T. Fritz, C. Wieners, G. Seemann, H. Steen, and O. Dössel. Simulation of the contraction of the ventricles in a human heart model including atria and pericardium : Finite element analysis of a frictionless contact problem. In Biomechanics and Modeling in Mechanobiology, vol. 13(3) , pp. 627-641, 2014
During the contraction of the ventricles, the ventricles interact with the atria as well as with the pericardium and the surrounding tissue in which the heart is embedded. The atria are stretched, and the atrioventricular plane moves toward the apex. The atrioventricular plane displacement (AVPD) is considered to be a major contributor to the ventricular function, and a reduced AVPD is strongly related to heart failure. At the same time, the epicardium slides almost frictionlessly on the pericardium with permanent contact. Although the interaction between the ventricles, the atria and the pericardium plays an important role for the deformation of the heart, this aspect is usually not considered in computational models. In this work, we present an electromechanical model of the heart, which takes into account the interaction between ventricles, pericardium and atria and allows to reproduce the AVPD. To solve the contact problem of epicardium and pericardium, a contact handling algorithm based on penalty formulation was developed, which ensures frictionless and permanent contact. Two simulations of the ventricular contraction were conducted, one with contact handling of pericardium and heart and one without. In the simulation with contact handling, the atria were stretched during the contraction of the ventricles, while, due to the permanent contact with the pericardium, their volume increased. In contrast to that, in the simulations without pericardium, the atria were also stretched, but the change in the atrial volume was much smaller. Furthermore, the pericardium reduced the radial contraction of the ventricles and at the same time increased the AVPD.
The contraction of the human heart is a complex process as a consequence of the interaction of internal and external forces. In current clinical routine, the resulting deformation can be imaged during an entire heart beat. However, the active tension development cannot be measured in vivo but may provide valuable diagnostic information. In this work, we present a novel numerical method for solving an inverse problem of cardiac biomechanics-estimating the dynamic active tension field, provided the motion of the myocardial wall is known. This ill-posed non-linear problem is solved using second order Tikhonov regularization in space and time. We conducted a sensitivity analysis by varying the fiber orientation in the range of measurement accuracy. To achieve RMSE <20% of the maximal tension, the fiber orientation needs to be provided with an accuracy of 10°. Also, variation was added to the deformation data in the range of segmentation accuracy. Here, imposing temporal regularization led to an eightfold decrease in the error down to 12%. Furthermore, non-contracting regions representing myocardial infarct scars were introduced in the left ventricle and could be identified accurately in the inverse solution (sensitivity >0.95). The results obtained with non-matching input data are promising and indicate directions for further improvement of the method. In future, this method will be extended to estimate the active tension field based on motion data from clinical images, which could provide important insights in terms of a new diagnostic tool for the identification and treatment of diseased heart tissue.
Models of cardiac mechanics are increasingly used to investigate cardiac physiology. These models are characterized by a high level of complexity, including the particular anisotropic material properties of biological tissue and the actively contracting material. A large number of independent simulation codes have been developed, but a consistent way of verifying the accuracy and replicability of simulations is lacking. To aid in the verification of current and future cardiac mechanics solvers, this study provides three benchmark problems for cardiac mechanics. These benchmark problems test the ability to accurately simulate pressure-type forces that depend on the deformed objects geometry, anisotropic and spatially varying material properties similar to those seen in the left ventricle and active contractile forces. The benchmark was solved by 11 different groups to generate consensus solutions, with typical differences in higher-resolution solutions at approximately 0.5%, and consistent results between linear, quadratic and cubic finite elements as well as different approaches to simulating incompressible materials. Online tools and solutions are made available to allow these tests to be effectively used in verification of future cardiac mechanics software.
Ventricular wall deformation is widely assumed to have an impact on the morphology of the T-wave that can be measured on the body surface. This study aims at quantifying these effects based on an in silico approach. To this end, we used a hybrid, static-dynamic approach: action potential propagation and repolarization were simulated on an electrophysiologically detailed but static 3-D heart model while the forward calculation accounted for ventricular deformation and the associated movement of the electrical sources (thus, it was dynamic). The displacement vectors that describe the ventricular motion were extracted from cinematographic and tagged MRI data using an elastic registration procedure. To probe to what extent the T-wave changes depend on the synchrony/asynchrony of mechanical relaxation and electrical repolarization, we created three electrophysiological configurations, each with a unique QT time: a setup with physiological QT time, a setup with pathologically short QT time (SQT), and pathologically long QT time (LQT), respectively. For all three electrophysiological configurations, a reduction of the T-wave amplitude was observed when the dynamic model was used for the forward calculations. The largest amplitude changes and the lowest correlation coefficients between the static and dynamic model were observed for the SQT setup, followed by the physiological QT and LQT setups.
