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General patterns and asymptotic dose in the design ofcoated stentsÉtienne Bourgeois a & Michel C. Delfour ba Département de mathématiques et de statistique, Université de Montréal, C. P. 6128, succ.Centre-ville, Montréal, Que., Canada, H3C 3J7b Département de mathématiques et de statistique, Centre de recherces mathématiques,Université de Montréal, C. P. 6128, succ. Centre-ville, Montréal, Que., Canada, H3C 3J7Published online: 06 Jun 2008.

To cite this article: Étienne Bourgeois & Michel C. Delfour (2008) General patterns and asymptotic dose in thedesign of coated stents, Computer Methods in Biomechanics and Biomedical Engineering, 11:4, 323-334, DOI:10.1080/10255840701700940

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General patterns and asymptotic dose in the design of coated stents

Etienne Bourgeoisa,1 and Michel C. Delfourb*

aDepartement de mathematiques et de statistique, Universite de Montreal, C. P. 6128, succ. Centre-ville, Montreal, Que., Canada

H3C 3J7; bDepartement de mathematiques et de statistique, Centre de recherces mathematiques, Universite de Montreal, C. P. 6128,

succ. Centre-ville, Montreal, Que., Canada H3C 3J7

(Received 4 June 2007; final version received 21 August 2007 )

Stents are used in interventional cardiology to keep a diseased vessel open. New stents are coated with a medicinal agent

to prevent early reclosure due to the proliferation of smooth muscle cells. It is recognised that it is the dose of the agent

that effectively controls the growth. This paper focusses on the asymptotic behaviour of the dose for general families of

coated stents under a fixed ratio between the coated region of the stent and the targeted region of the vessel and set

therapeutic bounds on the dose. It generalises the results of Delfour, Garon and Longo for stents made of a sequence of

thin equally spaced rings to stents with an arbitrary pattern. It gives the equation of the asymptotic dose for a normal

tiling of the target region using the theory of tilings, patterns and motifs on a cylinder.

Keywords: restenosis; atherosclerosis; bioactive material; dose; pattern modelling

AMS Subject Classification: 92C50; 35B27; 35B40

1. Introduction

Stents are used in interventional cardiology to keep a

diseased blood vessel open (Figure 1). Stents are

designed to be less invasive from the hemodynamic

point of view, but this strategy cannot prevent restenosis.

Stent coating is thus aimed at efficiently controlling the

fast proliferation of smooth muscle cells of the arterial

wall in the early stage following the implantation of the

stent. Such a proliferation is inhibited by a normal

endothelium functioning. However, the stent implan-

tation damages the endothelium, leading to both clotting

and restenosis. A typical system consists of a stent coated

with a thin layer of polymer which has been impregnated

with a molecule that impedes the rate of growth of

smooth muscle cells (Bertand et al. 1997; Julien 2000;

Julien et al. 2002).

From the mechanical viewpoint, once the contact

surface has been specified, it is sufficient to choose a

stent with the appropriate mechanical properties in order

to keep the vessel open. Ideally, that contact surface must

be distributed as uniformly as possible at the smallest

possible scale. A plain stent without holes is not

acceptable since it would be a barrier to the biochemical

exchanges between the blood and the wall of the vessel.

Thus, in the design process, the ratio r between the areas

of the contact region and the target region (for instance

approximately 21.4% for the original Wiktor stent) is

first specified. Then the problem at hand is again to

distribute the contact region as uniformly as possible and

at the smallest possible scale. Mechanically a flexible

pattern will be preferred to a rigid pattern.

From the biological viewpoint, it is generally

accepted that it is the effect of the dose, the time-integral

of the concentration of the molecule over all times, that

effectively controls the proliferation of the smooth

muscle cells in each point of the vessel. The concept of

the dose in the context of stents in a blood vessel seems to

have been introduced for the first time in Delfour et al.

(2005) and is now accepted in other works such as

(Migliavacca et al. 2007): ‘a significant parameter for the

therapeutic effect of the released drug is the drug dose

that is the accumulative concentration integrated over

time along the interval of interest’. The use of the dose in

such problems is further supported by their simulation-

s/computations of the concentration of the molecule as a

function of time and the residence time. Indeed, the dose

being an integral over an infinite time horizon, it is

pertinent to ask how quickly the concentration goes to

zero. They report that ‘the drug release process is

relatively fast in the stent configuration. Indeed after 1

day less than 10% of the initial drug load is left in the

stent coating . . . . The average residence time T10%,

which is defined as a time when the amount of the drug

within the vessel dropped down to 10% with respect to

the peak value, is 83 h’.

The other important design tool that was proposed by

Delfour et al. (2005) is the so-called asymptotic stent.

They show that for an academic stent consisting of a

ISSN 1025-5842 print/ISSN 1476-8259 online

q 2008 Taylor & Francis

DOI: 10.1080/10255840701700940

http://www.informaworld.com

*Corresponding author. Email: delfour@CRM.UMontreal.CA

Computer Methods in Biomechanics and Biomedical Engineering

Vol. 11, No. 4, August 2008, 323–334

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sequence of equally spaced thin flat rings with no

thickness, an asymptotic dose is quickly reached when

the width of the rings goes to zero and the number of

rings goes to infinity while keeping the total surface of

the rings constant. Extensive numerical simulations

indicate that, for the original Wiktor stent with 24 struts,

r ¼ 21.4% and realistic parameters for the lumen and the

wall, we are practically in the asymptotic region.

