The Shape of a Heartbeat: How Topology Is Revealing the Best Way to Rewire a Failing Heart
About 64 million people worldwide live with heart failure. For roughly one in three of them, the two ventricles — the heart's main pumping chambers — beat out of sync, a condition called dyssynchrony. The fix, cardiac resynchronization therapy (CRT), involves implanting a pacemaker with leads threaded to both ventricles, forcing them back into coordinated rhythm. It's one of cardiology's great success stories. And yet, up to 30% of patients who receive CRT don't improve. The device works. Something else is wrong.
The leading suspect has always been lead placement. Not just which ventricle, but where on the ventricle wall the electrical stimulus is delivered. Cardiologists have long suspected the answer matters enormously — but the data to prove it, and to explain why, has been stubbornly hard to extract from the noise of biological variation. A new study by Ferrà Marcús et al. (2026) takes a radically different mathematical approach to this problem — one borrowed not from medicine or statistics, but from abstract topology — and finds striking, statistically significant differences in how the heart responds to pacing depending on exactly where the lead sits.
The Science
The paper's central tool is the Mapper algorithm, a method from topological data analysis (TDA). To understand why it's useful here, you need a brief detour into what topology actually is.
Topology is the branch of mathematics concerned with the shape of data — not precise distances, but qualitative structure: what is connected to what, what loops exist, what clusters emerge. The Mapper algorithm takes a cloud of high-dimensional data points and compresses it into a simplified graph — a network of nodes and edges — that preserves the essential "shape" of the data without flattening everything into a single average. Think of it as a technique for making a rough but faithful map of a landscape: it won't tell you the exact height of every hill, but it will faithfully show you whether there are two mountains or one, whether they are connected by a ridge or separated by a valley.
This makes Mapper particularly powerful for biological data, which is rarely clean, rarely Gaussian, and rarely well-described by a single summary statistic. Standard statistical tests tell you whether two group means are different. Mapper tells you whether the entire shape of the distribution is different — whether the data sprawls loosely or clusters tightly, whether it forms one coherent blob or several disconnected islands.
The dataset comes from a previous experimental study on cardiac resynchronization in pigs. The animals were given pacing-induced non-ischemic cardiomyopathy — a form of heart failure caused by chronic abnormal pacing rather than blocked arteries, and one of the most common varieties in human patients. Hemodynamic measurements were taken across different pacing configurations: electrodes placed at the base, mid-wall, and apex of the left ventricle, and on both the epicardium (the outer surface of the heart) and the endocardium (the inner surface lining the chambers). Four key hemodynamic variables were tracked across these conditions.
The innovation of Ferrà Marcús et al. (2026) was not just to apply Mapper to this data, but to go further. Mapper has traditionally been a qualitative tool — it produces graphs that experts interpret visually. The researchers introduced three new quantitative indices to make those graphs statistically testable:
- Self-connectivity: how tightly interconnected the nodes of the Mapper graph are — a high score means data points cluster into a coherent, well-linked structure.
- Scattering: how spread out the graph is — a high score suggests high variability or fragmentation.
- Homogeneity: how uniformly the data is distributed across the graph's structure.
This trio of indices is what allows the team to move from "the graph looks different" to "the graph is different, with p < 0.01."
What They Found
The most striking result sits in a three-number comparison. When pacing was delivered to the basal region of the left ventricle — the upper portion, closest to where the major vessels enter and exit the heart — the self-connectivity index was 0.57. When pacing was delivered to the mid-wall region, the index dropped to 0.14. At the apex (the pointed bottom tip of the heart), it was 0.24. Both differences from the basal value were statistically significant at p < 0.01
Self-Connectivity Index by Pacing Region
Self-connectivity index values for left ventricular pacing at basal, mid-wall, and apical regions. Higher values indicate more coherent, clustered hemodynamic responses. Differences between basal and mid/apical are statistically significant (p < 0.01).
| Label | Value |
|---|---|
| Basal | 0.57 |
| Mid | 0.14 |
| Apical | 0.24 |
.
What does a self-connectivity score of 0.57 versus 0.14 actually mean in cardiac terms? A higher self-connectivity index means that the hemodynamic response to pacing is more coherent — the heart's mechanical behavior is clustering together in a consistent, structured way across the four variables being measured. Pacing the base appears to produce a response that is, in a topological sense, tighter and more unified. Pacing the mid-wall produces a response that is fragmented, dispersed, harder to characterize as a single coherent state.
This is biologically plausible in a satisfying way. The basal region of the left ventricle is where the bulk of the heart's mechanical work originates in a normally functioning organ. The conduction system — the electrical highway that coordinates contraction — is densely present near the base. Stimulating there may be closer to mimicking the heart's natural electrical architecture. Pacing the mid-wall or apex, further from the natural origin of the electrical wave, may introduce a kind of hemodynamic "noise" — producing improvement on average, but with more variation and less coherence.
The second major finding concerns the type of pacing, not just the location. When comparing endocardial stimulation (leads placed inside the chamber, touching the inner wall) versus epicardial stimulation (leads placed on the outer surface, the current standard in most CRT implants), endocardial pacing at lateral sites significantly amplified the contrast between basal and non-basal pacing
Topological Profile by Pacing Region
Comparison of the three novel topological indices — self-connectivity, scattering, and homogeneity — across basal, mid, and apical pacing sites. Basal pacing shows a distinctly different profile from mid and apical sites.
| Label | Value |
|---|---|
| Self-Connectivity | 0.57 |
| Mid Apical Contrast | 0.24 |
| Coherence (Basal) | 0.57 |
. In other words, endocardial stimulation makes the regional differences more pronounced — the advantage of the basal site becomes sharper, and the disadvantage of mid or apical sites becomes clearer.
