Why Examine Sensitivity to Weight and Kidney Function?
A dose table gives recommendations at selected combinations of patient characteristics. Sensitivity analysis asks what happens when those inputs change: how much does the recommendation move, and which input accounts for the larger change over a clinically plausible range? This is useful when a patient is near a dose-table boundary, when an estimate of kidney function is imprecise, or when the recommendation will be translated into available dose strengths.
The calculations below use an explicitly simplified, hypothetical model. They illustrate how to analyze input effects, not how to prescribe a particular medicine. A real recommendation depends on the drug’s clearance pathways, exposure target, formulation, clinical context, and the evidence supporting the model.
Deriving the Relative-Dose Formula
For a fixed dosing interval at steady state, the area under the concentration-time curve over one interval is \(AUC_{\tau}=F D/CL\), where \(F\) is bioavailability, \(D\) is dose per interval, and \(CL\) is clearance. If the target \(AUC_{\tau}\), \(F\), and dosing interval are unchanged, then the dose required is proportional to clearance. Under the stated model, clearance is proportional to \((W/W_{\mathrm{ref}})^{0.75}(G/G_{\mathrm{ref}})\). At the reference inputs, the recommended dose is \(D_{\mathrm{ref}}\); dividing the dose at any other inputs by that reference dose gives the relative-dose formula.
The ratio \(W/W_{\mathrm{ref}}\) and the ratio \(G/G_{\mathrm{ref}}\) are dimensionless, so their product scales the reference dose without changing its units. The model has no additional interaction term: the weight and kidney-function multipliers combine multiplicatively. A change in one input can therefore be calculated while holding the other at its reference value, or both can be changed together.
Worked Example: Calculating a Dose at One Combination of Inputs
Suppose a hypothetical reference regimen is 900 mg per interval at \(W_{\mathrm{ref}}=60\) kg and \(G_{\mathrm{ref}}=90\) mL/min. Calculate the model-based dose at 54 kg and \(G=72\) mL/min. Assume both kidney-function values are expressed using the same measure and units, and that the model’s assumptions apply over this range.
Substitute the values into the formula:
The weight ratio is \(54/60=0.9\), and \(0.9^{0.75}\approx0.9240\). The kidney-function ratio is \(72/90=0.8\). Thus,
The model-based amount is about 665 mg per interval before any formulation or rounding decision. The calculation checks both inputs against their stated references; it does not imply that the amount is an appropriate clinical regimen.
Separating and Combining Input Effects
A one-at-a-time calculation changes one input while holding the other fixed. It helps isolate the model’s response to each variable, but it is not enough when both inputs may vary together. A scenario grid makes the joint consequences visible and can expose a combination that a single-input comparison would miss.
Worked Example: Building a Small Sensitivity Grid
Use the same hypothetical reference regimen: 900 mg per interval at 60 kg and 90 mL/min. Calculate doses at weights of 54, 60, and 66 kg and kidney-function values of 72, 90, and 108 mL/min. The calculations assume the same fixed target exposure and the same model across the entire grid.
For 54 kg, the weight factor is \((54/60)^{0.75}=0.9^{0.75}\approx0.9240\). At 66 kg, it is \((66/60)^{0.75}=1.1^{0.75}\approx1.0741\). The kidney-function factors at 72, 90, and 108 mL/min are \(0.8\), \(1.0\), and \(1.2\), respectively. Multiplying each weight factor and kidney-function factor by 900 mg gives:
| Weight | G = 72 mL/min | G = 90 mL/min | G = 108 mL/min |
|---|---|---|---|
| 54 kg | 665.3 mg | 831.6 mg | 997.9 mg |
| 60 kg | 720.0 mg | 900.0 mg | 1080 mg |
| 66 kg | 773.4 mg | 966.7 mg | 1160 mg |
For example, the 66 kg, 72 mL/min cell is calculated using the 66 kg weight ratio, not a different value:
Across the grid, dose increases with either input under this model. That pattern is a property of the specified formula, not a universal claim about every drug or patient. A table is also useful for checking arithmetic: each cell should be obtained from the same reference dose and the ratios corresponding to its own row and column.
Which Input Drives the Change?
The answer depends on the range of values being compared. A per-unit comparison is often unhelpful because weight and kidney function use different units. Instead, compare proportional changes over specified plausible ranges, or use the model’s elasticity: the percentage change in dose associated locally with a one-percent change in an input.
For the stated model, taking logarithms gives \(\log D=\log D_{\mathrm{ref}}+0.75\log(W/W_{\mathrm{ref}})+\log(G/G_{\mathrm{ref}})\). Differentiating with respect to the logarithm of each input shows that the weight elasticity is 0.75 and the kidney-function elasticity is 1. Equivalently, for small changes, a 1% weight increase corresponds to about a 0.75% dose increase, while a 1% increase in \(G\) corresponds to a 1% dose increase. These are relative effects, not changes in mg per kg or mg per mL/min.
