Stiffening of the cardiac wall by coronary blood volume increase : a finite element simulation

Stiffening of the cardiac wall by coronary blood volume

increase : a finite element simulation

Citation for published version (APA):

Huyghe, J. M. R. J., Arts, M. G. J., Campen, van, D. H., & Reneman, R. S. (1990). Stiffening of the cardiac wall

by coronary blood volume increase : a finite element simulation. In Biomechanical transport processes / Ed. F.

Mosora (pp. 23-30). (NATO ASI seires. Series A, life sciences; Vol. 193). Plenum Press.

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Published: 01/01/1990

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SflTFF'N\IING OF TTIE CàRDIAC !{ALL BY @ROM\RY BIOOD VOII]ME TNCREASE: A fliNITE EI.ANNW SÏM]LAIION

Jacques M. Ilt4rghel, Ttreo Arts2, Dick H. rran canpen3, ard Roberts S. Renernan4

Departlents of Movernerrt Sciencesl, Bioptrysics2 ard Physiologf

University of Limburrl, I4aa.stric*rt, tJ'.e Netlrerlards ard Detrnrbnent of

Mechanical Hrgheerfug3, Eirldtlcnren UniversiQr of Tbchnology

Eirdhoven, tlre Netlrerlards

ABSTRACI

A porous nredir:n finite elenrerrt nrodel of ttre beatingr left ventricle is usea Co simul-ate the jlfluence of tlre irtracororanlr blood voltmre on left

ventricular mectranics. fhe sporgy rnaterial is ccrposed of ilconqrressiJcle

solid (myocardial tissue) ard incorpressible fltrid (corornrlr blood). fhe

model is-axislznunetric ard allcrus for finite deforrnation, ilcludiJrg torsion around tlre axis of slnrnet4z. Ttre total stress in ttre tissue is ttre sun of

the intramyocardial pre.ssr:re, effective trnssive stress due to myocardial

defornation ard the óntr:actile fi-ber stress. Ihe ncdel is able to si:nulate

a ful1 _{carrCiac rycle.} Ttrree-dilerrsional erd-systolic deforrnation conptrted

relative to tlre erd-diastolic state is strcrrn to be consisterrt wittr

o<peri:nerrtal data frorn tlre literiature. The direction of rnaxirnal shortening variea less tlran 30' from erdocarrdiun to epicardir.un raÈriIe fiber direction varied by nrore ttnn lOO'. ït is strcnm that tlre verrtricul-ar nrodel Qdli-bits

diastolió stiffenirg folJ,o^rirg an jrcrease of intracoronary blood voltune.

Enddiastolic left ventricrrlar pressure increases frqn 1.5 kPa to 2.0 kPa

wtren r:aisirg intracoronarlr blood voIurc from 9 to L4 ml per 1-00. g

myocaadial Lissue. the rncdel si-rnul-ation suggests _lfnt the nrectranisn

underlybg tlp increase in erddiastolic pressure at higher coronarT blood

volune!, ís an ilcrease il trnssive stiffness of the rnyocarrdial fj-bers. this i:rcreased stiffness is the combiJred result of an c /er:alI increase in strain

in myocarrlial tissue ard tlre non-ljnear stress-strain relationstrip of

myocarrCial tissue. Keyuords

r-eft ventricle / poror:s rediun ,z mixture / erectile properti-es / díasí-:ore / coronanY perfusion.

I}IIROUJSIION

The carriiac wa1l is a conpler< biological stnrcture coneosea of

different conqrcnerrts: muscle cells, corona4r vessels, collagen filcers,

Biomechanicql Transport Procxses, Edited by F. Mosora eÍ 4/. Plenum Press. New York, 1990

i-ntracel1ular ard interstitial f1uid, 11nph ard blood. For tie sake of sinplicity, rnany autlrors of cardiac ncdels assurre tire cardiac waII to be a

hornogenous contirrlnun of sol-id rnatter. Tlre aim of t'tris sbudy is to

irrvesti-gate to vÈridr extent a rpre detailed nrcdel description of ttre

myocardiai tissue, jrcludirq a solid ard a fluid ccnponent, is able to

descri-be the stiffenirg of the cardiac walt by coronaq/ blood volune

ilcrease.

