bain model

8/12/2019 bain model

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Three-Dimensional Brain Model

Laboratory for Product and Process Design

Director: Andreas A. Linninger

LPPD-Project Report: 04/30/2004

Autors: !aade"a#arat.R. $o%ayaji& !. 'enos& $.

(ondapa))i& P. Royce*ic+ and A. A. Linninger

Re"ision ,.0 - icago& ,0/0/2004

Re"ision 2.0 - icago& ,2/,/200

Re"ie*ers:

,


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Contents

$u%%ary ....................................................................................................................................3

,,.,.1ntroduction.........................................................................................................................4

,,.2.eo%etry Reconstruction too) !i%ics .0 .....................................................................

,,.3. !a5ing tree-di%ensiona) objects in !i%ics................................................................Po)y)ines fro% !i%ics...........................................................................................................6

De"e)op%ent of tree-di%ensiona) geo%etry using a%bit 2.,................................................6

,,.$i%u)ation of tree-di%ensiona) brain geo%etry using 7)uent..........................................,07)uent 8 a finite "o)u%e so)"er ...........................................................................................,0

9a"ier-$to5es 9-$ ;uations............................................................................................,,

$i%p)e A)gorit%.................................................................................................................,,7inite


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Summary

=e purpose of tis project is to capture te brain geo%etry of indi"idua) patients *it ig

accuracy. =e brain is co%posed of porous tissues gray and *ite %atter as *e)) as f)uidscoup)ed *it co%p)e> f)uid structure interactions cerebrospina) f)uid-

"entricu)ar/subaracnoid space and "ascu)ar syste%. 1n order to represent te tree-

di%ensiona) geo%etry of te nor%a) u%an brain& data a"e been co))ected fro% a ig po*er3 =es)a !R1%acine in co))aboration *it te Depart%ent of 9eurosurgery and Depart%ent

of Radio)ogy& ?ni"ersity of icago. =e e>act di%ensions of te brain geo%etry *idt suc

as "entricu)ar si+e and subaracnoida) space are needed to de"ise patient specific treat%entstrategies in "arious %edia) prob)e%s diseases.

1n order to estab)is co%p)e> f)o* patterns of te cerebrospina) f)uid $7& static pressure

fie)ds and to ana)y+e te f)uid transport troug porous surfaces in te brain parency%a& *esou)d re)y on nu%erica) si%u)ation tecni;ues. 1t is ard for !R ana)ysis to foresee te rea)

tree di%ensiona) ydrodyna%ics and te conditions of te brain in nor%a) and under se"ere

disorders. !R ana)ysis cannot predict infor%ation beforeand *itout rea) in-fie)d ana)ysis as

against si%u)ation tecni;ues. 1t cannot incorporate defor%ation effects of so)id brain %atri>&due to te reso)ution of te i%ages generated @,%%. n te oter and& co%putationa) f)uid

dyna%ics 7D can e)aborate tis situation by so)"ing te go"erning e;uation for te f)uidand te defor%ation of te soft tissue.

?nder te abo"e considerations& te !R i%ages a"e been inserted into a geo%etryreconstruction too) !i%ics .0 in order to represent te co%p)e> structure of te "entricu)ar-

subaracnoid syste% and of te brain %atri>. =ese too)s can pro"ide us *it te accurate

geo%etry of te *o)e organ and can re"ea) te intricacies of eac brain structure ta5ing into

consideration te "ariation bet*een indi"idua) u%an subjects. =e tree-di%ensiona)geo%etry is transfor%ed into co%putationa) %eses needed for nu%erica) ana)ysis by 7D

so)"ers based on te finite "o)u%e discreti+ation. =is *as acie"ed by inserting te

continuous geo%etry tat is representati"e of a ea)ty brain in state-of-te-art grid generators)i5e a%bit 2.,& a pre-processor to te nu%erica) so)"er. =e %ode) *as ten si%u)ated using

7)uent ., so)"er to study te $7 f)o* "e)ocity and static pressure profi)es using different

boundary conditions.

=e objecti"e of te present si%u)ations is to study te pysics of $7 f)o* into te

"entricu)ar syste% and in te subaracnoida) space $A$ in a nor%a) situation and to obtain a

better insigt into te pu)sati)e $7 f)o* patterns as a first step. An e>tension of tis *or5*ou)d be te ana)ysis of an abnor%a) situation e.g. Bydrocepa)us& tat arises due to $7

"o)u%e bui)d-up in te brain parency%a and is one of %ost co%%on pato)ogica) situations.