This matters clinically because endocardial left ventricular pacing is a newer, more invasive approach that requires threading a lead through the heart's septum. It has been proposed as an option for patients who don't respond to conventional CRT — the so-called "non-responders." If endocardial pacing is more sensitive to lead location, that cuts both ways: positioned well (at the lateral-basal wall), it may be more effective; positioned poorly, it may be worse. Topology, in this reading, is giving us a more precise map of the stakes.
The researchers also examined the scattering and homogeneity indices, which together paint a complementary picture. High scattering in the mid and apical groups further confirms that pacing from those sites produces more variable hemodynamic outcomes — the heart's response is not just weaker on average, it's more unpredictable, more spread across different physiological states. High homogeneity, by contrast, characterized the basal pacing data, consistent with the interpretation that stimulating closer to the natural pacemaker region keeps the heart's mechanics more uniformly organized.
Self-Connectivity: Basal vs. Mid vs. Apical (p-values)
Statistical significance of differences in self-connectivity index between basal pacing and the other two regions. Both comparisons reach p < 0.01.
| Label | Value |
|---|---|
| Basal vs. Mid | 0.57 |
| Basal vs. Apical | 0.57 |
Why This Changes Things
To appreciate the significance of these results, it helps to understand the current state of CRT optimization. Standard clinical practice places the left ventricular lead on the epicardium, typically via a small side branch of the coronary veins — wherever the coronary anatomy allows, rather than wherever physiology demands. It's a bit like installing a traffic signal wherever there's a convenient wall to mount it, rather than where cars actually need to be guided. Imaging-guided lead placement — using echocardiography or MRI to identify the latest-contracting segment — has improved outcomes, but the "non-responder" problem persists.
What makes the Mapper approach genuinely novel is that it doesn't just compare averages. It compares the structure of the hemodynamic response — its shape, its coherence, its internal consistency. Two pacing sites might produce the same mean cardiac output while differing enormously in the variability and reliability of that output. Standard statistics would call them equivalent. Topology would not.
This distinction has real consequences for clinical trials. If regional differences in CRT response are as structure-dependent as these results suggest, then trials that randomize patients to "basal vs. non-basal" pacing without accounting for the shape of individual responses may be washing out real signals in averaged noise. The Mapper framework offers a way to detect those signals.
There is also a deeper methodological contribution here. The three new indices — self-connectivity, scattering, and homogeneity — transform Mapper from a visualization tool into a hypothesis-testing framework. That's not a minor upgrade. The traditional knock on TDA in biomedical research has been that it produces beautiful pictures that are hard to submit to a journal's statistics reviewer. Ferrà Marcús et al. (2026) have taken a meaningful step toward bridging that gap.
The swine model is itself worth noting. Pigs are the gold standard for cardiac electrophysiology research — their heart size, coronary anatomy, and electrophysiological properties closely mirror those of humans. Non-ischemic cardiomyopathy, the specific condition modeled here, is caused by chronic dyssynchronous pacing rather than a heart attack, and is directly analogous to the human condition that CRT was designed to treat. The biological fidelity of the model strengthens confidence that these topological signals reflect genuine cardiac physiology, not experimental artifact.
What's Next
The study is, by design, a proof of concept. The sample comes from a pre-existing experimental dataset, not a prospectively designed trial. The pig cohort is not large. And while the Mapper results are statistically significant and biologically coherent, they have not yet been validated in human patients — the necessary and much harder next step.
Several open questions follow naturally from the findings. Do individual anatomical differences between animals (or patients) modulate the topological signature of pacing response? The heart is not a standard shape; the geometry of one patient's left ventricle differs from another's, and those differences may interact with pacing site in ways the current study cannot fully resolve. Future work might combine Mapper analysis with cardiac imaging to build individualized topological maps of pacing response before and after CRT implantation.
There is also the question of which of the four hemodynamic variables is doing the most work in driving the topological differences. The paper treats the four variables together as a joint distribution — a reasonable starting point — but a variable-by-variable decomposition might reveal which hemodynamic dimension is most sensitive to pacing location, and potentially guide the choice of what to measure in clinical monitoring.
The endocardial-versus-epicardial finding is particularly ripe for follow-up. The current result — that endocardial lateral pacing sharpens regional contrasts — is provocative, but the mechanism is unclear. Is this a property of the electrical wavefront geometry? The local myocardial architecture? The proximity to the conduction system? Answering these questions will likely require electro-anatomical mapping combined with topological analysis, a technically demanding but tractable project.
Perhaps most importantly, the new quantitative indices themselves deserve further development and validation. Self-connectivity, scattering, and homogeneity are conceptually compelling, but they are new. Their sensitivity and specificity as markers of hemodynamic coherence need to be tested across more datasets, more conditions, and more species before they can anchor clinical decisions. That validation work is the kind of collaborative, multi-institution effort that takes time — but the foundation laid here is methodologically solid.
What Ferrà Marcús et al. (2026) have demonstrated, ultimately, is that the shape of data carries information that its center does not. In a field where the difference between a responding and a non-responding patient can mean years of better life or continued decline, that is not a trivial claim. Topology, the mathematics of connected things, may be precisely the tool that a poorly understood connectivity problem — a heart whose chambers have forgotten how to beat together — has been waiting for.