Myocardial material parameter estimation: a non-homogeneous finite element study from simple shear tests

Advancing clinical translation of cardiac biomechanics models: a comprehensive review, applications and future pathways

Cardiac mechanics models are developed to represent a high level of detail, including refined anatomies, accurate cell mechanics models, and platforms to link microscale physiology to whole-organ function. However, cardiac biomechanics models still have limited clinical translation. In this review, we provide a picture of cardiac mechanics models, focusing on their clinical translation. We review the main experimental and clinical data used in cardiac models, as well as the steps followed in the literature to generate anatomical meshes ready for simulations. We describe the main models in active and passive mechanics and the different lumped parameter models to represent the circulatory system. Lastly, we provide a summary of the state-of-the-art in terms of ventricular, atrial, and four-chamber cardiac biomechanics models. We discuss the steps that may facilitate clinical translation of the biomechanics models we describe. A well-established software to simulate cardiac biomechanics is lacking, with all available platforms involving different levels of documentation, learning curves, accessibility, and cost. Furthermore, there is no regulatory framework that clearly outlines the verification and validation requirements a model has to satisfy in order to be reliably used in applications. Finally, better integration with increasingly rich clinical and/or experimental datasets as well as machine learning techniques to reduce computational costs might increase model reliability at feasible resources. Cardiac biomechanics models provide excellent opportunities to be integrated into clinical workflows, but more refinement and careful validation against clinical data are needed to improve their credibility. In addition, in each context of use, model complexity must be balanced with the associated high computational cost of running these models.

A multi-scale computational model for the passive mechanical behavior of right ventricular myocardium

We have previously demonstrated the importance of myofiber-collagen mechanical interactions in modeling the passive mechanical behavior of right ventricle free wall (RVFW) myocardium. To gain deeper insights into these coupling mechanisms, we developed a high-fidelity, micro-anatomically realistic 3D finite element model of right ventricle free wall (RVFW) myocardium by combining high-resolution imaging and supercomputer-based simulations. We first developed a representative tissue element (RTE) model at the sub-tissue scale by specializing the hyperelastic anisotropic structurally-based constitutive relations for myofibers and ECM collagen, and equi-biaxial and non-equibiaxial loading conditions were simulated using the open-source software FEniCS to compute the effective stress-strain response of the RTE. To estimate the model parameters of the RTE model, we first fitted a 'top-down' biaxial stress-strain behavior with our previous structurally based (tissue-scale) model, informed by the measured myofiber and collagen fiber composition and orientation distributions. Next, we employed a multi-scale approach to determine the tissue-level (5 x 5 x 0.7 mm specimen size) RVFW biaxial behavior via 'bottom-up' homogenization of the fitted RTE model, recapitulating the histologically measured myofiber and collagen orientation to the biaxial mechanical data. Our homogenization approach successfully reproduced the tissue-level mechanical behavior of our previous studies in all biaxial deformation modes, suggesting that the 3D micro-anatomical arrangement of myofibers and ECM collagen is indeed a primary mechanism driving myofiber-collagen interactions.

Can machine learning accelerate soft material parameter identification from complex mechanical test data?

Identifying the constitutive parameters of soft materials often requires heterogeneous mechanical test modes, such as simple shear. In turn, interpreting the resulting complex deformations necessitates the use of inverse strategies that iteratively call forward finite element solutions. In the past, we have found that the cost of repeatedly solving non-trivial boundary value problems can be prohibitively expensive. In this current work, we leverage our prior experimentally derived mechanical test data to explore an alternative approach. Specifically, we investigate whether a machine learning-based approach can accelerate the process of identifying material parameters based on our mechanical test data. Toward this end, we pursue two different strategies. In the first strategy, we replace the forward finite element simulations within an iterative optimization framework with a machine learning-based metamodel. Here, we explore both Gaussian process regression and neural network metamodels. In the second strategy, we forgo the iterative optimization framework and use a stand alone neural network to predict the entire material parameter set directly from experimental results. We first evaluate both approaches with simple shear experiments on blood clot, an isotropic, homogeneous material. Next, we evaluate both approaches against simple shear and uniaxial loading experiments on right ventricular myocardium, an anisotropic, heterogeneous material. We find that replacing the forward finite element simulations with metamodels significantly accelerates the parameter identification process with excellent results in the case of blood clot, and with satisfying results in the case of right ventricular myocardium. On the other hand, we find that replacing the entire optimization framework with a neural network yielded unsatisfying results, especially for right ventricular myocardium. Overall, the importance of our work stems from providing a baseline example showing how machine learning can accelerate the process of material parameter identification for soft materials from complex mechanical data, and from providing an open access experimental and simulation dataset that may serve as a benchmark dataset for others interested in applying machine learning techniques to soft tissue biomechanics.

On the structural origin of the anisotropy in the myocardium: Multiscale modeling and analysis

Due to structural heterogeneities within the tissue, the myocardium displays an orthotropic material behavior. However, the link between the microstructure and the macroscopic mechanical properties is still not fully established. In particular, if it is admitted that the cardiomyocyte organization induces a transversely isotropic symmetry, the relative role in the observed orthotropic symmetry of cardiomyocyte orientation variation and perimysium collagen "sheetlet" structure, two mechanisms occurring at different scales, is still a matter of debate. In order to shed light on this question, we designed a multiscale model of the myocardium, bridging the cell, sheetlet and tissue scales. More precisely, we compared the macroscopic anisotropy obtained by homogenization of different mesostructures consisting in cardiomyocytes and extracellular collageneous layers, also taking into account the variation of cardiomyocyte and sheetlet orientations on the macroscale, to available experimental data. This study confirms the importance of sheetlets layers in assuring the tissue's anisotropic response, as cardiomyocytes-only mesostructures cannot reproduce the observed anisotropy. Moreover, our model shows the existence of a size effect in the myocardial tissue shear properties, which will require further experimental analysis.

