Eindhoven University of Technology

MASTER

Study of the contribution of shear induced red cell rotation to gas transport andmeasurements on blood and hemoglobin solutions using a membrane oxygenator withtangential flow

Teirlinck, Harry C.J.M.

Award date:1977

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STUDY OF THE CONTRIBUTION OF SHEAR INDUCED

RED CELL ROTATION TO GAS TRANSPORT, AND

MEASUREMENTS ON BLOOD AND HEMOGLOBIN

SOLUTIONS USING A MEMBRANE OXYGENATOR

WITH TANGENTlAL FLOW

MASTER 1 S THESIS AFSTUDEERRAPPORT

H.C.J.M. TEIRLINCK

EINDHOVEN, August 1977

THIS STUDY HAS BEEN COMPLETED UNDER THE

GUlDANCE OF Ir. J.M.M.OOMENS~ AND THE

SUPERVISION OF PROF.DR. P.C.VEENSTRA

AT THE EINDHOVEN UNIVERSITY OF TECHNOLOGY,

THE NETHERLANDS

•lr.J.M.M.Oomens holds the authority over the data as mentioned in figures 1 up to 28, chapter V inclusive. Reproduetion of these data, in any form whatsoever without written consent is forbidden

Mijn afstudeerwerk heb ik verricht in de vakgroep Productie

Technologie van Prof.Dr. P.C. Veenstra, in de sectie

medische techniek. Bijzondere dank ben ik verschuldigd

aan ir. J.M.M. Oomens, die mij gedurende de afstudeer

periode dagelijks heeft begeleid. Dr.ir. H.J. v.Ouwerkerk

dank ik voor zijn theoretische begeleiding. Binnen de

sectie medische techniek bedank ik th. v.Duppen en

J.Cauwenberg voor hun technische begeleiding.Verder heb

ik altijd bijzonder prettig in de groep gewerkt. Ook de

discussies met Dr. P. Stroeve (Universiteit Nijmegen,

afd. fysiologie) zijn zeer gewaardeerd.

Harry Teirlinck

Table of contents

List of symbols 2

References 5 Summary 7 Samenvatting 8

Chapter I Introduetion

Short historical review of the membrane

lung project and the purpose of the

present investigation

C hapte r I I Th e o re t i ca 1 mode 1 s f o r o x y gen t rans f e r in flowing blood layers

2.1 introduetion to and fundamentals of oxygen

9

transfer 10

2.2 gas transfer in a blood layer and the advancing front theory 12

2.3 transport equations for the membrane oxyge-nator 17

2.4 oxygen transfer analysis: theory and experiment 21

Chater I I I Mass transfer due to shear induced ce11 mouvement

3~1 introduetion

3.2 gas transfer with stationary, isolated particles

3.3 gas transfer in stationary layers with spherica1 particles

3.4 gas transfer in flowing blood layers;

theoretica1 models:

3.4.1 Petschek,H.E. and Weiss,R.F.

3.4.2 Ke11er,K.

3.4.3 Hyman,W.A.

3.4.4 Antonini ,G.c.s.

3.4.5 Leal ,L.G.

25

27

31

34

36

37 38

39

40

3.4.6 Nir,A. 41

3.4.7 summary of literature 42

3.5 new modeis for shear induced oxygen transfer 44

Chapter IV Experimental techniques and sett-up for measuring gas transfer in flowing blood and hemoglobin solutions

4 • 1 introduetion

4.2 description of the new experimental

4. 3 the measuring circuit

4.4 photometer

4.5 principles and techniques of the quantitative determination

4.6 estimation of sourees of errors and influence of errors on the effective diffusion coefficient

Chapter V experiments and discussion

5.1 introduetion

5.2 experiments with blood in the second membrane oxygenator

se tt-up

5.3 experiments with hemoglobin solutions using the second membrane oxygenator

5.4 hemoglobin experiments using the first prototype of the membrane oxygenator

5.5 discussion

Conclusions

Suggestions for further research

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Appendix F

Determination of the oxygen partial pressure P at the membrane-blood . f 0 Inter ace

Solution of the transport equation (2.5)

Determination of the corrected fractional saturation change f . m

The terms iC/~t and 4blS/èt in the saturation calculation for flat duet and spherical qeometry, calculated with the advancing front approximation

Solution of equation 3.2, 3.3, 3.4

List of accumulated literature with Dutch summary

47

48

51

53

57

60

64

65

75

94

96

1 0 1

102

1 0 3

105

108

1 1 0

1 1 3

1 1 7

List of symbols

a

A

B

b

b. - I b

c

cv cvo

D m

D c

D p

0Hb d

d m

dS

f m

f{q)

Hb

Hb0 2 [Hb]

F

2

ratio of effectlve and Brownlan diffusion

coefficient { Deff/ D ) 2

m a re a

width of the cylinder

hemoglobin concentratien

Hb concentratien number

mean value of all b.'s I

concentrat ion

= [Hb] )

concentratien in the fluid layer

oxygen concentratien in the fluid

layer at the fluid-membrane interface

oxygen concentratien in the membrane

at the fluid-membrane interface

concentratien of deoxyhemoglobin

concentratien of oxyhemoglobin

optical density

oxygen diffusion coefficient

oxygen diffusion coefficient in

the fluid layer

oxygen diffusion coefficient in

the membrane

oxygen diffusion coefficient in

the continuous phase

oxygen diffusion coefficient

inside the partiele

hemoglobin diffusion coefficient

layer thickness

thickness of the membrane

drift in S during hemoglobin exp.

fractional saturation change

dimensionless velocity function

hernoglob in

oxyhemoglobin

hemoglobin concentratien

facilitation factor

m

gmol/m 3

gmo1/m 3

gmo1/m 3

gmol/m 3

gmo1/m 3

gmo1/m3

gmo1/m 3

gmo1/m 3

gmo1/m 3

2 m /s

2 m /s

2 m /s

2 m /s

2 m /s 2 m /s

m

m

gmo1/m 3

H i H

h

I . I

I 0

k.

k

M

N

n

I

~ dip

p

Pe

q

r

R

R

s s

s s .

I

T

t

3

hemiglobin as a fraction of the total ~~ ratio of remaining 02 uptake capacity

of the enterinq blood and the concen­

tration difference between the ente-

ring blood and the gas

height

integral with index i from the

advancing front equations

intensity of the incoming light beam

intensity of the transmitted light

beam

current

constant with index

constant

sphere of influence

dimensionless channel length

lengthof the light path

relative membrane reststance

total number of b.'s I

revoluttons per second (r.p.s.)

distance of the oxygenation front

dipole-strength of a field caused

by one partiele

dipole-strength of a field caused

by an assembly of particles

oxygen partial pressure

Péclet number;it gives the ratio

of transport by convection and the

transport by diffusion

dimensionless penetration depth of

the oxygenation front

radial length coordinate

radius of a sphere

resistance

mean outlet saturation

oxygen saturation

initiai oxygen saturation

temperature

time

m

A

m

m

- 1 s

m

2 gmol /s.m.atm

2 gmol /s.m.atm atm

m

m

s

t 0

t~ 0

V V

V

V

vo V ( y)

x x,y,z

y

e:

w

Cf LED

grad

[ ] th

exp

4

oxygenation time

dimensionless oxygenation time

voltage

velocity

mean velocity

velocity of the cylinder surface

velocity profile

length coordinates of a rectangular

coordinate system

dimensionless length

solubi1ity of the continuous ph a se

solubility of the dispersed phase

solubility of oxygen in blood

solubility of oxygen in water

solubility of oxygen i n a

hemoglobin salution

solubility of oxygen i n the membrane

solubil ity of oxygen i n the f 1 u i d

1ayer

shear ra te ( = dV/dy )

extinction coefficient

extinction coefficient of deoxy­

hemoglobin

extinction coefficient of oxy­

hemoglobin

viscosity

angle

standard deviation

angular velocity

hematocrit

flow

flux light emitting diode

a TI+ y_.'V

concentratien brackets

subscript meaning theory

subscript meaning experiments

s

V

m/s

m/s

m/s

m/s

m

gmol/m~atm gmol/m~atm gmol/m~atm gmol/m~atm

gmol/m~atm gmol/m~atm

gmol/m~atm -1

s

m2 /gmol

2 m /gmol

2 m /gmol

kg/m.s

-1 s

2 gmol/m.s

5

References

Antonini,G.,Guiffant,G.,Quemada,D.:effect du mouvement induit

des hematines sur letransport plaquettaire

Biorheology,Vol .12,pp 133-135(1975)

v.Assendelft,O.W.:spectrofotometry of haemoglobin derivates

Van Gorcum, Ltd., Assen, The Netherlands (1970)

Co 1 ton , C • K. , Sm i t h , K. A. , Me r r i 1 , E • R. , a n d Fr ie d man , S • , : d i f f u s ion

of urea in flowing blood, AIChE Journal Vo1.71,pp.800-808

(1971)

Colton,C.K.:Artificial lungs for acute resporatory failure.

Theory and Practic, W.M.Zapol and J.Qvist, eds.,Academic

Press,N.Y.(1976)

Cox,R.G.,Zia,I.Y.Z.,and Mason,S.G.:particle motion in sheared

suspensions. Journal of colloid and interface science,Vol 27

No.1,7-18 (1968)

Fesler,R.and Clerbaux,Th.: simple,accurate solid-state diode

fotometer for use in measuring oxygen saturation of whole

blood.Clinical chemistry,20,1135 (1974)

Goldsmith,H.L.,Mason,S.G.: the microrheology of disperslons

Rheology-theory and appllcations,Vol IV (chapt.ll) ed.by

F.R.Eirich, Academie Press,N.Y. (1967)

Hayashi,A.,Suzuki,T. and Shin,M.: an enzym reduction system for

metmyoglobin and methemoglobin,and its application to functi­

onal studies of oxygen carriers.Biochimica et Biophysica Acta

310,309-316 (1973)

Hyman,W.A.:augmented diffusion in flowing blood. ASME,73-WA­

Bio-4 (1973)

Kats,P.:internal report Eindhoven University of Technology,

medica) technic (W) (1976)

Ke11er,K.: effect of fluid shear on mass transport in flowing

blood. Federation proceedings vo1.30,No.5,sept.-okt (1971)

leal,L.G.:on the effective conductivity of a dilute suspension

of spherical drops in the limit of low partiele Peelet

number.Chem.Eng.Commun. ,vo1.1 ,pp.21-31 (1973)

Links,P.G.:de invloed van Tay1or-wervels en turbulentie op de

gasoverdracht in de membraan oxygenator met tangentiële flow

Master thesis Eindhoven University of Technology (1976)

6

Maxwell,J.C.:a treatise on electricity and magnetism,Vol.l.

p.440,Clarendon Press,london,England (1881)

Nir,A.:studies on the mechanics and termal properties of

sheared suspensions.Thesis,Stanford University,Ph.D. (1973)

Oomens,J.M.M.:membraan oxygenator met tangentiële flow.

ontwerp-modelvorming-meting.Master thesis, Eindhoven

University of Technology (1973}

Oomens,J.M.M. and Spaan,J.A.E.:a generalized advancing front

model descrihing the oxygen transfer in flowing blood.

2 nd international symposium on oxygen transport to tissue

Mainz (1975)

Oomens,J.M.M.,Spaan,J.A.E. and Donders,A.P.P.: annular membrane

oxygenator with tangential flow.Oxygen transfer analysis and

sealing rules.Physiological and clinical aspectsof oxygena­

tor design,edited by S.G.Dawids and H.C.Engell,publ ished by

Elsevier/North Holland Biomedical Press (1976)

Overcash,M.R.: couette oxygenator.Ph.D.Thesis,University of

Minnesota (1972)

Petschek,H.E. and Weiss,R.F.: hydrodynamic problems in blood

coagulation. AIAA paper No.70-143 (1970)

Schmid-Schönbein,H.,Wells,R.: fluid drop-like transition of

erythrocytes under shear. Science,vo1.165.pp.288-291 (196~)

Smith,K.A.,Meldon,J.H.,Colton,C.K.:an analysis of carrier­

facilitated transport, ibid,J1,102 (1973)

Spaan,J.A.E.:transfer of oxygen into hemoglobin solutions

Pflugers Arch.342,pp.289-306 (1973)

Spaan,J.A.E.:oxygen transfer in layers of hemoglobin solutions

Thesis, Eindhoven University of Technology (1976)

Stein.T.R.,Martin,J.C.,Keller,K.H.:steady-state oxygen trans­

port through red blood cell suspensions.Journal of applied

Physiology ,vo1.31,No.3,sept. (1971)

Stroeve,P.,Colton,C.K.,Smith,K.A.:steady-state diffusion of

oxygen in red blood ce11 and model suspensions. AIChE

Journa1 (vo1.22,No.6) p.1133-nov. (1976)

Stroeve,P: diffusion with irreversible chemical reaction in

heterogeneaus media;J.theor.Biol. 64,237-251 (1977)

7

Summary

This master 1 s thesis deals with theoretica! and experTmental

research on the contributTon to gas transfer of red cell

rotatien in a couette flow.ln a membrane oxygenator constructed

for this purpose gas transfer rneasurements have been carried

out with blood and hemoqlobin solutions.We use hemoqlohin as

a reference fluid to determ gas transport in the absence of

particles.However,since hemoglobin solutions and blood differ

essential ly,comparative experiments between blood and hemo­

qlobin solutions will not qive the effect of partiele rotatien

only,but a total effect.ln shear flow and also when no flow

occurs,the gas transfer in hemoqlobin solutions is eescribed

by rather simple differentlal equations,which may be solved

approximately using an advancing front theory.When the same

equations are used for red cell suspensions one must expect

to find an effective diffusion coefficient,which depend on

the shear rate.

The theoretica! part of this research consists of a

fundamental study of the influence of red cells under shear

flow conditions on mass transfer.From literature several models

are known,which derive an effective diffusion coefficient for

transport as a result of partiele rotation.Comparision of those

models with our experTmental results shows,that of mutual ly,

strongly differing models only one model correlates with the

experiments.

For the part of the experiments we can say,that we cannot

yet obtain definite conclusions about the hemoglobin experi­

ments,because of scattering in the measuring points.The blood

experiments show an increase of the effective diffusion

coefficient with shear rate,but these experiments do not agree

with earl ier experiments by Oomens(1976) and other investi­

qators.

8

Samenvatting

Mijn afstudeerwerk bestaat uit een theoretisch en experi­

menteel onderzoek naar de bijdrage van de beweging van de

rode cel in een couette stroming aan massa transport in

deze stroming in een richting loodrecht op de stroming.

