@-/@g@-<0>=1+5c/2
|c<.18;@g{m}|->(dv/d@g{x})|c<.12]
But for small spherical droplets, G.I. Taylor replaced the concentration's
multiplier by
@equation[(@g@-<0>+5@g@-<*>/2)/(@g@-<0>+@g@-<*>),]
which may be multiplied by a geometric factor which surpasses unity
for assymetric particles.@foot[pp.155-157, Caro, C.G., Pedley, T.J.,
Schroeter, R.C., & Seed, W.A., @i< The Mechanics of the Circulation>,
Oxford(1978);Weinbaum, S., @i,
Fall Semester, 1981 Course in The City College of the City University
of New York, ME5506.] A correlation has been determined for yield stress
@equation{t@-<0>=[(HCT-.1)(C@- +.5)]@+<2> H(HCT-.1)
|C@-(=[.21,.46] & HCT>.1,}
with C@- representing fibrinogen concentration and HCT the hematocrit,
or volume ratio of solids in blood, which is usually about .45. H is
Heaviside's unit step function, which is zero for arguments less than zero
and elsewise unity. @foot[p.53, Cooney, David O., @i, New York:Dekker(1976)]
@chap(Erythrocyte Rouleau Aggregation Phenomenon)
Normal whole blood, being a heterogenous fluid, has red cells
which aggregate at low shear rates,
forming "rouleaux", erythrocytes which are assembled as
a roll of coins [see figure.] This is believed to result in viscosity
increases. This is important for many normal and pathological situations.
@begin(figure) @blankspace(178mm)
@caption{Rouleau Phenomenon,
[Goldsmith & Mason @i;
Unpublished Computer Approximations,
Columbia Bioengineering Institute,
all rights reserved by institute.]}
@end(figure) @foot[ In August, 1980, Professor Richard Skalak,
Director of the CColumbia University Bioengineering Institute, gave me
permission to utilise the rouleau output graphs on a paper from which
segments of this paper are derived. No permission for this reutilisation has
been granted, and it is expected that this shall be treated with due
confidentiality until such grant. ]
These rouleaux separate at larger velocity gradients @foot(or
velocity changes with respect to direction). At shear rates greater than
50/sec. none remain and blood viscosity ceases to change considerably. @foot{
becoming asymtotic.R Skalak, P R Zarda, K-M Jan, S Chien, "Theory of Rouleau
Formation", in @i, @b<71>, [1977],
pp.299-308;
D O Cooney, @i, [New Yourk & Basel:Marcel
Dekker, l976], pp.45-47;Goldsmith, H.L., & Mason, S.G., "The Microrheology of
Dispersions," in @iF.R.Eirich, ed., vol.4, [N.Y.:Academic, l967];
S Middleman, @i, [New York:
Wiley-Interscience, 1972].pp.82.}
A number of adhesive agents may be responsible for the rouleau
phenomenon. The hydrophilic-hydrophobic phospholipid bilayer membrane
may produce electrostatic forces that contribute to both attraction
and repulsion. @foot(Prof. Stuart Keller, Private Communication,
August, 1980) It has also been suspected that the blood clotting
molecule, fibrinogen, may be involved. Other explanations, however,
may include the globular protein,
@g(b)-globulin.@foot(Middleman,@i,pp.84-86.)
Furthermore, in the words of Professor Skalak and his colleagues: "It
is clear that electrostatic repulsion due to surface charges on the
red cell membrane acts to prevent aggregation. The absorption of long
chain molecules such as dextran onto the sufaces of adjacent red blood
cells provides the bonds which hold rouleau together. ... The basic
concept is that the binding energy of the adhesive agent must be able
to supply sufficient energy to account for the strain energy that the
elastic shell [cell itself] acquires during rouleau formation."
@foot{Skalak, @i}
@chap(Fahraeus-Lindqvist Effect)
Robin Fahraeus and Torsen Lindqvist from the Pathological
Institute in Uppsala, Sweden stated @begin(quotation)
Nearly one hundred years have passed since the French
physician Poiseuille @foot(Poiseuille, @i, vii, 50[l843]; @i, xvi, 60[1843]) took up for consideration the important
problem of the resistance of the bloodstream in the narrow parts of
the vascular system. As experimental difficulties arose with blood,
his fundamental investigations were confined to experiments with water
and different fluids in glass capillaries. He found, as is well known,
that the time of efflux of a given volume of fluid is directly as the
length of the tube, inversely as the difference of pressure at the two
ends and inversely as the fourth power of the diameter. ....with
water... agreed very well with the law of
Poiseuille...@i @foot{R
Fahraeus, & T Lindqvist, The Viscosity of Blood in Narrow Capillary
Tubes, @i, @b<96>, [1931], pp.562-568.} @end(quotation)
@begin(figure) @blankspace(41mm) @caption{Aparatus of
Fahraeus and Lindqvist,
[@i.]} @end(figure)
This is known as the Fahraeus-Lidqvist or axial accumulation
effect. This effect occurs only at high rates of shear. It may be
related to the occurance of a cell-free layer near the tube wall.
This has been verified via high-speed photography of blood flow in
glass tubes (@i) and in blood vessels of mammals (@i) Blood accumulates at the axis so pronouncedly that if blood is
not drawn from the center of the vessel, the hematocrit, or volume
fraction of cells in blood, may differ by a fourth.
@foot(Cooney, @i, pp.54-58.) @# @foot(This axial accumulation
could limit interaction of the cells to their surroundings, such as the
transport of oxygen and carbon dioxide.) This may be explained via the
cell-free marginal layer model @equation{
Q=[@g(p)R@+<4>@gP/8L@g@-

