Computational models of cardiac mechanics have been shown to capture the general mechanical function of the heart, and have been parameterized based on in vivo patient data to accurately reproduce the heart function of individual patients. Pre- vious attempts for creating such patient-specific problem have typically been based on solving a deterministic inverse problem, which minimizes the misfit between cer- tain model outputs and measured values. While this approach is robust and efficient, and has shown good agreement between model results and patient data, it does not provide any information about the uncertainty in the resulting model parameters. We here present a parameter estimation framework based on a Bayesian approach, which estimates probability density functions (PDFs) of material parameters based on uncertain input values. The method is based on sampling the parameter space and solving the associated forward model for each sample, which may lead to a substantial computational problem if multiple parameters are considered. However, the model also offers additional information in the form of univariate or multivariate PDFs for the estimated parameters. We investigate the potential of the methodology by solving a simple parameter estimation problem in passive left ventricular mechan- ics, and show that the results are in agreement with previous results obtained with a deterministic parameter estimation method.

The mechanical function of cardiac tissue results from the complex interplay of
contracting cardiomyocytes and the passive extracellular matrix. Most computational
models of cardiac mechanics are based on a continuum approach, where the tissue
is viewed as a continuous and homogeneous mix of cells and extracellular material.
This approach has been successfully applied in numerous studies,
and will undoubtedly remain a cornerstone of computational biomechanics.
However, the extensive homogenization limits the models' ability to give detailed
insight into the mechanical forces experienced by individual cardiac cells, and
to delineate the mechanical contributions of myocytes and the extracellular matrix.
More detailed models exist, in the form of continuum mechanics models
of individual myocytes, but these are typically limited to a single myocyte and
do not consider the extracellular matrix.

Computational models of cardiac electrophysiology are typically based on the
same continuous and homogenized concept as the mechanics model, with the
bidomain model being the reference model for several decades. More recently,
models have been developed that explicitly represent the cells, the membrane,
and the domain between the cells. The approach is commonly referred to as the EMI
(Extracellular-Membrane-Intracellular) model, and has been applied in
studies of cardiac and neuronal electrophysiology. Natural extensions of the
EMI framework include detailed models of intra- and extracellular ion concentrations
and electro-diffusion, as well as coupling to models for cell contraction and mechanics.

We present a mechanical analogue of the EMI model, i.e., a continuum based
cardiac mechanics model that explicitly represents a three-dimensional network
of cells embedded in an extracellular matrix. Both the intra- and extracellular domains
are modeled as hyperelastic, with constitutive models based based on
the Holzapfel-Ogden model for passive myocardium. The two domains
have different passive mechanical properties, and the intracellular domain is
actively contracting while the extracellular domain is completely passive.
The active contraction is incorporated through a so-called active strain model,
and in this first version of the model it is assumed to be synchronous and homogeneous
across the entire intracellular domain. The model was parameterized using
publicly available experimental data for stretching and shear experiments,
and was used in preliminary explorations of mechanical interactions between
the intra- and extracellular domains. In particular, we studied how myocyte
stress and strain are affected by the mechanical properties of the two domains,
during passive stretching and active contraction. The results show considerable
spatial variations in both stress and strain, and indicate that the detailed
geometrical representation of the cells may give improved insight into certain
mechanisms of biomechanics and mechano-biology. Further development of
the model should include a more detailed exploration of material parameters,
cell geometry, and the mechanical coupling between the cells and the surrounding
matrix, as well as extending the framework to consider inhomogeneous contraction
and potentially coupling of contraction to cell electrophysiology and calcium diffusion.

During early-stage pulmonary arterial hypertension (PAH), the right ventricle (RV) undergoes anatomical adaptation in the form of a thickened RV free wall, and material adaptation in the form of altered passive stiffness and contractility. The early-stage compensatory adaptations may help to preserve cardiac output but can later evolve into maladaptive remodelling and organ failure. The transition from compensatory to detrimental remodelling remains poorly understood.

Using in vivo hemodynamic and morphological data from normotensive and hypertensive rats, we built idealized bi-ventricular finite element models representing different disease stages. We built a total of eight models, with RV wall thickness based on ex vivo measurements and passive material properties prescribed based on planar biaxial mechanical data obtained from the same animals. A previously developed inverse finite element framework was then used to fit the biventricular models to the in-vivo hemodynamic data, resulting in passive stiffness and contractility optimized to provide the best possible fit with the combined hemodynamic and biaxial data.

Model simulations were then used to study how the presence of PAH affects the stress distribution in the right ventricular free wall, and how wall stress and RV function are altered by the induced changes in morphology, passive stiffness and contractility. Further numerical experiments were conducted to gain insight into the relative contribution of geometric and material adaptation to maintaining RV function in early-stage PAH, as well as the role of the septal wall. Such insights can facilitate a more comprehensive understanding of the compensatory remodeling that occurs during the disease progression.

Mathematical models of the cardiovascular system have come a long way since they were first introduced in the early 19th century. Driven by a rapid development of experimental techniques, numerical methods, and computer hardware, detailed models that describe physical scales from the molecular level up to organs and organ systems have been derived and used for physiological research. Mathematical and computational models can be seen as condensed and quantitative formulations of extensive physiological knowledge and are used for formulating and testing hypotheses, interpreting and directing experimental research, and have contributed substantially to our understanding of cardiovascular physiology. However, in spite of the strengths of mathematics to precisely describe complex relationships and the obvious need for the mathematical and computational models to be informed by experimental data, there still exist considerable barriers between experimental and computational physiological research. In this review, we present a historical overview of the development of mathematical and computational models in cardiovascular physiology, including the current state of the art. We further argue why a tighter integration is needed between experimental and computational scientists in physiology, and point out important obstacles and challenges that must be overcome in order to fully realize the synergy of experimental and computational physiological research.

Atrial cardiomyocytes have a less well-developed T-tubule system than ventricular cells, resulting in intracellular calcium waves propagating from the membrane to the center via centripetal calcium diffusion. Failure of centripetal calcium-wave propagation (‘calcium silencing’) has been observed in a rabbit model of rapid atrial pacing and in patients with atrial fibrillation, but the underlying mechanisms remain incompletely understood. The goal of this study was to develop a novel computational model of the rabbit atrial cardiomyocyte that incorporates detailed compartmentalization of intracellular calcium dynamics, which can be used to investigate the mechanisms underpinning calcium silencing.
We incorporated ion-current formulations reflecting rabbit electrophysiology into a previously published human atrial cardiomyocyte model. The model was validated with published experimental data and the effects of altered rate of calcium diffusion between the calcium-release-unit space and the cytosol (τdiff) were investigated. τdiff modulates local calcium levels available to activate neighboring calcium-release sites, affecting wave propagation.
Simulation results showed that calcium-wave propagation was highly sensitive to τdiff during normal pacing at 2 Hz. We observed impaired calcium-wave propagation for a range of values of τdiff, with full calcium-wave propagation for values of τdiff exceeding 12.7 ms, and calcium silencing for values of τdiff below 10.6 ms due to insufficient local positive feedback for calcium-induced calcium release to maintain centripetal wave propagation. We also observed calcium alternans between propagating and non-propagating waves for intermediate values of τdiff.
This study provided new insight into the mechanisms of calcium-wave propagation failure in rabbit atrial cardiomyocytes and motivates further investigation of the effects of altered calcium diffusion on wave-propagation abnormalities and calcium-dependent arrhythmogenesis. Moreover, the newly developed model will be a useful tool for studying conditions which permit restoration of normal calcium wave propagation.