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A mathematical model for irreversible damage of the collagen scaffold in the myocardium

This paper proposes and validates a variational, quasi-static mathematical model for irreversible collagen damage in anisotropic myocardial tissue, demonstrating that post-infarction load redistribution within the left ventricle drives passive, dissipative damage consistent with experimental observations.

Original authors: Irena Radišić, Francesco Regazzoni, Luca Dede', Alfio Quarteroni

Published 2026-09-15

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CC BY 4.0

Original authors: Irena Radišić, Francesco Regazzoni, Luca Dede', Alfio Quarteroni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The human heart is a tireless pump, but its strength relies on more than just the muscle cells that contract to push blood. These cells are embedded in a rigid, intricate scaffolding made of collagen, a tough protein that holds the tissue together and gives it shape. Think of this collagen network as the steel rebar inside a concrete building; without it, the structure would collapse under pressure. When a heart attack occurs, blood flow to a section of the heart is cut off, causing the muscle cells in that area to die and stop working. In the days and weeks that follow, the heart undergoes a complex transformation known as remodeling. The damaged area often thins out and bulges, stretching the remaining healthy tissue. Scientists have long suspected that this stretching causes the collagen scaffold itself to break down, but pinning down exactly how and when this structural failure happens has been difficult to observe directly inside a living organ.

A team of researchers at Politecnico di Milano in Italy has developed a new mathematical model to visualize this hidden process of structural decay. Instead of trying to measure the damage in a living patient, they built a virtual simulation of the heart's mechanics, focusing specifically on the collagen fibers that separate layers of muscle cells. Their work suggests that the very act of the heart trying to pump blood after a heart attack creates a specific kind of stress that irreversibly tears apart the collagen in the damaged zone. By simulating the heart's behavior over time, they found that this damage is not just a side effect of cell death, but a direct mechanical consequence of the extra load placed on the weakened tissue.

The researchers began by creating a simplified digital model of heart tissue, shaped like a flat block, to test how their new equations worked under controlled conditions. They applied a slow, steady pressure to the surface of this virtual block, mimicking the force of an indenter pushing into the tissue. In this test, they discovered that the direction of the collagen fibers mattered immensely. When the fibers were oriented in a way that allowed the layers of tissue to slide past one another easily, the pressure caused the collagen to degrade rapidly. However, when the fibers were aligned differently, the tissue resisted the pressure, and the damage remained minimal. This initial test confirmed that the model could accurately capture how the orientation of the internal structure influences the point at which the material begins to fail.

With the model validated on a simple block, the team applied it to a realistic, three-dimensional digital reconstruction of a human left ventricle, the main pumping chamber of the heart. They simulated a heart attack by turning off the active contraction in a specific region of the ventricle, representing the loss of muscle function. In a healthy heart, the muscle cells contract in unison to squeeze blood out. When a portion of the muscle stops working, the remaining healthy tissue must work harder to compensate, and the dead zone begins to bulge outward during the squeeze. The researchers watched how this change in shape affected the collagen scaffold.

The simulation revealed that the damage to the collagen scaffold begins almost immediately after the loss of contraction. As the heart beats, the infarcted area stretches and bulges, placing a heavy load on the collagen fibers in the cleavage planes—the thin layers of tissue that separate the muscle sheets. The model showed that this stretching causes the collagen to degrade in a way that is permanent and irreversible. The damage is most severe during the systolic phase, the moment when the heart contracts and the pressure inside is highest. By the time the heart relaxes, the damage has already occurred, and the scaffold does not heal itself. The simulation showed that this degradation is concentrated precisely in the area where the heart attack happened, matching what has been observed in laboratory experiments on animal models.

The study also looked at how this structural damage affects the heart's overall ability to pump blood. The researchers compared three scenarios: a healthy heart, a heart with a heart attack but no structural damage to the collagen, and a heart with both the heart attack and the collagen damage. They found that the loss of muscle contraction alone caused a significant drop in the heart's efficiency, measured by how much blood it could pump with each beat. The addition of collagen damage made the situation slightly worse, causing the heart to stretch a bit more and pump slightly less effectively. However, the simulation suggested that the primary cause of the heart's failure is the loss of the muscle cells themselves, rather than the subsequent breakdown of the collagen. The damage to the scaffold appears to be a contributing factor that exacerbates the problem, rather than the sole driver of the heart's decline.

This work offers a new way to understand the mechanical aftermath of a heart attack. Previous theories often focused on biochemical signals or the gradual loss of mass as the main reasons for the heart's changing shape. This model, however, demonstrates that purely mechanical forces—the extra stretching caused by the healthy tissue compensating for the dead tissue—are sufficient to break down the collagen structure. The researchers noted that their model is a simulation, and while it aligns with experimental observations, it relies on specific assumptions about how the tissue degrades. They also pointed out that the model currently focuses on the early stages of damage and does not yet account for all the complex biological processes that occur over weeks or months.

Despite these limitations, the findings provide a clear mechanical explanation for why the heart wall thins and expands after an infarction. The study suggests that the redistribution of load within the heart creates a specific type of stress that the collagen scaffold cannot withstand, leading to its irreversible failure. This insight could help researchers develop better strategies to protect the heart's structure in the critical days following a heart attack, potentially by finding ways to reduce the mechanical stress on the damaged area before the scaffold begins to tear. The work stands as a bridge between the abstract world of mathematical modeling and the physical reality of heart disease, showing how the laws of physics govern the fate of biological tissue.

