Choosing Among Absorption Structures

Why the Absorption Structure Matters

After an extravascular dose, drug must enter the systemic circulation before its concentrations can be interpreted as evidence about disposition. The absorption structure describes when and how quickly drug enters. A poor choice can distort estimates of other model parameters: for example, an absorption delay may be mistaken for a change in disposition if the input process is not represented adequately.

A concentration-time profile does not show the absorption rate directly. Concentrations reflect both input and disposition. Therefore, selecting an absorption structure means comparing plausible descriptions of input while keeping the disposition model and the observation model consistent across candidates. The earlier material on separating absorption from disposition and on dose-dependent oral bioavailability is relevant: changes in systemic exposure do not, by themselves, identify which absorption mechanism changed.

Definition: An absorption structure is a mathematical description of the rate at which drug becomes available to the systemic disposition model after an extravascular dose. Let $D$ be the administered dose and $F$ the fraction available systemically. The total available amount is $FD$, and $r_{\mathrm{in}}(t)$, measured in amount per unit time, denotes its systemic input rate. The structure specifies how $r_{\mathrm{in}}(t)$ changes over time; it does not alone specify the resulting concentration-time profile.

Candidate Input Structures

Each candidate encodes a different assumption about the input process. The equations below use a single dose, a fixed $F$, and time $t$ measured from dosing unless a lag is specified. They describe input to the systemic model; they do not imply that concentrations rise or fall at the same rate.

First-order input: A depot amount $A_g$ is depleted at a rate proportional to the amount remaining. With $A_g(0)=FD$ and absorption rate constant $k_a$, $$ \frac{dA_g}{dt}=-k_aA_g,\qquad r_{\mathrm{in}}(t)=k_aA_g(t)=FDk_a e^{-k_at}. $$ The model assumes a single exponential depletion process with a constant $k_a$. Input is largest at time zero and declines thereafter.

Lag-time input: No systemic input occurs before a fixed lag $T_{\mathrm{lag}}$; after that time, first-order input begins. For $T_{\mathrm{lag}}\geq 0$, $$ r_{\mathrm{in}}(t)= \begin{cases} 0, & t<T_{\mathrm{lag}},\\ FDk_a e^{-k_a(t-T_{\mathrm{lag}})}, & t\geq T_{\mathrm{lag}}. \end{cases} $$ This idealization assumes a sharp onset at a specified positive delay when $T_{\mathrm{lag}}>0$, followed by first-order depletion. It does not describe gradual input before that onset.

Transit input: In a model with $n$ sequential transit stages, each with rate constant $k_{\mathrm{tr}}$, the systemic input rate is $$ r_{\mathrm{in}}(t)=FD\frac{k_{\mathrm{tr}}^n t^{n-1}e^{-k_{\mathrm{tr}}t}}{(n-1)!},\qquad t\geq 0, $$ where $n$ is a positive integer. For $n>1$, input starts at zero, rises to a peak, and then declines. For $n=1$, the expression reduces to first-order input, $FDk_{\mathrm{tr}}e^{-k_{\mathrm{tr}}t}$, which is nonzero at time zero and declines. Thus, the gradual build-up is a feature of $n>1$, not of every transit model.

Zero-order input: A fixed amount enters per unit time for a finite duration $T$, after which input stops: $$ r_{\mathrm{in}}(t)= \begin{cases} FD/T, & 0\leq t<T,\\ 0, & t\geq T. \end{cases} $$ This assumes a constant systemic input rate during the delivery interval and a sharp start and stop. It can represent an idealized finite-rate delivery process, but does not imply that all formulations or absorption processes are zero-order.

Saturable input: A depot amount $A_g$ is absorbed at a capacity-limited rate: $$ \frac{dA_g}{dt}=-\frac{V_{\max,a}A_g}{K_{m,a}+A_g},\qquad r_{\mathrm{in}}(t)=\frac{V_{\max,a}A_g(t)}{K_{m,a}+A_g(t)}. $$ Here $V_{\max,a}$ has units of amount per time and $K_{m,a}$ has units of amount. The rate is approximately $V_{\max,a}$ when $A_g$ is much greater than $K_{m,a}$, and approximately $(V_{\max,a}/K_{m,a})A_g$ when $A_g$ is much smaller than $K_{m,a}$. The model assumes capacity-limited absorption with these parameters held constant.

Deriving the Transit-Input Peak

The transit equation makes the distinction between $n=1$ and $n>1$ explicit. For $n=1$, its time dependence is $e^{-k_{\mathrm{tr}}t}$, which decreases from its initial value. For $n>1$, differentiate the logarithm of the time-dependent part $t^{n-1}e^{-k_{\mathrm{tr}}t}$:

\frac{d}{dt}\log\!\left(t^{n-1}e^{-k_{\mathrm{tr}}t}\right) =\frac{n-1}{t}-k_{\mathrm{tr}}.