=e resu)ts obtained fro% tese si%u)ations are in good agree%ent *it !R1 %easure%entsfor $7 f)o*. !ore precise)y it *as obser"ed tat te igest "e)ocity occurs in te a;ueduct

of $y)"ius. =e static pressure drop bet*een $A$ and te "ascu)ar syste% *as found to be

463 Pa *en porous boundary conditions *ere enab)ed.

3


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11.1. Introduction

=e brain is co%posed of porous tissues gray and *ite %atter as *e)) as f)uids "ascu)ar

syste%& $7. =e so)id structure of te brain is ca))ed parency%a and is co%posed fro%neuron and g)ia) ce))s. #et*een and around tese so)id structures tere are co%part%ents

*ere te $7 f)o*s "entricu)ar syste%/subaracnoid space. =e "entricu)ar syste% in te

center of te brain consists of four %ain co%part%ents tat are connected to eac oter bytiny a;ueducts s%a)) canne)s. =e )atera) "entric)es are considered as te biggest ca%bers

in te "entricu)ar syste%. =ese are connected troug te fora%ina of !onro *it te tird

"entric)e. =e tird "entric)e is connected *it te fourt troug te a;uaduct of $y)"ius*ic is te narro*est of a)) canne)s 7igure ,,.,. 7ina))y te *o)e "entricu)ar syste% is

connected *it te subaracnoid space troug te fora%ina of !agendie and Lus5e at te

end of fourt "entric)e. =e subaracnoid space surrounds te *o)e parency%a and )ies into

te space bet*een so)id brain and te craniu% bone C,&C2.

Cerebrospinal Fluid: =e brain parency%a is

ydrau)ica))y protected by te cerebrospina) f)uid

$7& *ic is produced in a pu)sating %anner atte coroids p)e>us in te center of te brain. =e

p)e>us is for%ed as a resu)t of info)ding of teependy%a into te ca"ities by b)ood "esse)s of te

pia %atter. =e rate of $7 for%ation "aries

bet*een 0.3 to 0. %)/%in C3 at te p)e>us andcircu)ates fro% te "entricu)ar syste% to te

subaracnoid space in te parency%a 7igure

,,.,.

Functions of CSF: $7 is a bio%edica) f)uid& a *atery so)ution containing ions and oter

species and ser"es as an intracerebra) transport %ediu% for nutrients. 1t as se"era) functionssuc as pysica) transport& e>cretory function& transport of bio%edica) e)e%ents& buffering of

e>trace))u)ar f)uids and participation in brain signa)ing C,. $ince $7 is responsib)e for

intracerebra) species transport& te f)uid is a)so usefu) for c)inica) researc.

=e %ost pro%pting researc

is te $7 f)o* dyna%ics and

is responsib)e for a nu%ber ofbrain disorders& te for%ation

of an ede%a C4& C. =e

5no*)edge of $7 f)o*dyna%ics in a nor%a) brain

*ou)d )ay foundation for

researc in predictingabnor%a) situations C4& C

and to propose appropriate

7igure ,,.,. #rain Pysio)ogy

7igure ,,.2. !R1 of te u%an brain )eft.


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%etods for treat%ent. 1ntracrania) pressure 1P rises if production rate of $7 e>ceeds

absorption rate. =is occurs if $7 is o"erproduced& resistance to $7 f)o* is increased& or

"enous sinus pressure is increased. $7 production fa))s as 1P rises. Eene"er an i%ba)ancearises bet*een te $7 production and absorption rate& an Fede%aG tat is an increase in brain

"o)u%e resu)ting fro% increased sodiu% and *ater content is for%ed. =ere are se"era) types

of ede%a:

Vasogenic: Arises due to )oca) disruption of te b)ood brain barrier ###. Hydrocephalic: ccurs *en tere is an accu%u)ation of e>trace))u)ar f)uid )eading to

Bydrocepa)us.

Cytotoxic: Arises due to ce)) s*e))ing a)ong *it reduction in te f)uid "o)u%e in tee>trace))u)ar space $.

Hydrocephalus:Bydrocepa)us can be defined broad)y as a disturbance of for%ation& f)o*& or

absorption of cerebrospina) f)uid $7 tat )eads to an increase in te "o)u%e occupied by

tis f)uid in te centra) ner"ous syste% 9$. =is condition a)so cou)d be ter%ed a

ydrodyna%ic disorder of $7. 1n ydrocepa)ic ci)dren& te rate of $7 for%ation isnor%a) or s)igt)y reduced& accounting to 0.30 %)/%in C2& C. =ere are se"era) types of

Bydrocepa)us and it *ou)d be appropriate to reference te%:

Communicating hydrocephalus: =is occurs *en fu)) co%%unication e>ists bet*een te"entric)es and subaracnoid space. 1t is caused by o"erproduction of $7 rare)y& defecti"e

absorption of $7 %ost often& or "enous drainage insufficiency occasiona))y.

