Measuring coronary lesions severity by virtual Fractional ...€¦ · Conclusions and perspectives Bibliography Measuring coronary lesions severity by virtual Fractional Flow Reserve

Clinical backgroundGeneralized Non Newtonian flow model

Numerical resultsFractional flow reserve computation

Realistic results and 3D caseConclusions and perspectives

Bibliography

Measuring coronary lesions severity by virtualFractional Flow Reserve issued fromnon-Newtonian blood flow simulation

PhD seminars INRIA : 29 October 2018

K. Chahour, R. Aboulaich, A. Habbal, C. Abdelkhirane andN. Zemzemi.

1/ 61 PhD seminars INRIA : 29 October 2018 Measuring coronary lesions severity by virtual Fractional Flow Reserve issued from non-Newtonian blood flow simulation




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Outlines

1 Clinical background

2 Generalized Non Newtonian flow modelNon Newtonian flow modelCoupling model FSI

3 Numerical results

4 Fractional flow reserve computation

5 Realistic results and 3D case

6 Conclusions and perspectives





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StatisticsPathology : AtherosclerosisFractional flow reserve (FFR)

Statistics

Cardiovascular diseases (CVDs) are the number 1 causeof death globally, killing more than 17.9 million worldwide.Therefore, 31% of total global mortality is due tocardiovascular diseases. An estimated 7.4 million are dueto coronary heart disease and 6.7 million to a stroke(2015). World health organisation (WHO).





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Statistics

Each year cardiovascular disease (CVD) causes 3.9million deaths in Europe.

It does not occur especially in developed countries. Morethan three-quarters of cardiovascular related deaths occurin low- and middle-income countries.





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Atherosclerosis is a chronic inflammatory disease that affectsthe entire arterial network and especially the coronary arteries.

. Available online from :https ://stanfordhealthcare.org/content/dam/SHC/conditions/blood-heart-circulation/images/anomalouscoronaryartery-diagram-heart.gif





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Atherosclerosis

It is an accumulation of lipids over the arterial surface, thatgives rise to a lesion or a plaque.The presence of the plaque causes a decrease or aninterruption of the supply of oxygen to the various organscalled ischemia.





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Atherosclerosis : Treatment

Revascularization can be established using differentangioplasty techniques, the most used is implantation of stents.

. Available online from : http ://tpe2016nddcoeur.e-monsite.com/pages/i-la-miniaturisation-au-service-du-cardiologue/le-stent-bon/





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Diagnosis phase : Angiography

Angiography is an X-ray technique to show the inside of bloodvessels, in order to identify vessel narrowing : stenosis.

FIGURE – Angiography image with the position of the plaque.





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Diagnosis phase : Angiography

Based on angiography, the clinician identifies the degree ofstenosis R = Diameter near the lesion

Original arterial diameter .

If the problem remains simple for non significant lesions(R ≤ 40%) or very severe ( R ≥ 70%), a very importantcategory of intermediate lesions must benefit from afunctional evaluation to determine the strategy oftreatment.





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Diagnosis phase : The fractional flow reserve (FFR)

The FFR consists on introducing a special guide-wire tomeasure the ratio : Pdistal

Paortic.

FIGURE – Left, the invasive FFR technique. Right, a typical exampleof FFR measurement.





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FFR value of 0.80 or less identifies ischemia-causingcoronary lesions with an accuracy of more than 90% .An FFR value higher than 0.8 indicates a lesion that is nothemodynamically significant.





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Many studies have demonstrated its effectiveness in improvingthe patients prognosis, by applying the appropriate approach :FAME study.

. For the figure, see V. Nunen and al. : Fractional flow reserve versusangiography for guidance of PCI in patients with multivessel coronary arterydisease (FAME) : 5-year follow-up of a randomised controlled trial (2015).11/ 61 PhD seminars INRIA : 29 October 2018 Measuring coronary lesions severity by virtual Fractional Flow Reserve issued from non-Newtonian blood flow simulation




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FAME study prove that the FFR approach contributed inreducing the mortality rate.

