A level‐set method for the evolution of cells and tissue during curvature‐controlled growth

Abstract

Most biological tissues grow by the synthesis of new material close to the tissue's interface, where spatial interactions can exert strong geometric influences on the local rate of growth. These geometric influences may be mechanistic or cell behavioural in nature. The control of geometry on tissue growth has been evidenced in many in vivo and in vitro experiments, including bone remodelling, wound healing, and tissue engineering scaffolds. In this paper, we propose a generalisation of a mathematical model that captures the mechanistic influence of curvature on the joint evolution of cell density and tissue shape during tissue growth. This generalisation allows us to simulate abrupt topological changes such as tissue fragmentation and tissue fusion, as well as three dimensional cases, through a level-set-based method. The level-set method developed introduces another Eulerian field than the level-set function. This additional field represents the surface density of tissue-synthesising cells, anticipated at future locations of the interface. Numerical tests performed with this level-set-based method show that numerical conservation of cells is a good indicator of simulation accuracy, particularly when cusps develop in the tissue's interface. We apply this new model to several situations of curvature-controlled tissue evolutions that include fragmentation and fusion.

Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size

“…While the Porous-Fisher model allows proliferation to occur wherever cells occupy space with lower density than the carrying capacity, in effect, geometric control of cell proliferation is concentrated in a region near the tissue interface where there are large density inhomogeneities. The Porous-Fisher model is similar in this respect to the cellular models of surface growth of Refs [24,22,25]. A common feature in these cell-based models is that an influence of local curvature on tissue growth rate is not necessarily indicative of changes in cell-level behaviour.…”

“…An often overlooked factor of the geometric control of tissue growth is the mechanistic influence of tissue crowding or spreading in confined spaces. The progression rate of the tissue interface is determined both by how much new tissue is produced locally, and by the availability of space around where this tissue is produced [24,22,25]. The former is related to cell behaviour, but the latter is a purely geometric influence.…”

“…In contrast, there is more space available to accommodate new tissue in a convex region, so that tissue progression in the normal direction is slower as the new tissue needs to fill space laterally too. When new tissue production is confined in a very narrow band near the tissue interface (a situation called 'surface growth', such as that which occurs during bone formation, and the growth of seashells [47]), the mechanistic influence of space constraints on growth rates lead mathematically to a type of hyperbolic curvature flow in which the normal acceleration of the tissue is proportional to curvature [24,22,25]. In the Porous-Fisher model considered here, proliferation is not strictly confined to the tissue interface.…”

“…Several mathematical modelling strategies have been developed to explore the interplay between tissue growth and curvature effects [16], such as models based on the idea of mean curvature flow, that borrow ideas from fluid mechanics and the role of surface tension, but that do not consider cells explicitly [11][12][13][14][15]. Other modelling approaches consider the effects of cell-level behaviour and tissue crowding and spreading during surface growth, which lead to hyperbolic curvature flow models [24,22,25].…”

Modelling cell guidance and curvature control in evolving biological tissues

“…Sophisticated techniques have been developed to simulate the evolution of interfaces in three-dimensional complex systems , Tryggvason et al, 2001, Glimm et al, 2001, Shin and Juric, 2002, Du et al, 2006. While the level-let-like method developed in (Alias and Buenzli, 2019) for curvature-controlled tissue growth may be suitably adapted to include tangential cell velocity, the CBPM of used in this work is also applicable to three-dimensional interfaces.…”