In the present paper, we generalise the results of

Delfour et al. (2005) on the asymptotic behaviour of the

dose to very general families of coated stents with an

arbitrary and not necessarily periodic motif under a fixed

ratio r between the coated region of the stent and the

targeted region of the vessel. We use the general concepts of

tilings and motifs on a cylinder as defined in Schattschnei-

der (1978) and Roman (1969). We assume that the stent is a

normal tilingmade up ofgood tileswhose diameter is of the

order 1 and where the area of the motif of the stent isr times

the area of the tile and a set of exceptional tiles whose total

area is of the order of 1 (for instance tiles at the edges of the

stent where the ratio of the area of the motif to the area of the

tile is not exactly r. It turns out that, for a completely

general normal tiling of the target region, the equation of

the asymptotic dose as 1 goes to zero, and hence the

asymptotic stent, is the same as the one in Delfour et al.

(2005). In practical terms, it means that the motif of the

stent is not a critical variable in the design process as long as

1 and hence the tiles, are sufficiently small.

2. Specification of the stent via a characteristic

function

2.1 Simple model of the vessel and the target region

Consider a cylindrical section of vessel of length H . 0

where the stent will be deployed. For the sake of

simplicity, assume that the vessel is made up of two

homogeneous regions: the lumen and the wall. More

realistic multilayer models of the wall can be considered

(Manseau, 2002), but transport data are still lacking,

particularly on a major factor such as the perfusion and

drainage by vasa vasorum of the external wall part.

Let

Cr¼def

ðx1; x2; zÞ : x21 þ x2

2 , r 2; 0 , z , H� �

ð2:1Þ

denote the truncated open cylinder of radius r . 0 and

length H . 0. The lumen is assumed to be the truncated

open cylinder

~Vl¼defCR ð2:2Þ

of radius R . 0 and the wall of radial thickness E . 0 is

the open domain

~Vw¼def CRþE

�CR

; ð2:3Þ

between the truncated open cylinder CRþE and the closed

cylinder �CR. Mathematically ~Vl and ~Vw are open

domains in R3.

A stent of zero thickness and length Ls, 0 , Ls , H,

will be deployed in the target region

~S¼def

ðx1; x2; zÞ :x2

1 þ x22 ¼ R2;

H2Ls

2# z # HþLs

2

8<:

9=;; ð2:4Þ

at the interface between the lumen and the wall. By

construction, the region ~S is centered in H/2 at equal

distance

z0¼defðH 2 LsÞ=2 . 0; ð2:5Þ

from the boundaries of CRþE in z ¼ 0 and z ¼ H which

are artificial boundaries introduced for the analysis of

the problem. The length H of the section of the vessel is

assumed to be sufficiently larger than Ls so that the effect

of introducing an artificial boundary in z ¼ 0 and z ¼ H

is negligible. It also means that the region ~S does not

touch the boundaries of the cylinder CRþE in z ¼ 0 and

z ¼ H.

2.2 Specification of the stent by a characteristicfunction in the target region

The zero-thickness stent is mathematically defined as a

measurable subset, ~Ss, of the target region ~S.

For the moment assume that ~Ss is a nice closed subset~S such that the trace of an H 1-function on ~Ss is well-

defined. The zero-thickness stent is coated with a zero-

thickness polymer impregnated with an inhibitor drug

that will effectively control the growth of smooth muscle

cells. Coating can exists on both sides of the stent.

The problem at hand is the evolution of the distribution

of the concentration c(x, t) of the medicinal agent initially

impregnated in the coating of the stent as a function of the

position x [ CRþE and the time t . 0. However, the

complexity of this time-varying problem can be consider-

ably reduced from the fact that it is not the concentration

that effectively control the proliferation of smooth muscle

Figure 1. Wiktor stent as drawn in US patent No. 4,886,062.

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cells, but the dose that is a biological entity met in many

other such problems (Chalifour and Delfour 1992). The

dose at a point x [ CRþE of the vessel is the cumulative

concentration time the contact time. Mathematically, it is

the integral of the concentration with respect to the time t

from 0 to infinity, that is

qðxÞ¼defð1

0

cðx; tÞ dt: ð2:6Þ

As shown in Delfour et al. (2005), the introduction of

the dose leads to a substantial mathematical simplifica-

tion that comes from the fact that once the diffusion-

transport equations for the concentration are integrated,

we end up with a set of partial differential equations that

are independent of the time t. Moreover, Delfour et al.

(2005), already introduced a variational equation

formulation for the dose that only depends on the

characteristic function

x ~SsðxÞ¼

def1; if x [ ~Ss

0; if x [ ~Sn ~Ss;

8<: ð2:7Þ

of the stent.

It is the design of the set ~Ss that is the ultimate objective

of the analysis. There are several aspects to this design. For

instance, the stent has to be mechanically strong enough to

keep the lumen open. In this paper, we assume that the

mechanical design has been taken care of by specifying the

mechanical parameters of the material and the total support

surface of the stent ~Ss. So we concentrate on the delivery of

the product to the wall by playing with the distribution of

the stent in the target region ~S while keeping the total area

of the stent (contact region) constant.

The boundary ~Gl ¼ › ~Vl of the lumen ~Vl is made up

of four parts:

. ~Ss, the stent;

. ~Glw, the contact interface between ~Vl and ~Vw2;

. ~Gli, the part of the boundary of ~Vl where the blood

comes in (inlet);. ~Glo, the part of the boundary of ~Vl where the blood

comes out (outlet).

The boundary ~Gw ¼ › ~Vw of the wall ~Vw is made up of

five parts:

. ~Ss, the stent;

. ~Glw, the contact interface between ~Vl and ~Vw;

. ~Gwi, the part of the boundary of ~Vw where z ¼ 0;

. ~Gwo, the part of the boundary of ~Vw where z ¼ H;

. ~GRþE, the outer lateral boundary of the cylinder of

radius R þ E.

