Engelbert Tijskens

A particle-based model to simulate the micromechanics of single-plant parenchyma cells and aggregates

This paper is concerned with addressing how plant tissue mechanics is related to the micromechanics of cells. To this end, we propose a mesh-free particle method to simulate the mechanics of both individual plant cells (parenchyma) and cell aggregates in response to external stresses. The model considers two important features in the plant cell: (1) the cell protoplasm, the interior liquid phase inducing hydrodynamic phenomena, and (2) the cell wall material, a viscoelastic solid material that contains the protoplasm. In this particle framework, the cell fluid is modeled by smoothed particle hydrodynamics (SPH), a mesh-free method typically used to address problems with gas and fluid dynamics. In the solid phase (cell wall) on the other hand, the particles are connected by pairwise interactions holding them together and preventing the fluid to penetrate the cell wall. The cell wall hydraulic conductivity (permeability) is built in as well through the SPH formulation. Although this model is also meant to be able to deal with dynamic and even violent situations (leading to cell wall rupture or cell-cell debonding), we have concentrated on quasi-static conditions. The results of single-cell compression simulations show that the conclusions found by analytical models and experiments can be reproduced at least qualitatively. Relaxation tests revealed that plant cells have short relaxation times (1 micros-10 micros) compared to mammalian cells. Simulations performed on cell aggregates indicated an influence of the cellular organization to the tissue response, as was also observed in experiments done on tissues with a similar structure.

Analysis of initial cell spreading using mechanistic contact formulations for a deformable cell model.

Adhesion governs to a large extent the mechanical interaction between a cell and its microenvironment. As initial cell spreading is purely adhesion driven, understanding this phenomenon leads to profound insight in both cell adhesion and cell-substrate interaction. It has been found that across a wide variety of cell types, initial spreading behavior universally follows the same power laws. The simplest cell type providing this scaling of the radius of the spreading area with time are modified red blood cells (RBCs), whose elastic responses are well characterized. Using a mechanistic description of the contact interaction between a cell and its substrate in combination with a deformable RBC model, we are now able to investigate in detail the mechanisms behind this universal power law. The presented model suggests that the initial slope of the spreading curve with time results from a purely geometrical effect facilitated mainly by dissipation upon contact. Later on, the spreading rate decreases due to increasing tension and dissipation in the cells cortex as the cell spreads more and more. To reproduce this observed initial spreading, no irreversible deformations are required. Since the model created in this effort is extensible to more complex cell types and can cope with arbitrarily shaped, smooth mechanical microenvironments of the cells, it can be useful for a wide range of investigations where forces at the cell boundary play a decisive role.

Mechanisms of soft cellular tissue bruising. A particle based simulation approach

This paper is concerned with modeling the mechanical behavior of cellular tissue in response to dynamic stimuli. The objective is to investigate the formation of bruises and other damage in tissue under excessive loading. We propose a particle based model to numerically study cells and aggregates of cells described on to subcellular detail. The model focuses on a parenchyma cell type in which two important features are present: the cells interior liquid-like phase inducing hydrodynamic phenomena; and the cell wall, a viscoelastic-plastic solid membrane that encloses the protoplast. The cell fluid is modeled by a Smoothed Particle Hydrodynamics (SPH) technique, while for the cell wall and cell adhesion a nonlinear discrete element model is proposed. Failure in the system is addressed to either cell wall rupture or to debonding of the middle lamella. We show that the model is able to reproduce experimental data of quasistatic compression, and investigate the role of the protoplasm viscosity and the cellular structure on the dynamics of the aggregate system. This indicates that a high viscosity causes better guidance of mechanical stresses through the tissue and can result in a higher penetration of damage, whereas low values will cause more local bruising effects.

Solving microscopic flow problems using Stokes equations in SPH

Abstract Starting 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 non-inertial 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 plasma.

Automatic differentiation for solving nonlinear partial differential equations: an efficient operator overloading approach

By resorting to Automatic Differentiation (AD) users of nonlinear PDE solvers can be relieved from the extra work of linearising a nonlinear PDE system and at the same time improve on the computational efficiency. This paper describes the main AD techniques and discusses how the operator overloading approach of AD can be extended to eliminate the overhead generally incurred with operator overloading. A recent AD system FastDer++, specially designed for this purpose, is integrated into a Least Squares solver. The necessary modifications to the general FEM algorithms. Code fragments and timing results demonstrate that (1) integrating AD with nonlinear PDE solvers leads to highly flexible code with a close resemblance to the mathematical expression of the problem, (2) coding and debugging efforts are greatly reduced, and (3) the computational efficiency is improved.

Quantifying the mechanical micro-environment during three-dimensional cell expansion on microbeads by means of individual cell-based modelling.

