Worked Case In Individualized Dose Adjustment

From Concentration History to an Individual Regimen

Individual dose adjustment is a sequence of linked decisions: use patient characteristics to inform the pharmacokinetic parameters, compare the model’s forecasts with appropriately timed concentrations, and then determine whether a practical regimen can meet the specified exposure targets. A dose calculation is only as credible as the model, the data, and the assumptions that support it.

The case below is entirely hypothetical. It concerns an adult receiving an oral drug with linear, one-compartment pharmacokinetics and first-order absorption. The prescribed target window is an AUC over each 12-hour dosing interval of 90–110 mg·h/L, a predose concentration of 5–8 mg/L, and a peak concentration no greater than 14 mg/L. These are stipulated targets for this example, not general therapeutic recommendations.

Model and assumptions: At steady state, the drug follows a one-compartment model with first-order absorption and elimination. For a dose $D$ given orally every $\tau$ hours, bioavailability $F$, apparent volume $V$, absorption rate constant $k_a$, and elimination rate constant $k=CL/V$, the concentration $t$ hours after a dose, for $0\le t\le\tau$, is $$ C_{ss}(t)=\frac{F D k_a}{V(k_a-k)} \left[ \frac{e^{-kt}}{1-e^{-k\tau}}- \frac{e^{-k_a t}}{1-e^{-k_a\tau}} \right]. $$ This expression assumes constant parameters and dose timing, complete adherence, linear disposition, and a patient at steady state. It does not account for time-varying clearance, nonlinear absorption, or missed doses.

Use Covariates and Concentration History

Suppose the population model predicts clearance according to $CL_{\mathrm{pop}}=8.0(WT/70)^{0.75}(eGFR/90)^{0.5}$ L/h, where weight $WT$ is in kilograms and estimated glomerular filtration rate $eGFR$ is in mL/min/1.73 m². The patient weighs 70 kg and has an eGFR of 50 mL/min/1.73 m². After incorporating the patient’s concentration history using Bayesian forecasting, the working individual parameter estimates are $CL=6.0$ L/h, $V=60$ L, and $k_a=1.0$ h$^{-1}$. For this worked calculation, take $F=1$.

Worked Example: Translating Covariates Into an Individual Model

First calculate the population clearance prediction from the stated covariate model:

CL_{\mathrm{pop}} =8.0\left(\frac{70}{70}\right)^{0.75} \left(\frac{50}{90}\right)^{0.5} =8.0(1)(0.7454) =5.963\ \text{L/h}.

The concentration-informed individual estimate, 6.0 L/h, is close to this covariate-based prediction. This agreement is reassuring but does not establish that either estimate is correct. With $V=60$ L, the individual elimination rate constant is

k=\frac{CL}{V}=\frac{6.0\ \text{L/h}}{60\ \text{L}}=0.1000\ \text{h}^{-1}.

The covariates provide a population-informed starting point; the concentration history contributes patient-specific information. The estimates must still be checked against the sampling times, dose record, and model forecasts.

Assume the patient has been taking 500 mg every 12 hours long enough to reach steady state. Two concentrations were collected with documented timing: 2 hours after a dose and immediately before the next dose. The hypothetical measured values are 9.8 mg/L and 4.0 mg/L, respectively. These samples can be compared with predictions at those exact times, not with an untimed “peak” or “trough.”

Worked Example: Checking the Existing Regimen Against Concentrations

For $\tau=12$ h, the concentration multiplier for the specified model is

\frac{F D k_a}{V(k_a-k)} = \frac{(1)(500\ \text{mg})(1.0\ \text{h}^{-1})} {(60\ \text{L})(1.0-0.1000)\ \text{h}^{-1}} =9.259\ \text{mg/L}.

At 2 hours, the bracketed part of the steady-state equation is approximately

\frac{e^{-0.1000(2)}}{1-e^{-0.1000(12)}}- \frac{e^{-1.0(2)}}{1-e^{-1.0(12)}} = \frac{0.8187}{0.6988}-\frac{0.1353}{1.0000} \approx 1.036.

Thus the forecast is $9.259(1.036)=9.595$ mg/L, compared with the measured 9.8 mg/L. Immediately before the next dose, set $t=12$ h:

\frac{e^{-1.2}}{1-e^{-1.2}}- \frac{e^{-12}}{1-e^{-12}} \approx 0.4310, \qquad C_{ss}(12^-)\approx 9.259(0.4310)=3.991\ \text{mg/L}.

