Why Therapeutic Drug Monitoring Needs a Model
For a narrow-therapeutic-index drug, a concentration measurement may help decide whether to change a patient's dose. But a measured concentration does not, by itself, say what dose is appropriate: the interpretation depends on the regimen, sample timing, pharmacokinetics, and the patient's clinical state. Therapeutic drug monitoring (TDM) uses concentration data together with a pharmacokinetic model to estimate an individual's relevant parameter, then uses that estimate to support a dosing decision.
This tutorial develops a deliberately simple case: a hypothetical drug given by continuous intravenous infusion. At steady state, concentration is determined by infusion rate and clearance. A population model supplies a distribution of plausible clearances before the patient's measurement is considered; the measurement updates that distribution. The resulting estimate can guide a new infusion rate, but it is not a substitute for clinical judgment or follow-up measurement.
Model, Assumptions, and Bayesian Update
Let \(R\) be the continuous infusion rate in mg/h, \(CL\) the patient's clearance in L/h, and \(C\) the steady-state concentration in mg/L. For linear pharmacokinetics, the steady-state mass-balance equation is \(R=CL\,C\), so \(C=R/CL\). This relation describes steady state; it should not be applied to a concentration collected before steady state has been reached.
For the worked case, suppose the population model gives a lognormal prior distribution for individual clearance. Equivalently, \(\log(CL)\) is normally distributed with mean \(\mu_0\) and variance \(\omega^2\). Assume that the measured concentration has multiplicative, lognormal error: \(C_{\mathrm{obs}}=(R/CL)\exp(\epsilon)\), where \(\epsilon\sim N(0,\sigma^2)\). Transforming the concentration observation gives \(z=\log(R/C_{\mathrm{obs}})=\log(CL)-\epsilon\). Thus, conditional on the patient's clearance, \(z\) is normally distributed around \(\log(CL)\), with variance \(\sigma^2\).
The update relies on several substantive assumptions. The infusion rate and units must be known; the patient must have received that rate long enough to be at steady state; the concentration must be collected and measured correctly; clearance must be reasonably stable over the relevant period; and the population prior and error model must be suitable for this patient and assay. It also assumes linear, time-invariant clearance over the concentration range in question. If these conditions are doubtful, the simple update may be misleading.
This result follows by multiplying the prior density by the likelihood and retaining terms involving \(x=\log(CL)\). The exponent of the resulting density is \(-\frac{1}{2}\left[(x-\mu_0)^2/\omega^2+(z-x)^2/\sigma^2\right]\). Expanding the squares, the coefficient of \(x^2\) is \(1/\omega^2+1/\sigma^2\), and the coefficient determining the center is \(\mu_0/\omega^2+z/\sigma^2\). Completing the square therefore gives variance \(v_1\) and mean \(\mu_1\) above. The posterior mean of clearance itself is \(\exp(\mu_1+v_1/2)\); it differs from the posterior median because clearance is lognormally distributed.
From an Observed Concentration to an Individual Estimate
Worked Example: Updating Clearance From One Concentration
All values in this example are hypothetical. A patient receives \(R=12.0\) mg/h by continuous infusion. After steady state is verified, the measured concentration is \(C_{\mathrm{obs}}=6.00\) mg/L. The applicable population model assigns clearance a lognormal prior with median \(1.50\) L/h and geometric standard deviation \(1.25\). Assume a 10% coefficient of variation for log-scale concentration error.
The prior parameters are \(\mu_0=\log(1.50)=0.4055\) and \(\omega=\log(1.25)=0.2231\), so \(\omega^2=0.04979\). A 10% concentration coefficient of variation corresponds to a log-scale standard deviation close to \(0.10\); for this calculation set \(\sigma=0.100\), so \(\sigma^2=0.0100\).