Controlled in vitro three-dimensional cell expansion requires culture conditions that optimise the biophysical micro-environment of the cells during proliferation. In this study, we propose an individual cell-based modelling platform for simulating the mechanics of cell expansion on microcarriers. The lattice-free, particle-based method considers cells as individual interacting particles that deform and move over time. The model quantifies how the mechanical micro-environment of individual cells changes during the time of confluency. A sensitivity analysis is performed, which shows that changes in the cell-specific properties of cell–cell adhesion and cell stiffness cause the strongest change in the mechanical micro-environment of the cells. Furthermore, the influence of the mechanical properties of cells and microbead is characterised. The mechanical micro-environment is strongly influenced by the adhesive properties and the size of the microbead. Simulations show that even in the absence of strong biological heterogeneity, a large heterogeneity in mechanical stresses can be expected purely due to geometric properties of the culture system. Supplemental data for this article can be accessed online.

A cell based modelling framework for skeletal tissue engineering applications

In this study, a cell based lattice free modelling framework is proposed to study cell aggregate behaviour in bone tissue engineering applications. The model encompasses cell-to-cell and cell-environment interactions such as adhesion, repulsion and drag forces. Oxygen, nutrients, waste products, growth factors and inhibitors are explicitly represented in the model influencing cellular behaviour. Furthermore, a model for cell metabolism is incorporated representing the basic enzymic reactions of glycolysis and the Krebs cycle. Various types of cell death such as necrosis, apoptosis and anoikis are implemented. Finally, an explicit model of the cell cycle controls the proliferation process, taking into account the presence or absence of various metabolites, sufficient space and mechanical stress. Several examples are presented demonstrating the potential of the modelling framework. The behaviour of a synchronised cell aggregate under ideal circumstances is simulated, clearly showing the different stages of the cell cycle and the resulting growth of the aggregate. Also the difference in aggregate development under ideal (normoxic) and hypoxic conditions is simulated, showing hypoxia induced necrosis mainly in the centre of the aggregate grown under hypoxic conditions. The next step in this research will be the application of this modelling framework to specific experimental set-ups for bone tissue engineering applications.

The Ionic Model: Extension to Spatial Charge Distributions

In this paper the validity of the classical ionic model, using a Madelung term and a Born-Mayer repulsive term, is investigated quantatively for systems with a considerable overlap of the electron clouds of neighbouring ions, such as silicates with a high degree of polymerisation. A modified ionic model is presented which takes into account the spatial extent of the ions within the approximation of spherical atoms. Both models are tested against quantum mechanical electron densities and energies for SiO44--clusters. The data demonstrate the validity of the spherical atom approximation, producing a fit of 99.995%, and the importance of manybody effects maintaining the spherical symmetry of the electron clouds as contraction/expansion of the ions and charge transfer between ions. Although the new interaction potential is physically more plausible than the classical Born-Mayer model, both models reproduce the quantum mechanical potential surface with numerical accuracies of the same order of magnitude. The new model provides an improved tool for judging between ionic and non-ionic effects and for analysis of the quantum mechanical electron densities and interaction energies.

Strategies for contact resolution of level surfaces

Particles in granular matter can have very different and irregular shapes. The computational treatment of nonspherical objects is a major difficulty in the simulation of granular flows. In this paper, two basic strategies for contact resolution between objects described by level surfaces are presented and analyzed. They are based on the iterative solution of systems of nonlinear equations. The major difficulties are pinpointed and necessary steps toward a generic algorithm are proposed. A test case of colliding cardioids in two dimensions is used to demonstrate the algorithms and illustrate common pitfalls.

FastDer++, efficient automatic differentiation for non-linear PDE solvers

FastDer++ is a C++ class library for automatic differentiation designed for use in situations where a set of dependent variables and their gradients are to be evaluated in a large number of points. Typical settings constitute non-linear systems of partial differential equations (PDEs) and ODEs. Although automatic differentiation is traditionally considered to slow for implementation in non-linear PDE and ODE solvers, it has recently been demonstrated [E. Tijskens, H. Ramon, J. De Baerdemaeker, Efficient operator overloading AD for solving non-linear PDEs, in: G. Corliss, C. Faure, A. Griewank, L. Hascoet, U. Nauman (Eds.), Automatic Differentiation of Algorithms--From Simulation to Optimisation, Springer, Verlag, 2002; Num. Algorithms 30 (2002) 259] that thanks to an extension called vectorised AD and careful design handcoded derivatives, finite differencing and state of the art AD tools can be outperformed in common situations. In addition, the user gains the advantage of directly dealing with the non-linear equations rather than with its linearised counterpart. This paper describes the FastDer++ library and its underlying principles in detail, both from the point of implementation and of user programming.