Adaptation and survival of surface-deprived red blood cells in mice

Ryan C. Murdock, Christopher Reynolds, Ingrid H. Sarelius, and Richard E. Waugh

Department of Pharmacology and Physiology, University of Rochester Medical Center, Rochester, New York 14642


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

The consequences of lost membrane area for long-term erythrocyte survival in the circulation were investigated. Mouse red blood cells were treated with lysophosphatidylcholine to reduce membrane area, labeled fluorescently, reinfused into recipient mice, and then sampled periodically for 35 days. The circulating fraction of the modified cells decreased on an approximately exponential time course, with time constants ranging from 2 to 14 days. The ratio of volume to surface area of the surviving cells, measured using micropipettes, decreased rapidly over the first 5 days after infusion to within 5% of normal. This occurred by both preferential removal of the most spherical cells and modification of others, possibly due to membrane stress developed during transient trapping of cells in the microvasculature. After 5 days, the cell area decreased with time in the circulation, but the ratio of volume to surface area remained essentially constant. These results demonstrate that the ratio of cell volume to surface area is a major determinant of the ability of erythrocytes to circulate properly.

circulation; spherocytosis; erythrocytes; senescence; deformability


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

THE MECHANICAL PROPERTIES of the red blood cell (RBC) have important effects on blood perfusion and oxygen delivery. The shape and deformability of RBCs have been shown to be significant factors in pathologies such as sickle cell anemia (13, 14) and hereditary spherocytosis (29), and, in many cases, the origin of RBC disorders has been traced to specific molecular lesions in the structural proteins of the RBC membrane (20). In the present study, we examine the specific issue of how loss of cell surface area affects the ability of the cell to circulate and what adaptations occur in vivo to improve the cell's chances for survival.

The deformability of the RBC has been well studied (5, 30). The limited ability of the membrane area to change, combined with constraints on cell volume resulting from the low permeability of the membrane to cations, results in a strict limitation on the ability of the RBC to deform and negotiate small vessels and constrictions within the microvasculature. Within this constraint, the shear elasticity, viscosity, and the bending resistance of the membrane determine the response of the cell to applied forces. However, the forces required to deform the cell in shear or bending are very small compared with the cell resistance to change in area and change in volume. Thus the ratio of membrane surface area to cell volume is a critical determinant of cellular deformability, and it is expected to have a major influence on the ability of cells to circulate.

Over the past 50 years, there have been a number of studies examining the relationship between RBC deformability and circulation. A number of these studies provide evidence that, in many hemolytic disorders, the membrane, or more specifically, the ratio of the surface area and volume, is the significant factor that affects the ability of cells to circulate. As early as the 1940s, investigators observed that highly spherical RBCs that formed as a result of thermal injury disappeared from the circulation of burn victims over a period of ~18 h (9). In a more recent study (8), investigators made RBCs less deformable by using sulfhydryl reagents to cross-link the membrane proteins and increase the shear rigidity of the cells. Interestingly, they found no relationship between increased shear rigidity and the removal of the cells from the circulation. However, they did find that, when the cells were heated for 15 min at 50°C, a process that decreases the effective surface area of the cell and thus increases the volume-to-surface area ratio, the cells were rapidly removed from the circulation (half-time = 10-30 min).

In a recent study from our own laboratories, the effect of reduced surface area on the short-term survival of RBCs was examined (32). Cell surface area was reduced by inducing endocytosis of the membrane, and the cells were labeled with a fluorescent marker to distinguish them from the rest of the cells in the circulation after reinfusion into a mouse. As expected, a rapid decrease in the circulating fraction of highly spherical cells was observed over a period of ~30 min. Somewhat surprisingly, ~20% of the modified cells remained in the system for up to 4 h, although most of the highly spherical cells were rapidly removed from the circulation. These surviving cells showed evidence of reductions in cell volume, as if to compensate for the reduction in surface area and maintain a more normal ratio of volume to surface area. In this present study, we extend this earlier study to examine long-term effects of lost membrane area on cell circulation, focusing specifically on the disappearance of spherical cells from the circulation and changes in the dimensions (area and volume) of the surviving population.


