Digital rheometer twins: Learning the hidden rheology of complex fluids through rheology-informed graph neural networks.

SignificanceScience-based data-driven methods that can describe the rheological behavior of complex fluids can be transformative across many disciplines. Digital rheometer twins, which are developed here, can significantly reduce the cost, time, and energy required to characterize complex fluids and predict their future behavior. This is made possible by combining two different methods of informing neural networks with the rheological underpinnings of a system, resulting in quantitative recovery of a gel's response to different flow protocols. The platform developed here is general enough that it can be extended to areas well beyond complex fluids modeling.

Role of Microtubule Network in the Passive Anisotropic Viscoelasticity of Healthy Right Ventricle.

Cardiomyocytes are viscoelastic and key determinants of right ventricle (RV) mechanics. Intracellularly, microtubules are found to impact the viscoelasticity of isolated cardiomyocytes or trabeculae; whether they contribute to the tissue-level viscoelasticity is unknown. Our goal was to reveal the role of the microtubule network in the passive anisotropic viscoelasticity of the healthy RV. Equibiaxial stress relaxation tests were conducted in healthy RV free wall (RVFW) under early (6%) and end (15%) diastolic strain levels, and at sub- and physiological stretch rates. The viscoelasticity was assessed at baseline and after the removal of microtubule network. Furthermore, a quasi-linear viscoelastic (QLV) model was applied to delineate the contribution of microtubules to the relaxation behavior of RVFW. After removing the microtubule network, RVFW elasticity and viscosity were reduced at the early diastolic strain level and in both directions. The reduction in elasticity was stronger in the longitudinal direction, whereas the degree of changes in viscosity were equivalent between directions. There was insignificant change in RVFW viscoelasticity at late diastolic strain level. Finally, the modeling showed that the tissue's relaxation strength was reduced by the removal of the microtubule network, but the change was present only at a later time scale. These new findings suggest a critical role of cytoskeleton filaments in RVFW passive mechanics in physiological conditions.

Entropy Analysis of Implicit Heat Fluxes in Multi-Temperature Mixtures.

We propose new implicit constitutive relations for the heat fluxes of a two-temperature mixture of fluids. These relations are frame-indifferent forms. However, classical explicit forms of the stress tensors and the interaction forces (specified as explicit forms of constitutive relations) as given in mixture theory are used. The focus here is to establish constraints imposed on the implicit terms in the heat fluxes due to the Second Law of Thermodynamics. Our analysis establishes that the magnitude of the explicit entropy production is equal to or greater than that of the implicit entropy production.

Strain energy density as a Gaussian process and its utilization in stochastic finite element analysis: application to planar soft tissues.

Data-based approaches are promising alternatives to the traditional analytical constitutive models for solid mechanics. Herein, we propose a Gaussian process (GP) based constitutive modeling framework, specifically focusing on planar, hyperelastic and incompressible soft tissues. The strain energy density of soft tissues is modeled as a GP, which can be regressed to experimental stress-strain data obtained from biaxial experiments. Moreover, the GP model can be weakly constrained to be convex. A key advantage of a GP-based model is that, in addition to the mean value, it provides a probability density (i.e. associated uncertainty) for the strain energy density. To simulate the effect of this uncertainty, a non-intrusive stochastic finite element analysis (SFEA) framework is proposed. The proposed framework is verified against an artificial dataset based on the Gasser-Ogden-Holzapfel model and applied to a real experimental dataset of a porcine aortic valve leaflet tissue. Results show that the proposed framework can be trained with limited experimental data and fits the data better than several existing models. The SFEA framework provides a straightforward way of using the experimental data and quantifying the resulting uncertainty in simulation-based predictions.

Constitutive Modeling with Single and Dual Internal Variables.

Phenomenological constitutive models with internal variables have been applied for a wide range of material behavior. The developed models can be classified as related to the single internal variable formalism based on the thermodynamic approach by Coleman and Gurtin. The extension of this theory to so-called dual internal variables opens up new avenues for the constitutive modeling of macroscopic material behavior. This paper reveals the distinction between constitutive modeling with single and dual internal variables using examples of heat conduction in rigid solids, linear thermoelasticity, and viscous fluids. A thermodynamically consistent framework for treating internal variables with as little a priori knowledge as possible is presented. This framework is based on the exploitation of the Clausius-Duhem inequality. Since the considered internal variables are "observable but not controllable", only the Onsagerian procedure with the use of the extra entropy flux is appropriate for the derivation of evolution equations for internal variables. The key distinctions between single and dual internal variables are that the evolution equations are parabolic in the case of a single internal variable and hyperbolic if dual internal variables are employed.

Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations.

Data-driven methods are becoming an essential part of computational mechanics due to their advantages over traditional material modeling. Deep neural networks are able to learn complex material response without the constraints of closed-form models. However, data-driven approaches do not a priori satisfy physics-based mathematical requirements such as polyconvexity, a condition needed for the existence of minimizers for boundary value problems in elasticity. In this study, we use a recent class of neural networks, neural ordinary differential equations (N-ODEs), to develop data-driven material models that automatically satisfy polyconvexity of the strain energy. We take advantage of the properties of ordinary differential equations to create monotonic functions that approximate the derivatives of the strain energy with respect to deformation invariants. The monotonicity of the derivatives guarantees the convexity of the energy. The N-ODE material model is able to capture synthetic data generated from closed-form material models, and it outperforms conventional models when tested against experimental data on skin, a highly nonlinear and anisotropic material. We also showcase the use of the N-ODE material model in finite element simulations of reconstructive surgery. The framework is general and can be used to model a large class of materials, especially biological soft tissues. We therefore expect our methodology to further enable data-driven methods in computational mechanics.

Modeling Atomistic Dynamic Fracture Mechanisms Using a Progressive Transformer Diffusion Model.

Dynamic fracture is an important area of materials analysis, assessing the atomic-level mechanisms by which materials fail over time. Here, we focus on brittle materials failure and show that an atomistically derived progressive transformer diffusion machine learning model can effectively describe the dynamics of fracture, capturing important aspects such as crack dynamics, instabilities, and initiation mechanisms. Trained on a small dataset of atomistic simulations, the model generalizes well and offers a rapid assessment of dynamic fracture mechanisms for complex geometries, expanding well beyond the original set of atomistic simulation results. Various validation cases, progressively more distinct from the data used for training, are presented and analyzed. The validation cases feature distinct geometric details, including microstructures generated by a generative neural network used here to identify novel bio-inspired material designs for mechanical performance. For all cases, the model performs well and captures key aspects of material failure.

In Vitro Mechanical Characterization and Modeling of Subcutaneous Adipose Tissue: A Comprehensive Review.

Mechanical models of adipose tissue are important for various medical applications including cosmetics, injuries, implantable drug delivery systems, plastic surgeries, biomechanical applications such as computational human body models for surgery simulation, and blunt impact trauma prediction. This article presents a comprehensive review of in vivo experimental approaches that aimed to characterize the mechanical properties of adipose tissue, and the resulting constitutive models and model parameters identified. In particular, this study examines the material behavior of adipose tissue, including its nonlinear stress-strain relationship, viscoelasticity, strain hardening and softening, rate-sensitivity, anisotropy, preconditioning, failure behavior, and temperature dependency.

Interpretation of Experimental Data is Substantial for Constitutive Characterization of Arterial Tissue.

The paper aims at evaluation of mechanical tests of soft tissues and creation of their representative stress-strain responses and respective constitutive models. Interpretation of sets of experimental results depends highly on the approach to the data analysis. Their common representation through mean and standard deviation may be misleading and give nonrealistic results. In the paper, raw data of seven studies consisting of 11 experimental data sets (concerning carotid wall and atheroma tissues) are re-analyzed to show the importance of their rigorous analysis. The sets of individual uniaxial stress-stretch curves are evaluated using three different protocols: stress-based, stretch-based, and constant-based, and the population-representative response is created by their mean or median values. Except for nearly linear responses, there are substantial differences between the resulting curves, being mostly the highest for constant-based evaluation. But also the stretch-based evaluation may change the character of the response significantly. Finally, medians of the stress-based responses are recommended as the most rigorous approach for arterial and other soft tissues with significant strain stiffening.

Adaptive Remodeling in the Elastase-induced Rabbit Aneurysms.

