Effects of Shear Rate on the Diffusion and Adhesion of Blood Platelets to a Foreign Surface Eric F. Grabowski, Leonard I. Friedman, and Edward F. Leonard* Artificial Organs Research Laboratory, Columbia rniversity, New York, N . E’. 10027
The purpose of this paper i s to develop a mathematical model for platelet diffusion and adhesion to foreign surfaces exposed to flowing blood. The model i s then applied to parallel experiments in order to determine values and confidence limits for both the platelet diffusion coefficient and the rate constant for platelet adhesion to glass. It i s found that the effective platelet diffusion coefficient is an order of magnitude or more larger than one calculated from Brownian motion considerations alone and i s shear rate dependent. The existence and predominance of diffusion induced b y erythrocyte motions (i.e., rotation) i s therefore confirmed. As for the mechanism of platelet adhesion to glass, given the accuracy of the experimental data i t could not b e determined whether the adhesion i s a diffusion limited or an “intermediate kinetics” process. Certainly, however, i t i s not an adhesion-limited process.
O n e of the most challenging problems confronting designers of vascular prostheses and artificial organs is the minimization in these devices of thrombus formationwithout the use of strong anticoagulants. For example, even with chronic anticoagulation, thromboembolism has remained a serious, if not deadly, late complication in more than onefifth of Mayo Clinic operations for insertion of prosthetic heart valves (Duvoisin, et aZ., 1967). Moreover, anticoagulation itself can present complications. I n a recent study of 32 patients having chronic renal failure and undergoing home dialysis, Remmers, et al. (1970), found anticoagulants responsible for all but one hemmorrhagic disorder and for six out of ten drug-related disorders. Guidelines for developing flow conditions and surface properties resulting in a nonthrombogenic environment for blood in vascular prostheses and artificial organs are thus needed urgently. Of paramount importance in determining these guidelines is the study of the interactions of flowing blood with “foreign” surfaces. Convective, diffusive, and kinetic phenonema resulting from the dynamics of blood flow and from blood and surface chemistry are believed to be involved. Such study can help to clarify the specific kinds of these phenomena present during thrombus formation and to indicate means by which they may be controlled. Of the many possible interactions of blood with foreign surfaces, the particular interaction known as platelet adhesion merits special attention. Platelet adhesion is not only the first directly observable step in the formation of a thrombus, but also an accelerator of the clott’ing process (Seegers, 1967). The purpose of this paper is twofold: to develop a mathematical model for platelet diffusion and adhesion to foreign surfaces exposed to flowing blood; to estimate, by application of the model to parallel experiments, values for both the platelet diffusion coefficient and the rate constant for platelet adhesion to glass. I n particular, in accordance with the suggestion of Blackshear, et al. (1966), and I3ernstein, et al. (1967)! that the diffusion of solutes in flowing blood may be augmented by red cell motions (including rotation) induced by fluid shear, the platelet diffusion coefficient is presumed possibly to depend on shear rate. Because of its relevance to the vascular system, intravascular 224
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
devices used to test the thromboresistance of materials, and cannulae, a circular cylindrical geometry will be studied. Theoretical Development
Basic Equation and Boundary Conditions. T h e blood flow of this study will be assumed homogeneous, laminar, steady, fully developed, and asisymmetric. For the range of shear rate of interest here (20-80 sec-I), Bugliarello, et a / . (1965), indicate that blood can be considered a pseudoplastic fluid obeying the “power law.” Under these conditions the convection equation for platelets in a circular, cylindrical tube takes the form
