Joint Optimization of Dose and Dosing Interval

Why Dose and Interval Must Be Chosen Together

A maintenance dose and its dosing interval jointly shape the exposure profile between doses. Raising the dose tends to raise concentrations throughout the interval; lengthening the interval gives more time for concentrations to fall before the next dose. Consequently, evaluating dose amount alone—or evaluating average exposure without checking the interval profile—can miss whether clinical targets are met at the times that matter.

The introductory material on decision objectives and constraints for regimen optimization establishes why an objective and its constraints must be specified. Here, the objective will be to use the longest feasible interval, thereby reducing dosing frequency, while keeping the steady-state trough above an efficacy target and the peak below a safety limit. A different objective could produce a different preferred regimen; there is no unique optimum without a stated objective.

Steady-State Profile Under a Simple PK Model

Begin with a linear, one-compartment model with instantaneous intravenous bolus doses. Let \(D\) be the dose in mg, \(\tau\) the dosing interval in hours, \(V\) the apparent volume of distribution in L, and \(k\) the first-order elimination rate constant in h\(^{-1}\). Let \(F\) be the fraction of the dose reaching the systemic circulation; for intravenous dosing, \(F=1\). Assume the same dose is given every \(\tau\) hours, pharmacokinetics are time invariant, elimination is first order, and the patient has reached steady state. These assumptions exclude absorption delay, nonlinear clearance, changing physiology, and missed doses.

Immediately after a dose at steady state, the concentration includes the residual concentration from all previous doses. Each previous dose contributes a fraction \(e^{-k\tau}\) of its preceding contribution. The resulting geometric series gives the profile for \(0\leq t<\tau\), where \(t=0\) is immediately after a dose.

The series converges because \(k>0\) and \(\tau>0\), so \(0<e^{-k\tau}<1\). Concentration declines between instantaneous doses. Thus the maximum occurs at \(t=0\), just after dosing, and the minimum occurs just before the next dose, at \(t=\tau\).

The trough expression follows by evaluating the profile at the end of the interval: multiply the peak by \(e^{-k\tau}\). The average follows because the area under the concentration curve over one steady-state interval is \(FD/CL\); dividing this interval AUC by \(\tau\) gives the average concentration. Peak, trough, and average answer different questions. Meeting an average-exposure target does not establish that a trough or peak constraint is met.

Turning Target Constraints Into Dose Bounds

For fixed \(\tau\), both peak and trough are proportional to dose. Suppose the required trough is at least \(C_{\mathrm{trough,target}}\), while the permitted peak is at most \(C_{\mathrm{peak,max}}\). Rearranging the trough and peak formulas gives a lower and an upper dose bound. A feasible dose exists only if the lower bound does not exceed the upper bound.

These bounds are meaningful when \(F>0\), \(V>0\), \(k>0\), and \(\tau>0\). A dose can satisfy both targets at that interval if and only if \(D_{\min}(\tau)\leq D_{\max}(\tau)\). Equality leaves exactly one dose satisfying both constraints; strict inequality leaves a range of feasible doses. This test avoids selecting a dose first and discovering afterward that the interval makes the targets incompatible.

Joint Optimization for the Longest Feasible Interval

The dose bounds can be used to optimize the interval instead of checking only a few candidate schedules. For the stated objective—maximize \(\tau\) subject to both exposure constraints—feasibility requires

Since \(V/F>0\), cancel it. Set \(x=e^{k\tau}\), which is greater than 1 for positive \(k\) and \(\tau\). Then \(1-e^{-k\tau}=(x-1)/x\), so the inequality becomes \(C_{\mathrm{trough,target}}(x-1)\leq C_{\mathrm{peak,max}}(x-1)/x\). Multiplying by \(x>0\) and dividing by \(x-1>0\) gives \(C_{\mathrm{trough,target}}x\leq C_{\mathrm{peak,max}}\). Thus, when the peak limit exceeds the trough target, the largest feasible interval under this model is

At this limiting interval, the lower and upper dose bounds coincide. For any shorter positive interval the ratio test permits both constraints to be met, subject to any additional dose restrictions; for any longer interval it does not. The result depends on the assumed objective and model. It is not a universal recommendation to choose the longest possible interval.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to check the profile's internal ordering: with positive dose and elimination, the peak must exceed the trough, and the interval average must lie between them. If a computed trough exceeds its peak, inspect whether the interval was placed in the exponent correctly. Also check units: \(k\tau\) must be dimensionless, \(D/V\) has concentration units, and the dose bounds must have dose units. The limiting interval formula requires \(C_{\mathrm{peak,max}}>C_{\mathrm{trough,target}}\); if the two targets are equal or the peak limit is lower, no positive interval satisfies both in this declining-profile model.

A common mistake is to calculate a dose from the trough target and assume the peak will consequently be acceptable. The trough calculation gives only a lower dose bound. The peak target gives an upper bound, and both must be checked at the same interval. Another error is to use \(e^{k\tau}\) where the peak formula requires \(1-e^{-k\tau}\), or to omit the subtraction of 1 in \(e^{k\tau}-1\). Keeping the derived lower and upper bounds visible makes these errors easier to catch.

An average-exposure-only approach is simpler when the clinical target is genuinely defined by interval AUC or average concentration, but it cannot guarantee a minimum trough or maximum peak. Conversely, this analytic calculation is convenient for a one-compartment linear bolus model; a more detailed PK model can calculate steady-state profiles numerically when absorption, multiple compartments, nonlinear elimination, or complex dosing makes the closed-form equations inappropriate. The model should be checked against observed data and the assumptions should be defensible before its calculated regimen is used to support a clinical decision.

Finally, the calculation is deterministic: the same parameters imply the same profile. A regimen meeting the targets for a typical parameter set need not meet them for every patient. Incorporating population variability or uncertainty requires additional analyses; this calculation alone does not establish a population success rate. Its proper role is to make the dose–interval consequences and the feasibility of specified targets explicit under a stated model.