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 cycle. 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. This article is protected by copyright. All rights reserved.
Over the last decades, computational models have been applied in in-silico simulations of the heart biomechan- ics. These models depend on input parameters. In particular, four parameters are needed for the constitutive law of Guc- cione et al., a model describing the stress-strain relation of the heart tissue. In the literature, we could find a wide range of values for these parameters. In this work, we propose an optimization framework which identifies the parameters of a constitutive law. This framework is based on experimental measurements conducted by Klotz et al.. They provide an end-diastolic pressure-volume relation- ship. We applied the proposed framework on one heart model and identified the following elastic parameters to optimally match the Klotz curve: 𝐶 = 313 Pa, 𝑏𝑓 = 17.8, 𝑏𝑡 = 7.1 and 𝑏𝑓𝑡 = 12.4. In general, this approach allows to identify optimized param- eters for a constitutive law, for a patient-specific heart geome- try. The use of optimized parameters will lead to physiological simulation results of the heart biomechanics and is therefore an important step towards applying computational models in clinical practice.
The contraction of the heart is a complex process involving the interaction of the passive properties of the tissue and the active tension development, which is elicited by the electrical activation of the cells. In this study, the electro-mechanical delay (EMD) was investigated as well as its dependence on the length of the sarcomeres, which are the contractile units within the cell. EMD was defined as the time offset between the electrical activation of the cell and the time of maximal tension. On a simple bar geometry with unidirectional fibre orientation and a linear local activation time distribution, the EMD proved to be inhomogeneous. The contraction of the early activated regions caused an elongation of the sarcomere (stretch) in the neighbouring regions, which ware electrically activated at a later time. The tension in the stretched region reached twice the value of the cells in the not-stretched, early activated region . Furthermore, the EMD in the early electrically activated region was more than 0.2 s, which was about twice the EMD of the stretched regions. In conclusion, the stretched region developed higher tension within a shorter time interval compared to the early activated region. Future studies will investigate how the inhomogeneous EMD affects cardiac output.
Individualized computer models of the geometry of the human heart are often based on mag- netic resonance images (MRI) or computed tomography (CT) scans. The stress distribution in the imaged state cannot be measured but needs to be estimated from the segmented geometry, e.g. by an iterative algorithm. As the convergence of this algorithm depends on different geometrical conditions, we system- atically studied their influence. Beside various shape alterations, we investigated the chamber volume, as well as the effect of material parameters. We found a marked influence of passive material parameters: increasing the model stiffness by a factor of ten halved the residual norm in the first iteration. Flat and concave areas led to a reduced robustness and convergence rate of the unloading algorithm. With this study, the geometric effects and modeling aspects governing the unloading algorithm’s convergence are identified and can be used as a basis for further improvement.
A variety of biophysical and phenomenological active tension models has been proposed during the last decade that show physiological behaviour on a cellular level. However, applying these models in a whole heart finite element simulation framework yields either unphysiological values of stress and strain or an insufficient deformation pattern compared to magnetic resonance imaging data. In this study, we evaluate how introducing an orthotropic active stress tensor affects the deformation pattern by conducting a sensitivity analysis regarding the active tension at resting length Tref and three orthotropic activation parameters (Kss, Ksn and Knn). Deformation of left ventricular contraction is evaluated on a truncated ellipsoid using four features: wall thickening (WT), longitudinal shortening (LS), torsion (Θ) and ejection fraction (EF). We show that EF, WT and LS are positively correlated with the parameters Tref and Knn while Kss reduces all of the four observed features. Introducing shear stress to the model has little to no effect on EF, WT and LS, although it reduces torsion by up to 3◦. We find that added stress in the normal direction can support healthy deformation patterns. However, the twisting motion, which has been shown to be important for cardiac function, reduces by up to 20◦.
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.
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.
The human heart is an organ of high complexity and hence, very challenging to simulate. To calculate the force developed by the human heart and therefore the tension of the muscle fibers, accurate models are necessary. The force generated by the cardiac muscle has physiologically imposed limits and depends on various characteristics such as the length, strain and the contraction velocity of the cardiomyocytes. Another characteristic is the activation time of each cardiomyocyte, which is a wave and not a static value for all cardiomyocytes. To simulate a physiologically correct excitation, the functionality of the cardiac simulation framework CardioMechanics was extended to incorporate inhomogeneous activation times. The functionality was then used to evaluate the effects of local activation times with two different tension models. The active stress generated by the cardiomyocytes was calculated by (i) an explicit function and (ii) an ode-based model. The results of the simulations showed that the maximum pressure in the left ventricle dropped by 2.3% for the DoubleHill model and by 5.3% for the Lumens model. In the right ventricle the simulations showed similar results. The maximum pressure in both the left and the right atrium increased using both models. Given that the simulation of the inhomogeneously activated cardiomyocytes increases the simulation time when used with the more precise Lumens model, the small drop in maximum pressure seems to be negligible in favor of a simpler simulation model.