Modelling the blood flow in a carotid J. Hron 1 , H. vihlov 1 , M. - - PowerPoint PPT Presentation

Modelling the blood flow in a carotid

• J. Hron1, H. Švihlová1, M. Čapek1, J. Hrnčíř1, A. Matajová1
• A. Hejčl2

1Faculty of Mathematics and Physics, Charles University, Prague 2Masaryk’s Hospital, Usti nad Labem

Carotid stenosis

☞ Carotid artery disease

■ Plaque deposits can grow larger and

larger, severely narrowing the artery and reducing blood flow to the brain.

■ Plaque deposits can roughen and

deform the artery wall, causing blood clots to form and blocking blood flow to the brain.

https://mayfieldclinic.com/pe-

☞ blood flow - mostly laminar, incompressible, isothermal,

viscoelastic behaviour, mechanical interaction with surroundings

• FSI, chemical reactions, clot formation....

Navier-Stokes solver

■ FEM discretization P2/P1 or MINI element (FEniCS) ■ solvers PETSc - parallel composable solvers ■ coupled direct solution - MUMPS, projection methods IPCS,

preconditioned iterative solvers GMRES + PCD (FENaPack https://github.com/blechta/fenapack) or LSC (PETSc) strong and weak

Carotid stenosis

■ given CT/MRI segmentation, mesh the domain (VMTK), compute

flow

■ outflow boundary conditions? (outflow mean pressure, flux by

Murry’s Law)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 time [s] 200 300 400 500 600 700 800 900 1000 peak velocity [mm/s]

ANIMATION - not included

Carotid stenosis

Look for correlation between histology and CFD results

Wiedermann ACI dx

A B C 20 mm pod bifurkací V místě bifurkace 3 mm nad bifurkací

Blood properties

■ Where is the non-Newtonian model really needed? ■ There are some models available...

III. II. I. µ0 µ

8

µ Shear Rate Viscosity Newtonian Shear−Thinning

Anand M, Rajagopal K (2017) A Short Review of Advances in the Modelling of Blood Rheology and Clot Formation.

Microstructure based model

Owens RG (2006) A new microstructure-based constitutive model for human blood Moyers-Gonzalez MA, Owens RG (2008) A non-homogeneous constitutive model for human blood.

Simplifying assumptions:

■ Newtonian solvent ■ basic (micro)structure - dumbbell

Specific assumptions:

■ dilute solution - dumbbells do interact, aggregation at rate

a(

.

γ),disaggregation at rate b(

.

γ)

■ Nk - the number density of k-aggregates ■ N0 := P✶

k=1 kNk number density of red blood cells

■ M := P✶

k=1 Nk number density of rouleaux

Microstructure based model - Owens

■ stochastic differential equations for elements of q -the end to

end vector of a dumbbell. dq dt = ru ✁ q – 2H ✏ q – s 4kBT ✏ dW(t), where ✏ is so-called friction coefficient, W(t) is a multidimensional Wiener process, H is a spring constant, kB is Boltzmann constant, T the temperature.

■ ✮ Fokker-Planck ✮ with ✥(q, t) the probability density function. ■ Kramers expression for the extra-stress tensor

T = TS + τ = 2✑SD + nH < qq > –nkBTI, where n = N0

M is the average dumbbells/aggregates size

Microstructure based model - Owens

Complete system for unknowns (v, p, τ, σ, N0, M): ✪❅v ❅t + div (✪v ✡ v) = div(T + τ) + ✪f div v = 0 τ + ¯ ✖

r

τ –Dtr¯ ✖(∆τ + (rr : τ)I) = N0(kBT + ✔)¯ ✖

.

γ σ + ¯ ✖

r

σ –Dtr¯ ✖(∆σ + (rr : σ)I) = M(kBT + ✔)¯ ✖

.

γ DN0 Dt – Dtr∆N0 = Dtr (kBT + ✔)rr : τ DM Dt – Dtr∆M = Dtr (kBT + ✔)rr : σ – a(

.

γ) 2 M2 + b(

.

γ) 2 (N0 – M)

Microstructure based model - Owens

■ it captures non-homogeneity of red blood cell distribution, i.e. it

results into non-constant hematocrit across vessel

■ it can capture for example Fahraeus effect ■ implemented by M. Čapek using deal.II

(https://www.dealii.org/)

Microstructure based model - Owens

Recent results of some computations: agregates density RBC density