From a Continuous Relationship to a Usable Schedule
A population pharmacokinetic model can describe clearance as a continuous function of kidney function. A clinical dosing schedule, however, often needs a small number of dose options rather than a different dose for every possible kidney-function value. Turning a smooth relationship into categories is therefore a translation step: it makes a model-based result easier to use, but it also replaces individual values with ranges and creates compromises within each range.
The aim here is to make that translation explicit. A dose category should be traceable to a continuous dose target, a stated kidney-function range, and a rule for choosing the dose within that range. The categories and numbers below are hypothetical teaching examples, not clinical recommendations.
The mapping depends on the continuous exposure-based target. In the examples, \(CL(K)\) denotes clearance at kidney-function value \(K\). We assume a maintenance-dose setting in which average exposure is proportional to dose divided by clearance, and in which the dosing interval is held fixed. These assumptions allow dose to be scaled in proportion to clearance when matching average exposure. They do not establish that a particular category scheme is clinically appropriate.
Deriving the Continuous Dose Target
Suppose a reference dose \(D_{\mathrm{ref}}\) is used at kidney function \(K_{\mathrm{ref}}\), with corresponding clearance \(CL(K_{\mathrm{ref}})\). Under the assumptions above, the exposure ratio for a dose \(D(K)\) relative to the reference is
Matching the reference exposure sets this ratio to one. Rearranging gives the continuous target dose:
This target is a function of the individual value \(K\). Categorization replaces it with one selected dose \(D_j\) for everyone in category \(j\). One practical method is to choose a representative value \(K_j\) for each category, calculate \(D^*(K_j)\), and then select a deliverable dose. The representative value, rounding rule, and category boundaries must all be stated: each affects the resulting schedule.
Worked Examples of Building and Assessing Categories
Worked Example: Calculating a Continuous Dose Target
Consider a hypothetical model for a drug in which \(K\) is a prespecified kidney-function measure in mL/min and clearance is
Assume this linear relationship is being used only over \(15 \leq K \leq 100\) mL/min. A hypothetical reference dose is 600 mg per day at \(K_{\mathrm{ref}}=100\) mL/min. Calculate the continuous dose target at \(K=45\) mL/min.
First calculate the clearance at the reference value and at \(K=45\):
The proportional exposure-matching target is therefore
The units of clearance cancel, leaving a dose in mg/day. Under the stated assumptions, 352.5 mg/day is the continuous target at \(K=45\) mL/min. It is not yet a category recommendation; selecting a practical dose requires a category scheme and a dose-rounding rule.
Worked Example: Assigning Doses to Four Categories
Use the same hypothetical clearance model and reference dose. Suppose a proposed schedule has categories \(90\)–\(100\), \(60\)–less than \(90\), \(30\)–less than \(60\), and \(15\)–less than \(30\) mL/min. These cutpoints are illustrative. For the calculation, use representative values of 100, 75, 45, and 22.5 mL/min, respectively, and round each target to the nearest 50 mg/day.
At the four representative values, the clearances are \(8.00\), \(6.50\), \(4.70\), and \(3.35\) L/h. For example, \(CL(75)=2.0+(0.060)(75)=6.50\) L/h. The corresponding unrounded targets are:
Rounding as specified gives category doses of 600, 500, 350, and 250 mg/day. The table makes the mapping explicit; the selected values are hypothetical and should not be interpreted as prescribing advice.
| Kidney-function category (mL/min) | Representative \(K\) (mL/min) | Continuous target (mg/day) | Rounded category dose (mg/day) |
|---|---|---|---|
| 90–100 | 100 | 600 | 600 |
| 60–less than 90 | 75 | 487.5 | 500 |
| 30–less than 60 | 45 | 352.5 | 350 |
| 15–less than 30 | 22.5 | 251.25 | 250 |
The model was assumed applicable throughout the stated range, and the dose calculation holds the interval and formulation fixed. Values below 15 or above 100 mL/min are not covered by this example: applying the linear equation there would be extrapolation, not a justified extension of the category schedule.
Worked Example: Quantifying Within-Category Exposure Differences
A category dose is a compromise across the values in its range. For the 30–less-than-60 mL/min category above, the selected dose is 350 mg/day. Compare relative exposure at the two ends of that range with the reference exposure from 600 mg/day at \(K=100\).