The subscripts i and o respectively stand for the inlet

surface at z ¼ 0 and outlet surface at z ¼ H of the vessel.

3. Mathematical model for the dose

3.1 The velocity u of the blood in the lumen

The blood in the lumen is assumed to be an

incompressible fluid, that is

div u ¼ 0 in ~Vl; ð3:1Þ

where u is the velocity of the blood. Further, assume that

u·nl # 0 on ~Gli and u·nl $ 0 on ~Glo ð3:2Þ

u·nl ¼ 0 or u ¼ 0 on ~Gl < ~Gw: ð3:3Þ

Condition (3.2) and (3.3) means that the blood is entering

through the inlet cross-section ~Gli and exiting through the

outlet cross-section ~Glo. The velocity u and the pressure p

will also verify the Navier–Stokes equation with the

condition u ¼ 0 on ~Ss < ~Glw. Yet the diffusion-transport

equations will still make sense under the weaker

condition u·nl ¼ 0 on ~Ss < ~Glw. This would correspond

to a questionable non-Newtonian viscosity model which

is not the purpose of this paper.

In order to integrate the concentration equation with

respect to time, the actual velocity is replaced by an

effective velocity that is independent of time. There are

many ways to justify this construction. In fact the flow

in the artery is pulsative, but the Strouhal number

corresponding to the pulsation frequency with respect to

the time of simulation or design horizon is very small.

Hence it is not necessary to retain the non-stationarity in

the coefficients in the Navier–Stokes equation. This is

part of the standard modelling leading to the equations

for the dose, but it is not the purpose of the present paper

that builds on the model already introduced in Delfour

et al. (2005).

3.2 Some preliminary definitions

In this section, we generalise the variational equation

formulation of Delfour et al. (2005) for the dose to a stent

specified by a measurable characteristic function in ~S in

the target region. Consider the open domain

~V¼defCRþE\ ~S; ð3:4Þ

that is the open cylinder CRþE minus the closed two-

dimensional target region ~S. It can be viewed as a

domain with a ‘cylindrical crack’. As in Delfour et al.

(2005), associate with CRþE and ~V the Sobolev

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subspaces

VðCRþEÞ¼def

v [ H 1ðCRþEÞ : vj ~Gli¼ 0

n owith condition ð3:2Þ– ð3:3Þ on u

8<:

H 1ðCRþEÞ

with condition ð3:7Þ– ð3:3Þ on u

(

8>>>>>>><>>>>>>>: ð3:5Þ

Vð ~VÞ¼def

v [ H 1ð ~VÞ : vj ~Gli¼ 0

n owith condition ð3:2Þ– ð3:3Þ on u

8<:

H 1ð ~VÞ

with condition ð3:7Þ– ð3:3Þ on u;

8<:

8>>>>>>>><>>>>>>>>: ð3:6Þ

where

’b . 0 such that 2 u·nl $0 on ~Glin ~gli

b on ~gli , ~Gli

8<:

and u·nl $ 0 on ~Gloð3:7Þ

and ~gli is some fixed subarea of the cross-section ~Gli

around its centre with strictly positive capacity.

Condition (3.7) is a strengthened version of condition

(3.2). The first case, with v ¼ 0 on ~Gli corresponds to

the assumption that ~Gli is chosen sufficiently far from the

target region ~S that the concentration c and hence the

dose on ~Gli can be taken as zero. The second case

involving a set of boundary conditions on u at the entry~Gli of the lumen corresponds to a transparent condition at~Gli on the dose similar to the ones used in Chalifour and

Delfour (1992). It allows for a possible backward

diffusion of the dose through the interface ~Gli.

The model of Delfour et al. (2005) can readily be

extended to a very large class of stents by starting from

any measurable characteristic function x defined in ~S,

that is

x [ L2ð ~SÞ such that xð1 2 xÞ ¼ 0 a:e: in ~S;

and specifying the stent as the measurable set

~SsðxÞ¼def

{x [ ~S : x ðxÞ ¼ 1}: ð3:8Þ

Since ~SsðxÞ is not even closed, the trace on ~SsðxÞ may not

be well-defined and the corresponding Sobolev space of

solution has to be specified in a special way. Associate

with ~SsðxÞ the following closed subspace of V(V)

Vð ~V; xÞ¼def

{v [ Vð ~VÞ : ð1 2 xÞ½v� ¼ 0 a:e: on ~S};

where the jump across ~S is defined from the traces of the

restrictions

vþ¼defvj ~Vw

and v2¼defvj ~Vl

ð3:9Þ

on each side of the interface ~S as

½v�¼defvþ ~S2 v2

�� ��~S: ð3:10Þ

Note that for any measurable characteristic function

x on ~S,

VðCRþEÞ , Vð ~V; xÞ , Vð ~VÞ: ð3:11Þ

It is also important to note that this generalisation of

the space of solution does not explicitly use a classical

boundary or interface condition on the arbitrary

measurable set Ss(x). As defined by (3.8) the stent ~SsðxÞ

can be quite ‘wild’. When it is assumed to be ‘sufficiently

nice’, we can say that we have ‘continuity of the trace’

across ~Sn ~SsðxÞ and thatVð ~V; xÞ ¼VðCRþEn~SsðxÞÞ, but in

all cases CRþEn~SsðxÞ is definitely not a Lipschitzian

domain.

