European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:30   MS50: Numerical methods to advance mathematical biology research (Part 1)
Chair: Alf Gerisch
25 mins
An ALE-method for simulating axisymmetric elastic surfaces in flow
Sebastian Aland, Marcel Mokbel
Abstract: We develop an axisymmetric ALE model to simulate the dynamics of elastic surfaces in flows. As a feature of the method, we take into account the in-plane elasticity of the surface. In consequence, we cannot only simulate inextensible lipid bilayer membranes, but arbitrary elastic shells, for instance the surface of mammalian cells, which due to an elastic cortex, can be significantly stretched. We present efficient and robust numerical stategies to couple the surface elasticity to the flow. A series of numerical tests illustrates the high efficiency and accuracy of the model. We simulate an experimental setting of a cell squished between two parallel plates. The comparison of numerical results with experimental data allows to extract the mechanical parameters of the cortical cell surface for the first time.
25 mins
Modelling the mechanics of the cell nucleus: a poroelastic model
Raquel Barreira, Anotida Madzvamuse
Abstract: The nucleus is the largest organelle of eukayotic cells and, although it was thought that the gene expression was a purely chemical phenomenum, it has been shown that mechanical cues can modify the expression of certain proteins. Also, due to the nucleus being relatively rigid when compared to the rest of the cell body, it is a limiting factor during important processes such as cell migration, changing its size and shape due to forces generated by the cell and the environment. Thus, understanding its dynamical mechanics is of key importance but some aspects of it are still unclear. Different models have been proposed to model the nucleus’ mechanics, such has viscous, elastic or viscoelastic models but there are some experimental evidences that it presents a poroelastic behaviour. By considering the nucleus to be composed by a porous solid medium (the chromatin mesh within the nucleus) and a fluid phase (the nucleoplasm) we propose a poroelastic model for the nucleus and present an overview of numerical methods that have been used for solving poroelastic models in other contexts, discussing their applicability to this particular application.
25 mins
Spatio-temporal heterogenities in a mechano-chemical model of collective cell migration
Andreas Buttenschoen, Leah Edelstein-Keshet
Abstract: Small GTPases, such as Rac and Rho, are well known central regulators of cell morphology and motility, whose dynamics also play a role in coordinating collective cell migration. Experiments have shown GTPase dynamics to be affected by both chemical and mechanical cues, but also to be spatially and temporally heterogeneous. This heterogenity is found both within a single cell, and between cells in a tissue. For example, sometimes the leader and follower cells display an inverted GTPase configuration. While progress on under standing GTPase dynamics in single cells has been made, a major remaining challenge is to understand the role of GTPase heterogenity in collective cell migration. Motivated by recent one-dimensional experiments (e.g. micro-channels) we introduce a one-dimensional modelling framework allowing us to integrate cell biomechanics, changes in cell size, and detailed intra-cellular signalling circuits (reaction-diffusion equations). Using this framework, we build cell migration models of both loose (mesenchymal) and cohering (epithelial) tissues. We use numerical simulations, and analysis tools, such as local perturbation analysis, to provide insights into the regulatory mechanisms coordinating collective cell migration. We show how feedback from mechanical tension to GTPase activation lead to a variety of dynamics, resembling both normal and pathological behaviour.
25 mins
Computational optimization and its role in improving the long-term stability of acetabular implants
Fernando Perez Boerema, Pawel Tomaszewski, Alexander Meynen, Jeroen Pellens, Emin Semih Perdahcioglu, Dennis Janssen, Lennart Scheys, Mattias Schevenels, Nico Verdonschot, Liesbet Geris
Abstract: Osteoarthritis (OA) is a disease characterized by the deterioration of the joint cartilage and underling bone that affects about half the world’s population over the age of 65.[1] One of the most commonly affected joints is the hip joint. When OA progresses to the state it cripples the patient there is no other recourse than to perform a hip arthroplasty, i.e. a surgical procedure where the dysfunctional joint is resected and replaced with an orthopedic implant. One of the main complications experienced after hip arthroplasty is aseptic loosening, i.e. failure of the implant-bone interface in the absence of infection, result of the mismatch between the bone and the implant stiffness. The only solution is to perform a more complicated revision surgery that needs to take into account developed bone defects and comes with a worse prognoses. The complex environment under which orthopedic implants have to operate makes computational optimization approaches ideal for their design. We have implemented two optimization approaches. The first approach, a surrogate based optimization approach, was used to optimize a standard acetabular implant, while the second approach, a topology optimization (TO) approach, was used to optimize an acetabular revision implant for large bone defects. Both implemented optimization approaches have as objective improving long-term implant stability by minimizing implant-induced stress shielding, a trigger for bone resorption and the leading cause of aseptic loosening. Constraints were placed on the maximum allowed stresses in the bone and implant. To predict the stresses required to compute the objective and constraints two patient-specific finite element (FE) models of the hip joint were derived for each optimization case from a computerized tomography scan of the pelvic region of the patient: one of the intact joint and one of the joint post-surgery. The hemi-pelvis was fixed at the sacroiliac joint and at the articular surface of the pubis. Loads were taken from the Hip98 database[2] and applied at the center of rotation of the hip joint. Bone material properties were derived from the CT-scan Hounsfield units. To optimize the standard implant first a surrogate model was created in Matlab (The MathWorks Inc., Natick, MA, USA) based on a limited set of FE-model evaluations. The FE-models, constructed in Marc/Mentat (MSC Software Corporation, Newport Beach, CA, USA), had ~250000 linear elastic solid tetrahedral elements with a layer of membrane elements lying on the external surface of the bony parts modeling the cortical layer. The surrogate was obtained by first reducing the size of the post-surgery FE-model output space, i.e. the stresses used to compute the stress shielding levels, by means of a proper orthogonal decomposition. Afterwards radial basis functions were used to perform an interpolation with regularization in this reduced output space. The surrogate model was finally used in a multi-start local optimization routine using sequential quadratic programing as local optimizer. The change in elastic modulus across the metal implant cup was described by a paraboloid whose coefficients were taken to be the design variables. The implant for large acetabular bone defects was optimized using a TO approach with and without penalization for intermediate densities. The TO approach allows, unlike the surrogate based approach, to handle a large number of design variables without assumptions on how the design should look. Optimization was performed using the method of moving asymptotes. A voxel mesh created in Matlab was used instead of the previously used tetrahedral mesh and implementation took place in the TopOpt in PETSc framework.[3] Stress shielding levels in the optimized standard acetabular implant decreased by ~56% compared to the solid titanium implant while maintaining an implant safety margin of factor three. The topology optimization approach has been implemented and is currently undergoing benchmarking and validation with artificially created cases before being applied to a real clinical case.