On fibre dispersion modelling of soft biological tissues: a review

Personalized evaluation of the passive myocardium in ischemic cardiomyopathy via computational modeling using Bayesian optimization

Biomechanics-based patient-specific modeling is a promising approach that has proved invaluable for its clinical potential to assess the adversities caused by ischemic heart disease (IHD). In the present study, we propose a framework to find the passive material properties of the myocardium and the unloaded shape of cardiac ventricles simultaneously in patients diagnosed with ischemic cardiomyopathy (ICM). This was achieved by minimizing the difference between the simulated and the target end-diastolic pressure-volume relationships (EDPVRs) using black-box Bayesian optimization, based on the finite element analysis (FEA). End-diastolic (ED) biventricular geometry and the location of the ischemia were determined from cardiac magnetic resonance (CMR) imaging. We employed our pipeline to model the cardiac ventricles of three patients aged between 57 and 66 years, with and without the inclusion of valves. An excellent agreement between the simulated and the target EDPVRs has been reached. Our results revealed that the incorporation of valvular springs typically leads to lower hyperelastic parameters for both healthy and ischemic myocardium, as well as a higher fiber Green strain in the viable regions compared to models without valvular stiffness. Furthermore, the addition of valve-related effects did not result in significant changes in myofiber stress after optimization. We concluded that more accurate results could be obtained when cardiac valves were considered in modeling ventricles. The present novel and practical methodology paves the way for developing digital twins of ischemic cardiac ventricles, providing a non-invasive assessment for designing optimal personalized therapies in precision medicine.

Shape or size matters? Towards standard reporting of tensile testing parameters for human soft tissues: systematic review and finite element analysis

Material properties of soft-tissue samples are often derived through uniaxial tensile testing. For engineering materials, testing parameters (e.g., sample geometries and clamping conditions) are described by international standards; for biological tissues, such standards do not exist. To investigate what testing parameters have been reported for tensile testing of human soft-tissue samples, a systematic review of the literature was performed using PRISMA guidelines. Soft tissues are described as anisotropic and/or hyperelastic. Thus, we explored how the retrieved parameters compared against standards for engineering materials of similar characteristics. All research articles published in English, with an Abstract, and before 1 January 2023 were retrieved from databases of PubMed, Web of Science, and BASE. After screening of articles based on search terms and exclusion criteria, a total 1,096 articles were assessed for eligibility, from which 361 studies were retrieved and included in this review. We found that a non-tapered shape is most common (209 of 361), followed by a tapered sample shape (92 of 361). However, clamping conditions varied and were underreported (156 of 361). As a preliminary attempt to explore how the retrieved parameters might influence the stress distribution under tensile loading, a pilot study was performed using finite element analysis (FEA) and constitutive modeling for a clamped sample of little or no fiber dispersion. The preliminary FE simulation results might suggest the hypothesis that different sample geometries could have a profound influence on the stress-distribution under tensile loading. However, no conclusions can be drawn from these simulations, and future studies should involve exploring different sample geometries under different computational models and sample parameters (such as fiber dispersion and clamping effects). Taken together, reporting and choice of testing parameters remain as challenges, and as such, recommendations towards standard reporting of uniaxial tensile testing parameters for human soft tissues are proposed.

Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

A data-driven computational model for engineered cardiac microtissues

Engineered heart tissues (EHTs) present a potential solution to some of the current challenges in the treatment of heart disease; however, the development of mature, adult-like cardiac tissues remains elusive. Mechanical stimuli have been observed to improve whole-tissue function and cardiomyocyte (CM) maturation, although our ability to fully utilize these mechanisms is hampered, in part, by our incomplete understanding of the mechanobiology of EHTs. In this work, we leverage experimental data, produced by a mechanically tunable experimental setup, to introduce a tissue-specific computational modeling pipeline of EHTs. Our new modeling pipeline generates simulated, image-based EHTs, capturing ECM and myofibrillar structure as well as functional parameters estimated directly from experimental data. This approach enables the unique estimation of EHT function by data-based estimation of CM active stresses. We use this experimental and modeling pipeline to study different mechanical environments, where we contrast the force output of the tissue with the computed active stress of CMs. We show that the significant differences in measured experimental forces can largely be explained by the levels of myofibril formation achieved by the CMs in the distinct mechanical environments, with active stress showing more muted variations across conditions. The presented model also enables us to dissect the relative contributions of myofibrils and extracellular matrix to tissue force output, a task difficult to address experimentally. These results highlight the importance of tissue-specific modeling to augment EHT experiments, providing deeper insights into the mechanobiology driving EHT function. STATEMENT OF SIGNIFICANCE: Engineered heart tissues (EHTs) have the potential to revolutionize the way heart disease is treated. However, developing mature cardiomyocytes (CM) in these tissues remains a challenge due, in part, to our incomplete understanding of the fundamental biomechanical mechanisms that drive EHT development. This work integrates the experimental data of an EHT platform developed to study the influence of mechanics in CM maturation with computational biomechanical models. This approach is used to augment conclusions obtained in-vitro - by measuring quantities such as cell stress and strain - and to dissect the relevance of each component in the whole tissue performance. Our results show how a combination of specialized in-silico and in-vitro approaches can help us better understand the mechanobiology of EHTs.

