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Capillary blood flow
MICROVASCULAR
RESEARCH
2,420-433 (1970)
Capillary II. Deformable
Blood
Model
Flow
Cells
in Tube
Flow1
S. P. SUTERA, V. SESHADRI, P. A. CROCE, AND R. M. HOCHMUTH Departments of Mechanical and Aerospace Engineering, Chemical Engineering and Radiology, Washington University, St. Louis, Missouri 63130 Received May 4,197O
Capillary blood flow is investigated using large-scale deformable models of the red blood cell. Cell velocity and the additional pressure drop caused by a single cell are measured over a wide range of volume flow and at undeformed cell-to-tube diameter ratios of 1.0, 1.3, and 2.0. The stability and orientation of individual cells and the effect of cell spacing on the additional pressure drop is also investigated. Apparent relative viscosities characterizing the flow of blood through capillaries of various diameters are calculated on the basis of the pressure-drop data. A relationship between average hematocrit in a capillary and the associated “cup-mixing” value of hematocrit is theorized.
INTRODUCTlON The first paper in this series (1) reports observations of the deformation of human erythrocytes in glass capillaries of diameter less than 10 TV.Beyond such observations we wish to know the relationships between (i) cell velocity and average plasma velocity, and (ii) pressure drop and flow, both for blood flow through a capillary. In both cases erythrocyte deformability is an intrinsic factor. Owing to the microscopic scale of the capillaries, it would be most difficult, if not impossible, to study these relationships either in vitro or in vivo. Hence, some investigators have been led to construct large-scale dynamically similar models in order to obtain quantitative data on these flow characteristics. Previous model studies have resorted to air bubbles (2, 3) and rigid discoids (4). Because a gas-liquid interface presents a boundary condition which is fundamentally different from the no-slip condition prevailing on the membrane of an erythrocyte, we cannot expect the bubble model to yield quantitatively valid information. On the other hand, the rigid model gives the right boundary condition and will yield relevant data as long as the model cell is geometrically similar to the deformed erythrocyte as the latter is propelled through a particular capillary. Of course, a rigid model cannot respond to changes in flow, as does the real cell, with passive changes in shape (1). For this reason experiments with rigid cells provide no insight to the fundamental relationship between cell shape and the instantaneous state of flow in a capillary. A realistic model study of that relationship requires deformable model cells structurally similar to the real erythrocyte. Also required is a similar ratio of the hydrodynamic forces acting to deform the cell to some appropriate stiffness modulus ’ This work was supported by the USPHS, NIH, through Grant No. FR-06115 to Washington University. 420
CAPILLARY BLOOD FLOW II
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characteristic of the cell. Lee and Fung (5) have simulated capillary blood flow using rubber liquid-filled model cells. The observed deformations indicated that the membranes of their model cells were considerably stiffer (relative to the cell size) than that of the human erythrocyte. The model cells employed in the present investigation, although similar in design to those of Lee and Fung, have substantially thinner and more flexible membranes. Moreover, a wider range of flow parameters is covered. DETAILS OF THE EXPERIMENTS Apparatus. A schematic diagram of the model apparatus is given in Fig. 1. Except for minor modifications it is similar to that used by Sutera and Hochmuth (4). For a detailed discussion of the design philosophy the reader is referred to the original paper (4).
FIG. 1. Schematic diagram of the apparatus. A, reservoir; B, precision-bore glass tube, i.d. = 1.974 f .0005 cm; C, full circumference pressure taps; D, mirror; E, l&mm movie camera; F, Plexiglas box; G, black anodized aluminum panel; H, pistons (2); J, movable carriage; K, lead screw driven by a motor; L, guide rails; M, optical compensating fluid mixture; N, oil-water interface; P, bypass valve; Q, pressure transducer; R, transducer indicator; S, zero suppression; T, recorder; U, drain valve.
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Briefly, the capillary vessel itself is modeled by a vertical precision-bore glass tube (internal diameter 1.974 * 0.0005 cm, total length 100 cm, approximately) in four segments. The segments are connected at three, full circumference pressure taps; the upper pair of taps are nominally 10 diameters apart, the lower pair 20 diameters. The upper reservoir is open to atmosphere and provides the access for introduction of the model cells. Flow through the test section, either up or down, is driven by a constant speed infusion-withdrawal pump which is effectively a flow source for the range of pressure encountered in these experiments. Silicone oil (Dow-Corning 200 Series, kinematic viscosity (v) 600 stokes, specific gravity (S.G.) 0.973) is the working fluid. Distortion-free viewing of the cells in the test section is accomplished by surrounding the tube with a solution (60% by volume dibutyl phthalate, 40% UCON oil LB1715) whose index of refraction matches that of the glass wall. Velocity of the model cells is obtained by timing their passage over known distances on steel scales mounted behind the tube. A plane mirror oriented at 45” to one of the viewing planes permits simultaneous orthogonal views of a moving cell. Pressure differentials are measured with a transducer (Pace P7D) having a full-scale displacement volume of 0.005 cc. The transducer is filled with water to reduce response time. Full-scale displacement of the transducer diaphragm, in response to a step input, occurs in about 8 sec. The transducer indicator has a null-balancing circuit which permits nulling of the steady Poiseuille component of the pressure drop and subsequent amplification of the additional pressure drop caused by the model cells. Calibration against a high sensitivity manometer revealed the transducer to be linear within 0.5 0;. As a check on the accuracy of the pressure-measuring system, pressure drops were measured in steady flows of a pure liquid of known viscosity. The measured drops were always within &l x1 of the values predicted by Poiseuille’s law. Model Cells. Fabrication of a model cell begins with a solid model cast from Cerrobend, a low melting-point (I 58°F) alloy. The solid model is cast with a cylindrical stem by which it is held and dipped into an aqueous solution of latex rubber. After dipping, the cell is rotated by its stem until the latex is partially cured (about 45 min). Several dips are usually required to produce a membrane of uniform thickness. The resulting membrane thickness depends on both the number of dips and the concentration of the latex solution. Thicknesses on the order of .004-.005 times the diameter of the model cell, which ranged from 24 cm, nominally, were achieved. The membrane is allowed to cure at room temperature for about 24 hr, after which the metal core is melted in boiling water and forced out of the membrane through the hole left by the stem. The now hollow membrane is checked for leaks, dried thoroughly, and then filled with silicone oil (Dow Corning 200 series, v = 125 stokes, S.G. = 0.973). The cell is sealed by either tying off the stem opening or by “welding” it closed with a drop of the original latex solution. By means of a simple tension test it was found that the latex rubber membrane has a linear stress-strain relationship for extensional strains less than 60%. The corresponding Young’s modulus is 1.66 x 10’ dyn/cm2. Poisson’s ratio is 0.5. These mechanical properties and the membrane thickness are found to be unaffected by immersion in silicone oil for periods as long as 1 week. Moreover, no aging was apparent over the duration of experiments conducted with individual cells.
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Technique. Pressure drops were always measured between the upper two taps (Fig. 1). A model cell is placed in the upper reservoir and drawn into the tube by the pump. Before the cell reaches the first tap, the pressure drop associated with the Poiseuille flow is recorded. The corresponding electrical signal is then nulled electrically in the transducer indicator. Subsequently the additional pressure drop caused by the cell while it is located between the two taps is recorded. If necessary,the recorder gain may be increased to magnify the additional signal. Becausethe transducer response is linear, the ratio of the two electrical voltages it generates,eDoisand e*, corresponding to the Poiseuille and the additional components of the pressure drop, respectively, is exactly the ratio of the two pressure drops (4): Ap*
e*
The Poiseuille drop can be written (2)
where pf is the dynamic viscosity of the working fluid (silicone oil) at the operating temperature, 0 the average velocity in the tube (volume flow rate divided by crosssectional area), D, the tube diameter, and L the distance between the two taps. Combining Eqs. (1) and (2) we obtain a dimensionless additional pressure drop in terms of the electrical signals, ‘!*
(ELIu/D,)
=32(k)
(2).
