1. What Is Uncertainty Quantification?
Uncertainty quantification (UQ) is the process of characterizing how uncertainty in model inputs, parameters, assumptions, and observations affects model outputs.
In quantitative systems pharmacology (QSP), this is especially important because QSP models often contain many mechanistic components. A model may connect drug exposure to target binding, signaling, cell populations, biomarkers, disease progression, and clinical outcomes. Each layer can introduce uncertainty.
QSP uncertainty quantification asks how uncertainty in the model and its inputs propagates to quantities predicted by the model.
2. Why Is Uncertainty Quantification Important in QSP?
QSP models are often used to explore questions where direct experimental evidence is incomplete. This makes it important to distinguish what the data strongly constrain from what remains uncertain.
For example, a QSP model might be used to predict the response to a dose that has not yet been studied clinically. The model may provide a useful prediction, but the prediction depends on quantities such as receptor abundance, binding affinity, turnover rates, signaling parameters, disease-state parameters, and relationships between biomarkers and outcomes.
| Source of uncertainty | Example in QSP | Potential consequence |
|---|---|---|
| Parameter uncertainty | Unknown or imprecisely estimated binding or turnover parameter | Different parameter values produce different trajectories |
| Biological uncertainty | Incomplete knowledge of a disease mechanism | Alternative mechanistic explanations may remain plausible |
| Measurement uncertainty | Assay variability or noisy biomarker measurements | Estimated parameters may be uncertain |
| Structural uncertainty | Competing model representations of a pathway | Different model structures may produce different predictions |
| Population variability | Subjects differ in target expression or physiological parameters | Individual responses may differ even under the same treatment |
| Input uncertainty | Uncertain baseline disease state or dose-response relationship | Predictions vary across plausible inputs |
These sources are related but should not be treated as interchangeable. Parameter uncertainty concerns what value a parameter has within a specified model, whereas structural uncertainty concerns whether the model representation itself is appropriate.
3. The Main Types of Uncertainty
A useful first step in QSP UQ is to identify what is uncertain before deciding how to quantify it.
Parameter uncertainty
Suppose a parameter \(k\) is estimated from experimental data. If the data do not determine \(k\) exactly, then \(k\) has a distribution or uncertainty interval rather than a single known value.
Structural uncertainty
Structural uncertainty occurs when multiple plausible mathematical representations of a biological process exist. For example, a biomarker turnover process might be represented using a linear production model, a saturable production model, or a feedback mechanism.
Observation uncertainty
Experimental measurements are subject to residual error. Concentration, biomarker, cell-count, and clinical measurements therefore generally should not be treated as exact observations of the underlying biological state.
Population variability
Variability and uncertainty are conceptually different. Variability describes real differences between individuals or biological systems. Uncertainty describes incomplete knowledge about quantities or processes.
4. Parameter Uncertainty in QSP Models
QSP models typically contain parameters describing biological rates, affinities, capacities, initial conditions, and other mechanistic quantities.
Suppose a simple biomarker model is:
The model contains at least two parameters, \(k_{\mathrm{in}}\) and \(k_{\mathrm{out}}\). If these are estimated from data, their uncertainty can be represented using parameter distributions or covariance information.
For example, if a parameter estimate is:
that number does not imply that the true value is exactly \(0.20\). An uncertainty analysis asks which alternative values remain reasonably compatible with the available evidence.
Parameter uncertainty can arise from limited sample size, sparse sampling, noisy measurements, weak experimental design, parameter correlations, or insufficient information about a particular biological process.
5. How Is Parameter Uncertainty Represented?
One common approach is to represent uncertain parameters as probability distributions.
For a parameter vector \(\theta\), Bayesian modeling can represent uncertainty using a posterior distribution:
Here, \(y\) represents observed data, \(p(\theta)\) represents prior information, and \(p(y\mid\theta)\) represents the likelihood or data model.
In a frequentist analysis, uncertainty may instead be summarized through a covariance matrix, confidence intervals, profile likelihoods, bootstrap distributions, or other inferential approaches.
| Representation | Typical interpretation | Potential use in QSP |
|---|---|---|
| Confidence interval | Frequentist interval associated with a specified inferential procedure | Summarizing uncertainty in estimated parameters |
| Covariance matrix | Approximate joint uncertainty and parameter correlation | Sampling parameter vectors for local uncertainty propagation |
| Profile likelihood | Likelihood-based evidence across parameter values | Assessing weakly identified parameters |
| Bootstrap distribution | Empirical variation across resampled datasets | Propagating sampling uncertainty |
| Posterior distribution | Updated uncertainty after combining prior information and data | Bayesian parameter and prediction uncertainty |
The choice of representation should match the estimation method, available information, and scientific purpose.
