Why Simulate Repeated Doses?
A drug-drug interaction (DDI) can change as treatment continues. A perpetrator drug may accumulate between doses, changing reversible inhibition as its concentration rises and falls. If it also changes enzyme abundance, that response develops through enzyme turnover rather than appearing instantly. The victim drug therefore experiences a clearance that can vary throughout a dosing interval and across successive intervals.
A dynamic simulation represents these linked changes over time. It is useful to keep three quantities distinct: perpetrator concentration, enzyme abundance or activity, and victim concentration. A perpetrator dose can cause an immediate concentration jump, but enzyme abundance remains continuous; victim concentration may jump at its own dose and otherwise changes through disposition.
A Compact Model for the Time Course
The following model is deliberately simple enough to calculate by hand. It assumes linear one-compartment elimination for the perpetrator, instantaneous intravenous bolus doses, first-order enzyme turnover, and linear victim disposition. These assumptions are not universal PBPK truths; they define the example. A PBPK implementation may instead represent multiple tissues, absorption, transport, or organ-specific enzyme abundance.
For doses that each increase the perpetrator concentration by \(D_P/V_P\), with first-order elimination rate constant \(k_P\), the concentration between doses is the sum of the surviving contributions from all previous doses:
At the instant of a dose, the new contribution is added to the concentration immediately before dosing. This produces a jump in \(C_P\), not in enzyme abundance. In the example below, enzyme abundance follows a first-order turnover equation toward a target level \(E_{\mathrm{target}}\):
Here \(E\) is relative to baseline enzyme abundance, so \(E=1\) denotes baseline. For a constant target over an interval of length \(\Delta t\), solving this differential equation gives:
This solution shows why enzyme change is delayed: the difference between the current value and the target shrinks exponentially, rather than disappearing at once. In a richer model, the target can depend on perpetrator concentration through an induction or inactivation mechanism. The constant-target case is used here to keep the update transparent.
For a victim drug with baseline elimination rate constant \(k_0=CL_0/V_C\), fraction metabolized by the affected enzyme \(f_m\), and reversible competitive inhibition constant \(K_i\), one illustrative effective rate is:
This expression assumes the unaffected fraction of baseline clearance remains unchanged, while the affected pathway scales with enzyme abundance and is reduced by reversible inhibition. It also assumes the concentration used for inhibition is the relevant unbound concentration at the enzyme site; a real PBPK model must justify that choice. Between victim doses, the model is \(dC_V/dt=-k_V(t)C_V(t)\). If \(k_V\) is held constant for one numerical interval, the update is \(C_V(t+\Delta t)=C_V(t)\exp[-k_V\Delta t]\).
Worked Example: Perpetrator Accumulation and Enzyme Turnover
Worked Example: Perpetrator Accumulation and Enzyme Turnover
Suppose a hypothetical perpetrator is given as an intravenous bolus at times 0 and 24 hours. Each dose increases concentration by 1.00 mg/L, and its half-life is 12.0 hours. Thus \(k_P=\ln(2)/12.0=0.05776\ \mathrm{h}^{-1}\). Assume \(E(0)=1\), \(k_{\mathrm{deg}}=0.0500\ \mathrm{h}^{-1}\), and a constant saturated induction target \(E_{\mathrm{target}}=7/3\) during the 48-hour trace.
At 24 hours, the first dose has undergone two half-lives. Therefore \(C_P(24^-)=1.00(2^{-2})=0.250\) mg/L, and the second dose gives \(C_P(24^+)=0.250+1.00=1.25\) mg/L. At 48 hours, before any further dose, the contributions are \(1.00(2^{-4})\) and \(1.00(2^{-2})\), so \(C_P(48^-)=0.0625+0.250=0.3125\) mg/L.
The enzyme update over the first 24 hours is:
The same constant-target update over the next 24 hours gives:
Thus the correct rounded value at 48 hours is \(E(48)\approx2.212\), not 2.213. Enzyme abundance has moved toward the target but has not reached it. At 24 hours the perpetrator concentration jumps from 0.250 to 1.25 mg/L, while \(E\) remains 1.93174 immediately across the dose event.
Worked Example: Victim Concentration Across the Second Interval
Worked Example: Victim Concentration Across the Second Interval
Continue the hypothetical simulation. The victim is on a 24-hour dosing schedule, with doses at 0 and 24 hours. For this interval calculation, specify the simulated pre-dose state \(C_V(24^-)=1.00\) mg/L; the second victim dose increases concentration by 1.00 mg/L, so \(C_V(24^+)=2.00\) mg/L. There is no victim dose at 48 hours; the trace ends at 48 hours before any further dose. Let \(k_0=0.100\ \mathrm{h}^{-1}\), \(f_m=0.400\), and \(K_i=0.0517\) mg/L.
