Oral PK Sampling for Absorption and Elimination

Why Oral Sampling Times Need to Cover Different Phases

After an oral dose, observed concentrations reflect both drug entering the systemic circulation and drug leaving it. A sample taken while concentrations are rising can be informative about absorption, while samples after absorption has largely finished can be informative about elimination. Concentration magnitude also carries information about volume. A sparse schedule must therefore place samples where the model prediction responds to the parameters of interest, rather than simply spread samples evenly across time.

The objective here is to use time-specific sensitivities to construct and assess candidate schedules for a one-compartment model with first-order absorption and elimination. This is a local design exercise: sensitivities depend on nominal parameter values, so a schedule that is informative at one assumed parameter set may be less informative elsewhere.

Model and Assumptions

Consider a single oral dose $D$, administered at time zero. Let $k_a$ be the first-order absorption rate constant, $k_e$ the first-order elimination rate constant, $V$ the apparent volume of distribution, and $F$ the bioavailability. Assume $F$ is known. The dose enters a single gut depot and is absorbed into a one-compartment central system; there is no lag time or transit compartment. Assume linear pharmacokinetics, constant parameters over the observation period, and $k_a\ne k_e$.

Definition: Under these assumptions, the concentration at time $t>0$ after one oral dose is $$ C(t)=\frac{FD}{V}\frac{k_a}{k_a-k_e}\left(e^{-k_e t}-e^{-k_a t}\right). $$ Here $k_a$ and $k_e$ have units of inverse time, $V$ has units of volume, and $C(t)$ has units of concentration. For a local sensitivity calculation, the nominal values of $k_a$, $k_e$, and $V$ are specified at which the model is evaluated.

The assumption that $F$ is known matters. If $F$ is also unknown, the concentration scale contains the ratio $F/V$; a single-dose concentration profile does not, by its scale alone, distinguish $F$ from $V$. The sensitivity to volume below is therefore interpreted with $F$ held fixed.

For comparability across parameter units, use log parameters $\log k_a$, $\log k_e$, and $\log V$. A sensitivity to a log parameter is the derivative with respect to that log parameter; for example, $\partial C/\partial\log k_a=k_a\,\partial C/\partial k_a$. These derivatives measure concentration change per unit change in the log parameter, with concentration units retained.

Deriving the Time-Specific Sensitivities

Write $A=FD/V$ and $h(t)=e^{-k_e t}-e^{-k_a t}$, so that $C(t)=A\,k_a h(t)/(k_a-k_e)$. Differentiating this expression gives the three local sensitivity functions:

\begin{aligned} S_a(t) &=\frac{\partial C(t)}{\partial\log k_a} =A k_a\left[ -\frac{k_e h(t)}{(k_a-k_e)^2} +\frac{k_a t e^{-k_a t}}{k_a-k_e} \right],\\ S_e(t) &=\frac{\partial C(t)}{\partial\log k_e} =A k_e\left[ \frac{k_a h(t)}{(k_a-k_e)^2} -\frac{k_a t e^{-k_e t}}{k_a-k_e} \right],\\ S_V(t) &=\frac{\partial C(t)}{\partial\log V}=-C(t). \end{aligned}

For example, the absorption derivative follows by differentiating both the factor $k_a/(k_a-k_e)$ and the term $e^{-k_a t}$. The first derivative contributes $-k_e/(k_a-k_e)^2$; differentiating $e^{-k_a t}$ contributes $t e^{-k_a t}$. Multiplying the resulting derivative by $k_a$ converts it to the derivative with respect to $\log k_a$. The elimination expression follows by differentiating with respect to $k_e$ in the same way. Since concentration is inversely proportional to $V$, its log-volume sensitivity is exactly $-C(t)$.

These formulas apply when the rates differ. At equal rates, the concentration expression has a finite limit, $C(t)=A k_a t e^{-k_a t}$ when $k_a=k_e$; the corresponding sensitivities should be evaluated by taking limits or by using a numerically stable equivalent expression, not by directly dividing by $k_a-k_e$.

