The review’s central argument is that venous valve biomechanics operates as a strict dependency pipeline, and that a weakness at any stage corrupts everything downstream. Biaxial mechanical testing feeds extracellular matrix characterization; matrix architecture explains mechanical anisotropy; anisotropy determines which constitutive model form is appropriate; and model parameters set the material boundary conditions for fluid-structure interaction, or FSI, simulation. Before 2017, not a single experimentally measured material parameter for venous valve tissue existed. Every computational model of venous valve function published up to that point borrowed linear isotropic stiffness values from arterial tissue or material constants derived from aortic or mitral valve leaflets, structures whose extracellular matrix architecture and loading environments differ from venous valves in ways that no simple scaling factor can correct. The review quantifies just how wrong those borrowed numbers were: fitted constitutive coefficients for venous valve tissue differ from cardiac valve values by factors of 5 to 100 across parameters, meaning the parameter spaces do not even overlap in a way that permits extrapolation.

The experimental foundation of the review rests on biaxial testing, a technique that applies simultaneous tensile loads along two orthogonal axes and captures the coupled mechanical response that uniaxial testing cannot resolve. Venous valve leaflets experience exactly this kind of biaxial loading in vivo: they billow open during forward flow and coapt under retrograde pressure, with the ratio of circumferential to radial stress varying continuously through the cardiac-phasic venous pressure cycle. The only uniaxial study of human tissue, performed in 1985 on femoral vein valve leaflets, established a breaking stress of roughly 11.5 megapascals, about twice that of the adjacent sinus wall, but could say nothing about the coupled deformation behavior that governs leaflet kinematics. Kaul and Huang’s group applied force-controlled biaxial protocols at multiple circumferential-to-radial loading ratios to bovine jugular and saphenous valve leaflets, producing the only systematically characterized nonlinear anisotropic material parameters for venous tissue published to date.

The results revealed a tissue that is strongly anisotropic and highly nonlinear. Jugular valve leaflets reach a peak circumferential tangent modulus of 27 plus or minus 9 megapascals under equibiaxial stretching, versus 9 plus or minus 6 megapascals radially, an anisotropy ratio of approximately 3 to 1 driven by preferential circumferential alignment of collagen fibers. Saphenous leaflets, the tissue directly implicated in lower-limb CVI, are stiffer still, with circumferential peak tangent modulus rising from 62.7 plus or minus 34.9 megapascals proximally to 92.1 plus or minus 27.9 megapascals distally. Strikingly, the two valve types show opposite proximal-to-distal stiffness gradients that track opposing collagen concentration gradients, a crossover that establishes jugular and saphenous valves as mechanically distinct tissues rather than interchangeable surrogates. For comparison, aortic valve leaflets exhibit circumferential moduli of 10 to 15 megapascals with an anisotropy ratio near 10 to 1, ruling out direct cross-tissue material substitution. Intra-valve variability of 30 to 40 percent in tangent modulus further means that single-leaflet sampling systematically underestimates the population-level mechanical range.

Why does venous valve tissue behave this way? The answer lies in its extracellular matrix architecture, which the review treats not as incidental background but as the mechanistic explanation of mechanical behavior. Imaging reveals a clear bi-layer organization: on the parietal, wall-facing side, collagen fibers align preferentially along the circumferential axis with angular deviations of less than 10 degrees in proximal jugular leaflets, resisting the tensile stress that peaks at full valve closure. On the luminal, blood-facing side, a crosslinked radial mesh of elastin dominates, providing the compliance that lets the leaflet billow toward coaptation without tearing. Collagen crimp, the sinusoidal undulation of unstressed fiber bundles, was quantified by second-harmonic generation microscopy at a wavelength of 38.46 plus or minus 8.06 micrometers and an amplitude of 4.51 plus or minus 1.65 micrometers, corresponding to a fiber recruitment threshold of approximately 12 percent true strain. Because physiological circumferential strains during valve closure are estimated at 10 to 20 percent, a substantial fraction of the leaflet’s working range sits precisely within this crimp-recruitment transition, the region where matrix degradation most severely compromises function.

