Basic Static Reversible-Inhibition Equations

From Reversible Inhibition to Exposure Change

A static drug–drug interaction (DDI) calculation uses a specified inhibitor concentration to approximate how much a victim drug’s metabolism is reduced. The basic calculation has two parts: first, quantify inhibition of an enzyme pathway; then, translate that pathway change into a change in the victim drug’s total clearance and exposure. Keeping those parts separate prevents a common error: treating inhibition of one enzyme as if it necessarily inhibited all victim-drug clearance.

The equations below describe reversible competitive inhibition. They are static approximations: the inhibitor concentration is represented by a selected value rather than a concentration that changes over time. The earlier material on static and dynamic DDI models frames this distinction; here the focus is the equation and the conditions that make it interpretable.

Definition: The inhibition ratio, $IR$, is the ratio of the victim-drug clearance through a specified enzyme pathway without inhibitor to that pathway’s clearance with inhibitor, under the model assumptions. For low-substrate, reversible competitive inhibition, $IR=1+[I]_u/K_i$, where $[I]_u$ is the unbound inhibitor concentration at the enzyme site and $K_i$ is the inhibition constant expressed in concentration units consistent with $[I]_u$.

Deriving the Inhibition Ratio

For a competitive inhibitor, inhibitor binding reduces the enzyme available to bind substrate, represented by an increase in the apparent Michaelis constant from $K_m$ to $K_m(1+[I]_u/K_i)$. The model assumes reversible binding, a specified inhibitor concentration, and the same substrate concentration with and without inhibitor. The Michaelis–Menten rates are then:

v_0=\frac{V_{\max}[S]}{K_m+[S]}, \qquad v_I=\frac{V_{\max}[S]}{K_m(1+[I]_u/K_i)+[S]}.

Here $v_0$ is the pathway rate without inhibitor, $v_I$ is the rate with inhibitor, and $[S]$ is substrate concentration. At low substrate concentration relative to $K_m$, each rate is approximately proportional to substrate concentration. Dividing the uninhibited rate by the inhibited rate gives:

\frac{v_0}{v_I} = \frac{K_m(1+[I]_u/K_i)+[S]}{K_m+[S]} \approx 1+\frac{[I]_u}{K_i} \quad \text{when } [S]\ll K_m.

In that low-substrate regime, pathway clearance is proportional to rate per unit substrate concentration, so the same ratio applies to pathway clearance. Thus inhibitor exposure equal to $K_i$ gives $IR=2$: the pathway clearance is predicted to be half its uninhibited value. If substrate concentration is not small relative to $K_m$, the exact rate ratio in the displayed derivation is smaller than $1+[I]_u/K_i$ for positive $[S]$. The simple inhibition ratio should therefore not be used as though it were exact under substrate-saturating conditions.

The concentration and inhibition constant must refer to compatible conditions. In this equation, $[I]_u$ means unbound inhibitor at the enzyme site, not automatically the total concentration measured in plasma. $K_i$ is the inhibitor concentration associated with the competitive binding model; it is not a victim-drug concentration or an enzyme abundance. If an experiment reports a different inhibition measure, it should not be substituted without establishing that it is appropriate for this model.

Worked Example: Inhibition of a Single Enzyme Pathway

Suppose a hypothetical victim drug is cleared entirely through one enzyme pathway. At the enzyme site, the assumed unbound inhibitor concentration is $0.30\ \mu\text{mol/L}$, and the competitive inhibition constant is $0.10\ \mu\text{mol/L}$. Assume low substrate concentration, reversible competitive inhibition, and linear victim-drug pharmacokinetics.

The inhibition ratio is:

IR=1+\frac{[I]_u}{K_i} =1+\frac{0.30\ \mu\text{mol/L}}{0.10\ \mu\text{mol/L}} =1+3=4.

