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Worked Forecast From a Sparse Dosing Record

Work through how dose history, covariates, and sparse concentration data combine to produce an individual posterior and a future concentration forecast.

Advanced 12 min read

What You'll Learn

  • Specify the dose, covariate, and concentration assumptions behind an individual forecast.
  • Calculate a normal posterior for an elimination rate from sparse log-concentration observations.
  • Forecast concentration after multiple recorded doses using posterior parameter uncertainty.
  • Distinguish a posterior interval for a latent concentration from a future-observation predictive interval.
  • Check fitted residuals, dose-history arithmetic, and the limits of a plug-in forecast.

From a Sparse Record to a Forecast

An individual concentration forecast is a chain of conditional calculations. The recorded doses and their times determine the concentration model; covariates inform the individual’s prior parameter distribution; observed concentrations update that prior; and the resulting posterior distribution is propagated through the future dose history. A mistake at any link—such as treating an uncertain dose time as exact or reporting only a forecast at the posterior mean—can conceal important uncertainty.

The worked calculation below uses a deliberately simple one-compartment model with intravenous bolus doses and first-order elimination. This makes the update and forecast transparent. It is a hypothetical example, not a clinical dosing recommendation.

Model and assumptions: For a dose \(D_j\) given at time \(t_j\), the contribution to concentration at time \(t\ge t_j\) is \((D_j/V)\exp[-k(t-t_j)]\). Contributions from recorded doses add by linear superposition. Here \(V\) is volume of distribution and \(k=CL/V\) is the elimination rate constant. The example assumes known dose amounts and times, fixed \(V\), linear one-compartment disposition, and independent normally distributed residual errors on the log-concentration scale. These assumptions are for calculation; they are not guaranteed by sparse data.

Worked Example: Updating the Individual Elimination Rate

Worked Example: Updating the Individual Elimination Rate

Consider a hypothetical 70-kg patient with a documented 100-mg intravenous bolus at time 0. The covariate model gives \(V=49.02\) L and a typical clearance of \(4.902\) L/h, so the covariate-based prior mean for \(k=CL/V\) is \(0.1000\) h\(^{-1}\). For this calculation, use the normal prior \(k\sim N(0.1000,0.0300^2)\) h\(^{-1}\). Concentrations are observed at 8 and 16 h: \(0.8221\) and \(0.4836\) mg/L. Let the log-scale residual standard deviation be \(0.1526\). Assume the patient received no other dose before either sample.

The predicted concentration immediately after the bolus is \(C_0=D/V=100/49.02=2.040\) mg/L. In the single-dose portion of the record, the log concentration relative to this value satisfies \(y_i=\log(C_i/C_0)=-t_i k+\epsilon_i\), with \(\epsilon_i\sim N(0,0.1526^2)\). The log observations, calculated before rounding, are approximately \(y_8=-0.9088\) and \(y_{16}=-1.43945\). In particular, \(\log(0.4836)=-0.726497\), or \(-0.7265\) to four decimal places; subtracting \(\log(2.040)\approx0.712950\) gives \(y_{16}\approx-1.43945\).

For a normal prior and this normal linear likelihood, the posterior is normal. Its precision is the sum of prior precision and data precision, and the data-weighted term in the posterior mean is \(-\sum_i t_i y_i/\sigma^2\), where \(\sigma=0.1526\) h-independent log-error SD. Thus:

$$ s_{\text{post}}^2= \left(\frac{1}{0.0300^2}+\frac{8^2+16^2}{0.1526^2}\right)^{-1} \approx 0.0000673273\ {\rm h}^{-2} $$

The posterior standard deviation is \(s_{\text{post}}\approx0.00820532\) h\(^{-1}\). The posterior mean is

$$ m_{\text{post}}=s_{\text{post}}^2 \left[ \frac{0.1000}{0.0300^2} -\frac{8(-0.9088)+16(-1.43945)}{0.1526^2} \right] \approx 0.09509\ {\rm h}^{-1}. $$

Small differences in the last displayed digit can result from rounding the log observations; the calculation uses the unrounded values. The data-weighted rate estimate in the likelihood is about \(0.09469\) h\(^{-1}\), so the posterior lies between that estimate and the prior mean, as expected. The update is not a simple average: the relative precision of prior and data determines how much each contributes.