where (Skelland, 1967)
usThe transient term in this equation arises from the fact that time dependence will be introduced t,hrough one of the boundary conditions. However, this term will be formally shown to be of negligible importance and is retained here for that purpose only. As for t’he platelet diffusion coefficient, its dependence on shear rate, if any, is unknown. Blackshear and Bernstein have proposed a linear dependence for blood solutes whose Brownian diffusion is small compared to that due to the small scale mass convection associated with red cell motions. Such a model has subsequently been applied to platelet diffusion and adhesion by Petschek, et al. (1968a,b). Qualitative agreement was found with esperimental results. Petschek and Keiss (1970) have discussed nonlinear models. Finally, Turitto, et al. (1972), have measured the platelet diffusion coefficient by means of a procedure which presumes this coefficient to be independent of shear rate. (A posleriori justification is given.) I n this study the platelet diff usion coefficient will be assumed t,o bear a “power law” relationship to shear rate. (Formally the relationship ma; be regarded as one with re-
spect t o the second invariant of t h e rate-of-deformation tensor.) Hence
where the esponent rn is expected to take on some value between zero and unity. For the assumed velocity profile, then
with
and where the symbol Pe denotes the Peclet number, Ca,/D,. The boundary and boundedness conditions become
(3) where Deviations of the constant /3 from unity characterize t h e importance of non-Sewtonian effects. Certain boundary and boundedness conditions are required. O n foreign surfaces exposed to flowing blood, platelets have been observed initially to adhere as an apparent partial nioiiolayer (Friedman, 1972; Friedman et al., 1970; Petschek, et al., 1968a,b; Petschek and Keiss, 1970); masimum platelet surface densities correq~onding to such a configuration have been achieved. i t seems rea3oiiable to suppose, therefore, that a t the tube surface the net platelet flus can be equated to a rate of platelet adhesion which is first order n-ith respect to both the platelet concentration very near the surface and the fractional surface area uncovered by platelets (at a particular iiistaiit of time)
(4) where
Initially the process of platelet adhesion locally appears t o be iiidistinguishable from a random one ( n u t t o n , et al.,1968; Petschek, et al., 196Sa,bj Petschek and Weiss, 1970) ; however, the same caiiiiot be said for the advanced stages of platelet adhesion, for which coherent patterns in the platelet surface density have been observed (Friedman, 1972). (The process eventually becomes necessarily nonrandom insofar as platelets are not free to adhere to surface sites occupied by previously adherent platelets.) The above boundary condition presumes, nevertheless, t h a t the effects of these depart'ures from randomness can be neglected. The authors believe this presumption to be reasonable for 1017 to moderate degrees of surface coverage (for example, degrees of coverage which, over the leiigth of the tube of interest, average less than about 50% of maximum). A t higher degrees of surface coverage it does not seem clear how best to model the process of platelet adhesion, ai1 approach based on contiiiuum theory perhaps being inapplicable. At the tube inlet the platelet concentration will be taken to be uniform cI2=o =
co
(5)
Finally, a necessary boundedness condition is c bounded for all t; 0
5 r 5 a,1: 2
0
arid F.t-0
=
1
bounded for all f; 0 _< i: _< 1, ? 2 0
(9) (10)
The symbol K signifies the dimensioiiless adhesion rate const,ant, k a ,'D,. Important simplifications ill the convectioii equation result if the Peclet number is taken to be large compared to unity, in n-hich case the effects of diffusion are coilfined to a narrow boundary layer region. Such a n assumption iz often valid for diffusion in a flowing liquid (Levich, 1962). (That the Peclet number is indeed large compared to unity in this study will be shown a posteriori in .Ippendis -1.) Consistent with this observation, a "stretched" coordinate (Van Dyke, 1964),17, is introduced
