Why Formulation Can Change the Shape of a Concentration Profile
Two formulations can deliver the same dose and the same total bioavailable amount yet produce different concentration-time profiles. One formulation may place more of the available dose on a rapidly absorbed pathway, while another may allocate more to a slower pathway. A formulation can also change the absorption rates themselves. These changes can shift early concentrations and peak timing without necessarily changing total exposure.
The model below represents two parallel first-order absorption pathways: a fast pathway and a slow pathway. It extends the earlier material on parallel first-order pathways by allowing the pathway fractions, the pathway-specific absorption rates, or both to depend on formulation. The central distinction is between how much dose is available systemically and how that available dose is divided over time.
Concentration Model and Assumptions
Consider a single extravascular dose \(D\) at time zero, a one-compartment disposition model with volume \(V\), and first-order elimination with rate constant \(k_e\). The dose entering the fast absorption depot is \(F_rDq_{f,r}\); the dose entering the slow depot is \(F_rDq_{s,r}\). For either pathway, drug leaves its depot by first-order absorption. Absorbed drug enters the central compartment, where it is eliminated.
For the model to have this interpretation, the dose, formulation, and disposition parameters are constant over the interval considered; the two absorption pathways act in parallel; the fractions are nonnegative and sum to one; and elimination is linear and first-order. The equations also assume that \(F_r\) accounts for the overall fraction of the administered dose that becomes systemically available, and that the allocated pathway doses are absorbed completely according to their respective first-order processes. The standard expression below applies when each absorption rate differs from \(k_e\); the limiting expression is used if a rate equals \(k_e\).
The equation is the sum of two single-pathway concentration contributions. Each pathway contributes in proportion to its allocated fraction, and each has its own absorption rate. If \(k_{a,j,r}=k_e\) for pathway \(j\), its factor \(k_{a,j,r}(e^{-k_et}-e^{-k_{a,j,r}t})/(k_{a,j,r}-k_e)\) is replaced by \(k_et e^{-k_et}\), the continuous limit as the two rates approach equality.
Why the Contributions Add
The result follows by solving for the amount in each depot and then accumulating the absorbed input in the central compartment. For pathway \(j\), where \(j\) is fast or slow, the depot amount at time \(t\) is \(F_rDq_{j,r}e^{-k_{a,j,r}t}\). Its input rate to the central compartment is therefore \(F_rDq_{j,r}k_{a,j,r}e^{-k_{a,j,r}t}\). An amount entering at time \(u\) contributes to the central amount at time \(t\) after surviving elimination for \(t-u\), with survival factor \(e^{-k_e(t-u)}\).
Thus the concentration contribution from pathway \(j\) is the convolution of its input rate with the elimination process:
The two inputs enter the same linear disposition system, so their central amounts add. Summing the fast and slow contributions gives \(C_r(t)\). This also shows why the fractions must represent parts of the total bioavailable dose: adding two full-dose pathway profiles would count the dose twice.
Worked Examples
Worked Example: Changing the Pathway Fractions
Suppose two hypothetical formulations each contain a 100 mg dose. For both, \(F=0.80\), \(V=20\) L, and \(k_e=0.20\ \text{h}^{-1}\). Both use \(k_{a,f}=2.0\ \text{h}^{-1}\) and \(k_{a,s}=0.50\ \text{h}^{-1}\). Formulation A has \(q_f=0.70\), while formulation B has \(q_f=0.30\). Find and compare their concentrations at one hour.
The slow fractions are \(q_s=1-q_f\), so they are 0.30 for A and 0.70 for B. The common scale factor is \(FD/V=(0.80)(100\ \text{mg})/(20\ \text{L})=4.00\ \text{mg/L}\). At \(t=1\) h, the fast pathway factor is \( [2.0/(2.0-0.20)](e^{-0.20}-e^{-2.0})=1.1111(0.81873-0.13534)=0.7593\). The slow pathway factor is \( [0.50/(0.50-0.20)](e^{-0.20}-e^{-0.50})=1.6667(0.81873-0.60653)=0.3537\).
