Discrete element framework for modelling extracellular matrix, deformable cells and subcellular components

This paper presents a framework for modelling biological tissues based on discrete particles. Cell components (e.g. cell membranes, cell cytoskeleton, cell nucleus) and extracellular matrix (e.g. collagen) are represented using collections of particles. Simple particle to particle interaction laws are used to simulate and control complex physical interaction types (e.g. cell-cell adhesion via cadherins, integrin basement membrane attachment, cytoskeletal mechanical properties). Particles may be given the capacity to change their properties and behaviours in response to changes in the cellular microenvironment (e.g., in response to cell-cell signalling or mechanical loadings). Each particle is in effect an 'agent', meaning that the agent can sense local environmental information and respond according to pre-determined or stochastic events. The behaviour of the proposed framework is exemplified through several biological problems of ongoing interest. These examples illustrate how the modelling framework allows enormous flexibility for representing the mechanical behaviour of different tissues, and we argue this is a more intuitive approach than perhaps offered by traditional continuum methods. Because of this flexibility, we believe the discrete modelling framework provides an avenue for biologists and bioengineers to explore the behaviour of tissue systems in a computational laboratory

Efficient solution methods for modelling slowly evolving mechanical phenomena in cells and tissues using the discrete element method

Cells and tissues exhibit complex mechanical behaviour, including large deformations, migration, growth, cell proliferation and death, as well as changes in behaviour due to external factors (e.g. chemical signals). The discrete element method is well suited for modelling such complicated behaviour, but computational efficiency is difficult to achieve due to the large number of particles needed for discretisation. Most of the mechanical behaviour of cells and tissues takes place slowly enough that it can be considered a quasi-static process. Taking this into account, we developed very efficient algorithms which ensure solution convergence and greatly reduce the computation time; these include an efficient neighbour search algorithm, explicit time integration with dynamic relaxation and stability control using mass scaling. In this paper we describe these algorithms and evaluate their performance using several numerical experiments. We demonstrate how some complex phenomena (constant tension membrane, growth, tissue degradation) can be easily modelled using the proposed methods

A new method for essential boundary conditions imposition in explicit meshless methods

In this paper we present a new technique of enforcing Essential Boundary Conditions (EBC) in Meshless Methods (MMs) based on the Element Free Galerkin (EFG) principles. Imposing EBC is a fundamental issue in MMs. The imposition of prescribed displacement values on the boundary in MMs based on approximating shape functions is not as straightforward as in the Finite Element Method (FEM) because the meshless shape functions are generally not interpolating at nodes. Furthermore, many techniques of enforcing EBC are not compatible with explicit time integration schemes. This paper describes a new method of imposing EBC in EFG based MMs suitable for explicit time integration, named Essential Boundary Conditions Imposition in Explicit Meshless (EBCIEM). The effectiveness of the proposed method is demonstrated using both 2D and 3D numerical example

Strong-and weak-Form meshless methods in computational biomechanics

Meshless methods (MMs) were introduced in the late 1970s to solve problems in astrophysics. In MMs the spatial domain is represented by a set of nodes (cloud of points) and not discretized by elements as in most of the mesh-based methods (finite difference method, finite element method, finite volume method); consequently, there is no need for predefined connectivity between the nodes. In this chapter we are going to give an overview of applications, advantages, and disadvantages of various MMs developed and applied in the context of computational biomechanics. Strong and weak formulations will be presented, focusing on the novel interpolation schemes such as modified moving least squares and discretization correction particle strength exchange method, along with the meshless total Lagrangian explicit dynamics method. The applicability of the methods in multiscale problems and their inherent parallelization will be depicted through various applications, along with their advantages over the traditional mesh-based numerical methods. MMs can be considered as mainstream numerical methods able to tackle demanding engineering applications. Intensive and rigorous research in the field will make MMs robust enough to be used by industry

Strong-and weak-Form meshless methods in computational biomechanics

Meshless methods (MMs) were introduced in the late 1970s to solve problems in astrophysics. In MMs the spatial domain is represented by a set of nodes (cloud of points) and not discretized by elements as in most of the mesh-based methods (finite difference method, finite element method, finite volume method); consequently, there is no need for predefined connectivity between the nodes. In this chapter we are going to give an overview of applications, advantages, and disadvantages of various MMs developed and applied in the context of computational biomechanics. Strong and weak formulations will be presented, focusing on the novel interpolation schemes such as modified moving least squares and discretization correction particle strength exchange method, along with the meshless total Lagrangian explicit dynamics method. The applicability of the methods in multiscale problems and their inherent parallelization will be depicted through various applications, along with their advantages over the traditional mesh-based numerical methods. MMs can be considered as mainstream numerical methods able to tackle demanding engineering applications. Intensive and rigorous research in the field will make MMs robust enough to be used by industry