    MATERIALS AND METHODS
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Cell preparation. Donor mice (C57BLJ, 2-3 per experiment) were anesthetized (pentobarbital sodium, 70 mg/kg ip), and cardiac puncture was performed with the use of a 1.0-ml syringe with a 25-gauge needle. The dead space between the tip of the syringe and the needle was filled with heparin before the blood was withdrawn. The blood was then placed in 1.5-ml Eppendorf microcentrifuge tubes and spun for 4 min at 3,700 rpm (Eppendorf model 54152). The plasma and buffy coat were aspirated, and the cells were washed three times in 290 mosM PBS (160 mM NaCl, 25.0 mM Na2HPO4, and 6.2 mM KH2PO4), pH 7.4. An 8.0 mM stock solution of lysophosphatidylcholine (LPC; Avanti Polar Lipids, Alabaster, AL) in 1:1 chloroform/methanol was prepared. RBCs were suspended in ~100 ml of 290 mosM PBS (1.0% vol/vol). LPC stock solution was added to 100 ml of 290 mosM PBS to an LPC concentration of 0.23 µM and sonicated for 20 min at 40°C (Branson model 2210). The LPC-PBS solution was cooled to room temperature, combined with an equal volume of the RBC-PBS solution, and then allowed to rotate for 10 min at 120 rpm. LPC-RBC-PBS solution was then divided equally, such that the solutions could be placed into an even number of 50-ml centrifuge tubes, and centrifuged for 6 min at 2,000 rpm (IEC model HN-SIIC). After centrifugation, the RBC pellets were collected and resuspended in a 50-ml centrifuge tube and then washed in RBC storage solution [RBCSS (in g): 0.225 glucose, 2.330 KCl, 0.263 NaCl, 4.279 sucrose, and 1.247 HEPES hemisodium salt dissolved in 250 ml of deionized H2O], pH 7.4, 340 mosM. The suspension was then spun for 8 min at 2,000 rpm, and the supernatant was removed. The pellet of RBCs was added to 35 ml of HEPES-buffered salt solution, pH 7.4 (in mM: 22.8 sodium bicarbonate, 8.8 HEPES sodium salt, and 11.2 HEPES acid), 320 mosM, and centrifuged for 8 min at 2,000 rpm. The supernatant was removed and discarded.

After incubation, the cells were labeled with substituted tetramethylrhodamine isothiocyanate using previously published protocols (23, 26). After cells were labeled, they were stored overnight at 4°C in RBCSS.

Reinfusion and sampling of modified cells. On the morning after cell incubation and labeling, the cells were washed twice in the overnight storage solution, followed by two washes in an excess of HEPES-buffered physiological salt solution containing 0.5% bovine serum albumin at pH 8.1 and then resuspended in storage solution at pH 7.35. All wash solutions were at an osmolarity of 320 mosM. After the last wash, 0.15 ml of packed cells were drawn into a 0.5-ml syringe and injected and into the tail vein. To inject the bolus of cells into the mouse, the animal was placed in a holding device that allowed accessibility to the tail.

To collect blood samples for fraction analysis and area and volume measurements, the mice were sedated with 0.15-0.17 ml of ketamine; blood was collected from a small nick made in a toe pad. The blood droplets were collected using a hematocrit capillary tube, suspended in a solution of PBS + 2.0% fetal calf serum (FCS), and then diluted to the desired concentration as determined by inspection. Samples, which could be used for both the circulating fraction and the area and volume measurements, were taken at day 0, day 1, and day 5 and then approximately every 7 days thereafter until the circulating fraction fell below 0.1%.

The circulating fraction was determined using fluorescent light microscopy and video image analysis software (1). The number of cells counted did not exceed 10,000 and was such that the coefficient of variation (CV) of the circulating fractions for all but the last two samples for each mouse was <20% (24). For the last two fractions, the CV was typically >25% because of the very small number of labeled cells remaining in these samples.

Micropipette measurements. Cell surface area and volume were measured using micropipettes according to established procedures (29). Micropipettes were prepared by fracturing off the tip of a glass capillary pulled to a needle point. The pipettes had an inside diameter between 1.2 and 1.8 µm. Cells were suspended at low (<1.0%) hematocrit in PBS plus 2.0% (vol/vol) FCS adjusted to an osmolarity of 316-322 mosM, pH 7.35. Cells were placed in a U-shaped chamber on the stage of an inverted microscope, and the micropipette was introduced via the open side of the chamber. The aspiration pressure at the tip of the pipette was controlled by adjusting the height of a water-filled reservoir connected to the back of the pipette via water-filled tubing. The image from the microscope was observed via a television camera and recorded on videotape for subsequent analysis. To measure cell surface area and volume, cells were aspirated at a pressure of ~1,200 Pa (12 cmH2O) to ensure that the membrane was fully extended into the pipette and that there were no folds or creases in the membrane projection (Fig. 1). The membrane area (A) and the cell volume (V) were calculated from measurements of the outer cell radius (Rc) and the length of the projection in the pipette (Lp)
A=2&pgr;R<SUB>c</SUB><FENCE><IT>R</IT><SUB>c</SUB><IT>+</IT><RAD><RCD><IT>R</IT><SUP><IT>2</IT></SUP><SUB>c</SUB><IT>−R</IT><SUP><IT>2</IT></SUP><SUB>p</SUB></RCD></RAD></FENCE><IT>+2&pgr;R</IT><SUB>p</SUB><IT>L</IT><SUB>p</SUB> (1)

