CEREBRAL CIRCULATION DURING ACCELERATION STRESS

Srdjan Cirovic

d thesis siibmitted in conformity with t hc requireriicnts

for the clegree of Doctor of Philosophy

Graduate Department of Aerospacc Science and Engineering

University of Toronto

@ Copyright by Srdjan Cirovic 2001

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Abstract

CEREBRiL CIRCCLATIOS Dt'RISG ACCELER,iTIOS STRESS

Doctor of Philosophy. 2001

Srdjan Cirovic

Graduate Department of ;\erospace Science and Engineering

University of Toronto

-4 niatheniatical niodel of the cerebro~ascular systeni has bwri clrvelopcd to rsaniine

the influence of acceleration ori ccrebral circulation. The objective is tii ciist ingiiish

the niain factors that lirnit ccrebral blood How in pilots siibjected to acc~leratioris

ahich esccecl the gravitational acreleration of the earth ( G D 1).

The cerrbrovascular system was approsimatcd ty ari opcri-loop n~twork of rlast ic

tubes aricl the Aow in blood \wscls nas nioclcle~l accorcling to a one-tlirricrisiotial tlicory

of flow in coliapsible tubes. Since liriear arialysis s h o ~ v d tliat the spccci of piilsc

propagation iri the intracranial vcsseis shotild not he niotfifid by t lie skiill const raint . the samc govcrning eqiiations wcre iiscd for the int racranial vcsscls as for t tic rest of

the rietwork. The steacly and pulsatile coniponents of the cerchrospinal Hiiid prrswire

w r e detcrmined froni t he coridit ion t liat t lie cranial volunie nirist l x consc rvd M e r

the qualitat ive aspects of the moclcl rcsiilts w r e verified espcrimcntally. t hc operi-loop

geornetry \vas incorporatecl into a global niathematical mociel of the cardiovnscular

systern.

Both the niat hematical rnoclels and the esperirnent show t hat ccrebral blood flow

dimiriislies for Gz>l due to an increiise in the rcsistance of the large veins in the

neck. which collapse as soon ,as the venoiis pressure becornes negative. In contrast.

the conservation of the cranial volume requires that the cerebrospinal and venoiis

pressure always be approsirnately the same. and the vessels containecl in the crariial

cavity do not collapse. Positive pressure breat hing provicies protection by elevating

blood arterial and venous pressures at the heart. t hus prewnting the venous collapse

and maintaining the normal cerebral vascular resistance.

Acknowledgment s

1 would like to express my gratitude to my supervisors Dr. Colin Walsh and Dr. Philip

Sullivan. \Vit hout their expert guidance and support the successful complet ion of this

work would not have been possible.

This thesis is a part of a larger project which has been initiated and enthusiast ically

guided by Slr. Bill Fraser at Defence and Civil Institute of Environmerital hleclicine.

His insight into the subject of G stress. based on years of experience in the acceleration

physiologv research. was crucial for setting this stiidy on the right track.

Contents

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . 1.1 P hyiological Barkgroiincl

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The hcart

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Blood

. . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Ttic vasciilar systrni

. . . . . . . . . . . . 1.14 The cont rol of the cartliot'i1scular systerri

. . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Ccrcbriil ctirr.tilat ion

1.1.6 The cffcct of acceleration on the ciirdiovasriilu- systcrri . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Carcliovascrilar rtioclcls

. . . . . . . . . . . 12.1 '\lo(icls of pulse propagation in t h artcries

. . . . . . . . . . . . . . . . . . . 12.2 Noclels of vascular rietworks

. . . . . . . . . . . . . . . . . . . 1 .'2 .3 Ccrebral circii t at ion niodcls

. . . . . . . . . . . . . . 1.3 Thcsis Ot~jectives . Scopc . and Organization

. . . . . . . . . . . . . . . . . . . 1.3.1 Thcsis objectives aricl scope

. . . . . . . . . . . . . . . . . . . . 1.32 Organization of the t liesis

2 The Speed of Pulse Propagation in Constrained Vessels

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 llethods

. . . . . . . . . . . . . . . . . . . 2.2.1 Physiological approsimat ion

. . . . . . . . . . . . 2.3 Go~erning Equations and Bounday Conditions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . '2.3.1 Solution

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Case 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Case2 45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Disctission 30

Cerebral Circulation and CSF Pressure 52

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction 52

.3.2 Methocls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .: '2

. . . . . . . . . . . . . . . . . . . 3.2.1 P hysiological approsiniat ion 5 2 - - . . . . . . . . . . . . . . . . . . . . .3.2.'2 \Iathernatical forniulation 9 .:,

. . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Boiinciary conclit ions 58

. . . . . . . . . . . . . . . . . . . 3.2.4 Solution for the steady Row 59

. . . . . . . . . . . . . . . . . . 3.23 Solution for the ptilsatilc How 62

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Rcsitlts 65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 .1 Steacly Row 6.5

3 . 3 Pulscitilc Aow . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Discussion 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 S tead- flow 73 -- . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Piilsatile How i i

4 Cerebral Autoregdation. Venous Properties. and Central Blood

Pressures 79

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 h t h o d s 82

4.2 1 P hysiological approsimat ion alid m i t tiemat ical forrriiilatioti . . 82

. . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Solution procedure 85

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.13 Simiilations 85

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results 8-3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Discussion 93

5 The Mechanical Model of Cerebral Circulation 96

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction 96

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .J.2 Uethods 9 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Mode1 97

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . 2 Esperiments 99

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Results 103

. . . . . . . . . . . . . . . . . . . . . 3 1 Preliminary Esperiments 103

. . . . . . . . . . . . . . . . . . . . . . . . 3 . 3 2 Nain Esperirnents 106

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 .-l Discussion 112

6 The Simulation of Cardiovascular Performance 117

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 htrociuction 117

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 llcthods 111

. . . . . . . . . . . . . . . . . . . 6.2.1 Physiological approsimat ion 111

. . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 G-suit anci PPB 120

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 W v e rnociel 120

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Heart niocieI 120

. . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Initial coridit ions 1 2

. . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Boiiiiclnry conditions 122

. . . . . . . . . . . . . . . . . 6 . 7 Split cocfficicnt niatris met hod 123

. . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 liass coriscrvation 124

. . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Rcsiilts and Disc.iission 125

7 Conclusions and Recommendations 133

7.1 The liechmical Dctcrminants of Cerebral

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circiilation 1.33

. . . . . . . . . . 7.1.1 Cerebral vasciilar resistance diiring Gz stress 133 - i . 1.2 The significance of negative wnoiis and CSF pressures for niain-

taining consciousness iirickr +Gz . . . . . . . . . . . . . . . . 134

. . . . . . . . . . . . . . . . . . . . . . 7-12 Central blood pressures 135

. . . . . . . . . . . . . . . . . . . . . 7.2 Cerebral Circulation and GLOC 136

. . . . . . . . . . . . . . . . . . . 7.3 Cranial Ycssels ancl CSF Dynamics 136

. . . . . . . . . . . . 7.4 Contributions and Suggestions for Futtire Kork 137

. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.41 Contributions 131 - . . . . . . . . . . . . . . . . . . . 4 . 2 Suggestionsfor futurework 135

A Numerical Methods for Closed-Loop Cardiovascular Simulation 140

. . . . . . . . . . . . . . . . . . . . . . . . ... 1 Methoci of Characteristics 1-40

1. 1.1 Characteristic variables . . . . . . . . . . . . . . . . . . . . . . 140

4 . 1.2 Sumerical solution . . . . . . . . . . . . . . . . . . . . . . . . 142

.A.? Split Coefficient Matris Nethocl . . . . . . . . . . . . . . . . . . . . . 14%

4 . 1 Split coefficient matris . . . . . . . . . . . . . . . . . . . . . . 142

A.2.2 Discretc equations . . . . . . . . . . . . . . . . . . . . . . . . 113

2 . 3 1-01 urne conservation . . . . . . . . . . . . . . . . . . . . . . . 144

A.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

.4.3.1 Scwton's niet hod . . . . . . . . . . . . . . . . . . . . . . . . . 146

.A. 3.2 Two segrnerit jurictions . . . . . . . . . . . . . . . . . . . . . . 116

vii

List of Tables

Some pliysical properties of the circtilation . . . . . . . . . . . . . . .

h i t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Parameters iiscd in the study of pulse propagutiori in the craniiirn. . .

Mean flow speed and Aow rates for the mode1 of piilse propagation iri

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t tic crrîniiini.

Parameter valiics iised in t h nlodcl of ccrebral circiilation. . . . . . .

Foiirior coriiporirrits of ttie ilipiit artcriid piilse. . . . . . . . . . . . . .

Paranictcr valiics uscd in ttie nioclel of ccrrbral circulation with sirri-

plificd gometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The arialogy bet\veen the pressures in t tic niecliariical nioclel and the

pressiirrs in the niathemetical niodel of cerebral circulation. . . . . .

\'ascular network used i ~ i the closecl-loop carciiovascular simulation. .

. . . . . . . . . . . . . . . . . . Pararrieters tiseci in the heart motiel. 1'11

viii

6.1 Closed loop: vascular network for response to mechanicd effects of Gz . 119

. . . . . . . . . . . . . . . . . 6.2 Ventricular elastance: rectifieci sinusoid 121

. . . . . . . . . . . . . . . . . . . . . . . . . 6.3 llethod of characteristics 122

. . . . . . . . . . . . . . . . . . . . . 6.4 Split coefficient matris niethoci 1'73

. . . . . . . . . . . . . . . . 6.5 Open loop: ccrebral Aow with wrying Gz 126

. . . . . . . . . . 6.6 Open loop: cerebral Flow with rnrying PPB pressure 126

. . . . . . . . . . . . . . 6 . 7 Closed loop: Gz and PPB pressure schediiles 127

. . . . . . . . . . . . . . . . . . . 6.8 Closecl loop: central ürterial pressure 128

. . . . . . . . . . . . . . . . . . . 6.9 Closecl loop: central rerious pressure 1'29

. . . . . . . . . . . . . . . . . . . . . . . . 6.10 Closed loop: carcliac output 130

. . . . . . . . . . . . . . . . . . . . . 6.1 1 Closecl loop: cerehral t~loocl flow 131

6.1'2 Closed loop: jiigiilar arra profiles with arici withoiit lifc support . . . . 132

. . . . . . . . . . . . . . . . . . . . . .LI Cc11 iised For volunic correction 14;

Nomenclature

Symbols

-4 Cross-sectional rirra ( pp. 17. cq. ( 1.9))

A The .lacohian matris (pp. 124. rq. ( 6 . 3 ) )

-40 Cross-sc>ctional are;\ iit zero traiisiriiiral prrssiirf> (1111. LS. NI. (1.1 1 ) )

(1 Prrturhation of (:ross-scrtiorial arra (pp. X)

BI. B" Constants iri t tic soliit iori o f liricarizcd Savitlr-S tokrs quat iotis

(pp. 33. eq. (2.9))

Ck Charartcristic paths (pp. 123)

c \ h v e spcecl (pp. 1-4)

C o \Vwe specd clerivetl froni tube law (inviscici) (pp. 18. cq. (1.17))

C o 1 \lave speed for Young's niode (inviscid. unconstrained) (pp. 1.3. eq. ( 1.3) )

(-'oz \\are speed for Lamb's niode (inviscicl. iinconstrainecl) (pp. 1.3. eq. ( 1.3))

C I \\ave speetl for Young's mode (pp. 40. ecl. (2.26))

cz \Vwe speetl for Lamb's mode (pp. 40. cc!. (2.26))

Dlt Dl1 Constants in the solution of Iinearized Xavier-S tokes equations

(pp. 33. eq. (2.9))

d Length scale (pp. 5. eq. (1.1))

Yoiing's modiilus of elasticity (pp. 15. eq. (1.3))

The Bus vector in 1-D formulation (pp. 18. eq. (1.12))

Stiffness constant for linearizetl tube law (pp. 56)

Elastance (pp. 20. eq. (1.21))

Heart clastance (pp. 120. eq. (6.1))

SIasiiiiuni heart elastance (pp. 120. cq. (6.2))

SLnirniini hmrt elastance (pp. 120. cq. (6.2))

Ttie source vcctor in 1-D formulation (pp. 18. eq. (1.1'2))

\\bniersley's fiiriction (pp. 16. rq. (1.8))

Sliciir riio~liiltis (pp

Radial actbelcrat ion

.lccelcration of the earth (pp. 11)

Elevatioii (pp. 39. eq. (3.16))

Heart to head distance (pp. -33. Fig. 3.1). arid the length

of the arrns in the niechanical mode1 ( pp. 98. Fig. 5 . 1 )

\Val1 thickness to rscliiis ratio of a tube (pp. 15. cq. (1 .3))

Distance from the zero Iewl in the esperinicntal mode1 (pp. 98. Fig. Z. 1)

J

Bessel function of the first kind. of order n (pp. 16)

Stiffness constant for collapsed tube in patctied tube law (pp. '20. ccl. (1.22))

Stiffness constant in the tube law (pp. 18. eq. (1.16))

\éssel length (pp. 34. Table 3.1)

The matris of eigenwctors in 1-D forinulation (pp. 140. cc1 (-4.2))

Lcngth of a vesse1 segrrient (pp. 39)

Airiplitude of the asial vcssel niotioii (pp. 3.5. cq. (2.13))

.~iriplittidc of the r;iclial vcssel rilotion (pp. 3Z. rq. (2.1:3))

\\'orrierslcy nuniber (pp. 1.5. ccb ( 1.5))

Surtibcr of vrssels in paralld in an ~qiiiviilrnt vesscl (pp. 18. cq. ( 1.1 1))

Pressure in 1-D foriniilation (pp. 17. NI. (1.10))

Fiinction of relative area in the tube law (pp. 18. eq. (1.16))

Pressures in the mechaniral mode1 (pp. 98. Fig. 3.1)

Artcrial pressiire at the level of tlie head (pp. 71)

Central arterial pressure (pp. 58)

Cerebrospinal Auid pressure (pp. S)

Pressure elevation at the heart (pp. 8;). and pressure elwntion

a t the base of the mechanical mode1 (pp. 101. Tablc -3.1)

\%nous pressure at the lerel of the head (pp. 71)

Central venous pressure (pp. 59)

Critical transrnural pressure (pp. 57. eq. (3.11)).

and pressure in the mechanical mode1 of the cranium (pp. 101. Table 5.1)

Esternal pressure in 1-D formulation (pp. 18)

Perfusion pressure (pp. 71. eq. (3.49)). and the clriving pressiire

in the mechanical model. (pp. 101. Table 5.1)

CerebraI perfusion pressiire (pp. 82)

Pressure thop across the resistor (pp. 104. Table .5.4 Trmsmural pressure (pp. 18)

Left ïentricular pressure (pp. 6.1. eq. (6.1))

Pressure in linearized Xavier-Stokes ecpations (pp. 2.1. rq. (2.1)).

and pressure perturbation in 1-D Foritiiilation (pp. 55)

Esternal pressure perturbation in 1-D formulation (pp. 5 5 )

Transniiiral pressure pertiirhation in 1-D formulation (pp. 3.35. eq. (3.35))

Flow rate (pp. 4. and rc r~hra l blood flow (pp. 70. cq. (3.49)) -

Flow predicteci by a siphon (pp. i 4. eq. (3.33))

Flow prcdirtetl t)y a watrrfall (pp. 75. cc!. (3.34))

Flow pertiirbation in 1-D Forrriiilatiori (pp. GR. rq. (3.35))

Radius of a vesse1 (pp. 15. eq. ( 1 . ; 3 ) )

The risroiis friction terni (pp. 18. eq. (1.11))

Mitral resistance (pp. 120. Table 6 .2 )

Art erial resist ance (pp. 70. eq. (3.45)). and the resistaricc

of t he ascendirig arni (pp. 109. eq. ( 3 . 5 ) )

Cranial vasciilar resistance (pp. 70. q. (3.46)). anci the resistaricc

of the niodel of the craniuni (pp. 109. eq. (.5.6))

Reynolds number (pp. 3. eq. (1.1))

Resistance of a vesse1 segment (pp. 59. eq. (3.21))

Resistance of the mechanical resistor (pp. 100. Table -5.4)

Total cerebral vascular resistance (pp. 70. eq. (3.48)). and the total

resistance of the mechanical rnodel (pp. 109. eq. (5.8))

Yenous resistance (pp. 70. eq. (3.47)): and the resistance

of the descending arm (pp. 109. ecb ( 5 . 7 ) )

Radial coordinate (pp. 28. Fig. '2.1)

Speed iiides (pp. 60. eq. (3.23))

Cardiac periotl (pp. 120. t'cl. (6 .2) )

The chiration of systole (pp. 120. cq. (6.2))

Tirne (pp. 18. ecl. (1.9))

Fliiicl velocity in 1-D fornidation (pp. 1.9. eq. (1.9))

Fliiid wlority pcrt iir\xit ion i r i 1-D fortiiiilat ion ( pp. G.i)

Radial displarenient in the ~qiiiitioris o f rriotiori of the solid (pp. 3 1. q. ( 2 . 4 ) )

Axial disp1;icfmriit in the rqii;itioris of i~iotiori of t hr solid (pp. 31. cc!. (2 .4 ) )

The rnatris of primitive variables in 1-D forniulation (pp. 1.12. ~ 1 . ( 1.12))

\kntriciilar volunic at zero pressure (pp. 120. rq. ( 6 1 ) )

Cranial artcrial bloocl volutrie (pp. 84)

Cranial venous blootl volurrie (pp. 84)

Cranial blood voliime (pp. -33)

Radial fluid velocity in linearized Xavier-Stokes ecliiatioris (pp. 30. eq. ('2.2))

Asial Ariid velocity in lineariztvl Xavier-Stokes ecliiations (pp. 30. rq. ( 2 . 2 ) )

Characteristic variables ( pp. 123. eq. (6.4))

Son-dimensional paraniet er in dispersion equat ion for the

pulse speed (pp. 16. eq. (1.7))

Asial coordinate (pp. 17. eq. (1.9))

Bessel function of the second kintl. of ortler n (pp. 33)

Radial coordinate normalized with respect to tube

radius (pp. '28. Fig. 3.1)

Relative radius of the rigid ttihe in the mode1 of

pulse propagation (pp. 28. Fig. '2.1)

Relative inner radiiis of the viscoelastic solid in the nioctet of

pulse propagation (pp. '28. Fig. 2.1)

Relative miter radius of the viscoclastic solitl in the nioclel of

piilse propagation (pp. 28. Fig. 2.1)

;\sis alignecl with the spirie (pp. 10. Fig. 1 .3)

RcIativc cross-sectiorial arca (pp. 18)

Critical rchtive cross-srctional arca (pp. 57. eq. (3.13))

Sori-dirncrisional paraniet cr in the soliit ion of 1ine;irizrd

Xavier-Stokes cquations (pp. 33)

Son-climensional parameter in the soliit ion of solid eqiiations ( pp. 34).

and tilt-angle in the espcriniental rriodel (pp. 98. Fig. Z1)

The ratio of fliiid to tube inertia [pp. 15. eq. (1.3))

Poissori's coefficient (pp. 15. eq. (1.3))

Radial tube clisplacenieiit (pp. 30. eq. ('2.1))

Son-dimensional parame ters in t lie model of

pulse propagation (pp. 36. eq. (2.1s))

Son-diniensional paramet ers in the model of

piilse propagation (pp. 36. eq. (2-17))

The angle betn-een a vesse1 and z asis (pp. 17. eq. (1.10))

Son-dimensional parameters in the mode1 of

pulse propagation (pp. 33)

The matris of eigen\alues in 1-D formulation (pp. 140. eq. (-4.2))

Wave length (pp. 33)

Dynamic viscosity (pp. 18. eq. (1.11))

Iiiriematic viscosity (pp. 5 . eq. (1.1))

;\sial tube displacenient (pp. 30. eq. (2.1))

Finite pressure in the ecp t ions of niot ion of the solid (pp. 31. eq. (2.4))

Density ;incl Aiiid density (pp. 17. eq. ( 1.10))

Density of the vesse1 w l l (pp. 1-3. eq. (1.3))

Density of the viscoelastic solicl in the niodel of

piilsc propagation (pp. 31. eq. ('2.4))

Radial stress (pp. 30. cq. (2.1))

Xsial stress (pp. 30. cq. (2.1))

Radial rsternal force per unit itrea (pp. 30. t q (2.1))

Axial csternal force prr unit arm (pp. 30. rq. (2.1))

Son-diniensional paranieter in the liiiearizcd 1-D ptilsc

propagation niodrl (pp. 63. rq. (3 .34))

J: Circiilar frequenry (pp. 13. cq. (1.2))

Subscripts

Forward propagat ing

Backward propagating

\Aue of a variable at the inlet of a tube segnirnt

The difference of values of a variable at the inlet and

Yalue of a variable at the outlet of a tube segrnerit

the outlet of a tube segment

Arterial. at the level of the head (craniurn)

Arterial. a t the level of the heart

At the level of the head (cranium)

At the 1eveL of the heart

Steady cornponent in linearized 1-D equat ions

Venous. at the level of the heatl (craniiim)

\énous. at the level of the heart

Fliiicl

At the inlct of the craniurn

Left of the jiinction

Somirial (riorrnal) value

At the outlet of the craniiini

Right of the jiinction

Solid

Superscripts - ( 1 r\mplitiide

( Y - ) EricicIttit wave

( ) ( - ) ~ c ~ c c t NI wavc

( Y 1 iciisiircci

( 1" Associatcd with the fliiicl in t hc annuliis. in thc rnorlt~l of

pulse propagation

( 1'' n-th tinic! stcp

( Y .\ssociatetL with the fiiiid in the tube. in thc modd of

pulse propagation

Acronyms

AGSM Anti-G straining maneuvers (pp. 13)

CSF Cerebrospirial fluid (pp. 8)

GLOC Cr-induced loss of consciousness (pp. 1)

PPB Posit ire pressure breat hing (pp. 13)

Chapter 1

Introduction

.-\rcelcratioris gen~ratecl iii high perforitiaiice aircraft riiancuvers niau scrioiisly clisriipt

vital physiological fiinctioris of the pilot ancl niay lead to iriipiiirc~cl vision itri<l loss of

coiisriotisness (GLOC). -4lttioiigh GLOC is rewrsible. whm it occiirs diiririg corribat

iiianciiwrs it poses gravr risks to t tic i i ircrw arid airclraft (Biiriks rt al. l<3!X). ;\part

froni a lirriited ~iiinibcr of stiidies whirti siigg~st tliat GLOC is cii~isccl II>- riiwlianiral

stresses iri the brain (Bctiiii~oiirt 1992). thr riiajority of rescarctiers in t t i v ficM o f

accrlcratiori physiology rrgard l o s of coriscioiisriess as a phcnonicnori rchtrcl to in-

ntlrcliiate hloocl siipply to the brain (l\oocl 1987: \\érrtian WH). .\ccrlrration strrss

causes clrastic changes in hlood pressures and carciiac pcrforriiarice cliir to a rtiagni-

fied hylrostntic prcssiirc gradient a i t hiri the vasciilatiirr. and t o t hc stiniiilat ion of

the cardiovascular refleses: the changes in both arterinl ancl venoiis pressures in thc

vessels above the heart may result in inadequatc rcrebral perfiisiori. Th<. p o t rct ive

measlires clesigned to prevent impaired vision ancl GLOC fociis on couritering the

cfkct of acceleration on the cartliovascular systeni. Currently. t lie eniptiasis is on

developing automatically rontrolled life support systems t h c m provitlc protection

cluring complex changes in the acceleration vcctor.

Slathematical models of the carcliovascular system are a iiseful tool in the design

and testing of advanced aircrew life support systems. since the! provide an insight

in the mechanical and physiological factors responsible For GLOC. and are capable

of simulating quantitative changes in major cardiovascular parameters caused by the

accelerat ion stress.

1.1 P hysiological Background

The cardiovascular system transports the materials neetLed for ce11 hinct ion and the

resulting waste products arouncl the bocly. Blood is the transport medium: the heart

is the piimp: ancl the bloocl vessels are the conduits. The circulation is tlivitled into the

systemic circulation and the pulmonary circulation. The systeniiç circulation supplies

al1 the tissues of t lie body wit h blood and is also called the peripheral circulation. The

pulnionary circulation is responsihle for tleliveririg hlood to the liings. where carboii

diosiclc is releasecl and cjsygcri taken up. ancl retiirriirig the osygcnated blood to the

hcart.

Figure 1.1: The cardiovascular system.

1.1.1 The heart

The henrt can be dirided into a right heart. that puriips the blooct through the

pulmonary blood vessels. and a left heart that piinips the blood throrigh the systeiiiic

bloocl vessels. Bcth the left and the right tieart are composed of two clianibers:

an atrium and a ventricle. The primary function of the atrium is to collert the

bloocl ancl move it into the ventricle. ahich supplies t lie niain force that propels the

blood throiigh the circulation. The patli of bloocl iri the carclioi-asciilar systcni is the

following: left reritriclc. systemic circulation. rigtit atriiini. right ventricle. piiltrionary

circiilatiori. left atriurri. L l r e s prevent back How Iwttvccn atria and ventricl~s. aiid

veut riclcs and artcries.

The tieart contracts and re lues cyclically clrirc.cn acitoriiati(d1y by clcctric sigtials

frorii spccializcd rscitatory and con(liictire cclls i n the Iieart riiuscle. The period of

rrlasation ciilring uliich thc hrart fills n i t h blood is c;illcd diastole and the pcriod of

coiitractiori diiririg whicti the bloocl is cspclled into circulation is cailed systole. For

ii Iiralthy aclult. lcft systolic prcssiirc is typically 123 mrrifig iri the wntriclr m c l the

aort ic root. while left cliastolic pressure is approsirnately 3 rrim, Hg iri t h wiit riclc

and 75 rnriiHg in tlic aortic root. The corr~sporidirig \nliirs i r i the right Iiriirt ;ire:

a systolic pressure of 2.5 r r m H g in thc teritricle ancl niairi piilnioriary artcry. and

cliastolic pressures of 3 rrirrtHg in thc ventricle and 10 mrr2Hg in the piilnionary

artery (SIilnor 19T4a. pp. 842). The pressure in the ri& atrium is approsirtiately

zero to several millinietcrs of nicrcury.

Cardiac output is the quantity of bloocl piinipecl into tlic spteniic circulation

in one minute. Cacciiac output varies greütly with the lcvel of actirity of tlic body.

It is cletermined by the niiniber of ventricular contractioris per minute (heart rate)

and the volunie of blood espelled per heart beat (stroke volume). Venous return

is the quanti- of blood flowing froni the mins into the right atrium in one niinutc.

The cardiac output depends on the renom rettirn: according to t lie Frank-Starling

mechanism of the heart. the same amount of blood stipplied to the heart by the veins

ivill be automatically espelled into the circulation by the heart. providing that the

limit for the cardiac output has not been reached (Guyton and Hall 1996. pp. 239

- 240). Therefore. in normal circurnstances cardiac output and vcnous return are

equal.

- - - - - -

Table 1.1 : Sonie pliysical propert ies of t lie circiilat ion at differcrit lomt ions in the vasciiiar

t rw.

1.1.2 Blood

Blood is a suspension in plasma of platelets (thronibocytcs). d i i t c cclls (lctikocytes)

arid r d cells (~rytlirocytcs) (Corilcy 1974). Tlic r d cells arp 1iy far the niost nu-

rneroiis of t lie tliree spccies ancl dorriiriatc t lie tricclianicai p r o p r t ics of idiolc blood.

The prrcentage of rcd cc11 to ~vtiole blood voltmie is callrd the heiri:itocrit. This is

approsiniately 40% for the entire blood. hiit varies strongly in the vasciilar heds of

different organs. Blootl plasnia is a soliitiori of large rrioleciiles that ciin be rrg;irrlcti

as a Scwtoniari Huid of viscosity 0.00 12-0.0016 kglrris at body tmipcrat tire 37°C.

Howwr . the viscosity of ~viiolc blood iricreascs norilinrarly wi t ki t lie limiatocri t. ancl

the measiirctl viscosity is slicar rate clepentient at shcar rate of less tliari 100 s - ' .