In een speciaal voor dit doel ontworpen oxygenator zijn metin­

gen gedaan aan het gas transport in stromende bloed en hemo­

globine oplossingen. We gebruiken de hemoglobine oplossing

als referentie vloeistof om het gas transport te bepalen in

een vloeistof zonder deeltjes. We moeten echter goed beseffen,

dat bloed en hemoglobine oplossingen in vele opzrshten ver­

schillen. Wanneer we vergelijkende metingen doen met bloed

en hemoglobine oplossingen, dan zal een gemeten verschil in

gas overdracht niet alleen veroorzaakt worden door het deel­

tjes karakter van bloed, maar een resultaat zijn van alle,

voor het merendeel onbekende,verschillen tussen bloed en hemo­

globine oplossingen.

Het gastransport in hemoglobine oplossingen wordt beschre­

ven door een differentiaal vergelijking, die met behulp van

een·advancing front theorie benaderend kan worden opgelost.

Als we dezelfde vergelijking gebruiken voor bloed, verwachten

een effectieve diffusie coefficient te vinden, die een functie

~is van de snelheids gradient.

Het theoretisch gedeelte van dit onderzoek bestaat uit een

fundamentele studie van de invloed van de rotatie van de rode

bloed cel, die zich in een veld bevin~met afschuiving, op het

transport in het bloed in een richting'loodrecbt op de stroming.

Uit de literatuur zijn verschillende modellen bekend, die een

effectieve diffusie coefficient afleiden voor dit verhoogde

transport.Een vergelijking van deze modellen met experimenten

toont aan, dat van de onderling sterk verschillende modellen

er één correleert met onze meetresultaten.

Wat betreft de hemoglobine experimenten kunnen we opmerken, dat

we nog geen conclusies kunnen trekken vanwege de grote spreiding

in de meetresultaten. De experimenten met bloed leveren een

effectieve diffusie coefficient op, die toeneemt met oplopende

snelheids gradient.Deze experimenten correleren niet met de

experimenten van Oomens(1976).

9

Chapter Introduetion

Short historica! review of the membrane lung project and the

purpose of the present investigation

During several years a sub-division of the sectien Medica1

Techniques has been werking on the development of a new type

of rotating membrane oxygenator for extracorporeal circulation.

Besides the construction of a medically applicable oxyqenator

a secondary purpose of the project was the study of the effect

of red blood ce11 rotatien in shear flows on oxygen transfer.

The first prototype of the membrane oxygenator served the

two purposes.lt permitted a study of the app1icabi lity of the

set-up as an oxygenator device and also served as a basis for

a theoretica! model for the flow behaviour and the oxygen take­

up of the system.As aresult sealing rules fora clinical oxy­

genator could be formulated.ln the series of experiments a

shear-induced augmentation of oxygen transport was measured.

There appeared an increasing relationship between an 11 effective 11

diffusion coefficient and shear rate.

The subject of my master 1 s thesis is a further study of the

contribution of red ce11 rotatien to mass transfer in the

membrane oxygenator and in a couette flow in general.For that

purpose the latest experimental sett-up is used.With this appa­

ratus it is possible to measure the contribution of shear­

induced diffusion to mass transport more accurately.Next to the

experimental part the fundamentals of red cell rotatien will

also be regarded.

Chapter I I

1 0

Theoretica! models for oxygen transfer in flowing

blood layers

2.1 introduetion to and fundamentals of oxygen transfer

The transport of oxygen in blood occurs mainly through hemoglo­

bin.Hemoglobin is a macromolecule (M=64500) which is present in

the red ce11s with concentrations of about 35 %wt .The molecule

'consists of a folded structure of four chains (two a- and two

8-chains).Besides the usually present tetrameric structure

(Hb 4 ) also dimer (Hb 2 ) and monomer (Hb) structures are found

in dilute hemoglobin solutions.Each subunit Hb of the molecule

is a polipeptide chain containing a protein named globin and

and a pigment heme.Heme consistsof a flat ring of four nitro­

gen atoms, at its center a Fe 2+ion.The Fe 2+ion is linked to the

four N-atoms of the heme and also with one nitrogen atom of a

histidine in the globin chain.The sixth site of the Fe 2+ion is

empty.Oxygen can bind to this site.lf oxygen is bound to the

heme group the iron ion retains its charge and the binding ~f O>CY~( >"4\..-hon.

oxygen is not like an oxidation proces.lt is called ~xyijatie~.

Hemoglobin also reacts with CO,NO,co 2 ,H+ and 2,3-DPG.CO and NO

occupy the site of 02 • As the affinity of the heme forCO is

about a hundred times that for o2 even low concentrations of

CO een block the oxygen transport by hemoglobin.ln addition

to the ferrous state of the iron ion within the heme the iron

can also be oxidized to the ferric state.ln this form oxygen

is bound strongly to the heme group.lt is known as hemiglobin.

or methemoglobin.This hemiglobin is no Jonger an active oxygen

carrier.

In relation to oxygen transport in blood and hemiglobin

solutions several basic concepts are used:

The oxygen binding capacity of blood is defined as the maxi­

mum amount of oxygen that hemoglobin can bind.One gram hemo­

globin can bind 1.36 ml oxygen.ln human blood there is an ave­

rage hemoglobin concentratien of about 15 %wt.The oxygen

binding capacity is 7.5 gmol/1.

The oxygen saturation is defined as the ratio of the concen-

1 1

tratien of bound oxygen and the oxygen binding capacity.

The concentratien of freely dissolved oxygen is determined

by the oxygen partial pressure P.The relation between the par­

tial pressure and the concentratien of a species X as given

by Henry's law is

[x]= Ct • p x x

where ~ is the solubility of X (gmol/m~atm) x

( 2 • 1 )

At equilibrium there is a certain relationship between the

oxygen partial pressure and the saturation called the oxygen

saturation curve.This curve is sigmoid in shape and its posi­

tien is affected by temperature,pH,PCO and the 2,3-DPG concen­

tration.At low values of the pH and hi~h co 2partial pressures

the dissociation curve wilt shift to the right;less oxygen wilt

be bound to the hemoglobin at the same o2 partial pressure.An

increase in temperature wilt shift the curve to the right also.

In the body these two effeGts are favorable both for oxygen

take-up in the lungs and for the oxygen release in the tissue.

Figure 1 shows examples of saturation curves.

Cc.c.AV8o p~ T~

I l.1 n·c. 1. n·c.. 3 l·C.C ll~ '1 't.t a~-c.

20 30 '10

Figure t. Saturation curves(fractional oxygen saturation

as a function of oxyqen partial pressure ) of normal human

blood at several values of pH and temperature.

1 2 ..

2.2 Gas transfer in a blood layer and the advancing front

theory

Transport of oxygen in stationary or flowing layers of blood

and hemoglobin solutions takes place in several ways.Oxygen

is bound to the hemoglobin molecule.ln flowing layers the

transport of hemoglobin wiJl result in the transport of oxygen.

A second transport mechanism is the diffusion of physica11y

dissolved oxygen.This effect is much smaller .At an oxygen

partial pressure of 100 mmHg the total oxygen content of blood

is 19.8 %wt.Of this amount only 0.3% wt is physically dissol­

ved.The rest is present as oxyhemoglobin.According to Henry's

law the amount of physically dissolved oxygen is proportional

to the oxygen partial pressure.The third possibility for oxygen

transport in hemoglobin solutions is the diffusion of oxyhemo­

globin,the so-called facit itated diffusion.

The diffusion coefficient of oxygen in hemoglobin solutions

is a function of the hemoglobin concentration.

J)xLO' CJtJ-16 -------------------.

l

10

,, ---....... __ -.[H\ol- i:_~-

Figure 2. Compilation by Kreuzer(1970) of the d~ta available

fo- the diffusion coefficient of oxygen in hemoglobin solu­

tions,normalized toa value of De in saline of 2.07 10 -9 m2/s

according to Go1dstick{1966). ~ Spaan(1976)

1 3

Figure 2 gives a compilation by Kreuzer of data available for

the diffusion coefficient of oxygen in hemoglobin solutions.

Note the almost linear relationship between D and the hemoglo­

bin concentratien which exists for Hb concentrations lower than

20% wt. Our experiments have been done with solutions having

Hb concentrations less than 15% wt. From figure 2 follows a

relationship between D and Hb

D = 2.07 10 -9- 0.271 10 -9 .[Hb] (2.2a)

where: D (m 2/s) and[Hb]

The diffusion coefficient of oxygen in blood is not exactly

known. lt can be approximated by -the effective diffusion co­

efficient of oxygen in an agar suspension of red blood cells.

The results of Stein c.s. (1971) are given in figure 3.

In our experiment the diffusion coefficient of o2 in blood

will bedescribed by the equation :

(2.2b)

where: D (m 2/s) and [Hb]

The solubi lity coefficient of oxyqen in hemoglobin solutions

~Hb and the solubility of o2 in blood ~Bare also functions

of the hemoglobin concentration.

S pa a n ( 1 9 7 6 ) ob t a i n e d t he de p e n de n c e o fOlH b-. on t h e s o 1 u b i 1 i t y

of oxygen in water, on the hemoglobin concentratien and on

the concentratien of NaCl in the solution.

O(Hb = O(H O (1 + 0.00312 [Hb] )- ..:1oL[_NaCl). 2

( 2. 3)

where: 40<= 4.310

-7 per mol NaCl and [NaCl] approximately

equals 0.07 mol/1)

The solubility of oxygen in blood follows from

( 2. 4)

Fig.3

1 4

Effective diffusion coefficient of oxygen in an agar

suspension of red blood cells at 25t as a function

of volume fraction of cel15. Solid line is tHbretical

predietien for x/y = 0.20, dashed line is th~retical

predietien for x/y = 1.0. x/y means the axis ratio

of the 1 mode1 1 spheroides. ~ Stein 1971

The diffusion coefficient of hemoglobin is smaller than that

of oxygen. This is rather obvious, the oxygen molecule is much

smaller than the hemog1obin molecule.

Figure 4 is a new comptlation of the date, available for the

diffusion coefficient of hemoglobin given by Spaan (1976).

In our experiments with the membrane oxygenator we can neglect

the contribution to gas transfer by the diffusion of hemoglobin.

The equations, which describe the oxygen transport in the membrane

oxygenator, as given by Oomens c.s. (1976), take into account

the oxygen chemically bound to hemoglobin and the physical ly

dissolved oxygen. These equations wilt be discussed in the

next section.

1 5 ,~·~------------------------------------,

cm2 I si-~~~.11"'--

T

5 10 15 20 25 30 35 (Hfil ---.. g%

Figure 4. Compilation of data available for the diffusion coefficient of hemoglobin by Spaan(1976)

•final experiments Spaan Oearlier experiments Spaan • Ke 1 1 er et a 1 • ( 1 9 71 ) •Riveros-Morena and Wittenberq (1972) Ot1o11 (1966) ..

However we have to make several assumptions about the trans­

port mechanism.Blood is a fluid with a non-Newtonian flow

behaviour because of its non homogenious character.Therefore

we suppose the.hemoglobin to be distributed uniformly over the

fluid.The oxygen transport can bedescribed now with differen­

tlal equations.Because the oxygen saturation appears in these

equations,which is related to the oxygen concentration acear­

ding to a sigmoid shaped curve ,these cquations can only be

solved numerically.

With the aid of the advancing front approximation we can

evate this difficulty.lnstead of the actual saturation curve we

assume a stepwise draqe of the saturation with oxygen concentra­

tion,which results in the presence of a sharp boundary between

regions containing oxygenated and deoxygenated hemoqlobin,as

can be seen in figure 5.

16 ·-·-·r--- ---

I I ., I I I I

Figure 5. Saturation curve(solid 1ine) and saturation step function according to the advancing front approximation(dashed 1 ine)

lf we compute the concentratien and saturation profile in an

immobile slab of blood with thickness d it is found that diffe­

rences between the numerical solutions and the advancing front

solutions are small (see figure 6)

js ~---..r·-·-----· ·-·-

o.~-

\ I I I

I \

si ---·------4-'~,---+

0.1·

0

1c

~ ~

Ci. ·-·-·-. ___ .. \::''----

0~-----4------~---

Figure 6. Oxygen saturation and concentratien profile as a function of the distance y in the fluid layer. solid llnes result from A.F.approximation,dashed lines are found wi~h numerical solutions

; ' • ' ~ ' ;. ~ '. j -· ; l.. . ' ,- : ·~ '

The advancing front approximation supposes that in the region

where the hemoglobin is fully saturated the oxygen diffuses under

a constant concentration gradient.

1 7

2.3, Transport equations for the membrane oxygenator

We wi11 derive the equations,that describe the steady-state

oxygen take-up in a straight parallel flow channel.As said be­

fore we take in account both chemica! reaction and physical

dissalution of oxygen.The derivation is done for an unknown

velocity profile inside the channel and the presence of a

membrane is taken in account:The membrane offers an additional

resistance to the oxygen flux into the channel. For the derivation of the equations we use the definition

sketch of figure 7.

x

0

d. /

/'cll'l) /

/• Cv

. .. '- . .. .. . ... . .. . .. : ... .... :... . , .. ~. '· ... ·• -, rY\~\c:,~ .... :. . . - -

~cL S=l

p /

..........._~======- Af>

Figure 7. scheme for the calculation of the oxygen up-take i n a s t ra i g h t , pa ra 1 1 e 1 f 1 ow c ha n n e 1

where: V (y) = flow velocity x

Hb = hemoqlobin concentratien S = saturation ~ = oxygen partial pressure d = thickness of the channel d • thickness of the membrane pm = distance from the blood-

membrane interface to the the oxygenation front

m/s

qmol/m 3

atm m m

m

1 8

We are interested in situations where:

1. the flow is laminar,the velocity profile is continuous

2. the concentratien CvO is higher than the concentratien

belonging to S=l

The differentlal equation for this problem is given by:

( _y.grad.) C = D ó.C + Q (2.5)

where:Q is a reaction term

Equation 2.5 can be written as:

(v.4- + v.l- + v.l-) c = o XaX yay ZdZ

To solve this eq~ation w~ have to make several assumotions:

1 Th · 1 d · f f · · h d · · a2c . o • e re 1 s o n y 1 u s 1 on 1 n t e y - 1 re c t 1 on , s o ä'X2.. =

2.The processes do not depend on the z-coordinate,the problem a 'Je

is two dimensional ,so Vzä'Z and a? equat 0

3.Chemica1 reaction takes place at the oxygenation front.

So Q=O,except on the front

lf we remark that V equals O,'+'e can simplify (2.6): y

vll xax

Since D/V x

=

and

As a result of this,ac;ay=const.