BACKGROUND: Rupture of brain aneurysms is associated with high fatality and morbidity rates. Through remodeling of the collagen matrix, many aneurysms can remain unruptured for decades, despite an enlarging and evolving geometry. OBJECTIVE: Our objective was to explore this adaptive remodeling for the first time in an elastase induced aneurysm model in rabbits. METHODS: Saccular aneurysms were created in 22 New Zealand white rabbits and remodeling was assessed in tissue harvested 2, 4, 8 and 12 weeks after creation. RESULTS: The intramural principal stress ratio doubled after aneurysm creation due to increased longitudinal loads, triggering a remodeling response. A distinct wall layer with multi-directional collagen fibers developed between the media and adventitia as early as 2 weeks, and in all cases by 4 weeks with an average thickness of 50.6 ± 14.3 µm. Collagen fibers in this layer were multi-directional (AI = 0.56 ± 0.15) with low tortuosity (1.08 ± 0.02) compared with adjacent circumferentially aligned medial fibers (AI = 0.78 ± 0.12) and highly tortuous adventitial fibers (1.22 ± 0.03). A second phase of remodeling replaced circumferentially aligned fibers in the inner media with longitudinal fibers. A structurally motivated constitutive model with both remodeling modes was introduced along with methodology for determining material parameters from mechanical testing and multiphoton imaging. CONCLUSIONS: A new mechanism was identified by which aneurysm walls can rapidly adapt to changes in load, ensuring the structural integrity of the aneurysm until a slower process of medial reorganization occurs. The rabbit model can be used to evaluate therapies to increase aneurysm wall stability.

Ex Vivo Regional Mechanical Characterization of Porcine Pulmonary Arteries.

BACKGROUND: Regional mechanical characterization of pulmonary arteries can be useful in the development of computational models of pulmonary arterial mechanics. OBJECTIVE: We performed a biomechanical and microstructural characterization study of porcine pulmonary arteries, inclusive of the main, left, and right pulmonary arteries (MPA, LPA, and RPA, respectively). METHODS: The specimens were initially stored at -20°C and allowed to thaw for 12-24 hours prior to testing. Each artery was further subdivided into proximal, middle, and distal regions, leading to ten location-based experimental groups. Planar equibiaxial tensile testing was performed to evaluate the mechanical behavior of the specimens, from which we calculated the stress at the maximum strain (S 55), tensile modulus (TM), anisotropy index (AI), and strain energy in terms of area under the stress-strain curve (AUC). Histological quantification was performed to evaluate the area fraction of elastin and collagen content, intima-media thickness (IMT), and adventitial thickness (AT). The constitutive material behavior of each group was represented by a five-constant Holzapfel-Gasser-Ogden model. RESULTS: The specimens exhibited non-linear stress-strain characteristics across all groups. The MPA exhibited the highest mean wall stress and TM in the longitudinal and circumferential directions, while the bifurcation region yielded the highest values of AI and AUC. All regions revealed a higher stiffness in the longitudinal direction compared to the circumferential direction, suggesting a degree of anisotropy that is believed to be within the margin of experimental uncertainty. Collagen content was found to be the highest in the MPA and decreased significantly at the bifurcation, LPA and RPA. Elastin content did not yield such significant differences amongst the ten groups. The MPA had the highest IMT, which decreased concomitantly to the distal LPA and RPA. No significant differences were found in the AT amongst the ten groups. CONCLUSION: The mechanical properties of porcine pulmonary arteries exhibit strong regional dissimilarities, which can be used to inform future studies of high fidelity finite element models.

Spontaneous Negative Entropy Increments in Granular Flows.

The entropy inequality, commonly taken as an axiom of continuum mechanics, is found to be spontaneously violated in macroscopic granular media undergoing collisional dynamics. The result falls within the fluctuation theorem of nonequilibrium thermodynamics, which is known to replace the Second Law for finite systems. This phenomenon amounts to the system stochastically displaying negative increments of entropy. The focus is on granular media in Couette flows, consisting of monosized circular disks (with 10 to 104 disks of diameters 0.01 m to 1 m) with frictional-Hookean contacts simulated by molecular dynamics accounting for micropolar effects. Overall, it is determined that the probability of negative entropy increments diminishes with the Eulerian velocity gradient increasing, while it tends to increase in a sigmoidal fashion with the Young modulus of disks increasing. This behavior is examined for a very wide range of known materials: from the softest polymers to the stiffest (i.e., carbyne). The disks' Poisson ratio is found to have a weak effect on the probability of occurrence of negative entropy increments.

A Two-Dimensional Model of Hypertension-Induced Arterial Remodeling With Account for Stress Interaction Between Elastin and Collagen.

We propose a novel structure-based two-dimensional (2D) mathematical model of hypertension-induced arterial remodeling. The model is built in the framework of the constrained mixture theory and global growth approach, utilizing a recently proposed structure-based constitutive model of arterial tissue that accounts for the individual natural configurations of and stress interaction between elastin and collagen. The basic novel predictive result is that provided remodeling causes a change in the elastin/collagen mass fraction ratio, it leads to a structural reorganization of collagen that manifests as an altered fiber undulation and a change in direction of the helically oriented fibers in the tissue natural state. Results obtained from the illustrative simulations for a porcine renal artery show that when remodeling is complete the collagen reorganization might have significant effects on the initial arterial geometry and mechanical properties of the arterial tissue. The proposed model has potential to describe and advance mechanistic understanding of adaptive arterial remodeling, promote the continual refinement of mathematical models of arterial remodeling, and provide motivation for new avenues of experimental investigation.