Technical Summary: A Mathematical Model for Irreversible Damage of the Collagen Scaffold in the Myocardium

Problem Statement
Following a myocardial infarction, the left ventricle undergoes structural and functional changes known as post-infarction remodelling. A critical aspect of this process is "infarct expansion," where the infarcted wall thins and expands, often attributed to collagen degradation and cell slippage. While biochemical mechanisms (e.g., matrix metalloproteinase activation) are known to drive collagen degradation, the mechanical drivers of this damage remain less understood. Existing mathematical approaches to tissue remodelling, such as the theory of evolving reference configurations (multiplicative growth) and constrained mixture theory, often rely on phenomenological evolution laws and fail to explicitly capture constitutive-level changes like material softening due to damage. Furthermore, these methods often struggle to distinguish between mass loss and structural rearrangement. This work addresses the need for a model that explicitly describes the irreversible, dissipative damage of the collagen scaffold in the myocardium's cleavage planes as a consequence of increased mechanical loads following a loss of contractility.

Methodology
The authors propose a dissipative, irreversible damage model for anisotropic media undergoing large deformations, derived from variational principles.

Variational Framework: The model is grounded in the minimization of a total mechanical energy functional. The system seeks configurations that minimize energy subject to displacement constraints and an irreversibility constraint on the damage variable.
Damage Variable: A scalar field $\alpha \in [0, 1]$ represents the damage state, where $\alpha=0$ is undamaged and $\alpha=1$ is fully degraded. The stored energy density $\psi_d$ is degraded by a function $g(\alpha)$ , leading to material softening.
Irreversibility: To ensure damage is irreversible ( $\alpha_n \geq \alpha_{n-1}$ ), the model employs a history variable $\xi_n$ , defined as the maximum of the dissipated energy density $\psi_{diss}$ over the loading history. This replaces the instantaneous dissipated energy in the damage evolution equation, guaranteeing monotonic damage growth.
Constitutive Modeling:
- Passive Mechanics: The myocardium is modeled as an orthotropic, quasi-incompressible hyperelastic material using a modified Holzapfel-Ogden strain energy density. This captures the distinct mechanical behavior along fiber ( $\hat{f}$ ), sheet ( $\hat{s}$ ), and sheet-normal ( $\hat{n}$ ) directions.
- Damage Localization: Damage is specifically modeled to occur in the collagen fibers within the cleavage planes (oriented along $\hat{n}$ ). The dissipated energy term $\psi_{diss}$ is isolated to the $\hat{n}$ -direction invariant ( $I_{4n}$ ).
- Anisotropic Diffusion: A damage propagation tensor $K$ is introduced to reflect that damage propagates more easily along the cleavage planes than across them.
- Active Mechanics: Cardiac contraction is modeled via an active stress approach, where active tension is modulated by an ischemic function $f_{isch}$ , representing the loss of contractility in the infarcted region.
Numerical Scheme: The coupled electromechanics-damage problem is solved using a staggered, loosely coupled finite element scheme. The displacement field is solved via elastodynamics, while the damage field is solved via a linear reaction-diffusion equation, utilizing a history variable to enforce irreversibility.

Key Contributions

Variational Damage Formulation: The paper derives a quasi-static differential model for irreversible damage evolution starting from energy minimization principles, contrasting with phenomenological growth theories.
Explicit Collagen Scaffold Damage: Unlike previous models where softening is a byproduct of mass loss or growth tensors, this model explicitly links damage to the degradation of the collagen scaffold in the cleavage planes, allowing for material softening without explicit mass loss.
Coupling with Electromechanics: The damage model is successfully coupled with a full left ventricular electromechanics simulation to study the onset of damage following an acute infarction.
First Application to Post-Infarction Remodelling: To the authors' knowledge, this is the first work to describe post-myocardial infarction remodelling within a damage mechanics framework.

Results
The model was applied to two test cases:

Indentation of a Passive Slab: A rectangular slab was subjected to an indenter-like load. Results showed that damage magnitude and evolution are highly dependent on fiber orientation. Damage occurred primarily where the load induced shearing of the sheetlets relative to one another. The model successfully demonstrated that damage leads to increased deformation under the same load during unloading, indicating material softening.
Acute Infarction in a Left Ventricle: The model simulated a left ventricle with a localized loss of contractility.
- Damage Onset: Damage evolved within the infarcted region, primarily during systole when the infarcted area bulged and experienced increased stretch due to the lack of active contraction.
- Spatial Distribution: The damage was concentrated in the infarcted area, consistent with experimental observations of collagen degradation in that region.
- Functional Impact: The inclusion of damage led to a slight increase in end-diastolic volume and a marginal further reduction in ejection fraction compared to a model with only contractility loss. However, the authors note that the damage itself had a relatively small impact on the pressure-volume loop shape compared to the loss of contractility. In healthy tissue, a reduction in stiffness in the $\hat{n}$ -direction (simulating damage) slightly improved function, whereas in the infarcted tissue, it contributed to a slight worsening.

Significance and Claims
The paper claims that passive, load-dependent dissipative phenomena are implicated in post-infarction remodelling. The results suggest that the redistribution of load following a cardiac infarction can produce damage to the collagen scaffold in the infarcted area consistent with experimental findings, even without invoking complex biochemical signaling for the initial damage onset. The model provides a mechanical explanation for the "cell slippage" and wall thinning observed in infarct expansion, attributing it to the degradation of the collagen scaffold rather than solely to mass loss.

The authors remain modest regarding the magnitude of the effect, noting that in their simulations, the damage field remained relatively low and that the primary driver of functional deterioration was the loss of contractility. They acknowledge limitations, such as the use of a scalar damage variable (rather than a tensorial one) and the restriction of damage to the cleavage planes. Future work is proposed to explore inelastic slippage of sheetlets and the inclusion of more complex dissipation functions to better capture the deterioration of cardiac function.

Technical Summary: A Mathematical Model for Irreversible Damage of the Collagen Scaffold in the Myocardium

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