(3)
This is actually independent of the length L as long as the cell is more than a few tube diameters away from either tap. For a given cell it depends only on the ratio of cell-totube diameter and the flow rate (0). Cell velocity U, is determined by measuring the time required by the cell to travel a known distance. The average velocity i7 is determined for each setting of the pump control by measuring the velocity of a tightly fitting disc as it is pushed through the tube. All velocity measurementswere reproducible within f 0.25 %. Both pressure-drop and velocity measurements were found to be independent of the direction (up or down) of the flow, indicating an absenceof gravitational effects in the experiments. Similarity Considerations. The subject of similitude, as it pertains to modeling of blood flow in a capillary, has been carefully examined elsewhere(4, 5). In the present experiments two primary quantities are measured: cell velocity, U,, and the additional pressure drop due to a single cell, Ap*. These quantities are presented as dimensionless ratios, the referencefactors being i? and pf o/D=, respectively. A combination of dimensional analysis and prior experience (4, 5) leads us to expect correlations of the form (4)
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ET AL.
and
where E, is the elastic “constant”2 of the cell membrane, D, the diameter of the undeformed cell, DT the internal diameter of the tube, and t, the thickness of the cell membrane. The correlations (4) and (5) are shown as incomplete. Missing are an unknown number of geometrical groups which would fully characterize the deformed shape of the cell in the vessel. The exact shape assumed by the cell in the tube cannot be controlled in a model experiment. Nevertheless, the experimental evidence presented here indicates that additional parameters describing the details of cell shape (e.g., the degree of axial asymmetry, and the contour of the rear cavity) would be of secondary importance in the correlations (4) and (5). The primary geometrical parameters are the two ratios shown. The group pfijlDTEc is the ratio of a characteristic viscous stress acting on the cell surface to the membrane’s modulus of elasticity. Thus it can be interpreted as a characteristic tensile strain in the membrane of a given cell. Two important assumptions which are tacitly contained in the correlations (4) and (5) are (i) Reynolds member is unimportant, and (ii) the axial position of the cell in the tube is unimportant. Both of these assumptions are valid because of the extremely low Reynolds number (10m6< pijDT/pf < 10m2)characteristic of capillary blood flow. Provided the Reynolds number achieved in the model study is of the same order, its exact value is not crucial. In low Reynolds-number tube flow the development or entrance length is small, less than one tube diameter. As a result, if a cell is farther than DT from either extremity of the tube, its speed and the additional pressure drop it causes will both be independent of its exact axial station. Range of Parameters Investigated. The exact range of tube-Reynolds number covered in the present model experiments is 8.2 x 10p5-1.65 x 10-3. The corresponding range of average velocity 0 is approximately 0.025-0.5 cm/set. The diameter ratio D,/D, was given values of 1.0, 1.3, and 2.0 while the ratio t,/D, varies between .004 and .005. The deformability parameter pf n/D,E,, which is varied through the average velocity i?, ranges from 4.3 x lo-’ to 8.7 x 10-6. RESULTS
OF THE MODEL
EXPERIMENTS
Cell Orientation and Stability. We will refer to the cell’s orientation (as it moves through the tube) as normal if the natural axis of revolution of the cell is coincident with or parallel to the tube’s axis. At the other extreme, corresponding to perpendicularity or near perpendicularity of these two axes, the orientation will be called edge-on. The observations made of the three different sizes of model cells are summarized below. (i) D,/DT = 1.0. At low flows (0 < 0.1 cm/set) the cell is essentially undeformed. No orientation appears to be stable as the cell is often seen to rotate slowly as it passes * Recent experiments performed in our laboratory indicate strongly that the cell membrane is not a linear elastic material. Here EC is interpreted as the modulus at infinitesimal strain.
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BLOOD FLOW II
through the tube. As the average velocity is increased the cell becomesstable in the edge-on orientation with a concavity on the rear side. This observation is in qualitative agreement with those of Skalak and Brinemark (7). Accompanying the formation of the cavity at the rear is an inflation of the whole forward portion of the membrane. As a result, the forward profile of the cell may appear parabolic even when viewed parallel to the original “plane” of the cell. See,for example, Fig. 1A of Ref. (6). The initial orientation of the cell, as it enters the tube, has no effect on the edge-on stability; a cell happening to enter in the normal orientation is highly likely to rotate into the edge-on attitude. The only perceptible effect of higher velocity (0 > 0.2 cmjsec) is to smooth the wrinkles in the forward portion of the cell membrane, indicating a rising membrane tension.
i J 1.0
.3
mi
/ .6
i 1
I
3
6
IO
30
60
100
x106-
FIG. 2. for O+m.
Cell velocity asa functionof strainparameter.Comparisonwith data of Lee and Fung (5)
(ii) DC/D, = 1.3. SeeFig. 2A of Ref. (6). The cell enters the tube only in the edge-on orientation and cannot be forced to do otherwise. Moving through the tube, the cell is stable in the edge-on orientation. The rear side of the cell is concave at all velocities, but the depth of the concavity increases with velocity. Smoothing of the forward portion of the membrane is again seento accompany increasing velocity. (iii) DC/D, = 2.0. See Fig. 3A of Ref. (6). Again only the edge-on orientation is observed. The deformed cell strongly resemblesa cylinder with one convex and one concave base. The lubrication layer between cell and tube is practically imperceptible at low flows but becomesmore visible as the flow is increased. Cell Velocity. In Fig. 2 the ratio of cell velocity to average flow velocity, UJO, is plotted against the strain parameter, pf ij/D,E,, which in this case may be viewed as a convenient dimensionless flow rate. The diameter ratio DC/D, is a parameter of the plot. For DC/D,, = 1.0 and 1.3 (two smallest cells), the cell velocity is greater than the
426
SUTERA
ET AL.
average velocity by approximately 40 and 207& respectively, but does not change appreciably over the range of U covered. In the case DC/D, = 2.0 (largest cell) a clear dependenceof UCon u is seen, UC/D increasing with flow. This nonlinearity can be explained as a manifestation of decreasing cell cross section (hence an increasing peripheral gap) owing to a changing distribution of surface stressesacting on the membrane. As D becomesvanishingly small the cell expands to nearly fill the lumen while UC-+ D. Similar behavior has been observed in the caseof erythrocytes in vitro (1).
It should be expected that the smaller the fraction of the lumen occupied by the cell the greater will be the ratio UC/D,with an upper bound of 2.0. The data of Fig. 2
200
5
IO
20
50
1 3
x 106-
FIG. 3. Additional pressure drop as a function of strain parameter. Comparison with data of Lee and Fung (5).
