Numerical Methods for Two-Dimensional Stem Cell Tissue Growth.

Growth of developing and regenerative biological tissues of different cell types is usually driven by stem cells and their local environment. Here, we present a computational framework for continuum tissue growth models consisting of stem cells, cell lineages, and diffusive molecules that regulate proliferation and differentiation through feedback. To deal with the moving boundaries of the models in both open geometries and closed geometries (through polar coordinates) in two dimensions, we transform the dynamic domains and governing equations to fixed domains, followed by solving for the transformation functions to track the interface explicitly. Clustering grid points in local regions for better efficiency and accuracy can be achieved by appropriate choices of the transformation. The equations resulting from the incompressibility of the tissue is approximated by high-order finite difference schemes and is solved using the multigrid algorithms. The numerical tests demonstrate an overall spatiotemporal second-order accuracy of the methods and their capability in capturing large deformations of the tissue boundaries. The methods are applied to two biological systems: stratified epithelia for studying the effects of two different types of stem cell niches and the scaling of a morphogen gradient with the size of the Drosophila imaginal wing disc during growth. Direct simulations of both systems suggest that that the computational framework is robust and accurate, and it can incorporate various biological processes critical to stem cell dynamics and tissue growth

Multiscale Modelling Of Platelet Aggregation

During clotting under flow, platelets bind and activate on collagen and release autocrinic factors such ADP and thromboxane, while tissue factor (TF) on the damaged wall leads to localized thrombin generation. Toward patient-specific simulation of thrombosis, a multiscale approach was developed to account for: platelet signaling (neural network trained by pairwise agonist scanning, PAS-NN), platelet positions (lattice kinetic Monte Carlo, LKMC), wall-generated thrombin and platelet-released ADP/thromboxane convection-diffusion (PDE), and flow over a growing clot (lattice Boltzmann). LKMC included shear-driven platelet aggregate restructuring. The PDEs for thrombin, ADP, and thromboxane were solved by finite element method using cell activation-driven adaptive triangular meshing. At all times, intracellular calcium was known for each platelet by PAS-NN in response to its unique exposure to local collagen, ADP, thromboxane, and thrombin. The model accurately predicted clot morphology and growth with time on collagen/TF surface as compared to microfluidic blood perfusion experiments. The model also predicted the complete occlusion of the blood channel under pressure relief settings. Prior to occlusion, intrathrombus concentrations reached 50 nM thrombin, ~1 μM thromboxane, and ~10 μM ADP, while the wall shear rate on the rough clot peaked at ~1000-2000 sec-1. Additionally, clotting on TF/collagen was accurately simulated for modulators of platelet cyclooxygenase-1, P2Y1, and IP-receptor. The model was then extended to a rectangular channel with symmetric Gaussian obstacles representative of a coronary artery with severe stenosis. The upgraded stenosis model was able to predict platelet deposition dynamics at the post-stenotic segment corresponding to development of artery thrombosis prior to severe myocardial infarction. The presence of stenosis conditions alters the hemodynamics of normal hemostasis, showing a different thrombus growth mechanism. The model was able to recreate the platelet aggregation process under the complex recirculating flow features and make reasonable prediction on the clot morphology with flow separation. The model also detected recirculating transport dynamics for diffusible species in response to vortex features, posing interesting questions on the interplay between biological signaling and prevailing hemodynamics. In future work, the model will be extended to clot growth with a patient cardio-vasculature under pulsatile flow conditions

Modelling cell movement and chemotaxis pseudopod based feedback

A computational framework is presented for the simulation of eukaryotic cell migration and chemotaxis. An empirical pattern formation model, based on a system of non-linear reaction-diffusion equations, is approximated on an evolving cell boundary using an Arbitrary Lagrangian Eulerian surface finite element method (ALE-SFEM). The solution state is used to drive a mechanical model of the protrusive and retractive forces exerted on the cell boundary. Movement of the cell is achieved using a level set method. Results are presented for cell migration with and without chemotaxis. The simulated behaviour is compared with experimental results of migrating Dictyostelium discoideum cells

An Unstructured Mesh Convergent Reaction-Diffusion Master Equation for Reversible Reactions

The convergent reaction-diffusion master equation (CRDME) was recently developed to provide a lattice particle-based stochastic reaction-diffusion model that is a convergent approximation in the lattice spacing to an underlying spatially-continuous particle dynamics model. The CRDME was designed to be identical to the popular lattice reaction-diffusion master equation (RDME) model for systems with only linear reactions, while overcoming the RDME's loss of bimolecular reaction effects as the lattice spacing is taken to zero. In our original work we developed the CRDME to handle bimolecular association reactions on Cartesian grids. In this work we develop several extensions to the CRDME to facilitate the modeling of cellular processes within realistic biological domains. Foremost, we extend the CRDME to handle reversible bimolecular reactions on unstructured grids. Here we develop a generalized CRDME through discretization of the spatially continuous volume reactivity model, extending the CRDME to encompass a larger variety of particle-particle interactions. Finally, we conclude by examining several numerical examples to demonstrate the convergence and accuracy of the CRDME in approximating the volume reactivity model.Comment: 35 pages, 9 figures. Accepted, J. Comp. Phys. (2018