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Flow behavior of neonatal and adult erythrocytes in narrow capillaries
MICROVASCULAR
RESEARCH
37, 267-279 (1989)
Flow Behavior
of Neonatal and Adult Narrow Capillaries
Erythrocytes
in
A. STADLER AND 0. LINDERKAMP Department
of Pediatrics,
Division of Neonatology, University Federal Republic of Germany Received
May
of Heidelberg,
18, 1988
This study was designed to analyze the flow behavior of red blood cells (RBCs) in circular vessels with diameters of 3 to 6 pm by means of a mathematical model. According to this model, the RBC flow velocity is 1 mm/set, RBCs assume axisymmetric shape, and the gap between the RBC and the vessel wall allows sufficient lubrication. The flow resistance depends on the surface area and volume of RBCs, the plasma viscosity, and the vessel diameter. Surface area and volume of RBCs from 10 term neonates and 10 adults were determined by means of a micropipet system and plasma viscosity was measured using a capillary viscometer. Neonatal RBCs had larger volumes (107 ? 6 fl vs 90 2 4 fl) and surface areas (154 ? 7 pm’ vs 137 2 7 pm*) than adult RBCs (P < 0.005). Plasma viscosity was lower in neonates than in adults (1.04 ? 0.10 CP vs 1.26 2 0.13 cP; P < 0.005). The flow model leads to the following conclusions: During the passage of 3- to 6grn vessels, the large neonatal RBCs are more elongated than the smaller adult RBCs. In vessels with diameters of less than 3.3 pm, the rear of neonatal RBCs becomes convex, whereas this critical vessel diameter is 3.1 pm for adult RBCs. If the cells are suspended in the same medium, neonatal RBCs require a 31% higher driving pressure than adult RBCs to achieve the necessary elongation for passing through a narrow capillary. However, both cell types require similar driving pressures, if the cells are suspended in the corresponding plasma. The tube/discharge hematocrit ratio of neonatal RBCs is 1 to 6% higher than that of adult cells. Relative viscosity of neonatal RBCs is approximately 7% higher compared with adult RBCs, whereas the blood viscosity (relative viscosity times plasma viscosity) is 12% less in neonates than in adults. We conclude that the large size of neonatal RBCs may cause impaired flow in narrow vessels with diameters below the critical value of 3.3 pm. In vessels with diameters of 3.3-6.0 pm, the disadvantage of the large size of neonatal RBCs appears to be completely compensated for by the lower plasma viscosity in the neonate. 0 1989 Academic Press, Inc.
INTRODUCTION Hyperviscosity in neonates is a frequent and potentially serious condition, which may cause circulatory failure and impaired microcirculation in various organs (Linderkamp, 1987). Cerebral hemorrhage, renal failure, and necrotizing enterocolitis have been observed in newborn infants with increased blood viscosity. Neonatal hyperviscosity is due mainly to a high hematocrit which in turn may be the result of a large placenta-fetal transfusion, to chronic intrauterine hypoxia, or to hyperinsulinism (Linderkamp, 1987). However, decreased red blood cell (RBC) deformability, increased RBC aggregation, and increased plasma viscosity can also contribute to an increase in blood viscosity in the neonate. 267 G026-286X39 $3.00 Copyright Q 1989 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.
268
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Blood flow in capillaries with diameters of 3 to 6 pm is little influenced by the hematocrit because red blood cells flow in a single file (Secomb and Skalak, 1982; Secomb and Gross, 1983; Secomb et al., 1986). Blood flow in narrow capillaries is mainly dependent on RBC deformability (Gaehtgens, 1981), RBC volume (Linderkamp et al., 1986b), plasma viscosity (Papenfuss and Gross, 1977), and the driving pressure (Secomb and Gross, 1983; Secomb et al., 1986). Cellular deformability of neonatal and adult RBCs has been studied by several methods. Neonatal and adult RBCs show no difference in deformability when studied under defined shear forces by means of a rheoscope (Linderkamp et al., 1986) or an ektacytometer (Coulombel et al., 1982). However, neonatal RBCs are less filterable (Linderkamp, et al., 1986a) and require higher aspiration pressures to enter micropipets with a diameter of 3.3 pm (Linderkamp et al., 1986b) than adult cells because of the larger volume of neonatal RBC. This suggests impaired microcirculatory dynamics of neonatal RBCs in narrow capillaries. On the other hand, plasma viscosity in the neonate is markedly less than in the adult as a result of the low total plasma protein and low fibrinogen concentration in the neonate (Linderkamp et al., 1984). It is unknown which role the large RBC size and the low plasma viscosity play for the blood flow in narrow capillaries of the neonate. Secomb and Gross (1983) and Secomb et al. (1986) developed a three-dimensional model to describe blood flow in narrow capillaries with internal diameters of 3 to 6 pm. The model considers RBC surface area and volume, plasma viscosity, and capillary diameter. It assumes that RBCs have an axisymmetric geometry and move with a velocity of 1 mm/set. The model uses lubrication theory to characterize the plasma flow in the narrow gap between the RBC and the vessel wall. The present study was designed to measure surface area and volume of neonatal and adult RBCs and to determine viscosity of plasma from newborn infants and adults. The results were used to analyze the flow behavior of RBCs and of plasma in narrow capillaries by means of the model developed by Secomb and Gross (1983) and Secomb et al. (1986). Moreover, this model was used to determine the pressure gradient and the hematocrit reduction (Fahraeus effect) and the viscosity for neonatal and adult blood in narrow capillaries.
THEORY In vessels with diameters of more than 500 pm, blood flow depends principally on factors summarized in the law of Hagen-Poiseuille. In capillaries with diameters of less than 500 pm, additional effects such as the Fahraeus (Fahraeus, 1929) and the Fahraeus-Lindquist effect (Fahraeus and Lindqvist, 193 1) play an important role for blood flow. In capillaries of less than 10 pm in diameter, RBCs are more and more pressed against the capillary wall and the surrounding plasma acts as a lubricant between the RBC and the capillary wall (lubrication theory). In capillaries where the erythrocytes can move relatively freely, the RBC membrane shows the so-called “tank-treading” motion (Schmid-Schonbein and Wells, 1968). In narrow vessels with diameters less than that of resting RBCs, the cells have to flow in a single tile without tank-treading and are deformed to an axisymmetrical or so-called “projectile” shape (Ehrly, 1975). Cell flow in these narrow capillaries
FLOW
BEHAVIOR
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269
can be analyzed by a three-dimensional model which is based on the theory of Secomb and Gross (1983) and Secomb er al. (1986). In this model the following assumptions are made: 1. The vessel diameter ranges from 3 to 6 pm. 2. The ratio of vessel radius to gap width (b/h) is between 5 and 14 when the diameter decreases from 6 to 3 pm. The gap width is determined by the model independent of the cell characteristics (Fig. 2). 3. The membrane stress can be approximated by an isotropic tension (T). Near the front of the cell, T is constant; at the concave trailing end of the cell T has decreased to zero. Membrane shear and bending resistance are neglected. 4. The driving pressure (P) is zero in front of the cell and constant directly behind the cell. 5. Pressure changes perpendicular to the direction of blood flow are negligible. 6. Cell flow velocity (u) is 1 mm/set. At lower velocities, shear and bending resistance become more important and lead to changes in cell geometry and increased relative viscosity. 7. Surface area (A) and volume (V) of a RBC are constant since the membrane can dilate by only 2% (Evans et al., 1976) and the cytoplasm is an incompressible fluid. 8. The apparent viscosity varies linearly with the tube hematocrit. 9. Surfa,ce area and volume of RBC and the viscosity of the suspending are known.
medium
As a result of these conditions a system of four nonlinear differential equations with the unknowns h (gap width), 8 (angle between arc length and vessel wall), P, and T has been derived. These four unknowns depend on the arc length (a’s). The boundary conditions are 0 = rr/2 and h = b when s = 0 and 8 + 0 as s + 00 downstream. In this model, the leakback (q) represents the flow rate per unit circumference in the gap relative to the cell. For the computation powerlaw relationships were fitted to approximate the numerical results from the four nonlinear differential equations for h, 0, p, and T (Secomb and Gross, 1983; Secomb et al., 1986),
kz 1 -=k,; b 0 bp=k pu
e!’ ‘b0
(1) (2) (3)
where k, = 1.22, k2 = 0.82, k3 = 2.59, k4 = 1.26, and u = 1 mm/set. When surface area (A) and volume (V) are given, there is a numerical approximation which determines the geometrical parameters (r, 1,, rl, 1), the driving pressure (P) and the pressure gradient (P/L).