verifies this expectation since, at a fixed value of pr U/D,E,, UC/D increases with decreasing DC/D,. Pressure Drop. The dimensionless additional pressure drop due to a single cell is plotted as a function of the strain parameter in Fig. 3. For each of the three cell sizes investigated the additional pressure drop is seen to decreasewith increasing strain parameter. The latter, in these experiments, is varied only through the flow rate. In each case the pressure drop approaches an asymptotic value with increasing flow, indicative of maximum cell deformation. The rate of decreaseof additional pressure drop with flow is most dramatic in the caseDC/D, = 2.0, where the cell assumesan elongated, nearly cylindrical form separatedfrom the tube wall by a thin lubricating layer. The pressure-drop data points carry a maximum uncertainty of f2% in the cases DC/DT = 1.O and 1.3. For DJ D., = 2.0, however, a greater uncertainty, indicated by
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the vertical line segments attached to the plotted points, was encountered. This is, due undoubtedly to a wider variation in the detailed pattern of membrane deformation, i.e., location and extent of wrinkles and larger folds, and the significant influence of these details on the average gap width which is very small in this case. Becauseof the sensitivity of pressure drop to gap width (8), a small change in the latter can result in a significantly different pressure drop. This uncertainty is attenuated at higher velocities which are accompanied by higher levels of tensile stress in the membrane. Comparison with Other Experiments and Theory. The only experimental results to which the present findings can be directly compared are those of Lee and Fung (5). The latter investigators have given correlations for the velocity ratio of the form where CLand k are curve-fitting parameters which were apparently determined for the best straight-line fit. For 0 small the velocity ratio predicted by this equation is very sensitive to the parameter CL(a negative value may even occur) which is arbitrary, in a sense,since it depends on the kind of curve presumed. This feature confuses any comparison attempted in the range of the strain parameter covered by our data (0.410 x 10-6). Thus we indicate on Fig. 2 only the values predicted by the Lee and Fung correlation for large 0, i.e., UC/Ox k. On this basis we seethat our measurementsare consistently higher at comparable diameter ratios. Working from the curves in Fig. 16 of the paper by Lee and Fung we have performed a similar comparison for the additional pressure drops. The result is shown in Fig. 3. The two sets of curves do not have similar slopes and barely overlap, but where they do overlap (pr U/D, E, sz lops), it is clear that our data are substantially lower at comparable diameter ratios. (Note D,/D, z 1.0 vs. 0.98, 1.3 vs. 1.36.) Except for the membrane-thicknessratio (t,/D,) all geometrical and flow parameters were of the same order in the two model studies. Our thickness ratio is roughly half that of the cells used by Lee and Fung. The thinner membrane results in greater cell deformability which should, in turn, lead to higher cell velocity and lower additional pressure drop as observed, The caseof large rigid spheresin tube flow has beeninvestigated both experimentally and theoretically as a model for capillary blood flow (8, 9). To test the applicability of a rigid sphere model we have compared our data to corresponding data for rigid spheres. For diameter ratios D,/DT = 1.O and 1.3 and the highest flow used, D = 0.5 cm/set, where, according to the pressure-drop data (Fig. 3) the asymptotic deformation is attained, we compared the diameter ratio actually observed for the deformed cell (i.e., the ratio of maximum lateral dimension of the cell in any given view to the tube diameter) to that required for a rigid sphere to give (i) the same relative velocity and (ii) the sameadditional pressuredrop. (The comparison was not extended to the largest cells, DJD, = 2.0, becausethe deformed shape of these cells is much closer to cylindrical than spherical.) Agreement within 2 % was found. On this basis we would submit that the rigid sphere model can be used to predict erythrocyte velocity and pressure drop in 66IO-pm capillaries, provided that the diameter ratio for the sphere is matched to that of the deformed, erythrocyte in the capillary. However, it is worth emphasizing again that the true nonlinear character of the pressure-flow relationship for capillary blood flow cannot be modeled with
428
SW-ERA ET AL.
rigid cells of one size. This can only be accomplished indirectly through a seriesof rigid cell experiments as suggestedby Sutera and Hochmuth (4). Efict of Cell Spacing. The data displayed in Figs. 2 and 3 apply to single, noninteracting cells. In the physiological situation erythrocytes frequently pass through capillaries in close successionand/or in contact (rouleaux). In order to determine the effect of axial proximity on the additional pressure drop due to a pair of cells, experiments are conducted with two cells of diameter ratio D,/D, = 1.3. It is found that the pressure drop due to the pair is essentially equal to twice the single cell pressure drop given by Fig. 3 as long as the spacing between the cells is not less than DT, approximately. This means that the cells do not interact when they are more than DT apart. [This result could be expected on the grounds that the development length in low Reynolds-number tube flow is small, of the order of the tube diameter, with a lower limit of about 0.65 DT in the limit Re + 0 (lo).] At the other extreme, i.e., when the two cells are in contact, the measuredadditional pressure drop is only about 5 % less than the no-interaction value. This result can be used to estimate the total additional pressure drop due to a long “rouleau” of cells in contact. Let Ap* be the additional pressure drop caused by a single cell moving independently through the capillary. Then two such noninteracting cells (of the same diameter ratio) would cause an additional pressure drop of 2Ap*. Now let dpZC* denote the additional pressure drop caused by two cells in contact, and 6p* the difference between 2Ap* and Ap 2C*.According to the measurements -%!T z ().05. 2Ap”
If we now consider placing a third cell, of the same size, in contact with two others already in contact, it seemsreasonable to assumethat the additional pressure drop for the rouleau of three would be Ap3c* = Apzc* + (Ap” - Sp*), = (2Ap* - Sp*) + (Ap* - 6p*), and = 3Ap” - 26~“.
By inductive reasoning this may be extended to a rouleau of N cells. Then, Ap,,* = NAP” - (N - 1) 6p* or For N large, Ap.wc*
NAP*
According to this simple result the total additional pressure drop due to a long rouleau of N cells in contact, where N B 1, would be about 10% less than N times the single cell drop. Apparent Viscosity of Blood in a Capillary. Apparent viscosity in tube flow is simply
CAPILLARY
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a measure of the total pressure drop through the tube per unit of volumetric flow. As these experiments demonstrate, the total pressure drop through an individual capillary passing blood will depend on the number, spacing, and size distribution of erythrocytes contained in the capillary instantaneously. These same factors are determined by the instantaneous red cell volume in the capillary and hence the average hemotocrit n in the capillary. The apparent viscosity will also depend on the rate of flow through the capillary. In this section we usea simplified theoretical model to estimate the apparent viscosity for blood flow through capillaries of different diameter, as a function of flow rate, and for a particular hematocrit H,, in the blood actually entering the capillary. Consider a steady flow of cells through a capillary and assumethat the diameter ratio is the same for all cells. Thus all the cells will be traveling at the same speed U,. Because U, > 0 (seeFig. 2) the averagehematocrit in the capillary 6 will be lessthan H,,. The relation between 14 and Ho in this hypothetical case is derived in the Appendix; it is simply
Ho_ uc H
0'
(6)
Thus, the hematocrit ratio can be taken directly from the curves of Fig. 2. For DC/D, = 1.0 (corresponding to a capillary of diameter 8.5 pm) and for values of the strain parameter in the range 0.5-9 x 10m6,R E 0.74 H,,. Also derived in the Appendix for this model is an expression for the relative apparent viscosity pr, which is just the ratio of the apparent viscosity to the viscosity of the liquid phase (plasma) : (7) where V is the mean erythrocyte volume and dp* is the additional pressure drop due to a single cell. Now assumeD, = 8.5 pm and V= 87 pm3 for a human erythrocyte. The model-cell data of Figs. 2 and 3, covering diameter ratios DC/D== 1.0, 1.3, and 2.0, are then applicable to capillaries with diameters 8.5,6.6, and 4.25 pm, respectively. Using these data in Eq. (7), we can construct the curves shown in Fig. 4 giving pLIas a function of the strain parameter (or, equivalently, a dimensionless flow rate). The hematocrit Ho was taken to be 0.40 in all cases.The results predict that the relative apparent viscosity will decreasewith increasing flow rate in a given capillary and will be greater the smaller the capillary at a given flow rate. In these calculations the possibility of a spacing correction, as described in the preceding section, was deliberately ignored. According to this simple model blood would flow easily through capillaries as small as 4 pm if the flow rate is high. For a capillary of this size pr < 4 when the strain parameter exceeds about 7 x 10e6. Let us calculate the corresponding D for a human capillary 4 pm in diameter. From ____ = 7 x 10-6 Ec
we get
DT
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SUTERA ETAL.
I 060
2
A
6
I 8
x 106-
FIG. 4. Calculated apparent relative viscosity of blood in capillaries. Ho = 40%.
