Compartmental Identifiability Through Transfer Functions

Why Transfer Functions Clarify Compartmental Identifiability

A two-compartment model can have several rate constants even when only one input and one output are observed. The transfer function summarizes the exact input-output relationship under linear, time-invariant dynamics and zero initial conditions. Its coefficients reveal which parameter combinations the ideal behavior can distinguish—and which different parameter values produce precisely the same behavior.

Here, “ideal input-output behavior” means the transfer function itself is known exactly, as if the system could be probed with sufficiently informative inputs and observed without error. This is a structural question, not a claim that finite, noisy data will estimate every identifiable parameter precisely.

Model and Assumptions

Let \(x_1(t)\) and \(x_2(t)\) be amounts in a central and a peripheral compartment. An input \(u(t)\), measured in amount per unit time, enters the central compartment. Drug leaves the central compartment by elimination at rate constant \(k_{10}\), moves to the peripheral compartment at rate constant \(k_{12}\), and returns at rate constant \(k_{21}\). Each rate constant has units of inverse time.

Assume the rates are constant and nonnegative, the dynamics are linear, the input enters exactly as shown, and the initial amounts are known to be zero for the transfer-function calculation. We first consider observation of central concentration \(y=x_1/V_1\), where the central volume \(V_1\) is known and positive. We will also consider observation of peripheral amount \(y=x_2\). These are different observation choices and need not have the same identifiability properties.

Deriving the Transfer Functions

With zero initial amounts, taking Laplace transforms gives two algebraic equations. The first equation expresses the central amount in terms of the input and peripheral amount; the second expresses the peripheral amount in terms of the central amount.

The determinant of this system is \((s+k_{10}+k_{12})(s+k_{21})-k_{12}k_{21}\). Expanding and collecting powers of \(s\) yields the common denominator below. Solving the equations for each output then gives the two transfer functions.

The numerator matters as much as the denominator. For central concentration, the numerator contains \(k_{21}\); for peripheral amount, it contains \(k_{12}\). The denominator alone supplies the coefficient combinations \(k_{10}+k_{12}+k_{21}\) and \(k_{10}k_{21}\). A transfer function is understood here in its reduced form: the examples below have no cancellation between numerator and denominator.

What the Coefficients Identify

Write the denominator as \(D(s)=s^2+As+B\), with \(A=k_{10}+k_{12}+k_{21}\) and \(B=k_{10}k_{21}\). For central concentration, the numerator gives \(k_{21}\) directly, assuming the model’s input and output scales are known. Once \(k_{21}\) is known and positive, \(B\) gives \(k_{10}=B/k_{21}\), and \(A\) then gives \(k_{12}=A-k_{10}-k_{21}\). Thus the exact central transfer function generically determines all three rates, not merely the denominator combinations.

This is a direct uniqueness argument: any parameter set producing the same central transfer function must have the same numerator coefficient \(k_{21}\), then the same \(k_{10}\) from \(B\), and finally the same \(k_{12}\) from \(A\). The conclusion depends on observing the full transfer function, including its numerator. It does not follow from the denominator coefficients alone.

When Parameter Assignments Remain Ambiguous

With peripheral amount as the only output, the transfer function has numerator \(k_{12}\) and denominator \(s^2+As+B\). If \(k_{12}\) is known, the denominator determines the sum \(k_{10}+k_{21}=A-k_{12}\) and the product \(k_{10}k_{21}=B\). A sum and product determine the two values as an unordered pair. Unless some additional information distinguishes their roles, exchanging \(k_{10}\) and \(k_{21}\) leaves this peripheral transfer function unchanged.

The two denominator roots are also sometimes described by two macro-rate constants. They are an unordered pair: switching their labels does not change the polynomial. Those pole labels are not, by themselves, compartment rate constants. The parameter-exchange ambiguity above is a separate issue: it concerns swapping \(k_{10}\) and \(k_{21}\) in a peripheral-output model when the denominator supplies only their sum and product.

Scaling, Conditions, and Diagnostics

The conclusions rely on the observation and input scales being specified. If \(V_1\) is unknown, the central transfer function’s overall gain contains the unknown factor \(1/V_1\), so the scale cannot simply be treated as known. The numerator’s zero still indicates \(k_{21}\) when the transfer function is known in an appropriate reduced form, but recovering all rate constants and the unknown volume requires accounting for that extra parameter. Likewise, an unknown input scaling factor can confound transfer-function gain with output scaling.

Check the recovered parameters against the model constraints. For positive rates, \(A>0\), \(B>0\), and the inferred \(k_{12}=A-B/k_{21}-k_{21}\) must be nonnegative. A negative inferred rate is not a valid positive-rate realization of the stated model; it can indicate that the assumed model, coefficient interpretation, or data-to-transfer-function calculation is inconsistent. A cancellation between a numerator zero and a denominator pole can also remove information from a reduced transfer function, so the generic recovery formulas should not be applied blindly at such exceptional parameter values.

In finite experimental data, estimating a numerator zero or distinguishing nearby rates may be difficult even when the exact transfer function uniquely determines the parameters. Limited input variation, noisy or sparse observations, and an uncertain observation scale can impair practical estimation. Structural uniqueness is a statement about exact ideal behavior; it does not guarantee narrow uncertainty intervals in an experiment.

How the Main Alternatives Differ

The transfer-function approach is especially useful for exposing input-output equivalences algebraically. An alternative is to fit the original compartment differential equations directly to time-course observations. Under the same model and ideal conditions, both approaches describe the same input-output behavior; direct fitting does not create information absent from the transfer function. It can, however, use the actual sampling schedule and observation model rather than first summarizing behavior through exact transfer-function coefficients.

A third useful perspective is to compare coefficients or pole locations without assigning each one to a microscopic rate. This can establish identifiable combinations, such as \(A\) and \(B\), when individual rates remain ambiguous. The limitation is that combinations alone may not answer which biological process a rate represents. Use numerator information and the specified observation to decide whether the microscopic assignment is actually resolved.