Writing Titration Rules as State-Update Equations

From Protocol Language to a Dose-Update Rule

A titration protocol describes how treatment changes as new information becomes available. In prose, the instruction may sound straightforward: increase the dose if response remains inadequate, but do not increase it when a safety condition is present. Yet implementing that instruction requires precise answers to several questions. When is the decision made? Which measurement counts as the response? Does a threshold include equality? What happens if two conditions apply at once? Writing the protocol as a state-update equation makes these choices visible.

The purpose here is to represent a stated protocol, not to determine the clinically appropriate adjustment size or decision interval. Those choices require separate evidence and design work. A mathematical rule is useful only when its inputs, timing, and action for every relevant condition are defined.

Definition: Let $k$ index scheduled titration decisions. Let $D_k$ be the dose in effect immediately before decision $k$, $R_k$ the response information available at that decision, and $S_k$ the patient state relevant to the protocol. A titration rule is a function $F$ that specifies the next dose: $$ D_{k+1}=F(D_k,R_k,S_k). $$ The state $S_k$ may include such information as a safety flag, a patient subgroup, or whether the required observation is available. The rule also requires a definition of when each input is assessed.

The index $k$ counts decisions, not necessarily equal periods of calendar time. A protocol should separately state the assessment occasion or event that makes decision $k$ occur. The notation $D_{k+1}$ means the dose to be used after the decision; it does not imply that a dose is taken immediately or that the next decision occurs after a fixed interval.

Specify the Inputs Before Writing the Rule

A usable rule needs operational definitions for its inputs. The response $R_k$ might be a measured concentration, a score, or a summary computed from observations. State the measurement scale, units, source, and assessment occasion. The patient state $S_k$ should contain only information the protocol actually uses, but it must include every condition that can change the action. If the protocol refers to “adequate tolerability,” for example, it must define how that status is recorded rather than leave it as an unstructured judgment.

The dose $D_k$ also needs a clear meaning. It could be an amount per administration, a total daily amount, or another specified quantity. The next-dose rule is ambiguous if its output uses a different dose definition from its input. Include units in both the rule and its implementation.

A verbal instruction often contains three components: a condition, an action, and a priority. Conditions define which branch applies; the action defines the next dose; priority determines what to do when conditions overlap. A decision table is a useful intermediate step between prose and an equation.

Protocol element	Question to answer	Example of a precise specification
Decision occasion	When are inputs evaluated?	At the scheduled assessment after the current dosing period
Response	What measurement and scale are used?	Score $R_k$, in points, recorded at the assessment
Threshold	Which side includes equality?	$R_k \geq 5$ is classified as inadequate response
Patient state	Which status changes the action?	$S_k=1$ denotes a safety restriction
Action	What dose follows each condition?	Increase by the specified protocol increment, or maintain
Priority	Which rule applies if conditions overlap?	The safety restriction takes precedence over response

Writing the Rule as a Piecewise Function

A piecewise equation states the action for each defined condition. The branches should be mutually exclusive and collectively cover every allowed combination of inputs. If a condition is not covered, the equation does not determine the next dose for that case. If two branches can apply at once, the rule needs an explicit priority or a revised set of conditions.

Key representation: If the protocol has an increase branch and a maintain branch, one possible form is $$ D_{k+1}= \begin{cases} D_k+\Delta, & \text{if the specified increase condition holds},\\ D_k, & \text{otherwise}, \end{cases} $$ where $\Delta$ is the increment already specified by the protocol and has the same units as $D_k$. This equation represents the protocol; it does not establish that $\Delta$ is clinically suitable.

Conditions must be stated using the protocol’s actual logic. “Response is inadequate and no safety restriction is present” is a conjunction: both statements must hold. “Response is inadequate or a safety restriction is present” is a disjunction: either statement is enough. Confusing these connectors can reverse the intended action.

Worked Example: Translating a Response Threshold With a Safety Priority

Hypothetical protocol. A patient receives a drug in milligrams per day. At each scheduled assessment, a symptom score $R_k$ is recorded; higher scores indicate more symptoms. If $R_k \geq 5$ and no safety restriction is recorded, increase the daily dose by 5 mg. Otherwise, maintain the current dose. A safety restriction takes precedence, even when the score is high.

