Representing Dose Schedules as Optimization Variables

From a Regimen Description to Decision Variables

A dose recommendation is not always a single number. A regimen can specify how much drug is given, how often it is administered, when administration begins, and how long an infusion lasts. If an optimization is allowed to choose any of these features, they must be represented explicitly as decision variables. The practical rules governing which choices are permitted must also be stated; together, those rules define the feasible set.

This representation is an important step between describing a treatment in clinical language and evaluating candidate regimens with a pharmacometric model. It is not yet a choice of an optimal regimen: no efficacy, safety, or other objective is calculated here. The goal is to make the allowed decision space precise enough that a later evaluation cannot accidentally compare regimens that could not be prescribed or administered.

These quantities describe different aspects of a schedule. \(D\) is an amount per administration, not an amount per day. \(\tau\) is elapsed time between successive administration starts. \(t_0\) is a clock-time phase that locates a schedule within the day; it is not the same quantity as \(\tau\). \(T_{\mathrm{inf}}\) is the duration of an infusion, not its rate. If a regimen uses bolus administration, it can be represented by \(T_{\mathrm{inf}}=0\); if only infusions are allowed, a positive lower bound is needed instead.

For a regular schedule with administrations indexed by integers \(k\), the start time of administration \(k\) is \(t_k=t_0+k\tau\), measured on an elapsed-time scale. Clock time repeats every 24 hours, so the clock-time representation of a start is \(t_k\) modulo 24. This indexing describes the schedule without making any claim about the concentration profile produced by repeated doses.

Writing the Feasible Set

A feasible set is commonly built by intersecting several kinds of restrictions. Continuous bounds restrict a variable to an interval. Discrete restrictions permit only certain choices, such as dose increments or a menu of approved intervals. Relational constraints connect variables, for example by limiting the total amount administered per day or requiring an infusion to finish within a permitted administration window.

Here \(\mathcal{W}\) is the set of permitted start times. The bounds must be chosen for the formulation and use case under consideration; they are not universal pharmacological constants. If dose amounts must be multiples of an increment \(q\), a precise restriction is \(D=mq\) for an integer \(m\), together with the dose bounds. If interval choices must come from a specified menu, write \(\tau\in\mathcal{T}\), where \(\mathcal{T}\) is that menu. Replacing such rules with continuous bounds alone would incorrectly allow intermediate choices.

A daily amount limit can be written in more than one way, depending on the schedule definition. For a regular schedule, the long-run average administered amount per 24 hours is \(24D/\tau\), with units of amount per day. Thus an average daily limit \(M\) can be expressed as \(24D/\tau\le M\). If the rule instead limits the amount in each fixed calendar day, the exact count and timing of administrations in that day matter, especially when starts can cross midnight. The formulation should match the actual rule rather than treating those two interpretations as automatically identical.

Deriving Constraints from Operational Rules

Translating a written restriction is a small but important derivation. First identify the quantity the rule limits. Then state its units and the range of values that meet the rule. If a restriction depends on more than one decision variable, preserve that dependence rather than applying separate bounds that do not ensure the rule.

For example, suppose each administration contributes \(D\) mg and the maximum allowed long-run average is \(M\) mg per 24 h. A regular schedule has \(1/\tau\) administrations per hour on average, so its average amount per hour is \(D/\tau\) mg/h. Multiplying by 24 h gives \(24D/\tau\) mg per 24 h. The limit therefore becomes

This is a coupled constraint: a larger amount per administration may be permitted when administrations are farther apart, and vice versa. Applying only \(D\le M\) would not enforce the daily limit because it compares an amount per administration with an amount per day. The unit check exposes that error.

Time Windows and Infusion Duration

Clock-time restrictions require an explicit convention. Here \(t_0\) is the start time on a 24-hour clock, represented as hours after midnight, so 08:30 is \(8.5\) h. If starts are allowed from 06:00 through 20:00, then \(6\le t_0\le20\). When a rule also requires the infusion to finish before a cutoff on the same day, the start-time and duration restrictions must be checked together. A start inside the allowed window does not, by itself, guarantee that the infusion ends in time.

For instance, if a start must be no later than 20:00 and the infusion must finish by 22:00 on that day, then \(t_0+T_{\mathrm{inf}}\le22\), with both times in hours after midnight. That inequality assumes the start and finish are being compared within the same day. If schedules are allowed to cross midnight, the time-window rule needs an explicit wraparound convention rather than an unexamined application of this inequality.

Assumptions, Diagnostics, and Common Formulation Errors

A feasible-set formulation is only as valid as its interpretation of the regimen instructions. It assumes that the variables capture the schedule features being chosen, that units and time references are consistent, and that the restrictions have been translated without omitting dependencies. Before using \(\mathcal{F}\), check the formulation against those assumptions:

• Check units. A per-administration amount, an amount per day, an interval, and a duration are different quantities. In a sum or inequality, the compared quantities must have compatible units.
• Check discreteness. If tablets, vials, or approved options restrict choices to increments or a finite menu, verify that the allowed set is represented explicitly.
• Check time conventions. Confirm whether a time means administration start or finish, whether it is a clock time or elapsed time, and how midnight is handled.
• Check joint restrictions. Test constraints involving multiple variables, such as an average daily limit or a same-day completion rule, rather than relying only on individual variable bounds.
• Check for an empty set. Restrictions may conflict. If no vector satisfies all conditions, \(\mathcal{F}\) is empty and the stated regimen rules cannot be met together.

Common mistake: “The dose is below the daily maximum, so the schedule is feasible.” This reasoning compares the wrong quantities if the candidate amount is per administration. The correction is to calculate the schedule-level amount using the rule’s intended definition. For a regular schedule and a long-run average daily limit, compare \(24D/\tau\) with the daily maximum; for a calendar-day cap, count the administrations assigned to that day according to the stated timing convention.

The main alternative to a constrained decision vector is to list a finite menu of complete regimens and evaluate each one. A menu can be clear and robust when only a few regimens are permitted, and it naturally avoids accidentally choosing values between approved options. A variable-based feasible set is more compact when many combinations are possible and makes the restrictions visible as mathematical conditions. Neither representation removes the need to check that every listed regimen or permitted variable combination obeys the operational rules.

A feasible set also does not determine whether a schedule is clinically useful. It says which schedules pass the stated restrictions, not which schedule has desirable exposure, target attainment, safety, or other outcomes. Those evaluations require an appropriate model and decision criterion. Keeping feasibility separate from evaluation helps make clear whether a candidate failed because it was disallowed or because its predicted performance was inadequate.