Comparing Transit, Lag-Time, and First-Order Inputs

Why Compare Absorption Inputs?

The same dose and systemic disposition can produce different early concentration profiles depending on how drug enters the systemic circulation. A profile that begins rising immediately, one that remains flat for a while, and one with a gradual delayed rise may call for different descriptions of the absorption process. The purpose of comparing input models is not to select a model from one visual feature alone, but to understand which timing patterns each model can represent.

The comparisons below hold dose, bioavailability, volume, and elimination constant. This isolates the effect of the absorption input. It also makes a key distinction clear: the time course of input into the systemic circulation is not the same thing as the time course of plasma concentration. Concentration reflects both input and elimination.

A Common Concentration Framework

Definition: Let $D$ be an oral dose, $F$ the fraction reaching the systemic circulation, $V$ the apparent distribution volume, and $k$ the first-order elimination rate constant. Let $g(s)$ be the rate of systemic input per unit of the total systemically available dose, with units of inverse time. For a single dose and one-compartment linear disposition, $$ C(t)=\frac{FD}{V}\int_0^t g(s)e^{-k(t-s)}\,ds. $$ Here $g(s)\geq 0$ and its integral over time is 1 when all systemically available drug eventually enters the circulation. The exponential term is the fraction of drug that entered at time $s$ and remains in the body at time $t$.

This convolution assumes a single dose, constant $F$ and $V$, linear first-order elimination, and no delay between systemic entry and the measured concentration. It also assumes the absorption model specifies the input-time distribution $g$. If disposition is nonlinear, or if the compared models use different $F$ or disposition parameters, the simple interpretation of differences as absorption-shape differences no longer follows.

For immediate first-order absorption with rate constant $k_a>0$, the input density is $g(s)=k_a e^{-k_a s}$ for $s\geq 0$. A fixed lag $T_{\mathrm{lag}}$ shifts that same input in time. A transit model instead represents absorption time as a distribution with a delayed, spread-out input.

Definition: In the transit convention used here, $N$ transit compartments precede a final absorption stage. The resulting absorption-time density is an Erlang distribution with shape $N+1$ and rate $k_{\mathrm{tr}}=(N+1)/\mathrm{MTT}$, where $\mathrm{MTT}$ is the mean transit time. Thus $$ g_{\mathrm{tr}}(s)=\frac{k_{\mathrm{tr}}^{N+1}s^N e^{-k_{\mathrm{tr}}s}}{N!},\qquad s\geq 0. $$ Its mean is $(N+1)/k_{\mathrm{tr}}=\mathrm{MTT}$. When $N=0$, this reduces to a first-order input with $k_a=1/\mathrm{MTT}$.

What the Input Mechanisms Can Represent

Input model	Input timing	Characteristic early concentration feature	Important limitation
Immediate first-order	Input is greatest at time zero, then declines exponentially.	Concentration begins rising immediately from zero after an oral dose.	It has no hard onset delay and cannot make the input rate rise after time zero.
Fixed-lag first-order	No input before $T_{\mathrm{lag}}$; first-order input begins after that time.	Concentration stays zero up to the lag, then follows a shifted first-order profile.	One fixed lag does not represent a spread of onset times across the dose.
Transit	Input times are distributed; for $N>0$, the input rate first rises and then falls.	Concentration can show a gradual delayed rise without a single hard lag.	A smooth distributed delay does not create a true period of exactly zero input.

For first-order absorption, substituting $g(s)=k_a e^{-k_a s}$ into the common concentration equation gives, when $k_a\ne k$,

C_{\mathrm{FO}}(t)=\frac{FD}{V}\frac{k_a}{k_a-k}\left(e^{-kt}-e^{-k_a t}\right),\qquad t\geq 0.

For fixed-lag first-order absorption, concentration is zero for $t\leq T_{\mathrm{lag}}$, and for $t>T_{\mathrm{lag}}$ it is the immediate first-order profile evaluated at the elapsed time $t-T_{\mathrm{lag}}$. With $N>0$, the transit input density is zero at time zero, rises to a mode at $s=N/k_{\mathrm{tr}}$, and then declines. The plasma concentration need not peak at that same time: it accumulates while input exceeds elimination and falls when input no longer compensates for elimination.

Worked Example: Immediate First-Order Absorption

Suppose a hypothetical oral dose has $FD/V=10.0$ mg/L, $k=0.200$ h$^{-1}$, and $k_a=1.00$ h$^{-1}$. Assume the common one-compartment, linear-disposition conditions above. Predict concentration at $t=0.500$ h.