Numerical investigation on circular and elliptical bulge tests for inverse soft tissue characterization

The acquisition of insights concerning the mechanobiology of aneurysmatic aortic tissues is an important field of investigation. The complete characterization of aneurysm mechanical behaviour can be carried out by biaxial experimental tests on ex vivo specimens. In literature, several works proposed bulge inflation tests as a valid method to analyse aneurysmatic tissue. Bulge test data processing requires the adoption of digital image correlation and inverse analysis approaches to estimate strain and stress distributions, respectively. In this context, however, the accuracy of inverse analysis method has not been evaluated yet. This aspect appears particularly interesting given the anisotropic behaviour of the soft tissue and the possibility to adopt different die geometries. The goal of this study is to provide an accuracy characterization of the inverse analysis applied to the bulge test technique using a numerical approach. In particular, different cases of bulge inflation were simulated in a finite element environment as a reference. To investigate the effect of tissue anisotropic degree and bulge die geometries (circular and elliptical), different input parameters were considered to obtain multiple test cases. The specimen deformed shapes, resulting from the reference finite element simulations, were then analysed through an inverse analysis approach to produce an estimation of stress distributions. The estimated stresses were, at last, compared with the values from the reference finite element simulations. The results demonstrated that the circular die geometry produces a satisfactory estimation accuracy only under certain conditions of material quasi-isotropy. On the other hand, the choice of an elliptical bulge die was proven to be more suitable for the analysis of anisotropic tissues.

Location specific multi-scale characterization and constitutive modeling of pig aorta

The mechanical and structural behavior of the aorta depend on physiological functions and vary from proximal to distal. Understanding the relation between regionally varying mechanical and multi-scale structural response of aorta can be helpful to assess the disease outcomes. Therefore, this study investigated the variation in mechanical and multi-scale structural properties among the major segments of aorta such as ascending aorta (AA), descending aorta (DA) and abdominal aorta (ABA), and established a relation between mechanical and multi-structural parameters. The obtained results showed significant increase in anisotropy and nonlinearity from proximal to distal aorta. The change in periphery length and radii between load and stress free configuration was also found increasing far from the heart. Opening angle was significantly large for ABA than AA and DA (AA/DA vs ABA; p = 0.001). Mean circumferential residual stretch (ratio of mean periphery length at load and stress free configurations) was found decreasing between AA and DA, and then increasing between DA to ABA and its value was significantly more for ABA (AA vs DA; p = 0.041, AA vs ABA; p = 0.001, DA vs ABA; p = 0.001). The waviness of collagen fibers, collagen fiber content, collagen fibril diameter and total protein content were found significantly increasing from proximal to distal. Pearson correlation test showed a significant linear correlation between variation in mechanical and multi-scale structural parameters over the aortic length. Residual stretch was found positively correlated with collagen fiber content (r = 0.82) whereas opening angel was found positively correlated with total protein content (TPC) (r = 0.76).

Characterization and modeling of the in-plane collagen fiber distribution in the porcine dermis

The structural arrangement of collagen fibers in the plane of the dermis layer plays a critical role in accurately predicting the mechanical behavior of skin tissues. This paper combines a histological analysis with statistical modeling to characterize and model the in-plane collagen fiber distribution in the porcine dermis. The histology data reveals that the fiber distribution in the plane of the porcine dermis is non-symmetric. The histology data forms the basis of our model, which employs a combination of two π-periodic von-Mises distribution density functions to create a non-symmetric distribution. We demonstrate that a non-symmetric in-plane fiber distribution is a significant improvement over a symmetric distribution.

