Representing Covariate and Dose Histories as Time-Indexed Processes

Why Time-Indexed Representation Matters

A PK/PD model may depend on a patient characteristic that changes over time, such as renal function, or on doses administered at particular times. These inputs are not adequately described by a single value per subject when their timing matters. A covariate value may describe a state at one time but be measured or reported later; a dose may be ordered at one time but administered at another. Treating those times as interchangeable can make a model use information that was not yet available, or place an input at the wrong point on the time axis.

The goal here is to define the basic representations precisely. A time-indexed process describes what value or input applies over time. A record stream describes events or measurements, including the times attached to them. Both may be needed: the process expresses the model-facing input, while the records document how that input was observed or administered. Neither representation alone determines how to fill gaps between measurements or how an input affects a PK/PD parameter.

The notation \(C_i(t)\) is a representation of the covariate input, not a claim that the value is known continuously or that it stays constant between measurements. A record is evidence about a covariate at its observation time. To define values at other times, an analysis must specify an additional rule, such as a justified interpolation or carry-forward convention. Those rules are substantive modeling choices, not automatic consequences of having a time-indexed process.

Covariate Values, Observation Times, and Availability

Three notions should be kept separate. Effective or observation time says when the covariate value describes the subject, according to the measurement or recording protocol. Availability time says when the analysis or prediction system could actually use that value. Model time is the time at which the model needs an input. Depending on the task, a retrospective analysis may use records available by a specified data cutoff, while a prospective prediction at time \(t\) must use only information available by \(t\).

For a prospective prediction at time \(t\), a record with \(t^{\mathrm{avail}}_{ij}>t\) is not an available input at that time, even if its observation time is earlier. A convenient formal statement is to define \(\mathcal{F}_i(t)\) as the information available for subject \(i\) by time \(t\). An input used for prediction at \(t\) must be determined by information in \(\mathcal{F}_i(t)\). This is an information constraint; it does not itself specify a covariate interpolation rule.

This distinction prevents a form of information leakage: a model intended to mimic decisions at an earlier time must not use a result that was only reported later. It also prevents a different error, temporal mislabelling: placing a delayed laboratory result at its report time can make it appear to describe the patient's state at that later time when the measurement protocol says otherwise.

Dose Histories as Event Streams or Rate Inputs

A dose history can be represented as a stream of administration events. For subject \(i\), let the \(j\)th administration have actual administration time \(\tau_{ij}\) and amount \(D_{ij}\). The pair \((\tau_{ij},D_{ij})\) is a marked event: \(\tau_{ij}\) locates the event on the time axis, and \(D_{ij}\) gives its dose amount. Other marks may describe route or formulation when relevant, but the essential time-and-amount representation is already distinct from a list of prescribed doses.

An instantaneous bolus can also be written as an input rate using a Dirac impulse. This is a mathematical representation, not a claim that a physical administration has zero duration. If the administered amount is \(D_{ij}\), its rate input is \(D_{ij}\delta(t-\tau_{ij})\), where \(\delta\) is the Dirac delta. Integrating the rate over an interval containing the administration returns the dose amount. By contrast, an infusion over a nonzero interval is naturally represented by a rate function \(u_i(t)\), with dose amount equal to the integral of the rate over the infusion interval.

This expression assumes bolus events are represented as impulses and uses the stated interval-boundary convention \(a\leq \tau_{ij}\leq b\). A continuous infusion instead has \(u_i(t)=r_{ij}\) during an infusion interval, where \(r_{ij}\) is the administration rate; its administered amount over that interval is the rate multiplied by elapsed time when the rate is constant. The event representation and rate representation are complementary ways to state an input, chosen according to how the dose was administered.

For dosing records, distinguish at least the administration time from times such as order entry, planned administration, or data entry. Only the administration time locates the actual input in a model of administered dose. A record can also have an availability time: for example, an administration may occur at 09:00 but be entered into a system at 09:20. That later entry time does not move the administration to 09:20, although it determines when the event is known to a prospective system.

A Short Justification: Why the Event Integral Returns Dose

The impulse representation is useful because it puts bolus and infusion inputs into a common rate-based form. The defining property of a Dirac delta at \(\tau\) is that its integral over an interval containing \(\tau\) is one, and its integral over an interval excluding \(\tau\) is zero. Multiplying it by \(D\) therefore makes the integrated input equal to \(D\) when the event is included. For a finite collection of administrations, integration distributes over the sum:

Thus the event stream and its impulse input encode the same administered amounts and times under the stated boundary convention. This is a representation identity; it does not specify absorption, distribution, elimination, or a response mechanism.

Choosing and Checking a Representation

The appropriate representation depends on what the input is and what information the analysis needs. A covariate observed at discrete times is naturally stored as a record stream; a model-facing function \(C_i(t)\) requires an explicit rule for times without observations. Bolus dosing is naturally stored as events, while an infusion is naturally represented by a rate over its duration. In either case, the record's availability time matters when the intended use is prospective.

Useful checks are straightforward but consequential. Confirm that times use a common origin and time zone, that units are coherent, and that the time attached to a covariate means what the data source says it means. For dose records, verify whether a timestamp represents planned, actual, or recorded administration. For prospective use, check that every input was available by the prediction time. Finally, inspect whether the representation silently fills gaps or converts an infusion into a bolus; those choices require justification.

A function is concise when a defensible rule assigns values across model time, but it can conceal how the values were obtained. An event or observation stream preserves the recorded timing more transparently, but it may leave the model input undefined between records. Neither format is universally superior: the key is to preserve timing semantics and state any additional assumptions explicitly.

A common mistake is to set the covariate time equal to the time its laboratory result was reported, or to use a dose order time as though it were an administration time. The correction is to retain separate fields for the time the value pertains to or the dose was administered and the time the information became available. A second mistake is to assume that the most recent measurement automatically applies until the next one. That is a carry-forward assumption, not a property of the record, and must be justified rather than smuggled into the representation.