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

Baron, L.

Dössel, O.

Kovacheva, E.

Müller, A.

Schuler, S.

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Effects of local activation times on the tension development of human cardiomyocytes in a computational model

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Armin Müller*, Ekaterina Kovacheva, Steffen Schuler, Olaf Dössel and Lukas Baron Effects of local activation times on the tension development of human cardiomyocytes in a computational model 1 Introduction Studying the human heart and its potential diseases as well as possible therapeutic approaches are challenging tasks which often require high-precision computational models. From these simulations, valuable information can be gathered for developing new treatment options. Simulations of high-precision computational models are very complex and therefore require a lot of computing power. Adding a new feature to such a computational model, which improves the simulation’s precision, might result in higher complexity. For this study a new feature was added to involve the local activation times of human cardiomyocytes. Without this, all cardiomyocytes contract homogeneously resulting in a larger tension development. By applying local activation times to the model, we create an inhomogeneous and thus physiologically more correct excitation that increases the precision of the simulation results at the expense of computational complexity. 2 Methods 2.1 Tension models The passive material properties of the atria were provided by the model described by Mooney and Rivlin [1]. For the passive material properties of the ventricles the model proposed by Guccione was applied [2]. To simulate the active myocardial tension during the contraction of the heart, two different tension models were combined with local activation times (LAT). The first model is the double Hill model (DHM) which uses a function to define the active tension as proposed by Stergiopulos et al. [3]: 𝐄𝐄(𝐭𝐭) =𝑷𝑷(𝒕𝒕)𝑽𝑽(𝒕𝒕)−𝑽𝑽𝒅𝒅(1) 𝒆𝒆(𝐭𝐭) =𝑬𝑬(𝒕𝒕)− 𝑬𝑬𝒎𝒎𝒎𝒎𝒎𝒎𝑬𝑬𝒎𝒎𝒎𝒎𝒎𝒎−𝑬𝑬𝒎𝒎𝒎𝒎𝒎𝒎=𝟏𝟏𝒌𝒌⋅ (𝒈𝒈𝒄𝒄𝟏𝟏𝟏𝒈𝒈𝒄𝒄) ⋅ (𝟏𝟏𝟏𝟏𝟏𝒈𝒈𝒓𝒓) , (2) Abstract: 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.Keywords: Local activation time, LAT, tension development, active stress, human cardiomyocytes, computational model, whole heart simulation. https://doi.org/10.1515/cdbme-2018-0060 ______ *Corresponding author: Armin Müller: Institute of Biomedical Engineering, Karlsruhe Institute of Technology (KIT), Kaiserstr. 12, 76131 Karlsruhe, Germany, e-mail: publications@ibt.kit.edu Ekaterina Kovacheva, Steffen Schuler, Olaf Dössel, Lukas Baron: Institute of Biomedical Engineering, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany Open Access. © 2018 Armin Müller et al., published by De Gruyter.  This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 License. Current Directions in Biomedical Engineering 2018; 4(1): 247 – 250Bereitgestellt von | Karlsruher Institut für Technologie KIT-BibliothekAngemeldetHeruntergeladen am | 08.02.19 10:37with 𝑔𝑔𝑐𝑐 = (𝑡𝑡′/𝜏𝜏𝑐𝑐)𝑚𝑚𝑐𝑐, 𝑔𝑔𝑟𝑟 = (𝑡𝑡′/𝜏𝜏𝑟𝑟)𝑚𝑚𝑟𝑟, 𝑡𝑡′ = 𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡 − 𝑡𝑡0, 𝑇𝑇) and 𝑘𝑘 = max {𝑘𝑘, 𝑒𝑒(𝑡𝑡)}. The time course of the active stress is assumed to be equal to the chamber elasticity 𝑬𝑬(𝒕𝒕), 𝑽𝑽(𝒕𝒕) is the ventricular volume and𝑽𝑽𝒅𝒅 is the unloaded (unpressurized) ventricular volume. 