A new approach to calculating fiber fields in 2D vessel cross sections using conformal maps

An arterial vessel has three layers, namely, the intima, the media and the adventitia. Each of these layers is modeled to have two families of strain-stiffening collagen fibers that are transversely helical. In an unloaded configuration, these fibers are coiled up. In the case of a pressurized lumen, these fibers stretch and start to resist further outward expansion. As the fibers elongate, they stiffen, affecting the mechanical response. Having a mathematical model of vessel expansion is crucial in cardiovascular applications such as predicting stenosis and simulating hemodynamics. Thus, to study the mechanics of the vessel wall under loading, it is important to calculate the fiber configurations in the unloaded configuration. The aim of this paper is to introduce a new technique of using conformal maps to numerically calculate the fiber field in a general arterial cross-section. The technique relies on finding a rational approximation of the conformal map. First, points on the physical cross section are mapped to points on a reference annulus using a rational approximation of the forward conformal map. Next, we find the angular unit vectors at the mapped points, and finally a rational approximation of the inverse conformal map is used to map the angular unit vectors back to vectors on the physical cross section. We have used MATLAB software packages to achieve these goals.

Hyperelastic continuum models for isotropic athermal fibrous networks

Many biological materials contain fibrous protein networks as their main structural components. Understanding the mechanical properties of such networks is important for creating biomimicking materials for cell and tissue engineering, and for developing novel tools for detecting and diagnosing disease. In this work, we develop continuum models for isotropic, athermal fibrous networks by combining a single-fibre model that describes the axial response of individual fibres, with network models that assemble individual fibre properties into overall network behaviour. In particular, we consider four different network models, including the affine, three-chain, eight-chain, and micro-sphere models, which employ different assumptions about network structure and kinematics. We systematically investigate the ability of these models to describe the mechanical response of athermal collagen and fibrin networks by comparing model predictions with experimental data. We test how each model captures network behaviour under three different loading conditions: uniaxial tension, simple shear, and combined tension and shear. We find that the affine and three-chain models can accurately describe both the axial and shear behaviour, whereas the eight-chain and micro-sphere models fail to capture the shear response, leading to unphysical zero shear moduli at infinitesimal strains. Our study is the first to systematically investigate the applicability of popular network models for describing the macroscopic behaviour of athermal fibrous networks, offering insights for selecting efficient models that can be used for large-scale, finite-element simulations of athermal networks.

Reconstruction of ovine axonal cytoarchitecture enables more accurate models of brain biomechanics

There is an increased need and focus to understand how local brain microstructure affects the transport of drug molecules directly administered to the brain tissue, for example in convection-enhanced delivery procedures. This study reports a systematic attempt to characterize the cytoarchitecture of commissural, long association and projection fibres, namely the corpus callosum, the fornix and the corona radiata, with the specific aim to map different regions of the tissue and provide essential information for the development of accurate models of brain biomechanics. Ovine samples are imaged using scanning electron microscopy combined with focused ion beam milling to generate 3D volume reconstructions of the tissue at subcellular spatial resolution. Focus is placed on the characteristic cytological feature of the white matter: the axons and their alignment in the tissue. For each tract, a 3D reconstruction of relatively large volumes, including a significant number of axons, is performed and outer axonal ellipticity, outer axonal cross-sectional area and their relative perimeter are measured. The study of well-resolved microstructural features provides useful insight into the fibrous organization of the tissue, whose micromechanical behaviour is that of a composite material presenting elliptical tortuous tubular axonal structures embedded in the extra-cellular matrix. Drug flow can be captured through microstructurally-based models using 3D volumes, either reconstructed directly from images or generated in silico using parameters extracted from the database of images, leading to a workflow to enable physically-accurate simulations of drug delivery to the targeted tissue.

Imaging and reconstruction of sheep brain axonal cytoarchitecture provides insight for brain biomechanics models that simulate drug delivery and other biological processes governed by interstitial fluid flow and transport.

