Tissue growth controlled by geometric boundary conditions: a simple model recapitulating aspects of callus formation and bone healing

Abstract

The shape of tissues arises from a subtle interplay between biochemical driving forces, leading to cell growth, division and extracellular matrix formation, and the physical constraints of the surrounding environment, giving rise to mechanical signals for the cells. Despite the inherent complexity of such systems, much can still be learnt by treating tissues that constantly remodel as simple fluids. In this approach, remodelling relaxes all internal stresses except for the pressure which is counterbalanced by the surface stress. Our model is used to investigate how wettable substrates influence the stability of tissue nodules. It turns out for a growing tissue nodule in free space, the model predicts only two states: either the tissue shrinks and disappears, or it keeps growing indefinitely. However, as soon as the tissue wets a substrate, stable equilibrium configurations become possible. Furthermore, by investigating more complex substrate geometries, such as tissue growing at the end of a hollow cylinder, we see features reminiscent of healing processes in long bones, such as the existence of a critical gap size above which healing does not occur. Despite its simplicity, the model may be useful in describing various aspects related to tissue growth, including biofilm formation and cancer metastases.

The Emergence of Complexity from a Simple Model for Tissue Growth

“…When this growth is slow compared to local remodelling rates of the extracellular tissue, shear stresses in the matrix are expected to relax fast enough to reach the mechanical equilibrium status of a fluid before the volume significantly increases. Under such conditions, the shape of the tissue "droplet" corresponds at any moment in time to the equilibrium shape governed by the surface stress and corresponding energy state [9]. Despite the extreme simplicity of such a model, fluid drops with remarkably complex shapes can emerge simply by interacting of the surface with three-dimensional substrates of complex geometry, see e.g.…”

“…Despite the extreme simplicity of such a model, fluid drops with remarkably complex shapes can emerge simply by interacting of the surface with three-dimensional substrates of complex geometry, see e.g. [9][10][11].…”

“…In order to get better insight into the development of geometric complexity, we further investigate a simple model introduced in earlier work [9], where growth is described as the time derivative of the tissue volume, V , being proportional to − K − K c V, with K being the total surface curvature (i. e. the trace of the surface curvature tensor) of the tissue aggregate. K is twice the mean curvature and should not be confused with the Gaussian curvature which plays no role in the following (we use the symbol K to be consistent with our previous papers).…”

“…In phase separation models, R c depends on the chemical potential difference between the phases and, thus, on the supersaturation of the solution from which the particle grows [19]. In our biological tissue growth model, the meaning is analogous and R c describes the driving force for tissue growth [9]. In general, R c may depend on time through the interaction with other growing or shrinking particles.…”

“…The wetting surface then represents a boundary which confines the growth in certain directions, leading to a deviation from spherical droplet shapes. In a previous paper we investigated this for the tissue growth model assuming that the tissue wets planar annulus-like substrates [9], an idealized geometry motivated by healing of the fracture gap in long bones [33,34]. Here we expand this approach, exploring the increased structural complexity based on wetting of non-planar surfaces.…”