Worked Design: Meeting an AUC Target With a Fixed Daily Dose

Designing the Interval When the Daily Dose Is Fixed

When the total amount of drug given each day is fixed, changing the dosing interval does not generally change long-run average daily exposure in a linear pharmacokinetic model. It does change how that exposure is distributed over time: a longer interval usually produces a higher peak and a lower trough. A useful design therefore separates two questions: does the regimen achieve the AUC target, and does its peak–trough profile meet the additional concentration requirements?

The calculations below use a hypothetical drug given as repeated intravenous boluses. They compare candidate intervals while holding the total daily dose constant. This is a controlled comparison under a specified model, not a general claim that every regimen with the same daily dose will have the same exposure in real patients.

Model and assumptions: The drug follows a one-compartment model with linear elimination, constant clearance \(CL\), volume of distribution \(V\), and elimination rate constant \(k=CL/V\). Each dose is an instantaneous intravenous bolus with bioavailability \(F=1\). Doses are administered regularly, and concentrations are evaluated at steady state. The daily dose is divided in proportion to the candidate interval: for interval \(\tau\) hours, \(D_{\tau}=D_{\text{day}}\tau/24\). These assumptions exclude absorption, time-varying clearance, nonlinear elimination, missed doses, and changes in patient characteristics.

For a fixed daily dose, the long-run average daily AUC is determined by the amount administered per day and clearance. The interval-specific peak and trough, however, depend on the amount in each dose and on how much drug is eliminated between doses.

Key result: With linear, time-invariant pharmacokinetics and a constant daily dose, the long-run average daily AUC is \(F D_{\text{day}}/CL\). An exact AUC over a 24-hour window equals this value when 24 hours contains an integer number of dosing intervals; the phase-averaged 24-hour AUC also equals this value. For repeated boluses at interval \(\tau\), the steady-state peak immediately after a dose and trough immediately before the next dose are \(C_{\max,ss}=F D_{\tau}/[V(1-e^{-k\tau})]\) and \(C_{\min,ss}=C_{\max,ss}e^{-k\tau}\).

Why AUC Is Fixed but the Profile Changes

Between doses, concentration declines exponentially under the stated model. At steady state, the concentration just before a dose is the fraction \(e^{-k\tau}\) of the concentration just after the preceding dose. Giving a new bolus adds \(F D_{\tau}/V\) to the concentration. The accumulation of residual drug from earlier doses gives the peak formula; multiplying that peak by the fraction remaining at the end of the interval gives the trough.

The AUC result follows from mass balance over a period containing an integer number \(m\) of dosing intervals, \(T=m\tau\). At steady state, the amount of drug in the body returns to its starting value over that period, so the amount eliminated equals the bioavailable amount administered, \(mF D_{\tau}\). Under linear elimination, the elimination rate is \(CL\,C(t)\), so integrating it over the period gives \(CL\) times the AUC over that period. Scaling this AUC to a 24-hour average gives:

CL\cdot AUC_{T,ss}=mF D_{\tau} \qquad\Longrightarrow\qquad \overline{AUC}_{\text{daily},ss} =\frac{24}{T}AUC_{T,ss} =\frac{F D_{\text{day}}}{CL}, \qquad T=m\tau,\quad D_{\tau}=D_{\text{day}}\tau/24.

This argument requires steady state over the period considered and constant clearance. It does not say the concentration is constant, nor that AUC within every individual dosing interval is the same. The long-run average daily AUC remains fixed; an exact 24-hour AUC equals this value when 24 hours contains an integer number of dosing intervals, and the phase-averaged 24-hour AUC has this value. Otherwise, an exact 24-hour window's AUC can depend on its phase relative to dosing.

Worked Example: Deriving the AUC Target From the Daily Dose

Suppose a hypothetical patient receives 600 mg/day of a drug with \(F=1\) and \(CL=5\) L/h. The design target is a steady-state AUC of 120 mg·h/L over 24 hours. Check whether the fixed daily dose meets that target.

The steady-state mass-balance formula is \(AUC_{24,ss}=F D_{\text{day}}/CL\). Substituting the stated values gives:

AUC_{24,ss} = \frac{1\times 600\ \text{mg/day}}{5\ \text{L/h}} = 120\ \text{mg·h/L}.

The units are consistent because 600 mg/day divided by 5 L/h is \(120\ \text{mg·h/L}\) when the daily amount is interpreted as the total input over the 24-hour period. The computed AUC equals the hypothetical target. For intervals that divide 24 hours, changing the interval while preserving 600 mg/day will preserve this 24-hour AUC; it will not preserve the peak or trough.