Noncommunicating hydrocephalus: =is occurs *en $7 f)o* is obstructed *itin te

"entricu)ar syste% or in its out)ets to te subaracnoid space& resu)ting in"entricu)ar/subaracnoid space non co%%unication.

Obstructive hydrocephalus:=is resu)ts fro% obstruction of te f)o* of $7 intra"entricu)ar

or e>tra"entricu)ar. !ost ydrocepa)us is obstructi"e& and te ter% is used to contrast te

ydrocepa)us caused by o"erproduction of $7.

Arrested hydrocephalus: =is is defined as stabi)i+ation of 5no*n "entricu)ar en)arge%ent&probab)y secondary to co%pensatory %ecanis%s. =ese patients %ay deco%pensate&

especia))y fo))o*ing %inor ead injuries.

$7 osci))atory f)o* resu)ts fro% cardiac pu)sations C4. =e pattern and ti%ing of $7

%otion as been studied e>tensi"e)y *it !R1 tecni;ues CH. 9e"erte)ess& te causes and

%ecanica) princip)es under)ying intracrania) dyna%ics are contro"ersia). $o%e autors )in5te $7 f)o* to te brain %otion. =is ypotesis is supported by carefu) %easure%ents of

brain %otion using 19 pase-contrast !R1 CH. A co%peting "ie* attributes $7 %otion to

a trans%ission of te arteria) pressure in te coroid p)e>us C& C6. =e si%u)ations in tis

*or5 are based on te assu%ptions tat te pu)sati)e %otion of te $7 is due to tepu)sations of te coroid p)e>us& neg)ecting te %o"e%ent of te brain tissue surrounding te

"entricu)ar syste%.


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11.2. Geometry Reconstruction tool (Mimics 8.!

!i%ics is a po*erfu) progra% designed to create co%p)e> geo%etries fro% standard !R1

i%ages 7igure ,,.2. An e>a%p)e of a tree di%ensiona) %ode) of te cerebrospina) f)uidsyste% %ade using !i%ics is so*n in 7igure ,,.3. !R1 s)ices of a ea)ty u%an subject

*ere i%ported into !i%ics. $)ices *ere disp)ayed of te fo))o*ing cuts:

orona)superior to inferior $agita) )eft to rigt A>ia)anterior to posterior

=e s)ices *ere co)ored %anua))y to a))o* te progra% to distinguis bet*een different parts

of te brain. =e resu)ting infor%ation *as e>ported as tree di%ensiona) %ode)s and

po)y)ines. =e fo))o*ing sections of te brain *ere %ode)ed:

7igure ,,.3. 3-D %ode) of te "entricu)ar syste% anda)f of te subaracnoid space.


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7igure ,,.4. 3-D reconstruction of te $7

pat*ays: "entricu)ar/subaracnoid syste%.

7igure ,,.. 3-D reconstruction of te u%an brain:

gray and *ite %atter.

H


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11.". Ma#in$ three-dimensional o%&ects in Mimics

ac object in !i%ics %ust be dra*n onto te s)ices tat te object spans. 7or eac object& a

separate I)ayerJ %as5 %ust be used *it a different co)or fro% oter )ayers& 7igure ,,..

Dra*ing on a )ayer %ay eiter be doneauto%atica))y or %anua))y by specifying a range

of sades fro% te !R i%age to be co)ored. =e

I=reso)dingJ button is used to specify terange of sades to be incorporated into a %as5

by se)ecting te upper and )o*er )i%its of

dar5ness in greysca)e. A)) dra*ings for tedifferent "entricu)ar sections *ere dra*n

auto%atica))y. !anua) dra*ing *as ten

re;uired since !i%ics *as not ab)e todifferentiate in so%e regions specific areas of te

brain.