Why the Posterior Calculation Has This Form

The posterior formula follows by multiplying the prior density by the likelihood and completing the square in \(k\). The prior contributes a quadratic term \((k-m_0)^2/\tau_0^2\); each observation contributes \((y_i+t_i k)^2/\sigma^2\). Collecting the terms involving \(k^2\) gives the posterior precision, and collecting the linear terms gives its mean.

$$ \frac{(k-m_0)^2}{\tau_0^2}+\sum_i\frac{(y_i+t_i k)^2}{\sigma^2} = \left(\frac{1}{\tau_0^2}+\sum_i\frac{t_i^2}{\sigma^2}\right)(k-m_{\text{post}})^2+\text{constant}. $$

Consequently,

$$ \frac{1}{s_{\text{post}}^2}=\frac{1}{\tau_0^2}+\sum_i\frac{t_i^2}{\sigma^2}, \qquad m_{\text{post}}=s_{\text{post}}^2 \left(\frac{m_0}{\tau_0^2}-\sum_i\frac{t_i y_i}{\sigma^2}\right). $$

This result requires the stated normal prior and normal likelihood with known residual variance, and the log-concentration model must be linear in \(k\). Here it is linear because the doses, times, and \(V\) are treated as known and the observations follow one dose without intervening doses. With multiple preceding doses, the concentration is a sum of exponentials; its logarithm is generally not linear in \(k\), so this conjugate update cannot be used unchanged.

Worked Example: Forecasting After the Next Recorded Dose

Worked Example: Forecasting After the Next Recorded Dose

Suppose the record schedules another 100-mg bolus at 24 h. Forecast the concentration at 30 h, six hours after that dose. At 30 h, the contributions come from the bolus at 0 h and the bolus at 24 h:

$$ C(30\mid k)=\frac{100}{49.02}\exp(-30k)+\frac{100}{49.02}\exp(-6k) =2.040\left[\exp(-30k)+\exp(-6k)\right]\ {\rm mg/L}. $$

Using the posterior mean \(k=0.09509\) h\(^{-1}\) gives the plug-in forecast \(2.040[\exp(-2.8527)+\exp(-0.57054)]\approx1.271\) mg/L. This is the forecast at one parameter value, not the posterior mean concentration. Because \(k\) is uncertain and the concentration is a nonlinear function of \(k\), those quantities differ.

For \(k\sim N(m_{\text{post}},s_{\text{post}}^2)\), the normal moment-generating function gives \(E[\exp(-ak)]=\exp(-a m_{\text{post}}+a^2s_{\text{post}}^2/2)\). Therefore the posterior mean of the latent model concentration is

$$ E[C(30)\mid\text{data}] =2.040\left[ \exp\left(-30(0.09509)+\frac{30^2(0.00820532)^2}{2}\right) +\exp\left(-6(0.09509)+\frac{6^2(0.00820532)^2}{2}\right) \right] \approx1.276\ {\rm mg/L}. $$

A central 95% posterior interval for \(k\) is \(0.09509\pm1.96(0.00820532)\), or approximately \(0.07901\) to \(0.11117\) h\(^{-1}\). The concentration function decreases monotonically with \(k\), so transforming the interval endpoints reverses their order. The corresponding interval for the latent concentration at 30 h is approximately \(1.120\) to \(1.461\) mg/L. Conditional on the stated model and observed data, this interval contains 95% posterior probability for the latent model concentration. It does not include the additional variability of a future measured concentration from the residual-error model.