6. How Does Uncertainty Propagate Through a QSP Model?
Suppose the QSP model produces an output \(Y\) from parameters \(\theta\) and inputs \(x\):
If \(\theta\) is uncertain, then \(Y\) is also uncertain. The goal of uncertainty propagation is to characterize the resulting distribution of \(Y\).
A simple Monte Carlo procedure is:
- Specify distributions for uncertain parameters and inputs.
- Draw a parameter vector from those distributions.
- Run the QSP model using that parameter vector.
- Store the resulting prediction.
- Repeat the process many times.
- Summarize the distribution of predictions.
The resulting simulations form a distribution of possible model outputs rather than a single deterministic trajectory.
Monte Carlo propagation converts uncertainty in inputs and parameters into a distribution of model predictions.
7. Prediction Intervals and Uncertainty Bands
After propagating uncertainty, the model can produce an ensemble of trajectories. At each time point, these trajectories form a distribution of predicted values.
A central summary might be the median prediction:
An uncertainty interval can then be constructed from selected quantiles of the simulated distribution.
For example, the 2.5th and 97.5th percentiles define a central 95% simulation interval:
These intervals describe the uncertainty represented by the specified simulation procedure. They should not automatically be interpreted as statements about every possible source of real-world uncertainty.
8. Sensitivity Analysis and Uncertainty Quantification
Sensitivity analysis asks how strongly model outputs respond to changes in inputs or parameters. It is closely related to UQ, but the two questions are different.
Consider:
A local sensitivity coefficient for parameter \(\theta_j\) can be written as:
In practice, normalized sensitivities are often more useful because parameters can have very different units and scales:
Sensitivity tells us which parameters can strongly influence an output. UQ tells us how uncertainty in those parameters translates into uncertainty in the output.
| Question | Primary concept |
|---|---|
| Which parameters influence this output? | Sensitivity analysis |
| How uncertain is each parameter? | Parameter uncertainty |
| How uncertain is the final prediction? | Uncertainty propagation |
| Which uncertainties contribute most to prediction uncertainty? | Global sensitivity / variance decomposition |
9. Global Sensitivity Analysis
Local sensitivity analysis examines behavior around a particular parameter set. QSP models can be highly nonlinear, so the sensitivity may change across the parameter space.
Global sensitivity analysis evaluates model behavior across a range of plausible parameter values.
One family of methods decomposes output variance into contributions from uncertain inputs. In a simplified independent-input setting:
where \(V_j\) represents contributions associated with individual parameters and \(V_{jk}\) represents interaction contributions.
Methods such as Sobol sensitivity indices can quantify these contributions. For example, a first-order Sobol index for parameter \(\theta_j\) can be written conceptually as:
A total-effect index includes the parameter's interactions with other uncertain inputs.
10. Why Parameter Correlation Matters
QSP parameters are often correlated. Two parameters may compensate for each other so that substantially different parameter combinations produce similar model predictions.
Suppose the parameter vector is:
Its uncertainty can be represented by a covariance matrix:
where \(\rho\) is the correlation between the parameters.
Sampling each parameter independently can therefore create parameter combinations that were not supported by the original estimation process.
11. Uncertainty and Parameter Identifiability
Uncertainty quantification is closely connected to identifiability.
A parameter is identifiable when the available information can sufficiently distinguish its value from alternative values. In QSP, weak identifiability can occur because several mechanisms produce similar observable behavior.
For example, if an observed biomarker depends on two parameters through their ratio:
the data may provide strong information about the ratio while providing relatively weak information about the individual parameters.
This can produce a characteristic situation: individual parameter estimates are highly uncertain, but certain model predictions remain relatively stable.
| Situation | Possible interpretation |
|---|---|
| Individual parameters uncertain, prediction stable | Parameters may be correlated or only combinations may be identifiable |
| Parameters precise, prediction uncertain | Prediction may depend strongly on uncertain inputs or extrapolation |
| Parameters and prediction both uncertain | Data may provide limited information for the scientific question |
| Parameters precise under multiple models, predictions differ | Structural uncertainty may dominate |
Therefore, QSP UQ should examine uncertainty in the quantities that matter scientifically, not only uncertainty in individual parameters.
12. Structural Uncertainty in QSP
Parameter uncertainty assumes that the model structure is sufficiently appropriate. Structural uncertainty challenges that assumption.
Suppose two models describe a disease biomarker:
Both models might fit the observed data reasonably well over the available experimental range. However, they may produce different predictions when extrapolated to a new disease state or treatment condition.
This is an example of structural model uncertainty. It cannot be completely addressed by narrowing the parameter distributions within only one model.
Approaches to structural uncertainty can include:
- Comparing scientifically plausible alternative model structures.
- Performing model averaging or ensemble modeling when appropriate.