3.3 Variational equation for the dose

Consider the continuous linear form

‘ðx; vÞ¼def

ð~S

csxvþdx; cs¼

defm=

ð~S

dS ð3:12Þ

on Vð ~VÞ, where m is the total mass of product and cs is

the surface density of the product in kg/m2. As in Delfour

et al. (2005), introduce the following continuous bilinear

form

aðq; vÞ¼def

ð~V w

Dw7q·7v dxþ

ð~Vl

ðDl7q2 quÞ·7v d x

þ

ð~Glo

u·nlqv dGð3:13Þ

on Vð ~VÞ or, equivalently by lumping together the

diffusion constants,

aðq; vÞ ¼

ð~V

7q·D7v d x2

ð~Vl

qu·7v d x

þ

ð~Glo

u·nlqv dG; ð3:14Þ

where the space-dependent diffusion is defined almost

everywhere on ~V

DðxÞ¼def

Dw; if x [ ~Vw

Dl; if x [ ~Vl:

8<: ð3:15Þ

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The bilinear form a is not symmetrical, but it is coercive

on Vð ~VÞ under the two boundary conditions (3.2) and

(3.3) on the velocity field u and v ¼ 0 on ~Gli and under the

two boundary conditions (3.7) and (3.3) on the velocity

field u for the transparent condition on the dose q. This

will be sufficient to invoke the Lax–Milgram theorem

and get existence of a unique solution. Indeed let a . 0

be the minimum of Dw, and Dl. Therefore, using the fact

that div u ¼ 0 and condition (3.3) and (3.2), we get

aðq; qÞ $ a

ð~V

j7qj2

d x21

2

ð~Gli

u·nljqj2

dG

þ1

2

ð~Glo

unljqj2

dG

$ a

ð~V

j7qj2

dx:

Using condition (3.3) and (3.7), we get

aðq; qÞ $ a

ð~V

j7qj2

d x21

2

ð~Gli

u·nljqj2

dG

þ1

2

ð~Glo

u·nljqj2

dG $ a

ð~V

j7qj2

d xþb

2

ð~gli

q2 d ~S

$ min a;b

2

� � ð~V

j7qj2

d xþ

ð~gli

q2 d ~S

� �:

But the last term on the right-hand side is an equivalent

norm on H 1ð ~VÞ and hence a is coercive on it and on all its

closed subspaces Vð ~V; xÞ and V(CRþE).

Theorem 3.1. Given any measurable characteristic

function x on ~S, the variational equation

;v [ Vð ~V; xÞ; aðq; vÞ ¼ ‘ðx; vÞ ð3:16Þ

has a unique solution q ¼ qðxÞ [ Vð ~V; xÞ, where the

respective continuous bilinear and linear forms are given

by (3.14) and (3.12).

Proof. By standard arguments using the Lax–Milgram

Theorem. A

3.4 Equations for q

Assume that the stent ~SsðxÞ defined by (3.8) is closed

and ‘sufficiently nice’ so that Vð ~V; xÞ ¼ VðCRþEn~SsðxÞÞ.

In that case we get the following set of equations for the

dose q(x) from the variational Equation (3.16)

divðDw7qÞ ¼ 0 in ~Vw ð3:17Þ

divðDl7q2 quÞ ¼ 0 in ~Vl: ð3:18Þ

The boundary conditions are

wall

›q›nw

¼ 0 on ~Gwi < ~Gwo < ~GRþE

Dw›qþ

›nw¼ cs on ~Ss

8<: ð3:19Þ

lumen

Dl›q›nl

2 u·nl q ¼ 0 or q ¼ 0 on ~Gli

›q›nl

¼ 0 on ~Glo

›q2

›nl¼ 0 on ~Ss:

8>>>><>>>>:

The condition at the interface is

wall=lumen Dw

›q

›nwþ Dl

›q

›nl¼ 0 on ~Glw: ð3:20Þ

4. Asymptotic dose for the stent made up of flat

rings

One of the purposes of the coated stent is to delay the

production of smooth muscle cells long enough to allow

the wall of the vessel to heal and to restore the regulating

functions of the local cells. This can be achieved by a

very uniform distribution of the dose at the wall/lumen

interface in the target region. At the same time, it is

desirable to keep the contact surface as small as possible

to preserve and maintain inasmuch as possible the

biochemical exchanges between the wall and the blood.

A stent with a surface completely covering the target

region would not be acceptable. So a key parameter in

the design is the ratio r between the contact surface and

the surface of the target region, say 20 or 50%. Once, r

has been chosen, the mechanical characteristics of the

material of the stent can be chosen to keep the vessel

open. Finally, the problem at hand is to distribute the

contact surface of the stent as uniformly as possible all

over the target region for a fixed ratio r.

The earlier work in Delfour et al. (2005) studied a

stent made up of flat rings with no thickness, a uniform

width, and equally spaced as shown in Figure 2.