( 2. 7)

4. so there is a 1 inear conce~tratton RrofJle inside the

membrane and the fluid layer behind the oxygenation front.

c c - c +C ( 2 • 8) = -m~u-m u m membrane:

fluid: c . - c

c I vO + c = p (x) V vO

19

The boundary conditions for the diffusion equation 2.7 are:

x<O and Vv C=C. I

x>O and y=O o (ac tay) = o (ac tay) V V m m

V x and y=d o (ac tay) = 0 ( 2 • 9) V V

V x and y=O c 0/0. = c /a. V .v mO m

Furthermore by integration of a volume element around the

front we can say,that the influx of hemoglobin(4bV.dy)and

free oxygen {o.actay.dx) equals at the outflux of oxyhemo­

globin (4bV.dy) and dissolved oxygen (VC.dy).

4b.Vd~ + vc d'a 1-----.

figure 8. lntegration of the diffusion equation around

the oxygenation front

In the appendices A and B these equations are solved.

lntegration over the thickness of the film leads to the

following dimensionless expression:

dx* = H (q+M) f(q)dq + 1/q( !qq'f(q')dq' )dq + 0 (2.10)

where:

M q q ••• +q+M { {f(q•)dq'- 1/q {q'f(q1)dq') dq

H = 4b(l-Si) a (P -P.)

V U I

a D • d M

V V m = b . a a m m

q = p/d

the ratio of rema1n1ng oxygen up-take capacity of the inflowing blood and the concentratien difference the entering blood and gas

the telative membrane resistance

the dimensionless penetratien depth of the oxygenation front

f(q) = V (q)/V x

20

the dimensionless velocity

at the front

lntegration of 2.10 over the penetration depthof the front

yieds:

= H ( 12 + Mfl) + 13 (Oomens 1976) (2.11)

whe re: L* = q dimensionless channel length

I = J: f(q 1 )dq 1 the saturated flow 1 ~

q qlf ( q I ) dq I I = 1 the saturated flow moment

2 0

q I 2+ MI 1

I = f dql the influence of physi-3 0 M + q I

ca 1 1 y dissolved oxygen

Equation 2.11 gives the dimensionless oxygenation length as a

function of the penetration depthof the oxygenation front.

Next we must determ the saturation increase for a given

penetration depth q. The fractional saturation change f is m

given by:

s-s. f

I = .,-:s:- = m I

q { f(q 1 )dq 1 = '1 (2.12)

-S is the mean outlet saturation of the oxygenator.

Since the physically dissolved oxygen concentration in the

area where the hemoglobin is fully saturated,is higher than

the value necessary for 100% saturation(P is about 1 atm),the u

flow average saturation (this is the average outlet saturation

of the oxygenator) wiJl be somewhat higherafter complete mix­

ing in the in the outlet of the oxygenator.Appendix C gives a

correction on 2.12 for this mixing effect:

f m = I 1 + 1 /H I 4 (2.13)

21

2.4 Oxygen transfer analysis:theory and experiment

With the aid of the advancing front equations 2.11 and 2.13

we are able to calculate the fractional saturation change as a

function of the system parameters of the oxygenator.ln this way

we are able totest if the approximating equations 2.11 and 2.13

describe the oxigen transfer in ~he oxygenator in a right way.

In practice we use the fo11owing procedure:

durinq an experiment all the system variables from equations

2.11 and 2.13 for one flow situation are measured.These va1ues

except the outlet saturation ,are substituted in the equations

2.11 and 2.13 .These equations then forma system of two equa­

tions with two unknown variables q and S .The penetratien depth u

is eliminated and we obtain the va1ue of the outlet saturation.

We call this value the theoretical outlet saturation S (th). u

We campare this with the measured outlet saturation S (exp). u

From earlier experiments (Oomens 1976) it could be concluded,

that the experimental value of the saturation increase AS is

h i g he r t h a n t h e va 1 u e 6 S f o u n d b y t h e a d. va n c i n g f r o n t t he o r y

(~S >~S h).This is demonstrated in figure 9. exp t

~ V ~

10 ç .!? ~~ ~~ ~<j ....

10 1

2.

0 0

[ftg : I~· "f !}/I C>O ..".)

\'ib % ~ si ~ ''~ o/o .]) -:: 14 \o-'S"vvt/j.

1

'v'o

_ À S t-C..eo'lit __ L'-S ~xpeu>""~~~

c~t~>

t

Figure 9. Theoretica\ and ex p e r i me n t a 1 5aturation change at two values of the flow (Oomens 1976)

22

Camparision of theory and experiment shows that the model under­

estimates the oxygen transfer.This cannot be explained,neither

by a difference between real and assumed velocity profile as has

been abol ished in the master's thesis of Oomens(1973) nor by

experimental errors.

One may try to fit Theory and experiment by an effective

diffusion coefficient Deff.As we will see in the next chapter

local blood mixing can be caused by shear induced red cell mo­

tion.Consequently the relation between Deff and shear rate dV/dy

has been studled as subject of this thesis.

Values for Deff calculated from the first experiments (Oomens

1976) are given in figure 10.

10

1.(8 ~s, ~ '>'f% C.0'1..1.e\c...ft·.,V'\

lic..!:.~ 3&S C.()(.f.fkte..vd· o.<:H-

T = l8°C. pfl, f.lit(

* a. I * ~

4~

* *

* * i * " * *

* t

------ --- --Ay .. .;:·')

0 cift 0

ll '1 .CJ

Figure 10. Ratio of effective and Brownian diffusion coefficient as a function of shear rate by Oomens(1976)

There is an increase in Deff of up to five times the Brownian

diffusion coefficient,when the shear rate is in the ranqe of

5.103 to 1,2 10 4 s- 1.This figure must be read with some reserve

as the scattering in data points is too great (correlation

coefficient 0.87) to conclude that there exists an explicit

relationship between Deff and shear rate.

23

Moreover these results do not agree with those of Overcash

(1972),who found the sametype of increase of Deff in the shear

rate region up to 3.103 s- 1 •

0 1000 3000

Figure 11. Effective diffusion coefficient of oxygen as a function of shear rate by Pvercash(1972)

Overcash also used a couette system,but the blood layer thick­

ness in his system was about seven times greater and the flow

through the system was nottangentlal but axial directed.

lt will be clear that a further study of the relation

between Deff and shear rate is important:we wi11 try to find

out if there is a general relationship between Deff and shear

rate and,if there is one which system parameters play a role

at this relationship.

We wi 11 do th is in two ways:

First we try to couple the parameters on microscopie scale,

which determ the shear induced cell mouvement,to the

macroscopie parameters,which are related to the overall

scaled effects such as oxygen transfer,for instance.This is

done in chapter 111.

24

Second1y we will do experiments in a tangential couette

system to determine macroscopie quantities,that cause an

enhancement of gas transfer as a result of microscopical

shear induced mixing effects.The experimental results are

given in chapter V.

25

Chapter I I I J:#

Mass transfer to shear induced cell movement

I I I • I introduetion

The description of oxygen transport in blood is a difficult

problem. As we saw in chapter I I the transport equations can

be solved approximately with the aiJ of the advanting front

theory. The main condition for this method is the assumption

that blood is a Newtonian fluid and that the hemoglobin is

homogeneously distributed over the fluid. We saw already, as

a logical result of these assumptions, that the saturation in­

crease.measured during experiments is higher than the satura­

tion change predicted by theory. By defining an effective

diffusion coefficient,theory and experiments can be fitted.

In this chapter we consider the blood no longer as a Newto­

nian fluid and note that blood consistsof red blood cells

suspended in plasma. The hemoglobin is inside the cells.

When~ the blood is subjected to shear stress, as in a cou-

e t te f 1 ow \ t he c e 1 1 s w i 1 1 st a r t mo v i n g • Sc h mi d - 5 c h ö n b e i n

(1969) pointed out, that in diluted suspensions of red cëlls

at low shear rates the cells are rotating. Another effect

of shear is that the cells are deformed into an elipsoid,

its longestaxis initially aligned at an angle em= rr;~

with a direction perpendicular to the flow. Both deformation

and Sm increase with )" (Goldsmith 1975).

I~ the shear stress exceed a certain maximum value the de­

formation wilt rupture the cells. In a couette flow several

types of deformation and rupture have been seen in the case

of water droplets in oil W and oil droptets in oil (b}

(figure 1}.

Figure 1

26

·l(ol cD 0 k37 $.~ ... I I z 3 ' 4 5

l(bl C1)J27~~g.,.g 2 3 4 5

2 3 4

,.

Changes in droplet shape seen during deformation

and breakup in couette flow.(a) Class A deformation

for system W (The ratio of the viscosity of water

and oil is smaller than 1). (b) Class B1 deformation,

in which necking-off results in stellite drops. (c)

Class B2 deformation resulting in a long cyl indrical

thread (The ratio of both viscosities is about 1).

(d) Class C deformation, in which no rupture is seen

in couette flow ( Goldsmith 1975).

Another effect which cells undergo in a couette flow is that of

repeated co11isions. As a result of these they can migrate in a

di~rection perpendicular to the flow. We11-known is also the pre­

senee of plasma skimming layers near the stationary walls,the

so-ca11ed Fahreus-Lindqvist effect.

When the shear rate is high all these effects eccur simultaneous­

ly in the membrane oxygenator.

lt will be clear that the combination of all these effects

makes a mathematically correct description of shear induced

diffusion nearly impossible. v»M~

lf we never~theless to predict an effective diffusion coefficient

by a mathmatica1 model,we have to simpl ify the whole flow situation again. We will consider possible geometries for the red ce11 in relation to saturation times of these geometries.

As a simplification we choose the red cell to be a rigid

sphere and consider gas transport in stationary and flowing

blood layers. Situations with one sphere and with more

27

spheres are reviewed.The conclusion wi11 be reached,that the

models found in literature mutually show large discrepances.

For high va lues of the shear rate only the theory of Ni r(1973)

is possibly satisfactory.

~ gas transfer with stationary,isolated particles

Consider an isolated sphere in a resting medium.Suppose,that

there is a concentratien difference of oxygen between the inner

of the sphere and the surrounding fluid.As a result of the

Brownian m~ement of the oxygen molecules there wi11 be a net

transport and a concentratien qradient of oxygen.This diffusion

process is described by Fick's first law:

whe re

cl> = -D.A.grad C

cl> = oxygen flux

D = Brownian diffusion coeff.

A = area

C = concentration

2 gmol/m.s 2

m /s 2

m

gmo1/m 3

( 3. 1 )

Spaan{1973) deriveè an extended diffusion equation for a

flat,resting blood film:

ie i>c os D ~-= ""$t + 4bat

where S = saturation t = time b = hemoglobin concentratien

s gmol/m 3

(3. 2)

The second term at the righthand side of 3.2 represents the

chemica1 reaction of hemoglobin with oxygen.

In the same way the extended diffusion equations for cylindrical

and spherical geometry can be derived for the case of radial

diffusion.

cylinder: D ( i) ( -rrr d c) \ r- J or = oe 4b~ ~ + ot (3. 3)

sphere:

. 0 ( 1 d ( 2 'àc) ) = rL 'à r r '})r'

28

vc 4b()s ~+ Tt (3. 4)

Equations 3.2 till 3.4 will be solved to estimate the oxygena­

tion time of a sphere,a rod and a disc.The oxygenation time is

defined as the time in which saturation is fully accomplished.

Figure 2. Oxygenation front

penetrating a sphere

The location of the oxygenation front is indicated by p. In the

interval p<.r<R we have: S=l and in O<::r<p : S=S .• I"

First we determine the position-dependant part of the concen-

tratien C in the interval p<r<R by supposing,that the effect

of accumulation of free oxygen is small (ë>C/ih~O).

The concentratien profile found in this way is substituted in

the mass balance at the oxygenation front to determine its

displacement.

In the case of a flat duet geometry the approximation ~C/Ût~O

can be avoided,because dC/ut,the accumulation of free oxygen,

is proportional to the oxygen binding by hemoglobin.

In the case of spherical geometry this proportionality does

not occur .In appendix 0 this is proven.

Neglecting the term oC/dt the diffusion problem is solvable

and we find the following expressions for the saturation times

in the different geometries.ln appendix E these expressions

are derived.Table 1 gives the results.

29

t~ t ( s) 0 0

sphere, H/6 (2q 3-3q 2+1) HR1 /60 r=R

2 2 HR 2/40 cylinder, H (!q ln(q) - k(q -1)) r=R

flat duet, H/2 2 Hd 2/20 q thickness d

Table 1. (dimensionless) oxygenation times for diffusion

in spherical,cylindrical and flat duet geometry

With these results we can compare different geometries of

particles.The red cell has a biconcave shape (figure 3).

The diameter is about 8Jlm,The thickness is about 2~m.

Figure 3. A red blood cell

Since calculations with this shape are too difficult,we have

to choose a modelpartiele .Three possibilities are: a sphere,

a rod and a disc.A di se with a diameter of 8 ~m and a thick­

ness of 2 ~m much resembles a red blood cell.lt has a volume

of about 50.3 10 -18 m3 .we choose this volume as the volume of

all the model particles considered.

We first compare the oxygenation time of a sphere and a disc.

The resu1ts are listed in table 2.

30 '

type remarks t~ t(lO~s)

sphere R=2. 2 9 )A-m H/6 1. 85

di se d=2 )4-m 1. 06 diffusion through H/2

flat planes only

di se R=4 rm diffusion through H/4 8.49 cylinder wall only

• Table 2. Oxygenation times for a sphere and disc

We can conetude from these results,that fora disc the diffusion

through the cylinder wall is smalt compared to the diffusion

through the flat planes.

However to make a fair comparision of oxygenation times between

model particles we have to choose particles with a same volume

and a samearea through which diffusion can take place.

Table 3 gives three of such particles and their oxygenation times.

type properties t~ t(lO~s)

sphere R=2.29)'lm H/6 1. 85

rod R=1.53f-m H/4 1. 24 h=6. 84 rm

di se R=3. 24 )Am H/2 0.62 h= 1 • 5 3 y.m

table 3. oxygenation time of a sphere,a disc and a rod with an equal area/volume ratio (=1.311 0-6)

lt is clear from table 3,that a disc is the best model partiele

fora red blood cell,but calculations are simplestfora sphere.

Therefore we choose a sphere with radius 2.2910

-6 as model

partiele for the red blood cell.

31

.1:.1 Gas transfer in stationary layers with spherical particles

The description of gas transport in suspensions starting from

the transport in each separate partiele is difficult.The

disturbance of one spherical partiele in a homogenious concen­

tratien field can be described,but the disturbance of a number

of particles is more complicated because the interaction of

the particles.For the present we wi11 neg1ect this interaction.

Maxwe11(1881) has calculated the electrical field of a number

of dipoles starting from the principle,that the dipoles do not

interact,but that the dipoles seen from a distance act together

as one dipole.