Alterations in the Geometry, Fiber Orientation, and Mechanical Behavior of the Lumbar Intervertebral Disc by Nucleus Swelling.

Soft tissues observed in clinical medical images are often prestrained in tension by internal pressure or tissue hydration. For a native disc, nucleus swelling occurs in equilibrium with osmotic pressure induced by the high concentration of proteoglycan in the nucleus. The objective of this computational study was to investigate the effects of nucleus swelling on disc geometry, fiber orientation, and mechanical behavior by comparing those of prestrained and zero-pressure (unswelled) discs. Thermoelastic analysis techniques were repurposed in order to determine the zero-pressure disc geometry which, when pressurized, matches the prestrained disc geometry observed in clinical images. The zero-pressure geometry was then used in simulations to approximately represent a degenerated disc, which loses the ability of nucleus swelling but has not undergone distinct soft tissue remodeling/disruption. Our simulation results demonstrated that the loss of nucleus swelling caused a slight change in the disc geometry and fiber orientation, but a distinct deterioration in the resistance to intervertebral rotations including sagittal bending, lateral bending, and axial torsion. Different from rotational loading, in compression (with a displacement of 0.45 mm applied), a much larger stiffness (3.02 KN/mm) and a greater intradiscal pressure (IDP) (0.61 MPa) were measured in the zero-pressure disc, compared to the prestrained disc (1.41 KN/mm and 0.52 MPa). This computational study could be useful to understand mechanisms of disc degeneration, and guide the future design of disc tissue engineering material and biomimic disc implants.

A generic physics-informed neural network-based constitutive model for soft biological tissues.

Constitutive modeling is a cornerstone for stress analysis of mechanical behaviors of biological soft tissues. Recently, it has been shown that machine learning (ML) techniques, trained by supervised learning, are powerful in building a direct linkage between input and output, which can be the strain and stress relation in constitutive modeling. In this study, we developed a novel generic physics-informed neural network material (NNMat) model which employs a hierarchical learning strategy by following the steps: (1) establishing constitutive laws to describe general characteristic behaviors of a class of materials; (2) determining constitutive parameters for an individual subject. A novel neural network structure was proposed which has two sets of parameters: (1) a class parameter set for characterizing the general elastic properties; and (2) a subject parameter set (three parameters) for describing individual material response. The trained NNMat model may be directly adopted for a different subject without re-training the class parameters, and only the subject parameters are considered as constitutive parameters. Skip connections are utilized in the neural network to facilitate hierarchical learning. A convexity constraint was imposed to the NNMat model to ensure that the constitutive model is physically relevant. The NNMat model was trained, cross-validated and tested using biaxial testing data of 63 ascending thoracic aortic aneurysm tissue samples, which was compared to expert-constructed models (Holzapfel-Gasser-Ogden, Gasser-Ogden-Holzapfel, and four-fiber families) using the same fitting and testing procedure. Our results demonstrated that the NNMat model has a significantly better performance in both fitting (R2 value of 0.9632 vs 0.9019, p=0.0053) and testing (R2 value of 0.9471 vs 0.8556, p=0.0203) than the Holzapfel-Gasser-Ogden model. The proposed NNMat model provides a convenient and general methodology for constitutive modeling.

Functional Grading of a Transversely Isotropic Hyperelastic Model with Applications in Modeling Tricuspid and Mitral Valve Transition Regions.

Surgical simulators and injury-prediction human models require a combination of representative tissue geometry and accurate tissue material properties to predict realistic tool-tissue interaction forces and injury mechanisms, respectively. While biological tissues have been individually characterized, the transition regions between tissues have received limited research attention, potentially resulting in inaccuracies within simulations. In this work, an approach to characterize the transition regions in transversely isotropic (TI) soft tissues using functionally graded material (FGM) modeling is presented. The effect of nonlinearities and multi-regime nature of the TI model on the functional grading process is discussed. The proposed approach has been implemented to characterize the transition regions in the leaflet (LL), chordae tendinae (CT) and the papillary muscle (PM) of porcine tricuspid valve (TV) and mitral valve (MV). The FGM model is informed using high resolution morphological measurements of the collagen fiber orientation and tissue composition in the transition regions, and deformation characteristics predicted by the FGM model are numerically validated to experimental data using X-ray diffraction imaging. The results indicate feasibility of using the FGM approach in modeling soft-tissue transitions and has implications in improving physical representation of tissue deformation throughout the body using a scalable version of the proposed approach.

