Dose and Exposure Scaling Symmetries

Why Exposure Scale and Potency Can Be Confounded

A pathway response may be measured accurately while the drug concentration reaching that pathway is not known on an absolute scale. For example, administered doses may be known, but conversion from dose to free concentration may depend on an unknown proportionality factor. If pathway activity depends on concentration relative to a potency parameter, changing both quantities by the same factor can leave every predicted pathway output unchanged. The outputs can then identify a combination of the parameters, but not their separate values.

This is a structural issue: it follows from the model equations, rather than from a small sample or noisy measurements. More precise pathway observations cannot resolve an ambiguity that leaves the predictions exactly invariant. The key question is whether any information in the experiment anchors exposure or potency on an absolute scale.

For a standard saturable effect, the fractional effect is \(E(t)=E_{\max}C(t)/(K+C(t))\). Substituting \(C(t)=a z(t)\) and dividing the numerator and denominator by \(K\) gives \(E(t)=E_{\max}\theta z(t)/(1+\theta z(t))\), where \(\theta=a/K\). The same reduction applies when this effect drives a pathway ODE, provided the pathway equations use exposure only through \(C/K\), and the other model components are unchanged.

The model assumptions matter. The scale factor \(a\) must be common across the relevant times and dose profiles; the exposure-to-dose relationship represented by \(z(t)\) must be specified; and there must be no other pathway term that depends separately on absolute \(C\) or on \(K\). If, for instance, concentration also drives a separate process with its own known concentration scale, the invariance need not hold.

The Invariant Combination

The dependence on \(\theta\) can be seen directly in the saturable effect equation:

For any \(\lambda>0\), replacing \(a\) by \(\lambda a\) and \(K\) by \(\lambda K\) leaves \(a/K\) unchanged. Consequently, it leaves \(E(t)\) unchanged at every time. If pathway states satisfy equations such as \(\dot{x}=f(x,E(t),t)\), with the same initial conditions and other parameters, then their trajectories and outputs also remain unchanged: the forcing \(E(t)\) is identical, so the same state trajectory solves the same equations.

Thus, even ideal continuous observation of the pathway output cannot distinguish parameter pairs along this scaling direction, under the stated model. If potency is instead written as \(P=1/K\), the invariant is \(aP\): the corresponding transformation is \(a\mapsto\lambda a\) and \(P\mapsto P/\lambda\). These are two notations for the same ambiguity.

A useful local check follows by differentiating the response with respect to log parameters. Since the response depends on \(a\) and \(K\) through \(\log a-\log K\), increasing \(\log a\) and \(\log K\) by the same amount produces zero first-order change. In fact, the finite transformation above shows that the invariance is exact, not merely a local approximation.

What Known Dose Information Does—and Does Not—Provide

A known administered dose is not, by itself, a known concentration at the site of action. The profile \(z(t)\) may be calculated from the known dose history, while \(a\) represents an unknown conversion from that profile to concentration. If the only route by which dose affects the pathway is \(C(t)=az(t)\), then knowing \(z(t)\) exactly still leaves the ratio \(a/K\) as the only combination in the response.

The same conclusion holds for several known dose levels if the same unknown proportional scale applies to all of them. Changing the dose changes \(z(t)\), and may provide information about \(\theta\), but it does not by itself separate \(a\) from \(K\). The assumption of a common scale is important: if dose-dependent absorption or another specified exposure mechanism changes the relationship, the model must be analyzed on its own terms rather than assumed to have this exact symmetry.

Information that does break the ambiguity must anchor at least one of the two quantities. Examples include an independently known exposure-scale factor \(a\), an independently known potency \(K\), or an absolute concentration measurement tied to a known \(z(t)\). A dose record breaks the symmetry only if it is accompanied by a known dose-to-concentration relationship that fixes \(a\); the dose record alone is insufficient under this model.

Diagnostics, Alternatives, and Limitations

A practical diagnostic is to inspect whether the model predictions change when \(a\) and \(K\) are jointly rescaled. If the equations satisfy the assumptions above, they will not. During estimation, this can appear as a flat ridge of equally good fits: many parameter pairs have the same ratio and the same predicted outputs. A numerical optimizer may return one pair, but its reported value should not be mistaken for evidence that the data identified the two parameters separately.

The main remedies answer different questions. Reparameterizing in terms of \(\theta=a/K\) expresses the quantity the pathway data can identify, but it does not recover separate values for \(a\) and \(K\). Fixing \(a\) or \(K\) at a justified, independently supported value makes it possible to estimate the other from \(\theta\), but the result depends on that external information. Adding an absolute concentration measurement can estimate the exposure scale directly, after which potency can be inferred from the pathway response. Adding more pathway outputs helps only if at least one output depends on \(a\) or \(K\) separately rather than solely on their ratio.

This scaling confounding is distinct from uncertainty caused only by finite, noisy observations. Even with perfect observations, the separate parameters remain indistinguishable when the exact invariance holds. Conversely, a model that includes a known absolute concentration scale or a separate process sensitive to concentration may not have this particular ambiguity. Whether that additional information actually identifies the parameters depends on the full model and the experimental design.

Before interpreting separate estimates, check the model’s dependence on concentration, verify whether the dose-to-exposure conversion is known or estimated, and ask whether any observation is on an absolute concentration scale. If not, report or analyze the identifiable combination rather than assigning unsupported meaning to one point on an equivalent parameter ridge.