Bounded Titration With Minimum and Maximum Doses

Why Dose Bounds Need to Be Part of the Rule

A titration rule may propose increasing a dose when response is below target and decreasing it when response is above target. But a mathematically valid proposal is not always an acceptable dose: it could fall below a required minimum or exceed a permitted maximum. A practical rule must specify what happens at those boundaries, rather than relying on a later informal correction.

This tutorial treats the adjustment proposal as an input to a bounded rule. The proposal might come from a proportional dose-adjustment rule, an additive step, or another previously specified update. The bounded rule then determines the actual next dose. This separation makes the boundary behavior visible and lets a sequence be checked step by step.

Definition: Let $D_k$ be the dose at decision $k$, and let $D_{\min}$ and $D_{\max}$ be fixed allowable dose limits, with $D_{\min} \le D_{\max}$. A raw proposed dose $D^{\mathrm{raw}}_{k+1}$ is the value produced by the adjustment rule before applying these limits. The clipped dose is $$ D_{k+1}=\min\!\left(D_{\max},\max\!\left(D_{\min},D^{\mathrm{raw}}_{k+1}\right)\right). $$ This operation, called clipping or projection onto the allowed interval, sets a proposal below the minimum to $D_{\min}$, a proposal above the maximum to $D_{\max}$, and leaves a proposal inside the interval unchanged.

The bounds should be stated in the same units as the dose, such as milligrams per administration or milligrams per day. The rule also needs a clearly defined decision point: $D_k$ is the dose in effect for the response $R_k$ used to make the next update. As in any response-guided rule, the response should be measured at an interval appropriate for interpreting the effect of that dose. The bounded update does not itself establish that the chosen interval is appropriate.

Clipping, Holding, and Reversing

Clipping, holding, and reversing are distinct operations. Clipping modifies an out-of-range proposal by setting it to the nearest allowed boundary. Holding means setting the next dose equal to the current dose, usually because a specified condition has been met. Reversing means changing the direction of the proposed adjustment when the response indicates that the previous direction is no longer appropriate. A protocol should make the order and conditions of these operations explicit.

For example, a rule can first check whether the response is in a target band. If so, it holds the dose. Otherwise, it calculates a raw proposal in the direction indicated by the response and clips that proposal to the allowable interval. This ordering means that a target-band hold takes precedence over an adjustment. At a boundary, clipping can also produce no change: if the dose is already at $D_{\max}$ and the raw proposal is higher, the clipped dose remains $D_{\max}$. That is a boundary-limited result, not necessarily a target-based hold.

To describe reversal, define the signed response error as $e_k=R^*-R_k$, where $R^*$ is the target response. With a positive adjustment factor $a$, an additive raw update could be $D^{\mathrm{raw}}_{k+1}=D_k+a e_k$. When $R_k<R^*$, the error is positive and the proposal increases the dose. When $R_k>R^*$, the error is negative and the proposal decreases it. A change in the sign of the error therefore reverses the proposed direction. Whether the magnitude of that change is appropriate depends on the adjustment rule and the response information; the bounds only constrain the resulting dose.

Worked Example: Clipping a Proportional Proposal at the Maximum

Suppose a hypothetical rule uses the current dose and response to propose $D^{\mathrm{raw}}_{k+1}=D_k[1+0.25(R^*-R_k)/R^*]$. At one decision, the dose is $280$ mg, the target response is $10$ response units, and the measured response is $6$ response units. The allowable dose range is $50$ to $300$ mg. Assume the response is valid for this decision and all quantities use the stated units.

The response error is $10-6=4$ response units. Substituting into the raw rule gives $$ D^{\mathrm{raw}}_{k+1} =280\left[1+0.25\frac{10-6}{10}\right] =280(1+0.10) =308\text{ mg}. $$

The raw proposal exceeds the maximum. Applying the bounded rule gives $$ D_{k+1}=\min(300,\max(50,308))=300\text{ mg}. $$

The next dose is therefore $300$ mg, not $308$ mg. The proposal has been clipped by $8$ mg relative to its raw value. This calculation enforces the stated upper limit; it does not show that $300$ mg will achieve the target response.