7or te brain parency%a - due to its undu)ating nature - eac s)ice fro% te !R i%ages *asco)ored in by and using a co%puter %ouse& and a set of too)s *itin %i%ics. =e too)s

inc)ude a brus *it "ariab)e si+e and sape& and a )asso *ic auto%atica))y fi))s in te area

*itin a dra*n border. =ese too)s can be used to dra* and erase fro% te se)ected )ayers. =ocon"ert te %anua) dra*ings into a 3-D %ode)& te )ayer %ust be c)ic5ed fro% te I!as5sJ

too)bar and te Ia)cu)ate 3DJ button %ust be se)ected. =ere are se"era) options to consider

regarding te reso)ution and te s%ooting of te tree di%ensiona) object. bjects *it aig reso)ution and )o* s%ooting *i)) appear "ery jagged and %ay be discontinuous. n te

oter and& infor%ation is )ost con"erting te dra*ings into )o* reso)ution and ig)y

s%ooted objects. =ere is no ob"ious coice for te best co%bination of reso)ution ands%ooting& but te user as to try for te best fit.

nce te 3-D objects a"e been created& tey can be e>ported for use *it oter progra%s as

I.st)J $=L fi)es using te $=LK !odu)e. =o e>tract an $=L fi)e& first se)ect te I$=LKJbutton fro% te I!as5sJ too)bar& and ten coose te source for te $=L. =e $=L fi)e can be

%ade direct)y fro% a %as5& or fro% an

e>isting 3-D object. =e 3-D objects *ereused as te source of te $=L fi)es. After

%a5ing te $=L fi)es& tey can be i%ported

bac5 into %i%ics and disp)ayed in te 3D

"ie*& 7igure ,,.H.

7igure ,,..=ypica) "ie* *itin !i%ics.

7igure ,,.H.$=L fi)e of te

#rain $te%.

7igure ,,..Po)y)ines of

te #rain $te% in I3DJ

"ie*.


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'olylines rom Mimics

ontours of te specific brain sections *ere e>tracted fro% te sagita) s)ices by se)ecting te

desired I)ayerJ and coosing te Ia)c Po)yJ option. A set of )ines ten appeared on tescreen. #efore te )ines cou)d be e>tracted tey ad to be first se)ected by dragging te

co%puter %ouse pointer o"er any part of te cur"e. =e po)y)ines cou)d ten be e>ported as

I.igsJ fi)es by se)ecting te I1ges >portJ button 7igure ,,.. $e"era) options e>ist *enfitting te po)y)ines to te dra*ings& and again tere is no best *ay to fit te po)y)ines.

De)elo*ment o three-dimensional $eometry usin$ Gam%it 2.1

=e ne>t step is to introduce te $=L or 1$ fi)es into progra%s tat can gi"e to te user tediscreti+ation of te entire geo%etry. =e process of discreti+ation in"o)"es transfor%ing

continuous infor%ation to discrete ones. A progra% tat is capab)e of acie"ing tis is a%bit

C,,. a%bit 2., is a *e)) 5no*n pre-processor grid generator to 7)uent& te so)"er and isused to reconstruct geo%etry gi"en by !i%ics to 7)uent understandab)e for%& ca))ed te %es

fi)e. btaining an ana)ytica) so)ution of te e;uations go"erning te $7 f)o* is i%possib)e.

=e on)y *ay to so)"e tese set of partia) differentia) e;uations PDs& ca))ed 9a"ier-$to5es&

is to di"ide te *o)e do%ain into subdo%ains. ac subdo%ain is ca))ed a FgridG or F%esG.

=e syste% is no* so)"ed by app)ying te discreti+ed 9a"ier-$to5es e;uations in eac grid byappropriate se)ection of te boundary conditions. =e discreti+ation sce%e is discussed in

9a"ier-$to5es e;uations section.

7igure ,,.6. a%bit 2., 1nterface.

6


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7igure ,,.6& so*s te

interface of te grid

generator tat *as used.7igure ,,.,0& so*s te

tree- di%ensiona)

geo%etry of te"entricu)ar syste% and

$A$ as reconstructed

fro% !i%ics and afterte introduction in te

post processor

a%bit.

=e 1$ fi)es obtained& need user *or5 in order to produce te 3-D geo%etry. nce teF"o)u%eG as been created by connecting FedgesG and FfacesG te *o)e 3-D geo%etry needsto be di"ided into grids. $pecifying te boundary conditions co%p)ete te pre-processing step.

=e boundary conditions specified for te $7 f)o* in te brain as *e)) as te %esed

geo%etry are so*n in 7igure ,,.,,.=e grid is constructed of tetraedra) e)e%ents and as640 tree-di%ensiona) finite "o)u%e e)e%ents and ,HH, nodes.

