Force Estimation during Cell Migration Using Mathematical Modelling

Abstract

Cell migration is essential for physiological, pathological and biomedical processes such as, in embryogenesis, wound healing, immune response, cancer metastasis, tumour invasion and inflammation. In light of this, quantifying mechanical properties during the process of cell migration is of great interest in experimental sciences, yet few theoretical approaches in this direction have been studied. In this work, we propose a theoretical and computational approach based on the optimal control of geometric partial differential equations to estimate cell membrane forces associated with cell polarisation during migration. Specifically, cell membrane forces are inferred or estimated by fitting a mathematical model to a sequence of images, allowing us to capture dynamics of the cell migration. Our approach offers a robust and accurate framework to compute geometric mechanical membrane forces associated with cell polarisation during migration and also yields geometric information of independent interest, we illustrate one such example that involves quantifying cell proliferation levels which are associated with cell division, cell fusion or cell death.

Keywords: cell migration; cell polarisation; geometric partial differential equations; mechanical membrane forces; optimal control.

Figure 1

The first row illustrates two adjacent frames from the T24 experimental data [30] that were taken 8 min apart. The second row shows the initial shape and computed solutions at 10 intermediate time steps accordingly. The dark shadow in the background shows the targeted shape as the objective, which is the shape of the cell from frame 4. Bars indicate 20 $\mathsf{\mu }$m.

Figure 2

(Top-left): The original image from experimental observation; (top-right), The segmentation of the T24 cancer cell from the image; (bottom-left): We define the interfacial region of the cell and its centroid position. Within this sub-figure, we continually overlay the cell shapes and positions as the cell migrates; (bottom-right): We use colour coding to identify red as protrusion and blue as retraction forces and the locations they are exerted on the cellular interfacial region. Bars indicate 20 $\mathsf{\mu }$m.

Figure 3

(The first image on the first row): The original image from experimental observation and the choice of the initial three cells which are used in the simulation. In this figure, it is the last frame of the data. The second image on the first row: The segmentation of the cells from the image on the left. (The first image on the second row): We define the interfacial region of the cell and its centroid position, within this sub-figure, we continually overlay the cell shapes and their positions as they migrate. (The second image on the second row): We use colour coding to identify red as protrusion, and blue as retraction forces and the locations they are exerted on the cellular interfacial region, the dark shadows in the background illustrate the targeted shapes that model (1) replicates. The bar on the right-hand side shows the maximum and minimum amount of forcing that the colour coding is illustrating. (The only image on the third row): We show the Euler number from Equation (5) computed at each $RTS$ and red circles indicate the events of cell division. Bars indicate 20 $\mathsf{\mu }$m.