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FIG. 1. Axisymmetrical model for a red blood cell (RBC) in a narrow capillary. 1 = length of a RBC; I, = depth of the rear concavity; r = radius of a RBC; r, = radius of the rear concavity; h = gap width between cell and vessel wall; b = radius of the capillary; s = arc length; 19 = angle between s and vessel wall.
Radius of the red blood cell: (b/h may vary between 5 and 14)
r=b-h
(4)
Depth of the rear concavity: 1 = 3 3(rA - 2v) 1 J
T
- 4r3 - r
(5)
Radius of the rear concavity: rz + 1; r’ = 21,
(6)
Length of the red blood cell: (7) Driving pressure: see Eq. (2) Pressure gradient: P/L
= ;
(8)
Surface area (A) and volume (V) of single RBCs and plasma viscosity (/A) have been measured for neonates and adults. Equations (4) to (8) can be solved numerically using the measured A, V, and p. This leads to the geometrical parameters, the driving pressure, and the pressure gradient as functions of the vessel diameter, as shown in Figs. 2 to 4. Figure 2 shows that in very narrow vessels, the depth of the rear concavity (I,) approaches zero (i.e., the cell assumes a flat rear surface) and the diameter of the cell assumes a critical value (critical diameter, D,). With a further decrease of the capillary diameter, the cell rear becomes convex and the cell diameter approaches the minimum cylindrical diameter
FLOW
BEHAVIOR
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271
CELLS
(D,). The gap between the cell and the vessel wall approaches zero. The cell loses its deformability and behaves like a rigid body. Without a sufficient layer of plasma there is marked friction between the cell and the vessel wall and the occurring phenomena cannot be explained by lubrication theory alone. The tube to discharge hematocrit ratio (&/H,,) can be deduced from estimates of the leakback (Eq. (3)) due to principles of conservation:
-=*2!LlIA HT
bu
HD
HT is assumed to be 0.45 l/l
(9)
k,l’
(i.e., 45%). Thus HT = 0.45 (HT/HD).
(10)
For calculation of the relative viscosity (Q,,) it is assumed that the apparent viscosity varies linearly with the tube hematocrit: %I
=
1 +
KTHT-
The constant KT represents the so called “apparent and Gross, 1983; Secomb et al., 1986): KT = ?!!!
P b2
(11)
intrinsic
-1
V [ Q-h - &lb) rrb= k312 =v [ 8b(HT/HD) - ’I ’
viscosity”
1
The viscosity of RBC in plasma suspensions has been calculated relative viscosity (nr,,) and plasma viscosity (p): 9 = %I. MATERIALS
P*
(Secomb
(12)
as product of (13)
AND METHODS
Placental blood samples from 10 healthy full-term newborn infants withgestational ages of 39-40 weeks and birth weights of 3310-3580 g were studied with approval of the Department of Pediatrics Human Subjects Research Committee. All infants had birth weights appropriate for their gestational age. Immediately after cord clamping and before delivery of the placenta, 10 ml of blood was collected into EDTA (1 mg/ml). Adult blood samples were collected from 10 healthy hospital personnel via venipuncture into EDTA. All measurements were made within 4 hr after collection (Linderkamp et al., 1986b). RBCs were isolated by centrifugation at 2000g for 10 min and, via gentle aspiration, the plasma was removed and the buffy coat discarded. Part of the isolated plasma was used for the measurement of plasma viscosity, total plasma protein, and fibrinogen concentration. Another part of the plasma was centrifuged a second time at 20,OOOg for 15 min to remove platelets. The isolated RBCs were suspended at a hematocrit of 0.1% in phosphate-buffered saline (PBS: 0.005 mole/liter KH2P04 + Na2HP0,, 0.153 mole/liter NaCl, pH 7.40, 300 mOsm/kg) containing 1 g/liter human serum albumin.
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Geometric properties of RBCs were measured by direct microscopic observations of RBCs with a micropipet system which consists of a microscope (Leitz, Wetzlar FRG, Model Diavert), a video system (Panasonic, Osaka, Japan), and a micromanipulator (De Fonbrune, Paris, France) (Linderkamp and Meiselman, 1982). One neonatal and one adult blood sample were studied on the same day using the same micropipet; 40 RBCs were tested from each donor. Micropipets with diameters of 1.9-2.1 pm were filled with platelet-depleted plasma of the donor and then flushed with the albuminated PBS used to suspend the cells. This procedure prevents RBC adhesion to the glass walls of the pipet. After the pressure in the pipet was brought to zero, a RBC was aspirated by adjusing the pressure to - 15 mm H20. One portion of the cell entered the pipet assuming a cylindrical shape, while the outer part became spherical. The surface area (A) and volume (V) were calculated from the diameter of the pipet (d,,), the length of the cell projection in the pipet (1,), and the diameter of the outer (spherical) portion (d,): (14) V = ~(6~,d2,
+ 4d3, - d3,).
(15)
These formulas do not consider the segment of the outer sphere, which is “cut off” by the tip of the pipet. This results in an error of less than 1% for surface area and volume. The resting diameter (D) of each RBC was measured before the cells were aspirated. The following parameters were calculated from the surface area (A), volume (V), and diameter (D) of the RBC: Mean thickness
= $
(16)
A
Surface area index =
.
(17)
4.84$? The minimum cylindrical diameter (D,) was calculated from the surface and volume of RBCs using equations for a cylinder with spherical end caps: A = ?rD,L v= Ehminating
n-D2L L+ 4
+ z-D’,,,
(18)
& 6
’
(19)
L yields v=>-dAD 4
rrD3 12 .
(20)
Red blood cell counts and mean corpuscular volumes (MCV) were measured electronically (Contraves, Zurich, Switzerland). Hemoglobin concentrations were measured spectrophotometrically by the cyanmethemoglobin method. Mean corpuscular hemoglobin (MCH) was calculated as ratio of hemoglobin concentration
FLOW
BEHAVIOR
OF RED
BLOOD
CELLS
273
and the RBC count. The mean corpuscular hemoglobin concentration (MCHC) is the MCH divided by MCV. Plasma. viscosity was measured by means of a capillary viscometer at 37” (Linderkamp et al., 1981). A glass tube of approximately 100 pm i.d. and 10 mm in length was connected to compressed air (pressure of 25 cm H20). The passage time for 0.2 ml plasma and of distilled water was measured. Plasma viscosity was calculated as the ratio of the passage times of the plasma to that of distilled water multiplied by both 0.6974 CP (viscosity of water at 37”) and by the ratio of the density of the sample to that of distilled water. Plasma densities were derived from nomograms (Van Slyke et al., 1950) using the measured plasma protein concentrations. One neonatal and one adult plasma sample were studied on the same day using the same tube. Total plasma protein concentration was measured by the Biuret test. Plasma fibrinogen concentration was determined via a radial-immunodiffusion technique (M - Partigen Fibrinogen Kit, Behring, Marburg, Germany).