Equivalently, the stent and its refinements can be

described by unwinding the target region onto the plane

as in Figure 3. They showed that, with a realistic choice

of parameters, the dose quickly converges toward the

asymptotic dose q0 that is the unique solution in V(CRþE)

of the variational equation

’q0 [ VðCRþEÞ; such that

;v [ VðCRþEÞ; aðq0; vÞ ¼ ‘0ðvÞ; ð4:1Þ

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where the continuous bilinear form a is given by

expression (3.13) and the linear form ‘0 by the expression

‘0ðvÞ¼def

ð~S

r csv dS: ð4:2Þ

When ~SsðxÞ is a nice closed set, the following set of

equations for the asymptotic dose q0 can be obtained

from the variational Equation (4.1)

divðDw7q0Þ ¼ 0 in ~Vw ð4:3Þ

divðDl7q0 2 q0uÞ ¼ 0 in ~Vl: ð4:4Þ

The boundary conditions are

wall›q0

›n w

¼ 0 on ~Gwi < ~Gwo < ~GRþE

�ð4:5Þ

lumen

Dl›q0

›nl2 u·nlq0 ¼ 0 or q0 ¼ 0 on ~Gli

›q0

›nl¼ 0 on ~Glo:

8<:

The condition at the interface is

wall=lumenDw

›q0

›nwþ Dl

›q0

›nl

¼0 on ~Gl > ~Gwn

~S

rcs on ~S

8<: ð4:6Þ

5. Bi-periodic patterns

We now extend the results of Delfour et al. (2005) to the

family of bi-periodic stents of Figure 4 and their

refinements. For such configurations the target region

appears as a tiling of the cylinder and the stent as a

periodic pattern specified by a motif defined on a nominal

tile. The asymptotic behaviour of the dose is reached

when the tile size quasi-uniformly goes to zero while

keeping the total contact surface quasi-constant or,

equivalently, the ratio quasi-constant.

It is convenient to work with the unwinded target

region onto the rectangle of height 2pR in the vertical

y-direction and width Ls in the horizontal z-direction.

In the y-direction the period is necessarily 2pR divided

by an integer N, that is ~b ¼ ð0; 2pR=NÞ; the other

periodic direction ~a can be chosen as any non-zero non-

colinear vector such as (1, 0). The pair ð~a; ~bÞ defines a

parallelogram that is the basic tile that will be

periodically repeated over the whole plane. In order to

construct the periodic tiling and pattern, we start from the

dimensionless parallelogram P specified by the angle u,

0 # u , p/2 and the dimensionless width ‘ . 0 with a

normalised area (Figure 5). Then we introduce a subset

M of P that will be referred to as the motif of the stent.

In a second step, we introduce a scaling parameter

1 . 0 to scale P to a parallelogram of width 1‘

and height 1=‘. This scaled parallelogram will be

bi-periodically repeated to cover the plane with tiles of

size 1. The associated repeated scaled motif will also

generate a periodic pattern in the plane (cf. Figure 6). Its

restriction to the flattened target region ~S will be the

stent. In view of the fact that the target region is a

cylinder of radius R, the scaling factor is necessarily

Figure 2. Periodic stent of radius R and length Ls with flatrings around the lumen.

Figure 3. Refinements of the unwinded stent onto therectangle of height 2pR and width Ls.

Figure 4. Bi-periodic hexagonal stent and its unwindedbiperiodic tiling and pattern.

Figure 5. Normalised parallelogram P of unit area, angle uand width ‘.

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of the form

1N ¼2pR‘

N

in order to get an integral number of tiles of height 1N=‘in the y-direction.

The stent will be specified from a measurable subset

M of the nominal tile P that will be referred to as the

nominal motif of the stent. Given the characteristic

function xM of the motif M , P, we construct the

following bi-periodic characteristic function x1 defined

in the whole plane

x1ðz;yÞ¼xMz

12‘

z

1‘

h i;y

12

1

‘

y2ztanu

1=‘

� �2‘tanu

z

1‘

h i� ;

ð5:1Þ

where [x ] denotes the integer part of the real number x.

At the 1N scale, the zero-thickness stent is specified as

~S1N

s ¼def

Rcosy

R;Rsin

y

R;z

�:

y[½0;2pR�

z[½z0;z0þLs�

andx1N ðz;yÞ¼1

8>>>><>>>>:

9>>>>=>>>>;, ~S:

Remark 5.1. Note that, at the edges of the stent, some

tiles may not be complete and that the global ratio

between the surface of ~S1N

s to the one of ~S may not be

exactly r. But this ratio will go to r as 1N goes to zero.

The asymptotic dose is the weak limit as 1 goes to zero

of the dose q1 associated with the characteristic function

x1 restricted to the target region ~S. Hence it is directly

related to the weak limit of x1 as 1 goes to zero.

Theorem 5.1. Given a measurable subset M of P, the

corresponding characteristic function xM [ L 2(P), and

the associated functions x1N:R2 ! R defined by (5.1),

1N ¼ 2pR‘=N, the whole family {x ~S1Ns

} is bounded in

L2ð ~SÞ and

x ~S1NsQ

ðP

xM dz dh in L2ð ~SÞ2 weak asN !1: ð5:2Þ

Moreover,

q ~S1NsQ q0 inVð ~VÞ2 weak; ð5:3Þ

where q ~S1Ns

is the dose corresponding to the stent ~S1N

s and

q0 is the asymptotic dose solution of the variational

Equation (4.1) for

r ¼

ðP

xM dz dh: ð5:4Þ

The integral of xM is precisely the area of the motif in

the nominal tile P of area 1. We omit the proof of the

above theorem since we shall prove a more general result

in the subsequent sections of the paper.

6. General families of stents

6.1 Tilings and patterns in the plane

Unfortunately, the bi-periodic theory of the previous

section does not extend to the helicoidal stent or stents

that may not present a neat bi-periodic structure. At the

same time, the conclusions of Theorem 5.1 remain true

for a much larger family of tile/pattern under weaker

assumptions (Figures 7 and 8).

In order to put everything on firm mathematical

grounds, we first recall a few notions from the theory of tiles

and patterns in the plane (cf. Grunbaum and Shephard

(1986)) and on a column (cf. Schattschneider (1978)).