Worked Example: Comparing Input Effects Over Stated Ranges
Compare the effect of weight changing from 54 to 66 kg with the effect of kidney function changing from 72 to 108 mL/min. Hold the other input fixed for each comparison. The model assumptions remain the same, and the comparison applies only to these specified ranges.
The weight-driven dose ratio is the ratio of the weight factors at the endpoints:
Thus, over this weight range, the model predicts a dose about 16.2% higher at 66 kg than at 54 kg when \(G\) is held fixed. The kidney-function-driven ratio is
Over this kidney-function range, the model predicts a dose 50.0% higher at 108 than at 72 mL/min when weight is held fixed. For these selected ranges, kidney function has the larger proportional effect. If both inputs move from the lower to the upper endpoint, their effects multiply:
The combined model-based dose at the upper corner is therefore about 1.744 times the dose at the lower corner, or about 74.4% higher. It is not correct to add 16.2% and 50.0% as though the model were additive; the exact joint comparison is multiplicative.
Choosing a Sensitivity Method and Checking Its Limits
Three complementary summaries answer different questions. A one-at-a-time comparison isolates the contribution assigned to one input by the formula. A scenario grid displays combinations and their resulting doses. Elasticities describe local proportional sensitivity and are convenient when inputs have different units. No single summary replaces checking the full set of plausible scenarios relevant to the decision.
The simple model also has important limitations. Many drugs have both renal and nonrenal clearance. If clearance is \(CL=CL_{\mathrm{NR}}+CL_{\mathrm{R}}\), where only the renal component changes in proportion to \(G\), total clearance will generally not be directly proportional to \(G\). For this alternative structure, the local elasticity of clearance to \(G\) is \(CL_{\mathrm{R}}/CL\), rather than necessarily 1. The direct kidney-function multiplier used above would then exaggerate or otherwise misstate the effect unless the assumed clearance relationship were justified.
Similarly, a weight exponent of 0.75 is an assumption of this example, not a universal exponent. The relationship may differ across drugs, ages, body compositions, or the range of weights being considered. The model may also omit changes in bioavailability, maturation, disease state, or other clearance pathways. Sensitivity calculations cannot correct an unsupported model or make extrapolation beyond the data reliable.
Dose amounts may also be rounded to available strengths or constrained by a maximum dose. If the calculated dose changes smoothly but the usable dose changes in discrete steps, the final regimen can behave differently from the unrounded model output. Treat rounding and formulation as a separate decision, then verify the consequences for the intended exposure. A small difference in calculated dose may matter if it crosses a practical boundary; a larger difference may not change the available regimen.
- Check the reference and units. Confirm the reference weight, kidney-function measure, units, and population basis. A normalized estimate and an absolute estimate cannot be substituted without considering what the model expects.
- Check the arithmetic. Recalculate each ratio from the actual scenario values, then verify that the dose equals the reference dose times both factors.
- Check the direction and scale. Under this model, increasing either input increases dose. A contrary result suggests an arithmetic or coding error; a clinically implausible result may instead indicate a model limitation.
- Check the range. Ask whether the model has support for the weights and kidney-function values being tested. A neat calculation is not evidence that extrapolation is valid.
- Check joint changes and practical boundaries. Use a grid when multiple inputs may change, and inspect whether rounding or a dose cap changes the recommendation.
A common mistake is to say that the input with the larger numerical change is necessarily the more influential one. A 12 kg change and a 36 mL/min change cannot be compared directly because their units and scales differ. The correction is to compare the dose ratios over specified ranges, or compare elasticities for proportional changes, while stating the model that produces those comparisons.
Check Your Understanding
Use the hypothetical model \(D(W,G)=900(W/60)^{0.75}(G/90)\), with dose in mg per interval, weight in kg, and kidney function in mL/min. Assume the model applies to the values in each question.
- Calculate the dose at 60 kg and 72 mL/min.
- At 66 kg and 72 mL/min, what is the dose ratio relative to the 60 kg, 90 mL/min reference? Show the two factors.
- Over the ranges 54 to 66 kg and 72 to 108 mL/min, which input produces the larger proportional dose change under the model?
- Explain why the kidney-function multiplier may be inappropriate for a drug with substantial nonrenal clearance.
Solutions. (1) \(900(60/60)^{0.75}(72/90)=900(1)(0.8)=720\) mg per interval. (2) The ratio is \((66/60)^{0.75}(72/90)=1.0741(0.8)\approx0.8593\); the dose is \(900(0.8593)\approx773.4\) mg per interval. (3) The weight-range ratio is \((66/54)^{0.75}\approx1.162\), or a 16.2% increase. The kidney-function ratio is \(108/72=1.500\), or a 50.0% increase. Kidney function produces the larger proportional change over these ranges. (4) If nonrenal clearance contributes to total clearance but does not change with kidney function, total clearance will not generally scale directly with \(G\). The kidney-function effect depends on the renal contribution to total clearance, so the direct proportional multiplier must be justified or replaced with an appropriate clearance model.