Severtheless. idiolc blood is coniinonly assumcd to be a honiogcrieoiis Xewtonian

fluid. Peclley (1980) gives the tlensity as 1.05 x 10%~glrr?. and viscosity 0.004 kg/rn.s.

nliile Nilnor (1974b) States that the viscosity of tdiole blood is iri the range 0.003 to

0.004 kglrns. Both authors assurne a terriperature of 37 OC and normal rdiies of the

Arteriole

40

0.003

lo-'

10-2

Capillaries

'2500

0.0006

10-~

10-3

hematocri t .

Artcries

'20

0.2

IO-'

1 oO

Totalarea [cm']

Dianieter [cm]

Ortler of magn.

of \Plocity [rn/.s]

Ortler of magn.

of R,

1.1.3 The vascular system

Iéiiiiles

2.50

0.004

IO-:{

10-2

Aorta

2

1.5

10-1

10:'

The systemic circulation begins with the aorta which receives blood from the left

ventricle and ends wit h the superior and inferior venae cane which return blood to

the right atrium. T h e blood vessels form a network that is predorninantly divergent

\éins

80

0.5

10-2

loO

\énae

C a n e

1.0

I O - '

102

on the arterial side ancl convergent on the venoiis side. The approsiniate total cross-

sectional area of the vasciilar tree is: 2.5 cm% the aortic root. '20 cm2 in the sniall

arteries. 40 m i 2 in the arterioles. '2500 crn2 in the capillaries. 950 m i 2 in the ventiles.

80 cm' in the veins. and 8 crn' in the venae c a n e (Guyton and Hall 1996. pp. 162).

Wiereas the total cross-sectional area of the arterial system and the nunit~cr of vessels

increase E'rorn aorta to the capillarics. the blood velocity and the sizc of iiidiviclual

vessels ckcrease. The iriean blood pressiire at the root of the aorta (heginning of t lie

systcmic circiilatiori) ccliials apprositnateiy 100 mrrlHg. This pressure is also known

as the central arterial pressure. At the capillary levd the pressure drops to 33

- 10 n m H g . wlicrcas it is approsimately zero i t i the largc reins and vrnae cavae.

The pressure in the right atriiini (end of the systerliic circulation) is also known as

the central venous pressure. The pressiirp drop in the vasciilar tree is cliic to

viscoirs forces that offer resistancc to blood flow. kriowi as vascular resistance. Tri

gerieral. vasciilar rcsist;incc is ;i fiiriction of the vrsscl diariictcr (cross-swtiorial arca)

lcrigt l i . T h Rcynolcls niini her

mcasiire of the ratio of inertial to viscuiis forces i r i a fliiid of kirierriatic viscosity

ith characteristic flow speed L' and length scale d . Re is of the ordrtr of 10'' iti th^

ta ancl IO-' or lcss in arteriolcs ancl capillarks. Tlirr~fore. in norriial circiinistariccs

the effect of visrosity is strongest in thc sniall vesscls \vliere most of the prcssiirt? cIrop

occurs.

The fiinction of the arteries is to transport blood iindcr high prcssiirc to thc

tissues. The' are characterizcd by strong tc'iills aricl provide little resistarice to Aow.

For the systernic arteries the ratio of \val\ tliickness to lunien radius varies frotn 0.1 iri

the aorta to 0.4 in the arterioles. The arteriolcs are the 1 s t branches of the arterial

system that release blood to the capillaries. The arteriolar wall is rich with srnooth

muscle alloiving an active change in the Lunien area. It is mainly a t the arteriolar lcvel

where the vascular resistance is rgulated. The function of the capilla. is to eschange

nu trients and ot her substances betaeen the blood and the surrounding tissue t lirough

diffusion. Therefore. the capillary walls are t hin ancl permeable. The venules collect

blood from the capillaries and converge into veins. The portion of the vascular system

beginning with arteriole and ending witli venules is called micro circulation. The

veins return blood to the heart and also serve as blood reservoirs. In contrast to

arteries. the venous thickness to radius ratio is only 0.05 in the vendes and 0.01 in

the venae cavae. Their thiii aalls allow the veins to eiisily change lumen area in

response to pressure changes. and store or espel volurries of blood. It is estiriiated

that the veins holtl approximately 80% of the systernic blootl (.\Iilrior 19T4a. p. 8-44

ancl (Pecllcy ~980. p. 2 5 ) . l lost large veins are collapsecl ininicdiately M o r e joirii~ig

renae cavae. either because of the pressure esertetl by surroiincling organs. or duc to

slightly negatire hlood pressure as in the case of the veins of the nerk (Guyton and

Hall 1996. pp. 177). Most veins are siipplied witli valves that prcvent back How.

1.1.4 The control of the cardiovascular system

In nianaging its response to the c h i g i n g estcrrial envirotinient and to interrial nictabolic

rieecls. the body reqiiires a control system tliat can rrgulate c*eiitral artcrial aritl w-

nous pressures. total card ix out put aiicl t tir clistribiitiori of t tiat out put to tlic varioiis

orgaris. This contra1 systeni niiist bc capable of nianaging roriirriori cvcnts si~ch as

cligestion. postiiral changc and esercise. and abnormal evcrits siicli as shock. limi-

orrhngc arici extrernes of terriperature. In principle the coritrol of the cardiovasciilar

performance involves the control of the carcliac output. rascular resistancr. mici \p

nous capacitance. This is acliievcd by rontrolling tieart rate ancl by actively changing

the Iiirnen of vessels. The tirne scales for cardiovasciilar stiniuli var. widely. Corrc-

spondingly. the body has a range of responses. iiicltiding slow chemically iriedinted

controls and ver- fast controls operating through the rierroiis systeni.

The nervoiis regulation of the circulation is mainly controlled by the vasoniotor

center ttirough the action of the autonoriiic nervous systeni. In partictilar. the sym-

pathetic nerve fibers innervate al1 blood vessels eacept capillaries. as \vell as the tieart

muscie. The nenous regulation is responsible for increasing the arterial pressure dur-

ing esercise. as well as for maintaining normal arterial pressure. The baroreceptors

are stretch sensors. located in the dis of several large systernic arteries. that proride

a measure of the arterial blood pressure. If the arterial pressure is lower than some

set point there will be peripheral vasoconstriction. increased heart rate. and increased

blood pressure (Gauer and Zuidema 1961). while the venous capacitance is reduced

to maintain venous return (Rothe 1953).

In addition to the nervous regulation. t here are also autoregulatory niechanisms

which operate on the b a i s of local vascular response to a change in the concen-

tration of metabolites in the siirrounding tissue. For csaniple. the caliber of the

vessels siipplying blood to the brain ciiarige iri response to changes in carbon cliosicle

and osygen concentration in bloocl so that the cerebral blood flow. of approsirnately -- 1 a0 cnz"lrnin. rciriains fairly constant when the arterial prpssiirc is betwecn 60 and

140 rnrn Hg (Guyton and Hall 1996. pp. 753 - 784).

1 . l . 5 Cerebral circulation

NlUM

Figure 1 2: Cerebral circulation.

The bloocl supply to the brain is mainly provided by two carotid and two verte-

bral arteries. The common carot id artery bifurcates into the int~rnal carotid artery.

which rnainly perfuses the brain. and the esternal carotid artery. which niainly per-

fuses the heatl tissiie outside of the skull. The major artcries to the hrain comniiinicate

with each other throiigh arterial anastomoses that forni the so calletl circlc of Killis.

Branches of the major arterics that enter the brain are siirroiinclctl by a spiice con-

taining cerebrospinal fluid (CSF). a clear liquitl siniilar to blootl plasnia. in ntiicti

the nervoas tissue is inirnerscd (Sullivan and Allison 198.5). Tlie blootl from the hraiii

is drainecl irito sinuses throiigh both tlic verious systeni on the siirface of the brairi

and the deep wnous systerri. Tlic biilk of the cerebral bloocl Ieaves the ski111 tliroiigli

the interna1 jugiilar wins.

A large portion of the cerebral circulation tiikes phcr in the cerebrospinal cav-

ity. This is confirietl t.y the skiill aiicl intervertrbral disks. and is fi llcci by the nervoiis

tissiic. I h o d vessels. and CSF. Tlic total voliirnii of tlir cavity is 1600 to 1700 crrr".

about 130 crnB of t h ~oliime is occiipieti by the CSF. LOO r ~ r i " II!. hlooci. aiid thr

reniaiiiclcr hy the brain and the spinal corti (Guyton and Hall 1996. pp. 753-759:

Takemae et al. 1987). llost uf t tic CSF is contiiincd within rescrvoirs kriown as the

ventricles of the brain. The CSF also occiipies the s p x e Iietwreri the pia rnatrr aritl

the archanoid nicnibrane of the hrain and spinal corcl (subardianoid space). the spacc

between t tic archanoid membrane and the diira (epidural spacc). m c l t lie space t ~ e -

twecn the blood vcsscls and the tiervoiis tissue ( pcrivasciilar space) ( 11-O Goss 19-48.

pp. 901-909). The pressure insidc the ccrchrospinal cavity is tliat of the CSF. and is

referrccl to as the cranial pressure or CSF pressure (PCSF). Ail the intracranial

cornponerits are largely coniprised of w t e r and are. therefore. cssentially inconiprcss-

iblc (Riian et al. 1991). Furtherniore the skull and intervertebral disks are effcctively

rigicl. and it is routinely assumcd that the total cranial volunie dors not change. This

is espressed through the 1Ionroe-Kelly principle. which statcs that the sum of the

blood. CSF. and tissue volume inside of the cranial cavity must alnays stay constant

(Takemae et al. 1987: Crsino 1988). At estreniely high values of PCsF factors such

as cranial bone mobility may be of importance (Heisley and Adams 1993). but these

values are substantially higher than the most pathologically elevated values of PCsF

in a live human.

Bearing in niind the incornpressibili ty of the intracranial coniponcnts. the SIonroe-

Iielly principle has a consequence that a- change in the intracranial blood voltinie

must be cornpensatecl by a simultaneous change in the CSF voliirne. In normal

circiinistarices the interaction betweeri the bloocl and CSF volunies occur tlirough

the process of CSF procliiction and absorption witli a typical rate of 10-' cni"/~tciri

(Crsino 1958). - ln abnormal increase in the CSF volunie results in raiscd PCSF and

a clrop in cerebral hlootl Row.

The gradient of PCsF change. with the rliartge in the CSF voliiriie is callet1 the

cranial cornpliance (Siillivari and Allison 19%). Cranial conipliarice is iis~ially

deterniincd by injcctirig a voliiiiie of the CSF into one of the wntriclcs. and iiicasiiring

the ctiange in the PCsF (Shriiiiiroii et al. 1973: Niirniarou et al. 1978). Howcwr. the

terniiriology " çraiiial corri pliarice" is sometvtia t niislcaclirig sirice the incrrasc iri the

CSF voliirne is not neccssarily associatecl with eitlier a <:ti;\ngc in ttic rranial voliirne

or the cotripression of ;in- of t hr intracranial corriponcnts. Rat hrr. t hr addit ional

voliinie of CSF is accorririiodatcd b . <-ollapsirig the vriris arid driving an ~qiiiil voliini~

of blood out of the crmial cavity (Slarniaroii et al. 1973).

Thrre is strong eviïlence ttiat ctwbr;il henioclynaniirs arid CSF dyriaiiiics are cboii-

pled anci shoiilcl not hc trcatrvl separately: thcre is a CSF pulse. ivli idi is a rrflec-

tion of the bloocl pressiire ptilse (Portnoy and Ctiopp 1994: Lakin m d Gross 1092):

pathologically raiscd PcSF is associat~ïl with R drop i r i the crrebral I h o d flow m d

an incrense in the PCçF pulsatility (Kety et al. 1947: Greenfielci aiitl Tindall 1965:

.Jotinston anci Rowan 1974: Piper et al. 1993): and a ctiange in the wioiis pressure

leacls to a siniiiltaneous ancl ecpivalent change iii PCsF (Hamilton et al. 1943: Raisis

et al. 1979). An important consequence of the cranial volume conservation rincl the

hlood-CSF coiipling is that the cranial vessels shoiilcl be protected froiri the collapse

even when the hlood pressure is significantly altered. It is esperinientally obser~cd

that the large intracranial veiiis do not collapse escept For a small portion 1-2 mm

prosimal to the junction with the clural sinuses (Yada et al. 1973: Sakagava et al.

193). In contrat . the jugular veins of an upright hiiman are usually significantly

collapsed (Guyton and Hall 1996. pp. 177).

1.1.6 The effect of acceleration on the cardiovascular system

7

Acceleration

Figiire 1.3: The steacty. levei. co-ordinatecl t iirn.

The acceleration stress or G-stress is caiis~ci by a net body force esperiencetl by ttic

pilot. This force is called G force arid is the suni of ttic inertial and gravitational

force. The total acceleration. or G acceleration. is the siini of thc gravitational and

inertial (usually centrifugai acceleration). The cornponent of the accelcration that

critically affects cardiovasciilar perforniance is the one aligned witti t h spine and the

niajor blood vessels. It is referred to as Gz acceleration. or simply Gz. The change

of Gz in time is referred to as a Gz-profile. Thr sirnplest Gz profile is thr one where

Gz is constant over a period of time and it is callcd sustained Gz. A practical

esample of the situation where the total acceleration is aligned a i t h the spine and

constant is the steady. level. CO-ordinated turn shonn in Figure 1.3. The aircraft is

in a horizontal turn round a circular path of radius Ra. at a constant speed \,. The

angle betaeen the vertical plane and the plane of symrnetry of the aircraft is s.. The

n-eight of the the aircraft (I \,) is balanced by the La cos nt component of the lift (La ) '

whereas La sin component provicies the centripetal force that keeps the aircr CI f t o n

the circular path. The magnitude of the total acceleration esperiencecl by the pilot is

(Clancy 1975). whcrc g is the gravitational acceleration of the eartti ( g = 9.8 1 rrt / .s2) .

For esaniple. for 7 = 30 degree. Gz = 1.15 x g = 11.28 [n2/.s2]. Khen considrririg

the ptiysiological effect of acceleration. it is ronvenient to express Gz as a niiiltiplt.

of g . rathrr theri espressing it i r i [ni/s2!. siricc this iriiriiediatrly rcvals ttir ratio of

G force to wciglit. Thiis. the acwlcration in the above csamplr is rtlfrrrrcl to ils

+ l . l X z . A ~ i ~ g a t i w Gz means that the acc ika t ion is clircctecl froni fwt to hcad.

For esamplr. - 1Gz is csperiencccl as tiangirig iipsidr donn.

The Gz ~ i l i i r s gcncratccl i r i aerial conihat rriariciivcrs iirr liirgr riioiigh to riiodify

the blood circiilatioii of the aircrcw. For Gz> 1 tlicrr is an incrcasr i r i t tic hydrostatic

prwmcl cliff~rrnct. hetivr~ri the hcart and t.ii(bii of tbcl orgaris. \-~ssels hclorv t h r Iiriirt

are distcricl~tl duc to incr~;ist.cl t~looci prrssiirc. blood pools iri t tic l o w r I~ody. aritl

vcrioiis return is <Icrrrasrtl. On tlir otticr Iiantl. ttic iit>iiorrrially low hlood prcssiirr in

the ricck and t tir heacl rnakes the cstracrariinl wsscls i ih«\.~ t h ~~~~~~t proricl to < ~ ( ~ l l i i l ) ~ ~ .

Thc iritracranial vrsscls riiay be protectt.ïl by tlicl skiill and th[. CSF. R i i sh i<~r et al.

(1947) nieasiir~cl ïenoiis and PCsF in cats csposccl to a range of positive aiid riegative

Gz and showed that t h CSF arici vciioiis prtassiire are the séirne crrry~vheri. in the

cerebrospinal cavity to the precisiori of t lie rsprrinicntal rrror. This siiggcsts t hat

cranial vcsscls stq* open during G strrss.

Positive Gz

The central arterial pressure is w l l regiilated during Gz variations. Inimediately

folloning an increase iri Gz. central arterial pressurp is niaintainecl by a rapid increase

in heart rate. This is followed by s lowr changes in the peripheral resistance and

venous resen-e voltirne that allow the heart rate to fa11 back to normal levels. The

refleses are fully mobilized after about 20 seconds (Howard 1963. pp. 584). In spite of

largely normal central blood pressures. bot h the cardiac out put and the st roke volume

are reduced (Howard 1968. pp. 595). AISO. for organs above the heart. particiilürly the

eyes and brain. an increased hydrostatic gradient iniplies a lowering of blood pressure.

which leads to recluced perfusion with increasing +Gz. The normal seqiience of events

following a large gradua1 increase in Gz is: peripheral vision loss (gray out). follo~vecl

bu total vision loss (blackou t ). and finally GLOC. In unprotectecl indiriclrials loss

of consciousness occiirs at roughly +4.5Gz. This approsimately corresponcls to the

point at whicb the arterial blood pressure at the lerel of the heacl tlrops to zero. tliat

is beconies approsiniately cqiial to the central venoiis pressiirc. However. the veins in

the neck are subjected to the satrie Iiyclrostatic pressure gradient as t hc arteries and. if

t hey stayed opened. the blood aoiild siphon t hroiigti the head l>y virtue of t ht> negative

vcnoiis pressiire at the lcvcl of the lwatl. Ttierefore. a recliictio~i in ccrehral blood How

shoiilcl be caiised by vesse1 collapse and increasctl wsciilar resistancc (Cirovic et al.

2000a). The vessels in the eyc arc more siisceptible to collapsc. sincc thc intraociilar

pressiire is approsimately 20 r r m Hg ahove t lie at rnosplieric. Coriseqiit~ntly. risiial

impairtiicrit occiirs beforr the loss of consrioiisiiess.

Negative Gz and Push-pull

Diiring ncgatire Gz. hyclrostatic effects raisc the blood priwire et the h c d and

increasr venoiis return. Ttic rcfles resporise to t his sitiiation is drrreasccl Iirart rate

and peripheral wsoclilation. In addition. a regular heart rtiytlirii is often tlisriipted.

Alttiocigh. the central arterial pressiirc rcrnains constant. the samc coritiot be saici

ahoiit the central venous prcssiire whicli tends to bc elewtecl. Elcvatcd blood pressure

in the tieacl may cause cerebral tieniorrhage. loss of vision (red-out) and even loss of

consciousness (Hotvarcl 1963. pp. 688-716). The occurrence of iinconscioiisness nia! be

esplainecl by the fact that the central venous pressure rises ivhile the central arterial

pressure remains constant. Therefore. the driving pressure clifference at the levrl of

heart is less than normal (Ganible et al. 1949).

Push-pull is a maneuver in which the pilot first esperiences a prolongcd period

of low or negative Gz that is follo~~ed by a rapid transition to nioderate or high Gz.

Since the refles response to negative Gz is of the opposite nature to that to positive

Gz. anci the esposure to -Gz is long enough t hat the reflex response be full. mobilizecl.

loss of consciousness may occur at +Gz much lower than normal (Banks and Gray

1994).

The methods of increasing tolerance to positive Gz

Gz tolerance of an individual is the Gz levcl at which specific pliysiologic systems

of that individual are significantly altered (Burton et al. 1974). The most conimonly

used criteria for quantifying Gz tolerarice are visual inputs such as periplieral or

central light loss. but other criteria such as auditory phcriornena or even GLOC

have also been used. The Gz tolerance is increased b!* coiinteririg the impact of C h

acceleration on the carcliovasciilar systcni. The physiological actions of ttir protectiw

nieasiirrs involve: prevention of blood pooling in the legs. changc of Imdy position

with respect to acceleration vector iri order to minimize the C h coniporient. r~cliiction

of the heart to lieatl distance. clevatiori of the central blood prrssiirrs. and prevrrition

of vesscl collapsc.

Ctiarige of posturc froni upright to supine ail1 transform Gz acceleratiori into

trarisverse acceleration. Esperinierits with r d i n i n g scats stiowr~l t hat + 10Gz tolrr-

aricr cari be adiievcd hy tilting the pilot S3 clcgr~cs froni tlic vertical (Ho~wrd 1965.

pp. 666). hnothcr aay of iising changr in posturc is to r*roiicli clown iii tiic scat

liarrirss. thus rrtlucing the heart-head c1ist;inct. by as miich as 20 cm. -4 scrioiis dis-

aclvantag~ of nicasiires bascrf on rhnngr of postiirr is tliat t t i ~ y ritstrict thc fit?ld of

vision to a greater csteiit t han t hat caiiscul by GL effects.

Anti-G siiits fiinctiori on the basis of applying a pressure tiigli~r t hiiri t tir xtmo-

spheric to the loiver hodu and abdorrieri. Apart froni prerenting blootl poolirig in the

legs. anti-G siiits also elevate the ceritral artcriiil pressure. An interrsting. though

impractical. m y of using the esternal pressure for Gz protection is water inirncrsion.

Siihjects immersed to the level of the eyes nerc able to withstand +16Gz before the

pressure on the îhest made breathing iriipossible (Howard 1963. pp. 667). Another

effective way of elevating blood pressure are anti-G straining maneuvers (AGSM)

such as muscle straining and eshaling iorcelully against a partinll- closed glottis

(forced \.alsalva maneuver). Properly eseciited XGSM can increase Gz tolerance by

as much as 4G (SlacDougall e t al. 1993). The main clisadvantage of the AGSSI is

that it requires a significant physical effort. An alternative to \klsalva is positive

pressure breathing (PPB). breathing air at pressure higher than the atmospheric.

Khen combined with anti-G suit to prevent blood pooling. and chest counter pres-

sure to facilitate breathing. PPB increases Gz tolerance by increasirig central blood

pressures (Buick et al. 1992: Burns 1988).

hl1 the mentioned protrctive measures act hy preventing a zero or siib-atmospheric

arterial pressure at the head level. This is niainly achieved by elevatirig central

bloocl pressures. Wood (1987) argiles t hat the effectiveness of anti-G siiits is directly

proportional to t heir capability of prodiicing hypertension at the heart level. Hoivever.

some stiidies show that applying suction to hend arid neck. withoiit elcwting hlood

pressure. also prorides some tlegree of protection (Glaistcr and Lerion 1987). This

clcarly points to mrious collapsc~ as the main cause of inaiicqiiate perfusion of the

tiead arid hrain during esposiire to +Gz.

1.2 Cardiovascular models

In this section we cliscuss niodclirig approiichcxs relevant for arialyzing I~lood How i r i

a port iori of t lie carrliovasciilar system. or t lie rarciio\risciiliir systerii es a tvliol~.

\\'hilr virt iially :il1 mat herriat ire1 niodt4s t ïiliit ing flow in v~ssrl nrtn-orks rdy ori ;i

one-diniciisional trratrrient of t hr bloorl flow. aiialyt ical rno(lcls of asi-syninitbt ric flotv

in chstic vrss~ls are important for corisiticring t tir ~ f f i ~ t s of t tic intrriict ion t>t~twrti

bloocl vcssels aiid t lie siirrounding p t-siological st riic tiircs.

1.2.1 Models of pulse propagation in the arteries

hnalytir rnockls of waw propagation in Riiid-fillecl elastic tubes tiare t m n iisrd rs-

tpnsively to describe pulse propagation in the arterial systeni. The piirposc is to

tleterniine the speed of pulse propagation. i.r. the wave speetl (c) iri the systeni.

Tlie general approach is to consider an infinite. straight clastic ttil~e fillrd with cit her

a viscous or inviscid incompressible Ruid. The fluid niotion is iisiialiy describeci by

a linearized forrn of the Xavier-Stokes and continuity eqiiations. while tube motlels

yary nidely in the lerel of complesity. The solution for the fliiid and tube niotion

is assiimed to be in the form of small-amplitude harmonic naves and the dispersion

equation follows from the boundary condit ions at the Huid-tube interface.

Lamb ( 1898) esamined wave propagation in an unconstrained elastic tube filled

nith inviscid. incompressible fluid. The tube \vas modeled as a cylindrical membrane

shell. \Vive speeds for two modes of aave propagation are given by

nhere E and c are the Young's niodiilus and Poissori's coefficient of the tube wdl.

respecti\.cly. p, ancl p are the fliiid and tube clensities. respcctively. R is radius of the

t ihe. Ir is the thickricss to radius ratio. and ci is the ratio of t lie t i i t~c to fliiid rriass

pw ilnit length.

The first rnocle is called \i>iing's niocle and reprrsrrits pressiirc. tvavcs propiigitirig in

the fluitl. while the scconcl. Lariib's mode. rcprtwrits \vares t r a d i n g wittiin thr \ d l

(Cos 1969). Xccorcling. to cquation ( 1.3) the wavr sprcd in the ;utcries is of ttic ortlrr

of nieter pcr srroricl. and is iricreasiiig clown thcl artrrial trrc. ;\lso. t l i ~ tt.;iv~ lerigth of

tlie piilse is of the order of nietcr and is rriiirli largcr than t h vrssel ratliiis. rrirariirig

tliat lorig wai-e approsirriation is miid for rlioclcling piilse propagatiuii in tlic arti.ries.

\i*orrirrsley (1955) estenclcd Lamb's riiotlcl for t hc case whcrr the fliiid occiipyiiig

tlie tiibt? is riscoiis and obtained a closcc1 forni soliitiori for tlie lorig wavc spwtl. The

aave propagation is dispersive. ancl the wavr spccd is a fiirictiori of tlir \\bnirrslc-

riiiirib~r (na,). Thr \\oniersley niinibcr is clrfi ned as

where .i is the circular freqoency: it rcpreserits the ratio of tlic transient iiiertia to

viscoiis force in the Ruid. In larger arteries of the systemic circulation the \\orrirrslcy

numbrr ranges approsimately froni 1 to 10. and it decreases doan the arterial tree.

The effect of viscosity is to decrease the wave speed and to int rodiice piilse at t cniiat ion.

The effect is strong for small n,. ancl aeakens with increasing nu,. For 12, + x the

solution reduces to that @en by Lamb. The dispersion equation is giren by

i = fi and .JO and J I are Bessel fiinctioris of the first kind. of order zero and one.

respect ively. Espancling eqiiat ion ( l .G) leads to a bicpaclratic eqiiat ion in ternis of

S. and c is calculated frorn eqiiation ( 1 .7).

Sunieroiis attempts bave hbeen nmle to iipgratlc \\'orriersley's soliition. .\ludi of

thr effort lias been fociised on vessel moclrlirig. citlier to introcliicc a thick-nnllcd

approsimation for the tube (Cos 1968). or to includc featiires siicli as initial strcss

and anisotropy (Atahek anci LW 1966: Atabck 1968). 1Iorc recently the criiphasis

has been on introclucirig the non-lineor Hiiid trrms into the arialysis (Ling arid Atabek

1972: \Yang and Tar bel1 l!XU). This type of riiotleling. howev~r. typir;dly rrqiiirrs a

niiniericèil solution. Piilst. propagation nioclrls cari Iw dividctl iiito Ircely nioving èind

constrairied. wit ti resprct to rvt1t.t her t hc interaction of t lie tiihc wit 11 t tir siirro~iiidirig

tissiie is ignorecl or not (Cos 1969). Coristrainetl trioilels vary in coniplc~sity. frorii thosc

wtiich siriiply a s s i i r ~ i ~ i hat the longitiidinal niotion is zero (Ling and .Atiit>ek 1072). to

thase which approsirnatc the siirrocincling tisslic as an atlclitional rriass (Konirrsley

1937h). or a mass-spring-clashpot system (Atat~ck 1968). .\II the rriotlcls meritioned

represcnt tlie siirro~iriciirig tissue as an arbitrarily chosen nierhanical systerri adclrd to

the original iiiiconstrainrd vessel witli the pararileters of the systerri being rtiosrn to

elirninate the discrepancy betwcen the t tieory and the csperirnent . To mir kno\vlrdgc.

the oriij- attenipt to represent the siirrounding anatornic striictiircs èis coritiniiurn

\-as made by Dinar (1975). a h o representecl the siirroiinding tissiie as an unboiiiiclccl

viscoelast ic solicl in imtnediate contact with the vessel. He conclucled t hat t lie wave

speed is affected mainly by the viscosity of the siirroiinding tissue. WC believe that

an analysis of the pulse propagation in the cranium rriiist be baseci on a rnoclel of

blood vessel. CSF. nenous tissue. and the skiill as a single system. and that the

consex-vation of the cranial volume must be incorporated into the niociel.