Data-driven computational models of ventricular-arterial hemodynamics in pediatric pulmonary arterial hypertension

Pulmonary arterial hypertension (PAH) is a complex disease involving increased resistance in the pulmonary arteries and subsequent right ventricular (RV) remodeling. Ventricular-arterial interactions are fundamental to PAH pathophysiology but are rarely captured in computational models. It is important to identify metrics that capture and quantify these interactions to inform our understanding of this disease as well as potentially facilitate patient stratification. Towards this end, we developed and calibrated two multi-scale high-resolution closed-loop computational models using open-source software: a high-resolution arterial model implemented using CRIMSON, and a high-resolution ventricular model implemented using FEniCS. Models were constructed with clinical data including non-invasive imaging and invasive hemodynamic measurements from a cohort of pediatric PAH patients. A contribution of this work is the discussion of inconsistencies in anatomical and hemodynamic data routinely acquired in PAH patients. We proposed and implemented strategies to mitigate these inconsistencies, and subsequently use this data to inform and calibrate computational models of the ventricles and large arteries. Computational models based on adjusted clinical data were calibrated until the simulated results for the high-resolution arterial models matched within 10% of adjusted data consisting of pressure and flow, whereas the high-resolution ventricular models were calibrated until simulation results matched adjusted data of volume and pressure waveforms within 10%. A statistical analysis was performed to correlate numerous data-derived and model-derived metrics with clinically assessed disease severity. Several model-derived metrics were strongly correlated with clinically assessed disease severity, suggesting that computational models may aid in assessing PAH severity.

A machine learning model to estimate myocardial stiffness from EDPVR

In-vivo estimation of mechanical properties of the myocardium is essential for patient-specific diagnosis and prognosis of cardiac disease involving myocardial remodeling, including myocardial infarction and heart failure with preserved ejection fraction. Current approaches use time-consuming finite-element (FE) inverse methods that involve reconstructing and meshing the heart geometry, imposing measured loading, and conducting computationally expensive iterative FE simulations. In this paper, we propose a machine learning (ML) model that feasibly and accurately predicts passive myocardial properties directly from select geometric, architectural, and hemodynamic measures, thus bypassing exhaustive steps commonly required in cardiac FE inverse problems. Geometric and fiber-orientation features were chosen to be readily obtainable from standard cardiac imaging protocols. The end-diastolic pressure-volume relationship (EDPVR), which can be obtained using a single-point pressure-volume measurement, was used as a hemodynamic (loading) feature. A comprehensive ML training dataset in the geometry-architecture-loading space was generated, including a wide variety of partially synthesized rodent heart geometry and myofiber helicity possibilities, and a broad range of EDPVRs obtained using forward FE simulations. Latin hypercube sampling was used to create 2500 examples for training, validation, and testing. A multi-layer feed-forward neural network (MFNN) was used as a deep learning agent to train the ML model. The model showed excellent performance in predicting stiffness parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_f$$\end{document}af and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f$$\end{document}bf associated with fiber direction (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2_{a_f}=99.471\%$$\end{document}Raf2=99.471% and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2_{b_f}=92.837\%$$\end{document}Rbf2=92.837%). After conducting permutation feature importance analysis, the ML performance further improved for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f$$\end{document}bf (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2_{b_f}=96.240\%$$\end{document}Rbf2=96.240%), and the left ventricular volume and endocardial area were found to be the most critical geometric features for accurate predictions. The ML model predictions were evaluated further in two cases: (i) rat-specific stiffness data measured using ex-vivo mechanical testing, and (ii) patient-specific estimation using FE inverse modeling. Excellent agreements with ML predictions were found for both cases. The trained ML model offers a feasible technology to estimate patient-specific myocardial properties, thus, bridging the gap between EDPVR, as a confounded organ-level metric for tissue stiffness, and intrinsic tissue-level properties. These properties provide incremental information relative to traditional organ-level indices for cardiac function, improving the clinical assessment and prognosis of cardiac diseases.

A viscoelastic model for human myocardium

Understanding the biomechanics of the heart in health and disease plays an important role in the diagnosis and treatment of heart failure. The use of computational biomechanical models for therapy assessment is paving the way for personalized treatment, and relies on accurate constitutive equations mapping strain to stress. Current state-of-the art constitutive equations account for the nonlinear anisotropic stress-strain response of cardiac muscle using hyperelasticity theory. While providing a solid foundation for understanding the biomechanics of heart tissue, most current laws neglect viscoelastic phenomena observed experimentally. Utilizing experimental data from human myocardium and knowledge of the hierarchical structure of heart muscle, we present a fractional nonlinear anisotropic viscoelastic constitutive model. The model is shown to replicate biaxial stretch, triaxial cyclic shear and triaxial stress relaxation experiments (mean error ∼7.68%), showing improvements compared to its hyperelastic (mean error ∼24%) counterparts. Model sensitivity, fidelity and parameter uniqueness are demonstrated. The model is also compared to rate-dependent biaxial stretch as well as different modes of biaxial stretch, illustrating extensibility of the model to a range of loading phenomena. STATEMENT OF SIGNIFICANCE: The viscoelastic response of human heart tissues has yet to be integrated into common constitutive models describing cardiac mechanics. In this work, a fractional viscoelastic modeling approach is introduced based on the hierarchical structure of heart tissue. From these foundations, the current state-of-the-art biomechanical models of the heart muscle are transformed using fractional viscoelasticity, replicating passive muscle function across multiple experimental tests. Comparisons are drawn with current models to highlight the improvements of this approach and predictive responses show strong qualitative agreement with experimental data. The proposed model presents the first constitutive model aimed at capturing viscoelastic nonlinear response across multiple testing regimes, providing a platform for better understanding the biomechanics of myocardial tissue in health and disease.