MEITION

l4aterial npdel. t{e assr-uned that myocardial tisstre r/as a q)orgy

stnrsture filled with intr:acor.ornry b1ood. Ihe rnaterial of vrtrictr the

sporgy strusture was rnade ard tirc intracororlarlr blood were both assured to be irrconpressilcle. Therefore, ctrrarges il voh:re of tlre solid fluid rnixtr:re were equal to the ancunt of blood beirg sqleezed out or sucl<ed in. ïhe

stress in tlre mixture was tlre sr-un of the intramyocardial pressure p

(present irr both fluid ard solid), effective _trnssive stress due to ttre

deforrnation of the porous strusture ard contractile fi-ber stress. As tlre

collagen weave is three-di:nensional, we assuned ttrat tlre passive effestive

stress was tfrree-dinrensional. Ttre contrastile fiber stress, hcwever, asted

only in the fiber direcbion, _{which charqed aqross ttre raral-I. The contrastile}

stress was tirne, str:ain ard strain rate deperdent. ïhe redistrilcution of

intramyocardial blood in tàe coronary bed was nrcdelled by Darqrrs 1aw: the

f1ow of irrtracoronar1l blood was proportional to tlre intramyocarCial

pressure gradient. ïhe proportionality constant is ttre pernreability of tlre

mediwn. In order to account for ttre drarqes il vascular resistance rrhen

irrtracoronary blood vo}.urp clurges, ttre perneability of tlre nrediun was

adapted proportiornlly to ttre square of ttre intramyocarraial coronary blood vol-r:nre in ttre course of the

corncrrtation-Fig. 1 Left: cross-secti-on of the rotationatl-v svnmetric fj-nite elsnent

nesh of a canirre left ventricle. ÀIl el&rerits contain contracti-l-e

flbers except ersrents 10, 11 and 12 whjlch represent the anrru-lus

fibrozus. Right: in a cross-sectj-on of the wa'l I of the teft

ventricle, the sequence of orrset of contraction is si-'rn:-l_ated to radiate frcrn a pojnt M (tinxe in rns).

Nunerical approactrr. The myocardial waII was subdivided into thirer

rirg-sha@ elenents (fiq. _{1). Each elenerrt trad eight nodes. rhe ctrrarge G}

position of-eactr node relative to the reference situation raras descriJced by a radial, axial ard circr-unferential displacenent at any tinre dr:rirg t]le

cardiac cycle. Quadratic intertrrclation yielded displacanents at

Íntermediate points. The intramyocardial pressure field was obtained by

linear interpolation from pressure values at the corïler nodes. Ttre total

nunber of degrees of freedom wàs 395 (351 displacenerrts ard 44 pressures).

fhite defornntion was accounted for by nreans of a totar ragrargian

z +

For tlre sake of rdiac wall to be a

his study is to

:scription of the

onent, is able to

:rary blood vol-une

;ue rdas a q)orgD/ ia1 of wtridt ttte re both asstuned to olid fluid nLórre

or sucked in. The

ardial pressure _P

stress due to the

:er stress. As the : passive effective ss, hcrnever, acted -1. fhe contracEife

redistrilcution of y Darcryts 1aw: the re intrarnyocardial nrneability of ttle

ar resistance whert

of tlre nredir.m was lial cororn4r blood

r i r F l n i * a o ' l m n f

mtain contractile

)resent the anrrulus

waIL of the left

ln is sinul-ated to

ivided into thirbY

cdes. The c}rarqe in on was descri-bed bY ny tinre dr:rirg the

dispJ-acenents at

Ld was obtained bY

r nodes. the total ard 44 pressures).

r total Iagrargian

\

\

a p p r o a c h . A n i - r p l i c i t - o < p t i c i t t i r r r e . i n t e g r a t i o n . s c t r e r r p w a s u s e d . w i t h i l 1

each tine step a noaitiá Newbon-Raphson ite::ative procedure was used to

Gunt for the non-ljlearities ilctuded il ttre rpdel'

The axial and circumferential displacenerÈ of t].e 7 top nodes of -the

n e s r r w e r e s r 4 p r e s s e d . N o b l o o d w a s a l l o m l e d t o c r o 6 s t } r e e r d o c a r d i a l s u r f a c e . A t t h r e ' ' e r d o c a r r c i a l s i d e o f e l e r r e r r t s ] . t o g a u n i f o r m intraventrj-cufar p-v was appfi-ta as an e>rtêrral load' rtre loads ocerted by

tlre papillarlz mlscles ard'-by the pericardir.un were negtected' Alorg the

epicardial sr:rface we al-loued fiee of blood between the

;Ëir""q1*.t if"1

"oro.rty vessels

ard tl.e epicarrlial coronaaY vessels'