1.2.2 Models of vascular networks

iksciilar network geomrtries fa11 into two broacl categories: closetl loop ancl open

loop. The closed loop treatinents incorporate a moclel of the heart (Snyder et al.

1968: H y d m a n 1973: Harcly et al. 1982: .Jaron et al. 1954: Sloore et al. 1992:

Samn 1993: Ozaiva 1996). and t tic esscrit ial goal is to simulate global cartliovasciilar

perforriiance tvith a set of gowrning ecpatioris arid material properties. Ttic open

loop treatments have no heart rnodcl. and it is tîssurned that ttie network tcrriiinal

boiinclary conditions are kriown. Consideratiori is rcstrictcd to tlic prrfiisiori of one

organ such as a single vesse1 (Gafiïe et al. 1991). the hcart (Colliris and Slatecva

1091) tlie lungs (Collins ancl llaccario 1979) or rven the entire systcniic circiilatiori

(\\ésterhof et al. 1969: Siicl et al. 1992: Sherig ct al. 1995).

Thr riiodcling approacties also fa11 into two hroacl cat egoriw cont iniiiini iiricl

liirripd parametrr. In corit iniium n~ocl~ls the basic network thticrit is ii tiihc \vit ti

t~oiiiidary conditions at its eiitls and ;i set of gowrning pirtiitl (liffrr~titièil rcliiatioris

that npply on the iritcrior of thc tiit~c. The liirnprd eqiiatiotis c m tw clrrivrcl froni

sri an;ilogiic rlert rical rictivork (Sriycltlr tlt al. 1965: Kester tiof et al. 1969: Hynclrrian

1973: .Jaron et al. 1984: Moore et al. 1992: Sanin 1993). or tlircctly frorti tlic goy-

erning nieclianical equations (Harcly et al. 1982). \\'litm the 1iirripi.d rlrnicnts arr

roriiiectcd. we obtain n set of coiipled ordiriary clifferential qiiatioris that rncodcs the

geompt ry arid boiinclnry in format ion.

One-dimensional approximation

Al1 nioclels of vascular networks rely on a one-dimensional (1-D) approsimatioii that

focuses on the hulk transport of blood mass and momenturn. The ccltiatioris governiiig

1-D unsteady flow of an incompressible Auid in a flexible tube are

X a c : ~ p - - at - --(y+-)-- a - p P

R(.') .AL* + Gz cos 0

(Cancelli and Pedley 1983: Gaffie et al. 1991: Sheng et al. 1995: Ozawa 1996).

Equation (1.9) is the continuity equation. while equation (1.10) is the momentum

equation. Here? t is tirne, x is the asial distance dong the segment. -4 the segment

cross-sectional area. L; the mean flow speed. P the blood pressure. p the blood tlensity.

antl 0 is the angle between the segment asis and the acceleration vector. Fliiid pressure

c m be espresseci as P = Pt + P,. whcre P, is esternal pressure. anci Pt is traiisniiiral

(interna1 minus cxternal) pressure. R is a laminar viscous friction terni. For Poiseuille . -

87ïp flow through a tube of circiilar cross-scctional area R = - (Constantiiiesku 1995.

-4- pp. 123). For a collapsible tube it is more appropriate to iise the forni which takes

into account the non-circular shapc of the cross-section

(Pd ley et al. 1996). Iri the cquation. .do is the segment cross-wctional arca for

Pt = O. antl nt, is the nimber of vessols in the srgniciit in thc case atirri an rqiii\alent

vesse1 is used to rrpresent a nunhrr of vess~ls iri paraIlcl. Eqiiations ( 1.9) ancl ( 1. IO)

cari also be writteri iri vector forrn

nhere E and F are the flux and source ternis. respectively.

Ecpations (1.9) id (1.10) have t hree unknowns A. Cg. arid P. A third cqiiatiori

is provided by a tube law nhich expresses Pt as the prodiict of a scaling constant

Ii,. with the diniension of pressure. with an increasing function of the relative area

.\hthernatically the tube law closes the system of equations. and physically it defines

a non-dispersive long aave speed co as

Such a tiibe law represents a drastic simplification of the vascuiar structural clynarn-

ics. Howver. it is a good approximation for long wave propagation in elastic tubes

(Light hi11 1978).

Figure 1.4: A typical tube law for a t tiin-walled collapsible tiibe. and cross-sectional shapes

for different stages of collapsc. The function P ( a ) is the one giveri in cqiiation ( 1.23).

.A number of general staternents can be made about the betiaviour of the collapsible

tubes. U'hen Pt is large and positive the tubes are inflated. circular in cross-section.

and Pt is supported by tensile membrane stresses. In this state. the tubes are relatively

stiff. in the sense that a unit change in pressure protiiices only a small change in area

(zone 1 in Figure 1.4). As Pt decreases through zero. the membrane stresses become

compressive. and the tiibe buckles to a non-circular cross-section that is generally

taken to be elliptical. with Pt supportecl by bentling stresses (zone II in Figure 1.4). In

this state the tube is relatively cornpliant in the sense that a unit change in pressure

produces a large change in area. As Pt is recluced further. wall contact occiirs at

approsiniately a = 0.21. The flow is then confinecl to a nuniber of scparatc lobes.

typically two (zone III in Figure 1.4). In this state Pt is still siipportetl by bencling

stresses. but the m l 1 contact causes a sitdden recliictiori iri the effective clianictw. and

a corresporicling jump in stiffness. As the collapse progrcsst.s. the effective rliaineter

reiluçes flirt her. ancl t lie stiffness continues to increase. For st rai& rtsisyninictric

thin-wallcd tubes biickling occurs at Pt 2 O. Thick nallecl tiibcs. on the other hancl.

c m support consiclerable compressive riienibrane stresses bcforc buckling.

.A major contril~iition to this probleni \vas proviclecl hy Flaherty et al. ( 1072). ivho

dcrivccl a similarit- soliition for the collapse of a thin-walled tiibc wticri n <_ 0.21:

Frorn tliis esprwsioii rvc sec tliat <r caiinot br rrrliicrd to zrro by a firiitt! trarisrriiirel

prrssiire. Eqiiatiori ( 1.18) tloes not apply to the collapsr of t liick vai il cd vcssrls siidi

as artcries. Howver. the siniilarity solution is typically applicrl to t hi) srvwr colliipsc

of an' tiibe. rcgarrllrss o f \.al1 thicknrss. T h laws hasctl on niodifi cd foriiis of

cqiiation (1.13) have bcen prescnted hy a tiitniber of arittiors. Bertrarii and P~c1lr.y

(1982) introdiirecl ii tube laa t liat patchcs t lie sindarit- forrri ( 1.18) for collapscd

tubes to a linear esprcssiori for inflatetl tiibes

This form prorides a good fit to the data on thin-walled rubber tubes. but it is not

suitable for numerical integration of the gowrning equat ions since most algorit hms

require al1 functions to be differentiable. Therefore. a ncimber of aiithors have iised

smoothing fiinctions at the patch to guarantee continuity of the slope. Peclley et al.

( 1996) have proposecl

P ( a ) x ctn - a -3/2 (1.23)

for the jiigular vein of the giraffe. The espoiierit n is assuniecl to have a value in the

range 10 - 20.

The connection between the wave speed derived from 2-D models of pulse

propagation and 1-D continuum models

The speecl of sniall-amplit iiclt.. rion-tlispersive wave propagatioris can he dct ermined

from eqiiation (1.1;). For a rion-collapsetl tiibe of circiilar cross-sertionnl arca it is

appropriate to use a linear t uhe laa givcn by eqiiation ( 1.19). The wavt speecl is t lien

ivherc El is kriown as cliwtaricr and is ctcfiricd ils

Thth iwve speed given in ecpirtion (1.24) is iclrritical to thkit detcrrriincd by Lamb

(18%). and given in cqiiation ( 1.3). provicling t tiat t hc Htiid aricl t il he wi l l clrnsities

are t t i ~ sarnc. wliicli is a comniorily iiscd assiiriiption dirr i tlir hloocl wssels arr

corisiclerecl. It is important to renienil~er t liat Lanib ohtairiecl t lie solu tiori for ari

iincoiistrninecl tube. Therefore. it is irnpliecl iri eqiiiition ( 1.1'7) t hat the collapsihlr

tube is not constraincd. This rises a question whetlier it is appropriate to iise the

1-D cquations ( 1 3) . (1.10). and ( 1.16) to dcterniine t h the piilse propagation speecl

for the vessels in the cranium.

Solution methods for 1-D continuum approach

The systeni forrned of equations (1.9). (1.10). and (1.16) is hyperbolic. and the clorni-

nant physical phenomenon is nave propagation (Hirsch 1990a: Hirsch 1990b). There

is a strong analogy with the equations of compressible gas ancl open channel flows

(Shapiro 1977). This class of equat ions presents numerical difficiilties because the

simplest diffcrencing schemes are al1 iinstable. However. this problcm is well under-

stood in computational Buitl dynaniics. and a nuniber of effective stable algorithms

are available (Anderson et al. 1984: Hirsch 1990a: Hirsch 1990h): the rrietliod of

characteristics (Collins and lfaccario 1979: Collins and '\lateeva 1991). fliis splitt ing

(Gaffie et al. 1991). split coefficient iriatris nietbods (hiiderson et al. 1984. sectiori

6-4). ancl the '\IacCorniack metliocl (0zatt.a 1996: Shcng ~t al. 1995). The solution

is cornplicatecl by the fact tliat the georrietry is iisiially coniples ancl therc arc niany

interna1 boundaries. Hoivever. the nietliod of characteristics pro~.itles a poiwrfiil tool

for tiandling the boiinclary conclitions.

Another useful approach is to linrarizc the governing eqiiations hy assiiniirig ttiat

the blood vclocity and pressure are coniposed of a steatly state coiiiponrnt. or1 whirh

a tinie ilcpenclmt perturbation is siiperiniposecl (Fting 1996. pp. 151-132). This

leads to a linear wm-e equation for thc piilsatile corriporieiit of th^ t)l»otl How (Mc-

Donald 1974: Light liill 1973: Pccllcy l!lSO). The liricarizctl rqiiatioris govcrriing the

iinsteacly romporierit of hlood Row arc ;irialogoiis t O t ransiriission lincl rqiiat ions arid

are particiihrly siiitahlc for arialyziiig branctiing rictivorks of vrwcls (Diian ; m l Za-

mir 1903: Diiari and Zaniir 1995). Foiirier tlecortiposition of t h piilsv cari hr applirtl.

and. in the casc when vessel propcrtirs iirr ilriiform. an ariiilytical soliitiori is üvail-

able. Cltirriately. by applyirig the consrrwtiori of niass arid continiiity of pressure ;it

the branching poirits ir i the net\.ork. the arialysis is rccliicccl to solving it systcni of

linpar algehraic equations. Fiirther siniplificntioris arc possible by iisirig nrtwork rp-

duction niethods (Helal 1903) or by taking irito accotint the synimetry of the rictwork

(llayer 1996). The equations gowrnirig the steady conipoiient of thc blood flow can

be combinecl into a single ordinary clifferential equation (Shapiro 1977) tliat can he

integrateci using a standard nunierical technique such as Riinge-Iiiit ta niet liod (Press

et al. 1986). A furt ticr simplification can be introducetl by assiiniing the conwctive

term in the rnomenttim equation can be neglected. either becaiise Re << 1. or berause

the changes in the cross-sectional area are rery graduai (Rubinow ancl Keller 197 '2).

Then. pressure drop in a vessel is only due to viscous resistance. and an analogy can

be established between the steady blood flow in vessel networks and a current flow

in resistive eleetric circuits (Mayer 1996: Rubinow and Keller 197'2). It has to be

remembered. however. that the "conductance" of a vessel segnient depends on the

vessel diameter which is. in turn. a function of the blood pressure. Therefore. the

network analysis involves solving a systern of nonlinear. rather than linear algebraic

eqiiat ions.

1.2.3 Cerebral circulation models

Khilc distributcd 1-D rnoclels of cerebral circulatiori n-it h dctailctl trcatrticrits of the

vasciilar network are available (Zagzoule aricl Slarc-Vcrgries 1986: Slicng ct al. 1993).

the' iisually do not take blood-CSF coupling irito accourit. The CSF and the fised

craiiial volume are consiclered by Gaffié et al. (1993). However. their treatnicrit is

restrictcd to tletermining the ccrebritl stress disrrihiition cltic to accelt'ratiori. This is

iised to deterniinc vessel collapse \vithout allowing the change in vesscl cliiinirters to

motlify the cerebral stresses or the volume of ccrcbrospirial Riiicl.

On t hc ot hcr hantl. rrio(1eis t bat iricorporate CSF clynarriics arr elrct ric arialogiie

liinipccl-piiraiiictcr approsiriiatioiis witli cstrcriicly sitriplificd treatriithrits of tlic \.ils-

cular antl blood tlynaniics (Bekkcr ct al. 19%: Tiikcmac ilt al. 1987: Crsino 1988:

Crsiiio 1900: Crsino ancl Di Gianiniarco 1991). Typically. ttir wholr vasciilar nrtnork

is presrnt~cl by scvrral liiriipccl elrnirnts. t)looti intbrtia is rirg1rctt.d. alid t h vasciilar

resistance is oftcri takcn to bc inciepmclcnt of t tir vrssd volilnie. In addition. the

estracranial veins are eithcr not incorporatetl iri the nioclel or are rcprtwiitccl by a

sin& liirnped t~l~met i t siibjectecl to a coiistarit crnt ral verioiis pressiircb.

The work report cd by Stoiitjesdij k (199.5) is physiologically det ailecl. but rriecliari-

ically simple. Cerebral perfusion is controlled by a Iumped ccrehrovasciilar resistarice.

whicli is siibject to a numhcr of iiiito regillatory nieclianisms. Tliere is no treatirient

of accaleration forces or regional flow tlistribiitiori. The effect of the CSF and of the

const raints imposed b\- the fisecl cranial volurne on risodilat ion antl rasoconst rict ion

are discussed. but not modeled (Stoutjesclijk 1993. pp. 9. 13-14). In summary. the

models of cerebral circulation currently in use either ignore the coupling of bloocl and

CSF dynamics in the cranium. or provicle an oversimplified treatment of the large

veins in the neck.

1.3 Thesis Objectives, Scope, and Organization

1.3.1 Thesis objectives and scope

The objective of this study is to develop a matliematical niodel of the cerebrorasriilar

systeni in order to itlentify the crucial Factors that limit cerebral blood Bow chiring Gz

stress. esamine how t lie protect ive nieasures help to restorc adcquate cerehral blood

flow. and to determine the level of coniplcsity with which the cerebral vascular tree

shoiild be represented in a global niociel of the cardiovascular systctii.

The atialysis is mainly basecl around ari open-loop georrietry represrnting the core-

bral rascular systeni and a 1-D continuum approsiniation for tlic Aos in elastic vrssels.

Thc fociis in sinidations is on the situation when the cardiovascular rcflcses arc fiilly

mobilizecl. i.c. the central blood pressures are either normal. or elcvntccl due to tlic

action of the protect ive rtwasures. Therefore. it is appropriate to iise the soliition a p

proach idiere the gowrnirig 1-D eqiiations are lincarizrd ancl the srtwly aritl piilsatil~

roinpon~nts of t Iir flow arc t rea trcl separately. Howver. rcsiilts arc also prrseiitrd

for '2.-D ~vave propiigiit ion s t iidy aricl for a 1-D closed-loop riiiriirricd siriiiilat ion of

cardiorasriilar pcrfornimce. The \vase propagat iori s t ucly ~sarnirics t lie r ffwt of t Iir

skiill constrnint a ~ i d CSF coupliiig ori tlic spcwl of cranial pulse propagation. The

closrd-loop simulation esamiries the cffcrt of incorpornting a siniple c e r e h l floiv

loop into a global cardiovascular systeni. Xlso. the ninin assumptions upon nhirti the

rnocleling is basccl. as well as the rrsul ts of t lie niat hema tical siinulat ions are testecl

espcrimcntally. using a niechanical modcl.

Special attention is paid to t w spwific issues: the circulation in the craniuni

ancl the dynaniics of the veins in the neck. The hypothesis ive used in analyzing

the coupling between blood flow and CSF is that the llonroe-Kelly principle applies

strictly. Therefore. the cranial volunie is constant at any instant of time. This

implies that the blood piilse in the cranium propagates without a change in the

cranial volume. Although the pulsatile component of the blood Rotv is not critical for

the brain perfusion. analysis of pulse propagation in the cranium has to be perforrned

in order to prove that the assumption about the cranial volume conservation does

not lead to physiologically implausible results for the blood and CSF pulses in the

cranium.

The empliasis on renoiis dynamics refiects the Fict that. in a circiiit beginning antl

entiing at the same elemtion. a change in Aoa due to a change in Gz. ni- oçcur only

if the resistance of the circuit is affccted by Gz. Our hypothesis is that the venoiis

resistancc is the niost likely to be affectecl by Gz. since the venoiis pressiire is the

lowest pressure in the systeni antl the thin-walled veins are likely to collepse as sooii

as the venous pressure bccomes siib-atmosplieric.

1.3.2 Organization of the thesis

Prior to consiclering the piilse propagation in ii tietnork of vesscls coiitairie(1 in thr

craniiini. it has to be determinecl how t h constrairit proviclecl 1- the ski111 and tlie

presericc of ttie CSF and hrain tissiie niotlifics the s p ~ e d of wave propagation iri ttic in-

tracrariial wssels. Therefore. an analyt ical rnodcl of piilsr propagnt ion iri rorist rairicd

vessels is developcrl in Chapter 2 and th^ infliimcc of the coristrairit is aiiiilyzcd for

various sitriat ions.

Aftcr concliiclirig that tlie 1-D governing cqiiatioiis r a i hc iisrcl in the original forni

for a11 the ccret)ral wssels. a mode1 of cw-ehral circii1;ition witli a (letailecl goorri<.try is

clcv~lopctl in Ctiaptrr 3. For the steacly flon*. the systrtri of vessrls is approsiriiiitrcl I)y

a resistiw network. nhile the piilsatile Aow is rsnrnincd iising a linpar t h ~ o r y of w r e

propagation derivccl from 1-D ccpations. The steady flow simiilatioris art1 prrfornicd

for Gz ranging froin -5 to +IO. and the effect on hlood pressure. vasciilar rrsistiince

and blood flow is esaniined. Since t h piilsatile flow is not esseritial for csaniining

the effects of Gz. the analysis of pulse propagation is performed orily for zero Gz.

Giiidecl by the results from Chapter 3. In Chaptcr 4 the arterial and cranial

port ions of the rnoclel geonietry are sirnplifiecl. while the flow in the jiigiilar vcins is

rnodeled by the full 1-D steady equations. Also ccrebral aiitoregiilation is inrorporated

into the model. Then the influence of venous properties. elentecl ceiitral vrnous

pressures ancl autoregulation are examineci.

In Chapter 5 a mechanical esperiment. based on an approxiniate analog of the

geometry used in Chapter 4. is designed to verify the basic physical assiirnptions

used in modeling. and evaluate the qualitative aspect of the results prorided by the

mat hemat icâl models.

Finally. the open-loop mode1 developed in Chapters 3 and 4 is incorporated in

Pressure [rnrnHg] = 0.1333 [kPa] ctn3

Yolumetric Flow Rate [-] = 0.0167 x 1 0 - ~ (-1 r r m s

mm s Hyclraulic Resistance [mn2 Hg-] = 8.0 x 10"kPa-]

CT TZJ n z T T ~ "

-4ccelerat ion [g ] = O.Si [-] S -

Table 1.2: Conversion factors for rriedicai and S.I. iinits.

a closed-loop moclel of the cardiovasciilar system in Chaptcr 6. -4 time-clt~ptwlent

siniiilation of the cardiovasciilar perforniance (Itiririg esposiire to a simple Gz profili)

is prrforniecl. and the effcct of applying various scenarios of G-protection nicasiircs is

esaminecl. Also. the resiilts of the open-loop ancl closed-loop siniiiliitions for sustaincd

Gz are conipared. The concliisions of tlie stutly arc presciitetl in Chapter 7 . Details o f

the nuinerical methods used in closetl-loop simulation of cardiovasciilar prrforniancc

arr siininiarizrd in Appcndis A.

Units

Ttirotiglioiit t tic t lirsis. most ptiysiological paramrt rrs stic ti as blood prrsslirr iiritl

hloocl How tare given in mrdical. rat lier thiin in S.I. iiriits. .\Itlioiigli i r i id1 of t h figiir~s

the results arc clisplayetl in hoth thr itiedir.al ancl S.I. iinits it nlay ht. hrnr+icial for

t h readcr that the conversion factors be surnrnarizcd in Table 1.2.

Chapter 2

The Speed of Pulse Propagation in

Constrained Vessels

2.1 Introduction

The »t)jrctivc is to dcvdop a niociel of piilsc propagatioii that will t a h iircoiint of

t ht! roriservatioii of voltirrie in t hc crariiospiriiil ravit-. [ri orclor t hat i.iinsrrvat iori

of voliirne be irnposcd. the transverse dinicnsiori of the systrm rrprrsriit irig cranial

circiihtion lias to he finitc. Also. al1 of the intracranial corriporimts (blood. nprvoiis

tissiic. CSF) have to be r~presentecl in the motlel. and trt?ati?d as contiriiiiini. Thp

analysis is rrst ricted to smnll-amplitude harnionic tra~elirig wavcs of arbit rary navc

lcngth. nith the eniphasis oii the long-naw case. The fociis is on clcarly irlcritifying

tlie impact of the sktill constraint. ratlier then on tlevclopiiig a sophisticatcd tiioclcl of

the tube wail. or on esploring the full range of w r c niimbers and parameter values.

Therefore. ive choose a simple niodel for the vesse1 ivall thnt will alloir- LIS to directly

emluate the effect of the constraint by comparing the resiilts of the mode1 witli well

established solutions given in equations (1.3) and (1.6).

Case 1

Skull .

Case 2

Figure '2.1: The gcornetry used for the mode1 of pulse propagation in constraincd vessels:

An elastic tube of a unit relative radius and filled witti incornpressiblc fluid is enclosecl

wit hin a rigid tube of the relative radius y0 > 1. Case 1: the anniilus between the tubes is

filled with incompressible Auid only. Case 2: the annuliis between the tubes is filled wit h a

viscoelastic solid separated from the tubes by fluid layers of thickness y,~ - 1 and - fi?.

2.2 Methods

2.2.1 P hysiological approximation

\\é consicler wave propagation in a systeni forrriecl of coaxial circiilar tiibcs of infinite

length. The inner tiibe. which represerits a blood vessel. is flesit~le while the outcr

tiibe. wliich represents the skiill. is rigid. Both the flesiblc ttibe aritl tiie annulus

betwen t hc tubes arc filled wit h inconipressible continua. represmt irig blood. CSF

ancl brain tissue. Therefore. the net voliinie fliis throiigii an- cross-section of the

rrioclel rriiist bc zero.

\\é csaminc two cases. with respect to t hr contcrit of the atiriiiliis betaecri the

tiihes. In t Lie first case al1 of the t tic anriiiliis is tillecl \vit h fliiid rrpresciitirig the

CSF (Case 1 in Figiire 2.1). Sincc it ignores the prescticc of the iirmuiis tissue. tliis

geonietry is riot wry rcalistic. Severthelrss it is iiseliil for ubtairiing an ininiccliate

insigiit into the effect of the constraint on thr pi i ls~ .;perd. Iri t l i ~ sc.c.oiit1 ('iisr thil

anniilils is oixxpird ii viscorlastir solid. rrprcsrnting thr ricnoiis tissiiil. aliiA is

srpiratrd froni hoth tiibcs b* 1-rs of viscoiis Htiid. reprrwritirig the CSF (Cifith 2

in Figiire 2.1). The fltiicl l q e r hetwwn ttic rigicl tiihr arid t hc solicl (! jc2 < !/ < in

Figiire 2.1) rcpresents the siibarctianoici space. whcrcas the l a y r bt~tivoeri t h flesiblc

tube and tiic solid (ycI < u < 1. iri Figiire 2.1) reprrsents the pcrivasciilar space. In

botli cascs the Hesihle tiibe is occiipietl bu fliiid rcprcscnting blood.

2.3 Governing Equat ions and Boundary Conditions

In the system considered. wave propagation irivolws the interaction betwen tbe ves-

sel. the fluid contained witiiin the vessel. and the fl~iid and solict in the anriiilus.

Therefore the mat hematical statenient of the prohlerri involms t lie eqiiations govern-

ing the motion of the vessel. fluid. and solid. as well as the boundary conditions at

al1 of the interfaces. Since we are interestecl in small aniplitudc n v e propagation.

the governing equations and boundary conditions are given in their linearizecl forms.

The convective terms in the Xavier-Stokes equation are small compared to the other

terms. if the axial Auid velocity is much smaller than the wave speecl. and the ampli-

tude of the radial vessel motion is much smaller than the vessel radius (Pedle- 1980.

Equations of motion of the vesse1

We consicfer a thin cylintirical tube of undistiirbecl radius R and tliickiicss to radius

ratio h. The tube w l l is homogeneoiis. isotropie. and linearly clastic. nitti h i n g ' s

!dodulus E. dcnsity p,. and Poisson's coefficient e. Ttic tube is subjccted to action

of radial and axial esternal force per unit arca r, and r,. respectively. Ttie qiiations -

of motion are a " T, - A -

E (#< + c a i l ) + - -- i)t? p,,h p m ( l - c 2 ) ax2 R as

wtiere ((2. t ) ancl T I ( - . t ) are. respectively. asial iiiid ridial tlisplacenicnts of the tiibc

\val1 ( llorgan and Iiiely 1954: \\biricrsley N X a ) .

Fluid equat ions

Ué assiimc that l~loorl aritl CSF are iriconiprrssiblc Scwtoniari fluicls of clmsity p and

kincmèitic viscosity v. \Yc fiirther assume that the flow is iisially syrrinitvic.. body

forces are abserit. and t tic magnit iides of the velocity coniporierits and t hcir clcrivat iws

arc so srnall t tiat their procliicts can be neglectctl. Cncler ttiesc assiiniptioris the

Xavier-Stokes and coritiniiity eqiiations in tlie cylintlrical coorciinatcs rediice to

where p is the pressure arid c, and c, a ~ i a l and radial conipoiients of velocity. respec-

tivcly. The stresses in the fluid are

Equations of motion of the solid

The brain tissue is assiiriicd to be lincar. viscoelastic. isotropie and incoriiprcssiblc.

The eqiiatioris of rnotiori and condition of inconiprcssibility iri cylindriral coordinates

I L dur il, + - + - = O

& d r r

is pressiire. arid 11, and 11, are the asial and radial coniponrnts o f displacr~rricwt.

rcspectively. Ttie stresses iri the solid are

dur a,, = n - ic -

3r

Boundary Condit ions

The following boundar- concli t ions have t O be sat isfiecl.

1. Continuity of velocities at ail ff uicl-solid boundaries. and at the elastic tube n-dl.

in the case when viscous fluids are considered. In the case where inviscid fluids

are considered. the slip parallel to the boundary is allowed.

2 . Continuity of stresses at al1 Eluid-soiid boiindaries.