V<IT>=</IT><FR><NU><IT>2&pgr;</IT></NU><DE><IT>3</IT></DE></FR> <FENCE><IT>R</IT><SUP><IT>3</IT></SUP><SUB>c</SUB><IT>+</IT><FENCE><IT>R</IT><SUP><IT>2</IT></SUP><SUB>c</SUB><IT>+</IT><FR><NU><IT>R</IT><SUP><IT>2</IT></SUP><SUB>p</SUB></NU><DE><IT>2</IT></DE></FR></FENCE><RAD><RCD><IT>R</IT><SUP><IT>2</IT></SUP><SUB>c</SUB><IT>−R</IT><SUP><IT>2</IT></SUP><SUB>p</SUB></RCD></RAD><IT>+R</IT><SUP><IT>3</IT></SUP><SUB>p</SUB></FENCE><IT>+&pgr;R</IT><SUP><IT>2</IT></SUP><SUB>p</SUB>(<IT>L</IT><SUB>p</SUB><IT>−R</IT><SUB>p</SUB>) (2)
where Rp is the inside radius of the pipette. One measure of the volume-to-surface area ratio of the cell is the sphericity (S). This is a dimensionless quantity proportional to the ratio of the two-thirds power of the cell volume to the membrane area
S=<FR><NU>4&pgr;</NU><DE>(4/3&pgr;)<SUP>2/3</SUP></DE></FR>·<FR><NU>V<SUP><IT>2/3</IT></SUP></NU><DE><IT>A</IT></DE></FR> (3)
The coefficient is constructed such that the maximum value of the sphericity is 1.0 (a perfect sphere). The smaller the value of the sphericity, the greater is the "excess" surface area of the cell, that is, the area in excess of the area required to enclose the spherical volume of the cell. The percentage of excess area can be calculated as
A<SUB>ex</SUB><IT>=</IT>(<IT>1/S−1.0</IT>)<IT>×100%</IT> (4)
Approximately 100 cells were measured for each population that was sampled. Statistical significance of difference between samples was evaluated at the 95% confidence level.


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Fig. 1.   Video micrograph showing a mouse red blood cell aspirated into a micropipette. The dimensions of the cell used to calculate surface area (A), volume (V), and sphericity (S) are indicated. For the cell depicted, A = 80 µm2, V = 44.4 µm3, S = 0.752, and percentage of excess A = 33.0%. Rc, outer cell radius; Lp, length of the projection in the pipette.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Nine different mice received modified cells and were sampled over a minimum of 5 wk after reinfusion. For the presentation of the data, day 0 denotes a sample taken at ~3 h after the infusion of the modified cells into the mouse and day 1 denotes a sample taken at ~24 h after infusion. The remaining samples are labeled according to the number of days after the day 0 sample at which they were retrieved. On day 0, the areas, volumes, and sphericities of the modified cells were significantly different from controls (Fig. 2), although the degree of modification varied from one preparation to another. Generally, the area of the modified cells was significantly reduced (Fig. 2A), and the volume of the cohorts was slightly elevated (Fig. 2B), resulting in a substantial increase in the sphericity of cells in the modified population (Fig. 2C).


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Fig. 2.   Effects of lysophosphatidylcholine treatment on cell dimensions. A: distribution of cell surface areas shifts toward lower values. B: distribution of cell volume changes little, increasing very slightly. C: sphericity increases substantially, with many cells becoming nearly perfect spheres.

The percentage of modified cells remaining in the vasculature declined with time (Fig. 3). The time course appeared to be approximately that of a decaying exponential, but, for at least one-half of the mice studied, the rate of disappearance was more rapid during the first 5 days after reinfusion than during the subsequent time. This is illustrated in Table 1, in which the time constants are tabulated for first-order exponential curves fit to the measured circulating fractions for all days and for days 5-35. It is worthwhile to note that the uppermost curve in Fig. 3, that is, the case in which the cells disappeared from circulation most gradually (experiment 5 in Table 1), corresponds to the least modification in the sphericity of the modified cells. In this case, an increase in sphericity of only 8% relative to control was achieved, whereas, in most other cases, increases of 12-20% were obtained. An inverse correlation was also observed between the mean sphericity of the modified cohort measured on day 5 and the time constant for cell disappearance (R = -0.71). Thus smaller changes in sphericity appeared to have a relatively smaller effect on the ability of cells to circulate.


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Fig. 3.   Circulating fraction as a function of time after reinfusion. Nine mice were studied, and each is represented by a different symbol. Time course of disappearance appeared to be approximately exponential, with time constants ranging from 2.2 to 13.7 days.


                              
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Table 1.   Time constants for the disappearance of modified cells from the circulation

Unlabeled cells from the host mouse were used as matched controls for the labeled cohort in each sample. The absolute dimensions measured for these controls varied systematically depending on the day on which the measurements were made. That is, when two or more different mice were measured on the same calendar day, the values tended to agree with each other but were larger or smaller than controls measured for the same mice on different days. We attribute this to either error in the measurement of the pipette diameter, slight variations in the concentration of the suspending buffer, or differences in light or contrast levels in the recorded images on different days. To avoid introducing errors in parameter values for the cohort populations because of these systematic measurement fluctuations, data taken after the infusion of cells were normalized with respect to the mean value of the corresponding parameter for the unmodified (control) sample for the host mouse on that day.