What the Projection Guarantees

Clipping provides a simple mathematical guarantee: every output dose lies inside the specified interval. It also leaves feasible raw proposals alone and, when a proposal is outside the interval, changes it to the nearest permitted value. These properties follow directly from the definition.

Proposal below the minimum.
If $D^{\mathrm{raw}}_{k+1}<D_{\min}$, then $\max(D_{\min},D^{\mathrm{raw}}_{k+1})=D_{\min}$, so the clipped dose is $D_{\min}$.

Proposal inside the interval.
If $D_{\min}\le D^{\mathrm{raw}}_{k+1}\le D_{\max}$, the inner maximum returns the proposal and the outer minimum leaves it unchanged.

Proposal above the maximum.
If $D^{\mathrm{raw}}_{k+1}>D_{\max}$, the inner maximum returns the proposal and the outer minimum sets the dose to $D_{\max}$.

Thus, in all three cases, $D_{\min}\le D_{k+1}\le D_{\max}$. For an out-of-range proposal, the selected endpoint is also the feasible value with the smallest absolute difference from the raw proposal. For example, if a proposal is below the interval, every allowed dose is at least $D_{\min}$, so $D_{\min}$ is closest; the upper-bound case is analogous. This is a justification of the projection rule, not a guarantee that the resulting response is close to target.

Worked Example: Reversing Direction Within the Dose Bounds

Consider the hypothetical additive rule $D^{\mathrm{raw}}_{k+1}=D_k+10(R^*-R_k)$, where the adjustment factor is $10$ mg per response unit. Let $R^*=8$ response units and let the allowable dose range be $50$ to $250$ mg. Suppose the sequential responses associated with the current doses are $5$, $9.5$, and $7$ response units, respectively. For this arithmetic example, treat each listed response as the valid response at its stated decision.

Start at $D_0=150$ mg. At the first decision, $R_0=5$, so the response error is $8-5=3$ response units: $$ D^{\mathrm{raw}}_1=150+10(3)=180\text{ mg}, \qquad D_1=\min(250,\max(50,180))=180\text{ mg}. $$ The dose increases by $30$ mg.

At the next decision, $R_1=9.5$, so the error is $8-9.5=-1.5$ response units: $$ D^{\mathrm{raw}}_2=180+10(-1.5)=165\text{ mg}, \qquad D_2=\min(250,\max(50,165))=165\text{ mg}. $$ The proposal reverses direction and decreases the dose by $15$ mg.

At the third decision, $R_2=7$, so the error is $8-7=1$ response unit: $$ D^{\mathrm{raw}}_3=165+10(1)=175\text{ mg}, \qquad D_3=\min(250,\max(50,175))=175\text{ mg}. $$ The dose increases by $10$ mg. All three proposals are within the allowed range, so clipping does not alter them. The example shows how the signed response error determines direction; it does not assert that these hypothetical responses arise from a particular pharmacokinetic or pharmacodynamic model.

Adding an Explicit Hold Condition

Some rules specify a target band rather than requiring an adjustment for every response that differs from a single target value. Let $R_L$ and $R_U$ be lower and upper response limits, with $R_L\le R_U$. A simple rule holds the current dose if $R_L\le R_k\le R_U$; otherwise it calculates a raw proposal and clips it. The band and adjustment rule are separate design choices: the band defines when to hold, while the adjustment rule defines what to propose outside the band.

Worked Example: Hold in the Target Band and Clip Below the Minimum

In a hypothetical regimen, the target response band is $8$ to $10$ response units, the dose limits are $50$ to $220$ mg, and the rule proposes a $20$ mg increase when the response is below the band and a $20$ mg decrease when it is above the band. At the first decision, the current dose is $60$ mg and the response is $11$ response units.