3. Symmetry at y = O for the Ruid in the elastic tube.

The above conditions are linearizeci in the sense that the- are applicci at the iindis-

turbed rather than at the actual boundaries. The error introduced by tlic linearization

of the boundary condition is of the same order of magnitude as that cliie to lincariza-

tion of fluid equations (llorgan and IGely 1954). Slatheniatically. tlicsc conditions

arc C, = O

2.3.1 Solution

\ \é consider the propagation of w v e s which are harniotiic in t and r. Thercfore. an?

variable " s ( r . x. t )" is assunieci to be of the forrn:

r nhere y = - is the radial coordiiiate normnlizetl with respect to the radius of the

R flexible tube. Then. the fliiid equations reduce to

1 R*t nu, is the \\oriiersley nu~iihcr g i w i iii q ia t ior i (L.5). K = - is a pararrieter

C

proportiorial to the ratio of tlie Hesihle tiibe riidiits arid the aiiw l~rigtli ( A ) . aiid ,3 = Jni, i : j + K~

Thc solution of ecliiations 2.8 is

(Cos 1068) ~vtierr 1; and 1 ; are Bessel fiinctions of the srconcl kirid of orclrr zrro arid

one. respectiwly arid Bj. B;'. D; arid Dy are constarits. Csirig rqiiations (2.0) w~ cari

writc t hc Riiicl strcsses as

) = (5) { J + D:'b(:!,) + (A) [ B J + B : l d w ) 1

The vessrl di~pla~ciiicrits are of t hc forni

where .\! aiid S ;ire ttic aniplitiitlrs of the asial and ratliiil tube niotiori. rrsprctivdy.

Tlir esternal forces acting on the vcssel wil l are fliiid strrsscs at y = 1. aiid ciin bc

whcre superscripts " t " and "a" denote fluicis in the tube and the anniilus. respcctively.

The equations of motion of the clastic tube can now he rcwritten as

where 6 and S are given in equations (1.4) and (1.7). respectirely.

Dispersion equation

When equations (2.9) - (2.12) are siibstituted into the boiindary conditions. giwn iri

eqiiations (2.6). a systern of m - 2 algebraic eqiiations with m iiriknowis is obtained.

The unknowns are the constants in the espressions for Ruici velocitics and solid tlis-

placement. as well as the amplitucles of tlie vesse1 motion. Tlie r n - 1-th and [ri-th

equations. neccssary to close the systeni. are the ecpatioris of niotioii of the tube.

The system is of tlie forni

d ie re coeffkierits CL,, . i = 1 . r r t . j = 1. 111 arc hinctioiis of -*. .L K . arid A.. ancl CL , . , p - 2 are t lie rorist ants in t tir csprcssions for fli i icl ve1oc.i t ies and solid clisp1;icc-

nients. Thr order of tlic systrm cIcpc.ricis on the gronirtry iisecl and tlic assiimption

about ttie riatiire of thr fliiid in. and aroiirid t h v~ssrl. For the rriost conipliwtrcl

case. witli il viscorlastic solid aricl two I-crs of viscoiis fliiitl in t l i ~ mriiiliis (Case 2)

rrl = 18. Sincr the systrm is horiiogcneoiis. tlii? uriknowri constants cannot hr calcii-

letecl iriclep~ii~lcntly but t liey cari be intcrrdatecl (Cos 1968). This. t hc orclrr of tlic

systeni cari be rcduced to t1r.o if the first r n - 2 eqiiations arc used tu espress al1 of tlic

rcmaining constants in ternis of the amplitiitlcs of t hc vcsscl mot ion. The reniaining

eqiiations are equations (2.15). wiicre the esterrial forces can now he written as

The coefficients 6 are obtained by espressing the constants in the espressions for

fluid stresses in terms of .\f and .Y. In general the' are fiinctions of the geometric

parameters. and of the parameters -{. 3. K. and >. and have the dimension of stress

dirided by length. It is convenient to define the following non-dimensional coefficients

Thcn. the terms associatecl with forces in eqiiations (2.13) can he written as

Cpon

radial

siibstitiiti~

q i i a t ions

equation (2.19) into 1

vcsscl motion by - I id

ecliiation (2.1.3). and > . -

JI and x. respect ive1

of honiogerieous alge braic eqiiat ions is oht airicd

ng tlie asial arid

following systeni

For R non-trivial solution to rsist. the dcterrniriarit of the systcam lias t» IF zcro. By

espindirig tlic deterniinarit WC ohtairi a dispersion rclwitiori iri tcrrris of S. K ; i r d

consqiirritly of tlie wavp speed r . Thr 8 cocfficierits iiccoiirit for tlie iiiteractiori

hrtwcn ttic vcsscl arid tlie rrictiia in ancl aroiirid t lie vcsscl. For al1 t h O cocîficictits

rqiial zrro clisprrsiori quat ion rrdiices to tliat b r ari rriipty cyliritfriral nicmibranc~

frce of coristraint (Graff 1991. pp. 263).

The analyis is siiriplificcl by tlic h c t tliat thr situation relevant for ptilst~ propaga-

tion in the arterics is t he onr wlicre K « 1 (Cos 1968). Tlicn. the trrrris of the order

K' conipared wi t h t lie leacling ternis can br ignorecl aritl t lie clispersiori mpat ion is

1

Furtherniore. if a long twve approximation is usecl. i.e. if it is assiirnetl tliat the wave !ro R

length is much larger than the transverse dimensions of the systeni (- « 1). tlie X

following asympt ot ic expressions can be usecl

since KY « 1 (;\tabek and Lew 1966). Also. the terms in the espressions for vclocities

and stresses which are of the orcler (5)' conipared with the leatling tcrms can be

ignored (ihniersley 1957b). For the ~ & e 1 and inviscid fluitls. a simple closed forni

solution is available for the long wave speed. Khen viscoiis fliiids are consiclrred. a

closed form soliitiori is still available but is miich more cornples. For tlie Case 2 a

closed form soliition is not available. Since the shear riiotiulus of the brain tissue is of

the order 103 -10.' Pa (Fallenstein et al. 1969: Galford and SlcElhancy 1970: Sliiick

i n l v n i 1 ) :; is of the same ortlcr as K. aricl the ternis of the orcler ($ . ,

cornparecl to tlie Ieiidirig ternis can no loiigcr bc ignorecl. Furtlicwriore. wlieri go is

largr tlic cqiiations (2 .22 ) niay no longer hr acciirate for thr nircliiini siirroiiriding the

vesscl. cven if K « 1. Thcrcforc. wr lise ari itcrative nicthod to obtairi the solution

for the Case '2. and to test the valiclity of long wèiw approsirnatioii for tlir Caw I .

The solution proccclure is as follows:

1. ;\ri initial giicss for paranieter rc is nièiclr tisirig a lorig-wirc ;ipproxiniatiori.

2 . T l i ~ cocfficimts a, , : i = 1 . . . m. j = 1 . . . m in equatiori (2.16) arc ciilciilat~i

for the currctit value of ti and ttic iinkriowi constants ;irr i q m w ~ d iri ti1rnis

of the amplitiides of thc vesscl motion iising classical rrirtliods of lirieiir a lpbra

(\\'-lie aricl Barret t 1952. pp. 13w5-200).

3. .Y is calculatcd from eqtiation ('2.20) iising the ciment value of K . ancl t t w new

d u e s of S. c ancl ri are cletermineci.

The procedure is repeatrd until the valiies of tî froni two siiccessiw iterations are

siifficient 1- close.

2.4 Results

2.4.1 Case 1

We first consider the case in wliich both the tube and the annulus are filled with

inviscid fluid. The long Kave approximation leads to the folloning espressions for the

Figure 2.2: The wave speed in terms of parameter y0 for Case 1 and inviscid Ruids.

velocities and pressure in the elastic tube

and the annuiiis

Thc orily non-zero tmi i is E),,. and it can bc intcrprctcd as ari iricrrastl of the caffix-tivc

radial inwtia of t lie t iibc. as t lie irirrt ia terni in tlir trans\wsc cqiiatiun of t lir vrssrl ' > - rriotiori can bc writtcii as 8,, - ti-0 2: Clrr. The first trriri oii thp riglit haricl si&

of the c.sprrssiori for 0,, accoitnts for the infliiencc of tlie Hiiid iii the tiihe i i t i ~ l thc

seconcl for the inflii~nce of the Aiiid iri ttic annuiiis. The iriotl~l predicts t t i t a folloiving

waw speeds for Yoiirig's and Lamhs's riioclr

Therefore. the Lamb's mode is not affected by the presence of the fliiid arouncl the

elastic tube. while the speed of the propagation of the Yoiirig's niode clecreases due

to the constraint. For large !]o. CI approaches col i.e. the effect of the Riiid around the

tube becomes insignificant. Lamb (1898) came to the same conclusion I>v considering

the influence of an infinite iriviscid fluid media surrounciing the tube. Figure 2.2 shows

that the rate at which CL approaches coi. with increasing yo. depends on the ratio of

p; and p;. For p~siological applications. the most relevant case is the one where

p: = p j . - " z \r-L 1 - go . and wave speecls of constrained anci uriconstrained vessels Co 1

ciifFer bu less than 1% for y0 > 10.

Pressures anci velocities in the annulus c m be relatcd to those in the elastic tiibc

using the condition tliat the ratlial cornponcnt of fluid velocity rqiials the ratlial

component of tube velocity at 9 = 1. The ratio of velocities in the aniiiiliis and the

tube is giren by

The Rtiid in the annulus nioves in the direction oppositc to that of thr Huiti iii the

tube wit h a mlocity t h is inverselu proportional to g;i' - 1. It follo\rs frorri cc~iiiitioii

(2.27) that the volumctric Row rates iii the elastic tiibc and the iiriniiliis artB of the

sanie niagnituclc biit of ttic oppositc sign. i . ~ . the net voltirnetric flow rate iii rach Pa cross-section is alwq-s zrro. The ratio of thi: prrssiires is - tirries t liat of t l i ~ vclocity P"

ratio. Thereforc. iirilike t he voliiriie Hiis. tlic riiass Hiis is iiot zero iiiilcss pt = dl.

It c m bc swn from eqiiations (2.27) and ( 2 . 2 8 ) t h hotli the prrssiirr ;iiiti tlic v~lority

iri the arintiliis rapiclly clt.crease with !h.

\\'lien the ttibr iind the enniiliis iiïr occiipicd visroiis Hiiicls. t tic wavc. sp~cci

drpericls on A* ancl as well as !/o and p y p ; Howevrr. ive rcstrictetl th^ analysis

to the values of pararricters corresponding to t h blootl and CSF. Since the coniposi-

tion of the CSF is siriiilar to that of blood plasnia. it i w s assiirnrcl tliat the de~isity

anci viscosity of the CSF are tlic sanie as tliose of plasnia anci that p U / p t = 0 . 5 ancl

p",lp; = 1. Then. the t ~ a w speed is n fuiict ion of ri:. aritl y*. only.

The fluid velocities and stresses for the fluid in the tube are

(\\'oniersIey 1957a) and Bt = BI;

CL = ~ ' ; . ~ ~ ( n : ; . i ~ ~ ~ )

Flriicl velocities a i c i stresses in the anniiltis arc

11 .3/2,/)) 2 ( 1 ~ ( n ~ , i 3 / ' ! j o ) . J l ( n ~ i " / ' ! ~ ) - .Jo(r~~.i""!jo) I i ( f l l L . ~ ,

i: = (1) { [ i:1/2 ~ ~ ( ~ ~ , i : v ' ! j ~ ) ,Jo(nt,i"l- - ,Jo(rla i : ! l2! j0) 1&tt,i'"2)) - ILI IL'

(Ciroric et al. 2000b) where the n e r constants Ba and Ca are defiriecl as

The O coefficients arc

n-here the function Fio is giwn in eqtiation (1.8). and functioiis 3. Ç. ariti 2 are

ciefined as

1 ; ( r l t P t . ' j q R(!jo. n,) = 1 -

i:3,/2 u! !JO)

Again. the first terni on the right tiand side of cqiiatioris (2.36) - (2.35) iiccotirits

for the Riiitl in the flesil~le tiit~e arid corresponds to \\*orncrslty*s solution givrri iri

qtiation ( 1.6). ntiercas t lie srrotid terni arroiints for t h~ Hiiid in th^ aririiiliis.

The pliase speecl and transriiission per wivc length cari t>c calciilated Froni the

coniples 11-aw spe'cd (htahck 1968). \\i? shon the rcsiilts orily for t tic \biirig's riiotle

sirice it is the niodc relcviint for hloocl piilse propagatiori. Tlie pb;isc spc~cl and

transrriissiori prr x a l r Irrigtti ;ire shonn as fiirictions of B:, in Figiires 2.3 a i i d 2.4.

Tlie thick linr rrpresmts \\oiiicrsley's soliitiori for ii frcely riioving tiibr. Clrarly. thi.

effcct of t h constraint is to rctluce t h wwc spcctl and iricrcasc the atteriuatiori. This

effcct weakeris as !jo iiicrrases. and the solution approaclics that of \\imicrslcy. For

t hc parameters iised iri t hc stucly. the phase speed for t lie constrairied t iihe diffcrs

by lcss than 5% froni that predictecl IF Komersley for go 3 4. nliereas the influerice

of the constraint on the wave iitteriuation becomes insignificarit for go 2 2. -4s nt,

increases. the inviscicl soliit ion is spproached.

The relocity profiles c m be reconstructed by assiiniing that the ffow is tlriven

by a harnionic pressure in the vessel of arnplitiide Bt. -411 otiier constants can be

represented in terms of Bt usiiig the boundary conditions and the equation of motion

of the vessel. \Ve shon results [or half a cycle. when nt. = 2 and ri; = 10 in

Figure 2.5. Tlie velocity profiles are flatter for ri:. = 10. since the effect of riscosity

beconies weaker at higher \Voniersley nunibers. i.e. iriertial Forces dominate over

viscous forces. It is obvious from the figure that the bulk Ruid motions in the tube

and the annulus are in the opposite directions.

Figure 2.3: Phase velocity in ternis of CVorncrsley riiiniber r i t , for C u e 1 aiid viscoiis Htiicis.

The t liick lirie shows Woiriersley's soliit ion arici t tic t hiri liries soliit ions gis

for y0 rririging froin 1.2 to -1.0.

ln by the triodel

2.4.2 Case 2

Finallu. we consicier the geonietry whicli involves the prrserice of a viscoeliistic solici

representing the h a i n tissue. This relatively comples geornetry introduccs two new

geometric parameters. I/,, and .yc2. as \vell as physical properties of the solid: al1 thcse

can strongly influence the solution. .\loreover. by introducing another elastic structure

in the niodel. a e may espect that the dispersion equation has more than two roots.

However. since we are mainly intercsted in blood pulse propagation. WC restrict Our

attention to a physiologically plausible range of parameters and the Young's mode of

ware propagation. The brain tissue is modeled as a standard viscoelastic solid based

F i 2.4: Trarisrxiissioii per wnve length i r i ternis of \Votiiersley riiiriiber ri:, for Case 1

arid viscous Hiiids. The tliick lirw shows Worriersley's soliitiori aiid the tiiiri lirics solutioiis

given by the mode1 fur go ranging froni 1.2 to 2.0.

on the esperimerital results reportrci t ~ y Shuck niicl Advani (1972). The vesse1 radius

and clast ic propert ies arc chosen t o represent t lie larger craiiiiil and spinal art cries.

ancl are based oti tlic values reportcd by Hillen et al. (19S-l) and Hillen et al. (1986).

Zagzoiile and Marc-Vergnes (1986). and Sheng e t al. (1905). The typical wiclth of the

skiiil is approsimately 150 r r m (Kismans and van Oorschot 1993). whcreas the radius

of the largest intracranial arteries is approsimately 2-5 mm. \\é clioose !/O = 30 as a

pl~~siologicall~ plausible value that niasimizes the effect of the constraint. The value

of t ~ = / ~ î is based on the Fact that the subarchanoid space hosts some of the relatively

large veins. Cnfortunately. t here is lit t le quantitative informat ion in the li terat ure

about the dimensions of the perirascular space. Based on the qualitatire observations

F i r 2.5: Velocity profiles over a tialf-cycle for Case 1 aiid viscoiis fltiids. calciilatcd for

n i = 2 (top) ;irid r r t . = 10 (bottoiii). Ftill. thiii linc shows the profiles in tlic tiibe wtiilc ttic

broken line shows the profiles in ttic mriiiliis. The bolcl iirie rcpreserits tlic rigiil tiibe.

(Dnniasia 1995). it is assiimed that ,yc, = 1.5 is a rcasoiiable choicr. In arldition. it

vas shown in siitiulations thüt raryirig of ,yci frorri 1.2 to 2 has little cffect on ttir

soliition. .\Il the parameters iisecl in the siniiilation are siininiarizcd in Table 2.1.

The solution \vas obtainecl using the iteratire procediire out lined in section 2.:3.1.

Inspection of Figures 2.6 and 2.7. which displ- the phase speed and transmission

per waye length. respectively s h o ~ that tliere is rirtiially no ciiffererice betaeen the

predictions of the mode1 and the \\omersley solution. This means that. even with

the presence of the viscoelastic solid. large y, results in a negligible influence of

the constraint on the wave speed. It should be noted that replacing the stanclard

viscoelastic solid 11% h the Slawell solid or increasing the shear modulus by an order

of magnitude makes no visible difference. Again. a e can espress al1 the constants in

Figure 2.6: Phase veloçity iri ternis of Woiricrslcy iluniber T L : , for CLSC '1. The syiiibols

show WomersIey's salut ion.

terrils of Bt and calculate Htiid ancl solicl velocitics in ternis of the clriring prrssiirc in

the elastic tiibe. Table 2.2 gives area arerageci velocitics and Row rates for 11:. = 4.

The solicl and the fliiid laycr between the solid and the rigid tiibe hoth rrioïc niuch

slowr than eitlier the fiiiid iri ttie elastic tube or the fliiid betweeri the elastic tube

and the solid. However. due to the large cross-sectional area. the volume Ails of the

soiid is significant. The numerical summation of the fiow rates confirms t h the

consemtion of volume in the systeni is satisfied.

Table 2.1: Par;itiieters tised in the st iidy of piilse propagat iori in t hc c:r;iniiirii.

(1 Blood 1 0.7799 ( 0.7799 / '1.10 -- !I

Table 2.2: Average velocities and flow rates for diffcrent layers in the triodel of piilsc prop-

agation in the craniiim. The velocities are nor~nalized with respect to pressure amplitude

in the elastic vessel. and the flow rates with respect to flow rate givcn by the geornetry

including only inviscid fluids. The phase angle is caIculated with respect to the pressure in

the tube.

F i 2.7: Tra~istiiission per wavc lerigtli in ternis of Womersley iiiiniber ri:. for Case 2.

The synibols show Wmersley's solution.

2.5 Discussion

The irioclel shoivs t hat the mairi effect of the constrairit is to clecrease tlic wavc speed

for the Young3 mode. priniarily through the action of the riornial stresses on the

outer surface of the vessel. The magnitude of the normal stress in tiic anniiliis is

dictated by the conserwtion of volume in the system. This is e~iclerit from the

results for Ceonietry 1 and inviscid fluids (equations (2.28) and (2.27)). The action

of the normal stresses on the vessel is primarily espressed t hroiigh the O,, coefficient.

ahich can be interpreted as an increase of the effective radial inertia of the vessel dite

to the interaction with the surrounding media. The effect of the constraint weakens

rapidly with increasing y. since the crosçsectional area of the annulus becomes large

comparecl to that of the tube. and a sniall pressure gradient is stifficient to create a

volume Bus of the sanie magnitude as that in the tube. Kben the medium in the

annuliis is viscous. the effect of the continuum surrouiiding tlic vessel is not eliminated

e w n when the transverse dimension of the system is infinitc (Dinar 1973). Howcver.

the results for the Case 2 differ ver- little from the \Vomersley's soliition. \\é attribtitc

this to tlie k t that the viscosity of the CSF 1-r between ttic vessel and tlie hraiii

is relatiwly low.

The niotlel capt iires the conserntiori of ~olunie. characterist ic of t lie cranial blood

flow within a franiework of a classical liiicar arialysis of piilscl propagation in the

ürteries. Difficiilties do arisc in t hc clioice of paranieters. siich as the tliickricss of the

prrivasciilar spacc. Fortiinately. oiir siriiiilations showcl t h the s«liitioii is iiff~c.ted

only if it is assiiined that the perivascular spaw is an orclrr of rti;igriitiick sniallcr tliari

t hr vesse1 radius. The niodel ~Ierrionstrates: i ) t liat ttic s p ~ t d of piilsr propagatiori

in the irit racrariial ~esscls slioiilcl riot bc affectcd 11)- t lie ski111 coristraint. i i ) t hat

the s p e d at whicli a clisttirbanccl propiigatps in the CSF layrr is the sanicl as t h

speecl of t~loocl piilse propagatiori: aiid iii) that thr CSF p d w is a ronscqiiericc of the

interaction betw.w the propagatirig biood piilse aricl tlic siirrotindirig riidia.

Chapter 3

Cerebral Circulation and CSF

Pressure

Introduction

-4 grorriet rically detailet 1. opencd-loop niocld is tleveloped to ~siiri i ir i<. t hct passiix~

rpsponse of t hc crrcbral circiilat ion to G z ;icccleration. Followitig t ho riwilts prcwritctl

iri Ciiaptcr 2 . i t is asslinitd t k i t t lie spcrtl of piilsr propagat iori iii t l iv irit racmiii;il

r~sscls is not affrctrtl 11- the ski111 coristraint. m c l that tlir CSF ( a i l he rtiarartt~rized

by a spatially constant. tinie dcpenclerit pressure. The 1-D govertiirig rqiiatioris arr

lincarized and steady aiitl pulsatile cornponeiits of the Hoa ;ire trmteil s~paratcly. For

the stcady Hori-. the msciilar network is approsiniated a resist iw network. For t hr

pulsatile flotv. a srniill ariiplitude harriioriic rvnw propagation tlieory is i isrd.

3.2 Methods

3.2.1 P hysiological approximation

The vesse1 network representing the cerehral circulation is shown in Figure 3.1. The

parameter values are given in Table 3.1. The mode1 geornetry and rheological data

are taken from Hillen et al. (1984), Hillen et al. (1986). Zagzoule and llarc-\érgnes

(1986). and Sheng e t al. (1993). The extracranial portion of the network is repre-

sented b - carotid and vertebral arteries and by jugular veins. which are al1 assumed

Figure 3.1: The niodcl of cerebral vascular systeni. The niodel bcgins at thc root of

corim~on carotid and vertebral arteries, and ericis with the interrial jugiilar veins. The vessels

surrouridect by the broken line are cranid vessels subjected to PCSF. Arrows inclicate the

pntb used to illustrate the blood pressure drop for different values of Gz (Figure 3.3). The

parameter values rised in the st udy are given in Table 3.1.

Vessel

nimber

Vessel

Carot id arteries (ext racraiiial)

Vertebral arterics

Basilar artery

Carotiti arteries 1 (iritracranial)

Posterior ccrebral arteries 1

Posterior conini~iiiicat irig arterics

Carot id artcries I l (irit racrariial)

Xnterior cerebral arterics 1

Posterior cercbrai art cries II

Micldle cerebral art cries

.-\ritcrior ccrcbral srtcrics II

Main branches of

cercbral artcrics

Pial rietwork

Ititraccrcbral art cries

hlicro circtilat iori

Intraccrebral vciris

Pial vciris

Cercbral vcins I

Ccrebral vcins II

Lorigit ildirial siniises 1

Veins

Lorigituditial sinuses II

Veins

Trarwerse siririses

.Jugulas veins

Lerigt ti

L [on]

Table 3.1 : Parameter valties used in the mode1 of cerebral circulation.

to estend vertical15 from the heart level to the cranit~m. The intracranial vessels

are oriented horizontally and are al1 assumecl ta be at the eIevation of A H = 30 crn.

Smaller intracranial vessels are combined in parallel. ancl represented by eqiiivalent

wssels. The esternal pressure acting on the blood vessels in the crariitirii is that of

the CSF. A11 the vessels are assumecl to be iixiaily iiniforni circiilar elastic ttibes at

Pt = O.

3.2.2 Mathematical formulation

Linearization of the Governing Equations

Esprcss I'. -4. P. and P, in the governing quat ions (1.9) and ( 1.10) as L i , + IL. .As + a . Ps + p. ancl PeS + p,. rrspectivcly. Uppcr case variables witli s h c r i p t S tlcnotr

steüdy state qiiantit ies and loiver cas<$ va-iat~les dcnotc sniall perturbation qiiantitirs.

Stibstituting the esprcssioris for L-. A. P. ancl P, into t lie govrrriing ~qiiations ancl

rieglecting the second order ternis ar obtain

Xote that the linearizecl forni of the viscous friction term given b\. ecliiatiori (1.11) is

Assuming that the steady state component of the flow satisfies the governing cqiia-

tions. and that Gz # Gz(t) . the governirig equation can be separated into steacly and

unsteady component ( McClurken et al. 1980). The stcady state go~erning eqiiations

are

Denoting - as Es. ne can rewrite the iinsteady componcnt of the gowrn-

ing equations as

Figure 3.2: Tube law for t hick- walled and t hixi- walled elas t ic vessels. Pi is nornialized wit li

respect to El. Solid squares: thick-walled vessel ( h / R = 0.25). Hollow squares: thin-walled

vessel ( h / R = 0.1).

Tube law

The form of tube law used in the simulation is essentially the one slioivn in equa-

tion (1.19) (Bertrarn and Pedley 1982). modified to account for the fact that thick-

aalled vessels (arteries. arterioles) do not collapse until the transniural pressure

reaches a critical negative value (see Figure 3.2). Following Bertram (1987). for

thick-walled vessels the critical transniiiral pressure is defined as

P, = - Eh"

4 R"-

and R is calculatecl froni .A0. For thin-tvallecl wsscls. P, is assiinied to bc zero. Tlie

fiiriction P(a) is given b .

where n, is the relative area at the point whcre ;i v~sscl starts collapsirig

CSF pressure

Tlir ~riathcniatical forniiilation of thc iritcractiori t~rtwccri the hlood flow iri ttic cra-

riiiini arid the Pc.SF is t > i ~ ~ t d on the followirig iissiinipt ions:

Tlic interctiangc bctwen tlic blood end CSF voliimcs is slow iiiid ciin t)o rit>-

glectccl for t hc t imc scales of intrrcst . Th~reforr. the .\ Ioriror-Kelly priririplr

irriplies t hat the cranial bloocl ~oluriie is coriscrwi at aiiy instant of tinir.

PCçF is time dependent but tioes not Vary spatiiilly. rsccpt for t lie Iiylrostatic:

pressure clifferences. Since iii t h irio(1el d l the cranial vessels are at thc samr

The second point reqiiires fiirther esplanation becaiisc it rnay scerri that it is not

compatible with the results of Chapter 2 . In this case. the cranium contains a niiniber

of vessels in series and in parallel. and the CSF will iriteract with al1 of them rnther

with a single vessel. Also. unlike the geornetn used in Chapter 2. tdiere an infinite

rigid tube is iised to represent the skull. here we have to take into account t h the

skull dimensions are much smaller than the wave lengt h of t be piilse. Also. the no-slip

condition requires that the CSF velocity be zero at the skull. Therefore. it is unlikely

that significant spatial gradients of PCsF esist in the skull. Final15 the arteries and

reins are often located parallel to each other and the perturbation of PcsF due to

the arterial pulse will be immeciiately transmitted to the veins. rather than haririg to

travel along the w c u l a r tree.

The connection between PCsF and the blooci pressiirc in the rranium is obtained

froni the condition that the crririiiil blood volume is constarit. The conservation of

the cranial blootl voliime ( I ;) can be espresscd in two n a y s :

The siim of al1 the intracranial vessels voliinics is constarit.

I , = corist (3.14)

The rate of the hlood flow enteririg the skiill is c~liiiil to the rate of thc biood

flon leal-ing the skiill iit ;in' instant of tirrita.

T h two formtilatioris arr in fact cqiiiulerit. bit t h first fornidation is niorp cori-

vrnirnt for motlcling steiidy flow. and ttic srconcl is rriorc roriwriirrit fur r~iud~lirig

piilsat ile How. Sirice t tir vesse1 voliinie (cross-scctional anla) is relatecl t o t lie t rans-

niiiral prcssiire t liroiigh r tic t ii be Iaw. the first forniiilat ion provicies t hr conncrt ion

hetseen PCSF a r ~ d the blood prcssiire in the iiitracranial vrssels. Siniilarly. for t h0

piilse propagation. ttie blood How and prrssiirc arc linearly related. and tlic srcuritl

formulatiori provides tlic conncctiori bc twen the blood arici CSF pressure piilse.

