Modelling non-local cell-cell adhesion: a multiscale approach.

Cell-cell adhesion plays a vital role in the development and maintenance of multicellular organisms. One of its functions is regulation of cell migration, such as occurs, e.g. during embryogenesis or in cancer. In this work, we develop a versatile multiscale approach to modelling a moving self-adhesive cell population that combines a careful microscopic description of a deterministic adhesion-driven motion component with an efficient mesoscopic representation of a stochastic velocity-jump process. This approach gives rise to mesoscopic models in the form of kinetic transport equations featuring multiple non-localities. Subsequent parabolic and hyperbolic scalings produce general classes of equations with non-local adhesion and myopic diffusion, a special case being the classical macroscopic model proposed in Armstrong et al. (J Theoret Biol 243(1): 98-113, 2006). Our simulations show how the combination of the two motion effects can unfold. Cell-cell adhesion relies on the subcellular cell adhesion molecule binding. Our approach lends itself conveniently to capturing this microscopic effect. On the macroscale, this results in an additional non-linear integral equation of a novel type that is coupled to the cell density equation.

A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis.

Cellular adhesion provides one of the fundamental forms of biological interaction between cells and their surroundings, yet the continuum modelling of cellular adhesion has remained mathematically challenging. In 2006, Armstrong et al. proposed a mathematical model in the form of an integro-partial differential equation. Although successful in applications, a derivation from an underlying stochastic random walk has remained elusive. In this work we develop a framework by which non-local models can be derived from a space-jump process. We show how the notions of motility and a cell polarization vector can be naturally included. With this derivation we are able to include microscopic biological properties into the model. We show that particular choices yield the original Armstrong model, while others lead to more general models, including a doubly non-local adhesion model and non-local chemotaxis models. Finally, we use random walk simulations to confirm that the corresponding continuum model represents the mean field behaviour of the stochastic random walk.

Mathematical investigation of normal and abnormal wound healing dynamics: local and non-local models.

The movement of cells during (normal and abnormal) wound healing is the result of biomechanical interactions that combine cell responses with growth factors as well as cell-cell and cell-matrix interactions (adhesion and remodelling). It is known that cells can communicate and interact locally and non-locally with other cells inside the tissues through mechanical forces that act locally and at a distance, as well as through long non-conventional cell protrusions. In this study, we consider a non-local partial differential equation model for the interactions between fibroblasts, macrophages and the extracellular matrix (ECM) via a growth factor (TGF-$ \beta $) in the context of wound healing. For the non-local interactions, we consider two types of kernels (i.e., a Gaussian kernel and a cone-shaped kernel), two types of cell-ECM adhesion functions (i.e., adhesion only to higher-density ECM vs. adhesion to higher-/lower-density ECM) and two types of cell proliferation terms (i.e., with and without decay due to overcrowding). We investigate numerically the dynamics of this non-local model, as well as the dynamics of the localised versions of this model (i.e., those obtained when the cell perception radius decreases to 0). The results suggest the following: (ⅰ) local models explain normal wound healing and non-local models could also explain abnormal wound healing (although the results are parameter-dependent); (ⅱ) the models can explain two types of wound healing, i.e., by primary intention, when the wound margins come together from the side, and by secondary intention when the wound heals from the bottom up.

Mathematical models for cell migration: a non-local perspective.

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Generalized moving least squares approximation for the solution of local and non-local models of cancer cell invasion of tissue under the effect of adhesion in one- and two-dimensional spaces.

The main aim of this study is to solve numerically the mathematical models showing cancer cell invasion of tissue with/without considering the effect of cell-cell and cell-matrix adhesion. The mathematical models studied here are the systems of time-dependent reaction-diffusion-taxis equations in one- and two-dimensional spaces, which are formulated in the local and non-local forms. There are some difficulties in finding their solutions via numerical methods. The main difficulty is to compute the non-local term appearing in one of the studied models, which causes more CPU time during simulations. The current paper aims to overcome this problem, where a new meshless method, namely generalized moving least squares (GMLS) approximation in space and a semi-implicit backward differential formula of first-order (SBDF1) in time have been applied. Based on GMLS theory, the non-local term is approximated without any difficulties. Moreover, a simple method based on the GMLS technique is presented to implement the boundary conditions. The obtained discrete scheme for both mathematical models is a linear system of algebraic equations per time step. The biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower-upper (ILU) preconditioner is used to solve the obtained linear system at each time step. At the end of this paper, some simulation results are reported to show the behavior of cancer cell invasion in the local model, and the non-local model due to reduction of cell-cell adhesion and increasing cell-matrix adhesion in one- and two-dimensional spaces, where two different types of distribution points have been considered in the square domain. The computational algorithms of the GMLS approximation and the developed numerical method for solving the non-local (local) model are included in the Appendix.