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A Transcritical Bifurcation at Infinity Governs Plaque Stability in a Lipid-Structured Macrophage Model of Atherosclerosis

This study employs bifurcation analysis and global sensitivity methods on a lipid-structured macrophage model to demonstrate that a transcritical bifurcation at infinity governs the transition from stable to unstable atherosclerotic plaques, revealing that macrophage proliferation, emigration, and efferocytosis are the dominant regulators of plaque dynamics near the critical threshold.

Original authors: Endes, E. A., PELEN, N. N.

Published 2026-09-09

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Endes, E. A., PELEN, N. N.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Inside the walls of our arteries, a quiet war is constantly being waged. It is not a battle of swords, but of cells and fats. Over time, fatty deposits known as plaques can build up, narrowing the passageways for blood. These plaques are not static lumps; they are living, breathing ecosystems populated by immune cells called macrophages. These cells act as the body's cleanup crew, swallowing up excess cholesterol and debris. However, when this cleanup process goes wrong, the plaque can become unstable, rupturing and causing heart attacks or strokes. The central mystery for doctors and scientists has long been understanding exactly when a stable, manageable plaque tips over into a dangerous, runaway state. For years, researchers have debated whether the local multiplication of these immune cells helps heal the plaque by clearing more fat, or if it makes things worse by swelling the plaque until it bursts.

A team of mathematicians and biologists has now used a sophisticated computer model to settle this question, revealing a precise tipping point that governs the fate of these arterial deposits. By translating the complex biological interactions into a set of rules that a computer can follow, they discovered that the stability of a plaque depends entirely on a delicate balance between how fast macrophages multiply and how fast they leave the area. The researchers found that as long as the rate at which these cells leave the plaque is high enough to keep up with their reproduction, the system remains stable. But if the reproduction rate creeps just a little too high, the plaque enters a new, chaotic phase where the number of cells and the amount of fat they carry grow without limit. This transition is not a gradual slide into danger, but a sudden, mathematical shift that explains why some plaques remain quiet for decades while others suddenly become lethal.

The study focuses on a specific type of mathematical model that tracks not just the number of cells, but also how much fat each individual cell is carrying. This level of detail is crucial because the danger in atherosclerosis comes from the accumulation of this fat, which eventually forms a soft, necrotic core that weakens the plaque's structure. The researchers took an existing model of this process and pushed it to its logical extreme, asking what happens when the number of macrophages becomes very large. In this scenario, they found that the system behaves like a two-speed machine. The processes involving cell death and the removal of dead cells happen very quickly, snapping into a new state almost instantly. In contrast, the growth of the living macrophage population and the buildup of fat happen much more slowly, acting as the slow-moving engine that drives the entire system.

This separation of speeds allowed the researchers to simplify the complex equations and look directly at the core mechanism driving the plaque's behavior. They proved that there is a specific critical threshold where the balance between cell multiplication and cell departure flips. Below this threshold, the plaque settles into a stable state where the number of cells and the fat content remain manageable. The system finds a natural equilibrium. However, the moment the rate of cell multiplication exceeds the rate at which cells leave the plaque by even a tiny amount, the system loses its ability to hold back. The stable state disappears, and the model predicts that the number of cells and the fat they carry will grow indefinitely. This is what the researchers call a "transcritical bifurcation at infinity," a technical way of describing a point where the stable solution vanishes and is replaced by an unbounded explosion of growth.

The findings clarify a long-standing biological paradox. For a long time, scientists were unsure if encouraging macrophages to multiply was a good or bad idea for heart health. Some thought more cells meant better cleanup, while others feared it meant a larger, more fragile plaque. This study shows that the answer depends entirely on the context of the balance. If the cells can leave the plaque as fast as they are born, their multiplication is harmless and part of a healthy, regulated system. But if the exit rate cannot keep pace with the birth rate, the very act of multiplying becomes the driver of destruction. The researchers identified that the rate at which cells leave the plaque is the key regulator. If this rate is high, the system can tolerate a higher rate of multiplication without collapsing. If the exit rate is low, even a small increase in multiplication can trigger the runaway growth.