First transform the observation: \(z=\log(R/C_{\mathrm{obs}})=\log(12.0/6.00)=\log(2.00)=0.6931\). The posterior variance and mean are
The posterior median clearance is \(\exp(0.6451)=1.906\) L/h. Its posterior standard deviation on the log scale is \(\sqrt{0.008328}=0.09126\), giving a 95% credible interval of \(\exp(0.6451\pm1.96(0.09126))=(1.594,\ 2.280)\) L/h. Conditional on the stated model and data, the posterior probability that clearance lies in this interval is 95%; this is not a guarantee that the model or its assumptions are correct.
The concentration alone would imply \(CL=R/C_{\mathrm{obs}}=2.00\) L/h if it were treated as error-free. The population prior pulls the updated estimate toward its median of 1.50 L/h, while the relatively precise measurement gives more weight to the observed value. The posterior median, 1.906 L/h, lies between them.
Using the Estimate to Revise the Infusion Rate
A target concentration range must come from clinical evidence and the treatment context; it is not determined by the clearance calculation. For illustration, suppose this hypothetical drug has an agreed steady-state target range of 7.0–9.0 mg/L, with a nominal target of 8.0 mg/L. Given a clearance estimate, the infusion rate corresponding to the nominal target follows from \(R_{\mathrm{new}}=CL\times C_{\mathrm{target}}\). This predicts the steady-state concentration under the model, not the exact concentration that will necessarily be observed after a dose change.
Worked Example: Selecting a Revised Rate and Showing Its Uncertainty
Use the posterior clearance estimate from the preceding example and target \(C_{\mathrm{target}}=8.00\) mg/L. Using the posterior median clearance gives
The units reduce to mg/h. For comparison, the current rate of 12.0 mg/h would predict \(12.0/1.906=6.30\) mg/L at steady state using the posterior median clearance, below the hypothetical target range. Transforming the clearance interval through the same dose equation gives rates of \(1.594\times8.00=12.75\) mg/h to \(2.280\times8.00=18.24\) mg/h. This range reflects uncertainty in the estimated clearance, not a range guaranteed to contain the clinically optimal rate.
A model-based decision could therefore use about 15.3 mg/h as a provisional rate associated with the nominal target, while considering whether the size of the change is appropriate for this patient. Because clearance remains uncertain and the drug has a narrow therapeutic window, the plan should include suitable clinical monitoring and a repeat concentration at an interpretable time after the change. A calculated rate should not be mistaken for an instruction to make an unsupervised dose change.
A useful diagnostic is whether the model's predictions are compatible with the observed concentration and known dosing history. A concentration far from the population prediction may indicate a genuinely unusual clearance, but it may also reflect a non-steady-state sample, an incorrect infusion history, a sampling or assay problem, a change in clinical status, or a poor population model. Check those explanations before interpreting an extreme individual estimate as pharmacology.
Updating Again After a Dose Change
Repeated TDM can refine the estimate when the later observation is collected under conditions represented by the model. Under the assumptions here, each transformed observation contributes information about the same stable \(\log(CL)\). If the observation errors are independent, the posterior precision after several measurements is the prior precision plus the sum of the measurement precisions. A new result should not be counted twice: the prior for a later update is the existing posterior, not the original population prior again.
Worked Example: Incorporating a Follow-Up Concentration
Suppose the infusion is changed to \(15.0\) mg/h. After the new rate has reached steady state, a hypothetical follow-up concentration is \(7.20\) mg/L. Retain the same \(\sigma=0.100\), and use the first measurement's posterior as the prior for this update: \(\mu_1=0.6451\), \(v_1=0.008328\). The transformed follow-up observation is \(z_2=\log(15.0/7.20)=\log(2.0833)=0.7340\).
The updated variance is \((1/v_1+1/\sigma^2)^{-1}=(120.1+100.0)^{-1}=0.004544\), and the updated mean is \(0.004544(0.6451/0.008328+0.7340/0.0100)=0.6855\). Thus the new posterior median clearance is \(\exp(0.6855)=1.985\) L/h. The observation shifts the estimate upward because its rate-to-concentration ratio, 2.083 L/h, is higher than the earlier posterior median.
