Why Rate Confounding Matters
Oral concentrations reflect both the rate at which drug enters the systemic circulation and the rate at which it is eliminated. If the sampled times produce similar changes in concentration when either rate changes, the data may estimate the two rates imprecisely or ambiguously. A fitted curve can still look convincing: the difficulty is whether the observations distinguish the roles of \(k_a\), the first-order absorption rate, and \(k\), the first-order elimination rate.
This tutorial uses the one-compartment, first-order oral model from the earlier material on Oral PK Sampling for Absorption and Elimination. The diagnostic emphasis is on how sensitivities vary across time and whether a proposed set of times supplies distinct information about the two rates. A sensitivity or information calculation is local: it describes the model near specified parameter values, not every possible parameter combination.
Model, Assumptions, and Rate Sensitivities
For a single oral dose with first-order absorption and elimination, write the concentration as
where \(A\) collects the dose and scale terms, such as bioavailability and volume. Assume \(A>0\), \(k_a>0\), \(k>0\), and \(t>0\). The concentration is positive under these conditions. For the sensitivity calculations below, use log concentration and log rates, \(\log(k_a)\) and \(\log(k)\). This makes the rate sensitivities dimensionless and describes proportional changes in concentration associated with proportional changes in a rate.
With \(A\) held fixed, differentiating the model gives
For example, the first term of \(g_a(t)\) comes from differentiating \(\log[k_a/(k_a-k)]\) with respect to \(\log(k_a)\); the second comes from differentiating the log of the exponential difference. Keeping both terms is essential. Dropping either can materially distort the apparent sensitivity pattern.
A sensitivity plot places time on the horizontal axis and \(g_a(t)\) and \(g_k(t)\) on the vertical axis. If the two sensitivities have similar patterns at the proposed sampling times, changes in the two rates can have similar effects on the observations. If their values and signs differ across the selected times, the observations may separate the rates more effectively. A plot suggests geometry; a matrix calculation quantifies it.
From Sensitivities to a Local Information Diagnostic
Suppose log concentrations are observed with independent, normally distributed errors of common variance \(\sigma^2\), the scale \(A\) is known, and the nominal rates are fixed at values used to evaluate the design. For sampling times \(t_1,\ldots,t_n\), form a sensitivity matrix \(G\) with row \(i\) equal to \((g_a(t_i),g_k(t_i))\). The local information for the two log rates is proportional to \(G^\mathsf{T}G/\sigma^2\). For comparing geometry, it is convenient to inspect \(G^\mathsf{T}G\); its overall scale does not affect the condition number or column cosine.
The eigenvalues describe information along the matrix's principal directions. A small minimum eigenvalue means that some combination of the rates changes the predictions little at the chosen times. The 2-norm condition number is the largest eigenvalue divided by the smallest; a large value indicates an elongated information geometry and potential numerical or estimation instability. The column cosine, the off-diagonal entry divided by the product of the square roots of the diagonal entries, measures alignment of the two sensitivity columns. A cosine near \(1\) or \(-1\) signals strong alignment; a value near zero signals approximate orthogonality.
These measures must be read together. A low column cosine does not guarantee useful information if one column is almost zero. A condition number describes relative anisotropy, not the absolute precision available: even a well-conditioned matrix may contain little information if all sensitivities are small. In addition, the calculation assumes the scale is known. If \(A\) is estimated, it contributes a nuisance sensitivity (a constant for log concentration); rate information must then account for that parameter rather than treating it as known.
Worked Example: Calculate Sensitivities at Three Times
Consider a hypothetical oral profile with \(k_a=1.20\ \mathrm{h}^{-1}\) and \(k=0.150\ \mathrm{h}^{-1}\). Calculate the sensitivities at 1, 2, and 8 hours using the formulas above. These values are local derivatives at the stated rates.
At 1 hour, \(e^{-0.150}=0.86071\), \(e^{-1.20}=0.30119\), and the exponential difference is \(0.55952\). Thus,
and
At 2 hours, the exponentials are \(e^{-0.300}=0.74082\) and \(e^{-2.40}=0.09072\). Substitution gives \(g_a(2)\approx-0.14286+0.33491=0.1921\) and \(g_k(2)\approx0.14286-0.34187=-0.1990\). At 8 hours, \(e^{-1.20}=0.30119\) and \(e^{-9.60}=0.00006773\); substitution gives \(g_a(8)\approx-0.14286+0.00216=-0.1407\) and \(g_k(8)\approx0.14286-1.2009=-1.058\).
The signs and relative magnitudes change over time. At 1 hour, the absorption sensitivity is substantially larger in magnitude than the elimination sensitivity. At 8 hours, the elimination sensitivity dominates. This suggests that early and late observations may contribute different information, but the three individual values alone do not establish how well any complete schedule separates the rates.
Compare Candidate Schedules
For a direct comparison, consider two hypothetical two-sample schedules at the same nominal rates. Use the sensitivities at 1, 2, and 8 hours just calculated. The calculations below assume equal variance, known \(A\), and independent log-scale errors; they are a geometry comparison, not a claim that these are universally optimal times.