-. - - - - - . - -- ---

Table 3.2: Fourier componerits of the input arterial pulse. The heart rate is 72 beats per

minute.

3.2.3 Boundary conditions

-'

The boundary conditions are defined in terms of the blootl pressure at the nioclel

ends. The rnean blood pressure at arterial end ( P . 4 is 103 m m H g and the mean

Fourier cornponerit #

.-\nip1itucle[mrnHg]

Phase h g l e [deg.]

G

0.63

160

1

13-97

3

3.36

173

'> - 6.94

- I

0.63

232 4 1 0

4

0.64

'131

8

0.50

163

- 4

1 .

207

9

0.28

A

191 1

blood pressure at the renous end (PI - l r ) is 5 rnmHg. These values of the niean blood

pressures are chosen to represent central blood pressures and are maintaiiied constant

regardless of Gz. The pulsatile component of the blood pressure at the arterial end is

taken from Remington (1963) and tleconiposed into nine Fourier components whicti

are surriniarized in Table 3.2. The frequency of the first Fourier ronipotieiit of 1.2 Hz

corresponds to the hcart rate of 72 beats per minute. I t is assurncd that ttirre is no

puisatility at the renous end (Guyton and Hall 1996. pp. 163). Gz is varicd frorn -.j

to +10.

3.2.4 Solution for the steady flow

Steady flow in a vesse1 segment

Consitler stcady flow tiiroiigti a vmstll srgriient of lengt ti 1. Froni w r i t iriiiity. t lit. ratr

of Row QS = Ls.-ls is constant dong ttie length of thr scyyiimt. KP USP l ( l l m d 1 ( 2 1 to deiiotc the variahlrs kit tlic inlr t ;incl the oiitlct of tlicl srgnicbnt. rrsprc-tivrly. and

1:;; to deriote the M i w r i c ~ brt\vrrii tlic rnliirs of tlir t a r i a t h iit tlir i1ilc.t and t h !

oiitlrt. The rlevatioii (H) of a p i r i t in the scgmcnt is giwti hy

Csing equatiori 3.16 antl miiltiplyirig the rnomentiim rqiiation by p WC ohtain "Bcrrioiilli-

Poiseiiillc" rqiiat ion.

AI1 die terms in the hrackets on tlic left hand side tiare the dimension of pressure.

Pressure. Ps. is ülso called static pressure. ;incl pGzH arc rcferrrd to as dynaniic - and liydrostatic pressiires. respectiiely. antl the stim of the t h e is calleci total prcs-

sure. Equation (3.17) states that the drop in total pressure in the vesscl segnicnt is

due to action of viscous forces. The net loss of total pressure across the segment is

obtained by integrating equation (3.17) frorn zero to 1 with respect to r.

mhere the resistance of the segment ( R I ) is given by

d,-ls 9ow introdiice t t i ~ assiirnption that - is small. Le. the change in the cross-

dx sectional area along the vessel length is very gradua!. From the continiiity. tliis implies -

that CFs- is also small. Then. ecpations (3.17) ancl (3.19) reclucc to ~ L C

and 1 (Ps + p G z H ) (I:;

RI = 1 R(.Ls)d.r = (:3.2 1) Qs

The conclitioii iinder whicli the iibovc assuniption is rülicl can be clctcrniinrd corii-

bining eqiiations (3.6) and ( 1.17). arid espressing t tie coiiwctive terni in t hc moriieri-

The non-dimcnsional pararrirter S is referrecl to as the sprrd inclrs and (irfinccl as

(Shapiro 1977). wlicre co is g i ~ e n in eqiiatiori (1.17). It follows frorri eqiiation ( 3 . 2 )

that tlie convrctire tcrm cari hr nrglrrtc~i if S < 1. In the viisciilar systrni. it riiïi

bc espcctccl that S be niurti lrss than iinity for tlic iirtrrics. niicro-circtiliition. iion-

rollapsrcl veins. ancl severely collapsed vcins. Howcver. for niodoratcly collaps~d w i n s

this assriniption is lcss acciirate sirice L> ancl co nia\- be of the sanie orcler. This issiic

Thc probleni is further sirriplificd by assiirriing that in iiclciition to ilAs/& being

sniall. the segment is sufficientl!- short so that .As 1 ( 1 1 2 As /(., . Then.

and the flon rate through the segment is

To justify this assumption. the ressels in the network are represented as shorter

segments in series. The level of discretization is chosen based on the compliance.

length. and the oricntation of a vessel. The more compliarit ressels and the verti-

cally oriented vessels require finer cliscr~tization. since they are more likely to eshibit

sig~iificant changes in the cross-sectional area. Thus. the jugiilar rein is diridecl into

100 segments. the estracranial arteries iiito 25 segnients. the intracranial arterics and

micro circulation into *5 segments and the intracranial reins into 10 segments.

Steady flow in a network of vessels

In each junction of the ressel network consermtion of niass Iias to apply. For a

junction of nt rcssels this is espressed as

rri

-41~0. the pressiire at the junction is the sanie for ail t h vcsscls in the junrtion.

Tlicse conditions are analogoiis to Kirchhoff's laws for t tic) rwist i w elect rie circiii ts.

Conibinirig ecpatioris (3.2s) and (3.76) for a jiinction. we obtairi ail illgcbraic eqii;itiori

in ternis o f pressures ancl cross-sectiorial arras of t hc srgrrirnts iri t lie jririrt ion. The

eqii;itiori is rion-liriear. since the pressure in ;i vcsst)l scgriic3iit affwts its ;iïtaii arid thils

its rrbsistarice.

Conservation of the cranial volume

If JI is the total niiniber of vesse1 srgnients in the craniiini. tlicri c!qiiation (3.14) ciiii

be rcwri t tcn as .\ f rri

~ ( n s . l o l ) l = E(;lol)'

Also. CI^ c m bc espressecl ~splicitly in ternis of the transmiirai pressure iising eqtiation

Equations (3.27) and (3.25) can be cornbined into a single algebraic eqiiation.

Solution procedure

Applying conservation of mass and the continuity of pressure at each junction in the

network yields .V algebraic eqiiations. where 'i is the nurnber of junctions in the

network. Fiirthermore. pressures in the junctions are reiateci to flow aiicl vessel area

through equations (3.1'2) and (3.25). PCsF is calciilated from eqiiation (3.27) and

(3.28)-

The solut ion is obtairietl iterat ively. wit h each iteratioii contiiinirig t hree steps

1. Blood pressiires for cacli vessel segment are cleterniiried assiiniing that the ves-

sels are rigicl ancl with the cross-sectional areas det~riiiined in the previoiis itrr-

ation. This reqiiires solving a s - s t m i of S lincar iilgebraic ecpatioris ot~tairied

t- corribiriirig cqiiations (3 .25 ) aritl (3.26) at cach jiitictiori.

'2. PCTsF is det~rniinetl froiii the coriscrwtion of the rraniül hlood voliinic. iisiiig

the blood prrssiire froni the ciirrciit iteratioii. This anioiints to solving a single

rioti-litmir algcbraic cqiiation.

3 . S e a valiies of cross-scctiorial area arr coriipiitcd froni the tube law. tisitig the

ciirreiit valuias of Ps aiid Psr For tliis piirposi!. Pt i t i ii vess~l scgniilrit is (:;llcii-

1;itcrl iisirig tlic arit tiiiirtic nieaii of t lie prossi1rc.s iit t t ir iwls of t lie sclgriit'rit.

The process is repc;itetl iintil the cliffcrcnce ht~tiwt~ii tlir v a r i a t h corriputrd in t a o

3.2.5 Solution for the pulsatile flow

The speed of pulse propagation

\\C consider the propagation of small-amplitilde harnionic waves in a vessrl segiiicnt of as

a constant steady cross-sectional area As. Thercfore. - = 0. and. [rom t h steacly dx ars

statc continiiity eqiiation. - = O. Under t liese condit ions. eqiiat ions (3.8)-(3.10) ait.

mq- be rewritten as

Equations (3.29)-(3.31) ran be differentiated with respect to t and x and combin~d

into a single partial differential ecpation in terms of the area or velocity perturbation a p B(P - P.) E~ an

(note that - = - ax as - =~jz)-

;\ssiiniiiig that area and velocity are harnioriic in spacr iintl tinic. the folloaing cs-

pression for the ~ v a w spced can be cl~riretl froni eqiiatiori ( 3 . 3 2 )

Since ; is iiirers~ly proportional to the \ \bni~rslry niirnhw. it rtyrrwrits the ratio of

t hc viscotis forcc to t ransicn t iricrt i d forcc.

If ttic aiialysis is rcstricted to zero Gz. it c m hc ~ . s p c ~ c t i ~ l tliat al1 the vrssc!ls are

openecl aacl that S « 1 every~viicrr in the nctwork. In tliat casr. the ivaw spcd

The wwe propagation is dispersive. due to the visroiis friction. hiit the ivaw speecl

is not moclifiecl bu the CSF.

Pulsatile area and flow in a vesse1 segment

In a vesse1 segment the pulsatile component of the area and vrlocity ivill be a cornbi-

nation of incident and reflected waves:

where d i ) and a(-) are the amplitudes of the incident anci reflectecl area waws.

respectiwly and E(+) arid $-) are the aniplitiides of the incident and reflected velocity

nnves. respectively. The esternal pressure perturbation is

wliere Fe is the amplitude. Es

Since the perturbation of transmiiral pressure and flow rate are p, = -a. ii~itl -4 s

g = A s i r ; [>B. tliey can be witteri as

- 4 - 1 ~ i ( t - f i + j j i - ) e > i ( t - : ) - P t

At t Lie inlet arici t tic oiit let of a wssel segment. t lie followitig tiolcis

AL) u ( t - b ) + p p e l & . ( t + ; ) P /(4 -P. = Pt e

Lsing equations (3.38). (3.40). and (3.41) n e c m write the followirig

Pulsatile pressure and flow in a network of vessels

Siniilarly as in the case of the steady Row the conservation of mass reqiiires that

Also. the pressiire at the junction has to be t hc same for al1 the vessels in the jiiriction

(Lighthill 1978). Csing equation (3.42) for a junction. ive obtain an algebraic eqiiation

in ternis of pressures at the ends of the segments in the jiinction and of the estcrnal

pressure.

The conservation of the cranial volume

Equation (3.1.5) can be rcwritteii as

ahcre .y,, is the niinibrr of vesscls eritrring thc crariiiini ancl riiiriibcr of vrssrls

Solution procedure

Applying ronservatiori of rriass and continuity of pressilri. i it t w h jiinction iii the

nctu-ork yiclcls .V liriear algebraic equatioris. Ttir .Y + 1-th cyiiation nccrssary for

deterniining p, is eqiiation (3.44). Thc soliition of the systtwi of .Y+ 1 linear aigchraic

cqciat ions yields prcssiires in the j iinct ions and t hr piilsat ile corriporicrit of the PCSF.

The flow rate in the juiictions cari be calciilateci frorri eqiiatioii (3.42). and pressure

ancl Row at any point in the vesse1 segrrierit froni ecliiation (3.38). The proccss is

repeated for each Fourier cornponent and the solutions arc siipcrimposrcl.

3.3 Results

3.3.1 Steady flow

Blood and CSF pressure

Figure 3.3 shows the pressure drop along the vascular tree for three different values of

Gz. For the sake of clarity. the resrilts are chosen for an arbitrarily chosen path rather

Relative Distance

Figure 3.3: Blood pressiire clrop dong the cercbral vascular tree. Hollow trianglw zcro Gz.

Solid squares: -5Gz. Hollow scluares: +jGz. Tlie particulnr path diosexi to be clisplayed is

iriclicated by arrows in figiirc 3.1.

than for al1 the ressels in the network. This path is indicated by arrotvs in Figure 3.1.

For zcro Gz blood pressure remains positive everywhere in the system. Pressure loss

is only due to viscous forces and it occurs alniost rnainly at the lerel of c~rebra l micro

circulation. For -.3 Gz the pressure change in the caroticls and jugular reins is piirely

h-drostatic. inclicating that t hese vessels are fully openecl. Blood pressure is elerated

in al1 of the intracranial vessels. However. the pressure drop in the micro circulation

is of the same magnitude as in the case of zero Gz. For +5 Gz the blood pressure is

negative at the level of cerebral micro circulation. The pressure cume on the venous

Figure 3.4: XIean CSF and venoiis blood pressures. Soiid squares: PrSF . Hoilow squares:

blood pressure in the cranial veiris at the point where t hcy joiri the diirai siriilses.

siclc is no longer a straight line. indicating t h the estracranial veins are colliil~~e~l

and that their resistance is significant. h much smaller prwsiire clrop at the lewl of

micro circulation indicatcs t hat the blood How is decreasecl.

The relation betwen Gz and PrsF is displayed by the line nitti solicl squares in

Figure 3.4. The line a i th hollow squares shows the blood pressure in the intracranial

veins at the point d ie re they join the dura1 sinuses. For zero Gz the mode1 preclicts

a PCsF of 9 mmHg which is well mithin observed physiological limits (Takernae et al.

l98T). For negative Gz. the increase in PCsF is essent ially hylrostatic. Consequent ly

for -5 Gz PCsF is approsimately 100 mm&. For positive Gz. PCsF becornes negative

Figure 3.5: Cerebral vascii1:ir resistmce. Solicl scliiues: total ccrcbral vasculnr resistarice.

Hollow squares: the vasciilar resistancc of the jiigiilnr vciris. Solid triarigles: the v;tsciilar

resistarice of the cranial vessels. Hollow triangles: the vnscular resis tarice of t hc ext rricrniiial

arteries.

before Gz reaches +l and continues clecliiiing with increuing Gz. The rate of tlecline

is. hoiwver. much less steep than in the case of negatiw Gz. and the relation between

Pesa and Gz is non-linear. The venous and CSF pressures stay close togettier for the

whole range of Gz esamined. Since the renous pressure is the loi~est blood pressure

in the craniurn it follows that the intracranial vessels should not be collapsed.

Relative Distance

Figure 3.6: Relative area dong the length of tlic jiigiiiar vein at + X z . The zero relative

distarice corresponds to the Icvel of the lieail ;iricl rclative distance eqiial to iiriity to the

level of the heart.

Vascular resistance and blood flow

\\é esaniinect the total vasciilar resistance of the circuit. as twlI as ~ascular resistarices

of tliree broad groups of cerebral vessels: the first group includes al1 of the cstracranial

arteries. the second involws al1 of the intracranial vessels. arid the tliirtl grocip is the

estracranial (jugiilar) wins. nhich in this moclel. Tliese resistarices are definecl as

f0Ilows:

Figure 3.7: Cerebral blood How. Ttiick solid liiie: cerebral blood How predictecl by the

model. Fiiitit broken line: full pressure recovery in t Lie jiigi11;ir veins. Fairit solicl liiie: no

pressure recovery in the jugiilar veiris.

Estracranial arteries

a Cranial vessels

Est racranial (jugular) veins

Total

she re Q is cerebral blooc

R, = R, + R, t R,,

i flow. P.-ic is the arterial pressure a t the top O f the estracra-

nia1 arteries. and PiWC is the vcnoiis pressures a t the top of the jugiiiar wins. It c m

be tleduced from equations (3.43)-(3.48) that the cerebral blooti flow can he csprcssetl

wliere P, = P,iri - Pisri is perfusion pressure.

The vasciilar rcsistarice and crrcbral blootl How with varying Ch a r p rlisplayd iri

Figures 3.5 ancl 3.7. respcctively.

Scvcrd concliisioris can bc clraïvn froni t l i ~ rcsul ts:

( i ) For zero and ncgative Gz total sesistance is dorriiriütcd hj- ttic visroiis

friction in the c ~ r r b r d micro circiiiiition. For positivr Gz. colhpsr of

thr cstracranial vcins conics into play ( s w Figiirc. 3.6) aiici tlir resist;iriw

risrs esponentially with iricrrasiiig Gz. Consrqiiriitly. at f 4 . X ~ <*f>t+<>t>~id

t~loocl HOK is approsirriatcly 400 crrl"rr1irr. di ich is t~rlotv t h rriiriiriiiirii

recpired for niaintaining a riornial hraiti prrfiision (Finnerty et al. 1954:

I<cty et al. 1947).

(ii) The iritracranial \-essels arc siibjcctcci only to rriirior fiuctiiiitions o f ttir

transniiiral presslire when csposetl to Gz and ;irr protccted from collapsc.

Conseqiiently. the vascular resistance of the iritracranial vcssels is not

dependent on the mechanical effects of Gz.

(iii) The vascular resistance of the estracriinial arteries is srnall and ncakly

dependent on Gz. The arterial resistance never contributes more tlian 3 %

to the total cerebral vascular resistance. Thick arterial walls make the

collapse unlikely. For +10Gz the ixicrease in the arterial resistance is less

than 10%.

(iv) The jugular veins are the main contributor to the increase of vascular

resistance during positive Gz. The t hin nalled veins are highly susceptible

to collapse. Khen Gz is positive. the blood pressure esperierices its lowest

value at the point where tlural siniises drain into jiigular veiris. Of al1 the

sites in the system. this is the one where the collapse is most likely to

occiir. Figure 3.6 shows that at +5Gz for a siibstantial portion of ttic

jiigular vein ns < 0.2. Before Gz reaches +5. R , = R, aiid Rt is dout~led.

3.3.2 Pulsatile flow

Blood and CSF pulse

.-\II the resiilts for the pulsatile How werr obtairictl for zero Cz. Figiire 3.5 show

blood pressiire wave-forrns at several locations iri the rrioclcl. bcginiiirig witti the input

prrssiire at the root of carotid arid wrtebral ;irteries. The piilsatile coniporicnt o f the

bloocl pressiire is siiperiniposetl on the meari blood prcssiirr ohtairied frorii t ht. steatly

state nioclel. Thc blood piilsc sniears oiit as it progrrssrs clo\vri tlic tasciilar trrcl. At

the lwel of the capillarics. tlic piilsc is ver' i ~ ~ i \ k . Ttir PCsF pti ls~ is shoivri in Figiire

3.9 (solicl sqiiares). togettirr with the intrammial vrnotis piilsc (tiolloa scparrs). Thc

PCsF p~tlsr aniplit~idc of the orclrr of scvcral rtlrrcHg cliicli is ptiysiologirally plaiisil~lt~

(Crsino 1955). \érious ancl PCsF piilse are vcry similiir. ntiirli is also corisisterit with

observations reportecl in t hc litrrature (Hamilton ~t al. 1943).

The effect of the skull constraint on the arterial pulse

The effect of the skiill ronstrairit on the pulse waveforni t ~ a s esaniined by riinning

the nioclel withoiit the skull constraint and corriparing the resiilts with those obtainetl

nhen the llonroe-Kelly principle is npplied strictly Ki t hout constraint. p, is zero

everywhere. Figure 3.10 shows pressure rsaveforrns in t lie irit racrariial port ion of the

carotid artcry for a i th and without the skiil1 constraint. It is evitlent froni the that

these two cases yield almost ideritical results. rncanirig that the skiill constraint has

little influence on the piilse wveform.

40 - -

-

3

60 - œ

- -

40 -

2 A v

O 0 I 1 I 1 1 1 1 1

0.5 1 1.5 2 Time [SI

Figure 3.8: Blood presstire wavefortris. Solid squares: the cxtracrariial mrotid artery.

Hollow squares: t lie intracranial carot id artery. Solid t rinrigles: small i nt rxrania l arterirs.

Hollow triangles: capillaries. Solici <liaiiionds: iiitracranial wins. The hcart rate is 72 beüts

per niiriute.

3.4 Discussion

3.4.1 Steady flow

Vascular resistance

The mode1 shows that. eyen if the normal central blood pressures are maintained. an

increase in vascular resistance may lead to inadequate blood flotv diiring Gz stress.

2.5

2

n a L

1.5 L! 3 cn V)

1 ;

0.5

O

Time [SI

Figure 3.9: CSF and venoiis pressure pitlscs. Solicl squares: the CSF pressure piilse. Hollow

squares: the iritracranial venoiis pulse. The Iieart rate is 72 bcats per niinilte.

Therc is a substantial difference in the estent to wliicli different portions of the cerc-

brovascular systern respond to gra~itational stress. The only vessels significantly

affect cd are the est racranial reins. cliaracterized by t hin. compliant wlls t hat reatl-

ily collapse at negatim transmural pressures. Consequently. the wnoiis resistance is

strongly influenced by Gz. In contrast. the resistance of the intracranial vessels is

largely independent of Gz. tvtiile the resistance of the estracranial ürteries is insignif-

icant regardless of Gz.

A simple mathematical esample can be used to illustrate the significance of the

renous portion of the vascular systern. K i t h no resistire losses there is only a hy-

Time [s]

Figure 3.10: Ititracraninl carot id blood piilse. Solid squares: wi t h constr:ritit. Hollow

squares: withoiit cotistraint. The heart rate is 72 bents per miriiite.

drostatic pressure drop of pGzA H in t lie estracranial artcries. Flow t hroiigli t lie

constant cranial resistance causes a further pressure drop QRc. Finally. iii the es-

tracranial veins the blood esperiences a hydrostatic pressure rise of pGzAH and a

resistive drop of Q R,(Gz). Thus. the central renous pressure is given y .

Sote that the hydrostatic terms cancel and the equation can be rearrangcd to

For zero Gz the jugiilar resistance is approsirnatcly zero. and Q = 2. For Gz > O R,

P the jugular resistance becomes significant and Q < -. R,

The effect of the Gz-dependent jugular resistanee and its effect on cerebral blood

fiow can be disciissetl in tcrrns of the estent to wliich pressure recovers as the hlood

ciescerids towards the hcart. For this piirposc anotlicr Forni of tlic espression for

cerehral blooci flow cari be dcduceci from eciiiatiori 3.50

If the veins were rigitl. R , 2 O. Pimc = PieII - pGz1H. and tlic t~loocl flow ~vould hc

(the hrokcn fairit lirie in Figure 3.7) . This situation is rrfcrrcd to iis "siptiori". The

ot her est renie case is t tic oric in w1iic.h viscoiis friction ir1 t lie jiigiilar vciris csact ly p ~ z i H-

balance the hyclrostat ic prcssiirc coniponent . Ttirri. R,. = Q

. hloocl prcbssiirc is

Pi-[, ( w r y w h ~ r r in t l i ~ jiigiilar veins. and the hlood Bow is giwn hy

(the solicl faint

approsiriiately

liric in Figure 3.7). Iri this c ; ~ . t~lood Hnw to tlir txaiii is zero at

t4.3 Gz. Sinre Pi.lI 2 O this sitiiation rrsiilts in sirriilar flow ratr

as if tiie jugiilar veins w r r absent. ancl is referrrd to as "wtcrfall". Accorcling to pGzAH

the mode1 O 5 R, 5 . and for positive Gz. Pierr - pGzAH 5 Pi-c 5 P.\/[. Q

Therefore some gravitatiorial pressure recoverj- does esist c t m tdim t h t-cins ;ire

considerably narrowed and Q > O for Gz 2 +4.L Or1 the other hancl the pressure

recover- is not complete ancl Q decreascs with iricrcasing Cz. Conseqiiently. blood

Bow predicted by the mode1 is some~vhere in betweeri these tno estrcnie cases (tiiick

line in Figure 3.7)

The effect of fluid inertia

The analysis was based on the assumption that the conwctiw term in the momentum

equation can be ignored in al1 the vessels in the nettvork since the gradient of the .-. . 0.4s

change in cross sectional area is small. The results show tliat - is small for ail the dx

vessels escept For the partially collapsed jugular veins. At +Gz. the jugular veins are

collapsed a t the leve-el of the head. ancl slightly inflatecl at tlie level of the lieart. The

transition froni the collapsecl into infiated state occurs over ü relatively short distance

whcre the area gradient is steep. In ddi t ion . the R, in the jiigular veins is of the

order of 10- for positive Gz. Therefore. t lie effect of fliiid inertia i r i q be iniportant.

at least iri the segnient of the jugiilar veiiis where the sharp iiicreasc in area occiirs.

CSF pressure

The resiilts siiggest ttiat CSF and venoiis pressures i r i the cr;iniiini liiivr to be al>-

prosimatrly the sarrie in ordcr tbat the cranial voliinie tw c~oiiscrved. Tlie total t~lood

voliinie c m be conserved orily if soriic vessels are dist~nclrd ntiile othcrs arcb collapsccl.

I t is rrasonable to assutrie thiit the collapsr occtirs after the lcvel of niirro circiilatioii

wherc hloocl prrssure csprrierices a stil>staritiiil drop. Tticrcforr. P<+sF mist ~ I C 10wr

t hari t lie artcrial biit tiighcr t l i m t hr vcnocis prrssiirt3.

.\ niore prwisr asscssrnrrit of PcSSF in rclation to xrtcriiil and vmoiis prrssiircs cari

t x obtairiecl iq- considering tlic ~l i is t i r properties o f iiitriicrariiel blood vrsscls. Sinrc

thr artcries have thick walls. the total arterial blood voliirnc. will riot hc vc3ry srnsi-

tive to changes in the transniural presstire. 011 thr othcr liarid i-rins arc estrrrii~ly

corripliant and the venous voliinie is vrry srnsitivc to clianges in the trarisrtiiiral pres-

sure. Thcr~forr . if PCsF is to procliicr esactly the sanic vu-oliirrie diangr ori the tirterial

aiid vcrioiis siclc. it stioiild be rniich c los~r to thr vcnoiis tkiari to the arterial blooc1

pressure.

3.4.2 Pulsatile flow

The resiil ts show siirprisingly lit t le difference bct w e n the piilsc propagation in an

unconstrairied vesse1 network and in a network in which the consenxtion of volume

is iniposed. Essentially. the conservation of volunie is assurecl by srnall acijustnients

in the PCsF which are observed as the PCsF pulse. The similrrrity between PCsF and

renous pulses are esplained in the same manner as for the steacly state coniponent of

CSF and venous pressures. Being more conipliant t han the arteries the veins require

much smaller vdues of the transrnural pressure to esperience the same change in

volume.

The espcrirnent wit h the variable cranial volunie siiggests t hat t lie cranial com-

pliance is mainly due to the capacitance of the intracranial vessels. Thcreforc. a

strict application of the Slonroe-Kelly doctrine is a ralid approsiiiiation for future

rnocieling.

Chapter 4

Cerebral Autoregulation, Venous

Properties, and Central Blood

Pressures

4.1 Introduction

Thc resiilts p r w r i t d Chaptcr 3 show that the clynèirriics of the jiigiiliir wiris plays

a crucial rolc in drterniining cerebral blooti How iiiicler Gz. In wtitriist. the cffect OF

Gz oii the estracranial arteries and intracranial vesscls is wak. Thcrrforc. i t shotilcl

bc possible to tlrastically retlitce the lcvel of coniplcsity witti which the estrarranial

artcries aiid intracranial wssels are represented iii the modcl {vit lioiit affect irig t lie

accuracy of the rcsults. -4 simpler gcornctry nill. on the othcr Iiand. allow a more

clct ailcd tnodelirig of the estracranial v in s since t reüt ing t lie wiris as purcly resistiw

conduits may riot be sufficicntly accurate. Furtherniore. a sinipler geonietry will aliow

introcluction of a mode! of cerebral aiitoregiilation. and esaniinatiori of the effert of

varying central blood pressures. venoiis stiffness. and the heart-to-liead distance. The

analysis is restricted t O steady Row on!.

Figure 4.1 : Siniplified geonietry used for motleling the steaciy conipotient of cerebral cir-

culation. The pressure drop from the lieart to the head is che to hydrostatic effects only.