]
[1-(1-@g/R)@+<4>(1-@g@-

/@g@-)]}
@foot{R F Hayes, Physical Basis of the Dependence of Blood Viscosity on
the Tube radius, @i, @b<198>, [1960], 1195.}
Alternatively one could add the finitely approximated results and
obtaine the summation or "sigma effect" equation
@equation{Q=[@g

@gPR@+<4>/8L@g][1+@g/R]@+<2>.}
@foot(Cooney, @i) These last two equations are not dissimilar to
the Poiseuille equation verbalised by Fahraeus and Lindqvist above, which
normally does apply to water flow,
@equation{Q=@g

R@+<4>@gP/8L@g.}Q is the flow rate, R is the
radius, P pressure, L length, @g@-

the viscosity in a tube of
infinite radius, @g@- the viscosity where @g/R approaches
zero, @g the cell free layer thickness, and
@equation{@g=2@g(@g@- - @g@-

)/@g@-

.}
@chap(Effect on Artificial Organs)
@b
@comment[@comment{]
Artificial organs include external devices such as heart-lung and
kidney-dialysis machines, as well as internal organs such as
artificial kidneys and hearts. @foot( Cooney, @i< opere
citato >,pp.302-438.)
@comment[}]
@comment(NASA ARTIF ORG & BIOENGG H,S,B...?)
The Fahraeus-Lindqvist Effect, @foot[also descriptively called radial
drift, or axial accumulation -- accumulation at the center of a vessel] may be
accounted for under certain conditions. In the words of Professor
Leonard, who has just completed his three-year chairmanship of chemical
engineering here at Columbia, and a student of his: "the effects of
particle shape, deformability, and interaction with other particles
must be considered before predictions of the radial drift phenomenon
can be accurate enough quantitavely to be of real value in the design
of artificial organs and plasma cell separation devices." @foot{R V
Repetti, & E F Leonard, Physical Basis for the Axial Accumulation of Red
Blood Cells, in @i, E.F.Leonard, ed., @b<62>, no.66, [l966], pp.86.}
The effect of rouleaux on artificial organs is as follows: Normally,
in a small vessel, a small difference in pressure (.02 torr) is sufficient to
induce flow (e.g.:capillary 50 @g in diameter and 0.5 cm. in length). If,
however there are 500 such vessels in a system, 10 torr would be needed to
restart halted flow. This would result in injury due to lack of oxygen, which
is transported by erythrocytes.
@foot(Middleman, @i, pp.83-84.)
@comment{
@comment(weak conxn)
@Begin(quotation)
a pressure difference of less than 0.02 mm. Hg would be sufficient to
maintain flow in a capillary 50 @g in diameter and 0.5 cm. in
length. However, in a capillary @i equivalent to 500 such
vessels in series if the flow were suddenly halted by local
vasoconstriction at the venous collecting end of the network,
something like a 10 mm. Hg pressure drop would be required to restart
flow. If sufficient pressure is not available, local tissue injury
may occur as a result of a reduction in local oxygen supply.
@foot(Middleman, @i, pp.83-84.)
@End(quotation)
}
This must thus be compensated for in design.
Perhaps the ideas of this paper are best concluded by this rather
lengthy quote from artificial heart researchers at Goodyear:
@begin(quotation)
Considering that more people die from cardiovascular related
causes than all other causes combined, there seems to be little question
but what [@i] the artificial heart, when
fully developed, will be regarded as
@i major achievment in artificial organs in this century, and very
possibly in medicine generally. ....It pumps 4 to 6 liters/min. through
each of the two circulation systems with a very modest weight/unit of
throughput and at high energy efficiency, maintaining the blood in essentially
laminar flow. ...requiremnts of not causing excessive hemolysis...is
considered primarily to be in the area of fluid flow problems, that is,
the province of the chemical engineer. ... Basic research in fluid flow, as
it pertains to the requirements of the artificial heart, is urgently