Comparing Candidate Intervals

Now suppose the same hypothetical drug has \(V=50\) L as well as \(CL=5\) L/h. Then \(k=CL/V=0.100\ \text{h}^{-1}\). Consider intervals of 6, 8, 12, and 24 hours. The dose per administration is the daily amount multiplied by \(\tau/24\), so the doses are 150, 200, 300, and 600 mg, respectively. For each interval, calculate the exponential fraction remaining, the steady-state peak and trough, and the AUC in one dosing interval.

Worked Example: Calculating the Peak–Trough Profiles

Use \(C_{\max,ss}=D_{\tau}/[V(1-e^{-k\tau})]\), \(C_{\min,ss}=C_{\max,ss}e^{-k\tau}\), and \(AUC_{\tau,ss}=D_{\tau}/CL\). Here \(F=1\), \(V=50\) L, \(k=0.100\ \text{h}^{-1}\), and \(CL=5\) L/h.

For the 6-hour interval, \(D_{\tau}=600(6/24)=150\) mg and \(e^{-k\tau}=e^{-0.6}=0.5488\). Therefore:

C_{\max,ss}=\frac{150}{50(1-0.5488)}=6.649\ \text{mg/L}, \qquad C_{\min,ss}=6.649(0.5488)=3.649\ \text{mg/L}.

The interval AUC is \(150/5=30.0\ \text{mg·h/L}\).

For the 8-hour interval, \(D_{\tau}=600(8/24)=200\) mg and \(e^{-0.8}=0.4493\). Thus:

C_{\max,ss}=\frac{200}{50(1-0.4493)}=7.264\ \text{mg/L}, \qquad C_{\min,ss}=7.264(0.4493)=3.264\ \text{mg/L}, \qquad AUC_{\tau,ss}=\frac{200}{5}=40.0\ \text{mg·h/L}.

For the 12-hour interval, \(D_{\tau}=600(12/24)=300\) mg and \(e^{-1.2}=0.3012\). This gives:

C_{\max,ss}=\frac{300}{50(1-0.3012)}=8.586\ \text{mg/L}, \qquad C_{\min,ss}=8.586(0.3012)=2.586\ \text{mg/L}, \qquad AUC_{\tau,ss}=\frac{300}{5}=60.0\ \text{mg·h/L}.

Finally, for the 24-hour interval, \(D_{\tau}=600\) mg and \(e^{-2.4}=0.09072\), not \(0.1353\). Hence:

C_{\max,ss}=\frac{600}{50(1-0.09072)}=13.197\ \text{mg/L}, \qquad C_{\min,ss}=13.197(0.09072)=1.197\ \text{mg/L}, \qquad AUC_{\tau,ss}=\frac{600}{5}=120\ \text{mg·h/L}.

For each regimen, the interval AUC multiplied by the number of intervals in 24 hours is 120 mg·h/L: 120 for the 24-hour regimen, \(2(60)=120\) for the 12-hour regimen, \(3(40)=120\) for the 8-hour regimen, and \(4(30)=120\) for the 6-hour regimen. Each regimen therefore has the same daily AUC, while longer intervals produce larger peak-to-trough swings.

Interval \(\tau\) (h)	Dose per administration (mg)	\(e^{-k\tau}\)	Peak (mg/L)	Trough (mg/L)	Interval AUC (mg·h/L)
6	150	0.5488	6.649	3.649	30.0
8	200	0.4493	7.264	3.264	40.0
12	300	0.3012	8.586	2.586	60.0
24	600	0.09072	13.197	1.197	120

Using Concentration Limits to Choose Among AUC-Matched Regimens

AUC matching alone cannot distinguish these four candidates: each delivers the same hypothetical daily dose and achieves the 120 mg·h/L daily target under the model. Suppose, for illustration, the design also requires a steady-state peak no greater than 10 mg/L and a trough no lower than 2.5 mg/L. These are hypothetical design limits, not universal therapeutic thresholds.

Worked Example: Selecting the Longest Interval That Meets Both Limits

Check each row of the calculated profile table against \(C_{\max,ss}\leq 10\) mg/L and \(C_{\min,ss}\geq 2.5\) mg/L. The 6-hour profile has peak 6.649 mg/L and trough 3.649 mg/L, so it meets both limits. The 8-hour profile has peak 7.264 mg/L and trough 3.264 mg/L, so it also meets both. The 12-hour profile has peak 8.586 mg/L and trough 2.586 mg/L, so it meets both, although its trough is relatively close to the lower limit. The 24-hour profile has peak 13.197 mg/L and trough 1.197 mg/L, so it fails both limits.