. For the figure, see V. Nunen and al. : Fractional flow reserve versusangiography for guidance of PCI in patients with multivessel coronary arterydisease (FAME) : 5-year follow-up of a randomised controlled trial (2015).12/ 61 PhD seminars INRIA : 29 October 2018 Measuring coronary lesions severity by virtual Fractional Flow Reserve issued from non-Newtonian blood flow simulation




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Fractional flow reserve (FFR) : Motivations

The FFR is binding since it is invasive.This technique induces additional costs, which are notcovered by insurances in several countries.A realistic simulation of the vascular blood flow, in thepresence of stenosis, can provide an estimation of theFFR, without any surgery or additional costs.





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Non Newtonian flow modelCoupling model FSI

Non Newtonian flow model

In their work, S.Boujena, O.Kafi and al (2014) gave andadapted flow model in the presence of atherosclerosis, with theproof of existence. The blood flow and velocity in the artery aresolutions of the following problem :ρf∂u∂t

+ ρf (u.∇)u −∇.(2µ(s(u))Du) +∇p = 0, in Ωf × (0,T )

∇.u = 0, in Ωf × (0,T )

FIGURE – 2D simplified geometry.14/ 61 PhD seminars INRIA : 29 October 2018 Measuring coronary lesions severity by virtual Fractional Flow Reserve issued from non-Newtonian blood flow simulation




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Non Newtonian flow model

The shape of the plaque is modeled as a sinusoïdal function :

ωs(x) =

D × cos(π(x − xs)/2 ∗ δ) if xs − δ < x < xs + δ0 otherwise.

(1)

D the height of the plaque, xs the position of the center of theplaque and 2× δ its length. R = D/H indicates the degree ofstenosis, it varies by changing the value of D.





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Generalized Non Newtonian flow model

With :(s(u))2 = 2Du : Du = 2

∑i,j

(Du)ij(Du)ji

Du =12

(∇u +∇T u)

The blood viscosity is given by Carreau law :

µ(s(u)) = µ∞ + (µ0 − µ∞)(1 + (λs(u))2)(n−1)/2

Parameters :ρf = 1060Kg.m−3,µ0 = 0.0456Pa.s and µ∞ = 0.0032Pa.s,λ = 10.03s and n = 0.344.





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Boundary conditions

Inlet : sinusoïdal pressure-wave

2µ(s(u))Du.n − pn = h, on Γin × (0,T )

With :

h =

(Pmax × (1− cos(2πt/T ∗)),0)T , x ∈ Γin, 0 s ≤ t ≤ T ∗

(0,0)T , x ∈ Γin, T ∗ ≤ t ≤ T .(2)

Parameters :Pmax = 104 Pa,T ∗ = 5.10−3 s.





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Boundary conditions

Zero pressure at the outlet

2µ(s(u))Du.n − pn = 0, on Γout × (0,T )

No slip conditions are enforced on the lower and upperboundaries :

u = 0, on Γω1 ∪ Γω2 × (0,T )





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Initial condition

Steady Stokes initial condition, with a Poiseuille flow profileu0 at the inlet :

u0(y) = u0m × y/H × (1− y/H)

Parameters :H = 0.5 cm is the arterial diameter.u0m = 0.4 m/s.





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Variational formulation : Problem I

(∀v ∈ V ) such that V = v ∈ (H1(Ωf ))2 | ∇.v = 0 in Ωf ,v .n = 0 on Γω1 , v = 0 on Γω2 ρf

∫Ωf

∂u∂t

vdx + (Au, v) + ρf b(u,u, v) =

∫Γin

hvdσ,

u(t = 0) = u0, sur Ωf

(Au, v) =

∫Ωf

2µ(s(u))Du : Dvdx , (3)

b(u, v ,w) =2∑

i,j=1

∫Ωf

ui∂vj

∂xiwjdx (4)





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Coupling model : fluid structure interaction

