Chap. II: Macroscopic model for fibers interacting through alignment interactions

We then define a set of statistical quantifiers to describe the cell and fiber structures obtained at equilibrium. To this aim, a cell cluster is defined as a set of cells almost in contact and we compute the total number of cell clusters, their averaged elongation and the standard deviation of their shape anisotropy direction. Fiber clusters are defined as sets of neighboring quasi-aligned fibers, and we measure the number of fiber clusters, their elongation and their mean alignment. We then perform a statistical analysis by averaging the values of these quantifiers over numerous simulations and represent them as functions of the model parameters.

In order to compare the numerical simulations with the experimental data, we developed segmentation techniques on the biological images for (a) cell detection and (b) cell cluster detection. Adipocytes and lobules only were visualized in biological images at hand, therefore the statistical quantifiers for the cell structures only were accessible from biological images.

12.2 Chap. II: Macroscopic model for fibers interacting through alignment interactions

In this chapter, we are interested in the large-scale dynamics of one part of the microscopic model of chapter I. To simplify the model, we focus on the modeling of the fiber network only. The presence of the cells in the medium is reduced to an

external potentialW_ext(Y, θ) which depends on fiber positions and orientation an-gles. We are thus interested in the properties of a fiber network composed of fiber elements of fixed length which have the ability to connect together, disconnect, and are subjected to mechanical interactions. We also incorporate random fiber motion which may occur in association with the movements of the tissue through the use of an entropy term W_noise(Y, θ). Note that the derivation of macroscopic equations needs a continuous description of the agent’s motion and that the mini-mization procedure (12.18) is discrete in time in the microscopic model. Therefore, we consider a gradient descent for a quadratic penalization of the minimization problem (12.18). To this aim, the functional Ψ(Y, θ) for the equality constraints of maintain of fiber links is incorporated in the total free-energy of the system such that:

W(Y, θ) =W_ext(Y, θ) +W_align(Y, θ) +W_noise(Y, θ) + κ

2|Ψ(Y, θ)|²,

whereκ is the penalization factor. Fiber motion and rotation is then supposed to be in the steepest descent of the gradient of this energy:

dY

dt =−µ∇_XW_ext+W_align+W_links+W_noise (12.20) dθ

dt =−λ∂_θW_ext+W_align+W_links+W_noise. (12.21) where µand λ are mobility coefficients.

To derive a macroscopic model from this microscopic model, we use the kinetic equation associated with this particle dynamics. To this aim, we define the one-particle distribution function f(x, θ) describing the N individual fibers, and the two particle distribution functiong(x₁, θ₁, `<sub>1</sub>, x<sub>2</sub>, θ<sub>2</sub>, `₂) describing theKfiber links:

f^N(x, θ, t) =1 N

N

X

i=1

δ_{(Xi(t),θi(t))}(x, θ), g^K(x₁, θ₁, `<sub>1</sub>, x<sub>2</sub>, θ<sub>2</sub>, `₂, t) = 1

2K

K

X

k=1

δ_(X

i(k),θi(k),`<sup>k</sup><sub>i(k)</sub>,Xj(k),θj(k),`^k_j(k))(x₁, θ₁, `<sub>1</sub>, x<sub>2</sub>, θ<sub>2</sub>, `₂) +δ_(X

j(k),θj(k),`<sup>k</sup><sub>j(k)</sub>,Xi(k),θi(k),`^k_i(k))(x₁, θ₁, `<sub>1</sub>, x<sub>2</sub>, θ<sub>2</sub>, `₂), where δx(y) denotes the dirac function at x, i.e the distribution acting on test functions φ(y) such that < δ_x(y), φ(y)>=φ(x). By a simple closure relation, we successfully obtain the formal limit of a large number of individual fibers and links and prove the following theorem:

Theorem 12.1. The formal limit of Eqs.(12.20), (12.21)forK, N → ∞, ^K_N →ξ, where ξ > 0 is a fixed parameter reads:

df

Here, S(g) describes the dynamics of creation/deletion of fiber links and reads:

