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title: "Modeling tissues: numerical simulations and continuum mechanics" author: Guillaume Gay, CENTURI multi-engineering platform, Marseille subtitle: Part II - Numerical Simulations fontsize: 10pt width: 1080 height: 800 font-size: 10pt bibliography: tyssue.bib data-transition: none center: 1 abstract: "In this second part of the course, we will go over the various methods used to simulate tissues. We will start by showing a rough taxonomy of cell models in general and we'll briefly discuss the general framework of agent-based modelling. Then we will see in some details the three big classes of tissue modeling strategies: 1. Lattice based models rely on a descretized space to simulate cells. Each cell here occupies a set of pixels, and the physics of the system is solved locally. Those model are adapted to rapid assessment of tissue dynamics with mixed cell types, proliferation and differentiation models. 2. Cell-center based models. Here each cell is an individual sphere (maybe deformable) interacting in free space with it's immediate neighbours. This class of model is adapted to problems in cancer biology, involving high cell numbers. 4. Vertex-based models. Here cells are delinated by polygons or polyhedron, and the phyics is applied at the polygon vertices. This class of models is widely used for morphogenesis modeling. For each section, we'll look at published examples and point towards available implementations." ...

A rough taxonomy of tissue models

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::::::{.columns}::: :::{.column width=70%} \vspace{2cm} This courses relies a lot on Carlos Tamulonis'

PhD Thesis (2013) ::: :::{.column width=30%} ::: ::::::

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Lattice based models

Conway's Game of life

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• Not really cells, but Cellular Automata

• Classical 'emergent behavior' system

• See distill.pub/2020/growing-ca for a fun example of cellular automata

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The Graner Glazier Hogeweg model

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• The world is a fixed grid
• Each cell $\alpha$ occupies a set of pixels
• Pixels at the interface can swap cells :::: :::: :::{.column width=40%}:::: { width=60% } ::: :::::::

The Modified Metropolis Algorithm

The behavior is governed by the definition of a Hamiltonian $H$ governing the energy of the cells

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1. Choose randomly a site $(i, j)$
2. Compute the change $\Delta H$ if $(i, j)$ swaps cell
3. If $\Delta H < 0$ : swap cell
4. If $\Delta H \geq 0$ : swap cell with probability $\exp( - \Delta H / kT)$ (T is not a "real" temperature)

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Cellular Potts Model Hamiltonian

The game is now to define the Hamiltonian to better reflect our problem!

. . .

The simplest model: volume conservation and adhesion:

\begin{align*} H & = & H_V + H_i \ H_V & = & \frac{\lambda}{2} \sum_\alpha (V_\alpha - V_0)^2 \ H_i & = & \sum_{ij, i'j'} J \left( \tau(ij), \tau(i'j') \right) \end{align*}

$\tau(ij)$ type of cell at $ij$

$J\left(\tau(ij), \tau(i'j')\right)$ : bond energy

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Cell sorting

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A classical problem:

2 cell types $(1, 2)$ --- $0$ is the medium

\begin{eqnarray*} J(0, 0) & = & 0\ J(1, 1) & = & 1\ J(2, 2) & = & 8\ J(2, 1) & = & 16\ J(1, 0) & = & J(2, 0) = 32 \end{eqnarray*}

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Chemotaxis

Add a term for chemotaxis:

• chemoatractant distribution on the grid ($C(ij)$)

• Favor switch for increasing $C$:

$$ H' = H - \mu \left(C(ij) - C(i'j') \right) $$

• The chemoatractant can be produced by the cells (cAMP)

. . .

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Existing Software

• Chaste
• CompuCell3D
• Morpheus

Cells as spheres

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• Cells are defined by their position in free space
• Movement goverened by Newton's 2nd law: $$ \sum F = m a \approx 0 $$
• Forces:
  • Cell-cell interactions
  • Friction with the medium ::: ::: ::::::

Cells as dipoles

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• Cell-cell interactions:

$$ V^{CC}(r) = \begin{cases} A R_{cc}^5 / r_k^4 + B r_k &\quad r_k \leq R_{pp}\\ 0 &\quad r_k > R_{pp}\\ \end{cases} $$

• Self-interaction (growth):

$$ V^G(r) = \frac{B}{r_i + r_0} $$

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Due to proliferation and death, cell aggreagate behaves as a fluid

Modeling tumors

• Spheres with adhesion and repulsion

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• Same Metropolis alorithm as GGH:

$$ P(\delta r) = \min { 1, \exp \frac{- (V(t + dt) - V(t))}{F_T} } $$

Mixed resolution models

• Deformable cells at high resolution meshes mixed with cell-based model

• Continuous / fluid dynamics finite elements for cells
• Very "realistic" results
• High computational cost

PhysiCell (Mathematical Oncology)

Physicell is a very powerfull simulation toolkit

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• Friction and adhesion model
• Very multi-agent oriented
• Coupled with a powerfull reaction / diffusion solver, BioFMV
• open development, great community ::: ::: ::::::

Cells as polygons

The apical junctions meshwork plays a central role in many morphogenesis events [@lecuit_cell_2007] and are poorly rendered by cell center models.

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Mechanical Model formulations

2D Models

Influence of cell packing

Quasi-static solution of:

$$ E = \sum_\alpha \frac{K_A}{2}(A_\alpha - A_0)^2 + \Gamma P_\alpha^2 + \sum_{ij}\Lambda \ell_{ij} $$

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Rigidity transition

$$ \epsilon = \sum_\alpha (a_\alpha - 1)^2 +\frac{(p_\alpha - p_0)^2}{r} $$

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Apical junctions in 2.5D

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• total volume conservation
• weigthed sum for the perimeter
• Apico-basal traction during apoptosis

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$$ p'\alpha = \frac{1}{n}\frac {\sum{ij \in \alpha} w_{ij}\ell_{ij}}{\sum_{ij \in \alpha} w_{ij}} $$

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3D Models

Monolayer and the ECM

• [@bielmeier_interface_2016] Extrapolates to 3D the 2D formulation

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Monlayer and bulk tissues

Sophisticated expression for the friction in [@okudaThreedimensionalVertexModel2015]

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Cells as foam

Consider surface tension and Laplace foce balance

Topology problems

Problem: we can have oscillating T1 transitions

Each time this happens, we have to write the equations again

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• Solution 1 : The active vertex model : Consider the Delaunay triangulation instead of it's dual [@bartonActiveVertexModel2016], and compute it at every step.

• Solution 2 : Allow for more than 3 way vertices! [@fineganTricellularVertexspecificAdhesion2019], give a finite life time to those.

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Topology gets trickier in 3D

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[@okuda_reversible_2013] define rules for stable deformations

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Can we generalize Tara Finegan et al. solution?

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Open problems

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• Self collisions at high deformation
• True curvature (E. Moisdon thesis)
• Non-naive ECM
• Nuclei
• Many more (it's exciting!) ::: ::: :::{.column}:::

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Existing (public) implementations

• Chaste
• Tyssue

Tank you

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• Sophie Theis
• Magali Suzanne
• Cyprien Gay
• Audrey Ferrand & Florian Bugarin
• The tyssue contributors
• The Scipy and Python communities ::: :::{.column width=20% }:::
• ANRT
• ERC
• CIR ::: ::::::

References

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