At \(K=30\), \(CL(30)=2.0+(0.060)(30)=3.80\) L/h. The relative exposure is
Thus, under the proportional exposure assumptions, the category dose gives about 22.8% more exposure than the reference exposure at \(K=30\) mL/min. At \(K=60\), \(CL(60)=2.0+(0.060)(60)=5.60\) L/h, so
At that end of the category, the same dose gives about 16.7% less exposure than the reference. The continuous target changes smoothly between these values, but the category dose does not. This calculation describes the exposure mismatch implied by the example; it does not determine whether that mismatch is acceptable for a particular drug or patient population.
Why Cutpoints Matter
Categories introduce a boundary effect. In the illustrative schedule, \(K=59.9\) mL/min falls in the 30–less-than-60 category and maps to 350 mg/day, whereas \(K=60.0\) mL/min maps to 500 mg/day. Yet the continuous model gives nearly identical clearances:
The category dose changes by 150 mg/day across a 0.1 mL/min difference in the recorded value, even though the modelled clearance changes by only 0.006 L/h. The discontinuity is created by the rule, not by a discontinuity in the continuous relationship. If measurement uncertainty or rounding makes values near a cutpoint common, the selected category can be sensitive to small changes in the recorded value.
This does not mean that every schedule should use continuous individualized doses. A finite set of doses may be needed for practical reasons. It means that cutpoints should be treated as decision rules with consequences, rather than as biological thresholds implied by a smooth model. The category assignment, any measurement rounding rule, and the possible dose jump at each boundary should be visible to those reviewing the recommendation.
Information Lost, Diagnostics, and Alternatives
Once \(K\) has been replaced by a category label, distinct values within that category become indistinguishable to the dose rule. A person at \(K=31\) and a person at \(K=59\) receive the same example dose, although the continuous model predicts different clearances and different dose targets. This loss of detail can be assessed by calculating the dose-to-target or exposure ratio across each category, especially at its boundaries and at clinically relevant values within it.
Useful checks include confirming that categories cover the intended range without gaps or overlaps; verifying that the chosen dose corresponds to the stated representative value and rounding rule; and calculating the implied exposure differences across each category. Also check that the clearance model is being used only within its supported range, that dose ordering is sensible for the model, and that the category assignments do not create unexplained jumps. Large within-category discrepancies or a particularly large boundary jump indicate that the proposed mapping deserves review; they do not, by themselves, identify the clinically correct alternative.
The main alternatives differ in how much of the continuous information they retain. Individualized dosing uses the continuous target directly, when reliable measurements and feasible dose adjustments are available. More, narrower categories can reduce the range of continuous targets represented by any one dose, but increase complexity and may still create jumps at cutpoints. Fewer, broader categories are simpler to communicate but generally permit more variation in target mismatch within a category. A category schedule is therefore a balance between practical usability and fidelity to the continuous relationship, not a mathematical consequence that selects one uniquely correct set of cutpoints.
The mapping is limited by the model and assumptions used to generate it. A misspecified clearance relationship, a different exposure target, non-proportional dose–exposure behaviour, or changes in regimen features not represented in the calculation can make the resulting categories inappropriate. Categorization also does not remove variation between individuals who share the same recorded \(K\). The calculations here address the mean relationship implied by the specified model; they do not establish that every individual assigned a category will have the same exposure or response.
A disciplined mapping therefore keeps three objects separate: the continuous dose target \(D^*(K)\), the category rule that assigns a range, and the practical dose selected for that range. Showing all three makes the information loss and the effect of rounding inspectable rather than hiding them behind a category label.
Check Your Understanding
Use the hypothetical model \(CL(K)=2.0+0.060K\) L/h, reference dose 600 mg/day at \(K=100\) mL/min, and the assumptions stated above. The solutions follow the questions.
- Calculate the continuous dose target at \(K=50\) mL/min.
- For the 30–less-than-60 mL/min category with dose 350 mg/day, calculate the relative exposure versus reference at \(K=45\) mL/min.
- Explain why two people in the same dose category need not have the same predicted exposure.
- Why can a small change in \(K\) near a category boundary produce a large change in assigned dose even when the clearance model is smooth?
Solutions. (1) \(CL(50)=2.0+(0.060)(50)=5.00\) L/h, so \(D^*(50)=600(5.00/8.00)=375\) mg/day. (2) \(CL(45)=4.70\) L/h, giving relative exposure \(=(350/4.70)/(600/8.00)=350(8.00)/(600(4.70))=0.9929\), or about 0.7% below reference. (3) Their \(K\) values can differ within the category, so the model predicts different clearances; with the same dose, exposure proportional to dose divided by clearance differs. (4) The category rule assigns different fixed doses on opposite sides of the cutpoint. The abrupt assignment change comes from the categorization rule, even though \(CL(K)\) changes continuously.