For formulation A, \(C_A(1)=4.00[(0.70)(0.7593)+(0.30)(0.3537)]=4.00(0.6376)=2.550\ \text{mg/L}\). For formulation B, \(C_B(1)=4.00[(0.30)(0.7593)+(0.70)(0.3537)]=4.00(0.4754)=1.902\ \text{mg/L}\). The formulation with the larger fast fraction has the higher concentration at one hour in this example. This comparison is a statement about the specified time, not a claim that A must have a higher concentration at every time.
Worked Example: Changing Absorption Rates While Holding Fractions Fixed
Consider two hypothetical formulations with \(D=100\) mg, \(F=0.80\), \(V=20\) L, \(k_e=0.20\ \text{h}^{-1}\), and the same fractions \(q_f=0.60\), \(q_s=0.40\). Formulation X has rates \(k_{a,f}=2.0\ \text{h}^{-1}\) and \(k_{a,s}=0.50\ \text{h}^{-1}\). Formulation Y has rates \(k_{a,f}=1.0\ \text{h}^{-1}\) and \(k_{a,s}=0.30\ \text{h}^{-1}\). Compare concentrations at two hours.
The scale factor is again \(4.00\ \text{mg/L}\), and \(e^{-0.4}=0.67032\). For X, the fast and slow factors are \(1.1111(0.67032-e^{-4})=0.7245\) and \(1.6667(0.67032-e^{-1})=0.5041\), respectively. Therefore \(C_X(2)=4.00[(0.60)(0.7245)+(0.40)(0.5041)]=2.545\ \text{mg/L}\).
For Y, the factors are \(1.25(0.67032-e^{-2})=0.6687\) and \(3.00(0.67032-e^{-0.6})=0.3645\). Thus \(C_Y(2)=4.00[(0.60)(0.6687)+(0.40)(0.3645)]=2.188\ \text{mg/L}\). At two hours X has the higher concentration, even though both formulations have the same total bioavailability and the same pathway fractions. Their absorption rates differ, so their profiles differ.
Worked Example: Checking the Amount Entering Through Each Pathway
A hypothetical 120 mg formulation has \(F=0.75\), \(q_f=0.25\), \(q_s=0.75\), \(k_{a,f}=1.5\ \text{h}^{-1}\), and \(k_{a,s}=0.25\ \text{h}^{-1}\). Find the total systemically available dose and the amount absorbed by four hours, before accounting for elimination.
The bioavailable dose is \(FD=(0.75)(120\ \text{mg})=90.0\) mg. The fast pathway is allocated \(90.0(0.25)=22.5\) mg and the slow pathway \(90.0(0.75)=67.5\) mg. For a first-order depot, the fraction absorbed by time \(t\) is \(1-e^{-k_at}\). By four hours, the fast pathway has absorbed \(22.5(1-e^{-1.5(4)})=22.5(1-e^{-6})=22.44\) mg. The slow pathway has absorbed \(67.5(1-e^{-0.25(4)})=67.5(1-e^{-1})=42.67\) mg. The total absorbed amount is \(22.44+42.67=65.11\) mg. This is the amount that has entered the systemic circulation by four hours, not the amount still present in the body; elimination has not been subtracted in this calculation.
What Changes, and What Does Not
Under the stated model, the total area under the concentration-time curve after one dose is
To see the cancellation for one pathway, integrate its concentration factor: \(\frac{k_a}{k_a-k_e}(1/k_e-1/k_a)=1/k_e\). Weighting the two resulting pathway areas by \(q_f\) and \(q_s\), then using \(q_f+q_s=1\), gives the displayed result. Consequently, when dose, total bioavailability, and clearance are shared across formulations, changing fractions or absorption rates changes the profile shape but not the total AUC. If \(F_r\) differs, AUC changes in proportion to \(F_r\), assuming clearance is unchanged.