Definition 6.1. (Tiling)

i) A tiling of the plane is a countable family of closed

sets D ¼ {D1;D2; . . . } , R2 such that

[1i¼1

Di ¼ R2 and ;i – j; intDi > intDj ¼ Y:

Figure 6. The tiling over the target region ~S for N ¼ 4 and14 ¼ 2pR‘=4.

Figure 7. Periodic helicoidal stent.

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The sets Di are called tiles.

ii) A tiling is said to be normal if the following

conditions are verified

(a) each tile is homeomorphic to a disk;

(b) the intersection of two tiles is a connected set;

(c) the tiles are uniformly bounded.

Tilings are characterised by via their group of

symmetries.

Definition 6.2. (Symmetry)

i) An affine transformation of the plane is an affine

objective map from R2 onto R2.

ii) A symmetry s of a tiling D is an affine

transformation such that s ðDÞ ¼ D.

Theorem 6.1. Up to an isomorphism a symmetry of a

normal tiling is one of the following

i) e, the trivial group,

ii) cn, the cyclic group of order n, n $ 2,

iii) dn, the diedral group of order 2n, n $ 2,

iv) the seven frieze groups3,

v) the seventeen crystallography groups4.

Definition 6.3. A tiling is said to be periodic if its

group of symmetries is one of the crystallography

groups.

A periodic tiling can be generated by translation of a

parallelogrammic tile.

Definition 6.4. (Pattern) Given a non-empty subset M

of the plane, a pattern with motif M is a non-empty

family P ¼ {Mi : i [ I} of subsets of the plane indexed

by I such that the following conditions are verified:

i) the sets Mi are pairwise disjoint,

ii) for each i [ I, Mi is congruent5 to M,

iii) for each pair ðMi;MjÞ of copies of the motif M,

there exists a symmetry s of the plane such that

s ðPÞ ¼ P and sMi ¼ Mj.

The pattern is said to be non-trivial if it contains at least

two elements.

As for tiling an affine transformation s is a symmetry

of a pattern P if s ðPÞ ¼ P and we can talk about the

groups of symmetries of a pattern.

Theorem 6.2. The list of possible symmetries of a

pattern is the same as the one of a tiling in Theorem 6.1.

As for tiling, a pattern is said to be periodic if its group

of symmetries is one of the crystallography groups.

6.2 Tilings and patterns on a column

In this section, we specialise the definitions and results of

the previous section to tiling and patterns on an infinite

cylinderCof radiusR. By unwinding the cylinder on a plane

strip S of width 2pR and identifying the points on the

opposite edges, it is readily seen that the symmetries on the

cylinder will be a subgroup of the symmetries in the plane.

Definition 6.5. .

i) A family D of subsets of C is a tiling of the column

C if the unwinded family D is a tiling of S.

ii) A family D of subsets of C is a normal tiling of the

column C if the unwinded family D is a normal

tiling of S.

iii) Given a non-empty motif M in C, a family of subsets

P of C is a pattern of the column C of motif M if the

unwinded family is a pattern in S with unwinded

motif M. The definition for a non-trivial pattern on

C is analogous.

The possible groups of symmetries on a cylinder are

given by the next theorem.

Theorem 6.3. There are 17 possible groups of

symmetries for a column:

i) e, the trivial group,

ii) the seven frieze groups6,

iii) nine of the crystallography groups7.

6.3 A weak convergence theorem

Given a target region ~S on a cylinder C of radius R of

length Ls and a measurable subset D of C, introduce the

following notation

AðDÞ¼def

ðD

dC; diam ðDÞ¼def

supx;y[D

jx2 yj: ð6:1Þ

Theorem 6.4. Given 10 . 0, let { f e}0, 1,10,

f e : C! R, be a family of measurable functions and D1

be a normal tiling of the cylinder C verifying the following

conditions: there exist constants r, K1 and K2 such that

i) the family { f e : 0 , 1 , 10} is bounded in L2ð ~SÞ,

ii) for each 1, 0 , 1 , 10, there exist integers

1 , n(1) # N(1) , 1 and a subset {D1,1,D1,2,

. . . ,D1,n(1), . . . ,D1,N(1)} of D1 that covers ~S such

Figure 8. Refinements of the unwinded helicoidal periodicstent.

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that

;i; 1 # i # nð1Þ; diam ðD1;iÞ , K11

andÐD1;i

f 1 dC ¼ rAðD1;iÞ

8<: ð6:2Þ

;i; nð1Þ , i # Nð1Þ;

A <Nð1Þi¼nð1ÞD1;i

�# K21:

8<: ð6:3Þ

Then f 1 Q r in L2ð ~SÞ-weak.

Remark 6.1. In our context, the family of functions

{ f e}0, 1,10will be characteristic functions {xe}0, 1,10

that specify the corresponding family of stents~S1

s ¼ {x [ ~S : x1ðxÞ ¼ 1}. This formulation is much

more flexible than the bi-periodic one since the tiling

does not need to be periodic at all. Moreover, the

motif ~S1

s > D1;i in each tile D1;i does not need to be

homeomorphic to a unique nominal motif. The family of

characteristic functions {xe}0, 1,10is only required to

verify the integral conditions (6.2).