Based on the work of M.Fernandez and al (2015), we considera Koiter model, where the arterial wall is assimilated to a 1Dlayer with a thickness ε :

ρsε∂t η − c1∂2xη + c0η = −σ(u,p)n.n, on Γω2 × (0,T )

u.n = η,u.τ = 0, on Γω2 × (0,T )η = 0, on ∂Γω2 × (0,T )

With :σ(u,p) = −pI + 2µ(s(u))Du.

c1 =Eε

2(1 + ν); c0 =

EεR2(1− ν2)





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Coupling model : fluid structure interaction

Parameters : ρs = 1.1, ε = 0.1, E = 0.75.106 and ν = 0.5.

Remarque : In S.Boujena and O.Kafi (2014) they used a 2D-and 3D-3D FSI model. Since we are interested to calculate theFFR, we prefered to use a 2D-1D model to be generalized in3D-2D based on the work of M.Fernandez and al (2015).





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Results : simulation

The simulation is performed using finite element solverFreefem++, using a semi-implicit finite difference scheme.Fluid velocity and pressure are calculated at each timestep.The time step is δt = 5.10−3s. The mesh have nv = 2586vertices and nt = 4879 triangles. The computation isstopped at T = 0.8 s which represents the duration of acardiac cycle.





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Results : Non Newtonian flow model vs N.S

FIGURE – Velocity and pressure fields using N.S flow model.





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Results : Non Newtonian flow model vs N.S

FIGURE – Velocity and pressure fields using the generalized flowmodel.





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Fluid structure interaction

Fractional flow reserve computation

The aortic pressure Pa is calculated at each time step bythe spatial mean of the pressures of the points at 1 cmfrom the inlet. Whereas the distal pressure Pd is obtainedat 1 cm after the lesion.A temporal mean of Pa and Pd is performed during thecardiac cycles :

P =1Tc

∫ Tc

0p(t)dt

The FFR ratio is calculated by :

FFR =Pd

Pa





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In this case, the degree of stenosis is equal to 40% and theFFR is equal to 0.81.

FIGURE – FFR calculation : left for a healthy portion, right for a 40%stenosed portion.





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The VFFR value was calculated at each cycle, during 5consecutive cardiac cycles. We notice that starting from thethird cardiac cycle, this value becomes constant.

FIGURE – VFFR variation during 5 cardiac cycles for a lesion with70% stenosis.





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Fractional flow reserve variation : rigid model

FIGURE – Left, Pa, Pd and VFFR variation according to the degree ofstenosis R. Right, Pa, Pd and VFFR variation according to the lesionradius δ.





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FIGURE – VFFR variation for lesions with different radius according tothe degree of stenosis.





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The VFFR value decreases with respect to the degree ofstenosis.The VFFR is not set to a big change according to thelesion’s radius for important lesions : R ≥ 60%





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Mutli-stenosis patient case

We consider two identic lesions of 40% stenosis, with a spacingof 0.5 cm. VFFR value is equal to 0.729.

FIGURE – Velocity and pressure fields.





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We consider two identic lesions of 40% stenosis, with a spacingof 1.5 cm. VFFR value is equal to 0.81.






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We consider two different lesions : the upper one is 60%stenosed, and the lower is 40% stenosed. The spacingbetween the two is 1.5 cm. VFFR value is equal to 0.68.






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Fractional flow reserve variation : multi-stenosis case

Two identic lesions : 40% stenosis. The distance between thecenters of the two lesions is minimal, it equals to 2× δ in thegraphics.






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Fractional flow reserve variation : multi-stenosis case

The flow and pressure distributions are modified with respect tothe spacing between the two lesions. Thus, the VFFR value isalso modified according to this parameter.





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Results : FSI

With the FSI model, we obtain an FFR value of 0.75 for a 40%degree of stenosis lesion

FIGURE – Pressure distribution with the FSI model





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Fractional flow reserve computation : FSI model






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FIGURE – VFFR variation for lesions with different radius according tothe degree of stenosis.