The general trends for how the mean area, volume, and sphericity of the labeled cohorts changed with time in the circulation are shown in Fig. 4. The normalized mean values for all cohorts were averaged, and the averages were plotted as a function of time after reinfusion. In general, the mean membrane area increased during the first 5 days after reinfusion and then gradually decreased thereafter. The normalized area for day 5 was significantly larger (Student's t-test) than the normalized area on both day 0 and day 33. The mean volume decreased monotonically with time after reinfusion. The sphericity decreased rapidly during the first 5 days and then remained relatively constant for the duration of the study. The mean normalized sphericity decreased by 7.8% during the first 5 days and then by less than 2% during the next 7 days, and there was no significant change (Student's t-test) from day 12 through day 33.


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Fig. 4.   Changes in the mean dimensions of the modified cell populations as a function of time after reinfusion. All parameters are expressed relative to the mean of the corresponding control parameter. Points represent the average of 9 normalized population means, and error bars indicate the standard deviation. A: mean surface area of the surviving population generally increased in the first few days after reinfusion and then declined gradually as the cells aged. B: mean cell volume of the surviving population decreased monotonically with time after reinfusion. C: mean sphericity of the surviving cells decreased rapidly toward normal values in the first week after reinfusion and then remained relatively constant during the remainder of time in which cells were detectable in the circulation.

Single cell micropipette measurements enabled us to examine not only changes in mean values for the circulating cohort but also changes in the distribution of area, volume, and sphericity within the population. In most of the mice studied (7 of 9), there was a shift in the areas of the modified population toward control values during the first 5 days after infusion (Fig. 5A). (In the other 2 mice, the mean area of the cohort population did not change over the first 5 days.) This increase in mean cell area for the surviving population was accompanied by a slight decrease in the mean cell volume (Fig. 5B) in five of the nine mice. In three of the mice, the mean volume did not change, and, in one mouse, the mean volume actually increased slightly. In all mice, there was a dramatic shift in the mean cell sphericity for the surviving cells toward control values (Fig. 5C).


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Fig. 5.   An example of the change in the distribution of cell dimensions after reinfusion. Parameter values for each cell were normalized with respect to the mean value of the corresponding parameter for control population. A: distribution of cell areas shifted toward higher values. Control means were 86.0, 86.6, and 82.5 µm2 for days 0, 1, and 5, respectively. B: distribution of cell volumes changed little or decreased very slightly. Control means were 51.4, 52.9, and 45.7 µm3 for days 0, 1, and 5, respectively. C: distribution of sphericity decreased toward normal values. Control means were 0.776, 0.787, and 0.748 for days 0, 1, and 5, respectively. Note that for these values of the control sphericity the maximum normalized sphericity (i.e., the value for a perfectly spherical cell) was 1.27-1.34.

The change in the distribution of cell surface area and volume in the surviving population appeared to be due to a combination of preferential removal of spherical cells from the circulation and changes in the dimensions of the remaining cells toward more normal values. This conclusion was reached by comparing the distribution of cell sphericity within the modified population on successive days with each distribution multiplied by the corresponding circulating fraction (Fig. 6). Between day 0 and day 5, there is a decrease in the fraction of circulating cells with sphericities greater than 20% above normal; however, there is an increase in the fraction of circulating cells with sphericities near normal. (Note that, in Fig. 6, the solid bars for day 5 are shorter than the day 0 bars at the right side of the distribution but are taller than the day 0 bars at the left side of the distribution.) The decrease in the fraction of circulating cells with sphericities more than 20% above normal can be explained by the selective removal of cells with high sphericity, but the increase in the percentage of circulating cells with sphericities near normal can only be explained by the modification or adaptation of some labeled cells while they were in the circulation.


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Fig. 6.   Typical example of the distribution of sphericity multiplied by the corresponding circulating fraction (circ frac) for days 0-5. Note that the height of the bar in each bin is proportional to the circulating fraction of cells within that bin for the different days (bin width = 0.05). Note that the number of cells with sphericities near normal increased with time after reinfusion. Because there were no new cells introduced into the circulation over this period, these results demonstrate that the sphericity of individual cells was altered toward normal values after the cells were reinfused into the animal. Control means for this mouse were 0.736, 0.737, and 0.748 for days 0, 1, and 5, respectively. Note that the normalized value for a perfectly spherical cell was ~1.35.

Changes in the distribution of area, volume, and sphericity of the labeled cohorts from day 5 onward were compared by first normalizing the cellular values by the mean of the matched control population, multiplying these values by the circulating fractions, and then fitting the distribution with a Gaussian curve. The distributions for one of the mice used in the study are shown in Fig. 7. (Distributions for the other mice in the study were similar.) Note that each successive distribution falls within the boundary of the previous one. This result is consistent with the possibility that the shifts in the distribution could be occurring simply by the preferential removal of cells of a certain size. However, these observations do not rule out the possibility that there may be changes in the dimensions of individual cells within the distribution.


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Fig. 7.   Changes in the number and distribution of cells with different dimensions after 5 days in the circulation. Frequency distributions were multiplied by the corresponding circulating fractions so that the height of the fitted curves reflects the relative number of cells having different dimension. Fitted curves, rather than frequency histograms, are shown to avoid clutter. A: surface area. B: volume. C: sphericity. Line styles corresponding to different numbers of days after reinfusion are shown in the insets.