This distinction is useful when representing formulation as a categorical predictor. One formulation can have its own pathway fractions, its own absorption rates, or both, while total bioavailability is held common if that is the intended assumption. Fractions must remain between zero and one and sum to one; one convenient representation is to specify \(q_{f,r}\) and define \(q_{s,r}=1-q_{f,r}\). A separate total bioavailability \(F_r\) should not be confused with either pathway fraction.
| Formulation change | Direct model change | Expected implication if \(F\) and clearance are common |
|---|---|---|
| Pathway allocation | Change \(q_f\) and set \(q_s=1-q_f\) | Changes the relative contributions of fast and slow input; total AUC is unchanged |
| Absorption rate | Change one or both \(k_a\) values | Changes input timing and profile shape; total AUC is unchanged |
| Total bioavailability | Change \(F\) | Scales the total exposure and the profile, if other parameters are fixed |
Alternatives, Limitations, and Common Mistakes
A single first-order absorption rate is a simpler alternative, but it cannot separately represent fast and slow dose fractions. A lag-time model shifts input in time rather than dividing the dose between two simultaneous pathways. Transit absorption represents a distributed delay using a sequence of transit compartments; a zero-order component represents input at a constant rate over an interval. These alternatives describe different input mechanisms. The parallel first-order model is appropriate when the intended representation is a mixture of two fractions with distinct exponential input rates; it should not be chosen merely because a profile has more than one visible feature.
The model is a compact representation, not proof that a formulation literally contains two physical absorption routes. Its interpretation depends on the assumptions of linear disposition and complete absorption of the allocated bioavailable fractions. If absorption is incomplete in a pathway, that must be represented consistently in the total bioavailability or in an explicitly defined pathway-specific availability. Separating an allocation fraction from a pathway-specific availability requires enough model structure and information to identify both; multiplying both freely can leave only their product informed by the concentration data.
A common mistake is to assign \(F D\) to each pathway and then add the resulting profiles. That predicts a total available dose of \(2FD\), not \(FD\). The correction is to allocate the available dose using fractions that sum to one. Another error is to expect a changed absorption rate to change AUC under linear elimination and constant \(F\): the integrated input is still \(FD\), so the rate redistributes exposure over time rather than changing its total. Conversely, equal AUC does not establish that two formulations have the same concentration-time profile.
Signs that the model assumptions may be inadequate include systematic disagreement between observed and predicted early concentrations, a profile shape that cannot be represented by two first-order contributions, or estimates that require invalid fractions or rates. Such features motivate checking the chosen input representation and parameter constraints; they do not, by themselves, identify a unique alternative model. Detailed profile-based diagnosis requires considering the sampling design and other plausible sources of shape differences.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A formulation has \(q_f=0.42\). What must \(q_s\) equal, and why?
- Two formulations have the same dose, \(F\), and clearance, but different absorption rates. Must their AUC values differ?
- For \(D=80\) mg and \(F=0.75\), with \(q_f=0.25\) and \(q_s=0.75\), how many milligrams are allocated to each pathway?
- State two assumptions needed for the concentration equation in this tutorial.
- Why is it incorrect to use the full bioavailable dose in both pathway contributions?
Solutions. (1) \(q_s=1-0.42=0.58\), because the two pathway fractions must sum to one. (2) No. With linear elimination and common \(F\), dose, and clearance, both AUC values equal \(FD/CL\); the profiles may nevertheless differ. (3) The bioavailable dose is \(FD=(80)(0.75)=60\) mg, so the fast pathway receives \(60(0.25)=15\) mg and the slow pathway \(60(0.75)=45\) mg. (4) Examples include first-order absorption from each depot, linear first-order elimination, and nonnegative pathway fractions summing to one. (5) It would count \(FD\) twice, predicting a total allocated dose of \(2FD\) rather than \(FD\).