Microscopic model

The microscopic 2D model consists of N fibers of uniform and fixed length L, described by their centerXi ∈R² and their orientation angleθi. Asθi is an angle of line, θ_i ∈ [−^π₂,^π₂). The following biological features are considered: (i) Fiber growth and elongation: In order to model fiber growth and elongation, pairs of unlinked intersecting fibers have the possibility to link to each other at random times following a Poisson process. Each linkk of theKlinks at timetis associated to a unique pair of fibers (i(k), j(k)), i(k) < j(k). (ii) Mechanical external forces:

In order to model the contribution of external elements in the tissue, external mechanical forces are considered. This can model -for instance- the repulsion of fibers by growing confined adipocytes. (iii) Fiber resistance to bending: To model fiber resistance to bending, linked fibers are subjected to an alignment potential force at their junction, in order to increase their rigidity. This alignment potential is characterized by a stiffness parameter α > 0 playing the role of a flexural modulus. (iv) Movements of the tissue: To model tissue movements, we consider random fiber motion and fiber orientation changes of respective amplitude d_X, d_θ >0.

Phenomenon (i) : Fiber links. In order to model long biological fibers as sets of connected segments, intersecting fibers have the ability to link together and linked fibers to unlink at random times, following Poisson processes of frequencies νf and νd respectively. To define the expression of W_links, we consider a time at which no linking/unlinking process occurs. Then, the set of links is well-defined and supposed to have K elements. Let k∈ {1, . . . , K}be a given link and denote by (i(k), j(k)) the pair of indices corresponding to the two fibers connected by this

link. To make the labeling of the pair unique, we assume without loss of generality that the first element of the linked pair is always the one with lowest index, i.e.

i(k) < j(k). The link is supposed to connect two points X_i(k)^k and X_j(k)^k on fibers i(k) and j(k) respectively. These points are determined by the algebraic distances

`<sup>k</sup><sub>i(k)</sub> and`^k_j(k) to the centersX_i(k) and X_j(k) of the two fibers respectively; We thus have the relation:

X_i(k)^k =X_i(k)+`<sup>k</sup><sub>i(k)</sub>ω<sub>i(k)</sub>, X<sub>j(k)</sub><sup>k</sup> =X<sub>j(k)</sub>+`^k_j(k)ω_j(k),

where`<sup>k</sup><sub>i(k)</sub>,`^k_j(k)∈[−L/2, L/2] and where, for any fiberi, we letω_i = (cosθ_i,sinθ_i) be the unit vector in the direction of the fiber. All along the link lifetime, the link places a spring-like restoring force that attractsX_i(k)back toX_j(k)(and vice-versa) as soon as their are displaced one with respect to each other. This restoring force gives rise to a potential energy V(X_i(k), θ_i(k), `<sup>k</sup><sub>i(k)</sub>, X<sub>j(k)</sub>, θ<sub>j(k)</sub>, `^k_j(k)), with

V(X₁, θ₁, `<sub>1</sub>, X<sub>2</sub>, θ<sub>2</sub>, `₂) = κ

2|X₁+`<sub>1</sub>ω(θ<sub>1</sub>)−(X<sub>2</sub>+`₂ω(θ₂))|², (2.1) where κ is the intensity of the restoring force. Obviously, the larger κ, the better the maintainance of the link is ensured. The potential W_links is then assumed to be the sum of all the linked fiber spring forces:

W_links= 1 2

K

X

k=1

V(Xi(k), θ_i(k), `<sup>k</sup><sub>i(k)</sub>, X<sub>j(k)</sub>, θ<sub>j(k)</sub>, `^k_j(k)). (2.2) We stress the fact that the quantities `<sup>k</sup><sub>i(k)</sub> and `^k_j(k) remain constant throughout the link lifetime. They are determined at the time of the creation of the link.

Phenomenon (ii): External mechanical forces. The external potential W_ext is supposed to be the sum of potential forces U(θi) acting on each of the N fibers.

In this paper, we consider rotation external forces acting on the fiber orientations only and of the form:

U(θ) = c_usin²(θ−θ_u), (2.3) where c_u is the intensity of the force and θ_u a parameter of the model. This potential tends to force the fibers to be oriented with directional angle θ_u. Then, the total external potential is the sum of all the rotation interactions:

Wext=^XN

i=1

U(θi). (2.4)

Phenomenon (iii): Fiber resistance to bending. As previously described, linked fibers are submitted to an alignment force at their junction to increase their

rigidity. This force tends to align linked fibers i(k) and j(k) and has potential b(θ_i(k), θ_j(k)) which reads:

b(θ₁, θ₂) =α|sin(θ₁−θ₂)|, (2.5) where α is the flexural modulus. The binary alignment potential depends on the anglesθ₁ and θ₂ only, and the total alignment energyW_align is supposed to be the sum of all the binary alignment interactions :

W_align = 1 2

K

X

k=1

b(θ_i(k), θ_j(k)). (2.6) Phenomenon (iv): Tissue movements. Finally, in order to model random motion of biological elements, fiber positions as well as fiber orientations are submitted random changes supposed to be regular in time. This is modeled by an entropy term Wnoise:

W_noise =d

N

X

i=1

log( ˜f)(X_i, θ_i), (2.7) where ˜f is a ’regularized density’ :

f˜(x, θ) = 1 N

N

X

i=1

ξ^N(x−Xi)⊗η^N(θ−θi).

Here, ξ^N and η^N are regularization terms which allow to define the logarithm of f˜. We refer to [2] for the properties of these terms.

The total potential of the system is defined as the sum of all the previously described potentials (Eqs. (2.2),(2.4) (2.6) and (2.7)), and fiber motion follows the steepest descent of the gradient of this potential. Equations for fiber motion and rotation then read:

dX_i

dt =−µ∇_XiW_ext+W_align+W_links+W_noise dθ_i

dt =−λ∂_θiW_ext+W_align+W_links+W_noise.

where µ and λ are mobility coefficients. Using the fact that W_align and W_ext do not depend on the space variable (see Eqs. (2.6)-(2.4)):

dX_i

dt =−µ∇Xi

Wlinks+Wnoise

(2.8)

dθ_i

dt =−λ∂_θiW_ext+W_align+W_links+W_noise. (2.9)

Remark The IBM defined by Eqs. (2.8)-(2.9) corresponds to a gradient descent for a quadratic penalization of a minimization problem related to the model of [1].

These results are shown in Appendix 6, where a numerical comparison between both models is performed.

We now introduce the mean field kinetic equation which describes the time evolution of the particle system in the limit of a large number of fibers. This is generally performed using the one-particle distributionf(x, θ, t) which depends on the positionx∈R², orientation angleθ ∈[−^π₂,^π₂) and the timet. As shown in [2], the presence of fiber links oblige to keep track also of the fiber links distribution function g(x1, θ₁, `<sub>1</sub>, x<sub>2</sub>, θ<sub>2</sub>, `₂, t) which can be seen as a two-particle distribution

The two-particle distribution function is described by:

dg where S(g) is the term for creation/deletion of fiber links:

S(g) =ν_ff(x₁, θ₁)f(x₂, θ₂)δ`(x¯ 1,θ1,x2,θ2)(`₁)δ`(x¯ 2,θ2,x1,θ1)(`₂)−ν_dg, (2.13) where δ`¯(`₁) denotes the Dirac delta at ¯`, i.e. the distribution acting on test func-tions φ(`₁) such that hδ`¯(`₁), φ(`<sub>1</sub>)i=φ(¯`).