Stroeve(1976) and many others used this relation to calculate

the mass transport in a suspension.Given is a fluid film with

a linear oxygen concentratien gradient:

In this concentratien field we place a sphere( r=R, ~2 , o2).

lnside (2) and outside (1) the sphere we have to solve

LapJace's equation:

AC = 0 2 and ÄC

1 = 0 (3.6)

The boundary conditions are:

1 • · ( C 1 ) r .. -. = - C 0 • z

2 • 0(2 • ( C 1 ) r= R = 0(1 • ( C 2) r= R ( 3. 7)

3 : D 1 • i>c 1 I~ r ) r = R = D 2 • ë9 C 2 I~ r ) r = R

and the solution is:

( 3. 8)

32

The concentration field outside the sphere is the original

field with the field of a dipole at the centre of the sphere

superposed on it .The strength of this dipole is:

(3.9)

At a distance a swarm of N dipales will act as one dipole

with a strength equal to the sum of the individual dipole'

strengths.

(3.10)

Now we consider a spherical shaped swarm of N dipales as a

sphere of homogeneaus material with a volume of Vt and

material constants ~eff and Deff.This large sphere will

cause a field change correspondending to an apparent dipole

with strength

at i ts center.

lf Pd. lp equals "T" p

L dip i then

=

Further:

N ,

o(2D2+li(1D1+ 2 "t'"(e.<'2D2 -c(1D1)

cx2D2+2o<1 o,-,..,<~2o2 -o<1 o1)

~eff = 0(2'1'+ o(l ( 1 - "Y)

(3.11)

(3.12)

(3.13)

Formula 3.12 has been derived for sma11 values of the hemato­

critrt'. lf there is only diffusion of oxygen and if the

oxygen concentratien around each partiele is constant(after

passage of the oxygenation front),we can pose that:

33

(3.14)

The spheres are impermeable for oxygen then.Equation 3.12 can

be simplified to:

Deff = ( 1 - 3/2"t-) (3.15)

As will appear furtheron the effective diffusion coefficient

calculated in this way does not agree very well with practice.

Smith(1973) pointed out,that the effective diffusion coeffi­

cient in stationary layers depends on the diffusion of oxy­

hemoglobin too(the so-called facilitated diffusion).

Stroeve c.s.(1976) computed with this samemodel an effective

diffusion coefficient taking into account,in an approximate

way,the facilitated diffusion. He found:

=

(3.16)

cx2

o2 (1+F) + 20<1

0 1 -2t\f'(~1 o 1 -0<2 D 2 (1+F)) D('2o2 (1+F) + 20(10

1 + ry(0(1 D 1 -~o2 (1+F)

In formula 3.1G Fis the facilitation factor. Fis a measure

for the contribution at oxygen transfer in the continuous

phase (plasma) by the hemoglobin inside the cell.

0 I() ~0 40

[Hio] 0fb wt Figure 4. Oxygen dlffusion coefficient in hemoglobin

so1ution,25°c, pH 7.0 .OaGhed curve represents

recommended values of Kreuzer(1970). Solid curve

represents Kreuzer 1 s correlatlon adjusted to a d "ff . • 2 ' usrvrty of 2.2 10 -5 cm /s in isotonic saline.

Measuring points are from Stroeve(1976)

34

Figure 4 shows,that the effective diffusion coefficient

calculated according to formula 3.16, agrees reasonable wetl

with the results of other investigators.

3.4 Gastransfer in flowing blood layers.

We wil 1 now look at the gas transport in flowing suspensions.

Since there is an almest linear velocity profile in the flow

channel of the oxygenator {as long as the blood layers are

thin),we will only regard this flow situation.lf a sphere is

in a 1 inear velocity field, the sphere will rotate because of

the velocity difference between upper and lower side of the

sphere (figure 5)

figure 5. Sphere in a linear velocity field Yx<~)

The angular velocity ~of a sphere in this velocity field

depends on the velocity gradient according:

l.ü = -! rot V = -! d\Vdy (3.17)

He·re we suppose,that there is noslip at the sphere-liquid

interface.Because of the rotation of the sphere there wi11

be an enhanced oxygen transport, if the sphere is in a

concentratien gradient.

I

35

Colton(1976) distinquishes in a revew artiele two kinds of

this enhanced mass transfer:

l.the interactlón of particles produces net mass transfer

in a direction perpendicular to the velocity.

2.the rotation of each pa~t~cle enhances the transport in

the direction of the concentratien gradient.

The effect of partiele rotation on mass transport in blood

has been investigate·d in several studies.ln all cases a suspen­

sion of rigid,solid spheres was considered;An effective diffu­

sion coefficient was found of the form:

(3.20)

or

where Do = diffusion coefficient in the absence of flow D .. diffusion coefficient of the continuous phase De = partiele diffusion coefficient kp = constant

"'Y = hematocrit f ( "t') = function of the 2ematocrit Pe = Péclet number( 'tR /De)

The Péclet number is a:.measure of the ratio of transport by

conveetien and the transport by diffusion.

reference condition constant f(Pe) f (""")

Petschek and 't<0.2 k Pe ("f-) 4 I 3 Wei ss 1970 ( Re ) c e 1 1< < 1

Ke 1 1 er 19 71 0.2 Pe

hyman 1973 I k (Pe/192) 2 (ti-) 1/3

Leal 1973 Pe<< 1 I 3.0 (Pe)3/2 "Y I

Nir 1973 Pe)') 1 k (Pe)l/5 l 1--

Table 4. Survey of investiqators on shear induced transport

and their results.k is 0(1). According table 3 of Co1ton(1976)

36

Only Antonini (1975) found an expression of a different form:

D c +- . Pe

4'i'r (3.21)

Th e mode 1 s g i ven i n tab 1 e 1, w i 1 1 b e d i s c u ss e d i n d i v i d u a 1 1 y •

We will see whether a fundamental approach is used or not and

inhowfar the results may be compared.

3.4.1. Petschek,H.E. and Heiss,R.F. 1!10

The work of Petschek and Weiss mainly concerns the blood

coagulation.They pointed out,that in a shear flow the trans­

port of platelets,which are important for the coagulation

process,increase as a result of partiele rotation. In this model

the first-order effect of red cel! tumb~ling is estimated.

Complete random interactions between the platelets and the red

cell velocity fields wi11 be assumed,with a characteristic

frequency f and a 11 mean free time 11 t.For a random walk pro­

cess the "mean free path 11 is equal to:

and )_ ~q I t (3.22)

where q 1 is the characteristic velocity perturbation.

From equation 3.21 a partiele diffusion coefficient may be

derived:

4/3 D = k.D .('1-') .Pe p c (3.23)

The above result is an estimate which is valid,at best,at low

hematocrit,for unbounded flows and small red cell Reynold 1 s

numbers.

Figure 6 gives D /D p 0 as a function of~ at several values of

Y· Petschek and Weiss found,that the diffusion coefficient is

a good approximation to their experimental data at moderately

high concentrations,velocity qradients and partiele Reynolds

numbers.

/ t:~.2boo

" / / ~0 // / 1 'S'oo

/ // / /

/ / I

I /

Figure 6. Ratio of partiele and Brownian diffuslon coefficient as a function of the hematocrit,shear rate an~ partiele size b.(Petschek 1970)

3 • 4 • 2 • Ke 1 1 e r , K. 1 9 71

Keller 1 s model is a drag model.A rotating sphere is consi­

dered in a shear flow.There is a concentratien gradient

perpendicular on the flow.

Figure 7. Model system for determi"e the effect of

red ce11 rotatien on mass transport.

The velocity field around the sphere is given by Keiler as:

Within a 11 sphere of influence 11 of radius 1 1 fluid is dragged

by the rotating particle.As there is a concentratien gradient

in the fluid,oxygen is pumped from the plane y=l 1 with a

38

higher to the plane y=-1 1 with alower oxygen concentration.

Defining an extended Fick 1 s law

dC U1= -( D + D ) -l c p dy

and giving a derived)approaching flux as

dC rt>=- 2V1 1

-.., dy

the following expression is obtained for the partiele

diffusion coefficient D : p

D = 2Vl 1 = 0.2 D Pe p c

The radius of the 11 sphere of influence 11 1 1 is choosen

(3.25)

(3.26)

(3.27)

as 1 1 = 1.5 R .This derivation of D is crude,but as fi rst order p

calculation it is valuable,because the transport of proteins

in a shear flow can be explained by rotation of the red cells

with this model.Experiments of Singh(1968) agree rather well

with equation 3.26 •

3.4.3. Hyman,U.A. 1973

For the calculation of enhanced mass transport by partiele

rotation Hyman used a samemodel as Keiler did. lnstead of

using an approximate concentration field around the sphere,

as Keiler did,Hyman first calculated the concentration profile

around the sphere and then determs the flux~ by the expres­

si on:

dC <f=- D cry + V.C {3.28)

For the effective diffusion coefficient he derived the

equation:

(3.28)

39

In 3.28 k is the factor,which determs the influence sphere.

The factor k is of the order one.

Equation 3,28 has been compared with measurements of Colton

(1971).They are given in figure 8.

Figure 8. Camparision of the theory of Hyman ·{sol id line) with measurements of Colton(1971)

3.4.4. Antonini ,G.Guiffant,G.and Quemada,D. 1975

Antonini c.s. derived an expression for the effective diffu­

sion coefficient as a function of shear rate.They were mainly

interested in an effective diffusion coefficient for the

transport of platelets in shear flow.For high values of shear

rate their results are valid for the diffusion of oxyqen and

proteins too.

Intheir analysis they start from a rotating sphere in a shear

flow with~a concentratien gradient.Analogous to Hyman the flux

is calculated from the disturbance of the concentratien field

by the rotating sphere:

tp= - D.grad C (3.30)

They obtain an effective diffusion coefficient as fellows:

Their resu1t is applicable for oxygen transport starting -1

from shear rate values of 1000 s

40

3.4.5. leal ,L.G. 1973

leal considered the effective heat conductivity of a dilute

suspension of neutrally buoyant spherical drops,undergoing a

simple shear flow.This theory may be applied to diffusional

problems and can be used for the calculation of an effective

diffusion coefficient.The viscosity,solubility and the dif­

fusion coefficient of drop and bulk are supposed to be dif­

ferent.Although the model is restricted to smal! values of the

hematocrit and to small Péclet numbers,it is important for

calculations of an effective diffusion coefficient in blood

because of its fundamental ba~is and derivation.

leal calculates the concentratien field inside and outside the

sphere.With the aid of this field the mass flux through the

suspension is determined.ln this way an expression for the

effective diffusion coefficient is obtained:

• • +

where index 1 means the suspended phase

index 2 means the dispersed phase

+ ••

(3.32)

Equation 3.32 wiJl be used to derive the effective diffusion

coefficient of a flowing blood layer after the oxygenation

front has passed: at that time the hemoglobin inside the cell

is fully saturated and the effect of red cell rotatien will

become significant.The rotating ce11 wiJl drag some fluid

around it.As the red cell and its content are in a uniform

rotatien Lea1 1 s result may be used in the limit ') 2-+oo.

After all the hemoglobin has been saturated the transport of

oxygen in the cell is pure diffusion only.When we estimate

41

the diffusion coefficient of oxygen in the concentrated hemo­

globin solution inside the cell to be D2=.r-o 1 , we may use 3.32 to calculate

D = D r 1 + C\U [~ + (1 • 1 76 (~-1) 2 + 3 0 -e f f c l I }<+ 2 {f+ 2 ) 2 • 0 1 4~) P e

3 I 2

+. lJ • f--+2 J

(3.33)

Fora hemoglobin concentration of about 10 %wt,}t~1.07.

We can rewrite 3.33 then by:

(3.34)

Both the presence of better permeable spheres and the fact,

that each sphere drags fluid fram areas with h,tgh oxygen concen­

tration to areas with low oxygen concentration, enhances the

effective diffusion coefficient.

This theory is derived for low Péclet numbers.However, in the

oxygenator very high shear rates are reached.l t wi 11 be clear,

that this theory cannot be used to describe an effective diffu­

sion coefficient for the oxygenator.

3.4~6. Nir,A. 1973

Nir has studied the effective Thermal quantities and the proper­

ties of sheared suspensions.His analysis has been done for high

Péclet numbers(Pe)>1). This condition implies,as stated in the

foregoing,that convective transport in the direction of the

velocity gradient is much higher than the transport by Brownian

diffusion.He considers a sphere in a shear flow.From results of

Cox(1968) it follows,that around the sphere one finds a region

enclosed by a limiting streamline(figure 9),in which all stream-

1 i n e s a r e c 1 o s e d ·'

Once the hemoglobin inside the eelt has been saturated,there

will be an almost constant concentration on the closed stream­

lines.lf the fluid velocity is large enough,an effective dif­

fusion coefficient is found by Nir under these 1 imiting condi-

tions: (3.35)

42

Figure 9. Sphere and region of closed

streamlines (Cox 1968)

The presence of a rotating 11 particle 11 larger than the original

sphere will enhance the oxygen transport.

3.4.7. Summary of literature.

When we campare the results of all these publications we can

conclude,that there is no agreement about the relation between

the effective diffusion coefficient and shear rate.Figure 10 ~·

gives the ratio of effective and Brownian diffusion coefficient

for rigid spheres with a radius of 2.29 10 -6 m and for a hemato­

crit of 0.1 .In addition to curves correspondending to the

various expressions also the curves through the experimental

data of Overcash(1972) and Oomens(1976) are given.For the curves

of Hyman,Petschek and Nir the constant k has been choosen

equal to one.

--"

0 -

44

~ New models for shear induced oxygen transfer

Another possibil ity to describe mass transfer by rotating

spheres is to consider situations when the spheres are closely

spaced.The problem is,that the spheres wilt affect each other.

The velocity and concentratien profiles around the spheres

are difficult to describe now.lf we take approximations for

these profiles an effective diffusion coefficient can be derived.

There are several possibilities:

3.5 .1. gear-wheel model:

We place the spheres on distances of half the radius, in regular

ranges.The direction of rotation of two adjacent spheres is

opposite (in the y-z plane the direction of rotation is equal

for constant values of x).Figure 11 makes this plan clear.

A'

~k..t.ï""' A - A' Figure 11, se~up ot the gear-wheel modei.Ceiis rotate in a~opposite direction

45

Analogous to the model of Ke11er(1971) we can say:

dC oeff"dy = v•.c (3.36)

where V' is the average velocity at which fluid is transported

in either direction across the y=O plane.

Now we suppose,that the velocity field around each sphere is

described by:

(3.37}

and we suppose,that the velocity field between two spheres

is only influenced by the two spheres;the velocity field is

a superposition of the fields of the spheres approximately.