6
=
(1
- p)pe1/3
(11)
where the factor Pel, is required in order that leading terms of eq i , which account for effects due to axial coiivectioii and radial diffusion, be of the same order of magnitude n-ithiii the boundary layer. K i t h terms of order l>e& or smaller neglected, relations 7 through 10 iiow become
with
The symbol K 6 denotes a dimensionless adhesion rate constant based on the characteristic boundary layer thickness, 6. It can be shown (Levicli, 1962) that
(6)
Nondimensionalization a n d Simplification. K h e n each term in the above relations is replaced by the product of a characteristic dimensional value and a dimensionless quantity of unit order, one obtains
Inspection of boundary condition 13, however, indicates that, for the fractional wall area 1 to be affected by the time integral of the platelet flus to the wall, the parameter (CoD,Ttap,) must be of unit order. This fact suggests the choice
for the characteristic platelet adhesion time. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
225
I
0.6
-”i a7
I
T.0
I
I i i =8.0
I
1
Ka= m
0 , 4 ~ s a \ 1
0.2
1.0
2.0
3.0
4.0
t
Figure 1 . T i m e dependence of the platelet flux at the tube surface for various values of Substitution of eq 17 into relations 12 and 13 now leads to
where
Equation 18, with boundary conditions 14, 15, and 19, describes, in simplified form, the assumed platelet diffusion and adhesion. The significant dimensionless governing parameters can be seen to include 2 , t’he dimensionless axial coordinate; 6,the “stretched” distance inward from the tube surface; f, the dimensionless time (a function of the average fluid velocity, as well as time); ~ e - ~ ’ ~ ( a C o / pa~ measure ), of the relative importance of transient diffusion bo axial convection; K g , a measure of the relative importance of plat’elet adhesion to radial diffusion within the concentration boundary layer; p, a non-Yewtonianess factor; and m , the power of shear rate t’o which the platelet diffusion coefficient is proportional. (The characteristic platelet diffusion coefficient, D,, used in the nondimensionalization procedure, is a,function of m.) It is evident from eq 18 that the transient convection term is of negligible importance provided that the parameter Pe-1’3(uColpp)is small compared to unity. For the present, this parameter will be assumed small, but the resulting simplification Jvill be demonstrated correct a posferiori in Appendis R. It should be pointed out that neglect of the transient convection term does not render the problem time independent : boundary condition 19 still contains a time integral. The magnitude of t’he parameter K 8 cannot be anticipated, especially since the adhesion rate constant’, 12, is unknown. However, the fact that the rate of platelet adhesion has been found experimentally t’o depend on flon- rate (Friedman et al., 1970; Friedman, 19i2; Petschek, et al., 1968a,b; Petschek and Weiss, 1970) and, hence, diffusion, restricts the possibilities
K 8 >> 1; K 6 = O(1)
(20)
corresponding to diffusion-limited and “intermediate kinetics” situations, respectively. The power-law esponent n has a value of about 0.8 for normal hematocrits (Rugliarello, et al., 1965). Hence, the noli-Kewtonian factor p has a value of about 1.06, indicating that, lion-Xewtonianess slightly enhances the importance of convection in eq 18. The dependence of the diffusion coefficient upon shear rate is characterized by the exponent m. The work of Turitto, 226
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
K8
et ul. (1972), suggests that this quantity, if indeed nonzero, may have a value less than 0.5. Asymptotic Solutions. Solutions of eq 18, with the transient term neglected, will now be sought subject to the indicated boundary conditions. First, for values of K 6 large compared to unity the problem reduces to one for diffusionlimited platelet adhesion. The extension by Pigford (1955) of the solution of LBvkque (1928) to non-Newtonian fluids then applies. In particular, the platelet flus to the tube wall becomes