Conference Contributions (16)
T. Fritz, E. Kovacheva, G. Seemann, O. Dössel, and A. Loewe. The inverse problem of cardiac mechanics - estimation of cardiac active stress from endocardial motion tracking. In Computational & Mathematical Biomedical Engineering Proceedings, vol. 1, pp. 91-95, 2019
The heart acts as the pump of the cardiovascular system due to the active stress developed in individ- ual cardiac muscle cells. The spatio-temporal distribution of this active stress could contain relevant diagnostic information but can currently not be measured in vivo. We introduce a method to esti- mate dynamic cardiac active stress fields from endocardial surface motion tracking derived from e.g. magnetic resonance imaging data. This ill-posed non-linear problem is solved using Tikhonov regu- larization in space and time in conjunction with a continuum mechanics forward model. We present a proof-of-concept using data from a biophysically detailed multiscale model of cardiac electrome- chanics (7649 tetrahedral elements) in which we could accurately reproduce cardiac motion (surface error <0.4 mm) and identify non-contracting regions due to myocardial infarction scars (active stress error <10 kPa). This inverse method could eventually be used to non-invasively derive personalized diagnostic information in terms of dynamic active stress fields which are not accessible today.
Cardiac computer modeling can help to gain a deeper insight into the physiological processes of the heart. In this work we present a new electromechanical modeling framework which allows to simulate the contraction of the atria in a model of the whole heart with realistic bound- ary conditions. For the active tension development (TD) we used a model, which was originally developed to describe the TD of the ventricles. However, TD in the atria differs significantly from that of the ventricles. On that account, we adapted the TD model to the measurement data of the atria. The modeling framework allows to obtain a realistic motion of the atria during the contraction cycle.
T. Fritz, and O. Dössel. Simulating the beating heart within the pericardium using finite element analysis. In Biomedizinische Technik / Biomedical Engineering, vol. 57(s1) , pp. 330, 2012
T. Fritz, O. Jarrousse, D. Keller, G. Seemann, and O. Dössel. In silico analysis of the impact of transmural myocardial infarction on cardiac mechanical dynamics for the 17 AHA segments. In Proceedings of the 6th International Conference on Functional Imaging and Modeling of the Heart, vol. LNCS, 6666, pp. 241-249, 2011
The impact of transmural infarctions of the left ventricle on the cardiac mechanical dynamics is evaluated for all 17 AHA segments in a computer model. The simulation framework consists of two parts: an electrophysiological model and an elastomechanical model of the ventricles. The electrophysiological model is used to simulate the electrophysiological processes on cellular level, excitation propagation and the tension development. It is linked to the elastomechanical model, which is based on nonlinear finite element analysis for continuum mechanics. Altogether, 18 simulations of the contraction of the ventricles were performed, 17 with an infarction in the respective AHA segment and one simulation for the control case. For each simulation, the mechanical dynamics as well as the wall thickening of the infarct region were analyzed and compared to the corresponding region of the control case. The simulation revealed details of the impact of the myocardial infarction on wall thickening as well as on the velocity of the infarct region for most of the AHA segments
T. Fritz, G. Seemann, and O. Dössel. Analyzing transmural myocardial infarction of the left ventricle using computer modeling. In 4th Cardiac Physiome Workshop, vol. Poster, 2011
Elastomechanical modeling of the heart can help to gain a deeper insight into the mechanical dynamics of the heart. Furthermore it can help to enhance diagnostic strategies and to investigate new therapeutic approaches. Phase contrast magnetic resonance imaging allows to directly measure the velocity vector field of the myocardial motion over the cardiac cycle. Aim of this work was to analyze the impact of transmural myocardial infarction on the velocity vector field of the left ventricle in a numerical model of the heart. For this purpose a multi-scale electromechanical computer framework was used. It consisted of two parts: an electrophysiological model and an elastomechanical model. The electrophysiological model described the electrophysiological processes on cellular level, the excitation propagation as well as the tension development. It was was linked to an elastomechanical model which was based on nonlinear continuum mechanics using the finite element method. The computer framework was used to simulate the contraction of the heart with left ventricular transmural infarctions, differing in size, location and stiffness of the scar tissue. For each simulation, the velocity vector field of the infarct region was analyzed and compared to the corresponding region of the control case. The simulations revealed a direct impact of the myocardial infarction on the magnitude and orientation of the velocity vectors of the affected region.