Model calculations. Calculations were performed to estimate the magnitude of forces that might be generated in the membranes of circulating cells as a basis for evaluating possible mechanisms for cell removal or modification of cell volume. The rationale for these calculations is that spherical cells may not be able to negotiate the smallest capillaries in the vasculature, particularly at sites of constriction as might occur where nuclei of endothelial cells protrude into the vessel lumen. We were particularly interested in estimating the size of vessels or vessel constrictions that would trap cells of different dimensions and the magnitude of the membrane stresses that might develop in cells trapped in those constrictions. This analysis follows the work of Fischer (7) but uses a slightly simpler geometric model of a vascular constriction and extends the analysis to consider a wider range of cell dimensions. We specifically address the importance of volume-to-surface area ratio in determining cell entrapment and the resulting mechanical forces in the membrane.

A simple geometry of a cell trapped in a tapered tube was chosen (see Fig. 8). The RBC membrane was treated as a thin-walled shell with isotropic properties in the plane of the membrane (5). The exchange of water between the cell and its environment due to osmotic and hydrostatic pressures was assumed to be at steady state; thus the cell volume was taken to be constant. The upstream (Ru) and downstream (Rd) radii of the RBC are related to the slope (M) of the tube taper and radius of the conical tube at the middle of the conical portion of the cell [we will refer to this as the midradius (Rm)]
R<SUB>u</SUB><IT>=R</IT><SUB>m</SUB><IT>−M </IT><FR><NU><IT>L</IT></NU><DE><IT>2</IT></DE></FR> (5A)

R<SUB>d</SUB><IT>=R</IT><SUB>m</SUB><IT>+M </IT><FR><NU><IT>L</IT></NU><DE><IT>2</IT></DE></FR> (5B)
With the use of these relationships, the area and volume of the shape were approximated as
A=4&pgr;R<SUP><IT>2</IT></SUP><SUB>m</SUB><IT>+2&pgr;</IT><RAD><RCD><IT>1+M<SUP>2</SUP></IT></RCD></RAD><IT> R</IT><SUB>m</SUB><IT>L+&pgr;M<SUP>2</SUP>L<SUP>2</SUP></IT> (6)

V<IT>=</IT><FR><NU><IT>4</IT></NU><DE><IT>3</IT></DE></FR><IT> &pgr;R</IT><SUP><IT>3</IT></SUP><SUB>m</SUB><IT>+&pgr;R</IT><SUB>m</SUB><IT>M<SUP>2</SUP>L<SUP>2</SUP>+&pgr;R</IT><SUP><IT>2</IT></SUP><SUB>m</SUB><IT>L+&pgr; </IT><FR><NU><IT>M<SUP>2</SUP>L<SUP>3</SUP></IT></NU><DE><IT>12</IT></DE></FR> (7)
Force resultants in the membrane were determined by applying a balance of forces at the upstream and downstream ends of the cell. The force resultants in the membrane were assumed to be uniform, as was the pressure inside the cell. The isotropic force resultant [or "isotropic tension" in the membrane (T)] can then be written as
T<IT>=</IT><FR><NU><IT>&Dgr;</IT>P</NU><DE>2</DE></FR> <FENCE><FR><NU>1</NU><DE><IT>R</IT><SUB>d</SUB></DE></FR><IT>−</IT><FR><NU><IT>1</IT></NU><DE><IT>R</IT><SUB>u</SUB></DE></FR></FENCE><SUP><IT>−1</IT></SUP> (8)
where Delta P is the pressure difference between the upstream and downstream ends of the cell.


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Fig. 8.   Schematic showing the dimensions and shape of the model of a cell trapped in a vascular convergence. M, slope (taper) of the convergence; Ru, upstream radius; Rd, downstream radius; Rm, midradius.

Equations 5-7 completely describe the geometrical aspects of the problem. By specifying the cell area, the midradius, and the tube taper, the length, volume, and sphericity of the cell can be calculated, enabling us to evaluate how changes in cellular dimensions affect the size of apertures in which the cell would become trapped. For gradually tapering tubes ( |M|  < 0.1), this part of the problem becomes insensitive to the value of M, and dependence of the trapping tube radius on cell area and sphericity can be explored using approximate relationships (see APPENDIX). A particularly useful relationship is an expression for the midradius as a function of the cell area and the sphericity
R<SUB>m</SUB><IT>=</IT><RAD><RCD> <FR><NU><IT>A</IT></NU><DE><IT>&pgr;</IT></DE></FR></RCD></RAD> cos <FENCE><FR><NU>cos<SUP><IT>−1</IT></SUP>(<IT>−S<SUP>3/2</SUP></IT>)<IT>+4&pgr;</IT></NU><DE><IT>3</IT></DE></FR></FENCE> (9)
This expression is exact when M = 0, yielding the exact value of the minimum cylindrical tube radius through which a cell of the given dimensions can pass.

Values for the area, the midradius, slope, and pressure were chosen to approximate a range of physiological conditions. Five slopes for the capillary taper were chosen to model possible changes in capillary lumen diameter due to protrusions into the lumen. The largest slope (-0.3) is close to the value used in a previous model of RBC flow in capillaries with variable cross sections (25), and the lowest value was chosen to model small changes in capillary geometry. Estimates of the pressure drop across the capillaries of mice (16) and rats (2) range from <1.0 mmHg (16) to >7.0 mmHg (2).

