Identifiability of Clearance and Volume From Sparse Levels

Why Sparse Levels Can Leave Clearance and Volume Entangled

A measured concentration is the combined result of how much drug is present and how quickly the body eliminates it. In a one-compartment model after an intravenous bolus, volume of distribution \(V\) controls the initial concentration, while clearance \(CL\) controls the elimination rate through \(k=CL/V\). A single concentration at a positive time therefore reflects both parameters. The question is not simply whether a model can produce a fitted value for each parameter, but whether the observations distinguish one parameter pair from other pairs that predict the same concentrations.

Here, identifiability is considered first for exact, noiseless concentrations and a specified dose history. This separates a structural limitation of the model-data combination from practical uncertainty caused by measurement error. A parameter pair may be structurally identifiable and still be estimated imprecisely when the data are noisy or the measurements are nearly redundant.

Definition: A parameter is structurally identifiable from a specified observation design if distinct parameter values cannot produce the same exact observations, subject to the stated model and known inputs. Local identifiability means this uniqueness holds in a neighborhood of a parameter value. Practical identifiability concerns whether finite, noisy data determine the parameter with useful precision.

Model and Assumptions

Assume a one-compartment model with instantaneous IV bolus doses, linear disposition, and first-order elimination. Clearance and volume are positive and constant for the individual over the observation period. Dose amounts and administration times are known, and concentrations are measured without error for the structural calculations below. There is no absorption phase, infusion, or additional compartment in this model.

For a single dose \(D\) at time zero, the concentration at time \(t\geq 0\) is:

C(t)=\frac{D}{V}\exp\left(-\frac{CL}{V}t\right) =\frac{D}{V}e^{-kt}, \qquad k=\frac{CL}{V}.

For several known bolus doses \(D_j\) administered at times \(t_j\leq t_i\), linear superposition gives the concentration at observation time \(t_i\):

C_i=\frac{1}{V}\sum_j D_j e^{-k(t_i-t_j)} =\frac{A_i(k)}{V}, \qquad A_i(k)=\sum_j D_j e^{-k a_{ij}}, \qquad a_{ij}=t_i-t_j.

The ages \(a_{ij}\) are the elapsed times from each dose to the sample. This expression makes the roles of the two parameters visible: \(V\) scales concentration, while \(k=CL/V\) determines how the known dose contributions decay. Identifying both \(CL\) and \(V\) requires information that distinguishes this scale from this decay.

Worked Example: One Concentration Gives a Continuum of Fits

Suppose a hypothetical patient receives \(D=100\) mg by IV bolus, and a concentration of \(2.00\) mg/L is measured at \(t=2.00\) h. Under the stated model, show that this one level does not uniquely determine \(CL\) and \(V\).

For any chosen \(V\) that gives \(D/V>C\), rearrange the concentration equation:

k=-\frac{1}{t}\ln\left(\frac{CV}{D}\right), \qquad CL=kV.

If \(V=20.0\) L, then \(CV/D=(2.00)(20.0)/100=0.400\). Thus \(k=-\ln(0.400)/2.00\approx0.45815\) h\(^{-1}\), and \(CL=20.0(0.458145\ldots)\approx9.1629\) L/h. Substitution gives \((100/20.0)e^{-0.458145\ldots(2.00)}=2.00\) mg/L before rounding.

A different choice, \(V=25.0\) L, gives \(CV/D=0.500\), \(k=-\ln(0.500)/2.00\approx0.34657\) h\(^{-1}\), and \(CL\approx8.664\) L/h. It also predicts \(2.00\) mg/L. The two parameter pairs are distinct but fit the same single level exactly. A fitted pair from one level therefore needs an additional constraint or information; it is not uniquely learned from that concentration alone.

How Two Concentrations Can Separate the Parameters

For one bolus followed by two samples at distinct times \(t_1\) and \(t_2\), the ratio cancels the volume scale:

\frac{C(t_2)}{C(t_1)} =e^{-k(t_2-t_1)}.

If \(t_2>t_1\), both concentrations are positive, and the model assumptions hold, this ratio determines \(k\) uniquely. Either concentration can then determine \(V\), and \(CL=kV\). The time separation is essential: two readings at the same time do not give a decay ratio across time.