Define $S_k=1$ when the safety restriction is present and $S_k=0$ otherwise. The increase condition is $R_k \geq 5$ and $S_k=0$. The complete rule is:

D_{k+1}= \begin{cases} D_k+5\ \text{mg/day}, & R_k \geq 5 \text{ and } S_k=0,\\ D_k, & R_k < 5 \text{ or } S_k=1. \end{cases}

For a patient with $D_k=20$ mg/day, $R_k=6$, and $S_k=0$, the increase condition holds. Substitution gives $D_{k+1}=20+5=25$ mg/day. If the same patient has $S_k=1$, the safety condition takes precedence: $D_{k+1}=20$ mg/day, despite the score of 6. If $R_k=5$ and $S_k=0$, equality is included by the $\geq$ sign, so the next dose is again 25 mg/day.

This rule covers all cases: either $S_k=1$, which leads to maintenance, or $S_k=0$, in which case $R_k$ is either at least 5 or below 5. The two branches therefore do not conflict and leave no response value unclassified.

Why the Equation Is a State Update

The rule updates a treatment state using information available at a decision. The current dose alone is not enough to determine the next dose: two patients receiving the same dose may have different responses or different safety states. The function $F$ represents that dependence explicitly. If the protocol uses a patient characteristic that does not change across decisions, it can still be included in $S_k$; if a state variable changes, its value must be updated when the next decision is made.

A compact indicator form can be derived from a two-branch rule. Let $I_k=1$ when the increase condition is true and $I_k=0$ otherwise. Then the piecewise rule “add $\Delta$ if the condition holds, otherwise maintain” can be written as:

D_{k+1}=D_k+\Delta I_k.

If the condition holds, $I_k=1$, so the right side is $D_k+\Delta$. If it does not hold, $I_k=0$, so the right side is $D_k$. Thus the compact equation produces exactly the same two actions as the piecewise form. This derivation also shows why $I_k$ must be defined from the protocol’s full condition, including any priority rule; it cannot represent merely one convenient part of that condition.

In a more complete model, response measurements may vary even when the underlying patient state and dose are similar. The update equation remains a description of the decision rule: it maps the observed $R_k$ and recorded $S_k$ to an action. It is not, by itself, a model of how dose causes response. Keeping the decision rule distinct from the response-generating process prevents a protocol description from being mistaken for a pharmacologic effect model.

Worked Example: Letting Patient State Change the Action

Hypothetical protocol. At an assessment, a biomarker $R_k$ is measured in units/L, and higher values indicate a response below the desired level. If $R_k>12$ units/L, the protocol increases the dose by 4 mg/day for patients with $S_k=0$, and by 2 mg/day for patients with $S_k=1$. If $R_k\leq12$ units/L, the dose is maintained. Here $S_k=1$ denotes a specified patient subgroup; it is not a safety flag.

The state changes the increment, so the patient subgroup must be part of the rule rather than a note outside it:

D_{k+1}= \begin{cases} D_k+4\ \text{mg/day}, & R_k>12\ \text{units/L and } S_k=0,\\ D_k+2\ \text{mg/day}, & R_k>12\ \text{units/L and } S_k=1,\\ D_k, & R_k\leq12\ \text{units/L}. \end{cases}

Suppose $D_k=16$ mg/day and $R_k=13$ units/L. For a patient with $S_k=0$, $D_{k+1}=16+4=20$ mg/day. For a patient with $S_k=1$, $D_{k+1}=16+2=18$ mg/day. If $R_k=12$ units/L, the strict inequality $R_k>12$ is false, so either patient maintains 16 mg/day. The boundary behavior is therefore specified, not left to interpretation.

This example assumes the subgroup status is known at the decision and that the stated dose increments are permitted by the protocol. If subgroup status is missing, the equation has no defined branch for that patient; the protocol would need to specify what action to take when required state information is unavailable.