Setting this derivative to zero gives $t=(n-1)/k_{\mathrm{tr}}$. It is positive when $n>1$; the derivative is positive before that time and negative after it. Thus, input rises from zero to a peak and then declines for $n>1$. For $n=1$, the peak is at $t=0$, and input declines thereafter. This distinction matters when using a gradual rise in inferred input as a reason to consider transit stages.

Worked Example: Comparing First-Order and Lag-Time Input

Suppose a hypothetical oral dose is $D=100$ mg, $F=0.80$, and $k_a=0.80\ \mathrm{h}^{-1}$. Compare first-order input with a lag-time model having $T_{\mathrm{lag}}=0.50$ h. Treat these as candidate input models, not as direct predictions of concentration.

The systemically available amount is $FD=(0.80)(100)=80$ mg. Under first-order input, $$ r_{\mathrm{in}}(t)=(80\ \mathrm{mg})(0.80\ \mathrm{h}^{-1})e^{-(0.80\ \mathrm{h}^{-1})t}. $$ At time zero, the rate is $64.0\ \mathrm{mg/h}$. At $t=1.00$ h, it is $64.0e^{-0.80}=28.8\ \mathrm{mg/h}$ (to three significant figures). At $t=3.00$ h, it is $64.0e^{-2.40}=5.81\ \mathrm{mg/h}$.

For the lag-time candidate, input is zero before $0.50$ h. At $t=1.00$ h, the time since the lag ended is $0.50$ h, so $$ r_{\mathrm{in}}(1.00)=64.0e^{-(0.80)(1.00-0.50)} =64.0e^{-0.40}=42.9\ \mathrm{mg/h}. $$ The difference between $28.8$ and $42.9\ \mathrm{mg/h}$ at one hour illustrates how the candidates differ after the lag; the lag candidate also predicts exactly zero input before $0.50$ h. Whether concentration data support either pattern must be evaluated together with disposition and sampling times.

Worked Example: Checking a Transit Model, Including One Stage

Consider hypothetical transit input with $FD=80$ mg, $n=3$, and $k_{\mathrm{tr}}=0.70\ \mathrm{h}^{-1}$. The peak time is $$ t_{\mathrm{peak}}=\frac{n-1}{k_{\mathrm{tr}}}=\frac{2}{0.70}=2.86\ \mathrm{h}. $$

The rate at that time is $$ r_{\mathrm{in}}(2.86) =80\frac{(0.70)^3(2.86)^2e^{-(0.70)(2.86)}}{2!} \approx 15.2\ \mathrm{mg/h}. $$ Here $0.70^3=0.343$, $(2.86)^2\approx 8.18$, $e^{-2.00}\approx 0.135$, and $2!=2$. The rate is zero at $t=0$ because the factor $t^2$ is zero. It rises until approximately $2.86$ h and then falls.

If instead $n=1$ with the same $FD$ and $k_{\mathrm{tr}}$, then $$ r_{\mathrm{in}}(t)=80(0.70)e^{-0.70t}. $$ The initial rate is $56.0\ \mathrm{mg/h}$, and the rate decreases immediately. A one-stage transit model is therefore first-order input, not a gradual build-up from zero.

Worked Example: Distinguishing Finite-Rate and Saturable Input

Suppose a hypothetical depot contains $80$ mg of systemically available drug. Compare zero-order input at $20\ \mathrm{mg/h}$ with saturable input having $V_{\max,a}=20\ \mathrm{mg/h}$ and $K_{m,a}=20$ mg.

Under zero-order input, the assumed duration is $80/20=4.00$ h. The rate is $20\ \mathrm{mg/h}$ throughout that interval and then becomes zero. Under saturable input, the initial rate is $$ r_{\mathrm{in}}(0)=\frac{(20)(80)}{20+80}=16.0\ \mathrm{mg/h}. $$ When the remaining amount is $20$ mg, the rate is $(20)(20)/(20+20)=10.0\ \mathrm{mg/h}$. The input rate therefore declines as the depot is depleted; it is not constant at $20\ \mathrm{mg/h}$.

The saturable depot equation can be rearranged and integrated from $A_0=80$ mg to a remaining amount $A$: $$ t=\frac{A_0-A}{V_{\max,a}}+\frac{K_{m,a}}{V_{\max,a}}\ln\!\left(\frac{A_0}{A}\right). $$ At $A=4$ mg, $t=(80-4)/20+(20/20)\ln(80/4)=3.80+\ln(20)=6.80$ h. By then, $76/80=95.0\%$ of the available amount has been absorbed. Unlike the idealized zero-order model, this saturable model approaches complete depletion asymptotically rather than stopping at a fixed finite time.

