Solving microscopic flow problems using Stokes equations in SPH

International audienceStarting from the Smoothed Particle Hydrodynamics method (SPH), we propose an alternative way to solve flow problems at a very low Reynolds number. The method is based on an explicit drop out of the inertial terms in the normal SPH equations, and solves the coupled system to find the velocities of the particles using the conjugate gradient method. The method will be called NSPH which refers to the noninertial character of the equations. Whereas the time-step in standard SPH formulations for low Reynolds numbers is linearly restricted by the inverse of the viscosity and quadratically by the particle resolution, the stability of the NSPH solution benefits from a higher viscosity and is independent of the particle resolution. Since this method allows for a much higher time-step, it solves creeping flow problems with a high resolution and a long timescale up to three orders of magnitude faster than SPH. In this paper, we compare the accuracy and capabilities of the new NSPH method to canonical SPH solutions considering a number of standard problems in fluid dynamics. In addition, we show that NSPH is capable of modeling more complex physical phenomena such as the motion of a red blood cell in plasm

Modeling extracellular matrix viscoelasticity using smoothed particle hydrodynamics with improved boundary treatment

International audienceWe propose viscoelastic smoothed particle hydrodynamics (SPH) with extended boundary conditions as a new method to model the extracellular matrix (ECM) in contact with a migrating cell. The contact mechanics between a cell and ECM is modeled based on an existing boundary method in SPH that corrects for the well-known missing kernel support problem in Fluid Structure Interactions (FSI). This boundary method is here extended to allow the modeling of moving boundaries in contact with a viscoelastic solid. To validate the method, simulations are performed of tractions applied to a viscoelastic solid, Stokes flow around an array of square pillars, and indentation of a viscoelastic material with a circular indenter. By drop out of the inertial terms in the SPH equations of motion, the new SPH formulation allows to solve problems in a low Reynolds environment with a timestep independent of the particle spacing, permitting to model processes at the cellular scale (i.e. µm-scale). The potential of the method to capture cell–ECM interactions is demonstrated by simulation of a self propelling object that locally degrades the ECM by fluidizing it to permit migration. This should enable us to model and understand realistic cell–matrix interactions in the future

Quantitative agent-based modeling reveals mechanical stress response of growing tumor spheroids is predictable over various growth conditions and cell lines

Model simulations indicate that the response of growing cell populations on mechanical stress follows the same functional relationship and is predictable over different cell lines and growth conditions despite the response curves look largely different. We develop a hybrid model strategy in which cells are represented by coarse-grained individual units calibrated with a high resolution cell model and parameterized measurable biophysical and cell-biological parameters. Cell cycle progression in our model is controlled by volumetric strain, the latter being derived from a bio-mechanical relation between applied pressure and cell compressibility. After parameter calibration from experiments with mouse colon carcinoma cells growing against the resistance of an elastic alginate capsule, the model adequately predicts the growth curve in i) soft and rigid capsules, ii) in different experimental conditions where the mechanical stress is generated by osmosis via a high molecular weight dextran solution, and iii) for other cell types with different growth kinetics. Our model simulation results suggest that the growth response of cell population upon externally applied mechanical stress is the same, as it can be quantitatively predicted using the same growth progression function

SOLVING MICROSCOPIC FLOW PROBLEMS USING STOKES EQUATIONS IN SPH

status: accepte

Simulating tissue mechanics with Agent Based Models: concepts and perspectives

International audienceIn this paper we present an overview of agent based models that are used to simulate mechanical and physiological phenomena in cells and tissues, and we discuss underlying concepts, limitations and future perspectives of these models. As the interest in cell and tissue mechanics increase, agent based models are becoming more common the modeling community. We overview the physical aspects, complexity, shortcomings and capabilities of the major agent based model categories: lattice-based models (cellular automata, lattice gas cellular automata, cellular Potts models), off-lattice models (center based models, deformable cell models, vertex models), and hybrid discrete-continuum models. In this way, we hope to assist future researchers in choosing a model for the phenomenon they want to model and understand. The article also contains some novel results

Coupling of a cell migration model with an (N)SPH substrate to investigate dynamic generation of cellular tractions

Cell migration is crucial for tissue development, maintenance and repair. To migrate, cells apply traction on the extra-cellular matrix (ECM) and degrade it. At the same time, the response of the ECM is an important cue for the cell to modulate mechanotransduction processes. To investigate cell-ECM interaction in the context of active spreading and migration, a previously developed deformable cell model [1] has been extended by (amongst others) including a protrusive pressure at the leading edge of the cell. In this model, the membrane/actin cortex of the cell is discretized into deformable rounded triangles, over which contact pressures including adhesion are integrated between the cell and the ECM. The matrix represents a typical substitute ECM in in vitro experiments that is continuous at the cellular level and has tunable physical and mechanical properties. To be able to model the possibly large deformations and degradation of this hydrogel, we chose the Non-inertial, Smoothed-Particle Hydrodynamics (NSPH) [2] method to avoid frequent remeshing that is typically required in finite element methods. We will discuss the extent to which this method can capture typical hydrogel properties by comparing with FEM simulations of a simplified system. In a second step, we will highlight the effects of hydrogel properties as well as the properties of the hydrogel-cell interaction on active spreading and migration of the deformable cell. In particular, the model is able to capture stress magnitudes and distributions which correspond to typical traction force microscopy results for migratory epithelial cells, see Figure 1. REFERENCES [1] Odenthal, T., Smeets, B., et al. (2013). Analysis of initial cell spreading using mechanistic contact formulations for a deformable cell model. PLoS Computational Biology, 9 (10), e1003267. [2] Van Liedekerke, P., et al. (2013). Solving microscopic flow problems using Stokes equations in SPH. Comp. Phys. Comm., 184 (7), 1686-1696.status: publishe