RESULTS Table 1 presents the measured and calculated data for neonatal and adult blood. Compared to adult RBCs, the volume of neonatal cells was 18.6% larger, their surface was 12.0% greater, and their resting diameter was 11.4% wider. The minimum cylindrical diameter (D,) of neonatal RBCs was increased compared to adult cells. The plasma viscosity was significantly decreased in the neonates compared to the adults. The decreased viscosity of neonatal plasma was associated with decreased total plasma protein and fibrinogen concentrations. Figure 2 shows the geometrical parameters calculated for neonatal and adult RBCs as they flow through capillaries with diameters of 6 to 3 pm. The cell length increases and the cell radius decreases with decreasing vessel diameter. The cell length of neonatal RBCs is increased by approximately 14% compared to adult Icells over the whole range of vessel diameters (Table 2). The depth of the rear concavity (1,) decreases as the vessel diameter decreases and reaches zero at a defined vessel diameter, i.e., the critical vessel diameter, 2b,. In vessels with diameters above 4.6 pm, neonatal RBCs have a larger I, than adult cells. In smaller vessels, neonatal RBCs have a smaller 1, than adult cells. The critical vessel diameter where I, becomes zero is 3.28 pm for neonatal RBCs and 3.09 pm for adult cells. In these critical vessels, the corresponding cell diameter is 3.03 pm for the average neonatal RBC and 2.86 pm for the average adult cell, In Figs. 2 and 3 the driving pressure (P) is shown for a suspension viscosity of 0.7 cP. It is 31% larger for neonatal RBCs than for adult cells over the whole range of vessel diameters. Figure 3 also shows the driving pressure for neonatal and adult RBCs by using the measured plasma viscosities. The driving pressures for neonatal RBCs are now only 8% higher than those of adult cells (Table 2). Figure 4 shows the pressure gradients calculated as the ratio of the driving pressure to the cell length (P/L) in vessels of various diameters. If neonatal and adult RBCs are suspended in alb minated (0.1%) PBS with a viscosity of 0.7 cP, the calculated pressure gradien r s are 14% larger for neonatal RBCs compared
274
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TABLE RED
BLOOD
CELL
PARAMETERS
1 AND
PLASMA
VISCOSITY
Neonatal RBCs Geometric properties of RBC 105.7 t (fr Mean hemoglobin concentration (pg) 35.0 -t 32.9 k Mean corpuscular hemoglobin concentration (g/dl) 8.8 t Diameter (pm) (” 107.3 +Volume of aspirated RBCs (fl)
0.04* 0.08)t 0.05 0.11) 0.10 0.26) 0.14* 0.27)
89.5 t (2 29.8 + 33.2 iz 7.9 k (k 90.5 + (k 137.1 k (-c 1.51 k (-t 1.41 + (2 1.84 ” (? 2.83 +(k
3.9 14.0) 1.5 1.3 0.4 0.7) 4.4 13.2) 6.7 14.8) 0.04 0.06) 0.04 0.09) 0.13 0.28) 0.14 0.25)
Plasma viscosity and proteins 1.04 k 0.10* 5.6 + 0.7* 0.279 2 0.052t
1.26 ? 7.0 + 0.362 -t
0.13 0.8 0.061
Mean corpuscular volume (fl)
(-t
153.5 2 (2 1.43 k (” 1.41 -’ (!I 1.76 + (2 3.00 2 (2
RBC surface area (ym) Surface area/volume Surface area index Mean RBC thickness (pm) Minimum cylindrical diameter (ym)
Plasma viscosity (cP) Total plasma protein (g/d]) Plasma fibrinogen (g/d])
Adult RBCs
5.1* 24.9)t
1.s* 1.4 0.4” 0.8)t
5.6* 20.2)t
7.0* 20.7jt
Note. Neonatal and adult samples were studied. Data represent means ? 1 SD. SD reflects the spread of the sample means, whereas the values in parentheses of the entire RBC populations studied in each group. * P < 0.005, when compared with adult RBCs (two-tailed unpaired t test, 2n t P < 0.05 when compared with adult RBCs (two-tailed unpaired f test, 2n -
Note that the first reflect the spread - 2 df). 2 dj).
to adult cells. However, the use of the actual plasma viscosities result in slightly smaller pressure gradients for neonatal RBCs. Figure 5 shows the calculated ratio of tube hematocrit to discharge hematocrit (&I&) and Fig. 6 shows the calculated relative viscosity as a function of the vessel diameter. Both the tube hematocrit (Fig. 5) and the relative viscosity (Fig. 6) of neonatal and adult RBC suspensions increase when going from a 6-pm vessel to a 3-pm vessel. In vessels below the critical diameter, tube hematocrit and relative viscosity rise steeply. The H,/H,, ratio of neonatal RBCs is 1% (in 3.3~pm vessels) to 6% (in 6-pm vessels) higher than that of adult RBCs. Relative viscosity of neonatal RBCs is 5.5 to 8.4% higher compared with adult RBCs. However, the calculated blood viscosity (product of relative viscosity and plasma viscosity) is 11 to 13% less in the neonates than in adults (Fig. 6). DISCUSSION This study demonstrates that neonatal RBCs require a higher driving pressure than adult RBCs for passing through a narrow vessel with a given diameter at
FLOW BEHAVIOR OF RED BLOOD CELLS r(pm) I, (yml
Adult Neonatal
RBC RBC
-
-
-
-
275
-
P(mmHg1
-“5
VESSEL
DIAMETER
(pm)
FIG. 2. Geometrical parameters (r, I, I,) and driving pressure (P) of neonatal and adult RBCs as functions of the vessel diameter (suspending medium viscosity, F = 0.7 cP). 26, = critical vessel diameter.
CALCULATED
FLOW
PROPERTIES
OF NEONATAL WITH DIAMETERS
TABLE 2 (N) AND ADULT (A) RED OF 3.3 pm AND 6 pm
BLOOD
CELLS
3.3~pm vessel Parameter RBC length (pm) Driving pressure (mm Hgl Pressure gradient (mm Hg * lO’/pm) Tube/discharge hematocrit ratio Relative viscosity Blood viscosity (cP)
IN VESSELS
6-pm vessel
Suspending medium
N
A
N
A
B,P B P B P
15.26 0.705 1.048 46.21 68.65
13.35 0.539 0.971 40.40 72.72
8.05 0.033 0.049 4.06 6.03
7.05 0.025 0.045 3.55 6.39
BJ’ W’ B P
0.914 2.139 1.50 2.22
0.902 1.974 1.38 2.49
0.704 1.157 0.81 1.20
0.602 1.097 0.77 1.38
Note. The flow properties were calculated for different suspending media: B, PBS solution (/J = 0.7 cP); P, plasma (CL = 1.04 CP for neonatal and 1.26 CP for adult plasma).
