Ad ances in Elec ical and Elec onic Enginee ing
212
INVESTIGATION OF BRAIN ARTERIAL CIRCLE MALFORMATIONS
USING ELECTRICAL MODELLING AND SIMULATION
K. ápo á, V. Blazek, I. áp
Uni e si y o Zilina, Facul y o Elec ical Enginee ing, Velký diel, 010 26 Zilina, Slo akia
Tel.: +421-41-513 2100, Fax: +421-41-513 1519, E-mail: capo a@ el.u c.sk
Summa y: The pape deals wi h he ce eb al a e ial sys em in es iga ion by means o elec ical modelling and simula ions.
The main a en ion is paid o he b ain a e ial ci cle mal o ma ions (s enoses and aneu ysms) and hei de e mina ion and
e alua ion by compu e -aided me hods as ools o a non-in asi e diagnos ics. The compensa ion possibili ies o b ain a e ial
ci cle in case o p esence o conc e e a e ial mal o ma ions a e modelled and simula ed. The simula ion esul s o b ain
a e ies blood p essu es and olume low eloci ies ime dependences a e p esen ed and discussed unde a ious heal h
condi ions.
Keywo ds: b ain a e ial ci cle, a e ial mal o ma ion, elec omechanical analogy, elec ical modelling and simula ions, non-
in asi e diagnos ics, compu e -aided me hods, s a e a iables.
1. INTRODUCTION
Du ing he medical ea men o ce eb o- ascula
diseases i is o en necessa y o occlude one o he
b ain supplying a e ies. Due o he a e ial ci cle o
Willis (CAW – Ci culus A e iosus Willisii) his
in e en ion con ains no u he consequences o he
majo i y o pa ien s. Bu because o an un a ou able
essel ana omy o essel disease his in e en ion
causes an ischemia in co esponding pa s o he
b ain o some pa ien s who hen may su e om
s okes.
Un il now, his si ua ion can only be es ima ed by
in asi e and inaccu a e diagnos ics, which
hemsel es con ain he isk o ce eb o- ascula
acciden s.
Fo his eason p o ound insigh s in o he complex
haemodynamic in e ac ions wi hin he b ain we e
ied o enable by modelling he a e ial ci cle o
Willis. Replacing in asi e diagnos ics by
compu a ional simula ions, a eliable and non-
in asi e diagnos ic ool es ima ing he haemodynamic
e ec s o endo ascula essel occlusion is ob ained.
The model is based upon he analogy o
hyd odynamic and elec omagne ic s a e a iables
called as elec omechanical analogy, [1], [2], [3].
The essel p ope ies a e desc ibed by coupled
i s o de linea di e en ial equa ions, which a e
subsequen ly nonlinea ly ex ended. The ana omical
pa ame e s and bounda y condi ions o he
di e en ial equa ions a e de ined by non-in asi e,
clinically es ablished diagnos ics. By hese means
any gi en ana omical s uc u e o essels can be
designed. Time con inuous simula ions o di e en
s a es o all calcula ed s a e a iables, such as blood
p essu e and low in any gi en essel o he model,
become easible and online obse able. C i ical
occlusions can be simula ed and co esponding
changes in low o p essu e can be obse ed wi hou
exposing he pa ien o a isk o heal h, [4] o [7].
On he i s s age o de elopmen he indi idual
model o each pa ien mus be designed and alida ed
by a compa ison wi h measu ed da a. Al hough an
ag eemen be ween he modelled da a and hose
eco ded om obse a ion o he subjec s is no
eached in each case, because he s a e a iables
gene a ed by he model beha e physiologically. The
eac ion o indi idual ce eb o- ascula sys ems in
c i ical si ua ions as occlusions o he main b ain
a e ies is in es iga ed unde special condi ions.
Some de ia ions be ween modelled and measu ed
da a a e based on sys ema ic inaccu acies conce ning
he de e mina ion o he essels geome y.
Using he men ioned elec omechanical analogy
he equi alen elec omagne ic sys ems ha e been
de i ed and consequen ly he compu e modelling and
simula ion o CAW unde a ious condi ions we e
pe o med. The esul s o he CAW equi alen
elec omagne ic sys ems de i a ion and hei
modelling and simula ions we e published in he
p e ious wo ks [6], [7], [8]. Following om hese
wo ks he CAW signi ican mal o ma ions ha e been
in es iga ed and e alua ed by means o he elec ical
modelling and simula ions in his pape .
2. THEORY
The s anda d model o CAW is he main basis o
he indi idual modelling o essels ne wo ks o
indi idual pa ien s CAW. The p incipal opology o
CAW is illus a ed a Fig.1.
In es iga ion o b ain a e ial ci cle mal o ma ions…
213
Fig. 1. Topology o b ain a e ial sys em
The basic equa ions o one essel segmen modelled
by he equi alen elec ic wo-po elemen acco ding
o he elec omechanical analogy a e exp essed in
e ms o he haemodynamic s a e alues in di e en ial
equa ions, [4], [5]
[ ]
)0()()()()(
)(
1
)(
1
0iiiii
i
i
Id R I P P
L
Z +−−=
−