(Since the cube root of the non-Kewtonian factor /3 does not differ appreciably from unity, however, the non-Kewtonianess of blood has a negligible effect on the platelet flux.) Another limiting solution of eq 18 is that for very small degrees of platelet surface coverage. I n this case boundary condition 19 becomes
and the platelet flux to the tube surface can be found by adapting the solution of Solbrig and Gidaspow (1967)
Equation 23 reduces to eq 21 for large values of the parameter Kg.Although very small degrees of platelet surface coverage are not of chief interest here, the above solution is of value as a useful asymptote to solutions for finite degrees of surface coverage. Specifically, fluxes obtained from the latter solutions have initial values given by eq 23. Numerical Solution. For finite (unit order) values of t h e parameter K s and for finite degrees of platelet coverage, recourse to numerical solution is necessary because of the nonlinear boundary condition 19. The procedure adopted here is analogous to the approach which von Karman and Pohlhausen used to obtain the velocity profile in a momentum boundary layer. First, the convection equation (eq 18), without the transient term, is integrated with respect to 6 froni zero to 8, where 8 is the diniensionless concentratio11 boundary layer thickness a t any axial position and instant of time. The result, similar to that of Eckert and Drake (1959), can be expressed as
0.8 -
06
-
0.4
-
P
I
2 .o
I O
-
I
I
30
4 0
t
Figure 2. Time dependence of the time integral of the platelet flux a t the tube surface for various values of Ka
Next, the concentration in the boundary layer is assumed to be of the form C = ao f
a1
(i) + 6)' + (0" a2
a3
I O
5 1
o
I
I
/
t=05
07
(25) 04
where the a's are functions of axial position and time to be determined from boundary condition 19 together with the following three conipatibility conditions d?j'?j=O=
I
02
P
(26)
01
which strttes that very near the tube surface, where convection effects are small compared to diffusion effects, the diffusive flus is uniform
07
I
0.1
Fq,s
04
(27)
= 1
which is analogous to boundary condition 15; and 02
which specifies that the diffusive flux vanishes a t the edge of boundary layer. Use of these four relations yields
0 01
7
10
I6
x
Figure 3. Axial dependence of the time integral of the platelet flux a t the tube surface for various values of K6
I n order to ensure compatibility of the ensuing solutions with inlet boundarv condition 14. the houndarv rondition where the quantity K1, defined implicitly by
can be interpreted as a local dimensionless adhesion rate constant (based on the characteristic boundary layer thickness) for a particular instant of time. Upon substitution of eq 29 into eq 24 and evaluation of the integral, eq 24 becomes
(
z p %1 +"f' -KJ
)
is imposed. K i t h these solutions one can then calculate from eq 29 not only concentration profiles, but also platelet fluxes a t the tube surface and time integrals of these fluxes (33)
and =
+:K18)
(31)
Numerical integration of the simultaneous eq 30 and 31 can be carried out to determine the boundary layer thickness 6 and the quantity K I as functions of axial position and time.
(34) The dimensionless platelet surface density, p , is related by a known constant of proportionality t o the measurements Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
227
depicted in Figure 2 can be approximated in closed form by -COVER
PLATE
FOREIGN
(35)
SURFACE
8 L O O D CHANNEL
where the constant, y, is chosen such that eq 35 best represents, in the least-squares sense, the numerical solution over a range of K 6 for a particular axial location. For the range 10 2 K 6 2 0.5 and for 3 = 8.0, the constant y has been found to have the value 0.90, with maximum deviations of eq 35 from the numerical solution less than 4y0,This equation will prove to be of great convenience in the comparison of theory and experiment to be undertaken in a later section.
T U B I N G CONNECTOR
(0318~rnI 0 I
-TUBING
.