This architecture also explains how disease destroys the valve. Elevated venous pressure triggers an inflammatory cascade that upregulates matrix metalloproteinases, particularly MMP-2 and MMP-9, which selectively degrade fibrillar collagen and elastin. CVI specimens show collagen disorganization and loss of the preferential circumferential alignment that generates the healthy 3-to-1 anisotropy ratio. The consequence is not merely reduced stiffness: the leaflet converts from an anisotropic load-bearing structure into a mechanically isotropic, globally compliant sheet, impairing the directed coaptation force needed for competent closure independent of any change in total collagen content. The authors are careful to note that this interpretation should be read as a plausible proximate mechanical contributor rather than a definitively established causal chain, since direct evidence in human venous valve tissue remains limited, and reported elastin changes in CVI specimens remain contested, likely reflecting differences in disease stage and assay methodology.

On the modeling side, the review identifies a critical failure mode in current constitutive models. The May-Newman and Yin exponential form, fitted to jugular venous data with R-squared values of 0.90 to 0.97 under equibiaxial and near-equibiaxial loading, breaks down systematically under asymmetric conditions. At a 2-to-1 circumferential-to-radial force ratio, the model overpredicts circumferential stiffness and underpredicts radial stiffness, an off-axis prediction error that does not diminish with additional parameter fitting. The mechanism traces to collagen fiber reorientation under off-axis loading, which exponential formulations cannot capture because they lack a structural coupling term to redistribute load across fiber families. The error is most consequential precisely where it matters most: the 2-to-1 ratio at which the model fails coincides with the loading state that dominates physiological valve closure at peak hemodynamic load. The Holzapfel-Gasser-Ogden framework, with its angular fiber dispersion parameter, and the structural constitutive approach of Sacks, which integrates measured fiber orientation distributions directly into the strain energy function, offer principled resolutions, but neither has yet been applied to venous valve tissue.

Three unresolved gaps emerge from the synthesis, each experimentally tractable with existing techniques. First, the off-axis prediction error requires expanded biaxial testing at asymmetric ratios and, ideally, co-registered confocal imaging and mechanical testing on the same specimens, a pairing not yet achieved for venous valves. Second, no failure criterion or damage evolution law exists for venous valve tissue; all biaxial studies to date have operated below the failure threshold, conservatively estimated at 60 to 70 percent true strain, leaving bioprosthetic durability computationally unassessable. Third, no validated material parameters have been published for bioprosthetic candidate materials such as decellularized pericardium, electrospun polyurethane, or silk fibroin scaffolds, even though tissue-engineered venous valves have reached preclinical testing in ovine and porcine models. An isotropic scaffold, the review argues, is mechanically unsuitable as a native surrogate regardless of geometric optimization, because the anisotropy mismatch alone produces divergent closure kinematics and shear distributions.

Perhaps the most sobering caveat is one of species translation: every biaxial dataset currently available derives from bovine tissue, while the clinical target is human. Studies of aortic valves have shown substantial interspecies divergence across humans, pigs, and bovines, and there is no principled reason to assume venous valves behave differently. Human venous valve biaxial characterization remains, in the authors’ framing, the single most consequential missing experiment in the pipeline. The review also flags uncharacterized residual stress, untested viscoelastic effects at physiological loading rates, and an identifiability problem in which multiple parameter sets fit equibiaxial data equally well yet diverge sharply under off-axis prediction. The path forward, the authors conclude, requires no new theoretical frameworks, only disciplined application of established biaxial testing and structural imaging methods to larger and more anatomically diverse specimen sets, ultimately including human tissue. Until then, patient-specific simulation of venous valve function, however sophisticated the software, remains built on a foundation that is only now beginning to be measured.

Subject of Research: Biomechanics, extracellular matrix architecture, and constitutive modeling of venous valve tissue in chronic venous insufficiency

Article Title: From Tissue to Simulation: A Review of Venous Valve Biomechanics, Extracellular Matrix Architecture, and Constitutive Modeling in Chronic Venous Insufficiency

Article References: Kaul, N., & Huang, H.-Y. S. (2026). From Tissue to Simulation: A Review of Venous Valve Biomechanics, Extracellular Matrix Architecture, and Constitutive Modeling in Chronic Venous Insufficiency. Annals of Biomedical Engineering. https://doi.org/10.1007/s10439-026-04412-2

Image Credits: AI Generated

DOI: 10.1007/s10439-026-04412-2

Keywords: chronic venous insufficiency, venous valves, biaxial mechanical testing, extracellular matrix, collagen crimp, elastin, constitutive modeling, fluid-structure interaction, soft tissue biomechanics, tissue-engineered venous valves, matrix metalloproteinases, hyperelasticity

The research, conducted by Hadi Nosrati and Mehdi Shafieian of Amirkabir University of Technology in Tehran together with Kurosh Darvish of Temple University in Philadelphia, set out to answer a deceptively simple question: does the loading history of brain tissue change its subsequent mechanical response? To find out, the team performed multistage simple shear experiments on fourteen bovine brain specimens. Bovine tissue is a widely used surrogate in brain biomechanics because it is readily available, structurally similar in many respects to human brain matter, and can be obtained as waste material from slaughterhouses without any animals being euthanized specifically for research purposes.