needed, since no pump can be considered practical until it has been
demonstrated to keep the rate of red cell destruction (hemolysis) down
to tolerable levels over very long pumping periods. The non-Newtonian
character and peculiar flow properties of blood further complicate the
problem of obtaining precise data for fluid flow studies....That phrase born
of the aerospace age, the trade-off, will become commonplace jargon amongst
the artificial heart researchers. ("Do you think we could tolerate a 10% higher
hemolysis rate as a trade-off to pick up a 15% weight saving on the pump,
Doc?") And when we reach that happy time, we can truly say that the day
of the artificial heart has arrived.
@foot(R M Pierson, A F Finelli, M V Mathis, R C Martin, D L Gardner, &
G B Camp, The Portable
Artificial Heart:Designed Criteria and Prototype Evaluation, in @i,
pp.97, 98, l06.)
@end(quotation)
@Chapter(Haemoviscological Computations)
@sec(Correlation From Experimental Data)
@begin(figure)
@begin(verbatim)
LOG FILE IS MLAB.LOG.
LEAVE THIS SESSION WITH THE COMMAND 'EXIT'.
*FCT SQ(X)=X^2
*SQRTAU=LIST(.001,.2,.3,.45,.55,.75,.85)
*SQRTRATE=LIST(0,.2,.25,.5,.7,1.5,2.2)
*DATA COL 1= SQ ON SQRTRATE;DATA COL 2= SQ ON SQRTAU
*FCT STRESS(RATE)=MU*RATE^POWER+INISTRESS
*INISTRESS=0;MU=.3;POWER=.77
*FIT (MU,POWER,INISTRESS), STRESS TO DATA
[...DISCONTINUED FOR BREVITY....]
BEGIN ITERATION 4
BEGIN ITERATION 4.1, SUM OF SQUARES= .449290@@-2
FINAL NORMAL ERROR DEPENDENCY
PARAMETER VALUES: STANDARD ERRORS: VALUES:
.402009 .414405@@-1 .901354 MU
.410949 .481016@@-1 .766126 POWER
-.246513@@-1 .309788@@-1 .832861 INISTRESS
CONVERGED
RMS ERROR= .335083@@-1
FINAL SUM OF SQUARES= .449121@@-2
# OF ITERATIONS USED=4
*TTYDRAW DATA &'( STRESS ON DATA COL 1)
.+.....+.....+.....+.....+.....+.....+.....+.....+.....+......
.725 + *+
. .
. .
. .
.571 + A +
. B .
. .
. .
.417 + +
. .
. .
. A .
.264 + B +
. .
. * .
. .
.110 + B +
.BA .
.A .
.* .
.+.....+.....+.....+.....+.....+.....+.....+.....+.....+......
.000 .968 1.94 2.90 3.87 4.84
@end(verbatim)
@caption(Haemoviscology Curve-Fitting Computations)
@end(figure)
In the previous section, data for shear rate and stress was
correlated to a Binghamoid Power Law.@foot[ Bird, R.B., Stewart, W.E.,
& Lightfoot, E. N., @i, New York:John Wiley(1960),
pp. 10-11. Data read from: Fig. 5, p. 85, Chien, S., Usami, S.,
Taylor, H., Lundberg, J.S., & Gregersen, M., Effect Of Hematocrit And
Plasma Proteins On Human Blood Rheology At Low Shear Rates, @i, @B<21>(1966); Whole blood, HCT=51.7% 37 degr C.
Marquardt-Levenberg iterative curve fitting algorithm, in: Knott,
G.D., & Reece, D.K., MLAB:A Civilised Curve Fitting System,
@i, @b<1>, p. 497-526,
Brunel,UK(1972); Knott, G.D., @i< Computer Programs in Biomedicine>,
@b<10>(1979)271-280; Knott, G.D., & Reece, D.K., @i, interactive program in SAIL, for DEC PDP-10 systems,
Laboratory of Statistical and Mathematical Methodology, DCRT, NIH,
Bethesda, Md.(1980).] Although blood is known to follow this
law, alternatively appelated the Casson Law, we may inquire as to
whether such properties are neglectible. Should a relative linearity
be indicated, we may assume that a Newtonian model is applicable.
@foot( That the power be unity and the yield stress negligible.)
Yet the data utilised indicates a power of .41 which is
significantly different from the Newtonian unity, although the
yield stress of -.02 is sufficiently negligible.
@sec(Linearisation for Typical Shear Rates)
@begin(figure)
@begin(verbatim)
LOG FILE IS MLAB.LOG.
LEAVE THIS SESSION WITH THE COMMAND 'EXIT'.
*FCT STRESS(RATE)=.402009*RATE^.410949 -.246513@@-1
*FCT NEWTSTRESS(RATE)=MU*RATE+INISTRESS
*INISTRESS=2;MU=.01
*FIT(MU,INISTRESS),NEWTSTRESS TO POINTS(STRESS,100:800)
[....DISCONTINUED FOR BREVITY....]
FINAL NORMAL ERROR DEPENDENCY
PARAMETER VALUES: STANDARD ERRORS: VALUES:
.485191@@-2 .262935@@-4 .831793 MU
2.59994 .129734@@-1 .831793 INISTRESS
CONVERGED
