Why a Two-Compartment Design Needs Phase Coverage
After an IV bolus, a two-compartment concentration profile can decline rapidly at first and more slowly later. A sparse schedule must therefore do more than collect samples at distinct clock times: its observations need to distinguish the contributions and rates of both phases. A schedule concentrated early may measure the rapid decline well but provide little information about the terminal rate. A schedule concentrated late may have the opposite problem.
The worked calculations below use a local design: sensitivities are evaluated at specified nominal parameter values, and the resulting information describes precision near those values under the stated error model. This is not a guarantee that the same schedule remains best when the true parameters, assay behavior, or clinical constraints differ.
Log-Parameter Sensitivities and Local Information
Use the parameter vector \(\eta=(\ln A,\ln B,\ln\alpha,\ln\beta)\). This scale describes proportional changes in each positive parameter. The sensitivity of a concentration observation to a component of \(\eta\) is its partial derivative with respect to that component. By the chain rule, the sensitivity row at time \(t\) is
Each entry is measured in mg/L per unit change in a log parameter. The first two entries measure the contributions of the two exponential terms; the last two also reflect how the corresponding term changes with its rate. A row with a large concentration contribution does not necessarily have large sensitivity to its rate: the rate sensitivity includes the factor \(t\).
For a schedule \(t_1,\ldots,t_n\), stack these rows into the sensitivity matrix \(J\). Under the independent, constant-variance error assumptions, the local Fisher information matrix is
If \(n=4\) and \(J\) is square, the determinant has a particularly direct form. A positive determinant indicates full local rank: all four log parameters have distinguishable sensitivity directions in this linearized calculation. The D-information criterion compares schedules by \(\det(F)\); larger values correspond to smaller local joint uncertainty volume when the same parameterization and error model are used.
To justify the determinant relation, start from \(F=\sigma^{-2}J^\mathsf{T}J\). Multiplying a \(4\times4\) matrix by the scalar \(\sigma^{-2}\) multiplies its determinant by \((\sigma^{-2})^4=\sigma^{-8}\). Also, \(\det(J^\mathsf{T}J)=\det(J^\mathsf{T})\det(J)=\det(J)^2\). Combining the two identities gives the stated result. For more than four observations, \(J\) is rectangular and the relevant determinant is that of \(J^\mathsf{T}J/\sigma^2\), not the square of a determinant of \(J\).
Worked Example: Building a Four-Time Sensitivity Matrix
Consider hypothetical nominal values \(A=8\) mg/L, \(B=2\) mg/L, \(\alpha=1.2\) h\(^{-1}\), and \(\beta=0.12\) h\(^{-1}\). Consider the four-time schedule \(t=(0.25,1,4,12)\) h, with \(\sigma=0.200\) mg/L. First calculate the concentration terms and then the log-parameter sensitivities.
At \(t=0.25\) h, \(A e^{-\alpha t}=8e^{-0.3}=5.9265458\) mg/L and \(B e^{-\beta t}=2e^{-0.03}=1.9408911\) mg/L. The rate sensitivities are \(-0.3(5.9265458)=-1.7779637\) mg/L and \(-0.03(1.9408911)=-0.0582267\) mg/L. The same formula at each time gives the following matrix, with entries rounded for display; determinant calculations use unrounded values.
For example, the third row is evaluated at \(t=4\) h: its first entry is \(8e^{-4.8}=0.0658380\) mg/L, and its third entry is \(-1.2(4)(0.0658380)=-0.3160223\) mg/L. At 12 h, the fast-phase contribution is small, while the slow-phase contribution remains \(2e^{-1.44}=0.4738555\) mg/L. This contrast is the intended coverage of the late sample.
Using the unrounded sensitivity matrix gives \(\det(J)\approx-6.03\;(\mathrm{mg/L})^4\). Hence
The negative sign of \(\det(J)\) is immaterial to the information determinant, which is positive. The magnitude is meaningful only relative to another schedule evaluated with the same parameterization, units, nominal values, and error model.
Comparing Candidate Schedules
A determinant comparison gives a scalar summary of joint local information, but it does not explain which phase is driving the difference. Examine both the determinant and the rows of \(J\). In this example, the 0.25 h observation has a strong fast-term contribution, and the 12 h observation has a very small fast-term contribution but appreciable slow-term and slow-rate sensitivities. These rows provide different directions in parameter space.
Worked Example: Comparing With a Schedule That Repeats a Time
Keep the model, error standard deviation, and four observations, but consider the candidate schedule \((0.25,1,4,4)\) h. The two observations at 4 h are independent measurements, but their sensitivity rows are identical because they have the same model time and nominal parameters.