A Physiology-Guided Classification of Active-Stress and Active-Strain Approaches for Continuum-Mechanical Modeling of Skeletal Muscle Tissue

The well-established sliding filament and cross-bridge theory explain the major biophysical mechanism responsible for a skeletal muscle's active behavior on a cellular level. However, the biomechanical function of skeletal muscles on the tissue scale, which is caused by the complex interplay of muscle fibers and extracellular connective tissue, is much less understood. Mathematical models provide one possibility to investigate physiological hypotheses. Continuum-mechanical models have hereby proven themselves to be very suitable to study the biomechanical behavior of whole muscles or entire limbs. Existing continuum-mechanical skeletal muscle models use either an active-stress or an active-strain approach to phenomenologically describe the mechanical behavior of active contractions. While any macroscopic constitutive model can be judged by it's ability to accurately replicate experimental data, the evaluation of muscle-specific material descriptions is difficult as suitable data is, unfortunately, currently not available. Thus, the discussions become more philosophical rather than following rigid methodological criteria. Within this work, we provide a extensive discussion on the underlying modeling assumptions of both the active-stress and the active-strain approach in the context of existing hypotheses of skeletal muscle physiology. We conclude that the active-stress approach resolves an idealized tissue transmitting active stresses through an independent pathway. In contrast, the active-strain approach reflects an idealized tissue employing an indirect, coupled pathway for active stress transmission. Finally the physiological hypothesis that skeletal muscles exhibit redundant pathways of intramuscular stress transmission represents the basis for considering a mixed-active-stress-active-strain constitutive framework.

Right ventricular myocardial mechanics: Multi-modal deformation, microstructure, modeling, and comparison to the left ventricle

The right ventricular myocardium, much like the rest of the right side of the heart, has been consistently understudied. Presently, little is known about its mechanics, its microstructure, and its constitutive behavior. In this work, we set out to provide the first data on the mechanics of the mature right ventricular myocardium in both simple shear and uniaxial loading and to compare these data to the mechanics of the left ventricular myocardium. To this end, we tested ovine tissue samples of the right and left ventricle under a comprehensive mechanical testing protocol that consisted of six simple shear modes and three tension/compression modes. After mechanical testing, we conducted a histology-based microstructural analysis on each right ventricular sample that yielded high resolution fiber distribution maps across the entire samples. Equipped with this detailed mechanical and histological data, we employed an inverse finite element framework to determine the optimal form and parameters for microstructure-based constitutive models. The results of our study show that right ventricular myocardium is less stiff then the left ventricular myocardium in the fiber direction, but similarly exhibits non-linear, anisotropic, and tension/compression asymmetric behavior with direction-dependent Poynting effect. In addition, we found that right ventricular myocardial fibers change angles transmurally and are dispersed within the sheet plane and normal to it. Through our inverse finite element analysis, we found that the Holzapfel model successfully fits these data, even when selectively informed by rudimentary microstructural information. And, we found that the inclusion of higher-fidelity microstructural data improved the Holzapfel model's predictive ability. Looking forward, this investigation is a critical step towards understanding the fundamental mechanical behavior of right ventricular myocardium and lays the groundwork for future whole-organ mechanical simulations.

A whole blood thrombus mimic: Constitutive behavior under simple shear

Deep vein thrombosis and pulmonary embolism affect 300,000-600,000 patients each year in the US. To better understand the highly mechanical pathophysiology of pulmonary embolism, we set out to develop an in-vitro thrombus mimic and to test this mimic under large deformation simple shear. In addition to reporting on the mechanics of our mimics under simple shear, we explore the sensitivity of their mechanics to coagulation conditions and blood storage time, and compare three hyperelastic material models for their ability to fit our data. We found that thrombus mimics made from whole blood demonstrate strain-stiffening, a negative Poynting effect, and hysteresis when tested quasi-statically to 50% strain under simple shear. Additionally, we found that the stiffness of these mimics does not significantly vary with coagulation conditions or blood storage times. Of the three hyperelastic constitutive models that we tested, the Ogden model provided the best fits to both shear stress and normal stress. In conclusion, we developed a robust protocol to generate regularly-shaped, homogeneous thrombus mimics that lend themselves to simple shear testing under large deformation. Future studies will extend our model to include the effect of maturation and explore its fracture properties toward a better understanding of embolization.

In vivo estimation of passive biomechanical properties of human myocardium

Identification of in vivo passive biomechanical properties of healthy human myocardium from regular clinical data is essential for subject-specific modelling of left ventricle (LV). In this work, myocardium was defined by Holzapfel-Ogden constitutive law. Therefore, the objectives of the study were (a) to estimate the ranges of the constitutive parameters for healthy human myocardium using non-invasive routine clinical data, and (b) to investigate the effect of geometry, LV end-diastolic pressure (EDP) and fibre orientations on estimated values. In order to avoid invasive measurements and additional scans, LV cavity volume, measured from routine MRI, and empirical pressure-normalised-volume relation (Klotz-curve) were used as clinical data. Finite element modelling, response surface method and genetic algorithm were used to inversely estimate the constitutive parameters. Due to the ill-posed nature of the inverse optimisation problem, the myocardial properties was extracted by identifying the ranges of the parameters, instead of finding unique values. Additional sensitivity studies were carried out to identify the effect of LV EDP, fibre orientation and geometry on estimated parameters. Although uniqueness of the solution cannot be achieved, the normal ranges of the parameters produced similar mechanical responses within the physiological ranges. These information could be used in future computational studies for designing heart failure treatments. Graphical abstract.

