Deriving Steady-State Profiles for Repeated Bolus Dosing

Why Derive the Repeated-Bolus Profile?

A dose and a dosing interval do not, by themselves, reveal the concentration experienced across an interval. After repeated intravenous boluses, each new dose adds to drug remaining from earlier doses. Under a linear one-compartment model, the shape within each interval is exponential, but its starting concentration depends on the entire dosing history. Deriving the steady-state profile makes that history explicit and gives exact expressions for the post-dose peak, pre-dose trough, and interval-average concentration.

The derivation below concerns the concentration-time profile, not a complete regimen-optimization rule. The three summary concentrations describe different features of the profile: the peak and trough are endpoint values, whereas the average is an integral over time. They are not interchangeable measures of exposure or target attainment.

Model and Assumptions

Definition: Consider an intravenous bolus dose $D$ administered every $\tau$ time units to a one-compartment system with volume of distribution $V$ and first-order elimination rate constant $k$. Between doses, concentration declines according to $dC/dt=-kC$. Each bolus instantaneously increases concentration by $D/V$. The model assumes $D>0$, $V>0$, $k>0$, and $\tau>0$, with constant parameters and complete, instantaneous delivery of each dose.

The assumptions imply linear, time-invariant pharmacokinetics: concentration contributions from separate doses add, and the elimination rate does not change with concentration. The model excludes absorption delay, infusion duration, distribution into multiple compartments, time-varying clearance, and nonlinear elimination. In particular, a bolus is an instantaneous input; it is not a short infusion.

Let time $t=0$ denote immediately after a dose at steady state, and let $0\leq t<\tau$ denote time within the following interval. The steady-state profile is defined after the current dose has been given. At the next dosing time, the pre-dose value is the left-hand limit as $t$ approaches $\tau$; the next bolus then causes an instantaneous upward jump of $D/V$.

Deriving the Steady-State Profile

After one bolus, the contribution at time $t$ is $(D/V)e^{-kt}$. At steady state, the concentration at a given point in the current interval includes the current dose and all earlier doses. The current dose has age $t$; the dose one interval earlier has age $t+\tau$; the dose two intervals earlier has age $t+2\tau$, and so on. By superposition,

C_{ss}(t)=\frac{D}{V}\sum_{j=0}^{\infty}e^{-k(t+j\tau)} =\frac{D}{V}e^{-kt}\sum_{j=0}^{\infty}\left(e^{-k\tau}\right)^j, \qquad 0\leq t<\tau.

Since $k>0$ and $\tau>0$, $r=e^{-k\tau}$ satisfies $0<r<1$. The geometric series therefore converges to $1/(1-r)$. Substituting that sum gives the steady-state profile:

Steady-state profile: Under the stated assumptions, $$ C_{ss}(t)=\frac{D}{V}\frac{e^{-kt}}{1-e^{-k\tau}}, \qquad 0\leq t<\tau. $$

The denominator accounts for residual drug from earlier doses; the numerator describes elimination during the current interval. If $k=0$, then $e^{-k\tau}=1$, the series does not converge, and repeated positive boluses do not produce a finite steady state. Thus the positive elimination assumption is essential, not merely a convenient condition.

Peak, Trough, and Interval Average

Within an interval, differentiating the profile gives $dC_{ss}(t)/dt=-kC_{ss}(t)<0$. It therefore decreases continuously after the bolus. Its maximum on $0\leq t<\tau$ is the value at $t=0$, immediately after dosing. Because $\tau$ is excluded from this half-open interval, the decreasing profile does not attain a minimum there: its infimum is the left-hand limit at $\tau$, which is the pre-dose trough. At steady state, the next bolus raises that trough by $D/V$ back to the same post-dose peak.

C_{\max,ss}=C_{ss}(0)=\frac{D/V}{1-e^{-k\tau}}, \qquad C_{\min,ss}=C_{ss}(\tau^-)= \frac{D/V\,e^{-k\tau}}{1-e^{-k\tau}}.

The average concentration over one dosing interval is the area under the profile divided by the interval length. Integrating the exponential gives

\begin{aligned} C_{\mathrm{avg},ss} &=\frac{1}{\tau}\int_0^\tau C_{ss}(t)\,dt\\ &=\frac{1}{\tau}\frac{D/V}{1-e^{-k\tau}} \int_0^\tau e^{-kt}\,dt\\ &=\frac{1}{\tau}\frac{D/V}{1-e^{-k\tau}} \frac{1-e^{-k\tau}}{k} =\frac{D}{Vk\tau}. \end{aligned}

For this one-compartment model, clearance is $CL=kV$. Thus the same result can be written $C_{\mathrm{avg},ss}=D/(CL\,\tau)$. This follows from the profile integral; it is not obtained by taking the arithmetic mean of the peak and trough. The profile is exponential, so an endpoint midpoint generally does not equal its time average.