BLOOD
FLOW
Figure 4. The experimental flow chamber
Experimental Work
to be reported here. Results of calculations of the above platelet fluxes and their time integrals are shown in Figures 1, 2, and 3. Figures 1 and 2 indicate that, for a particular axial position, the platelet flux a t the tube surface approaches zero with increasing dimensionless time, while its time integral approaches unity. (The particular axial location chosen, f = 8.0, corresponds to the center of the experimental flow chamber, a t which the bulk of the measurements of this study were taken.) How-ever, the rate a t which zero or unity is approached increases with the parameter K 6 until limiting curves, corresponding to a n infinite value of this parameter arid a diffusion-limited situation, are reached. The limiting curve for the platelet flux is identical with that predicted by the Pigford-Lhv&que solution, eq 21. I n addition, for finite K 6 , initial values for the platelet flux agree well with those predicted by the second asymptotic solution, eq 23. Both quantities also have, of course, an axial dependence. Figure 3 illustrates, for instance, that the axial dependence of the dimensionless platelet density, p , is stronger for large values of K6 than for small ones, dimensionless time held fixed. It must be stressed that the above curves may have physical relevance only for low to moderate degrees of platelet surface coverage. For degrees of surface coverage which, over the length of the tube, average less than 50% of maximum, the results
Methods. Experiment's were conducted with 76 anesthetized mongrel dogs of either sex weighing from 15 to 25 kg. The femoral arteries of these animals were surgically exposed and cannulated with 10-cm lengths of thin-walled Teflon tubing, in turn connected to sections of medical grade vinyl tubing ranging in length from 10 to 15 cm. All tubing had a n inner radius of 0.16 cm. Heparin was administered intravenously, as ari anticoagulant, prior to cannulation and during the actual esperiments. The flow chamber, Figure 4, was adapted from an earlier design of Dutton, et al. (1968). The inset in this figure shows the E-shaped cross-section of the blood channel, the halfwidth of which was identical with the inner radius of the tubing used. Initially the chamber was filled with a physiologic electrolyte solution to prevent the formation of a n air-blood interface and possible denaturation of blood proteins. A glass cover slip 2 . 2 cm in diameter was then inserted into the chamber and the tubing attached to the inlet port. Two pumps were employed. For the lower flow rates and shorter exposure times a syringe pump was used to withdraw blood from the exit port of the chamber, while at the highest flow rate and longer exposure times the same task was performed by a roller pump. I n the latter case the blood leaving the pump was reinfused into the dog. A n ultrasonic flow meter monitored the flow rate. Each experiment consisted of the exposure of a glass cover slip to blood flowing through the chamber a t a known flow
I
FLdW R A T E :
3.8;
I
me /min
T T
0
0
IO
30
20
40
t . min.
Figure 5. Platelet surface density, with standard deviations, as a function of exposure time at the lowest flow rate 228
Ind. Eng. Chem. Fundom., Vol. 1 1 ,
No. 2, 1972
IO
FLO'W R A T E :
7.64' m8 1 min
i
"
0
10
20
30
t, min
Figure 6. Platelet surface density, with standard deviations, as a function of exposure time a t the middle flow rate
with exposure time until a maximum was attained. The higher the flow rate, the greater was the initial rate of platelet adhesion. It should be noted that the exposure time and actual platelet adhesion time may not coincide, for delays have been observed in the onset of adhesion in the presence of flowing blood (Friedman, 1972; Friedman, et al., 1970; Petschek, et al., 1968a,b; Petschek and Weiss, 1970). Xevertheless, it will be assumed that these two times do not differ appreciably. Apart from providing an estimate of the maximum platelet surface density possible, data for high degrees of surface coverage, it may be added, will not be used in this study because of the possible failure of the theoretical model for such degrees of coverage. Even if the model were valid a t high degrees of coverage, however, it is doubtful whether
