Why Multiple Targets Need an Explicit Order
A titration rule may need to respond to several outcomes at once. A regimen could provide insufficient efficacy on one measure while satisfying another, or improve efficacy while approaching a tolerability limit. Simply saying “adjust toward all targets” does not tell a clinician or algorithm what to do when the targets conflict. A usable rule must specify which limits are non-negotiable, which goals take priority, and what action resolves a remaining tie.
The rule developed here treats each possible next regimen as a candidate action. It first screens candidates against hard safety constraints. Among the candidates that pass that screen, it minimizes target violations in a stated priority order. If candidates remain tied, a predeclared tie-break determines the action. The result is a reproducible decision rule, not a claim that the chosen regimen is universally optimal.
Represent Targets and Their Violations
Let \(a\) denote a candidate action, such as selecting a dose from an allowed set. For each action, the rule uses the corresponding observed or predicted target measures. The measures and predictions must refer to the same assessment time and regimen definition. Their units, data source, and any assumptions used to predict them must be specified before the rule is applied.
A target can be written as a constraint \(g_i(a)\leq 0\), where satisfying the target gives zero violation. For an upper bound \(Y_i(a)\leq U_i\), use \(g_i(a)=Y_i(a)-U_i\). For a lower bound \(Y_i(a)\geq L_i\), use \(g_i(a)=L_i-Y_i(a)\). For an interval \(L_i\leq Y_i(a)\leq U_i\), the violation is zero inside the interval and positive by the distance outside it.
Here \(s_i\) is a prespecified positive scale in the same units as \(g_i\). It makes violations on different measurement scales comparable for ranking purposes; it does not change whether the target is met. The scale should be clinically interpretable and chosen before examining which action wins. For a lower-bound target with \(L_i>0\), a ratio-scale shortfall can use \(s_i=L_i\), giving \(\max(0,L_i-Y_i)/L_i\). That normalization is appropriate only when \(L_i>0\) and the target is meaningful on a ratio scale. If \(L_i=0\), if \(L_i<0\), or if a ratio interpretation is not meaningful, use a specified positive scale instead; never divide by \(L_i\) in those cases.
For a range, one possible violation is the distance beyond the nearer violated bound, scaled by \(s_i\):
The scale and the aggregation rule are part of the titration rule, not arbitrary after-the-fact conveniences. In this tutorial, targets within a priority tier are aggregated by their largest normalized violation. This “worst-target” score gives the tier’s most serious shortfall control of that tier’s ranking. A rule could instead specify a sum or another prespecified aggregation, but that would be a different decision rule.
Hierarchical Priorities and the Decision Procedure
A practical hierarchy separates safety screening from optimization of desired outcomes. Define the safety-feasible set as the candidates satisfying every hard safety constraint. If that set is empty, the rule must not silently choose the least unsafe candidate: it invokes a prespecified fallback, such as no escalation and clinical reassessment. Among safety-feasible candidates, efficacy goals can be assigned ordered tiers. A candidate is compared on the highest-priority tier first; a lower-priority tier matters only when the candidates tie on all higher tiers.
The selected candidate minimizes \(\mathbf{V}(a)\) lexicographically: compare \(V_1\) first; if tied, compare \(V_2\), then \(V_3\), and so on. “Tied” must be operationally defined. For exact model outputs it may mean equality at the rule’s stored precision; for rounded or noisy inputs the protocol should specify a tolerance. After comparing all tiers, a final tie-break—such as the smallest absolute change from the current dose—makes the action deterministic.
State which actions are available, which measurements or predictions are used, and when they are assessed.
Remove every candidate violating any hard safety constraint. If none remain, apply the explicit fallback and reassess rather than choosing a violating candidate by score.
Calculate each candidate’s tier scores using the predeclared violation scales and aggregation rule; minimize them in order, stopping at the first tier that distinguishes candidates.
Apply the stated final tie-break, including any preference for maintaining the current regimen, if that preference is part of the rule.
