Structural vs. Practical Identifiability in QSP

1. What Is Identifiability?

Identifiability asks whether the information contained in model outputs and observations is sufficient to determine the model parameters of interest. It is a fundamental question in quantitative systems pharmacology (QSP) because QSP models can contain dozens, hundreds, or even thousands of biological and pharmacological parameters.

A model may have many parameters, but that does not mean that every parameter can be learned from every dataset. Identifiability provides a framework for distinguishing parameters that can, in principle, be uniquely determined from those that cannot.

2. Why Is Identifiability Especially Important in QSP?

QSP models are designed to represent biological mechanisms across multiple scales. A single model may connect drug exposure to target engagement, signaling, biomarkers, cell populations, disease processes, and clinical outcomes.

This mechanistic richness is scientifically useful, but it also creates a challenging inverse problem: many combinations of parameter values may generate similar observable behavior.

For these reasons, identifiability should be considered during QSP model development rather than only after parameter estimation has failed.

3. Structural vs. Practical Identifiability

Two concepts are particularly important: structural identifiability and practical identifiability.

A parameter can therefore be structurally identifiable but practically unidentifiable. This is one of the most important ideas to understand when developing a QSP model.

4. What Is Structural Identifiability?

Structural identifiability is a theoretical question. Imagine that the model is exactly correct and that the observable outputs are known continuously and without measurement error over the relevant time interval.

The question is whether the mapping from parameters to model outputs is one-to-one.

Let a model be represented abstractly as:

where \(y(t)\) represents an observable model output and \(\theta\) is the vector of unknown parameters.

If two different parameter vectors, \(\theta_1\) and \(\theta_2\), always produce different observable trajectories, then the parameters may be structurally identifiable. If distinct parameter vectors can produce exactly the same observable trajectory, the parameters are structurally non-identifiable under the specified observation scheme.

When this equality can hold for \(\theta_1\neq\theta_2\), the observations cannot distinguish those parameter values, even with perfect data.

5. A Simple Structural Non-Identifiability Example

Consider a deliberately simple model:

The model contains two parameters, \(CL\) and \(V\). In this particular setting, the initial concentration depends on \(Dose/V\), while the elimination rate depends on \(CL/V\).

Suppose the dose is known and the complete concentration-time profile is observed. The initial concentration provides information about \(V\), while the decay rate provides information about \(CL/V\). Under the assumptions of this model, both quantities can be separated.

Now consider a different hypothetical model in which the only observable quantity is:

Here, \(CL\) and \(V\) enter only through their ratio. If \(CL\) and \(V\) are both multiplied by the same positive constant, their ratio remains unchanged.

Therefore, the observable output cannot distinguish among infinitely many pairs with the same ratio. The individual parameters are structurally non-identifiable even with perfect observations of \(y(t)\).

6. Identifiable Parameter Combinations

Structural non-identifiability does not necessarily mean that the model is useless. Sometimes a specific parameter combination is identifiable even though the individual parameters are not.

For example, if the model depends on \(CL\) and \(V\) only through the ratio \(CL/V\), then the ratio itself may be identifiable while \(CL\) and \(V\) separately are not.

This distinction is important in QSP because mechanistic parameters may appear in products, ratios, sums, or other combinations.

For QSP, this means that a useful identifiability analysis should ask not only whether individual parameters are identifiable, but also which combinations of parameters are supported by the available observations.

7. Identifiability in a Mechanistic QSP Pathway

Consider a simplified QSP pathway in which a drug binds a target and the resulting target engagement influences a downstream biomarker.

Suppose the target-engagement equation contains an association rate constant \(k_{on}\), a dissociation rate constant \(k_{off}\), and a target concentration \(T_0\).

If the experiment measures only a downstream biomarker and not target engagement, several parameter combinations may influence the biomarker in similar ways.

Adding a direct measurement of target engagement can change the identifiability problem substantially because it provides an additional observable linked more directly to the parameters governing binding.

8. Identifiability Depends on What You Observe

A QSP model can contain hundreds of states, but investigators may directly observe only a small subset of them.

For example, a model might contain:

• Drug concentration.
• Free and bound target.
• Receptor occupancy.
• Intracellular signaling species.
• Cell populations.
• Biomarkers.
• Disease burden.

If only plasma drug concentration is measured, parameters governing downstream biology may be weakly constrained or completely unidentifiable from that dataset alone.

Mathematically, the observation model can be written as:

where \(x(t)\) represents the underlying model states and \(h(\cdot)\) maps those states and parameters to measured quantities.

Identifiability therefore depends on both the dynamic model and the observation model.

9. What Is Practical Identifiability?

Practical identifiability asks a different question: given the data that are actually available, can the parameters be estimated with sufficient precision and stability?

Real QSP datasets have finite sample sizes, measurement error, missing observations, assay variability, sparse sampling, and sometimes substantial between-subject variability.

A structurally identifiable model can therefore still have very broad parameter uncertainty.

In practice, practical identifiability is often assessed using parameter confidence intervals, profile likelihoods, posterior distributions, bootstrap results, covariance structures, sensitivity analysis, or related diagnostics.

