Upscaling mass transfer in brain capillary networks

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This presentation is published in:

http://oatao.univ-toulouse.fr/20439

To cite this version:

Doyeux, Vincent and Davit, Yohan and Quintard, Michel and

Lorthois, Sylvie. Upscaling mass transfer in brain capillary

networks. (2018) In: ECCM-ECFD 2018 (6th European

Conference on Computational Mechanics (Solids, Structures

and Coupled Problems) (ECCM 6) and the 7th European

Conference on Computational Fluid Dynamics (ECFD 7),

11-15 June 2018 (Glasgow, United Kingdom). (Unpublished)

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Upscaling mass transfer in brain capillary networks

Vincent Doyeux, Yohan Davit, Michel Quintard, Sylvie Lorthois

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Hybrid continuous/network models for large scale simulation

Simulating Mass Transfer in the brain: a multi-scale problem

Multi-scale problem (arterio-venous, capillary, parenchyma)

Continuous representation of Parenchyma (Nicholson 2001, Kojic 2017, Holter 2017) Still need to explicitely resolve the vessels

Upscale the capillary bed, including vessels and parenchyma Coupling with explicit arterio-venous tree is different for the capillaries and parenchyma. Two concentrations necessary.

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1 Homogenization - Equations

2 Fictitious Domain Framework

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Mass transfer model at capillary scale

Hypotheses

Parenchyma and endothelium are purely diffusivedomains

Parenchyma + endothelium = tissue, Ωp∪ Ωe= Ωt

Dtspacially varying (small in Ωe) Red blood cells are neglected

Two-phase capillary bed model

∂τcβ+ u· ∇cβ −Dv∆cβ= 0 in Ωβ ∂τct −Dt∆ct= 0 in Ωt cβ = ct on ∂Ωβt Dv∇cβ· n = Dt∇ct· n on ∂Ωβt

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Upscaled representation of the capillary domain

Upscaled model for the two concentrations

εβ∂τhcβiβ+(εβhuiβ− uββ)· ∇ hcβiβ−uβt· ∇ hctit+τm(hcβiβ− hctit)

=_{∇ · (K}ββ∇hcβiβ)+∇ · (Kβt∇hctit)

εt∂τhctit−utt· ∇ hctit−utβ· ∇ hcβiβ+τm(hctit− hcβiβ)

=_{∇ · (K}tt∇hctit)+∇ · (Ktβ∇hcβiβ)

Equations solved on a domain without explicit representation of vessels Presence of capillary network taken into account through effective coefficients

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Volume averaging

Averaging equations

h∂τcβ+ u· ∇cβ −Dv∆cβi = 0 in Ωβ h∂τct −Dt∆cti = 0 in Ωt

Decomposition in averaged and deviation

cβ =hcβiβ+ceβ ct =hctit+cet

hceβi = 0 hceti = 0 We obtain equations on_hcβiβ andhctit

Equations depend on deviation termsc_eβ andcet Need closure equations onceβ andcet.

Subtracting equations on the mean to microscopic equations leads to equations onc_eβ andcet.

Deviation terms as linear combination of averaged values and their gradients

e cβ = bββ· ∇ hcβiβ+ bβt· ∇ hctit− sβ hcβiβ− hctit · ∇ hc β · ∇ hc t β t

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Volume averaging

Decomposition of deviation terms

e cβ = bββ· ∇ hcβiβ+ bβt· ∇ hctit−sβ hcβiβ− hctit e ct = btβ· ∇ hcβiβ+ btt· ∇ hctit−st hcβiβ− hctit

Effective coefficients associated

uββ= ˜usβ − Dβ V Z ∂Ωβt nβtsβdA τm= 1 V Z ∂Ωβt nβt· Dβ∇sβdA

Closure problem on sβand st coefficients:

−∇· (Dβ∇sβ) − εβτm+ u· ∇sβ = 0 on Ωβ −Dt∆st− εtτm = 0 on Ωt τm− 1 V Z ∂Ωβt n· Dβ∇sβ dV = 0 sβ−st = 1 on ∂Ωβt n· Dβ∇sβ− n· Dt∇st = 0 on ∂Ωβt sβ β = 0 hstit = 0 Take-home idea

Obtaining the effective coefficients necessitates to solve PDEs on a Representative Elementary Volume of the micro-structure

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Computational Framework

Solution proposed:

C++/Parallel Finite element library1

Feel++

Use a distance function field to capture the domain geometries

I Mesh adaptation from distance function

I Sub-domain localization to solve Stokes problem

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Network geometries

(150 µm)3 _network. Secomb (2000)

(300 µm)3_network.

Ross (2005), Tsai (2009), Kleinfeld (2011), Kaufhold 2012, Blinder (2013)

Segment networks extracted from biological images (center-line detection) Need to mesh inner and outer domains (vessels and tissue)

Difficult to mesh automatically with CAD software (design of bifurcations) Geometry representation needs to be robust and automated

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Levelset field

Defining φi, the distance field to centerline Siof a vessel of radius ri The distance field to the closest vessel boundary of the network is:

φ(x) =Nb Vesselsmin i=0 dist(x, Si)− ri Properties of φ: φ(x) < 0 if x∈ Ωβ φ(x) > 0 if x_{∈ Ω}t φ(x) = 0 if x_{∈ ∂Ω}βt

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Mesh adaptation from distance function

Target mesh size field calculated from levelset field

h(x) =    hmin, |φ(x)| ≤ R1, α|φ(x)| + β, R1≤ |φ(x)| ≤ R2, hmax, |φ(x)| ≥ R2

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Transport equations

Two domain problem

∂τcβ+ u· ∇cβ −Dv∆cβ= 0 in Ωv ∂τct −Dt∆ct= 0 in Ωt cβ = ct on ∂Ωβt Dβ∇cβ· n = Dt∇ct· n on ∂Ωβt Prescribed B.Cs for cβ on ∂Ωβ\ ∂Ωβt Prescribed B.Cs for ct on ∂Ωt\ ∂Ωβt

One domain problem with spacialized coefficients

∂τc+ u· ∇c − D(φ)∆c = 0 in Ω Prescribed B.Cs for c on ∂Ω

Velocity u is obtained by solving Stokes equation on the vessel domain only.