Putting
EC = lo6 dyn/cm2,
D, = 4 x 10m4 cm, and pr = .02 dyn-set/cm*,
we find
0 = .14 cmjsec Z 1 mmjsec. This value is quite typical of the maximum erythrocyte velocities observed in mammalian systems. The very high apparent viscosities predicted in Fig. 4, pr > 10, say, would be encountered only in the smallest capillaries when the erythrocyte velocity drops to 0.1 mm/set or lower. In these circumstances the gap between the cell membrane and the endothelial layer of the capillary would be extremely small, and the interaction of the protein coats on the two surfaces might assume an important role. The large-scale model used in this work, with its continuous “plasma” and smooth-walled tube, would be clearly incapable of simulating any effect due to proteins. It is interesting to note the trend, evident in Fig. 4, of relative apparent viscosity increasing with the cell-to-tube diameter ratio at a given value of the strain parameter. This trend seems to oppose that observed by Fahraeus and Lindqvist (11) in the flow of whole blood through tubes in the 40-500-pm range. On the other hand, it is in qualitative agreement with the findings of Gregersen et al. (12) for blood flow through 4.5 and 6.8~pm polycarbonate sieves. In the latter work, relative resistance was seen to increase as pore size decreased. Taken together the results of Fahraeus and Lindqvist and Gregersen et al. suggest a possible minimum in the apparent viscosity when regarded as a function of diameter ratio (or tube or pore size). The value or values of D,/DT which give a minimum apparent viscosity cannot be pinpointed here; but it must be emphasized that comparisons of apparent viscosity in the capillary regime will not be valid if proper consideration is not given to the dependence of apparent viscosity on the strain parameter (or average velocity). The significance of this remark is shown by the curves in Fig. 4. For DC/D, = 1.0 and
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FLOW
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pf o/E, DT = 0.5 x lO-(j, pLrz 2.0, while for D,/DT = 1.3 (a smaller tube) and pLrU/E, D, = 5 x 1O-6(and a higher flow) p., z 1.8.This comparison, at two significantly different values of the strain parameter, indicates a decreasein apparent viscosity with decreasingtube diameter. This is contrary to the true trend indicated by the full curves in the figure.
CONCLUDING
REMARKS
The observed behavior of the deformable model cells shows that the edge-wisemode of motion through capillaries 6-10 pm in diameter is the only stable mode. Consequently, the shape of the forward portion of the deformed cell is not generally axisymmetric. The formation of a deep concavity at the rear end of the cell is accompanied by an inflation of the forward portion, with the result that the nose profile appears quasi-parabolic from any side. Also, the minimum radius of curvature at the nose will be greater than the edge radius seenin a meridional section of the relaxed cell. Previous analysesof erythrocyte deformation in capillaries (13-I 5) have assumed an axisymmetric profile. The present experiments suggestthat some account of asymmetry should be attempted in future analysesfollowing the example of Fitz-Gerald (16). Confidence in the applicability of a large-scalemodel to capillary blood flow depends on the degree of agreement that can be seen between quantities observable on the actual scale (in vivo or in vitro) and their counterparts in the model. The first paper of this series (1) reports on the deformation of erythrocytes moving through glass capillaries and quantitatively characterizes the observed deformations. In the final paper (6) we compare these with corresponding observations on the large-scale deformable models. Appendix
APPARENT VISCOSITY OF BLOOD IN A CAPILLARY Consider the steady flow of blood in a capillary of diameter DT and length L $ DT. Assume that a large number of erythrocytes are moving through the capillary in single file. The spacing between successivecells is variable, but the total number of cells in the capillary at any instant is n. Then the average hematocrit in the capillary, R, is nV H = (irD34)
64.1)
L’
where V is the volume of one cell. Let Q be the volumetric rate of blood flow through the capillary; then the average velocity in the capillary (an average over the lumen) is defined as
If the velocity of erythrocytes is U,, the it cells which are distributed along the distance will exit from the capillary in a time interval L/U,. During this same interval the total volume (cells plus plasma) leaving the tube is
L
QW uc)
= u(77D:/4)
(Ll uc).
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ET AL.
The ratio of cell volume to this total efflux volume (the “cup-mixing” of cells in the receiving vessel) is denoted by H,, :
concentration
64.3) This concentration, or hematocrit, is identical to the average concentration blood actually entering the capillary. Combining (A.l) and (A.3) we find
I7 u Ho uc’
in the
b4.4)
Since U, > 0 (Fig. 2) this relation says that E ( Ho. The total pressure drop through the capillary is represented as a sum of a pure Poiseuille drop, associated with pure plasma flow at the average velocity 0, and an additional component due to the n cells in the capillary. The Poiseuille component is
(A.3 where pLfis the plasma viscosity. Neglecting possible interactions between successive cells, a correction of 10 ‘A at most, we can write the additional pressure drop as follows:
The bracketed factor is the dimensionless additional pressure drop caused by a single cell (Fig. 3). Using (A.3) to replace n we then obtain (A.6) The relative apparent viscosity can be expressed as the ratio of the total pressure drop to the Poiseuille drop at the same 0. Thus
Recall that this is the apparent relative viscosity of blood with hematocrit Ho being forced steadily through a capillary of diameter DT with an average velocity 8. The actual volume average of cells in the capillary at an instant is generally less than HO [according to (A.4)]. REFERENCES 1. HOCHMUTH,R. M., MARPLE,R. N., ANDSUTERA,S. P. (1970). Capillary blood flow. I. Erythrocyte deformation in glass capillaries. Microvasc. Res. 2,409. 2. PROTHERO, J., AND BURTON,A. C. (1961). The physics of blood flow in capillaries. I. The nature of motion. Biophys. J. 1, 565. 3. PROTHERO, J., AND BURTON,A. C. (1962). The physics of blood flow in capillaries. II. The capillary resistance to flow. Biophys. J. 2, 199. 4. SUTERA,S. P., AND HOCHMUTH,R. M. (1968). Large scale modeling of blood flow in the capillaries. Biorheology
5, 45.
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5. LEE, J. S., AND FUNG, Y. C. (1969). Modeling experiments of a single red blood cell moving in a capillary blood vessel. Microuusc. Rex 1,221. 6. SESHADRI,V., HOCHMUTH,R. M., CROCE,P. A., AND SUTERA,S. P. (1970). Capillary blood flow. III. Deformable model cells compared to erythrocytes in uitro. Microvase. Res. 2,434. 7. SKALAK, R., AND BRANEMARK,P-I. (1969). Deformation of red blood cells in capillaries. Science 164,717. 8. HOCHMUTH,R. M., AND SUTERA,S. P. (1970). Spherical caps in low Reynolds-number tube flow. Chem. Eng. Sci. 25, 593.
9. WANG, H., AND SKALAK,R. (1969). Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 75. 10. LEW, H. S., AND FUNG, Y. C. (1969). On a low Reynolds number entry flow into a circular cylindrical tube. J. Biomechanics 2, 105. 11. FKHRAEUS,R., AND LINDQVIST,T. (1931). The viscosity of blood in narrow capillaries. Amer. .I. Physiol. 96, 562.
12. GREGERSEN, M. I., BRYANT, C. A., HAMMERLE,W. E., USAMI, S., AND CHIEN, S. (1967). Flow characteristics of human erythrocytes through polycarbonate sieves. Science 157, 825, 1967. 13. CHARM,S. E., AND NELSON,F. (1966). Red cell deformation and flow in capillaries. 4th Eur. Con/I Microcir.
Cambridge.
14. BARNARD,A. C. L., LOPEZ,L., ANDHELLUMS,J. D. (1968). Basic theory of blood flow in capillaries. Microvas.
Res. 1, 23.
15. LIGHTHILL, M. J. (1968). Pressure forcing of tightly fitting pellets along fluid filled elastic tubes. .I. Fluid Mech. 34, 113.
16. FITZ-GERALD,J. M. (1969). Mechanics of red cell motion through very narrow capillaries. Proc. Roy. Sot. Ser. B. 174,193.
