Worked Predictions Across Meal Conditions

From Meal Condition to Concentration-Time Prediction

A PBPK prediction across meal conditions is a linked calculation: meal composition informs gastric emptying, gastric emptying determines the rate at which drug becomes available downstream, and the resulting input is combined with disposition to predict plasma concentration. A slower or delayed input often lowers and postpones the peak, but it does not automatically change total exposure. Whether AUC changes depends on whether the meal also changes the fraction reaching systemic circulation, clearance, or another part of the model.

The earlier material on fed-state physiology and gastric-emptying models provides the physiological context for choosing meal-specific parameters. Here, the aim is to carry those parameters through to quantitative predictions and interpret the comparison without attributing every difference to emptying alone.

Model and assumptions: A dose \(D\) is initially in the stomach. Following a meal-specific lag time \(L\), gastric drug empties by first-order kinetics with rate constant \(k_g\). A fixed fraction \(F\) of the emptied dose reaches the systemic circulation, represented as an input to a well-mixed one-compartment body with volume \(V\) and first-order elimination rate constant \(k_e\). The model assumes linear, time-invariant disposition; no dose-dependent dissolution or absorption; and no meal effects on \(F\), \(V\), or \(k_e\), unless those quantities are explicitly changed. Thus, it is an effective input-disposition model, not a detailed representation of every intestinal process.

Let \(u=t-L\) be time since the end of the lag. For \(t<L\), the model has no gastric input. For \(u\geq 0\), the amount remaining in the stomach and the gastric emptying rate are

G(t)=D e^{-k_g u}, \qquad R_g(t)=D k_g e^{-k_g u}.

The systemic input rate is \(F R_g(t)\). If \(A(t)\) is the amount in the central compartment, then \(dA/dt=F R_g(t)-k_eA(t)\), with \(A(L)=0\), and concentration is \(C(t)=A(t)/V\). This formulation treats \(F\) as the net fraction of the gastric-emptying dose that contributes to systemic input; it does not resolve intestinal absorption as a separate time course.

Concentration Profile and Peak-Time Result

For \(u\geq0\), solving the amount balance gives the concentration profile below. The expression is written for \(k_g\ne k_e\); the equal-rate case has a finite limiting form.

C(t)=\frac{F D}{V}\frac{k_g}{k_e-k_g} \left(e^{-k_g u}-e^{-k_e u}\right).

For a nonzero peak after input begins, differentiate with respect to \(u\). Setting the derivative to zero gives \(k_e e^{-k_eu}=k_g e^{-k_gu}\), so the post-lag peak time and the time measured from the meal are

u_{\max}=\frac{\ln(k_e/k_g)}{k_e-k_g}, \qquad T_{\max}=L+u_{\max}.

The formula gives a positive post-lag time for positive, unequal rate constants. When \(k_g=k_e=k\), the profile is \(C(t)=(FD/V)ku e^{-ku}\), whose peak occurs at \(u_{\max}=1/k\). The lag shifts \(T_{\max}\) but does not change the post-lag profile or its peak height in this model.

The AUC follows from integrating the concentration profile over time. Equivalently, the total systemic input is \(FD\), and for linear elimination the integrated amount eliminated is \(k_e\int A(t)\,dt\). Since the body starts and ends with zero drug amount, total elimination equals total systemic input. Therefore:

\mathrm{AUC}_{0-\infty} =\int_0^\infty C(t)\,dt =\frac{F D}{V k_e}.

This result assumes complete observation to the model’s terminal tail and constant linear disposition. It shows why changing only \(L\) or \(k_g\) changes the shape and timing of the profile but leaves AUC unchanged. If \(F\) or \(k_e\) changes between meal conditions, that conclusion no longer applies.

Worked Example: Comparing Three Meal Conditions

Hypothetical model. Consider a 100 mg oral dose, with \(F=0.80\), \(V=40\) L, and \(k_e=0.10\ \mathrm{h}^{-1}\) in every condition. Assign the following illustrative gastric parameters: fasting \(L=0\) h and \(k_g=0.80\ \mathrm{h}^{-1}\); low-calorie meal \(L=0.25\) h and \(k_g=0.35\ \mathrm{h}^{-1}\); high-fat meal \(L=0.75\) h and \(k_g=0.12\ \mathrm{h}^{-1}\). These values are chosen to demonstrate the calculations, not to represent a clinical study. The assumptions are that only gastric lag and emptying rate differ and that all conditions share the same systemic availability and disposition.