The average velocity V' is calculated as follows:

V' = 4 ! !V. r. dr. d lf

tot a 1 a re a

2. I . I

= 7.5R 2

where I. is the contribut ion of each sphere. I

JJ/.Il 3 I i = J w0;. cos cp • r • d r • d cp

o R r I

lf we substitute 3.39 in 3.38 we obtain for 3.36 the

equation:

D = 2V 1 1 1 = 0. 73 D Pe p c

(3.38)

(3.39)

(3.40)

Though this derivation is very crude,it may serve as a rough

first approximation to Deff"

In comparision with Keller's mode1(1971) this Oeffis a factor

1.5 times the primary results of Keiler.

We have to remark,that physically this model is not quite real:

looking at one sphere there is no preferenee to turn to the

left or turn to the right.

46

3,5.2. Other possible models.

Another way to regard at spheres in a s~ear flow is to place

the spheres insome regular pattern,but now we suppose all the

sphe~es to rotate in the samedirection (figure 12).

OOG Figure 12. Range of particles rotating in the same direction

This may be called a caterpillar-track model.Since the shear

rates in the oxyqenator are very high the red cells will be

deformated to spheroides.

In the caterpillar-track model the spheres can then be replaced

by elongated caterpillar-tracks,as shown in figure 13.

;~>::~~~/$(~

-~~~~~{((~­;~>e~~-Q}~~~

Figure 13. Caterpillar-track model with elongated cells

This model with elongated cell shapes is closely alike to

sheared blood.Schmid-Schönbein(1969) and many others observed

such red cell deformation at high shear stresses.

lnstead of mode11 inq withafluid layer with separate particles

one might further consider a layer with bands of plasma and

hemoglobin.ln the transport equation 2.5 a position-dependent

hemoglobin concentration must be introduced then.

47

Chapter IV Experimental techniques and sett-up for measuring

gas transfer in ftowing blood and hemoglobin

solutions

~ Introduetion

The investigation of the contribution of partiele rotatien

to gas transfer has been done with an experimental sett-up

basedon the first prototype of the Eindhoven membrane

oxygenator.This type of oxygenator is a couette system with

a tangentlal directed flow.lt cocsistsof a rotating cylinder.

There are two flow channels: a 11 circulation 11 and a 11 recircula­

tion'' channel .The system is described extensively by Oomens

(1976).Because of the presence of two flow channels there is

mixing at the inlet and outlet of the system.

Dependent on the thickness of the layers and the velocity of

t he c y 1 i n der se ver a 1 types of f 1 ow ca n a p p e ar i n bot h t he

"circulation'' and the 11 recirculation 11 channel.They may be

laminar flow,turbulent flow and flo~ with Taylor vertices.

In the last two types mixing-effects will be present by

secundary flows(Links 1976).As long as the thickness of the

blood layer d between the two cylindrical walls is~thin com­

pared to the radius R of the cylinder (d<<R) there is an

almast linear velocity rpofile in the 11 recirculation 11 channel

and only only mixing effects as a result of red ce11 rota­

tion occur.The contribution of thti red cell rotatien to

gas transfer cannot be determined very ~fficiently because

of total mixing in inlet and outlet of the system.

Therefore a new experimental sett-up has been constructed.

Since this secend sett-up has only one flow channel ,there is

no recirculation of flow through the system but only net

bhrough flow.This system,which is described in the next

section,allows us to measure Deffas a function of shear

rate more precisely.We can separate the transport by diffu­

sion and the enhanced transport by red cell rotation,if we

do comparative experiments between blood and hemoqlobin

solutions under the same experimental condltions.ln a

48

hemoglobin solution the oxygen carrier hemoglobin mas been

dissolved freely.ln blood the hemoglobin is enclosed in the

red cells.Two identical experiments with blood and a hemoglobin

solution will then give information about the part of gas

transfer by the presence of the red cel ls.We have to take a

look at the fact that blood and a hemoglobin solution are two

fluids with different properties:measured differences in gas

transfer represent a total effect and not an effect of cell 4lt tumbl1ng only.

4.2 Description of the new experimental sett-up

The second membrane oxygenator ,constructed in Eindhoven, is

shown in figure 1.

Figure 1. Second prototype of the Eindhoven membrane oxygenator

49

A cylinder is horizontally mounted between two flanges.

The flanges are separated by a bridge piece in such a way,

that the cylinder can rotate.The two flanges rest on a frame

and are fixed there.A membrane of silicone rubber is steched

on the surface of the cyl inder.and is clamped at the edges of

the flanges and the bridge piece.

Figure 2. side-view on the oxygenator

As can be seen in figure 2 inlet and outlet openings for the

blood flow are provided in the bridge piece.By a flexible

strip inlet and outlet are separated.The strip,which reclines

on the surface of the cylinder,forms the two borders of the

flow channel.To obtain a flowing blood film over the whole

lengthof the cylinder whithout areasof stagnation,blood is

supplied to and removed from the edges of the bridge piece.

at equal rates.

2 The dimensions of the system are: cylinder length 10 cm,

cylinder diameter lO.~cm and a membrane area of about 2700 cm • Blood flows through the system because of the viseaus shear

stresses exerted by the rotating cylinder on the blood or hemoglobin solution.Oxyqen oiffuses through the membrane into

the blood or hemoglobin layer as a result of the oxyqen partial

pressure between fluid and gas phase.ln this form the oxygenator

50

is meant for measurin~ the contribution of the presence of

particles to gas transfer.This sett-up is useful for several

reasons:

l.there is only one flow channel.

2.blood layers are thin (<500}-ûn)and shear rate is

we assume a 1 inear velocity profile.

- 1 high (~5000 s ) '

The hydrodynamic circumstances durin~ the experiment are such,

that there are no mixing effects by secundary flows.

Other characteristics are:

l.the cylinder has been made of stainless steel.

2.the rubber strip causes blood damage.A 1 ittle bit of blood is

slipping between the strip and the cylinder.That blood will

be sheared very stron~ly because of the extremely hi~h shear

ra te in the qap.

However,this experimental sett-up can be used in a other way

t o o • I n s te ad of on e b r i d ~ e p i e c e t wo on es ca n b e f i x a t·e d • 0 n e o f

them has the inlet,the other one the outlet of the system.

In th~t way the system is scaled up according to and equal at

the first prototype of the Eindhoven membrane oxygenator.Then

we are able to campare the behaviour of the oxygenator,which

is scaled up,with the farmer one.Now secundary flows and the

resultinq mixinq effects are present aqain.

Bridge piece of the oxynenator with inlet and outlet channel 1 divorced by a black strip.

51

i.:J. The measuring circuit

Figure 3. The experimental circuit

The experimental circuit is shown schematically in figure 3.

From a disc oxygenator the blood or hemoglobin solution .. is fl.\.~~ ' .

through the system by a roller pump.A buffer vessel is in-

corporated to damp the pulses from the roller pump.A filter

section filters out any productsof blood coagulation.Then the6locx;l

passes the cyl indrical oxygenator ,and returns to the disc

oxygenator.Before and after the cylindrical ox~genator connee­

tions are made with another roller pump to take samples from

the main flow.The oxygen saturation of these samples is de­

termined in a separate piece of equipment.When we do experiments

with blood this is the oxymeter (Philips) ,a standard device,that

gives both oxygen saturation and hemoglobin concentration.

52

In experiments with hemoglobin solutions the oxygen saturation

is determined by a Photometer device,a copy from the one deve­

loped by Fesler(1974).We will discuss this instrument in a later

section.

The quantities,which are determined experimentally,can be devi­

ded in groups according to the time of measurement:

a.Before each experiment we have to determine:

l.hemo- and hemiglobin concentratien

2.hematocrit

b.During an experiment we determine:

l.velocity of the cylinder

2.thickness of the fluid layer in the flow channel

3.oxygen saturation at inlet and outlet of the oxygenator

4.acidity(pH) and temperature(T) of the blood or hemoglobin

salution at the outlet of the disc oxygenator

c.After an experiment we determine:

l.flow through the oxygenator(calibration of the pump)

The principles and techniques of these measurements are

discussed in a later section.

For an experiment using blood we take fresh heparinized bovine

blood.We add 0.1 gram streptomycine sulfate per litre blood to

prevent the growth of bacteria.These bacteria will disturb the

oxygen saturation measurements,since they may consume oxygen

inside the flow channel of the oxyqenator.

When we do hemoglobin experiments we first prepare a hemoglobin

solution.The red blood cells of fresh heparinized bovine blood

are separated from the plasma by centrifuginq(6,000 g,lO minutes).

The packed red blood cells are rinsed three times with saline

(9 %wt NaCl) and separated by centrifuging(6,000 g,lO minutes).

The cells are 1yzed osmotically by adding 70 ml.destilled water

to 100 ml of packed cells.The red cel! stroma is removed by

adding another 40 ml of toluene(C 6 H8) to~the solution.After

shaking the mixture the toluene is separated by centrfuging

(6,000 g,20 minutes).The hemoglobin is drawn off from under-

neath the toluene and poured into specially designed tonometer

bottles(having a content of about 0.6 1 itre).The tonometer

botties are rotated continuously for one hour whi le being

53

flushed by a humidified gas mixture of 95% N2 and 5% C02

• The

mean purpose of this eperation is to drive out remnants of

toluene,another to keep the hemiglobin concentratien in the

solution low.As we always,meàsure a hemiglobin concentratien

different from zero during experiments,the hemoglobin salution

is not completely deoxygenated.The measured hemoblobin concen­

tratien is always corrected for hemiglobin.The tonometers

bottles arestoredat 4°C. All hemoglobin experiments have

been done using solutions not more than one day old .The cor­

rect hemoglobin concentratien is obtained by diluting the

hemoglobin solutions with destilled water.The hemoglobin solu­

tion is buffered at ph=7.4 by adding a mixture of Na 2Po 4 .H 20

and KHP0 4 ,a phosfate buffer described in Clinical Chemistry

( Henry,R.J.c.s. Clinical chemistry, principles and technics,

ed.Harper and Row, Hagerstown , Maryland, 2e edition,

1592(1974) ).

4.4 Photometer

To measure the oxygen saturation in experiments with hemoglobin

solutions ,a photometer,developed by Fesler(1974) ,has been

copied.The principle of the method is an absorption

measurement.

figure 4. Saturation measurement using a flow-through cell (hemoglobin experiments)

The hemoglobin solution flows through a cell of suprasil glass.

The parallel beam of a high-power 1 ight-emitting diode(type

Honsanto HV4H with a wavelengthof 670 nm) falls through the l~eL

54

of hemoglobin in the flow-through cell.A phototransistor

detects the transmitted light intensity.According to \~w

Lambert-Beer's the intensity of the light beam as detected by the phototransistor is:

where

I 0 = I • e -"-t:t

I = intensity of the transmitted beam 0

I = intensity of the incoming beam

~ = extinction of the Hb-solution

C = hemoglobin concentratien

t = lengthof the light path

( 4. 1 )

lux

lux 2 m /gmol

gmo1/m3

m

The principle of the photometer is,that the intensity of the

1 ight beam detected by the phototransistor is helt constant.

The power of the LED is a linear function of the current

through the LED;so the intensity is a linear function of i:

( k0 = constant ) (4.2)

The current i is measured over a resistance R of 50!l. s

lf we substitute 4.1 in 4.2 we obtain:

V . R = I /k ~CL p = •• 0 o· e ( 4. 3)

Subsequently the signal is fed into a logarithmic amplifier:

(4.4)

We assume,that the hemoglobin salution consits óf deoxy­

hemoglobin (with extinction ~ 1 and concentratien c1) and

oxyhemoglobin (with extinction ~2 and concentratien c2).

We can then write for the total hemoglobin concentration:

c = c1 + c2 (4.5)

The saturation of the hemoglobin is defined as:

s = c2

( 4. 6) c

lf we combine equations 4.4- 4.6 ,fellows:

ln V = ln p

which is of the farm:

55

+ é, • c 1 • t

where k1

and k2 are constants.

+ ( E 2 - E.1 ) • C • -t. S ( 4. 7)

( 4. 8)

The output of the photometer is a 1 inear function of the

saturation of the hemoglobln solution,which flows through

the cell.All constantsof eqs. 4.7 and 4.8 are known or can

be determined,so the output signal provides a measure of the

saturation.

The liniarity of the photometer has been tested by camparing

the output signal with that of another instrument,cal led

Lex-0 2-con.This is an instrument,which determs the total 02 content of blood.Figure 5 gives the relation found between

the results with the photometer and the Lex-0 2-con for hemo­

globin solutions.Note,that there is a small difference between

the total amount of oxygen and and the amount of oxygen bound

to the hemoglobin.No correction has been applied in the com­

pilation of figure 5,since at 100% saturation the physically

dissolved oxygen is only 1.3% of the total oxyg~n content.

lf we callibrate the photometer at the beginning of each

experiment,we can continuously measure the oxygen saturation

of hemoglobin solutions.Ve make a correction table for the

photometer.Assuming liniarity only readings corresponding to

S=O and S=l are needed to allow the determination of inter­

mediate values of the saturation.The level S=O is determined

by adding sodium hydrosulfate (Na 2s2o4)to a sample in the cell.

S=l is determined using hemoglobin oxygenated with ambient

air by tonometry.

Another possibility is to take a sample with a known photo­

meter value and to determine saturation with the Lex-0 2-con.

Some adding determinations with the Lex-0 2-con during the

course of an experiment allow a further check on the photo­

meter and the qwality of the hemoglobin solution.

r-----------~----------T-----------~-----------~

... ~ 0

"

--t I !

I

i __ , _____ ______,~ I ------- -------r-------------1-

1 I

i

---------------1------

I I I ·----------+ ··-~~ I I I

I

- I .... ~ I .....,

I 4 E

:r- "'? "":'( -c:)

~ 0 0 o ..

~ . s

j1 . j

0

0 0

0 1/)

0

-0 ::>

57

4.5 Principles and techniques of the quantitative determination

As mentioned in section 4.3 a number of quantities is measured

during each experiment.Table 1 lists these quantities with some

characteristics of the measuring principle.We wi11 discuss each

of these principles.

quantity measurinq device I

measuring.principlè 1 calibration

Hb

H i

V 0

tot a 1 o2 content

hematocrit

p , p. U I

d

s , s. U I

spectrofotometer

spectrofotometer

tachometer

Lex-0 -con 2

ultra­centrifuge

fotonic sensor

Philips oxymeter

photometer

absorption

absorption

mechanica! transmission

chemical

reflection

reflection

absorption

+

+

+

table 1. Measured quantities with some characteristics of the measurement

4.5.1 hemoglobin and methemoblobin concentratien

+

+

+

+

The total hemoglobin concentratien is determined by a standard

analyse on a spectrophotometer.The principle is an absorption

measurement.The hemoglobin concentratien is calculated aceer­

ding equation 4.1

The hemoglobin concentratien is determined using the HiCN­

method of Zylstra en van Kampen (v.Assendelft 1970).The prin­

ciple of this methad is,that all hemoglobins are converted to

HiCN,which has an absorption maximum at ~=540 nm.

