A Discrete Model to Study Reaction-Diffusion-Mechanics Systems
Citation: Weise LD, Nash MP, Panfilov AV (
A Discrete Model to Study Reaction-Diffusion-Mechanics Systems
Louis D. Weise 0
Martyn P. Nash 0
Alexander V. Panfilov 0
Michael Breakspear, Queensland Institute of Medical Research, Australia
0 1 Department of Theoretical Biology, Utrecht University , Utrecht , The Netherlands , 2 Auckland Bioengineering Institute and Department of Engineering Science, The University of Auckland , Auckland , New Zealand , 3 Department of Physics and Astronomy, Ghent University , Ghent , Belgium
This article introduces a discrete reaction-diffusion-mechanics (dRDM) model to study the effects of deformation on reaction-diffusion (RD) processes. The dRDM framework employs a FitzHugh-Nagumo type RD model coupled to a masslattice model, that undergoes finite deformations. The dRDM model describes a material whose elastic properties are described by a generalized Hooke's law for finite deformations (Seth material). Numerically, the dRDM approach combines a finite difference approach for the RD equations with a Verlet integration scheme for the equations of the mass-lattice system. Using this framework results were reproduced on self-organized pacemaking activity that have been previously found with a continuous RD mechanics model. Mechanisms that determine the period of pacemakers and its dependency on the medium size are identified. Finally it is shown how the drift direction of pacemakers in RDM systems is related to the spatial distribution of deformation and curvature effects.
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Funding: LDW is funded by the Netherlands Organization for Scientific Research (http://www.nwo.nl) (NWO Grant No. 613.000.604) of the research Council for
Physical Sciences (EW). MPN is supported by a James Cook Fellowship administered by the Royal Society of New Zealand (http://www.royalsociety.org.nz/) on
behalf of the New Zealand Government. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the
manuscript.
Competing Interests: The authors have declared that no competing interests exist.
Reaction-diffusion (RD) partial differential equations describe
important spatio-temporal phenomena, including waves and
patterns in a variety of chemical, physical, and biological systems.
Important examples of these phenomena include waves in the
Belousov-Zhabotinsky (BZ) reactions [1,2], waves of CO oxidation
on platinum surfaces [3], waves of spreading depression in nerve
tissue [4], and the morphogenesis of Dictyostelium discoideum (Dd)
[5,6]. In the heart, electrical waves of excitation propagate
through the tissue and initiate its contraction. RD-equations have
been successfully applied to model normal and abnormal wave
propagation in cardiac tissue, such as rotating spiral waves, whose
initiation may result in life-threatening arrhythmias [2,7]. In many
of the systems mentioned above, wave propagation is
accompanied by a deformation of the medium. Important examples include
the chemotactical motion of cells during Dd-morphogenesis [6],
the swelling and deswelling of a polymeric gel in the BZ reaction
[8] and the contraction of the cardiac muscle [9]. As the heart
contracts, its deformations feed back on the process of wave
propagation. This important phenomenon, called
mechanoelectrical feedback, has been extensively studied in cardiac
electrophysiology [10].
To model the effects of deformation on wave propagation in
RD systems, it is necessary to describe the underlying mechanical
phenomena in terms of the RD process. As such, a coupled
reaction-diffusion-mechanics (RDM) framework was introduced in
[11] and applied to study cardiac tissue. In particular, the RD
equations were combined with the equations of finite deformation
continuum mechanics. With this approach several important
effects of deformation on wave propagation were identified such as
self-organized pacemakers, spiral wave drift, and break-up of spiral
waves [12,13].
Continuum mechanics is among the most valuable and widely
used approaches in engineering and modeling studies, however, it
does not explicitly describe the particular micro-organization of a
medium, which might be important for certain aspects of RDM
systems. Cardiac tissue, for example, consists of individual cells
that form layers of muscle fibers, which are tightly packed and
organized by an extra-cellular matrix into branching sheet
structures [14,15]. To study how this affects the elastic properties
of the heart, discrete models with similar micro-structure need to
be developed. Discrete models are computationally efficient and
widely used in various applications such as computer graphics
[16], medical tissue visualization [17], and the development of
elasto-mechanical models of anisotropic materials [18] such as
heart tissue [19,20]. Discrete models are also used to describe
discontinuous deformations in the case of fracture, plastic
deformation, and mass mixing processes [21,22].
In this paper, discrete elastic (...truncated)