In vivo estimation of passive biomechanical properties of human myocardium

Identification of in vivo passive biomechanical properties of healthy human myocardium from regular clinical data is essential for subject-specific modelling of left ventricle (LV). In this work, myocardium was defined by Holzapfel-Ogden constitutive law. Therefore, the objectives of the study were (a) to estimate the ranges of the constitutive parameters for healthy human myocardium using non-invasive routine clinical data, and (b) to investigate the effect of geometry, LV end-diastolic pressure (EDP) and fibre orientations on estimated values. In order to avoid invasive measurements and additional scans, LV cavity volume, measured from routine MRI, and empirical pressure-normalised-volume relation (Klotz-curve) were used as clinical data. Finite element modelling, response surface method and genetic algorithm were used to inversely estimate the constitutive parameters. Due to the ill-posed nature of the inverse optimisation problem, the myocardial properties was extracted by identifying the ranges of the parameters, instead of finding unique values. Additional sensitivity studies were carried out to identify the effect of LV EDP, fibre orientation and geometry on estimated parameters. Although uniqueness of the solution cannot be achieved, the normal ranges of the parameters produced similar mechanical responses within the physiological ranges. These information could be used in future computational studies for designing heart failure treatments.

An integrated inverse model-experimental approach to determine soft tissue three-dimensional constitutive parameters: application to post-infarcted myocardium

Knowledge of the complete three-dimensional (3D) mechanical behavior of soft tissues is essential in understanding their pathophysiology and in developing novel therapies. Despite significant progress made in experimentation and modeling, a complete approach for the full characterization of soft tissue 3D behavior remains elusive. A major challenge is the complex architecture of soft tissues, such as myocardium, which endows them with strongly anisotropic and heterogeneous mechanical properties. Available experimental approaches for quantifying the 3D mechanical behavior of myocardium are limited to preselected planar biaxial and 3D cuboidal shear tests. These approaches fall short in pursuing a model-driven approach that operates over the full kinematic space. To address these limitations, we took the following approach. First, based on a kinematical analysis and using a given strain energy density function (SEDF), we obtained an optimal set of displacement paths based on the full 3D deformation gradient tensor. We then applied this optimal set to obtain novel experimental data from a 1-cm cube of post-infarcted left ventricular myocardium. Next, we developed an inverse finite element (FE) simulation of the experimental configuration embedded in a parameter optimization scheme for estimation of the SEDF parameters. Notable features of this approach include: (i) enhanced determinability and predictive capability of the estimated parameters following an optimal design of experiments, (ii) accurate simulation of the experimental setup and transmural variation of local fiber directions in the FE environment, and (iii) application of all displacement paths to a single specimen to minimize testing time so that tissue viability could be maintained. Our results indicated that, in contrast to the common approach of conducting preselected tests and choosing an SEDF a posteriori, the optimal design of experiments, integrated with a chosen SEDF and full 3D kinematics, leads to a more robust characterization of the mechanical behavior of myocardium and higher predictive capabilities of the SEDF. The methodology proposed and demonstrated herein will ultimately provide a means to reliably predict tissue-level behaviors, thus facilitating organ-level simulations for efficient diagnosis and evaluation of potential treatments. While applied to myocardium, such developments are also applicable to characterization of other types of soft tissues.

Simulation of Left Ventricular Dynamics Using a Low-Order Mathematical Model

The eventual goal of this study is to develop methods for estimating dynamic stresses in the left ventricle (LV) that could be used on-line in clinical settings, based on routinely available measurements. Toward this goal, a low-order theoretical model is presented, in which LV shape is represented using a small number of parameters, allowing rapid computational simulations of LV dynamics. The LV is represented as a thick-walled prolate spheroid containing helical muscle fibers with nonlinear passive and time-dependent active contractile properties. The displacement field during the cardiac cycle is described by three time-dependent parameters, using a family of volume-preserving mappings based on prolate spheroidal coordinates. Stress equilibrium is imposed in weak form and the resulting force balance equations are coupled to a lumped-parameter model of the circulation, leading to a system of differential-algebraic equations, whose numerical solution yields predictions of LV pressure and volume, together with spatial distributions of stresses and strains throughout the cardiac cycle. When static loading of the passive LV is assumed, this approach yields displacement and stress fields that closely match results from a standard finite-element approach. When dynamic motion with active contraction is simulated, substantial variations of fiber stress and strain through the myocardium are predicted. This approach allows simulations of LV dynamics that run faster than real time, and could be used to determine patient-specific parameters of LV performance on-line from clinically available measurements, with the eventual goal of real-time, patient-specific analysis of cardiac parameters.

Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics

With heart and cardiovascular diseases continually challenging healthcare systems worldwide, translating basic research on cardiac (patho)physiology into clinical care is essential. Exacerbating this already extensive challenge is the complexity of the heart, relying on its hierarchical structure and function to maintain cardiovascular flow. Computational modelling has been proposed and actively pursued as a tool for accelerating research and translation. Allowing exploration of the relationships between physics, multiscale mechanisms and function, computational modelling provides a platform for improving our understanding of the heart. Further integration of experimental and clinical data through data assimilation and parameter estimation techniques is bringing computational models closer to use in routine clinical practice. This article reviews developments in computational cardiac modelling and how their integration with medical imaging data is providing new pathways for translational cardiac modelling.

