Accumulation Under Repeated Dosing With Saturable Elimination

Why Repeated Doses Behave Differently Under Saturation

With repeated doses, each new dose arrives before the drug from earlier doses has necessarily been eliminated. Under Michaelis–Menten elimination, the elimination rate increases with concentration but approaches a maximum rather than increasing without bound. As a result, a large concentration peak can be eliminated at a rate close to capacity, while a low concentration is eliminated much more slowly. Accumulation and the peak–trough pattern therefore depend on both the dose and the concentrations reached during the interval.

The key calculation is a mass balance over one dosing interval. At steady state, the amount eliminated during an interval must equal the dose added at its beginning. Combining this balance with the integrated Michaelis–Menten equation gives a direct relationship between the steady-state peak and trough—and a condition for whether a finite steady state can exist at all.

Model and Interval Mass Balance

Assume a one-compartment model with concentration \(C\), constant volume \(V\), and Michaelis–Menten elimination. Doses are instantaneous intravenous boluses of amount \(D\), given every \(\tau\) units of time. Between doses, \(V_{\max}\) and \(K_m\) are constant, there is no other elimination pathway or input, and concentration remains positive. The elimination rate in amount per time is \(V_{\max}C/(K_m+C)\).

Between doses, concentration decreases according to \(V\,dC/dt=-V_{\max}C/(K_m+C)\). Integrating from a post-dose concentration \(C_{\mathrm{start}}\) to a later concentration \(C_{\mathrm{end}}\) after elapsed time \(\Delta t\) gives

The left side has units of amount: \(V\) times concentration. The right side is the maximum elimination rate multiplied by time, also an amount. The logarithmic term matters: unlike a linear-elimination model, the concentration decline over an interval is not described by a single exponential rate constant.

Steady-State Peaks, Troughs, and Feasibility

At steady state, write \(C_{\mathrm{ss}}^+\) for the post-dose peak and \(C_{\mathrm{ss}}^-\) for the next pre-dose trough. The bolus raises concentration by \(D/V\), while the amount eliminated between those concentrations is \(V(C_{\mathrm{ss}}^+-C_{\mathrm{ss}}^-)\). Steady-state mass balance therefore requires

Apply the integrated equation over the full interval, so that \(C_{\mathrm{start}}=C_{\mathrm{ss}}^+\), \(C_{\mathrm{end}}=C_{\mathrm{ss}}^-\), and \(\Delta t=\tau\). Substituting the mass-balance difference gives

For finite positive concentrations, the logarithm is positive. Thus \(D<V_{\max}\tau\) is necessary for a finite steady-state cycle. It is also sufficient under the stated model: the resulting trough and peak are positive and satisfy both interval elimination and dose mass balance. Define \(d=D/V\), the concentration increase at each bolus. Solving the equation for the trough gives

When \(D=V_{\max}\tau\), no finite positive trough can satisfy the equation: that would require the peak-to-trough logarithm to be zero even though the dose makes the peak higher than the trough. When \(D>V_{\max}\tau\), the equation would require a negative logarithm, also impossible. In either case, each interval adds at least as much drug as the model can eliminate, so concentrations cannot settle to a finite repeating cycle.

Accumulation Across Intervals and Stability

The peak–trough formula describes the eventual repeating cycle, not the concentrations after the first few doses. To see why the cycle is approached when it exists, let \(F_\tau(C)\) denote the trough after an interval of length \(\tau\) that begins at concentration \(C\). It is determined implicitly by the integrated equation:

After the next bolus, the concentration is \(F_\tau(C)+d\). Thus the post-dose concentrations follow the map \(C_{\mathrm{next}}=F_\tau(C)+d\). A steady-state peak is a fixed point of this map. Differentiating the implicit interval equation with respect to its starting concentration gives

Because elimination makes \(0<F_\tau(C)<C\), the denominator exceeds the numerator and \(0<F_\tau'(C)<1\). Near a finite fixed point, a small change in the starting peak is therefore reduced over the interval map. This gives a local stability justification: concentrations close to the steady cycle tend to move back toward it over subsequent intervals.

A finite cycle is also consistent with the direction of change at low and high concentrations. At very low concentrations, little drug is eliminated in an interval, so a positive dose raises the amount. At very high concentrations, elimination over an interval approaches \(V_{\max}\tau\); when that capacity exceeds the dose, the interval removes more than the dose and concentrations are pushed downward. Between those extremes, the steady cycle balances input and elimination.

How the Dose Interval Changes the Pattern

When the same total daily dose is divided into more frequent boluses, the individual dose increment \(D/V\) is smaller, and each interval provides less time for elimination. The formulas show the resulting trough and peak together. This does not mean that saturable elimination has become linear: the interval-specific trough still depends on \(V_{\max}\), \(K_m\), the dose, and the interval through a nonlinear exponential relationship.

Assumptions, Diagnostics, and Alternatives

The calculations require a one-compartment concentration, an instantaneous dose, and constant Michaelis–Menten parameters throughout each interval and across repeated doses. If absorption takes time, if dose timing varies, if volume or elimination capacity changes, or if another elimination pathway contributes, the derived peak–trough formula need not describe the data. It also predicts concentrations only at the peak and trough; it does not by itself establish that the model describes the full observed profile.

In practice, compare predicted and observed concentrations at known times after dosing, including both peaks and pre-dose troughs. Repeatedly rising troughs rather than a plateau may be consistent with insufficient elimination capacity, but can also reflect missed timing information, changing parameters, or model misspecification. A single peak or trough cannot establish a steady-state cycle. Confirm that sampling occurred at the intended times and that the regimen had been maintained long enough for the assumed cycle to be plausible.

The integrated interval equation is useful when doses are idealized as boluses and the assumptions hold; numerical integration of the differential equation is a flexible alternative when the input profile or parameters vary with time. A linear-elimination approximation may be useful only over concentrations where elimination is approximately proportional to concentration. It should not be used to infer a universal accumulation factor for a regimen spanning a range where elimination saturates.

• Using capacity as though it were the actual elimination amount. \(V_{\max}\tau\) is an upper bound over an interval, not the amount necessarily eliminated. At finite concentrations, the rate is below \(V_{\max}\).
• Assuming every regimen reaches a finite steady state. Check \(D<V_{\max}\tau\) before calculating a steady-state peak and trough.
• Applying a linear accumulation ratio. Saturable elimination does not generally permit superposition or one dose-independent accumulation factor.