Inverse problems in blood flow modeling: A review

Mathematical and computational modeling of the cardiovascular system is increasingly providing non-invasive alternatives to traditional invasive clinical procedures. Moreover, it has the potential for generating additional diagnostic markers. In blood flow computations, the personalization of spatially distributed (i.e., 3D) models is a key step which relies on the formulation and numerical solution of inverse problems using clinical data, typically medical images for measuring both anatomy and function of the vasculature. In the last years, the development and application of inverse methods has rapidly expanded most likely due to the increased availability of data in clinical centers and the growing interest of modelers and clinicians in collaborating. Therefore, this work aims to provide a wide and comparative overview of literature within the last decade. We review the current state of the art of inverse problems in blood flows, focusing on studies considering fully dimensional fluid and fluid-solid models. The relevant physical models and hemodynamic measurement techniques are introduced, followed by a survey of mathematical data assimilation approaches used to solve different kinds of inverse problems, namely state and parameter estimation. An exhaustive discussion of the literature of the last decade is presented, structured by types of problems, models and available data.

Mechanics of tension-induced film wrinkling and restabilization: a review

Wrinkling of thin films under tension is omnipresent in nature and modern industry, a phenomenon which has aroused considerable attention during the past two decades because of its intricate nonlinear behaviours and intriguing morphology changes. Here, we review recent advancements in the mechanics of tension-induced film wrinkling and restabilization, by identifying three major stages of its progress: small-strain (less than 5 % ) wrinkling of stiff sheets, finite-strain (up to 30 % ) wrinkling and restabilization (isola-centre bifurcation) of soft films, and the effects of curved configurations and material properties on pattern formation. Growing demand for fundamental understanding, quantitative prediction and precise tracking of secondary bifurcation transitions in morphological evolution of thin films helps to advance finite-strain plate/shell theories and sophisticated modelling methods. This progress not only promotes our insightful understanding of complex instability behaviour but also reveals novel phenomena and sheds light on developing wrinkle-tunable membrane structures and functional surfaces.

Biomechanics of mitral valve leaflets: Second harmonic generation microscopy, biaxial mechanical tests and tissue modeling

Collagen fibers are the main load carrier in the mitral valve (MV) leaflets. Their orientation and dispersion are an important factor for the mechanical behavior. Most recent studies of collagen fibers in MVs lack either entire thickness study or high transmural resolution. The present study uses second harmonic generation (SHG) microscopy in combination with planar biaxial mechanical tests to better model and examine collagen fibers and mechanical properties of MV leaflets. SHG in combination with tissue clearing enables the collagen fibers to be examined through the entire thickness of the MV leaflets. Planar biaxial mechanical tests, on the other hand, enable the characterization of the mechanical tissue behavior, which is represented by a structural tissue model. Twelve porcine MV leaflets are examined. The SHG recording shows that the mean fiber angle for all samples varies on average by ±12° over the entire thickness and the collagen fiber dispersion changes strongly over the thickness. A constitutive model based on the generalized structure tensor approach is used for the associated tissue characterization. The model represents the tissue with three mechanical parameters plus the mean fiber direction and the dispersion, and predicts the biomechanical response of the leaflets with a good agreement (average r²=0.94). It is found that the collagen structure can be represented by a mean direction and a dispersion with a single family of fibers despite the variation in the collagen fiber direction and the dispersion over the entire thickness of MV leaflets. STATEMENT OF SIGNIFICANCE: Despite its prominent role in the mechanical behavior of mitral valve (MV) leaflets, the collagen structure has not yet been investigated over the entire thickness with high transmural resolution. The present study quantifies the detailed through thickness collagen fiber structure and examines the effects of its variation on MV tissue modeling. This is important because the study evaluates the assumption that the collagen fibers can be modeled with a representative single fiber family despite the variation across the thickness. In addition, the current comprehensive data set paves the way for quantifying the disruption of collagen fibers in myxomatous MV leaflets associated with disrupted collagen fibers.

Effects of dispersed fibres in myocardial mechanics, Part II: active response

This work accompanies the first part of our study "effects of dispersed fibres in myocardial mechanics: Part I passive response" with a focus on myocardial active contraction. Existing studies have suggested that myofibre architecture plays an important role in myocardial active contraction. Following the first part of our study, we firstly study how the general fibre architecture affects ventricular pump function by varying the mean myofibre rotation angles, and then the impact of fibre dispersion along the myofibre direction on myocardial contraction in a left ventricle model. Dispersed active stress is described by a generalised structure tensor method for its computational efficiency. Our results show that both the myofibre rotation angle and its dispersion can significantly affect cardiac pump function by redistributing active tension circumferentially and longitudinally. For example, larger myofibre rotation angle and higher active tension along the sheet-normal direction can lead to much reduced end-systolic volume and higher longitudinal shortening, and thus a larger ejection fraction. In summary, these two studies together have demonstrated that it is necessary and essential to include realistic fibre structures (both fibre rotation angle and fibre dispersion) in personalised cardiac modelling for accurate myocardial dynamics prediction.