276
STADLER AND LINDERKAMP
1.2
0.;
FIG. 3. Driving pressure (P) of neonatal and adult RBCs as functions of the vessel diameter. Driving pressures were calculated for RBCs suspended in buffer solution (CL= 0.7 cP) and in plasma with viscosities of 1.04 CP (neonatal plasma) and 1.26 CP (adult plasma), respectively. 2b, = critical vessel diameter.
a given speed if the cells are suspended in the same medium (Fig. 3). However, both cell types require similar driving pressures if neonatal RBCs are suspended in neonatal plasma and adult RBCs are suspended in adult plasma. Thus, the lower plasma viscosity in the neonate outweighs the adverse effect of the large size of neonatal RBCs for blood flow in 3- to 6-pm vessels (Fig. 4). This finding indicates that blood flow in narrow vessels of the neonate is not hindered by the large size of their RBCs. However, when the vessel diameter decreases below the critical value (i.e., 3.3 pm for the average neonatal RBC and 3.1 pm for the average adult RBC), the cells have to assume the shape of a cylinder with hemispherical caps at both ends. These cells lose their deformability and require extremely high driving pressures (Secomb and Gross, 1983; Secomb et al., 1986). In tubes, the flow velocity increases from the tube wall to the axis. Thus, the plasma layer on the wall flows slower than the central cell core of the tube. Since the more rapidly flowing cells leave the tube faster than the plasma, the tube hematocrit decreases (Fahraeus effect). The decrease in tube hematocrit depends on the ratio of the plasma layer to the tube diameter. The ratio of the
FLOW
BEHAVIOR
OF RED
BLOOD
277
CELLS
i \ 3-
\ \ 3 Adult
RBC
(p = 1.26
cP)
I-
1
I
1
4 VESSEL
I
5 DIAMETER
I
i
(wn)
FIG. 4. Pressure gradient (pressure divided by cell length) as function of the vessel diameter. The pressure gradients are shown for neonatal and adult red blood cells (RBCs) suspended in buffer solution (viscosity, p = 0.7 cP), for neonatal RBCs suspended in neonatal plasma (CL = 1.04 cP), and for adult RBCs suspended in adult plasma (/.L = 1.26 cP).
plasma layer to the tube diameter increases and the hematocrit decreases when the tube diameter is decreased from 500 pm to about 8 pm (i.e., the resting RBC diameter). A further decrease in tube diameter causes a decrease in the ratio of the plasma layer to the tube diameter and the hematocrit increases (inverse Fahraeus effect, see Fig. 5). In a narrow tube with a given diameter and gap between cell and vessel wall, the hematocrit depends on the RBC length (Eq. (9)). This explains the higher hematocrit of the larger neonatal RBCs compared to adult cells in narrow tubes (Fig. 5). The increase in hematocrit results in increased relative viscosity of neonatal
278
STADLER
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LINDERKAMP
l.O-
0.6 -
VESSEL
DIAMETER
(,um\
FIG. 5. Tube/discharge hematocrit ratio (I&/H,) of neonatal and adult red blood suspensions as functions of the vessel diameter. D, = critical cell diameter, 2b, = critical vessel diameter.
2.5 3 9 c z -%% iis 55 g 8
2.0-
1.5-
i=O 4iii E l.O-
3
*b,
f VESSEL
, DIAMETER
&
FIG. 6. Relative viscosity (q,J and blood viscosity (7) of neonatal and adult blood as functions of vessel diameter. 2b, = critical vessel diameter.
FLOW
BEHAVIOR
OF
RED
BLOOD
CELLS
279
RBCs compared to adult cells if both cell types have been suspended in the same medium (Fig. 6). However, if neonatal RBCs are suspended in neonatal plasma and adult RBCs are suspended in adult plasma, viscosity of neonatal blood is decreased compared with adult blood (Fig. 6). We conclude from these results that the large size of neonatal RBCs may be associated with impaired flow in narrow vessels with diameters below the critical diameter (3.3 pm). However, in vessels with diameters of 3.3-6.0 pm, the disadvantage of the large size of neonatal RBCs appears to be completely compensated for by the lower plasma viscosity in the neonate. ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeinschaft (Research Grant Li 291/4). We thank Thomas Behler, M.D., and Thomas Weiss, BSc., for their invaluable advice. We are grateful to Mrs. Liselotte Kriiger for secretarial assistance and Vivian M. Vargas, M.S.B.A., for help in preparing the manuscript.
REFERENCES L., TCHERNIA, G., FEO, C., AND MOHANDAS, N. (1982). Echinocytic sensitivity and deformability of human newborn red cells. Biol. Neonare 42, 284-290. EHRLY, A. M. (1975). Veranderungen der FlieBeigenschaften des Blutes und der Erythrozyten bei arteriellen VerschlulJkrankheiten. Med. Welt 26, 1971-1972. EVANS, E. ,4., WAUGH, R., AND MELNIK, L. (1976). Elastic area compressibility modulus of red cell membrane. Biophys. J. 16, 585-595. FAHRAEUS, R. (1929). The suspension-stability of blood. Physiol. Rev. 9, 241-274. FAHRAEUS, R., AND LINDQVIST, T. (1931). The viscosity of blood in narrow tubes. Amer. J. Physiol. COULOMBEL,
96, 562-586.
GAEHTGENS, P. (1981). Distribution of flow and red cell flux in the microcirculation. Stand. J. Clin. Invest. 4:L(Suppl. 156), 83-87. LINDERKAMP,0. (1987). Blood rheology in the newborn infant. Zn “Bailliere’s Clinical Haematology” (G. D. 0. Lowe, Ed.), Vol. 1, pp. 801-825. Bailliere Trudall, London. LINDERKAMP, O., AND MEISELMAN, H. J. (1982). Geommetric, osmotic, and membrane mechanical properties of density-separated human red cells. Blood 59, 1121-l 127. LINDERKAMP, O., MEISELMAN, H. J., Wu, P. Y. K., AND MILLER, F. C. (1981). Blood and plasma viscosity and optimal hematocrit in the normal newborn infant. C/in. Hemorheol. 1, 575-584. LINDERKAMP, O., HAMMER, B. J., AND MILLER, R. (1986a). Filterability of erythrocytes and whole blood in preterm and full-term neonates and adults. Pediatr. Res. 20, 1269-1273. LINDERKAMP, O., NASH, G. B., Wu, P. W. Y., AND MEISELMAN, H. J. (1986b). Deformability and intrinsic material properties of neonatal red blood cells. Blood 67, 1244-1250. LINDERKAMP, O., VERSMOLD, H. T., RIEGEL, K. P., AND BETKE, K. (1984). Contributions of red cells and plasma to blood viscosity in preterm and full-term infants and adults. Pediatrics 74, 45-51. PAPENFUSS, H. D., AND GROSS,J. F. (1977). The interaction between transmural fluid exchange and blood viscosity in narrow blood vessels. Biorheology 14, 217-228. SCHMID-SCII~NBEIN, H., AND WELLS, R. E. (1968). Fluid drop-like transition of erythrocytes under shear. Science 165, 288-291. SECOMB, T. W., AND GROSS,J. F. (1983). Flow of red blood cells in narrow capillaries: Role of membrane tension. Znt. J. Microcirc. Clin. Exp. 2, 229-240. SECOMB, T. W., AND SKALAK, R. (1982). A two dimensional model for capillary flow of an axisymmetric cell. Microvasc. Res. 24, 194-203. SECOMB, T. W., SKALAK, R., &KAYA, N., AND GROSS,J. F. (1986). Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405-423. VAN SLYKE, D. D., PHILLIPS, R. A., DOLE, V. P., HAMILTON, P. B., ARCHIBALD, R. M., AND PLAZIN, J. (1950). Calculation of hemoglobin from blood specific gravities. J. Biol. Chem. 183, 349-360.
