Tutorials › Physiologically-Based Pharmacokinetics (PBPK) › Worked Comparison of Static and Dynamic DDI Predictions

Home  /  Tutorials  /  Static Versus Dynamic DDI Prediction

Static Versus Dynamic DDI Prediction · Tutorial 485 of 799

Worked Comparison of Static and Dynamic DDI Predictions

Calculate static and dynamic AUCRs for a hypothetical repeated-dose interaction, and see how concentration profiles and changing enzyme activity affect the comparison.

Advanced 12 min read

What You'll Learn

  • Define steady-state AUCR for repeated victim dosing.
  • Calculate a static AUCR from a representative inhibitor concentration and fraction metabolized.
  • Translate changing inhibitor concentrations into time-varying victim clearance.
  • Derive and calculate the dynamic repeated-dose AUC for a piecewise clearance profile.
  • Explain why evaluating inhibition at mean concentration differs from averaging enzyme activity.
  • Check a dynamic calculation using periodicity and mass balance.

Why a Worked Comparison Needs a Shared Basis

Static and dynamic DDI predictions are comparable only when they refer to the same victim drug, dosing regimen, interaction mechanism, and exposure measure. Here, both predictions are steady-state ratios of victim AUC over one dosing interval, with and without the perpetrator. The static calculation treats inhibition as constant; the dynamic calculation lets inhibition, and therefore victim clearance, change over the interval.

The hypothetical example uses repeated intravenous bolus doses of a victim drug and a repeatedly dosed inhibitor. It is deliberately simplified to isolate the effect of changing enzyme activity. It is not a prediction for a real drug or study.

Definition: For a repeated-dose regimen at periodic steady state, the victim AUC ratio is \(AUCR=AUC_{\tau,\text{with inhibitor}}/AUC_{\tau,\text{without inhibitor}}\), where \(AUC_{\tau}\) is the victim concentration integrated over one dosing interval of length \(\tau\). An \(AUCR\) above 1 indicates greater victim exposure with the inhibitor under the stated conditions.

The static calculation will use reversible competitive inhibition of one metabolic pathway. Let \(f_m\) be the fraction of baseline victim clearance through that pathway, \(I\) the unbound inhibitor concentration at the enzyme site, and \(K_i\) the inhibition constant. The remaining fraction, \(1-f_m\), is unaffected. Under these assumptions, the static AUCR is the reciprocal of the relative clearance.

$$ AUCR_{\text{static}} = \frac{1}{(1-f_m)+\dfrac{f_m}{1+I/K_i}} $$

This expression assumes linear victim pharmacokinetics, a fixed representative inhibitor concentration, and no induction or time-dependent loss of enzyme. The dynamic calculation below retains reversible inhibition but allows \(I\) to vary during each interval.

Worked Example: Static Prediction at Mean Inhibitor Concentration

Worked Example: Static Prediction at Mean Inhibitor Concentration

Suppose 80% of the victim's baseline clearance is through the inhibited pathway, so \(f_m=0.80\). Across a 24-hour inhibitor dosing interval, assume the unbound inhibitor concentration is \(3K_i\) for 12 hours and \(K_i/3\) for 12 hours. For a static prediction, use the time-averaged inhibitor concentration.

The mean concentration relative to \(K_i\) is \(I_{\text{mean}}/K_i=[3+(1/3)]/2=5/3=1.667\). Substituting into the static expression gives

$$ AUCR_{\text{static}} = \frac{1}{(1-0.80)+\dfrac{0.80}{1+1.667}} = \frac{1}{0.20+0.30} = 2.00 $$

The static prediction is an \(AUCR\) of 2.00: under this constant-concentration approximation, victim AUC over the interval is predicted to be twice the no-inhibitor AUC. The result is conditional on using the mean concentration as the representative static input; it does not account for when the inhibitor concentration is high or low.

Translating the Profile into Dynamic Clearance

For the dynamic calculation, the same reversible inhibition relationship is evaluated separately in each half of the interval. The inhibitor concentration is \(3K_i\) during hours 0–12 and \(K_i/3\) during hours 12–24. The fraction of baseline clearance remaining through the inhibited pathway is \(1/(1+I/K_i)\). Thus, relative to baseline total clearance, the victim's clearance is

$$ \frac{CL(t)}{CL_0} = (1-f_m)+\frac{f_m}{1+I(t)/K_i} $$

During the high-inhibitor half, the inhibited pathway retains \(1/(1+3)=0.25\) of its baseline activity. Total victim clearance is \(0.20+0.80(0.25)=0.40\) of baseline. During the low-inhibitor half, pathway activity is \(1/(1+1/3)=0.75\), and total clearance is \(0.20+0.80(0.75)=0.80\) of baseline.