Inflow boundaryconditions

Symmetry boundaryconditions

Outflow boundary

condition (porous medium)

Inflow boundaryconditions

Symmetry boundaryconditions

Outflow boundary

condition (porous medium)

7igure ,,.,,. #oundary conditions and %esed geo%etry of te "entricu)ar syste% and $A$

11.+ Simulation o three-dimensional %rain $eometry usin$ ,luent

,luent a inite )olume sol)er

7)uent is a nu%erica) so)"er based on finite "o)u%es approac tat so)"es f)uid f)o* prob)e%s

under a gi"en situation by app)ying te go"erning e;uations of f)uid f)o* C,,. =e go"erninge;uations are described by continuity conser"ation of %ass and 9a"ier-$to5es e;uations

conser"ation of %o%entu% and tus 7)uent app)ies tese e;uations to eac %es of te

7igure ,,.,0. =ree di%ensiona) geo%etry of te "entricu)ar and

$A$ of a nor%a) u%an brain fro% a%bit 2.,

,0


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geo%etry. =e post-processor for data ana)ysis& based on te penL tecno)ogy is in-bui)t in

7)uent and ta5es care of te "isua)i+ation of output "ariab)es. 7)uent is using te $1!PL

a)gorit% de"e)oped by Patan5ar and $pa)ding in order to so)"e te syste% of e;uations C,2.

a)ier-Sto#es (-S! /0uations

=e 9a"ier-$to5es e;uations are te %ate%atica) state%ent of 9e*tonJs second )a* of%otion for "iscous 9e*tonian f)uids. =ey represent tat te su% of a)) forces acting on f)uid

e)e%ent e;ua)s te %ass ti%es te acce)eration as app)ied to te f)uid e)e%ent for *ic te

sear stress is )inear)y re)ated to te cange of te sear strain C,3.

> > > > > > >

> y +

y y y y y y y

> y +

+ + + + +

> y +

2 2 2

2 2 2

2 2 2

2 2 2

2 2

2

" " " " " " "K" " "

t > y + > y +

" " " " " " "K" " "

t > y + > y +

" " " " "K" " "t > y + >

gx

x

gy

y

+ + = + + +

+ + = + + +

+ + = +

+ +2

2 2

" "

y +

g!!

+ +

,,.

,

=e )eft and side is te %ass per unit "o)u%e ti%es acce)eration of te f)uid. =e )oca)acce)eration is due to te unsteadiness of f)o* and te con"ecti"e acce)eration is a resu)t of

spatia) "ariation of te "e)ocity fie)d. =e rigt and side represents te net force per unit

"o)u%e acting on te f)uid. =ese forces are a resu)t of pressure& "iscosity and te forces o"er

unit "o)u%e tat co%es fro% an e>terna) force fie)d acting trougout te f)uid e)e%ent e.g.gra"itationa) forces. =ese e;uations are non-)inear PDs and are genera))y "ery difficu)t to

so)"e ana)ytica))y. Bo*e"er& by using %odern ig speed co%puters& ad"anced discreti+ation

sce%es and state-of-te-art so)"ers it is possib)e to obtain nu%erica) so)utions to 9-$

e;uations. =o do so& te continuous f)o* fie)d "e)ocity or pressure as a function of space andti%e is described in ter%s of discrete "a)ues at prescribed )ocations rater tan continuous

"a)ues. #y tis tecni;ue& te differentia) e;uations are rep)aced by a set of a)gebraice;uations tat a co%puter can understand.

Sim*le l$orithm

$1!PL stands for $e%i 1%p)icit !etod for Pressure Lin5ed ;uations. =is a)gorit% is an

iterati"e tecni;ue tat is used to so)"e te 9a"ier-$to5es e;uations.

2, . u

u u p u

t

+ = +

,,.

2

0u v "

t x y !

+ + + =

;uation 3 is te continuity e;uation and spea5s about te )a* of conser"ation of

%ass and cou)d con"enient)y be represented by ;uation 4.

,,.

3

. 0u = ,,.4

,,


12/21

=e $1!PL a)gorit% is %ost)y suited to steady state prob)e%s and cou)d a)so be e>tendedto so)"e unsteady prob)e%s. =e a)gorit% begins by so)"ing te %o%entu% e;uations for a

guess "a)ue of pressure p#. =e resu)ting "e)ocities are represented as u# and v# and "# in

tree di%ensions and are ca))ed Istar velocitiesJ as tey are based on a guess "a)ue of

pressureC,,&C,2. 1n addition& star "e)ocities are not true "e)ocities. =e %etod ten see5s tofind corrections to te guessed pressure. nce te pressure corrections are found& te pressure

and "e)ocities are updated and te pressure is used for initia) guess "a)ue for ne>t iteration.