RESEARCH
37, 267-279 (1989)
Flow Behavior
of Neonatal and Adult Narrow Capillaries
Erythrocytes
in
A. STADLER AND 0. LINDERKAMP Department
of Pediatrics,
Division of Neonatology, University Federal Republic of Germany Received
May
of Heidelberg,
18, 1988
This study was designed to analyze the flow behavior of red blood cells (RBCs) in circular vessels with diameters of 3 to 6 pm by means of a mathematical model. According to this model, the RBC flow velocity is 1 mm/set, RBCs assume axisymmetric shape, and the gap between the RBC and the vessel wall allows sufficient lubrication. The flow resistance depends on the surface area and volume of RBCs, the plasma viscosity, and the vessel diameter. Surface area and volume of RBCs from 10 term neonates and 10 adults were determined by means of a micropipet system and plasma viscosity was measured using a capillary viscometer. Neonatal RBCs had larger volumes (107 ? 6 fl vs 90 2 4 fl) and surface areas (154 ? 7 pm’ vs 137 2 7 pm*) than adult RBCs (P < 0.005). Plasma viscosity was lower in neonates than in adults (1.04 ? 0.10 CP vs 1.26 2 0.13 cP; P < 0.005). The flow model leads to the following conclusions: During the passage of 3- to 6grn vessels, the large neonatal RBCs are more elongated than the smaller adult RBCs. In vessels with diameters of less than 3.3 pm, the rear of neonatal RBCs becomes convex, whereas this critical vessel diameter is 3.1 pm for adult RBCs. If the cells are suspended in the same medium, neonatal RBCs require a 31% higher driving pressure than adult RBCs to achieve the necessary elongation for passing through a narrow capillary. However, both cell types require similar driving pressures, if the cells are suspended in the corresponding plasma. The tube/discharge hematocrit ratio of neonatal RBCs is 1 to 6% higher than that of adult cells. Relative viscosity of neonatal RBCs is approximately 7% higher compared with adult RBCs, whereas the blood viscosity (relative viscosity times plasma viscosity) is 12% less in neonates than in adults. We conclude that the large size of neonatal RBCs may cause impaired flow in narrow vessels with diameters below the critical value of 3.3 pm. In vessels with diameters of 3.3-6.0 pm, the disadvantage of the large size of neonatal RBCs appears to be completely compensated for by the lower plasma viscosity in the neonate. 0 1989 Academic Press, Inc.
INTRODUCTION Hyperviscosity in neonates is a frequent and potentially serious condition, which may cause circulatory failure and impaired microcirculation in various organs (Linderkamp, 1987). Cerebral hemorrhage, renal failure, and necrotizing enterocolitis have been observed in newborn infants with increased blood viscosity. Neonatal hyperviscosity is due mainly to a high hematocrit which in turn may be the result of a large placenta-fetal transfusion, to chronic intrauterine hypoxia, or to hyperinsulinism (Linderkamp, 1987). However, decreased red blood cell (RBC) deformability, increased RBC aggregation, and increased plasma viscosity can also contribute to an increase in blood viscosity in the neonate. 267 G026-286X39 $3.00 Copyright Q 1989 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.
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Blood flow in capillaries with diameters of 3 to 6 pm is little influenced by the hematocrit because red blood cells flow in a single file (Secomb and Skalak, 1982; Secomb and Gross, 1983; Secomb et al., 1986). Blood flow in narrow capillaries is mainly dependent on RBC deformability (Gaehtgens, 1981), RBC volume (Linderkamp et al., 1986b), plasma viscosity (Papenfuss and Gross, 1977), and the driving pressure (Secomb and Gross, 1983; Secomb et al., 1986). Cellular deformability of neonatal and adult RBCs has been studied by several methods. Neonatal and adult RBCs show no difference in deformability when studied under defined shear forces by means of a rheoscope (Linderkamp et al., 1986) or an ektacytometer (Coulombel et al., 1982). However, neonatal RBCs are less filterable (Linderkamp, et al., 1986a) and require higher aspiration pressures to enter micropipets with a diameter of 3.3 pm (Linderkamp et al., 1986b) than adult cells because of the larger volume of neonatal RBC. This suggests impaired microcirculatory dynamics of neonatal RBCs in narrow capillaries. On the other hand, plasma viscosity in the neonate is markedly less than in the adult as a result of the low total plasma protein and low fibrinogen concentration in the neonate (Linderkamp et al., 1984). It is unknown which role the large RBC size and the low plasma viscosity play for the blood flow in narrow capillaries of the neonate. Secomb and Gross (1983) and Secomb et al. (1986) developed a three-dimensional model to describe blood flow in narrow capillaries with internal diameters of 3 to 6 pm. The model considers RBC surface area and volume, plasma viscosity, and capillary diameter. It assumes that RBCs have an axisymmetric geometry and move with a velocity of 1 mm/set. The model uses lubrication theory to characterize the plasma flow in the narrow gap between the RBC and the vessel wall. The present study was designed to measure surface area and volume of neonatal and adult RBCs and to determine viscosity of plasma from newborn infants and adults. The results were used to analyze the flow behavior of RBCs and of plasma in narrow capillaries by means of the model developed by Secomb and Gross (1983) and Secomb et al. (1986). Moreover, this model was used to determine the pressure gradient and the hematocrit reduction (Fahraeus effect) and the viscosity for neonatal and adult blood in narrow capillaries.
THEORY In vessels with diameters of more than 500 pm, blood flow depends principally on factors summarized in the law of Hagen-Poiseuille. In capillaries with diameters of less than 500 pm, additional effects such as the Fahraeus (Fahraeus, 1929) and the Fahraeus-Lindquist effect (Fahraeus and Lindqvist, 193 1) play an important role for blood flow. In capillaries of less than 10 pm in diameter, RBCs are more and more pressed against the capillary wall and the surrounding plasma acts as a lubricant between the RBC and the capillary wall (lubrication theory). In capillaries where the erythrocytes can move relatively freely, the RBC membrane shows the so-called “tank-treading” motion (Schmid-Schonbein and Wells, 1968). In narrow vessels with diameters less than that of resting RBCs, the cells have to flow in a single tile without tank-treading and are deformed to an axisymmetrical or so-called “projectile” shape (Ehrly, 1975). Cell flow in these narrow capillaries
FLOW
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269
can be analyzed by a three-dimensional model which is based on the theory of Secomb and Gross (1983) and Secomb er al. (1986). In this model the following assumptions are made: 1. The vessel diameter ranges from 3 to 6 pm. 2. The ratio of vessel radius to gap width (b/h) is between 5 and 14 when the diameter decreases from 6 to 3 pm. The gap width is determined by the model independent of the cell characteristics (Fig. 2). 3. The membrane stress can be approximated by an isotropic tension (T). Near the front of the cell, T is constant; at the concave trailing end of the cell T has decreased to zero. Membrane shear and bending resistance are neglected. 4. The driving pressure (P) is zero in front of the cell and constant directly behind the cell. 5. Pressure changes perpendicular to the direction of blood flow are negligible. 6. Cell flow velocity (u) is 1 mm/set. At lower velocities, shear and bending resistance become more important and lead to changes in cell geometry and increased relative viscosity. 7. Surface area (A) and volume (V) of a RBC are constant since the membrane can dilate by only 2% (Evans et al., 1976) and the cytoplasm is an incompressible fluid. 8. The apparent viscosity varies linearly with the tube hematocrit. 9. Surfa,ce area and volume of RBC and the viscosity of the suspending are known.