Simulation of micro-scale porous flow using Smoothed Particle Hydrodynamics

Fluid flow in a porous medium is a well-studied aspect of applied mathematics with significant real-world application. The standard modelling approach for this type of flow is to homogenise the porous structure. A dual-scale model, with the smaller scale at the pore-scale, would possibly capture the fluid mechanical phenomena more faithfully than a volume averaged approach. We investigate the significance of the microstructure shape on the flux through the medium. We also evaluate whether smoothed particle hydrodynamics may be viable in a dual-scale model. We find that varying the shape of the porous structure causes the average flux to vary significantly. This contradicts the assumption commonly made that only the porosity is important. We conclude that there is significant information present in the dual-scale model that is lost by a volume averaged model. We also find that the smoothed particle hydrodynamics simulation is computationally intensive, but that there is a time-saving measure that may provide viability to the dual-scale model. References S. Alyaev, E. Keilegavlen, and J. M. Nordbotten. Analysis of control volume heterogeneous multiscale methods for single phase flow in porous media. Multiscale Model. Sim., 12(1):335–363, 2014. doi:10.1137/120885541 H. Brenner. Dispersion resulting from flow through spatially periodic porous media. Phil. Trans. R. Soc. A, 297(1430):81–133, 1980. doi:10.1098/rsta.1980.0205 L. Brookshaw. A method of calculating radiative heat diffusion in particle simulations. Proc. Astron. Soc. Aust., 6(2):207–210, 1985. http://adsabs.harvard.edu/abs/1985PASAu...6..207B. E. J. Carr and I. W. Turner. Two-scale computational modelling of water flow in unsaturated soils containing irregular-shaped inclusions. Int. J. Numer. Meth. Eng., 98(3):157–173, 2014. doi:10.1002/nme.4625 A. J. Chorin. A numerical method for solving incompressible viscous flow problems. J. Comput. Phys., 2(1):12–26, 1967. doi:10.1016/0021-9991(67)90037-X R. A. Gingold and J. J. Monaghan. Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Mon. Not. R. Astron. Soc., 181:375–389, 1977. doi:10.1093/mnras/181.3.375 S. Litvinov, M. Ellero, X. Y. Hu, and N. A. Adams. A splitting scheme for highly dissipative smoothed particle dynamics. J. Comput. Phys., 229(15):5457–5464, 2010. doi:10.1016/j.jcp.2010.03.040 L. B. Lucy. A numerical approach to the testing of the fission hypothesis. Astron. J., 82(12):1013–1024, 1977. doi:10.1086/112164 C. C. Mei, J. L. Auriault, and C.-O. Ng. Some applications of the homogenization theory. Adv. Appl. Mech., 32:278–348, 1996. doi:10.1016/S0065-2156(08)70078-4 J. J. Monaghan. Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys., 30:543–574, 1992. doi:10.1146/annurev.aa.30.090192.002551 J. J. Monaghan. Implicit SPH drag and dusty gas dynamics. J. Comput. Phys., 138(2):801–820, 1997. doi:10.1006/jcph.1997.5846 J. J. Monaghan. From stars to volcanoes: The SPH story. In Eduardo Ramos, Gerardo Cisneros, Rafael Fernandez-Flores, and Alfredo Santillan-Gonzalez, editors, Computational Fluid Dynamics, Proceedings of the Fourth UNAM Supercomputing Conference, pages 193–203, Singapore, 2001. World Scientific. doi:10.1142/4623 J. J. Monaghan. Smoothed particle hydrodynamics. Rep. Prog. Phys., 68:1703–1759, 2005. doi:10.1088/0034-4885/68/8/R01 J. P. Morris. A study of the stability properties of smooth particle hydrodynamics. Publ. Astron. Soc. Aust., 13:97–102, 1996. http://adsabs.harvard.edu/abs/1996PASA...13...97M. J. P. Morris, P. J. Fox, and Y. Zhu. Modeling low Reynolds number incompressible flows using SPH. J. Comput. Phys., 136(1):214–226, 1997. doi:10.1006/jcph.1997.5776 D. J. Price. Smoothed particle hydrodynamics and magnetohydrodynamics. J. Comput. Phys., 231(3):759–794, 2012. doi:10.1016/j.jcp.2010.12.011 S. Shao and E. Y. M. Lo. Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv. Water Resour., 26(7):787–800, 2003. doi:10.1016/S0309-1708(03)00030-7 R. E. Showalter. Microstructure models of porous media. In Homogenization and Porous Media, pages 183–202. Interdisciplinary Applied Mathematics. Springer New York, 1997. doi:10.1007/978-1-4612-1920-0_9 A. Szymkiewicz, J. Lewandowska, R. Angulo-Jaramillo, and J. Butlariska. Two-scale modeling of unsaturated water flow in a double-porosity medium under axisymmetric conditions. Can. Geotech. J., 45:238–251, 2008. doi:10.1139/T07-096 P. van Liedekerke, B. Smeets, T. Odenthal, E. Tijskens, and H. Ramon. Solving microscopic flow problems using Stokes equations in SPH. Comput. Phys. Commun., 184:1686–1696, 2013. doi:10.1016/j.cpc.2013.02.013 S. Whitaker. Flow in porous media I: A theoretical derivation of Darcy's law. Transport Porous Med., 1(1):3–25, 1986. doi:10.1007/BF01036523 S. Whitaker. Coupled transport in multiphase systems: A theory of drying. Adv. Heat Trans., 31:1–104, 1998. doi:10.1016/S0065-2717(08)70240-