RESEARCH
2,420-433 (1970)
Capillary II. Deformable
Blood
Model
Flow
Cells
in Tube
Flow1
S. P. SUTERA, V. SESHADRI, P. A. CROCE, AND R. M. HOCHMUTH Departments of Mechanical and Aerospace Engineering, Chemical Engineering and Radiology, Washington University, St. Louis, Missouri 63130 Received May 4,197O
Capillary blood flow is investigated using large-scale deformable models of the red blood cell. Cell velocity and the additional pressure drop caused by a single cell are measured over a wide range of volume flow and at undeformed cell-to-tube diameter ratios of 1.0, 1.3, and 2.0. The stability and orientation of individual cells and the effect of cell spacing on the additional pressure drop is also investigated. Apparent relative viscosities characterizing the flow of blood through capillaries of various diameters are calculated on the basis of the pressure-drop data. A relationship between average hematocrit in a capillary and the associated “cup-mixing” value of hematocrit is theorized.
INTRODUCTlON The first paper in this series (1) reports observations of the deformation of human erythrocytes in glass capillaries of diameter less than 10 TV.Beyond such observations we wish to know the relationships between (i) cell velocity and average plasma velocity, and (ii) pressure drop and flow, both for blood flow through a capillary. In both cases erythrocyte deformability is an intrinsic factor. Owing to the microscopic scale of the capillaries, it would be most difficult, if not impossible, to study these relationships either in vitro or in vivo. Hence, some investigators have been led to construct large-scale dynamically similar models in order to obtain quantitative data on these flow characteristics. Previous model studies have resorted to air bubbles (2, 3) and rigid discoids (4). Because a gas-liquid interface presents a boundary condition which is fundamentally different from the no-slip condition prevailing on the membrane of an erythrocyte, we cannot expect the bubble model to yield quantitatively valid information. On the other hand, the rigid model gives the right boundary condition and will yield relevant data as long as the model cell is geometrically similar to the deformed erythrocyte as the latter is propelled through a particular capillary. Of course, a rigid model cannot respond to changes in flow, as does the real cell, with passive changes in shape (1). For this reason experiments with rigid cells provide no insight to the fundamental relationship between cell shape and the instantaneous state of flow in a capillary. A realistic model study of that relationship requires deformable model cells structurally similar to the real erythrocyte. Also required is a similar ratio of the hydrodynamic forces acting to deform the cell to some appropriate stiffness modulus ’ This work was supported by the USPHS, NIH, through Grant No. FR-06115 to Washington University. 420
CAPILLARY BLOOD FLOW II
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characteristic of the cell. Lee and Fung (5) have simulated capillary blood flow using rubber liquid-filled model cells. The observed deformations indicated that the membranes of their model cells were considerably stiffer (relative to the cell size) than that of the human erythrocyte. The model cells employed in the present investigation, although similar in design to those of Lee and Fung, have substantially thinner and more flexible membranes. Moreover, a wider range of flow parameters is covered. DETAILS OF THE EXPERIMENTS Apparatus. A schematic diagram of the model apparatus is given in Fig. 1. Except for minor modifications it is similar to that used by Sutera and Hochmuth (4). For a detailed discussion of the design philosophy the reader is referred to the original paper (4).
FIG. 1. Schematic diagram of the apparatus. A, reservoir; B, precision-bore glass tube, i.d. = 1.974 f .0005 cm; C, full circumference pressure taps; D, mirror; E, l&mm movie camera; F, Plexiglas box; G, black anodized aluminum panel; H, pistons (2); J, movable carriage; K, lead screw driven by a motor; L, guide rails; M, optical compensating fluid mixture; N, oil-water interface; P, bypass valve; Q, pressure transducer; R, transducer indicator; S, zero suppression; T, recorder; U, drain valve.
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Briefly, the capillary vessel itself is modeled by a vertical precision-bore glass tube (internal diameter 1.974 * 0.0005 cm, total length 100 cm, approximately) in four segments. The segments are connected at three, full circumference pressure taps; the upper pair of taps are nominally 10 diameters apart, the lower pair 20 diameters. The upper reservoir is open to atmosphere and provides the access for introduction of the model cells. Flow through the test section, either up or down, is driven by a constant speed infusion-withdrawal pump which is effectively a flow source for the range of pressure encountered in these experiments. Silicone oil (Dow-Corning 200 Series, kinematic viscosity (v) 600 stokes, specific gravity (S.G.) 0.973) is the working fluid. Distortion-free viewing of the cells in the test section is accomplished by surrounding the tube with a solution (60% by volume dibutyl phthalate, 40% UCON oil LB1715) whose index of refraction matches that of the glass wall. Velocity of the model cells is obtained by timing their passage over known distances on steel scales mounted behind the tube. A plane mirror oriented at 45” to one of the viewing planes permits simultaneous orthogonal views of a moving cell. Pressure differentials are measured with a transducer (Pace P7D) having a full-scale displacement volume of 0.005 cc. The transducer is filled with water to reduce response time. Full-scale displacement of the transducer diaphragm, in response to a step input, occurs in about 8 sec. The transducer indicator has a null-balancing circuit which permits nulling of the steady Poiseuille component of the pressure drop and subsequent amplification of the additional pressure drop caused by the model cells. Calibration against a high sensitivity manometer revealed the transducer to be linear within 0.5 0;. As a check on the accuracy of the pressure-measuring system, pressure drops were measured in steady flows of a pure liquid of known viscosity. The measured drops were always within &l x1 of the values predicted by Poiseuille’s law. Model Cells. Fabrication of a model cell begins with a solid model cast from Cerrobend, a low melting-point (I 58°F) alloy. The solid model is cast with a cylindrical stem by which it is held and dipped into an aqueous solution of latex rubber. After dipping, the cell is rotated by its stem until the latex is partially cured (about 45 min). Several dips are usually required to produce a membrane of uniform thickness. The resulting membrane thickness depends on both the number of dips and the concentration of the latex solution. Thicknesses on the order of .004-.005 times the diameter of the model cell, which ranged from 24 cm, nominally, were achieved. The membrane is allowed to cure at room temperature for about 24 hr, after which the metal core is melted in boiling water and forced out of the membrane through the hole left by the stem. The now hollow membrane is checked for leaks, dried thoroughly, and then filled with silicone oil (Dow Corning 200 series, v = 125 stokes, S.G. = 0.973). The cell is sealed by either tying off the stem opening or by “welding” it closed with a drop of the original latex solution. By means of a simple tension test it was found that the latex rubber membrane has a linear stress-strain relationship for extensional strains less than 60%. The corresponding Young’s modulus is 1.66 x 10’ dyn/cm2. Poisson’s ratio is 0.5. These mechanical properties and the membrane thickness are found to be unaffected by immersion in silicone oil for periods as long as 1 week. Moreover, no aging was apparent over the duration of experiments conducted with individual cells.
CAPILLARY
BLOOD FLOW II
423
Technique. Pressure drops were always measured between the upper two taps (Fig. 1). A model cell is placed in the upper reservoir and drawn into the tube by the pump. Before the cell reaches the first tap, the pressure drop associated with the Poiseuille flow is recorded. The corresponding electrical signal is then nulled electrically in the transducer indicator. Subsequently the additional pressure drop caused by the cell while it is located between the two taps is recorded. If necessary,the recorder gain may be increased to magnify the additional signal. Becausethe transducer response is linear, the ratio of the two electrical voltages it generates,eDoisand e*, corresponding to the Poiseuille and the additional components of the pressure drop, respectively, is exactly the ratio of the two pressure drops (4): Ap*
e*
The Poiseuille drop can be written (2)
where pf is the dynamic viscosity of the working fluid (silicone oil) at the operating temperature, 0 the average velocity in the tube (volume flow rate divided by crosssectional area), D, the tube diameter, and L the distance between the two taps. Combining Eqs. (1) and (2) we obtain a dimensionless additional pressure drop in terms of the electrical signals, ‘!*
(ELIu/D,)
=32(k)
(2).