Worked Example: A Different Recorded Dose Amount

Worked Example: A Different Recorded Dose Amount

Keep the same patient, posterior, and dose at 0 h, but suppose the recorded dose at 24 h was 50 mg rather than 100 mg. At 30 h, the first dose still contributes \(2.040\exp(-30k)\) mg/L, while the later dose contributes half as much as a 100-mg dose:

$$ C(30\mid k)=2.040\left[\exp(-30k)+0.5\exp(-6k)\right]\ {\rm mg/L}. $$

At the posterior mean rate, the plug-in forecast is \(2.040[0.05769+0.5(0.5653)]\approx0.694\) mg/L. For the posterior mean concentration, use the same normal expectation as before: \(2.040[0.05947+0.5(0.5660)]\approx0.699\) mg/L (approximately \(0.699\) mg/L after rounding). The numerical difference between this and the plug-in result is small here, but the definitions remain different. This calculation also shows why a forecast must use the actual dose amounts and times in the record rather than a nominal regimen substituted for them.

Residual Checks, Interpretation, and Alternatives

A calculation can be internally consistent and still be a poor description of a patient. First verify the dose record, units, sample times, and covariate inputs. Then compare observations with fitted concentrations on the scale used by the error model. For the 16-h sample, the fitted log concentration at the posterior mean is \(\log(2.040)-16(0.09509)\approx-0.8085\). The observed log concentration is \(\log(0.4836)\approx-0.7265\), so the residual is about \(0.0820\), or \(0.0820/0.1526\approx0.54\) residual standard deviations. This is a modest in-sample discrepancy, not an independent validation: the same observation contributed to the posterior. Check all samples, not just one, and investigate systematic patterns or a large discrepancy rather than treating one standardized residual as proof that the model is correct or incorrect.

The posterior is \(N(0.09509,0.00820532^2)\) h\(^{-1}\). A normal distribution technically assigns probability to negative rates. Here its mean is \(0.09509/0.008207\approx11.59\) posterior standard deviations above zero, so the negative-rate probability is negligible under this approximation. That does not establish that the one-compartment model or the recorded inputs are correct.

Three common forecast summaries answer different questions. A population-typical forecast uses the covariate-based typical parameter and does not update to this patient’s concentrations. A plug-in individual forecast uses a point estimate such as the posterior mean, but omits parameter uncertainty. A posterior forecast propagates the distribution of individual parameters; a future-observation prediction also includes residual error. For example, if a future log observation includes a new independent error with SD \(0.1526\), its predictive distribution is wider than the latent-concentration interval above. Posterior simulation is a practical way to obtain that distribution: draw \(k\) from its posterior, calculate concentration from the dose history, draw future residual error, and summarize the simulated concentrations.

The simple conjugate calculation has important limits. It treats \(V\) and the covariate-based prior as fixed, assumes exact dose and sampling times, and uses a normal approximation for \(k\). Uncertainty in these inputs would need to be propagated for a fuller forecast. The log-linear update also does not apply directly when multiple doses contribute to each observed concentration. In those settings, a nonlinear likelihood and numerical Bayesian estimation are needed. Sparse data can leave the forecast strongly influenced by the prior; a narrow posterior from an assumed model should not be mistaken for evidence that every model assumption is adequate.

Key takeaway: A defensible individual forecast uses the recorded dose history and covariate-conditioned prior, updates that prior with the observed concentrations under an explicit likelihood, and carries the resulting posterior uncertainty into the future concentration calculation.

Check Your Understanding

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

  1. In the example, why is the concentration at 30 h the sum of two dose contributions?
  2. What are the approximate posterior mean and standard deviation for \(k\), and in what units?
  3. Does the 95% posterior interval for latent concentration include residual error for a future measured concentration?
  4. If the recorded dose at 24 h is halved, what happens to its contribution at 30 h?

Solutions. (1) Both boluses remain in the one-compartment system: at 30 h they have elapsed times of 30 and 6 h, and linear superposition adds their exponential contributions. (2) The posterior is approximately \(N(0.09509,0.00820532^2)\) h\(^{-1}\), so its mean is \(0.09509\) h\(^{-1}\) and its standard deviation is \(0.00820532\) h\(^{-1}\). (3) No. It describes posterior uncertainty in the latent model concentration; a future-observation predictive distribution also includes a new residual error. (4) The 24-h dose contributes half as much, because its concentration contribution is proportional to dose amount; at 30 h it changes from \(2.040\exp(-6k)\) to \(1.020\exp(-6k)\) mg/L.

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