariance reduces that difference's variance. (4) \(8.00(3.00/5.00)=4.80\) mg. (5) No. The interval describes the specified uncertainty in the projection; individual variability is included only if it is explicitly modeled.