Remark 6.2. Without loss of generality, the integers

n(1) and N(1) can be redefined in such a way that for all i,

1 # i # N(1), AðD1;i > ~SÞ . 0. The tiles in D1 verifying

condition (6.3) are exceptional tiles containing a special

motif for a highly non-uniform stent pattern. They can be

partially or completely contained in ~S. The tiles of D1

that verify (6.2) can be divided in two categories: the

ones contained in ~S and the ones that do not have a full

intersection with ~S. Since they are located along the

boundary of ~S and, by assumption (6.3), their diameter is

bounded by K11, the total area of the boundary tiles is

bounded by the area of two flat rings of width K11. Hence

the total area of the boundary tiles is bounded by twice

2pRK11. Hence, the boundary tiles verify condition

(6.3). In the sequel, it will be convenient to redefine n(1)

and N(1) in such a way that the indices i of all the

boundary tiles lie between n(1) þ 1 and N(1) and for all i,

1 # i # n(1), D1;i , ~S.

Proof. In view of Remark 6.2, we can asssume that for

all i, 1 # i # N(1), AðD1;i > ~SÞ . 0 and that for all i,

1 # i # n(1), D1;i , ~S. We want to show that

;w [ L2ð ~SÞ;

ð~S

f 1w dC! r

ð~S

w dC:

We first prove the convergence for w [ C 1ð ~SÞ

ð~S

f 1w dC ¼Xnð1Þi¼1

ðD1;i

f 1w dC

þXNð1Þ

i¼nð1Þþ1

ðD1;i> ~S

f 1wdC:ð6:4Þ

By assumption (i), the second term goes to zero since

XNð1Þi¼nð1Þþ1

ðD1;i> ~S

f 1w dC�����

�����# sup

x[ ~S

jw ðxÞj

ð~S

j f 1jx<Nð1Þi¼nð1Þþ1

D1;idC

# kwkCð ~SÞ

k f 1kL 2ð ~SÞA <Nð1Þ

i¼nð1Þþ1D1;i

�# cðK21Þ

1=2:

For the first term choose a point xi in each tile D1,i,

1 # i # N(1),

Xnð1Þi¼1

ðD1;i

f 1w dC ¼Xnð1Þi¼1

ðD1;i

f 1ðxÞ½w ðxÞ2 w ðxiÞ� dC

þXnð1Þi¼1

w ðxiÞ

ðD1;i

f 1ðxÞ dC

Xnð1Þi¼1

ðD1;i

f 1w dC2Xnð1Þi¼1

w ðxiÞ

ðD1;i

f 1ðxÞ dC�����

�����#

Xnð1Þi¼1

ðD1;i

j f 1ðxÞkw ðxÞ2 w ðxiÞj dC

#Xnð1Þi¼1

ðD1;i

j f 1ðxÞjcjx2 xijdC

# cXnð1Þi¼1

diam ðD1;iÞ

ðD1;i

j f 1ðxÞj dC

# cK11Xnð1Þi¼1

ðD1;i

j f 1ðxÞj dC

# cK11

ð~S

j f 1j dC! 0

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by the first Equation (6.2) of (ii). By the second Equation

(6.3) of (ii), the term

Xnð1Þi¼1

wðxiÞ

ðD1;i

f 1ðxÞ dC

¼ rXnð1Þi¼1

wðxiÞAðD1;iÞ

¼ rXNð1Þi¼1

wðxiÞAðD1;iÞ þ rXNð1Þ

i¼nð1Þþ1

wðxiÞAðD1;iÞ:

We recognise the Riemann integral

ð~S

w dC ¼XNð1Þi¼1

ðD1;i> ~S

w dC

that can be approximated by

XNð1Þi¼1

wðxiÞAðD1;i > ~SÞ

¼Xnð1Þi¼1

wðxiÞAðD1;iÞ þXNð1Þ

i¼nð1Þþ1

wðxiÞAðD1;i > ~SÞ

where, by assumption (6.3), the second term goes to zero.

Therefore for w [ C 1ð ~SÞ

Xnð1Þi¼1

wðxiÞ

ðD1;i

f 1ðxÞ dC! r

ð~S

w dC:

The final result is obtained by density of C 1ð ~SÞ in

L2ð ~SÞ. A

7. Asymptotic stent

In the design of the stent we are left with the following

parameters: the surface density of product

cs ¼ m=area of ~S, the length of the target region Ls and

the ratior ¼ area of ~Ss=area of ~S, 0 # r , 1, between the

two surface areas.

In this section, we fix a ratio r, 0 # r , 1 and

consider a family of stents specified by their measurable

characteristic function xe : C! R and a normal tiling D1

of the cylinder C verifying the conditions of Theorem 6.4.

We study the limit q0 of the family of solutions {q1} of

the variational Equation (3.16) as 1 goes to zero.

The main technical difficulty is that the space of

solution depends on 1 and that there is no monotony of

the spaces Vð ~V; x1Þ as a function of 1. All we have is

;1; VðCRþEÞ , Vð ~V; x1Þ , Vð ~VÞ ð7:1Þ

In general, even if we can find a uniform bound

(independent of 1) on the norms kq1kVð ~VÞ in the large

function space Vð ~VÞ, we will only be able to use test

functions in the smaller space V(CRþE) unless the

projection onto the 1-dependent solution spaces Vð ~V; x1Þ

is strongly continuous. This asymptotic problem is very

similar to the Neumann sieve studied by Damlamian

(1985), where the plane surface is replaced by the lateral

boundary of the cylinder of radius R. Fortunately here the

total surface of the holes is constant and different from

zero in the limiting process and there will be no

discontinuity of the trace of the asymptotic solution or

stange term coming from nowhere in the asymptotic

variational equation.

By construction, the continuous bilinear form (3.13),

a(w, v), defined on Vð ~VÞ is independent of x1 and

continuous on Vð ~V;x1Þ and V(CRþE) as closed subspaces

of Vð ~VÞ. Since Ls and r are kept constant in the limiting

process, we get under the assumptions of Theorem 6.4,

the weak convergence of the family x1 to r and

;v [ Vð ~VÞ; ‘ðx1; vÞ ¼ cs

ð~S

x1vþdS

! ‘0ðvÞ ¼ csr

ð~S

vþdS

and the linear form ‘0 is linear and continuous on Vð ~VÞ

and its subspaces.