Effects of dispersed fibres in myocardial mechanics, Part I: passive response

It is widely acknowledged that an imbalanced biomechanical environment can have significant effects on myocardial pathology, leading to adverse remodelling of cardiac function if it persists. Accurate stress prediction essentially depends on the strain energy function which should have competent descriptive and predictive capabilities. Previous studies have focused on myofibre dispersion, but not on fibres along other directions. In this study, we will investigate how fibre dispersion affects myocardial biomechanical behaviours by taking into account both the myofibre dispersion and the sheet fibre dispersion, with a focus on the sheet fibre dispersion. Fibre dispersion is incorporated into a widely-used myocardial strain energy function using the discrete fibre bundle approach. We first study how different dispersion affects the descriptive capability of the strain energy function when fitting to ex vivo experimental data, and then the predictive capability in a human left ventricle during diastole. Our results show that the chosen strain energy function can achieve the best goodness-of-fit to the experimental data by including both fibre dispersion. Furthermore, noticeable differences in stress can be found in the LV model. Our results may suggest that it is necessary to include both dispersion for myofibres and the sheet fibres for the improved descriptive capability to the ex vivo experimental data and potentially more accurate stress prediction in cardiac mechanics.

Patient-specific computational modeling of endovascular aneurysm repair: State of the art and future directions

Endovascular aortic repair (EVAR) has become the preferred intervention option for aortic aneurysms and dissections. This is because EVAR is much less invasive than the alternative open surgery repair. While in-hospital mortality rates are smaller for EVAR than open repair (1%-2% vs. 3%-5%), the early benefits of EVAR are lost after 3 years due to larger rates of complications in the EVAR group. Clinicians follow instructions for use (IFU) when possible, but are left with personal experience on how to best proceed and what choices to make with respect to stent-graft (SG) model choice, sizing, procedural options, and their implications on long-term outcomes. Computational modeling of SG deployment in EVAR and tissue remodeling after intervention offers an alternative way of testing SG designs in silico, in a personalized way before intervention, to ultimately select the strategies leading to better outcomes. Further, computational modeling can be used in the optimal design of SGs in cases of complex geometries. In this review, we address some of the difficulties and successes associated with computational modeling of EVAR procedures. There is still work to be done in all areas of EVAR in silico modeling, including model validation, before models can be applied in the clinic, but much progress has already been made. Critical to clinical implementation are current efforts focusing on developing fast algorithms that can achieve (near) real-time solutions, as well as ways of dealing with inherent uncertainties related to patient aortic wall degradation on an individualized basis. We are optimistic that EVAR modeling in the clinic will soon become a reality to help clinicians optimize EVAR interventions and ultimately reduce EVAR-associated complications.

Uncertainty quantification in subject-specific estimation of local vessel mechanical properties

Quantitative estimation of local mechanical properties remains critically important in the ongoing effort to elucidate how blood vessels establish, maintain, or lose mechanical homeostasis. Recent advances based on panoramic digital image correlation (pDIC) have made high-fidelity 3D reconstructions of small-animal (e.g., murine) vessels possible when imaged in a variety of quasi-statically loaded configurations. While we have previously developed and validated inverse modeling approaches to translate pDIC-measured surface deformations into biomechanical metrics of interest, our workflow did not heretofore include a methodology to quantify uncertainties associated with local point estimates of mechanical properties. This limitation has compromised our ability to infer biomechanical properties on a subject-specific basis, such as whether stiffness differs significantly between multiple material locations on the same vessel or whether stiffness differs significantly between multiple vessels at a corresponding material location. In the present study, we have integrated a novel uncertainty quantification and propagation pipeline within our inverse modeling approach, relying on empirical and analytic Bayesian techniques. To demonstrate the approach, we present illustrative results for the ascending thoracic aorta from three mouse models, quantifying uncertainties in constitutive model parameters as well as circumferential and axial tangent stiffness. Our extended workflow not only allows parameter uncertainties to be systematically reported, but also facilitates both subject-specific and group-level statistical analyses of the mechanics of the vessel wall.