The measured predose value, 4.0 mg/L, is also close to its forecast of 3.99 mg/L. These two comparisons do not prove the model is valid, but they show no large discrepancy at the observed times in this hypothetical case. A discrepancy much larger than expected measurement and residual error would warrant checking dose and sample times, adherence, assay units, and the model and parameter estimates before changing the regimen.

Calculate the Target Regimen

For a fixed dosing interval and fixed individual parameters, the concentration forecasts are proportional to dose. AUC over a steady-state interval has a particularly direct relationship to dose: under linear pharmacokinetics, the amount entering the systemic circulation in each interval is $FD$, and at steady state this equals the amount eliminated over that interval. Since elimination rate is $CL$ times concentration, the interval AUC is $FD/CL$.

Justification of the steady-state AUC: At steady state, the amount in the body at the beginning and end of an interval is the same. Therefore, the net amount eliminated during that interval equals the amount systemically available from the dose, $FD$. With linear clearance, the eliminated amount is $CL\int_0^\tau C_{ss}(t)\,dt$. Equating these gives $$ AUC_{\tau,ss}=\int_0^\tau C_{ss}(t)\,dt=\frac{FD}{CL}. $$ The result requires steady state and constant linear clearance; it is not a general formula for nonlinear or time-varying disposition.

Worked Example: Selecting a Dose From the AUC Target

With $F=1$ and $CL=6.0$ L/h, the 500 mg regimen gives

AUC_{\tau,ss}=\frac{(1)(500\ \text{mg})}{6.0\ \text{L/h}} =83.33\ \text{mg·h/L}.

This is below the stipulated lower target of 90 mg·h/L. The dose that gives the center of the target interval, 100 mg·h/L, is

D_{\mathrm{target}}= \frac{AUC_{\mathrm{target}}CL}{F} = \frac{(100\ \text{mg·h/L})(6.0\ \text{L/h})}{1} =600\ \text{mg per dose}.

Equivalently, the AUC limits imply a dose interval of $90(6.0)=540$ mg to $110(6.0)=660$ mg every 12 hours. This calculation addresses AUC only; it does not establish that the concentration constraints are met.

The peak for a repeated oral regimen is not generally the concentration at a fixed number of hours after dosing. It occurs where the concentration curve has zero slope. Differentiating the concentration equation and setting its derivative to zero gives, for the stated first-order absorption model,

t_{\max} = \frac{\ln\!\left[ \frac{k_a(1-e^{-k\tau})}{k(1-e^{-k_a\tau})} \right]}{k_a-k}.

This expression identifies the model-predicted peak time within the dosing interval. It depends on the rate constants and dosing interval, not on dose, when the model is linear.

Worked Example: Checking Peak and Predose Constraints

For $k=0.1000$ h$^{-1}$, $k_a=1.0$ h$^{-1}$, and $\tau=12$ h, the peak time is

t_{\max} = \frac{\ln\!\left[ \frac{1.0(1-e^{-1.2})}{0.1000(1-e^{-12})} \right]}{1.0-0.1000} = \frac{\ln(6.988)}{0.9000} \approx 2.160\ \text{h}.

At 600 mg, the multiplier is $600/54=11.11$ mg/L. Substituting $t=2.160$ h into the concentration equation gives a bracket of approximately 1.038 and a peak of about $11.53$ mg/L. At the predose time, the bracket is approximately 0.4310, giving $11.11(0.4310)=4.789$ mg/L. Thus the 600 mg regimen forecasts a peak below 14 mg/L, but its predose concentration is below the stipulated 5 mg/L minimum, so it does not meet all three constraints.

The peak constraint allows doses up to approximately $600(14/11.53)=728.5$ mg, while the predose constraint corresponds to approximately $600(5/4.789)=626.4$ mg through $600(8/4.789)=1002$ mg. Intersecting these with the AUC-allowed range of 540–660 mg gives approximately 626.4–660 mg per dose. The 600 mg dose lies outside this intersection because it falls short of the predose minimum. A dose of 650 mg every 12 hours lies inside it: its AUC is $650/6.0=108.3$ mg·h/L, its predose concentration is $650(0.4310)/54=5.19$ mg/L, and its peak is $650(1.0377)/54=12.49$ mg/L.