The experimental design was deliberately more sophisticated than the single-stage loading protocols that dominate the literature. Each specimen was subjected to successive loading stages combining ramp-and-hold tests, in which the tissue is sheared to a target strain and held there while stress relaxes, and cyclic loading, in which the tissue is sheared back and forth repeatedly. By varying the order of these protocols, the strain amplitudes, and the loading modes, the researchers could probe whether the tissue’s response depended on the path it had taken through its own mechanical history. All tests were conducted within a reported sub-damaging loading range, meaning the deformations were gentle enough to avoid gross structural failure while still revealing how the material evolves under repeated loading.

To quantify the mechanical behavior, the team fitted their data to a viscoelastic–hyperelastic constitutive framework, extracting a shear modulus and a set of Prony-series parameters that describe how the tissue’s stress relaxes over time. In such models, the hyperelastic component captures the instantaneous, rubber-like stiffness of the tissue, while the viscoelastic terms, here denoted g1 through g3 and g-infinity, describe the time-dependent dissipation and relaxation that arise from the brain’s complex microstructure of neurons, glial cells, axons, and interstitial fluid. These parameters are precisely the numbers that feed into the finite element head models used in crash safety research, sports equipment design, and injury prediction, which makes any systematic change in them potentially consequential for how we simulate real-world head impacts.

The headline finding is that prior cyclic loading generally reduced the peak stress and the shear modulus of the tissue in subsequent loading stages. From the initial to the final loading stage, the shear modulus decreased from 2.94 plus or minus 0.94 kilopascals to 2.49 plus or minus 0.63 kilopascals, a statistically significant softening. In other words, after experiencing a history of cyclic deformation, the brain tissue became measurably less stiff, even though the individual loading episodes remained within the sub-damaging range. This is a subtle but important result: it suggests that the tissue accumulates a form of mechanical memory even at loads that would not normally be classified as injurious.

The viscoelastic parameters told a more nuanced story. The parameter g2 increased from 0.14 plus or minus 0.19 to 0.19 plus or minus 0.17, while g3 decreased from 0.21 plus or minus 0.08 to 0.16 plus or minus 0.06, both shifts reaching statistical significance. Meanwhile, g1 and the long-term equilibrium modulus g-infinity showed no significant changes. This pattern implies that the softening is not a uniform degradation of all time-dependent mechanisms. Instead, specific relaxation timescales are being redistributed, indicating that repeated loading alters the balance between the fast and slow dissipative processes within the tissue without fundamentally changing its long-term equilibrium stiffness. For modelers, this means that a single set of viscoelastic parameters measured on pristine tissue may misrepresent the relaxation behavior of tissue that has already been cycled.

Perhaps the most intriguing result concerns what did not correlate. The researchers tested whether the cumulative strain experienced by each specimen predicted its final shear modulus, and found no significant relationship, with a Spearman correlation coefficient of minus 0.31 and a p-value above 0.05. What did matter was the sequence of the loading protocols: different orderings of ramp-and-hold and cyclic stages produced different final moduli. This is the hallmark of path dependence, a property familiar from complex materials science but rarely demonstrated so explicitly in brain tissue. The tissue does not simply tally up the total deformation it has endured; it responds to the specific choreography of how that deformation was delivered.

The implications for repetitive traumatic brain injury research are considerable. Epidemiological studies have long shown that individuals with a prior brain injury face elevated risks of re-injury, dementia, and death, and animal models have demonstrated that traumatic axonal injury can be exacerbated by repeated closed head impacts. Computational models attempting to capture these phenomena have been hampered by a lack of experimental data on how prior loading alters tissue mechanics. The new findings suggest that constitutive models for repeated loading scenarios should incorporate loading-history dependence, rather than treating each impact as an independent event acting on virgin material. A brain that has already been rattled, even sub-clinically, may deform differently under the next blow than the models currently assume.

