Global sensitivity analysis for cardiovascular models

Motivation, concepts, and intuition – Seminar at PoliMi 3–4 February 2026

Leif Rune Hellevik

NTNU and Politecnico di Milano (PoliMi)

Welcome & framing

Format

• Short conceptual introductions
• Interactive notebook demonstrations
• Discussion-oriented, not tool-heavy
• Slides: https://lrhgit.github.io/uqsa2025/

Welcome everyone, both students and faculty in Gliwice Poland, my postdoc Rahul in Trondheim, and to the group here at PoliMi (students, postdocs, and faculty), and in particular Francesco who is hosting me.

and I am Leif Rune Hellevik, professor in biomechanics at NTNU, Trondheim and currently at PoliMi at a sabbatical.

This seminar on GSA for cardiovascular models is meant to be practical and concept-driven: not a full course, not a software tutorial, but a guided exploration of how GSA helps us reason about uncertainty in models.

Emphasize: notebooks are used as a medium for explanation and exploration, not something participants are expected to run line-by-line during the seminar.

For you as attendants I encourage and hope for interaction and questions throughout the sessions.

Before we go any further, please open the seminar slides in your browser. https://lrhgit.github.io/uqsa2025/

That way you can follow links and jump to notebooks when we get to the interactive parts.

What we will do today

• Frame and motivate the need for GSA
• Revisit basic statistics
• Introduce variance-based SA
• Explore interactive notebooks

From a VUCA world to uncertainty-aware modelling

• VUCA (1990s → today): a world of Volatility, Uncertainty, Complexity, and Ambiguity where deterministic reasoning is insufficient
• Implication for modelling: robust decisions require both uncertainty quantification (UQ) and sensitivity analysis (SA)
• Models are essential tools for reasoning under uncertainty
  
• However, models are:
  • based on uncertain inputs
    
  • built on simplifying assumptions
    
  • often nonlinear and high-dimensional
    
• Uncertainty quantification (UQ) provides a systematic framework to represent, propagate, and interpret uncertainty in models

note to bullet 1 — vuca

VUCA was introduced in the early 1990s to describe the less predictable post–Cold War world. Today it captures key features of many engineering contexts: rapid change (volatility), imperfect information (uncertainty), interacting components (complexity), and multiple possible interpretations (ambiguity). In such settings, purely deterministic reasoning is often insufficient.

note to bullet 2 — implication for modelling

Models are essential in a VUCA world, but they are imperfect and depend on uncertain inputs and simplifying assumptions. Uncertainty quantification (UQ) tells us how uncertain model outputs are, while sensitivity analysis (SA) tells us which uncertainties matter most—making both crucial for robust decisions.

Models help us reason in this situation, but models are never perfect. They are based on assumptions, simplified descriptions of reality, and uncertain inputs. This means model results are always uncertain.

Uncertainty quantification (UQ) helps us describe and propagate this uncertainty. It tells us how uncertain the model output is.

But, as emphasized by Saltelli, this is not enough for decision-making. In practice, we also need to know which uncertainties matter most.

This is why sensitivity analysis is essential in a VUCA context: it helps us understand which inputs drive the uncertainty in the results, and where modelling effort, data collection, or attention should be focused.

From uncertainty-aware models to credible evidence

• Models increasingly support or replace experiments → in silico trials and virtual cohorts

• In silico trials aim to:

  • reduce animal testing
  • limit human exposure
  • explore infeasible or rare scenarios

• Key question: when is a computational result credible - and why should we trust it?

• Credibility requires understanding uncertainty

  • not all uncertainties matter equally

• Global sensitivity analysis (GSA) operationalises credibility

  • connects uncertainty to decision relevance
  • identifies dominant sources of risk
  • informs where validation and modelling effort matter

An important reason why UQSA analysis have become so important is the rise of in silico trials.

In many areas — especially medical devices and biomechanics — models increasingly support, or partially replace, physical experiments. This is driven by ethics, cost, time, and feasibility: we want to reduce animal testing, limit human exposure, and explore rare or infeasible scenarios.

Key question This raises the key question on this slide: when is a computational result credible — and why should we trust it? Here, credibility is used in the ASME sense. A result is credible when there is sufficient evidence that the model is adequate for its context of use. Trust is the consequence — we trust a result because we understand why it is credible.

Crucially, credibility is not about removing uncertainty everywhere.

It is about understanding which uncertainties matter for the decision, and which do not. This point is central in ASME V&V and in Aldieri’s work on credibility assessment.

UQ tells us how uncertain a model result is. GSA tells us why it is uncertain — by identifying the dominant sources of uncertainty and risk.

In this sense, GSA operationalises credibility: it links uncertainty to decision relevance and shows where evidence and modelling effort actually matter.

