Modelling cell guidance and curvature control in evolving biological tissues

Abstract

Tissue geometry is an important influence on the evolution of many biological tissues. The local curvature of an evolving tissue induces tissue crowding or spreading, which leads to differential tissue growth rates, and to changes in cellular tension, which can influence cell behaviour. Here, we investigate how directed cell motion interacts with curvature control in evolving biological tissues. Directed cell motion is involved in the generation of angled tissue growth and anisotropic tissue material properties, such as tissue fibre orientation. We develop a new cell-based mathematical model of tissue growth that includes both curvature control and cell guidance mechanisms to investigate their interplay. The model is based on conservation principles applied to the density of tissue synthesising cells at or near the tissue's moving boundary. The resulting mathematical model is a partial differential equation for cell density on a moving boundary, which is solved numerically using a hybrid front-tracking method called the cell-based particle method. The inclusion of directed cell motion allows us to model new types of biological growth, where tangential cell motion is important for the evolution of the interface, or for the generation of anisotropic tissue properties. We illustrate such situations by applying the model to simulate both the resorption and infilling components of the bone remodelling process, and to simulate root hair growth. We also provide user-friendly MATLAB code to implement the algorithms.

How osteons form: A quantitative hypothesis-testing analysis of cortical pore filling and wall asymmetry

“…This model is based on previous mathematical models of tissue growth [31, 33] and is suitably extended to describe the infilling of symmetric and asymmetric osteons. The model includes the crowding or spreading of osteoblasts as the bone surface evolves, due to changes in local surface area, as well as the tendency of osteoblasts to even out their density by diffusive or directed cell motion along the bone surface [31, 33]. The model also incorporates specific mechanisms by which asymmetric bone formation may occur around the osteon perimeter, and an elaborate calibration procedure is proposed based on real osteon data.…”

“…In these studies, irregular (asymmetric) osteons were reported to have no well-defined cavity radius and are outliers in such relationships [34]. Several mathematical models have been developed to understand the influence of the geometry of bone surfaces on the rate of bone formation during pore filling [27, 29–33, 38, 39]. These cell population models elucidate the influence of the radius of curvature of the infilling pore on formation rate and match double labelling data in regular osteons very well [29, 31].…”

Curvature dependences of wave propagation in reaction–diffusion models

“…There is recent interest in elucidating curvature-dependent velocities in moving fronts that represent the growth of biological tissues due to their application in tissue engineering [9,22,[27][28][29][30][31][32]. In this context, the eikonal equation represents an understanding of a biological growth law based on migratory and proliferative cellular behaviours [33][34][35][36][37].…”

Curvature dependences of wave propagation in reaction-diffusion models

“…There is recent interest in elucidating curvature-dependent velocities in moving fronts that represent the growth of biological tissues due to their application in tissue engineering [7,20,[25][26][27][28][29][30]. In this context, the eikonal equation represents an understanding of a biological growth law based on migratory and proliferative cellular behaviours [31][32][33][34][35].…”