Among these candidates, 12 hours is the longest interval satisfying both stated conditions. This conclusion follows from the specified limits and model calculations; it is not a claim that 12-hour dosing is generally preferable. If uncertainty in \(CL\), \(V\), adherence, or the concentration limits matters, a design based only on point estimates may not provide an adequate margin.

The profile can also be summarized by the peak-to-trough ratio \(C_{\max,ss}/C_{\min,ss}=e^{k\tau}\). In this example, that ratio is about 1.82 at 6 hours, 2.23 at 8 hours, 3.32 at 12 hours, and 11.02 at 24 hours. This gives a direct description of fluctuation, but it does not replace checking the absolute peak and trough when limits are specified in mg/L.

Assumptions, Diagnostics, and Common Pitfalls

Before using the calculations as a design, check whether the model and regimen match the intended application. The equations presume regular dosing at steady state, instantaneous intravenous administration, linear elimination, and constant \(CL\) and \(V\). They also presume that the proposed doses can actually be delivered. If the available dosage form cannot provide the calculated amount, the realized daily dose may differ and the AUC target must be recalculated.

A practical diagnostic is internal consistency. The elimination fraction must equal \(e^{-k\tau}\), and the trough must equal the peak multiplied by that fraction. The interval AUC should equal \(F D_{\tau}/CL\), while, for intervals that divide 24 hours, the summed interval AUC over 24 hours should equal \(F D_{\text{day}}/CL\). If any of these checks fails, revisit the interval, units, exponential, or dose allocation. For example, at 24 hours in this design, \(k\tau=0.100(24)=2.4\), so the fraction remaining is \(e^{-2.4}=0.09072\). Using \(e^{-2.0}=0.1353\) would incorrectly inflate both the predicted peak and trough.

A frequent mistake is to conclude that identical AUC means identical concentration profiles. AUC is an integrated exposure measure: different curves can have the same area. Here, the 24-hour schedule has the same daily AUC as the 6-hour schedule but a much higher peak and lower trough. The opposite error is to infer that changing the interval changes the long-run average daily AUC even when daily dose and clearance are fixed. That would contradict the steady-state mass balance under the linear model.

Compared with changing the daily dose, changing the interval while holding the daily amount fixed primarily reshapes the profile in this model; changing the daily amount changes AUC in direct proportion. Joint changes to dose and interval require evaluating both effects rather than assuming either is held constant. The earlier material on joint titration of dose amount and interval addresses that broader adjustment problem. These bolus calculations also do not establish how an infusion regimen would meet peak or trough limits; that design uses a different input profile.

The result has important limitations. Real absorption can delay and smooth peaks; nonlinear clearance can make AUC depend on dose distribution; and interindividual variability means a regimen meeting limits for typical parameter values may not meet them for every individual. If the dosing interval is not an exact divisor of 24 hours, the daily allocation formula still specifies the average daily input for a regularly repeated regimen, but the daily calendar and the period used to define AUC must be stated carefully. The present examples use intervals that divide 24 hours so the dose accounting is transparent.

Key takeaway: Under linear steady-state pharmacokinetics, a fixed daily dose and clearance determine the long-run average daily AUC. Interval selection is then a separate profile-design decision: calculate the peak and trough for each feasible interval, check them against the stated limits, and verify the assumptions behind the model.

Check Your Understanding

Use the hypothetical model above unless a question specifies otherwise. The answers are included for checking your reasoning.

If the daily dose is 600 mg and clearance is 5 L/h, what is the steady-state 24-hour AUC when \(F=1\)?
For a 12-hour interval, verify the dose per administration and calculate the fraction remaining by the next dose.
Why can the 6-hour and 24-hour regimens have the same daily AUC but different peaks and troughs?
Suppose clearance rises while daily dose and interval stay fixed. Under the linear model, what happens to AUC, and which model parameter relationship should be reconsidered if \(V\) remains fixed?
Give one reason why the calculated interval that meets hypothetical peak and trough limits might not meet those limits in every patient.

Solutions. (1) \(AUC_{24,ss}=600/5=120\) mg·h/L. (2) The dose is \(600(12/24)=300\) mg; the fraction remaining is \(e^{-(0.100)(12)}=e^{-1.2}=0.3012\). (3) AUC depends on total daily input and clearance, whereas the exponential decline between doses depends on \(k\tau\), so changing \(\tau\) reshapes the profile. (4) AUC decreases in inverse proportion to clearance; if \(V\) is fixed, \(k=CL/V\) rises as clearance rises, changing the decline rate as well. (5) Individual clearance or volume may differ from the assumed values, or adherence and dose delivery may differ from the modeled regimen.