Why Two Absorption Rates Can Be Hard to Separate
An oral plasma profile records the combined consequences of drug leaving the stomach, entering the intestine, and crossing the intestinal wall. When gastric emptying and intestinal absorption both limit the rate of systemic input, a delayed or broadened concentration profile does not automatically reveal which process is slow. A model may fit the observed profile well while assigning very different values to the two rates.
This is an identifiability question: do the available observations contain enough distinct information to determine the parameters separately? The issue is not merely whether an optimizer converges or whether a fitted curve looks plausible. The shape of the objective function and the information contributed by each data type determine whether the parameters can be distinguished.
A Minimal Model Reveals the Confounding
Consider a single oral dose \(D\) initially in the stomach. Let \(A_g(t)\) be the amount in the stomach and \(A_i(t)\) the amount in the intestine. Assume first-order gastric emptying at rate \(k_g\) and first-order intestinal absorption at rate \(k_a\). Both rates have units of inverse time. Assume no gastric degradation, no intestinal loss other than absorption, complete initial delivery to the stomach, and a fixed fraction \(F\) of absorbed drug reaching the systemic circulation. For the plasma observation model, assume one-compartment disposition with known volume \(V\) and elimination rate \(k_e\).
These equations imply \(A_g(t)=De^{-k_gt}\). Solving the intestinal equation by an integrating factor gives the intestinal amount. The rate of absorption from the intestine is \(k_aA_i(t)\); multiplying by \(F\) gives the systemic input rate.
The systemic concentration follows \(\frac{dC}{dt}=R_{\mathrm{sys}}(t)/V-k_eC(t)\), with \(C(0)=0\). Because that equation uses the systemic input as its forcing term, identical input functions produce identical plasma profiles when disposition is unchanged.
Worked Example: Two Rate Assignments, One Plasma Input
Worked Example: Two Rate Assignments, One Plasma Input
Suppose a hypothetical oral dose is \(D=100\) mg, with \(F=1\). Compare \(k_g=0.80\ \mathrm{h}^{-1}\), \(k_a=0.20\ \mathrm{h}^{-1}\) with the exchanged assignment \(k_g=0.20\ \mathrm{h}^{-1}\), \(k_a=0.80\ \mathrm{h}^{-1}\). Evaluate the systemic input rate at \(t=1\) h.
For the first assignment, the formula gives \(R_{\mathrm{sys}}(1)=100(0.80)(0.20)/(0.20-0.80)\,[e^{-0.80}-e^{-0.20}]\). Substituting \(e^{-0.80}\approx0.4493\) and \(e^{-0.20}\approx0.8187\), this is \((-26.67)(-0.3694)\approx9.851\) mg/h.
For the exchanged assignment, \(R_{\mathrm{sys}}(1)=100(0.20)(0.80)/(0.80-0.20)\,[e^{-0.20}-e^{-0.80}]\). This is \((26.67)(0.3694)\approx9.851\) mg/h as well. The equality holds at every time, not just at one hour: exchanging the rates leaves the full input function unchanged. Consequently, a plasma profile described by this model cannot label the two rates without additional information.
The equality also has a limiting case. If \(k_g=k_a=k\), direct evaluation of the limit as \(k_a\) approaches \(k_g\) gives \(R_{\mathrm{sys}}(t)=FDk^2te^{-kt}\). This case is not an exchange between two distinct rates, but it is important numerically: the general expression has a denominator approaching zero, so software should use the limit or a numerically stable equivalent near equality.
Reading Profiles and Sensitivities
Structural symmetry is one source of ambiguity. More commonly, practical confounding appears as a long, shallow ridge in the objective function: many combinations of \(k_g\) and \(k_a\) fit nearly equally well. A profile for one parameter is formed by fixing it at successive values and re-optimizing the other parameters at each value. A flat or broadly extended profile indicates that a range of fixed values can be accommodated with little loss of fit. In the symmetric model above, two exchanged solutions are exactly equivalent; an objective-function profile can reveal equivalent solutions but does not by itself assign physiological labels to them.
Local sensitivities provide a complementary diagnostic. Let \(\theta=(\log k_g,\log k_a)\), and let \(y_j\) be the model-predicted log concentration at observation time \(t_j\). The sensitivity matrix has entries \(S_{jr}=\partial y_j/\partial\theta_r\). Using log parameters expresses changes on a relative scale, which helps avoid a misleading comparison caused only by different parameter units. If the two sensitivity columns are nearly proportional, small changes in one rate can be offset by changes in the other, leaving predicted concentrations nearly unchanged.
Under a local, approximately linear model with independent observation errors of known standard deviations \(\sigma_j\), the information matrix is \(S^\mathsf{T}WS\), where \(W\) has diagonal entries \(1/\sigma_j^2\). Nearly dependent columns make this matrix poorly conditioned; in turn, parameter estimates can have large uncertainty and strong dependence. This approximation is a diagnostic, not proof of global identifiability. Profile-based checks are useful because they can show nonlinear ridges and multiple separated solutions that a local sensitivity analysis might miss.
Worked Example: Nearly Parallel Sensitivity Columns
In a hypothetical local sensitivity calculation, four log-concentration predictions have sensitivities to \(\log k_g\) of \((-0.20,-0.10,0.02,0.08)\), and sensitivities to \(\log k_a\) of \((-0.19,-0.11,0.01,0.09)\). Assess whether the parameters are locally well distinguished by these observations.
