Point Forces and Their Alternatives in Cell-Based Models for Skin Contraction

Abstract

During wound healing, contractions occur due to the pulling forces released by (myo)fibroblasts. We consider a cell-based approach in which the balance of momentum is used to predict the cellular impact on the mechanics of the tissue. To this extent, the elasticity equation and Dirac Delta distributions are combined. However, Dirac Delta distributions cause a singular solution. Hence, alternative approaches are developed and a Gaussian distribution is often used as a smoothed approach. Based on the application that the pulling force is pointing inward the cell, the smoothed particle approach is probed as well. In one dimension, it turns out that the aforementioned three approaches are consistent. In fact, we are aware that the similar transformation exists in three dimensional electric dipole moment. For two dimensions, the ratio of the force magnitude is only worked out in special case, but for the general case, the numerical results show consistency between the direct approach and the smoothed particle approach.

Point Forces in Elasticity Equation and Their Alternatives in Multi Dimensions

“…Following the same process in two dimensions [10] and thanks to the continuity of Gaussian distribution, as ∆x → 0, the force converges to f s = P (∆x) n ∇δ ε (x − x ).…”

“…[8] demonstrated the convergence of finite-element solutions by means of piecewise linear Lagrangian elements in multiple dimensions. In our earlier studies [9,10], we proved the convergence of solutions obtained by regularization of Dirac Delta distributions, the so-called smoothed particle approach and the so-called 'hole' approach to the solution obtained by Dirac Delta distributions in the one-and two-dimensional cases. In the one-dimensional case, for the sake of completeness, we start with the presentation of force equilibrium with point forces, the equations are given by…”

Upscaling between an Agent-Based Model (Smoothed Particle Approach) and a Continuum-Based Model for Skin Contractions

“…In [12], a smoothed particle approach (SPH approach) is developed as an alternative of the Dirac Delta distribution describing the point forces exerted by the biological cells, in the application of wound healing:…”

“…Regarding the elasticity equation with point forces, we realized that the solution to the partial differential equation is singular for dimensionality exceeding one. Hence, we developed various alternatives to improve the accuracy of the solution in [11,12].…”

“…In this manuscript, we investigate and discover the connections between these two models, in the perspective of modelling the mechanism of pulling forces exerted by the (myo)fibroblasts. As the consistency between the smoothed particle approach (SPH approach) and the immersed boundary approach has been proven both analytically and numerically [11,12], we select the SPH approach here due to its continuity and smoothness, to compare with the cell density model using finite-element methods.…”