10. Profile Likelihood as a Practical Diagnostic

A useful way to examine practical identifiability is to vary one parameter while re-estimating the remaining parameters.

Suppose the objective function is \(L(\theta)\). For a parameter \(\theta_j\), a profile likelihood examines how well the model can fit the data as \(\theta_j\) is fixed at different values while other parameters are optimized.

Here, \(\theta_{-j}\) denotes all parameters except \(\theta_j\).

A sharply peaked profile suggests that the data provide relatively strong information about the parameter. A broad or nearly flat profile indicates that a wide range of values may fit nearly equally well.

11. Parameter Correlation and Compensation

Practical identifiability can also be affected by strong correlations between parameters.

Suppose two parameters influence the model in similar ways. Increasing one parameter may be offset by decreasing the other, producing a similar predicted trajectory.

This phenomenon is sometimes described as parameter compensation.

In a QSP model, compensation can arise because different biological mechanisms produce similar downstream effects. For example, increased pathway activation might be offset by increased turnover, or reduced receptor abundance might be partially compensated by increased signaling efficiency.

12. Sensitivity and Identifiability

Sensitivity analysis asks how much model outputs change when parameters change. It is closely related to identifiability because parameters that have little influence on measured outputs are difficult to estimate from those outputs.

A local sensitivity coefficient can be represented as:

A scaled or normalized sensitivity may instead be written as:

If two parameters have very similar sensitivity profiles across the observed outputs and time points, distinguishing their effects may be difficult.

This is one reason sensitivity analysis can be useful during QSP model development: it can reveal which parameters influence the quantities that are actually measured.

13. Fisher Information and Local Practical Identifiability

For models with an explicit statistical observation model, local information about parameter precision can be summarized using the Fisher information matrix.

In a simplified setting, the information matrix can be represented as:

where \(J(\theta)\) is a sensitivity or Jacobian matrix and \(W\) reflects the observation-error structure.

When the information matrix is well-conditioned and sufficiently full rank, the data contain substantial local information about the corresponding parameter directions. Near-singularity indicates that some parameter combinations are poorly distinguished.

The important concept is not simply whether the model has many parameters. What matters is whether the observations provide independent information about those parameters.

14. How Can Structural Identifiability Be Studied?

Structural identifiability can be investigated using analytical and computational methods. The appropriate method depends on the mathematical form of the model.

For large QSP models, fully analytical structural-identifiability analysis may be computationally demanding. Model reduction, modular analysis, parameter grouping, and targeted analysis of identifiable parameter combinations can therefore be useful.

15. Experimental Design Can Improve Identifiability

Identifiability is not solely a modeling problem. The experimental design determines what information enters the model-fitting process.

Several design choices can improve practical identifiability:

• Measure informative biomarkers. Select observations that are directly influenced by parameters of interest.
• Sample at informative times. Early, intermediate, and late observations may constrain different dynamic processes.
• Use multiple doses. Different exposure levels can help distinguish nonlinear mechanisms.
• Measure different biological levels. Combining drug concentration, target engagement, biomarkers, and clinical outcomes can constrain different portions of a QSP model.
• Use perturbations when feasible. Mechanistically informative interventions can separate competing pathways.
• Combine complementary datasets. Independent experiments can provide information about different parameters or states.

16. Literature Values, Priors, and External Information

QSP models frequently incorporate parameter information from literature studies, in vitro experiments, animal studies, clinical pharmacology analyses, or previous model-development efforts.

External information can be extremely valuable when the clinical dataset alone cannot identify all parameters. In a Bayesian framework, prior information can be represented explicitly:

where \(p(\theta)\) represents prior information and \(p(y\mid\theta)\) represents the likelihood contributed by the observed data.

However, external information should not be confused with information generated by the current dataset. A parameter may be weakly informed by the data but strongly constrained by an informative prior or fixed literature value.

17. Worked Example: A Two-Parameter QSP Submodel

Consider a simplified mechanistic model in which a biomarker \(B(t)\) is influenced by a signaling rate \(k_s\) and a turnover rate \(k_d\):

Assume that the initial biomarker level is \(B_0\), and suppose the biomarker is observed over time.

Step 1: Solve the model

The solution is:

This equation immediately reveals two distinct features of the trajectory.

• The long-term equilibrium is \(k_s/k_d\).
• The rate of approach to equilibrium is governed by \(k_d\).

Step 2: Identify what the trajectory contains

If the complete time course is observed, the decay rate toward equilibrium provides information about \(k_d\), while the equilibrium level provides information about \(k_s/k_d\).

Step 3: Recover the parameters

If the equilibrium value \(B_{\infty}\) and the decay rate \(k_d\) can both be determined from ideal observations, then:

and therefore:

In this idealized setting, the two parameters can be separated.

Step 4: Add realistic measurement limitations

Now suppose the biomarker is measured only three times, all relatively late after treatment, with substantial assay variability.

The model may remain structurally identifiable, but the data may provide little information about the early trajectory and therefore little information about the turnover rate.

The resulting estimates of \(k_s\) and \(k_d\) may be highly correlated and uncertain.

