Platelet Diffusion in Flowing Blood Vincent T. Turitto,* Anthony M. Benis, and Edward F. Leonard Department of Chemical Engineering, Columbia Zlniversity, A;ew York, N. Y . IOOdY
Platelet diffusivity was measured in flowing blood b y a technique based on the classical experiment of Taylor. Values of diffusivity were calculated from experimental data with the aid of theoretical results obtained b y Ananthakrishnan, et a/.Values ranged between 0.5 and 2.5 X lo-’ cm2/sec for a series of hematocrits ( H = 0-50) and shear rates (f,” = 40-440 sec-I). These values imply significant enhancement of diffusivity due to red cell motion. The experiment was not sensitive enough to detect the dependence of diffusivity on either shear rate or hematocrit, although on the assumption of power-law dependence of diffusivity on shear rate the data indicated a power-law coefficient of less than 0.5. At high shear rates (>200 sec-‘) results were found to be influenced b y nonrandom cell migration; however, upon correction for this migration, these results were consistent with the results obtained a t low shear rate.
T h e occurrence of both cellular and protein reactions a t bloodsurface interfaces presents major difficulties in the employment of artificial orgaiis (Gott, 1968). A proteinaceous layer, probably fibrinogen (Vroman and Adams, 1967, 1969), is initially deposited on the surface, followed by the adhesion and aggregation of platelets and the subsequent release of their constituents (Mustard, 1968). Concurrently, a series of enzymatic reactions involving the plasmatic clotting factors occurs (OllendoriT, 1967). Both platelets and clotting factors interact, in a way not fully understood, to produce the complex mesh of blood cells and polymerized fibrin known as thrombus. The relative importance of the platelet and enzymatic mechanisms may be mediated by the flow dynamics. Since platelet-surface reactions are important in the initial stages of thrombosis, quantitative aspects of both convective and diffusive transport of platelets to the surface must be considered. Whereas molecular diffusion is normally determined by Brownian motion, the diffusion of platelets in flowing blood is thought to be dominated by the complex movement of red cells. The purpose of the present study was to measure the diffusivity of platelets in flowing blood by use of the method introduced by Taylor (1953) for the study of molecular diffusion. This procedure involves measurement of axial platelet dispersion in a tube under laminar flow conditions. The results indicate that platelet diffusion, which is important primarily in the radial direction, is significantly greater than that expected due to Brownian motion alone; however, the experiment was not sensitive enough to determine precisely the effect of hematocrit and shear rate on diffusivity. Method
The apparatus is shown schematically in Figure 1. ii length of polyethylene tubing (diameter 250-500 p , Becton Dickinson and Co.) was looped over an 8-in. diameter disk and was supported in a vertical position. The tubing inlet was connected to a 1-m syringe containing platelet-rich fluid (blood or plasma) agitated by Teflon-coated magnetic beads to prevent sedimentation of the cells. The plunger of the syringe was driven by a multi-speed pump (Harvard Apparatus Co.). The tubing outlet was coiiriected to an automatic platelet counter (Technicon Corp.) operating on a photometric 216
Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
principle; it permitted continuous recording of the platelet count in either whole blood or plasma. Reproducibility of platelet count was within *5%. The procedure was as follows: the tubing was filled with platelet-rich fluid and the syringe was driven to produce a selected flow rate, until the platelet concentration as observed on the recorder was steady. The flow was then stopped; the syringe was replaced with one filled with platelet-poor fluid, aiid the flow was restarted. The platelet concentration was recorded continuously as the platelet-rich fluid was being displaced; this recording will be called the washout curve. After the washout was complete, the tubing was rinsed with water in preparation for reuse. The mean transit time of the fluid in the tubing, defined as the time to fill the tubing a t a given flow rate, was determined with a stopwatch to within 0.1 sec. Blood Preparation
Canine blood was collected in 10-ml evacuated glass tubes prefilled with 0.08 ml of 15% E D T A (ethylenediaminetetraacetate) solution (Vacutainer; Becton, Dickinson and Co.). This reagent complexes calcium ions, thereby blocking enzymatic clotting reactions and inhibiting both platelet adhesion and aggregation. The preparation of four fluids was required from the collected blood : platelet-rich plasma and plateletpoor plasrna for use in the plasma washout experiments, and platelet-rich blood and platelet-poor blood for use in the blood washout experiments. One portion of the blood was centrifuged (International Equipment Co., hIodel CL) a t 1500 rpm for 10 min. The resulting platelet-rich plasma (PRP) and packed red cells were separated. The packed red cells were then adjusted to the desired hematocrit (30 or 50y0) with P R P to obtain platelet-rich blood (PRB). The remaining portion of original blood was centrifuged (hdams Sero-Fuge) a t 3400 rpm for 30 min giving platelet-poor plasma (PPP), packed red cells, arid a thin layer of packed platelets and white cells (the “buffy coat”). After removal of the packed platelets and white cells and separation of the PPP, the red cells were washed four times with saline aiid twice with PPP. The washed red cells were then adjusted with PPI’ to the same hematocrit as the PRB, giving the desired sample of plateletpoor blood (PI’B). Hematocrits were adjusted to within 0.5%,