medium
As a result of these conditions a system of four nonlinear differential equations with the unknowns h (gap width), 8 (angle between arc length and vessel wall), P, and T has been derived. These four unknowns depend on the arc length (a’s). The boundary conditions are 0 = rr/2 and h = b when s = 0 and 8 + 0 as s + 00 downstream. In this model, the leakback (q) represents the flow rate per unit circumference in the gap relative to the cell. For the computation powerlaw relationships were fitted to approximate the numerical results from the four nonlinear differential equations for h, 0, p, and T (Secomb and Gross, 1983; Secomb et al., 1986),
kz 1 -=k,; b 0 bp=k pu
e!’ ‘b0
(1) (2) (3)
where k, = 1.22, k2 = 0.82, k3 = 2.59, k4 = 1.26, and u = 1 mm/set. When surface area (A) and volume (V) are given, there is a numerical approximation which determines the geometrical parameters (r, 1,, rl, 1), the driving pressure (P) and the pressure gradient (P/L).
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FIG. 1. Axisymmetrical model for a red blood cell (RBC) in a narrow capillary. 1 = length of a RBC; I, = depth of the rear concavity; r = radius of a RBC; r, = radius of the rear concavity; h = gap width between cell and vessel wall; b = radius of the capillary; s = arc length; 19 = angle between s and vessel wall.
Radius of the red blood cell: (b/h may vary between 5 and 14)
r=b-h
(4)
Depth of the rear concavity: 1 = 3 3(rA - 2v) 1 J
T
- 4r3 - r
(5)
Radius of the rear concavity: rz + 1; r’ = 21,
(6)
Length of the red blood cell: (7) Driving pressure: see Eq. (2) Pressure gradient: P/L
= ;
(8)
Surface area (A) and volume (V) of single RBCs and plasma viscosity (/A) have been measured for neonates and adults. Equations (4) to (8) can be solved numerically using the measured A, V, and p. This leads to the geometrical parameters, the driving pressure, and the pressure gradient as functions of the vessel diameter, as shown in Figs. 2 to 4. Figure 2 shows that in very narrow vessels, the depth of the rear concavity (I,) approaches zero (i.e., the cell assumes a flat rear surface) and the diameter of the cell assumes a critical value (critical diameter, D,). With a further decrease of the capillary diameter, the cell rear becomes convex and the cell diameter approaches the minimum cylindrical diameter
FLOW
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271
CELLS
(D,). The gap between the cell and the vessel wall approaches zero. The cell loses its deformability and behaves like a rigid body. Without a sufficient layer of plasma there is marked friction between the cell and the vessel wall and the occurring phenomena cannot be explained by lubrication theory alone. The tube to discharge hematocrit ratio (&/H,,) can be deduced from estimates of the leakback (Eq. (3)) due to principles of conservation:
-=*2!LlIA HT
bu
HD
HT is assumed to be 0.45 l/l
(9)
k,l’
(i.e., 45%). Thus HT = 0.45 (HT/HD).
(10)
For calculation of the relative viscosity (Q,,) it is assumed that the apparent viscosity varies linearly with the tube hematocrit: %I
=
1 +
KTHT-
The constant KT represents the so called “apparent and Gross, 1983; Secomb et al., 1986): KT = ?!!!
P b2
(11)
intrinsic
-1
V [ Q-h - &lb) rrb= k312 =v [ 8b(HT/HD) - ’I ’
viscosity”
1
The viscosity of RBC in plasma suspensions has been calculated relative viscosity (nr,,) and plasma viscosity (p): 9 = %I. MATERIALS
P*
(Secomb
(12)
as product of (13)
AND METHODS
Placental blood samples from 10 healthy full-term newborn infants withgestational ages of 39-40 weeks and birth weights of 3310-3580 g were studied with approval of the Department of Pediatrics Human Subjects Research Committee. All infants had birth weights appropriate for their gestational age. Immediately after cord clamping and before delivery of the placenta, 10 ml of blood was collected into EDTA (1 mg/ml). Adult blood samples were collected from 10 healthy hospital personnel via venipuncture into EDTA. All measurements were made within 4 hr after collection (Linderkamp et al., 1986b). RBCs were isolated by centrifugation at 2000g for 10 min and, via gentle aspiration, the plasma was removed and the buffy coat discarded. Part of the isolated plasma was used for the measurement of plasma viscosity, total plasma protein, and fibrinogen concentration. Another part of the plasma was centrifuged a second time at 20,OOOg for 15 min to remove platelets. The isolated RBCs were suspended at a hematocrit of 0.1% in phosphate-buffered saline (PBS: 0.005 mole/liter KH2P04 + Na2HP0,, 0.153 mole/liter NaCl, pH 7.40, 300 mOsm/kg) containing 1 g/liter human serum albumin.
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Geometric properties of RBCs were measured by direct microscopic observations of RBCs with a micropipet system which consists of a microscope (Leitz, Wetzlar FRG, Model Diavert), a video system (Panasonic, Osaka, Japan), and a micromanipulator (De Fonbrune, Paris, France) (Linderkamp and Meiselman, 1982). One neonatal and one adult blood sample were studied on the same day using the same micropipet; 40 RBCs were tested from each donor. Micropipets with diameters of 1.9-2.1 pm were filled with platelet-depleted plasma of the donor and then flushed with the albuminated PBS used to suspend the cells. This procedure prevents RBC adhesion to the glass walls of the pipet. After the pressure in the pipet was brought to zero, a RBC was aspirated by adjusing the pressure to - 15 mm H20. One portion of the cell entered the pipet assuming a cylindrical shape, while the outer part became spherical. The surface area (A) and volume (V) were calculated from the diameter of the pipet (d,,), the length of the cell projection in the pipet (1,), and the diameter of the outer (spherical) portion (d,): (14) V = ~(6~,d2,
+ 4d3, - d3,).
(15)
These formulas do not consider the segment of the outer sphere, which is “cut off” by the tip of the pipet. This results in an error of less than 1% for surface area and volume. The resting diameter (D) of each RBC was measured before the cells were aspirated. The following parameters were calculated from the surface area (A), volume (V), and diameter (D) of the RBC: Mean thickness
= $
(16)
A
Surface area index =
.
(17)
4.84$? The minimum cylindrical diameter (D,) was calculated from the surface and volume of RBCs using equations for a cylinder with spherical end caps: A = ?rD,L v= Ehminating
n-D2L L+ 4
+ z-D’,,,
(18)
& 6
’
(19)
L yields v=>-dAD 4
rrD3 12 .
(20)
Red blood cell counts and mean corpuscular volumes (MCV) were measured electronically (Contraves, Zurich, Switzerland). Hemoglobin concentrations were measured spectrophotometrically by the cyanmethemoglobin method. Mean corpuscular hemoglobin (MCH) was calculated as ratio of hemoglobin concentration
FLOW
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273
and the RBC count. The mean corpuscular hemoglobin concentration (MCHC) is the MCH divided by MCV. Plasma. viscosity was measured by means of a capillary viscometer at 37” (Linderkamp et al., 1981). A glass tube of approximately 100 pm i.d. and 10 mm in length was connected to compressed air (pressure of 25 cm H20). The passage time for 0.2 ml plasma and of distilled water was measured. Plasma viscosity was calculated as the ratio of the passage times of the plasma to that of distilled water multiplied by both 0.6974 CP (viscosity of water at 37”) and by the ratio of the density of the sample to that of distilled water. Plasma densities were derived from nomograms (Van Slyke et al., 1950) using the measured plasma protein concentrations. One neonatal and one adult plasma sample were studied on the same day using the same tube. Total plasma protein concentration was measured by the Biuret test. Plasma fibrinogen concentration was determined via a radial-immunodiffusion technique (M - Partigen Fibrinogen Kit, Behring, Marburg, Germany).