(1)
[ ]
)(
),(
)(
)()(
)(
)(
3
2
)(
.1
2
2
P
C
Q Q
I IR
Q
I
l P
ex
i
iui
iidi
i
i
i
+
−
+−+








=
+
ω
ρ
(2)
[ ]

+−=
+
)0()()()(
1iiii
Qd I I Q , (3)
whe e I
i
is he blood low, P
i
blood p essu e and Q
i
blood olume in he i- h essel segmen , R
i
is a
iscose ic ion esis ance o lowing blood and R
di
esis ance co esponding o he ic ion losses inside
he essel wall o segmen i.
The essels pa ame e s in i- h essel segmen ,
esis ance R
i
and induc ance L
i
in longi udinal
di ec ion and he capaci ance C
i
and c oss-
conduc ance G
i
in ans e sal di ec ion, depend on
he essel geome y and he elas ici y o essel walls.
The o mulas enabling hei calcula ions a e gi en, by
ollowing exp essions, [2] and [3]
)(
8
)(
4
l
R
i
i
i
π
η
=, )(
)()(2
3
),(
2
5
0
hE
l
C
i
i
idi
i
i
ω
π
ω
=
)(
)(
2
l
L
i
i
i
π
ρ
=,
ii
ii
di
l
h
R
3
0
0
)()(
3
2
)(
π
η
=, (4)
whe e
η
is he blood iscosi y, l
i
is he leng h o a
essel segmen i,
i
essel in e nal adius,
0
he
essel in e nal adius wi hou d awing ou ,
ρ
is he
blood densi y, E
di
is he ela i e dynamic pa o
Young elas ici y modulus, h
i
is he essel wall
hickness o segmen i.
F om he p eope a i e CAW isk es ima ion poin
o iew he e is he ealis ic simula ion o he a e ial
mal o ma ions, especially s enoses, o he main
impo ance in his a ea. In o de o in es iga e he
in luences o an a e ial s enosis in e ms o he s a e
alues as blood p essu e o blood eloci y, he basic
ma hema ical model, (e.g. in [6]), should be aided by
he equa ion exp essing he mu ual con inui y be ween
he p essu e d op caused by he s enosis and he
blood low h ough he essel sec ion unde s enosis
in luence,
d
Id
Kl
I I
K
I
K
P
us
s
s enose
)(
)(
)()(1
2
)(
)(2
)(
2
2
2
2
0
4
0
23
π
ρ
π
ρ
π
η
+








−+=∆ , (5)
Ad ances in Elec ical and Elec onic Enginee ing
214
in which he ( ) is he in e nal adius o he essel
(pipe) un h ough by blood,
0
he in e nal adius in
he heal hy essel and
s
he in e nal adius in he
place o he highes s enosis (which means maximum
na owing o he essel).
The leng h o he s enosis is desc ibed by l
s
,
η
and
ρ
design he blood iscosi y and densi y
co espondingly. The coe icien s K
, K
u
and K
we e
s a ed empi ically, [5]. The shape (p o ile) o he
s enose has been desc ibed by he cosinus unc ion
)()(
0
x x −− , whe e 















+=
s
l
x
x
πδ
2
cos1
2
)(
and 




−= 2
,
2
s s
ll
x (6)
The
δ
is he maximum heigh o he s enosis, Fig. 2.
The s enosis pa ame e s o he simula ion a e also
gi en a he Fig. 2. The nonlinea i ies caused by
blood whi l, eq. (1), c ea e he main eason o he
in ensi ied p essu e d op in he essel unde s enosis
in luence.
Fig. 2. Vessel segmen wi h s enosis.
Acco ding o he p e ious wo ks he ollowing
s enosis pa ame e s o he equi alen elec ic sys em
o he essel segmen can be de ined by he ela ions
)(
)(
2
Kl
L
us
s
π
ρ
=, )(2
)(
3
K
R
s
π
η
=,
2
2
2
0
4
0
2
1
2
)(
)( 







