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
Link to publication
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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
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*
*
*
ï1 -. -:- ? 0 0 (Ï"'.
* *
f~t 1r. J.
()
~;)
**
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<.ll rf 0
~ -. ~~ ~
9 -+--~---ro
~ ,...,
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c 'j to * ~~
~r ~
."
~ 10., .. -~ -11
"' - CS'"' :-r. to
f' ..... f,
r
~ ~
61:..
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*
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11 Cl:)
t i,
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(./)
-----•....- f", 0 Ü" 0
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r, .... 1 ~ V'
r-,fY :x:: .,.. -(' ;r o + L.J 0 11 11 52-
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t éf;'
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,, " " t& l-6f fo-&1
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'~ t--0---t
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~ .... '_)
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
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",
~ ,> ....
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. ~ "' ...., -
~ 11 1\
t z-çr
-j_ .... ,J
t+-~ 2 rt-
"' ~ 3 ;+. 0
1 t:J'
er
~
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._./
"' -
0 ...>
,..........,_ I
'Vi ~ ~
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ll2 I 11'" I
lP - co 0 -"
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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.