Computational Model Reveals How Vascular and Tissue Stiffness Regulate Coronary Blood Flow

A new multi-scale computational framework developed by biomedical engineers demonstrates that coronary blood flow velocity is primarily regulated by the mechanical stiffness of vascular smooth muscle cells and passive myocardial tissue rather than myocardial fiber stiffness. Coronary blood flow supplies essential nutrients and oxygen to the heart muscle, yet quantifying how interacting biophysical factors govern intramyocardial hemodynamics has historically proved difficult using conventional experimental tools. In living heart tissue, cyclic myocardial contraction, anisotropic tissue deformation, fluid-structure interactions, and dynamic extravascular tissue pressures simultaneously modulate microvascular dimensions. Pathological conditions like type 2 diabetes and metabolic syndrome alter these biomechanical variables by inducing vascular wall thickening through inward hypertrophic remodeling, reducing vascular smooth muscle cell stiffness, and increasing passive myocardial stiffness. Because clinical measurement modalities such as Doppler echocardiography and ultrasound localization microscopy cannot isolate individual cellular and mechanical contributors during disease progression, the research team constructed a simplified proof-of-concept finite-element model coupling cardiac electromechanics with microvascular fluid dynamics to evaluate these biophysical mechanisms independently and concurrently.

The computational domain represents a small proof-of-concept block of left ventricular myocardium measuring one cubic centimeter, integrating realistic transmural fiber architecture and vascular components. Myocardial fibers run parallel to the epicardial and endocardial planes, with their orientation angle varying linearly from minus sixty degrees at the epicardial boundary to positive sixty degrees at the endocardial boundary. Within this tissue block, the researchers embedded a deformable arteriole and a rigid venule, each modeled with an initial diameter of two hundred micrometers and an axial length of four millimeters. Deeply branching microcapillary networks cannot be directly discretized at organ scales due to prohibitive computational costs; therefore, the surrounding microvascular bed was represented as a continuous porous media domain governed by Darcy flow equations. Two distinct node configurations bridge the one-dimensional blood vessels and the three-dimensional porous myocardium: myocardial connection nodes enforce deformation continuity and apply contractile extravascular pressure to govern vessel caliber, whereas inlet and outlet connection nodes enforce pressure continuity and fluid mass exchange between the macro-vessels and the capillary porous matrix.

To capture cardiac electromechanics, the investigators solved modified Fitzhugh-Nagumo equations to track transmembrane electrical potential and cellular refractoriness following a cyclic excitation current pulse applied at the apical face of the block. Electrical conduction was defined using an anisotropic conductivity tensor with a four-to-two-to-one ratio along the fiber, sheet, and normal-to-sheet directions, with repolarization parameter gradients established across the myocardial wall. Electrical excitation triggered active myocardial stress generation through a delayed differential threshold formulation calibrated to replicate human ventricular pressure profiles. The resulting mechanical deformations were simulated by treating the myocardium as a fiber-reinforced, nearly incompressible hyperelastic solid governed by the Holzapfel-Ogden strain energy density formulation. This constitutive law combines isotropic matrix strain energy, volumetric compressibility, and directional fiber strain energy, with fiber strain resistance set to zero under compressive deformation. Solid motion and biphasic poroelastic equations were coupled so that local tissue volume changes directly altered porous fluid pressure and tissue density.

Microvascular hemodynamics within the embedded arteriole were solved using one-dimensional viscous flow equations that incorporate dynamic cross-sectional area changes, fluid density, and dynamic blood viscosity under Newtonian assumptions with Stokes friction resistance. Because classical linear Laplace tube laws generate unphysical wall collapse or extreme dilation under large contractile forces, the researchers implemented a non-linear sigmoidal constitutive relationship linking vessel radius to transluminal pressure. This transluminal pressure represents the difference between internal blood fluid pressure and external extravascular pressure generated during myocardial contraction, offset by steady-state baseline tissue pressure to ensure that the arteriole began each cardiac cycle at exactly two hundred micrometers. The arteriole was subjected to a constant upstream pressure boundary condition of twenty millimeters of mercury, reflecting the attenuated pulsatility found at this network depth, while downstream venular pressure was set to zero millimeters of mercury. Baseline arteriole wall thickness was established at twenty micrometers, matching physiological wall-to-lumen proportions of ten percent.

Simulations were conducted using COMSOL Multiphysics version 6.3 on a thirty-two-thread workstation utilizing a segregated backward differentiation formula solver. A mesh convergence study examining five spatial discretizations demonstrated that a finite-element mesh comprising two thousand five hundred elements and fourteen thousand degrees of freedom achieved peak diastolic velocity values within three percent of a fine thirty-five-thousand degree-of-freedom mesh while solving two to three times faster. Electrophysiological variables were solved with a fixed ten-millisecond timestep to capture rapid depolarization, while mechanical deformations and fluid equations were resolved using adaptive timesteps across four consecutive one-hertz cardiac cycles. Numerical results demonstrated realistic electro-mechanical propagation: transmembrane potential depolarized from minus eighty millivolts to positive twenty millivolts within fifty milliseconds, traveling from apex to base, while peak active stress reached three kilopascals at two hundred sixty milliseconds before dissipating. The anisotropic active stress produced characteristic myocardial twisting and peak first principal strains of approximately fifteen percent, which closely matches clinical four-chamber echocardiography strain recordings.