T. Fritz, O. Jarrousse, G. Seemann, and O. Dössel. Analyzing the transmural electromechanical heterogeneity of the left ventricle in a computer model. In Biomedizinische Technik / Biomedical Engineering (Proceedings BMT2010), vol. 55(Suppl 1) , 2010
There is a large number of published studies analyzing the inhomogeneously distributed electrophysiological properties of the ventricles in a computer model. However only few of them deal with the impact on the hearts mechanics. In 2003 Cordeiro and colleagues  analyzed the influence of the transmural left ventricular electrophysiological heterogeneity on the myocardial mechanics. Therefore, they examined the unloaded cell shortening of sub-epicardial cells, sub-endocardial cells, and cells from the middle of the wall, isolated from canine left ventricle.In this work a heterogenous electromechanical model was used to reconstruct these experiments of Cordeiro et al. in the computer. A simulation framework, which is consisting of an electrophysiological cell model, a tension development model and an elastomechanical model was used to simulate the cell shortening. Two experiments with different heterogeneities had been conducted. The first experiment examined, how the heterogeneity of the membrane channels influences the cell shortening. In the second experiment the additional impact of the heterogeneity of the intracellular calcium handling was analyzed. The results of the simulations were compared qualitatively to the findings of Cordeiro et al.
T. Fritz, O. Jarrousse, and O. Dössel. Adapting a mass-spring system to energy density function describing myocardial mechanics. In Proceedings of the 4th European Congress for Medical and Biomedical Engineering 2008. 23-27 November 2008, Antwerp, Belgium, vol. 22, pp. 2003-2006, 2008
Numerical simulations are increasingly often in- volved in developing new and improving existing medical therapies. While the models involved in those simulations are designed to resemble a specific phenomenon realistically, the results of the interplay of those models are often not suffi- ciently validated. We created a plugin for a cardiac simula- tion framework to validate the simulation results using clinical MRI data. The MRI data were used to create a static whole- heart mesh as well as slices from the left ventricular short axis, providing the motion over time. The static heart was a starting point for a simulation of the heart’s motion. From the simula- tion result, we created slices and compared them to the clinical MRI slices using two different metrics: the area of the slices and the point distances. The comparison showed global simi- larities in the deformation of simulated and clinical data, but also indicated points for potential improvements. Performing this comparison with more clinical data could lead to person- alized modeling of elastomechanics of the heart.
Orientations of myocytes impact electric excitation propagation and mechanical contraction in the human heart. Measured fiber angles in experiments are obtained from different species (e. g. rat, canine, dog, human heart) and vary by various reasons. It is unclear to what ex- tent non-exact fiber angles impact the quality of computa- tional simulations. In this paper, mechanical simulations with different ventricular angles were performed and com- pared. The simulations covered the complete heart with both ventricles, both atria and the pericardium and were performed using finite element method. Helix angles were varied between 20\0 and 70\0 on endocardium and \070\0 and \020\0 on epicardium. Results showed that fiber ori- entations had only a minor contribution to the difference between endsystolic and enddiastolic pressure of < 8.3 %. The influence on stroke volume as well as AVPD is sig- nificant (changes by 34 % for SV and 241 % for APVD) , but it could not be observed that a higher AVPD yields a higher stroke volume. Concludingly, fiber orientations are important for reliable computational simulations of human hearts and should be incorporated with great care.
Electrocardiographic imaging (ECGI) facilitates the non-invasive reconstruction of electrical activity in the entire heart at once. ECGI requires both recordings of multi-channel ECG signals as well as an MRI-based model of the thorax. The model is used to solve the underlying Poissons problem, which relates the gradient of transmembrane voltages in the heart to the ECG and is a spatial differential equation. In ECGI, this relationship has to be established before starting inverse calculations, i.e. the forward problem has to be solved. It solution depends strongly on the spatial discretization of the model, as its resolution affects the representation of the source gradients. To study the convergence of resolution-related effects in the forward problem, we use a simplified thorax model which allows for very high resolutions. An ECG is produced for the excitation origin of a premature ventricular contraction in the apex. The study reveals that the greatest resolution-related effects vanish below a resolution of 5 mm of the cardiac tissue. At below 1 mm, resolution effects stabilize and only marginal effects from the spatial structure of the mesh persist down to a resolution of 0.25 mm.