The percentage of methemoglobin is determined by a method

described by Evelyn and Mallory(v.Assendelft 1970).1t is a

methad based on the fact,that methemoglobin has an absorption

maximum at a wavelengthof 630 nm,while HiCN absorbs a percen­

tage of the value of HiCN at that wavelength.

4.5.2 Hematocrit

The hematocrit is determined by fill inga glass micro-tube

with blood. At one end the tube is closed. Then the red cells

are separated by centrifuging.The ratio of the volume of the

cells to the total volume gives the hematocrit.

4.5.3 Velocity of the cylinder

This quantity is determined by measuring the rotaticnat speed

of the cylinder. Since the circumference of the cylinder is

one meter, the units of the tachometer scale (in r.p.s.) give

the velocity of the cylinder directly in m/s •

4.5.4 Thickness of the blood layer

The thickness of the blood layer can be measured by a photonic

sensor ( Kats 1976 ). lt uses fibre optics.Half the fibres of

a bundle are used to send 1 ight to the blood layer.This 1 ight

is reflected at a strip of aluminium foil and returned to the

photodiode by the other part of the bundie of glass fibers.The

diode signal is processed by a special unit. The output of this

unit is a signal which is proportional to the distance between

fibre tip and the surface, when the distance is in a certain

range.See figure 6.

59

1

0 0.1 0.1. 1.9..

d..l~M'n\)

Flgure 6. Characteristic behaviour of the photonic sensor

The thickness of the fluid layer d can also be calculated

from the flow~ through the system, the mean velocity V and

the width of the channe1 B.

~ = V.B.d (4.9)

4. 5 0 5 lnlet and outlet saturation

The oxygen saturation can be determined in several ways, it

depents on the type of experiment, which is done.

In case of blood experiments we use a standard device, the

Philips oxymeter. In case of hemog1obin experiments we make use of the photo-

meter, described in section 4.4 .The methad is sensitive and

the errors are very sma11,. The d'evice can be used in case of

blood experiments. The flow-through ce11 has to be thinner then.Blood is turbid in comparislon with a hemoglobin salution

because of the presence of plasmaand dispersed cells.There are

several problems with such thin flow-through cells: they are

fragile, the resistance for the flow is high and contamination

60

by protein and clots is severe.

As a second possibility to measure oxygen saturation we use

the Lex-0 2-con. A sample of 20 ~~ Hb-solution with unknown

saturation is injected in a water circuit ;oxygen free gas is

bubbling through this circuit and forces the water to circu­

late.The oxygen dissolved in the water diffuses into the gas

bubbles. The gas flows toa fuel cell.At this cell each 02 molecule reacts and two electrans are released.

I ~ the fuel ce 11 the fo 11 ow i ng reaction takes place:

cathode: 4e + 02 + 2H 2o ;! 40H -at the (4.9)

at the anode: 2Cd + 40H- ~ 2Cd(OH) 2 + 4e

The sum of all electrons, which are released into the fuel

ce11, is proportional to the total oxygen content of the

sample. An advantage of this method is, that it is very pre­

cisely. A disadvantage is, that the method is laborious:

each analyse takes five minutes.

4.6 Estimation of sourees of errors and influence of errors

on the effective diffusion coefficient

For the determination of the effective diffusion coefficient

it is important to knowhow experimental errors influence

Deff obtained. Table 2 lists the quantities, which are measured

and for each of these the possible causes of errors, the esimate

of that error and the relative error.

To determine the error in Deff we refer to the advancing front

equations 2.11 and 2.13

(4.11)

"' Oomens(1976) has shown, that the contribution of 13

at L

and 14/H at fm is sma11.Therefore these terms are omitted in

61

Table 2. Revrew of errors rn measured quantrtres quantity

H b , H i

s. ' s I U

Lex-0 -2 con

s. ' s I U

Philips oxymeter

s. ' s I U

photo­meter

d

w

p. , p I U

sourees of errors

preparatien of reaction f 1 u i d

use of pipettes

contaminated sample cell in photometer

varying temperature

cal i bration

sample preparatien

injection of the sample

wrong fuel cell

dirt in flow-through cell

temperature effects

calibration

dirt in flow-through cell

cal i bration

drift

preparatien of Hb solution

calibration

drift

optical isolation

condensation on the fibre ; ~ip

meaöurinq with an angle >0 with regard to the

axis

sample pump

pressure in the system

est i mate

reading of volume ratio

estimated total me as u ring error

0.003

0 1 vol /oo

0.005

0.002 V

0. 0 1 V

0.02 t/s

0.1 mmHg

0.5 %

relative error

1 %

3 %

1 %

<0.5 %

10 %

1-4 %

3 %

1 , 5 0 %

3 %

62

4.11 • In this formula the effective diffusion coefficient

is introduced ncw by multiplying in 4.11 the terms containing

the Brownlan diffusion coefficient with a factor'a', which

equals the ratio of effective and Brownian diffusion coeffi­

cient: K

a.L = H ( 12 + M.a.l1

) (4.12)

f m = 11

According to Oomens (1976) the integrals 11 and 12

can be

calculated in case of a linear reference velocity profile

as:

12 = /q'.f(q') .dq 1 = 2/3 q 3

J!V 2

I 1 = f ( q') • d q' = q CS

where q is the dimensionless front depth

Equation 4.12 can be rewritten then as fellows:

K a.L = H { 2/3 q 3 + M.a.q2

)

2 f = q

m

'a' may be found from 4.14 as:

a = 2/3 H. f;J K

L -M.H.fm

When in this equation the dimensionless quantities are

expressed in terms of the original parameters,we get:

16 D (Hb-Hi) ~ 2 (s -S.)3/ 2 am m -v u t

a =

•• -4d (Hb-Hi)~(S -S.)} m u 1

(4.13)

(4.14)

(4.15)

(4.16)

note,that in this formula Hb and Hi stand for the hemoglobin

and hemiglobin concentration.

Generally,it should be possible to estimate the error in'a'

in two different ways;however,one way will not work here.

63

4.6.1 forma! method

According 4.16 1 a 1 is a function of several parameters:

a= a( Hb, Hi, ~. S., S , V, P , P. ) I U U I

(4.17)

The error in 1 a 1 is given by:

2 2 2 (8a) = (8aHb) + (8aHi) + ••• • • • • {4.18)

where for instanee 8aHb is the error in 8a as a result of

the measuring erro!H~n Hb. For 8aHb we can write the

expression:

(4.19)

Doing this for all parameters of 1 a 1 we can rewrite ~.18 as:

(4.20)

Because of the fact,that 1 a 1 is a very complicated function

of its parameters,expression 4.20 will not be used to

determine Aa •

4.6.2 staight forward method

We regard 1 a 1 as a function of all its parameters,still

according 4.17 • Again we write the error in Aa 1 ike:

(4.18)

For the error AaHb we write now:

=a( Hb+AHb, Hi, ~. S., S ,V,P ,P. ) -I U U I

a( Hb, Hi, ~' S., S, V, P., P) I U I U

(4.21)

Now the e~~or. AaHb is caused only by the error in Hb.ln this

way we can substitute both possitive and negative values of

AHb • For all parameters of 1 a 1 this analysis can be done

to get the possitive and negative error Aa.

This straight-forward method has been choosen, because it is

applied very easily.

64

Chapter V Experiments and discussion

~ introduetion

To determ the relation between the effective diffusion

coefficient and shear rate in the Eindhoven membrane oxyge­

nator, experiments have been done with blood.Because of total

mixing in the inlet and outlet of the first prototype of the

oxygenator, a second one has been constructed. This type has

only one flow channel.As long as the blood layers are thin,

no mixing effects by secondary flowscan happen and there wil 1

be a 1 inear velocity profile in the channel.The rotatien of

red blood cel ls in this velocity field wi 11 enhance gas trans­

fer.

in this oxygenator experiments with hemoqlobin solutions have

been done too. As a hemoglobin solution is a homogeneaus fluid

whithout particles, we expect, that the relation between rhe

effective diffusion coefficient and shear is independent of

shear rate. lf a hemoglobin solution is a Newtonian fluid, we

may expect, that 1 a 1, the ratio of effective and Brownian

diffusion coefficient, is about one: the gas transfer in the

hemoglobin solution is described adequately then by the advan­

cing front equations 2.11 and 2.13 •

A comparision of blood and hemoglobin experiments, done under

the same conditions, in order to estimate the effect of red

cell rotatien on gas transfer, is not possible. Blood and a

hemoglobin solution differ essentially. Blood consistsof red

blood cells, containing the oxyqen carrier hemoglobin, suspended

in plasma. In a hemoglobin solution hemoglobin is dissoluted in

destilled water, buffered at pH=7.4 • The preparatien of a hemo­

globin solution from blood changes two characteristics of the

blood: the plasma is removed from the ce11s and the red cell

membranes are broken. Among other things this results in a

disturbance of the enzyme reduction system of the cel 1.

Supplying an artificial reduction system is possible, but few

good results have been obtained( Hayashi 1973).

So, a measured differences in 1 a 1 between a blood and a hemo­

globin experiment, carried out under the same conditions, is

65

a re s u 1 t of a 1 1 t he, f o r t he most part u n k n ow n , d i f f erences

between blood and hemoglobin solutions and not only caused by

the rotatien of red cells in a shear flow.

These hemoglobin experiments have been done both with the

former and the present used membrane oxygenator. In this way

we can compare the behaviour of the first oxygenator with that

of the scaled-up one, both concerning the gastransfer in hemo­

globin solutions and concerning gas transport in blood {the

results of blood experiments with the first oxygenator are

still available)

The blood experiments with the second oxygenator will be ex­

pected to yield values of 1 a 1 which depend of the shear rate.

We try to pinpoint those parameters, which influence the rela-·

tion between 1 a 1 and rate of shear.

5.2 Experiments with blood in the second membrane oxygenator

Several experiments have been done with blood.Though each time

fresh, heparinized bovine blood was used, only two experiments

yielded useful results. ~ecause of the aggressiveness of the

system coagulates appeared in the flow channel of the oxygenator

quickly. These clots disturb the laminar, flat flow of blood in

the channel. The use of a blood filter resulted in a clot-free

channel.The experiments dated,18-0S-1977 and 05-01-1977,are

represented in this report.

Experiment 18-05-1977

Figure 1 gives the fractional saturation increase f as a m function of L~, the dimensionless channel length, both as it

is measured and as it is calculated with the advancing front

eqs. 2.11 and 2.13 •

Figure 2 and 3 give the measured and calculated saturation

increase (S-S.) as a function of the velocity of the cylinder U I

v0 at four values of the flow~·

Figure 4 gives 1 a 1 as a function of shear rate ,also at four

flow rates.

'

ó 0 0

I i' I

I

::.:-0

'"

Lr 0

ó

,..,~ ~ oJ

~ ')-

t! " ~ -~

~ ...

~ f -.J)

'o ~ r.

0 til

oJ

'l! 11

19+ 0 ~ oi ~ 11

~ 1'81 ...

* -.!

*

..

&7-

~ ~ ...,

~~

,::,-

~ ,,

161

~

...

-1

f 0

u ~

J.

Q;------0

0 0

0

~ <S1 -' lP

0 I

"" -\-~ "'! Ê~ -~ ~ q_ 0 ,.ll

~ er:' ~ 0

"''< f

"' lg ".. Ot>

ct" 11

1-&1

~

~

;te E v '!

,.... ct>

~

*

*

... 'e

c:D 0 0 -"()' ,, ,, ,. tl

g ,...., ~ 1-f}t /Eh - ~ u c::!>ó:r.

0 ,., 0 ..

*

\11

""' lp

(» 0

0 -11

1-ê-1

~ • ~ 1 v ~ ~

~ p-\I

1&1 -. c:J -r

* *

0

"" ó

b8 "(VI ...,

ID

P.:) 0 ó -••

f.&\ '

-e:é

~ E

'>

I 0 lP. ->1

_r-.cl

i 1~ ~:L 8

~

i~ ~ 7 j ::;,

~1 0

~~ ~

if en~ ~

.2. 0 D

1 1----*~-

*

1;!~

i i T:;t CH-~ D

:2 J fo ~ _;4::

..... rd ó

1

", 0 ...

70

For this experiment the independent variables are:

(Hb] 1. 80 gmo1/m 3 D 1.4810-9 2

= = m /s V 3

"""' = 0.33 o<.v = 1 • 2 0 2 gmol/m.atm

T = 25.0 oe 0( D = 0.221 gmol/s.atm.M m m p. = 0.034 atm dM."; It. =1- ,.- I. V"'\

I p = u 0.95 atm

The values of the used flows a re:

~ = 1 11.7410-6 m3/s

1 = 2 10.0810-6 m3/s

l3= 9.0110-6 m3/s

l4= 7.8710-6 m3/s

Experiment 05-01-1977

Figure 5 gives the fractional saturation increase as a function

of the dimensionless lengthof the flow channel.Both measured

and calculated values of f are given. m Figure 6 gives the theoretical and experimental values of the

saturation difference (S -s.) as a function of the velocity of U I

the cylinder v0 at three values of the flow~.

Figure 7 shows 1 a 1 as a function of shear rate for three values

0 f t he f 1 OW r. The independent variables of this experiment are:

[H b) =

C\f' =

T =

D = V

1. 4 7 gmo1/m 3

0.28

2 7. 1 oe

1.5410-9 2 m /s

d""" ":1: 11.~ 11)_, n'\

~ = 1.194 gmol/m~atm V

~mom= 0.221 10 -7 gmol/s.atm.m

P. = 0.034 atm I

P = 0.95 atm u

The three values of flow, applied during the experiment, are:

i1= 11.26 10 -6 m3/s

~2 = 8.67 10 -6 m3/s

!3= 6.31 10 -6 m3/s

~~

~ ' ""'. ~ •

' . ..... ~.~.

'~ ••

' ..... ó

*•' * ó

"""- -- -i --~-- ·------·-

1

0 ~..J)

• à ~ à

J_ .0 d

0 I

•

0 ~ - • ••

I ---1 •

• 0 •

•

_J.. ~ .... r--__;___ .o

0

*

*

*

ï1 -. -:- ? 0 0 (Ï"'.

* *

f~t 1r. J.