The calculations indicate that the trapping radius has a relatively weak dependence on cell area but a stronger dependence on sphericity (Fig. 9A). For example, at sphericity = 0.85, a cell with an area of 65 µm2 has midradius of 1.18 µm and a cell with the same sphericity but an area of 105 µm2 has a midradius of 1.66 µm. Thus an ~60% change in area results in an ~40% change in midradius. The same change in midradius can be produced by just a 15% change in sphericity (area held constant). For the results shown in Fig. 9A, the chosen value of M was -0.1. The effect of changing the tube taper on the relationship between the trapping radius and sphericity is illustrated in Fig. 9B. Note that the relationship is essentially independent of slope when  |M|  < 0.1. 


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Fig. 9.   Calculations of the midradius at which a cell is trapped as a function of sphericity. A: dependence for cells having different areas in a tube with taper M = -0.1. B: dependence for cells with a surface area of 85 µm2 in tubes with different tapers.

The dependence of the cell membrane tension on these geometrical parameters was also investigated. There is strong dependence of tension on sphericity, but the dependence of tension on area is slight, as evidenced by the overlap of the points for the different areas in Fig. 10A. (The tension scales linearly with pressure; thus tensions at pressure differences other than 10 mmHg can be determined readily from the values shown in Fig. 10.) There is also a strong dependence of the membrane tension on the tube taper (Fig. 10B). As the taper becomes more gradual, the membrane tensions can become very large. This is easily explained by inspection of Eq. 8. As the upstream and downstream radii approach a common value, the denominator in Eq. 8 approaches zero. Thus, for gradual tapers, large membrane tensions result from moderate pressures for trapped cells of any size, and, as cells become more spherical, tensions are large at all tapers because the upstream and downstream regions are separated by shorter distances and thus become similar in radius. It is important to note that, for cells with elevated sphericities (S > 0.9) in vessels with a slight taper (M = -0.01), tensions that are large enough to lyse the cell can result, even for very small pressure differences (1.0 mmHg).


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Fig. 10.   Dependence of membrane tension on cell and tube geometry at a transcellular pressure of 10 mmHg. (Note that the tension scales linearly with pressure so that tensions at other pressures can be calculated by a simple proportionality.) A: for a tube taper of slope = -0.1, tension becomes large for sphericity >0.9 for cells of all surface areas. B: as taper decreases, tensions at all values of sphericity increase significantly (cell area = 85 µm2).

The calculations presented so far do not account for the ability of the cell to increase its area or decrease its volume in response to applied forces. When large but sublethal forces are applied to cells, fractional area changes of 1-2% and changes in volume of similar magnitude are possible. (This corresponds to changes in sphericity of 2-4%.) Such changes might enable the cells to escape local constrictions in capillaries and continue to circulate. Therefore, it is of interest to consider how much axial travel might occur past the initial point of trapping if stress-induced changes in sphericity occur. In Fig. 11, the displacement along the tube axis resulting from a decrease in sphericity of 3% is plotted as a function of the taper for cells with different initial sphericities. These results show that cells can move the farthest when the taper is low and that the distances are significant. It is interesting that, although small tapers are most "dangerous" for cells because membrane tensions tend to be large, cells also have a better chance to escape from local, slowly tapering constrictions by small changes in area or volume.


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Fig. 11.   Effect of changing taper on the distance a cell would travel down a tapering capillary as a result of a 3% decrease in sphericity. Curves represent travel as a function of the tube taper for 5 different starting values for the sphericity. At small tapers, cells can travel significant distances as a result of small changes in sphericity (S), improving their chance to "escape" from entrapment and continue in the circulation.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

The importance of RBC deformability for RBC survival and for proper cell flux and oxygen delivery has been widely accepted for many years, but there has been little quantitative information published about the precise consequences of alterations in cell deformability on the ability of cells to circulate. The present study provides quantitative information about how specific changes in RBC area and volume affect long-term cell survival in the circulation. The results point to the ratio of volume to surface area as a major determinant of the ability of a cell to circulate properly. The tolerance of the circulation for abnormalities in this parameter is extremely low. Although cells having a wide range of different sizes circulate equally well, the data indicate that a cell with abnormal volume-to-surface area ratio is either removed from the circulation or its dimensions are altered to bring the ratio of volume to surface area to within 5% of the normal range.

