Bayesian Forecasts at Steady State

Why Steady-State Forecasts Need Patient-Specific Updating

At steady state, concentrations still rise and fall during each dosing interval; steady state means that this repeated profile no longer changes from one interval to the next. A population model supplies a starting distribution for an individual’s pharmacokinetic parameters. Measured concentrations can update that distribution, and the updated distribution can then predict the individual’s future peaks, troughs, and other concentrations under a specified regimen.

This tutorial derives the profile for a one-compartment model with first-order elimination and regularly repeated intravenous doses. The derivation assumes linear, time-invariant pharmacokinetics: clearance and volume do not change with concentration or over the intervals considered. Those assumptions make concentrations proportional to dose and allow contributions from successive doses to be added.

Definition: Steady state is reached when, for a fixed regimen and fixed pharmacokinetic parameters, the concentration–time profile over one dosing interval repeats in each subsequent interval. For repeated doses separated by \(\tau\), \(C_{\mathrm{ss}}(t+\tau)=C_{\mathrm{ss}}(t)\), where \(t\) is measured from the same point in each interval.

Deriving the Repeated-Dose Profile

For an intravenous bolus dose \(D\) into a one-compartment volume \(V\), with first-order elimination rate constant \(k\), the concentration from one dose after time \(t\) is \((D/V)e^{-kt}\). After many doses at intervals of \(\tau\), the concentration at time \(t\) after the most recent dose is the sum of the residual contributions from that dose and all earlier doses:

C_{\mathrm{ss}}(t)=\frac{D}{V}e^{-kt}\left(1+e^{-k\tau}+e^{-2k\tau}+\cdots\right),\qquad 0\leq t<\tau.

Since \(k>0\) and \(\tau>0\), the terms form a convergent geometric series with ratio \(e^{-k\tau}<1\). Its sum is \(1/(1-e^{-k\tau})\). Thus the steady-state concentration immediately after the bolus is the peak, and the concentration just before the next dose is the trough:

C_{\mathrm{peak,ss}}=\frac{D/V}{1-e^{-k\tau}},\qquad C_{\mathrm{trough,ss}}=C_{\mathrm{peak,ss}}e^{-k\tau}.

These expressions are linked: under this bolus model, the trough-to-peak ratio is \(e^{-k\tau}\). The average concentration over a steady-state interval is \(D/(CL\tau)\), where \(CL=kV\). To see why, the area under the concentration curve over an interval is the dose divided by clearance, \(D/CL\); dividing this area by the interval length \(\tau\) gives the average. A change in dose scales the entire profile proportionally, but a change in \(k\) changes both accumulation and the shape of the profile.

Worked Example: Calculating a Bolus Peak and Trough

Suppose a hypothetical patient receives 400 mg intravenously every 12 hours. Assume \(V=20\) L and \(k=0.150\ \mathrm{h}^{-1}\). Under the stated one-compartment, linear model, calculate the steady-state peak and trough.

The elimination rate over one interval is \(k\tau=(0.150)(12)=1.80\). The peak is

C_{\mathrm{peak,ss}}=\frac{400\ \mathrm{mg}/20\ \mathrm{L}}{1-e^{-1.80}} =\frac{20.00\ \mathrm{mg/L}}{0.8347} \approx 23.96\ \mathrm{mg/L}.

The trough is the peak multiplied by the fraction remaining after 12 hours:

C_{\mathrm{trough,ss}}=(23.96\ \mathrm{mg/L})e^{-1.80} \approx 3.960\ \mathrm{mg/L}.

The ratio \(3.960/23.96\approx0.1653\) agrees with \(e^{-1.80}\). Clearance is \(kV=(0.150)(20)=3.00\ \mathrm{L/h}\), so the average steady-state concentration is \(400/(3.00\times12)=11.11\ \mathrm{mg/L}\). The average lies between the calculated trough and peak, as expected for this repeating profile.

For an infusion of duration \(T\) within each interval, the end-of-infusion concentration and the concentration immediately before the next infusion are different points in the profile from a bolus peak and trough. With infusion rate \(R_0=D/T\) and \(CL=kV\), summing the residual contributions from previous infusions gives:

C_{\mathrm{end,ss}}=\frac{R_0}{CL}\frac{1-e^{-kT}}{1-e^{-k\tau}},\qquad C_{\mathrm{pre,next,ss}}=C_{\mathrm{end,ss}}e^{-k(\tau-T)}.

The exponential decay from infusion end to the next pre-infusion sample lasts \(\tau-T\), not \(\tau\). The formula requires regular infusions, \(T\leq\tau\), and the same linear one-compartment assumptions. Use the bolus equations only when the administration is appropriately treated as a bolus; do not silently apply them to a prolonged infusion.