()

~;)

**

*

<.ll rf 0

~ -. ~~ ~

9 -+--~---ro

~ ,...,

r~ 0 0

c 'j to * ~~

~r ~

."

~ 10., .. -~ -11

"' - CS'"' :-r. to

f' ..... f,

r

~ ~

61:..

*

*

*

~ * ~J

11 Cl:)

t i,

r-

* J ":::-.."' ..

i, ....

(./)

-----•....- f", 0 Ü" 0

r ~ " ,..... é,> -

r, .... 1 ~ V'

r-,fY :x:: .,.. -(' ;r o + L.J 0 11 11 52-

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0 ï

V' C'·

.. ~ ~

IQ. ,, 11'

"' -~. ,...

t éf;'

1"1-,+-<SI -

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-§~ ... _, ...... '.!! 'e '!

~rt-t%> -...11 ...JI c:-1 ,...

""' ~ ~ ~ . - o:> -.J - 0 -1~~

,, " " t& l-6f fo-&1

--:t::r. • 0 0 dJ I-J

1\i ...

'~ t--0---t

~-----~ ... ~ ~ ~ 0

~ .... '_)

1 t1 0 J

'4=-f j a.rt---.1_ ~

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1 ~ ~

·~ 1 ~

. j 1 ~$ ~ d

r' ~ d

-I \fl

...._;

r

r')

Q

~g

0

• • •

lil j.... ,.... e-1 c:-1 ." . 0 0

" tl

~ -Y c) c.)

~ ::J: ::r

' ~ -v f ,... ,.....

·:. "... ,...

(r) I:P Q) - -I)_ I I

~ - l/')

0 0 I

I

0 IJl (2)

g 0 --ti2 * •

~

'1-1../

• •

* * , .;;,. •

.",.

.* • • • • * * •• * • * • * ...... **•* • • Jlf

* •* * ** l4

* ~

*~

* *

I I I

I

* --------

~ ' . ~ ~ -s Si ~ ~ ~ 0

"' 0 '.J ..)

7r J....s? ~ a.+-o i ~1 "' ~ " ~ .~~

s: ~ s . } ~ s

i~ll lL g ~ ~

-

.

.

~

. r

.

.

-

.

0

I

t..l' '--···

3" 0

ooi 0

75

Figure 8 gives a combination of the two blood experiments.

The ratio of effective and Brownian diffusion coefficient 1 a 1

is given as a function of shear rate.

5.3 Experiments with hemoglobin solutions using the secend

membrane oxygenator

A second serie of experiments has been done with hemoglobin

solutions.Those experiments toke two days each, one day to

prepare the hemoglobin solution, the other day to do gas

transfer measurements in the membrane oxygenator. At the

beginning of each experiment the hemoglobin salution was clear

and free of any visible sediment.

During the course of the experiments the hemoglobin salution

became increasingly turbid • As a result the saturation mea­

surement with the ph9tometric method was disturbed.Therefore

both photometric instrument and the Lex-o 2-con are used

continuously during each experiment.

Three experiments ~sing hemoglobin solution are represented

in this report.

Experiment 30-11-1976

Figure 9 shows the theoretical and experimental fractional

saturation change as a function of the dimensionless channel

length L"'.

Figure 10 gives the theoretical and experimental saturation

difference as a function of the velocity of the cylinder v0 at two values of the flow ~.

Figure 11 gives the relation between the ratio of effective

and Brownian diffusion coefficient •a• and shear rate.

The independent variables a re: dtw\ ':1. l2.~ ur' ho\

[H b] 1. 77 gmo1/m 3 1. 33 .$

= ~V = gmol/m.atm

H i = 0.072 tX D = 0.22110-7 gmol/s.atm.M m m

T = 25.0 oe p. = 0.034 atm I

1.5910-9 2 p 0.95 D = m /s = atm

V u

......s ,... 0"' -I

i I

---~---------- --+-- -+ l I

....., --ó

•

I

• . I . I I I

~~----------t------------- +---~: I • . I I • • I

I • I

I •

I •

"" - ó

! N -------r----~4~--~·~~----------~o

I

I •

! I

I - ei

i I

I I I I V. ::r I"') .-.i 0 0 ó 0 0 .0

" -l&t

~ 0

0

•

0 c c:{

0

'1-9

I •

L

f..--- ---·-···-·--· - --- - ----· ~ -- ... ·- --------- --

I ~

-

I •

I L-....

,j -· ---- _"_-

I

I ! -+

I

~ ~

,..) ,...,

tû ~ I

c

~ ,.4- }-

I er

I

-! I ~ - ~

rt-·- - 0 '

_g I

~ 0 ....:. 0

en cP

,, ,. 0 ,--.

l~ _o ::t

._, :r

::r:-<5 L....

I I c-t") ei

- I

I

.. .,

- --~--- --.--- ·--

• I I

I

- - - -

--- ----- ~-----·-

-- ~· -----~--------

-s ~ -+ 1 t;j (

.J ~

~ -~

4 '-'

~ ~ v ~

.., oJ 0 ~ C4- \)

~ "'()

'-'6 c ~ .2

C> IJ>

i J:. ( 0 ct

~ .:! û

j__ -' - d

.~~ •-' ~ ~ d 0

I..L C' .ft ~ ~

_, 0

:r 0

,.., 0 -

eJ 0 -

The two applied flows are:

J 1= 11.26 10 -6 m3/s

i 2= 7.54 10 -6 m3/s

Experiment 29-12-1976

79

Figure 12 gives the fractional saturation change f as a function m

of the dimensionless channel length L~ bath according theory and

measured values.

Figure 13 shows the saturation increase S -S. as a function U I

of the velocity of the cylinder v0 •

Figure 14 shows 1 a 1 as a function of shear rate •

The independent variables were:

[Hbl = 1.64 gmol/m 3 r:l. D = 0.22110-7 gmol/s.atm.m m m

H i = 0.158 p. I

= 0.034 atm

T = 23.0 oe p = u 0.95 atm

D 1.6310-9 2

~ 11 • 2 6 m3/s = m /s = V

1 • 3 2 7 3"

12710-6 r:;<.v = gmol/m.atm d = m m

Experiment 12-05-1972

Figure 15 shows a calibration curve of the photometer.Numbers

indicate the measuring sequence.

Figure 16 gives the calculated drift of the photometer as a

function of the measuring sequence,for values of the saturation

measured bath with the photometer and the Lex-0 2-con. We have

to remark, that the time intervals between the mearuring points

are estimated.The values of the Lex-0 2-con are supposed to be

true. From the calibration curve(figure 15) the real photo­

meter value is determined. The difference between this value

and the measured photometer value is called dS. In figure 16

dS is given as a function of the sequence in measuring. An

" ' "": ' ..

""" ..

' ..

""" , .. 'Z I I \

J ~ ~ ", ~ 0 0 f ~

4-l ~ t ·;s oJ 0 Cl- ol

__ L c---±-.. •

~ rt- .. • <P -I

~ - .. . es" cl I -l.

I ~ I

•

·~ I

i ....

I t ~ "'~ 0 I ! ~ I

I

(J'- ...... r-~ 'g ·- ;)"IJ:) ...9 ....0 V' .....,

0 . - ~ I -7J..:> 0 I ,, - i -0 ï"T ,, I

~ -'> ,, i

•.J

I :r:.:r.tet ~

LJ ::c

j

eo

Q.. .t;f ~

E Ir~

- 0

:r. • 0

1'2 0

..J ó

~""" ó

;;

•

I .. I I

I

I

I 1 I I

i

I

I I I I

•

..

-·- ~-------··--~---

•

I

tO 0

~

I

j I I

t-I

·I I I

d .... -l ~

d -::!

r -IJ"-~

d &) - 0

tJ ""t) ó""" ~ s: s: d J -~ ()

{ ~ 0

.:.t u

J s: -.g rn ~

"' VI

~ ,_ -

1 ~

t)

1 :! ··-j..1.. o,J-

0

t ( /)

~ LL.j_

• - ó

0

&f

1 -.,) t- . ..., ".--...

~ V} ..,. s I è 'i ~ j

V}

~ -.......... ~

i r 0 0 • ....,

.S .Ç ~ 0 :s J ~ .1

--' • • l..f') ::n v 0

0 .;a

--1. .;

! -:1. Q)

E ~ 'i 0

_j-~ ....

J ~ 0

1 ~ ~ CP

~ 6 I

.)_ c.J -

? I

ër' c-1 :.s: Cltr--1-

~ 0 + 0 5 \J ~

~ 0 ~ ~ 'i~

o,/)

j_ .;;-... i

~ ~ .....-',g 6 0 co

3. ~ ~ ~ -..J

"' ·- ......0 eJ u. d -S.l _:.. 0 0 ---<JO (\

0 r-'1 rl

•' ~ -0 ·-> Q) '::I::::r:r&l

":l: L-l

•.> ~ ..n 0

·~ ~ <n

D 0 ,.,. ~ ëi ei 0

~ I

~

I • I ~~ ~ "1 . ..

~

I· • . .

t • I

' .. • ..,.

,

I

....si rt-<!'

('()

.!2 ~

I -e -J,.

t1l ~ -~ J Q)

1 ~

1 ·~ .. ~

f ...! ~ ~ CU-

~ ~

l~ ,., ., ~

<.) C))

-.J) ~ ~ 4

'c ~

:r (/) J_ ·- ~ s::

V, ......0 UI ~

-./) - c-1 ..L ,- ....:.

6

1 0 - ~ "'r\-- t (I - ..... 0

-""' ~ 11

0 .:t:

î (I

0

:I ~~

~ ~ J_

l.-J

d

.:r:. , ... ~

3 6 ~ ./1

c!2 d

-, T oi.Q

ro 0

ns .... ~---

83

interpolation gives the unknown values of dS. Since thJs · is

a rough estimation, the value of Aas in the error analysis is

0.02 instead of 0.01 •

Figure 17 shows an estimate of the effect of the drift of the

photometer on the theoretica! saturation difference (Su-Si)th

as a function of~ (v0 is constant). Corrected · and not

corrèçted: saturation values are used.

Figure 18 gives an estimation of the effect of the drift of the

photometer on the theoretically calculated saturation áiffe­

rence (Su-Si)th as a function of v0 at two values of 1! . Corrected · and not corrècted' saturation values are used for

the ca1cu1ation.

Figure 19 gives the ca1culated va1ues of the ratio of effective

and Brownian diffusion coefficient 1 a 1 as a function of shear

rate for corrected and incorrected saturation va1ues.

Figure 20 shows the corrected fractional saturation change f ~ m

as a function of the dimension1ess channel length L ,for

experimental and calculated va1ues

Figure 21 shows,at two values of i change s -s. as a function of vo·

U I

experimental results are given.

Figure 22 gives the calculated and

changeS -s. as a function of;r". U I :t

of f • m , the corrected saturation

Both calculated and

expertmental saturation

Figure 23 shows the corrected ratio of effective and Brownian

diffusion coefficient as a function of shear rate.

For this experiment the independeot variables we re:

[Hb] = 1. 74 gmo1/m 3 o(m 0m = 0.22110-7 gmo 1 /s. atm.m

Hi = 0.055 12710-6 dm = m

T = 27.5 oe 0.034 p. = atm

2 I

D = 1.6010-9 m /s p = 0.95 atm V u

1 '32 9 3

~. = 7.0110-6 m3/s D{v = gmol/m.atm

~= 1. 8.6710-6 m3/s

Figure 24 shows how the ratio of effective and Brownian

diffusion coefficient 'a' depends on shear rate for the

three hemoglobin experiments together.

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94

5.4 Hemoglobin experiments using the first prototype

of the membrane oxygenator

A third serie of experiments has been done with hemoglobin

solutions in the first membrane oxygenator. Gas transfer

measurements have been carried out in the recirculation chan­

nel of the system. A laminar flow was present in both channels.

Besides the quantities, measured during experiments with the

second oxygenator, also the thickness of the blood layer is

determined.

The experimental conditions of the two measurements, which

wilt be reported, were about the same.

(H~ = 2. 1 3 gmol/m 3 T = 24.0 oe

H i = 0.035 d = 4010-6 m 2 m

D .. #= 1.5010-9 m /s '<m D = 0.22110-7 gmol/s.atm V I m

I)( V = 1. 02 gmol/m.atm P.=0.034 atm I

PH = 7.09 Pu=0.95 atm

Figure 25 gives the•ratio of effeative and Brownian diffusion

coefficient as a function of shear rate for these two experi­

ments.

.... 9'5" Ir> 0 --ï

en .._,/

> \~ -o

r 0

• 0

0

• • ~ • 0

• 0

0 :r

- • ·-·-• •• -0

0 0 • •• •

0 • • •

.

~ ,_ -~~~--~J~---JJ----~J----Jr--~r---~----1-----"1 ..J J i! -1

hl Discussion

With regard to the blood experiments we can conetude from

figure 8, that the effective diffusion coefficient increases

with rate of shear.However, the scattering in measuring points

is to large to conclude, that the relationship is a 1 inear one.

When we compare the experiments using blood with the available

theory ( figure 26 ), we conetude in the first place, that the

new experiments give a complete different result from those of

Oomens(1976) and those of Overcash(1972). Probably system

variables, which play a role in the relationship between Deff

and shear, have remained unnoticed. With regard to the expe­

rimentsof Oomens (1976) it is supposed, that the small oxy­

genation length, which is only one-tenth of the present value,

have influenced the relationship between Deff and shear rate.

By varying the oxygenation length in the new sett-up the

influence of the oxygenation length is being studied. Results

of these experiments are not yet available. Furthermore with

respect to the experimentsof Overcash(1972) it is supposed,

that each system has its own transfer characteristics. lt is

probably not allowable to campare relations between 1 a 1 and

shear rate in these different systems, whithout regarding its

characteristics. Which factors possibly determine these

characteristics can be studled by a comparision of the results

of blood experiments done with the first and second prototype

of the Eindhoven membrane oxygenator.

When we compare the experiments with blood with the available

calculated curves , we note , that there is some agreement

of experiments with the theory of Nir(1973). Figure 27 shows

at four values of the hematocrit 1-- the theoretica} lines of

Nir. According his theory 1 a 1 increases, when the values of the

hematocrit increase • lt appears as if this tedency is also

present in ouc experimental results, but blood experiments with

stronger deviating values of the hematocrit have to confirm

this behaviour.

•

i

• • i •

] • • •

•

v-~ <r . _. t ·-(

_f· ~

ct

•• • • •

0 ,.. !:

~

j d!.

_c:'.J

i:: {)-

........ :::>

••• ~0 ,· ••• • ••

'· ...........

' "

(!)0

•

" \

0

•

",

~ ,> ....

2

0 0 •

0

0

,.. ~

CIÓ

,......._ -I \ft

..._;

I

. ....