𝑬𝑬(𝒕𝒕) can also be approximated by Eq. 2 as proposed by Stergiopulos et al. [3]. The parameters 𝒎𝒎𝒄𝒄, 𝒎𝒎𝒓𝒓, 𝝉𝝉𝒄𝒄 and 𝝉𝝉𝒓𝒓 represent contraction rate, relaxation rate, contraction time offset and relaxation time offset respectively in accordance to measurements from Mynard and Senzaki [4, 5]. The time 𝑻𝑻 equals the length of the heartbeat. The atria’s active tension was simulated with the DHM. The second model is the Lumens model (LM) which uses the myofiber mechanics proposed by Lumens et al. [6]. There the active tension is calculated using two differential equations: 𝒅𝒅𝑳𝑳𝒔𝒔𝒄𝒄𝒅𝒅𝒕𝒕= (𝑳𝑳𝒔𝒔−𝑳𝑳𝒔𝒔𝒄𝒄𝑳𝑳𝒔𝒔𝒔𝒔,𝒊𝒊𝒔𝒔𝒊𝒊− 𝟏𝟏) ⋅ 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 ,  (3) 𝒅𝒅𝒅𝒅𝒅𝒅𝒕𝒕=𝟏𝟏𝝉𝝉𝑹𝑹⋅ 𝒅𝒅𝑳𝑳(𝑳𝑳𝒔𝒔𝒄𝒄) ⋅ 𝑭𝑭𝒓𝒓𝒊𝒊𝒔𝒔𝒔𝒔(𝒕𝒕) +𝟏𝟏𝝉𝝉𝑫𝑫⋅𝒅𝒅𝒓𝒓𝒔𝒔𝒔𝒔𝒕𝒕−𝒅𝒅𝟏𝟏+𝒔𝒔(𝑻𝑻(𝑳𝑳𝒔𝒔𝒄𝒄)−𝒕𝒕)/𝝉𝝉𝑫𝑫 , (4) 𝝈𝝈𝒇𝒇,𝒎𝒎𝒄𝒄𝒕𝒕 = 𝝈𝝈𝒎𝒎𝒄𝒄𝒕𝒕 ⋅ 𝒅𝒅 ⋅ (𝑳𝑳𝒔𝒔𝒄𝒄 − 𝑳𝑳𝒔𝒔𝒄𝒄𝒔𝒔) ⋅𝑳𝑳𝒔𝒔−𝑳𝑳𝒔𝒔𝒄𝒄𝑳𝑳𝒔𝒔𝒔𝒔,𝒊𝒊𝒔𝒔𝒊𝒊 .  (5) 𝑳𝑳𝒔𝒔𝒄𝒄 is the length of the contractile cardiomyocyte part,  is the sarcomere length, 𝑳𝑳𝒔𝒔𝒔𝒔,𝒊𝒊𝒔𝒔𝒊𝒊 is the length of the isometric stressed elastic cardiomyocyte part, 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 is the sarcomere shortening velocity with zero load, 𝒅𝒅 describes the time course of mechanical activation, parameters 𝝉𝝉𝑹𝑹 and 𝝉𝝉𝑫𝑫 are scaling the rise and decay time,  is the time and 𝒅𝒅𝒓𝒓𝒔𝒔𝒔𝒔𝒕𝒕 is the diastolic resting level of the activation. Functions 𝒅𝒅𝑳𝑳, 𝑭𝑭𝒓𝒓𝒊𝒊𝒔𝒔𝒔𝒔 and 𝑻𝑻 describe the increase of activation with sarcomere length, the rise of mechanical activation, and the decrease of activation duration with the decrease of sarcomere length respectively and were defined according to [3]. 𝑳𝑳𝒔𝒔𝒄𝒄𝒔𝒔 is the length of the contractile cardiomyocyte part without load and 𝝈𝝈𝒎𝒎𝒄𝒄𝒕𝒕 is a scaling factor. In contrast to DHM, the LM uses an ODE-based formulation and is able to dynamically react on pre-stretch and a changing fiber length. It respects the physiologically imposed maximum sarcomere shortening length, which is at 84% of the sarcomere resting length [7, 8]. This length dependency results in a spatially heterogeneous tension distribution, whereas the DHM a uniform tension distribution provides. 2.2 Computational model The heart geometry was created from MRI data of a healthy 28 year old volunteer and was kindly provided by the Universitätsklinik Heidelberg. The resulting mesh consists of 39 k nodes and 61 k tetrahedral ten-node-cells and contains the left and right ventricles (LV, RV) as well as the left and right atria (LA, RA), the left and right atrioventricular rings and the left and right atrioventricular planes. 2.3 Simulation settings To evaluate the effects of the LAT on the active tension, four simulation settings were created. The first two settings used the DHM. One was activated homogeneously, the other heterogeneously. The two settings using the LM were created analogously. Figure 1: The used LATs from 0 ms (blue, at the hearts apex) to 120 ms (red, at the hearts base).  In Figure 1 the activation time for each cell is visualized. These were obtained with the fast marching method. The excitation starts at the apex and moves upwards to the base over a time of 120 ms, resembling the QRS-Complex length. To create an additional average pre-stretch of about 15% of the sarcomere resting length, both ventricles were inflated to their end-diastolic pressures before the contraction began. 3 Results Each setting was simulated for the duration of five heartbeats à 800 ms. The atrial contraction reached a maximum tension of 35 kPa. The simulations with the DHM resulted in a maximum tension of 80 kPa in the left ventricle, with and without using LAT. The values were reached every 800 ms due to the periodicity of the DHM. Since the right ventricle has a physiologically imposed lower tension development than the left ventricle and therefore a lower maximum tension, different scalings for the ventricles were used. Thus the right ventricular tension reached only 60 kPa. This resulted in the tension distribution seen in Figure 2. 