The intracranial vessels are represented by a liiniped resistance iridependent of Gz. The

extracranial veins are modeled as 1-D distributed coIlapsible tubes.

n

U

Elevatioii ( AH, ) 30 C T I I 1

Central arterial pressure ( P.., rrn )

/ / Cciitral venoiis pressure ( Pi- )

I I Cranial arterial voliitnc ( v!.~~*,, ) 15 ~711"

LO5.O rnrnHg (14.0 k P n )

5.0 rnrnHg (0.67 k P n )

+

Cranid venoiis voliirric ( Li .c,) 60 cm'' I

Stiffness coristarit

(cranid nrtcrics) (h',,,( , \ C s , )

1 I

S tiffriess coristarit

(cranid veiris) (Kpnl, .cl)

S t iffness constant

(jtigular veins) -, , ) C ross-sec t ional arca

(jugiilar veiris) (.-Io)

! l

3.6.5 rrlrrlHg ( 0 3 k P a )

3.0 m m H g (0.265 k P a )

0.86 LT~'

rt

Table 4.1: Paranieter values iised in the niodel of cerebral circulation with siniplified

Viscosity of blood ( p )

gcometry. Siibscript rt indicates normal values.

Derisity of blood (p)

kg 0.004 - nl s

kl 1000 , TI L t

4.2 Methods

4.2.1 Physiological approximation and mathemat ical formu-

lation

The siniplified geonietry is displaycc1 in Figure 4.1 and the values of the paranieters

iisecl in siniulaticins arc siirnniarizcd iii Table 4.1. The rnat ticniat ical forrniilatiori of

the probleni is as follows:

Extracranial arteries

Sincc the viscous resistance in the est racriininl art cries is iiisigriificant. t tie prrssiirc

rlrop front the ticart to the heatl is dile to gravitational cffrcts only. Ttir arterial

pressure at the lcvel of the ticad is t hcii giwri 1-

Intracranial vessels

Sincc the viscoiis prcssiire clrop in the craiiiiirii occiirs iriainly at tlir Icvcl of t tic) rriic*ro

circulation. the crariial resistancr is Iiiniped brtween intracrariial iirteries am1 vrins.

Thcrcfore. the vcrioiis prcssiire at tlir lcvcl of the tirad is giren by

(note ttiat Q = Qs since al1 the wssels are in scrics).

Autoregulat ion

Rc is taken to be constant when iiiitoreguiation is not iticliided. Otherwise it is

deterniined from

(Crsino 1988). The subscript n denotes nornial values (see Table 4.1). The mode1

!vas developed by Crsino (1988) and is based on the assumption that the cranial

resistance is a function of cerebral perfusion pressure P,, = P.4c - Piec Although in

Figurr 4.2: Aiitoregiilat ion: craiiial v.?sctilar rcsistarice CU a ftirict ioii of cercbrnl prrf~ision

pressure.

1.2

1 al O C a 0.8 .-

V) a K

0.6 N .- - a E 0.4 O 7

0.2

O0

redit- ttie mectiariism of aiitoregulotioti relies on the concentration of osygen. carbon

tliositle. and tiylrogen ions in the blood. rathcr than clirectly on P,,. the rrsults of thr

niat heniat ical rnoclel are consistent wit h the cspcrimcntal resiilts obtained hy Kontos

et al. (1978).

Autoregulation involves changing the cross-sectional area. and consecliiently thr

volume. of the resistance vessels (small arteries and arterioies). In order that con-

semition of voliime be satisfiecl. this has to be cornpensated hy a clecreasc in the

cranial venous volume. Honewr. a significant change in resistance c m be obtairied

by a small change in the crosssectional area of the resistance vessels. Furthermore.

the venoiis volunie is mucli larger than that of the resistance ~essels. and a very small

compression of the veins is sufficient to compensate for the dilation of the resistance

-

-

-

-

-

-

1 I

0.5 1 Normalized Pressure Difference

vessels. Therefore. it is assumed that the intracrariial venous resistance and PCsF are

not affectecl by autoregulation.

Jugular veins

Following Shapiro (1977). the 1-D gorerning eqiiations for steacly flow can bc rom-

bined into a single orclinary differential eqiiation.

L> and co cari be eliniinatcd f'roni cqiiatiori (4.4) hy iising the tiitw law and the fact Q that L> = - . Tlic forrn of tube laa usccl is the one giveii in cqiiiition ( 1.23)

.4,,os

This forrii of tube law lias a coritiriiioiis first cleriwtiw. aricl is appropriate for niodclirig

pressiirc-arca behavior of thc veins. Fiirthrrriiorr. t liis forni of tiibr law r.arici the oric

iisetl for t hc. t ttin w l l vcsscls i r i the groriirtrically coriiplcs mode1 arc iilrriost idrat ical

for riqytivc transmiiral pressures. Eqiiatiori (4.1) can tlicn br rri~vrit tcn as:

CSF pressure

Though the CSF tlynaniics is irriplicitly incorporatecl into the niocle1 by assuniing

that the cranial resistance is indepenclerit of Cz. PCsF c m still be cletertiiined t)y

representing intracranial arterics and wins as luniped capacitances withoiit resistance.

and Forcing the conservation of the cranial volunic. The lunipeci intracranial arterial

ancl senous \-ohmes (I,lc and IiSc. respertively) are functioris of the transmural

pressure.

Since P..ic and Pl -c can be calculatecl wit hout esplicit ly considering CSF clynamics.

PCsF follows directly from equat ion (-1.7).

4.2.2 Solution procedure

Eqiiation (4.6) can be solved numerically. using a Ruiige-Iiutta niethocl. However

the Row rate is riot known and therefore a "shooting" methotl has to be usccl. The

solution procediire is qs lollows:

2. Pi-c is tleterniinctl frorn eqiiation (4.2). If the ccrehral aiitorcguliition is incor-

porated. Rc tias to be dcterminccl froni ecpation (4.3).

3. The solution for tlifferential eqiiation (4.6) is obtainctl for the giiesscd mluc of

Q and Pinc as the boiinclary coriclition. Pi.lr ol~tairicd 11y solving cqiiatiori (-4.6)

is comparecl ivith the assigncd value of Pi-rr aricl a iicw vaaliie of Q is giicssed.

4. The proceclurc is repeatetl iintil t h e valiic of Pl-lr ohtainecl by solvirig t\qiiiition

(4.6) and the assignet1 value of Pi-lr arc close ctioiigh.

. Once ;ml PL-[[ are kriowri. PçSF can hc rletr?rniincd frorn (4.7).

4.2.3 Simulations

In tlic first siniiilatiori Gz \vas varicd from - 2 to +7 for the niocle1 n-ith or \vitholit

aiitorcgiilation. AI1 the pararnetcrs tvcre assigned values givcn in Tablc 4.1. csccpt

i r i the case ivhere the effect of varying 1 H arid fi, w s csarnined. Ir1 the second

simulation Gz nas kept at +.5 wliilc the influence of elrvated central hloocl prrssiire

on cerebral circulation was esarnined by increasing ancl PI-ir by the sanie arrioiirit

(mf 1.

4.3 Results

\\'hile the linear change of P.-lc tvit h Gz is a direct consequence of t lie mat hematical

formulation for the flow in the arteries. P I p c displays the same non-linear behaviour

as predicted by the geometricaily more comples rnotlel. P, diminishes with Gz (see

Figure 4.3). The changes in cerebral blood Row and vascular resistance are of the

same nature as those given by the comples niodels: the floiv rate drops with increasing

F i 4 . 3 Blood presstires at the level of tlic heacl. Solid scliinrcs: nrterial presstire.

Hollow squares: veiioos pressure. Faint linc withoiit synibols: cliffcrence (cerebral perfusion

press tire).

Gz as the jugular resistance incrcascs. Ttiere is an escellent agrremcrit hetween the

flow rates predicted by the two niodcls (see Figure 4.4). It is obviotis froni Figure

4.5 rhat the rise of total rascular resistance is esclusively due to an increasecl jugular

resistance. Sote that the jugular resistance is nos giren by

where Ic and l a denote cranial and heart elevations. respectively. When autoreg-

ulation is included. cranial ~ascular resistance initially drops with +Gz but then it

Figure 4.4: Cerebral blood Row. Solid squares: mode1 witti aiitoregiilatiori. Hollow sqtiares:

mode1 wit lioiit autoregiilatiori. Fnint h i e wit hout synibols: diff'ercrice. Hollow trimglcs: t iic

niorle1 wit h complex georrietry.

stabilizes a t a value nioclerately lowr thari nornial. The cffect of iiiitoregiilation on

cerebral blood flow can be seen in Figure 4.4. \\'hile aotoregiilation has a significant

effect at O < Gz < 3. it has a small effect at Gz > 5.

Figures 4.6 and 4.7 clearly show t hat shorter heart-to-liead tlist ance ancl st iffer

veins reduce the influence of +Gz on the blood Bon. n-hereas the opposite is the case

for larger heart-to-head clistance and more conipliant veins. It is also obvious from

the figures that these two parameters have no importance at negative Gz. The thick

lines in both figures indicate results obtained n-hen nominal parameter valiies n-ere

Figure 4.5: Ccrcimd v;\sciiInr resistance. Soiid sqiinres: total ceretxal resistarice. Hoilow

scpiares: veiioiis resistance. Solid triangles: rranial vnsciilnr rcsistancc.

iiscd.

Elevatcd ccntral blood pressures caiise the rasciilar resistancc to retiirn to its

normal \allie. This primarily means tliat the wnous resistance is elirninated by

prerenting collapse of the jiigular wins (see Figure 4.10). \\é see Ironi Figure 4.8 -- that the resistance is brought to normal when l P r I 2 i a rnmHg. At the same

point. the cerebral blood flow is fully restorecl (sec Figure 4.9). Fiirther pressure

increase has no effect on either flow or resistance.

Figure 4.G: Tlic c f f~ct of the lieart-to-liead distancc oii wret~ral blooci flow iiiicler Gz. The

tliick liric corrcsporitls to iiorrnnl distririce (AH,, = 30 cm).

2 Kpn

Figure 4.7: The effect of the venous stiffness on cerebral blood flow under Gz. The thick

Iine corresponds to normal stiffness (Kpn = 2 mrnHg).

Figure 4.8: Cerebral vascular resistance at +5 Gz as a furiction of blood pressure increase.

Solid squares: total cerebral resis t ance. Hollow squares: venous resis tance. Solid triangles:

cranial vascular resist ance.

Figure 4.9: Cerebral blood flow at +5 Gz as a funcrion of blood pressure increase.

Relative Distance

Figure 4.10: Relative area of tlie jugiilar vein at +G Gz. Bold lirie: APII = 60 rnrnHg.

Faint iine: APcr = O

4.4 Discussion

The significance of fiuid inertia

The resiilts suggest that the inertial cffects in the estracranial veiiis are of no sigiiifi-

cance for cleterniining cercbral blood flow. This can be csplainecl by consiclering the

cross-sectional arca of a partially coilapsecl jiigiilar vein sliowri iri Figiirc 4.10. The

area of the win is fairly constant. both in the region tvhere ttie vessel is collapsecl. and

in the rcgion wlicre it is inflated. Tlie transition froni collepsecl to iriffatcd state occurs

over a wry small distance. and the location of the trarisitiori is rriairily cletermined

bj- PL-,[. In the rcgion where the wsscl is coliapscd the area change is sniall hficaiist!

the hyclrostatic coniponent of the prcssiire gradient is balancecl t>v viscoiis forces (scr

Figiirc 4.11). ntiercas in the rcgion niiere thr? vcsscl is iriflatid the arca change is 8 L ->. ilAs

snia11 bccaiisc tlic vcsscl bccorrit~s stiff. Sincc - is siriall wticn - is sniail. t h 3s as

convrctivr trrrii will bc signific;irit only at the locatiori of tlic jiinip. Howwr. it cari

bc scrn iri Figiirr 4.1 1 tliat rirar the iiït l i i jiinip. t h prcssiire gr;iciicrit cliic to the

conrcct ivc terni is miich srnaller t han th^ hyclrostat ic prPssurr gritdi~rit.

?'lit. How rate is niainly cietcrniincd by t h rrsistancc offcrd by t h (~olliips~cl

portion of the vesscl. wlier~ t hc inert i d r f fects are wak. .Ut hoiigh Htiid irirrt ia nia!-

affect dctails of the solution ncar the locatiori of the areü jiinip. or iritrociticc niirior

changes of the location whrre the junip ocriirs. ttiis d l not sigriifiîaritly affect t h

Row rate. However. fliiid incrtia may becoriie iniportant if t h bloocl rriocity rcaclics

tlie local wave speecl üt sonie point iti t tic vessel. Tlirn. iiccorcliiig to rqtiation (4.4)

the spatial derivative of the vessel area beronies infinite. so that tlie iiiodel breaks

clown. and the flot\- retiirns to sub-cri tical t tirougli a cliscoritiniiity rcferrctl to as an

"ehst ir junip". The plot of thc blood velocity arici local waw speed iri ttie jugular

rein is shown in Figure -1.12. -4lthough the Bow is sub-critical everywhere in the

vessel. the flow speed and m v e speed are of the same order of rnngnitiicle. and it is

not impossible that supercritical Bon- does occur in practice. either due to non-stcady

transient phenomena or because the jugular veein has a higher conipliance than nas

assumed in the simulation.

Autoregulation, heart-to-head distance, and venous stiffness

The mode1 shows that cerebral autoregulation is an important niechanisni for main-

taining normal cerebral blood flow at O < Gz < 1. Howcvcr at Iiigh +Gz the effcct of

aiitoregiilation is irisignificarit. There are tao reasons for t liis: first. the niccliariisni of

autoregdation is ineff~ctive once the cerebral perfiision pressiire tlrops to less t h r i a

tialf of its normal value (set. Figure 4.2): second. at ver? high +Gz tlic jiigiilar vriris

are the niain source of vasciilar rcsistance. and. et-en if the cranial vasciilar resistarice

droppecl to zero. ttie total resistance ~ o u l d be significaritly iricreascd.

The siniiilation shows t h t bot h the hcart-to-hcacl c t istancc antl vcnoils stiffness

significaritly infliience cerebral circulation uricier Gz stress. Tticrc are signifiriirit in-

cliviciual clifferencrs in ttie lieart-to-head distarictl. Fiirtlierriiorc. posturr lias a sig-

nificilrit effcct on this parariicter. The stiffn~ss of t t i ~ tiiiriiari jiigiilar wiiis i r i vivo

lias not heeii dcterrriiricrl cspcrinirntally. Howvrr. i t is possible tliiit st rong varia-

tioris iri veriotis propclrt ics cxist forrii pimon to pmsori. Thr rc.iiil ts. t liereforc. siiggrst

that intliviclual tliffcrcriccs in vrnoiis propcrtirs niay hr ii lactor t h irifiiiriiccs Cz

toleraricc.

The effect of elevated central blood pressures

By clevating central blooci pressures. n e simulate ctiangrs iri the crritral artpria1 antl

rcnotis pressures causcd by protcctive niraslires iisrd t o rrcliicr the cffcct of Gz stress

on the cart1iovascul;ir fiinction of fightcr pilots. For esample. PPB iricreases Gz

tolerance by elevating botli central arterial and central vcnoiis pressure (Burns 1988:

Biiick et al. 199'2: Gooclrrian et al. 1992). \\è espiairi the positive rffcct of thc

elevated bloocl pressure on cerebral blootl flow in the follou-irig way: elcvating the

bloocl pressure in the system recluces the narrorving of the jugular vriri. nhich is t h

main contributor to the Gz-inducetl rasciilar resistance: if the pressure is elevatecl

enough the jugular veins are no longer be collapsed at an' point and the normal

blood flow is restored: further increase in blood pressure has little effect on the total

cerebrovascular resistance since the resistance of the full' opened jugular veins is very

small.

Relative Distance

Figure 4.1 1: Pressure gradient in the jiigiiinr vcin at +jGz n~ id APlr = O. Bold hie:

Relative Distance

Figure 4.12: Blood velocity and wave speed in the jugular vcin at + lGz. Solid squares:

wave speed. Hollow squares: blood velocity.

Chapter 5

The Mechanical Mode1 of Cerebral

Circulation

5.1 Introduction

A triechanical cspcrimr.nt lias brrn clrsigrird to siniillate ttir tiiost irriportant ptiysiciil

aspects of the crrrt~rovasciilar systcrri subjcctccl to gravit-. To a large estent. ttiis

is a riicctianical analog of the rriatheniatical niotlel prrsrritrd in Cliaptrr 4. \YiiiI~

ac do not attempt to reproducr qiiantitatively wciiratc resiilts in trrnis of t~lood

pressures and Row rate. al1 the parameters Iiaw plaiisiblr physiological viiliics. As in

t lie mat hemat ical model. the action of the G-protcrt ion is acldrcssrd indirect ly. by

rrinnipulatirig the pressures at t h riiode1 ends. Firially. the constriiint imposed or1 the

cranial vcssels by the skull and the cerebrospinal Atiicl is represetited iri the niodel.

The objectives of the esperiment can be summarized in the folloning way:

1. To demonstrate that in a conduit that begins and ends at the same elevation

gravitation affects the flow only if it causes a portion of the conduit to collapse.

t hus. changing the total hydraulic resistance of the conduit.

2. To test the hypothesis that the closeness of the cerebrospinal and venous pres-

sures originates from the condition that the cranial volume must al-s remain

constant.

3. To estimate the extent to which collapsetl t hin-walled vessels niau help the blood

siphon tlirough the head when the pressure is zero or negative.

4. To ciemonstrate that elevating central blood pressures helps restoring normal

blood flo\v.

5.2 Methods

The niodel consists of a system of tiibes in series witti a tiigli p s s u r e or asceniling

atm. reprcsenting carotid artcries. a low presstire or clcsccriding arni rrpreseritirig

jiigular vcins. and a horizontal model of the cranial circulation (Figiirr 5.1). For

the ascericling arrti use a clear. t hick-wallcd. Tygori tiihe (Cole-Piirnicr: 0.95 c m

inner diamctrr. 0.16 ( - rn wall thickncss). For t hc clesccricling arrti 1i.r iisr rit h<lr a t hiri-

aalled. siirgiciil clrairi tiibing (Perry: 1;itcs Pcnrosr tiibirig 1.27 rrn iniier climic~trr)

or t tic sairic Tygori tiibing iis is i i s d for r he ;isc~nrlirig srtii. Tlir arnis iiïcl hl = I I : ~ =

A H = 14.5 r w i long. arid the roots of hoth arnis w r r sitiiatrcl at tlir siirml lrvrl. wtiirli

rvc cleriotcl "zero l~vcl". \\C also deriote thr root of thc asccnrling arrn as thr high

prcssiirr eiicl of the rnoclcl and the root of the clrsccnding arni as the low prwsiirt. rwl

of the niodel. Since drain tiibing is not rriaiiiifact~irccl in lcrigtbs siifficient to c'over

t tic ent ire lerigt h of the drsccriciing ami \vit ti a single peacc of t ihirig. t hrce pirrcs arc

connectccl \vit h 2 ml long Plesiglüs tubes acconinioclntecl tri t h prcssiirc taps.

The craniiini mode! consists of a .j nn long Tygon tube (crimial artcries). a resistor

(micro circulation) and a 10 cm long Pcnrose tube (crimial wiris) in series. The tiibcs

are enclosecl witliin a w t e r filled. coaxial. Plesiglas tube scalecl n i t h rubber pliigs.

to represent the constraint imposed on the cranial vesscls by the skiill and CSF (see

detail 'X in Figure 5.1). The resistor is obtainecl by filling a 2.3 cm long Plesiglas

tube of the 1.27 cm inner diameter n i th granular material to provide a pressure drop

of 100 mmHg for a flow rate of 1100 m 3 / m i r i .

The Aow of triter through the systern of tubes is obtained by connecting the

high pressure and low pressure ends of the mode1 to the top and bottom constant

level containers. respectively. By adjusting the elevation of the containers. desired

. - - Resistor - " - - P&,

Tilting . ,

board

- Horizontal board

Water collection

Detail "A"

Figure 5.1: Experimental model.

pressures can be obtained at the niodel eiids. The niodel is attaclied to a plate ahich

can be tilted from :. = -10 to 7 = 90 degree. where is ttie tilt angle defiriecl as

s h o w in Figure 5.1. In this case Gz is ttie component of the gravitational acceleration

aligneci with the arms of' the model. Tiius. a raiigc from -0.17 to +1 Gz is comreti

by tilting the model froni miniriial to niasimiil 7 . Iii teriris of t h gravitational cffect

at the top of the model this translates iiito hyclrostatic pressure change frorri + L U

rnrnHg to -106.6 rnrnHg. In Iiuriiaiis. idicre t hc clifferencr iii ele\rition hetiv~cn the

lieart arid t lie Iieaci is approxirnately 30 un . t ht. siilne hydrostatic cffcfcct woiild resiilt

froni Gz changing lrom -0.8 to +-l.S.

Figure 3.2: The setiip user1 for detcrnii~iiiig tube law of the ttiin-waIlccl siirgical drain

tubirig.

5.2.2 Experiments

Preliminary experiments

Two preliminary esperiments itwe carried out. The first to cliaracterize the resistor

used in the model of the cranium. and the second to cletermine the tube l a - of the

Penrose drain tube.

To characterize the resistor. it was connected to the top and bottom constant

l e ~ e l container and the pressure drop \vas variecl by changing the elevation of the top

container. Pressiires at the inlet and the outlet of the resistor were nieasrired ivitli

water manometers and the Row rate was cletermiried by nieasuring the tinie needed

to collect orie liter of \rater overflowing froni the bottom contairicr.

To determine the tube 1aw of the Penrosc tube. a 33 crn long piece of tiibirig.

filletl with water. m s sccured to the sidcs of a irater fillecl container. The tiibc Kas

connecteci to a syringe at one siclc. aiid a s a t e r nianoriicter. a t t h othcr side (see

Figure 5 . 2 ) . Csing ttie syringe. a voliinic of watcr \vas irijectcd or rrniowl froni the

tube. thus caiising a change in tube voliinie ancl a change in the pressiire in the water

occupying the tiibc. The transniural pressiire is pgA HI,, where 1 H,, is the tliffrrcricc

betwern the lewl of the water in the rnaiionictcr and the coritairier. Ttic climge in

tube volume equais Ali -AH,.l,,. wtirre AI. is ttic voliinic injrctrd froni ttie syirigo

niid lH,;L, is the voliinii. iiecclrcl to ch;ingc t h ~ level of the water in the rriariomctrr.

A,, being t tie cross-sectional arca of t tir ni;inonicntcr. For a vcry corilpliarit tiibe the

cncl effccts chie to nttaclinient are sigiiificant orily if t h distiinre frorri t h point of

at tactirnent is srnallrr t han t tir cliariieter of t tic tiibr (Brxrtrarii 1987). Siiiw t lie lengt I i

to diarritttw ratio of the tube wis 27.6. we assiinicd that the arca cari be calciilatecl

by clividing the tiihe voliinir hy thil tube lcrigtli.

Main experiments

\ \ é pwfornid two sets of cspcrinicnts: tlir first set (iqerirrirnt 1) w t i r r ~ Gz wris

raripcl while the prpssiires at the zero lewl fised. ;ml the second (cspwiriimt I I )

ivhere the pressures at the zero lrwl W C ~ P e lcvat~d whilc Gz {vas fisrcl. Ir1 rspcrirrirrit

I . t hc pressure a t the base of the ascenclirig arni ( P l ) \vas 102 rrlnr Hg (140 n n H 2 0 )

and the pressure iit the base of the descenciing arm ( P 6 ) {vas 0.7 rnn1Hg (1 c-rrd&O).

The tilt angle was variecl from -10 to +90 clegrccs. Tlie clescenciing arrri was eitiier

a Penrose (1,) or a Tygori tiibe ( I b ) . A slightly positive p6 ivas chosen to reprrscnt

the central renous pressure. This also avoids a sudden espansion of the Penrose tiibe

at the point of attachment. In esperiment I I . the tilt angle cvas fiscd at 30 tlegree

(+O..jGz) and the Penrose tube \.as used for the rlescencling a m . The pressure \vas

increased at both rnodel ends by the same arnount h m the values used in esperiment

I which ive denote as Pl, and

Pressures nere measured at six points in the network as well as in the water

surrounding the tubes in the cranium model. The locatioiis of pressure measurements

Es~eriment 1 Nathematical niodel /I

Table 5.1: The analogy between the pressures in the tncclianical ttiodel aiitl the pressures

iri t lit? mat hetiiat ical niodcl of ccrebral circiilation.

wcre: the hast. of the asccnding arrii. the top of the asrcmclirig artti (6). tlic top o f

the clcscending arm ( P I ) . points whcre t tic piecrs of tlic Periros~ t iibing arr connwtccl

iri tlic clescetitling arni ( P I aricl Pi). tlie base of the tlcswnding m i l . ;incl t iir rrariiiini

rriocl~l at tlii. 1rvt.l of tht. t i i b ~ s rrpriwnting the crariiiil vcssrls ( Pr). Tlic Iomtions of

the points wherc prcssiiros PI aritl P; w r c rnresiirrd w r r Ir . ! = 91 r w t arid h i = 46 m l

froiti t h zero level. resprctivcly as indiciitid in Figiirr 3.1. The arialogy betuwn thr

prmsiirrs i r i ttic ni~chanical nioiiel and tlir prrssiircs in tlic riiattirniatic.al nioclrl of

ccrehral circulation is siimniarized iri tat~le -5.1. For the ciriring prcsstirr t l i f f~r~nrr at

the tx~sc level arici pressiire rlevatiori at the t ~ a s ~ lcvcl. WC use tlie snrric syrnbols as

we clic1 for the perfusion pressiire and the rlevation of crritral t~loocl prrss1iri.s in t h

mat licniat ical rnocfcl.

\\'c uscd ra ter manometers for al1 pressure nieasiiremeiits. escept in the case of

P,. wlicre a strain gaiige pressure transdiicer hncL to he usecl to avoitl violating th^

corisemation of volume principle. The pressure taps wrre placecl in the rigid tu lm

at a distarice 61 = 1 cm from the point at cvhich the Resible tube aas attachecl (see

Figure .5.3). Assuming that the resistance of the rigid segment is sniall. the pressiire

at the level of attachment can be dctermined as

p = PA! , pGz 61 (3. i )

where PA* is the measured pressure and dl is positive when the pont of attachrnent

is below the point of measiirement. Since the maximal R, based on the diameter of

Figure 5.3: Pressure ~iicasiircnicnt.

the rigitl scgrncnts is approsiniately 2 x 10". it caii he c?spectctl t h t tic $C' trrni - is signifi carit ancl t tiat the pressure in tlie collapsccl Pcrirosr tube inirricciiatcly iiftcr

the point of attachnient is l o w r tliari the prcssiirc at th^ point of attarlirnrnt. Tho

ctorrcct valw of t hr prpssiirr in t h r ~ol l i ips~t l Pmrosr* t iibr cari t ~ e cirttttrniintd ;is

Sirice u drpericls on P. cquation ( 3 . 2 ) has to br solvrcl iteratiwly. Prcssiir<!s calcil-

latecl froni rqiiation (5.2) and t hosr c.alculatcd froni cqiiat ion (5.1) cliffer by lcss t han

1 r r m H g for csperiment I and less than 2 rnrnHg for cspcrirnerit II. Fiirtticrniorr.

thcre is some anatomicnl jiistificatiori in rneiisiiring pressure at the rigid segmcrits.

since the jugiilar vein rcccives blood frorn t lie st iff dura1 sinilses. T h e r ~ fore. prmiircs

tvere deterrnined using equation (5.1)-

The How rate was measuretl by water collrction. Al1 mcasurrments nere repeateil

ten tinies and niean values and standard tieviations wcre calculated for each of the

meaured values. The iargest standard deviation found nas for pressure Pi in esIler-

iment 1. and it amounted to 10% of the calculatcd mean. For al1 other nieaswetl

values the standard deviation \vas Less than .j% of the mean.

Pressure Drop [kPa]

Pressure Drop [mmHg]

F i r 4 : Pressure-Row characteristic of the rcsistor iised in the niorle1 of ttie cranial

circulation. Syrribols: expcrit~ieiital data. Tliick solicl line: fittcd ciirvc.

5.3 Results

5.3.1 Preliminary Experiments

The relation between the pressure drop across the resistor (AP,) and Q is of the

general form

IP, = CIQ + C?Q~ (.3.3)

where CI and C2 are constants (Idelchik 1984. pp. 3: pp. 393). Ke assume that. for

the range of interest. the following law can be used for the pressure chop across the

Figure -5.5: Tube law for the ttiiri-walled siirgical cirai11 tubing. Syrribols: cxperinieiitnl

data. Thick solid linc: fitted cime.

resist or

AP, = R,Qn

nhere R, and n arc constants. ancl 1 5 n 5 2 (see Figure -7.4). From a least scluares

fit Ive obtain n = 312 and R, = 0.0024 mmHg or 0.15 x 10' kPa (3)n. The correlation coefficient bet~veen 4.5 rneasured and fitted values of the flow rate is

0.94.