Digital twins need global sensitivity analysis (GSA)

• DT rely on models for monitoring, prediction, feedback and decision support

• They operate continuously under uncertainty

  • uncertain and evolving data
  • uncertain model parameters
  • changing system states

• Decisions must be made despite uncertainty

  • not all uncertainties can be reduced

• GSA supports credible digital twins

  • identifies decision-relevant uncertainties
  • assesses robustness to uncertainty
  • supports credibility for the intended context of use

• Without GSA, digital twins risk being precise but not reliable

  • apparent accuracy may hide sensitivity to decision-relevant uncertain inputs

Digital twins are a good example of why credibility under uncertainty matters.

A digital twin is a model that is used continuously to monitor a system, make predictions, and support decisions. Because of this, it always operates under uncertainty — in the data, in the model parameters, and because the real system itself may change over time.

At the same time, decisions still have to be made. We cannot remove all uncertainty, so the key question becomes: which uncertainties actually matter for the decisions we care about?

In the ASME sense, a digital twin is credible only if it is robust for its intended context of use — that is, if it is fit for purpose. Looking precise is not enough.

This is where global sensitivity analysis (GSA) comes in. GSA helps us identify the uncertainties that are decision-relevant and where the model is fragile. In this way, GSA supports credibility, not just numerical accuracy.

So, without GSA, a digital twin may look very precise but still be unreliable.

Transition: Digital twins are just one example — this issue applies to any complex model used for decision-making.

What problem sensitivity analysis solves?

• Models are used under uncertainty

  • Inputs are not known exactly

• Complexity defeats intuition

  • Nonlinearity and interactions matter

• Decisions require prioritisation

  • Not all uncertainties are equally important

• Sensitivity analysis = relevance

  • Identifies what really drives uncertainty

We’ve already touched on the first two points before.

We use models under uncertainty — inputs are never known exactly. And as models become nonlinear and high-dimensional, our intuition often fails. These points are important, but not very controversial.

The real motivation for sensitivity analysis is in the last two points.

In practice, decisions require prioritisation. We don’t have unlimited time, data, or resources, so we cannot reduce all uncertainties. We have to focus our effort where it actually matters.

That is why sensitivity analysis is essentially about relevance. It helps us identify which uncertainties truly drive the uncertainty in the model output, and which ones matter less for the decision at hand.

So, sensitivity analysis is not about explaining every detail of a model — it is about understanding what matters for decision-making.

Main objectives of sensitivity analysis

• Factor prioritisation
  Identify which uncertain inputs contribute most to output uncertainty
• Factor fixing (screening)
  Identify non-influential inputs that can be fixed without affecting results
• Understanding model behaviour
  Reveal nonlinear effects and interactions between inputs
• Supporting robust decisions
  Focus modelling, data collection, and calibration effort where it matters

Now that we know the problem, let me briefly summarise what sensitivity analysis is actually used for.

First, factor prioritisation. We want to find which uncertain inputs matter most for the output. These are the parameters that deserve better data, calibration, or closer attention.

Second, factor fixing, or screening. Here we look for inputs that have little influence. If an input is unimportant, we can fix it to a constant value and simplify the model without changing the results.

Third, understanding model behaviour. Many models are nonlinear and include interactions between inputs. Sensitivity analysis helps reveal these effects and gives insight into how the model really behaves.

Finally, supporting robust decisions. Instead of trying to improve everything, sensitivity analysis helps us focus modelling, data collection, and validation effort where it actually matters.

These objectives will guide the rest of the seminar — different sensitivity analysis methods serve different purposes, and there is no single “best” method for all problems.

From local to global sensitivity analysis

• Local (derivative-based) measures
• One-factor-at-a-time (OAT)
• Why they fail for nonlinear models

The key message of this slide is very simple:

Local sensitivity looks at the model in one point. Global sensitivity looks at the model over the whole input space.

Local sensitivity methods use derivatives or small changes around a single reference point. They can be useful when the model is almost linear and uncertainty is very small.

But in most real problems:

• parameters are uncertain over wide ranges
• models are nonlinear
• parameters interact with each other

In these cases, local methods can be misleading. They may say an input is unimportant just because the derivative is small at that one point.

Global sensitivity analysis takes uncertainty seriously. Inputs are varied over their full ranges, and we study how they affect the output variability.

So the important shift here is: we move from “What happens if I change one parameter a little bit?” to “Which uncertainties matter most for the overall behaviour of the model?”

Variance as a measure of importance

• Output variability as information
• Conditional expectation and variance
• Total variance decomposition

Variance is central in GSA because it measures uncertainty in the model output.

If the output variance is large, the prediction is uncertain. If it is small, the prediction is more stable. So variance gives us a clear way to quantify how uncertain our model results are.

The key question in sensitivity analysis is: How much of this output variance comes from each input?

This is why variance is so useful — it can be decomposed into contributions from individual inputs and from their interactions, which is a core idea in Saltelli’s work.