The study also speaks to a persistent methodological problem in brain biomechanics. The field has long struggled with inconsistencies between ex vivo mechanical measurements, partly because different laboratories use different preconditioning protocols, the preliminary loading cycles applied to a specimen before formal data collection. The authors’ own prior work has catalogued these inconsistencies in detail. The present results suggest that preconditioning is not a mere technical nuisance to be standardized away; it is itself a manipulation of the tissue’s mechanical state that can shift the measured parameters. Two laboratories testing ostensibly identical tissue with different preconditioning histories may legitimately obtain different answers, because the tissue itself has been changed by its history.

Certain caveats frame the scope of these findings. The experiments used bovine tissue under ex vivo conditions, and the reported loading range was sub-damaging, so the results speak most directly to modeling of repeated mild loading rather than severe injury. The authors note that the findings may inform constitutive models for repeated loading scenarios, including those relevant to repetitive traumatic brain injury, which is a careful and appropriately measured claim. Translating the results to living human brains, with their differences in scale, vasculature, and age-related variation, will require further work. Nevertheless, by demonstrating that brain tissue mechanics are path- and sequence-dependent within a controlled experimental framework, the study provides both a caution and an opportunity: a caution against relying on history-blind material models, and an opportunity to build the next generation of head injury simulations on a more faithful description of how soft neural tissue actually behaves when it is loaded again and again.

Subject of Research: Path-dependent viscoelastic and hyperelastic mechanical response of brain tissue under repeated multistage shear loading

Article Title: Path-dependent viscoelastic–hyperelastic response of brain tissue under multistage shear loading

Article References: Nosrati, H., Shafieian, M., & Darvish, K. (2026). Path-dependent viscoelastic–hyperelastic response of brain tissue under multistage shear loading. Medical & Biological Engineering & Computing. https://doi.org/10.1007/s11517-026-03692-z

Image Credits: AI Generated

DOI: 10.1007/s11517-026-03692-z

Keywords: brain tissue mechanics, traumatic brain injury, viscoelasticity, hyperelasticity, shear loading, loading path dependence, repetitive head impact, constitutive modeling, biomechanics, Prony series, ex vivo testing, soft tissue rheology

Constitutive models are the connective tissue of solid mechanics. They are the mathematical rules that translate stress, strain, temperature, and strain rate into predictions of how a real object will bend, dent, stretch, or shatter. Every crash simulation of a car body, every structural analysis of a bridge, and every design study for a battery electrode depends on them. Hao Xu, a postdoctoral researcher at EIT and lead author of the study, explains that the traditional paradigm, while enormously successful, carries an inherent ceiling. Researchers derive a mathematical form based on physical intuition and then calibrate its parameters with data, but the predetermined equation structure restricts what the model can ultimately describe and predict. If the true material behavior involves a functional dependence the chosen formula cannot express, no amount of parameter fitting will recover it. The new framework, called GraphED, inverts the workflow: data comes first, and the equation emerges from it.

The central technical obstacle to such data-driven discovery is representation. Machine learning systems can only optimize what they can encode, and mathematical equations are notoriously awkward objects for algorithms to manipulate. Yuntian Chen, EIT associate professor and co-corresponding author of the study, describes the challenge as efficiently encoding both equation architectures and material-specific parameters into a searchable format for computational algorithms. Conventional symbolic regression approaches typically represent expressions as trees, with operators branching from a root node down to leaves. Trees, however, make it difficult to share structure across different materials and to embed tunable parameters cleanly. The EIT team’s solution is to abandon the tree entirely and encode each candidate equation as a directed graph. In this graph architecture, nodes correspond to mathematical operators and physical variables, while directed edges define the logical and computational connections between them.

The graph representation is more than a cosmetic change, because the edges themselves become carriers of physical meaning. An edge can hold a fixed physical constant, such as a universal exponent, or a tunable material-dependent parameter that varies from one alloy or polymer to another. This design allows GraphED to do two things simultaneously that earlier methods treated separately. It can identify a universal mathematical structure shared across diverse materials and experimental conditions, and it can adaptively calibrate personalized parameters for each individual material scenario. The framework then iterates: it generates candidate graph-structured equations, evaluates how well each one reproduces the experimental data, and optimizes both the topology of the graph and the values of its parameters. The output is a constitutive law that is physically consistent, mathematically compact, and, crucially, interpretable by a human engineer who can read the equation and understand what each term contributes.