Interpret the Decision, Not Just the Calculation

Under the stated model and point estimates, 600 mg every 12 hours is not an acceptable regimen: its predicted AUC is 100 mg·h/L and its predicted peak is about 11.53 mg/L, but its predicted predose concentration is 4.789 mg/L, below the 5 mg/L minimum. A dose between about 626.4 and 660 mg satisfies all three constraints; 650 mg every 12 hours is one candidate, with a predicted AUC of 108.3 mg·h/L, predose concentration of 5.19 mg/L, and peak of 12.49 mg/L. The conclusion is conditional on the assumptions and targets; it is not a guarantee that a particular patient will attain those concentrations.

The calculation also explains why an adjustment should not be based on one concentration alone. A trough-based proportional adjustment might raise the dose because the current predose concentration is below the stipulated lower bound. But that approach alone does not check the AUC or peak, and its validity depends on comparable sampling, steady state, and a suitable relationship between trough and the desired exposure. Here, the model-based forecast considers all three constraints together. Conversely, an AUC-only dose calculation would also be incomplete until peak and predose forecasts were checked.

The dose-proportional calculations above apply when the dosing interval, bioavailability, and model parameters remain fixed. Changing the interval changes accumulation and the shape of the concentration-time profile; the concentrations must then be recalculated for that interval rather than scaled by dose alone. A model-based point forecast also does not describe all uncertainty in an individual’s future exposure. The selected regimen should therefore be interpreted with clinical context and an appropriate plan for follow-up concentration measurement.

Common pitfall: Replacing $V$ with $CL/k$ in the concentration multiplier without retaining the factor $k$ gives an incorrect forecast. Since $V=CL/k$, $$ \frac{F D k_a}{V(k_a-k)} = \frac{F D k k_a}{CL(k_a-k)}. $$ For the 600 mg case, this is $600(0.1000)(1.0)/[6.0(0.9000)]=11.11$ mg/L, not a value obtained by using $CL$ in place of $V$ while omitting $k$. A dimensional and numerical check of a transformed equation can catch this error before it affects the dose decision.

Diagnostics and Limits of the Case

Before applying the result, check that recorded doses and sample times are internally consistent, that the samples were collected after a stable regimen, and that the assay result and units are plausible. Compare each observed concentration with the forecast at its actual time. A systematic discrepancy across samples may indicate a poor parameter estimate or model structure; a single discrepancy may also reflect timing, adherence, or residual variability. Do not label a large mismatch as ordinary noise without checking its scale and context.

The calculation is limited by its assumptions: constant clearance and volume, first-order absorption, known bioavailability, and linear dose proportionality. The covariate equation is a population relationship, not a direct measurement of this patient’s clearance. Bayesian individual estimates depend on both that population model and the concentration data, so they can be affected by misspecified covariates, poor sampling, or inaccurate dose histories. If no dose satisfies all constraints, the appropriate conclusion is that the current model and regimen do not identify a feasible solution—not that one constraint can be silently ignored. Review the inputs, consider whether the target window and model are appropriate, and obtain clinical or pharmacometric review.

Check Your Understanding

Use the case assumptions above. The questions are followed by concise solutions so that the arithmetic and interpretation can be checked.

What is the population clearance prediction for a 70 kg patient with eGFR 50 mL/min/1.73 m²?
For 500 mg every 12 hours, what is the predicted steady-state AUC, and is it within the specified AUC target?
Why must a concentration sampled 2 hours after dosing be compared with the model prediction at 2 hours?
If $F$, $CL$, and the 12-hour interval remain fixed, what dose gives an AUC of 90 mg·h/L?
State the three predicted exposure measures for 600 mg every 12 hours and whether the regimen meets the stipulated window.

Solutions. (1) $8.0(70/70)^{0.75}(50/90)^{0.5}=5.963$ L/h. (2) $500/6.0=83.33$ mg·h/L, below the 90–110 mg·h/L target. (3) Concentration changes over the dosing interval, so a sample at 2 hours is not interchangeable with a peak, trough, or value at another time. (4) $90(6.0)/1=540$ mg per dose. (5) AUC is 100 mg·h/L, predose concentration is about 4.789 mg/L, and peak is about 11.53 mg/L. The AUC and peak meet their constraints, but the predose concentration is below 5 mg/L, so 600 mg does not meet the full window; doses of about 626.4–660 mg do.