A finite element method for growth in biological development

Abstract

We describe finite element simulations of limb growth based on Stokes flow models with a nonzero divergence representing growth due to nutrients in the early stages of limb bud development. We introduce a "tissue pressure" whose spatial derivatives yield the growth velocity in the limb and our explicit time advancing algorithm for such tissue flows is described in de tail. The limb boundary is approached by spline functions to compute the curvature and the unit outward normal vector. At each time step, a mixed hybrid finite element problem is solved, where the condition that the velocity is strictly normal to the limb boundary is treated by a Lagrange multiplier technique. Numerical results are presented.

The Role of Spatially Controlled Cell Proliferation in Limb Bud Morphogenesis

“…This imaging revealed a highly dynamic nature of the filopodial protrusions—contracting and re-extending in a manner reminiscent to migrating fibroblasts (Figure 7F). Considering that almost all labelled mesenchymal cells display this complex array of dynamic filopodia, these observations considerably alter the classical view of the limb bud mesenchyme which has been repeatedly modelled as if cell divisions and cell growth represented the main source of force-generation [22],[23],[25],[32]. As before, a single preferential direction for filopodial activity was not evident, arguing against a simple model of chemotactic migration towards the AER.…”

“…Previous models (which have all been 2D) have included a variety of approaches, either considering a space larger than the limb bud itself (using the immersed-boundary method to represent the limb within the larger space [22]) or representing the 2D shape of the limb bud itself with an irregular triangular mesh, which can be used as the framework for a finite element method [32] or a spring-lattice method [23]. For our 3D model we used a tetrahedralised mesh to approximate the geometric domain.…”

“…The first FEM of limb development (2D) therefore employed the Navier-Stokes equations to represent the mesenchyme as a viscous incompressible fluid whose volume increases corresponding to a distributed source term, s , which represents the patterns of cell division [22]. Due to the small size of the limb bud, and the extremely low velocities involved (∼75 µm/h, described above) we chose a modified version of the Navier-Stokes equation to describe the movement of mesenchyme in our 3D model, in agreement with other cases where convection is negligible [32],[34],[35]. This also agrees with the recent lattice model proposed by Morishita [23].…”

“…Equation (4) describes fluid continuity, which usually represents the conservation of mass. In our case (following Dillon and Othmer [22] and Murea and Hentschel [32]) we alter this equation to allow a distributed material source term s which can vary arbitrarily across space and time:In this way, the s field (tissue growth) can drive the velocity (tissue movements) of the system. Positive s values result in tissue growth at the position x , y , z at time t .…”

Untitled

“…Within our model, cells are allowed to secrete diffusible chemicals which can either directly regulate the growth rate of cells ('growth regulator'), or indirectly affect growth by changing the secretion rate or the effect of other growth regulators. We obtain the growth of the overall cell cluster by treating it as an incompressible fluid, an approach that has been used to model a wide variety of systems [28][29][30][31][32][33][34][35][36][37] including the development of multicellular tissues [28][29][30][31][32][33][34]. Within this simple class of models, we explore what is possible and how complex the regulatory schemes need to be when there is just a single cell type whose growth rate is regulated by diffusible morphogens alone.…”