For positive elimination, the profile decreases from its peak toward its trough, so the interval average lies between them. This ordering is a useful check, but it does not mean every concentration is close to the average or that the average meets any particular clinical target.

Worked Example: Deriving All Three Metrics From a Half-Life

Suppose a hypothetical patient receives $D=100$ mg IV every $\tau=12$ hours. Let $V=20$ L and the elimination half-life be 6 hours. Calculate the steady-state profile, peak, trough, and interval average.

The model assumptions are constant volume and first-order elimination, with an instantaneous bolus every 12 hours. Convert the half-life to the elimination rate constant:

k=\frac{\ln 2}{6\ \mathrm{h}}=0.115525\ \mathrm{h}^{-1}, \qquad e^{-k\tau}=e^{-(0.115525)(12)}=0.2500.

The bolus increment is $D/V=100\ \mathrm{mg}/20\ \mathrm{L}=5.000\ \mathrm{mg/L}$. Therefore, for $0\leq t<12$ hours,

C_{ss}(t)=\frac{5.000e^{-0.115525t}}{1-0.2500} =6.6667e^{-0.115525t}\ \mathrm{mg/L}.

The post-dose peak is $C_{\max,ss}=5.000/0.7500=6.6667\ \mathrm{mg/L}$. The pre-dose trough is the left-hand limit at 12 hours: $C_{\min,ss}=6.6667(0.2500)=1.6667\ \mathrm{mg/L}$.

The interval average is $C_{\mathrm{avg},ss}=D/(Vk\tau)=100/[(20)(0.115525)(12)]=3.6067\ \mathrm{mg/L}$. Equivalently, $CL=kV=(0.115525)(20)=2.3105\ \mathrm{L/h}$, so $D/(CL\tau)=100/[(2.3105)(12)]=3.6067\ \mathrm{mg/L}$. The computed average lies between the peak and trough, as expected for a decreasing profile.

Worked Example: Evaluating Concentration at a Specific Time

For a hypothetical regimen, $D=250$ mg is given every 24 hours, $V=35$ L, and the half-life is 8 hours. Find the steady-state peak, trough, average, and concentration 6 hours after a dose.

The assumptions are the same linear, one-compartment bolus assumptions. Here $k=\ln(2)/8=0.0866434\ \mathrm{h}^{-1}$, and $e^{-k\tau}=e^{-3\ln 2}=1/8=0.1250$. The bolus increment is $250/35=7.14286\ \mathrm{mg/L}$. Hence,

C_{ss}(t)=\frac{7.14286e^{-0.0866434t}}{0.8750} =8.16327e^{-0.0866434t}\ \mathrm{mg/L}.

The peak is $8.16327\ \mathrm{mg/L}$, and the pre-dose trough is $(8.16327)(0.1250)=1.02041\ \mathrm{mg/L}$. The interval average is $250/[(35)(0.0866434)(24)]=3.435\ \mathrm{mg/L}$.

At 6 hours, the concentration is $C_{ss}(6)=(8.16327)e^{-(0.0866434)(6)} =(8.16327)(0.594604)=4.8539\ \mathrm{mg/L}$. It is above the interval average and below the peak, consistent with the declining profile. All concentrations have units of mg/L because the dose-to-volume ratio has those units and the exponential factor is dimensionless.

Worked Example: Choosing a Dose for a Specified Interval Average

A hypothetical design uses $V=30$ L, a 10-hour half-life, and a 12-hour interval. What dose gives a steady-state interval average of $5.00\ \mathrm{mg/L}$, and what peak and trough does that dose imply under the model?

The assumed profile remains linear and first-order, so the average formula can be rearranged for dose. First, $k=\ln(2)/10=0.0693147\ \mathrm{h}^{-1}$. Then

D=C_{\mathrm{avg},ss}Vk\tau =(5.00\ \mathrm{mg/L})(30\ \mathrm{L})(0.0693147\ \mathrm{h}^{-1})(12\ \mathrm{h}) =124.767\ \mathrm{mg}.

