Why Measurement Times Matter
In pharmacokinetic data, covariates such as renal function, body weight, or a biomarker may be measured less often than drug concentrations. Their measurement times may not coincide with pharmacokinetic sampling times. A model still needs a rule for determining which covariate value applies at each sample time, and sometimes throughout the interval between samples.
This is a time-alignment problem, not simply a matter of matching rows in a data table. A rule reconstructs a usable covariate history from the recorded measurements. Different rules can give different histories, and therefore different predictions, even when the dose history, model parameters, and observed data are unchanged.
The recorded times and the pharmacokinetic sample times should be kept distinct. A covariate observation at time \(s_{ik}\) is a measurement at that time; it does not, by itself, specify values at every other time. The rule used to extend or connect measurements supplies that additional definition. It does not establish that the reconstructed history is the subject’s exact, continuously observed biological history.
Common Alignment Rules and Their Support
For measurements at \(s_{i1},\ldots,s_{iK_i}\), three useful rules are forward-hold, linear interpolation, and nearest measurement. Each has different support and makes different assumptions about the covariate between observations.
- Forward-hold: the most recent measurement is used until the next measurement. On the interval \([s_{ik},s_{i,k+1})\), \(z_i^A(t)=z_{ik}\). At the exact time \(s_{i,k+1}\), the new measurement applies: \(z_i^A(s_{i,k+1})=z_{i,k+1}\). The basic rule is defined from the first measurement through the last; values before the first or after the last require an explicitly stated extension.
- Linear interpolation: between two successive measurements, values are joined by a straight line. For \(s_{ik}\leq t\leq s_{i,k+1}\), the rule is \(z_i^A(t)=z_{ik}+\frac{t-s_{ik}}{s_{i,k+1}-s_{ik}}(z_{i,k+1}-z_{ik})\). Its basic support runs from the first to the last measurement; extrapolation beyond that range is a separate choice.
- Nearest measurement: at each time, use the value whose measurement time is closest. The support can be defined over all times by assigning each time to a nearest measurement, but a tie rule must be specified. For instance, ties can be assigned to the earlier measurement. Any boundary restrictions on the time range should also be stated.
A rule should be chosen to reflect the measurement process and the scientific question. Forward-hold describes a covariate assumed unchanged until it is measured again. Linear interpolation encodes gradual change between observed values. Nearest measurement assigns a value by temporal proximity without asserting a linear trend. None is automatically correct merely because it is easy to implement.
Worked Example: Aligning Three Samples to Sparse Measurements
Suppose a hypothetical subject has a dimensionless covariate measured at times 0, 5, and 11 hours, with values 0.8, 1.0, and 0.7, respectively. Pharmacokinetic samples are collected at 3, 8, and 10 hours. Find the aligned values using each of the three rules. For nearest measurement, assign ties to the earlier measurement.
Forward-hold. At 3 hours the latest measurement is the value at 0 hours, so the aligned value is 0.8. At 8 and 10 hours the latest measurement is at 5 hours, so both aligned values are 1.0. Thus the values at 3, 8, and 10 hours are 0.8, 1.0, and 1.0.
Linear interpolation. At 3 hours, the interpolation fraction between 0 and 5 hours is \(3/5\), giving \(0.8+(3/5)(1.0-0.8)=0.92\). At 8 hours, the fraction between 5 and 11 hours is \(3/6\), giving \(1.0+(3/6)(0.7-1.0)=0.85\). At 10 hours, the fraction is \(5/6\), giving \(1.0+(5/6)(0.7-1.0)=0.75\).
Nearest measurement. At 3 hours, the measurement at 5 hours is nearer than the one at 0 hours, so the value is 1.0. At 8 hours, measurements at 5 and 11 hours are both 3 hours away; the stated tie rule selects the earlier measurement, giving 1.0. At 10 hours, the measurement at 11 hours is nearest, giving 0.7.
All three rules have values at these sample times because they lie between the first and last covariate measurements. The resulting aligned values differ because the rules encode different histories, not because the recorded data changed.
Using an Aligned History in an NLME Model
An aligned covariate can enter a structural model at observation times or affect the model’s dynamics over time. A general representation for a continuous pharmacokinetic state \(A_i(t)\) is
Here \(D_i(t)\) represents the dose history, \(\eta_i\) the subject-specific random effects, \(\theta\) the population parameters, and \(\epsilon_{ij}\) the residual error. This notation makes an important distinction: if the covariate affects the state dynamics through \(g\), the history may be needed at every time in the integration interval, not just at the times when concentrations were sampled.
A comparison of alignment rules should hold the other model inputs fixed. In particular, use the same dose history, parameter values, random effects, initial conditions, and observation times. Then differences in predictions can be attributed to the alternative reconstructed covariate histories. Refitting the model under each rule may also change parameter estimates, so it answers a different question.
Worked Example: Comparing Concentration Predictions Under Two Histories
Consider a hypothetical one-compartment model after an intravenous bolus, with initial concentration \(C(0)=10\) mg/L and time-varying elimination rate \(k(t)=0.2[1+0.25z(t)]\) h\(^{-1}\). The dimensionless covariate is measured as \(z(0)=0\) and \(z(4)=1\). Compare the predicted concentration at 4 hours under forward-hold and linear interpolation. Assume no additional dose occurs during this interval.
For a one-compartment model with \(dC(t)/dt=-k(t)C(t)\), separating variables and integrating gives \(C(4)=C(0)\exp[-\int_0^4 k(u)\,du]\). Under forward-hold, the covariate is \(z(t)=0\) on \([0,4)\), and \(z(4)=1\). The point value at 4 hours is the new measurement; it does not change the integral over the interval.
