6 Supporting Information
6.3 Statistical quantifiers
µ2 30 Lagrange multiplier actualization imax 2500 Iteration number
r 10−5 Convergence test tolerance 6.2.6 Decreasing the computational time
The simulations are performed on a 2D-domain Ω = [xmin, xmax]×[ymin, ymax]. In order to reduce the computational time, the domain is divided into sub-squares whose side lengthLsis a measure of the maximal distance of the agent interactions.
The goal is to compute each interaction potential element with the agents located in neighboring sub-squares of the domain only. The procedure is classical and details are omitted. Periodic boundary conditions are set by creating ghost numerical boxes of length Ls at each boundary of the domain.
6.3 Statistical quantifiers
This section is devoted to the computation of statistical quantifiers used to describe cell and fiber structures in both numerical simulations and biological images. A cell cluster is defined as a set of cells almost in contact. Let ∼a be the reflexive and symmetric relation:
j ∼ai ⇔j ∈ Ni,
where Ni is the set of cell i neighbors:
Ni ={j ∈[1, Na], j 6=i, | |Xi−Xj| ≤(Ri+Rj+a)2}, (6.15) where a is the maximal allowed distance up to which two cells not in contact are defined as neighbors and is set to 50% max(Ri, Rj). The equivalence relation ∼A
then reads:
j ∼Ai⇔∃n ∈N∗,∃(a1..an)
such that j ∼aa1 ∼a...∼aan ∼ai.
Cells i and j belong to the same cluster if and only if i∼A j. Fig. 1.16 shows an example of cell cluster separation.
Figure 1.16: Example of cell cluster detection. Left: original numerical simulation.
Cells are represented as 2D black spheres, fibers as black segments. Right: After cell cluster detection. Cells which belong to the same cluster are indicated with the same color.
The statistical quantifier NC is defined as the total number of cell clusters which have more than 5 adipocytes per 100 adipocytes.
The statistical quantifier E measures the mean elongation of the cell clusters, and is defined as the number of cells at the boundary of the clusters normalized by the total number of cells in the clusters. As the parameter E is irrelevant for clusters with less than 5 cells, its computation is restricted for clusterscsuch that nc >5 and reads:
E = PNc=1C Card(R ∩ Cc)
PNC
c=1nc .
Here, Cc is the set of indices of the cells belonging to cluster c,nc is the number of cells in cluster c and R is the set of indices of all cells with less than 5 neighbors:
R={i∈[1, Na], Card(Ni)≤5}, where Ni is defined by Eq. (6.15).
Finally, in order to determine if the cell clusters have anisotropic shape with a preferred direction, we define the SQ Θcas the angle of clustercshape anisotropy direction. For this purpose, let Xc be the center-of-mass of cluster c, i.e:
Xc = 1
The maximal eigenvalue λ+c of Pc gives a measure of the shape anisotropy of cell cluster c and its associated eigenvector uc = (uc1, uc2) gives the shape anisotropy direction. Then, Θc is defined as:
Θc = arctan(uc2 uc1).
Note that Θc ∈ [−π2,π2]. The SQ Θ is then defined as the circular standard deviation of all the angles Θc for all clusters:
Θ =q−2 ln( ¯R),
Finally, the mean ¯Θ of Θc over all the cell clusters reads:
¯Θ = 12argXNC
c=1
e2iΘc,
which ensures that ¯Θ∈ [−π2,π2]. Note that large Θ corresponds to fully isotropic cell cluster organization, while small Θ indicates that cell clusters have a prefer-ential direction.
In order to describe the fiber structures, we define a fiber cluster as a set of neighboring quasi-aligned fiber elements and Λ as an estimate of the average curvilinear length of such fiber clusters. Finally, A measures the mean alignment
of the fibers of a cluster. For this purpose, let us defineMk as the set of neighbors of fiberk, quasi-aligned with fiber k. Then:
Mk ={m∈[1, Nf], m6=k,
min(df(Yk, Ym), df(Ym, Yk))≤0
and |sin(θk−θm)|<sin(π 4)}, where df(Yk, Ym) reads:
df(Yk, Ym) =d(Yk, Ym−) +d(Yk, Ym+)−2
s
(Lf
2 )2+ (τfLf)2.
Here, Ym± =Ym± L2fωm and d(X, Y) is the distance of point X to point Y. Note that df(Yk, Ym) ≤ 0 (resp. df(Ym, Yk) ≤ 0) if the center of fiber k (resp. m) is contained in the ellipse of focii Ym± (resp. Yk±) with semi minor axis of length τfLf and semi major axis of length q(L2f)2+ (τfLf)2. We chose τf = 13, which means that a fiber detects a neighboring fiber up to a distance L3f in its orthogonal direction. This allows us to define fiber clusters as sets of quasi-aligned neighboring fibers. Let us define the reflexive and symmetric relation ∼f by:
k∼f m ⇔m∈ Mk. Define the equivalence relation ∼F such that:
k ∼F m⇔∃n∈N∗,∃(a1..an)
such that k∼f a1 ∼f ...∼f an∼f m.
Then, we say that fiberskand mbelong to the same cluster if and only ifm∼F k.
Fig. 1.17 shows the results of fiber cluster detection from the numerical simulation displayed on Fig. 1.16.
Figure 1.17: Example of fiber cluster detection, corresponding to simulation of Fig. 1.16. Fibers which belong to the same cluster are indicated with the same color.
The mean elongation of fiber clusters is estimated by Λ. Given a fiber clustercf and a division of the simulation domain into squares of side lengthLf, the length of cf is estimated by LfΛcFf, where ΛcFf is the number of squares which contain the centers of at least one fiber of cf. Then, the dimensionless mean fiber cluster elongation Λ is defined as the mean ofLsΛcFf over all the fiber clusters, normalized by the maximal cell diameter:
Λ = Lf 2RmaxNTf
X
1≤cf≤NTf
ΛcFf,
where NTf is the total number of fiber clusters. The longer the fiber clusters, the larger the Λ.
Finally, we define the SQ A to quantify the mean alignment of the fibers of a cluster. Given a fiber cluster cf, the mean alignment of its fibers is defined as the maximal eigenvalue λ+cf of the mean projection matrix defined by Eq. (6.6), where the set B(Vk, Ral) is replaced by the set of all fibers of cluster cf. Then, A is defined as the mean of the fiber cluster alignment, weighted by the number of fibers in the cluster:
A= 1 NF
X
1≤cf≤NTf
λ+c
fncf, where ncf is the number of fibers in cluster cf.