For the same nominal target of 8.00 mg/L, the new point estimate implies \(1.985\times8.00=15.88\) mg/h. This is a model-based estimate, not an automatic second dose increase: the follow-up concentration of 7.20 mg/L is already within the illustrative 7.0–9.0 mg/L range. The prescriber should weigh the target, measurement uncertainty, clinical response, and the risk of changing a regimen that is currently producing an in-range concentration.
Diagnostics, Alternatives, and Limitations
A successful calculation can still be inappropriate if the input conditions are wrong. Before acting on a TDM result, verify the actual administered rate and duration, sample time, steady-state status, assay units, and any recent interruption or change in treatment. Review clinical factors that may have altered clearance since the population model was developed. A large difference between the concentration and the model's expected range, or a sequence of measurements that cannot be reconciled with a stable clearance, calls for investigation rather than blind dose adjustment.
The example's one-compartment steady-state infusion relation is intentionally narrow. For intermittent or oral dosing, a sample depends on dose timing and may depend on absorption, distribution volume, bioavailability, and the interval since the last dose. Those cases require a model that represents the actual regimen and sample time; substituting a trough concentration into \(C=R/CL\) is invalid. Likewise, changing renal function, nonlinear elimination, interacting drugs, or time-varying clearance may require a more appropriate model or a new set of measurements.
| Approach | What it contributes | Main caution |
|---|---|---|
| Population covariate-based dose adjustment | Uses known patient characteristics and population relationships to inform a dose before or without a concentration. | May not capture an individual's unexplained clearance difference. |
| Direct concentration-to-clearance ratio | For this steady-state infusion example, \(R/C_{\mathrm{obs}}\) gives a simple individual estimate. | Treats the measurement as exact and ignores population information and observation error. |
| Model-informed TDM update | Combines a population distribution and individual concentration data, making the relative influence of prior knowledge and measurement uncertainty explicit. | Can be biased or overconfident if the model, prior population, error assumptions, or sample conditions are unsuitable. |
The population model is not a universal prior: a patient unlike the model's development population may not be well represented. The posterior interval also describes uncertainty under the specified model; it does not include every source of model misspecification. For a narrow-therapeutic-index drug, TDM is most useful as a disciplined link between data and a monitored clinical decision, not as a mechanical dose calculator.
Common Pitfall
- Wrong reasoning: “The measured concentration is 6.00 mg/L and the target is 8.00 mg/L, so increase the dose by exactly one third.” This assumes concentration changes proportionally with dose for this patient, ignores uncertainty and the patient's clearance, and may also ignore whether the sample represents steady state.
- Correction: Check the regimen and sampling assumptions, interpret the measurement through a suitable PK model, combine it with the population information, and calculate a target rate from the updated individual estimate. Then account for uncertainty and plan appropriate monitoring.
Check Your Understanding
Answer each question using the infusion model and assumptions developed above. Solutions follow the questions.
- At steady state, what clearance is implied by an infusion rate of 10.0 mg/h and a measured concentration of 5.00 mg/L if the measurement is temporarily treated as exact?
- Why is the relation \(C=R/CL\) not enough to interpret a sample collected before steady state?
- For the first worked example, why does the posterior median clearance differ from the direct ratio \(12.0/6.00\) L/h?
- Using a clearance estimate of 2.00 L/h and a target concentration of 8.00 mg/L, calculate the corresponding infusion rate and state its units.
- Name two reasons a concentration might disagree with a population model prediction that do not require the patient to have unusual clearance.
Solutions. (1) The direct ratio is \(10.0\ \text{mg/h}/5.00\ \text{mg/L}=2.00\) L/h. (2) Before steady state, concentration depends on the dosing history and the approach to steady state, so the steady-state mass-balance equation does not describe that sample. (3) The posterior combines the observed ratio with the population prior and allows for measurement error; in the worked example the posterior median is 1.906 L/h rather than 2.00 L/h. (4) \(R=2.00\ \text{L/h}\times8.00\ \text{mg/L}=16.0\) mg/h. (5) Examples include an incorrect or incomplete dosing history, an incorrectly timed sample, an assay or unit error, and a clinical change affecting the observation or model conditions.