Adjoint multi-start-based estimation of cardiac hyperelastic material parameters using shear data

Cardiac muscle tissue during relaxation is commonly modeled as a hyperelastic material with strongly nonlinear and anisotropic stress response. Adapting the behavior of such a model to experimental or patient data gives rise to a parameter estimation problem which involves a significant number of parameters. Gradient-based optimization algorithms provide a way to solve such nonlinear parameter estimation problems with relatively few iterations, but require the gradient of the objective functional with respect to the model parameters. This gradient has traditionally been obtained using finite differences, the calculation of which scales linearly with the number of model parameters, and introduces a differencing error. By using an automatically derived adjoint equation, we are able to calculate this gradient more efficiently, and with minimal implementation effort. We test this adjoint framework on a least squares fitting problem involving data from simple shear tests on cardiac tissue samples. A second challenge which arises in gradient-based optimization is the dependency of the algorithm on a suitable initial guess. We show how a multi-start procedure can alleviate this dependency. Finally, we provide estimates for the material parameters of the Holzapfel and Ogden strain energy law using finite element models together with experimental shear data.

In vivo epicardial force and strain characterisation in normal and MLP-knockout murine hearts

The study's objective is to quantify in vivo epicardial force and strain in the normal and transgenic myocardium using microsensors.Male mice (n = 39), including C57BL/6 (n = 26), 129/Sv (n = 5), wild-type (WT) C57  ×  129Sv (n = 5), and muscle LIM protein (MLP) knock-out (n = 3), were studied under 1.5% isoflurane anaesthesia. Microsurgery allowed the placement of two piezoelectric crystals at longitudinal epicardial loci at the basal, middle, and apical LV regions, and the independent (and/or concurrent) placement of a cantilever force sensor. The findings demonstrate longitudinal contractile and relaxation strains that ranged between 4.8-9.3% in the basal, middle, and apical regions of C57BL/6 mice, and in the mid-ventricular regions of 129/Sv, WT, and MLP mice. Measured forces ranged between 3.1-8.9 mN. The technique's feasibility is also demonstrated in normal mice following afterload, occlusion-reperfusion challenges.Furthermore, the total mid-ventricular forces developed in MLP mice were significantly reduced compared to the WT controls (5.9  ±  0.4 versus 8.9  ±  0.2 mN, p < 0.0001), possibly owing to the fibrotic and stiffer myocardium. No significant strain differences were noted between WT and MLP mice.The possibility of quantifying in vivo force and strain from the normal murine heart is demonstrated with a potential usefulness in the characterisation of transgenic and diseased mice, where regional myocardial function may be significantly altered.

Towards causally cohesive genotype-phenotype modelling for characterization of the soft-tissue mechanics of the heart in normal and pathological geometries

A scientific understanding of individual variation is key to personalized medicine, integrating genotypic and phenotypic information via computational physiology. Genetic effects are often context-dependent, differing between genetic backgrounds or physiological states such as disease. Here, we analyse in silico genotype-phenotype maps (GP map) for a soft-tissue mechanics model of the passive inflation phase of the heartbeat, contrasting the effects of microstructural and other low-level parameters assumed to be genetically influenced, under normal, concentrically hypertrophic and eccentrically hypertrophic geometries. For a large number of parameter scenarios, representing mock genetic variation in low-level parameters, we computed phenotypes describing the deformation of the heart during inflation. The GP map was characterized by variance decompositions for each phenotype with respect to each parameter. As hypothesized, the concentric geometry allowed more low-level parameters to contribute to variation in shape phenotypes. In addition, the relative importance of overall stiffness and fibre stiffness differed between geometries. Otherwise, the GP map was largely similar for the different heart geometries, with little genetic interaction between the parameters included in this study. We argue that personalized medicine can benefit from a combination of causally cohesive genotype-phenotype modelling, and strategic phenotyping that captures effect modifiers not explicitly included in the mechanistic model.

Parameter estimation in a Holzapfel-Ogden law for healthy myocardium

A central problem in biomechanical studies of personalized human left ventricular (LV) modelling is to estimate material properties from in vivo clinical measurements. In this work we evaluate the passive myocardial mechanical properties inversely from the in vivo LV chamber pressure-volume and strain data. The LV myocardium is described using a structure-based orthotropic Holzapfel-Ogden constitutive law with eight parameters. In the first part of the paper we demonstrate how to use a multi-step non-linear least-squares optimization procedure to inversely estimate the parameters from the pressure-volume and strain data obtained from a synthetic LV model in diastole. In the second part, we show that to apply this procedure to clinical situations with limited in vivo data, additional constraints are required in the optimization procedure. Our study, based on three different healthy volunteers, demonstrates that the parameters of the Holzapfel-Ogden law could be extracted from pressure-volume and strain data with a suitable multi-step optimization procedure. Although the uniqueness of the solution cannot be addressed using our approaches, the material response is shown to be robustly determined.

A computational pipeline for quantification of mouse myocardial stiffness parameters

The mouse is an important model for theoretical-experimental cardiac research, and biophysically based whole organ models of the mouse heart are now within reach. However, the passive material properties of mouse myocardium have not been much studied. We present an experimental setup and associated computational pipeline to quantify these stiffness properties. A mouse heart was excised and the left ventricle experimentally inflated from 0 to 1.44kPa in eleven steps, and the resulting deformation was estimated by echocardiography and speckle tracking. An in silico counterpart to this experiment was built using finite element methods and data on ventricular tissue microstructure from diffusion tensor MRI. This model assumed a hyperelastic, transversely isotropic material law to describe the force-deformation relationship, and was simulated for many parameter scenarios, covering the relevant range of parameter space. To identify well-fitting parameter scenarios, we compared experimental and simulated outcomes across the whole range of pressures, based partly on gross phenotypes (volume, elastic energy, and short- and long-axis diameter), and partly on node positions in the geometrical mesh. This identified a narrow region of experimentally compatible values of the material parameters. Estimation turned out to be more precise when based on changes in gross phenotypes, compared to the prevailing practice of using displacements of the material points. We conclude that the presented experimental setup and computational pipeline is a viable method that deserves wider application.