=is iterati"e procedure ta5es p)ace unti) te true so)ution is obtained at te point ofcon"ergence. At tis point& te residua) error bet*een te true so)ution and te appro>i%ate

so)ution is a)%ost +ero *it a difference gi"en by te to)erance "a)ue tat is usua))y in te

order of ,0-,0. A reasonab)e re;uire%ent for con"ergence is satisfy te ;uation ,,.3.

,inite olume discreti3ation scheme

A "ery ad"anced %etod for tediscreti+ation of te 9-$ e;uations is te

7inite


13/21

du

dx (

=

,,.

;uation,,.is te one-di%ensiona)-DarcyJs La*. =e "ariab)es and para%eters in"o)"ed

are )isted in =ab)e ,,.,.

aria%le 'hysical Descri*tion ,ield 4nits

static pressure CPa

d

dx

static pressure drop across te %ediu% CPa

) f)uid "iscosity CPa sec( %ediu% per%eabi)ity %ateria) property C%2

u fi)tration f)uid "e)ocity C%/sec

=ab)e ,,.,.tended to anisotropic %ediu% too& constant f)uid density& non-

defor%ab)e so)id %atri>& f)o* at )o* Reyno)ds nu%bers and for a"erage porosity up to 0C,H.

on-Darcy eects

=e )inear re)ationsip bet*een pressure drop and "e)ocity during f)uid f)o* troug porous

%edia is affected by f)o* rate C,H& C,& C,6. At ig f)o* rates& te pressure drop predictedby DarcyJs )a* beco%es in"a)id and tis peno%enon is 5no*n as non-Darcy f)o*

peno%enon. 1n suc cases& te f)uid "iscosity is )o* and te f)uid "e)ocity is ig. =erefore&

inertia) effects cannot be neg)ected& )eading to te addition of non)inear ter%s to te Darcye;uation. =e %ost co%%on)y accepted corre)ation is 7orcei%erMs )a*& *ic introduces a

second order ter% for "e)ocity as *e)) as a ne* roc5 property& te 7orcei%er coefficient or

te inertia) para%eter or te non-Darcy f)o* coefficient C,6.

7or an isotropic porous %edia& 7orcei%erJs )a* is:2uu

(dx

d

+= ,,.

1n %any reser"oir roc5s& te per%eabi)ity depends on direction. ften& te "ertica)

per%eabi)ity is %uc )ess tan te ori+onta) per%eabi)ity because of sa)e )ayers or oter

sedi%entary features C,H& C,. =e per%eabi)ity %ay "ary ori+onta))y because of stresses&

,3


14/21

fracturing& or depositiona) processes. =e resu)t is tat one %ust consider different

per%eabi)ities for f)o*s in tese different directions.

1n a tree di%ensiona) syste%& te co%ponents of "e)ocity are:

+

+

=

+

+

=

+

+

=

+

p5

y

p5

>

p5

N

,u

+

p5

y

p5

>

p5

N

,u

+

p5y

p5>

p5N

,u

+++y+>+

y+yyy>y

>+>y>>>

,,.

H

1n anisotropic syste%s& per%eabi)ity is a tensor gi"en by te re)ation 5O

+++y+>

y+yyy>

>+>y>>

555

555

555

Momentum /0uations or 'orous Media

Porous %edia are %ode)ed by te addition of a %o%entu% source ter% to te standard f)uid

f)o* e;uations. =e source ter% is co%posed of t*o parts: a "iscous )oss ter% Darcy& te terigt-and side of ;uation ,,.& and an inertia) )oss ter% te second ter% on te rigt-and

side of ;uation ,,..

3 3

, ,

,

2

i i* * i* mag *

* *

S + v C v v

= =

= +

,,.

*ere iS is te source ter% for te ith >& y& + %o%entu% e;uation& + and Care prescribed

%atrices C,,. =is %o%entu% sin5 contributes to te pressure gradient in te porous ce))&

creating a pressure drop tat is proportiona) to te f)uid "e)ocity or "e)ocity s;uared in te

ce)).

=o reco"er te case of si%p)e o%ogeneous porous %edia

2

,

2i i mag iS v C v v

= +

,,.

6

*ere is te per%eabi)ity and 2C is te inertia) resistance factor& si%p)y specify + and

Cas diagona) %atrices *it,

and 2C & respecti"e)y& on te diagona)s and +ero for te

oter e)e%ents.