t, min
Figure 7. Platelet surface density, with standard deviations, as a function of exposure time a t the highest flow rate
rate for a specified period of time. After each exposure, the chamber was rinsed with saline and the cover slip removed and cheniically fixed. Phase microphotographs (500 X ) were then taken of platelets adherent to the center of the cover slip, with platelet surface densities obtained by averaging the number of platelets on five neighboring 900-p2 areas of each microphotograph. Red cell and bulk platelet concentrations were also determined. The former concentration, in terms of hematocrit (volume per cent), ranged between 30 and 50, while the latter varied between 2.0 X lo8 and 4.0 X 10s platelets/cma. Three flow rates (3.82, 7.64, and 15.3 ml/min) and six exposure times (2.5, 5.0, 7.5, 15, 30, and 45 min) were used. The flow rates have corresponding shear rates on the glass cover slip of approximately 20, 40, and 80 sec-l, respectively. Dye injection studies indicated that all flows were laminar and essentially fully developed. Because of the noncircular cross section of the flow chamber, however, the theory developed previously does not strictly apply. Some approsimation will therefore be involved in the comparison of experiment and theory to follow. Keverthelesb, for measurements taken equidistant from the chamber sides, as was the case here, the error introduced by this approximation can be shown to be negligible (Grabowski, 1972). Further details of equipment and procedure have been given by Friedman (1972), and Friedman, et al. (1970). Results. Figures 5, 6, and 7 portray the platelet surface density as a function of exposure time for the various flow rates. For a particular flow rate, this surface density increased
much further information could be obtained from many of these data owing to their relative insensitivity to changes in exposure time and flow rate. Comparison of Theory and Experiment
In this section the unspecified constants in the theoretical model, c y , m, and k , will be estimated, with confidence limits, from the data for degrees of platelet surface coverage less than 50% of maximum. First, eq 35 can be expressed in a form involving these unspecified constants and, hence, more convenient for a regression analysis
Dimensionless platelet surface densities have here been normalized with respect to real time, thereby removing time as an independent variable. For a fixed axial position, eq 36 constitutes a relation in three parameters and a single independent variable, C (proportional to the surface shear rate). X regression analysis employing the above equation would be, however, a nonlinear, transcendental one, for which there exists no general theory for estimating confidence limits (Rfarquardt, 1964). I t is therefore a more efficient procedure, for the purposes of this paper, to estimate the unspecified constants and their confidence limits from consideration of certain limiting "linear" cases. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
229
from which it is readily apparent that
1
1
KS= m
UPPER 9 5 PERCENT CONFIDENCE LIMIT
For a value of U of unity, relation 39 indicates that CY
1
Further, upon taking logarithms on both sides of relation 39 and rearranging, one obtains
which, when U is taken to approach infinity, yields
m 1 m, Another limiting case, which appears to provide a lower bound for k , is that for value of m of unity, a plausible upper bound for m. For such a value of m eq 36 becomes I 0-81
1
10
I
1
I
I
I
I
r = a , sec-‘
Figure 8. Platelet diffusion coefficient evaluated at the tube surfaces as a function of shear rate
Table 1. Results of Linear Regression Analyses Based on Eq 36 with the Exponent m Assumed k. cm/sec X 104
m
Index of determinationa
Lower 95% confldence limit
Expected value
Upper 95% confldence limit
1.0 0.253 0.32 0.68 m 0.92 0.255 0.33 0.78 W 0.62 0.257 0.41 2.6 rn 0.32 . . .b 0.67 W W a An indicator of how well a part’icular regression line fits the observed dat’a, defined as SSR/SST = 1 - (SSE/SST) where SSR, SST, and SSE denote the regression, total, and error sums of squares, respectively. Analysis failed to yield an index of determination in this instance.