That last quality, interpretability, is what separates this work from the black-box models that dominate much of modern machine learning. A neural network trained on material data may predict well, but it offers no equation, no insight, and no transferable knowledge. GraphED delivers the opposite: an explicit formula that a researcher can inspect, question, embed in existing simulation software, and carry into new contexts. The team validated this promise across three very different solid material systems, each chosen for its distinct mechanical character and its historical resistance to empirical modeling. The breadth of the test cases matters, because a discovery method that only works on one material family is a curiosity; one that generalizes across steels, metals, and elastomers is a tool.

The first test case was alloy steel, the workhorse of structural engineering, whose behavior under high strain rates and large plastic deformation is governed by strain-rate dependence and strain hardening. These are precisely the phenomena captured by the Johnson–Cook model, one of the most widely adopted constitutive formulations in impact and crashworthiness analysis. GraphED discovered explicit equations governing both strain-rate dependence and strain hardening directly from experimental data, and when the team integrated these data-driven equations into a complete constitutive model, the result delivered more precise mechanical predictions than the Johnson–Cook benchmark. For a field in which the incumbent model has been refined over decades, the demonstration that an autonomously discovered equation can beat it on predictive accuracy is a significant result.

The second material pushed the method into far harder territory. Lithium metal is a critical component of next-generation energy devices, including high-energy-density batteries, but its mechanical behavior is notoriously difficult to model. Its plastic flow is highly sensitive to both temperature and strain rate, and the interplay of these dependencies has made traditional empirical formulations unreliable. GraphED yielded concise plastic-flow constitutive equations that achieve superior agreement with experimental measurements compared with conventional empirical models. For battery designers, whose simulations of electrode deformation and dendrite-related failure depend on accurate mechanical descriptions, the ability to extract a trustworthy equation directly from data could shorten development cycles and improve the fidelity of electrochemical-mechanical coupled models.

The third validation targeted filled rubbers, a class of elastomers whose hyperelastic response changes with filler composition and temperature. Rubber components in tires, seals, and vibration isolators experience large, reversible deformations, and their stiffness depends on both what they are made of and how hot they are. GraphED captured a compact hyperelastic constitutive equation that maintains robust accuracy across varying material compositions and temperature conditions, meaning a single discovered formula could serve a family of rubber compounds rather than requiring a bespoke model for each formulation. Together, the three case studies span rate-dependent plasticity, temperature-sensitive metal plasticity, and finite-strain hyperelasticity, covering a remarkably wide slice of solid mechanics with one framework.

The researchers see the implications reaching well beyond computational mechanics. Dongxiao Zhang, Chair Professor at EIT and corresponding author of the study, notes that many complex material mechanical behaviors cannot be well described by existing empirical constitutive models, and that equation discovery via graph-based AI provides a powerful new paradigm to assist researchers in deriving rigorous mathematical descriptions when traditional model forms fall short. The phrase that matters is rigorous mathematical description: the goal is not a surrogate that mimics data, but a law, expressed in symbols, that captures the underlying behavior. Because the framework requires only experimental or observational data as input, its potential applications extend to any discipline seeking interpretable physical laws, from soft matter physics and biomechanics to geophysics and beyond, wherever data exists but the governing equation does not.

The study also signals a broader shift in how science may handle the equation-discovery step itself. Symbolic regression and equation learning have been growing fields, but the graph-based encoding developed here resolves a key bottleneck by enabling simultaneous optimization of equation topology and material parameterization, a combination that makes the search tractable for realistic, multi-material datasets. If the paradigm spreads, the role of the mechanician may evolve from proposing formulas to curating data, specifying physical constraints, and interpreting the equations that the algorithm returns. The authors declare no competing interest, and the research was published as a peer-reviewed contribution in Science Advances. For a field built on equations handed down through generations of researchers, the prospect of machines that read raw experimental data and hand back compact, readable laws of material behavior is a genuinely new way of doing physics, and the steels, lithium, and rubber that proved it are only the beginning.

Subject of Research: Graph-based AI equation discovery of constitutive laws in solid materials from experimental data

Article Title: AI discovers interpretable constitutive laws in solids directly from data

Article References: AI discovers interpretable constitutive laws in solids directly from data. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: constitutive models, solid mechanics, graph-based equation discovery, symbolic regression, alloy steels, lithium metal, filled rubbers, machine learning, Science Advances, interpretable AI, strain hardening, hyperelasticity