The exponent over one interval is $k\tau=0.831776$, so $e^{-k\tau}=e^{-0.831776}=0.435275$. The bolus increment is $124.767/30=4.15890\ \mathrm{mg/L}$. Therefore,

C_{\max,ss}=\frac{4.15890}{1-0.435275} =7.3644\ \mathrm{mg/L}, \qquad C_{\min,ss}=(7.3644)(0.435275)=3.206\ \mathrm{mg/L}.

The calculated average is $5.00\ \mathrm{mg/L}$ by construction, and it lies between the implied peak and trough. This dose calculation only matches the specified average under the stated model; it does not establish that the peak and trough are clinically acceptable.

Checks, Alternatives, and Limitations

Several checks can expose algebra or implementation errors. The exponent $kt$ must be dimensionless, so $k$ must have inverse-time units and $t,\tau$ must use compatible time units. $D/V$ and each concentration must have concentration units. For $D,V,k,\tau>0$, the peak must exceed the trough, and the average must lie between them. Also, as the interval becomes long relative to the half-life, $e^{-k\tau}$ approaches zero and the steady-state peak approaches the single-dose increment $D/V$. These checks do not verify the pharmacokinetic assumptions themselves.

The geometric-series expression is the closed-form approach. A numerical alternative is to add contributions from a finite number of prior doses. That approximation is useful for checking code or handling dose histories that are not yet periodic, but it must include enough prior doses to make the omitted residual negligible. For a genuinely repeated regimen under the stated model, the closed form is exact and more direct. A single-dose calculation alone is not a steady-state calculation: it omits residual drug from earlier administrations.

The main limitation is model dependence. If elimination is nonlinear, parameters vary over time, distribution is not adequately represented by one compartment, or the input is not an instantaneous bolus, this profile may be inappropriate. Real patients may also differ in $V$ and $k$; the formula for one parameter set does not automatically describe a population distribution. Finally, the interval average can conceal clinically important high or low portions of the profile. A dose chosen from an average target should therefore be checked against whatever other exposure constraints the decision requires.

Common Mistakes

Using a single-dose curve as the steady-state curve. The correction is to sum the residual contributions from previous doses, or use the geometric-series denominator $1-e^{-k\tau}$.
Calling the trough an attained minimum on $[0,\tau)$. On that half-open interval the profile decreases toward, but does not attain, its infimum. The pre-dose trough is the left-hand limit $C_{ss}(\tau^-)$.
Taking the endpoint midpoint as the average. The average is the integral over time divided by $\tau$; for an exponential profile it is generally not $(C_{\max,ss}+C_{\min,ss})/2$.
Confusing an average target with a full-profile target. Matching $C_{\mathrm{avg},ss}$ does not guarantee an acceptable peak, trough, or concentration at every time.

Key takeaway: For repeated IV boluses in a one-compartment model with first-order elimination, the steady-state profile is a convergent geometric sum. Its peak is the post-dose value, its trough is the pre-dose left-hand limit, and its interval average is $D/(CL\,\tau)$.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

For what values of $k$ and $\tau$ does the geometric sum for a finite steady-state profile converge?
A hypothetical dose is $80$ mg, $V=16$ L, and $e^{-k\tau}=0.20$. Calculate the steady-state peak and trough.
For that same regimen, if $k\tau=1.20$, calculate the interval average in terms of $D/V$, and explain why the average must lie between peak and trough.
Why is the pre-dose trough described as a left-hand limit when the profile is defined on $0\leq t<\tau$?
State one model feature that would make the repeated-bolus formula inappropriate.

Solutions. (1) The sum converges when $0<e^{-k\tau}<1$, which holds for $k>0$ and $\tau>0$. (2) The bolus increment is $80/16=5.00\ \mathrm{mg/L}$. Thus $C_{\max,ss}=5.00/(1-0.20)=6.25\ \mathrm{mg/L}$, and $C_{\min,ss}=6.25(0.20)=1.25\ \mathrm{mg/L}$. (3) Since $C_{\mathrm{avg},ss}=(D/V)/(k\tau)=5.00/1.20=4.167\ \mathrm{mg/L}$, it lies between $1.25$ and $6.25\ \mathrm{mg/L}$; the profile declines continuously from peak toward trough, and its time average is between those bounds. (4) The interval excludes $t=\tau$, so the decreasing profile approaches the pre-dose value without attaining it within the interval; that value is $C_{ss}(\tau^-)$. (5) Examples include nonlinear elimination, a multi-compartment distribution model, time-varying parameters, or an infusion rather than an instantaneous bolus.