Using the first-order formula, $$ C(0.500)=10.0\frac{1.00}{1.00-0.200}\left(e^{-(0.200)(0.500)}-e^{-(1.00)(0.500)}\right). $$

The exponentials are $e^{-0.100}=0.904837$ and $e^{-0.500}=0.606531$. Therefore, $$ C(0.500)=10.0(1.25)(0.904837-0.606531)=3.72883\ \text{mg/L}. $$

The positive prediction is consistent with the model: input begins at time zero, and concentration rises from zero as systemic input accumulates. The input rate itself is not increasing; it begins at its maximum and declines.

Transit Input: A Distributed, Rising-Then-Falling Input

A transit model can describe a distribution of absorption times rather than one common onset time. For $N>0$, its input density starts at zero, rises to a mode, and then falls. Increasing $N$ while holding $\mathrm{MTT}$ fixed makes the distribution more concentrated around its mean; it does not move the mean absorption time. The model therefore offers a way to represent a smoother delay than a fixed lag, while retaining a defined average transit time.

The mean and mode follow directly from the gamma density. For shape $N+1$ and rate $k_{\mathrm{tr}}$, the mean is $(N+1)/k_{\mathrm{tr}}$. Differentiating the log density for $N>0$ gives $$ \frac{d}{ds}\log g_{\mathrm{tr}}(s)=\frac{N}{s}-k_{\mathrm{tr}}. $$ Setting this to zero yields $s=N/k_{\mathrm{tr}}$, the input-density mode. Since $k_{\mathrm{tr}}=(N+1)/\mathrm{MTT}$, that mode is $N\,\mathrm{MTT}/(N+1)$, earlier than the mean. This describes the input, not necessarily the plasma-concentration peak.

Worked Example: Transit Absorption With One Transit Compartment

Use the same hypothetical $FD/V=10.0$ mg/L and $k=0.200$ h$^{-1}$. Let $N=1$ and $\mathrm{MTT}=1.00$ h. Then $k_{\mathrm{tr}}=2.00$ h$^{-1}$ and $g_{\mathrm{tr}}(s)=4s e^{-2s}$. Calculate concentration at $t=0.500$ h.

Substitute this input density into the convolution. Factoring out the terms that do not depend on $s$ gives $$ C(0.500)=10.0(4)e^{-0.100}\int_0^{0.500}s e^{-1.8s}\,ds. $$

Integration gives $\int_0^t s e^{-as}\,ds=[1-(1+at)e^{-at}]/a^2$. Here $a=1.8$ h$^{-1}$, so $$ C(0.500)=10.0\frac{4}{1.8^2}e^{-0.100}\left[1-(1+1.8(0.500))e^{-1.8(0.500)}\right]. $$

Using $e^{-0.9}=0.406570$, the bracket is $1-1.9e^{-0.9}\approx 0.227518$. Thus $$ C(0.500)\approx 10.0\frac{4}{3.24}(0.904837)(0.227518)=2.54156\ \text{mg/L}. $$

The input-density mode is $N/k_{\mathrm{tr}}=1/2=0.500$ h, while its mean is 1.00 h. The concentration at 0.500 h is lower than in the immediate first-order example, but this single-time comparison does not establish that transit input is always lower. Input is distributed differently over time, and elimination acts throughout.

Fixed Lag: A Shift, Not a Distributed Delay

A fixed lag is appropriate when the model requires no systemic input before a specified time and then uses a chosen post-lag absorption process. For fixed-lag first-order absorption, the input density is $0$ before $T_{\mathrm{lag}}$ and $k_a e^{-k_a(s-T_{\mathrm{lag}})}$ afterward. Consequently, the concentration is also zero through the lag, and the post-lag profile is shifted rather than reshaped. This sharp onset is a substantive assumption, not merely a convenient way to make a profile rise later.

Worked Example: Shifting the First-Order Profile by a Fixed Lag

Continue with the hypothetical parameters $FD/V=10.0$ mg/L, $k=0.200$ h$^{-1}$, and $k_a=1.00$ h$^{-1}$, and set $T_{\mathrm{lag}}=0.500$ h. At $t=0.500$ h, the elapsed time after the lag is zero, so the concentration is zero. At $t=1.00$ h, the elapsed post-lag time is $0.500$ h, giving $$ C(1.00)=10.0\frac{1.00}{0.800}\left(e^{-0.100}-e^{-0.500}\right)=3.72883\ \text{mg/L}. $$

At $t=2.00$ h, the elapsed time is $1.50$ h: $$ C(2.00)=10.0(1.25)\left(e^{-0.300}-e^{-1.50}\right) =12.5(0.740818-0.223130)=6.47110\ \text{mg/L}. $$

The lag shifts the first-order profile by 0.500 h without changing its post-lag shape. It does not create a gradual distribution of onset times; the model assigns the same fixed onset to the whole dose.