A useful timing landmark is the concentration peak. Differentiating $C(t)$ with respect to time and setting the derivative to zero gives $k_a e^{-k_a t}=k_e e^{-k_e t}$. For distinct positive rates, this yields

t_{\max}=\frac{\log(k_a/k_e)}{k_a-k_e}.

The peak time is not itself a complete sampling rule. Rather, it helps locate the transition between the rising and falling parts of the profile. The sensitivity functions show how strongly each proposed observation responds to each parameter.

Worked Example: Computing Sensitivities at Three Times

Suppose a hypothetical subject receives $D=400$ mg, with $F=1$, $V=40$ L, $k_a=1.2$ h$^{-1}$, and $k_e=0.20$ h$^{-1}$. Calculate concentration and the three log-parameter sensitivities at $t=0.5$, $2$, and $8$ h. These are illustrative model values, not observations from a study.

Here $A=FD/V=10$ mg/L and $k_a-k_e=1.0$ h$^{-1}$. At $t=0.5$ h, $h=e^{-0.1}-e^{-0.6}=0.356025$, so

C(0.5)=10\frac{1.2}{1.0}(0.356025)=4.272\ \text{mg/L}.

Substitution into the sensitivity formulas gives

\begin{aligned} S_a(0.5)&=12[-0.20(0.356025)+1.2(0.5)e^{-0.6}] =3.097\ \text{mg/L},\\ S_e(0.5)&=2[1.2(0.356025)-1.2(0.5)e^{-0.1}] =-0.2313\ \text{mg/L},\\ S_V(0.5)&=-4.272\ \text{mg/L}. \end{aligned}

Repeating the same substitutions at $2$ h gives $h=e^{-0.4}-e^{-2.4}=0.579602$, $C(2)=6.955$ mg/L, $S_a(2)=1.222$ mg/L, $S_e(2)=-1.827$ mg/L, and $S_V(2)=-6.955$ mg/L. At $8$ h, $h=e^{-1.6}-e^{-9.6}=0.201829$, so $C(8)=2.422$ mg/L, $S_a(8)=-0.4766$ mg/L, $S_e(8)=-3.392$ mg/L, and $S_V(8)=-2.422$ mg/L.

The values illustrate different roles for the times: the early sample has a relatively large absorption sensitivity, while the late sample has a larger-magnitude elimination sensitivity. Volume sensitivity follows the negative concentration at every time. These patterns motivate considering samples across the profile rather than concentrating all observations in one phase.

From Sensitivities to a Sparse Schedule

For independent observations with additive, normally distributed errors of known standard deviation $\sigma$, the local Fisher information matrix for the log parameters is the sum of sensitivity outer products divided by the error variance. This gives a direct way to assess how a candidate set of times informs the three parameters.

Definition: For observations at times $t_1,\ldots,t_n$, define the sensitivity vector $s(t_i)=(S_a(t_i),S_e(t_i),S_V(t_i))^\mathsf{T}$. Under independent errors with variance $\sigma^2$, the local Fisher information matrix is $$ I=\sum_{i=1}^{n}\frac{s(t_i)s(t_i)^\mathsf{T}}{\sigma^2}. $$ This expression assumes the stated error model and evaluates each sensitivity at the chosen nominal parameter values. A larger determinant of $I$, for the same parameters and error assumptions, indicates greater local joint information under the D-optimality criterion.

A practical construction begins by selecting feasible candidate times over the absorption, peak, and declining portions of the profile. At each candidate time, calculate the sensitivity vector, then evaluate the information matrix for schedules of interest. A schedule should not be called optimal merely because it includes a peak sample or appears well spread: optimization requires comparing the information criterion across the allowed schedules.

Worked Example: Assessing a Three-Sample Candidate Schedule

Use the hypothetical parameters and sensitivities from the previous example. Consider the candidate times $0.5$, $2$, and $8$ h, with independent additive errors having $\sigma=0.20$ mg/L. Show how the schedule's information can be summarized without claiming it is optimal.

