Mathematical Models of Cardiac Cells Arrangements: The Bidomain Model

2014 ◽  
pp. 77-122
Author(s):  
Piero Colli Franzone ◽  
Luca F. Pavarino ◽  
Simone Scacchi
Keyword(s):  
Mathematical Models ◽  
Cardiac Cells ◽  
Bidomain Model

Irregular Dynamics of Excitation in Biologic and Mathematical Models of Cardiac Cells

1990 ◽  
Vol 601 (1 Electrocardio) ◽  
pp. 281-298 ◽  
Author(s):  
ALAIN VINET ◽  
DANTE R. CHIALVO ◽  
JOSE JALIFE
Keyword(s):  
Mathematical Models ◽  
Cardiac Cells

Numerical solution of the bidomain equations

2009 ◽  
Vol 367 (1895) ◽  
pp. 1931-1950 ◽  
Author(s):  
S. Linge ◽  
J. Sundnes ◽  
M. Hanslien ◽  
G.T. Lines ◽  
A. Tveito
Keyword(s):  
Mathematical Models ◽  
Cardiac Electrophysiology ◽  
Mathematical Reasoning ◽  
Full Potential ◽  
Processing Unit ◽  
Bidomain Model ◽  
Central Processing ◽  
Cellular Models ◽  
Bidomain Equations ◽  
The Past

Knowledge of cardiac electrophysiology is efficiently formulated in terms of mathematical models. However, most of these models are very complex and thus defeat direct mathematical reasoning founded on classical and analytical considerations. This is particularly so for the celebrated bidomain model that was developed almost 40 years ago for the concurrent analysis of extra- and intracellular electrical activity. Numerical simulations based on this model represent an indispensable tool for studying electrophysiology. However, complex mathematical models, steep gradients in the solutions and complicated geometries lead to extremely challenging computational problems. The greatest achievement in scientific computing over the past 50 years has been to enable the solving of linear systems of algebraic equations that arise from discretizations of partial differential equations in an optimal manner, i.e. such that the central processing unit (CPU) effort increases linearly with the number of computational nodes. Over the past decade, such optimal methods have been introduced in the simulation of electrophysiology. This development, together with the development of affordable parallel computers, has enabled the solution of the bidomain model combined with accurate cellular models, on geometries resembling a human heart. However, in spite of recent progress, the full potential of modern computational methods has yet to be exploited for the solution of the bidomain model. This paper reviews the development of numerical methods for solving the bidomain model. However, the field is huge and we thus restrict our focus to developments that have been made since the year 2000.

scholarly journals A note on discontinuous rate functions for the gate variables in mathematical models of cardiac cells

2010 ◽  
Vol 1 (1) ◽  
pp. 945-950 ◽  
Author(s):  
Monicaa Hanslien ◽  
Nina Holden ◽  
Joakim Sundnes
Keyword(s):  
Mathematical Models ◽  
Cardiac Cells ◽  
Rate Functions

scholarly journals Operator Splitting and Finite Difference Schemes for Solving the EMI Model

2020 ◽  
pp. 44-55
Author(s):  
Karoline Horgmo Jæger ◽  
Kristian Gregorius Hustad ◽  
Xing Cai ◽  
Aslak Tveito
Keyword(s):  
Finite Difference ◽  
Mathematical Models ◽  
Individual Cell ◽  
Operator Splitting ◽  
Cardiac Cells ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Coupled Equations ◽  
Computational Mesh ◽  
Large Numbers

Abstract We want to be able to perform accurate simulations of a large number of cardiac cells based on mathematical models where each individual cell is represented in the model. This implies that the computational mesh has to have a typical resolution of a few µm leading to huge computational challenges. In this paper we use a certain operator splitting of the coupled equations and showthat this leads to systems that can be solved in parallel. This opens up for the possibility of simulating large numbers of coupled cardiac cells.

Confocal imaging of calcium in intact ventricular muscle provides a new view of excitation-contraction coupling

1996 ◽  
Vol 54 ◽  
pp. 738-739
Author(s):  
W.G. Wier
Keyword(s):  
Local Control ◽  
Cardiac Tissue ◽  
Cell Function ◽  
Isolated Heart ◽  
Cardiac Cells ◽  
Confocal Imaging ◽  
Ion Concentration ◽  
Heart Cell ◽  
Free Calcium ◽  
Heart Cells

A fundamentally new understanding of cardiac excitation-contraction (E-C) coupling is being developed from recent experimental work using confocal microscopy of single isolated heart cells. In particular, the transient change in intracellular free calcium ion concentration ([Ca2+]i transient) that activates muscle contraction is now viewed as resulting from the spatial and temporal summation of small (∼ 8 μm3), subcellular, stereotyped ‘local [Ca2+]i-transients' or, as they have been called, ‘calcium sparks'. This new understanding may be called ‘local control of E-C coupling'. The relevance to normal heart cell function of ‘local control, theory and the recent confocal data on spontaneous Ca2+ ‘sparks', and on electrically evoked local [Ca2+]i-transients has been unknown however, because the previous studies were all conducted on slack, internally perfused, single, enzymatically dissociated cardiac cells, at room temperature, usually with Cs+ replacing K+, and often in the presence of Ca2-channel blockers. The present work was undertaken to establish whether or not the concepts derived from these studies are in fact relevant to normal cardiac tissue under physiological conditions, by attempting to record local [Ca2+]i-transients, sparks (and Ca2+ waves) in intact, multi-cellular cardiac tissue.

Handbook of Mathematical Models for Languages and Computation

2019 ◽  
Author(s):  
Alexander Meduna ◽  
Petr Horacek ◽  
Martin Tomko
Keyword(s):  
Mathematical Models
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