The tube law for the Penrose tube is displayed in Figure 5.3 . The relative area

is calculated by normalizing the tube cross sectional area by the tube area at zero

transmural pressure calculated from the nominal tube diameter. The Penrose tube

eshibits behaviour characteristic of collapsible silicone tubes. The niost significant

reduction in cross-sectional area occurs for Pt hettveen zero and -5 mniHg wlien n

changes from 1 to less than 0.1. \Ve use the Forni of tube Iaw given in ecliiations (1.19)

and (1.20) and ohtain El = 219.4 rnrnHg or 29.2 k P n . and IGl = 0.34 rrunHg or

0.045 k Pa from the least squares fit. The correlat ion coe Aicient bctween '100 nieasiircd

and fittecl chta is 0.99.

Figure 3.6: Flow rate for experiment I . Hollow squares: experimerit 1,. Solid squares:

experirnent Ib.

Figure 5.7: Pressures in t lie triocieI for cxperiment 1,. tIoIlow squares: Pl. Soiid squares:

f i . Hollow triangles: Pi. Solid triangles: Pj.

5.3.2 Main Experiments

Experiment I

In esperiment I, the t liin-walled descending arni collapses when tilt arigles are posi-

tive. aiid the flou- rate drops. The collapse is approsimately uniform dong the length

of the descending ami. escept in the imniecliate vicinity of the low-pressure end of the

model. and becomes more serere with increasing tilt angle. For the full tilt (A! = 90

degree)! the flon rate is significantly reduced. but is still non-zero. Pnlike the tubes

of the descending arm. the Penrose tube in the craniurn mode1 rernains opened re-

Figure 5.8: Pressures in the mode1 of the craniiini for expeririient 1.. Hollow squares: P,.

Solid squares: pj.

gardlcss of the tilt angle. In espeririierit Ib tilting docs riot produce an' visible change

in the flot\-. As in csperiment Ia . the tubes in th[! crnnium mode1 are never collapsed.

Figure 5.6 clearly sliows that the gravitu affects the flow only if a collapse of the

wssels conducting the fluid is involwd. as occurs in esperirne~it 1,. The Aow rate for

the full tilt is less than one fifth of that for zero tilt in esperiment I,. Howewr. for

the full tilt pressures at the top of the mode1 are sub-atrnospheric. ancl a non-zero

flow rate means t hat the siphon functions with a collapsed descending arrn to some

degree. The flow rate is unaltered for the negative tilt both in esperiments 1, and Ib.

Figure 5.7 shows pressures Pro P3' P4, and for esperiment I,. The pressure at

LOS

Figtirc 5.9: Pressures in the mocicl for experiment Ib. HoIlow squares: P2. Solici squares:

f'3. Hollow triangles: P,.

the top of the ascencling arm changes nlniost linearly witli Gz which mcans that the

hydrostatic effect is cloniinant. while the riscoiis friction is negligiblr. Conseqiient ly

fi is approsimately -3 mnzHg for the Full tilt. In contrast. the pressures dong the

descending arm is a non-linear function of Gz. Alsa P1 2 Pl = Pi. meaning that

riscous Friction counterbalances hydrostatic pressure gradient in a substantial portion

of the descending arm. For the full tilt f i is approsimately -10 mmHg as shown in

Figure 5.8. Hence P3 is lower than fi. but is much less sub-atmospheric than in

experiment Ib where it drops below -100 m m H g (see Figure 5.9). Figure 5.8 also

gives the cornparison between P, and P3: where P, is corrected to account for the

Figure -5.10: Hydrnulic resistance for expcrirtient 1,. Hollow sqiixes: Rt . Solici sqiiares:

Ra. Hollow triarigles: R,. Solid triangIes: R,.

difference in elewtion at which the two werc measured. The pressure in the fliiitl

surroiincling the tubes closely follo~r-s pressure changes in t lie r tiin wallecl t iibe. t hiis.

preventing vesse1 collapse and a change iri volume. It can be seen froni Figure 5.9

that. in esperimerit Ib. changes in P2 and PR are ptirely hydrostatic. The samc figure

shows that. as in esperirncrit I.. P3 arid P, always stay close.

As in Chapters 3 and 4. ive esamine the effect of Gz on the resistance. Since Re is

of the order of 103. equation (3.19) should be used to determine hydraulic resistance.

However. al1 the pressures were measured in the rigid tubes of the same cross-sectional P area so that al1 the ;;c' terms cancel out. Therefore the resistances can be defined -

as:

Ascending arm (arterial)

Cranial mode1 (intracranial vesscls)

Drscending arrn (est racranial reins)

Total

Eq~iatioris ( 3 . 5 ) - (5.8) are arialogoiis to cqiiiitions (3 .43) - (i3.45) i r i Cliaptw 3 . It cari

be secn frorn Figiiro 3-10 t h the total rrsistanre rises rapidly witli incrtuing Gz.

This is i~scliisi~ely dite to ;in incrrasrd rrsistaricr of the rollapsctl clescriiiling tirni.

On tlic ot hcr tiiirid. Ra is vrry stria11 iliid constant. wliile R, rlrops witl i Gz iliic to a

non-lincar natiirc of the resistor in the cranial irioclcl. T h vrsscls in the niorlcl of t h

craniiini alwsays reiiiain oprn m d do not contribiitc sigiiificaritly to the ovrrall cranial

resistancc. It is cas- to dcdiice froni the ciefiriition of the total rrsistanre and frorri

Figure 3.6 that iri esperinient Ib the rcsistance st-s coiistant.

Experiment I I

In a systern of rigid tiibes the flow rate would depend only oii P, rt.g;trtlless of the

absolute values of Pl and P6. When collapsiblc tiihes are preserit in thc circuit. the

results are much lrss intuitive. Figure .5.l l shows that the flow rate increnscs \vit h

1% up to the point where the flow approsimately equals that obtained for the

zero tilt in esperimerit 1.. then the Bos rate starts leveling off. Pressure ele~at ion

results in opening up a lower portion of the descending arni. approsimately up to the

point ahere APH is counterbalanced by the hydrostatic pressure change. Figure 5-12

whirh gives pressures P3. Pd. and Pi shows two distinct regions in terms of the

Figure 5.11: Flow rate far experinient I I .

pressure betiavior: in the region whcre th^ t ubc is opetied the pressure rsperi~nces

ii hydrostatic change n i th elemtion antl tliere is a one to one relation bctneen 1P/[

and a pressure change at a certain point in the tube: in the region abow thr point

of collapse the pressure is essentially constant. The point of col laps^ nioves up with

increasing APIf. antl eventually the whole ciescencling arni opens up. Ki th al1 the

tubes opened. the flow becomes independent of the further increase in l P H .

As in esperiment I . we esamine the resistance in order to gain a better insight

into flow behavior. This time the focus is on R , alone. since that is the component

of resistance which is strongly affected by pressure changes at the zero level. Fig-

ure 5.13 shows t hat R,. drops significant ly ni t h iricreasing 1 Pfr . At approsimately

Figure 5.12: Pressiirrs in t tic descending ami for experii~icrit II. Hollow squares: Pj. Soiid

squares: Pi. HoIlow triariglcs: Pj.

APH = 50 n21nHy. R, is negligiblc. Thereforc the floa is restored by openirig tlir

rlcsceiidirig arm of the niotlel. nhile mairitaining P, constarit.

5.4 Discussion

Thc esperirnental resul ts are consistent wit h the predict ions of the mat hcniat ical

models: esperiment I demonstrates that the rate of fion through a system of tubes

beginning and ending at the same elevation is affected by gravit- only if the low

pressure portion of the network is collapsible. i.e. the resistance of the circuit changes

Figure 5.13: Hylraiilic resistance of the cimcenditig ami for cxpcrimerit II.

with Gz: esperinient II demonstrates that the rate of Aow throiigh tlic circuit siib-

jected to +Gz increases whcn the pressiires at the zero lewl are tka t ec l . -1s in

Chapter 3 a e will esaniine possible theoretical models of Cz dependrnt Row t tiroiigh

the esperimental niodel iising a simple mat liernatical arialysis based on pbysical ap-

proximations of the collapsible descending arm and qiialitative observations provided

by the experiment .

Since the tubes in the mode1 of the craniurn are a lwqs opened. and provide

minimal resistance to flow. the pressure drop (Pi - P3) is given by the esperimentally

determined characteristic of the resistor and equals R,Q3/'. Furthermore. 6 can be

eliminated by assurning that the pressure drop in the ascending arm is only due to

Figure 5.14: Prcdictions of rliffererit tlicoretical riiodels for the How rate in expcritiicrit I,.

Syrribols: experiinental deta. Thiri broken linc: the cfesccricliiig artri is approxiniated t>y a

rigid tube. Thin solid lirie: the clescenclitig arni is absent. Tliick solid iirie: n niodel o f the

flow i r i collapsible t iibcs t~ased oii the experirnental observnt ioris is iised for the descendiiig

ar m.

the hyclrostatic effect. This leads to the following espression for the Bou- t hroiigh the

system = ( P , - pGï_\H -

" ) 21.3

(5.9) R r

Eqiiation (3.9) corresponcls to equation (3.32) in Chapter 3. The pressure P3 in eqiia-

tion (-5.9) depends on the physical approximation chosen to represeiit the descending

arm of the esperimental model. and it critically influences the estimated flow rate.

\\é examine three cases. In the first case. the descencling arrri is approsiniated by a

stiff tube. In the second case the descencling arni is absent. so ttiat the nater esiting

the mode1 of the crnnium clischarges into atniosphere. Finally. in the t hird case LW

propose a simple approsimation for the Bon iri the collapsible descending arni which

is basecl on esperirnental observations. If the clescencling arni is approsimatecl by a

stiff tube (siphon). P3 = PF, - pGzAH and the fiow rate is given by

Obvioiisly. in tliis case the Hon is coniplctrly irid~prnclcnt of t hr gravitatioiial rffect.

On the other hancl. when ttie cIcscendirig arrii of the nioclcl is ahserit (waterhll).

P3 = O. ancl cqiiation (3.0) rcdiires to

(Pl - Q.WS = ( ~ 1 1 )

In tliis case. tlic rate of fiow is zero nheii Pl = p G z l H . Khcri tlic tlcsc~riding iirni

is collapsiblc. the presslire Pi rriiist be rlctt.rrriiricd as part of the soliitiori.

A relativcly siniplc approsiriiatim cari bib riiade if W. rcsstrict t l i i b anal+ to tlitl

case when at least a portion of the descmciing tirni is collaps~d. Tiicn. L w cari rcwitr

wtiere n~ is the relative cross-sectional area at the top of tlir cirscrndirig arni. 111

ordcr to calculate Q we necd to clctermirir 0:) basrd on an approsiniatr nio(le1 of the

Flow in collapsible tube. For ttiat piirpose we use the observation tliat pressure in

ttie collapscd portion of the descending arni is approsiniately constant and eqiials Pl.

Since pressure ancl area are related throiigh ttie tube law. the area of the collapsed

portion ülso has to be constant and the hydrostatic pressure gradient is halaricecl by

the ïiscoiis resistance. This can be espresseci as

Equations (3.13) and (5.12) can be cornbined into a single non-linear algebraic equa-

tion in terms of a3.

The relative area cari be determined by solving equation (2.14). Theri. the pressure

and the flow rate follow froni the esperimentally determined tube lan and q u a -

tioii (5.13).

Figure 5.14 shows t h the approximation basecl on the eqiiation (.LI?) is the

closest to tlie Aon rate iri esperinient I,. which lies betnecn the valtics pretlicted by

equations (5 .1 1) itnd (Z.10). Froni equation (3.12) ~ v e sec that Q clcpentls on 4. i.e. or1 the stiffriess of the tube. Sincc (1 - fi:;"') is negativr. a larger li, resiilts

in a higher flow rate. For a ver- sniall li, the vitliies prctlictcd by cquatioris (3.12)

aritl (5.1 1) will be close. wit h the cscep tiori t hat cqiiat ion (.Y 12) never preclicts a zero

flow since the forni of the tube law cliaracteristic for the thin-walled silicorie tiihes

does not allow a zrro cross-sectional area. For cspwir~ient Ib ttie Aow rate is giveri tjy

ccpat iori (2.10).

Esperinierit II siriiiilates chariges in central hlood prrssiirrs. In parti(w1ar. c spr i -

merit II. reprociiicrs t tie cffrcts of PPB on central blood pressures. Onr intrrprrtat iori

of tlie risc in Row rate in csperiiiicnt II, is tliat tlic rrsistance of ttie cltwriidirig arni

is rrcliiced. whilr ttic ciriving pressure P, is mairititiricd at a constant viiliio. A scw~iirl

iriterpretatioa is thiit PI reniains Fairly constant in spitr of olcviitrcl P6. iirid tliat

elevating Pi resiilts iri iiicreasing the clriving pressiirc ciifference. PL - el. across the

nioclel of the craiiiiirri. Tlie two interpretat ions arc not cwnt radictory i r i any srnsc. biit

rather siipplenient eacli otlier in clcscribing the niotlrl beliavior. The raparity of thc

collapsible (Icsccncling arni to maintain a constant pressure in the collapsed portiori.

rcgardlcss of the pressure at the zero level. c m be lirikecl to observations tliat tlir

peripheral venoiis pressure is not affected by nioderate changes in the cmtral bloocl

pressures (Ho1 t l9-Ll).

-4s predicted by the niathematical rnodels. the vessels in the craniuni motlel a lwqs

reniain open. Thi. aater-filiecl PIesiglas tuhe imposes the consemation of voiurne on

the Ruid in and around the tubes representing the cranial wssels. Since only the

volume of the t hin-aalled tube is significantly affectecl by clianges in t lie t ransrnural

pressure. the pressure in the aater around the tubes has to track the pressure in the

Penrose tube in order tIiat the total volume reniains constant.

Chapter 6

The Simulation of Cardiovascular

Performance

6.1 Introduction

Iri t h Ctiapter t lie gcoriirtry tlevcloprd in Chaptrbr 4 is iricorporatcd iiito a riosrcl-

loop nioclrl of cartliovasciilar systern irivolvirig t h Iirart. wliich wis ~Irvdopcd at

Defcncr arici Civil Institiitr. of Envirorinicntal S[cdiririr (Kalsti aiid Cirovic 1998:

\\iilsh and Ciroïir 1999). The closecl-bop nioclcl rplies on niinicriral solution of t h

1-D rontiniiiiiri governing cqiiations. and is ciirrriitly able to simiilatv t h rt\sponsr of

the carcliovascttlar systcrn to t hr rncchariicd effect of t inic varyirig G z and lifc support

intcrverit ioris.

6.2 Methods

6.2.1 Physiological approximation

The parameter values for the vasciilar netv-orks are giïrn in Table 6.1. Figure 6.1

illusrrates the geometn considereti. The circle denotcs the heart. and the " >" sym-

bols denote val~es. The line elements are blood vessels connecting the heart to the

capillary beds represented in the diagrani by resistor symbols. K e assunie a linear

tube law based on equation (1 .KI) for al1 vessels escept for the jugular vein. A linear

tube lai- is a valid approsimation for arteries' and it is appropriate to use it for in-

- -

Table 6.1: Tube data for the three-capillary bcd tietwork iised to investigrite the rcsponse

to the rnechariical effects of Gz variation and life support intervention.

Tube

.

1

-1

L

[cm]

Left

.Join

a

i,

c

d

e

f

tr O

h

1

j

k

1

II

Thickiress

[rrrin]

Right

Join

E

:\

B

C

D

. J G

H

Angle

[ k g ]

.Ao

[ c m 2 ]

X

B

C

D

E

D

Tt1 bc law rd

\r'oiing's SIodtlltis

[k Pu] - Sec Table 6.2 for h a r t data.

0.0

180.0

0.0

180.0

180.0

H I I I

6.697 ' 2.224

0.86

24.0

20.0

16.0

12.0

4.0

1.62

1.25

0.1

270.0

270.0

8.0

30.0

24.0

0.8

0.37

1.27

0.39

0.37

F 1 2.945

Li near

Lincar

Li near

Linear

Linear

0.999

7.260

5.597

7.548

1.260

I

B

F

N

8.0

8.0 G

-!00.0 l

400.0

400.0

100.0 1 400.0 1

?3

H

J

F

90.0

180.0

0.0

12.078

7 . 4

0.244

1.17 1.498

Liricar 1 I 100.0

'30.0

180.0

Liricar

Lincar

Linear

50x1- Liiicar

8.0

4.0

400.0

120.0 h

400.0 1

27-73 I

Liriear 0.1

0.39

100.0

Liricar 1 200.0 I

4

flated veins. Therefore the simulations have to he restricted to positive Gz in ordcr

that pressure in the veins below the heart be positive. For the jiigular rein iise

the tiibe law given by equation (1.23).

PPB

Figure 6.1 : The t hree capillary bed network iisecl for closed Ioop siriiiilat ions. The iipper

case let ters are the jiinction names, and the lower case lctters t tie tube nanics. Ttiese are

also t lie iiarnes that are used in Table 6.1. where the quantitative description of t his network

is given. .A chevron at a jiinction inclicates a valve and the rcsistor synibol a resistance.

The jugular vein. vessel d. bold line. is the only vessel with a nonlinear tiibe law.

The right heart. pulmonary circulation. and left atrium are absent from the heart

mode1 indicated in Figure 6.1. Therefore. it is assumed that the balance between the

piilmonan and systemic blood floiv is not affected by Gz stress. Le. the same amount

of blood returned by the systemic veins to the right atrium is suppliecl to the left

atrium by the piilrnonary veins. Inclusioti of the neglected eleriients would great 1-

increase the coniplesity of the iriotlel and a niiniher of additional asstiniptions would

be required regarcling the properties of the new circulatory elernents.

6.2.2 G-suit and PPB

The G-suit niodel is basrd on the original Franks Flying Suit (Sacicr 1992. p. 3 2 ) : a

aater-filled garnicnt cowring the l o w r t~ody. abdomen. and chest iip to the k a r t .

The G-suit rserts an esternid pressiire cq~ial to ttic hylrostatic prcssiirr rrlatirr to

th(? licart on al1 vcssels helow the heart. Tlir elcvatecl tlioracic prrssiirc ciiiisrtl hy

PPB is sinidatecl by applyirig estcrnal pressiire cqiiiil to thc PPB prcssiirr or1 vrssels

b arid e and the licart.

6.2.3 Valve model

It is assiinied t h t al1 ulws arc passive. Thry open iristantan~oiisly iiricler a forwarcl

pressiirp griiclicnt. ancl iirp closed iristarit;tiiroiisly bark How. rc~gardlrss of thfi

prtwm-c gradient. \Vheri closecl. thc ~ i l v e s do iiot Irak. Ttiiis. t h ~ i l v r s l1iit.e no

ecpatioris of motion. biit entc3r t hc gowrriing rqiiat ions t hroiigli t lie st!giticnt boiinclary

conditions. An open valse has no effect oii tlic flow. A closeci valvr isolates thr

scgments aricl imposes a zero flow condition a t the rcspwtive scgrncnt ends.

6.2.4 Heart model

The left ventride is represented by variable elastaricc rtiodcl in whicli the vmtriciilar

pressiire ( P m ) is defined by

(Ozalva 1996) where E , ( t ) is a time-varying elastance. 1 , the ventricular volume. ancl

l the vent ricular volume at zero pressure. E, ( t ) is given by

1 Hear t

I Perioti (T) 1 1.0 s II

Maximal heart elastance (E,,,) 30 mmHg (4.0 k P n ) 1

Table 6.2: Paranieters usecl in t lie hart rriodel.

Mininial heart elastarice (Ernin) 1 1 m m H g (0.1 kP«) I I

Mitral resistance ( R L I - )

Emin

VilTt S 0.0179 rntriHg7 (0.14 x 1 0 ~ ~ r r ~ ) 1

i

Heart t period

Figiire 6.2: A variable elastancc trioclel for vctitriçirlar fiirictioti.

nhere the carcliac parariieters are the rriininium and masirriiirri clastances Emdn ancl

Emax respccti~ely. the carcliac period T. the cliiration of systole T,.. ancl t hc beat niirn-

ber r r . This functiori is illustratccl in Figiire 6.2 and tlic parameters arc siininiarized

in Table 6.2.

Diastole occiirs when tlic central arterial pressure is greater than PL,. T h . the

valve between the atrium and ventricle (Nitral valve) opcns. the ventricle fills pas-

sively t hrough a resistor R L I . which rcpresents mitral resistance. Systole occurs when

PL, is greater than central arterial pressure. Then t hc aortic valve opcns and the heart

empties actively. rvith tlie ejection fiow speed determined by a mornent urn ecpation.

At al1 times the heart volume is determined by the rate of filling or ejection.

6.2.5 Initial conditions

The initial state for each cardiovascular simulation is = O everynhere. with

determined as follo~vs: select appropriate initial blood pressures for the root of

each of the systemic and pulmonary arteries and veins: establisli separate tiydrostatic

eqiiilibria for the veins and arteries branching froiii tlicir respective roots: calculate

the corrcsponcling and assign volumes to cadi of the Iicart chanibcrs. 111 the

ciment forniiilatioti. t t i ~ initial pressures ;ire 80 n2rriHg at the aortic root and 10

n m H g in the central veitis. Tlie initial voliinie of the left vcritriclc is 110 crri".

6.2.6 Boundary conditions

The boiinclary condit iotis are applicd tisirig t lie riict liod of rliarac terist irs followiiig

Hirsch (Hirsch 1990b. Chapter 19). Here ive dcscribc how tlie nirthocl of ctiarac-

Figure 6.3: Boiiiiclary riodes and methocl of characteristics for a two segrrient boiinclary.

The nodes at xi ancl x, are on two cliffererit tube segt~iciits. but at the samc spatial locatiori.

The t-axis is broken bccause the distance r~ieasiircd aloiig a tube scgriie~it lias no absoliitc

significance. ancl it is coritiriuotis only ori a tube seginexit.

teristics is applied at a tu-o-segment hountlary for the case of sub-critical How. For

a niore det ailed description of the borindary treat nient. the reader shoiild refer to

hppendis A.1.

\\é assume that tno segments join with the positive direction for each segment

being left to righi. The subscript "1" denotes boundary quantities from the left

tube. and subscript "r" quantities from the right tube. Sote that the sizes of the

spatial g i d on the two tubes. XI - rl-, and x,-I - x, diffcr. but that the tinie step

tn+l - tn is the sanie for both tubes. Following Hirsch (1990b. section 16-4) we can

define characteristic variables II-* that are governed by first order or di na^ differential

ecpations on characteristic paths Cr as

with C= defincd by dx - = L'Ica dt

Thus a pair of characteristic paths rneet at the boiiiitlary at tinie tn-' . aitli I L l prop-

agatirig from t be left tube dong Cl. and I L , propagatirig froni the right t iibc almg

C-,. To first order the pat hs are st raiglit lines. and WC cari use linear iriterpolation to

cleterniine the I I = at tirne tn . and thcri solve t hc goverriirig ODE'S to obtain cstitnatcs

of the II, at time FL. Sow. using the ttvo estirnates for IF=. AOW coiitirluity. aiid

the riionimtuni cquatioii. ae cari solvc fur the values of . - I l . L;. A,. and L i . In grnrral

tticse cqu;itions arc noriliriear and canriot t ~ e solved ;irialyticaliy. Fortiiriatrly. tlic

wliirs of the iirikriowns at tiriic t" provide an c.s(*ellerit iipprosirn;iti»ri to tlic mlues

soiiglit a t t n - l . and Sewtoii's met hocl cmverges w r y qtiickly.

6.2.7 Split coefficient matrix method

Figure 6.4: In the split coefficient matrix method. ive split the coefficient matrix into

addit ive components associated wit h the forwarcf and backward propagating chztracteristics.

We then use backward spatial differences for the forward propagating signais. and forward

spatial differences for the backward propagat ing signals.

In the split coefficient matris (SCII) rnethotl (.Anderson et al. 1984. section 6-4) the

equation (1.12) ciin be wi t t en as:

\Tc c m now d~ te rmine the eigenvalues of the coefficient rriatris A. iirid thrrrby split

it irito iitirlitive corriponents A, t liat arc iissociatetl \vit li signds propagiit irig in t hc

positive arid negiitivr directions along the tube segnierit. Iri ortlcr to xliicve stal~ility.

ii backwartl clifference is usecl for the spatial clcrivativtl of forwartl propagat ing coni-

ponerit. ancl a Fornard clifference for t lie spatial tlrrivatiw of bat*kwartl propagiting

corriponcnt . The triincation crror for t liis niet lioci cleptwls or1 t tic ordw of the cliffcrrricr 1)s-

pressions arid t lie tinie integrator usecl. ;\ srcorid ordcr ttirt h l ~ i r i hc rorist riictecl.

Ho\vrver. it woiiltl irivolv~ two spatial riocies on rithcr side of ttir rioclc of iritcrrst.

and special I>oiiiitliir~ trcatrrierits woiiltl he ricwlrtl for the tmiiriclary nocies aiid cv-

ery node adjacent to a hoiindary. Figure 6.4 illiistratrs titr case For ii first orcl~r

met hocl. in wiiidi spcciül trratnient is recpirtd only for tlip bounclary riotlrs. X niorr

tletailcd description of the split coefiïciciit niatris n i e t h l iisrrl in tliis st iidy is gircn

iii Appcndis A.?.

6.2.8 Mass conservation

This SCXI methocl is not conservatire. and in a closecl loop simiilatiori an! drift i r i the

total blood volume ail1 lead to spurious drifts in the blood pressure. T h d o r e . at eacti

tinie step. nhen Vn and Vn'l are both still availüble. we correct eüch segment voliinie

by emluating r he continuity equation using cent red t ime and space differences. K i t h

this procedure the total blood volume varies by less than O.OO3X tluririg 16 cardiac

cycles. Details of the analysis can be found in Appenclis A.?.

6.3 Results and Discussion

The numerical solution schenie used in the closed-loap simiilations ivas testetl on t hc

opened-loop geonietry usecl in Chapter 4. The resiilts iii Figiires 6.5 and 6.6 show

coniparisons between the preclictions of the steady siniulation clevelopecl in Chapter 4

(solid lines). and of the iinsteady simulation (synibols). Figure 6.: shows the predicted

steady state response of ccrebral flow to the niechmical cffects of Ch nit li no PPB.

Figure 6.6 shows the predictecl steady state response of cerct~ral flow to t hr nicchanical

cffccts of PPB at +.jCz. In hoth cases. the two methocls prrclict thr sanie rcspunse.

For tlic closecl-loop simiilation we iiscd the Gz and PPB schcdiiles showri in Fig-

ure 6.7. Sotc that. when the G-suit is sn~itchccl on. its rcsponsc is hylrostatic arid

tracks Gz instantancoiisly. Thrrr cases are corisiclcrcd: no lifc support. G-suit only.

and G-suit plus PPB. !\.lien PPB is applicd wit hoiit t he G-sui t carclio~asciilar prr-

forniance is macle worsc. hecaiise tlie large ccritral vr.rioiis pressiirtbs hincler vcrioiis

ret urn.

In cach of the following five figures. rcwilts arc prtwritrd for tlic t1irt.i' rits<.s c-ciri-

siclered. \\é esairiine the follonirig piiranietrrs as iiidiîators of the cardiov;isciilar

performance:

0 Ccnt ral art erial pressurc

0 Central ïerious prrssiirc

0 Cardiac outpiit

Cercbral bloocl fiow

Because of the long sinidation tirne. the instantaneoiis values are not tielpfiil. and

only the per beat time averages are shown.

