Distinguishing Structural Alternatives from Parameter Variants

Why the distinction matters

Two QSP formulations can use different biological language and different equations yet make exactly the same predictions for every measured output under every allowed input. In that case, the formulations are alternative representations of one input–output model, not empirically distinct hypotheses. Conversely, two structures may agree at baseline or at one dose and disagree when the system is perturbed or followed over time. Agreement on a limited set of observations is not, by itself, proof of equivalence.

The distinction matters when interpreting model comparisons. A structural difference is meaningful for evidence only if it can change predictions for observables under conditions of interest. If a parameter transformation makes the predictions identical throughout the comparison domain, fitting the two formulations separately cannot provide evidence between their biological explanations. Earlier material on nested and non-nested QSP model relationships and shared assumptions for fair comparisons provides useful context; here the focus is on determining whether a claimed difference is genuinely present in observable behavior.

A parameter variant changes parameter values within the same specified model structure. A structural alternative changes features such as the way an input enters a rate law, the direction of a pathway, or the relationship between states. That distinction is not settled by comparing equation appearance alone. A change of state variable can make equations look different while preserving all observable behavior; different-looking biological explanations can also imply the same equations for the measured variables.

An exact equivalence from changing the state scale

Consider a state \(X(t)\), measured in concentration units, with constant production and first-order loss. Let \(k_{\mathrm{in}}\) have units of concentration per day and \(k_{\mathrm{out}}\) have units of inverse days. Suppose the measured output is a scaled state, \(y=qX\), where \(q>0\) is dimensionless.

Define a new state \(Z=qX\). Since \(q\) is constant, differentiating gives \(dZ/dt=q\,dX/dt\). Substituting \(X=Z/q\) yields the alternate formulation below, whose measured output is simply \(Z\).

The parameter mapping is \(k_{\mathrm{in}}'=qk_{\mathrm{in}}\), \(k_{\mathrm{out}}'=k_{\mathrm{out}}\), and the initial-state mapping is \(Z(0)=qX(0)\). The output is identical at every time because \(Z(t)=qX(t)\) by construction. The production parameter in the transformed model still has units of concentration per day, since \(q\) has no units. If the original parameters and initial state are otherwise unrestricted, either formulation generates the same output trajectories as the other. Thus, for the stated observation and parameter domains, they are reparameterizations rather than distinct observable mechanisms.

This conclusion depends on what is measured. If \(X\) itself is independently observed in addition to \(y\), or if \(q\) varies with time or treatment, the simple equivalence need not hold. Always include the observation definitions, input conditions, and initial conditions when asserting equivalence.

When different mechanisms agree at one condition

Now compare two biological hypotheses for a normalized biomarker \(B\). In one formulation, drug concentration suppresses production. In the other, drug concentration increases loss. Let \(C\) be concentration and \(C_*\) a positive concentration scale with the same units as \(C\). The dimensionless concentration effect is \(f(C)=C/(C_*+C)\), defined for \(C\geq0\). This scale is essential: adding a concentration directly to a dimensionless number would be dimensionally invalid.

Assume \(B\) is dimensionless, is directly observed, and experiences constant drug concentration during each simulation. The numerical rate constants below are in inverse days; \(B(0)\) is specified in each calculation.

For a constant concentration, setting the derivative to zero gives the steady states \(B_{\mathrm{ss,P}}=k_0[1-\eta f(C)]/k_{\mathrm{loss}}\) and \(B_{\mathrm{ss,D}}=k_0/[k_{\mathrm{loss}}+k_Cf(C)]\). These formulas describe steady-state predictions, not transient behavior.

Transient predictions can distinguish the models even where their steady states match. With constant \(C\), each equation has the form \(dB/dt=a-rB\), whose solution is \(B(t)=B_{\mathrm{ss}}+[B(0)-B_{\mathrm{ss}}]e^{-rt}\), where \(B_{\mathrm{ss}}=a/r\). For the production-inhibition model, \(r=k_{\mathrm{loss}}\); for induced clearance, \(r=k_{\mathrm{loss}}+k_Cf(C)\). The exponential rate is therefore generally different even when the steady state is identical.

A practical comparison workflow

A useful assessment combines algebra and simulation. Algebra can establish equivalence over a broad domain, while simulation can reveal differences at concrete inputs and times. Simulation alone cannot prove equivalence: a finite grid of conditions can miss a difference elsewhere. The following sequence keeps the claim tied to the conditions being compared.

Diagnostics, alternatives, and limitations

Algebraic analysis is the strongest route when a transformation can be written explicitly and verified by substitution. Simulation is useful when the equations are nonlinear, the mapping is difficult to find, or realistic inputs and observations need to be compared. A calibration-based comparison asks whether fitted models can reproduce the data under shared assumptions; the earlier material on controlling parameter-flexibility bias and information criteria addresses aspects of that comparison. Good fit alone does not show that two formulations are distinct, and different fitted parameter values do not establish different mechanisms.

Several warning signs deserve attention. If apparent discrepancies shrink markedly when solver tolerances are tightened, numerical error may be involved. If models match only at equilibrium, the transient rates may still differ. If a conclusion changes when an output is rescaled, verify that the observation model and units were transformed consistently. If the models agree only for one dose or one initial condition, report that restricted agreement rather than calling them equivalent.

There is also a distinction between structural equivalence and limited practical discrimination. Two models can be structurally different yet make predictions so similar over the observed range that available measurements cannot reliably separate them. Conversely, a simulated difference is not evidence that the biology follows one model; it shows only that the formulations make different predictions under the specified conditions. Sparse or indirect measurements can further limit what is observable, as discussed in the earlier material on mechanism testing with sparse and indirect measurements.

The scope of any conclusion must therefore be explicit. An algebraic result depends on the allowed parameter ranges, input histories, initial states, and outputs. A simulation result depends on the tested values and numerical accuracy. A fitted-model result depends additionally on the data and observation assumptions. None of these alone guarantees that a biological mechanism has been uniquely established.