,4


15/21

11.+ Results and Discussion

=e crania) ca"ity is a c)osed space and tere e>ists e;ui)ibriu% bet*een te rate of for%ationand te rate of $7 absorption. !ost of te $7 absorption ta5es p)ace in te aracnoid "i))i

tat is a eterogeneous anisotropic porous %ediu%. 1n case 2& *e i%pose porous boundary

conditions to si%u)ate $7 absorption troug te aracnoid "i))i. =is $7 absorptionresistance to f)uid f)o* i%poses a pressure drop bet*een $7 and te "enous pressure tat

is about 00 Pa in te u%an brain.

Ee i%pose a pu)sating in)et boundary

condition for $7 "e)ocity at te )eft& rigt&

tird and fourt "entric)es *it user defined

functions ?D7 see 7igure ,,.,,. ?D7pro"ides te abi)ity to prescribe a ti%e

dependent boundary condition in te so)"er

7)uent. =is boundary condition gi"es to

te $7 te abi)ity to f)o* troug te"entric)es to te subaracnoid space troug

te tird and fourt "entric)e *it apu)sating %anner si%i)ar to te *ay tat

$7 f)o*s in te !R i%ages. =e pu)sati)e

nature of $7 f)o* is ta5en care of by tesine *a"e input defined in ?D7. $7 f)o*s

out of te subaracnoid space troug te

porous %ediu% and te resistance to f)o*

depends on te per%eabi)ity of te %ediu%.=is obstruction causes an additiona) pressure drop for f)o*. Ee a"e ta5en care to i%pose a

pressure drop of 463 Pa by carefu))y defining te "a)ues for porous %ediu% properties suc as

porosity& per%eabi)ity and inertia) factor in 7)uent.

=e properties of te porous %ediu% deter%ine te conducti"ity of te f)uid across its )engt.Ee a"e %ode)ed te brain as:

i. Bo%ogenous %ediu% in *ic te %ediu% properties suc as porosity is sa%e

e"ery*ereii. 1sotropic *erein te per%eabi)ity is sa%e in a)) direction >& y& + in te preset study

=e "a)ues for te properties are so*n in te #oundary conditions pane) =ab)e ,,.2 and7igure ,,.,3. =e

in"erse of per%eabi)ity is ca))ed as te "iscous resistance and 7)uent uses tis ter% forca)cu)ations.

7igure ,,.,3. Porous %edia boundary

conditions enab)ed in 7)uent .,

Geometric 'ro*erty aria%le aluePer%eabi)ity 5 3.3-,2 %2

Porosity 0.4

=ab)e ,,.2.


16/21

Case 15 'resence o 'orous medium as a re-a%sor%ance Boundary condition

Ee *i)) present resu)ts concerning te contours of "e)ocity %agnitude& y-"e)ocity and static

pressure. 1t is i%portant to "isua)i+e te y-"e)ocity tan te > and +-"e)ocities because itJs te

basic $7 f)o* direction and gi"es infor%ation for te re"ersa) in "arious co%part%ents of

te brain. =e input sine *a"es tat i%part te pu)sati)e nature to te f)o* are so*n in 7igure

,,.,4. 1n tis grap *e represent te di)ations of te coroid p)e>us in te )atera)


17/21

7igure ,,.,H.


18/21

As referenced pre"ious)y tere are so%e areas *ere f)o* recircu)ation occurs at eac cardiac

cyc)e and eddy %otion is obser"ed in tese regions. =is %otion is ta5ing p)ace at te end of

eac cardiac cyc)e and at te beginning of te ne>t one fro% ti%e 0.6 to ,.2 sec and atspecific )ocations. 7or "isua)i+ing tese areas *e *i)) use te "e)ocity "ectors co)ored by te

y-"e)ocity since tis is te basic $7 f)o* direction.

7igure ,,.20. Areas of bac5f)o* 7igure ,,.2,. #ac5f)o* areas in te "entricu)ar

syste%

1n 7igure ,,.20*e i))ustrate *it te b)ac5 arro*s te areas *ere bac5f)o* occurs. Ee can

see tat te recircu)ation ta5es p)ace inside te fora%ina& a;ueduct of $y)"ius and in a s%a))

region near te re-absorption area. Eit red *e "isua)i+e te positi"e and *it b)ue tenegati"e "e)ocity in te do%ain. 1n

7igure ,,.2, *e +oo% in te

"entricu)ar syste% for a better )oo5 of

te bac5 f)o* in te fora%ina of!onroe and in te a;ueduct of

$y)"ius. =e %a>i%u% "e)ocity in

tese regions is 3.0,0-4%/sec for teend of cardiac cyc)e *ere te f)uid

dece)erates and fina))y re"erses.