38 s a

........._

~ 0

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ó

'

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f V'

~ 11

" r- •r V

>r~ .."~

lr V. ~ "' ...., -

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99

With respect to the hemoglobin experiments we cannot draw any

firm conclusions: the scattering in measuring points is to

large to conclude, that the ratio of Effective and Brownian

diffusion coefficient is shear rate independent.lf we regard

the measuring points( figure 26 ) there is a tendency for the

results to be above the line 1 a 1 = 1 • lf a hemoglobin salution

behaves as Newtonian fluid, we should expect the measurements

to be closer to the line 1 a 1 =1. Because of scattering in the

measuring points we cannot confirm this deviating behaviour.

The hemoglobin experiments in the first membrane oxygenator

are shown to be on the same level as the blood experiments

of Oomens(1976) in figure 28.Note the large scattering in

the experimental results. In figure 26 can be seen, that the

hemoglobin experiments , done with the two memb.oxygenators,

give completely differing results

• • ~

0

I CliO

•

* ** * • • 0

•

0 0

0

I

~

0 I

l ~

I "..

•

•

I ~

0

0

* * .,. 0

0 0* * * -• •• * • • .. • • *** • •

*

i ,,..

1 0 1

Conclusions

More experiments have to be done to determine the parameters,

which influence the relationship between the effective

diffusion coefficient of blood and shear rate. A higher

accuracy in the saturation measurement will reduce the error

in the effective diffusion coefficient.

Also the causes of the large scattering in resuts of

hemoglobin experiments wilt have to be determined. lt would

also bedesirabie if the theoretica1 description of the

effect of shear induced cell rotation on mass transfer were

extended: Nir 1 s theory agrees tosome extent only with our

present experimental results.

102

Suggestions for further research

1. Todetermine system variables in the relationship between

Deff and shear rate blood experiments have to be done with

varying values of the oxygenation length. This can be done

by varying the length of the gas chamber.

2. To determine the influence of the parameter '*'on the

relation between Deff and shear rate experiments are proposed

with varying values of the hematocrit ( for instanee ~=0.10

and 'f-/=0.40 )

3. A possibi 1 ity to determine explicit the contribut ion of

partiele rotatien to mass transfer is to use a hemoglobin

salution as basic fluid and to add micro spheres of polystyrene

( \'!Ïth a diameter of about B j.J.Jn ). lf enhanced mass transport

is measured then, this is a result of the rotatien of particles

only.

4. Other possibilities to estimate the cell rotation are to

do experiments with blood and hemoglobin solutions as we did

before, but to use inert gases like He and H2 insteadof 02 •

The influence of the hemoglobin is eJiminated then, but the

saturation length will be much reduced.

1 0 3

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11 7

Appendix F List of accumulated literature with Dutch summary

Antonini ,G.,Guiffant,G.,Quemada,D.: effect du mouvement induit

des hernaties sur le transport plaquettaire.

Biorheology.Vo1.12,pp. 133-135 (1975)

Een convectie-diffusie analyse van verhoogd plaatjes trans­

port t.g.v. de rotatie van de rode bloedcel.Er wordt een

uitdrukking afgeleid voor de effectieve diffusie coefficient: . 2

D = .!liQAU + R dU eff 2TT dy Ti1T dy

Antonini G.: transport de masse induit par la rotatien d'une

partiele dans un fluïde en mouvement.

Journat de chimie Physique,1974.no.7-8,p.1123

Afleiding van eerder genoemde diffusie coefficient

A.S.A.I.O.: progress report:subcommittee for blood gas exchan­

gers. ASAIO, trans.Amer.soc.Artif.lnt.Organs,ll:545,1971

Overzicht van de ontwikkeling van kunstmatige organen

Blackshear,P.L.,Watters,C.: observation of red blood ce11s

hitting solid walls. Advances in bioengineering; chemical

engineering progress symposium series No.114,vo1.67

Beschouwing van de interactie van de rode cellen met de wand

en de invloed hiervan op adhesie aan de wand

Blackshear,P.L.jr: mechanical hemolysis in flowing blood

Biomechanics,chpt.19

Overzicht artikel van mechanische hemolyse van bloed

B r ow n , C • H • , L e m u t h , R • F • , He 1 1 u m s , J • D • , Leve re t t , L • B • , A 1 f re y , C • P • :

response of human platelets to shear stress

vo1.XXI Trans.Amer.Soc.Artif.Organs,1975

studie met een rotational viscameter van het effect van

schuifkrachten op plaatjes en de stolling daardoor

Castonguay,Y.: determination d 1 une importante cause d'hemolyse

1ors de chirurgie cardiaque avec circulation extra-corpore11e

Canadian journal of medical technology ]i 1972

1 1 8

Empirisch onderzoek van hemolyse tijdens hartoperaties

Chow,J.C.: Blood flow:theory,effective viscosity and effects

of partiele distribution. Bulletin of mathematical bio1ogy

volume 37, 1975

Studie van de bloed stroming met de veronderstelling,dat

het bloed bestaat uit een suspensie van cellen in plasma

~rown,C.H.,Leverett,L.B.,Lewis,C.W.,Alfrey,C.P.,Hellums,J.D.:

Morphological,biochemical and functional changes in human

p1ate1ets subjected to shear stress. The journai of labora­

tory and clinical medicine,vo1.86,No.3,pp.461-471,sept.1975

Studie met een rotational viscometer van plaatjes onder

schuifkrachten i.v.m.stolling in hartkleppen

Colton,C.K.,Smith,K.A.,Stroeve,P.,Merril,E.W.: laminar flow mass

transfer in a flat duet with permeable wa11s.

AIChE journal,vo1.17,No.4,ju1y 1971

massa transport bij het afscheidings proces in kunstnieren

Di11er,T.E.Mikic,B.B.,Drinker,P.A.: The effect of red blood

cell motion on oxygen transfer in blood.

ASME Bicengineering Conference,nov. 1975

Meting van een effectieve diffusie coefficient als functie

van verschillende parameters

Dobell ,A.: Biologie evaluation of blood after prolonged

recirculation through film and membrane oxygenators

Annals of surgery,april 1965,vo1.161,No.4

Forstrom,R.J.,Blackshear,P.L.,Keshaviah,P.,Dorman,F.D.:

Fluid dynamic lysis of red ce11s. Advances in bicengineering

chemical engineering progress symposium series,No.114,vo1.67

toevoeging van saline ter voorkoming van hemolyse t.g.v.

wand interakties en hemolyse t.g.v. injectie van bloed door

een naald

Frojmovic,M.M.: Rheo-optical studies of blood ce11s. Biorheo­

logy, 1975, vo1.12,pp.193-202. Pergamon press

Persentatie van een nieuw apparaat en vergelijk met bestaande

methoden

11 9

Goldsmith,H.L.Mason,S.G.: The microrheology of dispersions

theory and applications,vol.IV (chapt.ll},ed.by F.R.Eirich

Academie press,N.Y. 1967

-rotatie van starre bolletjes in een shear flow

-krachten op deeltjes in een laminaire shear flow

-beweging van deeltjes in niet uniforme shear velden

-kinetiek van stromende suspensies

-viscositeit van suspensies

-traagheids effecten

Goldsmith,H.L.,Mason,S.G.: Some model experiments in hemodyna­

mics-V:microrheo1ogica1 technics. Biorheology 1975, vo1.12

pp.181-192, pergamon press

bestudering van het gedrag van model deeltjes,ghosts en

verharde bloedcellen in een Poiseuille stroming en plug

stroming.in een buis

Goldsmith,H.L.,Skalak,R.: Hemodynamics. Annual revew of fluid

mechanics vol.7, 1975

Revew artikel betreffende het stromings gedrag van vol bloed

-stromingsgedrag van aparte cellen en roulleaux

-stromingsgedrag van vol bloed

-theorieên over bloed stroming

-micro circulatie

Go1dsmith,H.L.,Mar1ov,J.: Flow behaviour of erythrocytes.l

Rotatien and deformation in dilute suspensions

Proc.Roy.Soc.Lond. 8182, 351-384. 1972

Bestudering van het gedrag van individuele rode cellen en

roulleaux in suspensies zowel in Poiseuille als in

couette flow.Bij shear<<0.1 N/m2 volgt de rotatie de theorie

van starre balletjes.Bij shear >0.1 N!m2 resulteert een cel

orientatie onder vaste hoek en het membraan lijkt te roteren

om de celinhoud.

Goldsmith,H.L.: Red ce11 motion and wa11 interactions in tube

flow. Federation Proceedings, vo1.30, No.S, sept.okt. 1971

120

Hyman,W.A.: Augmented diffusion in flowing blood

ASME publ ication, paper No.ll ,WA-Bio-4

Brownse diffusie en verhoogde diffusie t.g.v.celrotatie.

Gegeven wordt een effectieve diffusie coefficient als

funktie van de schuifspanning

Jeffery,G.B.: The motion of e11ipsoida1 particles immersed in

a viscous fluid. Proc.R.Soc,Lond.A.102,161-179, 1922

Elementaire berekening van de beweging van en de krachten

op ellipsoides

Keller,K.H.: Effect of fluid shear on mass transport in flowing

blood. Federation Proceedings,vol.30,No.5,sept.oct. 1971

Model voor de bepaling van een effectieve diffusie coeffi­

cient als funktie van schuifkrackten voor de rotatie van

een rode bloedcel in een stroming met schuifkrachten.

Keller,K.~.: Development of a couette oxygenator

Ontwerp van een couette oxygenator ,metingen,bloedbeschadi­

ging en deeltjes rotatie

Kirk,B.W. c.s.: A simplified method for determing the P50

of

blood. Journal of applied Physiology,vo1.38,No.6,june 1975

Leal,L.G.: On the effective conductivity of a dilute suspen­

sion of spherical drops in the limit of low partiele Peelet

number. Chem.eng.commun.1973,vo1.1,pp.21-31

Berekening van de effectieve warmte geleidbaarheid van een

suspensie van bolvormige druppels in een shear flow met

lineaire temperatuur gradient.

Lee,W.H.: Denaturation of plasma proteins as a cause of

morbidity and death after intra cardiac operations.

Surgery 2Q 1961

Bloedbeschadiging in oxygenatoren wordt in verband gebracht

met de denaturatie van plasma eiwitten aan het bloed-gas

oppervlak( disc,bubble and screen oxygenatoren )

1 2 1

MacCallum,R.N. a.s.: Fragility of abnormal erythrocytes evalu­

ated by response to shear stress. Journal o- laboratory and

clinical medicine,St.Louis, vol.85,No.1,pp.67-74, jan 1975

Cellen van personen met sikkel anaemie blijken zeer gevoelig

voor shear stress.Cellen van personen met Fe-deficiency en

andere ziekten vertoonden dit gedrag vanaf een bepaalde

drempel waarde ( onderzoek met een cylinder viscometer)

Marsden,N. G.Q.:some theoretical considerations on the measure­

ment of the kinetics of hemolysis in individual red cells.

Upsala J.Med.Sci 78:12-18, 1973

Correctie berekening bij de lichtmeting van hemolyse vanwege

niet wegdiffunderende hemoglobine buiten de cel.

Nishizawa,E. e.s.: Non-thrombogenic surface inhibiting platelet

adherence. Vol.XIX Trans.Amer.Soc.Artif.Organs, 1973

Oomens,J.M.M.,Spaan,J,A.E.,Donders,A.P.P.: Annular membrane

oxygenator with tangential flow.Oxygen transfer analysis

and sealing rules. Physiological and clinical aspectsof

oxygenator design. ed.by S.G.Dawids and H.C.Enge11,

published by Elsevier/North Holland Biomedica1 Press 1975

Theoretische basis en eerste resultaten van de Eindhovense

membraan oxygenator

Oomens,J.M.M.,Spaan,J.A.E.: A general advancing front model

descrihing the oxygen transfer in flowing blood.

2 nd International symposium on oxygen transport to tissue,

Mainz, March 12-14, 1975

Afleiding advancing front theorie voor vlakke plaat en buis

geometrieën

Petschek,H.E.,Weiss,R.F.: Hydrodynarnic problerns in blood coagu­

lation. AIAA paper No.70-143, 8 th aerospace science meeting

1970

Plaatjes transport ten gevolge van de rotatie van de rode cel:

afleiding van een effectieve diffusie coefficient

1 2 2

Richardson,E.: Deformation and haemolysis of red cells in shear

flow. Proc.R.Soc.Lond.A.338,129-153, 1974

Theoretisch model voor hemolyse in een uniform shear veld

Richardson,E.: Appl ications of a theoretica! model for haemoly­

sis in shear flow. Biorheology, 1975,vo1.12,pp.27-37,

Pergamon press

Onderzoek van de toepasbaarheid van een theoretisch model

voor hemolyse op "in vitro" experimenten

Rothstein,A. a.s.: Mechanism of anion transport in reà blood

cells. Role of membrane proteins

Satterfield : Mass transfer in heterogeneous catalysis(thesis)

Diffusie coefficient ,diffusie en reaktie in porous

catalysts

Schmid-Schönbein H.,Wells,R.: Fluid drop-like transition of

erythrocytes under shear. Science,vo1.165,pp.288-291, 1969

Vervorming van cellen onder schuifkrachten tot ellipsoides

Stein,T.R.,Martin,J.C.,Keller,K.H.: Steady-state oxygen trans­

fer through red blood cel suspensions. Journal of applied

Physiology,yo,11'31,No.3, Sept. 1971

Zuurstof diffusie in suspensies van deeltjes;bepaling van

een effectieve diffusie coefficient.Verder wordt verwaar­

lozing van de membraan weerstand aangetoond.

Formule voor Deff voor elliptische deeltjes.

Sutera,S. e.s.: Deformation of erythrocytes under shear.

Blood ce11s 1.369-374 (1975)

Studie van rode bloedcellen onder shear in een concentri­

sche cilinder viscameter

Schmid-Schönbein,H. a.o.:a counter-rotating "Rheoscoop chamber 11

for the study of the microrheology of blood ce11 aggregation

by microscopie observation and microphotometry

Microvascular research, 6, 366-376, 1973

Technisch rapport

i 2 3

Woolgar,A.,Morris,G.: Some combined effects of hypotonic solu­

tions and changes of temperature on posthypertonic hemolysis

of human red blood cells. Cryobiology 10,82-86; 1973

Hemolyse ten gevolge van koeling

Zander,R.,Schmid-Schönbein,H.: lntracellular mechanism of

oxygen transport in flowing blood.

Respiration Physiology 1973 19,279-289

Zuurstof transport ten gevolge van o2-convectie e~ o2-

diffusie enHb02-convectie en Hb02-diffusie in de cel.