Worked Example: Diagnose Two Two-Sample Schedules
For samples at 1 and 2 hours, the rows of \(G\) are approximately \((0.5031,-0.08789)\) and \((0.1921,-0.1990)\). Summing the row products gives
Its eigenvalues are approximately \(0.3154\) and \(0.0220\), so its condition number is \(0.3154/0.0220\approx14.35\). Its column cosine is \(-0.08244/\sqrt{0.2900(0.04733)}\approx-0.704\). The columns are fairly aligned in opposite directions, and one information direction is much weaker than the other.
For samples at 1 and 8 hours, the corresponding rows are approximately \((0.5031,-0.08789)\) and \((-0.1407,-1.058)\). The resulting matrix is
Its eigenvalues are approximately \(1.140\) and \(0.2602\), giving a condition number of about \(4.38\). Its column cosine is approximately \(0.189\). In this local, equal-variance comparison, the 1-and-8-hour schedule has less aligned sensitivity columns and a less ill-conditioned information matrix than the 1-and-2-hour schedule. The early and late observations respond differently to the two rates.
This does not prove that the second schedule yields precise estimates under every error model or parameter value. Its total information, practical feasibility, scale uncertainty, and sensitivity to nominal parameters still matter. In particular, condition numbers alone cannot compare absolute precision across matrices.
Simulated Profiles and a Fundamental Ambiguity
Sensitivity diagnostics are local, so it is useful to inspect model profiles as well. The next example shows a limitation that adding more sample times cannot necessarily fix. If the scale \(A\) is free, the oral model has a rate-label symmetry: exchanging \(k_a\) and \(k\) changes the profile only by a constant multiplicative factor.
Worked Example: Compare Profiles Under Swapped Rates
Take a hypothetical profile with \(k_a=1.20\ \mathrm{h}^{-1}\), \(k=0.150\ \mathrm{h}^{-1}\), and \(A=100\ \mathrm{mg/L}\). At 1, 2, and 8 hours, the dimensionless shape \(k_a/(k_a-k)(e^{-kt}-e^{-k_at})\) is approximately \(0.6395\), \(0.7430\), and \(0.3441\), respectively. The predicted concentrations are therefore about \(64.0\), \(74.3\), and \(34.4\ \mathrm{mg/L}\).
Now swap the rates: \(k_a=0.150\ \mathrm{h}^{-1}\) and \(k=1.20\ \mathrm{h}^{-1}\). The shape becomes \(0.150/(0.150-1.20)(e^{-1.20t}-e^{-0.150t})\), which is exactly one eighth of the original shape at every \(t\). Choosing \(A=800\ \mathrm{mg/L}\), eight times the original scale, reproduces the same concentrations at all times, not only at the three listed times.
Thus, if the scale is unconstrained, concentration observations alone cannot determine which of the two rate labels is the faster one from this model symmetry. External information, justified parameter constraints, or other data may be needed. A local sensitivity matrix at one parameter ordering does not reveal every global alternative; inspect plausible parameter combinations and fitted profiles as well.
Diagnostics, Alternatives, and Limitations
A useful diagnostic sequence is to plot both rate sensitivities over the feasible sampling window, evaluate them at candidate times, and calculate the information matrix using the intended error model. Then inspect the eigenvalues, condition number, and column cosine alongside the predicted concentration profile. If a design looks satisfactory only at one nominal rate pair, repeat the calculation over a scientifically plausible range of rates. Mark times at which predicted concentrations are so small that the assumed error model or assay performance is implausible.
These diagnostics are not interchangeable with fitting simulated data. A sensitivity plot is a visual local check. The information matrix summarizes local linearized information under stated error assumptions. Simulated profiles can expose curvature, practical measurement limits, and alternative parameter combinations, but a few simulated curves do not establish identifiability or precision by themselves. A formal design search is a separate task; here, the aim is to diagnose rate separation, not to assert an optimal schedule.
Common failures include using sensitivities with respect to raw rates in one column and log rates in the other, omitting a derivative term, or interpreting a small condition number as proof of high precision. A further mistake is to assume that an early sample always identifies absorption and a late sample always identifies elimination. Their value depends on the nominal rates, the error model, and any unknown scale parameters. If \(k_a\) is close to \(k\), direct formulas involving \(k_a-k\) can also suffer numerical cancellation; evaluate the model and derivatives with stable numerical methods in that region.
Check Your Understanding
Try each question, then compare your reasoning with the solutions.
- If two sensitivity columns have a column cosine of \(-0.98\), what does that indicate, and what does it not establish?
- For a two-rate information matrix with eigenvalues \(0.40\) and \(0.010\), calculate the condition number and interpret it.
- Why might an early and a late sample separate \(k_a\) and \(k\) better than two nearby early samples?
- Under what condition can swapping \(k_a\) and \(k\) leave an oral concentration profile observationally equivalent?
Solutions. (1) The sensitivity columns are strongly aligned in opposite directions, suggesting that rate changes can produce similar local changes with opposite signs. It does not establish adequate absolute information, precise estimates, or global identifiability. (2) The condition number is \(0.40/0.010=40\), indicating substantial relative weakness along one information direction; it does not by itself give standard errors. (3) Early and late observations may have different relative sensitivities to absorption and elimination, reducing column alignment; the actual result must be checked at the nominal rates and under the error model. (4) When the scale \(A\) is free, swapping the rates changes the profile shape by a constant multiplicative factor that can be absorbed into \(A\).