RMS ERROR= .140875
FINAL SUM OF SQUARES= 13.8723
# OF ITERATIONS USED=2
*TTYDRAW POINTS(NEWTSTRESS,100:800)&'(STRESS ON 100:800)
.+.....+.....+.....+.....+.....+.....+.....+.....+.....+......
6.39 + AAA+
. AAA**BB.
. A***BBB .
. A****B .
5.62 + B****B +
. BB***A .
. BB***AA .
. BBB**AA .
4.85 + BB**AAA +
. BBB**AA .
. BBB*AAA .
. BB**AAA .
4.08 + B**AA +
. B**AA .
. A***A .
. A***B .
3.31 + AAA**B +
.AAABBB .
. BB .
.BBB .
.+.....+.....+.....+.....+.....+.....+.....+.....+.....+......
100. 240. 380. 520. 660. 800.
*EXIT
@caption(Haemoviscological Linearisation for Artificial Organ Shear Rates)
@end(verbatim)
@end(figure)
Since rouleau were previously cited as not existing at shear
rates greater than 50/s , we must seek shear rates greater than this.
We may riskily extrapolate the previous correlation for rates upto 5/s
upto the 800/s region, where it approaches a straight line. But this
is not so bizzarre since rouleau have been postulated to create the
low shear nonlinearities and the aforementioned high shear rate
Fahraeus-Lindqvist effect behaves in a @comment(...) .Cooney
@foot[@i, p.76; citing Whitmore, R.L., @i, Oxford:Pergamon(1968).] tells us of shear rates in
the human varying from a mean of 40/s in the veanae cavae to 800/s at
the capillary walls. @foot(Mean is exactly two thirds of maximum
.) Furthermore, he states that
@begin(quotation) non-Newtonian viscosity characteristics of blood ...
are confined to shear rates of less than about 50 sec@+<-1> , it
is clear that under the actual physiological conditions of the human
body, blood may be regarded as Newtonian @end(quotation)
A common physiology text @foot[@i, Selkurt, Ewald
E., Boston:Little Brown(1976),4ed., p.257.] indicates haemoviscosities
between twice and fifteen times that of water. Five times the
viscosity of water is given by the Columbia @i
@foot[ @i, Columbia University in the City of New
York, College of Physicians and Surgeons(1981), 5ed., p.199.] Caro
@foot[ @i, p.176.] indicates shear rates of 1000/s as
typical @i, at which he states the insignificance of
non-Newtonian properties, and a viscosity of 3-4mNsm@+<-2>.
The linear model approximated in this paper gives a viscosity of
4.85191 mNsm@+<-2>, which justifies the extrapolation.
This also has particular significance for artificial organs, in
which order of magnitude calculations indicate shear rates within
the plotted region.
It may be derived from the Navier-Stokes equation that the
Poisseuilloid shear rate is of the form of a volumetric flow rate
divided by some characteristics area and length. For a Poisseuille
capilary flow, the dividing factor is the area times the radius; for a
falling film, it is one third the thickness squared times the width;
and for a slit it is one and a half the width time the half-thickness
squared. Typical artificial organ flow rates are between one fifth
and two liters per minute (3.3@@-5 to 3.3@@-6 cu. m/s). @foot[Cooney,
@i, from various problems, pp. 337,437.] Taking one
liter per minute and thickness or radius of one tenth of a milimeter
and width of about a tenth of a meter, we obtain rates from ten (in
the capillary flow) to five million (in the film) reciprocal seconds.
But rarely do we have such small capilaries with such flow, nor such
thick films. Even though, they almost fit our assumptions. Thus, most
cases conform quite well to the Newtonian model, although a preliminary
calculation for shear rate should be made. @foot[ Should the non-Newtonian
assumption need to be retained, one can find generalised derivations
of many fluid equations in @i, Elrod, Harold. G.,
Columbia University in the City of New York(1981).]
One should note that the shear rate, being the derivative of the
velocity, is zero at the center line, since this is where velocity is
maximum. This is not as serious though, since we are usually interested
in the overall and not point flow.