Structural Identifiability of Two-Compartment Parameters

Why Identifiability Is a Question About the Whole Curve

A biexponential equation can fit a concentration profile, but fitting the curve and identifying the underlying compartment parameters are different tasks. Structural identifiability asks whether two different parameter sets could generate exactly the same observations under ideal conditions. For a two-compartment IV-bolus model, the key question is whether the entire, exact central concentration curve uniquely fixes both its exponential components and the rate constants that produced them.

Here, “ideal data” means the central concentration is known without error continuously for all \(t\geq 0\), and the bolus dose and concentration scale are known. This is a mathematical benchmark, not a claim about what can usually be recovered from a finite clinical sampling schedule.

Model and Assumptions

Consider a linear two-compartment mammillary model after an IV bolus of known dose \(D\) into the central compartment. Let \(A_c(t)\) and \(A_p(t)\) be central and peripheral amounts. The central and peripheral volumes \(V_c,V_p\) are finite and positive. Clearance from the central compartment is \(CL>0\), and intercompartmental clearance is \(Q>0\). Define the microconstants \(k_{10}=CL/V_c\), \(k_{12}=Q/V_c\), and \(k_{21}=Q/V_p\), all with units of inverse time. Concentration is observed in the central compartment, so \(C(t)=A_c(t)/V_c\).

The standard shared-clearance parameterization matters: \(k_{12}V_c=k_{21}V_p=Q\). It links the two exchange rates through the same \(Q\); they are not independent in this model. The introductory material on two-compartment amplitudes establishes the central IV-bolus concentration’s biexponential form. Write it with an ordering convention:

The rates \(\alpha,\beta\) and coefficients \(A_{\alpha},A_{\beta}\) are macroparameters of the observed curve. They are not, by themselves, the physiologic rate constants. Under the positive, connected model above, both coefficients are positive.

From Macroparameters Back to Microconstants

The initial concentration is \(C(0)=D/V_c=A_{\alpha}+A_{\beta}\). Differentiating the central concentration at the instant after the bolus gives \(C'(0)=-(k_{10}+k_{12})D/V_c\), because the peripheral amount is initially zero. Differentiating the biexponential representation gives a second expression for the same slope. Thus the initial decline rate, defined here as \(r_0=-C'(0)/C(0)\), is determined by the curve:

The characteristic polynomial of the amount-balance system has roots \(\alpha\) and \(\beta\). Its sum is \(k_{10}+k_{12}+k_{21}\), and its product is \(k_{10}k_{21}\). These relationships, together with the initial slope, yield the inverse mapping.

These formulas give a unique inverse when the denominators are nonzero and the result is admissible. In particular, for a strictly connected model require \(k_{10}>0\), \(k_{12}>0\), and \(k_{21}>0\), as well as the stated positive volumes and clearances. The dose and concentration units must also be compatible: for example, mg divided by mg/L gives L.

Why the Inverse Is Unique

First, the exact positive biexponential curve uniquely determines its ordered rates and amplitudes. Since both coefficients are positive and \(\alpha>\beta\), the slower exponential dominates at late times:

Subtracting \(A_{\beta}e^{-\beta t}\) from the known curve leaves \(A_{\alpha}e^{-\alpha t}\), which determines \(\alpha\) and \(A_{\alpha}\). This proves uniqueness of the ordered biexponential representation under these conditions. Next, the displayed inverse equations determine \(V_c,k_{21},k_{10},k_{12}\) successively from \(D,A_{\alpha},A_{\beta},\alpha,\beta\). Finally, the shared-clearance relations determine \(CL,Q,V_p\). Thus, for an admissible strictly connected model with known dose, the ideal curve uniquely identifies these parameters.

The macroparameters must also be compatible with a positive connected model. One useful check is that \(r_0\) lies strictly between the two exponential rates. To see why, set \(r_0=k_{10}+k_{12}\). The polynomial with roots \(\alpha,\beta\) is \(f(x)=x^2-(k_{10}+k_{12}+k_{21})x+k_{10}k_{21}\). Substitution gives \(f(r_0)=-k_{12}k_{21}<0\). A monic quadratic is negative strictly between its two distinct real roots, so \(\beta<r_0<\alpha\). This condition alone is not a substitute for checking all recovered microconstants, but it is a useful necessary check.

Boundary Cases and What the Curve Cannot Reveal

Strict positivity is important to the uniqueness result. If the compartments are disconnected in the standard shared-clearance model, then \(Q=0\). With finite positive \(V_c\) and \(V_p\), the relations \(k_{12}=Q/V_c\) and \(k_{21}=Q/V_p\) require \(k_{12}=k_{21}=0\). The central curve is then a single exponential, and the peripheral volume does not affect the observation.

Another limitation arises if the bolus dose is unknown. The curve still determines its biexponential rates and amplitudes under ideal conditions, and hence the rate constants through the slope and root relationships. But \(V_c=D/(A_{\alpha}+A_{\beta})\) cannot be calculated without \(D\). Scaling \(D,V_c,CL,Q,V_p\) by the same positive factor leaves the concentration curve unchanged while preserving the rate constants. Thus absolute volume and clearance parameters require a known dose and concentration scale.

Diagnostics, Alternatives, and Scope

A structural calculation is useful partly because it exposes failures of interpretation. Check that the two rates are distinct and positive, the amplitudes sum to a positive initial concentration, \(k_{21}\neq0\), and all recovered microconstants and derived parameters meet the model’s positivity constraints. Confirm units at every conversion: rates have units of inverse time, \(CL\) and \(Q\) have volume per time, and volumes have volume. A failed check means either the curve is incompatible with the assumed model and conditions or a boundary/scale ambiguity applies; it does not justify forcing a physiologically impossible parameter value.

The exact-curve inversion is not an estimation method for ordinary sparse observations. With finite noisy data, exponential rates and amplitudes must be estimated, and the inverse formulas can magnify estimation error. Those are practical-identifiability and sampling questions, distinct from whether the ideal mapping is one-to-one. The main alternative conceptual route is to estimate the microconstants directly by solving the compartment equations and comparing predicted concentrations with data. That approach may be more suitable for data analysis, but it does not remove structural ambiguities: if two parameter sets generate the same ideal curve, direct fitting cannot distinguish them. Conversely, the macroparameter inversion makes the algebraic source of uniqueness and boundary failures especially transparent.

The conclusions rely on a linear, time-invariant two-compartment model, an IV bolus into the central compartment, observation of central concentration, a known dose and scale, and the stated shared-clearance relationships. They do not establish that the model is biologically correct or that real samples can determine its parameters precisely. Structural uniqueness is a property of the specified model under ideal observation, not evidence that every estimated parameter will be reliable in practice.