The response is above the band, so the raw proposal is $60-20=40$ mg. Clipping gives $$ D_{k+1}=\min(220,\max(50,40))=50\text{ mg}. $$ The dose decreases by $10$ mg rather than the proposed $20$ mg because the minimum limit is reached.

Suppose the next valid response at $50$ mg is $9$ response units. Since $8\le9\le10$, the hold condition applies: $$ D_{k+2}=D_{k+1}=50\text{ mg}. $$ There is no new raw dose adjustment in this step. The dose remains at the minimum because the response is in the specified band, not because a below-minimum proposal was clipped. If instead a subsequent response were $7$ response units, the rule would propose an increase to $70$ mg, reversing the previous direction and moving away from the lower boundary.

Choosing and Checking Boundary Behavior

Hard clipping is not the only possible boundary policy. Under a boundary-blocking hold, a proposal outside the interval may be rejected entirely, leaving the current dose unchanged rather than setting it to the nearest endpoint. For example, if the current dose is $60$ mg, the minimum is $50$ mg, and the raw proposal is $40$ mg, clipping yields $50$ mg while boundary-blocking yields $60$ mg. The choice has a real consequence and must be stated in the rule. A hold is not interchangeable with clipping.

A proportional proposal can also be altered before applying bounds, for example by reducing its adjustment factor. That changes the proposal rule rather than the boundary operation. Whatever policy is chosen, check the following:

Verify the bounds and units. Confirm $D_{\min}\le D_{\max}$, and ensure both limits use the same dose basis as $D_k$ and the proposal.
Check the error sign. Confirm that below-target and above-target responses produce the intended adjustment directions.
Record raw and final doses separately. Repeated differences between them reveal how often bounds are shaping the actual rule.
Check boundary cases explicitly. Test proposals below, at, within, and above the allowed interval, including a hold condition at the same time.
Investigate repeated boundary contact. Frequent clipping can mean that the adjustment rule often proposes infeasible values or that the chosen bounds make the target difficult to reach. It is a reason to review the design, not evidence that the target has been attained.

Dose limits enforce dose constraints, but they do not guarantee that the response will be safe, effective, or within its target range. A sequence can remain at a boundary while the response remains outside the target band. Clipping can also change the effective size of an adjustment: near a boundary, a large proposed increase may become a small actual increase, or no increase at all. If the response continues to call for movement beyond the bound, the rule alone cannot resolve that conflict; the regimen and its constraints require clinical and model-informed review.

Common mistake: “The raw proposal is too high, so the algorithm should hold the current dose.” That conclusion does not follow unless the rule explicitly uses boundary-blocking. Under clipping, an above-maximum proposal becomes $D_{\max}$; under boundary-blocking, it may leave the dose unchanged. Specify which operation applies, and do not describe a dose held in the target band as if it had been clipped.

Check Your Understanding

Answer each question, then compare with the solutions below.

With dose limits of $40$ to $180$ mg, a raw proposal is $32$ mg. What is the clipped dose, and how much does clipping change the proposal?
A rule proposes $D^{\mathrm{raw}}_{k+1}=D_k+5(R^*-R_k)$. If $D_k=100$ mg, $R^*=12$, and $R_k=14$, what is the raw adjustment direction and the raw proposed dose?
A patient is at the minimum dose and the response is inside a specified hold band. What is the next dose under a rule that checks the band before calculating a proposal? Explain why this is not an example of clipping.
In one sentence, state a guarantee provided by clipping and one conclusion it does not guarantee.

Solutions. (1) The clipped dose is $\min(180,\max(40,32))=40$ mg, which is $8$ mg above the raw proposal. (2) The error is $12-14=-2$, so the raw adjustment is $5(-2)=-10$ mg and the raw proposal is $100-10=90$ mg, a decrease. (3) The next dose equals the current minimum dose because the hold condition takes precedence; no out-of-range proposal is being projected. (4) Clipping guarantees that the resulting dose lies between the stated limits, but it does not guarantee that the response reaches its target.