S(g) = νff(x1, θ₁)f(x2, θ₂)δ`(x¯ 1,θ1,x2,θ2)(`1)δ`(x¯ 2,θ2,x1,θ1)(`2)−νdg, (12.25) The main novelty of this kinetic equation is that it is a simple way of keeping track of the two-particle interactions. These pair of interactions can be seen as a way of describing a random graph of the links of the fibers. Indeed, as the links are located on the fibers, they are convected and follow the motion of the fibers. At the same time, they constrain the linked fibers to move together, so they directly impact their motion. The restoring potential generated by the maintain of the links, V, is expressed as non local forces F₁ and F₂ in Eqs. (12.22),(12.23). The second and fifth terms describe transport in physical and orientational spaces due to the external potentialU, while the fourth and seventh terms are diffusion terms of amplitude λd or µd which represent the random motion of the fibers. The kinetic counterpart of the alignment force between linked fibers b is comprised in the forceF₂ and only acts on the orientation of the fibers. The right hand sideS(g) of equation describes the Poisson processes of linking/unlinking at frequencies ν_f

and ν_d, respectively. The transition between the particle dynamics to the kinetic equation is only formal.

Once we have the kinetic equation Eqs. (12.22)-(12.23), we use an hydrodynam-ics scaling in order to derive a macroscopic model. More precisely, we introduce the new macroscopic variables:

˜t=εt , x˜=√ εx,

and show that in a regime of fast linking/unlinking, the two-particle distribution function g^ε can be written as a product of two one particle distributions f^ε: g^ε(x1, θ₁, `<sub>1</sub>, x<sub>2</sub>, θ<sub>2</sub>, `₂) = νf

ν_df^ε(x1, θ₁)f^ε(x2, θ₂)δ`(x¯ 1,θ1,x2,θ2)(`1)δ¯`(x2,θ2,x1,θ1)(`2)+O(ε²).

This closure relation simplifies system (12.22) and we are able to obtain the large scale limit ε →0. We show that the equilibrium solutions of the kinetic equation obtained are of form

f(x, θ) = ρ(x)Mθ0(x)(θ), M_θ0(x)(θ) = 1

Ze^{−rcos 2(θ−θ0(x))}

wherer is a model parameter,Z a normalization function such thatf is a density distribution, ρ(x) is the local density of fibers and θ₀(x) their local orientation angle. By the use of the concept of generalized collision invariant ([23]), we obtain the macroscopic limit of our kinetic equation. When ε goes to 0, we obtain a system of two equations, one for the fiber density ρ:

∂_tρ− ∇_x.(∇_xU⁰ρ)−d∆xρ= 0, (12.26) where we have supposedU(x, θ) = U⁰(x)+U¹(θ), whereU⁰is the external potential acting on the fiber positions andU¹(θ) acting on fiber orientations. The evolution equation for the fiber mean direction angle θ₀(x) reads:

ρ∂_tθ₀−ρ∇_xU⁰.∇_xθ₀−2α₂∇_xρ.∇_xθ₀−α₂ρ∆xθ₀

+α3(ρ∇²_xθ₀+∇xθ₀⊗ ∇xρ+∇xρ⊗ ∇xθ₀) : [ω0⊗ω₀−ω₀^⊥⊗ω₀^⊥]

+2ρα₃∇_xθ₀⊗ ∇_xθ₀−α₄∇²_xρ: [ω₀⊗ω₀^⊥+ω^⊥₀ ⊗ω₀] +α₅ρ < ∂_θU¹ >= 0, (12.27) where ω0 = (cosθ0,sinθ0) denotes the 2D directional vector of norm 1 associated to the angle θ₀, ω₀^⊥ denotes its orthogonal vector and < h >= ^π/2R

−π/2

h(θ)M_θ0(θ)^dθ_π for any function h of θ ∈ [−^π₂,^π₂). The coefficients α₁, α₂, α₃, α₄, α₅ are fully determined by the model parameters.

12.3 Chap III: Macroscopic model for linked fibers with