a mathematical model for blood flow and mass transport by...

A mathematical model for blood flow and masstransport by a drug eluting stent

Elas Gudino

Adelia Sequeira

Departamento de Matematica, CEMAT/IST, Universidade de Lisboa.

December 2014

E. Gudino CEMAT/IST 1

Contents

1 Introduction

2 Modeling blood flow

3 Modeling drug release

4 Finite element discretization for blood flow

5 Finite element discretization for drug delivery

6 Numerical experiments


Introduction

A stent is an expandable metallic tube used to open a narrowed orclogged artery.


Introduction

During a procedure called angioplasty, the stent is inserted into acoronary artery and expanded using a small balloon.


Introduction

Drug eluting stents: polymeric coating loaded with drugs that inhibitsmooth muscle cells proliferation.


Modeling blood flow

Contents

2 Modeling blood flow


Modeling blood flow

Assumptions

The stented vessel region is a straight cylinder with axis l.

Flow is stable and laminar in the lumen (characteristic Reynoldsnumber is low).

The wall is a single homogeneous layer (the fluid-wall model).


Modeling blood flow

Domain

Figure : Physical domain


Modeling blood flow

System of equations

The model is defined by the following system of equations,

ul (ul )ul pl = 0 in l (0, T ), ul = 0 in l (0, T ),

1uw +pw = 0 in w (0, T ), uw = 0 in w (0, T ),

where is the blood dynamic viscosity and 1 is the inverse permeabilityof the arterial wall.


Modeling blood flow

System of equations

We denote by p =ki=1

(i)p, the polymeric coating of the stent, then,

1up +pp = 0 in p (0, T ), up = 0 in p (0, T ),

where 2 is the inverse permeability of polymeric coating.


Modeling blood flow

Boundary conditions

Coupled with boundary conditions,

ul = uin on in,

pl = p0 on out,

ul l = 0 on l,uw w = 0 on cut,

pw = p0 100 on adv,

where uin : R2 R2 is a given function and p0 R is a constant.


Modeling blood flow

Interface conditions

We consider the continuity of the pressure and the continuity of thenormal component of the velocity,

pw = pl, ul l = uw w on int,pp = pw, up p = uw w on p,w,pl = pp, ul l = up p on p,l.


Modeling drug release

Contents

3 Modeling drug release



Assumptions

The coating of the drug eluting stent is made of a porous polymericmatrix that encapsulates a therapeutic drug in solid phase.

Drug transfer does not affect blood and plasma flow.

The drug is present in two states (dissolved and undissolved).

Drug release is controlled by non-Fickian diffusion of plasma into thepolymeric matrix, by advection activated by trans-mural pressuregradients and dissolution.



Non-Fickian diffusion

Ficks classic law:

C

t= JF (C)

JF (C) = D(C)C

Modified flux:

J = JF (C) + JNF () JNF (C) = Dv(C)

Non-Fickian diffusion equation:

C

t= (JF (C) + JNF ())




Ficks classic law:

C

t= JF (C) JF (C) = D(C)C

Modified flux:



C

t= (JF (C) + JNF ())




Ficks classic law:

C


Modified flux:

J = JF (C) + JNF ()

JNF (C) = Dv(C)


C

t= (JF (C) + JNF ())




Ficks classic law:

C


Modified flux:



C

t= (JF (C) + JNF ())



Non-Fickian behavior

We consider a generalized Maxwell-Wiechert model with m+ 1 arms inparallel. The stress associated to solvent uptake and exerted by thepolymer is defined as

(cp) =

(mk=0

Ek

)f(cp) +

t0

(mk=1

Ekke tsk

)f(cp)(s)ds ,

where k =kEk

are the relaxation times associated to each of the mMaxwell fluid arms



System of equations

The evolution of solvent penetration, drug diffusion and dissolution in thepolymeric matrix are described by the following equations on the domainp and for t > 0,

cpt

= (Dp(cp)cp +Dv(cp)(cp) pupcp) ,

cdt

= (Dd(cp)cd + ((cp) dup)cd) +Kp(ds cpds

)cpH(s) ,

s

t= Kp

(ds cpds

)cpH(s) ,

where (cp) = Dv(cp)(cp)cp

and H is the heaviside step function.



System of equations

The concentration of dissolved drug in the lumen and in the wall can betracked with the following equations

cwt

= (Dw(cw)cw wuwcw) ,

clt

= (Dl(cl)cl ulcl) ,

where Dw and Dl denote the diffusion coefficients of the drug in thelumen and in the wall respectively. The constant 0 < i 1 denotes thehindrance coefficient accounting for penetrant-media frictional effects.