248 — A. Müller et al.: Effects of local activation times on the tension development of human cardiomyocytes ... Bereitgestellt von | Karlsruher Institut für Technologie KIT-BibliothekAngemeldetHeruntergeladen am | 08.02.19 10:37The simulations with the Lumens model without LATs reached a maximum active tension of 104.3 kPa in the left ventricle. Using LATs, the maximum tension of the left ventricle dropped by 0.5% to 103.8 kPa in the first heartbeat. Figure 2: Changes of the active tension for the DHM. In the first and third column with, in the second and fourth column without LATs. The active tension ranged between 0 Pa (dark blue) and 80 kPa (red). Figure 3: Changes of the active tension for the LM In the first and third column with, in the second and fourth column without LATs.  The active tension ranged between 0 Pa (dark blue) and 100 kPa (red). In Figure 3 we can also clearly see the spatially heterogeneous distributed tension of the LM.  The blood volumes and the pressures generated from the different settings can be seen in Table 1.    DHM without local activation time DHM with local activation time LM without local activation time LM with local  activation time LV Stroke Volume 76.98 74.44 59.88 55.75 LV Ejection Fraction 21.01% 21.21% 16.80% 16.02% RV Stroke Volume 65.81 59.91 50.01 45.21 RV Ejection Fraction 33.96% 32.11% 18.25% 16.77% Total Stroke Volume 142.79 134.36 109.89 100.96 Total Ejection Fraction 25.49% 24.99% 17.43% 16.65% Maximum LV Pressure 133.97 130.86 100.69 95.27 Minimum LV Pressure 2.69 2.46 2.92 2.87 Maximum RV Pressure 23.96 23.25 19.80 19.25 Minimum RV Pressure -0.19 -0.18 0.71 0.73  Table 1: Pressure and volume data. The values were extracted from each simulation’s last heartbeat. Pressures are given in mmHg and volumes are given in ml. 4 Discussion and Outlook When using cell activation delayed by a local activation time, the cells reached their tension maximum one after another and not simultaneously, which led to a drop in the maximum pressure in the ventricles for both tension models. The pressure drop in the left ventricle was 2.3% for the DHM and 5.4% for the LM compared to the homogeneous case. The right ventricle maximum pressure dropped by 3% and 2.8% respectively. Also the volumes decreased in both ventricles using LATs. Due to the lower volumes the overall ejection fraction was also decreased and dropped to 16.7% when the more precise LM was used. This is well below the physiological values for the ejection fraction, however ejection volumes are normal or even slightly above the normal range. One problem might be the limited resolution of the mesh, which allows only a coarse representation of the real geometry. For instance the papillary muscles and the trabeculae weren’t modelled. Their volume contributes to the blood volume instead, which reduces the computed ejection fractions. We also inflated our mesh for the initial tension calculation to a better a pre-stretch. Therefore the volume of the whole heart is expanded which additionally decreases the ejection fractions.  