RESULTS Table 1 presents the measured and calculated data for neonatal and adult blood. Compared to adult RBCs, the volume of neonatal cells was 18.6% larger, their surface was 12.0% greater, and their resting diameter was 11.4% wider. The minimum cylindrical diameter (D,) of neonatal RBCs was increased compared to adult cells. The plasma viscosity was significantly decreased in the neonates compared to the adults. The decreased viscosity of neonatal plasma was associated with decreased total plasma protein and fibrinogen concentrations. Figure 2 shows the geometrical parameters calculated for neonatal and adult RBCs as they flow through capillaries with diameters of 6 to 3 pm. The cell length increases and the cell radius decreases with decreasing vessel diameter. The cell length of neonatal RBCs is increased by approximately 14% compared to adult Icells over the whole range of vessel diameters (Table 2). The depth of the rear concavity (1,) decreases as the vessel diameter decreases and reaches zero at a defined vessel diameter, i.e., the critical vessel diameter, 2b,. In vessels with diameters above 4.6 pm, neonatal RBCs have a larger I, than adult cells. In smaller vessels, neonatal RBCs have a smaller 1, than adult cells. The critical vessel diameter where I, becomes zero is 3.28 pm for neonatal RBCs and 3.09 pm for adult cells. In these critical vessels, the corresponding cell diameter is 3.03 pm for the average neonatal RBC and 2.86 pm for the average adult cell, In Figs. 2 and 3 the driving pressure (P) is shown for a suspension viscosity of 0.7 cP. It is 31% larger for neonatal RBCs than for adult cells over the whole range of vessel diameters. Figure 3 also shows the driving pressure for neonatal and adult RBCs by using the measured plasma viscosities. The driving pressures for neonatal RBCs are now only 8% higher than those of adult cells (Table 2). Figure 4 shows the pressure gradients calculated as the ratio of the driving pressure to the cell length (P/L) in vessels of various diameters. If neonatal and adult RBCs are suspended in alb minated (0.1%) PBS with a viscosity of 0.7 cP, the calculated pressure gradien r s are 14% larger for neonatal RBCs compared
274
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TABLE RED
BLOOD
CELL
PARAMETERS
1 AND
PLASMA
VISCOSITY
Neonatal RBCs Geometric properties of RBC 105.7 t (fr Mean hemoglobin concentration (pg) 35.0 -t 32.9 k Mean corpuscular hemoglobin concentration (g/dl) 8.8 t Diameter (pm) (” 107.3 +Volume of aspirated RBCs (fl)
0.04* 0.08)t 0.05 0.11) 0.10 0.26) 0.14* 0.27)
89.5 t (2 29.8 + 33.2 iz 7.9 k (k 90.5 + (k 137.1 k (-c 1.51 k (-t 1.41 + (2 1.84 ” (? 2.83 +(k
3.9 14.0) 1.5 1.3 0.4 0.7) 4.4 13.2) 6.7 14.8) 0.04 0.06) 0.04 0.09) 0.13 0.28) 0.14 0.25)
Plasma viscosity and proteins 1.04 k 0.10* 5.6 + 0.7* 0.279 2 0.052t
1.26 ? 7.0 + 0.362 -t
0.13 0.8 0.061
Mean corpuscular volume (fl)
(-t
153.5 2 (2 1.43 k (” 1.41 -’ (!I 1.76 + (2 3.00 2 (2
RBC surface area (ym) Surface area/volume Surface area index Mean RBC thickness (pm) Minimum cylindrical diameter (ym)
Plasma viscosity (cP) Total plasma protein (g/d]) Plasma fibrinogen (g/d])
Adult RBCs
5.1* 24.9)t
1.s* 1.4 0.4” 0.8)t
5.6* 20.2)t
7.0* 20.7jt
Note. Neonatal and adult samples were studied. Data represent means ? 1 SD. SD reflects the spread of the sample means, whereas the values in parentheses of the entire RBC populations studied in each group. * P < 0.005, when compared with adult RBCs (two-tailed unpaired t test, 2n t P < 0.05 when compared with adult RBCs (two-tailed unpaired f test, 2n -
Note that the first reflect the spread - 2 df). 2 dj).
to adult cells. However, the use of the actual plasma viscosities result in slightly smaller pressure gradients for neonatal RBCs. Figure 5 shows the calculated ratio of tube hematocrit to discharge hematocrit (&I&) and Fig. 6 shows the calculated relative viscosity as a function of the vessel diameter. Both the tube hematocrit (Fig. 5) and the relative viscosity (Fig. 6) of neonatal and adult RBC suspensions increase when going from a 6-pm vessel to a 3-pm vessel. In vessels below the critical diameter, tube hematocrit and relative viscosity rise steeply. The H,/H,, ratio of neonatal RBCs is 1% (in 3.3~pm vessels) to 6% (in 6-pm vessels) higher than that of adult RBCs. Relative viscosity of neonatal RBCs is 5.5 to 8.4% higher compared with adult RBCs. However, the calculated blood viscosity (product of relative viscosity and plasma viscosity) is 11 to 13% less in the neonates than in adults (Fig. 6). DISCUSSION This study demonstrates that neonatal RBCs require a higher driving pressure than adult RBCs for passing through a narrow vessel with a given diameter at
FLOW BEHAVIOR OF RED BLOOD CELLS r(pm) I, (yml
Adult Neonatal
RBC RBC
-
-
-
-
275
-
P(mmHg1
-“5
VESSEL
DIAMETER
(pm)
FIG. 2. Geometrical parameters (r, I, I,) and driving pressure (P) of neonatal and adult RBCs as functions of the vessel diameter (suspending medium viscosity, F = 0.7 cP). 26, = critical vessel diameter.
CALCULATED
FLOW
PROPERTIES
OF NEONATAL WITH DIAMETERS
TABLE 2 (N) AND ADULT (A) RED OF 3.3 pm AND 6 pm
BLOOD
CELLS
3.3~pm vessel Parameter RBC length (pm) Driving pressure (mm Hgl Pressure gradient (mm Hg * lO’/pm) Tube/discharge hematocrit ratio Relative viscosity Blood viscosity (cP)
IN VESSELS
6-pm vessel
Suspending medium
N
A
N
A
B,P B P B P
15.26 0.705 1.048 46.21 68.65
13.35 0.539 0.971 40.40 72.72
8.05 0.033 0.049 4.06 6.03
7.05 0.025 0.045 3.55 6.39
BJ’ W’ B P
0.914 2.139 1.50 2.22
0.902 1.974 1.38 2.49
0.704 1.157 0.81 1.20
0.602 1.097 0.77 1.38
Note. The flow properties were calculated for different suspending media: B, PBS solution (/J = 0.7 cP); P, plasma (CL = 1.04 CP for neonatal and 1.26 CP for adult plasma).