Updating the Individual Profile With Bayesian Evidence

For a patient-specific forecast, let \(\theta\) denote the individual parameters, such as \(k\) and \(V\). A prior distribution \(p(\theta)\) represents the parameter uncertainty before the new concentrations are used. Given observed concentrations \(y\), the posterior distribution is proportional to the prior multiplied by the likelihood:

p(\theta\mid y)=\frac{p(y\mid\theta)p(\theta)}{p(y)}.

The likelihood must match the sampling times and the assumed concentration model. In the example below, the peak and trough errors are assumed independent and normally distributed with a common known standard deviation. That is a simplifying model assumption, not a universal property of therapeutic drug monitoring data. A posterior forecast for a future time is obtained by averaging the model-predicted concentration over the posterior distribution of \(\theta\), rather than automatically substituting a single estimate.

Worked Example: Updating a Discrete Prior With Peak and Trough Data

Consider a hypothetical patient receiving 500 mg by intravenous bolus every 12 hours. Suppose \(V=25\) L is fixed for this calculation, while the prior for \(k\) assigns probabilities \(0.20\), \(0.50\), and \(0.30\) to \(0.080\), \(0.120\), and \(0.160\ \mathrm{h}^{-1}\), respectively. At steady state, a measured peak is 23.5 mg/L and a measured trough is 3.6 mg/L. Assume independent normal measurement errors with standard deviation \(\sigma=2.5\ \mathrm{mg/L}\), and use the bolus model.

The dose-to-volume ratio is \(500/25=20.00\ \mathrm{mg/L}\). Applying the peak and trough equations gives the following predictions (rounded):

\(k\) (\(\mathrm{h}^{-1}\))	Prior probability	Predicted peak (mg/L)	Predicted trough (mg/L)
0.080	0.20	32.41	12.41
0.120	0.50	26.21	6.210
0.160	0.30	23.44	3.436

For example, at \(k=0.160\ \mathrm{h}^{-1}\), \(e^{-k\tau}=e^{-1.92}=0.1466\), so the peak is \(20.00/(1-0.1466)=23.44\ \mathrm{mg/L}\), and the trough is \(23.44(0.1466)=3.436\ \mathrm{mg/L}\). For each candidate \(k_i\), the sum of squared residuals is \(SSE_i=(23.5-C_{\mathrm{peak},i})^2+(3.6-C_{\mathrm{trough},i})^2\). The three values are approximately 157.0, 14.16, and 0.03106 \((\mathrm{mg/L})^2\), respectively.

Under the stated independent normal-error model, the likelihood is proportional to \(\exp[-SSE_i/(2\sigma^2)]\). Relative to \(k=0.160\), the likelihood ratio for \(k=0.120\) is \(\exp[-(14.16-0.03106)/(2(2.5)^2)]\approx0.323\). Multiplying by the prior odds \(0.50/0.30\) gives posterior odds of about 0.538 for \(k=0.120\) versus \(k=0.160\). The relative weight for \(k=0.080\) is about \(2.4\times10^{-6}\). Normalising these weights gives posterior probabilities approximately 0.000002, 0.350, and 0.650 for the three candidate values.

The posterior mean of \(k\) is approximately \((0.000002)(0.080)+(0.350)(0.120)+(0.650)(0.160)=0.1460\ \mathrm{h}^{-1}\). But a forecast can retain the full discrete posterior. The posterior-weighted peak is \((0.350)(26.21)+(0.650)(23.44)\approx24.41\ \mathrm{mg/L}\), and the posterior-weighted trough is \((0.350)(6.210)+(0.650)(3.436)\approx4.407\ \mathrm{mg/L}\). These are posterior means of predicted concentrations; they need not equal the concentrations predicted by plugging the posterior mean \(k\) into the nonlinear peak and trough formulas.

Using the Updated Profile for a Regimen Forecast

Once the patient-specific posterior is available, a proposed regimen can be evaluated by recalculating the steady-state profile at each plausible parameter value. This carries the updated uncertainty into the forecast. For a dose change with the same interval and unchanged linear pharmacokinetics, concentrations at each corresponding time scale in direct proportion to dose.

Worked Example: Forecasting a Higher Dose From the Posterior

Use the posterior distribution from the preceding example to forecast steady-state peak and trough concentrations for 600 mg every 12 hours, still by intravenous bolus. This is a hypothetical calculation, not a treatment recommendation. The dose ratio is \(600/500=1.20\). For the \(k=0.120\) scenario, the predictions become \(1.20(26.21)=31.45\ \mathrm{mg/L}\) and \(1.20(6.210)=7.452\ \mathrm{mg/L}\). For \(k=0.160\), they become \(1.20(23.44)=28.13\ \mathrm{mg/L}\) and \(1.20(3.436)=4.123\ \mathrm{mg/L}\). The negligible posterior weight on \(k=0.080\) contributes negligibly to the posterior means.