One such limiting case, which provides lower bounds for CY and m, is that for large values of k ( K , large compared to unity). I n this instance eq 36 reduces to (37) for which a linear analysis may be performed if In ( p l t ) is assumed to be normally distributed. The resulting approximation is often sufficiently close (Crow, et al., 1960). Figure 8 illustrates the results for this case in terms of calculated values of the platelet diffusion coefficient for various shear rates. The lower 95% confidence limit form, is 0.32; the expected value, 0.62. That the values of C Y , and vi,-and, consequently, the values for D,-so obtained are indeed lower bounds can be seen from equating the right-hand sides of eq 36 and 37 and solving for the quantity (cYL”)*”, which involves “true” values of CY and m. The result is
230
(43)
10’
-cl$2
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
A linear analysis of eq 43 is possible if l/(p/t) is assumed to be normally distributed. Table I summarizes the results of this and similar analyses for other values of m. (The additional values of m chosen are the expected and confidence limit values determined in the first limiting case.) A minimum value for k can indeed be seen to occur for a value of m of unity. I n the calculations which led to Table I, in addition, no values for the platelet diffusion coefficient, D,, were found which exceeded the upper 95% confidence limit of Figure 8. As a result, this limit is apparently also the upper 95% confidence limit for “trueJ1values of D,. Conclusions
A calculation of the platelet diffusion coefficient based on Brownian motion considerations alone, as in Appendix D, yields a value of the order of 10-9 cm2/sec. It is clear from the foregoing analyses, however, that the effective platelet diffusion coefficient in the vicinity of a foreign surface, D,, is, for the range of shear rates investigated, an order of magnitude or more larger and shear rate dependent. These findings tend to confirm the existence and predominance of diffusion induced by red cell motions in the shear flows examined experimentally. Consequently, increasing shear rate may increase the flux of platelets to a foreign surface not only by diminishing the thickness of the platelet concentration boundary layer but also by simultaneously augmenting the platelet diffusion Coefficient. Further, the bounds established for the expected values of and %yo confidence limits for this diffusion coefficient, as portrayed in Figure 8, agree well with the results of the work of Turitto, et al. (1972). Turitto measured the axial spreading in a tube of a platelet step function in flowing blood, from which, using a modification of the dispersion theory due to Taylor, the diffusion coefficient for platelets mas inferred. H e found this diffusion coefficient to lie, with 95% confidence, between 0.5 and 2.5 x 10-7 cm2jsec for a hematocrit of 50 and for a wide range of wall shear rates (40 to 200 sec-1). H e could not, hom-ever, determine whether shear rate dependence existed. As for the mechanism of platelet adhesion to glass, it is not certain whether the adhesion is a diff usion-limited process or one in which the finite rates of diffusion and surface adhesion are of comparable magnitude. The calculated dimen-
sionless adhesion coefficients, K,, ranged, with 95y0 confidence, from 0.25 to infinity. Although the dimensional adhesion rate constant, k , could similarly be effectively infinity, its apparent loiver bound is of the order of l o p 4cm/sec. From another viewpoint, in a least-squares sense no improvement in the correlation of the data resulted from assuming K , to be finite rather than infinite. I n fact’, as appears from the trend in the indices of determination of Table I, a diff usion-limited model correlated the data slight’ly better than an “intermediat’e kiuetics” one. Accordingly, a diffusionlimited model appears to suffice a t present in the interpretat’ionand correlation of platelet adhesion data obt,ained on glass esposed to flowing blood. Appendix A
A Posteriori Determination of the Magnitude of the Peclet Number, I;a/D,. I n this study t h e quantities U , a, and D, have had, or have been found to have, the following values: I: = 0.9, 1.8, and 3.6 cm ’sec; a = 0.16 em; and D, = O(lO-7) to O(10-8) cm2/sec. Hence l’e
=
0(10+6) to 0(10+7)
(A.1)
The boundary layer approximations of the Theoretical Development section. which assumed that this parameter is large compared to unity. are therefore correct. Further, the error introduced by these approximations is of the order of (A.2)
= o(10-2)
Appendix B