Model calculations suggest that cell trapping is a likely mechanism both for cell removal and for the modification of cell dimensions to improve the chances of survival. For cells with average surface areas and elevated sphericities, the calculations indicate that a cell will become trapped in passages with diameters on the order of 2-3 µm. Average capillary dimensions in the mouse are ~5.0 µm (18), but constrictions caused by endothelial nuclei and other irregularities may reduce the local capillary diameter to <3.0 µm and in some cases to <1.0 µm (17, 21). Indeed, transient (~20 s) trapping of RBCs in such constrictions has been observed experimentally (21). Calculations show that cells trapped in such constrictions may develop lytic or near-lytic membrane stresses when they become lodged in constrictions and subjected to transcellular pressures of magnitudes found in capillary networks in vivo (7). Past experiments have shown that membrane tensions of 3-4 mN/m do not produce lysis, but membrane tensions of 6 mN/m or greater result in lysis within ~10 s (6). Thus cells may be lysed as a result of membrane stresses generated by the pressure in the capillary and the geometry of the trap, or the stresses induced in the membrane may cause changes in membrane permeability, leading to a loss of intracellular cations and reductions in cell volume.

The evidence that mechanical stress can cause changes in RBC membrane cation permeability comes from experiments in which cation leak was measured in cells subjected to fluid shear forces in a viscometer. Under these conditions, cells become elongated and undergo a tank-treading motion around the cell interior. Although there is disagreement on the precise mechanism by which the increased permeability occurs, in several studies it has been shown that there is significant loss of potassium in cells subjected to shear rates on the order of 300 s-1 (10-12). Unfortunately, the complexity of the cell motion and its interaction with the suspending fluid make it difficult to determine the precise mechanical forces generated in the membrane under these conditions. Approximate analyses indicate that, when the fluid shear rate approaches this magnitude, cells approach maximum deformation, and membrane force resultants increase rapidly with further increases in shear rate. Thus, at shear rates in excess of 300 s-1, at which significant cation loss is known to occur, membrane force resultants are in excess of 0.1 mN/m (27), well within the range that can be generated in the membranes of trapped RBCs (7).

This mechanism could account for the increase in cell density that is associated with RBC senescence (3). If a cell loses membrane area, either through a mechanical or chemical mechanism, then it could become transiently trapped in a narrow passage in the microvasculature. The resulting membrane stresses could lead to a transient increase in cation permeability, loss of intracellular cations, and a decrease in cell volume. Studies of stress-induced cation loss show that these losses occur without loss of hemoglobin; therefore, as the cell volume decreases, the density of the cell would increase, as the concentration of hemoglobin within the cell increases. Such a scenario is consistent with findings from an earlier, short-term study of surface-deprived RBCs in the circulation (32). The volume of the modified cells decreased over a period of 4 h, during which the reduced membrane area remained constant. The scenario is also consistent with observations that senescent cells in humans are not only more dense but are also smaller than younger populations of cells (15, 19, 31). The present study is consistent with these former studies in that the initial response of the cells was to decrease volume (while the mean cell area of the population actually increased), leading to a decrease in the mean sphericity of the population. In the long term, after the volume-to-surface area ratios of the cells in the population returned to nearly normal values, the population of surviving cells decreased in size while maintaining a requisite ratio of volume to surface area.

During this study we obtained some evidence that the ability of cells to adapt and survive after surface loss may depend on the metabolic state of the cell. In one strategy developed to create cells with reduced membrane area, cells were incubated at 39°C in the absence of glucose for 3-4 h in an attempt to facilitate echinocytosis and shedding of membrane vesicles. On the three occasions in which this strategy was employed, the modified cells all disappeared from the circulation within 5 days, regardless of their volume-to-surface area ratio (data not shown). Thus, although volume-to-surface area ratio is an important determinant of cell survival, it is clear that other factors can contribute to the removal of abnormal cells from the circulation. Energy depletion can affect many cellular processes; thus a number of mechanisms exist that might have contributed to the disappearance of these cells. One possibility is that the cells lost their ability to regulate their volume because of reduced activity of the Na-K-ATPase or other transport proteins in the membrane. Another is that cells may have lost their ability to maintain lipid asymmetry because of reduced activity of the lipid translocase. The consequent appearance of charged lipids in the outer leaflet of the membrane could trigger removal of cells by the reticuloendothelial system.

The recognition that factors other than the ratio of volume to surface area can affect cell survival does not take away from our conclusion that this ratio is a critical determinant for circulation. There is evidence from prior studies that the fluorescent label used in the present study has no effect on the ability of cells to circulate normally (22, 23). Furthermore, the present study demonstrates that cells that have undergone treatment to reduce area, but which have reacquired a normal volume-to-surface area after 5 days, will continue to circulate for up to 35-42 days, the full lifetime of normal mouse erythrocytes (4, 28). Thus the present procedures do not shorten cell lifetime as long as cells can achieve a requisitely small ratio of volume to surface area.

In conclusion, the ratio of RBC volume to surface area (sphericity) is a key factor in determining the ability of RBCs to circulate and function in the living vasculature. Cells with sphericities more than 5% above the normal range were either removed from the circulation or modified within a period of 5 days after infusion. Model calculations support the hypothesis that both modification (probably by a reduction in cell volume) and removal could result from mechanical trapping of cells at constrictions within the microvasculature. Preliminary evidence suggests that the modification of the cells may involve an energy-dependent mechanism, inasmuch as cells incubated at body temperature for 4 h in the absence of nutrients before reinfusion did not survive past 5 days in the vasculature. The present results are consistent with prior studies of RBC senescence in that, after the 5-day period of adjustment, cells became progressively smaller as they aged but maintained a requisite ratio of volume to surface area.