A. Müller et al.: Effects of local activation times on the tension development of human cardiomyocytes ... — 249Bereitgestellt von | Karlsruher Institut für Technologie KIT-BibliothekAngemeldetHeruntergeladen am | 08.02.19 10:37Since the drop in maximum pressure is not particularly high, LATs might be neglected in favor of a simpler model. The DHM is not capable of respecting the limits of physiological myocardial fibers or incorporating effects like an increased tension due to pre-stretch. A tension model like the LM might be more suitable to reproduce not only a correct global ventricular elasticity, but also respect the tension-length-dependency. For more sophisticated modeling one can consider using other models such as the model proposed by Sander Land in 2017 [9]. The measurements of the cardiomyocytes used to create this model were made at body temperature rather than at the often used room temperature. Additionally the created model is designed for a four chamber model. A simulation of an inhomogeneous activation while using the Land model could show the effects of local activation times more precisely. Then it might be possible to obtain a different result regarding the importance of inhomogeneous activation times.  Author Statement Research funding: The author state no funding involved. Conflict of interest: Authors state no conflict of interest. Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use complies with all the relevant national regulations, institutional policies and was performed in accordance with the tenets of the Helsinki Declaration, and has been approved by the authors' institutional review board or equivalent committee. References [1] R. Rivlin, “Large elastic deformations of isotropic materials IV. Further developments of the general theory,” Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Band 241, 835th ed, p. 379–397, 1948.  [2] J. Guccione, A. McCulloch, & L. Waldman, “Passive material properties of intact ventricular myocardium determined from a cylindrical model,” vol. 113, p. 42–55, 03 1991. [3] N. Stergiopulos, J. J. Meister, & N. Westerhof, “Determinants of stroke volume and systolic and diastolic aortic pressure,” American Journal of Physiology - Heart and Circulatory Physiology, vol. 270, 6th ed, p. H2050–H2059, 1996. [4] J. P. Mynard, J. J. Smolich, “One-dimensional haemodynamic modeling and wave dynamics in the entire adult circulation,” Annals of Biomedical Engineering, vol. 43, 6th ed, p. 1443–1460, 2015. [5] H. Senzaki, C.-H. Chen, D. A. Kass, “Single-beat estimation of end-systolic pressure-volume relation in humans. A new method with the potential for noninvasive application,” Circulation, vol. 94, 10th ed, p. 2497–2506, 1996. [6] J. Lumens, T. Delhaas, B. Kirn, et al., “Three-wall segment (triseg) model describing mechanics and hemodynamics of ventricular interaction,” Annals of Biomedical Engineering, vol. 37, 11th ed, p. 2234–2255, Nov 2009. [7] D. Allen & J. Kentish, “The cellular basis of the length-tension relation in cardiac muscle,” Journal of Molecular and Cellular Cardiology, vol. 17, 9th ed, p. 821–840, 1985. [8] C. Vahl, T. Timek, A. Bonz, et al., “Length dependence of calcium- and force-transients in normal and failing human myocardium,” Journal of Molecular and Cellular Cardiology, vol. 30, 5th ed, p. 957 – 966, 1998. [9] S. Land, S.-J. Park-Holohan, N. P. Smith, et al., “A model of cardiac contraction based on novel measurements of tension development in human cardiomyocytes,” Journal of Molecular and Cellular Cardiology, vol. 106, p. 68–83, 2017. 250 — A. 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Effects of local activation times on the tension development of human cardiomyocytes in a computational model

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