The baseline model successfully reproduced the characteristic morphology of human coronary blood flow velocity waveforms, exhibiting sharp deceleration during early systole, sustained flow suppression through peak ventricular contraction, rapid diastolic acceleration, and gradual late-diastolic decay. Quantitative waveform evaluation focused on seven key parameters: systolic duration, diastolic duration, diastolic acceleration, peak diastolic velocity, diastolic deceleration, systolic deceleration, and the total velocity-time integral. Under baseline parameters, the simulated peak diastolic velocity reached approximately twenty-six centimeters per second, comparing favorably with clinical adult Doppler echocardiography norms of nineteen plus or minus five centimeters per second. Examining local structural mechanics revealed that the proximal arteriole compressed rapidly during the first two hundred milliseconds of systole and underwent approximately one hundred micrometers of axial displacement. As active contractile stress subsided, elastic recoil of the vessel wall caused a transient rebound in lumen diameter beyond baseline dimensions, providing a major mechanical driver for diastolic flow acceleration.

Through systematic parameter variation studies, the authors investigated two functional categories: material properties consisting of vascular smooth muscle cell wall stiffness, passive myocardial stiffness, and myocardial fiber stiffness; and excitation-contraction and geometric properties comprising active stress magnitude, electrical conductivity along fibers, heart rate, and arteriole wall thickness. A normalized dimensionless sensitivity analysis showed that coronary flow parameters were at least ten times more sensitive to vascular smooth muscle cell stiffness and passive myocardial stiffness than to myocardial fiber stiffness. Specifically, reducing vascular smooth muscle cell stiffness led to a two-fold increase in peak diastolic velocity alongside deeper systolic flow suppression, while reducing passive myocardial stiffness produced a one-and-a-half-fold increase in peak diastolic velocity. Conversely, altering myocardial fiber stiffness modified peak diastolic velocity by less than ten percent, even when simulated at different transmural depths or when compressive fiber resistance was artificially incorporated into the strain energy formulation.

Analysis of electromechanical and geometric parameters further highlighted the physical forces governing coronary perfusion patterns. Increasing myocardial active stress magnitude caused greater systolic vessel compression, thereby suppressing systolic velocity while augmenting diastolic elastic recoil and peak velocity. Elevating electrical conductivity along myocardial fibers generated an approximate thirty percent rise in peak diastolic velocity with minimal impact on systolic flow. Elevating heart rate shortened diastolic phase duration three-fold and substantially reduced diastolic deceleration and the velocity-time integral, matching well-established clinical observations of tachycardia. Increasing arteriole wall thickness reduced both diastolic acceleration and peak velocity by approximately one-and-a-half-fold, mirroring the flow impairments seen in microvascular hypertrophy. Dimensionless sensitivity metrics confirmed that heartbeat frequency, wall thickness, and active stress were the dominant secondary modulators of coronary acceleration and total flow volume, whereas electrical conduction velocity exerted more limited overall control over waveform morphology.

To evaluate biomechanical interactions under pathological conditions such as type 2 diabetes, the investigators conducted concurrent parameter sweeps where vascular smooth muscle cell stiffness was varied from two to thirty-two kilopascals across three distinct passive myocardial stiffness environments, representing baseline tissue, five-fold elevated stiffness, and ten-fold elevated stiffness. Under healthy myocardial stiffness, decreasing vascular smooth muscle cell stiffness substantially enhanced peak diastolic flow and overall velocity-time integral, functioning as an effective compensatory mechanism to preserve tissue perfusion. However, when passive myocardial stiffness was elevated ten-fold, the regulatory influence of vascular smooth muscle cell compliance was markedly blunted, and the sensitivity of all waveform parameters dropped significantly. These computational predictions align closely with recent ex vivo experimental findings where pharmacological relaxation of vascular smooth muscle cells using Latrunculin B augmented coronary flow significantly more in healthy rodent hearts than in diabetic hearts with preexisting myocardial stiffening, underscoring how surrounding tissue mechanics constrain vascular compensation.

The authors identified several simplifying assumptions within their proof-of-concept formulation that provide clear directions for future biomechanical research. The current model examined an isolated cubic-centimeter myocardial slab subjected to idealized basal displacement constraints and non-pulsatile boundary pressures rather than anatomical full-ventricle geometries supplied by cyclic epicardial coronary pressures. In addition, blood flow was approximated using Stokes resistance with an assumed parabolic velocity profile, fluid bulk modulus was not dynamically coupled to vascular volume fractions, venular dimensions were held rigid, and active metabolic autoregulation mechanisms were omitted. Future extensions will incorporate automated vascular network synthesis algorithms, patient-specific bi-ventricular geometries, detailed cellular calcium handling dynamics, and metabolic feedback to scale these coupled one-dimensional and three-dimensional formulations to whole-heart clinical models. Nonetheless, this multi-physics approach provides an efficient computational framework demonstrating that cellular vascular mechanics and passive myocardial stiffness jointly dictate intramyocardial blood flow, offering vital mechanistic insights into microvascular dysfunction in metabolic disease.

Subject of Research: Multi-physics finite-element modeling of coronary microcirculation biomechanics and intramyocardial hemodynamics

Article Title: Development of a Simplified Multi-Physics Computational Model of the Coronary Microcirculation

Article References: Tomizawa, Y., McCallinhart, P. E., Ueyama, Y., Trask, A. J., & Ghadiali, S. N. (2026). Development of a Simplified Multi-Physics Computational Model of the Coronary Microcirculation. Annals of Biomedical Engineering. https://doi.org/10.1007/s10439-026-04421-1

Image Credits: AI Generated

DOI: 10.1007/s10439-026-04421-1

Keywords: Coronary Circulation, Microvascular Disease, Biomechanics, Multi-Physics Modeling, Finite Element Analysis, Poroelasticity, Myocardial Stiffness, Cardiovascular Dynamics, Development, Simplified, Multi-Physics, Computational