This procedure assumes the target definitions and priority order are clinically justified, candidate predictions are applicable to the patient and assessment time, and the inputs are available and valid. It is deterministic conditional on those inputs. It does not itself account for uncertainty in an estimated outcome; if uncertainty should affect decisions, the rule must explicitly define how (for example, by using a prespecified conservative prediction).
Worked Examples
Worked Example: Safety Gate Before an Efficacy Target
Consider a hypothetical titration rule with a required efficacy measure \(E\geq 10\) response units and a hard peak-concentration limit \(C_{\max}\leq 8.0\) mg/L. The efficacy target has a positive ratio-scale threshold, so use \(s_E=10\) response units. Candidate doses and their hypothetical predictions are:
| Dose (mg) | Predicted \(E\) (response units) | Predicted \(C_{\max}\) (mg/L) |
|---|---|---|
| 40 | 8.0 | 7.0 |
| 50 (current) | 9.2 | 7.6 |
| 55 | 9.8 | 7.9 |
| 60 | 10.4 | 8.2 |
The 60 mg candidate fails the hard safety constraint because \(8.2>8.0\), so it is excluded even though its predicted efficacy exceeds the target. The remaining candidates are safety-feasible. Their normalized efficacy shortfalls are:
Thus 55 mg has the smallest efficacy violation among safety-feasible choices. It is selected by this rule, although it still does not meet the efficacy target. The conclusion is not that 55 mg meets both targets; rather, it is the best available candidate under the stated safety gate and efficacy objective. The predictions and thresholds are hypothetical and would require clinical validation before use.
Worked Example: Prioritizing Two Efficacy Targets
Suppose a hypothetical rule has already screened candidates against its hard tolerability constraints. It now compares two efficacy targets: a primary response measure \(R\geq30\) percentage points, followed by a secondary exposure measure \(A\geq40\) mg·h/L. The targets are in ordered tiers, and their positive ratio-scale thresholds are used as scales. The candidate predictions are:
| Candidate | \(R\) (percentage points) | \(A\) (mg·h/L) |
|---|---|---|
| A | 28 | 50 |
| B | 30 | 37 |
| C | 31 | 36 |
The primary-tier violations are \(v_R=\max(0,30-R)/30\): candidate A has \(2/30=0.0667\), while B and C each have \(0\). Because B and C tie at the primary tier, compare the secondary-tier violations \(v_A=\max(0,40-A)/40\). Candidate B has \(3/40=0.0750\), and C has \(4/40=0.100\). Candidate B is selected. Candidate A’s secondary target is met, but that cannot compensate for its primary-tier shortfall. Candidate C has a slightly larger primary response, but the rule treats both B and C as having met the primary target and therefore distinguishes them at the secondary tier.
This illustrates why the hierarchy must be stated before calculating scores. If the two outcomes were instead combined into one weighted sum, the result could depend on the chosen weights and might select a different action.
Worked Example: A Nonpositive Lower Bound and a Final Tie-Break
A hypothetical signed biomarker target is \(B\geq-2\) units. Because the lower bound is negative, dividing the shortfall by \(-2\) would give a negative number for a genuine shortfall. The rule therefore specifies a positive scale \(s_B=5\) units. Two safety-feasible candidates have the same primary efficacy score; their biomarker predictions and dose changes are:
| Candidate | Predicted \(B\) (units) | Absolute dose change (mg) |
|---|---|---|
| D | -3 | 10 |
| E | -4 | 5 |
The biomarker violations are \(v_B(D)=\max(0,-2-(-3))/5=1/5=0.200\) and \(v_B(E)=\max(0,-2-(-4))/5=2/5=0.400\). Candidate D therefore wins at this priority tier; the smaller dose change for E is not considered because the candidates are not tied on the target score. If instead both candidates had \(B=-3\), their scores would tie at \(0.200\), and the stated final tie-break—smallest absolute dose change—would select E. The scale is positive and prespecified, so a shortfall remains nonnegative and interpretable as a scaled distance below the target.