Figures 6.8 and 6.9 show the variation in the niean central blood presstires. Kith-

out Iife support. there is a moclerate drop in the pressures following the O to +G

transition and a severe chop in pressure lolloaing the transition to +.Xk \Yitli the

G-suit switched on. the central pressures are returnecl to values that are only some-

wliat lower than normal. With both the G-suit switched on and PPB applied the

central blood pressures are elevated approsimately by the same amount as the tho-

racic pressure. However. PPB causes a slight additional fa11 in the driving pressure

F i e 6 . : Tlie open loop prcdictioiis of the stcady statc rcspotisc of ccrct~ral How to

varyitig Gï.. witli rio PPB. Solid hic: tlic resiilts froixi Ciiapter 4. Syt~ibols: ShIC iiictliod.

Fiyre 6.6: The open loop predictions of the steady state response of cerebral flow to PPB

at t5Gz. Solid line: the resiilts from Chapter-L. Symbols: SMC method.

I l O 10 20 30 40

Time [SI

Figure 6.7: Tlit? Gz and PPB schediile iiscd for closcd-loop siniulirtioris. Yotc that. when

the G-suit (Frariks Flying Suit) is switclicd on. its hydrostatic responsc tracks Gz iristanta

neo tisly.

across tlie heart. because it tiinders venotis retiirn. Figtirr 6.10 show Iiow tliis trans-

lates irito cardiac output. \\'ittioitt protection. ttierc is a dramatic faIl in t h cardiac

oiitpiit with transition to +.5Gz. On the other hand. both G-suit and ttie conibination

of G-suit and PPB rnaintain the cardiac output a t the lewls moderately lowcr than

normal. G-suit alone is slightiy more successful due to tlie ~f fec t of PPB on renous

return.

Figures 6.11 and 6.12 show. respectively. the variation in mean cerebral flow cluring

the simulation. and the jugular cross-sectional area at the end of the simulation. The

response of the cerebral flow to increasing Gz nitli no life support is qualitatirely

similar to that of the central pressures. but the fa11 is much more severe. This is

Figure 6.8: Tlie variation in the trieaii. per hcart beat. central artcrial prcssiire avcraged

per hcnrt bcat . Solid sqtiares: No protection. Hollow sqiiiircs: G-sui t. Hollow triangles:

G-suit arid PPB. The resiilts are for the schediiles siiown iri Figilrc 6.7.

becaiise tlie pressure across the heart lias fallen. arid the jugiilar resistarice lias risen.

With the G-suit switched on. there is a large improvement in cerebral ff ow. but it

rises to only slightly more than half its normal value. It is irnniediately apparent from

Figures 6.10 and 6.11 that PPB has a markeclly different effect on carcliac output and

cerebral blood fiow. Alt hough PPB sliglit ly loners t lie cardiac out put. it clramatically

improves cerebral florr. raising it to approsimately 530 cmvlmin. This behavior is

esplained by the area cun-es in Figure 6.1'2. For the simulations with no life support

and with G-suit only the jugular rein is collapsed over most of its length. Tberefore

its resistance to floiv is large. and. even with a normal pressure drop across the heart.

Figure 6.9: Tlic variation in the niean. pcr hcart herit. central vcnoiis pressure avcragcd

pcr lienrt bcat. Solici squares: No protection. Hollow sqii;ires: G-suit. Hollow triangles:

G-suit and PPB. The resiilts are for the s(i.tiediilt!s stmwn in Figiirc 6.7.

cerebral fiow is si1 bst ant ially recliiced. Sotc. hoaever. t hat iii acldit ion t O iniproving

venous return and cardiac output. the G-suit also raises the central pressures and

reduces slightly the jiigular collapse. The most clraniatic cffect on jiigular collapse.

honerer. is due to the PPB. whicii opens the vesse1 conipletely.

The results presented above demonstrate the integration. into the unsteady closed

loop simulation. of the ideas developed in Chapters 3 and 4 iising the steacly open loop

simulation. They also sliow that the roles of estracranial vein collapse in rcducing

cerebral blood flow. and of PPB in restoring cerebral blood flow. are the same in

the unsteady and steady simulations. The closed-loop simulation demonst rates t hat

k'igure 6.10: The vnrintioti iri the nieari. per tieart bcat. i:,udiac oittpiit nvtngcd per lieart

bent. Solid sqiinrcs: 210 protection. Hollow squares: G-suit. Hollow triangles: G-suit nrid

PPB. The resirlts arc for t iie scheditles shown in Figure 6.7'.

6000 1 O0

maintaining nornial cardiac output is a neccssary. but not a siifficient. condition for

ensuring normal blood siipply to the brain. Aciequate crrcbral circiilation is achieveci

only if the veins in the neck reniain opened.

It has to be strcsseci that the effect of +Gz on central blood pressures is over-

estirnatecl in the siniulation. since the mode1 does not incorporate refles control.

Ho~vever. we believe that the purely niecliariical response of the cardiovascular sys-

tem to Gz and prot ective measures is presented accuratel-

80 5000

* - n c

a- - O *Ë 4000 - 7

-È 7

- 60 cn O .. 0 3000 - 3 u

Ci a a CI

CT -

2000 - P

LL

1000 - -

O0 i 1 I 1 l I 1

IO 20 30 4 8 Time [s]

- -

- -

Time [SI

Figure 6.11: The variation in the mean. per Iieart beat. cerebral blood How averageci per

heart bcat. Solid squares: Xo protection. Hollow squares: G-suit. Hollow triangles: G-suit

and PPB. The resrilts are for the schedules shown in Figure 6.7.

Relative Distance

Figure 6.1'2: Profiles of the jugular area at tirne -40 secorids. with and without lire support.

Solid squares: No protection. Hollow squares: G-suit. Hollow triangles: G-suit and PPB.

The results are for the schedules shown in Figure 6.7.

Chapter 7

Conclusions and Recommendat ions

7.1 The Mechanical Determinants of Cerebral

Circulation

Ttii. siriiiilations arid esperinirnts coniplct~d i r i this stiidy iiIl support the coric.liision

that ttic niccliariiral factors coritrollirig cerchral perftisiori dtiriiig rlrv;itctl G z ;irr

~asciilar rcsistaricc and cardiac perforiiiaiicr: cerebral hlootl flow (lrops with +Gz

owing to o r i ~ or botti of tno factors: incrrasc of the rcsistanw of t h flow circuit

stipplying ttie brai11 diie to ~stracranial vrnoiis collapsc. ancl dccrrasc of the caniiac

oiitpiit duc to inadrqiiate venous rct iirri. The hiilk of t htl t hmis lias bwn drcli(:ated

to aniilyziiig whet hrr and how cerebral vacular rcsistance affects blootl siipply to the

brnin. The effect of Gz on cardiac pcrforniarice \vas acldressecl. to sonic estent. in

Chaptcr 6 where a closed-Ioop cardiovasctilar sinidation mis performd.

7.1.1 Cerebral vascular resistance during Gz stress

Three groiips of cerebral vessels can be clistingiiishetl. with respect to the resistarice

t hat t tiey provide to blood flou?: the est racranial arteries. the int racranial vessels.

and the est racranial veins.

Arterial resistance

The estracranial arteries are characterizcd by relatively large cross-scctional area and

sti& t hick walls. The arterial ~111s arc not likely to collapse iintler negative t rarisiriiiral

pressures caiised by the range of Gz stresses esperiencecl hy pilots. Ttiercfore the

arteriül resistance is both very small and inticpendent of Gz.

Cranial vascular resistance

The ressels in the craniiini do not collapse whcri t lie hlood prpssure is rirgatiw. This.

although ttie cranial resistance domiriates iintlrr riorrriai conditions ( O 2 Gz 5 1). it

cloes not increase whcn Gz>l. In fact. the auto-rcgiilatory nierhariisni will iict once

the ccrchral perfusion prrssiire is cliministietl due to +Gz. iirid the cranial rrsistmw

will clrop. Howcvrr. tliis is not siifficient to retiirri t h r total cerehral wsciilar rt.sistitnce

to nornial values sirice t h niairi site of viscoiis pressiire loss shifts to the estracrariial

reins. whicli lack atiy riicchanisni of protcctiori agairist rollapse.

Jugular resistance

The estracranial vcins arc characterizccl by wry roriipliant walls. For rit!gativr or

zero Gz t tiese art) clistcnclecl and the wrioiis rrsistiirirc is negligi Me. For positivr Gz.

liowrver. t lie reins collapsc as soon as the t ransmiiral pressiirr hcîornrs riegat i v ~ .

This sigriificaritly increases t heir resistance: at iipprosiniately + - l . X z t tiis qiials

the resistarice of the rest of the vasciilar nctwork siipplyirig the brairi n-itli blood.

Therefore. ttie verioiis msistarice is the only portion of the rasciilar resistanw that is

affectecl by the piirely niechanical effects of Gz. and the veins in the neck are the triain

contributor to overall rise in rcsistancc of thc vcssels above the hcart. It shoiilrl be

saicl. hoaever. that the hydrostatic pressure recovery in the veiris is never conipletely

eliniinated. and the mnoiis blood pressiire at the Iieacl level is negative for positive Gz.

7.1.2 The significance of negative venous and CSF pressures

for maintaining consciousness under +Gz

It has been s h o m in t his study that the flow does nut stop nhen the arterial pressure

at the head level is zero since the estracranial wins are able to sustain a certain

negative transmural pressure. Experimental evidence of the negative jiigiilar pressure

\-as provicled by Henry et al. (1931) nho nieasiired pressure betiwen -20 n m H g ancl

-60 n m H g at the top of the jugular vein in subjects esposed to i4Gz. This shoiild

provide a significant niargin of protection once the arterial pressure at the eye lewl

approaclics the atniospheric level. and should esplain wliy sorrie siibjrcts esposecl to

sustainecl Gz between t4 and +6 clo not lase coiiscioiisricss in spitc of the zcro arterial

pressure at the hcad lcvel (\Voocl 1987). Tlicrefore. a ncgatiw vcnoiis pressure in-

be a significant factor in rnaintaining corisciousriess iirider riiotlrratp t G z > 1. in sonic

animal species the capability of the veiiis to prociiice tlic siphon cffect scrnis to bc

greater t han in Ilriniaris. Thiis. rabbits si1 bjectctl to +2OCz iiiairitaiii riorr~ial r e r r t d

blood flow in spite of the arterial prtwiires in tlic range of -7 to -13 rr t rr t H g (Floreric~

et al. 1994).

Thr stiicly shows that tlie vcssels iriside th[> crariiospinal cavity do iiot collapse

duc to ronsewation of the cranial roliirric. This provides a certain clcgrre o f atldi-

tional protectiori to iritracranial circiilation in (-oniparison to t hc cirriiliitiori in t h e

estr;icraniiil tissiie of the Iieatl. h i n i a l rsprririicnts show a niiicli grratrr clecrrasr

in the blood flow in ttir t>stracranial vrssrls ttiari in tlir vcssrls siipplyirig the brain

chiring esposurc to liigh +Gz (Laugliliri et al. 1879: \\'crclian et al. 1994). An cqiial

tiyclrostatic pressure gradient in thc t>loocl and the CSF stio~ilci also rliniiriiitc regioniil

hydrostatic diff~renccs \vit tiin the craniospirial cavity. \ \ id i rar i et al. fourid t liat t tic

regiorial clifferences in blood f l o ~ to the hrain of baboons siibjcctrd to higli siistairied

acceleration cannot be associatecl \vit h a vertical gradient (\\érclian et al. lg!N).

Finally since PCsF is approsimately eqiial to venous pressiire regarcllcss of Gz. the

capillary bloocl pressurc in the craniiim ni11 always be higher than the tissiie pressiire

which is iniportarit for the perfusion of the brain tissiic.

7.1.3 Central blood pressures

In an open-loop model. the central blood pressures are t lie only parameters reflecting

cardiac performance and the action of protective mechanisms. The niain conclusion

of the study is that the absolute values of the central blood pressures are as important

as the driving pressure difference between the central arterial ancl central venous p res

sure. \Yhen Gz> 1. it is not siifficient to maintain normal central pressures at normal

level. since the cerebral bloocl flow will cirop cliic to increased vasciilar resistance. The

normal blood Aow can be restored. eittier by increasing the driving pressure clifference

at the heart. or by reversiiig the changes in t hc ccrebral vasciilar rcsistancc caiisecl by

Gz. \Ve believe that tlie protective nieastires. and PPB in particiilar. act by simiil-

taneolisly elevating central arterial and venoiis pressures. thiis. rrciiicirig the venoiis

resistarice while maintaining the clriring pressure clifference at tlie Iieart.

7.2 Cerebral Circulation and GLOC

\\liilc tliis stiiciy cioes riot provide a clefmite ariswcr about the rsart iiirchiiiiisni of

GLOC it reinforces the helief that loss of conscioiisn~ss is riiairily diir to sigriifiraritly

cliniinishecl blood flow to tlic brain. Sirice ttiis is caiisccl by collapsc of tlic vesscls

oiitsiclc of the craniiini it seriris tliat tlie effcct of Gz on t h fiiiictioriing of ttir brèiin

is intlircct. ttiiit is it is iiot caiisrd by tlir liytirostatic: ~ffccts withiri thc rr;iriiuni.

hccording to tlir niodcl resiilts. the tlrop in ccwhral hloocl flon nith incrriisirig +Cz

stiould he l a r g ~ crioiigti to bc rcgarclrcl as a priiicipal causr of CLOC. At +3 Cz. ivliicli

is t lie lcvrl of iicccl~rat ion ofton iissocia ttid wi t li loss of conscioiisrirss iii il ri pro t w t rd

intliviciuitls. thc rriocirl prcdicts a cl~crrasc in ccrrbral blood Hoa o f ;ipprosirriatcly

.50%. whicli stioiilcl br siifficient to cause cerrbral isctirrriia (Fiiiiierty ilt ai. 1054).

Tlierefor~. i t appears t hat GLOC ocriirs ivhcri rerebriil blood Row fèills bclow a certain

threstiolci lcrel. which r i i q he cfiffercnt for cadi iriclivicliial. -4ccortiing to \\krchari

(1991) this thrcstiold lcvel shoiild bc below 60% of the nornial vaiiic. or Iess tlian

approsirnatcly 430 crn"lmin. Esperimerital cvidcwce that cormlates inackqiiatc biood

flow with the loss of conscioiisness \vas provicleci by Clerc et al. (1990) ivtio reportecl

a case of GLOC ahere the blood flon in the niiclclle cerebral artery of the siibject

measiired with transcraninl Doppler \vas approsiniately 40% of the riornial valii~.

7.3 Cranial Vessels and CSF Dynamics

CSF pressure

The fact t hat CSF is subjected to the same hydrostatic gradient as the blood in the

cranial vesseis. is not sufficient to esplain the closeness of the cerebrospinal and venous

pressures. Simultaneoiis changes in venous and CSF pressures are observed not only

wtien caused by gravitat ional effect. but also cliiring cougtiing ancl sneezing (Hamilton

et al. 1943). \\-hile a hydrostatic pressure gradient is always present in a resting HiiicL

in the gravitational field. an additional condition is reqiiirrcl for deterrnining the

absolute value of the pressure. e.g. the bounclary condition at the frce siirface of a

fiiiid coritaincd in a mssel. That condition is provicled 1- the consernition of voliirne.

itiiposecl on the clastic vcssels in the the craniiim. Since the volunic of tlie ttiin-

wallecl vrins reacts strongly to an? change iri transniiiral pressiirc. the CSF pressiirc

has to rcniain closc to the vcnotis pressure iri ordcr that the corisc.rwtiori of volume

hc sat isfictl.

CSF and blood pulse in the cranium

\ \ k m a rigid coristraiiit is iriiposeil on a n clastic fliiicl-fillccl t i i t m its rffcctirc iiiertia

incrcascs chic to norriial strcsscs rirccssary to niaintain the roristarit ~o11irnc iri ttic

systmi. The i r i rr~as~ iri t+Fcctivc incrtia is proportional to ttio r a t i o of thr f l i i i t l rtixss

in arid aroiind t lic ttihta. For the piilsr propagiitiori iri t t ic* m i r i i i i t i i . t lic tlitritvisioiis of

the skiill coriipard to that of a typical bloocl vcsc.1 slioiilcl bc largr erioiigti t o riiaki. t tir

impact of the cmstraint wak. Ttirrrforr. tvc t~elieve that the skiill cotistraint does

not affwt ttie specd of pulse propagation in tlic craniiim. Ttir CSF prrssiire piilse is ;i

reflection of blood pulses in al1 intracranial vesscls. The fact that it strorigly rrsrriiblcs

the vcnoiis pulse is. again. dictated by ttir rorisrrution of the cranial \-oliirric

7.4 Contributions and Suggestions for Future Work

7.4.1 Contributions

This study points to the significance of the venous mechanics for iinclcrstancling the

effect of Gz stress on the cardiovascular system. It has been shown that i t w i n s

cannot be adecpately represented as lumped elements when simulating the effect of

accderation. Noreover. it has becn dernonstrateci t hat the veins shoultl be rnodeled

in significant ümount of detail. ivhereas the rest of the vasculature can be represerited

by relatirely simple elements. The study also shows that neither the siphon nor the

saterfail effects are adequate physical models for the flow in vertically orientecl. par-

tially collapsed vessels. siich as the jugular win of ari upright human. The sigiiificance

of venous dynaniics is not limitecl to the collapsc of the veins in the rieck. since bloocl

pooling associatccl with increasecl blood pressiire below tlie heart is mainly cliie to the

capacity of the l o w r bocly reins to accept large qiiantitics of blood.

To oiir knowledge. t his is the first niodcl of bloocl-CSF coiipling whcrr t lie hlood

vessels are represented as distributetl c l i ~ ~ t i c tubes. -Uso. we are iiot a w r e of a

study where the effect of skull constraint on the speeci of piilse propagation in the

craniuni is acl(lressecl. or a study where the CSF tiynaniics is consitlerrd as ;I Factor

of importance cliiring Gz stress. In Iiis nioclel of the rarrliovasciilar systrm unclcr

+Gz Chia-Lin (195-1) assiinied tliat PCsF always rqiials vrnoiis prtlssiirc. but dit1

not esplain t lie mcctianisni t hrough nhich t liis occiirs. Oiir nioclcl (Ici~ionst rates

t tiat the rlosciicss of CSF and verioiis prcssiirps origiriates frorri a piirrly ~tir(.liariical

condition t h tlie voliinic has to be conservccl. rathcr tlicri froni iritriwtc iictiori of t tic

c;irdiov;isciilar rcfleses iind ;i~itoregiiiatiori. as lias hc~r i proposrd i i i tlir p s t ('L'ah

et id. 1973: Sakagawa et al. 1974: Raisis et al. 1079). Firiiillj-. the rtsiilts stiow t h

a strict iiriplenieiitation of the the cranial voltirnc cotiservation is ii \.;ilid nio(lc1irig

assiirnption that can bc iisecl beond the scopr of accelcratioii physiology.

7.4.2 Suggestions for future work

111 the models relying on open-loop geonietry. the infliience of the rcst of the cardiovas-

ciilar system. and the protective measiires. cilil hr accoiirit.ed for otily by nianipiilating

the boiinclary condit ions. Also. the mat lieniat ical analysis applicd in t lie open-loup

rnodels is not appropriate for predicting tlie transient responsr of the crrebro\ascular

system. Iri summary. the open-loop rnodels are valid only for sustaincd Gz. This is-

sue is addressed to a certain degree by incorporating the simplifieci niodel of cerebral

circulation into a closed loop model of the cardiovascular model. However. althoiigh

the closed loop model does simulate purel! niechanical aspects of the transient be-

havior. it does not accoiint for the action of the cardiovascular refleses. The refles

response is essential in ana ly ing complex Gz profiles. in particular push-pull where

the refles action can be detrimental. Future analyses should incorporate models of

baroreceptors and the refles cont rol of heart rate. heart contract ili ty. and peripheral

vascular resistance into a closed-loop mode1 of the cardiovascular systeni.

As demonstrated in Chapter 1. cerebral circiilation strongly depencls on mechan-

ical propert ies of the veins sucli as vetioiis stiffness. Cnfort iinately. the pu blished

mechanical properties of the veins are often unreliable. and proride little inforniation

on hon given factors vary from incliridual to indivitlual. Therefore. a stiidy t h

\voiild estahlish the properties of the hiinian jiigular vcins in vivo. nould he of great

heriefit for acciiratcly predicting the impact of Gz stress on cercbral c.irculatiori.

Appendix A

Numerical Met hods for

Closed-Loop Cardiovascular

Simulation

A.1 Method of Characteristics

A.1.1 Characteristic variables

Consider the governirig 1-D ecpations in the lorni

Follo~ving (Anderson et al. 195-4. pp. 285-302) and Hirsch (1990b. pp. 1.57476). we

express A in terms of the eigenvalues. C * co. and the matris of left eigenvectors.

L-l . as:

IV here

A =

Sow. ori preniultiplying by L-'

characteristic variables. W. as:

ecluatiori ( A l ) can bc rewritteri in terms of the

Sirice the wave spccd is inclcprnclent of L'. WC c m ohtain an rspr~ssioti for W:

and

Finally on characteristic paths C= defined by

in cornponent forrn this is drr -* - - Co - -FI + F2

dt -4

( A . 10)

(-4.13)

(-4.14)

(A. 1 *5)

mhere Fi are the components of F.

A.1.2 Numerical solution

Ré can obtain estimates at time level tn" by first iising the valiies of -4 aiid L- at

tinie Icvel t n to estinlate the slopes of C= at (.ri. t n i l ) as C - I c o . Then Ive can estimate

t lie s-location of the characteristic paths C=. at tirne tn . as:

Ué c m tlien estiriiate the vdiies of II" - at (r?. t n ) by linex interpolation on the

points closcst to i=. Finally. tve estirriate II-:-' as

Sote that in gcncral if cliaracteristics rcacli a riode froni hotli iipstrtwii ancl clown-

Stream locatioris. the ttvo calciilat ions i~iiist I)r perfornied wit h t hr appropriate i i p

st rcam and (Io\vrist reani t iibe pararrirters. Fiirthtmtiorr. at inlets. out l~ts . ancl cm-

ricctions to lieart chanibers. cliarnctcristics \vil1 riot ~ s i s t on orir side of t tic tmiriclary.

Ratlicr. wt. tvill tiave sonic inforrriation rcgarclirig tlie valiics of -4 oiid L* iiiiposrd thrrc.

Cnfortiiriatrly. tlie intcgral iii cqiiatioii (-4.12) will gcwrally rcyiiirc* ~iiitrirrical

c\aliiation. and the noriliricèir rclatioti bct\verri tlir ctiarxteristic aritl priniitiw wri-

ablcs will gcnerally recliiir~ a nunicrical soliition of tlie rqiiations. This can Iw clone by

Sewton's niethod iisirig valiies at ttic prwious t ir i i~ levels as initial g i i ~ s t ~ s . Hoiwwr.

it is clcar tlint applying the niethocl of cfiaractrristics iit wcry no& in the prohlem

niay be conipiitationally espensive. In genrrnl. ne propose to use t h tiidioti of

characteristics at botinclary nodes ahere t h conipiitatiorial espense is jiistified. Sote

that secoiid order interior niettiods rvill not lose acctiracy if the boiiridary trpatnient

is first orcler (Gustafssoii 1915: Kirsch 1990b. Ch. 19).

A.2 Split Coefficient Matrix Method

A.2.1 Split coefficient matrix

Using the diagonalisation given in Section A. 1. ive split A in equation (A. 1) into

additive components Ai that are associated with signals propagnting in the positive

and negat ive direct ions respect ively.

where

For siipercritcal floa. L - 2 co

A; = A

for siipcrcritcal Row. L' 5 -ro

A, = O

and for suimitcd How. I' < 6:

Equatioti (-4.1) can riow be rcwritteri as

(A. 17)

(A. 18)

(A. 19)

(-4.20)

A.2.2 Discrete equations

For the discrete problem ae use the notation V:. nhere the siiperscript indicates thc

time level and the siibscript the nocle nurnber. tn-' = t n + At. shere the tinic step It

is the same for al1 segments. On an- segment the iiodes are numbered sequentially

t h + = r, + Ar. where the the space step I r is constant oti any segnient.

but can Vary frorn segnient to segment. Haviiig split A into components associatecl

with signals propagating in the positive and negatiw directions. we use a backward

spatial difference for the A+ terril. and a forward spatial difference for the A- term.

Difference equations can be obtained by using first order difference approximations

to the derivatives.

The triincation error for t his discretisation can he obtairietl froni Taylor Scries cspari-

sions of the primitive variable aboiit the nodc (i. n ) .

now t hc tiiscrctc cqtiat ion (.LX) rcdiices to:

and siricr Ai - A- # O

Thus. t lie niet liod is a lwys first urtlcr.

A.2.3 Volume conservation

Since the SCSI methotl is not conscrvativr. rnass arid niomeiitlini are only iipprosi-

niately coriservccl by the basic difference srhcnie. The failiire to consrrw niomcntum

does not appear to be a problem becatise of the vcry strong dissipation in the c a p

illary becls. However. the error in the continuity eqiiation is the only roliinie source

term and it leads to a systernatic accurniilatiori of voliirne in the circulatory systern.

This volume increase leads to a systernatic risc in the vasciilar pressures. K e correct

for the volume errors at the end of each time step jiist before the global vector of

primitive variables is updated. At that stage 1ve still have the variable values at both

time lerel tn and t n + l . These allow us to make an accurate. O(1x + At)'. estiniate

of the volume error and correct the time lerel t"" values of -4 before they are stored.

Consider the cornputational ce11 illustrated in Figure A. 1. During the penod

tn 5 t 5 tnCL the volume of the ce11 increases by approsimately It1xtX4fût. and the

Figure A.1: Cc11 usctl for wliinie corrtxtioii.

there is a net voliiriie inflow of approsirriatcly - l t l r i )Q/ i ) r . Tlicrcforr.

whrre LI is the lrrigtti of ttie j t h tube on aliicti t h crlls considerrd lir. Ttic siini-

niatiori is orcr the rells or1 the j th tiitw. \\*c rstiniatc t h trnipor;il and spiitid

derivatiws at (.Y,. T n ) as:

A.3 Boundary Conditions

Here n e consider the treatment of bounciary at two segment junctions. For the in-

formation 011 the treatment of other types of bouridary. such as: branches from one

vessel to two. the convergence of two vessels to one. heart artery junctions. ancl vein

heart junctions. the reader should refer to (Kalsh and Cirovic 1998). The characteris-

tic variables. I?*,below. have been estimated at the vessel bounclaries as described in

Section A. 1. \té obtain boundary conditions from them by eqriating these estiniates

t O the locally defined values.

A.3.1 Newton's method

In each case the boiintiary conditions ni11 be espressed as

f (s) = O (-4.32)

where s is the wctor of iinknown primitive variables at the boundary and f is a

vector valuecl fiinction that also clepencls on paranieters sucii ils t. R (ttie rrsistaiicc

to flow at the jiinction). ancl I \ = . The subscript . denotes qiiantities rvaliiated iit the

ciirrerit approsiniation to tlie solution to e<!iiiition (A.32). Thcri an inipro~ed soliitiori

est iniate is proridecl by

Iri cvery ciise the i-aliie froni thc prcvioiis tirlie levrl is usctl as the initial rstiniate

of tlie soliitiori. and the riietliotl is applicd itcratively until ii r«nvrrgPri(:r critrrion is

niet or a niasirriiini iiiiniber of iterations is escee(1rd. In the latter ciisc a w;irriing is

printcd.

A.3.2 Two segment junctions

The iinkriowns at this type of botindary arc

M' is tlii. transpose of the a r r q M. and t lie siibscripts 1 arid r. tlenote the srgmmts

to the lcft and right of ttie jiiriction rcspectivt~ly. Left anci riglit arc dcterminrd bu

the positive sense of Aow clefiried in the gcomctry data.

The four boundary equations are tlctcrmineti by II*=. contiriuity of fiow. and a

pressure equat ion.

The Riiid pressures. Pi aiid Pr. are calciilated iisirig the tube lans and rsteriial pres-

sures appropriate to cadi scgriicnt. Sote that the rsternal prcssiire may riot bc con-

tiniious at a juriçtion corrrspontlirig to a division betwerri two body cotiipartnierits.

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