(3)
This is actually independent of the length L as long as the cell is more than a few tube diameters away from either tap. For a given cell it depends only on the ratio of cell-totube diameter and the flow rate (0). Cell velocity U, is determined by measuring the time required by the cell to travel a known distance. The average velocity i7 is determined for each setting of the pump control by measuring the velocity of a tightly fitting disc as it is pushed through the tube. All velocity measurementswere reproducible within f 0.25 %. Both pressure-drop and velocity measurements were found to be independent of the direction (up or down) of the flow, indicating an absenceof gravitational effects in the experiments. Similarity Considerations. The subject of similitude, as it pertains to modeling of blood flow in a capillary, has been carefully examined elsewhere(4, 5). In the present experiments two primary quantities are measured: cell velocity, U,, and the additional pressure drop due to a single cell, Ap*. These quantities are presented as dimensionless ratios, the referencefactors being i? and pf o/D=, respectively. A combination of dimensional analysis and prior experience (4, 5) leads us to expect correlations of the form (4)
424
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ET AL.
and
where E, is the elastic “constant”2 of the cell membrane, D, the diameter of the undeformed cell, DT the internal diameter of the tube, and t, the thickness of the cell membrane. The correlations (4) and (5) are shown as incomplete. Missing are an unknown number of geometrical groups which would fully characterize the deformed shape of the cell in the vessel. The exact shape assumed by the cell in the tube cannot be controlled in a model experiment. Nevertheless, the experimental evidence presented here indicates that additional parameters describing the details of cell shape (e.g., the degree of axial asymmetry, and the contour of the rear cavity) would be of secondary importance in the correlations (4) and (5). The primary geometrical parameters are the two ratios shown. The group pfijlDTEc is the ratio of a characteristic viscous stress acting on the cell surface to the membrane’s modulus of elasticity. Thus it can be interpreted as a characteristic tensile strain in the membrane of a given cell. Two important assumptions which are tacitly contained in the correlations (4) and (5) are (i) Reynolds member is unimportant, and (ii) the axial position of the cell in the tube is unimportant. Both of these assumptions are valid because of the extremely low Reynolds number (10m6< pijDT/pf < 10m2)characteristic of capillary blood flow. Provided the Reynolds number achieved in the model study is of the same order, its exact value is not crucial. In low Reynolds-number tube flow the development or entrance length is small, less than one tube diameter. As a result, if a cell is farther than DT from either extremity of the tube, its speed and the additional pressure drop it causes will both be independent of its exact axial station. Range of Parameters Investigated. The exact range of tube-Reynolds number covered in the present model experiments is 8.2 x 10p5-1.65 x 10-3. The corresponding range of average velocity 0 is approximately 0.025-0.5 cm/set. The diameter ratio D,/D, was given values of 1.0, 1.3, and 2.0 while the ratio t,/D, varies between .004 and .005. The deformability parameter pf n/D,E,, which is varied through the average velocity i?, ranges from 4.3 x lo-’ to 8.7 x 10-6. RESULTS
OF THE MODEL
EXPERIMENTS
Cell Orientation and Stability. We will refer to the cell’s orientation (as it moves through the tube) as normal if the natural axis of revolution of the cell is coincident with or parallel to the tube’s axis. At the other extreme, corresponding to perpendicularity or near perpendicularity of these two axes, the orientation will be called edge-on. The observations made of the three different sizes of model cells are summarized below. (i) D,/DT = 1.0. At low flows (0 < 0.1 cm/set) the cell is essentially undeformed. No orientation appears to be stable as the cell is often seen to rotate slowly as it passes * Recent experiments performed in our laboratory indicate strongly that the cell membrane is not a linear elastic material. Here EC is interpreted as the modulus at infinitesimal strain.
CAPILLARY
425
BLOOD FLOW II
through the tube. As the average velocity is increased the cell becomesstable in the edge-on orientation with a concavity on the rear side. This observation is in qualitative agreement with those of Skalak and Brinemark (7). Accompanying the formation of the cavity at the rear is an inflation of the whole forward portion of the membrane. As a result, the forward profile of the cell may appear parabolic even when viewed parallel to the original “plane” of the cell. See,for example, Fig. 1A of Ref. (6). The initial orientation of the cell, as it enters the tube, has no effect on the edge-on stability; a cell happening to enter in the normal orientation is highly likely to rotate into the edge-on attitude. The only perceptible effect of higher velocity (0 > 0.2 cmjsec) is to smooth the wrinkles in the forward portion of the cell membrane, indicating a rising membrane tension.
i J 1.0
.3
mi
/ .6
i 1
I
3
6
IO
30
60
100
x106-
FIG. 2. for O+m.
Cell velocity asa functionof strainparameter.Comparisonwith data of Lee and Fung (5)
(ii) DC/D, = 1.3. SeeFig. 2A of Ref. (6). The cell enters the tube only in the edge-on orientation and cannot be forced to do otherwise. Moving through the tube, the cell is stable in the edge-on orientation. The rear side of the cell is concave at all velocities, but the depth of the concavity increases with velocity. Smoothing of the forward portion of the membrane is again seento accompany increasing velocity. (iii) DC/D, = 2.0. See Fig. 3A of Ref. (6). Again only the edge-on orientation is observed. The deformed cell strongly resemblesa cylinder with one convex and one concave base. The lubrication layer between cell and tube is practically imperceptible at low flows but becomesmore visible as the flow is increased. Cell Velocity. In Fig. 2 the ratio of cell velocity to average flow velocity, UJO, is plotted against the strain parameter, pf ij/D,E,, which in this case may be viewed as a convenient dimensionless flow rate. The diameter ratio DC/D, is a parameter of the plot. For DC/D,, = 1.0 and 1.3 (two smallest cells), the cell velocity is greater than the
426
SUTERA
ET AL.
average velocity by approximately 40 and 207& respectively, but does not change appreciably over the range of U covered. In the case DC/D, = 2.0 (largest cell) a clear dependenceof UCon u is seen, UC/D increasing with flow. This nonlinearity can be explained as a manifestation of decreasing cell cross section (hence an increasing peripheral gap) owing to a changing distribution of surface stressesacting on the membrane. As D becomesvanishingly small the cell expands to nearly fill the lumen while UC-+ D. Similar behavior has been observed in the caseof erythrocytes in vitro (1).
It should be expected that the smaller the fraction of the lumen occupied by the cell the greater will be the ratio UC/D,with an upper bound of 2.0. The data of Fig. 2
200
5
IO
20
50
1 3
x 106-
FIG. 3. Additional pressure drop as a function of strain parameter. Comparison with data of Lee and Fung (5).