For a hand calculation, use four 6-hour intervals. In each interval, hold \(C_P\) and \(E\) at their values at the interval start when calculating the victim elimination rate. Update \(E\) exactly at each 6-hour step using the constant target above. The resulting rate and concentration updates are:
| Interval (h) | \(C_P\) at start (mg/L) | \(E\) at start | \(k_V\) (h\(^{-1}\)) | \(C_V\) at end (mg/L) |
|---|---|---|---|---|
| 24–30 | 1.250 | 1.93174 | 0.06307 | 1.370 |
| 30–36 | 0.8839 | 2.03583 | 0.06450 | 0.930 |
| 36–42 | 0.6250 | 2.11293 | 0.06646 | 0.624 |
| 42–48 | 0.4419 | 2.17006 | 0.06909 | 0.412 |
For example, in the first interval:
Then \(C_V(30)=2.00\exp[-0.06307(6)]=1.370\) mg/L. Applying the same update successively gives \(C_V(36)\approx0.930\), \(C_V(42)\approx0.624\), and \(C_V(48^-)\approx0.412\) mg/L. The value at 48 hours is pre-dose: adding a dose there would be a third victim dose and would create a different post-dose concentration.
The table is a numerical approximation because it holds the changing perpetrator concentration and enzyme abundance fixed within each 6-hour interval for the victim update. Shorter intervals, or direct numerical integration using their continuously varying values, would generally improve the approximation. The table's rounded values should not be treated as exact model outputs.
Immediate Inhibition and Delayed Enzyme Change
The model separates two mechanisms by their timing. Reversible inhibition depends directly on \(C_P\), so a perpetrator dose can alter the affected clearance pathway immediately. Enzyme abundance \(E\), by contrast, changes according to a turnover equation and remains continuous at the dose time. In the worked example, increasing \(C_P\) from 0.250 to 1.25 mg/L at 24 hours changes the inhibition denominator from \(1+0.250/0.0517\) to \(1+1.25/0.0517\), while \(E\) stays at 1.93174 at that instant.
An increasing enzyme abundance does not necessarily mean that total victim clearance increases at every moment. If perpetrator concentration is also high, reversible inhibition can offset the enzyme-related increase. The net rate depends on both terms in the stated clearance equation. This is why plotting only enzyme abundance, or only perpetrator concentration, is not enough to understand the victim profile.
The constant-target update used above is a special case of the first-order turnover equation, not a claim that induction is always saturated. If the target varies with perpetrator concentration, the model must update \(E\) using that changing target. A numerical implementation can do this at each time step; the step size and the target-function assumptions then matter to the result.
Diagnostics, Alternatives, and Limitations
A useful simulation should pass basic checks before its exposure predictions are interpreted. The perpetrator concentration must not become negative, and at a dose it must increase by exactly the modeled dose increment. Enzyme abundance must remain continuous at dose events and should move toward its specified target at the rate implied by \(k_{\mathrm{deg}}\). Victim concentration should change continuously between doses under the differential equation, remain nonnegative, and show a dose-related jump only when a victim dose is actually administered.
- Check timing and units. Use the same time units for half-life, dose interval, and turnover rate. A rate in \(\mathrm{h}^{-1}\) cannot be multiplied by a time in days without conversion.
- Check event ordering. Distinguish pre-dose from post-dose concentrations. A dose at 48 hours must not be included in a value labeled \(48^-\).
- Check the numerical step size. Repeat the simulation with a shorter step. Large changes in predicted concentrations indicate that the chosen time step is too coarse for the changing rates.
- Check the mechanism. Reversible inhibition and enzyme turnover should not be collapsed into a single instantaneous multiplier when the model intends to represent both.
A static model can be a practical alternative when the question is adequately represented by a selected exposure level and the relevant mechanism is treated as constant over the evaluation period. Dynamic simulation is preferable when the timing of doses, accumulation, turnover, or changing victim concentrations is central to the question. Neither approach removes the need to justify the concentration used at the enzyme site or the assumed clearance pathways.
The compact example has important limitations. It uses a single well-mixed perpetrator concentration, assumes bolus input, treats victim elimination as linear, and represents enzyme abundance with one turnover state. Real PBPK models may need absorption, tissue distribution, unbound concentrations, several enzyme pathways, multiple organs, or other mechanisms. Model complexity should be added only when supported by the question and available information; more states do not automatically make a prediction more reliable.
Check Your Understanding
Answer each question using the assumptions and notation stated in this tutorial.
- A perpetrator has a 12-hour half-life and a 1.00 mg/L dose increment at time 0. What is its concentration immediately before a second dose at 24 hours?
- Why does enzyme abundance not jump when a perpetrator bolus is administered?
- In the worked victim trace, is the value \(C_V(48^-)=0.412\) mg/L before or after a 48-hour dose?
- If the numerical step is reduced from 6 hours to 3 hours, what should be checked before accepting the revised prediction?
Solutions. 1. Two half-lives have elapsed, so \(C_P(24^-)=1.00(2^{-2})=0.250\) mg/L. 2. Enzyme abundance follows a turnover differential equation; a concentration jump changes its driving conditions, but not its state instantaneously. 3. It is before any 48-hour dose; the example administers no dose at that time. 4. Confirm the dose-event timing and units remain unchanged, then compare the predictions. Agreement supports numerical stability for this step-size check; a large difference calls for a smaller step or a more appropriate integration method.