A mathematical model for blood flow and mass transport by...
Transcript of A mathematical model for blood flow and mass transport by...
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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
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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
E. Gudino CEMAT/IST 2
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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
E. Gudino CEMAT/IST 2
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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
E. Gudino CEMAT/IST 2
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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
E. Gudino CEMAT/IST 2
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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
E. Gudino CEMAT/IST 2
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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
E. Gudino CEMAT/IST 2
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Introduction
A stent is an expandable metallic tube used to open a narrowed orclogged artery.
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Introduction
During a procedure called angioplasty, the stent is inserted into acoronary artery and expanded using a small balloon.
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Introduction
Drug eluting stents: polymeric coating loaded with drugs that inhibitsmooth muscle cells proliferation.
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Modeling blood flow
Contents
2 Modeling blood flow
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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).
E. Gudino CEMAT/IST 7
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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).
E. Gudino CEMAT/IST 7
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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).
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Modeling blood flow
Domain
Figure : Physical domain
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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.
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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.
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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.
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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.
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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.
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Modeling drug release
Contents
3 Modeling drug release
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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.
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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.
E. Gudino CEMAT/IST 14
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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.
E. Gudino CEMAT/IST 14
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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.
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Modeling drug release
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 ())
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Modeling drug release
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 ())
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Modeling drug release
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 ())
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Modeling drug release
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 ())
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Modeling drug release
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 ())
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Modeling drug release
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
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Modeling drug release
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.
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Modeling drug release
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.
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Modeling drug release
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) .
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Modeling drug release
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.
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Modeling drug release
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.
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Modeling drug release
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.
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Finite element discretization for blood flow
Contents
4 Finite element discretization for blood flow
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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.
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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 ).
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Finite element discretization for blood flow
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 ).
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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.
E. Gudino CEMAT/IST 27
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Finite element discretization for blood flow
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.
E. Gudino CEMAT/IST 27
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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,
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Finite element discretization for blood flow
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,
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Finite element discretization for blood flow
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,
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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.
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Finite element discretization for blood flow
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.
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Finite element discretization for drug delivery
Contents
5 Finite element discretization for drug delivery
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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.
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Finite element discretization for drug delivery
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.
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Finite element discretization for drug delivery
Bilinear form
For any ch, vh Vh we define,
e(ch, vh) =
(chE{D}wJchKJdhK + {uh }{ch}wJdhK
{Dch }wJdhK {Ddh }wJchK) .
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Finite element discretization for drug delivery
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.
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Finite element discretization for drug delivery
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.
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Finite element discretization for drug delivery
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.
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Numerical experiments
Contents
6 Numerical experiments
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Numerical experiments
Figure : Simplified domain
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Numerical experiments
Figure : Computational domain, NT = 10314 and NV = 5306
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Numerical experiments
Figure : Velocity field
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Numerical experiments
Figure : Pressure
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Numerical experiments
Figure : Mass of drug in the wall vs the lumen
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Figure : Computational domain, NT = 10314 and NV = 5306
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Figure : Mass of drug different stent profiles
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Figure : Computational domain, NT = 10314 and NV = 5306
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Figure : Mass of drug in the wall plaque vs no plaque
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Figure : Computational domains, NT = 10314 and NV = 5306
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Numerical experiments
Figure : Mass of drug in the wall different positions
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Numerical experiments
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.
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[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.
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Numerical experiments
[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.
E. Gudino CEMAT/IST 56
IntroductionModeling blood flowModeling drug releaseFinite element discretization for blood flowFinite element discretization for drug deliveryNumerical experiments