The common concentration scale is \(FD/V=(0.80)(100\ \mathrm{mg})/(40\ \mathrm{L})=2.00\ \mathrm{mg/L}\). For fasting,

u_{\max} =\frac{\ln(0.10/0.80)}{0.10-0.80} =2.971\ \mathrm{h}, \qquad T_{\max}=0+2.971=2.971\ \mathrm{h}.

Substituting into the profile, retaining the negative denominator, gives

C_{\max} =2.00\left(\frac{0.80}{0.10-0.80}\right) \left(e^{-0.80(2.971)}-e^{-0.10(2.971)}\right) =1.486\ \mathrm{mg/L}.

For the low-calorie meal,

u_{\max} =\frac{\ln(0.10/0.35)}{0.10-0.35} =5.011\ \mathrm{h}, \qquad T_{\max}=0.25+5.011=5.261\ \mathrm{h},

and

C_{\max} =2.00\left(\frac{0.35}{0.10-0.35}\right) \left(e^{-0.35(5.011)}-e^{-0.10(5.011)}\right) =1.212\ \mathrm{mg/L}.

For the high-fat meal,

u_{\max} =\frac{\ln(0.10/0.12)}{0.10-0.12} =9.116\ \mathrm{h}, \qquad T_{\max}=0.75+9.116=9.866\ \mathrm{h},

and

C_{\max} =2.00\left(\frac{0.12}{0.10-0.12}\right) \left(e^{-0.12(9.116)}-e^{-0.10(9.116)}\right) =0.8038\ \mathrm{mg/L}.

In all three conditions, \(AUC_{0-\infty}=(0.80)(100\ \mathrm{mg})/[(40\ \mathrm{L})(0.10\ \mathrm{h}^{-1})]=20.0\ \mathrm{mg\cdot h/L}\). In this particular comparison, slower emptying and longer lag progressively delay and lower the peak, while AUC remains equal because \(F\), \(V\), and \(k_e\) are held fixed.

Checking the Input and Its Mass Balance

A concentration profile should be consistent with the gastric input that generated it. With first-order emptying, the cumulative amount emptied by time \(u\) after the lag is \(D(1-e^{-k_gu})\). The instantaneous input rate is not the same quantity: it is \(Dk_ge^{-k_gu}\), with units of amount per time. These checks help catch errors in time origin, units, and interpretation.

Worked Example: Low-Calorie Gastric Input at Two Hours

Use the low-calorie parameters from the comparison: \(D=100\) mg, \(L=0.25\) h, and \(k_g=0.35\ \mathrm{h}^{-1}\). At \(t=2.00\) h after the meal, time since the lag is \(u=2.00-0.25=1.75\) h. The instantaneous gastric emptying rate is

R_g(2.00) =(100\ \mathrm{mg})(0.35\ \mathrm{h}^{-1})e^{-(0.35\ \mathrm{h}^{-1})(1.75\ \mathrm{h})} =18.97\ \mathrm{mg/h}.

The fraction still in the stomach is \(e^{-0.6125}=0.5420\), so the remaining amount is \(100(0.5420)=54.20\) mg. The amount emptied is \(100-54.20=45.80\) mg, which also follows from \(100(1-e^{-0.6125})=45.80\) mg. The gastric rate is not the systemic rate: under the specified \(F=0.80\), the model’s systemic input rate at that instant is \(0.80(18.97)=15.18\) mg/h.

This arithmetic is internally consistent: the rate has units mg/h, whereas the cumulative emptied and remaining quantities have units mg. The mass balance also confirms that emptied plus remaining drug equals the original 100 mg dose.

When Exposure Also Changes

A meal may affect more than gastric emptying. For example, it can alter dissolution, intestinal availability, or presystemic loss. Those processes can change the net systemic fraction \(F\), although the simple input model does not explain the mechanism. The earlier material on intestinal availability is relevant when deciding whether such a change is supported. A useful comparison therefore distinguishes a gastric-emptying-only prediction from one in which systemic availability also differs.

Worked Example: Adding a Hypothetical Change in Availability

Keep the high-fat gastric parameters from the first example, but suppose a separate, explicitly hypothetical model assumption sets \(F=0.72\) rather than \(0.80\). Keep \(D=100\) mg, \(V=40\) L, and \(k_e=0.10\ \mathrm{h}^{-1}\). Because \(F\) multiplies the whole concentration profile, changing only \(F\) scales every concentration by \(0.72/0.80=0.900\) but does not change the time of the peak. Thus \(T_{\max}=9.866\) h, and

C_{\max,\mathrm{new}} =(0.900)(0.8038\ \mathrm{mg/L}) =0.7234\ \mathrm{mg/L}.