−=
s
dp
IK
R
π
ρ
(7)
Then he equa ion (5) can be exp essed by ela ions
[ ]
d
Id
L I R R P
s dps s enose
)(
)()()()()( ++=∆
( )
[ ]

++−∆= )0()()()()(
)(
1
)( I d R R I P
L
I
dps s enose
s
(8)
The esis ance R
s
( ) is simila o he s a iona y
longi udinal esis ance in Hagen - Poisseuille law,
[1], and i exp esses he ac ha wi h he inc easing
a essel na owing he iscose ic ion also g ows
up. I s alue is s ongly in luenced by he s enosis
geome y. The coe icien K
is no mo e cons an , bu
i is a unc ion dependen on he s enosis shape. The
e m R
dp
desc ibes he esis ance which a ises
unde he in luence o he pos s eno ic u bulences
and i depends only a li le on he s enosis geome y.
The equa ion (8) is he equi alen one o he
di e en ial equa ion (5), which desc ibes he blood
s a e alues o a essel segmen wi hou s enosis. In
he case o a essel segmen wi h he s enosis i he
ollowing di e en ial equa ion is used o he blood
lows calcula ions
( )
[ ]
)0()()()()()()(
)(
1
)(
1idps iiii
s
i
I d R R R I P P
L
I +++−−=

−
, wi h )(
8
)(
4
l
R
i
i
i
π
η
= (9)
By he alue L
s
he e ec o he blood ine ial mass
has been in ol ed in o he calcula ions.
In o de o espec be e he s enosis in luence on
he CAW haemodynamics he s enosis deg ee SD has
been conside ed. The o mula o he SD
de e mina ion is
10011001
2
0
2
0
×








−=×







−=
A
A
SG
s s
(10)
whe e A
0
is he heal hy essel c oss-sec ion and A
s
x [cm] 1
2
3
0,04
(x)
δ
0
s
l
s
In es iga ion o b ain a e ial ci cle mal o ma ions…
215
is he c oss-sec ion o he essel unde he s enosis.
Fo he modeling easons he s enosis has been
placed in he A e ia ca o is in e na , conc e ely 5 cm
abo e he spli ing o A e ia ca o is communis in he
A e ia ca o is in e na and ex e na, Fig.1. The inpu
signal has been c ea ed by blood low eloci ies in
he igh A e ia e eb alis and he le A e ia
ca o is in e na.
The s enosis model is gi en a Fig. 3, whe e he
modeled p es eno ic essel shun is shown.
Fig.3. Topology o he modelled p es eno ic shun .
P
1
P
2
I
1
P
1
’= P
1
P
3
< P
2
I
1
I
3
I
2
I
1
= I
2
+ I
3
90
70
50
30
10
-10
Deg ee o s enosis
100
90
70
80
60
50
A. ce eb i media dex.
100
90
70
80
60
50
A. commun. pos . dex.
100
90
70
80
60
50
0 1 2 3 4 5 6
A. basila is
A. ce eb i media sin.
A. ca o is in . sin.
A. pe icallosa sin. (A2)
A.ce eb i pos .sin. (P2)
100
90
70
80
60
50
100
90
70
80
60
50
100
90
70
80
60
50
100
90
70
80
60
50
0 1 2 3 4 5 6
P essu e [kPa x 130]
Time [s] Time [s]
P essu e [kPa x 130]
%
Fig. 4. The blood p essu e ime dependence in CAW in he case o s enosis in A e ia Ca o is In e na
Ad ances in Elec ical and Elec onic Enginee ing
216
Because o s enosis he educed blood low I lea es
he p es eno ic essel pa (I
3
< I
1
). In o de o keep
he con inui y p inciple he aided blood low I
2
lea es he modeled essel segmen h ough he shun .
The blood p essu e in he p es eno ic pa ha dly
inc eases while he conside able p essu e d op P in
he pos s eno ic essel pa is ema kable (P
3
< P
2
).
In he p inciple he low I
2
co esponds o a
p es eno ic blood oundabou way in o o he essels,
in his case bene i o A e ia ca o is ex e na o
A e ie subcla ia and i leads o a less conges ion o
he pos s eno ic a eas.
3. RESULTS
The ollowing simula ion esul s acco ding o he
CAW opology, Fig. 1, and wi h a slowly con inuous
inc ease o s enosis deg ee a e p esen ed in Fig. 4
and Fig. 5. The g aphs ep esen he blood p essu e
and blood low ime dependences in a ious
in ac anial a e ies (o CAW).
The essel segmen (A e ia ca o is in e na sinis a)
pa ame e s, [4], [5], used o he simula ion a e gi en
by ollowing way, adius
0
= 0,192cm, leng h l =
5cm, elas ici y E
l
= 1,1652x10
6
g/cm.s
2
and pulse
wa e eloci y PWV = 925cm/s. The maximum
s enosis deg ee was SG
m
= 83%, he s enosis leng h
was l
s
= l = 5cm. The mean a e ial p essu e
co esponds in all essels be o e he s enosis
in oduc ion o he mean p essu e alues gi en in he
li e a u e, [4], [5], Fig. 1.
In he case o blood p essu e he signi ican
changes occu as a as he s enosis deg ee alue is
abou 70%. The small blood p essu e d op is by he
educed conges ion in he CAW e iden and in
acco dance wi h he p inciple o con inuous pipes he
p essu e d op in all o he essels is also possible o
explain.
Also in he case o blood low, a nonlinea pa o
he whole esis ance R
dp
becomes mo e impo an
han he linea one R
s
mainly wi h he g ow h o he
s enosis deg ee. The mos dis inc is he educing o
he blood low in he igh A e ia ca o is in e na
behind he s enosis, while he low in he le A e ia
90
70
50
30
10
-10
Deg ee o s enosis
A. pe icallosa dex. (A2)
100
110
70
50
160
110
60
10
500
400
300
200
100
0 1 2 3 4 5 6
A. ce eb i an . dex. (A1)
A. ca o is in . dex.
A. ca o is in . sin.
A. pe icallosa si
n. (A2)
A. commun. an e io
A. ce eb i an . sin. (A1)
110
90
70
50
80
60
40
20
0
-20
160
110
60
10
500
400
300
200
100
0 1 2 3 4 5 6
Volume low [ml/min)
[%]
Volume low [ml/min)
Time [s] Time [s]
Fig. 5. The blood low ime dependence in CAW in he case o s enosis in A e ia Ca o is In e na
.