276
STADLER AND LINDERKAMP
1.2
0.;
FIG. 3. Driving pressure (P) of neonatal and adult RBCs as functions of the vessel diameter. Driving pressures were calculated for RBCs suspended in buffer solution (CL= 0.7 cP) and in plasma with viscosities of 1.04 CP (neonatal plasma) and 1.26 CP (adult plasma), respectively. 2b, = critical vessel diameter.
a given speed if the cells are suspended in the same medium (Fig. 3). However, both cell types require similar driving pressures if neonatal RBCs are suspended in neonatal plasma and adult RBCs are suspended in adult plasma. Thus, the lower plasma viscosity in the neonate outweighs the adverse effect of the large size of neonatal RBCs for blood flow in 3- to 6-pm vessels (Fig. 4). This finding indicates that blood flow in narrow vessels of the neonate is not hindered by the large size of their RBCs. However, when the vessel diameter decreases below the critical value (i.e., 3.3 pm for the average neonatal RBC and 3.1 pm for the average adult RBC), the cells have to assume the shape of a cylinder with hemispherical caps at both ends. These cells lose their deformability and require extremely high driving pressures (Secomb and Gross, 1983; Secomb et al., 1986). In tubes, the flow velocity increases from the tube wall to the axis. Thus, the plasma layer on the wall flows slower than the central cell core of the tube. Since the more rapidly flowing cells leave the tube faster than the plasma, the tube hematocrit decreases (Fahraeus effect). The decrease in tube hematocrit depends on the ratio of the plasma layer to the tube diameter. The ratio of the
FLOW
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277
CELLS
i \ 3-
\ \ 3 Adult
RBC
(p = 1.26
cP)
I-
1
I
1
4 VESSEL
I
5 DIAMETER
I
i
(wn)
FIG. 4. Pressure gradient (pressure divided by cell length) as function of the vessel diameter. The pressure gradients are shown for neonatal and adult red blood cells (RBCs) suspended in buffer solution (viscosity, p = 0.7 cP), for neonatal RBCs suspended in neonatal plasma (CL = 1.04 cP), and for adult RBCs suspended in adult plasma (/.L = 1.26 cP).
plasma layer to the tube diameter increases and the hematocrit decreases when the tube diameter is decreased from 500 pm to about 8 pm (i.e., the resting RBC diameter). A further decrease in tube diameter causes a decrease in the ratio of the plasma layer to the tube diameter and the hematocrit increases (inverse Fahraeus effect, see Fig. 5). In a narrow tube with a given diameter and gap between cell and vessel wall, the hematocrit depends on the RBC length (Eq. (9)). This explains the higher hematocrit of the larger neonatal RBCs compared to adult cells in narrow tubes (Fig. 5). The increase in hematocrit results in increased relative viscosity of neonatal
278
STADLER
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l.O-
0.6 -
VESSEL
DIAMETER
(,um\
FIG. 5. Tube/discharge hematocrit ratio (I&/H,) of neonatal and adult red blood suspensions as functions of the vessel diameter. D, = critical cell diameter, 2b, = critical vessel diameter.
2.5 3 9 c z -%% iis 55 g 8
2.0-
1.5-
i=O 4iii E l.O-
3
*b,
f VESSEL
, DIAMETER
&
FIG. 6. Relative viscosity (q,J and blood viscosity (7) of neonatal and adult blood as functions of vessel diameter. 2b, = critical vessel diameter.
FLOW
BEHAVIOR
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RED
BLOOD
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279
RBCs compared to adult cells if both cell types have been suspended in the same medium (Fig. 6). However, if neonatal RBCs are suspended in neonatal plasma and adult RBCs are suspended in adult plasma, viscosity of neonatal blood is decreased compared with adult blood (Fig. 6). We conclude from these results that the large size of neonatal RBCs may be associated with impaired flow in narrow vessels with diameters below the critical diameter (3.3 pm). However, in vessels with diameters of 3.3-6.0 pm, the disadvantage of the large size of neonatal RBCs appears to be completely compensated for by the lower plasma viscosity in the neonate. ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeinschaft (Research Grant Li 291/4). We thank Thomas Behler, M.D., and Thomas Weiss, BSc., for their invaluable advice. We are grateful to Mrs. Liselotte Kriiger for secretarial assistance and Vivian M. Vargas, M.S.B.A., for help in preparing the manuscript.
REFERENCES L., TCHERNIA, G., FEO, C., AND MOHANDAS, N. (1982). Echinocytic sensitivity and deformability of human newborn red cells. Biol. Neonare 42, 284-290. EHRLY, A. M. (1975). Veranderungen der FlieBeigenschaften des Blutes und der Erythrozyten bei arteriellen VerschlulJkrankheiten. Med. Welt 26, 1971-1972. EVANS, E. ,4., WAUGH, R., AND MELNIK, L. (1976). Elastic area compressibility modulus of red cell membrane. Biophys. J. 16, 585-595. FAHRAEUS, R. (1929). The suspension-stability of blood. Physiol. Rev. 9, 241-274. FAHRAEUS, R., AND LINDQVIST, T. (1931). The viscosity of blood in narrow tubes. Amer. J. Physiol. COULOMBEL,
96, 562-586.
GAEHTGENS, P. (1981). Distribution of flow and red cell flux in the microcirculation. Stand. J. Clin. Invest. 4:L(Suppl. 156), 83-87. LINDERKAMP,0. (1987). Blood rheology in the newborn infant. Zn “Bailliere’s Clinical Haematology” (G. D. 0. Lowe, Ed.), Vol. 1, pp. 801-825. Bailliere Trudall, London. LINDERKAMP, O., AND MEISELMAN, H. J. (1982). Geommetric, osmotic, and membrane mechanical properties of density-separated human red cells. Blood 59, 1121-l 127. LINDERKAMP, O., MEISELMAN, H. J., Wu, P. Y. K., AND MILLER, F. C. (1981). Blood and plasma viscosity and optimal hematocrit in the normal newborn infant. C/in. Hemorheol. 1, 575-584. LINDERKAMP, O., HAMMER, B. J., AND MILLER, R. (1986a). Filterability of erythrocytes and whole blood in preterm and full-term neonates and adults. Pediatr. Res. 20, 1269-1273. LINDERKAMP, O., NASH, G. B., Wu, P. W. Y., AND MEISELMAN, H. J. (1986b). Deformability and intrinsic material properties of neonatal red blood cells. Blood 67, 1244-1250. LINDERKAMP, O., VERSMOLD, H. T., RIEGEL, K. P., AND BETKE, K. (1984). Contributions of red cells and plasma to blood viscosity in preterm and full-term infants and adults. Pediatrics 74, 45-51. PAPENFUSS, H. D., AND GROSS,J. F. (1977). The interaction between transmural fluid exchange and blood viscosity in narrow blood vessels. Biorheology 14, 217-228. SCHMID-SCII~NBEIN, H., AND WELLS, R. E. (1968). Fluid drop-like transition of erythrocytes under shear. Science 165, 288-291. SECOMB, T. W., AND GROSS,J. F. (1983). Flow of red blood cells in narrow capillaries: Role of membrane tension. Znt. J. Microcirc. Clin. Exp. 2, 229-240. SECOMB, T. W., AND SKALAK, R. (1982). A two dimensional model for capillary flow of an axisymmetric cell. Microvasc. Res. 24, 194-203. SECOMB, T. W., SKALAK, R., &KAYA, N., AND GROSS,J. F. (1986). Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405-423. VAN SLYKE, D. D., PHILLIPS, R. A., DOLE, V. P., HAMILTON, P. B., ARCHIBALD, R. M., AND PLAZIN, J. (1950). Calculation of hemoglobin from blood specific gravities. J. Biol. Chem. 183, 349-360.