Post on 31-Aug-2019
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
E. Gudino CEMAT/IST 2
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
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
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
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
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
Introduction
A stent is an expandable metallic tube used to open a narrowed orclogged artery.
E. Gudino CEMAT/IST 3
Introduction
During a procedure called angioplasty, the stent is inserted into acoronary artery and expanded using a small balloon.
E. Gudino CEMAT/IST 4
Introduction
Drug eluting stents: polymeric coating loaded with drugs that inhibitsmooth muscle cells proliferation.
E. Gudino CEMAT/IST 5
Modeling blood flow
Contents
2 Modeling blood flow
E. Gudino CEMAT/IST 6
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
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
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
Modeling blood flow
Domain
Figure : Physical domain
E. Gudino CEMAT/IST 8
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.
E. Gudino CEMAT/IST 9
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.
E. Gudino CEMAT/IST 9
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.
E. Gudino CEMAT/IST 10
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.
E. Gudino CEMAT/IST 11
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.
E. Gudino CEMAT/IST 12
Modeling drug release
Contents
3 Modeling drug release
E. Gudino CEMAT/IST 13
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
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
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
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
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 ())
E. Gudino CEMAT/IST 15
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 ())
E. Gudino CEMAT/IST 15
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 ())
E. Gudino CEMAT/IST 15
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 ())
E. Gudino CEMAT/IST 15
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 ())
E. Gudino CEMAT/IST 15
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
E. Gudino CEMAT/IST 16
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.
E. Gudino CEMAT/IST 17
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.
E. Gudino CEMAT/IST 18
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) .
E. Gudino CEMAT/IST 19
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.
E. Gudino CEMAT/IST 20
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.
E. Gudino CEMAT/IST 21
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.
E. Gudino CEMAT/IST 22
Finite element discretization for blood flow
Contents
4 Finite element discretization for blood flow
E. Gudino CEMAT/IST 23
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.
E. Gudino CEMAT/IST 24
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.
E. Gudino CEMAT/IST 25
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.
E. Gudino CEMAT/IST 25
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 ).
E. Gudino CEMAT/IST 26
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 ).
E. Gudino CEMAT/IST 26
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
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
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
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
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
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.
E. Gudino CEMAT/IST 28
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.
E. Gudino CEMAT/IST 28
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,
E. Gudino CEMAT/IST 29
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,
E. Gudino CEMAT/IST 29
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,
E. Gudino CEMAT/IST 29
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.
E. Gudino CEMAT/IST 30
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.
E. Gudino CEMAT/IST 30
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.
E. Gudino CEMAT/IST 30
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.
E. Gudino CEMAT/IST 31
Finite element discretization for drug delivery
Contents
5 Finite element discretization for drug delivery
E. Gudino CEMAT/IST 32
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.
E. Gudino CEMAT/IST 33
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.
E. Gudino CEMAT/IST 34
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) .
E. Gudino CEMAT/IST 35
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.
E. Gudino CEMAT/IST 36
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.
E. Gudino CEMAT/IST 37
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.
E. Gudino CEMAT/IST 38
Numerical experiments
Contents
6 Numerical experiments
E. Gudino CEMAT/IST 39
Numerical experiments
Figure : Simplified domain
E. Gudino CEMAT/IST 40
Numerical experiments
Figure : Computational domain, NT = 10314 and NV = 5306
E. Gudino CEMAT/IST 41
Numerical experiments
Figure : Velocity field
E. Gudino CEMAT/IST 42
Numerical experiments
Figure : Pressure
E. Gudino CEMAT/IST 43
Numerical experiments
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E. Gudino CEMAT/IST 44
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Figure : Mass of drug in the wall vs the lumen
E. Gudino CEMAT/IST 45
Numerical experiments
Figure : Computational domain, NT = 10314 and NV = 5306
E. Gudino CEMAT/IST 46
Numerical experiments
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E. Gudino CEMAT/IST 47
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Figure : Mass of drug different stent profiles
E. Gudino CEMAT/IST 48
Numerical experiments
Figure : Computational domain, NT = 10314 and NV = 5306
E. Gudino CEMAT/IST 49
Numerical experiments
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E. Gudino CEMAT/IST 50
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DrugP2.mp4Media File (video/mp4)Numerical experiments
Figure : Mass of drug in the wall plaque vs no plaque
E. Gudino CEMAT/IST 51
Numerical experiments
Figure : Computational domains, NT = 10314 and NV = 5306
E. Gudino CEMAT/IST 52
Numerical experiments
Figure : Mass of drug in the wall different positions
E. Gudino CEMAT/IST 53
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.
E. Gudino CEMAT/IST 54
Numerical experiments
[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.
E. Gudino CEMAT/IST 55
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