    APPENDIX
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

For cases in which the capillary taper M is small and terms of order M2 are neglected, an approximate expression for the cell surface area is obtained
A<IT>=4&pgr;R</IT><SUP><IT>2</IT></SUP><SUB>m</SUB><IT>+2&pgr;R</IT><SUB>m</SUB><IT>L</IT> (A1)
The corresponding approximate expression for volume is
V<IT>=</IT><FR><NU><IT>4</IT></NU><DE><IT>3</IT></DE></FR><IT> &pgr;R</IT><SUP><IT>3</IT></SUP><SUB>m</SUB><IT>+&pgr;R</IT><SUP><IT>2</IT></SUP><SUB>m</SUB><IT>L</IT> (A2)
(These simplified forms are accurate for tapered tubes to order M 2 because no terms of order M appear in the expressions. For straight tubes, the expressions are exact.) From the definition of the sphericity (Eq. 3), the following expression for the volume is obtained
V<IT>=</IT><FR><NU><IT>S<SUP>3/2</SUP>A<SUP>3/2</SUP></IT></NU><DE><IT>6</IT><RAD><RCD><IT>&pgr;</IT></RCD></RAD></DE></FR> (A3)
Combining Eqs. A1---A3 to eliminate L and V, a cubic equation in Rm is obtained
<FR><NU>2</NU><DE>3</DE></FR> &pgr;R<SUP><IT>3</IT></SUP><SUB>m</SUB><IT>−</IT><FR><NU>A</NU><DE><IT>2</IT></DE></FR><IT> R</IT><SUB>m</SUB><IT>+</IT><FR><NU><IT>S<SUP>3/2</SUP>A<SUP>3/2</SUP></IT></NU><DE><IT>6</IT><RAD><RCD><IT>&pgr;</IT></RCD></RAD></DE></FR><IT>=0</IT> (A4)
There are three roots to this equation
x<SUB>1</SUB>=2<RAD><RCD>−<FR><NU><IT>a</IT></NU><DE><IT>3</IT></DE></FR></RCD></RAD> cos <FENCE><FR><NU><IT>&phgr;</IT></NU><DE><IT>3</IT></DE></FR></FENCE>  and <IT> x<SUB>2</SUB>=2</IT><RAD><RCD>−<FR><NU><IT>a</IT></NU><DE><IT>3</IT></DE></FR></RCD></RAD> cos <FENCE><FR><NU><IT>&phgr;</IT></NU><DE><IT>3</IT></DE></FR><IT>+</IT><FR><NU><IT>2</IT></NU><DE><IT>3</IT></DE></FR><IT> &pgr;</IT></FENCE>  (A5)

and <IT> x<SUB>3</SUB>=2</IT><RAD><RCD>−<FR><NU><IT>a</IT></NU><DE><IT>3</IT></DE></FR></RCD></RAD> cos <FENCE><FR><NU><IT>&phgr;</IT></NU><DE><IT>3</IT></DE></FR><IT>+</IT><FR><NU><IT>4</IT></NU><DE><IT>3</IT></DE></FR><IT> &pgr;</IT></FENCE>
where
a=−<FR><NU><IT>3A</IT></NU><DE><IT>4&pgr;</IT></DE></FR>  and <IT> b=</IT><FR><NU><IT>S<SUP>3/2</SUP>A<SUP>3/2</SUP></IT></NU><DE><IT>4&pgr;<SUP>3/2</SUP></IT></DE></FR> (A6)
and
cos (<IT>&phgr;</IT>)<IT>=</IT>−<FR><NU><IT>b</IT></NU><DE><IT>2</IT></DE></FR> <FENCE><RAD><RCD>−<FR><NU><IT>a<SUP>3</SUP></IT></NU><DE><IT>27</IT></DE></FR></RCD></RAD></FENCE> (A7)
After substitution and simplification, the physically relevant root takes the form
R<SUB>m</SUB><IT>=</IT><RAD><RCD> <FR><NU><IT>A</IT></NU><DE><IT>&pgr;</IT></DE></FR></RCD></RAD> cos <FENCE><FR><NU>cos<SUP><IT>−1</IT></SUP> (−<IT>S<SUP>3/2</SUP></IT>)<IT>+4&pgr;</IT></NU><DE><IT>3</IT></DE></FR></FENCE> (A8)


    ACKNOWLEDGEMENTS

We thank Pat Titus, Donna Brooks, and Richard Bauserman for technical assistance.


    FOOTNOTES

This work was supported by National Heart, Lung, and Blood Institute Grant PO1-HL-18208.

C. Reynolds was a recipient of a Summer Undergraduate Research Fellowship from the Strong Children's Fund at the University of Rochester.

Address for reprint requests and other correspondence: R. E. Waugh, Dept. of Pharmacology and Physiology, Univ. of Rochester Medical Center, 601 Elmwood Ave., Box 711, Rochester, NY 14642 (E-mail: waugh{at}seas.rochester.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received 4 February 2000; accepted in final form 12 April 2000.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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