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The VFFR value decreases with respect to the degree ofstenosis.In contrary to the rigid model, the VFFR is not set to a bigchange according to the lesion’s radius for small lesions :R ≤ 50%

This could be explained by the fact that the elastic propertyof the upper wall compensates the over-pressure beforethe lesion and thus modifies the pressure distribution in thewhole domain.





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Realistic 2D geometry reconstructionSimulation in realistic 2D domain

3D flow model

Realistic 2D domain : Image segmentation

We started from a 2D patient specific angiography.Different filters were used to improve the contrast of theoriginal image. Then opening/closing Matlab functionswere used to extract a black and white image that containsonly the coronary tree of interest.Starting from this black and white image, the segmentationand the meshing were made later using a Freefem++ codebased on the isoline function already implemented underFreefem++.





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3D flow model

Realistic 2D domain : Image segmentation

FIGURE – From left to right : The original angiography image, thecoronary tree of interest is framed with red. The Black and whiteimage. The 2D meshed domain.





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3D flow model

Realistic 2D domain : Inlet BC

Since the image treated corresponds to a left coronary artery,we used sinusoïdal functions to approach the inlet flowdistribution into the left coronary artery.

FIGURE – Left, spline function approaching left coronary blood flow.Right, the flow function prescribed at the inlet I(t).





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3D flow model

Realistic 2D domain : Inlet BC

Considering that Tsys is the period of systole, ts the start of thesystolic phase of the current cardiac cycle and td the start ofthe diastolic phase, the periodic function I(t) is given by :

I(t) =

(Ip + I0 ∗ sin(π ∗ (t − ts)/Tsys),0),0s ≤ t ≤ Tsys(Ip + Ic ∗ sin(π ∗ (t − td)/(Tc − Tsys)),0),Tsys ≤ t ≤ Tc .

Parameters :Ip = 10 cm/s represents the dominant flow,I0 = 10 cm/s and Ic = 10 cm/s,Tsys is considered equal to 0.33s.





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3D flow model

Realistic 2D domain : Outlet BC

In order to take into account the effect of arterial complianceand total peripheral resistance, a 2 elements Windkesselmodel is incorporated in the outlets.

The arterial compliance C is represented as a capacitorwith electric charge storage properties. Peripheralresistance of the systemic arterial system R is representedas an energy dissipating resistor.The flow of blood from the heart I(t) is analogous to that ofcurrent flowing in the circuit and the blood pressure Pd (t)is modeled as a time-varying electric potential.





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3D flow model


The theoretical modeling as seen in the electrical analogy isgiven by :

I(t) =Pd (t)

R+ C

dPd (t)dt





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3D flow model


The downstream domain is incorporated into the model byway of the two operators : M = [Mm, ~Mc] and H = [Hm, ~Hc].The following terms are add to the fluid variationalformulation : ∫

Γout

v .(Mm(u,p) + Hm(u,p)).~nds∫Γout

q.( ~Mc(u,p) + ~Hc(u,p)).~nds

Each one of M and H is composed of a momentum and acontinuity operator respectively.





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3D flow model

Realistic 2D domain : Outlet BC∫Γout

v .Mm(u,p).~nds = −∫

Γout

v .~n (R∫

Γout

u.~nds +∫ t

0

e−(t−t1)/δ

C

∫Γout

u(t1).nds.dt1 + ~n.τ.~n) ds +

∫Γout

v .τ.~nds∫Γout

v .Hm(u,p).~nds =

−∫

Γout

v .~n ((P(0)− R∫

Γout

u(0).~ndΓ

− Pd (0))e−t/δ + Pd (t)) ds∫Γout

q.( ~Mc(u,p) + ~Hc(u,p)).~nds =

∫Γout

q.u.~nds





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3D flow model


Pd (t) is a solution to the electrical equality.

Parameters :δ = RC is the time decay constant of the Windkesselmodel.R = 0.95mmHg.s/cm3 the resistance andC = 1.06cm3/mmHg the capacitance.