Initiation of

"o.,tt Jioí was not simultaneous for all

sarcollpres (fiq' 1) ' Ihe depolarisatj-on wave noveo from erdocar"dir-un to epicardiun ard frcm the apical region t o,"J trr" basal region. Ttre wave r€eded abort 40 ms to reactrl ;Ë;d"'ieft ventricul,ar r,rall.- rhe jnitial penteability of T" P":gf nedirun was derived tiÀ-aaga on tine constants of the coron;lrlr circufati-on

ard equal-s 2 nrn, iÀ:rr-r. Àe initial porosity of ttre rnedir:rn (= the

o e r . c e n t a q e o f i n t r a m y o c a r r c i a l s t r n c e o c c r : p i e d b y c o r o n a r l r b r o o d a t 0 k P a perfusion Pressure) is 6?'

The transm'ral variation of fijcer argle was derived from e4teriÍrental d a t a o f s t r e e t e r u ' ' a g u ' " ^ ( L 9 7 3 ) . T t r e S a r c o n e r e r r p d e l u s e d i n t } r e sjmulations is descrjlced elsevtrere (Htry$e' 1?891 ' Ttre trnssive constitutive betravior:r of tl.re *yo"utai-f tissue t""- á +r"=i-1íFár viscoelastic law wittt

an e4:onentiar eraJtic ,"+r"=", \,hictr *u= ritt a to ope,rimerrtal data of v-t ià,-:tqo, et a1 (1982), ard Yil et a1 (l-987) '

T h r e e c o l l r l t a t i o n s w e r e p e r f o r r r e d . E a c h c o r y l t a t i o n s t a r t e d w i t h a n

increase ot :ntracoiornry blóod volr.lre. The i'crease eq*alled 3% of the

myocardial vol.one io tfr" first .o*prrtiiio.t, 5,5eo in the.seconn corp,tation

ard gZ i5 the tfriá-conputation. Aiter ttre jlsrease of intracoronary blood

voltune, the i-ntraverrtricular pressure was ilcreased up to 2.5 kPa in all ttrree conputati".r=.-iit^fiy ttre tfrira corq>'tation yas. repeated, loadirq the ventricle onfy re to t kfá, ard tlren consecutive beats were initiated' The duration of ttre cardiac qgcles was 0'55 s'

A connercial post-processirq package I-DEA'S Í?F"tE -"1 Qgrnmics

Research Corporatioi| produced coÍor-coaeá plots of different local ortput

variables. To facilitate the ccnparison of conquted strails with

;re"tin*rt"l strajn data, lrcal trrree<inereiornl- Greerl strain tensors s/ere conputed witrr rereience- to tne erddiastolic state ard were irrter-preted in terrns of tneir

"ilerwalues (the principal

s!r?ins) a$ eiOervectgrl _$ne

principal axes of

-strai')

. 'ihe ttrree principal strains were rariked from snall-est (nrost negative) to larrEest (nrost positive) '

RESUT,TS

Diastolic stiffness. The il"lcrease in irrtracoronarY b1d voltme

il1a'rffif wall thicrsre.ss. lïre pressure-vorure curves

resultirg frorn tr," uo." conprtations (fis. 2)- sho^r that diastolic

stiffness i'creased wit1. i'creasirq coronai vascrrlar volute' At a givert

left ventricurar vor'trme left ventricular pressure increased frcm t'5 kf^to

2. O kpa wtren raisirq blood- voh.une frorn 9 to 14 rnl per 1o0 g

left verrtricle.

Ttre cardiac cvcle. Rjection fractions for the three caldiac qlcJ-es were 5g2,55? ard 5 Z. a1,,e ijr,r" co.,r=e of the radial ard axial displaceuent confonent of all nodes of the nrestr is strcx'in in fig' :'

1 2

o]uronuiuuo*,i, ,, ó ivtrvrlv)rvol/

Fig- 2 Sjmulated pressure-volure relations of the passive left ventricle

at different revels of intr:amyocardiar brood volure. The

ventricul-ar__!ía11 qqiffens at ilsreased intramyocaruial blood

vo}.:nre. voLV ard vLV are tl:e initial ard currerrt intracavitanz voltune of tlre left ventricr:.lar urodel respectively. nB i= t]-e ratió of coronarlz vascular volune over total rnyocardial voh.une.

