The Spherical Design Algorithm in the Numerical Simulationof Fibre-Reinforced Biological Tissues

Melania Carfagna1, Alfio Grillo1

1DISMA G.L.Lagrange Politecnico di Torino

October the 13th, 2016Comsol Conference - Munich

Articular Cartilage

Living Cells(proteoglycans,chondrocytes)constituting the ECM.

Fibres of Collagen statisticallydistributed

Fluid (mainly water) passing throughand escaping from the tissue

2 / 14

Fibres Distribution

Ψ̂(ξ,Θ) =1Z(ξ)

exp(− [Θ− Q(ξ)]

2

2[σ(ξ)]2

)S2XB := {M ∈ TXB : ‖M‖ = 1}

M = sin(Θ) cos(Φ)E1 + sin(Θ) sin(Φ)E2 + cos(Θ)E3

3 / 14

Mechanical Behaviour - Unconfined Compression

J̇ + Div(−K̂(F) Grad p

)= 0

Div(−Jp g−1F−T + P̂sc(F)

)= 0

K̂(F) = Ki(C, ξ) + α(C, ξ)Ẑ(C)

Z = Ẑ(C) =∫S2B

Ψ(M)M ⊗M

I4(C,M).

P̂sc(F) = Pi(F) + FSa

Sa = 2φ1sRc∫S2B

Ψ(M)H(I4 − 1)[I4 − 1]A.

in which A = M ⊗M, I4 = C : A. 4 / 14

The Spherical Design Algorithm

Let f be any (scalar, vector, or tensor) function defined over S2B.Then f (X,M) = f̂ (X,Θ,Φ) with (Θ,Φ) ∈ D = [0, π]× [0, 2π].∫

S2Bf (M) =

∫∫D

f̂ (Θ,Φ) sin(Θ)dΘdΦ

∫∫D

f̂ (Θ,Φ) sin(Θ)dΘdΦ ' 4πN

m∑i=1

n∑j=1

f̂ (Xij)

with Xij = (Θi,Φj) ∈ D, with i = 1, . . . ,m and j = 1, . . . , n, N = mn.

5 / 14

The Proper Choice of the Integration Points

(Z0)11 = (Z0)22 = π∫ π

0Ψ̂(ξ,Θ)[sin(Θ)]3dΘ,

(Z0)12 = (Z0)13 = (Z0)23 = 0,

(Z0)33 = 1− 2(Z0)11.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

ξ

Z011

equispaced points

increasing N ∈ [10,50]equidistributed points

6 / 14

The Proper Choice of the Integration Points

(Z0)11 = (Z0)22 = π∫ π

0Ψ̂(ξ,Θ)[sin(Θ)]3dΘ,

(Z0)12 = (Z0)13 = (Z0)23 = 0,

(Z0)33 = 1− 2(Z0)11.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ξ

Z011

41 Sloane points240 Sloane pointsNew Set10 equidistributed points50 equidistributed points30 equidistributed pointsMatlab

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

8

9x 10

−3

mIn

tegr

al E

rror

Equidistributed points21 Sloane points120 Sloane pointsNew Set

7 / 14

Benchmark Tests

w(t) = −0.2L tT.

J̇ + Div(−K̂(F) Grad p

)= 0

Div(−Jp g−1F−T + P̂sc(F)

)= 0

χ3 = H + w, Q.E3 = 0, ∀X ∈ Γu− p = 0, P.N = 0, ∀X ∈ Γwχ3 = 0, Q.(−E3) = 0, ∀X ∈ Γl

8 / 14

Benchmark Tests

Name Performed integration N

Sim-1 SDA 50 equidistributedSim-2 SDA 200 equidistributedSim-3 Matlab Integration, external call to Matlab //Sim-4 SDA 21 Sloane pointsSim-5 SDA 120 Sloane pointsSim-6 SDA 41, (Θ,Φ) ∈ I × J

Θ ∈ I ={

0,π

6,π

4,π

3,

2π5,π

2

},

Φ ∈ J ={

0,π

4,π

2,

3π4, π,

5π4,

3π2,

7π4

}.

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Results - Radial Permeability

0 0.2 0.4 0.6 0.8 1.00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ξ

Z011

(t=

0)

Sim−1Sim−2Sim−4Sim−5Sim−6Matlab

0 0.2 0.4 0.6 0.8 1.00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ξ

Z11

(t=

20)

Sim−1Sim−2Sim−4Sim−5Sim−6Matlab

Sim-1 Sim-2 Sim-4 Sim-5 Sim-6

Comp. Time 5 min 15 s ≈ 40 s 55 s 2 min 20 s ≈ 30 s

Memory [Gb] 2.52 2.54 2.53 1.44 1.21

Sim-3 Computational time: 3 h 40 minSim-3 Memory: 2.4 Gb 10 / 14

Results - Axial Permeability

0 0.2 0.4 0.6 0.8 1.00

0.5

1.0

1.5

0

0.5

1.0

1.5

ξ

Z33

(t=

0)

Sim−1Sim−2Sim−4Sim−5Sim−6Matlab

0 0.2 0.4 0.6 0.8 1.00

0.5

1.0

1.5

0

0.5

1.0

1.5

ξ

Z33

(t=

20)

Sim−1Sim−2Sim−4Sim−5Sim−6Matlab

Sim-1 Sim-2 Sim-4 Sim-5 Sim-6

Comp. Time 5 min 15 s ≈ 40 s 55 s 1 min 20 s s 18 s

Memory [Gb] 2.52 2.54 2.53 1.44 1.21

11 / 14

Results - Stress

0 0.2 0.4 0.6 0.8 1.00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ξ

S11

[MP

a]

Sim−1Sim−2Sim−4Sim−5Sim−6Matlab

0 0.2 0.4 0.6 0.8 1.00

0.005

0.010

0.015

0.020

0.025

ξ

S33

[MP

a]

Sim−1Sim−2Sim−4Sim−5Sim−6Matlab

Figure: Radial (left) and Axial (right) stress components related to fibres

12 / 14

Conclusions

If the quadrature points are properly chosen, an internal implementation of the SDA is ingeneral preferable to an external Matlab call.

To validate the point set I × J for a more general computational and mathematical setting,we need to test it on a wider range of benchmark problems and constitutive laws.

Some CitationsAspden, R.M., Hukins, D.W.L.: Collagen organization in articular cartilage, determined by X-ray diffraction, andits relationship to tissue function, Proc. R. Soc. B, 212:299–304, 1981.

Carfagna, M., Grillo, A., Federico, S., The Spherical Design Algorithm in the numerical simulation of biologicaltissues with statistical fibre-reinforcement, submitted.

Hardin, R.H., Sloane, N.J.A.: McLaren’s Improved Snub Cube and Other New Spherical Designs in ThreeDimensions, Discrete and Computational Geometry, 15:429–441, 1996.

Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs, Geometriae Dedicata, 6:363–388, 1977.

Spherical t-designs: http://neilSloane.com/sphdesigns/dim3/

http://neilSloane.com/sphdesigns/dim3/

Thank YouFor Your Kind

Attention