The AUC is now

AUC_{0-\infty,\mathrm{new}} =\frac{(0.72)(100\ \mathrm{mg})} {(40\ \mathrm{L})(0.10\ \mathrm{h}^{-1})} =18.0\ \mathrm{mg\cdot h/L}.

Relative to the fasting prediction with AUC \(20.0\ \mathrm{mg\cdot h/L}\), this is a 10.0% reduction: \((18.0-20.0)/20.0=-0.100\). The reduction follows from the assumed change in \(F\), not from the high-fat gastric parameters alone. It is a model-based comparison, not evidence that a real meal produces this change.

Interpretation, Alternatives, and Limitations

The results should be read as conditional predictions. In the first example, high-fat emptying is slower and more delayed than fasting emptying by construction. The later, lower peak follows from that parameterization under the common disposition assumptions. Equal AUC is not a general property of fed and fasted dosing; it follows specifically because the model holds \(F\) and \(k_e\) constant. Likewise, a meal-related change in \(T_{\max}\) is not uniquely attributable to gastric emptying if other input processes also change.

The single first-order emptying model is useful for transparent calculations, but it is only one representation of gastric input. A lagged first-order model provides a simple delay and exponential release. A transit-time distribution can represent a broader range of emptying times rather than a single exponential time course. A more physiological PBPK implementation can couple gastric release to intestinal segments, dissolution, absorption, and gut metabolism. Those alternatives may be preferable when the data or mechanistic question requires them; they also introduce more parameters and require evidence to support them.

Several checks are useful before interpreting a prediction:

Confirm that \(u=t-L\) is nonnegative when using the post-lag equations and that \(T_{\max}=L+u_{\max}\).
Check units: \(k_g\) and \(k_e\) must use the same inverse-time units, and \(Dk_g\) must have units of amount per time.
Check signs in the peak-height expression. When \(k_g>k_e\), both \(k_e-k_g\) and \(e^{-k_gu}-e^{-k_eu}\) are negative at positive \(u\), making their ratio positive.
Check mass balance: gastric amount remaining plus amount emptied must equal \(D\), and the integrated systemic input must equal \(FD\).
Check that the predicted AUC changes only when a parameter affecting total systemic exposure changes under this model, such as \(F\) or \(k_e\), rather than when only lag or emptying rate changes.

A common mistake is to interpret a delayed, lower peak as proof that AUC must also be lower. That conclusion does not follow: with complete input and linear disposition, changing only gastric lag or emptying rate redistributes the concentration profile over time but leaves the integrated exposure unchanged. Another frequent error is to substitute \(k_g-k_e\) in the denominator while retaining the exponentials in the order \(e^{-k_gu}-e^{-k_eu}\). Use the displayed expression consistently, or reverse both the denominator sign and the order of the exponential terms.

Key takeaway: Build each meal-condition prediction from explicit gastric parameters and disposition assumptions. Compare \(T_{\max}\), \(C_{\max}\), and AUC separately: gastric timing and rate govern the profile shape here, while systemic availability and elimination govern total exposure.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If only the lag time increases, which of \(T_{\max}\), \(C_{\max}\), and AUC changes in this model?
For a 100 mg dose with \(k_g=0.20\ \mathrm{h}^{-1}\), what fraction remains in the stomach 3.00 h after emptying begins?
If \(F\) falls by 20% while all other parameters remain fixed, how do AUC and \(T_{\max}\) change?
Why must the concentration peak formula retain a negative denominator when \(k_g>k_e\) and the exponentials are written as \(e^{-k_gu}-e^{-k_eu}\)?
Name one limitation of treating gastric emptying as a single first-order process with an effective systemic fraction.

Solutions. (1) \(T_{\max}\) increases by the increase in lag; \(C_{\max}\) and AUC are unchanged because the post-lag profile is merely shifted. (2) The remaining fraction is \(e^{-(0.20)(3.00)}=e^{-0.600}=0.5488\), or 54.88%; the remaining amount is 54.88 mg. (3) AUC falls by 20% because it is proportional to \(F\); \(T_{\max}\) is unchanged because \(F\) scales the profile without changing its shape. (4) At positive time with \(k_g>k_e\), \(e^{-k_gu}<e^{-k_eu}\), so the exponential difference is negative; the denominator \(k_e-k_g\) is also negative, giving a positive concentration. (5) The model does not resolve intestinal transit, dissolution, absorption, or gut metabolism as separate time-dependent processes, so it may not capture conditions where those processes materially shape input.