A Posteriori Determination of the Magnitude of the Transient Convection Term. T h e importance of t h e transient convection term has been shown to be governed by the parameter Pe-’/’(aCJp,). I n this qtudy the quantity Pe-‘ ’ haq the magnitude given by eq A.2, while the quantities a, Po, and pp have the values: a = 0.16 em; Co = 3.0 X 108 platelets/cm3; and pP = 6.5-7.0 X lo6 platelets,’cm3. Consequently
Equation B.1 indicates t h a t the transient term, while perhaps not entirely negligible, is, a t worst, of marginal importance. A better estimate of the magnitude of this term is possible, however. First, the integrated boundary layer equation, eq 24, with the transient term retained, has the form
4P
E dZ
(1
-
E ) ijd: =
dE, Y
31 i j = O
(B.2)
The first term in this relation, if indeed small, can be accurately estimated from the solution to eq 30 and 31. I n particular
Results of calculations based on eq €3.3indicate that -
s,” (g)
d6
=
O(l0-l)
from which, together with eq B.1, i t can be concluded that -pe-~/3
(f3 s,” (g)
dij
=
0(10-*)
(B.5)
Therefore, the transient convection term, a t least for the integrated boundary layer equation, is negligible. Appendix C
Axial Dependence of Experimental Data. For one combination of flow rate (3.82 mlimin) and exposure time ( 5 min) the microphotographs from several experiments were also examined at four equally spaced axial positions. The observed platelet surface densities in every instance were found to diminiqh with axial position, in qualitative agreement with the predictions of Figure 3 for values of the parameter K s of unit order or larger. Appendix
D
Calculation of a Platelet Diffusion Coefficient Based on Brownian Motion. T h e Bronnian diffusion coefficient for “large” spherical particles or molecules in liquids is generally calculated from the Stokes-Einstein equation (see, for example, Bird, et al., 1960)
Equation D.1 assumes the particles or molecules to be “large” in the sense that the liquid medium appear. to the diffusing species as a continuum. The diffusing species, further, is assumed to be i n dilute concentration. For platelets a t physiologic temperatureq, the following valueq may be chosen for quantitieq appearing on the righthand side of the above relation: k g = 1.381 X 10-16 g cmz/ sec2; T , = 31OOK; R , = 1.0 X cm; pf = 1.0 X IOe2 g/cm sec. The viscosity value stated 15 that of plasma (Merrill and Wells, 1961), not that of whole blood, sinre the theory requires that the fluid medium considered be a continuum. Any effect of the presence of red cells on the Brownian motion of platelets cannot be considered within the framework of this theory. Substitution of these values into eq D.l
DB = 1.6 X 10-9 cm2/sec
D.2)
platelet concentration, platelets/cm3 platelet diffuqion coefficient, cni2/sec axial coordinate, em radial coordinate, em time, see (unlesq otherwise indicated) axial velocity component, cm/sec average a v i d velocity, cm/qec tube radius, em power-law exponent const a n t proportionality constant, rm2/set(' - m) non-Sewtonian factor = (3n 1)/4n adhesion rate constant, em ’see fractional area of tube surface not covered hy platelets a t any instant of time maximum observed platelet surface density, platelets/cm2 bulk platelet concentration (experimental average), platelets/cm3 characteristic adhesion time, see platelet diffuqiori coefficient evaluated a t the tube surface, cni2/qec = a [ p ( 4 C ) / a ] n dimensionless platelet conrentration = c ’Co
+
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
231
dimensionless time = t / T = Pel/~(CoD,t/a) dimensionless axial coordinate = x/a dimensionless radial coordinate = r;a dimensionless axial velocity component = uc‘ dimensionless datelet diffusion coefficient =
D/D,
Peclet number = Y a / D , dimensionless adhesion rate constant = ka 1D. “stretched” dimensionless coordinate = (1 7) Pe‘,’3 dimensionless adhesion rate constant based on the characteristic boundary layer thickness = IZGjD, characteristic boundary layer thickness, em = a . pe-’/a dimensionless boundary thickness = J1/6 boundary layer thickness (local), cm functions of axial position and time local dimensionless adhesion rate constant (based on the characteristic boundary layer thickness) dimensionless platelet surface density = p / p p platelet surface density, platelets1 em2 constant constant, cm3/’platelet = l/yCo functioii of axial position only cn11/3(2m+ll) 3az; a(m+1) -
platelet
co
2(4)?m/3 \
-
/
aluei of CY arid m,respectively, corresponding to values of Ks large compared to unity value5 of CY and k , reqpectively, corresponding to a \ d u e of ?n of unity 13roniiiaii platelet diffusion coefficient, cm*/sec 13oltzmann’s conatant, g tin* sec2 absolute temperature, “K mean effective platelet radius, cm plasma T. iscosit> , g / em see \
literature Cited
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