In es iga ion o b ain a e ial ci cle mal o ma ions…
217
ca o is in e na is nea ly he same in spi e o he
s enosis, Fig. 2. F om he Fig. 2 he e is possible o
show he CAW as he esponsible and eliable one
o he ascula iza ion o he b ain. The simula ion
esul s show how because o s enosis in one essel
segmen he low dis ibu ion in CAW de elops in
o de o ensu e a su icien blood supply o he b ain.
In he le A e ia ce eb i an e io (A1) he blood
low inc eases so ha no only he le one bu also
he igh (A2) h ough he A e ia communicans
an e io is supplied by blood. Simul aneously he
blood low in he A1 dec eases and so signi ican ly
mo e blood can low om he s eno ic A e ia ca o is
in e na in o he igh A e ia ce eb i media.
4. CONCLUSION
The ob ained simula ion esul s o he blood
p essu e and mainly o he blood low show he
compensa ion possibili ies o CAW in case o one
b ain a e ial occlusion. The c i ical occlusions o an
indi idual ce eb al model we e simula ed and
co esponding changes in low and p essu e we e
obse ed wi hou exposing he pa ien o a isk o
heal h. The eac ions o he indi idual a e ial sys em
in c i ical si ua ions we e in es iga ed and compa ed
wi h he eco ded da a om obse a ions o he
subjec . The elabo a ed me hod o a non/in asi e
p eope a i e isk/es ima ion can be assessed o be
e icien . The e is hope ha his lexible, ime sa ing
and ep oducible me hod makes a aluable
con ibu ion o a oid complica ions o induced essel
occlusion du ing medical ea men by an imp o ed
ope a ion planning.
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Oscilla o y Flow Impedance in Elec ical Analogy o
A e ial Sys em, Ci cula ion Resea ch 16 (1965),
121-133.
[2] Gaelings, E.W.: Nume ische Simula ion
Haemodynamische P ozesse in Vascula en Ne zen,
Shake Ve lag Aachen, 1996, ISBN-3-8265-1509-9.
[3] ápo á, K, Blazek, V., áp, I., Bukuliako á, L.:
Physiological Fluid Sys em Modelling and
Visualiza ion, In e na ional Jou nal o Applied
Elec omagne ics and Mechanics, 14 (2002), IOS
P ess, pp.377/380.
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ze eb alen Ge äßeing i en, Fo sch i Be ich e VDI,
VDI Ve lag, GmbH Düsseldo , 2005.
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[7] ápo á, K., Blazek, V., áp, I.: Pa hological
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[8] Viedma, A., O iz, C.J., Ma co, V.: Ex ended Willis
ci cle model o explain clinical obse a ions in
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(1997), pp. 265-272