Quantization and Rounding in Titration Algorithms

From a Continuous Recommendation to a Usable Dose

A titration rule may calculate a dose such as 52 mg/day, while the available tablet strengths or administration constraints permit only doses on a coarser grid. The prescribed dose must then be selected from the feasible options. This is not merely a matter of presentation: rounding can change the achieved exposure or response, stop an iterative titration before its continuous target is reached, or contribute to oscillation between dose levels.

The key is to distinguish two operations: rounding a final, continuously calculated target dose once, and rounding after every update in an iterative algorithm. They need not produce the same trajectory. The analysis below uses a deterministic dose-response model and dose updates; practical decisions also need to account for measurement error, patient safety, and the actual available formulations.

For a uniform dose grid, write \(G=\{d_0+jh:j\text{ is an allowed integer}\}\), where \(d_0\) is the grid origin and \(h\) is the spacing. The spacing alone does not identify the available doses: both \(d_0\) and \(h\), as well as the bounds, matter. If there are irregular combinations of tablets or other feasibility restrictions, the set \(G\) may not be a uniform grid; the nearest feasible dose must then be found from the actual set.

Assumptions and the Effect of One Rounding Step

Suppose a dose-response relationship \(R(D)\) is known and sufficiently accurate for the decision, and a target response \(R^*\) corresponds to a continuous target dose \(D^*\). For the simple calculations below, assume \(R(D)\) is increasing and locally linear, \(R(D)=a+bD\), with \(b>0\). Thus \(D^*=(R^*-a)/b\). The model is treated as deterministic: the examples do not include response measurement noise, parameter uncertainty, or a delay between changing the dose and measuring the response.

If the continuous target is rounded once to the nearest dose on an unbounded uniform grid of spacing \(h\), the dose error is at most half a grid step. This follows because every real number lies within \(h/2\) of a grid point. With the linear response assumption, the corresponding response error is bounded by the slope times that dose error.

These bounds assume the nearest grid dose is feasible. At a minimum or maximum dose, a bound can still be calculated by comparing the target with the feasible set, but it need not be \(h/2\): the target may lie outside the allowable range. With a nonuniform set of doses, the relevant error is the actual distance to the selected feasible dose, not half of a presumed spacing.

Rounding at Every Iteration

An iterative algorithm can be affected differently because it does not necessarily calculate \(D^*\) and round that value once. Consider the proportional dose update \(D_{k+1}^{\mathrm{cont}}=D_k+\lambda(D^*-D_k)\), where \(D_k\) is the current dose, \(D_{k+1}^{\mathrm{cont}}\) is the continuous next-dose recommendation, and \(\lambda\) is a positive adjustment gain. If the dose actually administered must be feasible, the implemented rule is \(D_{k+1}=Q_G(D_k+\lambda(D^*-D_k))\).

Before quantization, the dose error \(e_k=D^*-D_k\) obeys a simple recurrence. Substituting the continuous update gives \(e_{k+1}=D^*-D_k-\lambda(D^*-D_k)=(1-\lambda)e_k\). In the absence of rounding, the error therefore shrinks in magnitude when \(0<\lambda<2\), remains unchanged in magnitude at the boundary \(\lambda=2\), and grows for \(\lambda>2\). This continuous result alone does not establish convergence of the rounded rule.

For a uniform grid, write the quantization discrepancy at update \(k+1\) as \(q_{k+1}=D_{k+1}-(D_k+\lambda e_k)\). If the recommendation is within the grid bounds and nearest-dose quantization is used, \(|q_{k+1}|\leq h/2\). The rounded error consequently satisfies \(e_{k+1}=(1-\lambda)e_k-q_{k+1}\), and hence \(|e_{k+1}|\leq |1-\lambda||e_k|+h/2\). This inequality shows that rounding can leave a residual error even when the continuous rule would continue making progress. It is a bound on one update, not a proof that the rounded sequence converges.

A particularly important case occurs when an update rounds back to the current grid dose. If \(D_k\) is on a uniform grid, the continuous recommendation moves by \(\lambda(D^*-D_k)\). When the magnitude of this move is less than \(h/2\), nearest-dose rounding returns \(D_k\), assuming the current dose is feasible and the recommendation is not affected by a bound. Thus an iterative rule can stop changing while the continuous target remains unreached. At exact midpoints, the result depends on the tie rule.

For \(\lambda>0\), this no-change region corresponds to a dose error smaller than \(h/(2\lambda)\). Under the linear response model, its response discrepancy from the target is smaller than \(bh/(2\lambda)\). This bound describes a possible stopping band for the iterative rule, not the error from rounding the final target once. In particular, a smaller gain can make the stopping band wider.

When Rounding Produces a Dose Cycle

Rounding does not always make an iterative update settle at one dose. If the continuous update gain is large enough to cross the target repeatedly, quantization can turn the resulting movements into a repeating cycle. A sequence that alternates between two feasible doses does not converge to a single dose, even if the unrounded error recurrence would shrink for that gain.

Diagnostics and Practical Choices

When implementing or reviewing a titration algorithm, inspect the actual sequence of feasible doses, not only the continuous recommendations. For each update, record the current dose, the continuous recommendation, the selected feasible dose, and the predicted target metric. The following checks help reveal when quantization is materially changing the rule:

• Verify the feasible set. List dose strengths, permitted combinations, dose bounds, and any limits on splitting or administration. A grid spacing without an origin or a complete feasible set is insufficient to determine a rounded dose.
• Check the rounding rule. State whether doses are rounded to nearest, always down, always up, or chosen by another clinically justified rule. Specify how exact ties are handled.
• Look for repeated doses and cycles. A dose that remains unchanged despite a nonzero target error may indicate that updates fall inside a rounding deadband. Alternation or repeated dose patterns may indicate quantization-induced cycling.
• Evaluate the consequence in response units. Convert the difference between selected and target doses through the response or exposure model. A small dose difference is not necessarily clinically small if the response is sensitive to dose.
• Reassess near bounds and with uncertain inputs. Clipping at a minimum or maximum dose changes the available choices. Measurement noise or uncertainty in the model can also make recommendations jump between neighboring strengths, so deterministic calculations should not be treated as guarantees about individual outcomes.

The main alternatives answer different design questions. Rounding the continuous target once by nearest dose gives a simple bound of \(h/2\) in dose units on a uniform grid, but does not ensure that every interim dose follows a prescribed stepwise titration. Rounding each iterative recommendation preserves the update structure but can create a stopping band or cycle. Directional rounding, such as always rounding toward the target, can avoid some small steps being rounded away, but may systematically bias the dose and does not by itself prevent overshoot. A finer feasible grid can reduce rounding error, but may not be available or appropriate. These choices should be evaluated against dose safety constraints and the target metric, rather than selected solely for mathematical convenience.

Quantization analysis has important limits. The simple half-step and deadband results require nearest-dose rounding on a uniform grid and, for response-error calculations, a locally linear response relationship. Nonlinear response requires evaluating the model at the actual selected doses; one cannot infer a response difference from dose spacing alone. If the response is noisy, delayed, or uncertain, observed changes may not follow the deterministic trajectory. A discrete update rule should therefore be checked in its full operating range, including boundaries and plausible measurement variation.