To ensure their mathematical conclusions matched biological reality, the team ran thousands of computer simulations, testing how the system responded to changes in different factors. They confirmed that the rate of cell multiplication and the rate of cell departure are the dominant forces controlling the size of the plaque and the amount of fat it contains. Other factors, such as how efficiently the cells clean up dead neighbors, play a role in the details of the plaque's composition but do not change the fundamental tipping point. The simulations showed that when the balance is tipped, the system does not just grow a little larger; it enters a regime of uncontrolled expansion. This mirrors the clinical reality of advanced atherosclerosis, where a plaque suddenly becomes unstable and prone to rupture.

The study also revealed a surprising decoupling between the number of cells and the size of the dangerous fatty core. While the total number of cells is driven primarily by the balance of birth and departure, the size of the necrotic core—the soft, fatty center that causes the most damage—is heavily influenced by how well the living cells can clear away dead ones. This means that even if a plaque is growing in size, the specific threat of a rupture might be managed by improving the efficiency of the cleanup crew, provided the overall balance of cell numbers is not already broken. However, once the critical threshold is crossed and the system enters the unbounded growth phase, no amount of cleanup efficiency can stop the explosion of cell numbers.

This work provides a rigorous mathematical explanation for a phenomenon that was previously observed but not fully understood. It moves beyond simply describing what happens in a plaque to explaining why it happens and exactly when the danger begins. The researchers have identified a precise condition where the system shifts from a state of controlled inflammation to one of chronic, unresolved crisis. This insight suggests that therapeutic strategies aimed at stabilizing plaques should focus on maintaining the balance between cell production and cell removal. By ensuring that the rate at which macrophages leave the plaque keeps pace with their reproduction, it may be possible to keep the system in the stable, safe zone. The study does not offer a new drug or a specific treatment, but it offers a clear map of the terrain, showing exactly where the line between safety and disaster is drawn.

In the end, the story of the atherosclerotic plaque is a story of balance. It is a system that can remain stable for a long time, provided the forces of growth and removal are kept in check. The researchers have shown that this balance is fragile. A small shift in the rate of cell multiplication can push the entire system over the edge, leading to a state where the plaque grows without limit. This mathematical discovery offers a new perspective on why some plaques remain silent while others become deadly, pointing the way toward a deeper understanding of heart disease and the potential for interventions that target the fundamental dynamics of cell growth and removal.

Technical Summary: A Transcritical Bifurcation at Infinity Governs Plaque Stability in a Lipid-Structured Macrophage Model of Atherosclerosis

Problem Statement
Atherosclerotic plaque stability is critically influenced by macrophage dynamics, yet the specific conditions under which stable plaques transition to unstable, unbounded growth remain unclear. While macrophage proliferation is known to drive plaque development, its role is paradoxical: it may enhance lipid clearance or exacerbate necrotic core expansion. Previous work by Chambers et al. (2024b) proposed a lipid-structured continuum model incorporating macrophage proliferation via a non-local pantograph term. Their steady-state analysis suggested that proliferation could reduce necrotic core lipid content by distributing the lipid load. However, the dynamic pathways leading to these states, particularly the conditions under which stability is lost, were left unexplored. This paper addresses the gap in understanding the dynamical mechanisms governing the transition from stable, bounded plaque dynamics to unbounded macrophage accumulation.

Methodology
The authors perform a rigorous dynamical systems analysis on the dimensionless ordinary differential equation (ODE) subsystem derived from the lipid-structured partial integro-differential equation (PIDE) model of Chambers et al. (2024b). The analysis proceeds in three stages:

Equilibrium and Stability Analysis: The authors identify equilibrium points and compute the Jacobian matrix to analyze local stability. They investigate the eigenvalue trajectories as the macrophage proliferation parameter ( $\rho$ ) varies, specifically looking for the critical threshold where the system loses hyperbolicity.
Asymptotic Fast-Slow Decomposition: Recognizing that the system exhibits distinct time-scales in the asymptotic regime where the macrophage population becomes large ( $M \to \infty$ ), the authors introduce compactifying variables ( $u = 1/M$ , $a_m = A_M/M$ ). This transformation reveals a singular limit where apoptotic and necrotic variables ( $P, A_P, N$ ) relax on a fast time-scale relative to the macrophage population and lipid load.
Center Manifold Reduction and Bifurcation Proof: Utilizing Fenichel's invariant manifold theorem, the full five-dimensional system is reduced onto a two-dimensional slow invariant manifold near the singular limit ( $u \to 0^+$ ). The authors then apply center manifold theory to rigorously prove the nature of the bifurcation occurring at the critical threshold.
Global Sensitivity Analysis: To complement the analytical findings, the authors conduct a global sensitivity analysis using Partial Rank Correlation Coefficients (PRCC) with Latin Hypercube Sampling. This identifies the dominant parameters influencing system outputs ( $M, P, A_M, A_P, N$ ) across biologically relevant parameter ranges.

Key Contributions and Results

Identification of a Critical Threshold: The analysis identifies a critical proliferation-emigration threshold defined as $\rho_c = 1 + \gamma$ , where $\gamma$ is the macrophage emigration rate. Numerical evidence shows that the determinant of the Jacobian vanishes and a critical eigenvalue crosses zero at this threshold.
Transcritical Bifurcation at Infinity: The paper proves that the positive steady state loses stability through a transcritical bifurcation at infinity at $\rho = \rho_c$ . Unlike standard finite bifurcations, this transition involves an exchange of stability between the finite positive equilibrium branch and an equilibrium at infinity. As $\rho$ approaches $\rho_c$ from below, the finite equilibrium grows unbounded ( $M^* \to \infty$ ), effectively colliding with the boundary equilibrium at infinity.
Regulatory Decoupling: The sensitivity analysis and targeted parameter sweeps reveal a regulatory decoupling between macrophage accumulation and necrotic core growth. While macrophage proliferation ( $\rho$ ) and emigration ( $\gamma$ ) are the dominant regulators of the macrophage population size ( $M$ ) and live macrophage lipid content ( $A_M$ ), the efferocytosis rate ( $\eta$ ) primarily regulates apoptotic burden ( $P, A_P$ ) and necrotic core size ( $N$ ). Crucially, the bifurcation threshold $\rho_c$ is independent of $\eta$ ; efferocytosis can modulate the severity of the necrotic core within the stable regime but cannot prevent the unbounded growth transition once $\rho > \rho_c$ .
Fast-Slow Structure Validation: The study validates that in the large-macrophage limit, the system possesses a fast-slow structure where apoptotic and necrotic components rapidly adjust to quasi-steady states determined by the slower macrophage dynamics. This structure allows for the rigorous reduction of the system, facilitating the bifurcation proof.

Significance and Claims
The authors claim that this work provides a rigorous dynamical systems explanation for a transition previously identified but unexplored in the original study by Chambers et al. (2024b). By characterizing the loss of plaque stability as a transcritical bifurcation at infinity, the paper offers a precise mathematical criterion ( $\rho_c = 1 + \gamma$ ) for the onset of uncontrolled plaque growth.

The significance lies in resolving the paradox of macrophage proliferation: the effect of proliferation is not inherently pathological or protective but depends entirely on its balance with emigration. When the proliferation rate exceeds the combined loss rates (apoptosis and emigration), the system transitions to a regime of chronic, unresolved inflammation characterized by unbounded growth. The authors suggest that therapeutic strategies targeting the proliferation-emigration balance may be more effective for restoring stability than targeting efferocytosis alone, particularly in the asymptotic regime where the bifurcation occurs. Furthermore, the identification of this threshold provides a potential framework for investigating whether biological systems operate near such critical balances and for assessing patient-specific risks of plaque destabilization based on the proximity of proliferation and emigration rates to $\rho_c$ .

The paper concludes by noting that while the current model assumes linear proliferation, future work could incorporate density-dependent mechanisms (e.g., logistic growth) to bound the unbounded regime, and extend the framework to include multiple macrophage subpopulations, such as smooth muscle cell-derived macrophages, to further refine the understanding of plaque composition and stability.

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