This example illustrates the central distinction: structural identifiability asks whether the parameters can be separated in principle; practical identifiability asks whether the actual experiment provides enough information to separate them reliably.

18. What Can You Do When a QSP Model Is Not Identifiable?

Non-identifiability is not automatically a reason to discard a QSP model. Several strategies can address different sources of the problem.

The appropriate response depends on whether the problem is structural, experimental, statistical, or biological.

19. Why a Good Model Fit Does Not Prove Identifiability

Suppose two parameter sets produce nearly identical model predictions across all measured observations.

If both parameter sets fit the data similarly well, the dataset provides limited evidence for choosing one set over the other.

This can occur even when the visual fit appears excellent.

For QSP, this distinction is particularly important because a mechanistic interpretation may depend on individual parameter values. If several parameter combinations explain the observations equally well, mechanistic conclusions based on one particular parameter estimate should be interpreted cautiously.

20. Identifiability and QSP Model Validation

Identifiability is one component of a broader model-development process. A model can be structurally identifiable yet biologically misspecified, or it can fit historical data well while making poor predictions in a new experimental setting.

A practical QSP workflow should therefore consider:

1. Structural plausibility. Does the model represent the biological mechanisms relevant to the scientific question?
2. Structural identifiability. Can the parameters or parameter combinations be uniquely determined from the specified observations in principle?
3. Practical identifiability. Do the available data constrain those parameters sufficiently?
4. Parameter plausibility. Are estimated values consistent with known biology and external evidence?
5. Goodness of fit. Does the model reproduce the observations adequately?
6. Predictive validation. Does the model make useful predictions for data or experiments not used for calibration?
7. Uncertainty analysis. How does parameter uncertainty propagate into predicted outcomes?

These questions address different aspects of model credibility and should not be reduced to a single fit statistic.

21. A Practical QSP Identifiability Workflow

1. Define the scientific question. Identify which mechanisms, parameters, or predictions actually matter.
2. Specify the observables. Determine which states and outputs are measured and with what precision.
3. Examine structural identifiability. Determine whether the parameters of interest can theoretically be distinguished.
4. Identify parameter combinations. Determine whether the model identifies ratios, products, sums, or other combinations rather than individual parameters.
5. Perform sensitivity analysis. Determine which parameters influence the measured outputs.
6. Assess practical identifiability. Use profile likelihoods, confidence intervals, posterior distributions, covariance diagnostics, or related methods.
7. Inspect parameter correlations. Look for compensating parameter combinations and poorly constrained directions.
8. Evaluate experimental design. Ask whether additional measurements, doses, time points, or perturbations would provide new information.
9. Incorporate external information carefully. Distinguish data-driven information from fixed parameters and prior knowledge.
10. Quantify prediction uncertainty. Propagate parameter uncertainty into the QSP predictions that support decisions.

22. Structural vs. Practical Identifiability at a Glance

23. Identifiability and QSP Prediction

Parameter identifiability matters because uncertainty in parameters can propagate into uncertainty in model predictions.

However, an important distinction is that individual parameter identifiability and predictive identifiability are not always identical.

Two parameter sets may differ substantially while generating nearly identical predictions for a particular endpoint under the conditions of interest.

For example:

where \(g(\theta)\) represents a prediction of scientific interest.

This means that a model may be uncertain about individual biological mechanisms while still making relatively stable predictions for some outputs. Conversely, a prediction that depends strongly on poorly identified parameters may have substantial uncertainty.

24. Common Identifiability Mistakes in QSP

Mistake 1: Equating a good fit with identifiable parameters

A model can reproduce observations closely while multiple parameter sets produce essentially the same fit.

Mistake 2: Assuming every parameter needs to be estimated from the same dataset

QSP models often combine information from different experimental systems. Some parameters may be informed by in vitro experiments, others by animal studies, and others by clinical data.

Mistake 3: Ignoring the observation model

Parameters associated with unobserved states may be weakly constrained even when those states are central to the mechanistic model.

Mistake 4: Treating a fixed parameter as an identified parameter

Fixing a parameter can make estimation more stable, but the current dataset has not necessarily identified that parameter.

Mistake 5: Collecting more data of the same type without checking information content

More measurements do not automatically solve an identifiability problem if they are redundant with existing observations.

Mistake 6: Ignoring parameter combinations

A model may identify a ratio, product, or effective pathway strength rather than the individual mechanistic parameters.

Where to Go Next

A natural progression is to study parameter estimation and uncertainty in QSP, followed by sensitivity analysis, profile likelihood, Bayesian calibration, parameter uncertainty propagation, and optimal experimental design.

The next tutorial can build directly on these ideas by showing how local and global sensitivity analysis reveal which QSP parameters influence model outputs, how parameter correlations arise, and how sensitivity information can be used to design more informative experiments.

References

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3. Villaverde, A. F., Barreiro, A., & Papachristodoulou, A. (2016). Structural identifiability of dynamic systems biology models. PLOS Computational Biology, 12(10), e1005153.
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Identifiability methods should be selected according to the mathematical structure of the QSP model, the observation model, and the scientific question. No single diagnostic establishes model adequacy on its own.