Worked Example: Two Times After One Bolus

Consider a hypothetical \(100\) mg IV bolus. Exact model-generated concentrations are recorded at \(t_1=1.00\) h and \(t_2=4.00\) h: \(C_1=5e^{-0.300}=3.70409\) mg/L and \(C_2=5e^{-1.200}=1.50597\) mg/L. These values correspond to \(V=20.0\) L and \(k=0.300\) h\(^{-1}\). Recover the parameters from the two levels.

The concentration ratio is \(C_2/C_1=e^{-0.900}=0.406570\). Therefore:

k=-\frac{\ln(C_2/C_1)}{t_2-t_1} =-\frac{\ln(0.406570)}{4.00-1.00} =0.300\ \text{h}^{-1}.

Then \(V=De^{-kt_1}/C_1\). Using the unrounded model-generated concentration, \(V=100e^{-0.300}/(5e^{-0.300})=20.0\) L, so \(CL=kV=(0.300)(20.0)=6.00\) L/h. The displayed concentrations are rounded decimals; using them in place of their exact generating values produces estimates close to, rather than exactly equal to, these parameters.

A Sensitivity Condition for General Known Dose Histories

With multiple doses, two concentrations need not identify \(k\) just because they occur at different times. Their sensitivity to a change in \(k\) depends on the ages and sizes of the doses contributing to each sample. A local calculation makes this dependence explicit. Write the parameters on a logarithmic scale, \(\log CL\) and \(\log V\), so small changes are interpreted proportionally. For observation \(i\), define:

q_i=\frac{\partial\log C_i}{\partial\log k} =-k\frac{\sum_j D_j a_{ij}e^{-ka_{ij}}} {\sum_j D_j e^{-ka_{ij}}}.

The fraction in this expression is the dose-contribution-weighted mean age of the drug contributing to that concentration. Thus \(q_i\) is minus \(k\) times that weighted mean age. Since \(C_i=A_i(k)/V\) and \(\log k=\log CL-\log V\), the sensitivities are:

\frac{\partial\log C_i}{\partial\log CL}=q_i, \qquad \frac{\partial\log C_i}{\partial\log V}=-q_i-1.

For two observations, the Jacobian of their log concentrations with respect to \((\log CL,\log V)\) has determinant:

\begin{vmatrix} q_1 & -q_1-1\\ q_2 & -q_2-1 \end{vmatrix} =q_2-q_1.

A nonzero determinant means that, at that parameter value, the two observations respond differently enough to small changes in \(CL\) and \(V\) for the Jacobian to have full rank. This supports local identifiability under the model. A zero determinant means the two local sensitivity rows are redundant, so the pair cannot locally separate both parameters by this criterion. The condition is local: a nonzero determinant does not by itself establish global uniqueness for every possible dose history.

Worked Example: Sensitivities With a New Dose Between Samples

Suppose a hypothetical patient receives \(100\) mg at time \(0\) and another \(100\) mg at time \(10.0\) h. Consider samples at \(t_1=2.00\) h and \(t_2=12.0\) h, evaluated at \(V=20.0\) L and \(k=0.200\) h\(^{-1}\) (so \(CL=4.00\) L/h). The second dose has not yet been given at the first sample.

At \(t_1\), the only contributing dose is \(100\) mg with age \(2.00\) h. Thus \(q_1=-k(2.00)=-0.400\). At \(t_2\), the dose ages are \(12.0\) and \(2.00\) h. The weighted mean age is:

\frac{12.0e^{-0.200(12.0)}+2.00e^{-0.200(2.00)}} {e^{-0.200(12.0)}+e^{-0.200(2.00)}} \approx 3.19203\ \text{h}, \qquad q_2\approx-(0.200)(3.19203)=-0.638406.

The concentrations at these parameter values are \(C_1=5e^{-0.4}\approx3.35160\) mg/L and \(C_2=5(e^{-2.4}+e^{-0.4})\approx3.80519\) mg/L. The Jacobian determinant for the raw concentrations with respect to \((\log CL,\log V)\) is \(C_1C_2(q_2-q_1)\). Using unrounded model values gives approximately \((3.35160)(3.80519)(-0.238406)=-3.041\) \((\text{mg/L})^2\), which is nonzero. The two samples therefore have distinct local sensitivities at this parameter point. This calculation supports local separation in this example; it is not a guarantee of precise estimates from noisy levels.