Structure-based finite strain modelling of the human left ventricle in diastole

Finite strain analyses of the left ventricle provide important information on heart function and have the potential to provide insights into the biomechanics of myocardial contractility in health and disease. Systolic dysfunction is the most common cause of heart failure; however, abnormalities of diastolic function also contribute to heart failure, and are associated with conditions including left ventricular hypertrophy and diabetes. The clinical significance of diastolic abnormalities is less well understood than systolic dysfunction, and specific treatments are presently lacking. To obtain qualitative and quantitative information on heart function in diastole, we develop a three-dimensional computational model of the human left ventricle that is derived from noninvasive imaging data. This anatomically realistic model has a rule-based fibre structure and a structure-based constitutive model. We investigate the sensitivity of this comprehensive model to small changes in the constitutive parameters and to changes in the fibre distribution. We make extensive comparisons between this model and similar models that employ different constitutive models, and we demonstrate qualitative and quantitative differences in stress and strain distributions for the different constitutive models. We also provide an initial validation of our model through comparisons to experimental data on stress and strain distributions in the left ventricle.

A murine experimental model for the mechanical behaviour of viable right-ventricular myocardium

Although right-ventricular function is an important determinant of cardio-pulmonary performance in health and disease, right ventricular myocardium mechanical behaviour has received relatively little attention. We present a novel experimental method for quantifying the mechanical behaviour of transmurally intact, viable right-ventricular myocardium. Seven murine right ventricular free wall (RVFW) specimens were isolated and biaxial mechanical behaviour measured, along with quantification of the local transmural myofibre and collagen fibre architecture. We developed a complementary strain energy function based method to capture the average biomechanical response. Overall, murine RVFW revealed distinct mechanical anisotropy. The preferential alignment of the myofibres and collagen fibres to the apex-to-outflow-tract direction was consistent with this also being the mechanically stiffer axis. We also observed that the myofibre and collagen fibre orientations were remarkably uniform throughout the entire RVFW thickness. Thus, our findings indicate a close correspondence between the tissue microstructure and biomechanical behaviour of the RVFW myocardium, and are a first step towards elucidating the structure–function of non-contracted murine RVFW myocardium in health and disease.

Why SIT works: normal function despite typical myofiber pattern in Situs Inversus Totalis (SIT) hearts derived by shear-induced myofiber reorientation

The left ventricle (LV) of mammals with Situs Solitus (SS, normal organ arrangement) displays hardly any interindividual variation in myofiber pattern and experimentally determined torsion. SS LV myofiber pattern has been suggested to result from adaptive myofiber reorientation, in turn leading to efficient pump and myofiber function. Limited data from the Situs Inversus Totalis (SIT, a complete mirror image of organ anatomy and position) LV demonstrated an essential different myofiber pattern, being normal at the apex but mirrored at the base. Considerable differences in torsion patterns in between human SIT LVs even suggest variation in myofiber pattern among SIT LVs themselves. We addressed whether different myofiber patterns in the SIT LV can be predicted by adaptive myofiber reorientation and whether they yield similar pump and myofiber function as in the SS LV. With a mathematical model of LV mechanics including shear induced myofiber reorientation, we predicted myofiber patterns of one SS and three different SIT LVs. Initial conditions for SIT were based on scarce information on the helix angle. The transverse angle was set to zero. During reorientation, a non-zero transverse angle developed, pump function increased, and myofiber function increased and became more homogeneous. Three continuous SIT structures emerged with a different location of transition between normal and mirrored myofiber orientation pattern. Predicted SIT torsion patterns matched experimentally determined ones. Pump and myofiber function in SIT and SS LVs are similar, despite essential differences in myocardial structure. SS and SIT LV structure and function may originate from same processes of adaptive myofiber reorientation.

Deciphering the structure-function relation in healthy hearts is important to understand cardiac pathologies. In the structure-function relation, the myofiber orientation patterns play a central role. Between people with normal organ arrangement (Situs Solitus, SS) this pattern is strikingly similar. Such consistency in myocardial structure might be the result of an adaptation process to accommodate for homogeneous distribution of myofiber strain across the wall and for optimal pump function. The heart of people with a mirror-imaged position of their organs (Situs Inversus Totalis, SIT) has a modified myofiber orientation pattern with respect to SS: normal at the LV apex, but mirrored at the base. Hence, studying SIT hearts provides a unique possibility 1) for understanding adaptation mechanisms related to myofiber orientation and mechanical load, and 2) to gain additional insights into the structure-function relations of the LV. Through mathematical modeling of LV mechanics, we have found that myofiber orientation pattern in both SS and SIT may originate from same processes of adaptive myofiber reorientation. After reorientation, pump and local myofiber function were found to be similar between SS and SIT as well: a remarkable finding when considering the large difference in myofiber orientation pattern.

Kinematics of cardiac growth: in vivo characterization of growth tensors and strains

Progressive growth and remodeling of the left ventricle are part of the natural history of chronic heart failure and strong clinical indicators for survival. Accompanied by changes in cardiac form and function, they manifest themselves in alterations of cardiac strains, fiber stretches, and muscle volume. Recent attempts to shed light on the mechanistic origin of heart failure utilize continuum theories of growth to predict the maladaptation of the heart in response to pressure or volume overload. However, despite a general consensus on the representation of growth through a second order tensor, the precise format of this growth tensor remains unknown. Here we show that infarct-induced cardiac dilation is associated with a chronic longitudinal growth, accompanied by a chronic thinning of the ventricular wall. In controlled in vivo experiments throughout a period of seven weeks, we found that the lateral left ventricular wall adjacent to the infarct grows longitudinally by more than 10%, thins by more than 25%, lengthens in fiber direction by more than 5%, and decreases its volume by more than 15%. Our results illustrate how a local loss of blood supply induces chronic alterations in structure and function in adjacent regions of the ventricular wall. We anticipate our findings to be the starting point for a series of in vivo studies to calibrate and validate constitutive models for cardiac growth. Ultimately, these models could be useful to guide the design of novel therapies, which allow us to control the progression of heart failure.

