Colloquium: Modeling the dynamics of multicellular systems: Application to tissue engineering

Tissue engineering is a rapidly evolving discipline that aims at building functional tissues to improve or replace damaged ones. To be successful in such an endeavor, ideally, the engineering of tissues should be based on the principles of developmental biology. Recent progress in developmental biology suggests that the formation of tissues from the composing cells is often guided by physical laws. Here a comprehensive computational-theoretical formalism is presented that is based on experimental input and incorporates biomechanical principles of developmental biology. The formalism is described and it is shown that it correctly reproduces and predicts the quantitative characteristics of the fundamental early developmental process of tissue fusion. Based on this finding, the formalism is then used toward the optimization of the fabrication of tubular multicellular constructs, such as a vascular graft, by bioprinting, a novel tissue engineering technology.

Figure 1

Cell interactions with their environment. In the body as well as in vitro, the fate and function of living cells are regulated by the entire context of the cellular microenvironment. Diffusing, cell- and extracellular-matrix- (ECM-) derived morphogens and physical forces will determine the expression of genes and lead to a range of biological outcomes: from cell renewal, differentiation into specific lineages, synthesis of new biological material, migration, to apoptosis (cell death). At the same time, cells actively respond to their environment by secreting molecular factors, making and breaking extracellular matrix and generating physical forces. From 74.

Figure 2

Tissue engineering is largely based on the design of biometric (nativelike) environments established by an integrated use of scaffolds and bioreactors, designed to recapitulate the regulatory milieu that cells normally encounter in the body, during development, adult life, or regeneration.

Figure 3

(a) Capillary micropipettes, as printer cartridges, containing bioink units in the form of either spherical cellular aggregates (left) or cellular cylinders (right). The inner radius of the micropipettes shown is 500 μm which determines the dimensions of the bioink units. (b) The fully automated Novogen MMX bioprinter (Organovo, San Diego, CA) employed to build 3D cellular structures, in particular, tubular ones. The printer has two printing heads. One prints cellular material, the other the temporary supporting units (typically composed of cell inert hydrogels such as agarose).

Figure 4

Schematics to print tubular structures by layer-by-layer deposition with spheroidal (a) and cylindrical (b) bioink units and the cylindrical supporting units. The supporting units (e.g. made of cell inert agarose) surrounding the bioink units are removed once the latter fuse. Thus the supporting units are not part of the final structure.

Figure 5

Maturation of a vascular construct. Left panel: cannulated vessel in a custom-built perfusion bioreactor. The direction of the applied flow through the vessels is from right to left. Right panel: engineered tubular grafts of different diameter ready for application. The structures shown were prepared from a single cell type (porcine smooth muscle cells) and to use them as vascular grafts they need to be endothelialized. From 55.

Figure 6

Upper panel: representation of a cell in the model of 57 as an elongated body with hexagonal cross section. Each face and body diagonal of the model cell is represented by a viscoelastic body (rectangle) composed of a dashpot and a spring. The cell changes shape through “apical constriction,” as illustrated by changes in the apical surface, caused by cortical tension generated by the apical actin cortex. Changes in the overall shape of the model cell is the consequence of assuming conservation on volume. Lower panel: representation of a multicellular system (composed of 13 cells) used in the Glazier-Graner cellular Potts model. Each colored square is a “spin.” Contiguous spins with the same color form one cell. Cells adhere along their perimeters and change their shape by the rearrangement of their spins. As a result, the multicellular system changes its configuration from the one on the left to the one on the right. From 24.

Figure 7

Cellular particle (CP) model of a cell. (a) For biomechanical description, a cell can be regarded as a continuum material object with defined volume and adaptive shape. (b) The cell is coarse grained into a finite number of equal volume elements. (c) Each volume element is represented by a pointlike CP situated at its center of mass.

Figure 8

Close-up view of two cells in contact (bottom) within a model spheroidal multicellular aggregate (top). The CPs in the two highlighted cells are colored differently for clarity. $V_1$ and $V_2$ represent the pairwise intracellular and intercellular interaction energies between CPs.