When Two Levels Still Fail to Add Information

Worked Example: Proportional Dose Histories Are Redundant

Imagine two hypothetical, otherwise identical occasions. On the first, a \(100\) mg bolus is given and a sample is taken \(2.00\) h later. On the second, a \(200\) mg bolus is given and a sample is also taken \(2.00\) h later. Assume the same \(CL\) and \(V\) apply to both occasions.

The predicted concentrations are \(C_1=(100/V)e^{-2k}\) and \(C_2=(200/V)e^{-2k}=2C_1\). Their ratio is fixed by the dose ratio and contains no information about \(k\). Both observations have the same \(q=-2k\), so \(q_2-q_1=0\); their sensitivity rows are proportional. Even though there are two recorded levels, this design supplies only one distinct parameter-dependent concentration pattern. It cannot identify both \(CL\) and \(V\) from these levels alone.

The same issue can arise less obviously when different dose histories produce nearly the same weighted age of contributing drug. Then \(q_1\) and \(q_2\) are close, and the determinant is small. The model may technically be locally identifiable, but small concentration errors can then produce large changes in the fitted parameter pair. Structural identifiability is not a promise of precision.

Diagnostics, Alternatives, and Limitations

For a proposed sparse design, a useful diagnostic is to compute or inspect the sensitivity of each prediction to the parameters across plausible parameter values. Identical or nearly identical sensitivity patterns indicate that the observations contribute little independent separation. Also check that the dose amounts and administration times used in the calculations are known accurately; an incorrectly reconstructed dose history can change both predicted concentrations and their sensitivities.

There are several ways to address non-identification, but they answer different questions:

Add informative observations. Samples at distinct post-dose times can reveal decay; additional measurements after a meaningful change in dose history may add independent information.
Fix or externally constrain one parameter. If volume is specified from reliable external information, a concentration may be used to estimate clearance within the model. The result then depends on that constraint, not on the concentration alone identifying both parameters.
Use population information. A population prior can regularize an individual fit and yield a finite estimate or posterior distribution. It does not make the individual data structurally informative about a parameter combination they cannot distinguish; the result may be strongly influenced by the prior.

Sensitivity calculations depend on the assumed model. If the drug has multicompartment behavior, absorption after extravascular dosing, time-varying clearance, or inaccurate dose times, a one-compartment IV-bolus analysis may attribute model mismatch to \(CL\) or \(V\). A nonzero determinant only addresses local parameter separation inside the specified model. It does not validate that model, prove global uniqueness, or show that noisy patient data are sufficiently precise.

Common Pitfall

Key takeaway: Counting concentration records is not the same as counting independent pieces of parameter information. Ask whether the dose histories and sample times create distinct sensitivities to \(CL\) and \(V\). One level generally leaves a continuum of compatible pairs; two levels can resolve the pair in favorable designs, but redundant or nearly redundant levels may not.

A common error is to say, “There are two unknown parameters and two concentrations, so the parameters are identifiable.” The number of equations alone is not enough. In the proportional-dose example, the second concentration is exactly twice the first for every \(CL\) and \(V\). The correction is to examine whether the model predictions change in independent ways as the parameters change, using the dose-history equations and, locally, the sensitivity determinant.

Check Your Understanding

Answer each question, then compare with the solutions.

For a single bolus, why can one concentration at a positive time generally be matched by multiple \(CL,V\) pairs?
For two samples after one bolus at times \(1.00\) h and \(4.00\) h, the exact concentration ratio is \(e^{-0.900}\). Find \(k\).
Two samples have \(q_1=-0.50\) and \(q_2=-0.50\). What does the two-observation sensitivity determinant indicate?
Does using a population prior make a structurally non-identifiable individual data design structurally identifiable? Explain.

Solutions. (1) Choose any admissible \(V\) with \(CV<D\), then set \(k=-\ln(CV/D)/t\) and \(CL=kV\); this produces the same observed concentration, so there is a continuum of compatible pairs. (2) The time separation is \(3.00\) h, so \(k=-\ln(e^{-0.900})/3.00=0.300\) h\(^{-1}\). (3) The determinant is \(q_2-q_1=0\), so the sensitivity rows are locally redundant and do not separate both parameters by this criterion. (4) No. A prior can regularize the fit and influence the estimate, but it does not add individual-data information that distinguishes parameter pairs under the likelihood.