Assume a one-compartment victim model with instantaneous intravenous bolus dosing and first-order elimination. Let \(A=D/V\) be the increase in victim concentration caused by one dose, and let \(k_0=CL_0/V=0.10\ \text{h}^{-1}\) be the baseline elimination rate constant. Then the dynamic elimination rate constants are \(k_1=0.40k_0=0.04\ \text{h}^{-1}\) and \(k_2=0.80k_0=0.08\ \text{h}^{-1}\). The model assumes the inhibitor profile repeats each day, inhibition responds immediately to concentration, clearance is constant within each 12-hour segment, and victim disposition remains linear.

Worked Example: Dynamic AUC at Periodic Steady State

Worked Example: Dynamic AUC at Periodic Steady State

Let each daily victim dose produce \(A=1.00\ \text{mg/L}\) of concentration immediately after dosing. The victim is dosed every 24 hours. Calculate the AUC over a dosing interval at periodic steady state, when the concentration profile at the start of each day repeats.

Let \(C_0\) be the victim concentration immediately before the daily dose. Immediately after dosing, concentration is \(C_0+A\). In the first 12 hours, the fraction remaining is \(r_1=e^{-k_1(12)}=e^{-0.48}=0.6188\). In the next 12 hours, the fraction remaining is \(r_2=e^{-k_2(12)}=e^{-0.96}=0.3829\). The fraction remaining across the whole interval is \(q=r_1r_2=e^{-1.44}=0.2369\).

At periodic steady state, the concentration just before the next dose equals the fraction of the post-dose concentration remaining after 24 hours:

$$ C_0=q(C_0+A) \qquad\Longrightarrow\qquad C_0=\frac{qA}{1-q} = \frac{0.2369(1.00\ \text{mg/L})}{1-0.2369} = 0.3105\ \text{mg/L} $$

For first-order elimination over a segment with rate constant \(k\), integrating an exponential concentration profile that starts at \(C_{\text{start}}\) gives \(C_{\text{start}}(1-e^{-k\Delta t})/k\). Applying that result to both segments, with the second segment starting at \((C_0+A)r_1\), gives:

$$ \begin{aligned} AUC_{\tau,\text{with inhibitor}} &=(C_0+A)\frac{1-r_1}{k_1} +(C_0+A)r_1\frac{1-r_2}{k_2}\\ &=(1.3105\ \text{mg/L}) \left[ \frac{1-0.6188}{0.04\ \text{h}^{-1}} + 0.6188\frac{1-0.3829}{0.08\ \text{h}^{-1}} \right]\\ &=(1.3105\ \text{mg/L})(9.530+4.773\ \text{h})\\ &=18.74\ \text{mg}\cdot\text{h/L} \end{aligned} $$

Without inhibitor, the rate constant is \(k_0=0.10\ \text{h}^{-1}\) throughout. At periodic steady state, the AUC per interval is \(A/k_0\), or \(1.00\ \text{mg/L}/0.10\ \text{h}^{-1}=10.00\ \text{mg}\cdot\text{h/L}\). Therefore,

$$ AUCR_{\text{dynamic}}=\frac{18.74}{10.00}=1.874 $$

The dynamic prediction is an \(AUCR\) of 1.874, compared with 2.00 from the static calculation. In this example, the static prediction is higher; that direction is not guaranteed for other concentration profiles or victim models.

Why the Two Calculations Differ

The static calculation first averages inhibitor concentration and then evaluates the inhibition function at that mean. The dynamic model instead evaluates inhibition at each time and carries the resulting victim concentration forward. Those operations need not give the same result: the inhibition function is nonlinear, and victim concentration reflects the history of clearance as well as its value at the current time.

The worked dynamic model also illustrates why simply taking the reciprocal of a mean clearance ratio is not generally an AUC calculation. With constant clearance, dose divided by clearance determines the steady-state AUC for a linear victim. With time-varying clearance, the concentration changes throughout the interval, and the relevant mass balance involves the product of clearance and concentration at each time.

Derivation: Over one interval at periodic steady state, the victim concentration returns to its starting value after the dose has been added and eliminated. For the bolus model, the concentration increment from the dose is \(A=D/V\). Integrating the differential equation between doses gives \(\int_0^\tau k(t)C(t)\,dt=A\), where \(k(t)=CL(t)/V\). This constrains the clearance-weighted concentration, not the unweighted AUC \(\int_0^\tau C(t)\,dt\). When \(k(t)\) varies, the AUC therefore depends on the timing of the concentration profile relative to the changing elimination rate.