Weighting by the posterior probabilities gives a forecast peak of \((0.350)(31.45)+(0.650)(28.13)\approx29.29\ \mathrm{mg/L}\), and a forecast trough of \((0.350)(7.452)+(0.650)(4.123)\approx5.288\ \mathrm{mg/L}\). The posterior uncertainty remains important: the two plausible parameter values imply noticeably different troughs, even though both are supported by the observations. A decision would also require the relevant clinical target and safety context; these predicted values alone do not establish that the higher dose is appropriate.

Diagnostics, Alternatives, and Limitations

A useful internal check for a bolus profile is whether the observed trough-to-peak ratio is compatible with \(e^{-k\tau}\) for the inferred \(k\). In the Bayesian example, the measured ratio is \(3.6/23.5\approx0.1532\), close to \(e^{-1.92}=0.1466\) for \(k=0.160\ \mathrm{h}^{-1}\). A large mismatch across candidate parameter values, or systematic residuals in which both measurements lie on the same side of predictions, warrants scrutiny of the dosing and sampling times, the assumed model, and the error assumptions. A good fit to two observations does not prove that the structural model is correct.

The main alternative to averaging predictions over the posterior is to forecast at one selected parameter estimate, such as a maximum a posteriori (MAP) estimate or posterior mean. A single estimate is simpler to report, but it suppresses parameter uncertainty; with nonlinear equations, its predicted concentration can differ from the posterior-averaged prediction. Conversely, a posterior mean forecast is not itself a prediction interval and does not show the range of plausible future concentrations. The choice depends on the question: a central forecast can summarize expected exposure, whereas decisions sensitive to uncertainty require the distribution of forecasts.

The calculations here have important limits. They assume steady state has actually been reached, the dose history is known and regular, and the individual follows a one-compartment model with constant \(k\) and \(V\). The discrete-prior example fixes \(V\) and treats the error standard deviation as known, so it does not represent uncertainty in those quantities. Real data can also include correlated or heteroscedastic errors and deviations from the assumed structural model. These are reasons to assess the assumptions, not reasons to interpret an apparently precise posterior as proof of a precise forecast.

Common mistake: treating the measured trough as if it were a new dose’s concentration, or multiplying the peak by \(e^{-k\tau}\) for a sample taken before the end of an infusion. Correction: identify the exact dosing and sampling times. For bolus dosing, the post-peak decay lasts \(\tau\); for an infusion ending \(T\) hours after its start, the end-infusion-to-pre-next-infusion decay lasts \(\tau-T\).
Another pitfall: reporting a prediction at the posterior mean parameter as though it includes all posterior uncertainty. Correction: distinguish plug-in predictions from posterior averages and from predictive distributions.

Key takeaway: Derive the steady-state profile for the actual administration method, update the individual parameter distribution with a likelihood matched to the sample times, and forecast by carrying that posterior uncertainty into the future concentration profile.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

For bolus dosing, why does the steady-state peak contain the factor \(1/(1-e^{-k\tau})\)?
A hypothetical bolus regimen uses \(D=300\) mg, \(V=15\) L, \(k=0.10\ \mathrm{h}^{-1}\), and \(\tau=8\) h. Calculate its steady-state peak and trough.
For a 4-hour infusion repeated every 12 hours, what elapsed time separates the end-infusion peak and the pre-next-infusion trough?
Why might a posterior-averaged peak differ from the peak calculated at the posterior mean of \(k\)?
What does a substantial mismatch between the observed peak–trough ratio and the ratio predicted by the fitted bolus model suggest?

Solutions. (1) It is the sum of the geometric series of residual concentrations from the current and all preceding doses; the ratio between successive contributions is \(e^{-k\tau}\). (2) Here \(D/V=20.00\ \mathrm{mg/L}\) and \(k\tau=0.80\). Thus \(C_{\mathrm{peak,ss}}=20.00/(1-e^{-0.80})\approx36.32\ \mathrm{mg/L}\), and \(C_{\mathrm{trough,ss}}=36.32e^{-0.80}\approx16.32\ \mathrm{mg/L}\). (3) The time is \(\tau-T=12-4=8\) hours. (4) The peak is a nonlinear function of \(k\), so averaging the function over parameter values generally differs from evaluating the function at the average parameter. (5) It suggests checking timing, dosing assumptions, error assumptions, and model compatibility; the mismatch alone does not identify which assumption failed.