Since, from (7.1), for all 1 . 0, VðCRþEÞ ,Vð ~V; x1Þ , Vð ~VÞ, it is easy to show that the solutions q1are uniformly bounded in the norm of the bigger fixed

space Vð ~VÞ. There will be a weakly convergent

subsequence to some q0 [ Vð ~VÞ and q0 will be a solution

of the equation

;v [ VðCRþEÞ; aðq0; vÞ ¼ ‘0ðvÞ: ð7:2Þ

However, this equation is incomplete since the test

functions belong to the smaller space V(CRþE) which does

not see the cylindrical crack ~S.

Theorem 7.1. Given the constants Ls, 0 , Ls , H,

and r, 0 # r , 1, and a family of measurable

characteristic functions x1 and of normal tilings D1 of

C that verify the assumptions of Theorem 6.4, the

corresponding sequence of solutions q1 [ Vð ~V;x1Þ to

the variational Equation (3.16) converges in Vð ~VÞ2

weak to q0 which is the unique solution in V(CRþE) of the

variational equation

;v [ VðCRþEÞ; aðq0; vÞ ¼ ‘0ðvÞ; ð7:3Þ

where the continuous bilinear and the linear forms a and

‘0 are given by the expressions (3.13) and (4.2).

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Remark 7.1. When the molecule is applied on the

inner and outer surfaces of the stent, the asymptotic

model predicts that the two sides contribute to increase

the total dose in the wall and the linear form will be

‘0ðvÞ ¼ r

ð~S

cþs vþ þ c2s v

2dS

where cþs and c2s are the respective surface densities on

the outer and inner surfaces of the stent.

Proof. By weak convergence of {q1} in Vð ~VÞ ,H 1ð ~VÞ the jump ½q1�¼

defqþ1 j ~S 2 q21 j ~S [ H 1=2ð ~SÞ of q1

across ~S converges in L2 ð ~SÞ-strong:

½q1�! ½q0� inL2ð ~SÞ2 strong;

since the trace v 7! ðvþ; v2Þ : H 1ð ~VÞ! H 1=2ð ~SÞ £

H 1=2ð ~SÞ is linear and continuous. By the definition of

the space Vð ~V; x1Þ

;w [ L2ð ~SÞ;

ð~S

ð1 2 x1Þ½q1�w dG ¼ 0:

Since, x1 Q r in L2ð ~SÞ � weak and for all w [ L2ð ~SÞ,

[q1] w ! [q0]w in L2ð ~SÞ2 strong, then

0 ¼

ð~S

ð1 2 x1Þ½q1�w dG

!

ð~S

ð1 2 rÞ½q0�w dG ¼ ð1 2 rÞ

ð~S

½q0�w dG

) ;w [ L2ð ~SÞ; ð1 2 rÞ

ð~S

½q0�w dG ¼ 0:

Hence, for 0 # r , 1

¼ 0 along ~S ) q0 [ H 1ðCRþEÞ ) q0 [ VðCRþEÞ:

Combining q0 [ V(CRþE) with Equation (7.2) and the

coercivity of a on V(CRþE), we conclude that q0 is the

unique solution of the variational Equation (7.3). A

8. Conclusions and remarks

At this juncture, it is interesting to make a few conclusive

comments in the context of the design of coated stents.

Firstly, the notion of dose considerably simplifies the

analysis by removing the time from the equation.

The pertinence of the dose in this context has recently

been backed up by the nice space-time computations of the

concentration (Migliavacca et al. 2007) that shows release

time of the order of one day. This means that infinity in the

definition of the dose is of the order of one or two days.

Secondly, as indicated in the introduction, the earlier work

(Delfour et al. 2005) has shown that, since the asymptotic

dose is very quickly reached, it is a quick and convenient

tool to evaluate the required concentration cs and the total

drug mass m of medicinal agent. Thirdly, the asymptotic

analysis in the present paper shows that the choice of a stent

pattern is notcritical as long as the actualdesign is uniformly

distributed at the smallest practical scale. The choice of the

actual pattern/motif of the stent would be more dictated by

mechanical properties such as flexibility.

Finally, it is generally accepted that it would take

about a week to regenerate the endothelial cells that

actually control the growth of smooth muscle cells.

In view of (Migliavacca et al. 2007) and the comments in

the introduction, two or three releases of the molecule at

48 h intervals would be sufficient to prevent restenosis.

The strategy is similar to the one for the control of black

fly larvae by periodic treatments of rivers with larvicides

(Chalifour and Delfour 1992). As suggested in

Migliavacca et al. (2007), such sequential releases can

be achieved by using properly timed reservoir stents.

Acknowledgements

This research has been supported by a discovery grant from

National Sciences and Engineering Research Council of Canada.

Notes

1. Email: toutvabien25@hotmail.com.2. In fact ~Glw ¼ ~Gl > ~Gwn

~Ss:3. cf. Grunbaum and G.C Shephard (1986) p. 39.4. cf.Grunbaum and G.C Shephard (1986) p. 40–42 and Schattschneider

(1978).

5. A subset of R2 is congruent to M if there exists a symmetry s of the

plane such that s (Mi) ¼ M.

6. cf. Grunbaum and Shephard (1986) p. 39.7. cf. Cf. Roman (1969) p. 30 and Schattschneider (1978) p. 449.

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