Abstract
We apply POD-DEIM model order reduction to a 0D/1D model used to simulate the propagation of action potentials through the myocardium or along skeletal muscle fibers. This corresponding system of ODEs (reaction) and PDEs (diffusion) is called the monodomain equation. 0D sets of ODEs describing the ionic currents flowing across the cell membrane are coupled along muscle fibers through a 1D diffusion process for the transmembrane potential. Due to the strong coupling of the transmembrane potential and other state variables describing the behavior of the membrane, a total reduction strategy including all degrees of freedom turns out to be more efficient than a reduction of only the transmembrane potential. The total reduction approach is four orders of magnitude more accurate than partial reduction and shows a faster convergence in the number of POD modes with respect to the mesh refinement. A speedup of 2.7 is achieved for a 1D mesh with 320 nodes. Considering the DEIM approximation in combination with the total reduction, the nonlinear functions corresponding to the ionic state variables are also approximated in addition to the nonlinear ionic current in the monodomain equation. We observe that the same number of DEIM interpolation points as the number of POD modes is the optimal choice regarding stability, accuracy and runtime for the current POD-DEIM approach.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Snapshots may be subsequent time steps or only a selection of time steps. In general, snapshots may be based on parameters independent of time. Therefore, we do not use parenthesis for the superscripts denoting the snapshots as we do for time step data.
- 2.
References
Pullan, A.J., Buist, M.L., Cheng, L.K.: Mathematically Modelling the Electrical Activity of the Heart: From Cell to Body Surface and Back Again. World Scientific Publishing Company, Singapore (2005)
Röhrle, O., Davidson, J.B., Pullan, A.J.: A physiologically based, multi-scale model of skeletal muscle structure and function. Front. Physiol. 3, (2012)
Heidlauf, T., Röhrle, O.: Modeling the Chemoelectromechanical Behavior of Skeletal Muscle Using the Parallel Open-Source Software Library OpenCMISS. Computational and Mathematical Methods in Medicine 1–14 (2013)
Heidlauf, T., Röhrle, O.: A multiscale chemo-electro-mechanical skeletal muscle model to analyze muscle contraction and force generation for different muscle fiber arrangements. Front. Physiol. 5(498), 1–14 (2014)
Mordhorst, M., Heidlauf, T., Röhrle, O.: Predicting electromyographic signals under realistic conditions using a multiscale chemo-electro-mechanical finite element model. Interface Focus 5(2), 1–11 (2015)
Heidlauf, T., Klotz, T., Rode, C., et al.: A multi-scale continuum model of skeletal muscle mechanics predicting force enhancement based on actin-titin interaction. Biomech. Model. Mechanobiol. 15(6), 1423–1437 (2016)
Heidlauf, T., Klotz, T., Rode, C., Siebert, T., Röhrle, O.: A continuum-mechanical skeletal muscle model including actin-titin interaction predicts stable contractions on the descending limb of the force-length relation. PLOS Comput. Biol. 13(10), 1–25, 10 (2017)
Miller, W.T., Geselowitz, D.B.: Simulation studies of the electrocardiogram. i. the normal heart. Circ. Res. 43(2), 301–315 (1978)
Tung, L.: A bi-domain model for describing ischemic myocardial DC potentials. PhD thesis, Massachusetts Institute of Technology (1978)
Bradley, C.P., Emamy, N., Ertl, T., et al.: Enabling detailed, biophysics-based skeletal muscle models on HPC systems. Front. Physiol. 9 (2018)
Clayton, R.H., Bernus, O., Cherry, E.M., et al.: Models of cardiac tissue electrophysiology: Progress, challenges and open questions. Prog. Biophys. Mol. Biol. 104(1–3), 22–48 (2011)
Kellems, A.R., Chaturantabut, S., Sorensen, D.C., Cox, S.J.: Morphologically accurate reduced order modeling of spiking neurons. J. Comput. Neurosci. 28(3), 477–494 (2010)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952)
Boulakia, M., Schenone, E., Gerbeau, J-F.: Reduced-order modeling for cardiac electrophysiology. Application to parameter identification. Int. J. Numer. Methods Biomed. Eng. 28(6–7), 727–744 (2012)
Mitchell, C.C., Schaeffer, D.G.: A two-current model for the dynamics of cardiac membrane. Bull. Math. Biol. 65(5), 767–793 (2003)
Yang, H., Veneziani, A.: Efficient estimation of cardiac conductivities via POD-DEIM model order reduction. Appl. Numer. Math. 115, 180–199 (2017)
Mordhorst, M., Strecker, T., Wirtz, D., Heidlauf, T., Röhrle, O.: POD-DEIM reduction of computational EMG models. J. Comput. Sci. 19, 86–96 (2017)
Rogers, J.M., Mc Culloch, A.D.: A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41(8), 743–757 (1994)
Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)
Wirtz, D.: Model Reduction for Nonlinear Systems: Kernel Methods and Error Estimation. epubli GmbH (2014)
Heidlauf, T.: Chemo-electro-mechanical modelling of the neuromuscular system. Institut fur Mechanik (Bauwesen), Lehrstuhl fur Kontinuumsmechanik, Research Group on Continuum Biomechanics and Mechanobiology, Universität Stuttgart (2015)
Shorten, P.R., O’Callaghan, P., Davidson, J.B., Soboleva, T.K.: A mathematical model of fatigue in skeletal muscle force contraction. J. Muscle Res. Cell Motil. 28(6), 293–313 (2007)
Acknowledgements
This research was funded by the Baden-Württemberg Stiftung as part of the DiHu project of the High Performance Computing II program and the Cluster of Excellence for Simulation Technology (EXC 310/1).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Emamy, N., Litty, P., Klotz, T., Mehl, M., Röhrle, O. (2020). POD-DEIM Model Order Reduction for the Monodomain Reaction-Diffusion Sub-Model of the Neuro-Muscular System. In: Fehr, J., Haasdonk, B. (eds) IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018. IUTAM Bookseries, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-030-21013-7_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-21013-7_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21012-0
Online ISBN: 978-3-030-21013-7
eBook Packages: EngineeringEngineering (R0)