Physically-based structural modeling of a typical regenerative tissue analog bridges material macroscale continuum and cellular microscale discreteness and elucidates the hierarchical characteristics of cell-matrix interaction

This paper presents a comprehensive physically-based structural modelling for the passive and active biomechanical processes in a typical engineered tissue - namely, cell-compacted collagen gel. First, it introduces a sinusoidal curve analog for quantifying the mechanical response of the collagen fibrils and a probability distribution function of the characteristic crimp ratio for taking into account the fibrillar geometric entropic effect. The constitutive framework based on these structural characteristics precisely reproduces the nonlinearity, the viscoelasticity, and fairly captures the Poisson effect exhibiting in the macroscale tensile tests; which, therefore, substantially validates the structural modelling for the analysis of the cell-gel interaction during collagen gel compaction. Second, a deterministic molecular clutch model specific to the interaction between the cell pseudopodium and the collagen network is developed, which emphasizes the dependence of traction force on clutch number altering with the retrograde flow velocity, actin polymeric velocity, and the deformation of the stretched fibril. The modelling reveals the hierarchical features of cellular substrate sensing, i.e. a biphasic traction force response to substrate elasticity begins at the level of individual fibrils and develops into the second biphasic sensing by means of the fibrillar number integration at the whole-cell level. Singular in crossing the realms of continuum and discrete mechanics, the methodologies developed in this study for modelling the filamentous materials and cell-fibril interaction deliver deep insight into the temporospatially dynamic 3D cell-matrix interaction, and are able to bridge the cellular microscale and material macroscale in the exploration of related topics in mechanobiology.

Efficient computational modelling of smooth muscle orientation and function in the aorta

Understanding the mechanical effects of smooth muscle cell (SMC) contraction on the initiation and the propagation of cardiovascular diseases such as aortic dissection is critical. Framed by elastic lamellar sheets in the lamellar unit, there are SMCs in the media with a distinct radial tilt, which indicates their contribution to the radial strength. However, the mechanical effects of this type of anisotropy have not been fully discussed. Therefore, in this study, we propose a constitutive framework that models the passive and active mechanics of the aorta, taking into account the dispersed nature of the aortic constituents by applying the discrete fibre dispersion method. We suggest an isoparametric approach by evaluating various numerical integration methods and introducing a non-uniform discretization of the unit hemisphere to increase its computational efficiency. Finally, the constitutive parameters are fitted to layer-specific experimental data and initial computational results are briefly presented. The radial tilt of SMCs is also analysed, which has a noticeable influence on the mechanical behaviour of the aorta. In the absence of sufficient experimental data, the results indicate that the active contribution of SMCs has a remarkable impact on the mechanics of the healthy aorta.

Predictive constitutive modelling of arteries by deep learning

The constitutive modelling of soft biological tissues has rapidly gained attention over the last 20 years. Current constitutive models can describe the mechanical properties of arterial tissue. Predicting these properties from microstructural information, however, remains an elusive goal. To address this challenge, we are introducing a novel hybrid modelling framework that combines advanced theoretical concepts with deep learning. It uses data from mechanical tests, histological analysis and images from second-harmonic generation. In this first proof of concept study, our hybrid modelling framework is trained with data from 27 tissue samples only. Even such a small amount of data is sufficient to be able to predict the stress–stretch curves of tissue samples with a median coefficient of determination of R² = 0.97 from microstructural information, as long as one limits the scope to tissue samples whose mechanical properties remain in the range commonly encountered. This finding suggests that deep learning could have a transformative impact on the way we model the constitutive properties of soft biological tissues.

A Hybrid Microstructural-Continuum Multiscale Approach for Modeling Hyperelastic Fibrous Soft Tissue

The heterogeneous, nonlinear, anisotropic material behavior of biological tissues makes precise definition of an accurate constitutive model difficult. One possible solution to this issue would be to define microstructural elements and perform fully coupled multiscale simulation. However, for complex geometries and loading scenarios, the computational costs of such simulations can be prohibitive. Ideally then, we should seek a method that contains microstructural detail, but leverages the speed of classical continuum-based finite-element (FE) modeling. In this work, we demonstrate the use of the Holzapfel-Gasser-Ogden (HGO) model [1, 2] to fit the behavior of microstructural network models. We show that Delaunay microstructural networks can be fit to the HGO strain energy function by calculating fiber network strain energy and average fiber stretch ratio. We then use the HGO constitutive model in a FE framework to improve the speed of our hybrid model, and demonstrate that this method, combined with a material property update scheme, can match a full multiscale simulation. This method gives us flexibility in defining complex FE simulations that would be impossible, or at least prohibitively time consuming, in multiscale simulation, while still accounting for microstructural heterogeneity.