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3D flow model

Results : Realistic 2D domain

FIGURE – Velocity and pressure fields with Windkessel outlet BC attime t = 0.25s.





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3D flow model

FFR computation : Realistic 2D domain

In this case, the degree of stenosis is equal to 68% whichmakes it an intermediate lesion. The FFR is equal to 0.6168,and then the lesion is classified as hemodynamically significant.

FIGURE – FFR calculation.51/ 61 PhD seminars INRIA : 29 October 2018 Measuring coronary lesions severity by virtual Fractional Flow Reserve issued from non-Newtonian blood flow simulation




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3D flow model

3D case : flow model

In a 3D geometry, the Navier stokes equation can be written asfollows :

ρf∂u∂t

+ ρf (12∇|u|2 + curl u × u)−∇.(2µ(s(u))Du)

+∇p = 0, in Ωf × (0,T )∇.u = 0, in Ωf × (0,T )

(5)

Using the identity : u · ∇u = 12∇|u|

2 + curl u × uThe expression of s(u) is the same as in the 2D case.





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3D flow model

3D case : flow model

The considered 3D domain Ωf modelize a stenosed arterialportion :

FIGURE – Considered geometry for the problem.





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3D flow model

3D case : Boundary conditions

The equation is completed with suitable boundary conditions onΩf ( n is the normal ). We consider the total stress tensor :

σtot = −(p +ρf

2|u|2)I3 + 2µ(s(u))Du.

σtot (u,p).n = 0, sur Γout × (0,T )

u = 0, sur Γω × (0,T )





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3D flow model

3D case : Boundary conditions

σtot (u,p).n = h, sur Γin × (0,T )

With :

h = Pmax × (0.8− 0.2× cos(2π(0.2− t)); x ∈ Γin, 0s ≤ t ≤ T

Remarque : The values of parameters for the 3D model are thesame as in 2D.





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3D flow model

3D case : Numerical results

FIGURE – Top, 2D slices of 3D velocity at t = 0.4s and at t = 0.7s.Bottom, 2D slices of 3D pressure at t = 0.2s and at t = 0.5s.





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Conclusions and perspectives

A 2D angiography based reconstruction is not the bestreproduction to the physiological domain. This is principallydue to the branchs’ torsion and to the transient movementsinduced by the respiratory system during acquisition :Angiography based 3D image reconstruction, like in J.Yang and al (2009).Considering patient-specific boundary conditions : modelintegrating measurable clinical Parameters :with simulationperformances.





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Conclusions and perspectives

Our aim for the futur works is to compare the results frommodeling with in vivo measurements recorded during aninterventional procedure, based on 2D or 3D geometriesissued from image reconstruction techniques.





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Bibliography

S. Boujena, N. El Khatib and O. Kafi : GeneralizedNavier-Stokes Equations with Non-standard Conditions forBlood Flow in Atherosclerotic Artery (2014).O.Pironneau : Simplified fluid-structure interaction forhemodynamics (2013).Miguel Angel Fernandez, Mikel Landajuela, MarinaVidrascu : Fully decoupled time-marching schemes forincompressible fluid/thin-walled structure interaction(2015).





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Bibliography

Toni Lassila, Andrea Manzoni, Alfio Quarteroni, GianluigiRozza : A reduced computational and geometricalframework for inverse problems in haemodynamics (2013).Jian Yang, Yongtian Wang, Yue Liu, Songyuan Tang,Wufan Chen : Novel Approach for 3-D Reconstruction ofCoronary Arteries From Two Uncalibrated AngiographicImages (2009).K. Chahour, R. Aboulaich, A. Habbal, C. Abdelkhirane andN. Zemzemi : Simulation of blood flow in a stenosed arteryand fractional flow reserve computation. PICOF (June2018).





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Measuring coronary lesions severity by virtual Fractional ...€¦ · Conclusions and perspectives Bibliography Measuring coronary lesions severity by virtual Fractional Flow Reserve