The forward-hold integrated rate is therefore
Thus \(C(4)=10\exp(-0.8)=4.493\) mg/L.
With linear interpolation, \(z(t)=t/4\) for \(0\leq t\leq4\). Since \(\int_0^4 z(t)\,dt=\int_0^4(t/4)\,dt=2\) hours, the integrated rate is
The corresponding prediction is \(C(4)=10\exp(-0.9)=4.066\) mg/L. Under these fixed parameters and initial conditions, the linear-interpolation prediction is 0.428 mg/L lower, or about 9.52% lower relative to the forward-hold prediction. This difference is a consequence of the specified alignment rules and model, not evidence that either rule is correct for an unobserved covariate history.
Support, Boundaries, and Model Checks
Before computing predictions, check that the alignment rule defines the covariate wherever the model needs it. For a sample-level effect, check every \(t_{ij}\). For a covariate in a differential equation, check the whole interval from the state’s initial time to the prediction time, including any intervals across which doses or covariate values change.
Boundary conventions matter. Under forward-hold, a measurement at time \(s_{ik}\) applies at that time and after it, until the next measurement. Thus the interval immediately before that measurement uses the earlier value, while the value at the measurement time is the newly recorded value. A value at a single point does not alter an ordinary time integral, but it can affect a model evaluated directly at that point. Software implementations should follow the stated convention rather than relying on row order or implicit defaults.
Useful checks include plotting the recorded covariate values and the aligned history on the same time axis, listing aligned values at every pharmacokinetic sample, and confirming that all required intervals are supported. Review abrupt jumps, implausible interpolation slopes, ties in nearest-measurement assignment, and any extrapolation at the beginning or end of follow-up. These checks can reveal coding errors or a mismatch between the chosen rule and the measurement schedule.
A common mistake is to use the value measured at time 4 hours throughout the interval from 0 to 4 hours under forward-hold. That is not forward-hold: the value at 4 hours applies at \(t=4\), whereas the earlier value applies on \([0,4)\). For a model depending on the covariate only through an integral, the endpoint value alone does not change that interval’s integral; for a model evaluated at exactly 4 hours, the endpoint value does matter.
The main alternatives have distinct trade-offs. Forward-hold produces a piecewise-constant history and does not create intermediate values. Linear interpolation is continuous between observations but imposes a straight-line path. Nearest measurement is straightforward at sample times but can switch values at midpoints and needs a tie convention. A joint model for the covariate process could represent its evolution and uncertainty, but requires additional assumptions and data and is not equivalent to simply choosing an alignment rule.
Alignment is a modeling choice with limitations: sparse observations cannot reveal the true path between measurements, and a deterministic rule treats its reconstructed values as fixed inputs. Results can depend on that choice. When the measurement schedule or plausible covariate behavior leaves genuine uncertainty, compare scientifically defensible rules and report how the predictions change rather than presenting one reconstruction as directly observed fact.
Worked Example: Quantifying a Second Prediction Comparison
Suppose a hypothetical subject receives a 100 mg intravenous bolus at time 0, has volume \(V=20\) L, and follows \(dC(t)/dt=-[CL(t)/V]C(t)\). Clearance is defined as \(CL(t)=5[1+0.2(z(t)-1)]\) L/h. The covariate is measured as \(z(0)=1.2\) and \(z(8)=0.6\). Compare predictions at 2 and 6 hours under forward-hold and linear interpolation. Use the same dose and parameters for both rules.
The initial concentration is \(100/20=5\) mg/L. Under forward-hold, \(z(t)=1.2\) on \([0,8)\), so \(CL(t)=5[1+0.2(0.2)]=5.2\) L/h over both prediction intervals. Hence \(C(t)=5\exp[-(5.2/20)t]\). At 2 hours this gives \(5\exp(-0.52)=2.973\) mg/L, and at 6 hours it gives \(5\exp(-1.56)=1.051\) mg/L.
Under linear interpolation, \(z(t)=1.2-0.075t\) for \(0\leq t\leq8\), so \(CL(t)=5.2-0.075t\) L/h. At 2 hours, the integrated clearance is \(\int_0^2(5.2-0.075t)\,dt=10.25\) L. Dividing by \(V=20\) L gives an integrated rate of 0.5125, so \(C(2)=5\exp(-0.5125)=2.995\) mg/L. At 6 hours, the integrated clearance is \(31.2-1.35=29.85\) L, giving an integrated rate of 1.4925 and \(C(6)=5\exp(-1.4925)=1.124\) mg/L.
The predictions are close at 2 hours but differ by about 0.073 mg/L at 6 hours. At 8 hours, the forward-hold history takes the newly measured value \(z(8)=0.6\); this does not retroactively change its values on \([0,8)\). Both predictions through 6 hours are supported by the measurements under the stated rules.
Check Your Understanding
Answer each question using the stated alignment conventions.
- A covariate is measured at 0 and 6 hours, with values 2 and 5. Under forward-hold, what value applies at 5 hours and exactly at 6 hours?
- For the measurements in question 1, what is the linearly interpolated value at 3 hours?
- A PK sample occurs at 9 hours, but the last covariate measurement is at 6 hours. Is the sample supported by basic linear interpolation? What would be needed to use that rule there?
- Why can a dynamic model require more covariate support than a model that uses the covariate only at PK sample times?
Solutions. (1) The forward-held value is 2 at 5 hours, and the value at exactly 6 hours is 5. (2) The interpolation fraction is \(3/6=0.5\), so the value is \(2+0.5(5-2)=3.5\). (3) No. Basic linear interpolation is supported only through the last measurement time, 6 hours; an explicit extrapolation rule or a later measurement is needed. (4) A dynamic model may use the covariate throughout the state-evolution interval, so its history must be defined over that entire interval, not only at the observation times.