An explicit meshless point collocation method for electrically driven magnetohydrodynamics (MHD) flow

In this paper, we develop a meshless collocation scheme for the numerical solution of magnetohydrodynamics (MHD) flow equations. We consider the transient laminar flow of an incompressible, viscous and electrically conducting fluid in a rectangular duct. The flow is driven by the current produced by electrodes placed on the walls of the duct. The method combines a meshless collocation scheme with the newly developed Discretization Corrected Particle Strength Exchange (DC PSE) interpolation method. To highlight the applicability of the method, we discretize the spatial domain by using uniformly (Cartesian) and irregularly distributed nodes. The proposed solution method can handle high Hartmann (Ha) numbers and captures the boundary layers formed in such cases, without the presence of unwanted oscillations, by employing a local mesh refinement procedure close to the boundaries. The use of local refinement reduces the computational cost. We apply an explicit time integration scheme and we compute the critical time step that ensures stability through the Gershgorin theorem. Finally, we present numerical results obtained using different orientation of the applied magnetic field

Controlling seepage in discrete particle simulations of biological systems

It is now commonplace to represent materials in a simulation using assemblies of discrete particles. Sometimes, one wishes to maintain the integrity of boundaries between particle types, for example, when modelling multiple tissue layers. However, as the particle assembly evolves during a simulation, particles may pass across interfaces. This behaviour is referred to as ‘seepage’. The aims of this study were (i) to examine the conditions for seepage through a confining particle membrane and (ii) to define some simple rules that can be employed to control seepage. Based on the force-deformation response of spheres with various sizes and stiffness, we develop analytic expressions for the force required to move a ‘probe particle’ between confining ‘membrane particles’. We analyse the influence that particle’s size and stiffness have on the maximum force that can act on the probe particle before the onset of seepage. The theoretical results are applied in the simulation of a biological cell under unconfined compression

Biomechanical model for computing deformations for whole-body image registration: A meshless approach

Patient-specific biomechanical models have been advocated as a tool for predicting deformations of soft body organs/tissue for medical image registration (aligning two sets of images) when differences between the images are large. However, complex and irregular geometry of the body organs makes generation of patient-specific biomechanical models very time-consuming. Meshless discretisation has been proposed to solve this challenge. However, applications so far have been limited to 2D models and computing single organ deformations. In this study, 3D comprehensive patient-specific nonlinear biomechanical models implemented using meshless Total Lagrangian explicit dynamics algorithms are applied to predict a 3D deformation field for whole-body image registration

An implicit potential method along with a meshless technique for incompressible fluid flows for regular and irregular geometries in 2D and 3D

We present the Implicit Potential (IPOT) numerical scheme developed in the framework of meshless point collocation. The proposed scheme is used for the numerical solution of the steady state, incompressible Navier-Stokes (N-S) equations in their primitive variable (u-v-w-p) formulation. The governing equations are solved in their strong form using either a collocated or a semi-staggered type meshless nodal configuration. The unknown field functions and derivatives are calculated using the Modified Moving Least Squares (MMLS) interpolation method. Both velocity-correction and pressure-correction methods applied ensure the incompressibility constraint and mass conservation. The proposed meshless point collocation (MPC) scheme has the following characteristics: (i) it can be applied, in a straightforward manner to: steady, unsteady, internal and external fluid flows in 2D and 3D, (ii) it equally applies to regular an irregular geometries, (iii) a distribution of points is sufficient, no numerical integration in space nor any mesh structure are required, (iv) there is no need for pressure boundary conditions since no pressure constitutive equation is solved, (v) it is quite simple and accurate, (vi) results can be obtained using collocated or semi-staggered nodal distributions, (vii) there is no need to compute the velocity potential nor the unit normal vectors and (viii) there is no need for a curvilinear system of coordinates. Simulations of fluid flow in 2D and 3D for regular and irregular geometries indicate the validity of the proposed methodology