Structural remodeling and mechanical function in heart failure

The cardiac extracellular matrix (ECM) is the three-dimensional scaffold that defines the geometry and muscular architecture of the cardiac chambers and transmits forces produced during the cardiac cycle throughout the heart wall. The cardiac ECM is an active system that responds to the stresses to which it is exposed and in the normal heart is adapted to facilitate efficient mechanical function. There are marked differences in the short- and medium-term changes in ventricular geometry and cardiac ECM that occur as a result of volume overload, hypertension, and ischemic cardiomyopathy. Despite this, there is a widespread view that a common remodeling "phenotype" governs the final progression to end-stage heart failure in different forms of heart disease. In this review article, we make the case that this interpretation is not consistent with the clinical and experimental data on the topic. We argue that there is a need for new theoretical and experimental models that will enable stresses acting on the ECM and resultant deformations to be estimated more accurately and provide better spatial resolution of local signaling mechanisms that are activated as a result. These developments are necessary to link the effects of structural remodeling with altered cardiac mechanical function.

Modeling of cardiac growth and remodeling of myofiber orientation

The heart has the ability to respond to long-term changes in its environment through changes in mass (growth), shape (morphogenesis) and tissue properties (remodeling). For improved quantitative understanding of cardiac growth and remodeling (G&R) experimental studies need to be complemented by mathematical models. This paper reviews models for cardiac growth and remodeling of myofiber orientation, as induced by mechanical stimuli. A distinction is made between optimization models, that focus on the end stage of G&R, and adaptation models, that aim to more closely describe the mechanistic relation between stimulus and effect. While many models demonstrate qualitatively promising results, a lot of questions remain, e.g. with respect to the choice of the stimulus for G&R or the long-term stability of the outcome of the model. A continued effort combining information on mechanotransduction at the cellular level, experimental observations on G&R at organ level, and testing of hypotheses on stimulus-effect relations in mathematical models is needed to answer these questions on cardiac G&R. Ultimately, models of cardiac G&R seem indispensable for patient-specific modeling, both to reconstruct the actual state of the heart and to assess the long-term effect of potential interventions.

Simple shearing of soft biological tissues

Shearing is induced in soft tissues in numerous physiological settings. The limited experimental data available suggest that a severe strain-stiffening effect occurs in the shear stress when soft biological tissues are subjected to simple shear in certain directions. This occurs at relatively small amounts of shear (when compared with the simple shear of rubbers). This effect is modelled within the framework of nonlinear elasticity by consideration of a class of incompressible anisotropic materials. Owing to the large stresses generated for relatively small amounts of shear, particular care must be exercised in order to maintain a homogeneous deformation state in the bulk of the specimen. The results obtained are relevant to the development of accurate shear test protocols for the determination of constitutive properties of soft tissues. It is also demonstrated that there is a fundamental ambiguity in determining the normal stresses in simple shear when soft tissues are modelled as incompressible hyperelastic materials owing to the arbitrary nature of the hydrostatic pressure term. Two physically well-motivated approaches to determining the pressure are presented here, and the resulting hydrostatic stresses are compared and contrasted. The possible generation of cavitational damage owing to critical hydrostatic stress levels is briefly discussed.

A finite element study of invariant-based orthotropic constitutive equations in the context of myocardial material parameter estimation

A previous study investigated a number of invariant-based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue. The study was based on the assumption of a homogeneous deformation. Here, we extend the previous study by performing an inverse finite element material parameter estimation. This ensures a more realistic deformation state and material parameter estimates. The constitutive relations were compared on the basis of (i) 'goodness of fit': how well they fit a set of six shear deformation tests and (ii) 'variability': how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive relations. It was found that a specific form of the polyconvex type as well as the exponential Fung-type equations were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.

Constitutive modelling of passive myocardium: a structurally based framework for material characterization

In this paper, we first of all review the morphology and structure of the myocardium and discuss the main features of the mechanical response of passive myocardium tissue, which is an orthotropic material. Locally within the architecture of the myocardium three mutually orthogonal directions can be identified, forming planes with distinct material responses. We treat the left ventricular myocardium as a non-homogeneous, thick-walled, nonlinearly elastic and incompressible material and develop a general theoretical framework based on invariants associated with the three directions. Within this framework we review existing constitutive models and then develop a structurally based model that accounts for the muscle fibre direction and the myocyte sheet structure. The model is applied to simple shear and biaxial deformations and a specific form fitted to the existing (and somewhat limited) experimental data, emphasizing the orthotropy and the limitations of biaxial tests. The need for additional data is highlighted. A brief discussion of issues of convexity of the model and related matters concludes the paper.