A matrix with two identical rows has linearly dependent rows. Therefore its rank is at most three, and its four-column sensitivity matrix has determinant zero. The information matrix is singular as well, so \(\det(F)=0\). Repeating a measurement can reduce uncertainty in some identifiable combinations, but it cannot create a new sensitivity direction. In this four-parameter local problem, the repeated-time schedule cannot identify all four log parameters.
By contrast, the schedule \((0.25,1,4,12)\) h has \(\det(F)\approx1.42\times10^7\), so it has full local rank at the stated nominal values. This is a strict determinant improvement over the repeated-time schedule. It does not, by itself, prove that the four-time schedule is best among all feasible schedules; that requires evaluating the other candidates under the same assumptions.
A less extreme alternative might use four distinct times but place them all early. Such a matrix need not be singular: its determinant may be positive. However, if the slow-term sensitivities are small or nearly proportional to other columns, the determinant can still be low. The practical diagnostic is to inspect the sensitivity columns and parameter correlations as well as the determinant. A nonzero determinant rules out exact local rank deficiency, but it does not rule out near-collinearity or unstable estimates.
Worked Example: Reading Information Diagonals Without Overstating Precision
For the four-time schedule above, the diagonal entries of \(F\) are the squared column lengths of \(J\), divided by \(\sigma^2=0.0400\;(\mathrm{mg/L})^2\). Using the displayed sensitivities at adequate precision gives approximately
For example, the first entry is approximately \((5.9265^2+2.4096^2+0.06584^2+0.00000446^2)/0.0400\approx1.02\times10^3\). The fourth entry is approximately \((0.05823^2+0.2129^2+0.5940^2+0.6824^2)/0.0400\approx21.7\). These are information in individual coordinate directions, not the diagonal of \(F^{-1}\), so they do not by themselves give marginal parameter standard errors.
If all other parameters were known, the conditional standard error for a log parameter would be \(1/\sqrt{F_{kk}}\). For \(\ln\beta\), this is approximately \(1/\sqrt{21.7}=0.215\) on the log scale. That conditional calculation is not the same as estimating all four parameters together. In joint estimation, correlations can make marginal uncertainty larger; the local covariance approximation is \(F^{-1}\), and its diagonal—not the reciprocal of \(\operatorname{diag}(F)\)—gives marginal variances.
Checks, Limitations, and a Common Pitfall
Before trusting a design comparison, check the assumptions and the numerical calculation. Confirm that times and rate constants use compatible units, that the parameterization is the same for all candidate schedules, and that the residual variance is applied consistently. Recompute determinants from unrounded sensitivities: rounded matrix entries can noticeably affect a determinant, especially when rows or columns are nearly dependent. Also inspect the sensitivity matrix for a tiny singular value or near-collinear columns; a positive determinant can coexist with poor practical precision.
The local approximation depends on the nominal values. If the true rates differ substantially, the fast and slow contributions at a proposed time can change, and so can the information ranking. A sensible design assessment therefore checks plausible nominal-value scenarios and considers whether the schedule covers the phases across them. The calculation also assumes an additive, constant-variance error model. If assay variability changes with concentration, or observations may be below quantification limits, the stated \(F\) is not the appropriate information calculation without modification.
Common mistake: “The schedule with the largest fast-term concentration has the most information about every parameter.” This is incorrect. A large concentration contribution can inform an amplitude, but rate information depends on time as well as concentration; one early sample may contribute little to distinguishing rates. The correction is to calculate all relevant sensitivity columns and compare the information matrices under the same assumptions. D-optimality summarizes joint information; it does not ensure that every individual parameter has adequate precision or that important phases are well covered.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For the stated model and nominal values, what is the fast-term contribution \(A e^{-\alpha t}\) at \(t=0.25\) h, and what is the sensitivity to \(\ln\alpha\)?
- Why does the schedule \((0.25,1,4,4)\) h have zero four-parameter D-information in the local calculation?
- If \(\det(J)=-2.00\;(\mathrm{mg/L})^4\) and \(\sigma=0.200\) mg/L, calculate \(\det(F)\) for four observations.
- Does a large value of \(F_{44}\) directly equal high marginal precision for \(\ln\beta\)? Explain.
Solutions. (1) \(8e^{-1.2(0.25)}=8e^{-0.3}=5.9265\) mg/L. The sensitivity is \(-\alpha t A e^{-\alpha t}=-0.3(5.9265)\approx-1.778\) mg/L. (2) The two 4 h observations have identical sensitivity rows, so the four rows are linearly dependent, \(\det(J)=0\), and consequently \(\det(F)=0\). (3) \(\det(F)=(-2.00)^2/(0.200)^8=4/0.00000256=1.5625\times10^6\). (4) No. \(F_{44}\) is information in that coordinate direction with the other parameters treated as known; joint marginal variance for \(\ln\beta\) is the fourth diagonal entry of \(F^{-1}\), which reflects parameter correlations as well.