Deriving the Quasi-Steady-State TMDD Reduction

Why Reduce the Complex Equation?

In target-mediated drug disposition (TMDD), drug binds a target to form a complex. That complex can dissociate or be removed through internalization. When complex formation and loss adjust rapidly compared with changes in the surrounding drug and target pools, it can be useful to approximate the complex as adjusting immediately to those pools. This removes one differential equation while retaining the effect of target binding and complex internalization on drug disposition.

The reduction is an approximation, not a claim that the complex is constant over the whole dosing interval. At each time, its value is recalculated from the current drug and target concentrations. The central derivation is to set the complex rate of change approximately to zero and solve the resulting algebraic relation.

Complex Formation and Loss

Let \(D\) denote free drug concentration, \(R\) free target concentration, and \(C\) drug–target complex concentration. Binding occurs at rate \(k_{\mathrm{on}}DR\), where \(k_{\mathrm{on}}\) has units of concentration\(^{-1}\) time\(^{-1}\). Dissociation occurs at rate \(k_{\mathrm{off}}C\), and internalization removes complex at rate \(k_{\mathrm{int}}C\). Thus, the complex equation is

The rate constants \(k_{\mathrm{off}}\) and \(k_{\mathrm{int}}\) have units of inverse time. The model treats dissociation and internalization as competing first-order losses from the complex. In particular, internalization is a loss term in the complex equation even though it is not dissociation back to free drug and target.

Deriving the QSS Binding Relation

Apply the approximation to the complex equation, not to the entire TMDD system:

Rearranging and dividing by the positive rate constant \(k_{\mathrm{on}}\) gives

The expression is dimensionally consistent: \(DR/K_{\mathrm{ss}}\) has units of concentration. It also shows what the approximation means locally: for the current free concentrations, the complex concentration is approximately the value at which complex formation balances dissociation plus internalization.

The composite constant is not generally the equilibrium dissociation constant \(K_D=k_{\mathrm{off}}/k_{\mathrm{on}}\). Here internalization contributes to complex loss, so \(K_{\mathrm{ss}}=K_D+k_{\mathrm{int}}/k_{\mathrm{on}}\). If \(k_{\mathrm{int}}>0\), then \(K_{\mathrm{ss}}>K_D\). Using \(K_D\) in this QSS relation would omit a modeled route by which complex is removed.

Expressing the Complex Using Total Pools

Often, a model or measurement is described in terms of total drug and total target rather than their free concentrations. For a local binding pool in which drug and target concentrations use the same volume basis, define \(D_{\mathrm{tot}}=D+C\) and \(R_{\mathrm{tot}}=R+C\). These definitions count each complex once in each corresponding total. Therefore \(D=D_{\mathrm{tot}}-C\) and \(R=R_{\mathrm{tot}}-C\). Substitution into \(C=DR/K_{\mathrm{ss}}\) gives

or, after expansion,

Solving the quadratic gives two algebraic roots. The physical root is the smaller one:

The other root is not an allowable complex concentration because a complex cannot exceed either total pool. To see why the smaller root lies in the physical range, suppose \(D_{\mathrm{tot}}\leq R_{\mathrm{tot}}\), with both totals positive. For the quadratic polynomial \(f(C)\), \(f(0)=D_{\mathrm{tot}}R_{\mathrm{tot}}>0\), while \(f(D_{\mathrm{tot}})=-K_{\mathrm{ss}}D_{\mathrm{tot}}<0\). Thus one root is between 0 and \(D_{\mathrm{tot}}\); the other lies beyond \(D_{\mathrm{tot}}\). The smaller root is the admissible one. The same argument applies with the roles reversed if \(R_{\mathrm{tot}}\leq D_{\mathrm{tot}}\).

What Makes the Approximation Plausible?