7ina))y& in 7igure ,,.22*e i))ustratete bac5f)o* in te s%a)) region near

te re-absorption area. 1n tis figure

*e can see b)ue negati"e "e)ocityand green positi"e "e)ocity arro*s

describing a recircu)ation of te f)o*in te specific )ocation.

11.6 Conclusions

=e report introduces a co%putationa) f)uid %ecanics %ode) of te cerebrospina) f)uid

f)o* inside te brain. =e e;uations of %otion for $7 f)o* in te "entricu)ar and

subaracnoida) pat*ays as *e)) as $7 seepage inside te porous brain parency%a *ere

7igure ,,.22. Recircu)ation near te re-absorptionarea

,


19/21

so)"ed. =e boundary conditions for te co%p)e> brain geo%etry *ere for%u)ated. =e

co%putations used patient-specific co%putationa) %eses obtained by i%age reconstruction

too)s. =e syste%atic procedure for conducting a detai)ed 7D f)o* ana)ysis for indi"idua)patients based on !R1 data is reported for te first ti%e. =e si%u)ations proposed in tis

*or5 accounts for indi"idua) differences in geo%etric features as *e)) as f)o* and pressure

patters in eac patient. $tate-of-te-art ine pase contrast !R1 tecni;ues pro"ided accurate$7 f)o* "e)ocity %easure%ents in tree di%ensions. =e c)inica) in-"i"o f)o* %easure%ent

*as used to "a)idate te proposed co%putationa) %ode). 1t %ay be possib)e in te future to

deter%ine $7 production and reabsorption rates by a co%bination of ine-!R1 $7%easure%ent *it te co%putationa) %ode). =ere is a)so te possibi)ity of inferring

bio%ecanica) properties of te brain suc as co%p)iance or per%eabi)ity using suc %etods.

ur $7 %easure%ents and si%u)ations presented a co%p)ete picture of te pu)sati)e $7

dyna%ics during te cardiac cyc)e in tree din%ensions. ine !R1 %easure%ents and 7Dsi%u)ations de%onstrated $7 f)o* re"ersa) in te "entricu)ar syste% as *e)) as in te pre-

pontine subaracnoida) space. =e e>cange of $7 *it te spina) $A$ and its inf)uence

spina) $A$ co%p)iance on intracrania) dyna%ics appears to be insufficient)y recogni+ed in te

e>isting %ode)s of te brain as a c)osed ca"ity !onroe-(e))y doctrine.1n addition to predicting te f)o* fie)d& our first princip)es approac accurate)y ;uantifies

te intracrania) pressure dyna%ics in a)) )ocations of te brain. =is 5no*)edge cou)d bei%portant for assessing te specific c)inica) state of a patient based on)y on a andfu) of

5no*n pysica) and bio%ecanica) properties. =e 1P f)o* patterns cannot be %easured by

!R1& yet %ay be i%portant in deter%ining te de"e)op%ent of nor%a) pressureydrocepa)us. ur 7D si%u)ations re"ea)ed s%a)) trans%ant)e pressure difference in te

nor%a) patient and ydrocepa)ic patient.

=e successfu) integration of te !R1-7D approac for studying intracrania) dyna%ics

presented in tis artic)e o)ds pro%ise for considering patient-specific obser"ations in tedesign of treat%ent options in te future.

,6


20/21

Reerences

C, L.P. Ro*)and& !.. 7in5& L. Rubin& erebrospina) 7)uid: #)ood-#rain #arrier& #rainde%a& and Bydrocepa)us& ed. . (ande) et a).& Princip)es of 9eura) $cience& App)eton

Q Lange 3rdedition& pp. ,00-,00& ,66,.

C2 $. A%%oura& A. Aroussi and !.


21/21

C, !. #raaten and E. $yy& FA study of reca)cu)ating f)o* co%putation using body-

fitted coordinates: consistency aspects and %es s5e*nessG&Numerical Heat &ransfer& 7&pp. 6-H4& ,6.

C,H T. #ear ,

U. #ac%at&2ntroduction to 3odeling of &ransport henomena in orous3edia& ed. T. #ear& & Dordrect: ()u*er Acade%ic Pub)isers& ,660.

C, 7.A.L. Du))ien, orous 3edia Fluid &ransport and ore Structure, Acade%ic Press&

19& ,6H6.

C,6 '. Eang& 7. =au"in& (.(. !oanty& Non/+arcy flo" through anisotropic porousmedia,e%ica) ngineering $cience& +& pp. ,6-,6& ,666.

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