The sensitivity matrix, with one row for each observation time and columns for $\log k_a$, $\log k_e$, and $\log V$, is approximately

S= \begin{pmatrix} 3.097&-0.2313&-4.272\\ 1.222&-1.827&-6.955\\ -0.4766&-3.392&-2.422 \end{pmatrix}.

For three observations and three parameters, $I=S^\mathsf{T}S/\sigma^2$. The determinant can be calculated using $\det(I)=\det(S)^2/\sigma^6$. From the displayed sensitivity values, $\det(S)\approx-39.4\;(\text{mg/L})^3$. Therefore,

\det(I)\approx \frac{(-39.4)^2}{(0.20)^6} =\frac{1552}{0.000064} \approx 2.43\times 10^7.

This determinant summarizes local joint information for this particular parameterization, nominal parameter set, schedule, and error variance. It is not a universal score: comparison with another schedule is meaningful only when those choices are held fixed. Nor does this calculation establish that these times maximize the determinant over all feasible schedules.

Design Checks, Alternatives, and Limitations

Before relying on a candidate schedule, check that the calculation reflects the intended study. Confirm dose, units, nominal parameter values, observation times, and the assumed error variance. Verify that each sensitivity has concentration units and that changing a parameter in the model numerically produces a local concentration change consistent with the derivative. If $k_a$ and $k_e$ are close, the displayed formulas can suffer numerical cancellation; use stable evaluation or finite-difference checks rather than trusting a calculation that changes sharply with minor numerical precision.

Several timing failures are especially easy to recognize. A sample at exactly $t=0$ has predicted concentration zero under this model, and its three sensitivities are zero, so it contributes no information about these parameters. Very late samples can be useful for elimination but may have concentrations so small that they are difficult to measure; the additive-error calculation alone does not account for every practical assay constraint. Conversely, samples confined to the early rising phase may provide little leverage on the terminal decline. A design should be checked against realistic collection windows and the intended observation-error assumptions.

Sensitivity-based local design is one approach among several. A simple phase-based schedule uses pharmacokinetic reasoning to include early, peak-region, and late samples; it is transparent but does not quantify the information trade-off. A formal local information criterion, such as the determinant used above, compares candidate schedules quantitatively but depends on nominal values. Bayesian information averages over uncertainty in parameters, while robust design criteria seek performance across a range of plausible values. These approaches answer related but distinct design questions; none removes the need to specify the error model and practical constraints.

The main limitation is locality: the sensitivities describe changes near the nominal parameter values, not the whole range of plausible individual profiles. The model also omits features such as lag time, nonlinear absorption, and multiple compartments. If such features are important, a schedule chosen from this model may not serve the actual inferential goal. In particular, a good determinant at one nominal point does not guarantee precise estimates in every parameter direction or reliable performance under model misspecification.

Key takeaway: For sparse oral PK sampling, calculate the parameter sensitivities at candidate times and seek coverage of distinct profile phases. Interpret any information score conditionally on the model, nominal parameters, parameterization, and error assumptions used to obtain it.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In the model, what happens to concentration sensitivity with respect to log volume if concentration at a time is $5.0$ mg/L?
For $k_a=1.2$ h$^{-1}$ and $k_e=0.20$ h$^{-1}$, calculate the peak time.
Why does a sample at $t=0$ contribute no local information under this model?
What assumptions are needed for the information matrix $I=\sum_i s(t_i)s(t_i)^\mathsf{T}/\sigma^2$?
Why is a larger determinant at one nominal parameter set not proof that a schedule is best for all individuals?

Solutions. (1) $S_V=-C=-5.0$ mg/L per unit change in $\log V$. (2) $t_{\max}=\log(1.2/0.20)/(1.2-0.20)=\log(6)=1.792$ h. (3) At $t=0$, the model predicts $C(0)=0$, and all three parameter sensitivities are zero, so its sensitivity outer product is the zero matrix. (4) The local model sensitivities must be evaluated at specified parameter values, and observations must have independent additive errors with common known variance $\sigma^2$. (5) Sensitivities and the resulting information depend on nominal parameters; different individuals or plausible parameter values can produce a different information ranking.

Oral PK Sampling for Absorption and Elimination

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