Bulk-Surface Electrothermodynamics and Applications to Electrochemistry.

We propose a modeling framework for magnetizable, polarizable, elastic, viscous, heat conducting, reactive mixtures in contact with interfaces. To this end, we first introduce bulk and surface balance equations that contain several constitutive quantities. For further modeling of the constitutive quantities, we formulate constitutive principles. They are based on an axiomatic introduction of the entropy principle and the postulation of Galilean symmetry. We apply the proposed formalism to derive constitutive relations in a rather abstract setting. For illustration of the developed procedure, we state an explicit isothermal material model for liquid electrolyte|metal electrode interfaces in terms of free energy densities in the bulk and on the surface. Finally, we give a survey of recent advancements in the understanding of electrochemical interfaces that were based on this model.

A homogenization model of the Voigt type for skeletal muscle.

A three-dimensional constitutive model for skeletal muscle incorporating microstructural characteristics is developed and numerically implemented in a general purpose finite element program. The proposed model takes into account explicitly the volume fractions of muscle fibers and connective tissue by using the Voigt homogenization approach to bridge the different length scales of the muscle structure. The model is used to estimate the active and passive homogenized muscle response. Next, the model is validated by experimental data and periodic three-dimensional unit cell calculations comprising various fiber volume fractions and mechanical properties of the constituents. The model is found to be in very good agreement with both the experimental data and the finite element results for all the examined cases. The influence of fiber volume fraction and material properties of constituents on effective muscle response under several loading conditions is examined.

A viscoactive constitutive modeling framework with variational updates for the myocardium.

We present a constitutive modeling framework for contractile cardiac mechanics by formulating a single variational principle from which incremental stress-strain relations and kinetic rate equations for active contraction and relaxation can all be derived. The variational framework seamlessly incorporates the hyperelastic behavior of the relaxed and contracted tissue along with the rate - and length - dependent generation of contractile force. We describe a three-element, Hill-type model that unifies the active tension and active deformation approaches. As in the latter approach, we multiplicatively decompose the total deformation gradient into active and elastic parts, with the active deformation parametrizing the contractile Hill element. We adopt as internal variables the fiber, cross-fiber, and sheet normal stretch ratios. The kinetics of these internal variables are modeled via definition of a kinetic potential function derived from experimental force-velocity relations. Additionally, we account for dissipation during tissue deformation by adding a Newtonian viscous potential. To model the force activation, the kinetic equations are coupled with the calcium transient obtained from a cardiomyocyte electrophysiology model. We first analyze our model at the material point level using stress and strain versus time curves for different viscosity values. Subsequently, we couple our constitutive framework with the finite element method (FEM) and study the deformation of three-dimensional tissue slabs with varying cardiac myocyte orientation. Finally, we simulate the contraction and relaxation of an ellipsoidal left ventricular model and record common kinematic measures, such as ejection fraction, and myocardial tissue volume changes.

A constitutive modeling interpretation of the relationship among carotid artery stiffness, blood pressure, and age in hypertensive subjects.

Aging has a profound influence on arterial wall structure and function. We have previously reported the relationship among pulse wave velocity, age, and blood pressure in hypertensive subjects. In the present study, we aimed for a quantitative interpretation of the observed changes in wall behavior with age using a constitutive modeling approach. We implemented a model of arterial wall biomechanics and fitted this to the group-averaged pressure-area (P-A) relationship of the "young" subgroup of our study population. Using this model as our take-off point, we assessed which parameters had to be changed to let the model describe the "old" subgroup's P-A relationship. We allowed elastin stiffness and collagen recruitment parameters to vary and adjusted residual stress parameters according to published age-related changes. We required wall stress to be homogeneously distributed over the arterial wall and assumed wall stress normalization with age by keeping average "old" wall stress at the "young" level. Additionally, we required axial force to remain constant over the cardiac cycle. Our simulations showed an age-related shift in pressure-load bearing from elastin to collagen, caused by a decrease in elastin stiffness and a considerable increase in collagen recruitment. Correspondingly, simulated diameter and wall thickness increased by about 20 and 17%, respectively. The latter compared well with a measured thickness increase of 21%. We conclude that the physiologically realistic changes in constitutive properties we found under physiological constraints with respect to wall stress could well explain the influence of aging in the stiffness-pressure-age pattern observed.