Multi-scale models of cell and tissue dynamics

Abstract

Cell and tissue movement are essential processes at various stages in the life cycle of most organisms. The early development of multi-cellular organisms involves individual and collective cell movement; leukocytes must migrate towards sites of infection as part of the immune response; and in cancer, directed movement is involved in invasion and metastasis. The forces needed to drive movement arise from actin polymerization, molecular motors and other processes, but understanding the cell- or tissue-level organization of these processes that is needed to produce the forces necessary for directed movement at the appropriate point in the cell or tissue is a major challenge. In this paper, we present three models that deal with the mechanics of cells and tissues: a model of an arbitrarily deformable single cell, a discrete model of the onset of tumour growth in which each cell is treated individually, and a hybrid continuum-discrete model of the later stages of tumour growth. While the models are different in scope, their underlying mechanical and mathematical principles are similar and can be applied to a variety of biological systems.

Figure 1.

Decomposition of deformation gradient.

Figure 2.

(a) Displacement boundary conditions for cell motility on a deformable substrate. (b) One set of attachment sites under consideration in the keratocyte model. (c) The second set of attachment sites used for the numerical experiments of the keratocyte.

Figure 3.

(a) Initial position of the keratocyte cell (light grey) and the substrate (dark grey). (b) The final position of the cell, viewed from above, for a given set of attachments. The initial position of the cell is outlined in black.

Figure 4.

(a) Displacement pattern resulting from extension–contraction function A₁ and the full set of adhesion sites shown in figure 2b. (b) Displacement pattern for function A₁ and only rear adhesions, as in figure 2c. (c) Displacement pattern for function A₂ and the full set of adhesion sites. In all three figures the displacement arrows are uniform in length and do not reflect the magnitudes of the displacements.

Figure 5.

(a) Numerically computed shear traction vectors (t_shear=〈σ_xz,σ_yz〉) magnified four-fold. (b) Magnitude of the numerically computed shear tractions ( in kPa) acting on the substrate at the cell–substrate interface for a substrate stiffness of 2 kPa. (c) Experimentally determined maximum shear traction vectors (i) and magnitudes (ii) from Doyle et al. (2004). (d) Time course of traction magnitudes as measured in Doyle et al. (2004). (1 kPa=1×10⁴ dyne cm⁻².)

Figure 6.

The growth rate function f(σ). From Kim et al. (2007), with permission.

Figure 7.

Tumour growth in the presence of adequate nutrients. (a–f) Tumour growth. Parameters σ⁻=−4.0 nN, σ⁺=800.0 nN, c⁺=5.16089× 10⁻⁹ mm (min nN)⁻¹ were used. (a) t=0 h, 7 cells; (b) t=60 h, 28 cells; (c) t=132 h, 179 cells; (d) t=168 h, 343 cells; (e) t=204 h, 650 cells; (f) t=276 h, 1614 cells. Diameter inside each figure is in the unit of 10 μm. (g) The occupancy (%) for each clone corresponding to (a–f). Notice that the occupancy by cells in the central clone (black cells) decreases significantly compared to other types due to the stress effect on growth. (h) Growth kinetics for different level of the compression parameter σ⁻=−400 nN (red circles), −4 nN (blue squares) and −0.04 nN (green crosses).

Figure 8.

Comparison of simulation results with experiments. (a) H³-labelled cells on the surface of EMT6 spheroids at 24 h (Dorie et al. 1982). (b) Cells remain at the surface in the simulations as observed experimentally in (a). (c) Internalized microspheres in spheroids at 48 h (Dorie et al. 1982). (d) Simulation results showing internalized discs. (e–f) Simulation results showing frequency of distance from the edge for (e) labelled cells at 0, 54, 114, 167 h and (f) discs (microspheres) at 0, 84, 198, 270 h. In the simulations, the labelled cells and discs were placed on the surface initially. ((a,c) Reprinted from Dorie et al. (1982), with permission.)

Figure 9.

A schematic showing the notation used for the subdomains, the representation of cells in the proliferating zone as ellipsoids, and the representation of the standard solid and growth elements that characterize the internal rheology of each cell in . (Reprinted from Kim et al. (2007), with permission.)

Figure 10.

An illustration of how force is transmitted between cells and the continua. From Kim et al. (2007), with permission, wherein details of the algorithm are given.

Figure 11.

(a) The effect of gel stiffness on tumour growth. Curves 1, 2, 3, 4 correspond to different Young’s modulus E^a of agarose gel, 10, 20, 80 and 200 MPa, respectively, while other parameters are fixed. The diameter of the tumour was defined as , where is the distance from the ith node point on the interface to the tumour centre and is the number of nodes on the interface. (b) The effect of gel stiffness on packing density: packing density at 137 h, computed as given in the text. (c) The average cell area A_c(t) (normalized) in the region for each case: , where is the normalized cell area and N_c(t) is the number of cells at time t. Curves as in (a). (d) Area distribution of proliferating cells corresponding to cases 1 (light grey), 3 (dark grey) and 4 (black) in (c) at 137 h.

Figure 12.

The linear relationship between the spheroid diameter and the diameter of the necrotic core for tumours in gels of increasing stiffness. (a–d) Young’s modulus E^a of the gel of 10, 20, 80 and 200 MPa, respectively, while other parameters are as in the tables in Kim et al. (2007). The solid grey line in each panel represents the best linear relation between the diameter of the necrotic core and the spheroid diameter (square boxes). The circles are for the region. The steps in the computational results arise from the manner in which cells are converted to continuum (Kim et al. 2007).