Anisotropic Mechanical Properties of the Human Uterus Measured by Spherical Indentation

The mechanical function of the uterus is critical for a successful pregnancy. During gestation, uterine tissue grows and stretches to many times its size to accommodate the growing fetus, and it is hypothesized the magnitude of uterine tissue stretch triggers the onset of contractions. To establish rigorous mechanical testing protocols for the human uterus in hopes of predicting tissue stretch during pregnancy, this study measures the anisotropic mechanical properties of the human uterus using optical coherence tomography (OCT), instrumented spherical indentation, and video extensometry. In this work, we perform spherical indentation and digital image correlation to obtain the tissue's force and deformation response to a ramp-hold loading regimen. We translate previously reported fiber architecture, measured via optical coherence tomography, into a constitutive fiber composite material model to describe the equilibrium material behavior during indentation. We use an inverse finite element method integrated with a genetic algorithm (GA) to fit the material model to our experimental data. We report the mechanical properties of human uterine specimens taken across different anatomical locations and layers from one non-pregnant (NP) and one pregnant (PG) patient; both patients had pathological uterine tissue. Compared to NP uterine tissue, PG tissue has a more dispersed fiber distribution and equivalent stiffness material parameters. In both PG and NP uterine tissue, the mechanical properties differ significantly between anatomical locations.

Fate of a bulge in an inflated hyperelastic tube: theory and experiment

Mechanical instability in a pre-tensioned finite hyperelastic tube subjected to a slowly increasing internal pressure produces a spatially localized bulge at a critical pressure. This instability is studied in controlled experiments on inflated latex rubber tubes, from the perspective of buckling observed in aneurysms and their rupture risk. The fate of the bulge under continued inflation is governed by the end-conditions and the initial tension in the tube. In a tube with one end fixed and a weight attached to the other freely moving end, the bulge propagates axially at low initial tension, growing in length, and the tube relaxes by extension without buckling. Rupture occurs when the tension is high. By contrast, the bulge formed in an initially stretched tube held fixed at both its ends can buckle or rupture, depending on the amount of initial tension. Experiments are reported for different initial tensions and boundary conditions (BCs). Failure maps in the stretch parameter space and in stretch–tension space are constructed by extending existing theories for bulge formation and buckling analyses to the experimentally relevant BCs. Failure maps deduced from the theory are compared against experiments, and the underlying assumptions are critically assessed. Experiments reveal that buckling provides an alternative route to relieve the stress built up during inflation. Hence, buckling, when it occurs, can be a protective fail-safe mechanism against the rupture of a bulge in an inflated elastic tube.

Passive mechanical properties in healthy and infarcted rat left ventricle characterised via a mixture model

During the cardiac cycle, electrical excitation is coupled with mechanical response of the myocardium. Besides the active contraction, passive mechanics plays an important role, and its behaviour differs in healthy and diseased hearts as well as among different animal species. The aim of this study is the characterisation of passive mechanical properties in healthy and infarcted rat myocardium by means of mechanical testing and subsequent parameter fitting. Elasticity assessments via uniaxial extension tests are performed on healthy and infarcted tissue samples from left ventricular rat myocardium. In order to fully characterise the orthotropic cardiac tissue, our experimental data are combined with other previously published tests in rats - shear tests on healthy myocardium and equibiaxial tests on infarcted tissue. In a first step, we calibrate the Holzapfel-Ogden strain energy function in the healthy case. Sa far, this orthotropic constitutive law for the passive myocardium has been fitted to experimental data in several species, however there is a lack of an appropriate parameter set for the rat. With our determined parameters, a finite element simulation of the end-diastolic filling is performed. In a second step, we propose a model for the infarcted tissue. It is represented as a mixture of intact myocardium and a transversely isotropic scar structure. In our mechanical experiments, the tissue after myocardial infarction shows significantly stiffer behaviour than in the healthy case, and the stiffness correlates with the amount of fibrosis. A similar relationship is observed in the computational simulation of the end-diastolic filling. We conclude that our new proposed material model can capture the behaviour of two kinds of tissues - healthy and infarcted rat myocardium, and its calibration with the fitted parameters represents the experimental data well.