The fluxes

They are defined as,

Jp = (Dp(cp)cp +Dv(cp)(cp) pupcp) ,Jd = (Dd(cp)cd + ((cp) dup)cd) ,Jw = (Dw(cw)cw wuwcw) ,Jl = (Dl(cl)cl ulcl) .



Initial conditions

We have,

cp(0) = 0 in p,

cd(0) = 0 in p,

s(0) = s0 in p,

cw(0) = 0 in w,

cl(0) = 0 in l.



Boundary conditions

Coupled with,

cp = 1 on p,l p,w,Jw w = 0 on adv cut,Jl l = 0 on in out,Jd p = 0 on str,Jp p = 0 on str.



Interface conditions

We consider the continuity of the drug concentrations and the continuityof the normal component of the fluxes,

cw = cl, Jw w = Jl l on int,cd = cw, Jd p = Jw w on p,w,cl = cd, Jl l = Jd p on p,l.


Finite element discretization for blood flow

Contents

4 Finite element discretization for blood flow



Preliminaries

We use the edge stabilization technique introduced by Badia and Codina[1], further explored by DAngelo and Zunino [2, 3] based on aH1-conforming approximation for velocities on each subdomain togetherwith Nitsches type penalty method for the coupling between differentsubproblems. This combination leads to a continuous-discontinuousGalerkin type penalty method.



Preliminaries

We will assume that each i is a convex polygonal domain, equippedwith a family of quasi-uniform triangulations Th,i made of affine trianglesthat are conforming on each interface.

On each subregion we consider the spaces,

Vh,i = {vh C(i) : vh|K Pk(K), K Th,i}, Vh,i = [Vh,i]2 ,Qh,i = {qh L2(i) : qh|K Pk1(K), K Th,i},

where k 1.



Preliminaries

As global approximation spaces, we define

Vh =

Ni=1

Vh,i, Qh =

Ni=1

Qh,i.

For the treatment of the nonlinear advective term of the steadyNavier-Stokes equation, we consider Picards iteration, thus we have that

ul (wl )ul pl = 0 in l (0, T ).



Preliminaries

We denote with

ij = i j , for all j = 1, .., N .

Gh,ij the intersection of the trace meshes on ij .

Bh,i the trace meshes at the boundary subdomains.

Fh,i the set of all interior edges of Th,i.

hE a piecewise constant function taking the value diam(E) on eachedge E.



Preliminaries

Let h represent a finite element function, we define the jump and theaverage across any (internal) face F of the computational domain in theusual ways,

JhK(x) = lim0

[h(x F ) h(x+ F )] , x F,

{h}(x) = lim0

1

2[h(x F ) + h(x+ F )] , x F.

We also define the weighted average on ij ,

{h}w = lh,i + wh,j , (1)

where = (l, w) are suitable weights such that l + w = 1.



Bilinear forms

For any uh, vh Vv and ph, qh Qh, we define,

a(uh, vh;wl) =

(uh : vh + (wl uh) vh + uh vh)

(uh vh +vh uh)

+

Bh

(1

2(|wl | wl ) +

uhE

)uh vh

b(ph, vh) =

ph vh +

phvh ,

d(ph, vh) =

{ph}wJvh K,



Bilinear forms

c(uh, vh) =

Gh

uhE{}wJuhK JvhK

({uh}w JvhK + {vh} JuhK) ,

ju(uh, vh) =

Gh

uhE

Juh KJvh K

+

Bh

uhE

(uh )(vh ),

jp(ph, qh) =

FhphEJphKJqhK.



Mixed formulation

Find (uh, ph) Vh Qh such that

a(uh, vh;wl) + c(uh, vh) + ju(uh, vh) + b(ph, vh)

+d(ph, vh) = F(vh), vh Vh,b(qh, uh) + d(qh, uh) jp(ph, qh) = 0, qh Qh,

where

F(vh) =

in

(1

2(|wl | wl ) +

uhE

)uin vh,l

+

in

uhE

(uin )(vh,l )

out

p0vh,l l

adv

(p0 100)vh,w w.


Finite element discretization for drug delivery

Contents

5 Finite element discretization for drug delivery



Preliminaries

We use the Implicit-Explicit (IMEX) finite element method introduced byFerreira, Gudino and de Oliveira [6, 7], in combination with the domaindecomposition method for advection-diffusion equations introduced byBurman and Zunino [4] on each subdomain.