Contribution of extracellular matrix to the mechanical properties of the heart

Extracellular matrix (ECM) components play essential roles in development, remodeling, and signaling in the cardiovascular system. They are also important in determining the mechanics of blood vessels, valves, pericardium, and myocardium. The goal of this brief review is to summarize available information regarding the mechanical contributions of ECM in the myocardium. Fibrillar collagen, elastin, and proteoglycans all play crucial mechanical roles in many tissues in the body generally and in the cardiovascular system specifically. The myocardium contains all three components, but their mechanical contributions are relatively poorly understood. Most studies of ECM contributions to myocardial mechanics have focused on collagen, but quantitative prediction of mechanical properties of the myocardium, or changes in those properties with disease, from measured tissue structure is not yet possible. Circumstantial evidence suggests that the mechanics of cardiac elastin and proteoglycans merit further study. Work in other tissues used a combination of correlation, modification or digestion, and mathematical modeling to establish mechanical roles for specific ECM components; this work can provide guidance for new experiments and modeling studies in myocardium.

Modelling passive diastolic mechanics with quantitative MRI of cardiac structure and function

The majority of patients with clinically diagnosed heart failure have normal systolic pump function and are commonly categorized as suffering from diastolic heart failure. The left ventricle (LV) remodels its structure and function to adapt to pathophysiological changes in geometry and loading conditions, which in turn can alter the passive ventricular mechanics. In order to better understand passive ventricular mechanics, a LV finite element (FE) model was customized to geometric data segmented from in vivo tagged magnetic resonance images (MRI) data and myofibre orientation derived from ex vivo diffusion tensor MRI (DTMRI) of a canine heart using nonlinear finite element fitting techniques. MRI tissue tagging enables quantitative evaluation of cardiac mechanical function with high spatial and temporal resolution, whilst the direction of maximum water diffusion in each voxel of a DTMRI directly corresponds to the local myocardial fibre orientation. Due to differences in myocardial geometry between in vivo and ex vivo imaging, myofibre orientations were mapped into the geometric FE model using host mesh fitting (a free form deformation technique). Pressure recordings, temporally synchronized to the tagging data, were used as the loading constraints to simulate the LV deformation during diastole. Simulation of diastolic LV mechanics allowed us to estimate the stiffness of the passive LV myocardium based on kinematic data obtained from tagged MRI. Integrated physiological modelling of this kind will allow more insight into mechanics of the LV on an individualized basis, thereby improving our understanding of the underlying structural basis of mechanical dysfunction under pathological conditions.

Determinants of left ventricular shear strain

Mathematical models of cardiac mechanics can potentially be used to relate abnormal cardiac deformation, as measured noninvasively by ultrasound strain rate imaging or magnetic resonance tagging (MRT), to the underlying pathology. However, with current models, the correct prediction of wall shear strain has proven to be difficult, even for the normal healthy heart. Discrepancies between simulated and measured strains have been attributed to 1) inadequate modeling of passive tissue behavior, 2) neglecting active stress development perpendicular to the myofiber direction, or 3) neglecting crossover of myofibers in between subendocardial and subepicardial layers. In this study, we used a finite-element model of left ventricular (LV) mechanics to investigate the sensitivity of midwall circumferential-radial shear strain (E(cr)) to settings of parameters determining passive shear stiffness, cross-fiber active stress development, and transmural crossover of myofibers. Simulated time courses of midwall LV E(cr) were compared with time courses obtained in three healthy volunteers using MRT. E(cr) as measured in the volunteers during the cardiac cycle was characterized by an amplitude of approximately 0.1. In the simulations, a realistic amplitude of the E(cr) signal could be obtained by tuning either of the three model components mentioned above. However, a realistic time course of E(cr), with virtually no change of E(cr) during isovolumic contraction and a correct base-to-apex gradient of E(cr) during ejection, could only be obtained by including transmural crossover of myofibers. Thus, accounting for this crossover seems to be essential for a realistic model of LV wall mechanics.

Myocardial material parameter estimation: a comparison of invariant based orthotropic constitutive equations

This study investigated a number of invariant based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue. A number of orthotropic laws based on Green strain components and one microstructurally based law have previously been investigated to fit experimental measurements of stress-strain behaviour. Here we extend this investigation to include several recently proposed functional forms, i.e. invariant based orthotropic and transversely isotropic constitutive relations. These laws were compared on the basis of (i) 'goodness of fit': how well they fit a set of six shear deformation tests, (ii) 'variability': how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive laws. It was found that a specific form of the polyconvex type as well as the exponential Fung-type law from the previous study were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.

Computational cardiac atlases: from patient to population and back

Integrative models of cardiac physiology are important for understanding disease and planning intervention. Multimodal cardiovascular imaging plays an important role in defining the computational domain, the boundary/initial conditions, and tissue function and properties. Computational models can then be personalized through information derived from in vivo and, when possible, non-invasive images. Efforts are now established to provide Web-accessible structural and functional atlases of the normal and pathological heart for clinical, research and educational purposes. Efficient and robust statistical representations of cardiac morphology and morphodynamics can thereby be obtained, enabling quantitative analysis of images based on such representations. Statistical models of shape and appearance can be built automatically from large populations of image datasets by minimizing manual intervention and data collection. These methods facilitate statistical analysis of regional heart shape and wall motion characteristics across population groups, via the application of parametric mathematical modelling tools. These parametric modelling tools and associated ontological schema also facilitate data fusion between different imaging protocols and modalities as well as other data sources. Statistical priors can also be used to support cardiac image analysis with applications to advanced quantification and subject-specific simulations of computational physiology.

A Computationally Efficient Optimization Kernel for Material Parameter Estimation Procedures

Estimating material parameters is an important part in the study of soft tissue mechanics. Computational time can easily run to days, especially when all available experimental data are taken into account. The material parameter estimation procedure is examplified on a set of homogeneous simple shear experiments to estimate the orthotropic constitutive parameters of myocardium. The modification consists of changing the traditional least-squares approach to a weighted least-squares. This objective function resembles a L2-norm type integral which is approximated using Gaussian quadrature. This reduces the computational time of the material parameter estimation by two orders of magnitude.