Constitutive modeling using structural information on collagen fiber direction and dispersion in human superficial femoral artery specimens of different ages

Arterial mechanics plays an important role in vascular pathophysiology and repair, and advanced imaging can inform constitutive models of vascular behavior. We have measured the mechanical properties of 14 human superficial femoral arteries (SFAs) (age 12-70, mean 48±19 years) using planar biaxial extension, and determined the preferred collagen fiber direction and dispersion using multiphoton microscopy. The collagen fiber direction and dispersion were evaluated using second-harmonic generation imaging and modeled using bivariate von Mises distributions. The microstructures of elastin and collagen were assessed using two-photon fluorescence imaging and conventional bidirectional histology. The mechanical and structural data were used to describe the SFA mechanical behavior using two- and four-fiber family invariant-based constitutive models. Older SFAs were stiffer and mechanically more nonlinear than younger specimens. In the adventitia, collagen fibers were undulated and diagonally-oriented, while in the media, they were straight and circumferentially-oriented. The media was rich in collagen that surrounded the circumferentially-oriented smooth muscle cells, and the elastin was present primarily in the internal and external elastic laminae. Older SFAs had a more circumferential collagen fiber alignment, a decreased circumferential-radial fiber dispersion, but the same circumferential-longitudinal fiber dispersion as younger specimens. Both the two- and the four-fiber family constitutive models were able to capture the experimental data, and the fits were better for the four-fiber family formulation. Our data provide additional details on the SFA intramural structure and inform structurally-based constitutive models.

Computational modeling for cardiovascular tissue engineering: the importance of including cell behavior in growth and remodeling algorithms

Understanding cardiovascular growth and remodeling (G&R) is fundamental for designing robust cardiovascular tissue engineering strategies, which enable synthetic or biological scaffolds to transform into healthy living tissues after implantation. Computational modeling, particularly when integrated with experimental research, is key for advancing our understanding, predicting the in vivo evolution of engineered tissues, and efficiently optimizing scaffold designs. As cells are ultimately the drivers of G&R and known to change their behavior in response to mechanical cues, increasing efforts are currently undertaken to capture (mechano-mediated) cell behavior in computational models. In this selective review, we highlight some recent examples that are relevant in the context of cardiovascular tissue engineering and discuss the current and future biological and computational challenges for modeling cell-mediated G&R.

A damage model for collagen fibres with an application to collagenous soft tissues

We propose a mechanical model to account for progressive damage in collagen fibres within fibrous soft tissues. The model has a similar basis to the pseudoelastic model that describes the Mullins effect in rubber but it also accounts for the effect of cross-links between collagen fibres. We show that the model is able to capture experimental data obtained from rat tail tendon fibres, and the combined effect of damage and collagen cross-links is illustrated for a simple shear test. The proposed three-dimensional framework allows a straightforward implementation in finite-element codes, which are needed to analyse more complex boundary-value problems for soft tissues under supra-physiological loading or tissues weakened by disease.

Effect of myofibre architecture on ventricular pump function by using a neonatal porcine heart model: from DT-MRI to rule-based methods

Myofibre architecture is one of the essential components when constructing personalized cardiac models. In this study, we develop a neonatal porcine bi-ventricle model with three different myofibre architectures for the left ventricle (LV). The most realistic one is derived from ex vivo diffusion tensor magnetic resonance imaging, and other two simplifications are based on rule-based methods (RBM): one is regionally dependent by dividing the LV into 17 segments, each with different myofibre angles, and the other is more simplified by assigning a set of myofibre angles across the whole ventricle. Results from different myofibre architectures are compared in terms of cardiac pump function. We show that the model with the most realistic myofibre architecture can produce larger cardiac output, higher ejection fraction and larger apical twist compared with those of the rule-based models under the same pre/after-loads. Our results also reveal that when the cross-fibre contraction is included, the active stress seems to play a dual role: its sheet-normal component enhances the ventricular contraction while its sheet component does the opposite. We further show that by including non-symmetric fibre dispersion using a general structural tensor, even the most simplified rule-based myofibre model can achieve similar pump function as the most realistic one, and cross-fibre contraction components can be determined from this non-symmetric dispersion approach. Thus, our study highlights the importance of including myofibre dispersion in cardiac modelling if RBM are used, especially in personalized models.