Checking a Rule Before Use

A written rule should be checked against both the source instruction and its possible inputs. For each branch, verify the condition, action, units, threshold direction, and precedence. Then test boundary values and combinations of state variables. A simple implementation table with representative cases is often enough to reveal a reversed inequality or an uncovered case.

Fix the decision occasion.
State what assessment event defines $k$ and which measurements are available then.

Name and define every input.
Specify the dose quantity and units, response measure and scale, and every patient-state variable used by the protocol.

Write conditions as logical statements.
Translate “and,” “or,” “at least,” and “below” into explicit conditions, preserving inclusive or strict thresholds.

Assign one action to every case.
Resolve overlaps with a stated priority and check that allowed input combinations are all covered.

Test the equation with cases.
Substitute ordinary values, threshold values, and relevant state combinations, then confirm that the outputs match the verbal instruction.

For example, an algorithm that uses $R_k\geq c$ should be tested at $R_k=c$, not just at values clearly above or below $c$. If the protocol says “greater than,” changing it to “greater than or equal to” changes the action at the boundary. Similarly, if a safety restriction overrides an increase, test a case where the response condition and safety condition are both true.

Comparisons, Limitations, and Common Pitfalls

An explicit state-update equation differs from a fixed schedule, which assigns doses by time or treatment stage without necessarily using current response. It also differs from informal clinician judgment, where an action may depend on relevant considerations that are not fully represented in a written algorithm. A rule-based titration is most transparent when the protocol intends specified inputs to determine specified actions. In practice, the appropriate approach depends on the clinical context and protocol; an equation does not remove the need for clinical oversight or validated safety procedures.

The main benefit of the equation is reproducibility. Two implementers applying the same defined inputs should obtain the same next dose. The main limitation is equally important: a precise equation cannot repair an incomplete or ambiguous protocol. It may faithfully encode a poor rule, or conceal an omitted exception if the analyst assumes that an unstated case means “maintain.”

Changing a threshold at equality. Treating “above 5” as “5 or above” changes the action when the response equals 5. Preserve the stated inequality.
Leaving precedence implicit. If an increase condition and a safety condition can both be true, state which governs. Do not rely on the order in which a reader happens to notice the clauses.
Using undefined patient state. A label such as “eligible” is not an operational input until the protocol says how eligibility is determined at the decision.
Mixing dose definitions or units. Adding an amount per administration to a daily dose is not a coherent update. Keep the input, increment, and output on the same dose scale.
Assuming an unspecified action. “No instruction to increase” does not necessarily mean “maintain.” Encode maintenance only when the protocol specifies it or an authorized rule resolves that case.

A useful final review is to ask whether someone who has only the stated inputs and the rule can determine exactly one next dose. If not, identify the missing definition, uncovered case, or priority decision before treating the equation as executable.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

A hypothetical rule increases dose when $R_k\geq 8$ and $S_k=0$, and otherwise maintains it. What happens at $R_k=8$ when $S_k=0$?
Why is it not sufficient to write “increase if response is inadequate” in a mathematical rule?
Write an equation for a hypothetical protocol that increases the dose by 3 mg/day when $R_k<4$, and maintains it otherwise. The response is measured in points.
A protocol says to increase when $R_k>10$, unless a safety restriction is present; the restriction overrides the increase. For $R_k=11$ and a restriction present, what is the action, and what should be clarified if the protocol does not state the alternative action?
What does the state-update equation describe, and what does it not establish about the relationship between dose and response?

Solutions. (1) Increase, because equality is included by $\geq$ and $S_k=0$. (2) The response scale, threshold, decision occasion, and action for other cases are unspecified, so the next dose may not be uniquely determined. (3) $D_{k+1}=D_k+3\ \text{mg/day}$ if $R_k<4$ points, and $D_{k+1}=D_k$ otherwise. (4) The restriction overrides the increase. The protocol must specify what action applies in that case; maintenance should not be assumed unless stated. (5) It describes how the protocol maps current dose, observed response, and patient state to the next dose. It does not establish a causal or pharmacologic model of how dose changes response.