Fig. 3 Simulated successive states of deforrnation of tàe left verrtrj-cie. A

conputer generated picture is shcnrn of the deforrnation of a

meridional section of the nrcdel (dotted line : refererce state,

contirruous lile defornred state). Fïom left to rigfrt:

"r-l diastole, besinnins ard erd of ejection ard begirrLirg of díastole.

the circunferential displacenent _{ccnqrcnent (not strov,m) strcrued rotation} of the aPo< relative to tlre base in countercloc]$^/ise direcÈion dqrirg the

ascendirg limbs of the ventricular pressure, _{wtrile ttre opposite napienea}

durirg ttre descerdins rinÈ. lhe model ccrputed ircreasirg ern-qlsloric

values of tfie three principal strains with insreasirq depth (fig. a).

Erd-systolic prilcipal strails _{equalled 0.45t -0.01 ard -0.24 at 2/3}

of tlre wa11 thic]<rress from the epicarditun ard.0.26, 0.00 ard -0.19 aE L/3

of the wa1l ttriclaness from the epicarrdiun. To arnryse the transrnrál

variation in the orientation of the prilc.itrui strain axes at tlre erd of

gjectior, we ccmputed the argle between tlre fi::st prilcipal strain axis

(i.e. _{the axis of rnaxi:na1 slrortenirq) ard ttre ci-rcrlnferential coordi5ate}

z o

í

d

;sive left ventricle

btood volune. The

cramyocardial blood

urerrt irrtr:acavitanr :1y. nB is ttre ratià a.]. vo}.ue. re left ventricie. A deforrnation of a = reference state, :ft to ri@t: erd Lnrdrg of diastole. wn) stroved rotation

lirection durirg the

,e opposite happened

'easirq _erd-systolic :ptà (fig. 4). )1 ard -0.24 at 2/3

00 ard -0.19 at 1-,23 yse the b:angnu::a]

axes at tfie erd of iJrcipal strail axis :erentia-]" coordi-nate

Fig. A rrànsrura1 distribution of erd-qlsto1ic prilcipal- str-aiJ]s as

-

predicted by tlre nrodel ard neasured bY l^taldnan et a1 (1985). In both nrodel

-ard

operi-nrent strajlls are given with respect to the

erd-diastotic

state-. EXPERIMENTAL DATA WALDIUAN ET AL (1988)

^.^ EXPERIMENTAL DATA WALDMAN ET AL(1985)

- _{FIBER DISTFilBUTION 1} - - _ FIBER DISTRIBUTION 2 --- _{FIBER DISTRIBUTION 3 A} a r r i _a

n

----]

^ [

^ r l

,^

-rffi

EPI

Fig.5 Ïhe arqle @ is the projection onto tlre epicardial.plane of the

argle betveen axis oi rnaxj:raf shortenirq ard tlre cirq.unfererrtial aiiection. The wariation of argle 4 with depth is plotted for three

fi-ber argle distributions, defiled jI table l-. Ttre two sets of

o,p"rimental data ( ^ a]ld a) are from two anjnals representirq tlre rarge observed in five dogs by t^tal&nan et a1' (1985) '

A I , A \ - : : - - - - A r

I

dj-restion. ïhe projection of this argle on tlre epicarrdial tarqent p1ale is refered to as the argle O ard is plotted as a funcÈion of á"ptrt at ttre irrterface _{of tlre elenrerrts 6t !8t 27 arrd. ttre elenrerrts 7, L3, Zf 1fig. S1 .} The angle O shifted from -37' epicardially to -6' erdocardiallv.

DISCÍISSION

This studlz shcn^p tlnt (1) a bipÈrasic nrcdel of left verrtricr:1ar

nrechanics is able to conptrte an ilcrease of diastolic stiffne.ss by cororra4r

blood voh:ne llcrease, provided tiat tlre non-linear rpture of tlre

constitutive betraviour of nyocardial tissue _{is tal<en into accor:nt a15 (2)}

the axisymnetric fhite elenrent nndel ccnputes qlstolic triaxial straiÉ

consisterrt with etperirnental data frqn ttre l-iter:ature.