The cosine between the columns is their dot product divided by the product of their lengths. The dot product is \(0.038+0.011+0.0002+0.0072=0.0564\). The squared lengths are \(0.0568\) and \(0.0564\), so the cosine is \(0.0564/\sqrt{0.0568\times0.0564}\approx0.9965\).
A cosine close to 1 means the sensitivity directions are nearly parallel. The data respond similarly to relative changes in either rate, so this local diagnostic indicates strong practical confounding. It does not show that either parameter is unimportant: both can affect the predictions substantially, yet their separate effects can be hard to tell apart.
Check diagnostics across the observed time range and against the observation-error scale. Sparse samples near the peak may miss informative differences in the early input and later tail. Residual patterns can also flag a more basic problem: if the assumed model cannot reproduce the observed profile, a narrow parameter interval from that model is not evidence that the physiological parameters are well identified. A visually good fit, a converged optimizer, or small standard errors from one local approximation should not be treated as sufficient evidence on its own.
Worked Example: An Independent Emptying Measurement
Worked Example: An Independent Emptying Measurement
Suppose a hypothetical plasma analysis under the serial first-order model supports two rate assignments: \(k_g=0.40\ \mathrm{h}^{-1}, k_a=0.15\ \mathrm{h}^{-1}\), or the exchanged assignment. A separate imaging measurement estimates the fraction of a gastric marker remaining at \(t=2\) h as \(0.45\). Assume the marker tracks gastric emptying for this illustration.
Under first-order emptying, the gastric fraction remaining is \(A_g(t)/D=e^{-k_gt}\). Therefore the imaging observation implies \(0.45\approx e^{-2k_g}\), so \(k_g=-\ln(0.45)/2\approx0.3993\ \mathrm{h}^{-1}\).
This measurement supports the assignment \(k_g\approx0.40\ \mathrm{h}^{-1}\), leaving \(k_a\approx0.15\ \mathrm{h}^{-1}\) as the corresponding absorption rate from the plasma fit. The imaging measurement adds information about gastric emptying directly; it is not simply another sample of the same systemic output. In an actual application, marker behavior, measurement error, and compatibility with the assumed emptying model would also need to be assessed.
Choosing Measurements That Add Information
Different data types answer different questions. More densely sampled plasma concentrations may improve practical precision by better resolving the profile’s timing and shape. But when the model has the exact exchange symmetry derived above, additional plasma samples alone cannot break it. The most useful additions measure one process through a distinct observation pathway.
| Data type | What it can inform | Key consideration |
|---|---|---|
| Gastric imaging or a validated gastric marker | Gastric emptying over time | Confirm the marker tracks the drug-containing material appropriately. |
| Intestinal or luminal drug measurements | Drug appearance in the intestine and subsequent loss | Sampling location and recovery affect interpretation. |
| Plasma concentrations with denser early and late sampling | Systemic input timing and profile shape | Helpful for practical precision, but not sufficient to resolve exact rate-exchange symmetry. |
| Independent disposition information | Elimination and distribution contributions to the plasma curve | Useful when disposition uncertainty is confounded with absorption timing. |
These approaches complement, rather than replace, careful model checking. An independent measurement may resolve which rate is gastric emptying, while uncertainty in model structure, marker behavior, or other parameters can remain. If a fitted rate is driven mainly by an assumed value or constraint rather than by observations, report that dependence rather than presenting the estimate as data-determined.
Common Pitfalls and Practical Interpretation
- Calling a good fit proof of unique parameter estimates. Different rate pairs can generate the same plasma input. Compare profiles or solutions, not only fitted curves.
- Assuming more plasma samples always solve the problem. They may help with noisy, sparse data, but cannot remove an exact symmetry in the model’s plasma predictions.
- Reading a high sensitivity as good identifiability. Two parameters may each strongly affect predictions while affecting them in nearly the same way. Their distinct sensitivity directions matter.
- Ignoring assumptions that create the ambiguity. The exchange result depends on the stated serial first-order structure and common downstream disposition. Other losses, transit structures, or direct measurements can change the identifiability question.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In the serial first-order model, why can exchanging \(k_g\) and \(k_a\) leave the plasma profile unchanged?
- A profile objective changes very little as \(k_g\) is fixed over a broad range and \(k_a\) is re-optimized. What does this suggest, and what does it not establish?
- Why might adding many more plasma samples fail to identify which of two fitted rates is gastric emptying?
- Name one measurement that can add information specifically about gastric emptying, and state one assumption or limitation to check.
Solutions. 1. The systemic input contains the product \(k_gk_a\) and the difference of the two exponential terms with a matching denominator; exchanging the rates leaves the complete input function unchanged. With the same disposition, the resulting plasma profiles are identical. 2. The broad, shallow profile suggests practical non-identifiability: a range of parameter combinations fits similarly. It does not prove structural non-identifiability for every possible model or show that the parameters have no physiological meaning. 3. If the model has exact rate-exchange symmetry, every additional plasma sample is still a prediction of the same systemic output and cannot label the rates. 4. Gastric imaging or a validated gastric marker can directly inform emptying. Check that the marker tracks the relevant drug-containing material and that its measurement is consistent with the assumed emptying model.