Worked Example: Averaging Activity Is Not the Same as a Dynamic AUC

Worked Example: Averaging Activity Is Not the Same as a Dynamic AUC

For the two equal-duration inhibitor segments, pathway activity is 0.25 and 0.75. Their arithmetic mean is \(0.50\). If this mean activity is used to create one constant clearance ratio, the result is \(0.20+0.80(0.50)=0.60\), giving a reciprocal-clearance estimate of \(1/0.60=1.667\).

This differs from the static result based on mean inhibitor concentration, \(2.00\), and from the dynamic interval simulation, \(1.874\). The value 1.667 is a constant-clearance approximation based on mean activity, not the dynamic AUC ratio. It omits the way victim concentration accumulates more during periods of reduced clearance and then experiences a different elimination rate later. Neither averaging activity nor averaging concentration substitutes, in general, for integrating the victim concentration trajectory.

Checks, Comparisons, and Limitations

A segmented dynamic calculation is useful when a representative constant concentration may hide meaningful changes within the dosing interval. A static calculation is simpler and can be informative when its concentration input and mechanism are appropriate. A more detailed PBPK simulation can represent additional compartments, absorption, distribution, enzyme-site concentrations, and other time-varying processes, but it also requires more structural and parameter assumptions. The simplified calculation here should not be mistaken for such a full model.

Several checks help identify arithmetic or modeling failures. First, verify that the inhibitor profile repeats at periodic steady state and that the victim concentration at the end of the interval matches the pre-dose concentration at its start. Second, verify the units: each integrated concentration term must have units of concentration multiplied by time. Third, check mass balance. In this example, the clearance-weighted AUCs of the two segments sum to approximately \(1.00\ \text{mg/L}\), matching the dose increment \(A\). A mismatch can signal an incorrect segment start concentration, missing accumulation, or inconsistent units.

The result is limited by its assumptions: one victim compartment, intravenous bolus dosing, linear elimination, two piecewise-constant inhibitor levels, immediate reversible inhibition, and no effects beyond the specified pathway. Real concentration profiles may vary smoothly, and inhibitor concentration at the enzyme site may not equal a chosen plasma concentration. The result also depends on \(f_m\), \(K_i\), and the assumed inhibitor profile. Consequently, the numerical difference between static and dynamic predictions is an illustration of the mechanism, not a general correction factor.

A frequent mistake is to conclude that the dynamic AUCR must equal the reciprocal of clearance evaluated at the average inhibitor concentration. That reasoning treats a time-varying clearance as though it were constant and ignores the evolving victim concentration. The correction is to define the exposure interval and dosing regimen, translate the inhibitor profile into clearance over time, solve the victim model to periodic steady state, and integrate the resulting concentration profile.

1
Establish a common comparison.
Use the same victim regimen and the same steady-state AUC interval for the with-inhibitor and without-inhibitor conditions.
2
Calculate the static estimate.
State the representative inhibitor concentration and inhibition assumptions before applying the static AUCR equation.
3
Simulate the dynamic profile.
Translate concentration changes into clearance changes, account for accumulation, and integrate victim concentration over the interval.
4
Check and interpret.
Confirm periodicity, units, and mass balance, then describe the difference as conditional on the modeled assumptions.

Check Your Understanding

Try each question, then check the solutions below.

  1. If \(f_m=0.60\) and a static calculation uses \(I/K_i=1\), what is the predicted static AUCR?
  2. In the worked dynamic example, what fraction of baseline total clearance remains during the high-inhibitor half?
  3. Why does the dynamic steady-state calculation require a pre-dose concentration \(C_0\), rather than assuming the victim concentration is zero at the start of each day?
  4. What does the dynamic AUCR of 1.874 mean in the hypothetical example?

Solutions. For question 1, \(AUCR=1/[(1-0.60)+0.60/(1+1)]=1/(0.40+0.30)=1.429\). For question 2, the inhibited pathway retains 0.25 of its activity, so total clearance remaining is \(0.20+0.80(0.25)=0.40\). For question 3, repeated victim doses leave residual drug from earlier doses; at periodic steady state that residual is represented by \(C_0\), determined by the repeated interval profile. For question 4, the modeled victim AUC over a 24-hour interval with inhibitor is 1.874 times the corresponding no-inhibitor AUC at periodic steady state.

Previous

Next

Back to Tutorials