verifies this expectation since, at a fixed value of pr U/D,E,, UC/D increases with decreasing DC/D,. Pressure Drop. The dimensionless additional pressure drop due to a single cell is plotted as a function of the strain parameter in Fig. 3. For each of the three cell sizes investigated the additional pressure drop is seen to decreasewith increasing strain parameter. The latter, in these experiments, is varied only through the flow rate. In each case the pressure drop approaches an asymptotic value with increasing flow, indicative of maximum cell deformation. The rate of decreaseof additional pressure drop with flow is most dramatic in the caseDC/D, = 2.0, where the cell assumesan elongated, nearly cylindrical form separatedfrom the tube wall by a thin lubricating layer. The pressure-drop data points carry a maximum uncertainty of f2% in the cases DC/DT = 1.O and 1.3. For DJ D., = 2.0, however, a greater uncertainty, indicated by
CAPILLARY
BLOOD FLOW II
427
the vertical line segments attached to the plotted points, was encountered. This is, due undoubtedly to a wider variation in the detailed pattern of membrane deformation, i.e., location and extent of wrinkles and larger folds, and the significant influence of these details on the average gap width which is very small in this case. Becauseof the sensitivity of pressure drop to gap width (8), a small change in the latter can result in a significantly different pressure drop. This uncertainty is attenuated at higher velocities which are accompanied by higher levels of tensile stress in the membrane. Comparison with Other Experiments and Theory. The only experimental results to which the present findings can be directly compared are those of Lee and Fung (5). The latter investigators have given correlations for the velocity ratio of the form where CLand k are curve-fitting parameters which were apparently determined for the best straight-line fit. For 0 small the velocity ratio predicted by this equation is very sensitive to the parameter CL(a negative value may even occur) which is arbitrary, in a sense,since it depends on the kind of curve presumed. This feature confuses any comparison attempted in the range of the strain parameter covered by our data (0.410 x 10-6). Thus we indicate on Fig. 2 only the values predicted by the Lee and Fung correlation for large 0, i.e., UC/Ox k. On this basis we seethat our measurementsare consistently higher at comparable diameter ratios. Working from the curves in Fig. 16 of the paper by Lee and Fung we have performed a similar comparison for the additional pressure drops. The result is shown in Fig. 3. The two sets of curves do not have similar slopes and barely overlap, but where they do overlap (pr U/D, E, sz lops), it is clear that our data are substantially lower at comparable diameter ratios. (Note D,/D, z 1.0 vs. 0.98, 1.3 vs. 1.36.) Except for the membrane-thicknessratio (t,/D,) all geometrical and flow parameters were of the same order in the two model studies. Our thickness ratio is roughly half that of the cells used by Lee and Fung. The thinner membrane results in greater cell deformability which should, in turn, lead to higher cell velocity and lower additional pressure drop as observed, The caseof large rigid spheresin tube flow has beeninvestigated both experimentally and theoretically as a model for capillary blood flow (8, 9). To test the applicability of a rigid sphere model we have compared our data to corresponding data for rigid spheres. For diameter ratios D,/DT = 1.O and 1.3 and the highest flow used, D = 0.5 cm/set, where, according to the pressure-drop data (Fig. 3) the asymptotic deformation is attained, we compared the diameter ratio actually observed for the deformed cell (i.e., the ratio of maximum lateral dimension of the cell in any given view to the tube diameter) to that required for a rigid sphere to give (i) the same relative velocity and (ii) the sameadditional pressuredrop. (The comparison was not extended to the largest cells, DJD, = 2.0, becausethe deformed shape of these cells is much closer to cylindrical than spherical.) Agreement within 2 % was found. On this basis we would submit that the rigid sphere model can be used to predict erythrocyte velocity and pressure drop in 66IO-pm capillaries, provided that the diameter ratio for the sphere is matched to that of the deformed, erythrocyte in the capillary. However, it is worth emphasizing again that the true nonlinear character of the pressure-flow relationship for capillary blood flow cannot be modeled with
428
SW-ERA ET AL.
rigid cells of one size. This can only be accomplished indirectly through a seriesof rigid cell experiments as suggestedby Sutera and Hochmuth (4). Efict of Cell Spacing. The data displayed in Figs. 2 and 3 apply to single, noninteracting cells. In the physiological situation erythrocytes frequently pass through capillaries in close successionand/or in contact (rouleaux). In order to determine the effect of axial proximity on the additional pressure drop due to a pair of cells, experiments are conducted with two cells of diameter ratio D,/D, = 1.3. It is found that the pressure drop due to the pair is essentially equal to twice the single cell pressure drop given by Fig. 3 as long as the spacing between the cells is not less than DT, approximately. This means that the cells do not interact when they are more than DT apart. [This result could be expected on the grounds that the development length in low Reynolds-number tube flow is small, of the order of the tube diameter, with a lower limit of about 0.65 DT in the limit Re + 0 (lo).] At the other extreme, i.e., when the two cells are in contact, the measuredadditional pressure drop is only about 5 % less than the no-interaction value. This result can be used to estimate the total additional pressure drop due to a long “rouleau” of cells in contact. Let Ap* be the additional pressure drop caused by a single cell moving independently through the capillary. Then two such noninteracting cells (of the same diameter ratio) would cause an additional pressure drop of 2Ap*. Now let dpZC* denote the additional pressure drop caused by two cells in contact, and 6p* the difference between 2Ap* and Ap 2C*.According to the measurements -%!T z ().05. 2Ap”
If we now consider placing a third cell, of the same size, in contact with two others already in contact, it seemsreasonable to assumethat the additional pressure drop for the rouleau of three would be Ap3c* = Apzc* + (Ap” - Sp*), = (2Ap* - Sp*) + (Ap* - 6p*), and = 3Ap” - 26~“.
By inductive reasoning this may be extended to a rouleau of N cells. Then, Ap,,* = NAP” - (N - 1) 6p* or For N large, Ap.wc*
NAP*
According to this simple result the total additional pressure drop due to a long rouleau of N cells in contact, where N B 1, would be about 10% less than N times the single cell drop. Apparent Viscosity of Blood in a Capillary. Apparent viscosity in tube flow is simply
CAPILLARY
BLOOD FLOW II
429
a measure of the total pressure drop through the tube per unit of volumetric flow. As these experiments demonstrate, the total pressure drop through an individual capillary passing blood will depend on the number, spacing, and size distribution of erythrocytes contained in the capillary instantaneously. These same factors are determined by the instantaneous red cell volume in the capillary and hence the average hemotocrit n in the capillary. The apparent viscosity will also depend on the rate of flow through the capillary. In this section we usea simplified theoretical model to estimate the apparent viscosity for blood flow through capillaries of different diameter, as a function of flow rate, and for a particular hematocrit H,, in the blood actually entering the capillary. Consider a steady flow of cells through a capillary and assumethat the diameter ratio is the same for all cells. Thus all the cells will be traveling at the same speed U,. Because U, > 0 (seeFig. 2) the averagehematocrit in the capillary 6 will be lessthan H,,. The relation between 14 and Ho in this hypothetical case is derived in the Appendix; it is simply
Ho_ uc H
0'
(6)
Thus, the hematocrit ratio can be taken directly from the curves of Fig. 2. For DC/D, = 1.0 (corresponding to a capillary of diameter 8.5 pm) and for values of the strain parameter in the range 0.5-9 x 10m6,R E 0.74 H,,. Also derived in the Appendix for this model is an expression for the relative apparent viscosity pr, which is just the ratio of the apparent viscosity to the viscosity of the liquid phase (plasma) : (7) where V is the mean erythrocyte volume and dp* is the additional pressure drop due to a single cell. Now assumeD, = 8.5 pm and V= 87 pm3 for a human erythrocyte. The model-cell data of Figs. 2 and 3, covering diameter ratios DC/D== 1.0, 1.3, and 2.0, are then applicable to capillaries with diameters 8.5,6.6, and 4.25 pm, respectively. Using these data in Eq. (7), we can construct the curves shown in Fig. 4 giving pLIas a function of the strain parameter (or, equivalently, a dimensionless flow rate). The hematocrit Ho was taken to be 0.40 in all cases.The results predict that the relative apparent viscosity will decreasewith increasing flow rate in a given capillary and will be greater the smaller the capillary at a given flow rate. In these calculations the possibility of a spacing correction, as described in the preceding section, was deliberately ignored. According to this simple model blood would flow easily through capillaries as small as 4 pm if the flow rate is high. For a capillary of this size pr < 4 when the strain parameter exceeds about 7 x 10e6. Let us calculate the corresponding D for a human capillary 4 pm in diameter. From ____ = 7 x 10-6 Ec
we get
DT
430
SUTERA ETAL.
I 060
2
A
6
I 8
x 106-
FIG. 4. Calculated apparent relative viscosity of blood in capillaries. Ho = 40%.