Conditional expectation and conditional variance help formalise this. If fixing one input greatly reduces the output variance, that input is important. If it makes little difference, the input is less important.

So variance is not just a statistical choice — it directly links uncertainty in the inputs to uncertainty in the output, and allows us to rank inputs by importance.

This provides the foundation for variance-based measures like Sobol indices, which we will look at next.

Sobol indices – idea, not formulas

• First-order effects
• Total effects
• Interaction effects

Sobol indices are the most widely used variance-based sensitivity measures.

The key point is not the formulas, but the idea behind them.

The first-order Sobol index tells us how much of the output variance is explained by one input alone. A large value means that input has a strong direct effect.

The total Sobol index measures the overall effect of an input — both its direct effect and all its interactions with other inputs. If the total index is much larger than the first-order index, interactions are important.

Sobol indices also let us quantify interaction effects, which one-at-a-time or purely local methods cannot do.

In short, Sobol indices answer the central question in sensitivity analysis: which inputs — alone or in combination — really drive the uncertainty in the model output?

Seminar structure

The seminar is organised around three main components:

• motivating the usefulness of Sobol sensitivity indices for model-based decision making,
• showing how Monte Carlo sampling and polynomial chaos expansion (PCE) can be used to compute them,
• demonstrating the methods on simple examples (benchmark functions and wall models), supported by interactive notebook-based exploration and discussion.

The detailed and up-to-date agenda is available here:

👉 https://lrhgit.github.io/uqsa2025/seminar/agenda.html

Notebooks and colab

Jupyter notebooks are interactive documents that combine

• short explanations (text and equations),
• executable code,
• figures and tables.

In this seminar, notebooks are used as a medium for explanation and exploration — not as a software tutorial.

Google Colab lets you run the notebooks in a browser, without local installation:

• 👉 open the notebook via the Colab-link
• or navigate to https://lrhgit.github.io/uqsa2025/

If you’re following along in the slides, click the link to the notebook index now…

How to use the notebooks in this seminar

To make a notebook your own in Colab:

• File → Save a copy in Drive
• add notes in text cells (your interpretation, reminders, questions),
• change parameters (including sliders) to explore “what if?” scenarios,
• run selected cells to reproduce key results.

You do not need to run everything line-by-line during the sessions.

The goal is to build intuition: how uncertainty propagates and why Sobol indices change.

Colab workflow (practical notes)

When opening a notebook in Colab:

• use File → Save a copy in Drive before making changes,
• if prompted, choose Runtime → Run all to initialise the notebook,
• re-run a cell if you change parameters or sliders above it.

If something breaks, simply reload the page and start from your saved copy.

These notebooks are meant to be exploratory and robust — not fragile.

What notebooks will show

• Scatterplots and Sobol indices Complementary ways to understand sensitivity

• How to compute Sobol indices with the Monte Carlo Method and Polynomial Chaos Expansions

• Simple linear model (Saltelli) Build intuition in the simplest case

• Benchmark nonlinear models G-function and maybe other model functions

• Applied example Wall model for blood flow simulation

The purpose of the notebooks is to show how the concepts we discussed work in practice.

We will mainly use scatterplots and Sobol sensitivity indices as two complementary tools. Scatterplots help us see input–output relationships directly, while Sobol indices give a quantitative measure of how important each input is for output uncertainty.

We start with a simple linear model, as used in the book by Saltelli. This is the simplest case and helps build intuition about variance and sensitivity.

Then we move to standard benchmark models: the G-function and the Ishigami function. These models are nonlinear and include strong interaction effects, so they clearly show why global sensitivity analysis is needed.

Finally, we look at an applied example: a wall model for blood flow simulation. This shows how the same ideas apply in a realistic engineering context.

In the notebooks, we will compute sensitivity indices using both Monte Carlo sampling and polynomial chaos expansion (PCE). Both methods estimate the same quantities: the output variance and the Sobol indices. The difference is efficiency, not interpretation.

The goal is not to teach software or implementation details. The goal is to build intuition about uncertainty propagation and about how sensitivity analysis supports modelling and decision-making.

Selected references

Foundations of global sensitivity analysis   Global Sensitivity Analysis: The Primer. Saltelli et al., Wiley, 2008.   Sensitivity estimates for nonlinear mathematical models. Sobol’, I. M., Math. Model. Comput. Exp., 1993.

Variance-based methods and practice   Variance-based sensitivity analysis of model output: Design and estimator for the total sensitivity index. Saltelli et al., Comput. Phys. Commun., 2010.

Credibility, uncertainty, and modelling practice   Why so many published sensitivity analyses are false. Saltelli et al., Environ. Model. Softw., 2019.   Model credibility assessment and uncertainty quantification in in silico trials. Aldieri et al. (ASME V&V).

General overview and terminology   Sensitivity analysis. Wikipedia — https://en.wikipedia.org/wiki/Sensitivity_analysis

Full reference list: Seminar references