The units cancel, so $IR$ is dimensionless. The pathway clearance with inhibitor is $1/4$ of its uninhibited value, or $25\%$. Because this hypothetical drug has no other clearance route, total clearance also falls to $25\%$ of baseline. With unchanged bioavailability and dose, its predicted AUC is four times baseline. This conclusion depends on the stated all-clearance-through-one-pathway assumption.

From Pathway Inhibition to Victim AUCR

For a victim drug with more than one clearance contribution, let $CL_{\text{base}}$ be its baseline total clearance, $CL_{\text{inh}}$ the baseline clearance through the inhibited pathway, and $CL_{\text{other}}$ the sum of unaffected clearance contributions. Assume these contributions are additive and first-order over the exposure range of interest. Define $f_m$ as the fraction of baseline total clearance attributable to the pathway being inhibited:

CL_{\text{base}}=CL_{\text{inh}}+CL_{\text{other}}, \qquad f_m=\frac{CL_{\text{inh}}}{CL_{\text{base}}}.

If inhibition divides clearance through that pathway by $IR$, while leaving other clearance unchanged, the new total clearance is $CL_{\text{inh}}/IR+CL_{\text{other}}$. Dividing by baseline clearance and substituting the definition of $f_m$ gives the clearance ratio:

\frac{CL_{\text{with inhibitor}}}{CL_{\text{base}}} = \frac{CL_{\text{inh}}/IR+CL_{\text{other}}}{CL_{\text{base}}} = \frac{f_m}{IR}+(1-f_m).

For linear pharmacokinetics, the AUC over a complete dosing interval is proportional to dose times bioavailability divided by clearance. Therefore, if dose and bioavailability are unchanged, the ratio of victim AUC with inhibitor to victim AUC without inhibitor is the inverse of the clearance ratio.

Definition: Under the assumptions of linear victim-drug pharmacokinetics, unchanged dose and bioavailability, additive clearance pathways, and inhibition confined to the pathway represented by $f_m$, the static victim area-under-the-curve ratio is: $$ AUCR=\frac{AUC_{\text{with inhibitor}}}{AUC_{\text{without inhibitor}}} =\frac{1}{(1-f_m)+f_m/IR} =\frac{1}{(1-f_m)+f_m/(1+[I]_u/K_i)}. $$

This result has useful checks. If $[I]_u=0$, then $IR=1$ and $AUCR=1$. If $f_m=0$, the inhibited pathway contributes no baseline clearance, so the equation also gives $AUCR=1$. If $f_m=1$, all baseline clearance is through the inhibited pathway and $AUCR=IR$. For $0\leq f_m\leq1$ and $IR\geq1$, predicted total clearance cannot increase, and $AUCR$ cannot be less than one.

Worked Example: Inhibiting One of Two Clearance Contributions

Consider a hypothetical victim drug for which $70\%$ of baseline clearance is through an enzyme pathway subject to competitive inhibition. Let $[I]_u=0.20\ \mu\text{mol/L}$ and $K_i=0.10\ \mu\text{mol/L}$. Assume the remaining clearance is unaffected, substrate is low relative to $K_m$, and dose and bioavailability are unchanged.

First calculate the pathway inhibition ratio:

IR=1+\frac{0.20}{0.10}=3.

Set $f_m=0.70$. The fraction of baseline clearance not assigned to this pathway is $1-f_m=0.30$. The predicted total clearance ratio is:

\frac{CL_{\text{with inhibitor}}}{CL_{\text{base}}} =(1-0.70)+\frac{0.70}{3} =0.30+0.2333 =0.5333.

Consequently:

AUCR=\frac{1}{0.5333}=1.875.

The model predicts an AUC with inhibitor about $1.875$ times the baseline AUC, or an $87.5\%$ increase, given the stated assumptions. It does not predict a threefold increase in total AUC: the inhibition ratio of 3 applies to the specified enzyme pathway, while the other $30\%$ of baseline clearance remains in the model.