Using Data to Choose Among Candidates

Model choice begins with mechanistic plausibility and the design, then asks whether the observed data can distinguish the proposed input patterns. Concentration-time evidence is indirect: a slow rise or late peak could reflect absorption, disposition, or both. A lag is most credible when the design and observations can support a genuinely delayed onset; a gradual build-up is consistent with transit input for $n>1$, but is not proof of it. A finite, nearly constant delivery rate suggests considering zero-order input. Dose-dependent changes in the rate or extent of absorption can motivate a saturable-input candidate, provided changes in bioavailability and other processes are also considered.

A practical comparison holds the dose, disposition structure, and observation model fixed while changing the absorption structure. Fit candidates to the same observations and assess parameter plausibility, uncertainty, residual patterns, and predictive performance. For models fitted by maximum likelihood to the same data, AIC is $ -2\log L+2k$, where $L$ is the maximized likelihood and $k$ is the number of estimated parameters; a lower AIC indicates a preferred fit-penalty tradeoff among the candidates considered, not proof that the model is mechanistically true. A likelihood-ratio test is not automatically appropriate for every pair: the models must have the required nested relationship and regularity conditions for its reference distribution to apply.

Candidate	Characteristic input assumption	Evidence that may motivate testing it	Important limitation
First-order	Rate proportional to amount remaining	Input is plausibly exponential without a distinct delay	Can be difficult to separate from disposition or from transit with $n=1$
Lag-time	No input before a fixed time, then first-order input	Evidence consistent with a distinct onset delay	Sparse early sampling may not identify the lag
Transit, $n>1$	Sequential stages produce a gradual rise and later decline in input	Input appears to build up before its peak	Stage number and rate may be weakly identified; $n=1$ is first-order input
Zero-order	Constant rate for a finite duration	Controlled or finite-rate delivery is plausible	A sharp stop and constant rate may be unrealistic
Saturable	Rate depends nonlinearly on the amount remaining	Input rate changes with depot amount or dose in a compatible way	May be hard to distinguish from first-order input over a narrow amount range

For a dose-ranging study, examine whether input-related features change with dose, but interpret exposure measures cautiously. A dose-related change in peak concentration or area under the curve can also reflect dose-dependent bioavailability or disposition; it does not identify saturable absorption on its own. Adequate dose range, sampling around the onset and peak, and repeated observations across relevant doses improve the chance of distinguishing structures.

Diagnostics, Failure Modes, and a Common Mistake

After fitting, examine whether predictions reproduce the timing and shape of the observed profiles across individuals and doses. Systematic residual patterns around the early rise, peak, or decline can indicate that the input timing or shape is inadequate, although misspecified disposition or observation error can produce similar patterns. Check whether estimated parameters are precise enough to support the proposed structure and whether predictions remain plausible over the observed dose range. A model that fits only because a parameter is at an implausible boundary, or whose candidates make nearly indistinguishable predictions at the sampled times, provides weak evidence for a particular mechanism.

Confusing concentration shape with input shape. A rising concentration does not mean the input rate is rising; concentration reflects input and disposition together.
Calling every transit model a gradual-onset model. That description is correct for $n>1$. At $n=1$, transit input reduces to first-order input and declines from a nonzero initial rate.
Using a lag to explain any late peak. A late concentration peak alone does not establish zero input before a fixed lag. Sparse early observations may leave lag time poorly identified.
Interpreting dose-related exposure as proof of saturable input. Dose-dependent bioavailability and other dose-related processes are alternatives that must be considered.

No candidate is universally best. First-order input is parsimonious when proportional depletion is adequate; added transit stages, a lag, or nonlinear input should be supported by the design, a plausible mechanism, and improved model behavior. If the data cannot distinguish the candidates, report that uncertainty rather than assigning mechanistic meaning to a small fit difference.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a transit model with $n=4$ and $k_{\mathrm{tr}}=0.50\ \mathrm{h}^{-1}$, when does the input peak? Does the input start at zero?
A lag-time model has $T_{\mathrm{lag}}=1.5$ h. What is its predicted input before $1.5$ h, and what assumption distinguishes it from transit input with $n>1$?
In a saturable-input model, what happens to the input rate as the depot amount becomes small compared with $K_{m,a}$?
Why is a higher observed peak concentration at a higher dose insufficient, by itself, to establish saturable absorption?

Solutions. (1) The peak is at $(n-1)/k_{\mathrm{tr}}=3/0.50=6.00$ h. Because $n>1$, the rate starts at zero and rises to that peak before declining. (2) Input is exactly zero before $1.5$ h; after the lag, it follows the specified post-lag process. Transit input with $n>1$ builds gradually from zero rather than waiting for a sharp positive lag. (3) When $A_g\ll K_{m,a}$, $r_{\mathrm{in}}\approx (V_{\max,a}/K_{m,a})A_g$, so it is approximately first-order in the remaining amount. (4) Peak concentration also depends on disposition, dose-dependent bioavailability, and sampling around the peak. The concentration difference is not a direct measurement of absorption rate and does not uniquely identify saturable input.