- Examining whether conclusions are robust across competing models.
- Collecting new data that distinguish the competing mechanisms.
- Explicitly documenting assumptions that are not strongly supported by data.
13. Measurement Error and Observation Uncertainty
QSP models distinguish the underlying biological state from what is actually measured.
Let \(X(t)\) represent the latent biological state and \(Y_i\) an observed measurement. An observation model might be written as:
where \(\varepsilon_i\) represents measurement or residual variability.
For positive biomarkers or concentrations, a proportional or log-scale error model may sometimes be more appropriate:
The observation model matters because uncertainty in measurements affects parameter estimation, and parameter uncertainty subsequently affects QSP predictions.
14. Bayesian Uncertainty Quantification
Bayesian methods provide a natural framework for combining prior knowledge with experimental evidence and propagating resulting parameter uncertainty into predictions.
Starting with a prior distribution \(p(\theta)\) and data \(y\), Bayes' theorem gives:
The posterior distribution \(p(\theta\mid y)\) can then be sampled to generate predictions.
For a future quantity \(Y_{\mathrm{new}}\), the posterior predictive distribution is:
This framework naturally separates parameter uncertainty from future observation variability when the model is specified accordingly.
Bayesian methods can be especially useful in QSP when external experimental knowledge is available, because prior distributions can encode previously established information. The assumptions used to construct those priors should nevertheless be documented and evaluated.
15. Scenario Analysis: A Complement to Probability Distributions
Not every uncertainty is naturally represented as a probability distribution.
Suppose the literature supports two biologically plausible mechanisms, but there is insufficient evidence to assign defensible probabilities to them. Treating one mechanism as 70% likely and another as 30% likely could introduce an arbitrary assumption.
Instead, QSP analysts may evaluate separate scenarios:
| Scenario | Assumption | Question addressed |
|---|---|---|
| Scenario A | Linear biomarker production | What happens under mechanism A? |
| Scenario B | Saturable biomarker production | What happens under mechanism B? |
| Scenario C | Strong feedback regulation | How sensitive are predictions to feedback? |
Scenario analysis is therefore particularly useful for epistemic uncertainty—uncertainty arising from incomplete knowledge—when assigning probabilities would not be scientifically justified.
16. Worked Example: Propagating Parameter Uncertainty
Consider a simple QSP biomarker model:
Suppose the parameters have been estimated with the following uncertainty:
- \(k_{\mathrm{in}}\): mean \(10\) units/h, standard deviation \(1\) unit/h.
- \(k_{\mathrm{out}}\): mean \(0.20\) h\(^{-1}\), standard deviation \(0.02\) h\(^{-1}\).
The baseline steady-state biomarker level is:
Step 1: Recognize that the prediction is uncertain
Because both \(k_{\mathrm{in}}\) and \(k_{\mathrm{out}}\) are uncertain, \(B_{\mathrm{ss}}\) is also uncertain.
Step 2: Sample plausible parameter values
For example, one Monte Carlo draw might produce:
giving:
Step 3: Repeat the calculation
Thousands of plausible parameter pairs can be generated from the assumed joint parameter distribution. Each pair produces a corresponding steady-state prediction.
Step 4: Summarize the prediction distribution
The resulting collection of \(B_{\mathrm{ss}}\) values can be summarized using its median, selected quantiles, standard deviation, or other appropriate summaries.
The important point is that the original point estimate of \(50\) is only one prediction. UQ provides information about the range of predictions compatible with the specified parameter uncertainty.
17. How UQ Can Guide Experimental Design
Uncertainty quantification is not only an endpoint of QSP analysis. It can also help identify what experiments would be most informative.
Suppose a prediction is highly uncertain because two parameters are poorly identified. An experiment designed to estimate those parameters may substantially reduce uncertainty.
Conversely, if the parameters remain uncertain but the prediction is insensitive to them, collecting more data on those parameters may have limited value for the scientific question.
| Finding from UQ | Potential experimental implication |
|---|---|
| Prediction dominated by one poorly known parameter | Prioritize experiments that inform that parameter |
| Prediction sensitive to parameter interactions | Consider experiments that distinguish parameter combinations |
| Predictions differ across model structures | Seek measurements that discriminate between mechanisms |
| Prediction robust despite parameter uncertainty | Additional parameter estimation may provide limited predictive benefit |
| Prediction highly sensitive to baseline state | Improve characterization of the relevant biological population |
This creates an important connection between QSP modeling and experimental design: the most valuable new experiment is often the one that reduces uncertainty in a prediction that matters scientifically.
18. A Practical QSP Uncertainty Quantification Workflow
- Define the scientific prediction. Specify exactly what output or decision the uncertainty analysis is intended to address.