Diastoric stiffness. when usirg quasi-rilear viscoelasticitlr,

stiffness ilsreases wit]" increasirg _{coronallr vascr:lar volture (fig. 3). Thi;}

result is consistent wittr eperimental _{data of olsen et ai_.} -(rgdr) _arxi

Vogel et al. (1982) (fiq. 6).

T X P E R I M E N T

tr r Fotasstum arresfed {Olsen et at.1981 ) N=10 d o [cntroi I ) í V o q e l e t a l 1 9 8 2 ) I ' t . 4 ^ a P c s i l s r h e m t r , -- / t i 5

*

\/:

R=,r^1,171Í,'

Fig. 6 The left panel strours ttre deperderry of erd-diastolic _reft

verrtricular pressure _{(LVp) on tlre coroÍraq/} perfusion pressure

(CPP) at a given cavitlz voh-une. Ttre right parrel _{shctrs ncdel}

results of left ventricular pressure es _{a funcAion of}

iatracoronary blood vo]ure at constant cavity volune. Stiffenilg

of the cardiac wall by insrease of corornry vascul_ar volune iÁ

more pronounced at higher caviQr _{voJ-r:nes, in both elperirnerrt}

(Ieft) ard nrodel (right).

on the basis of the e>peri:nental qr:antification _{of ttre relationstrin of}

coronarlz perfusion pressure ard intracoronary _{blood volure by Uorgenstern}

et a1. (1973), we ilfer tbat the ttree va}ues of irrtracoronanr blood volune chosen in fig. 3 correstrrcrd with coronarl/ pe.rfusion pressures-of _{6, l_l-, ard}

14 kPa. rhis i:rplies that the ncdel predicts ttnt at constant caviw

vohne, left ventricular pressure ilsreases _{frcrn 1.5 kpa to 2.0 kpa whó}

raisirgÍ perfusi-on pressure _{frorn 6 to 14 )<Pa. the sane strift of cor.on1nz}

2 A

( k P a J

ial targerrt plane is J-on of depttr at the 7 , L 3 t 2 8 ( f i g . 5 ) . ardia1ly.

)f lefb ventricular 'tiffness tryr corona4r

ear nature of ttre

i-nto account ard (2) Lic triaxial strains

lr viscoelasticiQr, ohnne (fig. 3). ïtris r et al. (l-981) ard "/"'4

-Ï--1

t

i

panel shcrus npdel

s a function of volune. Stifferdry wasorlar volune is in both operi:rent ttre relationstrip of 'ltrne by l,Íorrgenstern )ronarT blood voltune ;sures of 6, 11, ard

at constant cavity

cPa to 2.0 kPa lrihen : strift of cor.onarlz

perfusion pressure carlses an iscrease i1 lefb verrtrio.rl-ar pre-ssurê fr"on L'5 il-;.4-kl'.'in potassir-un arrested hearts accorràirg to tlre eryerlnertar !a! oi Ofro, et a1l (19Bl) , vÈri1e Vogel et a1. (1982) fird an i'crease of left veritricular pressure ii*, f b f.a kPa ard from 3 kPa to 3.8 kpa in intacb

;*r1" irdicatilq that ncdel re.sults are \,/ithin the l..r'ne of val-ues

Ë-="t a operinentally. The i'crease in !ta1} ttriclsress irduced by the

i[""="a i:rt

"""r"rrri blood volme

is consisterrt with tJle operirental

art^ of Morgenstern

"€ .f. Q973). One might be Fnetea

to attribute the

str:ift of the pressure-vol-r-ure cr:rrre to tfre láft in fig. 3 to ttre ilcrease in wa1l ttriclsress, on the basis of Iaplacers law. Hcn'rever, it is not eVident ifr.t f..ptu""ts law applies to biptnsic rnaterials. In a sirgle trfnse solid roai"riai, ilcrease

-in

thiclsresl of the strell irduces an increased

=titfrr."t of tlre she1l, because more naterial is available to take tp shell

forces. In the case of a solid-f1uid nLixture, addirtg nrcre fluid to t}re

miJ<b-rre, does not result i1 additiornl rnaterial- to take rp shell forces'