Why Lexicographic Priorities Work—and Where They Can Fail
The key property of lexicographic comparison is that a lower-priority score cannot reverse a difference at a higher-priority score. Suppose candidates \(a\) and \(b\) tie on tiers 1 through \(j-1\), but \(V_j(a)<V_j(b)\). The rule selects \(a\), regardless of the values at tiers after \(j\). This follows directly from the definition of lexicographic ordering: the first unequal component determines the ranking. In particular, a hard safety constraint is not traded against efficacy at all; it is applied as an eligibility gate before scoring.
A weighted sum is a common alternative, for example \(W(a)=\sum_i w_i v_i(a)\). It can make trade-offs explicit when all targets are intended to be compensable, but a sufficiently large improvement in one outcome can offset a safety-related violation unless safety is separately gated. Its result also depends on the weights and scales. A simple branch rule such as “reduce if toxicity is high; otherwise increase if response is low” may be easier to communicate, but overlapping conditions need an explicit order and boundary definitions. A hierarchical rule provides that order while retaining quantitative comparisons within tiers.
Several checks help reveal a poorly specified rule. Confirm that each candidate has the required input values, the units and assessment times match, every scale is positive, and every safety limit is checked before efficacy ranking. Inspect the target boundaries: equality should be classified consistently as meeting a bound when the constraint uses \(\leq\) or \(\geq\). Confirm that candidates tied at one tier are genuinely passed to the next tier and that the final tie-break always yields one action. If a target is missing or a prediction is outside the validated domain, the rule needs a stated fallback rather than an improvised score.
A hierarchy also has limitations. Its priorities encode clinical judgments; mathematics cannot establish that the selected ordering is appropriate. The worst-target aggregation can focus attention on one shortfall and ignore several smaller ones, while a sum can let multiple modest shortfalls accumulate. Predictions may be uncertain or biased, and a deterministic ranking does not represent that uncertainty unless the rule was designed to do so. The candidate set can constrain the outcome: if no available action meets the targets, the rule may select the least-violating feasible candidate, but it must report that the targets remain unmet.
Common Mistakes
- Letting efficacy compensate for a hard safety violation. A candidate that breaches a hard limit is excluded before efficacy scoring; it is not rescued by a high efficacy score.
- Normalizing by a zero or negative target. A lower-bound ratio shortfall using \(L_i\) is valid only when \(L_i>0\) and the ratio scale is meaningful. Otherwise specify a positive scale in the measurement’s units.
- Leaving “priority” vague. State the ordered tiers and the aggregation within each tier. A weighted sum and a lexicographic hierarchy encode different trade-offs.
- Failing to define ties or missing-data behavior. Specify the comparison precision, final tie-break, and fallback for unavailable or invalid inputs before applying the rule.
Check Your Understanding
Answer each question, then check the solutions below.
- A lower-bound target is \(Y\geq0\). Why is \(\max(0,0-Y)/0\) not a valid normalized shortfall, and what should be used instead?
- A candidate violates a hard tolerability limit but has zero efficacy shortfall. Another candidate passes the safety gate but has a positive efficacy shortfall. Which candidate is eligible for selection under the rule defined here?
- Two candidates tie on the primary efficacy tier. Candidate X has a secondary-tier score of 0.04 and candidate Z has 0.07. Which is selected, assuming lower scores are preferred?
- For a lower bound \(Y\geq12\) units with \(s=6\) units, calculate the normalized violation when \(Y=9\) units.
- Why must a final tie-break be specified even after priority tiers are defined?
Solutions. (1) The denominator is zero, so the ratio is undefined. Use a prespecified positive scale in the same units, such as \(s=5\) units, and calculate \(\max(0,0-Y)/s\). (2) Only the candidate that passes the hard safety gate is eligible; efficacy cannot override the safety exclusion. (3) Candidate X is selected because the candidates tie at the primary tier and \(0.04<0.07\) at the secondary tier. (4) The violation is \(\max(0,12-9)/6=3/6=0.500\). (5) Candidates can tie on every scored tier; without a final rule, the algorithm may return more than one action or depend on an unstated preference.