A Stabilized Mixed Hybrid Finite Element Framework for Large-Deformation Poromechanics with Torsion: Application to Transmural Perfusion and Vascular Hemodynamics

The human heart is a pump that does more than simply squeeze blood; it twists, stretches, and reshapes itself with every beat. Inside the thick muscular wall of the left ventricle, a complex fluid dynamics problem plays out. The heart muscle is not a solid block of tissue but a sponge-like material, roughly fifteen percent filled with fluid that moves through tiny channels. As the heart fills with blood and then twists to eject it, this internal fluid is squeezed and pushed, creating pressure that can either help or hinder the flow of blood through the heart's own tiny vessels. If the pressure inside the muscle becomes too high, it can crush these delicate vessels, starving the heart tissue of oxygen. Understanding how the twisting motion of the heart interacts with this internal fluid flow is critical, yet traditional computer models often struggle to simulate it accurately, frequently producing confusing errors or "locking" up when the tissue is nearly impossible to compress.

A researcher at Eindhoven University of Technology has developed a new way to simulate this behavior, creating a mathematical framework that captures the large-scale twisting and stretching of the heart wall without falling into these common traps. By treating the heart muscle as a fluid-saturated sponge and using a specialized method that tracks both the movement of the solid tissue and the flow of the fluid separately, the study reveals how the heart's natural twisting motion acts as a protective mechanism. The simulations show that when the heart is allowed to twist freely during filling, it significantly reduces the stress on the inner wall fibers and lowers the internal pressure that threatens to collapse the blood vessels. Conversely, when the heart wall becomes abnormally thick due to high blood pressure, the fluid gets trapped, and the pressure builds up dangerously, a condition that standard computer models often fail to predict correctly.

The core of this work lies in fixing a long-standing problem in how scientists model soft, wet materials like heart muscle. In standard computer simulations, when a material is nearly incompressible—meaning it cannot be squished into a smaller volume—the math often breaks down, creating unrealistic spikes in pressure or jagged, oscillating results that do not reflect reality. This is known as volumetric locking. To solve this, the researcher formulated a "mixed hybrid" approach. Instead of trying to calculate everything at once, this method breaks the problem into smaller pieces, enforcing strict rules that ensure fluid flows continuously across the boundaries of each piece, just as it must in the real body. This technique allows the computer to solve for the movement of the tissue and the pressure at the edges of the simulation blocks separately, ensuring that the fluid flow is conserved exactly and that the results remain smooth and physically realistic, even under extreme deformation.

To test this new framework, the researcher first derived a precise mathematical solution for a thick-walled cylinder that is inflated and twisted, serving as a perfect benchmark. They then applied the method to a model of the left ventricle, a cylindrical section of heart muscle with dimensions matching human anatomy. The simulations explored three specific scenarios. First, they looked at how different levels of blood pressure inside the heart's own coronary vessels affect the wall's stiffness. They found that when the vessels are well-filled with blood, the internal fluid pressure rises, which actually stiffens the relaxed heart wall. This phenomenon, known as the garden-hose effect, means the heart requires more pressure to fill, limiting the volume of blood it can take in. The new model accurately predicted this shift, showing that high internal perfusion pressure reduces the solid stress in the muscle wall from ten kilopascals down to three kilopascals, effectively sharing the load between the fluid and the tissue.

The second scenario examined the vital role of the heart's twisting motion. In a healthy heart, muscle fibers are arranged in a spiral that runs from the inner surface to the outer surface, changing angle as it goes. When the heart fills, this spiral architecture causes the wall to twist. The simulations compared a heart that was allowed to twist freely against one that was held rigid. When the twist was prevented, the stress on the muscle fibers at the inner wall skyrocketed to 57.20 kilopascals, a dangerous concentration of force. However, when the model was allowed to twist naturally, that peak stress dropped by 45.5 percent to 31.20 kilopascals. This redistribution of force also helped fluid drain more easily through the wall, lowering the internal pore pressure from roughly 128 millimeters of mercury to 122 millimeters of mercury. The study demonstrates that the heart's twist is not just a byproduct of contraction but a necessary mechanism to relieve stress and maintain blood flow to the inner layers of the muscle.

The third and most critical case study addressed the heart under the strain of severe hypertension, where the wall thickens significantly to withstand high pumping pressures. In this scenario, the inner radius of the heart was modeled at 2.00 millimeters and the outer radius at 5.00 millimeters, with a cavity pressure of 180 millimeters of mercury. Under these conditions, the thickened wall traps fluid near the inner surface because the fluid has a longer distance to travel to escape. The new mixed hybrid model showed that this trapped fluid creates a sustained high-pressure zone that persists even after the initial loading. In contrast, when the researcher ran the same scenario using standard computer modeling techniques, the results were chaotic. The standard method produced spurious, non-physical oscillations in pressure near the inner wall and failed to capture the true stress distribution, resulting in a massive error rate of 12.75 percent. The new method, however, produced a smooth, monotonic pressure curve with an error of only 0.01 percent, correctly identifying that the thick wall combined with the twisting motion creates a mechanical obstacle that slows drainage and sustains high internal pressure.

These findings suggest that the heart's ability to twist and untwist is a fundamental part of its hydraulic health. When the heart twists freely, it homogenizes the stress across the wall and facilitates fluid drainage, protecting the microvessels from collapse. When the wall thickens or the twist is restricted, the fluid becomes trapped, leading to high pressures that can compromise blood flow. The study confirms that the new computational framework is capable of capturing these subtle but life-critical interactions, offering a reliable tool for understanding how mechanical forces and fluid flow are coupled in the heart. While the current work focuses on a simplified cylindrical geometry to prove the concept, the success of this method in handling large deformations and complex fluid-structure interactions points toward a future where such models can be applied to more complex, three-dimensional heart shapes to better understand heart failure and the effects of high blood pressure.