Tensj-le forces can only be borne bY tl"re solid. _{_ Ïtre} il1crease in

Gt-^"ororrty blood volune- resulted in increased 1Ë11 thiclg1ess, krub not in ir-r*""a stliffness wtren we replaced the quasi-li-near viscoelastic law by isotropic linear elasticiQr !\]trille rnaintailirg Darqfts law as a description

;;;--ái" redistribution of intracorornry blood. These resul1s shcnr that

rtittorirg of the diastolic left ventricle by coronaaT blood voltme

increase strould not be irrterpreted in te11ls of Laplacets law' but rather as the conrbhed resul-t of an oir"r.ff increase in strail in myocard-ial tissue utra tn" non-linear stress-strail relationstrip of ttre myocaldial tissue.

Ttre cardiac cvc1e. The npdel conputes ilcreasirq values of sf;:ajn with incr"asn'tq depth. I,'Iany eperjnrental data frcrn the fiterature point in the *n* ait".tion. fn tif. a-we have used the operirnental data of l^Ialdnan et

aI. (l-985) to a.ssêss the transrn:ral distribution of principal strain as

preaitUeA by the nrodel. Althor-lgh tlre direction of rnaxj:rnl stress alrrpst -coinaic"=

wlt]- the fiber direclion, the direction of naxirnal strorterirg

(i.e., ttre third principal strain axis) does not. Asross tlre wall tlp

à,e"i.a d-irection of rnaxjrnal strortenirg does not v-ary. nearly as muctr,as

t]lJ muscle fjlcer direction (fig. 5). ttris firdfuIg is consistent wittr

eryerinrental data of killzen et aI. (1984) ard l{aldnan et a1. (1985). Ihe

taiter jrnrestigators fourd that ttre abclre defiled argle 4 of rnaxirnal

strortenirg equaled -22+2L" jl ttte jlner tralf of tlre wa1l (65!9? of ttre wall thiclsress from tlre epicardirm) . A conunon feab.rre of the nrcdel prediction

ard ïfatdnants data is tfre progressive rotation of ttre principal axis of

shortenirq tcnrards t].e circr.unferential direstion with increasirq depth. @NCIUSION

An axisrTnu'netric two-phrase finite elenrerrt nrodel is used to simrfate myocardial defornration a"iitq the card.iac rycle. ColÍqrÊed tr:ansrnrral straiJl dLstrj5ution is in agreemerrt witJ: e4rcri$enta1 data frcm t-l-e literature.

The ncdel irdicates ttat tle increase in diastolic stiffness by increase of

coronaay vascul-ar voh-une stroul-d be interpreted as a cclnbifred effect of an

overall- hsreased strail in the rnyocardial fi-bers ard tlre non-lirear

stress-strain relationships of ttre myocardial tissue.

Qrestion from t].e audience: rsnrt axislnmnetrlr a rattler rough appr:oxi:nation

@ of t]le left verrtricre?

tfu1ghe: We ctrose for an axiqarnretric rncdel il order to reduce conpr-rtation ti-t. fn:-s approxi:nation is irdeed a limitation of this nrodel. In decernber' our grol4) wili present a three-dinrensioral nrcdel of the left ventricle at the Wjnter Annual Meetirg of the

A,SI4E-Ë*

Q'rcstion _{from ttre audience: Hcnr do you analyse torsional- defornation} with an axislarmetric nrodel?

Hryghe: Ïhe finite _{el-enrent code &rat Íie usê has been r^rritten speciarly for}

finite deforrnation incl'dirg t"Ài"". _{bccept a rad'al ard}

axiaf

dirylacenrent, ttlere _{is al-so a-cirqmterentiar displacenent,}

\dtlictr is also axiqprunetric.

oddou: There is an essentiar d.ifference--b.pg, _{the poro's nrediun approach}

of your gïoup arxi our gn:oqp, b""""""

{1uid in y.rr -*}er

is ttre coronary blood, raÈrire in o'r arnlysi"'it _{i" trr. itt .titíJri"à}

ttuyghe: This is

"ï{^.1*!: fr: y". ÊrenCIrena have also verlz different tinre constants _{ard ttris is raÈry we feel i€ ir}

""""plJi"-i"-".gr".t intêr:stitial fluid _{flcni il our analysis.}

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Huy$re, J.M., Oomens, C.W., Van Caffi, _{D.H. arxtHeetlraar, R.M. , IgBg, Lo',r}

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í