- Identify uncertainty sources. Separate parameter uncertainty, biological uncertainty, measurement error, structural uncertainty, and population variability.
- Evaluate identifiability. Determine which parameters and parameter combinations are supported by the available data.
- Specify uncertainty distributions. Use appropriate confidence-based, likelihood-based, bootstrap, Bayesian, or other representations.
- Preserve important parameter correlations. Avoid independent sampling when the estimation procedure indicates substantial dependence.
- Propagate uncertainty through the QSP model. Monte Carlo simulation, Bayesian posterior prediction, or other computational methods can be used.
- Perform sensitivity analysis. Determine which uncertain inputs contribute most strongly to the output uncertainty.
- Evaluate structural uncertainty. Compare scientifically plausible model structures when model assumptions are uncertain.
- Summarize prediction uncertainty. Report distributions, intervals, uncertainty bands, and scenario ranges in a way that matches the analysis.
- Assess robustness. Determine whether the scientific interpretation changes under reasonable alternative assumptions.
- Identify information gaps. Use the analysis to determine which additional experiments or measurements could most reduce decision-relevant uncertainty.
19. Common Interpretation Mistakes
Mistake 1: Treating a point estimate as known
A parameter estimate is generally an estimate, not an exact biological constant.
Mistake 2: Reporting only parameter uncertainty
A QSP analysis can have apparently precise parameters while still producing uncertain predictions because of nonlinear interactions, uncertain inputs, or structural assumptions.
Mistake 3: Ignoring parameter correlation
Independent sampling can generate unrealistic combinations of parameters when the original estimation procedure identified strong correlations.
Mistake 4: Treating structural uncertainty as parameter uncertainty
Narrowing the uncertainty around parameters within one model does not address uncertainty about whether another model structure may be more appropriate.
Mistake 5: Confusing variability with uncertainty
Variation between patients is not necessarily evidence that the model parameters are uncertain. A population model may intentionally represent real between-subject variability.
Mistake 6: Assuming a wide interval means the model is useless
A wide prediction interval can be scientifically informative. It may reveal precisely where additional data are needed.
20. What UQ Does Not Automatically Solve
Uncertainty quantification improves transparency about uncertainty, but it cannot compensate for every modeling limitation.
- Uncertainty distributions can themselves be uncertain. The chosen distribution may depend on limited evidence.
- Unmodeled mechanisms remain unmodeled. A simulation cannot quantify a mechanism that has not been represented.
- Structural uncertainty can be difficult to quantify. Competing models may not have defensible probabilities.
- Extrapolation can be substantially more uncertain than interpolation. Predictions far outside the experimental domain depend strongly on model assumptions.
- Computational cost can be substantial. Large QSP models may require thousands or millions of simulations for global analyses.
- Numerical uncertainty can exist. Approximate solvers, convergence problems, and stochastic algorithms can introduce computational uncertainty that should be distinguished from biological uncertainty.
21. Key Takeaways
- Uncertainty quantification describes how uncertainty in QSP parameters, inputs, observations, and assumptions affects model predictions.
- Parameter uncertainty and biological variability are different concepts and should not automatically be represented in the same way.
- Structural uncertainty concerns uncertainty about the model representation itself, not merely uncertainty about parameter values.
- Monte Carlo simulation is a common way to propagate uncertain parameters through a QSP model.
- Prediction uncertainty can be summarized using distributions, quantiles, intervals, and uncertainty bands.
- Parameter correlations matter because independently sampled parameters may create combinations that are inconsistent with the original estimation.
- Sensitivity analysis identifies influential parameters, while UQ quantifies how uncertainty in those parameters affects predictions.
- Global sensitivity methods can help determine which uncertain inputs contribute most to prediction uncertainty, including interaction effects.
- Identifiability is important: some individual parameters may be poorly determined even when particular model predictions are relatively robust.
- Bayesian methods provide one framework for representing prior information, updating uncertainty with data, and generating posterior predictive distributions.
- Scenario analysis can be useful when alternative biological mechanisms are plausible but assigning probabilities to them would not be justified.
- UQ can guide experimental design by identifying which measurements or mechanisms contribute most to decision-relevant uncertainty.
- A narrow uncertainty interval is meaningful only relative to the uncertainty sources that were actually represented.
- The goal of QSP UQ is not to eliminate uncertainty, but to make uncertainty explicit, traceable, and scientifically useful.
Where to Go Next
A natural progression is to study QSP parameter identifiability and structural versus practical identifiability in more detail. These topics explain why some QSP parameters can be estimated precisely, why others remain weakly informed, and why prediction uncertainty can behave differently from parameter uncertainty.
From there, useful next topics include QSP sensitivity analysis, global sensitivity methods, Bayesian QSP modeling, model calibration, virtual populations, and model qualification.