O. Jarrousse, T. Fritz, and O. Dössel. Modeling breast tissue mechanics from Prone to supine positions with a modified mass-spring system. In Proceedings BMT 2010, 44. DGBMT Jahrestagung, 3-Länder-Tagung D-A-CH, Rostock, vol. 55(S1) , pp. 87-90, 2010
A volumetric mass-spring system, originally developed for myocardial mechanics modeling , is used to simulate the elasto-mechanical deformation of several breast datasets from prone to supine positions. Segmented MRI datasets of pa- tients in prone position, available from the online repository provided by Susan C. Hagness at the University of Wisconsin- Madison  were used in the biomechanical simulations. These models were considered to be consisting of two materi- als, fat and fibroconnective/glandular tissues. Each tissue is represented as a nearly incompressible Neo-Hookean elastic isotropic material. Each simulation was conducted in two steps: in the first step, the unloaded model is generated by apply- ing gravity forces to the original model pointing toward the body. The unloaded model is then used in the second step, by applying gravity forces. Eventually, the breast model in supine position is obtained.
Motivation: Anatomical models of the heart can be used to conduct multi-physics simulations. These simulations can aid basic and clinical research and are being translated into clinical practice nowadays.Problem statement: The human myocardium has very complex fiber structure, which has a strong impact on cardiac physiology. To understand and evaluate 3D fiber orientation in volumetric cardiac models, it is often necessary to project these onto printed pictures.Approach: Images of myocardial fibers using color-coded cylinders, color-coded streamlines and anaglyph methods are compared. Results: Streamlines provide a good distinction of myocardial bundles. Cylinders show the most accurate results. Color-coded representations reveal abrupt changes in fiber direction. Anaglyph visualizations give an illusion of depth in 2D prints and can display overlaying bundles. Conclusions: Streamlines are superior in imaging global fiber orientation, whereas cylinders give better results for local structures. Color-coding increases information where fiber structure is very complex, e.g. in the atria. Anaglyph images cause a loss in color information but help the viewer to understand the 3D object. Overall, it is necessary to choose the appropriate method of picturing fibers for specific tasks.
O. Jarrousse, T. Fritz, and O. Dössel. Implicit time integration in a volumetric mass-spring system for modeling myocardial elastomechanics. In IFMBE Proceedings World Congress on Medical Physics and Biomedical Engineering, vol. 25/4, pp. 876-879, 2009
A modified mass-spring system for simulating the passive and active elastomechanical properties of the myocardial tissue was presented in a previous publication. The previously presented results are combined with the method also published earlier to use continuum mechanics calculate passive forces in a mass-spring system directly starting from the energy density function of the stress-strain relation. An efficient method for volume preservation is presented and the implementation of an implicit time integration method for solving the systems equations of motion is described. The computational complexity of the system is analyzed and shown to be of O(n). At the end several simulations are conducted to demonstrate the method.
O. Jarrousse, T. Fritz, and O. Dössel. A volumetric mass-spring system for modeling myocardial elastomechanics. In The Cardiac Physiome: Multi-scale and Multi-physics Mathematical Modelling Applied to the Heart, 2009
A volumetric mass-spring system for simulating the passive and active elastomechanical properties of the myocardial tissue is presented. A 3D computer model containing information about the ﬁber, sheet, and sheet-normal directions and about the modeled objects physiological properties, is used to initialize the systems structure.Using an electrophysiology model and a force development model, contracting forces are introduced to the systems elements at each time step of the simulation loop.Using the methods of continuum mechanics, suitable springs functions were derived analytically from the energy density function of describing the hyperelastic properties of heart. That eliminated the need of springs parametrization. An efficient method for volume preservation is used to ensure the conservation of the model's volume under deformation.Implicit time integration is implemented to solve the equations of motion, that improves the stability of the simulation and allows larger simulation time steps. An iterative solver that take advantage of the sparsity of the system's matrices is used and the systems complexity is shown to be of O(n) where n is the the count of the models elements.
O. Jarrousse, T. Fritz, and O. Dössel. A modifed mass-spring system for myocardial mechanics modeling. In Proceedings of the 4th European Congress for Medical and Biomedical Engineering 2008. 23-27 November 2008, Antwerp, Belgium, vol. 22, pp. 1943-1946, 2008
T. Fritz. Biomechanical Modeling of the Human Heart - Modeling of the Ventricles, the Atria and the Pericardium and the Inverse Problem of Cardiac Mechanics. Dissertation. 2015
A biomechanical simulation framework has been developed which allows to simulate the contraction of the whole human heart. Further, an inverse solving algorithm has been developed, which works in an opposite manner and allows to reconstruct the active tension distribution from provided data of the motion of the heart surfaces, which can for example be extracted from medical imaging data. This allows for a personalization of the model based on clinical data.
Student Theses (1)
T. Fritz. Analyzing the electro-mechanical heterogeneity of the heart using computer modeling. Institut für Biomedizinische Technik, Karlsruher Institut für Technologie (KIT). Diplomarbeit. 2009