Setting \(dC/dt\) approximately to zero is plausible when the complex adjusts faster than the free pools or totals change. A local indication of the complex adjustment timescale can be obtained by holding \(D_{\mathrm{tot}}\) and \(R_{\mathrm{tot}}\) fixed during a short interval. Since \(D=D_{\mathrm{tot}}-C\) and \(R=R_{\mathrm{tot}}-C\), the complex equation becomes

The negative derivative of the right-hand side with respect to \(C\), evaluated at a candidate complex concentration, is

Under this fixed-total local calculation, perturbations near the QSS value relax on a timescale of roughly \(1/\lambda\). The QSS approximation is more credible when that timescale is short compared with timescales for changes in drug input, free target turnover, or other processes that alter the pools. This comparison is a diagnostic rather than a universal numerical cutoff: a rapidly changing bolus input, for example, can make the approximation less accurate during a transient even if binding is fast at other times.

Using and Checking the Reduced Model

In the full model, \(C\) has its own differential equation. In a QSS reduction, calculate \(C\) algebraically from the current free concentrations, or from the current totals using the smaller-root formula. The other model equations must still account for the consequences of binding and internalization. For example, under the usual convention that only free drug undergoes linear elimination and internalized complex removes drug, a local total-drug balance includes loss terms proportional to \(D\) and \(k_{\mathrm{int}}C\). The algebraic QSS expression does not, by itself, specify the rest of the reduced model.

Several checks help detect implementation or assumption problems:

• Units: \(K_{\mathrm{ss}}\) and every concentration in the algebraic expression must use the same concentration units. The ratio \((k_{\mathrm{off}}+k_{\mathrm{int}})/k_{\mathrm{on}}\) must have concentration units.
• Bounds: verify \(0\leq C\leq\min(D_{\mathrm{tot}},R_{\mathrm{tot}})\), and verify that implied free concentrations are nonnegative.
• Substitution: for a total-pool calculation, check that \(C\) agrees with \((D_{\mathrm{tot}}-C)(R_{\mathrm{tot}}-C)/K_{\mathrm{ss}}\), allowing for rounding.
• Transient behavior: compare the full model and QSS predictions where possible, especially around rapid dosing or when target pools change quickly. Large discrepancies suggest that complex adjustment is not fast enough there.
• Numerics: when \(K_{\mathrm{ss}}\) is small relative to large totals, subtracting nearly equal quantities in the smaller-root formula can lose precision. A solver implementation may use an algebraically equivalent, numerically stable root calculation and should still check the physical bounds.

Alternatives and Limitations

The direct alternative is to retain the differential equation for \(C\) and simulate the full TMDD system. This avoids assuming fast adjustment, but it adds a state and may require resolving a fast timescale. The QSS reduction is useful when its timescale assumption is supported and a simpler model is appropriate; it can be inaccurate during rapid transients or when complex formation and loss are not fast relative to changes in drug or target.

A separate approximation based on rapid binding equilibrium uses a different balance: it omits internalization from the association–dissociation balance and uses \(K_D=k_{\mathrm{off}}/k_{\mathrm{on}}\). QSS instead balances formation against both dissociation and complex internalization, giving \(K_{\mathrm{ss}}\). The two constants should not be interchanged. A further limitation is that the total-pool formula assumes drug, target, and complex are expressed on a compatible local concentration basis. If the species occupy different effective volumes or move between compartments, their amounts and compartment-specific concentrations must be accounted for before applying a total-pool relation.

Common Mistake

A frequent error is to set \(dC/dt=0\) and then write \(C=DR/K_D\) without checking the complex loss terms. That reasoning treats dissociation as the only route out of the complex. In the stated model, internalization also removes complex, so the correct QSS balance is \(k_{\mathrm{on}}DR=(k_{\mathrm{off}}+k_{\mathrm{int}})C\), giving \(C=DR/K_{\mathrm{ss}}\). Another error is to interpret \(dC/dt\approx0\) as saying that free drug, free target, and total pools are also constant. The approximation applies only to the complex equation.