Worked Example: Using an Unbound Concentration

In a hypothetical calculation, a total plasma inhibitor concentration is $2.0\ \mu\text{mol/L}$, and the unbound fraction is $0.050$. Assume, specifically for this example, that unbound plasma concentration is an appropriate proxy for unbound concentration at the enzyme site. Let $K_i=0.050\ \mu\text{mol/L}$, and let the inhibited pathway account for $50\%$ of baseline victim clearance.

First convert total to unbound concentration using the stated unbound fraction:

[I]_u=2.0\ \mu\text{mol/L}\times0.050 =0.10\ \mu\text{mol/L}.

Then calculate the inhibition ratio and victim AUCR:

IR=1+\frac{0.10}{0.050}=3, \qquad AUCR=\frac{1}{(1-0.50)+0.50/3} =\frac{1}{0.6667} =1.500.

The predicted AUC is $1.500$ times baseline. Substituting the total concentration directly would give $IR=1+2.0/0.050=41$, a very different result, and would be inconsistent with the example’s stated concentration assumption. In a real application, the relationship between unbound plasma concentration and enzyme-site concentration must be justified rather than presumed.

Assumptions, Diagnostics, and Alternatives

The equations are useful because they separate an enzyme-level effect from its consequence for total victim exposure. They are not a general solution for every DDI mechanism. Before interpreting a calculated AUCR, check that the intended mechanism is reversible competitive inhibition, that the concentration used is appropriate to the enzyme site, and that the $K_i$ and concentration use compatible units and binding conventions. Also check that the victim drug behaves approximately linearly over the relevant range, clearance contributions can be treated as additive, and the inhibitor does not independently alter bioavailability or other clearance routes.

Several warning signs suggest the simple calculation may be inadequate. A predicted AUCR below one despite an inhibitor-only model with $IR\geq1$ and unchanged bioavailability usually indicates an algebra or input error. An unusually large $IR$ can arise from using total instead of unbound concentration, mismatched concentration units, or an inappropriate inhibition constant. A prediction based on low-substrate kinetics is also questionable when the enzyme is substantially saturated. Finally, an inhibitor concentration that varies substantially during the period relevant to victim exposure cannot be represented faithfully by one constant value without an explicit justification.

A dynamic model is the main alternative when concentration changes over time matter to the question: it can represent time-varying perpetrator exposure and victim-drug concentrations rather than compressing the interaction into one concentration and one ratio. That added representation requires suitable concentration-time and mechanistic inputs. The static equations are simpler and transparent for screening or approximation, but they do not describe interaction onset, offset, or within-interval concentration changes.

Common pitfall: “The enzyme pathway has $IR=4$, so the victim AUC must increase fourfold.” This is correct only when all baseline victim clearance is through that pathway and the other AUCR assumptions hold. If the pathway contributes only part of baseline clearance, use $AUCR=1/((1-f_m)+f_m/IR)$; unaffected clearance limits the total exposure change.

Check Your Understanding

Use the equations and assumptions in this tutorial. The solutions are provided below the questions.

For $[I]_u=0.15\ \mu\text{mol/L}$ and $K_i=0.05\ \mu\text{mol/L}$, calculate $IR$ under the low-substrate competitive-inhibition assumption.
A victim drug has $f_m=0.40$ for the inhibited pathway and $IR=2$. Calculate its predicted AUCR, assuming unchanged dose and bioavailability.
What AUCR does the model predict if $f_m=1$ and $IR=5$? State why.
Give two reasons why substituting a measured total plasma inhibitor concentration directly for $[I]_u$ could be inappropriate.

Solutions. (1) $IR=1+0.15/0.05=4$, dimensionless. (2) $AUCR=1/((1-0.40)+0.40/2)=1/(0.60+0.20)=1.25$. (3) $AUCR=1/(0+1/5)=5$, because all baseline clearance is assigned to the inhibited pathway, so the total clearance changes by the pathway inhibition ratio. (4) Total concentration includes bound inhibitor, whereas the equation uses unbound concentration; additionally, unbound plasma concentration may not represent unbound concentration at the enzyme site. The relevance of each issue depends on the specific system and assumptions.

Basic Static Reversible-Inhibition Equations

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