Comparing the Predicted Profiles

The following values use the same hypothetical dose scale and elimination rate in all three models. The transit predictions use $N=1$, $\mathrm{MTT}=1.00$ h; the lagged predictions use $T_{\mathrm{lag}}=0.500$ h. Values are rounded to four significant figures.

Time (h)	Immediate first-order (mg/L)	Transit (mg/L)	Fixed-lag first-order (mg/L)
0.500	3.729	2.542	0
1.00	5.636	5.430	3.729
2.00	6.687	7.235	6.471

These predictions illustrate why one early observation cannot identify an input mechanism by itself: different mechanisms may yield similar concentrations at some times, while separating at others. A lagged profile is exactly zero before its onset in this idealized model. The transit profile instead starts with zero input but permits input to increase smoothly; its concentration is not forced to stay zero for a fixed interval. The immediate first-order model starts input at its maximum and often predicts a more prompt early rise.

For immediate first-order absorption with $k_a\ne k$, differentiating the concentration formula and setting its derivative to zero gives the concentration peak time $$ t_{\max}=\frac{\ln(k_a/k)}{k_a-k}. $$ For the hypothetical $k_a=1.00$ and $k=0.200$ h$^{-1}$, $t_{\max}=\ln(5)/0.800=2.0118$ h. The fixed-lag version peaks $0.500$ h later, at $2.5118$ h, because its entire curve is shifted. This peak-time expression is not a transit-input peak formula: transit concentration must be obtained from its input and disposition together.

Diagnostics, Alternatives, and Limitations

When judging whether an input mechanism can represent a profile feature, first ask whether that feature is predicted by the input definition itself. A clear period with no rise may be compatible with a fixed lag, but observed concentrations below quantification do not prove that input was exactly zero. A smooth delayed rise may be compatible with transit input, but it can also arise from other model structures or from sparse observations. A prompt rise is compatible with immediate first-order absorption, but does not establish that input began instantaneously in the underlying process.

Useful checks include plotting predictions alongside observations on the actual time scale, checking whether predicted onset and curvature match the data, and confirming that comparisons use the same dose, $F$, and disposition assumptions. Systematic early residuals—observations repeatedly above or below predictions—can indicate that the model's input timing is inadequate. They can also reflect misspecified disposition, dose timing, or measurement issues, so residual patterns should not be attributed to absorption automatically.

The models are alternatives with different structural commitments, not interchangeable curve labels. First-order absorption is parsimonious and represents an exponentially declining input from time zero. A fixed lag adds a sharp onset parameter but retains that post-onset shape. Transit absorption uses $\mathrm{MTT}$ and $N$ to describe distributed input timing and can represent an input rate that rises before falling. Their flexibility has limits: a single transit distribution does not by itself represent multiple distinct absorption pathways, and a single fixed lag does not represent a range of onset times. Such structures require separate model extensions.

Mistake: “Transit input peaks at $N/k_{\mathrm{tr}}$, so plasma concentration must peak then.”
Correction: $N/k_{\mathrm{tr}}$ is the mode of the absorption-time density. Concentration peaks when systemic input and elimination balance, so it depends on both absorption and disposition.
Mistake: “A lagged model just has slower absorption.”
Correction: A fixed lag imposes zero input before $T_{\mathrm{lag}}$; slow first-order absorption begins input immediately but spreads it over time.
Mistake: “The transit prediction is lower at 0.500 h, so it must predict lower concentrations throughout.”
Correction: The worked predictions cross in their relative ordering over time. Compare the full profiles, not a single time point.

Key takeaway: Immediate first-order input starts at once and declines, fixed-lag input starts only after a specified time, and transit input distributes absorption over time. The resulting concentration profile reflects each input together with disposition; input features and concentration features must not be conflated.

Check Your Understanding

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

Under the transit convention here, what is $k_{\mathrm{tr}}$ when $N=2$ and $\mathrm{MTT}=3.00$ h?
For $N=2$, where does the transit absorption-time density reach its mode, expressed in terms of $\mathrm{MTT}$?
What does a fixed-lag first-order model predict for concentration before the lag, and what assumption produces that prediction?
Why can the plasma-concentration peak occur later than the transit input-density mode?
If a transit model has $N=0$, what familiar input model does it reduce to?

Solutions. (1) $k_{\mathrm{tr}}=(N+1)/\mathrm{MTT}=3/3.00=1.00$ h$^{-1}$. (2) The mode is $N\,\mathrm{MTT}/(N+1)=2(3.00)/3=2.00$ h. (3) It predicts zero concentration before and through the lag because it assumes zero systemic input until the fixed onset. (4) Concentration also depends on elimination; it continues rising while input exceeds elimination, so its peak need not coincide with the input mode. (5) It reduces to first-order input with $k_a=1/\mathrm{MTT}$.