Putting
EC = lo6 dyn/cm2,
D, = 4 x 10m4 cm, and pr = .02 dyn-set/cm*,
we find
0 = .14 cmjsec Z 1 mmjsec. This value is quite typical of the maximum erythrocyte velocities observed in mammalian systems. The very high apparent viscosities predicted in Fig. 4, pr > 10, say, would be encountered only in the smallest capillaries when the erythrocyte velocity drops to 0.1 mm/set or lower. In these circumstances the gap between the cell membrane and the endothelial layer of the capillary would be extremely small, and the interaction of the protein coats on the two surfaces might assume an important role. The large-scale model used in this work, with its continuous “plasma” and smooth-walled tube, would be clearly incapable of simulating any effect due to proteins. It is interesting to note the trend, evident in Fig. 4, of relative apparent viscosity increasing with the cell-to-tube diameter ratio at a given value of the strain parameter. This trend seems to oppose that observed by Fahraeus and Lindqvist (11) in the flow of whole blood through tubes in the 40-500-pm range. On the other hand, it is in qualitative agreement with the findings of Gregersen et al. (12) for blood flow through 4.5 and 6.8~pm polycarbonate sieves. In the latter work, relative resistance was seen to increase as pore size decreased. Taken together the results of Fahraeus and Lindqvist and Gregersen et al. suggest a possible minimum in the apparent viscosity when regarded as a function of diameter ratio (or tube or pore size). The value or values of D,/DT which give a minimum apparent viscosity cannot be pinpointed here; but it must be emphasized that comparisons of apparent viscosity in the capillary regime will not be valid if proper consideration is not given to the dependence of apparent viscosity on the strain parameter (or average velocity). The significance of this remark is shown by the curves in Fig. 4. For DC/D, = 1.0 and
CAPILLARY
BLOOD
FLOW
II
431
pf o/E, DT = 0.5 x lO-(j, pLrz 2.0, while for D,/DT = 1.3 (a smaller tube) and pLrU/E, D, = 5 x 1O-6(and a higher flow) p., z 1.8.This comparison, at two significantly different values of the strain parameter, indicates a decreasein apparent viscosity with decreasingtube diameter. This is contrary to the true trend indicated by the full curves in the figure.
CONCLUDING
REMARKS
The observed behavior of the deformable model cells shows that the edge-wisemode of motion through capillaries 6-10 pm in diameter is the only stable mode. Consequently, the shape of the forward portion of the deformed cell is not generally axisymmetric. The formation of a deep concavity at the rear end of the cell is accompanied by an inflation of the forward portion, with the result that the nose profile appears quasi-parabolic from any side. Also, the minimum radius of curvature at the nose will be greater than the edge radius seenin a meridional section of the relaxed cell. Previous analysesof erythrocyte deformation in capillaries (13-I 5) have assumed an axisymmetric profile. The present experiments suggestthat some account of asymmetry should be attempted in future analysesfollowing the example of Fitz-Gerald (16). Confidence in the applicability of a large-scalemodel to capillary blood flow depends on the degree of agreement that can be seen between quantities observable on the actual scale (in vivo or in vitro) and their counterparts in the model. The first paper of this series (1) reports on the deformation of erythrocytes moving through glass capillaries and quantitatively characterizes the observed deformations. In the final paper (6) we compare these with corresponding observations on the large-scale deformable models. Appendix
APPARENT VISCOSITY OF BLOOD IN A CAPILLARY Consider the steady flow of blood in a capillary of diameter DT and length L $ DT. Assume that a large number of erythrocytes are moving through the capillary in single file. The spacing between successivecells is variable, but the total number of cells in the capillary at any instant is n. Then the average hematocrit in the capillary, R, is nV H = (irD34)
64.1)
L’
where V is the volume of one cell. Let Q be the volumetric rate of blood flow through the capillary; then the average velocity in the capillary (an average over the lumen) is defined as
If the velocity of erythrocytes is U,, the it cells which are distributed along the distance will exit from the capillary in a time interval L/U,. During this same interval the total volume (cells plus plasma) leaving the tube is
L
QW uc)
= u(77D:/4)
(Ll uc).
432
SUTERA
ET AL.
The ratio of cell volume to this total efflux volume (the “cup-mixing” of cells in the receiving vessel) is denoted by H,, :
concentration
64.3) This concentration, or hematocrit, is identical to the average concentration blood actually entering the capillary. Combining (A.l) and (A.3) we find
I7 u Ho uc’
in the
b4.4)
Since U, > 0 (Fig. 2) this relation says that E ( Ho. The total pressure drop through the capillary is represented as a sum of a pure Poiseuille drop, associated with pure plasma flow at the average velocity 0, and an additional component due to the n cells in the capillary. The Poiseuille component is
(A.3 where pLfis the plasma viscosity. Neglecting possible interactions between successive cells, a correction of 10 ‘A at most, we can write the additional pressure drop as follows:
The bracketed factor is the dimensionless additional pressure drop caused by a single cell (Fig. 3). Using (A.3) to replace n we then obtain (A.6) The relative apparent viscosity can be expressed as the ratio of the total pressure drop to the Poiseuille drop at the same 0. Thus
Recall that this is the apparent relative viscosity of blood with hematocrit Ho being forced steadily through a capillary of diameter DT with an average velocity 8. The actual volume average of cells in the capillary at an instant is generally less than HO [according to (A.4)]. REFERENCES 1. HOCHMUTH,R. M., MARPLE,R. N., ANDSUTERA,S. P. (1970). Capillary blood flow. I. Erythrocyte deformation in glass capillaries. Microvasc. Res. 2,409. 2. PROTHERO, J., AND BURTON,A. C. (1961). The physics of blood flow in capillaries. I. The nature of motion. Biophys. J. 1, 565. 3. PROTHERO, J., AND BURTON,A. C. (1962). The physics of blood flow in capillaries. II. The capillary resistance to flow. Biophys. J. 2, 199. 4. SUTERA,S. P., AND HOCHMUTH,R. M. (1968). Large scale modeling of blood flow in the capillaries. Biorheology
5, 45.
CAPILLARY
BLOOD FLOW II
433
5. LEE, J. S., AND FUNG, Y. C. (1969). Modeling experiments of a single red blood cell moving in a capillary blood vessel. Microuusc. Rex 1,221. 6. SESHADRI,V., HOCHMUTH,R. M., CROCE,P. A., AND SUTERA,S. P. (1970). Capillary blood flow. III. Deformable model cells compared to erythrocytes in uitro. Microvase. Res. 2,434. 7. SKALAK, R., AND BRANEMARK,P-I. (1969). Deformation of red blood cells in capillaries. Science 164,717. 8. HOCHMUTH,R. M., AND SUTERA,S. P. (1970). Spherical caps in low Reynolds-number tube flow. Chem. Eng. Sci. 25, 593.
9. WANG, H., AND SKALAK,R. (1969). Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 75. 10. LEW, H. S., AND FUNG, Y. C. (1969). On a low Reynolds number entry flow into a circular cylindrical tube. J. Biomechanics 2, 105. 11. FKHRAEUS,R., AND LINDQVIST,T. (1931). The viscosity of blood in narrow capillaries. Amer. .I. Physiol. 96, 562.
12. GREGERSEN, M. I., BRYANT, C. A., HAMMERLE,W. E., USAMI, S., AND CHIEN, S. (1967). Flow characteristics of human erythrocytes through polycarbonate sieves. Science 157, 825, 1967. 13. CHARM,S. E., AND NELSON,F. (1966). Red cell deformation and flow in capillaries. 4th Eur. Con/I Microcir.
Cambridge.
14. BARNARD,A. C. L., LOPEZ,L., ANDHELLUMS,J. D. (1968). Basic theory of blood flow in capillaries. Microvas.
Res. 1, 23.
15. LIGHTHILL, M. J. (1968). Pressure forcing of tightly fitting pellets along fluid filled elastic tubes. .I. Fluid Mech. 34, 113.
16. FITZ-GERALD,J. M. (1969). Mechanics of red cell motion through very narrow capillaries. Proc. Roy. Sot. Ser. B. 174,193.