TUBING
1
0 STRIP CHART RECORDER I
MAGNETIC STIRRER
I
I I
,
COUNT1NG STAGE
-
DILUTING AND LYSING STAGE
-
AUTOMATIC PLATELET
measurements being made by the microhematocrit method without correction for plasma trapping (Chien, et al., 1962). Theory
The diffusivity of spherical particles due to random molecular motioii can be described by the Stokes-Einstein equation (Sutherland, 1905)
D
=
KTI'G~TR~
(1)
where K is Boltzmanu's constant, T is the absolute temperature, p is the solvent viscosity, and R is the particle radius. For a single platelet, assumed to be a sphere of 2-p diameter, iii plasma of viscosity of 0.01 P,eq 1 gives D = 10-9 cm2/sec. This equation is considered applicable since platelets in normal concentrations of 2-5 x 105i'mm3occupy only 0.370 of the plasma by volume. The main objective of the present study was to determine if the complex motioii of the red cells could augment this diffusivity. Both rotational motion of red cells aiid their motion perpendicular to laminar streamlines have been observed in motion pictures made by Goldsmith and this motion has been discussed by Karnis, et al. (1966). It has been verified further that the rate of rotation of red cells is linearly dependent' on shear rate in dilute suspensions, but is retarded by cellular interactions in more concentrated suspensions (Goldsmith and RIason, 1968). Blackshear, et al. (1966), and Bernsteiii, et al. (1967), have used an assumption of linear dependence of red cell diffusivit'y on shear rate to correlate data for hemolysis due to collisioiis of red cells with walls for flow in a tube. Petschek, et al. (1968), and Petschek and Weiss (1970) have assumed that the diffusivity of platelets induced by red cell motioii varies linearly with shear rate in order to predict theoretical platelet adhesion rates in a laminar flow cell. Recently, Colliiigham (1968) measured the increased rate of mass transfer of helium aiid oxygen dissolved in a n aqueous solution due to the presence of neutrally buoyant suspensioiis of polystyrene spheres for laminar flow in a tube. His data correlated well lvith a mathematical model based on a shearrate dependent dsusivity. However, measurements of plate-
let diffusivity in flowing blood have not appeared in the literature. Measurements of diffusion coefficients in flowing liquids were conducted b y Taylor (1953), who flowed a solution of dye into a glass capillary prefilled with clear solvent. The combined effects of the parabolic velocity distribution aiid radial diffusion cause axial dispersion of the dye. Taylor showed theoretically and confirmed experimentally that measurement of dye concentration along the tube length allows calculation of the dye diffusion coefficient. Taylor's analytical solution requires that radial diffusion dominate both convection and axial diffusion. h a n t h a k r i s h n a n , et a/. (1965a), extended Taylor's theory to all flow regimes by solving numerically the complete equation describing diffusion and coiivectioii in a tube
(2)
with boundary conditions
ac
+
~
br
C+(O,x,r) = 0
f 3)
C+(f,O,r) = Co
(4)
C+(t, m ,r)
f 5)
(f,z,O) =
ac
=
+
~
br
0
(t,x,a)
=
0
where C+ is the concentration at any point in the tubing. The solution assumes a homogeneous, Newtonian fluid, as well as constant radial diffusivity. These authors obtained the point values of concentration, C+, and by integration, the velocityaveraged values
(7) as a function of time and distance along the tubing. Ind. Eng. Chem. Fundam., Vol. 11,
No. 2, 1972
217
Since these platelets are displaced by platelet-free fluid, the number of platelets in the washout, as determined by the area under the washout curve, must equal
N
z 0 t-
a a I-
z
w
0
z
8
J
lm C dt
=
Cora2L
This area, A , is a measure of the total amount of platelets originally in the tubing and must be equal to the measured mean transit time for the mass balance to be consistent. The integration was performed numerically by use of Simpson’s 3/8 rule.
FLOW RESUMED,
+
t
Results
=o
C F , FINAL CONCENTRATION cTIME, t
Figure 2. Washout curve for qW = 200 sec-l and H = 50. Recording of platelet concentration vs. time as platelet-poor blood displaces platelet-rich blood. Note that time increases from right to left
An analytical solution for purely convective flow, or negligible radial diffusion, has been derived by Rich and Goldman (1965), as
where Q is the volumetric flow rate, CO is the initial concentration in the tubing and the mean transit time is defined as t, = ?ra2L/’Q
(9)
Combination of eq 8 and 9 gives (10) In the present study the numerical results of Ananthakrishnan, et al. (196513) were interpolated with a two-dimensional LaGrange formula and then used to plot concentrationtime curves corresponding to different diffusivities. These curves approach the “purely conrective curve” calculated from eq 10 as the diffusivity approaches zero. Furthermore, under the flow conditions used in this study theroretical curves calculated for D