Figure 9

Potential energies (top) and forces (bottom) corresponding to the intracellular (index 1) and intercellular (index 2) interactions.

Figure 10

The shape of two fusing spherical cell aggregates can be quantified by the angle $\theta (t)$ and the radius $R(t)$.

Figure 11

(a) Snapshots of the fusion of two similar multicellular aggregates ($300\mu \mathrm{m}$ diameter; the upper fluorescently labeled) composed of $N$-cadherin-transfected Chinese hamster ovary (CHO) cells. All lengths are expressed in units of initial aggregate diameter. (b) Theoretical fit of ${sin}^2\theta =(r/R{)}^2$ vs time [cf. Eq. (14)] to the experimental data shown in (a). The fusion time constant extracted from the fit $\tau \approx 70\mathrm{h}$. (c) Experimental results and (volume conserving) theoretical prediction of $\rho (\theta )=R/R_0$ vs $\cos \theta$. The volume of the experimental system increases during the fusion process due to cell proliferation.

Figure 12

(a) Snapshots from CPD simulation of the fusion of two identical spherical cell aggregates. The CPs from different aggregates are colored differently to emphasize their limited degree of mixing. All lengths are expressed in units of the initial aggregate diameter. (b) Theoretical fit of ${sin}^2\theta =(r/R{)}^2$ vs time [cf. Eq. (14)] to the CPD simulation data shown in (a). The fusion time constant extracted from the fit ${\tau }_{\mathrm{CPD}}\approx 560$. (c) $\rho (\theta )=R/R_0$ vs $\cos \theta$ for CPD simulation data matches well the volume conserving theoretical prediction in Eq. (5).

Figure 13

Snapshots of CPD simulation of cylindrical cellular aggregates. Upper and lower rows show, respectively, the top and the side view of the fusing cylinders. Each cylinder contains 1164 cells.

Figure 14

Top: Cross-sectional snapshots from CPD simulation of the fusion of two identical cylindrical cell aggregates. The CPs from different aggregates are colored differently to emphasize the absence of mixing. Bottom: Comparison between the theoretical prediction for $\sin \theta =r/R$ vs $t/\tau$ for the fusion of cylindrical [solid curve; cf. Eq. (19)] and spherical [dashed curve; cf. Eq. (14)] aggregates. The corresponding data from the CPD simulation for cylindrical aggregates (circles; see text), performed with the characteristic fusion time ${\tau }_{\mathrm{CPD}}\approx 560$, determined previously from the fusion of spherical aggregates, matches almost perfectly the theoretical curve.

Figure 15

Cellular torus built with CHO spheroids. Top and bottom rows show, respectively, an experiment (40) and simulation. The left and right columns correspond to, respectively, the initial and final configurations. Neighboring spheroids were labeled with different colors to emphasize the lack of cell mixing between adjacent aggregates.

Figure 16

Top: shape evolution of ten spherical aggregates during their fusion into a ringlike structure. The snapshots in the top and bottom row are, respectively, from a CPD simulation and an experiment with $500\mu \mathrm{m}$ diameter spherical CHO aggregates. The times of the snapshots are indicated in hours. The size of the fusing ring structure is quantified by the mean diameter $d(t)$ of the closed circular (for CPD simulation) and elliptical (for experiment) curves. Bottom: $d(t)/d(t_0)$ vs time $t-t_0$ for the CPD simulation (solid curve) and the experimental snapshots in the top panel: open squares (circles) correspond to $t_0=0$ ($t_0=19\mathrm{h}$).

Figure 17

Representative snapshots from CPD simulation of tube formation through the fusion of (top row) four stacked rings, formed by six identical spherical cellular aggregates and (bottom row) six identical cylindrical cellular aggregates. The times of the snapshots are indicated in CPD time units $\mathcal{T}$. Only the unit periodic box along the axis of the tube is shown. The cylinder at the middle of the tube is a cell inert support unit that preserves the shape and size of the lumen. Cells from adjacent aggregates have been colored differently to emphasize the absence of their mixing during the fusion process.