Lattice-free agent-based models and Delaunay-Object-Dynamics
Intracellular and cellular models may require the representation of individual agents in a lattice-free environment. We are using molecular dynamics derived methods with interactions between agents defined by interaction potentials. Lenard-Jones potentials are suitable for cell-cell interactions because they inherently include hard-core repulsion and adhesion.
As for the lattice-based models, these models may be combined with a reaction-diffusion system which is solved on an overlaid grid. We applied this kind of approach to problems in biofilm formation, cancer, and dynamics of intracellular organelles.
The observation that densely packed cells acquire a polygonal shape initiated the development of a new modelling strategy that combines cell geometry with the philosophy of molecular dynamics. This modelling platform is called Delaunay-Object-Dynamics (DOD). In DOD every cell is represented as a node. Its neighbours are derived from the Delaunay triangulation of all nodes using the empty circumsphere criterion. This replaces the sometimes difficult search for interaction neighbours in molecular dynamics because in DOD, the interaction neighbours are unambiguously determined. The dual of the Delaunay triangulation is the Voronoi tessellation, which gives every cell object a shape, a size, and an interaction face to other cells. The size of this interaction face is also known by construction and enables application of classical Newton mechanics.
If the biomechanical parameters of the cells are known, this model is a quantitative representation of cell dynamics with a high level of predictive power. We have made this concept usable for biological problems with many cells by developing a sequential and parallel version of a code with local adaptation of the Delaunay triangulation rather than re-triangulation of the whole system.
Sebastian C. Binder, Jaber Dehghany, Michael Meyer-Hermann
Grise G, Meyer-Hermann M. Surface reconstruction using Delaunay triangulation for applications in life sciences, Comput Phys Commun 182 (2011) 967-977.
Grise G, Meyer-Hermann M. Towards sub-cellular modeling with Delaunay triangulation, Math Model Nat Phenom 5 (2010) 224-238.
Meyer-Hermann M. Delaunay-Object-Dynamics: Cell mechanics with a 3D kinetic and dynamic weighted Delaunay-triangulation. Curr Top Dev Biol 81 (2008) 373-399.
Schaller G, Meyer-Hermann M. Multicellular tumor spheroid in an off-lattice Voronoi/Delaunay cell model. Phys Rev E 71 (2005) 051910-1-16.
Beyer T, Schaller G, Deutsch A, Meyer-Hermann M. Parallel dynamic and kinetic regular triangulation in three dimensions. Comput Phys Commun 172 (2005) 86-108.
Schaller G, Meyer-Hermann M. Kinetic and dynamic Delaunay tetrahedralizations in three dimensions. Comput Phys Commun 162 (2004) 9-23.
- Systems Immunology- Prof. Dr. Michael Meyer-Hermann