Preliminaries

On each subregion we consider the space,

Vh,i = {vh C(i) : vh|K Pk(K), K Th,i},

where k 1.

As global approximation space, we define

Vh =

Ni=1

Vh,i.



Bilinear form

For any ch, vh Vh we define,

e(ch, vh) =

(chE{D}wJchKJdhK + {uh }{ch}wJdhK

{Dch }wJdhK {Ddh }wJchK) .



Plasma diffusion

Find ch,p Vh,p such that

ap(ch,p, vh) + e(ch,p, vh) = (cn1h,pt

, vh), vh Vh,p,

where,

ap(ch,p, vh) =

(ch,pt

vh +(Dp(c

n1h,p )ch,p

+Dv(cn1h,p )(c

n1h,p ) puh,pch,p

) vh

)+

Fh,p

uh2Euh,p L(E)Jch,p KJdh K.



Drug diffusion in the polymer

Find ch,d Vh,p such that

ad(ch,d, vh) + e(ch,d, vh) = (cn1h,dt

, vh), vh Vh,p,

where,

ad(ch,d, vh) =

(ch,dt

vh + (Dd(ch,p)ch,d

+((ch,p) duh)ch,d) vh

+Kd

(ds cn1h,d

ds

)ch,pH(s

n1h )vh

)

+

Fh,p

uh2Euh,p L(E)Jch,d KJdh K.



Drug diffusion in the lumen and wall

Find ch,i Vh,i such that

ai(ch,i, vh) + e(ch,i, vh) = (cn1h,it

, vh), vh Vh,i,

where ,

ai(ch,i, vh) =

i

(ch,it

vh +(Di(c

n1h,i )ch,i iuich,i

) vh

)+

Fh,i

uh2Euh,i L(E)Jch,i KJdh K.

for i = l, w.


Numerical experiments

Contents

6 Numerical experiments



Figure : Simplified domain



Figure : Computational domain, NT = 10314 and NV = 5306



Figure : Velocity field



Figure : Pressure



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DrugC2.mp4Media File (video/mp4)


Figure : Mass of drug in the wall vs the lumen



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DrugS2.mp4Media File (video/mp4)


Figure : Mass of drug different stent profiles



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DrugP2.mp4Media File (video/mp4)


Figure : Mass of drug in the wall plaque vs no plaque



Figure : Computational domains, NT = 10314 and NV = 5306



Figure : Mass of drug in the wall different positions



Acknowledgments

This work was partially supported by Center for Computacional andStochastic Mathematics (CEMAT) of the Universidade de Lisboa, fundedby the Portuguese Government through the FCT - Fundacao para aCiencia e Tecnologia under the project Mathematical and ComputationalModeling in Human Physiology PHYSIOMATH -EXCL/MAT-NAN/0114/2012.



[1] S. Badia, R. Codina, Unified stabilized finite element formulatons forthe Stokes and Darcy problems, SIAM J. Numer. Anal., 47 (2009),pp. 19712000.

[2] C. DAngelo, P. Zunino, A finite element method based on weightedinterior penalties for heterogeneous incompressible flows, SIAM J.Numer. Anal., 47 (2009), pp. 39904020.

[3] C. DAngelo, P. Zunino, Robust numerical approximation of coupledStokes and Darcys flows applied to vascular hemodynamics andbiochemical transport, Mathematical Modelling and NumericalAnalysis, 45 (2011), pp. 447476.

[4] E. Burman and P. Zunino, A domain decomposition method basedon weighted interior penalties for advection-diffusion-reactionproblems, SIAM J. Numer. Anal., 44 (2006), pp. 16121638.

[5] J.A. Ferreira, M. Grassi, E. Gudio, P. de Oliveira, A new look tonon-Fickian diffusion, Applied Mathematical Modelling, 39 (2015),pp. 194204.



[6] J.A. Ferreira, M. Grassi, E. Gudio, P. de Oliveira, A 3D model formechanistic control of drug release, SIAM Journal on AppliedMathematics, 74(3) (2014), pp. 620633.

[7] J.A. Ferreira, E. Gudio, P. de Oliveira, A second order approximationfor quasilinear non-Fickian diffusion models, Computational Methodsin Applied Mathematics, 13(4) (2013), 471493.

IntroductionModeling blood flowModeling drug releaseFinite element discretization for blood flowFinite element discretization for drug deliveryNumerical experiments

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