Diffusion coefficient spatialized to take into account the BBB.

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PDEs solved by our framework

The frameworks of fictitious network allows to solve:

The microscopic problem (DNS)

Closure equations to get the effective coefficients

Upscaled model

Monolitic solver used for the upscaled model

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Test Case, order 3 network.

Representative Elementary Volume, size (2_{× 2 × 2)}

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Comparison DNS - Homogenized model

Direct Numerical Simulation (DNS)

DNS

Number of elements (tetrahedrons): 8.5 × 106

Computational Cost: 1h on 100 cores

Darcy Scale

Number of elements (tetrahedrons): 3 × 104

Computational Cost: 10 min on 20 cores

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Simulation Visualization

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Simulation Visualization

2 9

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Simulation Visualization

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Simulation Visualization

4 9

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Simulation Visualization

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Simulation Visualization

6 9

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Simulation Visualization

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Simulation Visualization

8 9

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Simulation Visualization

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Quantitative comparison

0 2 4 6 8 10 12 t∗ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 hcβ i β REV− 2 REV− 1 REV 0 REV 1 REV 2 0 2 4 6 8 10 12 t∗ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 hcσ i σ REV− 2 REV− 1 REV 0 REV 1 REV 2 PeREV= 1, Km= 3× 10−2 Adimentional coefficients

P´eclet number PeREV= hui β zL REV Dv Membrane Permeability Km∗ = De Dv r e

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Quantitative comparison

0 2 4 6 8 10 12 t∗ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 hcβ i β REV− 2 REV− 1 REV 0 REV 1 REV 2 0 2 4 6 8 10 12 t∗ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 hcσ i σ REV− 2 REV− 1 REV 0 REV 1 REV 2 PREV e = 10, Km= 3× 10−4 Adimentional coefficients

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Anatomical Network

1 4

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Anatomical Network

2 4

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Anatomical Network

3 4

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Anatomical Network

4 4

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Quantitative comparison

0 1 2 3 4 5 6 7 8 9 t∗ 0 2 4 6 8 10 12 14 16 hcβ i β REV− 1 REV 0 REV 1 REV 2 0 1 2 3 4 5 6 7 8 9 t∗ 0 2 4 6 8 10 12 14 16 hcσ i σ REV− 1 REV 0 REV 1 REV 2 PREV e = 1, Km= 3× 10−2 Adimentional coefficients

P´eclet number PREV

e = hui β zL REV Dv Membrane Permeability Km∗ = De Dv r e

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Conclusion and perspective

Conclusion

Two-phase representation of the capillaries and tissue Sucessful upscaling of the capillary bed

Framework using fictitious domain for DNS and Closure equations

In the future...

Coupling with arterio-venous tree (1D-3D) or (3D-3D) Include the RBCs

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Use of the distance function

Distance function used to: define a sub-domain: Ωvh⊂ Ωh, Ωvh = [ i Ωei| π P_dh0 Ω_ei(φ) < 0

An example of projection function on a elements is given by: πPdh0

Ω_ei =

−1, if φ < 0 on at least one dof of Ωei

1 else

define continuously quantities having a different value in the domains thanks to a smoothed Heaviside function (level set method):

Hε(φ) =    0, φ_{≤ −ε,} smooth variations, _{−ε ≤ φ ≤ ε,} 1, φ_{≥ ε}

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All results, order 3

0 2 4 6 8 10 12 t∗ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 hcβ i β REV− 2 REV− 1 REV 0 REV 1 REV 2 0 2 4 6 8 10 12 t∗ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 hcσ i σ REV− 2 REV− 1 REV 0 REV 1 REV 2 PeREV= 0.1, Km= 3× 10−2 Adimentional coefficients

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All results, order 3

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All results, order 3

0 2 4 6 8 10 12 t∗ 0.00 0.02 0.04 0.06 0.08 0.10 hcβ i β REV− 2 REV− 1 REV 0 REV 1 REV 2 0 2 4 6 8 10 12 t∗ 0.00 0.02 0.04 0.06 0.08 0.10 hcσ i σ REV− 2 REV− 1 REV 0 REV 1 REV 2 PeREV= 10, Km= 3× 10−2 Adimentional coefficients

P´eclet number PREV

e = hui β zL REV Dv Membrane Permeability Km∗ = De Dv r e

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All results, order 3

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All results, order 3

0 2 4 6 8 10 12 t∗ 0.00 0.05 0.10 0.15 0.20 hcβ i β REV− 2 REV− 1 REV 0 REV 1 REV 2 0 2 4 6 8 10 12 t∗ 0.00 0.05 0.10 0.15 0.20 hcσ i σ REV− 2 REV− 1 REV 0 REV 1 REV 2 PeREV= 1, Km= 3× 10−3 Adimentional coefficients

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All results, order 3

0 2 4 6 8 10 12 t∗ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 hcβ i β REV− 2 REV− 1 REV 0 REV 1 REV 2 0 2 4 6 8 10 12 t∗ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 hcσ i σ REV− 2 REV− 1 REV 0 REV 1 REV 2 PREV e = 10, Km= 3× 10−4 Adimentional coefficients